self organization in nonequilibrrium systems - prigogine

Copyright \302\251 1977by John Wiley & Sons, Inc.

All rights reserved. Published simultaneously in Canada.

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Library ofCongress Cataloging in Publication Data:Nicolis, G 1939-

Self-organization in nonequilibrium systems.

\"A Wiley-Interscience publication.\"

Bibliography: p.Includes index.1.Self-organizing systems. 2. Chemistry,

Physical and theoretical. 3. Biological chemistry.4. Nonequilibrium Phenomena. 5. Fluctuation Theory.I.Prigogine, Ilya, joint author. II.Title.Q325.N5 001.533 76-49019

ISBN 0-471-02401-5

Printed in the United States ofAmerica.

1098765432

Contents

General Introduction

ART I. THE THERMODYNAMICBACKGROUND

1. Introduction 191.1.GeneralComments, 191.2.OpenSystems, 24

2. ConservationEquations 262.1.OpenSystemsat MechanicalEquilibrium, 262.2.The Mass-balanceEquations, 27

3. Thermodynamicsof IrreversibleProcesses:The Linear Region 313.1.Gibbs'sFormula:Entropy Production, 313.2.PhenomenologicalRelations:The Linear Range of

IrreversibleProcesses,363.3.Symmetry Propertiesof the Phenomenological

Coefficients, 393.4.Stationary Nonequilibrium States, 413.5.Theorem of Minimum Entropy Production, 423.6.Impossibilityof OrderedBehavior in the Linear

Range of IrreversibleProcesses,453.7.Diffusion, 46

4. NonlinearThermodynamics 494.1.Introduction, 494.2.The GeneralEvolution Criterion, 504.3.Evolution Criterion and Kinetic Potential, 514.4.Stability of Nonequilibrium States.

DissipativeStructures, 55

vi Contents

PART II. MATHEMATICALASPECTSOF SELF-ORGANIZATION:DETERMINISTICMETHODS

5. SystemsInvolving ChemicalReactionsand Diffusion-Stability 635.1.GeneralFormulation, 635.2.Lyapounov Stability, 655.3.Orbital Stability, 665.4. Structural Stability, 68

6. MathematicalTools 706.1.Introduction, 706.2.Theory of Bifurcations, 706.3.Stability Theory, 716.4. Theory of Catastrophes, 746.5.HomogeneousSystemsInvolving Two Variables, 766.6.Branchings,Bifurcations,and Limit Cycles, 83

7. SimpleAutocatalytic Models 907.1.Two Intermediates, 907.2. The Trimolecular Model(the

\" Brusselator\,") 937.3. Scaling,SteadyStates,and Boundary Conditions, 947.4. Linear Stability Analysis, 967.5. Bifurcationof Steady-stateDissipative

Structures:GeneralScheme, 1067.6. Bifurcation: FixedBoundary Conditions, 1097.7. Bifurcation: No-flux Boundary Conditions, 1137.8. Qualitative Propertiesof DissipativeStructures

in Vicinity of First Bifurcation, 1157.9. SuccessiveInstabilitiesand Secondary

Bifurcations, 1207.10.Comparisonwith ComputerSimulations, 1247.11.LocalizedSteady-stateDissipativeStructures, 1317.12.Bifurcation of Time-periodicDissipative

Structures, 1407.13.Qualitative Propertiesof Time-periodic

DissipativeStructures, 1477.14.TravelingWaves in PeriodicGeometries,1537.15.The Brusselatoras a ClosedSystem, 1567.16.ConcludingRemarks, 158

Contents

8. SomeFurther Aspectsof DissipativeStructures and

Self-organizationPhenomena 1608.1.Introduction, 1608.2.ConservativeOscillations, 1608.3.SimpleModelsGiving Rise to Limit Cycles, 1658.4.Multiple SteadyStatesand All-or-none

Transitions, 1698.5.Two-dimensionalProblems, 1788.6.SystemsInvolving Morethan Two Chemical

Variables, 1928.7. CoupledOscillators, 1958.8.HeterogeneousCatalysisand Localized

Transitions, 1978.9.SystemsInvolving Photochemical Steps, 2008.10.SomeFurther Methodsof Analysis of

Reaction-Diffusion Equations, 2028.11.ThermodynamicAspectsof Dissipative

Structures, 212

PART III. STOCHASTICMETHODS

9. GeneralComments 2239.1.Introduction, 2239.2.StochasticFormulation, 2249.3.Markovian Processes,2289.4.Equilibrium Limit, 2329.5.Fluctuations in Nonequilibrium Systems:

An HistoricalSurvey, 236

10.Birth-and-deathDescriptionof Fluctuations 23910.1.MasterEquation for Birth-and-death

Processes,23910.2.Limitations of Birth-and-death Formalism, 24110.3.SomeMethodsof Analysis of Birth-and-death

MasterEquations, 24210.4.Moment Equations, 25210.5.SimpleExamples, 25710.6.SystemsInvolving Two StochasticVariables:

The Lotka-VolterraModel, 26410.7.ConcludingRemarks, 272

vili Contents

11.Effectof Diffusion:Phase-spaceDescriptionandMultivariate MasterEquation 27311.1.Necessityfor a LocalDescriptionof

Fluctuations, 27311.2.Phase-spaceDescriptionof Fluctuations, 27411.3.A SimpleModel, 27611.4.Approximate Solutionof MasterEquation, 28011.5.MolecularDynamicsStudiesof Fluctuations, 28311.6.Discussion, 28411.7.Reduction to a Multivariate MasterEquation in

Concentration Space, 28511.8.The Multivariate MasterEquation in a Model

System, 28911.9.SpatialCorrelationsin the Trimolecular

Model, 29711.10.Critical Behavior, 30211.11.ConcludingRemarks, 309

12.A \" Mean-field\" Descriptionof Fluctuations:NonlinearMasterEquation 31312.1.Introduction, 31312.2.Derivation of Nonlinear MasterEquation, 31412.3.Further Propertiesand Moment Equations, 31712.4.Onsetof a Limit Cycle, 31912.5.Onsetof a SpatialDissipativeStructure, 32412.6.Multiple Steady-stateTransitions and

Metastability, 32712.7.Asymptotic Solutionsof Nonlinear Master

Equation, 33112.8.ConcludingRemarks, 334

PART IV. CONTROLMECHANISMSIN CHEMICALANDBIOLOGICALSYSTEMS

13.Self-organizationin ChemicalReactions 33913.1.Introduction, 33913.2.Belousov-ZhabotinskiReaction:

Experimental Facts, 33913.3.Mechanism, 34313.4.The\"Oregonator\",34513.5.OscillatoryBehavior, 34713.6.SpatialPatterns, 35113.7.Briggs-RauscherReaction, 352

Contents ix

14. RegulatoryProcessesat the SubcellularLevel 35414.1.MetabolicOscillations, 35414.2.The GlycolyticCycle, 35414.3.Allosteric Modelfor GlycolyticOscillations, 35814.4.Limit-cycleOscillations, 36914.5.Effectof External Distrubanceson Limit-cycle

Oscillation, 37114.6.Patterns of SpatiotemporalOrganization in

AllostericEnzymeModel, 37514.7.PeriodicSynthesisof cAMP, 37914.8.Reactions Involving Membrane-boundEnzymes, 38214.9.PhysiologicalSignificanceof Metabolic

Oscillations, 384

15.RegulatoryProcessesat CellularLevel 38715.1.Introduction, 38715.2.LacOperon, 38815.3.Mathematical Modelfor Induction of

M-Galactosidase,38915.4.All-or-noneTransitions, 39115.5.CataboliteRepression:SustainedOscillations

and ThresholdPhenomena, 39415.6.Controlof CellularDivision, 40215.7.Quantitative Model, 404

16.CellularDifferentiation and Pattern Formation 40916.1.Introductory Remarks, 40916.2.Positional Information, 41016.3.MechanismsInvolved in Positional Information, 41316.4.DissipativeStructuresand Onsetof Polarity, 41516.5.A Quantitative Model, 41616.6.Positional Differentiation, 42116.7.Applications, 424

PART V. EVOLUTIONAND POPULATIONDYNAMICS

17. Thermodynamicsof Evolution 42917.1.The Notion of Competition, 42917.2.PrebioticEvolution: GeneralPresentation, 42917.3.PrebioticPolymer Formation, 43017.4.BiopolymerCompetition and Hypercycles, 434

Contents

17.5.Evolution Viewedas a Problem of Stability, 43817.6.Evolutionary Feedback,44117.7.Energy Dissipationin SimpleReaction Networks, 44217.8.A BiochemicalIllustration, 446

18.Thermodynamicsof Ecosystems 44818.1.Introduction, 44818.2.BasicEquations, 44818.3.Exampleof OrderedBehavior:Organization in

InsectSocieties,45218.4.Evolution of Ecosystems, 45518.5.Structural Instabilitiesand Increaseof Complexity:

Division of Labor, 45918.6.Stability and Complexity, 462

Perspectivesand ConcludingRemarks 464

1. Introduction, 4642. Fluctuation Chemistry, 4643. Neural and Immune Networks, 4664. Immune Surveillanceagainst Cancer, 4695. SocialSystemsand EpistemologicalAspects, 472

References 475Addendum:Mathematical Problems 487Index 489

Symbols

LATIN

A,B,...

EecFfJA:eqkkl\"

LklLPSSoSTV

AF\\wp

wki

X, Y,...X

affinity of reaction pinitial and final productsin a reaction sequencediffusion coefficient of constituent idiffusion frequency of constituent iinternal energyinternal energy per unit volumefree energy; generating functionreducedgenerating functiongeneralized flux

equilibrium constantwave numberrate constantcharacteristic length

phenomenologicalcoefficientslinearized operator of a reaction-diffusionsystementropy production;probability distributionentropyentropy densityindependentvariable in generating function representationtemperature; periodof oscillationvolumevolume elementconstant of motionvelocity of reaction ptransition probability per unit timevariable intermediates in a reaction sequencegeneralized flux

GREEK

diffusion frequency of constituent ; in nonlinear master equa-equation formalism

xi

xjj Symbols

@ generalizedthermodynamic potentialA bifurcation parameter;cellsize in multivariate master equa-

equation

ft. chemical potential of constituent iv, stoichiometriccoefficient of constituent ; in reaction p

p massdensitya entropy-production density2 surfaced, cumulant generating function

w eigenvalue; oscillation frequency

Self-Organizationin NonequilibriumSystems

GeneralIntroduction

Our era is witnessinggreat advances in knowledgein the natural sciences.The dimensionsof the physical world that we can exploreat presenthaveincreasedto truly fantasticproportions.On the microscopicscale,elementaryparticle physicsrevealsprocessesinvolving physicaldimensionsof the orderof 10~15cm and timesof the orderof 10~22s.On the other hand, cosmologyleadsus to times of the orderof 1010years (the \"age\" of the universe)and,therefore, to distancesof the orderof 1028cm (the distanceto the eventhorizon, i.e.,the furthest distancefrom which physical signalscan be re-received). Finally, a number of novel features relating to the very character ofthe physicalworld have beendiscoveredrecently.

Classicalphysicshas emphasizedstability and permanence.We now seethat, at best,such a qualificationappliesonly to very limited aspects.Where-ever we look,we discoverevolutionary processesleading to diversificationand increasing complexity. This shift in our vision of the physical worldleadsus to investigatebranchesof physicsand mathematics that are likelyto beof interest in this new context.The main objectof this monograph is topresentan introduction into these new branchesfor a better understandingof the problemof self-organizationin physicalor biologicalsystems.

Classicalmechanicshas beenimmensely successfulin dealing with prob-problems concerning trajectories.The scopeof classicalmechanics has beengreatly enlargedby the formulation of quantum mechanics and relativity.But even when suitably generalized,the standard formulation of dynamicsmakesno distinctionbetweenthe future and the past.The motions \"forward\"

and \"backward\" in time are both possible.Yet, without introducing thedirection of time, we cannot describeprocessesinvolving evolution in anynontrivial way. Obviously,new toolsare necessary.

It is not our purposein this monograph to discussthe relation betweenthedirectionoftime and dynamics.This is a subjectofa highly technicalcharacter.

2 General Introduction

Sufficeit to mention that irreversibility, as related to the direction of time, isby no means in contradiction with the laws of dynamics;on the contrary, it

follows from these laws whenever a sufficient degreeof \"complexity\" isreached.*

Independentlyof dynamics, the idea of evolution was introduced in

physicsin the 19th century through the so-calledsecondlaw of thermo-

thermodynamics. It was noticed that from a purely phenomenologicallevela basicdistinction had to be introducedbetween \"reversible\"and \"irreversible\"

processes.An example of an irreversibleprocessis heat conduction,whichleads to a uniform distribution of temperature, if compatiblewith the

boundary conditions.An exampleofa reversibleprocessiswavepropagation,neglectingfriction and energy losses.The secondlaw dealswith this distinc-distinction between reversible and irreversible processes.It introducesa newfunction, entropy, which increasesas the result of irreversibleprocesses.

Clausiusgave a dramatic formulation to this law when he stated that

\"the entropy of the universe is increasing.\"Accordingto this statement, theuniverseis driven to the \" heat death.\" This is,however,not what we observearound us in the presentstage of the universeorwhat we can infer from itspast so far as it is known to us. There, as already mentioned, we observecontinuous diversificationand evolution toward complexity. Remarkably,the ideaof evolution that appearedin physicsthrough the secondlaw wasformulated almost simultaneously in the 19th century in biologyt andsociology.% However,as we shall see,the interpretation of this conceptwasquite different in these various fields.

Let us briefly outline the principal stagesof the development of thermo-thermodynamics. Accordingto the secondlaw an isolatedsystemreachesin time thestate of \"

thermodynamic equilibrium,\" which correspondsto a maximum

entropy. In its most general formulation this law appliesto both equilibriumand nonequilibrium situations. Nevertheless,most of the striking resultsofclassicalthermodynamicsasdevelopedin the 19thcentury referto equilibriumsituations. Well-known examplesare the Gibbsphaserule and the law ofmassaction, which have becomeintegral parts of every introductory text in

physicalchemistry.There may have beenvarious reasonsfor this. Nonequilibrium was con-

considered asa perturbation temporarily preventing the appearanceof structureidentifiedwith the orderat equilibrium.To growa beautiful crystalwerequirenear-equilibrium conditions,and to obtain a good yield from a thermal

* I.Prigogine, A. P.Grecos,and C.George,A976), Proc.Nat. Acad. Sci.(U.S.A.)73, 1802andCelestial Mech.,submitted. Seealsoa forthcoming monograph by I.Prigogine.t C.Darwin A859), The Origin ofSpecies,John Murray, London.+ H.Spencer A904), Study ofSociology, Paul Kegan, London.

General Introduction 3

engine we needto minimize irreversibleprocessessuch as friction and heatlosses.The situation changeddrastically with the appearanceof Onsager'sdiscovery of the \"

reciprocity relations,\"*which confirmed that, at least in

the neighborhoodof equilibrium, thermodynamic methods couldprovideuseful information.

This led to an extensionof classicalthermodynamics that may be appro-appropriately

called \"linear nonequilibrium thermodynamics\"and that coversthe range of situations in which the flows (or rates) of irreversibleprocessesare linear functions of the \"thermodynamic forces\"(e.g.,temperature orconcentration gradients).A situation to which this linear nonequilibriumthermodynamics applies is thermal diffusion. When we apply a thermalgradient to a mixture of two different gases,weobservean enrichment ofoneof the componentsat the hot wall, while the other concentratesat the coldwall.As a result, the entropy is generallylower than it would be in a uniformmixture. We see later that Boltzmann's order principle associateslow

entropy with orderand high entropy with disorder.Therefore,we have herean exampleof a situation where nonequilibrium may be a sourceof order.This observation was the starting point of the outlookpioneeredby theBrusselsschool.t

However, in the framework of linear nonequilibrium thermodynamicswe cannot really speak of new structures.We deal with the equilibriumstructures modifiedby the constraints preventing the system from reachingequilibrium.This was why our groupstarted the investigationof situationsfurther away from equilibrium, that, however,couldstill bedescribedin termsof macroscopic,thermodynamic variables.

The results have beenmost unexpectedand even spectacular.The mainsituation of interest for us here is the caseof chemicalreactions.The rate ofchemicalreactions is generallya nonlinear function of the variables involved(e.g.,concentration or temperature). As the result, a chemically reactingmixture is describedby nonlinear equationshaving, in general, more than

one solution (even when boundary and initial conditionsare taken intoaccount). Let us then considerthe solution correspondingto equilibriumconditions(this implies maximum entropy for isolatedsystems,minimumHelmholtz free energy for systems at given temperature and volume).Wecall this solution the \"thermodynamic branch.\" Supposewe now vary theconstraintssoasto forcethe systemfurther and further away from equilibrium.Then, nonequilibrium thermodynamics leads us to formulate a sufficient

condition for the stability of the thermodynamic branch.If this condition is

* L.Onsager A931), Phys. Rev. 37,405.11.Prigogine A945), Thesededoctorat, Universite Libre de Bruxelles. For a recent historicalaccount, seeI.Prigogine and P.Glansdorff A973), Bull. Acad. Roy. Belt/. Ci.Sci.59,672.


not satisfied the thermodynamic branch may becomeunstable, and thesystemmay evolvetoward a new structure involving coherentbehavior^

Suchbehavior has long beenknown to occurin hydrodynamics.A well-known exampleis a pan of liquid heated from below.When the temperaturegradient remains small in respectto somecharacteristic value, heat passesthrough the liquid by conduction.As the heating is intensified,however,at acertain well-definedtemperature gradient regular convection cellsappearspontaneously.Thesecorrespondto a high degreeof molecularorganizationand becomepossiblethrough transfer of energy from thermal motion tomacroscopicconvectioncurrents.This is the so-calledBenardinstability.

The essentialpoint is that beyond the instability of the thermodynamicbranch we may have a new type of organization relating the coherentspace-timebehavior to the dynamical processes(e.g.,chemical kineticsand convection)insidethe system.Only_ifjtpgro\302\243riate feedbackconditionsare satisfiedcan the thermodynamic branch becomeunstable at a sufficient

distancefrom equilibrium.The new structures that appear in this way areradically different from the \"equilibrium structures\" studied in classicalthermodynamics, such as crystals or liquids.They can be maintained in

far-from-equilibriumconditionsonly through a sufficient flow of energyandmatter. An appropriateillustration would be a town that can only surviveas long as it is a center of inflow of food, fuel, and other commoditiesandsendsout productsand wastes.

We have introduced the term \"dissipativestructures\" to contrast suchstructures from the equilibrium structures^Dissipative structures providea striking exampleof nonequilibrium as a sourceof order.Moreover,themechanism of the formation of dissipative structures has to be contrastedwith that of equilibrium structures basedon Boltzmann'sorderprinciple.

It was, indeed,Boltzmann* who first pointed out that entropy was ameasure of disorderand concluded,therefore, that the law of entropy in-increase was simply a law of increasingdisorganization.In a more quantitativefashion, Boltzmann related entropy to the \"number of complexions\"Pthrough the relation

S = kB\\ogP A)

where kB is Boltzmann's universal constant. This relation indicates that

thermodynamic equilibrium of an isolatedsystem (Smax) correspondstosituations where the number of complexionsis maximum.

We may extend theseconsiderationsto closedsystems(which exchange

* L.Boltzmann A896), Vorlesungen tiber Gastheorie, J. A. Barth, Leipzig.


energy but not matter with the outsideworld) at a given temperature. Thesituation remains similar exceptthat insteadof the entropy S we must nowconsiderthe free-energyfunction F definedby

F = E - TS B)

where E is the energy of the system and T the absolutetemperature (indegreesKelvin).At equilibrium the free energyreachesa minimum value.

The structure of Eq.B)reflectsa competition between the energy E andthe entropy S.At low temperatures the secondterm is negligibleand theminimum of F imposesstructures correspondingto minimum energy andgenerally to low entropy. At increasing temperatures, however, the systemshifts to structures of increasinglyhigh entropy.

Experiment confirms thesepredictionssinceat low temperatures we find

the solidstate characterized by an orderedstructure of low entropy, whileat higher temperatures we find the high-entropy gaseousstate. We may,therefore,concludethat equilibrium structuresare dominated by Boltzmann'sorderprincipleas expressedthrough Eqs.A) and B).*

However, Boltzmann's order principle is inapplicableto dissipativestructures.Consideragain the B6nard instability. Boltzmann's principlewould assignan almost zero probability to the occurrenceof a coherentconvection pattern involving more than 1020molecules.We then have tointroduce quite different considerations.We may imagine that there arealwayssmall convectioncurrents appearingas fluctuations from the averagestate, but below a certain critical value of the temperature gradient thesefluctuations are damped and disappear.On the contrary, above somecriticalvalue certain fluctuations are amplifiedand give riseto a macroscopiccurrent. A new molecular order appears that correspondsbasically to amacroscopicfluctuation stabilized by exchangesof energy with the outsideworld. This is the order characterized by the occurrenceof dissipativestructures.We call this order \"orderthrough fluctuations\" to contrast it

with the Boltzmann orderprinciple,which is basicfor the understanding ofequilibrium structures.

The aim of the presentmonograph can now be expressedas the study ofself-organizationin nonequilibriumsystems,characterizedby the appearanceofdissipativestructures through the amplificationofappropriatefluctuations.While relatively new,this field ofinvestigationencompassesalreadyat presenta widerangeofproblemsfrom chemistryto biologyand population dynamics.

* Or,alternatively, through the canonical distribution used in equilibrium statistical mechanics,seeL.D.Landau and E. M. Lifshitz A957), Statistical Physics, Pergamon, Oxford.


In 1971ProfessorP. Glansdorff and one of the authors (I. Prigogine)published a monograph entitled Thermodynamic Theory of Structure,Stability, and Fluctuations* The aim of this monograph was to outline thethermodynamic theory of nonequilibrium systems in the entire range ofmacroscopicdescription,starting from equilibrium and including non-nonlinear situations and instabilities. This monograph contains the thermo-thermodynamic criterion for the occurrenceof dissipative structures, as well asapplicationsto simplesituations in hydrodynamics and chemistry.A greatamount of work has sincebeendevoted to this type of problem.Importantprogresshas beenrealized mainly in the following directions:

(i) Wha^ happensbeyond the instability of the thermodynamic branch?What are the types of coherent behavior and how are they related to themolecular mechanisnisTnvo1ved;aTwelhas-tcrtheconstraints\"acting on thesysTem?(Theseare problemsrelated to \" bifurcation th^bTyT^which dealswith the appearance_of new types of solution of differential equationsatcritical orbifurcationpoints.)(ii) What is the kineticsof the growth of dissipativestructures, and how canfluctuation theory be applied to these nonlinear, far-from-equilibriumsituations?(iii) What are the situations to which we may usefully apply the conceptsof dissipativestructures and orderthrough fluctuations?

It would have beenimpossibleto incorporatethesenew developments in

the framework of a new edition of the book by Glansdorff and Prigogine.Therefore,wedecidedto write a new monograph.tTo avoid duplication wehavekeptthe thermodynamictheory to a minimum and expandedthe aspectsthat couldnot have beentreated adequately in Glansdorff and Prigogine'smonograph, asthey report the result of work performedsinceits publication.Let us briefly outline the structure of the presentwork.

Part I is devoted to \"the thermodynamic background.\"We attempt togive a concisebut self-containedpresentation of the basicprinciplesof non-equilibrium thermodynamics, with specialemphasison chemical reactionscoupledto transport processessuch as diffusion.

The term \"chemicalreactions\" is taken in a general, formal sense.Wemeet similar equations in different problems,such as the ecologicalorsociobiologicalproblemsin Part V.

*A971), Wiley, New York-London; A971), French ed.,Masson, Paris; A973), Russian

translation, Mir, Moscow, Japanese translation to appear 1976.t An interesting and compact review ofsome aspects of self-organization can alsobe found in

the recent mornograph by Ebeling (Strukturbildung bei irreversiblen Prozessen BSB B.G.Teubner Verlagsgesellschaft, 1976).


The time evolution of systems involving chemical reactionsand diffusion

is describedby a set of couplednonlinear partial differential equations.Thesolution of such equations,oncesuitable initial and boundary conditionsare provided, is notoriously difficult. In general, there existsmore than onesolution.It is here that thermodynamics singlesout a specialsolution,namely, that correspondingto the \"thermodynamic branch\" as definedabove. When we permit the system to deviate from equilibrium, new solu-solutions may or may not replacethe \"

thermodynamic branch.\" Distancefromthermodynamic equilibrium then becomesthe natural parameter in dis-discussing the appearanceof new types of solution.

The main problemdiscussedin Part IIof this monograph is the questionof how to construct solutionsthat occurwhen the thermodynamic branchbecomesunstable.We first discussthe conceptof stability, which plays suchan important rolein our wholeapproach(Chapter5).In equilibrium thermo-thermodynamics, stability theory was introducedby Gibbs.*Stability can beeasilyformulated and discussedin terms of thermodynamic potentials such asentropy or freeenergy.Onceweknow that the systemis in a state ofminimumfree energy, we may concludethat it is stable.Even if a fluctuation woulddrive it momentarily away from the equilibrium state, the system wouldrespondby loweringits freeenergytill it reachesagain the equilibrium value.The situation is radically different in the nonequilibrium situations of in-interest to us here, as there existsin general no potential whosevalue wouldcharacterize the state of the system.It is for this reasonthat stability theoryplays an important role.

The natural approachto the problemof the emergenceof new patternsis in terms of the bifurcationtheory.The purposeof this theory is to study the

possiblebranching of solutions that may ariseunder certain conditions.We have tried to presenta readableintroduction to this rapidly expanding

field and have not always presentedproofs of theorems that are easilyaccessiblein textbooks.

Our main emphasisis in physicalexamplesand in simplebut representa-representative models,and our aim is to give to reader an idea of the variety ofspace-timestructures that may arise through bifurcation. We may find

\"chemicalclocks,\"coherent statescorrespondingto time independentbut

spatiallyinhomogeneousdistribution ofmatter, or chemicalwavespresentinga considerablevariety. We are far from the situations studiedin linearmathe-mathematics where the equationslead,together with the boundary conditions,to asingle solution.This multiplicity of solutions in nonlinear systems corre-corresponds to a gradual acquisition of autonomy from the environment.

*J.W. Gibbs A928), CollectedWorks, Longmans, New York. For a moregeneral formulation,seethe monograph ofGlansdorff and Prigogine A971),Chapter 5.


Besidesbifurcationtheory,in Part IIwepresenta survey ofother techniquessuitable to study the appearanceof new structures, where the \"catastrophetheory\" due to R. Thorn is of specialinterest. However, for most of theproblemsthat we are concernedwith in this monograph this method is not

applicable,notably becausethe rate equationsgenerallydo not derive froma potential.

Part IIdealsentirely with \"deterministic\" methods.Theroleoffluctuationsis studiedin Part III.This is probablythe part of the theory where the pro-progress has been the most spectacularover the last years and where new

developmentsare expectedin the near future.

The systemsweconsiderconsistof a largenumber of units (e.g.,moleculesin a liquid, cellsin an organism, and neurons in the brain). Obviously,all ofthese units cannot be in the samestate; therefore, a macroscopicsystemspontaneously generates\"noise,\" which plays somewhat the roleof pertur-perturbations in stability theory. But here the perturbations are generatedby thesystem itself (in addition, we have fluctuations introducedby the externalmedium).

We have to expectthesefluctuations to play an especiallyimportant rolenear bifurcationpoints.Thus, the systemhas to \"choose\"oneof the possiblestablebranchesof the macroscopicequations.But nothing in the macro-macroscopic equationsjustifiespreferencefor any onechoice.Therefore,stochasticelements must be taken into account, and we need a finer descriptioninvolving fluctuations.

We have, therefore, to considerstandard techniques such as Markovchains orbirth-and-death processesto includefluctuations.However,in theproblemsof interest to us somequite specificnew featuresappear.The usualexamplesin Markov chains refer to processesfor which the transitionprobabilitiesare constant (i.e.,{to the \"left\" and i to the \"right\") or linearfunctions of the stochasticvariables.

On the contrary, here we have to considernonlinear chemicalequations,and the transition probabilitiesbecomenonlinear functions of the fluctuatingvariables.

The theory of fluctuations at equilibrium is well known.* Of specialimportance is Einstein'sformula for the probability distribution of fluctua-fluctuations, which correspondsto the inversion of Boltzmann's definition ofentropy. It expressesthe probability of a state (related to the number ofcomplexions)to its entropy.Einstein'sformula leadsto a Poissondistributionfor fluctuating variables in ideal systems.We show that this result remainsvalid for nonequilibrium situations when linear chemical equationsareinvolved. However, for nonlinear situations the whole picture changes.

* SeeL.D.Landau and E. M. Lifshitz, opcit.


The probability distribution of the fluctuations dependson their size andrange and on macroscopicconditionsthat prevail in the system. In a suffi-

sufficiently small scalewe obtain, indeed,a nearly Poissondistribution, whilewitrTmcreasingrange the fluctuations deviate from a Poissondistribution.Thesedeviationsbecomespeciallyspectacularnear a bifurcationpoint wherelong-rangespatial correlationsappear.The analogybetweennonequilibriuminstabilities and phasetransitions is striking. Much of the recent statisticaltheory of phasetransitions and critical behavior finds here a direct analog.Beyondthe instability, the statistical fluctuations increasein time and finally

drive the average values to their new macroscopicstate.We seehere veryclearlythe meaning of the conceptoforderthrough fluctuations that wehavealready mentioned in this introduction. We may considersuch processesasexpressinga breakdownof the \"law of large numbers.\" When this law issatisfied the average provides an adequatedescriptionof the system.Here,on the contrary, fluctuations drive the averages.

There are even classesof nonlinear phenomena involving no phasetransition where fluctuations may play a much more important role than

thus far supposed.Forexample,in explosionsdueto freeradicalpropagation,the conditionsfor the explosionmay be first satisfiedin a domain small fromthe macroscopicpoint of view while still containing a large number ofmolecules.This leads to a kind of new \"fluctuation chemistry\" which wediscussbriefly in the concludingchapter.

We now turn to Parts IV and V, which discussspecificexamplesof self-organization chosenin various fields of interest from chemistry to socio-biology.Again, the situation has greatlychangedsincethe publication of the

monograph by Glansdorff and Prigogine in 1971,when few exampleswereknown. Now somuch material has becomeavailable that we had to make achoicebetween the most important features of self-organization in non-equilibrium systems.*

Fora long time the existenceof oscillatingchemicalreactions in a homo-homogeneous phase was a matter of controversy. It is true that Lotkat andVolterraJ had introduceda kinetic schemeof an oscillatory reaction in theecologicalcontext of the competition between prey and predator (seeChapter18).However,as discussedin Chapter8, this schemeleadsto a setof periodicmotions, each with a different period.As a result, it cannot beusedto model chemicalreactionswith a well-definedperioddetermined by

* For instance, the stability of interfaces in the presenceof nonequilibrium phenomena is notdiscussed. SeeT.S.Sofensen, M. Hennenberg, A. Steinchen-Sanfeld and A. Sanfeld A976),Progr. Coll.Pol.Set,61,64.t A. Lotka A920), Proc.Nat. Acad. Sci.(U.S.A.),6,420.+ V. Volterra A936), Lemons sur la Theorie Mathemalique dela Lullepour la Vie, Gauthier-Villars,Paris.


the values of the characteristic parameterssuch as rate constants andtemperature.

Fortunately, today we know oscillatory chemical reactions such as theBelousov-Zhabotinskireaction, which presentsstriking features of self-organization and whosemechanismhas beenlargelyelucidatedthrough thework ofNoyesand his school.While vastly morecomplex,the kineticschemecorrespondingto the Belousov-Zhabotinskireaction presentsa qualitativebehavior closelyrelated to the simplemodelstreated in Part II,and thereis no doubt that the spatial and temporal self-organizationpatterns may beexplainedin terms of dissipative structures arising in far-from-equilibriumconditions.

A large portion of Parts IV and V is devoted to self-organization in

biologicalproblems.Even in the simplest cells, the metabolic function involves several

thousandsof coupledchemical reactionsand requires,therefore, delicatemechanisms for coordinationand regulation. In other words,we need anextremelysophisticatedfunctional organization. Furthermore, the metabolicreactions require specific catalysts, namely, the enzymes. Each enzymeperforms onespecifictask,and if we considerthe manner in which the cellperforms a complexsequenceof operations,we find that it is organized onexactly the samelines as a modern \"assemblyline\" (seeChapter14).Thereis a definite relation between structure as expressedin space-timepatternsand function.Biologicalorderis both architectural and functional.*But that

is preciselythe type of order characterizing dissipative structures, which

appearas the result of their function (asdescribedby the chemicalprocesses)in far-from-equilibriumconditions.

Therefore, it is of specialinterest to apply the approachdevelopedin this

monograph to biologicalproblems.Let us emphasizethat there is a strikingdifferencebetween the chemical compositionof even the simplestcell andits environment (remember that the average molecular weight of a proteinis ~ 105,while that of water is 18).We are in a sensein the positionof avisitor from another planet who, finding a suburban house,wishestounderstand its origin. Of course,the houseis not in conflictwith the laws ofmechanics,otherwise it would have fallen down.However,this is besidesthe

point, namely, what is of interest is the technology available to the housebuilders,the needsof the occupants,and soon.The housecannot beunder-understood outsidethe culture in which it is embedded.

This is preciselyour positionin respectto problemsin biology.While eachof the important biomoleculesdiscoveredin the recent years is obviously

*This point has alsobeen emphasized by some biologists. Seee.g.P.Weiss A967), in \"TheNeurosciences,\" Rockefeller Univ. Press.New York.

General Introduction 11formed accordingto the laws of physicsand chemistry,we have to identifythe mechanismsthat make the \"mass production\"of thesemoleculespossibleand coordinatetheir productionaccordingto the needsof the organism.It has often been stated that biologicalorganization requiresa seriesofstructures and functions of growing complexityand hierarchical character.One of our main concernsis to understand the way in which transitionsbetweenlevelsoccurand to relate the molecular levelto the supermolecularone,and the cellular to the supracellular.The examplesin Parts IV and V

show that this problemcan beformulated and in somecasessolvedusing themethods we have introduced.

We start with two chapters(Chapters14and 15)devoted to regulatoryprocessesat the subcellularlevel(e.g.,thoseinvolved in the glycolyticcycle)and at the cellular level(e.g.,thoseinvolved in the lac operon).

Glycolysisis a phenomenon of the greatest importance for the energeticsof living cells.It leads to the productionof ATP molecules(adenosinetriphosphate),which becauseof their high energycontent play an importantrolein the synthesis of biomolecules.Now glycolyticoscillations,that is,oscillationsin time of the chemical componentsinvolved in the glycolyticcycle,havebeendiscoveredby B.Chanceand his co-workersand extensivelyinvestigated since,notably by B. Hessand his school(seeChapter 14fordetailedreferences).It appearedimmediately that becauseof their repro-ducibility glycolyticoscillationscouldonly be modeledby kinetic schemespresentingdissipative structures (and not by the Lotka-Volterratype ofequations).This was, then, the first examplein biology where the roleofa temporal dissipative structure could unambiguously be recognized.*Hence,the glycolytic cycle takes place beyond the stability limit of the

thermodynamic branch.It is quite remarkablethat someof the most spectacularaspectsof bio-

biological activity, such as control of cellular division (seeChapter 15)orcellular differentiation and morphogenesis(Chapter16)can, at present,bemodeledin terms of dissipativestructures.Instability in respectto diffusion

plays an essentialrolehere as it may generate a privilegedaxis of polaritystarting from an isotropicstate.

The question of stability with respectto diffusion was first investigatedby

Turing A952)in a remarkablepaperon the chemicalbasisofmorphogenesis.This was one of the first examplesof dissipative structures ever studied.However, this work remained unknown outside the context of morpho-morphogenesis, and even there was severely criticized on various grounds (seeChapter 16).These criticisms originate from a somewhat unfortunate

*I.Prigogine, R. Lefever, A. Goldbeter, and M. Herschkowitz-Kaufman A969), Nature223.913.


choiceof someof Turing's examples(the obvious requirement that the rateequationsmust admit positive and boundedsolutionsis not satisfied).Weseenow that this criticismdoesnot touch the essentialpoint.In fact,Turing'sinstability is one of the most striking phenomena associatedwith the break-breakdown of the stability of the thermodynamic branch.

The spectacularprogress of molecular biology has been of utmostimportance in the formulation of our approachto self-organization.Indeed,a preliminary condition for the study of the relation between function andstructure is a detailed knowledgeof the chemicalmechanismsinvolved.

However,our approachemphasizessupplementaryaspects.Forexample,an instability can occuronly if suitable conditionson the sizeof the systemare satisfied.While the genomeproducesthe morphogens,the morphogensmay lead to the appearanceof a polarity only if certain conditionson thecellas a whole are satisfied.If so,they may then act backon the genomeandlead to a morphogeneticdifferentiation.

There is a deep and unexpectedrelation between the \"chemistry\" asstudiedby molecularbiologyand the \"space-timestructure\" as determinedby the mathematical propertiesof the kinetic equationswhen variousconditionsare satisfied.It isthis relation betweenchemistry,thermodynamicsand mathematics that is so fascinating in the study of self-organization.

There seemsto be no doubt that dissipative structures play an essentialrolein the function of living systemsas we seethem today. What was theroleof dissipative structures in evolution? It is very tempting to speculatethat prebioticevolutioncorrespondsessentiallyto a successionofinstabilitiesleading to an increasing levelof complexity.Thesequestionsare discussedin Chapter17,where the conceptof competition is formulated in molecularterms.Herewe make contact with the important work of Eigen on theevolution of interacting and self-replicatingmacromolecules.

In very general terms, evolution may beviewedasa problemin structuralstability. The fluctuations involved are not fluctuations in concentrations orother macroscopicparametersbut fluctuations in the mechanismsleadingtomodificationsof the kinetic equations.To clarify our point, let us considera chemicalschemedescribingsomepolymerizationprocesswherepolymersare constructedfrom moleculesA and B that are pumpedinto the system.Supposethe polymer has the following molecular configuration

ABAB...

Supposethe reaction producingthis polymer is autocatalytic. Then if anerroroccursand a modifiedpolymer appearssuch as

ABAABBABAB...


it may multiply in the system as a result of a modifiedautocatalytic mech-mechanism. Will this new productdie out, or will it take over (with the initial

polymer dying out)?This is the type of question in which fluctuations andstructural stability play a significant role.Similar questionsare essentialto the understanding of the evolution of ecosystems,the generation and thepropagationof \"innovations,\" and similar phenomena (seeChapter 18).Mostof the other examplesstudiedin Chapter18refer to the realization ofcollective tasks in insect societies(e.g.,nest building) and provide strikingillustrations of the conceptof \"orderthrough fluctuations.\"

Note that we may always introduce fluctuations that will lead to insta-instabilities and to new typesof function and structure. In other words,no systemis structurally stable, the evolution of a dissipative structure is a self-determiningsequenceaccordingto the scheme

Function <> Structure

Fluctuations

We would like to concludethis introduction with a few general remarks.As we have emphasized,our approachcombinesboth deterministic andprobabilisticelements in the time evolution of the macroscopicsystem.At

pointsfar from bifurcation,the deterministicequationssuffice,whereasnearthe bifurcationpointsthe stochasticelementsbecomeessential.

A similar distinction was introduced sometime ago by sociologists.Forexample,Carneiro,following Spencer,emphasizedthe distinction betweenquantitative and qualitative changesin culture. He distinguished cultural

development, in which new cultural traits are coming into being, fromcultural growth. In our terminology,cultural developmentwould correspondto instabilities in which stochasticeflectsplay a basicrole, while culturalgrowth correspondsto \"deterministicdevelopments.\"We do not go intofurther detail here, as theseaspectsare analyzed in another work, wheredetailedreferencesto the work of Carneiroand Spencerare given.*

Before any interpretation of functional orderof comparablegeneralityto equilibrium order was available, \"living\" processeswere in somesensepushed\"outsidenature\" and physical laws.Onewas tempted to ascribeanaccidental character to living organismsand to imagine the origin of life

as being the result of a seriesof highly improbableevents.Now, in classical

*I.Prigogine, P. Allen, and R. Herman A977),\"The evolution of complexity and the Laws ofNature,\" in Goalsfor Mankind Report to the Club ofRome,ed.by E. Laszlo and J. Bierman.


dynamics a sharpdistinction is made between \"events\" and \"regularities.\"The laws of dynamicsdealwith regularitiesbetweenevents but not with theeventsthemselves.The eventsare the initial conditionsabout which classicaldynamics can make no statement. At most, we could use Boltzmann'sprobabilisticinterpretation of the secondlaw of thermodynamicsto ascribea probability to each possibleinitial condition.Oncean initial condition is

specified,the system will be led to its most probablestate through an ir-irreversible process.

Lifeconsideredas a result of \"improbable\"initial conditionsis,therefore,compatiblewith the laws of physics(initial conditionscan be arbitrarilychosen)but doesnot followfrom the lawsof physics(which do not prescribethe initial conditions).This is the outlooksupported,for example,by Monod'swell known book.*Moreover, the maintainance of life would appearin this

view to correspondto an ongoing struggle of an army of Maxwelldemonsagainst the laws of physicsto maintain the highly improbableconditionsthat permit its existence.The results summarized in this book supporta different point of view. Far from beingoutsidenature, biologicalprocessesfollow from the laws of physics,appropriateto specific nonlinear inter-interactions and to conditionsfar from equilibrium. Thanks to these specificfeatures, the flow of energy and matter may be usedto build and maintainfunctional and structural order.

We have insistedon the variety of the situations involvedin self-organiza-self-organization phenomena.Inevitably,the toolsand the techniquesemployedto tackletheseproblemsare extremelydiverse,and so is also the backgroundof the

potential readersof this monograph.We think, therefore, that it would beuseful to make a number of suggestionsconcerningthe way this monographshouldbe studied.

(i) For readers interested primarily in applications(e.g.,experimentalchemists,biologists,or sociologists),the thermodynamic analysis (Part I),Chapters5, 6, and Chapter7 up to Section7.5,shouldprovide an adequatebackgroundfor switching immediately to Parts IV and V, devotedto concretephysicochemicaland biologicalproblems.(ii) Readersinterestedprimarily in the generaltheory and its physicochemicalimplications are encouragedto read all of Chapter7, as well as Chapter8and Part III.On the other hand, thesereadersmight not insist in many ofthe detailsof Chapters14,15,and 17.(iii) Most of the readersare likely to find Chapters11and 12technically* J. Monod A970), LeHasardet la Necessite, Seuil, Paris.


more complicatedthan the others.We feel this is an inevitablefeature of the

present state of fluctuation theory. We did not hesitate including theseChaptersin the monograph, in order to give to the reader an up-to-datepicture of this growing and increasinglyimportant aspectofself-organizationphenomena.(iv) Owing to the fact that many results describedin the monograph arestill the subjectof active research,we found it necessaryto switch from

time to time from a \"tutorial\" style appropriatefor presentinggeneralmethodsand techniquesto the style of \"progressreport.\"Readersshould befully aware of the fact that further progressin the field is likely to occurexplosively,in the form of a true instability!

Severalaspectsof the work developedin this monograph reflecta collectiveeffort, and in this respectthe membersof the statistical mechanics andthermodynamicsgroupsin the University of Brusselsand in the UniversityofTexasat Austin haveplayedan essentialrole.We are particularly indebtedto P. Allen, A. Babloyantz, J. L. Deneubourg,T.Erneux, R. Garay, A.

Goldbeter, M. Herschkowitz-Kaufman, J. Hiernaux, W. Horsthemke,L. Kaczmarek, K. Kitahara, R. Lefever,M. Malek-Mansour,A. Nazarea,J. Portnow, M. Sanglier,J. S. Turner, J. W. Turner, A. Van Nypelseer,and R. Welch.

Moreover, we gratefully acknowledgethe fruitful discussionsand thesuggestionsand encouragementsreceivedfrom J.F.G.Auchmuty of IndianaUniversity, R.Balescuofthe University ofBrussels,J.Chanu of the Universityof Paris VII, M. Eigen of the Max-PlanckInstitute at Gottingen, P.Glansdorffof the University of Brussels,B.Hessof the Max-PlanckInstituteat Dortmund, J. Kozak of the University of Notre Dame,R. Mazo of theUniversity of Oregon,J.S.Nicolisof the University of Patras,R. Noyesofthe University of Oregon,P.Resiboisof the University of Brussels,J.Rossof the MassachusetsInstitute of Technology,Albert and Annie Sanfeld ofthe University of Brussels,R. S. Schechterof the University of TexasatAustin, D. Thomas of the TechnologicalUniversity at Compiegne,R.Thomas of the University of Brussels,and D.Walls of the University ofWaikato.

The late Professor Aharon Katzir-Katchalsky has been an activeprotagonist in the field of self-organizationphenomena.We have greatlybenefitedfrom the stimulating exchangesof ideaswe had with him.

We have beengreatly encouragedin pursuing the line of approachsum-summarized in the monograph by the continuous interest and support ofMr. J. Solvay and of ProfessorA. Jaumotte, Presidentof the UniversiteLibrede Bruxelles.


We havealsoappreciatedthe numerouscontacts wehad with the membersof the ResearchLaboratories,GeneralMotorsTechnical Center,Warren,Michigan,particularly with VicePresidentDr.P.F.Chenea,Dr.R. Hermanand Mr.A. Butterworth.

Finally, it is a pleasureto thank Mrs.S.Dereumaux-Wellensand Mrs.L.Fevrier for the difficult task of typing the manuscript and Mr.P.Kinet forhis efficient technicalassistancein the work.

This researchhasbeensupportedby the FondsNational de la RechercheFondamentale Collective (Belgium), the Belgian Ministry of Education(Ministerede l'EducationNationale et de la Culture Francaiseet Ministerede l'EducationNationale et de la Culture Neerlandaise),the R. A. WelchFoundation (Houston,Texas),and the Fondscancerologiquede la CaisseGenerated'Epargneet de Retraite (Belgium).

Parti

The ThermodynamicBackground

Chapter1

Introduction

1.1.GENERAL COMMENTS

The objectof this monograph, as stated in the GeneralIntroduction, is topresenta unified formulation of self-organizationphenomena in complexsystems,that is, systems involving a large number of interacting subunits.Experimentalevidence,as well as daily observation, show that such systemscan present,under certain conditions,a marked coherent behaviorextendingwell beyond the scaleof the individual subunit. Biologicalorder,the genera-generation ofcoherent light by a laser,the emergenceofspatial or temporal patternsof activity in chemicalkineticsand in fluid dynamics,orfinally, the function-functioning

of an animal ecosystemor of a human society, provide somestrikingillustrations of the occurrenceof such self-organizationphenomena.

In addition to the pointsdevelopedin the GeneralIntroduction, wewant to

presentin this introductory chapter somefurther comments on biologicalorder,which certainlyconstitutes the prototype ofsystemspresentingvariouspatterns of self-organizationand coherent behavior. It is well known that thestatus of living beingsin respectto the laws of physics,particularly thermo-thermodynamics, is a very controversial matter that has given rise to passionatediscussion.It will, therefore,beuseful to presenta few introductory remarksin orderto make our positionmore precisebeforewe undertake the analysisof specificproblems.

Biologicalsystemsare extremelycomplexand orderedobjects.They havethe unique ability to preserveand transmit information, in the form ofstructures and functions acquiredin the past during a long evolution. Onthe other hand, the maintenance of life, even in its simplestbacterial form,implies a continuous metabolism and synthesis of macromolecular sub-substances, as well as the regulation of theseprocesses.Thesephenomena arepossiblethanks to a highly inhomogeneousdistribution of matter in living

cells,which is the result of severalthousandsof chemicalreactions involvingcomplexfeedbacksand taking placesimultaneously in a crowdedspaceofa few fi3 (cubicmicrons).Finally, the processof developmentof a fertilized

19

20 Introduction

eggto an adult organism implies the transmission of genetic informationover macroscopicdistancesand during macroscopictime intervals.Heretoo,regulation plays a primordial role.From the first cellular division to theformation of the adult organism, through the differentiationto specializedcellsand the tissueand organ formation, all of the events must occurat theright time and place,otherwise the result would be a completely chaoticbehavior resulting in death.

Modern molecular biology has been remarkably successful in inter-interpreting this coherent behavior of living systems in terms of the structure ofthe constituting molecules.Therefore, it regardsthe functional order pre-prevailing in living beingsas a result of an architectural order.It describesvital

phenomena in terms of information, of message,of code.It assimilatesthegenetic material of a cell to the magnetic tape of a computer.The programprinted therein describesa seriesof operationsto be carriedout, with thesharpnessof their successionin time\342\200\224 briefly, everything that is related to thesurvival of the organism and to reproduction.Nevertheless,not everythingis fixed equally sharply in the genetic program.Phenomena such as theregeneration of tissues,or even memory or learning in organismswith ahighly developednervous system,illustrate a certain plasticity that permitsthe organism to acquirea supplement of information coming from theenvironment.

Doubtless, the determination of the structure of biologicalmacro-moleculesand the discovery of the genetic codehave solved a great manyproblemsin biology and have helpedto poseseveralothers on a more firmand concretebasis.However,most of theseproblemsrefer to intracellularphenomena at the enzymaticor the genetic levels.Considernow a different

kind of phenomenon, such as the developmentof a fertilizedeggto an adult

organism, the functioning of the brain of higher mammals, the immune

responsein higher vertebrates,or the major problemof biology,namely, theevolution of biopolymersand the origin of life. In all of theseexamples,what

happensis the manifestation at a macroscopic,supermolecular(and evensupercellular)levelof a seriesof events originating at the microscopicscaleof individual molecules.

Here,physicsand physicalchemistry can provide new conceptsand newideas.Indeed,both disciplinesdeal with this continual interplay betweenmacroscopiclevel and propertiesat the atomic scale,to such an extent,in fact, that the explanation of the observablepropertiesof macroscopicbodies(heat capacity, thermal conductivity, etc.)in terms of the propertiesof atoms and their interactions is the object of statistical mechanics,aspecialbranch ofphysics.Forthe purposesofour discussion,oneof the mostimportant conclusionsof this type of analysis of macroscopicsystemsisthat the macroscopicdescriptionintroduces somenew qualitative aspects.

/./ General Comments 21

Think, for a moment, of the conduction of heat acrossa fluid submitted to atemperature differenceAT. Locally, each particle undergoesa disorderedthermal motion. And yet, on the whole,becauseof the contraints experiencedfrom the outsideworld, a macroscopicquantity of energy is propagatingthrough the fluid, which to a good approximation is simply proportionalto AT. Naturally, such macroscopicpropertiesare perfectly compatiblewith the microscopiclaws of motion of the individual particles.The onlypoint is that the microscopiclanguage is simply not adequateonceoneexceedsa certain scaleof phenomena.

Our approachto the study of biologicaland other complexsystemsiscarriedout preciselyin the above-definedcontext. We start from the resultsof molecular biology to provide the analog of what we refer to above asthe \"microscopiclevel.\" We are then interested in large-scalephenomenaand in constructing modelsthat coupleamong themselvesseveralvariables,each one referring to a particular molecular event. In this way we try toobtain a satisfactorypicture ofsomemacroscopicpropertiesof living systemsreflectingthe influenceof such factorsas the size,the geometry,the boundaryconditions,and soon (seealsoremarksin the GeneralIntroduction).Let usstressthat such a program can now be envisaged thanks to the impressiveprogressof molecularbiology.

What tools are we going to use in order to carry out our program?Naturally, our basicpostulate is that all vital phenomena can be studiedwith referenceto both the laws ofphysicsappropriateto the specificnonlinearinteractionswe have to considerand to the far-from-equilibriumconditions.Furthermore, the very fact that we are interested in the modelingof macro-macroscopic phenomena introducesa most drasticsimplification,permitting us toadopt a contracted descriptionin terms of a limited number of observablessuch as the concentrations (pu...,pn) of the cellular chemicalconstituentsand the temperature T,and ignore finer propertiessuch as the distribution ofinstantaneous velocitiesorrelativepositionsof the molecules.

In this respect,our objectis to understand biologicalorder by limitingourselvesto the laws of macroscopicphysics,that is,essentially,the laws ofthermodynamics. As usual, however, it is necessaryto supplement theselaws by equationsdescribingexplicitlythe time evolution of macrovariablessuch as pu ...,pn and T The justification of theseequations (discussedindetail in Chapters2,4, and Part II)is beyond the scopeof this bookas it is,typically, a problem of statistical mechanics.In general, they have thefollowing form:

^ j j, {V2Pj},...,T,\\T,V2T,...) A.1)

where in the most general caseF, is a nonlinear functional of its variables.

22 Introduction

Typical examplesof theseequations are the various conservation equationsof matter, energy, momentum, and so on.Theseequationsdescribeonlytime evolution of averages.In addition, it is often necessaryto discussthefluctuations of the macrovariables. The necessarytools are introduced in

Part III.Having now definedour viewpoint, let us retrogresssomewhatand analyze

somenew aspectsonemight encounter in this context.We want to explainthe origin and the maintenance of the extremelysophisticatedorderchar-characterizing living beings,using the laws of macroscopicphysics.Now a deepbelief of most physicistshas always been that the evolution of a physico-chemicalsystemshouldinvariably leadto an equilibrium state that is entirelydisorderedat the molecular level.

In an isolatedsystem,which cannot exchangeenergyand matter with thesurroundings,this tendency is expressedby the second law of thermo-thermodynamics. The latter is ascertaining the existenceof a function of the macro-macroscopic state of the system, namely, the entropy S,which increasesmono-tonically until it reachesits maximum at a state known as thermodynamicequilibrium:

\342\200\224->0 (isolatedsystem) A.2)at

On the other hand, since in physicsonealways associatesorder with adecreasein entropy (for a further discussion,seeChapters4 and 8),it followsthat Eq.A.2)rules out, in an isolatedsystem,the spontaneousformation oforderedstructures.As an example,considera gas in an initial state wherea certain degreeoforderhas beenimposed,suchasby confining it to one halfof a box.Obviously,as time increasesthe gaswill very rapidly tend to occupythe wholevolumeand destroy the initial order.Moreover,this tendency is anirreversibleprocessin the sensethat the inversephenomenon cannot occurspontaneously.

Considernext a system that is at equilibrium but that can now exchangeenergy with the outsideworld at a certain constant temperature T (closedsystem).Equilibrium thermodynamics showsthat its behavior is describedby a new state function, the Helmholtz freeenergyF given by the followingrelation [seealsoEq.B)of GeneralIntroduction]:

F = E - TS (closedsystem) A.3)

where E is the internal energy.At equilibrium F is minimum. This minimumhas a simplemicroscopicinterpretation. Supposethe system may be found

on variousenergylevels.Then, at equilibrium the probability that the system

/./. General Comments 23

is at a state of energy \302\243\342\200\236

isgiven by (Landau and Lifshitz, 1957)

A.4)

where kB is Boltzmann's constant. For sufficiently low T, the above twoequationsimply that only the levelsof low En will bepopulatedappreciably.As T increases,the contributions of E and S in F becomecomparable,andthe populationsof the various energetic levels tend to becomeequal.Weseethat in a nonisolatedsystem there existsa possibilityto form low-entropy ordered structures provided T is low. This orderingprinciple,referred to in the GeneralIntroduction as the Boltzmann orderingprinciple,is responsiblefor the appearanceof structures such as crystals and alsofor phasetransitions.

Is this principlesufficient to explain the origin of biologicalstructures?It is easyto realizethat the probability A.4)to have,at ordinary temperatures,the condensationof a macroscopicnumber of moleculesgiving risenot onlyto the highly orderedstructures (macromolecules,membranes,etc.)char-characterizing life but also to the coordinatedfunction of the thousandsofchemical reactionsper cell, is practically vanishing. As an example,let usconsidera biologicalmacromoleculesuch as a protein chain of about 100amino acids.In nature there exist20typesof amino acid.It is known that thephysiologicalfunction of a protein is fulfilled only if the arrangement of theconstituting amino acidsalong the chain followsa well-definedorder.Thenumber of permutations necessaryto obtain this arrangement starting froman arbitrary initial distribution is:

N ~ 20100^ 10130and we assumethat all correspondingconfigurations are equally probablea priori.Thus, even if a change of structure (e.g.,through a \"mutation\of this \"initial\" protein could occurevery 10~8s (which is certainly toorapid), one would need

t ~ 10122sto form the desiredprotein.As the age of the earth is \"only\" of 1017s, werealize that the spontaneousformation of this (rather small) protein must beruled out (Eigen,1971).

In conclusion,the apparent contradiction between biologicalorder andthe laws of physics\342\200\224particularly the second law of thermodynamicscannot be removed as long as onetries to understand living systemsby themethods of equilibrium thermodynamics.As we shall see,the very conceptof the probability of various statesas expressed,for example,through Eq.A.4),has to be reconsidered.

24 Introduction

1.2.OPENSYSTEMS

To advance further, onemust now analyze more closelycertain character-characteristic featuresofbiologicalsystems.We havealready insistedon the generalityof the phenomenon of metabolism.From the simplestbacterial cell to man,maintenance of life requiresa continuous exchangeof energy and matterwith the surrounding world.Living organismsbelong,therefore,to the classof open systems,whose thermodynamic theory has been developedbyDe Donder'sschoolin Brussels(Defay, 1929).Later von Bertalanffy A932)and SchrodingerA945)insistedon the necessityto treat biologicalsystemsas opensystems.Also in 1945,Prigogineformulated an extendedversion ofthe secondlaw applicableto both closedand opensystems.We discusshisapproachin subsequentchapters;herewe only presentsomeremarks.

Let us considerthe entropy change dSduring a time interval dt. We maydecomposeit into the sum of two contributions (seealsoFig.1.1):

dS= deS+ d,S A.5)

deSis the entropyflux dueto exchanges(ofenergyormatter) with the environ-environment and d,S is the entropy production due to the irreversible processesinsidethe system,suchasdiffusion, heat conduction, and chemicalreactions.The secondlaw implies:

d,S>0 (=0 at equilibrium) A.6)

Foran isolatedsystemdeS= 0, and A.6)yields

dS = diS>0 A.7)

Summarizing, opensystemsdiffer from isolatedonesby the presenceof flow

terms, deS, in the entropy change.Contrary to d{S,which can never be

Figure 1.1.Entropy flux and entropy production in an opensystem.

1.2.Open Systems 25

negative,theseterms donot havea definitesign.As a result,onemay imagineevolutionswhere the systemattains a state of lowerentropy than the initial

one:AS = dS<0

path

This state, which from the point of view of the equilibrium relation A.4)would be highly improbable,can be maintained indefinitely provided the

system is allowed to attain a steady state such that dS = 0 or

deS= -d,S<0 A.8)

Thus, in principle,if wegive a systema sufficient amount of negativeentropyflow we enableit to maintain an orderedconfiguration.As Eq.A.8)shows,this supply must occurunder nonequilibrium conditions,otherwise bothdtS and deSwould vanish. We can already feel the possibilityof a new,nonequilibrium order principle.Moreprecisely,nonequilibrium may be asourceof order.This is of obvious interest for living systems;the biosphereas a whole is a nonequilibrium system, as it is subjectto the flow of solarenergy. In the cellular level, cell membranes or the various biochemicalreaction chains are subject to concentration gradients of chemical con-constituents. Moreover it is quite common to have elementary reaction steps,such as thoseinvolving regulatory enzymes,that are practically irreversible(Atkinson, 1968).

Needlessto say, thesesimpleremarkscannot sufficeto solve the problemofbiologicalorder.Onewould likenot only to establishthat the secondlaw(rfjS >0) is compatiblewith a decreaseof the overall entropy (dS<0),but also to indicate the mechanisms responsiblefor the emergenceandmaintenance of coherent states.Our principal goal in this monograph is toprovide the elementsof a responseto this question.Oneof our conclusionsis that there existsystemsshowingtwo typesof behavior,namely, a tendencyto a disorderedstate under certain conditionsand a coherent behavior underothers.The destruction of order prevails in the neighborhoodof thermo-dynamic equilibrium. Creation of order may occurfar from equilibriumprovided the systemobeysto nonlinear laws of a certain type. In this case,thespontaneousappearanceof order is accompaniedby an instability of thestatesshowing the usual thermodynamic (i.e.,disordered)behavior. Tradi-Traditionally, thermodynamics was limited to equilibrium or near equilibriumand could only treat the first type of behavior. Recently an extension ofirreversiblethermodynamics to far-from-equilibriumconditionswas devel-developed that permits treatment of both \"destruction\"and \"creation\"of orderwithin the same formalism (Glansdorffand Prigogine, 1971).This theory isthe subjectof Chapters2-4.

Chapter 2

ConservationEquations

2.1.OPENSYSTEMSAT MECHANICAL EQUILIBRIUM

Beforewe undertake a detailed thermodynamic study of systems that mayexhibit orderedbehavior it is necessaryto define the classof situations in

which we are primarily interested in this monograph.Considera reactive mixture of n constituents Xl,...,Xn. The system is

open to the flow of chemicalsfrom the outsideworld that are converted toXu ...,Xn within the reaction volume V. In most of the subsequentchapterswe make the following assumptions:

(i) The systemis isothermal (T = constant).(ii) The system is at mechanical equilibrium (no mass flow) and is notsubjectto external fields.

(iii) The concentration gradients are not too high in the sense that the

compositionvariablesp1,...,pndonot vary appreciablywithin distancesofthe order of the mean free path. This restriction implies,in particular, theabsenceof interfacesinsidethe volume V.

(iv) The system is subjectto time-independentboundary conditions.

The motivation behind theserestrictionsis fairly obvious.The behaviorof living cellsis largelydeterminedby chemicaltransformationsand diffusion

phenomena of macromoleculesor of small metabolites such as ATP orcAMP. Most of these phenomena do not seem to involve convection.Moreover,living organismsoften developcomplexhomeostaticmechanismsmaintaining the temperature at a constant level. Naturally, certain phe-phenomena such as bloodcirculation in higher animals or the locomotion ofmicroorganismsare specificallymechanicaland are beyond the frameworkdenned by our previous assumptions.On the contrary, all fundamentalphenomena listed in Chapter1 can still be analyzed without great sacrificeseven if one adopts these simplifications.Referencesto the more general26

2.2. The Mass-balance Equations 27

situations involving, for example,the coupling betweenmechanicalmotion,thermal effects,and chemicalphenomena are presentedwheneverof interest.

What are the variables describingthe instantaneous state of a systemobeying the above conditions,and how can one calculate their values?Firstly, assumptions(i) and (ii) permit decouplingof the temperature and theconvection velocity from the compositionvariables (e.g.,partial densities)of X!,...,Xn. In orderto calculate the latter, we must establishequationsdescribingthe evolution of the quantity of matter insidea multicomponentreacting mixture. This point is developedin Section2.2.

2.2.THE MASS-BALANCE EQUATIONS

The principleof total mass conservation of an arbitrary (nonrelativistic)systemcan beexpressedin the following form (seeFig.2.1):

dm\342\200\236 \342\200\236 dem

\342\200\224 = massflow through surface \302\243= \342\200\224\342\200\224 B.1)

In other terms, there is no productionof massinside V, regardlessof theprocessesoccurring therein. The situation is different if onefocuseson themassof each chemicalconstituent separately.Onehas

dt

denij dt m-s~dt dt

(/= 1,\342\200\242\342\200\242\342\200\242,\302\253) B.2)

where

~y^ =productionof Xj resulting from chemicalreactions.

Figure 2.1.Open system exchanging sub-'

\342\226\240>pn stances 1....,n with external world through

surrounding surface 1,where jf, . . . , jj; and

(i\\, \342\226\240\342\226\240\342\226\240, pi represent, respectively, values ofdiffusion fluxes and concentrations of 1,...,n

on surface.

28 Conservation Equations

Supposethat constituent Xj participatesin r chemical reactionsand let

Wp (p = 1,...,r) be the individual rates.The latter can be related to the

compositionvariables {pj}provided one assumesthat the intermolecularinteractions do not enter explicitly in the expressionfor the frequency ofreactiveencounters.This assumption extendsthe notion of an ideal mixturefamiliar from equilibrium thermodynamics,accordingto which each chemi-chemical in the mixture followsthe laws of perfectsolutions.

As an example,for a homogeneousreaction in a small volume AV:

A + X: *'> A + B B.3a)

the rate will be:

Similarly, for

onewill have:

A -h 2Xj -

\\W2\\

= *i

k2

= k2

AVpAPj

-> A + B

AV pAp)

hC

B.3b)

B.4a)

B.4b)

We observethat onemoleculeof Xj disappearsin the first reaction, andtwo in the second.We shall say that the stoichiometriccoefficientsv of Xj in

thesereactionsare \342\200\224 1 and \342\200\224 2, respectively.The total massof Xj disappear-disappearing

in this caseis:

^= -kx AV PaPj - 2k2 AV pAp] B.5)

In general:

^\342\204\242l = Y v- Wdt ph,

Jp \"

and Eq.B.2)takesthe form:

dm, _ dems + yv w ^.6)dt dt p= i

JP p

It will beconvenientto switch to massdensity variables,which wedenoteas

Pj,by reducing Eq. B.6)per unit volume V Moreover,we introduce thereaction rate perunit volume,wp, through the relation

Wp= J dV wp B.7)

2.2. The Mass-balance Equations 29

We then have:

Note that onemay alsoexpressEq.B.8)in molar variables,particledensities,or mole or massfractions, that is, in terms of the ratios p/Z;Pj (Prigogineand Defay,1951).

In general,a chemicalsystemof the type weare interestedin hereisspatiallyinhomogeneous.The flow term demj/dt then describesthe distribution ateach point insideV, of the matter penetrating into V through the surface T,.Let jj be the correspondingdiffusion flux. Then

provided the normal vector n is oriented outward of V (Fig.2.1).The surfaceintegral can be transformed by the Gaussiandivergencetheorem:

J dLn \342\200\242j? = J dV div j, B.10)

Combining with B.8),onemay obtain a local equation for p-s by requiringequality of the integrands at each point of V. The derivatived/dt commuteswith the integral sign, thanks to the assumptionabout mechanicalequilibrium.Thus, we obtain (DeGrootand Mazur, 1962):

^=-divj;+\302\243v;,,W,, B.11)As we saw earlier in this section,the wp values are even in simplecases

nonlinear (polynomial)functions of {pj}values. Consequently,Eqs.B.11)necessarilyconstitute a system of nonlinear partial differential equations.In order to determine the solutionsof theseequationsit is necessarytosupplement them by suitable boundary conditions(Sneddon,1957).Theseconditions,which reflect the extrinsic constraints acting on the system,can assumea considerablevariety. In subsequentchapterswe are interestedmore specificallyin boundary conditionsprescribingthe fluxes {jy},or theconcentrations {pj}on the surface T,. Theseconditionsare familiar from

boundary-valueproblemsin fluid dynamics,and electromagnetictheory andwill be referredto, respectively,asNeumann and Dirichletconditions.

Now the characteristic of nonlinearity implies that, even in relativelysimple cases,the solution of Eqs.B.11)is an arduoustask.At the presenttime there is no general mathematical theory of these equations.It is,therefore, important to undertake a study of their qualitative behavior,

30 Conservative Equations

in other words,of that part of their propertiesthat is not strongly dependenton the detailsof the chemicalstepsand diffusion.

Part IIis devoted to the mathematical aspectsof this question.Forthetime beingwe investigate how far onecouldgo in the qualitative under-understanding of systems describedby Eq. B.11)by the methods of thermody-thermodynamics of irreversibleprocesses.

Chapter3

Thermodynamicsof IrreversibleProcesses:TheLinearRegion

3.1.GIBBS'SFORMULA: ENTROPY PRODUCTION

We have already insistedin the previous sectionson the specialstatus ofentropy in physics.Entropy expresses,in a very general way, the tendencyof isolatedphysicochemicalsystems to evolve irreversibly to equilibrium.Moreover, it provides a measure of the orderprevailing in the system.We

shall, therefore,extend the conceptof entropy in a way applicableboth toopen systemsand to situations far from equilibrium.

The first problemwe face in such an extension is to find the variables onwhich entropy depends.Naturally, the compositionvariables pl5...,pn isamong them. But it is quite conceivable to have situations in which the

thermodynamicstate would dependon additional variables.In a nonuniform

system,which interests us here primarily, pu ...,pn vary in spaceand time.Thus one couldconsidera functional that dependsnot only on {pjbut

alsoexplicitlyon their gradient in spaceand on their variation in time:

S = sL,...,p.;{VpJ;{V2pJ,.\342\200\242.; fei...;r,t) C.1)

However,the generalityof this Ansatz equation makesits usevery difficult.

As a matter of fact, there is no guarantee that an entropy function defined in

terms of such a relation has always a macroscopicmeaning arbitrarily farfrom equilibrium, that is, for arbitrarily high values of the gradients {Vp,}and of the time derivatives{dpjdt}.Consequently,in orderto selecta func-functional form of the entropy that is both meaningful and pertinent to the typeof problemwe must analyze in this monograph, we appealto someresultsfrom statistical mechanics of irreversible processes(Prigogine,1949b),which we simply enumerate without giving the proofs.

Onecan show that condition (iii) of Section2.1implies,from the micro-microscopic point of view, that the distribution functions of particle momenta and

31

32 Thermodynamics ofIrreversible Processes:The Linear Region

relative positionsin the system are maintained, locally, closeto the equi-equilibrium Maxwell-Boltzmann distribution. This means that dissipativeprocessesarising primarily from the frequent elastic collisionsbetweenmoleculesconstitute the dominant contribution to the moleculardistributionfunctions, which is only slightly perturbedby the macroscopicconstraints(e.g.,gradients) imposedon the boundaries.A similar restriction is imposedon chemicalreactions:weshallsupposethat reactivecollisionsbesufficientlyrare events (energyof activation sufficiently large) so that elasticcollisionsmay restorethe Maxwell-Boltzmanndistribution.

Under these conditionsthe entropy of a nonequilibrium state can bedefined,and contains a dominant contribution that is a local functional ofthe state variablespu ...,pn:

S = S(Pl(r,*),...,pn(r, 0) + Orfmeanf7ePfY1 C.2)LA macr length J J

The functional relation C.2)has the samestructure as in thermodynamicequilibrium, exceptthat now it has to be understoodlocally,rather than forthe entire system.Thus, entropy becomesan implicit function of timethrough p1,...,pnvarying accordingto the balance Eqs.B.11).

Let us indicate here a few systemsfor which theseassumptionsapply andthat still may exhibit typical far-from-equilibriumbehavior. Firstly, quitecomplexsystems of chemical reactionswith highly nonlinear kinetics canbe treated by a local theory provided the rate of elasticcollisionsremainslarger than the rate of reactivecollisions.This is generallytrue in a not-too-rarefiedsystem,in particular for all biologicalprocesses,that occurin densemedia.Similarly, all convectiveand transport effectsdescribedby the Navier-Stokes equations, including hydrodynamic instabilities, are within thedomain of validity of the localdescription.On the contrary, suchphenomenaas shockwaves,flows in rarefied systems,or plasticdeformations of solidsare at presentbeyond the range of this approach.Let us emphasize onceagain that no proof of existenceof a macroscopicentropy describingthesesituations has as yet beengiven.*

The results mentioned so far define the domain of validity of localthermodynamics(sometimesreferredto as localequilibrium thermodynamics)

*Nonequilibrium statistical mechanics leads, indeed, to the definition of a functional that

changes monotonously in time and may beused for the definition of entropy. However, this

functional is expressed in terms of the statistical distribution functions and not in terms ofthe macroscopicvariables such asthe {pj(Prigogine, George,Henin, and Rosenfeld, 1973).Thisresult generalizes the well-known Boltzmann kinetic interpretation of entropy. There also,entropy may bedefined in terms of the velocity distribution function. However, it is only when

locally the distribution functions arecloseto their equilibrium form that Boltzmann's quantitycan beexpressed in terms ofmacroscopicquantities (Prigogine, 1949b).

3.1.Gibbs's Formula: Entropy Production 33

or, equivalently, the domain of a local macroscopicdescriptionof matter. Inorder to expressthese results in a more quantitative form we define the

entropy density sv as

S= [dVsv C.3)

with

sv = sv(Pl(r, t),...,pn(r, t)) C.4)where sv is now an ordinary function. Taking time derivatives of C.4)weobtain:

dsc_ y fdsv\\ dpj~dt~ y \\dpj~dt

Now, using the assumptionsindicated above, we may introduce as in

equilibrium thermodynamicsthe relation

= -y C-5)

wherep}is the chemicalpotential per unit massof constituent Xj (Prigogineand Defay,1951).Thus

This relation is a particular caseof the celebratedGibbsformula, which atequilibrium has the form (Prigogineand Defay,1951):

TdS= dE + p dV \342\200\224

Yj Mi drtij C.7)i

where E is the internal energyand p the pressure.We may now combineEqs.C.6)and B.11)to deducea balance equation

for sv:

ds,,

iTSf, +div^j,.-Ej,-vf C.8)

The expression

MP=-ZWjP C.9)


occursfrequently in nonequilibrium thermodynamics. It is the affinity ofreaction p (DeDonder, 1936).To seethe meaning of this quantity it isinstructive to consider,as in Section2.2,an ideal mixture. According toequilibrium thermodynamics one has:

j C.10)The affinity becomes:

The quantity

K\302\260\302\253(T, p) = expf-X^njvj^C.11)

is known in chemistryas the equilibrium constant.Accordingto the law of massaction:

KeqG;P) = FIP^ec, C-12)j

where p, eq are the concentrations of the reactants at chemicalequilibrium.Finally, 01p takesthe form:

[\\py C12a)

At equilibrium, the law of massaction imposes(Xp= 0.Thus, the affinity

measuresthe deviation of a chemicalreaction from the state of equilibrium.It plays a rolesimilar to that of \\/ij, which measuresin transport theory thedeviation of the distribution of matter from the uniform state.

Comingbacknow to C.8)we obtain:

ot j T j I p I

This relation has a remarkablestructure. It may be decomposedinto two

parts:

-divJs = -divS^jj C.14a)j i

and

ffs -Vj(-V + y^w0 C.14b)

S.I.Gibbs's Formula: Entropy Production 35

We observethat a is a bilinear form of the quantities j,-, wp, which can bethought of as the flows, Jk associatedwith the various irreversibleprocesses,and of the quantitiesV(^/T),01.JT,which can bethought ofasthe generalizedforces,Xk giving riseto theseflows:

o= Y,JkXk C.15)k

with

Jff = j,; Xff= -V|J\302\253act

= wp; X\342\204\242\302\253= -^ C.16)

To better understand the differencebetween the Js and a parts of C,14),we integrate this relation over the volume. After a partial integration andapplicationof the Gaussiandivergencetheorem [seealsoEqs.B.9)and B.10)]we obtain, using alsothe condition of mechanicalequilibrium:........ ... . .\342\200\236 .

\\dV a\342\200\224 = _ fdFdiv Js + \\dV a = -\\ dl.n-Js+ \\ d

We arrive, therefore, at the decomposition,developedin Chapter1,of the

entropy change into a flux term and an entropy production term. As deS/dtis relatedto the constraints exertedby the outsideworld, we cannot imposeany restriction on this part ofdS/dt.In contrast, the secondlaw [seeEq.1.6]stipulates that the entropy producedinsidethe systemcan neverbenegative:

d{Sdt

The validity of the local formulation of irreversibleprocessesimplies,then:

a >0

This relation leadsto the following classification:

o>0:irreversibleprocessproducingentropy C.18a)cr = 0:reversibleprocessconnecting two equilibrium states C.18b)

In conclusion,we have expressedthe entropy balancein terms of flows Jkand forcesXk, which must beconnectedby inequality C.18a).To go further


it is necessaryto seehow the flows Jk, which in principleare unknown

quantities, are related to the forces Xk which are known functions of thecompositionvariables,provided local thermodynamicsremains valid.

3.2.PHENOMENOLOGICALRELATIONS:THE LINEAR RANGE

OF IRREVERSIBLE PROCESSES

We have already observedthat at thermodynamic equilibrium the general-generalized forcesXk vanish identically:

VI \342\200\224 I = 0 (system at uniform composition)

af = 0 (chemicalequilibrium) C.19)On the other hand, by definition at equilibrium there is neither macroscopictransport of matter nor an overall production of constituents \\,...,nfrom the chemicalreactions.We concludethat:

Jekq= 0 C.20)

From Eqs.C.19)and C.20)it is natural to postulate that near equilibrium,that is, when the generalizedforces remain sufficiently \"weak,\" onemight

expandthe flows in power seriesof Xk:

X, +I\302\243 y%-)XtXm + - C.21)

(/0 Z lm \\0A! (\"Wo

The first term of this expansionvanishes identically owing to Eq. C.20).The contributions of the third and subsequentterms can be neglectedprovided the systemremains near equilibrium.The remaining terms yield:

Jk = YJLklXl C.22)i

where we have set:

($)The above two relations define the linear range of irreversible processes.The phenomenologicalcoefficients Lkl are entirely determined by theinternal structure of the medium, independently of the constraints appliedon the system.They may depend,however, on the state variables such astemperature, pressure,and composition.

Thanks to the phenomenologicalrelations C.22),the mass-balanceequa-equations B.11)becomeentirely closed,enabling explicit computation of thevaluesof the generalizedforcesor, equivalently, thoseof the state variables.

3.2. Phenomenologkal Relations: The Linear Range ofIrreversible Processes 37

Note that the rigorousjustification of Eq.C.22)is beyond the frameworkof thermodynamics and belongs,rather, to statistical mechanics.Thelatter also permits better specification of the domain of applicability oflinear laws.One can show that all transport phenomena are describedsatisfactorilyby such relations,provided the macroscopicgradients vary ona scale lh, which is larger than the mean free path lr (Prigogine, 1949b):

/* > K C.24)We recover, therefore, the conditionsfor the validity of local thermo-thermodynamics itself, basedon the notion of local equilibrium (seeSection3.1).Alternatively, as far as transport phenomena are concerned,relations C.22)is as general as the local formulation of thermodynamics.

The situation is entirely different for chemical reactions.As an example,considerthe step

A <

*'> B C.25)

We have [seeEqs.C.13)and B.6)]:

C-26)or:

I\302\243)<(\302\243))C27)

In order to recovera linear law such asC.22)it is necessaryto assume

a<kBT

Then, expanding the exponential in Eq.C.27),onefinds:

i^ C.28)kBT

This condition restrictsthe validity of linear relations to situations cor-corresponding to the immediate vicinity of equilibrium or to reactionsproceedingwith an extremely low activation energy. In general, suchconditionsare not satisfied in realistic systems.Thus, in order to set up asatisfactory descriptionof chemical reactionsit is necessaryto extend the

theory to the nonlinear domain of irreversibleprocesses.This extension iscarriedout in Chapter4.Onecan alsoshow(Glansdorffand Prigogine,1971)that such a generalizationof linear theory is necessaryto tackleother prob-problems as well, in particular the spontaneousonset of convection, which isruled out by the linear relations C.22).


Let us now examinethe effectof the linear relations on the structure of theentropy production.From Eqs.C.15)and C.22)oneobtains:

kl

We want this inequality (imposedby the secondlaw) to remain valid for allpossiblevalues of the generalizedforces.We alsowant the equality sign tocorrespond to the state of thermodynamic equilibrium: {Xk} = {0}.Obviously, theserelations will imposeconditionson the phenomenologicalcoefficientsLkl. In different terms, the couplingsbetween irreversible pro-processescannot be arbitrary but must remain compatiblewith inequalityC.29).

Now, relation C.29)defines a positive definite quadratic form. Thisimposesthe restriction on the matrix of the coefficientsLk, to be positivedefinite, that is, to have eigenvalueswith positive real parts.An alternativecondition can beobtainedif one observesthat a quadratic form can alwaysbe symmetrized. Thus, the validity of inequality C.29)imposesthat alleigenvaluesof the symmetricmatrix:

LS= UL + Lt) C.30)

where LT is the transposedof L = (Lkl), must be positive.Among the explicit criteria assuringthesepropertieswe may quote the

following theorem (Beckenbachand Bellman,1965).

Theorem. A necessaryand sufficient condition that a symmetric matrix

Lsofmatrix elements/-7A < i,j< n) bepositivedefinite,is that the determi-determinants |Lf | satisfy the inequalities:

> 0 C.31)

In particular, this theorem impliesthat the diagonal elements /-, mustbe positive.According to C.30)it follows that the phenomenologicalco-coefficients L,,are themselvespositive:

Lit >0 C.31a)As an example,considerthe caseof two coupledirreversibleprocesses.

The entropy production,Eq.C.29),becomes:

kl

k

l\\ 2

'k2

= 1,...,n

Ilk

a = LXXX\\ + (L12+ L21)X1X2+ L21X\\\\

3.3. Symmetry Properties of the Phenomenological Coefficients 39

We regardthis expressionasa trinome with respectto the variable (XJX2).The well-knownalgebraiccondition for positivedefinitenessis,then, that thecoefficientof the highest power of (X1/X2)bepositive and the discriminantbenegative:

Ln > 0

(Li2+ L21J- 4LnL22<0

or, in terms of the symmetric matrix Ls:

/'\342\200\236

>0

Theseare preciselythe conditionsC.31)requiredby the general theorem ofpositivedefiniteness.

3.3.SYMMETRY PROPERTIESOF THE PHENOMENOLOGICALCOEFFICIENTS

The phenomenologicalcoefficientsC.23)introduce into irreversiblethermo-thermodynamics a great number of parameterswhosevalues cannot be obtainedwithin the framework of a macroscopictheory. Fortunately, it is possibletoreducethe number of independentphenomenologicalcoefficientsby appeal-appealing

to symmetry arguments. We examine, successively,the influence ofspatial and temporal symmetries.

SpatialSymmetries

Accordingto Eq.C.22),each Cartesiancomponent of a flow Jk may depend,in principle,on the Cartesiancomponentsof all generalizedforcespresent.On the other hand, going back to the explicit propertiesof the entropyproductiondiscussedin Section3.1,we seethat the irreversibleprocessesofinterest can be decomposedinto two quite different classes,namely, vectorphenomena (e.g.,diffusion) and scalarphenomena (e.g.,chemicalreactions).If thesetwo kinds of processeswerecoupled,then a chemicalreaction wouldbecapableof giving risespontaneously to an orienteddiffusion flux, even in

the absenceof a systematic initial concentration gradient. Intuitively, it isobvious that such a coupling is to be ruled out in an isotropicmedium. Inother words,the causesof different phenomena cannot have more elements


of symmetry th^rthe effectsproduced.*In fact, this property can bedemon-demonstrated explicitly on both purely macroscopicgroundsand kinetic theoryin the linear approximation.

It is important to realize that a systemdescribedby linear laws is alwaysisotropicas long as the equilibrium state itself is isotropic.In contrast,ifi the nonlinear range the property of isotropy is lost independentlyof thestructure of the medium at equilibrium. In this casethe Curie propertydoesnot apply.As weseein Part IIthis may leadto spectacular\"symmetry-breaking\" phenomena correspondingto the emergenceof spatial patternsin a previouslyuniform medium.

In conclusion,entropy productioncan bedecomposed,in the linear range,into two entirely uncoupledparts:

a =(Tch + ad C.32)

with separately positivecontributions:

C.33)

TemporalSymmetries:The OnsagerReciprocityRelations

Despite the spectacularreduction of the number of phenomenologicalcoefficientsachievedby the analysisof spatial symmetries,there still remainn2 independentcoefficientsfor diffusion and r2 coefficientsfor chemicalreactions.

In 1931,Onsager discoveredadditional relations between these co-coefficients. He showedthat in the linear domain of irreversibleprocessesthematrix of phenomenologicalcoefficientsis symmetrical:

Lu = L,k C.34)In other words,the increaseof flux Jk arising from a unit increaseof forceX, (keepingXj\302\261l constant) is equal to the increaseof flux J,arising from aunit increaseof Xk. Relations C.34)are known as Onsager'sreciprocityrelations.

In his original derivation Onsagerappealedto considerationsbasedonfluctuation theory and stochasticprocesses.In particular, he showedthat

* The influence ofspatial symmetries on coupling was first studied by Curie in relation to someproblems in crystal physics such as piezoelectricity. It was then applied to nonequilibriumsystems by one of the authors (Prigogine, 1947).

3.4. Stationary Nonequilibrium States 41

the reciprocity relations are rigorous consequencesof the property ofdetailed balance or, alternatively, of the time-reversal invariance of the

elementary stepsassociatedwith the various irreversible phenomena.Asthe proof of Onsager'sreciprocity relations is presentedin many textbooks(e.g.,De Groot and Mazur, 1962),we do not reproduceit here.However,additional comments are made in Part IIIin connection with the theory offluctuations around nonequilibrium states.

3.4. STATIONARY NONEQUILIBRIUMSTATES

We are now in a position to apply the propertiesof phenomenologicalcoefficientsto the analysis of nonequilibrium states.Among these,the so-calledsteady states play a privileged role.Indeed,in a great number ofphysicallyinterestingproblemsa systemeither evolvesin the neighborhoodof such statesor attains a steady state for sufficiently long times. Also, themathematicalanalysisof steadystatesis obviouslysimpler than that of time-dependentstates.

By definition, a system is at a steady state if the state variables\342\200\224in ourcasethe compositionvariables {pj\342\200\224 do not evolve in time. Within the do-domain of validity of local thermodynamics, this condition implies that thelocal state functions like the entropy density sv or the entropy productiondensity oare alsotime independent.The resultsof Section3.1lead,then, tothe following relation:

deS= -d{S<0 C.35)Thus, in order to maintain a steady nonequilibrium state it is necessarytopump, continuously, a negative flow of entropy of magnitude equal to thevalue of the internal entropy production.Note that in a systemat or near asteady state the entropy flux cannot generallybe imposedarbitrarily fromoutsidebut becomes,instead, a functional of the state of the system.

It is a matter of observation that a system subjectto time-independentconstraints reaches,after a sufficiently long time, a steady state.Considerfirst the caseof constraints, compatiblewith the maintenance of an equi-equilibrium state.The equilibrium state attained in this way is a particular caseofa steadystate.Imaginenext that the constraints are progressivelymodifiedto shift the state of the system away from equilibrium. By continuity, oneexpectsthe new state, attained asymptotically,to be,again, a steady state.

Now, in a system subjectto usual types of constraint, the only spatialinhomogeneities that can arise at equilibrium are those induced by thepresenceof external forcessuch as gravity and by the boundary conditions.In the absenceof such effects,the balanceequationsB.11)imply that the


contribution of chemical reactions and of diffusion to the evolution ofmacrovariables vanish separately.Thus, by the samecontinuity argumentas before, oneshouldexpectsteady states sufficiently closeto equilibriumto exhibit similar properties.As weseein Chapter4 and Part II,the situationmay change drastically for systemsmaintained beyond a critical distancefrom equilibrium.

3.5.THEOREMOF MINIMUM ENTROPY PRODUCTION

We want now to study somequalitative propertiesof steady nonequilibriumstates.The mass-balanceequations B.11),supplementedby the linear laws,takethe form:

|i= divELijV^+Ev1,W^ C.36)

with

Lu = Ljt, \\pp.= l,p C.37)

The total entropy productionbecomes*[seeEq.C.14)]:

[dv=\\dVa

= ~ [dvW L,jWiij+\302\243 lPP- MP aj\\ >0 C.38)

\342\226\240j'

\342\200\242> Lu pp' J

We are interestedin the way the systemevolvesto the steady state.To this

end,weevaluatethe derivativedP/dt.In performingexplicitlythe time deriva-derivative in Eq. C.38),we admit that the phenomenologicalcoefficientsLti, lpp.remain constant, in other words, that their values depend only on the

equilibrium parameters.This assumption is more restrictive than linearity.Supposefor a moment that LV] dependson a compositionvariable pa.Onewould have:

dpj itAccording to Eq.C.36),the time derivative dpjdt would introduce higherorder terms, rendering dP/dt a cubic function of the generalized forces.Supposingthat the latter remain weak, this contribution couldbeneglected.Alternatively, the assumption of constant phenomenologicalcoefficients,alsoreferredto as \"strict linearity\" condition, is equivalentto an assumptionof \"weak amplitudes,\" whereas the assumption of linear laws implies\" weak gradients.\"

* We recall that we areinterested here in isothermal systems at mechanical equilibrium.

3.5. Theorem ofMinimum Entropy Production 43

Bearing thesepoints in mind oneobtains,using the definition C.9)of thechemicalaffinity:

.-, d/l; ^ , _, dillI \\//i \\7 . \\ / fit \\)I j: \342\200\242t it; V ~~Z~ /

'\342\226\240no' o' in ^

Accordingto local thermodynamics ti{ = //,({p7}),that is,

-\302\243

=Z[\302\243-)

ir C-39)

Thus:

dP 2 r[\342\200\236 \\7^^A ^Pk V (^^l\\/7 ^Pj

In the first term we perform a partial integration. The divergenceterm istransformed to a surfaceintegral:

By imposingtime-independent boundary conditions\342\200\224in order to enablethe system to evolveto a steady state\342\200\224we cancelthis term automatically.The remaining contributions yield,using alsoEq.C.36):

dt T2J\"'jj\\dPjJdt dt

At this point wemay invokesomepropertiesofequilibrium state functions,also referred to as thermodynamic potentials. From classicalthermody-thermodynamics oneknowsthat in an open system at a state of stableequilibriumthe generalized thermodynamic potential 0 = 0G;V, {/i,})is a minimum

(Landau and Lifshitz, 1957):

(<50)eq= 0 (<520)eq>0 C.43)

Introducing the density </>\342\200\236 through

0 = Lv(j)L,

one obtains for an isothermal system in the absenceof convection:

S(f>v=

Yj Pi <>Hi C-44)i


and

= S6Pi:5fit = E (~) SPidP]>0 C.45)

where the inequality followsfrom the secondrelation C.43).Inequalities such as Eq. C.45)are generally referred as thermodynamic

stability conditions.We do not go into further detailshere concerning suchthermodynamic stability conditions.This theory has been initiated byGibbsand is presentedin many textbooks(for a recent presentation, seeGlansdorff and Prigogine, 1971).Let us only mention that in addition toEq.C.45),familiar thermodynamic stability conditionsare:

where the specificheat at constant volume and the isothermal compres-compressibility x are positivequantities.

Returning to Eq. C.45),we notice that the compositionvariations t)p,

appearingthere are arbitrary. They may, therefore,representthe variationsof p( arising during the time evolution of these quantities. This remark,together with the validity of local thermodynamicsdiscussedin Section3.1,leadto the conclusionthat the structures of the two quadratic forms C.42)and C.45)are identical.Thus, one may assertthat

Figure 3.1.Time evolution of entropyt production in linear range.

3.6. Impossibility ofOrdered Behavior in the Linear Range ofIrreversible Processes

in other words(Prigogine,1945,1947):

\342\200\224 <0 away from the steady state

\342\200\224 = 0 at the steady stateat

45

C.46)

Figure 3.1representsthe time variation of P.In summary, we have shown that linear systemsobey to a general in-

inequality implying that at a steady nonequilibrium state, entropy productionbecomesa minimum, compatiblewith the constraints appliedon the system.

3.6.IMPOSSIBILITYOF ORDEREDBEHAVIOR IN THE LINEAR

RANGE OF IRREVERSIBLE PROCESSES

A physical system is subject,inevitably, to perturbations of various kinds.Thesecan beeither external excitationsarising from a random or systematicvariation of the environmentalconditionsor internal fluctuations generatedby the systemitselfas a result of the molecular interactions and the randomthermal motion of the particles.As a result, the systemdeviatescontinuously

\342\200\224 although usually weakly\342\200\224from the macroscopicbehaviordescribedby thebalanceequations of the thermodynamic macrovariables.

Imaginenow a system initially at a steady state.The variousperturbationsacting on it induce deviations from the steady-state regime {p\302\260}

and driveit to time-dependentstates{pi+ Spk(t)} (seeFig.3.2).By virtue of the mini-minimum entropy-production theorem, the entropy-production density of the

p\302\260

Figure 3.2. Illustration of theorem of minimum entropy production; p\302\260

= steady-state valueof variable pk; dpk = perturbation from steady state.


latter statesis larger than the steady-statevalue a0.Accordingto inequality

C.46),the value<r({p\302\260

+ dpk(t)}) then evolves in time until the minimumvalue Gois attained or,equivalently, until the system is driven back to thereferencestate.The perturbation, therefore, regresses.By definition, we saythat the referencestate is asymptotically stable.We seethat the theorem ofminimum entropy production guarantees the stability of steady non-equilibrium states.The notion of stability introducedin this sectionbyintuitive arguments is developedfully in Chapter5.

We see,therefore, that in the linear range of irreversibleprocesses,theentropy productiona plays the sameroleas the thermodynamic potentialsin equilibrium theory. Nevertheless,the general tendency of systemsnearequilibrium to approacha stablesteady state does not necessarilyimplythat entropy is always increasing in thesesystems.Thus, it has beenfoundthat in simplesituations suchas the approachto the steady state for thermaldiffusion, entropy is decreasing.Also, its value at the nonequilibrium steadystate correspondingto a minimum of P may be smaller than at equilibrium,Pmin = 0 (Prigogine,1947).However, this decreaseof entropy does notreflect the emergenceof macroscopicorder of any kind, as it occursin acontinuous and smooth fashion as the constraint responsiblefor non-equilibrium is switchedon from its equilibrium value.

More generally, as pointed out in Section3.4, the steady states nearequilibrium are essentially uniform in space if permitted by the externalconstraints.The stability of thesestatesimplies,therefore, that in a systemobeying linear laws, the spontaneousemergenceof order in the form ofspatial or temporal patterns differing qualitatively from equilibrium-likebehavior is ruled out. Moreover, any other type of order imposedon thesystem through the initial conditionsis destroyedin the courseof theevolution to the steady state.

Finally, it isof interest to point out the existenceof a variational principleimpliedby the theorem of minimum entropy production.The great practicaladvantage of such a principleis to generate a convergent approximationscheme for calculating approximately the variables {p,}.The classicalexampleof this schemeis the Rayleigh-Ritz method familiar from mathe-mathematical physics(Kantorovich and Krylov, 1964).

3.7. DIFFUSION

When a local inhomogeneityis createdin an initially uniform system,say bya chemical reaction, the thermal motion of the particlestendsto damp this

inhomogeneity.This is the phenomenon of diffusion. In this way a macro-

3.7. Diffusion 47

scopicquantity of matter is transportedacrossa surface situated in theregion of inhomogeneity.

Diffusion is an extremelycommon phenomenon in nature. Quite often it

constitutes the \"rate determining\" step in a lot of processes.Thus, the rateofwaterevaporation from a freesurfaceat 20\302\260 is about 0.3g/cm2s~1,whereasthe rate of diffusion acrossan unstirred air layer of a width of 0.1mm is athousand times less.Thus, diffusion acrossthe gaseouslayer just above theliquid surfacedeterminesthe rate of evaporation of the liquid.

In many instances,diffusion manifests itself as a transient process.Inother cases,it is possibleto set up a steady state of diffusion by maintainingconstant constraints at the boundaries.This is particularly easy in diffusion

experimentsacrossmembranesor solidphases.In this sectionwe give a quantitative formulation of diffusion. According

to discussionin Section3.3we may neglect the effectof chemicalreactions,although the two types of effectwill inevitably be coupledin the equationsof evolution of the macrovariables. The entropy-production density takesthe following form:

ffdiff= ~t W\302\243 C-47)

Note that, by definition, the n compositionvariables {p,}are independent.In this respect,our treatment differs from thosebasedon classicalthermo-thermodynamics of closedsystems(Prigogine,1947;De Groot and Mazur, 1962).In the latter, one starts with a number of constituents that are coupledthrough the relation:

Ei= 0 C.48)i

The latter is necessaryin order to assure the compatibility between thebalanceequationsB.11)and the conservation equation (in the absenceofconvection):

dt

with

P = E Pi C-49)

The picture weadopthere is,instead,asfollows.We consideran opensystemof n chemicalconstituents in a medium (e.g.,inert gasor liquid solvent)which

is in excess.Thus, the fluxes j;are to beunderstoodas diffusion fluxes in themedium. If, in addition, the reactive mixture is dilute, then the couplingsbetweendiffusion fluxes of componentsi and j is negligible.


RelationsC.16)and C.22)imply, therefore:

C.50)T,p

where LH dependson the propertiesof the solvent as well as on the massfractions of i in the solvent.As /i,-

= n,({pj}):

Usingagain the assumption of a dilute mixture we can reduceEqC.51)to:h=-Dt\\Pi C.52)

where we introduced the diffusion coefficientD,:

Dt = %(f) C.53)

Relation C.52)is known as Fick'slaw. Combining with Eq. C.36),oneobtains:

\\-!r) =divD,Vp,- C.54)V Ot /diff

In many casesD{can beregardedasconstants,and C.54)can bereducedto alinear relation known as Fick'ssecondlaw (DeGrootand Mazur, 1962):

= Dt V2pt C.55)f

The diffusion laws analyzed in this sectionimply that the flux of matter is

always directedto the region of low concentration. This situation prevailswhen the referencestate (i.e.,the state before the appearanceof the inhomo-geneity) is homogeneousand isotropic.In nonhomogeneousor anisotropicmedia, such as in membranes or in fluid phasesin the presenceof highlynonlinear chemical reactions,the diffusion flux can be inverted. Matter isthen transportedagainst the concentration gradient. In this casethe systemis said to undergo an active transport (examplesare given in Chapters14and 16).

Chapter4

NonlinearThermodynamics

4.1.INTRODUCTION

The analysis of linear laws performed in Section3.2showsthat transportphenomena such as diffusion can be treated satisfactorilyby linear thermo-thermodynamics. On the other hand, a realisticdescriptionof chemical reactionsrequiresan extensionof the theory to the nonlinear range:

a >kBr D.i)As we have seen,this condition is compatiblewith the local descriptionofirreversibleprocessesoutlined in Section3.1as long as chemical reactionsremain slow phenomena in respectto elasticcollisionswithin the reactionmixture.

Nonlinear thermodynamics is, therefore, essentially a thermodynamicsof chemical reactions.*The state variables of the system satisfy the balanceequations B.11):

^= -divI +\302\243 vlpWf> D.2)p

and the entropy production takesthe form:

P= JdFff=Jdr[-Xj-V^+ 2>,,^1=jdV^JkXk D.3)

In Eq. D.2)it is understoodthat both j, and wp are related to ps valuesthrough a suitable set of nonlinear phenomenologicallaws.An exampleofsuch laws is provided by relation C.27).* We recall that we are interested here in isothermal systems in mechanical equilibrium. In

moregeneral situations, additional nonlinearities appear through the Reynolds stress term and

through the temperature dependence ofchemical rate constants.

49

50 Nonlinear Thermodynamics

4.2.THE GENERAL EVOLUTIONCRITERION

As in Section3.5,weare now interested in the way the systemevolvesto the

steady state. Unfortunately, by performing the derivative dP/dt one finds

that this quantity has no specialpropertiesoncethe domain of linear lawsand of Onsagerrelations is exceeded.Nevertheless,it is possibleto obtain ageneral inequality by decomposingdP/dt as follows:

dt J t dt J ^ dt dt dt

In the linear range, both contributions to dP/dt becomeidentical.Indeed,using Eqs.C.22)and C.34):

D.5)

The last inequality is a consequenceof the minimum entropy-productiontheorem.

It turns out that beyond the linear range, dP/dt does not exhibit anyproperty of general validity. However,asweshow presently,dxP/dtsatisfiesa general inequality extendingthe theorem of minimum entropy production.

We have:

<46)dt

The first contribution is transformed by a partial integration. The surfaceterm arising from this transformation vanishes for time-independent con-concentrations or for zero fluxes at the boundaries,by an argument identical tothat in Section3.5.Theseconditionscorrespondto an opensystem in com-communication with someexternal phasesthat are in a time-independent andspatially uniform state and are characterizedby given valuesof temperature,pressure,and chemical potentials.Next we express(Itl in terms of the fij

4.3. Evolution Criterion and Kinetic Potential 51

values and evaluate dfij/dt as in Section3.5,by taking into account the

dependencefij = Hj{{pk}).We finally obtain:

dt Tj <?\" dt

where the surface term vanishesthanks to the boundary conditions

Usingthe balanceequation D.2)we may transform the right-hand sideto:

w^)!^ D8)T J f/ \\dpj dt dt

Again, the arguments developedin Section3.5can beappliedhere to ensurethe definitecharacter of the quadratic form appearingin Eq.D.8).We con-conclude that (Glansdorffand Prigogine, 1954):

dxPdt

<0 (=0 at the steady state) D.9)

This inequality is asgeneral as local thermodynamics itself.*The derivationoutlined in this sectionshowsthat in the linear range it becomesidenticalto the theorem of minimum entropy production.Becauseof its generality,it has been calledthe universal evolution criterion.In Section4.3weexaminethe information obtained from this criterion on the propertiesof non-equilibrium steady states.

4.3. EVOLUTIONCRITERIONAND KINETICPOTENTIAL

Let{w\302\260}, {\\f}, {<%\302\260}, {/if}be the values of flows, forces, and chemical

potentials at the steady state.According to Eq.D.2),the compositionvari-variables {pf}at this state satisfy the relations:

-divj,<{pj})+ X vipw\302\273({py0})= 0 D.10)

p

We regardnow the steady state as a referencestate and set:Pi = pf + SPi

Wp=

w\302\260p

+ 3W(I D.11)

* For boundary conditions different from those considered in this section, such as constantfluxes on the surface E, one can still derive a general inequality. However, the result can no

longer becast in the form of Eq.D.9).


Relation D.6)becomes:

By performing the same transformations as in the previous section[Eq.D.6)-D.8)]and by taking Eq.D.10)into account, onecan seethat the first

bracketvanishes identically.Thus, dxPis a quantity of secondorder in theexcessflows and forcesaround the steady state:

Jkd5Xk<0 D.13)

Figure 4.1provides a schematicrepresentation of this inequality.In the immediate neighborhoodof the steady state oneexpectsthat

3JkJ\302\260k

bXk

Ak< 1

By analogy to C.22)it is natural to set:D.14)

D.15)

where the coefficients /tt. are time independent.Note that relation D.15)can be compromisedin the presenceof critical phenomena leading toself-organizingprocesses.Such phenomena are discussedin detail in PartsIIand IIIfrom standpointsof both macroscopicequationsand stochastictheory.

ExpressionD.13)becomes:

TdxP= D.16)kkf

In general, the matrix lkk. is not symmetrical:

'kk' ^ Ik'k

Nevertheless,it is alwayspossibleto decomposeit into a symmetric and an

antisymmetric part:

4.3. Evolution Criterion and Kinetic Potential

6X,

53

5X

-5X-

5X,

Figure 4.1.Geometrical interpretation of inequality D.13).Whereas angle between 8J and8X may belessthan 7t/2, angle between 8J and d&X is necessarily larger than n/2.

Thus:

f V f VJ kk' J kk'

=d\\ (dVY, '\302\253\342\200\242 &Xk &Xf + \\dvZ 'L-^ dd Xk. D.18)2 J kk' J kk'

The quantity appearingunder the differentiationsign can betransformedas:

\\dV X Ilk' ?>Xk SXk, =\\dV X /\302\253- <5Xt 5Xt.= {dV^SJkSXk D.19)J kk' J kk' J k

The last term representsthe excessentropy production,that is, the part ofthe entropy productionarising from the excessflows and forces.Considernext the quantity:

TSXP= (dV^JkSXk= (d [dV5Jk5Xk D.20)

We now show that the first term of the right-hand sidevanishesidentically.Indeed:

D.21)

\342\226\240^ Nonlinear Thermodynamics

In the first term weperforma partial integration and observethat the surfaceterm vanishes for boundary conditionsof the type discussedin Section4.2:

Combiningthe remaining terms and taking Eq. D.10)into account oneobtains:

D.23)k J i 1 L ip A

Returning to Eqs.D.19)and D.20),onemay write:

T 5XP =dK\302\243 5Jk SXk = excessentropy production D.24)

J k

and

SXk dSXk. <0 D.25)kk1

In summary, dxPwould be the differential of the state functional SXPif thecontribution of the antisymmetrical part llw could vanish. The situationwould be as in the linear range, where the differential dP of P has a well-defined sign, thanks to the theorem of minimum entropy production.Astraightforward extensionof the arguments of Section3.6would then permitthe asymptotic stability of steady nonequilibrium states to be established,provided 5XPis positive:

SXP> 0 D.26)

Conversely,the presenceofan antisymmetric contribution in the generalcaseraisesthe problemof stability of steady statesfar from equilibrium, which

isno longerensuredby inequality D.25).This important problemis examinedin detail in the remaining part of this chapter.We closethis sectionby recallingthat the reasoningfollowedso far has beenadapted to the casewhere thereference states are steady states.The evolution in the neighborhoodoftime-dependentstatesis discussedbriefly in Chapter8.

4.4. Stability ofNonequilibrium States. Dissipathe Structures 55

4.4. STABILITYOF NONEQUILIBRIUMSTATES.DISSIPATIVESTRUCTURES

The important role played in the evolution criterion by the excessentropyproduction suggeststhat a study ofstability ofnonequilibrium statescouldbebasedon the excessentropy-balance equation.To this end we introducethe entropy and the entropy productionvariations around the steady state:

AS = S({Pi})- S\302\260({pf})

C ^ C ^ <427>LSI \342\200\224 I il V 7 J I. y\\ I. \342\200\224 I If V 7 J i. y\\ i.

\342\200\242Ik J k

As discussedin Section3.6,these variations arise from either externaldisturbancesor random internal fluctuations inducing deviations of thestate variables{p,}from their steadyvalues

{p\302\260). Assuming that the deviations{dpi}of thesevariables remain small,* one can expandAS and AP in thefollowing form:

fplfp%2Y+'.'. D28)

with [seeEqs.C.5)and D.24)]:

<5S= [dVY\\^\\ SPi=-- [dVYfifSpt D.29a)J t \\8PiJo T J i

S2S= \\dVY\\4^) SPidPi D.29b)T J ij \\8Pj/o

SP= I*dV X (J\302\260 $Xk + X?SJk) ={dVY^X\302\260kdJk D.29c)

J k J k

j S2P= \\dV\302\243 SJk SXk = SXP= excessentropy production D.29d)

The point is that in the entire range where the local formulation of thermo-thermodynamics remains valid, the quadratic form S2S[seeEq. D.29b)]has thesamestructure as in equilibrium, and asa consequenceit is a negativedefinitequantity:

S2S= -~\\dV Y I^ ) 5PidPi <0 D.30)= -II* dV X (P) SPidPj <0

*As pointed out in Section 4.3,this assumption can break down in the presence of criticalphenomena (seePart III for a detailed discussion of these phenomena).


Let us now compute the time derivativeof the excessentropy S2S:

A\342\200\236 SSp: ,.,,,

-\\dpi\342\200\224\342\200\2241 D.31)

We have taken into account that the quantity

D.32)l dPjj0

representsa symmetrical matrix in the indices/ and j.The time derivative

dSpj/dtcan be substituted from the excessbalanceequations[seeEqs.D.2)and D.11)]:

ddPi

We obtain:

= -divdj,+ Ivi,\302\253Wp D.33)or p

= -1\\dvx(d\302\243) sPjT-divit,+ X VipsW

In the first term we perform a partial integration. The surface term vanishesfor constant concentration or for constant flux boundary conditions.Theremaining terms yield:

Introducing the affinity through Eq.C.9)we finally obtain:

=\\dV Yj <>Jk dXk = excessentropy production= SXP D.34)

\342\200\242J k

Supposefirst the referencestate is the state of equilibrium.Then

SJk = Jk, SXk = Xk

and

jt &2SU=jdV<r

= P>0 D.35a)

by the secondlaw, with [seeEq.D.30)]:(<52S)eq<0 D.35b)

Thus, (<52S)eqevolvesin time, as indicated in Fig.4.2.

4.4. Stability ofNonequilibrium States.Dissipative Structures 57

Figure 4.2. Timeevolution of second-order excessentropy (<52S)eq around equilibrium.

The argument, by now familiar, first developedin Sections3.5and 3.6in connection with the minimum entropy-productiontheorem, showsthat

as f -> oo the system evolves to the referencestate.Thus, we have demon-demonstrated the asymptotic stability of equilibrium in respectto small perturba-perturbations. Note that the argument about stability relieson the definitecharacterof (E2S)eq,which vanishes only at the reference state. Such functions arereferredto in analysis as Lyapounov functions.Lyapounov's stability theoryis discussedin somedetail in Chapter6.

As we have indicated in Chapter3, the equilibrium state is characterizedby the absenceof spatial or temporal order.The result wejust establishedimplies,therefore, that this disorderedsituation cannot bemodifiedas longas the systemdeviatesfrom equilibrium only through fluctuations or throughrandom disturbances.

Imagine now a processcausing a systematicdeviation from equilibrium,for instance, the increaseof a state parameter or of a constraint, X (affinityof an overall reaction, compositiongradient at the boundaries,etc.).Thecompositionvariables are subjectto the changesdescribedqualitatively in

Fig.4.3.According to the theorem of minimum entropy production,thesteady statescloseto equilibrium remain asymptotically stable [Fig.4.3,branch (a)].By continuity this branch of states\342\200\224subsequently referred to asthe thermodynamic branch\342\200\224maintains its stability in a finite neighborhoodof the equilibrium state.But beyond a critical value Xc one couldnot ruleout the possibilitythat the states on the thermodynamic branch becomeunstable [Fig.4.3,branch (b)].In this casethe least disturbancecompelsthesystemto evolveaway from this branch.The new stableregimeattained by


@

Figure 4.3. Branching of states as the distance from equilibrium increases:(a) stable part of

thermodynamic branch; (b) this branch becoming unstable; (c) new solution (dissipativestructure) emerging beyond instability of thermodynamic branch.

the system[Fig.4.3,branch (c)]can correspondto an orderedconfiguration.We say that at X = Xc wehave a phenomenon of bifurcation of a new branchof solutions.

In summary, near equilibrium the property SXP>0 is an immediateconsequenceof the secondlaw, whatever the perturbations {Sp^of the statevariables might be.Far from equilibrium, however, the time derivative of4<<52S)\342\200\224the excessentropy production5XP\342\200\224need not be positive definite.The possibilitiesthat might arisein this more general caseare indicated in

Fig.4.4.

F2S)

Stable

Marginallystable

Unstable

Figure 4.4. Time evolution of second-order excessentropy (S2S)in caseof (asymptotically)stable, marginally stable, and unstable situations.

4.4. Stability of' Nonequilibrium States.Dissipathe Structures 59

As we seefrom Eq.D.34),the nonequilibrium steady state becomesun-unstable as soon as the excessentropy productionSXPbecomes(and stays)negative for t > t0. (Glansdorffand Prigogine, 1971):

SXP<0 for t > t0 :unstable referencestate

SXP>0 for t > t0 :asymptoticallystablereferencestate D.36)Fora given system,one can modify the value of 5XP by changing a set ofparametersX measuring the distancefrom equilibrium. When a criticalvalue Xc of X isreachedthe sign of the inequality in Eq.D.36)will beinverted,and the referencestate loosesits stability. We referto this criticalsituation asthe state of marginal stabiliy.

SXP(/.C)= 0 for t > t0 :marginal stability D.37)

This relation enablescalculation of the values of the constraints capableofinducing an instability of the thermodynamic branch.

It is of interest to relate the thermodynamic stability conditionsD.36)and D.37)to the kineticsofchemicalreactions.Considerfirst a unimoiecuiarstepin a reaction chain:

X j==\302\261 A D.38)k

We

For

and

have [seeSections2.2and 3.1]J = w

a = kB

pA, T constant (and thus also

Sw =

sa=

=

T

k,*i

k,

ka

ln-

,k2

bp

T-

'a: \"

'<lP

=

X

$Px

Px

- k2pA

'x>A

const):

dxa = 5w 5-= kBk, \342\200\236'

>0 D.39)T px

We concludethat reaction D.38)tendsto stabilize the system.In contrast,in the autocatalytic process

A + X r~^ 2X D.40)


where the presenceof X enhancesits own production,onehas:

-k2p\\\\ Sw = (klpA-

2k2p\302\260x)Spx

BPx

and

&x\302\260

= - ^(kiPA -2k2p\302\260x)(dpxJ D.41)

Px

Undercertain conditions,this expressioncan becomenegative.We concludethat autocatalytic reactionsor, more generally,reactionsinvolving nonlinearsteps tend to destabilizethe system.Naturally, a singlereaction such asEq.D.40)cannot producean instability, as it will always evolveto equilib-equilibrium. On the other hand, such a processcan be a part of an open systemundergoinga wholesequenceofreactions(Glansdorff,Nicolis,and Prigogine,1974).Now, the most important biochemicalreactionsin cellular dynamicsare nonlinear, as they involve complexfeedbacks.Thus, the biologicalinterest of these considerationsis quite obvious. This point is discussedin detail in Parts IV and V.

In summary, we have found that the distancefrom equilibrium and the

nonlinearity may both besourcesofordercapableofdriving the systemto anordered configuration. A highly nontrivial connection between order,stability, and dissipationappears.To indicate clearly this relation we callthe orderedconfigurations that emergebeyond instability of the thermo-dynamic branch the dissipativestructures.

The analysis of this sectionshould already have clarified the importantroleof fluctuations in theseself-organizationprocesses.Onecan reallyspeakof order through fluctuations in connection with these transitions. Never-Nevertheless, it should,be emphasizedthat what has beendealt with here is thetime evolution of a macroscopicfluctuation, oncethe latter appearsin thesystem.The origin of such fluctuations and the a priori probability of afluctuation enjoyingspecificpropertiesisa profound and important problemthat will beapproachedin Part III.

Finally, oneshould be fully aware that the results of this chapter merelysuggestthe possibilityof self-organization processes.The real proof ofexistenceof such processesrequiresa detailed analysis of the balanceequations;this analysis isperformed in Part II.

PartII

MathematicalAspectsof Self-organization:DeterministicMethods

Chapter5

SystemsInvolving ChemicalReactionsand Diffusion.Stability

5.1.GENERAL FORMULATION

The thermodynamicanalysisof nonlinear systemsoutlined in Part I provedto bean extremelypowerful tool in that it sortedout the role of the distancefrom equilibrium as a sourceof order.We now turn to the kinetic aspectsofthis problem.Moreprecisely,the purposeof this part of the monograph is toillustrate various types of self-organizationprocessesand to analyze themechanismsof emergenceof the spatial or temporal patterns.The presenta-presentation we adopt is,of course,inspiredby the resultsof Part I.In addition, weappeal quite frequently to the techniques of nonlinear mathematics in

order to obtain quantitative information about the solutionsdescribingthe organized states.

We consideragain,as in Chapter2, an opensystemat mechanicalequilib-equilibrium involving n chemically reacting constituents X1;...,Xn. Under the

assumptions(i) to (iv) enunciated in Section2.1,the instantaneous macro-macroscopic state of the system is describedby the compositionvariables {/),}.The latter obey the massbalance equations[seeEq.F.11)]:

^= -divj,+ Iv/pW/) (i =],...,n) E.1)

In the nonlinear rangeofirreversibleprocesses,j, and wp becomecomplicatednonlinear expressionsinvolving p{.However, in many instancesin physicalchemistry and biology one deals with the evolution of intermediate sub-substances that are very dilute. According to Section3.7,the diffusion flux j,for such systemstakesthe form:

j;=-\302\243>iVpi (Fick'sfirst law) E.2)63

*\" Systems Involving Chemical Reactions and Diffusion. Stability

where the diffusion-coefficientmatrix has been taken diagonal and thecoefficientsDt are henceforthtaken constants.Equation E.1)takesthe form

^ = MPi})+ Dt V2Pi E.3)

where the nonlinear function/\342\226\240

describesthe overall rate of production of X,from the chemicalreactions.Accordingto Sections4.1and 2.2,for a physico-chemical system obeying the law of massaction at equilibrium, j\\ will benonlinear functions of{pj}of the polynomialtype.This makesthe differential

systemE.3)a systemof nonlinear partial differential equations.According to the usual classificationof theseequations(Sneddon,1957),

the presenceof first ordertime derivativesand second-orderspacederivativesconfers to Eq.E.3)a paraboliccharacter.This is a very general feature ofevolution equationsdescribingdissipative systems,such as Fick'ssecondlaw for diffusion or Fourier'slaw for heat conduction.

As emphasizedin Section2.2,in order to have a well-posedproblemit is

necessaryto prescribeappropriateboundary conditionsthat expresstheconstraints acting on the systemfrom the outsideworld.In most applicationswe dealeither with Dirichlet conditions:

{pf,...,pl)= {const} E.4a)or with Neumann conditions:

{n \342\200\242Sp\\,...,n \342\200\242Npl} = {const} E.4b)

or with a linear combination of both conditions.Sometimes,oneof theconstantsin Eq.E.4b)vanishes identically.In this casethe system is closedwith respectto exchangesof the correspondingchemical substance.Thiscondition appliesto most of the experimentsof the Belousov-Zhabotinskireaction, which is the bestknown chemicalexamplegiving riseto dissipativestructures, as we seein detail in Chapter13.

We may also recall from Section4.2,that the conditionsof fixed con-concentrations or of zero fluxes at the boundarieswas necessaryin deriving theexplicitform of the evolution criterion in terms of dxP/dt <0.

Beforeweclosethis sectionwewant to insist on the generalityofEqs.E.3)to E.4),as they apply in ordinary chemicalkineticsaswell as in biochemistryat the cellular or the supercellularlevel.With slight modifications they canaccount for thermal effects;it sufficesto add to Eq.E.3)the equation for the

internal-energyconservation:

c^= div2VT+X(-AHp)wp(T, {Pj}) E.5)at

5.2. Lyapounov Stability 65

wherec is the specificheat of the mixture, AHP the heat of reaction p, andwp dependson temperature in a highly nonlinear (usually exponential)fashionthrough the rate constants.Moreover,a simplechange in the interpre-interpretation of the variablespermits descriptionof such diversephenomena as thecompetition of populationsin an ecosystemor the electrical activity of thebrain.

The modeling of a concreteproblemamounts, therefore, to a judiciouschoiceof the variables, of the form of the \"chemical\"laws determining fhAHP, and wp in Eqs.E.3)to E.5)and of the valuesofthe parametersdescribingthe system.-Amongthe latter the diffusion coefficients{DJ,the rate constants{kip}, the sizeof the systemand, finally, the nature of the appliedconstraints,play a decisiverolein the behavior of the system.

5.2.LYAPOUNOV STABILITY

The thermodynamic theory developedin Part I suggeststhat an importantphenomenon accompanying a self-organization processis the loss ofstability of the thermodynamic branch.In this and subsequentsectionsofthis chapter wedefinemore rigorously this important notion in a number ofdifferent situations.

We first introduce the conceptof Lyapounov stability (Minorski,1962).Let Xi{r,t) be a solution of the differential system E.3),that is, a set offunctions {X^r,t)} dependingon the respective spaceand time variablesr and t, satisfying identically Eq. E.3),together with initial and boundaryconditions.We assumethat the motion isdefined in the space-timedomain{0< rx < lx, a = 1,2, 3;0 < t < oo} and that in this domain X^r, t) is

physically meaningful, that is,boundedand positive.

Definition 1. We say that X,(r,t) is Lyapounov stable if given e >0 andt = t0, there existsand tj = tj(E, t0) such that any solution Yj(r, t) for which

\\xi(r, to) - ^(r, to)I <1 satisfies also \\X;(r, t) - Yfc, t)\\ < e for t >t0.If no such

r\\ exists,the solution is unstable.We seethat stability in the senseof Lyapounov is equivalent to the property of uniform continuity (familiarfrom differential calculus)of X(r, t) with respectto the initial conditions.

Definition 2. If ^,-(r,t) is stable,and if

lim \\Xfa, t)- Yfc,t)\\ = 0 E.6)f-\302\273OO

we say that X((r, r) is asymptotically stable.

66 Systems Involving Chemical Reactions and Diffusion. Stability

x

-1

Figure 5.1.Stability of steady-state solutions of Eq. E.7).

In other terms a solution(ormotion)isasymptoticallystableif all solutionscoming near it approachit asymptotically.As an examplelet

E.8)

E.9)

This equation admits two steady-statesolutions

xOi = -1, x02 = +1as well as the family of solutions(seeFig.5.1)

x = th(f -to + k), k = arth x@)- 1 < x@) < 1

By applying Definitions 1 and 2 it is found that x01is unstable and x02 isasymptoticallystable.*

5.3.ORBITALSTABILITY

In many casesof interest in physical chemistry and biology onedealswith

systemssubjectto time-independent constraints.This is translated by thefact that the right-hand sideofEq.E.3)\342\200\224essentially the rate function/\342\200\224does

not dependon time explicitly.We call such systemsautonomous. Let Xt(r, t)bea solution of Eq.E.3).It is then clearthat any function X,(r, t + x) where

* Note that if we had considered stability for t -\302\273

\342\200\224 oo, x01 would beasymptotically stable andx02 unstable.

5.3. Orbital Stability 67

x is an arbitrary constant (the phase),is still a solution of Eq.E.3).In otherwords,autonomous systemsexhibit the property of translational invariance.These infinitely many solutions,differing from each other by the phase,define a trajectory (or orbit) of the system in an appropriatespace.In theabsenceofspacedependences,this phasespaceis an n-dimensionalEuclideanspacespannedby the n variables pu ...,pn and the trajectory becomesacurve in this space.In the more general casewe deal with infinitely dimen-dimensional spacesspannedby setsof functions,known as functional spaces.We

give examplesof orbitsin thesespacesin Chapter7.Supposeone choosesa certain \"metric\" defining the distancein phase

spaceand let C denotethe orbit.We say that C is orbitally stableif, givene >0, there existsn >0 such that if Xo is a representativepoint of anothertrajectory within a distancen from C at time t0, then X remains within adistancee from C for t >t0.Otherwise,C is orbitally unstable.

If C is orbitally stableand the distancebetweenX and C tendsto zero ast -* oo,C isasymptoticallyorbitally stable.

Lyapounov stability and orbital stability should not be confused (seeFig.5.2).

Figure 5.2. Comparison between Lyapounov stability and orbital stability. C,C\" aretwo orbitsof different periods. Although their distance remains bounded for all times, distance betweentwo points 1 and 1'on these orbits can increase in time owing to phase shift induced by differencebetween periods. Thus, state 1 need not beLyapounov stable, even if orbit Cis orbitally stable.

6* Systems Involving Chemical Reactions and Diffusion. Stability

A simpleillustration of the conceptof orbital stability is provided by the

following systemdue to Poincare(Minorski,1962):

^ = Y + X(\\- X2 - Y2)at

^-= -X+ 7A -X2 - Y2)at

Theseequationsadmit the steady-statesolution

Xo=Yo= 0

Switching to polarcoordinatesfor X and Ycenteredon this state:X = r cos6Y = r sin 6

we may transform the equationsinto the form:

?= <

dt

Theseequationsadmit a solution representinga circular orbit with a finite

radius:>\342\200\242

= 1

e = eo-tBy applying the definitionsone finds that this orbit isasymptoticallystable

and attracts the trajectoriesin the phaseplane as t -> oo.

5.4. STRUCTURALSTABILITY

Implicit in the definition ofLyapounovand orbital stabilitieswasthe assump-assumption that the structure of the right-hand side of the basicequationsE.3)remained unchanged.The various kinds of disturbanceacting continuouslyon the system (seeSection3.6and Chapter4) merely displacedthe instan-instantaneous state of the system from a \"reference\"state to a new \"perturbed\"state representedby someother point in the samephasespaceas for the un-unperturbed system.

In many physically interesting situations the evolution of the systemdependson a number of parameters,examplesof which are given in Section

5.4. Structural Stability 69

5.1.As the systemevolvesand is continuouslyperturbedby the outsideworld,someof theseparameterscan themselveschange smoothly or evenabruptly.New parameterscan be \"turned on,\" thus increasing the number of inter-interacting degreesof freedom. Finally, otherscan becomeextinguishedperma-permanently or momentarily. In each casethe change of the parametersgenerallychangesthe structureofthe equationsthemselvesby modifying the right-handsidesof Eq.E.3).Let e be a measureof the change of somerepresentativeparameter. If all solutionsof the modifiedsystemremain in a neighborhood0{e)of ordere of the solution of the \"initial system\" for e = 0, we say that

the latter is structurally stable.If no such neighborhoodexists,the system isstructurally unstable. Alternatively, in a structurally stable system thetopoiogicalstructure of the trajectoriesin phasespace(wheneverthe lattercan bedefined)remains unchanged.

A simpleillustration of the idea of structural stability is given by themotion of a pendulum. The usual mathematical model of this system is theharmonicoscillatorand is characterizedby an infinity ofperiodictrajectoriesdependingcontinuously on the initial conditions.In nature, however, apendulum is never a harmonic oscillator,as it is always subjectto friction.As a result, the system evolveseventually to a unique equilibrium state ofrest, reachedsooneror later dependingon the magnitude of friction. Weconcludethat the equationsdescribingharmonic oscillatorsare structurallyunstable with respectto friction.The practical functioning of a dependableclockrequires,therefore, a finite external driving force that compensatesthe effectof friction.

Structural stability has long been known to play an important rolein

electricalengineering.An extremelylucid descriptionof particular problemsinvolving this conceptis to be found in the monograph by Andronov, Vit,and Khaikin A966).Morerecently, Thorn drew attention to the generalityof this conceptand its implications in a variety of phenomena.We return tothis conceptin Chapter6 and in Part V.

Chapter6

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6.1.INTRODUCTION

The solution of the reaction-diffusionequationsE.3)and E.4)is an extremelyarduous task, primarily becauseof the nonlinear character of the ratefunctions.On the other hand, the thermodynamicanalysisof Part Isuggeststhat an important feature common to a large class of self-organizationprocessesshouldbe the lossof stability of the thermodynamic branch andthe subsequentevolution to a stabledissipativestructure. At the point wherethe exchange of stability takesplace,a branching of at least two solutionsofEqs.E.3)and E.4)\342\200\224the thermodynamicbranch and a dissipativestructure\342\200\224would take place.

To date there exist a number of general techniquesof nonlinear mathe-mathematics that cover such phenomena.We discuss two that seem to beparticularly well adapted to the analysis of pattern formation and thebiologicalimplications:(a) theory of bifurcations and (b) theory of catas-catastrophes. Both theoriescorrespondto two different ways of extending thework of Poincare,who was the first to put the qualitative theory of nonlineardifferential equationson a solidground.A third mathematical technique,namely, stability theory, alsoretains our attention as it constitutes, in manyways, a powerful alternative to someotherwise complicatedarguments in

bifurcation theory.

6.2.THEORYOF BIFURCATIONS

As we pointedout repeatedly,the right-hand sideof Eq.E.3)dependson asetofparametersX. Supposethat wechooseX such that for X sufficiently small,and with time independentconstraints, theseequationsadmit as t -* oo,asingle physical acceptable(i.e.,positive and bounded)solution.Accordingto irreversible thermodynamics, the latter lies necessarilyon the thermo-thermodynamic branch. It may be further assumed that this branch dependssmoothly on the parameter X.

70

6.3. Stability Theory 71

Now for certain criticalvaluesAc of X, the solutionson the thermodynamicbranch might no longer be unique and might even loosetheir stabilityproperties.In the neighborhoodof thesecritical points the system couldevolveto a new regimeexhibiting spatial or temporal order.We say that atthesecriticalpointsthere isbifurcationof the solution on the thermodynamicbranch.*In general, one expectsthat in the immediate neighborhoodof Ac

the new solutionswill dependon A in a nonanalytic fashion.The purposeof bifurcation theory initiated by Poincareand developed

further by Andronov and his school,Hopf, Krasnosel'skii,and others,isto developmethodsenablingone to:(a)demonstrate rigorously the existenceof branchingofsolutionsfor certain criticalvalueskc,and (b)construct, in anapproximate fashion, analytic and convergent expressionsfor certainimportant types of solution emergingat the bifurcationpoints.

We illustrate this approachdirectly on a solvable model in Chapter7.Herewe turn to the conceptof stability, which recurs frequently in thismonograph, and which is intimately connectedwith the very existenceofthe phenomenon of bifurcation.

6.3.STABILITYTHEORY

The mathematical definition of stability has already been developedin

Chapter5.Herewe would like to outline a number of techniques enablingone to assessthe stability of solutionsof the differential equations.

The \"Principle\"of LinearizedStability

As oneis interestedusually in the stability of a particular type of solution ofEq.E.3),it is often convenientto study the behavioroftheseequationsaroundthis particular reference state {XOi{r,t)}.We may also regard {Xoi}as anunperturbed solution that iscontinuouslyperturbedby the action ofexternaldisturbancesor by internal fluctuations x,(r,t). The latter lead from {XOi}

to the new solution

Z,(r,t) = Z0,(r,t) + x,(r,t) F.1)Onecan think of F.1)as denning a new coordinatesystem in phasespace,centeredon the referencestate rather than at the point @,...,0).

From Eqs.E.3)and F.1)one can deducethe systemof equationsfor {xj:

^ = MXoj+ xj})-MXOj})+ \302\243>, V2x,- F.2)* Note that the very occurrence of bifurcation implies nonlinear kinetics; otherwise, theevolution equations would admit a unique solution.

72 Mathematical Tools

or,setting

vt = MX0J+ Xj})-MXOj}), F.3)

dx

-\302\243

= v/L{xj}) + D, V2x,- F.4)

The stability of the referencestate {XOi}is now reflectedby the stability ofthe \"trivial solution\" {x,= 0}of Eq.F.4).

In general,the systemof Eq.F.4)isnonlinear and ascomplicatedto solveas the original system E.3).Quite frequently, however, the local behaviorof the solutionsaround the origin:

X;

X0l1 F.5)

turns out to provide extremely significant information about the behaviorof the completenonlinear system.To describethis local motion it sufficesto linearizeEqs.F.2)-F.4)with respectto x;.Onethen obtainsthe linearizedsystem, alsoreferredto as variational system:

where the elements of the coefficientmatrix\342\200\224the Jacobian\342\200\224are evaluatedat the referencestate.The enormousadvantage is that the linear systemF.6)can be studiedby such classicalmethodsof analysis as the normal modemethod.We illustrate thesemethods with the examplesin Chapter7.

The following important theorem, first proved for ordinary differentialequationsby Lyapounov(Minorski,1962;Sattinger,1973),links the stabilitypropertiesof the systemsF.6)and E.3).

Theorem. If the trivial solution of Eq.F.6)is asymptotically stable,then{XOi}is an asymptoticallystablesolution of Eq.E.3).If it is unstable, then{XOi}is alsounstable.

We note that the theorem givesno information in casethe trivial solutionof Eq. F.6)is (Lyapounov) stablebut not asymptotically stable.This so-calledcriticalcase(Sansoneand Conti, 1964)has to bestudiedseparately.Asa matter of fact, a state correspondingto the critical caseis structurallyunstable, as its propertiesare likely to change qualitatively on the action ofa disturbance,even if the latter ismade as small asdesired.The phenomenonof branching, mentioned repeatedly in previous sections,occursnecessarilyin this critical case.

6.3. Stability Theory 73

Lyapounov'sSecondMethod

The investigation of stability basedon the system of linearized equationsamounts to the direct integration of this system.Forsystemsinvolving manychemicalvariablesthis can often bean awkward way ofproceeding,especiallyif the referencestate isspaceand/ortime dependent.In contrast, the methodknown as Lyapounov'ssecondmethod providesstability conditionsthat are:(a) independentof the integration of the linearizedsystem,(b) applicabletoreferencesolutions{Xoi}of all kinds, including spaceand/ortime-dependentones,and (c)applicabledirectly to the nonlinear systemsE.3)or F.4).Thepriceone usually has to pay is that the method providesqualitative informa-information and does not enabledetermination of such quantities as relaxationtimes or oscillation frequencies.

We first considerthe nonlinear system F.4)with diffusion terms absent:

~ = v&{xj}) F.7)

We introduce the following:

Definition. Considera function of n variables V = V(xu ...,xn). We sayFisdefinite in a domain D in phasespacearound the origin (D:|x,|< n,where n is a positive constant), if it takes values having a single sign in Dand vanishes only for x l = \342\226\240\342\226\240\342\226\240= xn = 0.V is semidefinite if it takes thesame sign or vanishes in D.In all other casesV is indefinite.

Next weconsiderthe derivativeof V along a solution ofEq.F.4).Fortime-independent constraints systemF.7)becomesautonomous, and one has:

where v = (vu ...,vn) and

(\302\243\342\200\242\342\226\240\342\200\242\342\200\242\342\200\242\302\243

The first theorem of Lyapounov asserts:

Theorem1. The referencestate (xt = \342\200\242\342\226\240\342\226\240= xn = 0) isstablein a domain Dif one can determine in D a definite function V whose \"Eulerian derivative\"[Eq.F.8)]is either semidefiniteof sign oppositeto V or vanishesidenticallyinD.


Let us also state without proof two more fundamental theorems ofLyapounov:

Theorem2. The state (xi = \342\200\242\342\200\242 = xn = 0) is asymptotically stable if onecan determine a definite function V whoseEulerian derivative is definiteand has a sign oppositeto that of V.

Theorem3. The state (xt = \342\200\242\342\200\242\342\200\242= xn = 0) is unstable if one can determinea function V whoseEulerian derivativeisdefinite and V assumesin D valuessuch that V(dV/dt) >0.

Note that thesetheorems only provide sufficient conditionsfor stability.Indeed,in general the way of constructing a Lyapounov function V is notprescribedby the theory.

It is instructive to point out the closerelation between Lyapounov'stheoremsand the ideasunderlying equilibrium thermodynamics.In essence,whenever it can be constructed,a Lyapounov function plays the roleof apotential in the immediate vicinity of the reference state, whose extremalpropertiesdetermine the stability of this state.

The ideas outlined for the ordinary differential system F.7)have beenextendedto partial differential equationsby Zubov (e.g.,Zubov, 1961).Sufficeit to mention that in this caseone dealswith Lyapounov functionalsrather than with Lyapounov functions which are, typically, integrals overspaceof appropriatelocal quantities.

It is important to recall here that the entire discussionof Section4.4dealingwith the stability of nonequilibrium stateswas, in fact,an applicationof the above-mentioned Lyapounov theorems.The Lyapounov functional,whosepossibletime behaviors are illustrated on Fig.4.4,is in this casetheexcessentropy

E2SH= f dVE2sH= -~ f dV X(\342\200\224)

SPidPj F.9)J 1 J ij \\Cpj/o

6.4. THEORYOF CATASTROPHES

An essentialproperty of the rate equationsF.4)or E.3)is to contain explicitlya diffusion term in the right-hand side.The roleof this term is extremelyimportant. As we see in detail in the next several chapters,this term caninduce, under certain conditions,an instability of the homogeneousstateand establishwithin the systema regimebreakingspontaneouslythe initial

spatial symmetries.This exceptionalrichnessin the behavior of the solutionsis accomplishedat a considerableprice;no general classification of the

6.4. Theory ofCatastrophes 75

solutionsof such equationshas beenachieved as of now. At best,we canconstruct approximate expressionsfor specialtypes of solutionssuch assteady-state,periodic,or almost periodic ones, without being able toassertthat thesesolutionsexhaust all of the possibilitiesof bifurcation in the

system.In recent years,Thorn (e.g.,Thorn, 1972)has developeda powerfulmethod

enabling such a classification,provided the diffusion term can besuppressedin Eq.E.4).The entire spatial dependence\342\200\224that is, the entire morphologyof the solutions\342\200\224is accountedfor by a parametric dependenceof the ratesfi in spaceand time. The differential systemE.4)becomes,then, an ordinarysystem:

~ = MP]),l>) (i = 1\302\253) F-10)

with

fi = (t,t;X) F.11)In this way, the initial symmetriesare brokenby an unidentified externalaction rather than by the system itself.The spontaneouscharacter of self-organization processesis,therefore,not accountedfor explicitly.

A secondassumption that playsan important rolein the theory\342\200\224although

a great deal of current attempts aim to relax it\342\200\224is that Eq.F.10)derivesfrom a potential

dp,= _ dV({pj},,)dt dpt

where V is the \"potential\" function. We note that from the standpoint ofirreversiblethermodynamics (seeespeciallySection4.3),the dynamics of adissipative system can derive from a potential only in someexceptionalsituations such as systemsinvolving a singlevariable, or systemsoperatingin a small neighborhoodaround the state of thermodynamic equilibrium.Thus, Eq. F.12)is not representative of a system subjectto strongly non-equilibrium constraints.An important consequenceof this is that somevery widely spreadphenomena such assustainedoscillations,which becomepossiblebeyond the instability of the thermodynamic branch, cannot bedescribedby theseequations.

Underthesetwo assumptions,one may now undertake a general classifi-classification of the solutionsof Eq.F.12)basedon the breakdownin structuralstability, by lookingfor the pointswhere there is a change in the stability

propertiesof the steady statesdV/dpt = 0.The latter exist as long as Fdoesnot dependon time explicitly.Thesepoints\342\200\224called ensembledecatastrophesby Thorn\342\200\224are hypersurfaces in the parameter space along which either


a branching of solutionsof the equationstakesplace,or where V attains itsabsoluteminimum in at least two distinct points.In other words,by crossingthesehypersurfaces one switches from a region where a certain dynamicstakes placeto a region where the dynamics is qualitatively different. Ingeneral [seeEq. F.11)]a hypersurface defines a spatial domain having acertain morphology, separating different types of regime.In this way

dynamics and form becomeintimately related.We illustrate Thom'stheoryon an explicitexample in Section8.4.

6.5.HOMOGENEOUSSYSTEMSINVOLVING TWO VARIABLES

GeneralConsiderations

Before we proceedto the analysis of specificmodels,we illustrate in thissectionsomeof the conceptsdevelopedthus far on the simplest possiblenontrivial caseof systemscapable of undergoing cooperative behavior.Obviously, the simplestsuch examplein the context of chemical kineticsare systemsinvolving two variable concentrations.*Our main purposeis to seethe conditionsunder which such systemscan exhibit instabilities in

the steady-state solutions,without explicit reference to the detailsof thekinetics.To simplify even further we ignore the effectsof diffusion.

Let X, Y denotethe compositionvariables of the two intermediates.Therate equationstake the form:

Ht

F.13)Y)

Forsystemssubjectto time-independentconstraints, both fx and fY do not

dependon time explicitly.Thus the differential systemF.13)isautonomous.We assumethat fx,fY are continuous and satisfy the Lipschitzcondition

in a certain boundeddomain, D, of the phasespace(X, Y), in other words,

\\fx(X2, Y2) - fx(Xu rj <K|X2-X,|\\fy(X2,Y2)-fY(X1,Yl\\<K\\X2-Xl\\

* Systems involving a single variable areexcluded here becauseof their inability to give rise to

oscillatory behavior and/or spatial structures arising beyond instability of the thermodynamicbranch. They arestudied briefly in Section 8.4.

6.5. Homogeneous Systems Involving Two Variables 77

for every X,- = {X,, Yt) in D and for K>0.We observethat Eq. F.14)isalwaysfulfilled if the kineticsis polynomial,that is,if the individual chemicalstepssatisfy the law of massaction at equilibrium.

Undertheseconditions,an important theorem of analysisdue to Cauchyand Picard (Coddingtonand Levinson, 1955)assertsthat the solutionsofEq. F.13)correspondingto a certain initial condition in D exist and areunique for t within a boundedinterval @, T).

In orderto appreciatebetter the meaningof this theorem, let us eliminate t

between the two equalitiesin Eq.F.13).We obtain:

dXMX,Y) FJ5)

Thesolutionof this (nonlinear)first-orderequation providesa one-parameterfamily of trajectoriesin the phaseplane (X, Y). To each point on thesecurvescorrespondsa solution of the system F.13).Thus, time can be used toparameterizethe trajectories.Moreover,from the uniquenesstheorem statedabove, any intersection of the trajectoriescorrespondingto different initial

conditionsis ruled out. Another important point is that a closedtrajectory,C, in the (X, Y) plane describesnecessarilya periodicsolution of the dif-differential system.Indeed,from Eq.F.13):

dXdt =

fx(X, Y)

Integrating over the closedcurve C we obtain:A V*

At =\\ T7V~v\\

= const F-15a)

Thus, one can definea periodfor this motion.

The CharacteristicEquation

Let us return to Eq.F.15).Forevery point (X, Y) in the plane there cor-corresponds a well-definedvalue of the slopedY/dX of the trajectory, exceptfor the pointswhere

We call these the singular points.Note that by Eq.F.13)a singular point isa steady-state solution of the differential system. All other points in theplaneare calledregularpoints.Accordingto our previousanalysis,one of thesingular points correspondsto the extrapolation of the thermodynamicbranch in the far-from-equilibrium region.Thus, in order to investigate

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the possibilityof self-organizationprocesseswe must analyze the stabilityof this particular steady state {Xo,Yo). To this end we construct the cor-corresponding linearizedsystem [cf.Eq.F.6)]:

dx\342\200\224 = aux + al2y

F.16)dy\342\200\224 = a21x+ a22y

with

au =(j^\\ (etc) F.17)

and

XY=Yo + yF'18)

Owingto its linear structure, the systemF.16)admitssolutionsof the form:

x =xoe\302\260\",y

=yoe\302\260\" F.19)

Suchsolutionsare callednormal modes.Substituting back into Eq. F.16)weobtain a homogeneousalgebraicsystemof first degreefor the coefficientsx0,y0. The condition for having nontrivial solutionsof this system is

det|fly -(oS%\\= 0 F.20)liecharacteristicequation.In the presentcaseof a> the form

co2- Tco+ A = 0 F.21)

This equation isknown as the characteristicequation.In the presentcaseof atwo-variablesystem it takesthe form

where

are, respectively,the trace and the determinant of the matrix of the coeffi-coefficients.

In general, Eq. F.21)admits two distinct solutions(ou at2. Thus, the

solution of Eq.F.16)is of the form:

x-c^'+ CtS* F23)y = CiKte\"\" + c2K2ea2t


where ct, c2 are determined by the initial conditions,and the coefficientsKu K2,sometimesreferred to as distribution coefficients(Andronov et al.,1966),are the rootsof the equation [seeEq.F.16)]:

a12K2+ (au -a22)K- a21 =0 F.23a)From theseexpressionsone obtains straightforwardly the following stabilitycriteria:

\342\200\242 If both Re o)j <0 (i = 1,2), the steady state {Xo,Yo) is asymptoticallystable..

\342\200\242 If for at least one of the roots Re co^>0 (a = 1 or2), the state (Xo, Yo)is unstable.

\342\200\242 If for at least oneof the roots Re coa= 0 (a = 1 or 2) while the othersremain negative, the system is stable in the senseof Lyapounov, but not

asymptoticallystable.We call this situation marginal stability.

Now onceEq.F.21)is known, that is,oncethe dependenceof T and A

on the parametersisspecified,it isimmediatelypossibleto seewhich of thesepossibilitiesisactually realized.Moreover,from Eq.F.23)wecan determinehow the perturbedsystemevolvesbackor departsfrom the singular point.Thesevarious types of behavior are briefly compiledin the next subsection.

Classificationof the SingularPoints:SimpleSingular Points

Both RootsAre Real, T2- 4A > 0

(i) If in addition, A > 0, both roots a>; have the samesign.According toEq.F.23),this implies a nonoscillatory approachto\342\200\224or departure from\342\200\224

the singular point.*Onespeaksin this case,respectively,of a stable or of anunstable node. Figure 6Aa,b describesthe qualitative behavior of the tra-trajectories in thesecases.

Two particular cases,both correspondingto a doubleroot of the char-characteristic equation, are worth mentioning. The first correspondsto thesituation a12 = a21 = 0, an = a22 = a # 0:

dx\342\200\224 = axdt

F.24)dy

* We note that the behavior may well be nonmonototic for some finite time interval. Thisarises when the signs off, and c2(orof the Ku K2) arenot identical.

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(a) (b)

Figure6.1. (a)Asymptotically stable node; (b)unstable node.

The trajectories,besidesthe singular point S, are all rays issuing from S,in the direction of S if a <0, toward infinity if a >0.Point S is calledastellar node,stableor unstable, respectively(Fig.6.2a,b).

The secondcasecorrespondsto T2 \342\200\224 4A = 0.This yields:dxIt = ax + ay

F.25)

The solutionsare given by:x = x01eay = yOiea

ay01tea F.25a)

By eliminating t one sees that the trajectoriesare the curves y = 0 andx = y log |y | + cy, with c a constant.Thesecurves are representedin Fig.63a,b.Onespeaks,respectively,of a stableor unstable one-tangentnode.

(a) (b)

Figure 6.2. (a)Asymptotically stable stellar node; (b)unstable stellar node.


Figure 6.3. (a)Asymptotically stable one-tangent node; (b)unstable one-tangent node.

(ii) If A <0, then the two (real)rootsa>i have different signs.From Eq.F.23)one obtains:

e0\"'= ^ - K2)

y-Ktxc2(K2-K,

F.26)

Taking coj >0 and co2<0, one deducesthen the following equationsfor the trajectories:

y-K2x

cl(Kl-K2)y-KlX

c2(K2-K, F.27)

Theseare hyperboloidcurves with two asymptotes passing through thesingular point, which is calleda saddlepoint. Theseasymptotes correspondto the particular choiceof initial conditionsc2 = 0 or c,= 0.Their slopesare, therefore, K^ and K2, respectivelyand are the roots of Eq. F.23a).Figure 6.4representsthese trajectoriesin the phaseplane (x,y). We note that

Figure 6.4. Separatrices and configuration of tra-

trajectories around a saddle point.


(a) (b)Figure 6.5. (a)Asymptotically stable focus; (b)unstable focus.

the representativepoint on all trajectories,but for one of the two asymptotes,goesto infinity as t -* oo.Thus, a saddlepoint is always unstable.We callthe two asymptotes the separatricesof the saddlepoint.TheRoots are ComplexConjugate,T2 \342\200\224 4A < 0

(iii) If, in addition, T # 0, then the two rootshave nonvanishing real part.According to Eq. F.23),this implies an oscillatory approach to (T <0)or departure from (T >0), the singular point. One speaks,respectively,of a stableor an unstable focus.The trajectoriesare representedin Fig.6.5a,b.(iv) T \342\200\224 0, but A >0.The roots are purely imaginary, a>i

=\302\261 iy. Thus,

the representative point in phasespaceundergoesundamped oscillations^provided of course linearization constitutes a valid approximation.According to the considerationsdevelopedin the beginning of this section,the trajectoriesare necessarilyclosedcurvessurrounding the singular point,which is referred to as a center.The situation is describedin Fig.6.6.Thesingular points exhibits Lyapounov stability; however, neither S nor theorbitsare asymptoticallystable.

Figure 6.6. Closedtrajectories surrounding a center.

Multiple Singular Points

The classificationof the singular pointsgiven in the previoussubsectionwasbased on the existenceof two nonvanishing roots of the characteristicequation, that is, on the assumption A # 0 [seeEqs. F.21)and F.22)].

6.6. Branchings, Bifurcations, and Limit Cycles 83

Figure 6.7. Configuration oftrajectories in vicinity of a multiple singular point of saddle-nodetype.

Topologically,this implies that the singular point is then necessarilya pointof intersectionof the curves

fx(X0, Y0)= 0; fY(X0, Yo) = 0 F.28)

Suchsingular points are referred to as simple singular points.Multiple singular points, or points for which A = 0, are, on the other hand,

points of contact of the curvesF.28),dennedby

(dJ\302\243\\ I(8A\\ = (8A) l(dJi\\\\dxJol\\dXjo \\dYjol\\dYjo

Owingto this, for arbitrarily smallvariationsof the functionsfx, fY a multiplesingular point generallysplitsinto two or more singular points.Accordingtothe previous subsection,in systemsinvolving two variables a multiplesingular point is a borderlinecasebetween a nodeand a saddlepoint.An

analysisof the trajectoriesin the vicinity of this point givesthe imagedepictedin Fig.6.7(Andronov, Vit, and Khaikin, 1966).As we see,the stabilitypropertiesbecomequite complex.

In systemsinvolving several variables, new types of multiple singularpoints may appear.Nevertheless,they can always be viewedas coalescencepoints of simplesingular pointsas a certain set of parameterscrossessome\"critical\" value.

6.6.BRANCHINGS, BIFURCATIONS,AND LIMITCYCLES

As emphasizedrepeatedlyin Chapter5, the propertiesof a chemicalsystemdepend,in general, on a set of parametersX. In a homogeneoussysteminvolving two variables the latter act through the rate functions fx,fY that

S4 Mathematical Tools

dependexplicitlyon X. Accordingto the classificationof the singular pointsof Section6.5,for a sufficiently small variation of X the character of the

singular point may only vary when either A or T are zero:

A(X0(X), Y0(X), X) = Q (saddle-node)T(X0{X), Y0(X), X) = (center)

Theserelations provide us with a set of criticalvalues,Xc. A slight variationof X from Xc resultsin qualitatively different trajectoriesin the phaseplane.Thus, the two casescited above correspondto structurally unstable situa-situations. From the classificationof Section6.5,it is also quite clear that atX = Xc the singular point coexistswith trajectoriesthat passin its immediatevicinity without necessarilyemanating from it. Intuitively, therefore, oneexpectsthat A = Xc is a point ofbifurcation,wherebya branchingofsolutionsof the differential equationsF.13)occurs.

We now discussa number of general theorems stating the conditionsunder which bifurcation may occur.To this end we write the nonlineardifferential systemfor the perturbationsaround the thermodynamic branch(seebeginningof Section6.3)in the form

dx\342\200\224

or F-30)

J(X)x + N(X\\x) = 0

wherex isnow regardedas a vectorwhosecomponentsare (x1,...,xn),Listhe linearizedoperatorappearingin Eqs.F.6)orF.16),and N comprisesallcontributions that are nonlinear in x. Note that in the secondform ofEq.F.30)J contains both the time derivativesand the Laplaceoperators,inaddition to the contributions due to the chemicalreactions.

An equivalent way to formulate the problemis to define X in such a waythat J can be split into a A-independentpart Jo and a contribution pro-proportional to X:

Jox- Xx + N(X;x)= 0 F.31)Onecan then prove the following:

Theorem1. The number Xc can bea birfurcationpoint ofEq.F.31)only if it

is an eigenvalueof the operatorJo.The converse of this theorem neednot be true. In order to obtain some

information about thosepointsof the spectrumofLo,which, in fact,generate


branchesfor the nonlinear problem,we have to introduce the conceptofmultiplicity of an eigenvalue.Let us first illustrate this conceptin the casewhereL is a squarematrix, and let / be the unit matrix. We say that Xk is aneigenvalueof algebraicmultiplicity /, if:

det|L-A/| = (A -Xjh(X)

with F.32)h(Xk) # 0

More generally, consider the set of linearly independenteigenvectorscorrespondingto Xk, which belongto the so-callednull spaceof the operatorL \342\200\224 XkI. The multiplicity of Xk is then dennedas the dimensionof the unionof the null spacesof (L -

XkI)m, where m = 1,2, It can be shown (Rieszand Sz.Nagy, 1955)that as m increaseseach null spaceis strictly containedin the succeedingone until a certain index / is reached;thereafter, all null

spaceshave the same dimension,necessarilyfinite.

We are now in positionto state the following theorem due to Leray andSchauderA934):

Theorem2. If Xc # 0 is an eigenvalueof oddmultiplicity ofJoin Eq.F.31),then Xc is a bifurcation point for this equation.

The proof of this important theorem isbasedon topologicaldegreetheoryand isomitted (seeKrasnosel'skii,1964;Sattinger,1973).Note that there maybe,even under the smoothestassumptionson Joand N, a large or even aninfinite number of bifurcating solutions,and a completediscussionincludingalsotheir stability propertiesseemsto be impossible.If the eigenvalueis ofeven multiplicity, the additional element is that there may beno bifurcatingsolutionsat all. It is only in someparticular cases(McLeodand Sattinger,1973;Bauer, Keller,and Reiss,1975),that one can discussbifurcation froma multiple eigenvaluein full detail.

As an example of the roleof the multiplicity of the eigenvalue,considerthe simplealgebraicequation

J(X)x = mx \342\200\224 Xx = (m \342\200\224 X)x = 0

Onehas a nontrivial solution x # 0 if and only if m = X, and this obviouslycorrespondsto an eigenvalueof odd multiplicity of the \"operator\"Jo= m.In contrast, in the problem(Stakgold,1971)

xt + x\\-

Xxx = 0

x2 \342\200\224

x\\\342\200\224

Xx2 = 0

*\" Mathematical Tools

the linearized systemhas A = 1 as a doubleeigenvalue.No nontrivial real-valued branchesof solutionscan be issuedfrom this value as one seesimmediatelyby reducing this systemto the singleequation

x\\ + x\\ = 0

Two important casesfor which the theory isquite complete(Pimbley,1969;Sattinger, 1973)are those in which:(a) a simpleeigenvalueof the operatorL in Eq.F.30),that is an eigenvalueof multiplicity / = 1,crossesthe originor (b) a pair of simplecomplexeigenvaluesof L crossthe imaginary axis.In both caseswe assume

Re\342\200\224^^>0 F.30)where cocis oneof the critical eigenvaluesof the operatorL. This impliesbehavior for atc as shown in Fig.6.8a,b.Then, in both caseswe have atX = Xc bifurcation correspondingeither to a new branch of steady-statesolutionsor to a periodictrajectory. Fora two-variable system thesecasescorrespond,respectively, to A(AC)= 0 and to T(XC)= 0 [cf. Eq. F.29)].Moreover,one can show (Sattinger, 1973)that in the vicinity of A = Xcsolutionsthat bifurcate above critical level, that is, for X > Xc, are asymp-asymptotically stable,while solutionsbifurcating below criticality, that is, forX < Xc, are unstable. We illustrate these results in detail on the modelsdevelopedin Chapter7.

It is remarkablethat these highly nontrivial results are based on thepropertiesof the linearizedsystem.This isdue to the fact that one is interestedprimarily in the propertiesof the solutionsbelow orabove the bifurcationpoint.In contrast, at A = Ac, the principleof linearized stability (cf.Section6.3)doesnot apply, as the system is in a state of marginal stability. In this

(b)

Figure 6.8. Behavior of (simple) eigenvalue of operator L corresponding to Eq. F.30) in

vicinity ofa bifurcation point Xc: (a) real eigenvalues; (b)complex eigenvalues.


case,one needsthe full nonlinear system to obtain information about thecharacter of the criticalpoints,as well as about the possiblebifurcations.

The emergenceofmultiple steadystatesand asymptoticallystableperiodicorbitsis most important in terms of physicochemicalapplication.Considerfirst the caseof many steady states.Of particular interest is the situationwhere the system can exhibit more than one simultaneously stable stateof this kind. In this casea certain functional orderbecomespossiblethroughtransitions betweenthesestates,providinga self-regulationof the concentra-concentrations of the variouschemicals.Note,however,that in casesinvolving bifurca-bifurcation this need not always be the case.Forinstance, a single singular pointcan besucceededby a singlenew asymptoticallystablesolution.It may alsohappen that for certain initial conditionsthe systemdoesnot admit boundedsolutionsas t -* oo or that it even explodesat a finite value of t. As we seerepeatedly in the sequel,multiple steady statesarise in a great number ofmodelsof physicochemicalor of biochemicalinterest.

Comingto the bifurcation of time-periodicsolutions,we seethat we nowhave the possibilityof cooperative behavior in the form of a \"temporalorganization

\"in the system.Now temporal organization impliesnecessarily

structural stability. Therefore, two periodic(i.e.,closed)trajectoriesin the

phasespacearising via a bifurcation mechanism are necessarilyseparatedby a finite distancefrom eachother aswell asfrom the singular point.Accord-According

to Poincare,we call them limit cycles.In contrast, structurally unstablesystemsmay exhibit in a finite domain of phasespacean infinity of closedtrajectories.Their amplitudes and periods are determined by the initial

conditions,whereas in the limit-cyclecasethey are determined by the systemitself. Forthis reason,periodictrajectoriessurrounding a center are hence-henceforth dismissedas a model of chemicaloscillations(we return to this pointin Sections7.1and 8.2).

Note that a limit cycleneednot alwaysbeasymptoticallystable.Unstableor even semistablelimit cyclesare possible,as describedon Fig.6.9.

Figure 6.9. Asymptotically stable, unstable, and semistable limit cycles.

88 Mathematicals Tools

Let us mention, without proof, some important resultsestablishedforlimit cyclesin two-dimensionalphasespaces(Andronov, Vit, and Khaikin,1966):

(i) A closedtrajectory surroundsat least one singular point.When a singlesingular point is inside,this point can only be a focus,a center or a node.Itcan beneither a saddlepoint nor a multiple singular point,(ii) Forthe negativecriterion of Bendixson,if the expression

-> dfx dfY -> /fydnf^8X+

8Y Wkh/ =U

[cf.Eq.F.13)]doesnot changesign in a domain of the (X, Y) plane,there canbeno limit cyclesin this domain.This statement alsoshowsthat limit cyclescan only arise in nonlinear systems.(iii) Simpleasymptotically stablelimit cyclescan emerge smoothly from asingular point via the bifurcation mechanismoutlined earlier in this section.(iv) Stablelimit cyclescan emerge from multiple limit cyclesarising fromthe coalescenceof a stableand an unstable limit cycle.(v) Two or several limit cyclessurrounding a singular point are possible,as a result of a subcritical bifurcation of a periodictrajectory from thissingular point, arising for certain criticalvaluesof the parameters.A concreteexampleof this behavior isanalyzed in Section15.5.(vi) Morecomplicatedbifurcationsof limit cyclesare possiblein the presenceof separatriceloopsjoining two singular points,one of which is a saddlepoint, (Fig.6.10).

Ml

Multiple singular point(saddle-node)

Figure 6.10.Bifurcation of limit cycles in presenceof separatrices joining two singular points,one ofwhich is a saddle point (after Andronov et al.,1966).


As we seeshortly, a great number of chemicalsystemsmay exhibit limit

cycle behavior under conditions(c).However,modelsare now known (seeChapter 15)giving rise to the more complexbehavior describedunder (v)and (vi). Although we comment more extensively of the significanceoflimit cycleslater in the sequel,we may already stress that the ability toexhibit a sharp periodicity in the concentrations shouldbe an importantmeans for regulating chemical or biologicalactivities in a great variety ofsituations.

Having seen that the relatively simpletwo-variable systemscan alreadyexhibit complexbehavior, we now addressourselves to the conditionsthat

have to be fulfilled by the kinetics in order to get cooperative behavior. We

already know from Part I that the system must be open,driven far fromequilibrium and subjectto feedbacks.Our purpose,however, is to producemore precisestatements regarding the nature of the individual chemicalstepsinvolved in the reaction sequence.

Chapter 7

SimpleAutocatalyticModels

7.1.TWO INTERMEDIATES

We consider a chemical system involving two variable intermediates,together with a number of initial and final productswhoseconcentrationsare assumedto be controlledthroughout the reaction process.The reasonwe are interested in chemical systemsinvolving two variables is not onlyacademic.As we seein Part IV, an impressive number of biochemicalproblemscan be describedby modelsthat can be reducedto two variablesunder quite realisticconditions.

We now show the following:

Theorem. It is impossibleto have a limit cycle surrounding an unstablenodeorfocus in a reaction sequenceinvolving two variable intermediates if

the reaction stepsare only uni- and bimolecular.

Thus, a trimolecular or higher orderstep is necessaryfor the onset ofcooperative behavior in time. The theorem was first proven by HanusseA972)for spatially homogeneoussystems.Tyson and Light A973)rederivedthe theorem independently and extendedit to systems involving diffusion

as well.

Proof. Let {A} denotethe initial and final \"reservoir\"chemicals,and X

and Y the two intermediates.The system is taken to be isothermal andwith no convectivemotion. Denoting the concentration of X, Y by the samesymbolsas the chemicalsthemselves,wehave, in the notation of Section6.5:

~=fx(X,Y) ~ = fr(X, Y) G.1)90

7.1.Two Intermediates 91

The steady statesare given by:

A(*o.lo) = /y(*o, >o) = 0 G.2)

and their stability propertiesare determined by the linearizedsystem:

dx\342\200\224 = aux + a12y

dy\342\200\224 = a21x+ a22y G.3)

The characteristic equation of this systemreads:

a>2 - Tco+ A = 0 G.4)

with

\\dXj0+

\\3Yjo

A = ana22- ai2a2l G.5)

Assuming now that the system is subjectto a mass-actionkinetics,takingthe mixture to be ideal and consideringthat only uni- and bimolecularstepsare present,we may write:

fx = X + aX + bY + cXY+ dX2 + eY2

fY= Y + aX + /?y + yXY + SX2+ eY2 G.6)

whereX, fareconstants independent of X, Y. At the steady state:

X0(a + cY0 + dX0)= -(X+ bY0 + eY2)

UP+ yX0 + eY0) = -(?+ aX0 + 5X2) G.7)

The coefficientTin Eq.G.4)becomes:

T= a + cY0 + 2dX0 + p + yX0 + 2eY0

^ ^]G.8)

92 Simple Autocatalytic Models

Let us now compilethe various processesthat are possiblein the system(seeTable7.1):

Table7.1

Step

A-+X ->y->

x + y->

2y->

We seethat onealways

Contribution to

(if *0)

1>0a?

b >0c?

d < 0e>0

has:

i, 7, b, e, a, 5

Contribution to

dY/dt

(if ^ 0)

y> oa > 0

/J?y?

E>0\302\243 < 0

>0and G.9)

d, \302\243< 0

whatever the detailsof the kinetics.On the other hand, a, c, jS, y have nodefinite sign.Forexample,for a it is sufficient to realize that the reactionX->can produce,consumeor leave X unchanged:

XhXh

u A

i- A \342\200\224

X \342\200\224

\342\200\224> X +\342\200\224* 2X\342\200\224* E

Y aaa

= 0>0<0

Finally, we want Xo, Yo to be positive on physical grounds.Thus, weconcludethat expressionG.8)cannot becomegreater than zero.Combiningwith the classificationof the singular pointsin Section6.5we concludethat

it is not possibleto obtain, by removing the system from equilibrium,either an unstable focus or an unstable node.The formation of a saddlepoint, which remainspossible,is not interesting in the context of instabilitiesleading to coherent behavior. Indeed,as we saw in item (i) on p. 88, a limit

cycle can bifurcate only from an unstable focus or,more generally, it cansurround only a nodeor a focus.

7.2. The Trimolecular Model (The \"Brusselafor\") 93

7.2. THE TRIMOLECULARMODEL(THE \"BRUSSELATOR\

According to the theorem just proven, in order to destabilizethe thermo-dynamic branch one needsat least a cubicnonlinearity in the rate equations.A simpleway to realize such a nonlinearity is through the step

2X + Y \302\253=\302\261 3X

or through a symmetricalstep interchanging the rolesof X and Y. Let us,therefore,considerthe sequenceof reactions under open-systemconditions:

B + X < , ' Y + D

2X +Y

E G.10)

In order to achieve nonequilibrium constraints we may assumethat the\"final products\"D, E are removed from the reaction spaceas soonas theyare produced.In terms of the rate equationsthis is equivalent to takingL4= L2= 0, To simplify the analysis even further we then introducethe additional condition k_, ^ 0 (which is certainly justified as long as A

is in excess)and k_ 3~ 0.The schemeis then describedby the rate equations

[cf.Eq.E.3)]:

^ = M~- (k2B + k,)X + k3X2Y + b,V2X

PY-^= k2BX- k3X2Y+ D2V2Y G.11)6(

Accordingto the ideasdevelopedso far, we expectthis systemof equationsto feature various kinds of cooperative behavior.

It is instructive to comment on the meaning of the cubicnonlinearity in

this model, first proposedby Prigogine and Lefever in 1968 and oftenreferredto sinceas the trimolecular modelor the Brusselator(Tyson, 1973).Firstly, it is important to realize that the primary objective in studying amodel is to discoverthe types of qualitative behavior compatiblewith somefundamental laws, such as the laws of thermodynamics and chemical


kinetics.In this respect,regardlessof the reasonablenessor not of a tri-molecular reaction step,the Brusselatoris a perfectlyacceptablemodel forthe study of cooperative processesin chemical kinetics.It plays somewhatthe same roleas such modelsas the harmonic oscillator,orthe Heisenbergmodel in ferromagnetism,which have both beenwidely used to illustratebasic features of systemsdescribedby the laws of classicalor quantummechanics.Secondly,in most branchesofphysicsof cooperativephenomenasuch as plasma or laser physics,cubicnonlinearities translating multiplecouplingsbetween appropriately defined \"modes\" are, indeed,the first

nontrivial nonlinearities giving rise to cooperative behavior. Thirdly,although not nearly as common as bimolecular processes,trimolecularstepsare not completelyunknown in chemicalkinetics,even in the gaseousphase.Forinstance,one of the stepsof the Chapman sequenceof reactionsin the upperatmosphere(Nicolet,1964)involvesthe formation of ozone byatomic oxygenvia the triple collision:

O+ O2 + M > O3 + M

whereM isa third bodywhoserole isto dissipatethe excessenergyavailablein the reaction.Finally, the rate equationsin a number of biochemicalreactions involving enzyme catalysis can be reduced to cubic terms in

certain limiting cases.Onecaseis that of the glycolyticpathway (seeChapter14).Another exampleis the following sequenceof enzymatic reactions:

x +E,Xh

E,XYhEx>Ei

F

E, <=h Y ?=h X \302\253=

:2y \342\200\224

XY

i X

=> E=? E=^ E\342\200\224+ E\342\200\224> E\342\200\224+ E

iX,XYiX2YiXY + X

,X + X

i+Xwherethe enzymeEx is assumedto have at least three catalyticsitescapableof fixing two moleculesofX and one moleculeofY simultaneously.Providingthe rates of decompositionof the complexesare sufficiently large and theenzymes are presentin very small amounts, onecan easily show that this

sequencecan bereducedto a singleoverallstepgiving a nonlinear contribu-contribution in the rate equationsof the formX2Y.

7.3.SCALING,STEADY STATES,AND BOUNDARY CONDITIONS

Equations G.11)dependon as many as six chemical parameters,the twodiffusion coefficients,and the dimensionsof the reaction vessel.Forbetter

7.3. Scaling, Steady States, and Boundary Conditions 95

insight into the qualitative behavior we introduce the following scaledvariables (Erneuxand Herschkowitz-Kaufman,1975):

t = kjfU \\l/2

X = (k) *

fk2k \\1/2 H \\J\\A = (kp-) A G-12)

B = ^B

By substituting these relations into Eq. G.11)one finds that the reducedvariables X, Y satisfy the system:

_ = A - (B + l)X + X2Y +\302\243>!

V2X

G.13)\342\200\224 = BX-X2Y + D2V2Y

where the Laplaceoperatorrefersto the same spacecoordinater as before.Note that accordingto Eq.G.12),quite realisticvalues of the rate constantsand diffusion coefficientsmay correspond,in the reducedsystemG.13),tonumerical values of the coefficientsof the orderof unity. This point shouldbekeptin mind in view of the computer simulationsto bereported,in which

the values of A, B,Du and D2may seemunrealisticat first sight.We study the solutionsofEq.G.13)under the following two typesofbound-

boundaryconditions:

Xz = A

y1 = \342\200\224 (Dirichletconditions) G.14)A

fdXY fdYYn-f\342\200\224 =n'K- =\302\260 (no flux conditions) G.15)\\drj \\drj


With either of these two conditions,Eq. G.13)admits a single uniform

steady-statesolution:

Xo = A, Y0 =l G.16)A

Accordingto the ideasdevelopedin Part I,this solution lieson the thermo-dynamic branch. This can be proven explicitly by taking the reversibleschemeG.10)and gradually increasing the overall affinity of the reaction,to remove the system from equilibrium (Lefever, 1970;Glansdorff andPrigogine,1971).Note that if the boundary values G.14)had beenchosendifferent from A and B/A the thermodynamicbranch would not beuniform,

owing to the presenceof boundary-layer effects.However,theseeffectsarespuriousfrom the standpoint ofcooperativeprocessesand are not consideredexplicitlyin this chapter.

Following the general methodsestablishedthus far, we carry out a linearstability analysis of the thermodynamic branch G.16).We expectthat this

analysis will give interesting information regarding the possibilityof self-organization processeswithin the system.

7.4. LINEAR STABILITYANALYSIS

We setX = A + x

Y =|+ y G.17)

and construct, from Eq. G.13),the linearized system [seeEq. F.6)]for

(x, y). Note that (x,y) can be interpreted either as systematicexternal distur-disturbances or as internal fluctuations around the referencestate.In the lattercase,the problemwe addressourselves to for the time being is not thea priori probability for having such a fluctuation but, rather, its time evolu-evolution onceit has beenestablishedby somemechanismin the system.

It isconvenientto write the linearizedsystemfor x,y in the form:

ICK)subjectto:

x = y = 0 on I G.19a)or

n-Vx= n-Vy = 0 on I G.19b)

7.4. Linear Stability Analysis 97

fx\\Here the set of x,y is viewed as a vector I in someappropriatespace.\\yJ

After a few manipulations one finds the following form for the linearizedoperatorL:

fB - 1 +\302\243>!

V2 A2

-B -A2 + D2\\

This is a parabolicoperator(Sneddon,1957).To analyze the asymptoticbehavior of the solutionsof G.18)as t -* oo it is sufficient to find the eigen-

eigenvalues (om and the eigenvectorsIm

I of L:

B - 1 +\302\243>!

V2

-B -A2 +2 , r. V72 M .. /\342\200\224

^m\\ .. / G.21)

subjectto

ul = vl = 0 G.22a)or

n \342\200\242

V\302\253\302\243

= n \342\200\242

Vt\302\243

= 0 G.22b)

In terms of (\"m

1 the solution vector (*

1 is [cf.Eq.F.19)]:

G.23)

Thus, the reference state (A, B/A) is asymptotically stable if for all m the

eigenvaluesa>m obey

Re com <0

If for somem Re (om >0, then the solution on the thermodynamic branchis unstable.According to the Leray-Schaudertheorem (Section6.6),atRe com

= 0 onehas a bifurcation phenomenon provided the eigenvalueis ofoddmultiplicity. We are interestedespeciallyin simple eigenvalues.

In order to display the detailsof the calculation in a transparent way weconsiderin this chapter the caseof one-dimensionalsystems(higher dimen-dimensional systemsare treated in Chapter8).Note that throughout the analysiswe deal with boundedmedia.One reasonis that in chemistry, and to aneven greater extent in biology, boundariessuch as membranes play a veryimportant role.Another reasonis that the mathematical analysis becomessimpler as the spectrum of the operator L is then discrete,and onecan


construct analytic expressionsfor the solutions of the nonlinear systemsbeyond the instability.

Let / be the length of the system.The LaplaceoperatorV2 reducesto:

V2 = -r-2 0 < r < 1dr

whoseeigenvaluesare trigonometric functions.As V2 is the only differential

operatorappearin

sarily of the form:

operatorappearingin L the eigenvectors Im

I subjectto G.22)are neces-

for the boundary conditionsG.22a),and

m = o,l,2,... G.25)

for the no-flux boundary conditions.We now deal successivelywith the

eigenvalues,the eigenvectors,and the structure of the operatorL in a spaceof square-integrablefunctions, for which the set of normal modesin Eq.G.24)or G.25)provide a completebasis.

Eigenvalues

Introducing Eqs.G.24)and G.25)into Eq.G.21),oneobtainsa characteristicequation (seeSection6.5)expressing<x>m in terms of m and the parametersofthe system.After a few algebraicmanipulations one finds:

wm + (Pm\342\200\224

Xmfelm + A2B \342\200\224

am jSm= 0 G-26)

with

. m2n2

G.27)

i =K\302\253m -Pm\302\261 x/K+ /U2-4/12B] G.28)

The solution of this equation yields:


From this expressiononegetsthe following results:

(i) As m - oo, (am + pmJ -> (m*n*//*)(D2 -\302\243>iJ

-> ao. Thus, for finite 4and B a>* is necessarilyreal.Moreover, as D2 + Di > \\D2

-D^, com isalways negative. We seealready that all normal modesdo not play thesame role,as thosewith m -> oo tend to stabilize the system,(ii) com iscomplexif

(\302\253*+ /U2- *A2B <0,

or

B2 - 2(A2 + SJB+ (A2 - SmJ <0 G.29)with

Sm = l +\"}2j^(D1-D2) G.30)

The rootsof the trinome in Eq.G.29)are:

B\302\261

= A2 + Sm \302\261\302\261

Thus, one must have

Sm >0 G.31a)and

U -v^J2< B<(A+ ^/SJ2 G.31b)From Eqs.G.31a)and G.30)wededuce:

D2-Dl^ G.32)

In particular, if the value m = 0 can be excluded\342\200\224for instance, by takingDirichlet conditions\342\200\224one cannot have complexconjugate eigenvaluesfor

I2D2-D,.>^

(iii) A complexeigenvaluehas a positive real part if the coefficientof com in

Eq.G.26)isnegative,orif [seedefinitionsG.27)]:

B>A2 + 1 +m~^-{D1+ D2)

Figure 7.1.representsB as a function of m along the critical curve

Bm = A2 + l+ \"^(\302\243>!+ D2) G.33)


Figure 7.1.Linear stability diagram associated with bifurcation of time-periodic solutions,where m = wave number; Bm (m =0,1,...)= corresponding value of bifurcation parameter.At first bifurcation point Bo one always has appearance ofa spatially homogeneous limit cycle.

Obviously, the (purely imaginary) eigenvalues com are simplein this case.Thus, accordingto the theoremsdiscussedin Section6.6,the points on thecurve Bm correspondingto integer values of m are necessarilybifurcationpoints of time-periodicsolutions.As B increasesfrom a region of smallvalues,the first bifurcationcompatiblewith the boundary conditionsoccurseitherat m = 0 for no-flux conditions[seeEq.G.25)]orat m = 1for Dirichletconditions[seeEq.G.24)].(iv) If com is real, onemay have one positiveroot provided

ampm -A2B>0or

I2

Figure 7.2representsB as a function of m along the critical curve

zn2

D2mznzI2

D{tn2n~T2 G.34)

Fora given B the zero eigenvaluescom= 0 are generallysimple,exceptfor

someexceptionalvalues of the parameters(seebelow).Thus, the pointsonthe curve Bmcorrespondingto integer valuesof m are necessarilybifurcation

points ofsteady-statesolutions.As B increases,the first instability occursfor a

7.4. Linear Stability Analysis

B

101

Figure 7.2. Linear stability diagram associated with bifurcation of steady-state solutions.First bifurcation point Bc lies in vicinity of minimum Bu of marginal stability curve, although in

general Bc # B,,.

B = Bc correspondingto an integer mc closestto the minimum (jU,

the critical curve.Oneeasilyfinds:

Al2

of

G.35)

(v) Combining cases(iii) and (iv) we may derive conditionsconcerning thefirst instability to take place in the system as B increases.Obviously, this

happenswhen B crossesthe value (seeFigs.7.1and 7.2):

for no-flux conditions,and

for Dirichlet conditions.

B = min(Bc,Bo)

B = min(Bc,

ComparingEqs.G.33),G.34),and G.32)we find that for Dirichlet con-conditions the first bifurcation leadsto time-periodicsolutionsprovided


and

or

In particular, for D, = D2 =\302\243> oneobtains:

/I/2D <-^ G.37)

On the other hand, for Dl/D2~* 0 or for Dl/D2-> oo, inequality G.36)iscompromised.Thus, the bifurcation of time-periodicsolutionsis favoredwhen the diffusion coefficientsbecomevery close.Note,however, that forDirichlet conditionsthe bifurcation of steady-state solutionsis possiblefor

equal diffusion coefficients,provided the latter satisfy the inverse inequalityG.37).Figure 7.3describesvarious possibilitiesconcerning the first bifurca-bifurcation.

(vi) Forcompleteness,let us finally derive the conditionsunder which theeigenvaluecom in case(iv) isdoubly degenerate.Obviously,what isneededisthat Eq.G.34),which isregardedasan equation for m2, bewritten in the form:

(m2 - m\\)(m2 -m22)

= 0

wheremu m2 are two positiveintegers.In particular, if wetake m^ = mc < \\x,

then wenecessarilyhave m2~ mc + 1.The condition for doubledegeneracy

is,then, that the productof the rootsof Eq.G.34)bemc(mc + 1).By workingthis condition out explicitly one finds (Mahar and Matkowsky,1976;Auchmuty and Nicolis,1975):

^w^=m^ + 1) G38)

Subsequently,we assume that this rather exceptional condition is notsatisfied.We return to the problemof degenerateeigenvaluesin Section7.9.

Throughout this subsectionwe have been interested only in the firstbifurcation from the thermodynamic branch.The reasonis that, as we seeshortly, bifurcation theory can give information about the stability of the


B ,

50

40 -

30 -

20 -

10

, (c)_ I

I

\\

t

1

\\\\\\y

1 *4*l V*1\\

' 1i 1f' ' /

\"i\\

\342\200\242 1H /

1A /'if //' (a) /if /

B ,

20

15 -

10

1 i-1 '

1

J

1

V i

(all

f@ J/ \342\226\240

10 20 30 n 0 10 20 30 n

Figure 7.3. Linear stability diagrams resulting from Eqs. G.33)-G.35).Bifurcation parameterB is plotted against wave number n. (a) and (b)show region of complex eigenvalues; (b) shows

region corresponding to an unstable focus; (c)shows unstable region corresponding to a saddle

point. Vertical lines indicate allowed discrete values of n for a system submitted to zerofluxes orfixed boundary conditions. (A) shows A = 2,/=1,O,= 8.0x 10~3.D, = 1.6x 1(T3;(B)shows A = 2, / = 1, D, = 1.6x 10\" \\ D2 = 8.0x 10\023. Line limiting regions (a) and (b)have not been drawn fully in Fig. (B).

bifurcating branchesonly in this case.The problemof successiveinsta-instabilities is dealt with in Section7.9and Chapter8.

Eigenvectorsof OperatorL

Assuming that condition G.38)is not satisfied,we concludefrom the pre-preceding analysis that in the vicinity of B = Bcand in the caseof a steady-statebifurcation, the operatorL has a one-dimensionalnull spacespannedbythe functions (dependingon the boundary conditions):

Ci \\ . mnrsin -\342\200\224

G.39)

Cl)cosmnr

104 Simple Autocatatytic Models

Substituting into the linearized system G.21)oneobtains the value of theratio c2/cias follows:

c2 _ (\302\243>,m27t2//2)

A1

- BnG.40)

or,for m = mc ^ jx:

So far, Ci remains undetermined. One can fix it by requiring a specific

normalization for m .This is not needed,however; as we seelater, thewnonlinear analysis beyond the bifurcation point can remove this indetermi-indeterminacy.

If, on the other hand, one has a time-periodicstate bifurcation, thefx\\solutionsI I in the vicinity of Bu for example,feature both spacedepen-

dependence and time periodicity:

expi(lma>cf)sin^ G.42a)

Forno-flux boundary conditionsone has for the first bifurcation (mc = 0):

w =u-1)expi{lmoi^ G-42b>

The steady state (/I,B/A) behaves likea center in much the sameway as in

the systemdiscussedin Section6.5.Accordingto the theorems developedinSection6.6,weexpectthat for B > Bothis situation givesriseto a limit-cyclebehavior.

o /

Figure 7.4. Polar eigenfunction of linearized operator L for zero-flux boundary conditions.


An important feature of the spatial part of eigenfunctionscorrespondingto no-flux boundary conditionsis the possibilityto exhibit a macroscopicgradient acrossthe system,as shown on Fig.7.4.Thus, we have a generalmechanismfor the spontaneousonsetof polarity in a previouslyunorientedspace(Babloyantzand Hiernaux,1975).

The Adjoint Operator

In the sequelwe needthe propertiesof the adjoint L* of the operatorL, in

the functional spacedefinedearlier in this section.We have:

T1clrV = I I G.43)

<DlT1+ B - 1clr

Obviously,the eigenvaluesof this operatorare the sameas for L,whereasthe eigenfunctionsfeature the same spacedependence.Forinstance, in thecaseof Dirichlet conditionsand in the vicinity of a steady-state bifurcation

mnr

wherethe ratio d2/d1now has the value:

R 1'~WV

Dm \342\200\224 1 \342\200\224

I2G.44)

Note that under no conditionscan the operatorL be self-adjoint for thetrimolecular model. Stated differently, the linearized equations for thismodel cannot derive from a potential. We have here a striking confirmationof the resultsof Part I (seealso comments in Section6.4),accordingtowhich nonlinear systemsfar from equilibrium cannot, in general,bedescribedby the extrema of a state functional.

The analysis of the presentsection refers specificallyto the trimolecularmodel.Scriven and co-workers(Gmitro and Scriven, 1966;Othmer andScriven,1969)carriedout a formal linear-stabilityanalysisofgeneralreactionschemes,extending the results,describedin Section6.5.

We now study, successively,the two kinds of bifurcation suggestedby thelinear stability analysisof the trimolecular model.


7.5. BIFURCATIONOF STEADY-STATE DISSIPATIVESTRUCTURES:GENERAL SCHEME

We now placeourselvesin case(iv) of the previoussubsectionon eigenvalues.Our aim is to produceexplicitforms of the steady-statesolutionsbifurcatingbeyond the critical value Bc.To this end we insert the decompositionG.17)into the rate equationsG.13),keepingthis time the nonlinear contributionsin x and y. Moreover, we decomposethe operator L in G.20)in a waysimilar to Eq.G.31):

G.45)

where Lcis the operatorL evaluated at the critical point of the first bifurca-bifurcation. The main motivation for this decompositionis the property establishedin the previous section,that Lc is a parabolicoperator which admits a

single null eigenvector Im\"

Jand has otherwise eigenvalueswith negative

\\ mcjreal parts. Such operatorsare referred to as dissipativeoperators in theliterature. They are well known from stochasticprocessesand kinetictheory where the Fokker-Planckoperator provides the most familiar

example.We obtain in this way the differential system (setting alsodx/dt = dy/dt

= 0):

where

h(x, y) = (B -Bc)x+ 2Axy + \342\200\224 x2 + x2y G.47)

The boundary conditionsare,again, the sameas in Eq.G.22).Note that in

the excessvariablesx and y the nonlinearity isnot only cubicbut alsogivesriseto quadratic terms.

AATo calculate the solutions we observethat closeto the bifurcation

\\yj

( A \\point the correctionsto the thermodynamic branch I I should besmall.

\\d/AJ

7.5. Bifurcation ofStead-State Dissipative Structures: General Scheme 107

(x\\We expressthis by expandingboth and y = B -Bc in terms of a small

\\yj

parameter e*:

yj \\yoj \\yi,

y = B - Bc = ey, + e2y2 + \342\226\240\342\226\240\342\226\240 (IAS)

Thisexpansionismore flexiblethan the seeminglymore natural one wherebyx^

would be developedin powersof (B \342\200\224 Bc); in particular, it enables

one to determine straightforwardly a fractional powerdependenceof

on (B -Bc).We now introduce the expansioninto Eq.G.46)and identify equal powers

of e.We then obtain a set of relations of the form:

(Vj= f~ak\\fc = al G.49)

together with the boundary conditions

**@) = xk(l) = \342\226\240\342\200\242\342\226\240= 0or

dr dr

The first few coefficientsak are:

p~Jxo

IB+

x2+ 2Axoyt + Vi -7+ xly0 G.50)

A

As a result:/vA = 0 G.51)

Recently Lefever, Herschkowitz-Kaufman and Turner were able to carry out an exactbifurcation calculation in a simplified version of the trimolecular model. Their results arebriefly described in the Concluding Remarks.


In other words,I\302\260

I is proportionalto the zero eigenfunctionIme

I of Lc.

On the other hand, Eqs.G.49)for k > 1 constitute inhomogeneous systemsof linear equationswith constant coefficients.If the operator Lc were in-vertible, thesesystemswould be solved straightforwardly by constructingGreen'sfunction for the operator*Lc (Sneddon,1957)and by convertingthem to a set of linear integral equations.In our case,however, Lc is notinvertible as it admits one nontrivial null eigenfunction[seeEq.C.51)].A

somewhat analogoussituation familiar from linear algebrais that of a setof inhomogeneousalgebraicequationsof first degree,where the matrix of thecoefficientswould besingular, that is,it would havea vanishing determinant.

The following theorem, which constitutes the extension of a result first

proven in the context of problemsof linear algebra,specifiesthe conditionsunder which the systemsG.49)admit a solution (Sattinger, 1973).

/x \\

Theorem(Fredholm alternative). The vector Ik

I is a solution of Eq G.49)

provided the right-hand side Ik

I is orthogonal to the null eigenvector

of the adjoint operatorL*.In the problemwe are concernedwith \"orthogonality\" is defined by

meansof a scalarproduct that isa combination of the scalarproductfamiliarfrom vector analysis and the scalarproduct defined in functional spacessuchasa Hilbertspace(Rieszand Sz.Nagy, 1955).This leadsto the followingquantitative expressionof the theorem:

(x*, ?*)(\"*))= |odr (y* - x*) x ak({xk^m(r), yk-m(r)}) = 0

k= I,...;0 <m<k G.52)From section7.4,subsectionon the adjoint operator,we know that theeigenvector (x*,y*) has the samespacedependenceas the eigenvector ofLc and that the amplitudes du d2 of x*, y* are generally different. Thus,Eq. G.52)takes the explicit form (dependingon the choiceof boundaryconditions):

mcnr\\sin \342\200\224\342\200\224

IJodr

I

mcnr (ak({xk-m(r), yk-m(r)}) = 0cos

{ J

0 <m < k- k = 1,... G.52a)* In fact, since Lcis a differential-matrix operator it would be more appropriate to speak in

terms of Green'smatrix for this operator.

7.6. Bifurcation: Fixed Boundary Conditions 109

Theseconditions,together with Eq.G.50),determineentirely the coefficientsyt. Next, from the secondrelation G.48),onedeterminese as a function of(B -Bc).Inserting this calculated e into the first relation G.48)and solvingthe inhomogeneousequationsG.49),onehas an explicit expressionfor the

solution I I. It turns out that the behaviorof this solution dependscrucially

on whether mc is even or odd, so we treat thesecasesseparately.In thefollowing analysiswe calculateonly the first few terms in Eq.G.48),as thesealready give quite goodapproximationsto the solutionsobtained numeri-numerically.

We treat, successively,the caseof Dirichlet and of no-flux boundaryconditions.Further detailsare to be found in Nicolisand Auchmuty A974),NicolisA974b),Auchmuty and NicolisA975),Boaand CohenA976), andHerschkowitz-KaufmanA975).

7.6. BIFURCATION:FIXED BOUNDARY CONDITIONS

mc IsEven

When mc is even and oneusesthe expressionG.50)for ar in the solvabilitycondition G.52),together with Eq.G.39),onefinds:

Cl , . mcnr ( . mcnr Bc . , mcnr _\342\200\236. , mcnr\\dr sin ~V- yt sin -\302\261\342\200\224 + ~ ct sin2 -\302\261\342\200\224 + 2Ac2 sin2 -V~ = 0

Jo I \\ I A I I Jor

C' . , mcnr (Bc \\ [' . , mc7rrVt t/r sin2 \342\200\224\342\200\224 = ~

I -jCj + 2^c2 I rfr sin3\342\200\224^j\342\200\224Jo '

V^ /Jo 'The coefficientof yx on the left-hand sideis always positive. On the otherhand, sincemc is even,the integrand in the right-hand sideisan odd functionof r in the interval 0 < r < I.Thus, the right-hand sidevanishes.We concludethat:

yi = 0 G.53)Next, onesolvesthe equation

G.54)/by introducing the Fourierseriesexpansion:

sin


Substituting this into Eq.G.54)one seesthat

where

if m is even

if misodd G.55b)and

(x \\On inverting relation G.55a)one determines I1 and then, using once

Vyi/again the solvability condition\342\200\224this time for a2(r)~onefinds an explicitvalue of the coefficienty2 in the form:

y-\\= <b\\mc,A,Bc,-\302\261\\

When mc ^ /x, Bc ^ B,, (seeSection7.4)the function 4> can be expressedentirely in terms of A and D^/Dj'-

The explicitexpressionsare of no importance here.The point is to realizethat the sign of 4> determines the nature of the bifurcating solution.If

</>>0,

then one seesfrom Eq.G.48)that

4)'\"reb>b.

that is, that the solution is defined in the supercriticalregion.If, on the otherhand, 4> <0 ory2 <0, one has:

7.6. Bifurcation: Fixed Boundary Conditions 111and the solution is dennedin the subcriticalregion.When y2 = 0 one has tocontinue the calculations still further. However, this is a singular caseoflittle physicalinterest.

The bifurcating solutionsnear B \342\200\224 Bccan-becalculatedto a first approxi-approximation by inserting this value of e into the first expressionG.48)and by

retaining the first two terms.One finds that the arbitrary amplitude c,ofthe linear stability analysiscancels,and:

- Bc\\1/2 . mcnr B - Bc 8ro2/2H sm+-2A

. mnrsin\342\200\224\342\200\224

X JL (m2 -m2J m2 - 4mc2G\0256)

A similar expressionholdsfor y(r). Moreover,one can establishthe followingremarkable theorem (Sattinger,1973):

Theorem. In the vicinity of the critical point Bc,the new bifurcating solu-solutions are asymptotically stable in the supercritical region B > Bc(y2 >0).However, when y2 <0 the subcritical branchesare unstable.Note that

the very occurrenceof bifurcation here is compatible with the Leray-Schaudertheorem discussedin Section6.6,as the multiplicity of the null

eigenvalueof Lcis equal to one.

Theseresultscan adequately be summarized in the following bifurcationdiagrams:

The diagram in Fig. 7.5b is worth analyzing somewhat further. As

Auchmuty and Nicolishave shown (Auchmuty and Nicolis,1975),there is a

(a); (b)(e)

Id)

(f)

(a) (b)

Figure 7.5. Bifurcation diagram for trimolecular model in one spacedimension in caseofeven critical wave number. Full and dotted lines represent, respectively, stable and unstablesolutions, (a)shows supercritical bifurcation, (b)shows subcritical bifurcation.


minimum value of B belowwhich the completenonlinear rate equations forthe trimolecular model admit only one solution.Consequently,the sub-critical branches(b) and (c) have to \"stop\" at somepoint Bmin. By a well-known theorem of bifurcation theory either X -> oo at Bmin, orthe branchesturn and continue in the directionof increasing B values until B -> oo.Now, Auchmuty and Nicolishave found a priori boundsfor the steady-statesolutions.Thus, the only issueis the one shown on Fig.7.5b.Moreover,the new branches(e)and(f) are likely to bestable,aspointedout by SattingerA976)in a different context.

When B > Bc the system undergoesan abrupt transition to one of thesetwo branches,whereasfor B <Bcthere is one stablesolution on the thermo-dynamic branch and two stabledissipative structures.The particular solu-solution the system picksup dependson the initial conditionsand on the waythe parameter B has been varied. This phenomenon is very similar tohysteresis.

mc IsOdd

In this casethe solvability condition G.53)for ar yields [seecalculationsprecedingEq.G.53)]:

f dr sin* ^= -(\302\261 + 2A C-A fJo / \\A cj Joj I dr sin2 \342\226\240^L=-[^+ 2A^-) |dr sin3 '-^p G.57a)

The two spaceintegrations yield,for mc odd:f' . 2 mcnr I

Jc

' . , m,nr 4/dr sin3 -~-=

0 / 5mcn

Moreover, the ratio c^c-^can be expressedin terms of Bc and the otherparameters,thanks to Eq. G.40).Substituting into Eq. G.57a)we obtain,for Bm = Bc:

Vi = 8 [_2(D^n2/l2+ 1)-Bc]Ci 3mcn A

= g(mc,A,Bc,Dl/l2) G.57b)

Formc ^ fi, Bc ~ B^,one gets:

c,

7.7. Bifurcation: No-Flux Boundary Conditions 113

X

tude

AmpI

j\\(c)(a) 1 X^

1 1

1 I

I i

(e)^^(d)

Figure 7.6. Bifurcation diagram for tri-

molecular model in one spacedimension in

caseof odd critical wave number. Full anddotted lines denote stable and unstablesolutions, respectively, (a),(d) show thermo-

dynamic branch, (b)supercritical dissipativestructure, (c)subcritical dissipative structure,(e) finite-amplitude dissipative structure

emerging when (a)is subject to finite pertur-perturbations and Bmm < B < Bc.

(x \\As in the casewhere mc is even, the functions I

1 can bedetermined from

Eq. G.49)for k = 1 by means of a Fourier seriesexpansion.Combiningwith the two relations G.48)onefinally finds:

O((B-Bcf) G.58)

Again there is a similar expressionfor y(r).We see that in- the vicinity of Bc the bifurcating solution is defined for B

on both sidesof Bc.Accordingto the theorem quotedin the precedingsub-subsection, the new bifurcating solution is stableon the supercritical branchwhere B > Bc and unstable on the subcritical branch where B <Bc.Usingthe samearguments that ledto Fig.7.5b,one obtains the bifurcationdiagramshown in Fig.7.6.Thisdiagram featuresa hysteresisphenomenon along with

the possibilityof an abrupt transition on branch (e)for B near Bc.Auchmutyand NicolisA975)derived expressionsfor x{r) on branch (e) as well as forthe higher ordercorrectionson branches(b) and (c).

7.7. BIFURCATION:NO-FLUX BOUNDARY CONDITIONS

In the caseof no-flux boundary conditionsone has always(i.e.,whether mcis evenorodd):

owing to the presenceof cosineterms in the solvability condition G.57a).(x \\The coefficienty2 is computed by first determining I

1 I. The remarkable

point here is that, thanks to the particular structure of the inhomogeneousterm a^r), the Fourier seriesreducesto a sum up to the secondharmonic


of the unstable modem = mc. Otherwise,the procedureis quite similar tothat in Section7.6,and one finally finds:

(B- BA1/2 mcnr 2 B - Bc^^hH cosi-+9\342\200\224ir

rnrcr- G.59a)

^ + * ~Bc

IB- 1

G.59b)I Z. ip f\\

where

Note that this expressionrepresentsa periodicfunction with period2l/mc.As in the first part of Section7.6,when

</>>0, the bifurcation is super-

supercritical and one has a diagram similar to Fig.7.5a.When</>

<0 the bifurca-bifurcation is subcritical, and Fig.7.5bshowsthe phenomenon of hysteresis.Thestability propertiesof the bifurcatedbranchesare the same as in Section7.6.

An elegant approximate method of solution of nonlinear differentialequationsleading to periodicsolutions in spacehas beenproposedrecentlyby J.W. Turner A975).Forthe trimolecular reaction in a systemwith no-flux

boundary conditionsone recoversexpressionsG.59),which can be thoughtof as an approximate representation of a \"

spatial limit cycle.\"The methodis inapplicablein the caseof Dirichlet conditions,where the presenceof aninfinite seriesin Eq.G.56)destroys the periodiccharacter of the dominantcontribution proportionalto sin(mc7rr//).

7.8. Qualitative Properties ofDissipative Structures in Vicinity ofFirst Bifurcation 115

7.8. QUALITATIVE PROPERTIESOF DISSIPATIVESTRUCTURESINVICINITY OF FIRSTBIFURCATION

Symmetry Breakingand Critical Behavior

The most important property of the dissipative structures arising via thebifurcation mechanism outlined in the previous sectionsis their symmetry-breaking character.When a certain critical value of a parameter X (or of aset of parameters)is crossed,the most symmetric solution of the rate equa-equations ceasesto be stable,and the system evolves to a regime with a lesserspatial symmetry. In the caseof no-flux conditions,as well as in the caseofDirichlet conditionswith mc even, this symmetry breakingis accompaniedby a doubledegeneracyof the solutionstranslated by the \"criticalexponent\"1/2in Eqs.G.56)and G.59).Stateddifferently, beyond the transition the

system has equal a priori probability to evolve to two different solutions,dependingon the initial conditions.We discussthis important point in thestochasticanalysis of dissipativestructures in Part III.Note that in the no-flux casethe first symmetry breakingleadsto a Fourier serieswith a well-defined fundamental mode cos(mc7rr//). In contrast, in the caseof fixedboundary conditionssuccessiveapproximations in terms of the parameterB \342\200\224 Bc break successivelythe symmetry of the lower approximations andlead to spatially asymmetric solutions.A typical comparisonof xo(r) andcxj(r)given in Fig.7.7illustrates this point. In a sense,if the dissipativestructuresarising under no-flux conditionswere to be comparedto a crystal,those arising under Dirichlet conditionsshouldbe regardedas \"imperfectcrystals.\"

x(r)

x(r)

e sin 2m

Figure 7.7. Illustration of spatial asymmetry induced by presence of subharmonic terms in

Eq. G.56).


The situation is different in the caseof Dirichlet conditionswith mc odd.In this caseone cannot really speakof a spontaneoussymmetry breakingin the subcriticalcase,in the sensethat in orderto evolveto the branch (e)ofFig.7.6from the thermodynamic branch one would need perturbationsexceedingsome threshold value related to the preciselocation of the un-unstable branch (c).ForB > Bc,however,it is most likely that arbitrarily small(but still macroscopic)positive perturbations should lead to branch (e),whereasnegativeoneswould leadto branch (b).

Somepreliminary analogiesbetweenthe formation ofdissipativestructuresand phasetransitions can already bedrawn here, although a more completeview of this point is possibleonly in Part III,which is devoted to the stochastictheory of fluctuations.Thus, in the caseof the smooth transitions describedby Fig.7.5aor by the passageto branch (b) in Fig.7.6,one can speakof asecond-ordertransition involving a soft mode(Stanley, 1971),in the sensethat the amplitude of the bifurcating solution, which plays the role hereof the \" order parameter,\"goesto zero as B -> Bc.On the other hand, in

the caseof Fig.7.5b,as well as in the passageto branch (e) in Fig.7.6,onecan speakof a first-order transition. We may note, however, that this isa purely conventional classification which cannot possiblyprovide in-information about the nature of the critical fluctuations in the transitionregion.This important point is fully discussedin Part III.

Insofar as the purely formal analogy drawn above is acceptable,thestructure ofEqs.G.56)and G.59)is reminiscentofsomeaspectsof the Landautheory ofphasetransitions(Stanley,1971).Consider,for instance,expressionsG.59).To order (B \342\200\224 BcI/2x(r)and y{r) are directly proportional.Thus,the passagethrough the critical regime can be describedby a single-orderparameter, w, such that:

l|w||2=\\\\x\\\\2 +

\\\\y\\\\2 r-r^ G.60a)

where||\342\200\242|| is the norm in the interval 0 < r < / appropriatefor our functional

space.Onefinds:

,1 ,,2 B~Bc||wroc\342\200\224,\342\200\224-

t

that is,apart from an unimportant phasefactor:

w oc + (BjlAY2 G.60b)


Moreover, the dynamicsof w in the vicinity of the critical point is governedby a single differential equation obtained by combining the differential

equations for xand y, together with relation G.60a).This is the basisof thereductiveperturbation approachto chemicalinstabilitiesdevelopedrecentlyby Kuramoto and Tsuzuki A974, 1975)and of the adiabatic approximationused by Haken A975a,b) in the context of laser theoretical problems.Now, a singledifferential equation for a real variable alwaysderives from a

potential. To seethis on a simpleexample,consideran ordinary differential

equation of the form F.30):

-^ = A(X)w + JV(A; w)ot

Introducing the integral

F(w)=- f dZlA(W + N&;mJo

we write the equation in the form:

dw _ 3Fdt dw

Obviously, the extrema of F determine the steady-state solutions of theequation. In particular, the minima determine the stablesolutions arisingbeyond the bifurcation point, given by Eq. G.60b).This property remindsstrongly a familiar situation encounteredin equilibrium statisticalmechanics,where the critical behavior of the order parameter is determined by theextrema of the free energy, which in turn is expressedas a seriesof (even)powersof the orderparameter (Stanley,1971).In that sensethe potential Fdetermining the parameter </>

in Eq.G.60)is the nonequilibrium analog offree energy and can be thought of as an \"extendedLandau-Ginzburgfunctional\" (Graham and Haken;1971;Haken,1975a,b).

Note that this procedurecannot be extendedstraightforwardly to higherordersin B \342\200\224 Bc.In this casex(r) and y(r) are no longer proportional,andthe critical behavior cannot be characterized in a simple way by a singleorderparameter.Fora discussionof the higher orderadiabatic approxima-approximations, we refer to HakenA975b).

Amplitudes and Mean Values

From the explicitexpressionin Eqs.G.57)and G.59)oneseesthat, as a rule,the total amounts of X and Y in the systemare not conserved in the course

US Simple Autocatalytic Models

of the transition to the dissipative structure. In particular, by calculating

x =\\

dr x(r), y = \\ dr y(r)Jo Jo

one finds that:

(i) ForDirichlet conditionsand mc even 3c, y vanish to O((B-BcI/2)but

becomenonzero to the next order.(ii) Formc odd,X and Y are never conserved.(iii) Forno-flux conditions3c vanishes\342\200\224in other wordsX is conserved\342\200\224

independentlyof the degreeof approximation. Indeed,addingthe two equa-equations in Eq.G.46)one finds:

and

dy\\

Sincex and y are periodicfunctions in the interval @,/), the right-hand sidevanishes, and thus x = 0.In contrast, y in general remains nonzero.Forinstance,from Eq.G.59b):

_ _ 1 B - Bc -2(D1wc27r2//2+ 1)+ B,y~2~$~ a3

The fact that x and y are not conservedmay have somefar reaching con-consequences as far as the functional propertiesof the reaction sequenceareconcerned.To illustrate this point, considerthe trimolecular model underno-flux boundary conditionsand in a range of parameter values where y is

positive. According to Eq. G.10),the rate of the autocatalytic step of thesequenceis then enhanced.In other words,the system tendsto produceXthrough this steppreferentiallyin the right part of the reaction spaceunderthe conditionsof Fig.7.8.The ability to produceenhanced quantities ofcertain chemicalsin a limited region of spaceconfers to the system someremarkableregulatory properties.Moreover,owing to the polarity of thestructure in Fig.7.8,this regulation manifestsitselfin the form of a vectorialphenomenon, in the sensethat the catalytic step of the reaction proceedsin a well-defineddirection rather than in a completely isotropicmanner.The phenomenon of vector catalysis is a well-knownone in cellular meta-metabolism. It is striking to seea primitive form of this phenomenon alreadyin the simplestpossiblemodel exhibiting dissipativestructures.


A

B/A

0 r I

Figure 7.8. Vector catalysis induced by polar dissipative structure arising under zero-flux

boundary conditions.

Dependenceon Length

Hitherto we have analyzed the transition to a dissipativestructure by usingthe concentration of the initial product B as bifurcation parameter.Inproblemsinvolving gradual changesof shape or size of the system (e.g.,cellular growth or various other developmentalprocesses)it is of interest toconsiderthe influenceof the length / in the formation ofdissipativestructures(Hanson,1974a;Babloyantz and Hiernaux, 1975).To this endone switchesto the new spacevariable

P =

Then the rate equations become:dXdt

= A + X2Y-(B+l)X + ~D1d2XTipo<p< 1

The boundary conditionsremain the same as before, although they nowrefer to the interval @, 1).Thus changing / may be viewedas changing thediffusion coefficientsin our problem.Alternatively, sinceDt and D2measure,in a sense,the strength of the coupling (through diffusion) of neighboringspatial regions,an increaseof / tends to diminish this coupling.All of theprecedingresultshold and in particular Eq.G.34),provided B is kept fixedand / varies and plays the roleof the bifurcation parameter.The stability

diagram of Fig.7.2can be interpreted in the alternative form shown in

Fig.7.9,where m is taken to be one of the integers compatiblewith the


(m/l)

(m/l).

Figure 7.9. Linear stability diagram displaying dependence on length / constructed under sameconditions as Fig. 7.2.

boundary conditions(e.g.,mc). The point is that for a given B, the unstableregion couldbe reachedfor different values of m and /. Thus, the transitionto a dissipative structure involves explicitly the length. We need not beinterested in the detailsof this dependence.However, from Eq.G.34)it isobvious that the marginal stability condition couldnever be satisfied if / islessthan a critical value.Stateddifferently, a systemcan evolveto a dissipa-

dissipative structure only if its size exceedssomecritical value. As the size variesfurther from this critical value, new transitions are likely to occurcorre-corresponding to different values of m. This point is confirmed explicitly in

computer simulations (Hanson,1974a;Babloyantz and Hiernaux, 1975).We return to this in Chapter 16,which is devoted to development andmorphogenesis.

7.9.SUCCESSIVEINSTABILITIESAND SECONDARY BIFURCATIONS

When m is not equal to the critical number mc, there is bifurcation of a newbranch from the uniform solution each time B crossesa value B = Bm givenby Eq. G.34),at least when 0 is a simpleeigenvalueof the operatorLBm.Thesenew solutionsare similar to those bifurcating from B = Bc, whichhave beendescribedabove, but instead of beingdominated by the modesin(mcnr/l) or cos(mc7rr//), they contain the fundamental modesin(m7rr//) orcos(mnr/l).When m is even, the new branchesare degenerateand have acritical exponentof \\, while for Dirichlet conditionsand m odd the newbranch is similar to thosedescribedin the secondpart of Section7.6.

7.9. Successive Instabilities and Secondary Bifurcations 121

Figure 7.10. Bifurcation diagram displaying successive primary bifurcations that may arisefrom thermodynamic branch.

As in the vicinity of Bm > Bmc the operatorLBm has already at least onepositiveeigenvalue,the new bifurcating branchesare unstable in the neighbor-neighborhood of Bm. Unfortunately, we do not know in what regions,if any, thesenew branchescan becomestable.

When B is much greater than Bc, there are many possibledissipativestructures,someof which are stableto smallperturbations and others whichare not. The situation may bedepictedby the diagram in Fig.7.10.

In Fig.7.10branch A) bifurcates when B = Bc,but branchesB)and C)bifurcate when B > Bc.One sees that when B > Bd, there may be sevenpossiblesolutionsof the equations.

Now if after the first bifurcation the system is on branch A), there is noobvious reasonwhy for B > Bc it shouldmake a transition to branchesB)or C)bifurcating from the thermodynamic branch.In other words,onceonbranch A) the system enjoys additional possibilitiesof complexificationif

the branch A) itself loosesstability via new bifurcation phenomena.Suchcascadephenomena are usually referred to as secondarybifurcations.Theiroccurrencefor the trimolecular model has beenestablishedanalytically byMahar and MatkowskyA976)and inferred,independently,from the resultsof computer simulations by Herschkowitz-Kaufman A975).This is a mostimportant result, especiallyin the context of biologicalproblems.It is onlythanks to secondaryand higher orderbifurcations that an evolving systemcan give rise spontaneously to increasingly complexpatterns emergingthrough a successionof instabilities.

Let us briefly outline the main stepsof the Mahar-Matkowskymethodfor fixed boundary conditions.According to linear stability analysis thezero eigenvalueof the operatorLc is doubly degenerateprovided that (seeEq.G.38)]:

8= - A2l2/4+ m?(mc + IJ = 0


Thus, wheneverS deviates from zero it provides a measure of the distancebetween the two smallesteigenvaluescorresponding,respectively,say to mcand mc + 1.As Sapproachesto zero,the two eigenvaluescoalesceto form adoubleeigenvalue.With <5 / 0 and small, the first primary branch bifurcatingat Bmc correspondsto the mode sin(mcnr/l), whereas the secondprimarybranch bifurcating at Bmi + 1correspondsto the modesin((mc + l)nr/l).

One can now expandsystematicallyBmc, Bmc+1,in powersof 6 around avalue Bccorrespondingto a doubly degenerateeigenvalue.Next, one deter-determines each of the primary branchesby bifurcation theory as in the previoussections.To fix the ideaswe considerthe caseof mc odd.We have:

uj

J=0 \\vj.00

Bp = Bmc(S)+Z?^ G.61)

where the subscriptp denotes a primary branch. Using the solvabilityconditionsoneobtains:

Formc:

(xp\\=e/mr~with

BTp<= BJd)-e\342\200\224(-=^ + 2AAmc(S)) + 0(e2) G.62)

Formc + 1:

yP,

>mc+\\ _\342\200\224

2A(S2+ Amc+l(d)S1)\\ +0{\302\2433) G.63)

Herea and Am(S) are, respectively, the amplitude and the yjx ratio cor-corresponding to the null-spaceeigenfunction:

) G.64)

According to the linearizedequations,Am dependson S and can, therefore,be expandedin powersof this parameter around a value Am. Finally, Si

7.9. Successive Instabilities and Secondary Bifurcations 123

and S2are well-definedcoefficientscontaining an infinite-seriescontributionarising in the evaluation of the coefficienty2.

The next step is to calculate the points on the primary branch B\342\204\242c orgmc

+ i from which secondarybranchescan bifurcate. To this end oneobservesthat the primary branchesare functions of the parameterse and 6.Therefore,we treat e in the same way as B in the primary bifurcation andseekfor the \"nonlineareigenvalues\"en = en(d)of this problem.The latter arecomputed by a perturbation expansionin S(Bauer,Keller,and Reiss,1975).Thus, we set:

e = d(b0 + 8b!+ \342\226\240\342\200\242

\342\200\242)

I\"-\" G.65)

where it has beenassumedthat 8ljlis the appropriateexpansionparameter.Substituting these expressionsinto the rate equations linearized around

x \\\" and equating equal powersof 8, one obtains a set of relations, the first-V

of which represents the linearized problem at the double eigenvalue.Obviouslythe solution of this problemis:

fuo\\ . mcnr ( 1 \\ (mc + l)nr ( 1 \\= 0ism \342\200\224r- - + 02 sin 3 G.66)

Thecoefficientsfi^ and ft2 can bedetermined by the solvability conditionsofthe higher orderequations in the sequence.Onedisposesof two such con-conditions here becauseof the doubledegeneracy.Whenever a contributionwith /?2 # 0 arisesin Eq.G.66),weconcludethat a secondarybifurcationhas

fx\\taken place.Indeed,the solution I at this point jumpsfrom a contribution

containing the modesin(mc 7tr)/7 to one containing, in addition, the highermodesin((mc + l)nr/l).This situation is describedin the bifurcationdiagramof Fig.7.11.Finally, the coefficientb0 in Eq. G.65)is determined by the

solvability conditionsand by the condition of normalization of

Knowing b0 we can deduce8 in terms of e, that is, in terms of B \342\200\224 Bc,andestimate in this way the \"distance\"from Bc at which secondarybifurcationtook place.


11

Figure 7.11.Bifurcation diagram for trimolecular model displaying possible secondarybifurcations of steady-state solutions from a previously established dissipative structure.Dotted and full lines denote plausible stability properties of the various branches (dotted line

denotes instability and full line denotes stability).

The detailedconstruction of the secondarybranchesin the nonlinearregion beyond the secondarybifurcation proceedsstraightforwardly, justas for the primary branches,and is not discussedhere.

In summary, in the supercritical region beyond the first bifurcation the

system exhibits a striking multiplicity of qualitatively different solutions.Thesesolutionsare discreteand emergeonly when the parameter B crossessomecritical values.Onecan really speakhere of a \"macroscopicquantiza-quantization\" of the states(Hanson,1974b).Although the information concerningstability of these statesis rather scarce,one can expectthat at least someofthem would be stable and attract somespecificsets of initial conditions.Sucha property would conferto the system a primitive \"memory\" associatedwith the capacity to store \"information,\" through the initial conditionsattracted by the correspondingdissipative structure. This seems to beconfirmedby computer simulations,to which we now turn in somedetail.

7.10.COMPARISONWITH COMPUTERSIMULATIONS

LefeverA968) and Herschkowitz-Kaufman A973, 1975)have carriedoutextensive numerical simulations on the trimolecular model.Their results

7.10. Comparison with Computer Simulations 125

provide striking confirmations of the theoretical predictionsbasedon theanalytic calculations.Moreover, they suggest some fascinating trendsconcerningthe evolution of the systemin the supercriticalregion B $> Bc.

Genera]Behavior

Considerfirst the caseof Dirichlet conditions.The numerical valuesA = 2,I = 1,Di = 1.6x 10~3,and D2 = 6 x 10~3lead to a positive value of 4>

[seeEq.G.56)],which permits us to calculate x(r) and y(r) for B > Bc.Thesolutions obtained in this way are shown on Figs.7.12and 7.13and arecomparedwith those obtainedby numerical integration on a CDC-6500computer.ForB near Bc (Fig.7.12),the two solutions agreevery well.Both the predominant influenceof the critical mode,aswell as its distortionby the infinite seriescontribution, are clearly seen.One observesthat thedistortion tendsto enhance the successivemaxima as r increases.

ForB > Bc(Fig.7.13),the agreementbecomespoor.Indeed,the numericalsolution of the rate equations now indicates an abrupt transition to a statewhere,although the number of extrema remains the same,the maxima areenhanced(and the minima depressed)near the boundaries.A great number ofnumerical simulations with various initial conditionshave shown that this

type of asymmetry does not dependon the way the system is perturbedinitially.

0.5 1

Space (arbitrary units)

Figure 7.12. Steady-state dissipative structure for fixed boundary conditions and B ^ Bc.Dashed line, analytical curve given by Eq. G.56);solid line, result of numerical integration ondigital computer. The following numerical values of the parameters have been chosen:A = 2,1 = \\,Dl = 1.6x Kr3,ZJ = 6.0x 10*3,B= 4.17.The critical wave number is mf = 8andB, = 4.133.Boundary values for X and Y: X = A = 2, Y = B/A = 2.085.


0.5 1


Figure 7.13. Steady-state dissipative structure for fixed boundary conditions and B > Bc.Dashed line: analytical curve given by Eq. G.56).Solid line: result of numerical integrationon digital computer. B = 4.6;other parameters areas in Fig. 7.12.Boundary values for X andY: X = A = 2,Y = B/A = 2.3.

Theseresultsindicate strongly the occurrenceof a secondarybifurcationfrom the primary branch, in agreement with the analytic results presentedin Section7.9.The simulations also confirm the influence of the spacingbetween the values of B correspondingto the first successiveprimary bi-bifurcations on the occurrenceof the secondarybifurcation.

In contrast with theseresults,for no-flux conditionsthe agreementbetweenanalytic and numerical solutionsremains satisfactory, even for B con-considerably larger than Bc.Presumably, if a secondarybifurcation occurs,iteither occursmuch later or leadsto unstable branches.Figure 7.14describesthe concentration profiles.

Degeneracyand SpatialSymmetry

The doubledegeneracyof the first bifurcating solutionsfor no-flux con-conditions and for Dirichlet conditionswith mc even are well confirmedby thenumericalsimulations,as shown in Fig.7.15.On applying a single,localizedperturbation to the homogeneoussteady state as an initial condition, oneobservestwo different solutions,dependingon the sign and location of the

perturbation.The spatial asymmetry of the solution under Dirichlet conditionsand

their periodiccharacter in the caseof no-flux conditionsare alsoexhibitedclearly.

The caseof mc odd under Dirichlet conditionshas alsobeeninvestigated

7.10.Comparison with Computer Simulations

X,

127

0 0.5 1


Figure 7.14.Steady-state dissipative structure for zero-flux boundary conditions. Dashed lines:

analytical curves given by Eq. G.59).Solid lines: result of numerical integration on digital

computer. Curves have been established for A = 2, / = 1.B = 4. D[ = 1.6x 10~3,D2 =8.0x 1(T3.Critical wave number: mL.

= 8, B, = 3.602.

numerically.ForB > Bc,both degeneracyand spatial asymmetry are verifiedin the same way as for mc even. ForB < Bc,a transient nonuniform stateappears:on a finite amplitudeperturbation of the homogeneoussteadystate,the system first evolvestoward a structured state with a number of extremacorrespondingto mc. Subsequently,the system homogenizes slowly andreturns to the thermodynamic branch. This result supports the generalfeaturesof the bifurcation diagram depictedin Fig.7.6.

Multiplicity of Solutions

Computer simulations have shown that more than one coupleofdegeneratestructures,each correspondingto a different number of extrema,can appearsimultaneously.This occursunder the following conditions:

\342\200\242 The valuesof B are large.\342\200\242 At f = 0, the system is submitted simultaneously to different localized

perturbations whoselocation obeyscertain rules.

This situation is illustrated on Fig.7.16in the caseof no-flux boundaryconditions.Forthe samevaluesof the parametersas in Fig.7.14one observes


I,

IFigure 7.15. Degenerate steady-state dissipative structures obtained for samevalues of param-parameters by a localized initial perturbation of homogeneous steady state of same strength but

opposite sign (fixed boundary conditions). Arrows show for somepoints the sign of perturbationleading to corresponding spatial structure. Numerical values used: A = 2, 1=1,Dl =1.6x 10\023, D2 = 8.0x 10~3,B = 4.6.Homogeneous steady-state concentrations X = A,Y \342\200\224 B/A aremaintained at boundaries.

three different stable dissipative structures with 8, 9, and 10extrema,respectively,dependingon the initial conditions.Each of these structureshas alsoits \"symmetrical.\"

As weemphasizedin Section7.9,the stability of the successivebifurcatingsolutionscannot beguaranteed a priori within the framework of a perturba-tive analysis.However, various computer simulations for B $> Bc indicatethat there is a finite multiplicity of stablesolutionscorrespondingto thefastest growing unstable modes,accordingto the linear stability analysis

7.10. Comparison with Computer Simulations

X

5

129

(B)

0.5Space (arbitrary units)

X

5

4

3

2

1

00.5



Figure 7.16. Steady-state profiles obtained for same values of parameters but different initialconditions. For instance, one obtains spatial distributions with 8,9,or 10extrema shown herewhen length / is divided into 101equal intervals and a positive perturbation applied at point 9is combined with a second perturbation of same sign and strength, respectively, at points 21,48, or 72; 17.34,43;55or 70.Numerical values of parameters aresameas in Fig. 7.15.

around the thermodynamic branch.This can be seenon Fig.7.17,wherem = 7, 8, and 9 appearas the leading modesfor the numerical values cor-corresponding to our previous example.The existenceof unstable modeswith

amplification factors of the same orderof magnitude as that of the criticalone is due to the fact that the first few successiveinstabilitiesappearfor veryclosevalues of Bm, whereas for the others the spacingof the values of Btends to becomelarge.

PeriodicBoundary Conditions

In many instancesof biologicalinterest a reaction sequencetakesplaceon aclosedgeometry,suchasthe surfaceofa sphererepresentingthe membraneofa developingcell.The boundary conditionsin this caseare, obviously:

G.67)Y@) =

130 Simple Autocatalyric Models

1.0

0.5

B = 4.6

0 5 10 m

Figure 7.17.Real part of the eigenvalues con corresponding to unstable modesemerging from

thermodynamic branch for different values of B (zero fluxes and fixed boundary conditions).Numerical values used:A = 2,/ = 1,Dx = 1.6x 10\023, D2 = 8.0x 10~3.

where / is the length of a closedcoordinatecurve along the surface. Forsimplicity we only considerhere a ring of length /.

The calculations reported in the precedingsectionscan be repeatedstraightforwardly. The differenceis that instead of Eq.G.24)or G.25),onenow has a critical modeof the form:

e2ninr/l G.68)

RelationsG.33)and G.34)remain valid provided nn/I is replacedby 2nn/l.Numerical simulations (Herschkowitz-Kaufman, 1973,1975)have been

performed by dividing the ring into M equal intervals. ConditionsG.67)become:

X M+\\

Yo\342\200\224

As before, spatially nonuniform solutionsarise beyond instability, which

exhibit a sharp spatial periodicity.The structures seemto dependstronglyon the initial conditions.However,they are all superposableby translation,and in the limit of a continuous spaceone shouldobserve an infinity ofsolutionsdiffering only by the phase(seeFig.7.18).

7.11Localized Steady-State Dissipative Structures 131

r


Figure 7.18. Stationary concentration profiles of X for periodic boundary conditions

(XM + 1= Xt, VM + 1

=Vj) and different initial conditions. Straight line, initial perturbations

6X = +0.02at points 9 and 21;dashed line, initial perturbation SX = +0.02at point 21;broken dotted line, initial perturbation SX \342\200\224 +0.02at point 9.Numerical values used:A = 2,I = 1.01,B =4.6,\302\243>,

= 1.6x 10~\\D2= 8.0x 10 -'(Bc= 3.598and mc = 4).

7.11.LOCALIZEDSTEADY-STATEDISSIPATIVESTRUCTURES

Introduction

The spatial dissipativestructures describedin the precedingsectionsextendthroughout the whole system.This is a consequenceof assuming a uniformmedium where the concentrations of the initial productsA and B are main-maintained constant in space.Clearly, this is a highly idealized situation; in anactualphysicochemicalexperimentthe reactants are injectedinto the reactionvolume through the boundaries.At best,their concentrations or their fluxescan only becontrolledon theseboundaries.Thus, in the trimolecularschemeone expectsA and B to diffuse and react within the medium and establishin this way their own profile. In this sectionwe analyze the transitionphenomena when A is no longer imposedthroughout the system, but westill assumethat the concentration of B remains uniform. We see that this

\"spatial dispersion\"of A results in the localization of the dissipativestructures within natural boundaries(Auchmuty and Nicolis,1975;BoaandCohen,1976).

The rate equationsfor the intermediate productsX and Y are given by


Eq.G.13),where the concentration A depends,in this instance,on spaceandtime via the following equation:

r)A FJ A4-=- A+D^-i 0<r<l G.69)at or

We adopt for A the boundary conditions:

4@)= 4@ = A

and apply Dirichlet conditionsfor X and Y with boundary values

X = A, Y = \302\243 G.70)4

Forsimplicity we take / = 1.The solution of Eq. G.69)is straightforwardat the steady state and gives:

-cosha

with

/2G71)

Becauseof the spatial dependenceof Ao, the steady-state solutionsX0(r), Y0(r) on the thermodynamic branch can no longer be determinedexactly. Systematic approximation schemescan, however,be set up in thelimit

\302\243>-> oo

Utilizing these two ratios as well as \\/D as perturbation parametersonefinds to first approximation (Auchmuty and Nicolis,1975):

I ID1 ,

G.72)

7.11.Localized Steady-State Dissipathe Structures 133

B

100

75

50

25

III

D2M X 10

Figure 7.19.Stability diagram in spaceof parameters B and D2-A = 14,D = 0.195,D, =1.05x 10\023.1:region of stability of thermodynamic branch. II:region of emergence of (localized)time-independent dissipative structures. Ill:region of emergence of time-periodic solutions.

We want now to investigate the possibleexistenceof additional steady-state solutionsof the dissipative structure type. Again, becauseof spatialdispersion,the stability analysisof the thermodynamic branch G.72)cannotbecarriedout analytically.A variational technique,known as localpotential(Glansdorff and Prigogine, 1971),has been applied by Herschkowitz-Kaufman and Platten A971)to analyze theseproperties.The resultsaresummarized on Fig.7.19in the form of a stability diagram in the (B,D2)plane for fixed A, Du and D.This diagram reveals the existenceof a domainIIwhere perturbations around the thermodynamic branch tend to amplifyin a nonoscillatory fashion. Presumably, then, in this domain a steadysolution of the dissipativestructure type doesbifurcate.We now outline an

analytic construction of the solutionsto the linearizedequationsaround the

thermodynamic branch associatedwith this dissipativestructure. The non-nonlinear analysis via bifurcation theory proceedsexactly as in the previoussectionsand is not developedhere.

Analytic Constructionof LocalizedDissipativeStructures

We considerEq.G.18)with dx/dt = dyjdt = 0.Adding the two equations,one gets

Dxx\" + D2y\" -y = 0Let

z = Dxx + D2y


Then the equationsG.21)are replacedby the single fourth-order equation

D,D2z\"\"+ lBX0Y0-B-l)D2~D.Xly+ X\\z = 0 G.73)We know that if Xo, Yo are constant this equation admits solutionsof theform exp(ikr).Forspace-dependentcoefficientsthe most natural extensionis to assumea solution of the following form:

z(r) = exp@(r)) G.74)where (f>'(r) is a slowly varying function of r.

Neglectingall derivativesof 0exceptthe first (in a manner similar to theWKB approximation familiar from quantum mechanics)one obtains

DjZWL+ [_BX0 Yo-B-\\)D2-D.XlW2+ X20

= 0 G.75)The solution of this equation is

^{rJ =wx y + ^ x2\302\260 ~F{r)

I\302\261 wx ^{F{r)~ B+{r)){F{r)-^w)G.76)

where

F(r) = 2X0(r)Y0(r) - B

B\302\261=[x0(r) /gi\302\261 lj G.77)

From Eq.G.75)it is apparent that one can neverhave

(f>'(rJ = 0

If this were the case,then Xl(r)/DlD2= 0, which is impossible.However,4>'(r) can change from real to complexvalues when the right-hand sideofEq.G.76)changesfrom real to complexvalues. In fact, one seesfrom Eq.G.76)that 4>'(r)is real only when

F(r) < 1 + ^ X20{r)

and

F(r)>B+ (r) or F(r) < B_(r) G.78)

Forthis problemwe see that theseconditionshold in certain regionsofspacebut are violated elsewhere.The boundariesbetween the two types ofbehavior are called the turning points of the equation.On one side of a

7.11.Localized Steady-State Dissipative Structures 135

turning point the solutionsz(r) are monotonic[<\302\243'(>\") real],while on the other

sidez(r) is oscillatory [<j)'(r)complex].The condition for a turning point in this problemis somewhat different

from the definition for the second-orderequation (Erdelyi,1956):z\" + f(r)z= 0

Forthis equation a turning point must obey

4>'(r)= 0

which, as we have seen,is not the casefor our problem.When F(r) > B+(r),one observesthat the first Eq.G.78)is contradicted,

so that one cannot get a turning point in this manner and finds only delocal-izedstructures.

If F(r) < B_(r), then the first condition G.78)automatically holds.Theturning pointsare given by the following equation:

0(r)Y0(r) -B= [xo(r)l-

1J2X0(r)Y0(r) -B= [xo(r)l-

1J G.79)

Usingour approximationsG.72)for Xo, Yo, this becomes

r2 - r + $= 0

where

The rootsof this are given by

When 0 < P < \\ one seesthat 0 < r_ < r+ < 1,and r_ , r+ are symmetri-symmetrically

situated about r = \\. When 0<r<r_orr+ < r < 1,z(r) is mono-tonic, but in the middle, r_ < r < r+,it is oscillatory.

As P tends to 0, one seesthat these turning points are pushed to theboundariesr = 0 and r = 1.In terms of B,this requiresthat

The sizeof the dissipativestructure is approximately

-2/5 = 1 -~ | (/ - 1)-t^t | G.80)

136 Simple Autocatalytii Models

Figure 7.20. Spacevariation of B+, B_,and F in caseof very fast diffusion ofconstituent A.

and this is a decreasingfunction of k for large k. Thesevarious relations areillustrated in Fig.7.20[seealsoEq.G.72)].

To find the form of z(r) near the turning points,one has to approximate4>'(r)around thosepoints.Usingthe first few terms in the Taylor seriesexpan-expansions about r_, one finds (Auchmuty and Nicolis,1975):

z(r) ocexp[i(const)x (r \342\200\224 r_) G.81)and this is nonsinusoidal.The constant in the exponent dependson DuD2,B,and A.

However,near r = \\, one getsa very different form of the solution:

z(r) ocexp[i(const)x (r - |)] G.82)

which is approximatelysinusoidal.

ComparisonWith ComputerSimulations

The existenceof localizeddissipativestructures confinedwithin the reactionspaceby a turning point mechanism is confirmed entirely by the resultsofcomputer simulations (Herschkowitz-Kaufman and Nicolis, 1972;Herschkowitz-Kaufman,1973,1975).Figure 7.21gives a typical profile forthe intermediate X. Figure 7.22illustrates the dependenceof the positionofthe turning pointson the parameters.However,most of the numericalresultshave beencarriedout for values of Du D2,and D such that F(r) and B+ (r)vary considerablyin the interval @, 1),contrary to the situation depictedin

Fig.7.20correspondingto the theoretical calculation. As a result, thecondition F(r) < B+ (r) is never satisfied for all r in @, 1).The situationscorrespondingto the numerical computations are illustrated in Figs.7.23and 7.24.

7.11.Localized Steady-State Dissipative Structures 137

10

0 0.5 1


Figure 7.21.Localized dissipative structure for D = 0.1972,\302\243>,

= 1.052x 10\023, D25.26x 10~3,/ = \\,X = A = 14,B = 24, Y = B/A = 1.71.

b <b:= a ^ -case1 (Fig.7.23):

The correspondingprofile for X is shown in Fig.7.21.Comparisonwith

the theoretical analysis shows that although z(r) should be oscillatorywithin (r01,r02),the size of the numerically computeddissipative structureappearsto be much smaller than predicted.This couldbe explainedby thefact that the wavelength of the oscillatory solution G.82)varies with r in

X>

15

10

5

n

_V

A

J J

AA A

J

-

0 0.5 1

Space

Figure 7.22. Localized dissipative structure under same conditions as Fig. 7.2 except forB = 30, Y = B/A = 2.14.


Figure 7.23. Spacedependence of B+, B , F for values of parameters corresponding toFig. 7.21.

0 0.5Space (arbitrary units)

Figure 7.24. Spacedependence of B+, B_, F for values of parameters corresponding to

Fig. 7.22.

138

5 \342\200\224

X

15'

10

B = 25A A J

0 0.5 1

Figure 7.25. Duplication of dissipative structure obtained for successively higher values ofparameter B and for relatively low value of the diffusion coefficient of A :D = 0.026.

139


sucha way that it becomesvery long near the turning points.Thus, oscil-oscillations may appearto benegligible,and so the exact boundariesof the dis-sipative structure becomehard to identify.

case2 (Fig.7.24):

B > Bb =

The correspondingprofileof X is shown in Fig.7.22.While the theoretical calculations would predictan entirely delocalized

dissipativestructure, one observesfrom the numerical calculations that thesolution is apparently not oscillatory near the boundariesbut is, instead,confined approximately within the limits F(r) = B+(r).As previously, this

could be explainedby the nonuniformity of the wavelength so that theoscillationsbehave differently near the boundaries(long wavelength) andnear the middle (short wavelength).As a result, the solution appearsto benonoscillatorynear the boundaries.

In the caseof very low values of the diffusion coefficientD one observesa very pronounceddepressionof X near the middle,together with a duplica-duplication of the dissipativestructure (seeFig.7.25).An extensionof the theoreticalcalculationspresumablyleadsto the appearanceof additional turning pointsthat separatethe two dissipativestructure regions.Indeed,sincein this caseF(r) can vary considerablyin space,the observeddepressionin the middlecould correspondto the situation F(r) <B_(r)around r = \\, which cor-corresponds to a nonoscillatory behavior.

Finally, the localized dissipative structures exhibit both the degeneracyand the multiplicity observedin the caseof a uniform distribution of A.

The possibilityof spontaneousformation of natural boundariesconfininga dissipative structure is likely to have some far-reaching consequences.Obviously, localization is an efficient means of improving the stability

properties of the pattern with respect to environmental disturbances,thanks to the \"buffer zone\" constituted by the thermodynamic branch.Moreover,the ability to concentrate the chemicallyactivecomponentswithin

a limited spaceenhancesconsiderablythe probability of somekey reactionsthat may subsequentlygive riseto further evolution.

7.12.BIFURCATION OF TIME-PERIODICDISSIPATIVESTRUCTURES

According to the linear stability analysis outlined in Section7.4,when thebifurcation parameter B crossesthe critical value B given by Eq.G.33),thelinearizedoperatorL presentsa simplepair of purely imaginary eigenvalues.

7.12.Bifurcation of Time-Periodic Dissipative Structures 141

ForB larger than the first critical value of B compatiblewith the boundaryconditions,we expectthere to bea stable periodicsolution of the equations.Sucha phenomenon is already known for systemsof ordinary differentialequationswhere it givesriseto limit cycles,and is calleda Hopfbifurcation

(Hopf, 1942).In this sectionwe analyze the time-periodicsolutionsin the

presenceof diffusion. In a sense,therefore,we are lookingfor limit cyclesinsomeappropriatespaceof functions rather than in the usual phasespacefamiliar from ordinary differential equations.We assumethat conditionG.36)is fulfilled and that the first bifurcation from the thermodynamic branchleads,indeed,to time-periodicsolutions.

We write the rate equationsin a form similar to Eq. G.46),adding thetime-derivativeterms:

G.83)

with

We are interestedmore specificallyin the bifurcation occurring at Bo andSj.The boundary conditionsare as those in Eq. G.22).Let 2n/Q, Q =Q(B \342\200\224 Bm)be the periodof the solution.We introduce the scaledvariable

z = Qt G.84)and look for 27t-periodicsolutionsin this variable. In particular, we lookfor periodicsolutionsof small amplitude near B = Bm and of frequencynear icom, where com is the eigenvalueof the linearizedoperator.To this endwe expand,asin Section7.5,x,y, B,and Q in termsofa parameter \302\243 (Sattinger,1973):

'\342\226\240 \342\200\224

<x2

B = Bm + \302\2437i+ e2y2 + \342\226\240\342\200\242\342\226\240

Q =ixm + ecoi+ e2co2+ \342\226\240\342\226\240\342\226\240 G.85)

where

Hm = Im<x>m

Substituting into Eq.G.83)and equating equal powersof \302\243 we obtain:

\342\200\236> i G-86)


The coefficientsan involve the parametersyY to yn_l and the solutionsxu yx to xn-1,ya-i- The first few expressionsare [cf.Eq.G.50)]as follows,after taking into account the slight change in notation adoptedin this section:

cii = 0/ g

a2 = *i 7i + -f-Xi + 2Ayx\\ A

IB7i +

\342\200\224f~xi

2Axiy2 + ^ + xjyi G.87)A

Substituting into Eq.G.86)we find, to the first order:d

G.88)

Eq. G.88)has the same form as the linearized equation G.18)around the(x \\

thermodynamic branch. Thus, I'

I is proportional to the (complex)\\3V

eigenfunctionof Lm correspondingto the eigenvaluescom= + ijxm. We treat

the casesof Dirichlet and of no-flux boundary conditionsseparately.

DirichletConditions

We consideragain a line of length /. Taking Eq.G.42a)into account we have

^ eh G.89)J \\ \\c2

where k is an arbitrary constant.Forn > 1,Eq.G.86)is an inhomogeneouslinear equation.As discussed

in detail in Section7.5,it has a B7t-periodic)solutionprovideda compatibilitycondition holds.The Fredholmalternative (Sattinger, 1973)requiresthat

the right-hand sideof Eq.G.86)beorthogonal to the vectors:

/x*\\where the bar denotescomplexconjugation and I I is an eigenvectorof

\\y /the adjoint operatorL* [seeEq.G.43)]:

G.91)

7.12.Bifurcation of Time-Periodic Dissipative Structures 143

The scalar product here is defined to be the Hilbert space one,as in

Eq.G.52),followedby a time integration over the period2n of the solutionexpressedin terms of the scaledvariable t.

Combiningwith Eq.G.86)we may write the explicitterm of the Fredholmalternative as:

p2n /\342\200\242!

[_x*(r,t)-y*(r,t)-]an(r,T)drdTJo Jo

= T, ^ [x*Txm.k+ yk = i Jo Jo \\ ox

Foreach value of n, this equation providestwo real equationsthat togetherare sufficient for determining the parametersyn_1 and

\302\253\342\200\236_! appearing in

the expansionG.85).Usingthese values, one can then solve the second(x \\relation G.85)for e as well as Eq.G.86)for I

\" 1.Then, substituting into the

first relation G.85),one has a convergentseriesexpansionfor e sufficiently

small, which providesus with an analytic representation of the solution

yjLet us briefly outline the calculations up to second-orderterms in e.

We first construct the eigenvector

We choosea normalization such that cf = cf = 1.Then from the linearizedequations of Section7.4one finds:

d =BiPe\302\261ie

where


and[cf.Eq.G.28)]: [^^^|f ,,,2,The complexnormalized solution I # I of the linearized adjoint problem

is (for the choicec^):

\\-A2pe'7whereasthe real normalized solution of Eq.G.88)reads:

7The next stepis to study the solution of Eq.G.86)with n = 2:

From Eqs.G.87)and G.93b)the explicit value of a2 can be written in thefollowing form:

a2 = 7i cost sin \342\200\224 + I \342\200\224 cos2x + 2ABxp cost cos(t+ 0) Isin2 \342\200\224 G.95)I \\A ) 1

One can now evaluate explicitly the compatibility condition G.91).Wehave:

Cl nr C2nA + A2pe+

W) sin2-fdr e~hdxJo ' Jo

x yx cost + I ~ cos2t + 2ABiPcost cos(t+ 0) Isin \342\200\224

C1 nr C2n= aI sin2Tdr [-e\"iTsin x + BiA2p2e+iee~izsin(i + 0)]drJo ' JoG.95a)

Becauseof the quadratic dependenceof a2 in cost, the terms other than

7!cost in the left-hand sidevanish after time integration. Noting also that

cos2x dt = nisin(i + G)\\Jo (.'

sin t rfi in cos0cost dx }nsin 0

7.12. Bifurcation of Time-Periodic Dissipative Structures 145

we may reduceEq.G.95a)into the simpler form:

7t(l + A2p e+w)yl =@1-in\\_- 1 + BiA2p2 e2W~\\

Taking real and imaginary partsof this relation and noting that, accordingto Eq.G.92),1 + A2p ew ^ 0 we concludethat:

?1=0, w, = 0 G.95b)

Equation G.94)can now be solvedexplicitlyfor I2

I by Fourier expansion

methods.Substituting into the compatibility condition containing a3 (and thus

Xi, yi,x2,y2), we may then compute w2 and y2. Thesecalculations arelong and are not reproducedhere (seeAuchmuty and Nicolis,1976a).We

give, instead,the type of expressionone obtains finally on determiningc, <x>2, v2, and y2 and inserting into expressionG.85):

x(r, t)\\ (B-fliY'Y cosOr \\ . nr){ ) { )Smak + ak cosBOt+ ij/k)\\ . knr

sin ~T+fa iy \\bk + Bk si ;

(odd

n = 1*1+ ^ cu2 G.96)

Herep and 9 have beendefined in Eq.G.92)and 4>u to2,ak, bk, ak,bk,i//k, <pk

are well-definedexpressionsdependingon the coefficientsA, Du D2.Note that the

\302\261 sign in front of the first term is now superfluous,in viewof the changeof sign of the trigonometric functions during a period.Anotherremarkable point is that Eq.G.96)representsthe first terms of a harmonicserieswith a well-definedperiod2n/Q.Finally, when the coefficient

<j>in

Eq.G.96)is positive, the bifurcation leadsto supercritical branches;other-otherwise, the brancheswill besubcritical.The stability of these various solutionscan be analyzed by a suitable extension of Floquet'stheory, familiar fromordinary differential equations(Sattinger, 1973;Minorski,1962).One findsagain that supercritical branchesare stable,whereas subcritical branchesare unstable.

No-flux Boundary Conditions

As we have discussedrepeatedlyfollowing from Section7.4,the first birfurca-tion in this caseleadsto a spatially uniform limit cycle.The latter can be


constructedusing bifurcation theory along the lines of the previous sub-subsection. We first write the complexnormalized solution of the linearizedadjoint equation as: au>

The real normalized solution of the linearizedequation is,therefore:

cost

p and 6 now correspondsto a bifurcation at Bo[seeEq.G.92)with Dt = 0]:1

P =~

sin y =

with

B0 = A2 + \\

Now one can compute y^coi from the compatibility condition fora2.One finds, asbefore:

y, =0, co, = 0 G.99)(x \\

Thus, the equation for I2

I can be solved explicitlyand used in the com-

patibility condition for a3 to determine y2, co2.One finds that (Auchmutyand Nicolis,1976):\342\200\242 y2 is positive. Thus, there is a supercritical bifurcation leading to an

asymptoticallystableorbit.\342\200\242

<x>2 is alwaysnegative.Thus, the periodof the oscillationsincreaseswith

B near Bo.fx\\The final expressionfor reads:\\yj

x(t)\\ (B- 1 -A2\\l2( cosfity(t)J \\ <Po J \\B0pcos(Clt4

B - 1 - A2 ( acosBfif+<^o \\^o + b cosBf2f+ <Pi)

B-l-A2\\co2,co2<0 G.100)<Po

7.13.Qualitative Properties of Time-Periodic Dissipative Structures 147

The coefficients<j>0,a,b0,b,\\j/x,(pxare explicitfunctions of the parameter A

and are given in the paperby Auchmuty and NicolisA976a).

7.13.QUALITATIVE PROPERTIESOF TIME-PERIODICDISSIPATIVESTRUCTURES

Synchronizationthrough Diffusion Coupling

As wesaw in Section7.12,in the absenceof diffusion and as long asB > Bo,the concentrations in the trimolecular model oscillateat each point in space.When diffusion is switched on these local oscillatorsbecomecoupledviaa term proportionalto the \"coupling constant\" DJI2,as one can seefromthe third part of Section7.8.The fact that the system in the presenceof dif-diffusion reachesa unique asymptotic state implies that this coupling may,under certain conditions,synchronize these local oscillatorsand maintain

sharp phase relations between them, which are completely reproducibleafter eachperiod.

Clearly, this important property is a consequenceof having a boundedmedium. In an unboundedsystem no unique asymptotic state needsto bereached.Moreover, one can have situations where a pulse of chemicalactivity travels through the systemand leavesbehind it a quiescentmedium.True, in somecases,one can haveperiodicsolutionsin the form ofplanewavetrains (Kopelland Howard,1973a;Ortoleva and Ross,1973,1974a;Howard,1974).However, these are one-parameterfamilies of solutionswhere theamplitudecan bearbitrarily prescribed.Thiscontraststhe patterns describedby Eq.G.96)wherefrequency,amplitude,and wavelength are all determinedby the system itself independently of the initial excitation applied to themedium.

Another important differencebetween boundedand unboundedmediaconcernsthe role of diffusion. As we have seenrepeatedly in this chapter,the emergenceof a stablespace-dependentpattern in an initially uniformboundedsystem is necessarilythe result of an instability triggered by dif-diffusion. This instability can leadto small-amplitudeor relaxation-likespatio-temporal patterns,* dependingon the distanceof the bifurcation parameterfrom its critical value. In contrast, the investigationof plane-wavesolutionsin unbounded media and someexperimentalobservations led to the conceptof kinematic or pseudowaves(Kopelland Howard,1973a).The latter areregardedas the result of a loosecoupling of local oscillatorsof the limit

*The latter patterns are frequently referred to as trigger waves by Winfree and by Kopell andHoward.


cycle type, which isnot sufficient to synchronizethe oscillatorsand producea coherent regime.Hence,no instability involving diffusion occursin this

case,although the functioning of the local oscillatorsdoes require an in-instability of the steady state with respectto a limit-cycleformation. Clearly,in boundedsystemsthis type of behavior can only representa transientstate that, eventually, is going to evolveto one of the asymptotic solutionsevaluated in Section7.12.

Existenceof a Propagation Velocity

Equation G.96)describesa superpositionof standing wavesof two differentfrequenciesQ and 2Q.Thus, each term correspondsto an evolution of thechemical concentration in time in the form of a \"vibrating string.\" An im-important point is that under certain conditionsthis superpositioncan give riseis propagating fronts that can subsistduring part of the overall period.Thesituation isdescribedschematicallyin Fig.7.26.

Nevertheless, one does not have true propagating waves. Indeed,let xbe the ordinate of a wave front. Along this front x = x(r, f), and

dx dxdx = \342\200\224- dr + \342\200\224 dt = 0

dr ot

Thus

dt dx/dr

where v is the \"propagation velocity.\"One immediatelyseesthat v is a function of r and f. Forinstance,

u@, f) = v(l, f) = 0

and

lim v(r, f) = + oo G.102)r-M/2

to secondorderin c.Thus, the usual ideasassociatedwith propagating waves, as in electro-

magnetism or fluid dynamics,do not necessarilydescribethe time-periodicsolutions of reaction-diffusion equations.The term \"chemical waves\"should,therefore,beusedwith caution, at least in one-dimensionalsystems.Thesesolutionsare more similar to the synchronousoscillationsencounteredin many problemsin electrical engineering (Andronov, Vit, and Khaikin,

1966).

L r

(ii)

(iii)

Figure 7.26. Characteristic steps of evolution of spatial profile of X during one period ascomputed from Eq. G.96).(i)and (ii):modesm - 1 and m = 3predominate, (iii):m = 3 modevanishes and that with m = 5 takes over. Comparing (ii) and (iii), we seethat, effectively, thetwo fronts denoted by dotted lines have propagated to the middle; thus, a wavelike activity hasappeared in system during part of overall period.

149

ISO Simple Autocatalytic Models

Despitethese differences,the chemical signalsdescribedby Eq. G.96)are capableof transporting matter in much the sameway asordinary waves.To seethis onecomputesthe diffusion current of the X-componentat apoint r

dx D.nfB-B.X'2nr1 ' cos-cosQf

Averaging over a period,one seesthat

D,nB\342\200\224 B, ^ mnrL I ma cosJx(r)=--J T\342\200\224L I ma.cos\342\200\224 G-103)

In general, this is not identicallyzero,but onehas

Jx(l/2)= 0

and

D^B-B,

In this system,matter is transportedby the diffusion of the chemicalsinthe medium. This is different from transportation by particlesfollowing awave front during macroscopictime intervals, as is the casein usual wavepropagationin fluid dynamics.

From a purely mathematical point of view, onecan alsounderstand theorigin ofthesedifferences.The reaction-diffusionsystempropagatesinforma-information with infinite velocity,as it is a parabolicsystem.Thus, its behavior is

quite different from thoseof the classicalwave equations,which propagateinformation at finite speeds.Moreover,we are obtaining thesewavesas the

asymptotic behavior in time of the system for a wide range of initial con-conditions, whereasfor the usual waveequationsone getsdifferent wavesfor dif-different initial conditions.Forinstance, the frequency of an electromagneticwave in the vacuum is determined entirely by the frequency of the sourceemitting the wave.Thenotion ofpropagationvelocityin chemicalsystemshasalsobeenanalyzed in a recent seriesof papersby Othmer A975, 1976).

7.13. Qualitative Properties of Time-Periodic Dissipative Structures 151

SuccessiveInstabilities

As in the caseof steady-statedissipativestructures,each time the bifurcation

parameter B crossesa value Bm compatiblewith the boundary conditionsand the marginal stability condition G.33),a time-periodicsolution of therate equationsbifurcates from the thermodynamic branch.Moreover,the

possibility of secondarybifurcations from the primary branchescannotberuled out. Unfortunately, the stability of thesesolutionscannot beassertedin general,and each casehas to be studiedseparately.

Comparisonwith ComputerSimulations

The numerical solution of the rate equations(Herschkowitz-KaufmanandNicolis,1972;Herschkowitz-Kaufman and Erneux, 1975)confirms theexistenceof time-periodicsolutionsexhibiting sharp wave fronts (seeFig.7.27).In the courseof one periodthere appearwave fronts near the bound-boundaries that propagateinwards. After collisionbetween two fronts the wavecharacter is annihilated and the concentration pattern varies slowly like avibrating string until the wave fronts are formed again and the entirephenomenon repeatsitselfwith a sharpperiodicity.

The simulations alsoconfirm the multiplicity of dissipative structures in

the supercritical region B > Bc.Forinstance, for zero-fluxboundary con-conditions onecan find, in addition to the uniform limit cyclesolution analyzedin Section7.12,solutionswith a nontrivial spacedependencepersistingforover 40 periodswithout any tendency to return to homogeneity. Again,these solutionsfeature a front that propagatesin a polarizedfashion (e.g.,from left to right). Wave trains are also found under certain conditions.

In concluding this subsection,let us stressagain someremarkableprop-properties of the time-periodicsolutionsin chemical kinetics.Thus, the abilityto produceappreciablequantities of chemicalswithin a limited region ofspaceand at regular time intervals endowschemicaloscillatorswith powerfulregulatory properties.Moreover, the high velocity of propagationof thewave fronts\342\200\224which can exceedby ordersof magnitude the rate ofspreadofaconcentrationfront by diffusion\342\200\224provides an efficient meansof transmittinginformation over the macroscopicdistancesand during macroscopictimeintervals.The biologicalimplications of this type of behavior are analyzedin Chapter14.Critical Length of the Transport-dominatedRegion

In Section7.8weinsistedon the important roleof the sizeof the systemin theoccurrenceof bifurcations from the thermodynamic branch.When time-Periodicsolutionsare availableto the system the individual local oscillators


t = 0 t = 0.68

1 0X t = 1.10

3

2

1

0

t = 1.88

1 0X t = 2.04 t = 3.435

0 10 1

Figure 7.27. Successive steps of evolution of spatial profile of concentration of constituent Xin trimolecular model for fixed boundary conditions. Numerical values of parameters used insimulations are:/ = 1,A = 2,B = 5.45,0,= 8 x 1(T3, D2 = 4 x \\03.

are coupled,aswepointed out in the first subsection,via a couplingconstantthat is inversely proportional to the squareof the size parameter /. As aresult of this coupling,coherent spatiotemporalstructures can arise andpersistindefinitely.

The remarkablepoint is that this is possibleonly if the system is abovesomecritical size.The latter can be defined as that sizeup to which thereexistsexactly oneunique time-independent solution, which is completelytransport dominated.This solution characterizesthe couplingof the reactivemodesto diffusional transport.In a sense,the coupling between local oscil-oscillators in this regime is so strong that the oscillatory behavior is entirely

y.J4. Traveling Waves in Periodic Geometries 153

quenched.In contrast,when this criticalsizeisexceededthe couplingbecomeslooserand a time-periodicactivity is superimposedto a spatial structure(Goldbeter,1973).Recently,Nazarea A974) was ableto derive a theoreticalestimateof the criticalsize,/*, by making useonly of the parameter-dependentperiodof the oscillatorymodeand of the diffusion coefficients.

Heobtains

/* ~

where T is the periodand S*is the maximum column sum of the inversediffusion matrix.

Moreover,he provideshydrodynamicestimatesof /* for reactiveprocessesin solutionsin which an arbitrary number of inert componentsare presentin addition to the bufferedand unbuffered reactants (Nazarea, 1975).Theseestimates shouldbe applicableto complexreaction systemsof biologicalinterest.

Intuitively, the expressionfor /* is quite clear.Written in the equivalentform

it expressesthe equality betweenthe periodof the oscillatory reactivemodeand the characteristic time of diffusion. When these two time scalesareequal, oneshould,indeed,expectthat new kinds of effectscouldarisefromthe competition between the chemicalreactionsand the diffusion.

7.14.TRAVELING WAVES IN PERIODICGEOMETRIES

The oscillatory solutionsstudied in the previous sectionsare subjecttoboundary conditionsof either the Dirichlet or the Neumann type. An

important roleof theseconditionsis to prevent the appearanceof true wavefronts propagating at constant speed(Auchmuty and Nicolis,1976a).Thereis, however,another type of boundary condition ofconsiderableinterest,andwhich ariseswhen one solves the equationsin a medium that can be repre-represented by a closedcurve in a two-dimensionalspace,or a closedsurface in

three-dimensionalspace.The simplestcasesis a ring, a circle,or the surfaceof a sphere.The biologicalimportance of such geometriesis quite obvious.In many biologicalprocessesrelated to metabolism,to motion, or embryonicdevelopment, the cellular membrane plays a decisiverolein monitoringthe overall behavior.

In this sectionwe study the caseof a ring of length 2n in two dimensions.The caseof a circleis briefly discussedin Chapter8.


Forthe trimolecular model the thermodynamic branch is still given by

Eq.G.16).The boundary conditionsare now periodic:

X@, t) = XBn,t); ^{0,t)=~ Bn,t)

7@,t) = YBn, t); ^@, t) = ~ Bn,t) G.104)

where 9 is the coordinatealong the ring.We now show that a new type of solution can bifurcate from the thermo-

thermodynamic branch, namely traveling wave solutions.To seethis, we set

\302\243

= 6 - vt G.105)where t> is a (constant)propagationvelocity,and lookfor solutionsX, Y ofthe reaction-diffusion equationsG.13)that are functions of

\302\243 only. Therate equationsbecome[setting X' =

(dX/d\302\243,), etc.]:DXX\"

- (B+ l)X + X2Y + A = -vX'D2Y\" + BX - X2Y= -vY'

subjectto

X(\302\243)

= XBn + ^), 7E) = YBn +\302\243)

for all 0 < ^ < 2n. We want to find thosevalues of v for which there arenonconstant solutionsof these equations.Alternatively, introducing theexcessvariables x and y around the thermodynamic branch we seek fornonzero 27i-periodicsolutions of the system[cf.Eq.G.83)]:

Dyx\" + (B - l)x + A2y + h{x,y) = -we'

D2y\"-Bx-A2y- h(x, y) = -vy' G.106)

The only pointswherenew solutionsmay bifurcatefrom the trivial solutionx = y = 0 of theseequationsare at thosevalues of B for which the linearsystemof equations

D,x\"+ (B- l)x + A2y = ~vx'D2y\"

-Bx- A2y = ~vy' G.107)has nontrivial 27r-periodicsolutions.

Any such solution must have the form

) = (\\ G.108)W

7.14.Traveling Waves in Periodic Geometries 155

and there is sucha solution if and only if the following characteristicequationadmits solutionswith m # 0 (seealsoSection7.4):

detB - 1 -D^m2 + imv A2

-B -A2-D2m2+ imv= 0 G.109)

The new feature here is that this is a complex-valuedequation giving risetotwo conditions.Oneof them fixesthe valuesof B along the marginal stability

curve, in much the sameway as in Eq.G.33):B = Bm= 1 + A2 + m2(Dl + D2) G.110)

The secondcondition givesthe valueof v in the form:

v2 = vi =\342\200\2242

+ A\\D, -D2)- D\\m2 G.111)m

Oneseesthat vm is real provided

< A 2(Dl -D2) G.112)m

In particular, there can only be a finite number, M, of possiblebifurcationpoints for traveling wavesolutionswhereM is the largest integercompatiblewith

IDJ2 G.113a)

There can be no such bifurcation points if Eq. G.112)doesnot hold with

m = 1.That is, if

D2>\342\200\224\\-l+ ll+\342\200\2242(i+Dl\\ G.113b)\302\273\342\226\240]

Note that the expressionG.11)for vm isa decreasingfunction of m. Thus, themost rapid modesare those having a long wavelength. In particular, thelimit cycle(m = 0) correspondsto an infinite propagationvelocity.

The actual construction of thesetraveling wave solutionsproceedsin amanner similar to that of the Hopf bifurcation. As is apparent from Eq.G.108),for each allowable Sm the eigenspaceis two-dimensional, but forthis type of bifurcationonealsohas to find two unknowns, so in general, theequations are soluble.Throughout the calculation the velocity of the

traveling wave plays a roleanalogousto the frequency Q of the periodicsolution in the Hopfbifurcation. The resultsfeature the samedependenceon the criticalparameter (B \342\200\224 Bm)and on the variable^ asthosecorrespond-corresponding

to the Hopfbifurcation under zero flux conditions[cf. Eq.G.100)]and

'56 Simple Autocatalytic Models

are not reproducedhere.Fordetails,the readeris referred to the paperbyAuchmuty and NicolisA976a).

As in the analysis of Section7.12in the caseof zero-fluxboundary con-conditions, the space-independentsolution correspondingtorn = 0 bifurcatesfor a value of B lower than any solution with finite m. Thus, the stability ofthe traveling wavesolutionscannot beascertainedin general.The computersimulations reveal,however,the existenceof space-dependenttime-periodicsolutionsin rings, which subsistduring severalperiodswithout any tendencyto go back to spatial homogeneity (Herschkowitz-Kaufman, personalcommunication).

7.15.THE BRUSSELATORAS A CLOSEDSYSTEM

Hitherto we have analyzed the rate equations for the trimolecular model by

assuming that the concentrations of certain reactant speciesare maintainedconstant within the reactionspace,or at leastare controlledat the boundaries.In many instancesan actual physicochemicalexperiment is performed, for

practical reasons,under closed-systemconditions.If the initial compositionof the mixture is far enough from equilibrium, the concentrations of theintermediatesmay present,for a limited time interval, a behavior reminiscentof a dissipative structure, such as repeatedoscillationsaround a quasi-steady state.Eventually, this state evolves to equilibrium and all types ofcooperative behavior collapse.

Recently, Noyes A976a,b) has examined the conditionsunder which atwo-intermediate model such as the trimolecular schemecan provide asatisfactory descriptionof these transient cooperative phenomena whenoperating as a closedsystem.He studiedthe possibilityof repeatedoscilla-oscillations and pointed out that a necessarycondition for this to be possibleisthat the \"initial products\"A and B are depletedby only a small fractionduring each cycle.

Considerthe rate equationsG.11)in a well-stirredsystem, that is, in theabsenceof diffusion terms. The steady-state concentrations for the inter-intermediates X and Ybecome:

G.U4)ik3

Moreover,the condition for the emergenceof sustainedoscillationsbeyondthe instability of this steady state becomes[seeEq. G.33)with m = 0 and

7.1S. The Brusselator asa ClosedSystem 157

with rate constants different from unity]:k2k A2

k2B>k4+ ^f- G.115)

If a closedsystem is to generate this steady state, onemust usually have the\"initial\" productsA, B in largeexcesswith respectto X, Y. If the oscillationamplitudearound Eq.G.114)is not very large, then the instantaneous valuesof X, Y can be approximatedby Xo, Yo. Thus, the following inequalitiesmust besatisfied:

G.116a)or, taking the explicit

Introducing the three

i

form G.114)-id

l-id

-id

ratios p,q,r-id1-id

1

4 >B J> Yo

into account:

i ^

iA

c2

iB

kA

ky

1

k3A

k2

= P

= r

G.116b)

G.117)

one can write inequalitiesG.116b)as

p>r> 1

Moreover, inequality G.115)can be expressedin terms of p, q, r and theratio B/A. Oneobtains the following form:

(q>\\) G.118)The conditionsimposedby these inequalities becomemore stringent if

\302\260ne requires,in addition, that the periodTof the oscillationbeof the orderof


a minute. Now, for the rate constants different from unity the period ofoscillationsdeducedfrom linear stability analysisbecomes:

Taking A and B to beboth about 0.1M,onethen finds that fc, cannot belessthan about 10\024 s\021. If p/r is about 100,then r must be still larger sinceA

and B are comparable.Thus, pqr can hardly be much lessthan 107.Now,accordingto G.117)

pqr = k3AB

This givesfor k3 the estimate:

A third-orderreaction of small moleculeswithout activation energy mightwell have rate constantslarger than this value.Thus, the requirement that theBrusselatorperforms well as a closedsystem is that the trimolecular steptake placewith a remarkably low activation energy.

Theserestrictionscan be further relieved if onepostulatestwo additionalrapidequilibria between A, B, and two speciesM and N, such that M P A,N ^ B:

This model would require four intermediates,although the rapid equilibriapermit reduction of the rate equationsto only two differential equations.

The conclusionseemsto be,therefore, that only under certain conditionscan the Brusselatormodel repetitive oscillationsin a closedsystem.Noyesfurther showsthat systemsinvolving three variables,suchasthe Field-Noyesmodel of the Belousov-Zhabotinskireaction (seeChapter 13),can modelmore naturally closedsystems in which the major reactants are depletedonly by small amounts.

7.16.CONCLUDINGREMARKS

From the analysis of the simplestmodel capableof exhibiting instabilitiesof the thermodynamic branch we may concludethat open_physicochemicalsystemsfar from equilibrium and involving nonlinear feedbackprocessesare endowedby the capacity to undergo self-organizationprocesses.Theseprocessesgive rise to dissipative structures showing an unusual flexibilityand variability, along with some remarkableregulatory properties.Such

7.16.Concluding Remarks 159

unexpectedphenomena as the spontaneousformation of spatial compart-compartments, the generation of burstsof chemical activity, and the ability to storepast information through somekind ofprimitive memory, appearto bemereconsequencesof sometimesslight changesof either the physicochemicalparametersof the system,sizeof the domain, or boundary conditions.It is,therefore, legitimate to concludethat more complexreaction sequencesrepresentingreal processesgive rise a fortiori to a similar behavior. Theobjective in Parts IV and V is to substantiate this conjecturein the contextof various concretephysicochemicaland biophysical problems.Before wediscussthis, howeverweanalyzein Chapter8 somefurther modelspresentinga cooperativebehavior.Finally, in Part IIIwediscussthe onsetof instabilitiesleadingto large-scaleordervia a stochastictheory of fluctuations.

Chapter8

SomeFurtherAspectsofDissipativeStructuresandSelf-organizationPhenomena

8.1.INTRODUCTION

In Chapter7 weillustrated most of the mechanismsleadingto a cooperativebehavior on a simplechemical system\342\200\224the trimolecular model.On theother hand, chemical oscillationsand related phenomena are now widelyspreadin physical chemistry,chemical engineering,biology,and elsewhere(Nicolisand Portnow, 1973;Noyesand Field,1974).Although they werefirst dismissedas curiositiesor even as artifices,sincethe early 1960stheyhave becomean important part of experimental research,thanks to thediscovery of sustainedoscillationsin biochemical pathways. At the sametime, modelshave been proposedto account for this type of phenomena in

such diverse fields as ecology,the functioning of chemical reactors,orbiology.The purposeof the presentchapter is to give a general account ofvarious modelsgiving riseto dissipativestructures and related phenomena.We insist primarily on the variety and the wide range of applicabilityof the

conceptofself-organizationand lesson the technicaldetails,asthe latter havebeendiscussedin sufficient depth in Chapter7.Thus, certain parts of thepresentchapter have the character of a reviewmore than of a comprehensiveanalysis.

8.2.CONSERVATIVE OSCILLATIONS

The first theoretical model predicting sustainedoscillationswas reportedby LotkaA920)and may bedescribedas follows:

k,> ZA

(8.1)

160

8.2. Conservative Oscillations 161The concentration of substancesA and B are maintained constant anduniform insidethe reaction volume.Thus, the system is open.Becauseof thetwo autocatalyticsteps,the reactionrates are quadratic in X and Y. Themass-balanceequationsread:

\342\200\224 = kxAx-k2XY (82)

They admit a singlenontrivial solution

Xo =^, y0 = M (8.3a)K2 *C2

and the trivial solution

Xo =Yo

= 0 (8.3b)The motion around thesestatescan be investigated by means of a linearstability analysis, which follows the lines of Chapter6 and of Section7.4.One finds (Glansdorff and Prigogine, 1971;Nicolis,1971)that the trivial

state is a saddlepoint (and thus always unstable), whereas the nontrivialone is a center.Smallperturbations around Eq.(8.3a)are, therefore,periodic,with a universal frequencyco0dependingon the parametersdescriptive ofthe system:

co0=\302\261i(kikiABI12 (8.4)

In orderto study large-amplitudemotions around Eq.(8.3a)weintroduce thenew variables(Volterra, 1936;Kerner,1957):

u = log X, v = log 7 (8.5)which satisfy the equations:

j=kyA- k2 e\" =(8.6)

dv

Multiplying both sides by (k2 e\" - k3B),(k2 e\" - kxA), respectively,andadding, we obtain the important result that Eq. (8.6)admits a constant ofmotion, which is additive over the different constituents:

r = k2(e\" + ev) - k3Bu -k^Av

= k2(X + Y) - k3 B log X - k,A log 7 = const (8.7)

162 SomeFurther Aspects ofDissipative Structures and Self-organization Phenomena

Clearly,relation (8.7)defines an infinity of trajectoriesin phasespacecor-corresponding to different initial conditions.ForX and Y closeto the steadystate (caseof small perturbations) these trajectoriesbecomeconcentricellipses.Indeed,expandingEq. (8.7)around Xo, Yo and retaining terms tofirst nontrivial orderwe obtain:

(X -XoJ (Y - YoJ 1(8.8)

For finite X - Xo, Y -Yo, the trajectoriesare deformed but remain

closed,provided -f doesnot exceedsomecritical value (Davis, 1962).Thus,finite perturbations around (Xo, Yo) are also periodic(seeFig.8.1).Thecalculation of the periodT of thesetrajectorieshas recentlybeenperformedanalytically by FrameA974). One can show that each trajectory has adifferent perioddependingon the constant \"f~, that is,on initial conditions.

i.o-

0.5-

0.2 0.5 2.5

Figure 8.1.Lotka-Volterra cycles obtained for successively higher values of constant of motionV and for k3B/k2 = klA/k2 = 1.Sdenotes steady state.

8.2. Conservative Oscillations 163

The Lotka-Volterramodel has, therefore, a continuous spectrum of fre-frequencies associatedwith the existenceof infinitely many periodictrajectories.This is a very important point, as it implies the lackof asymptotic orbitalstability [cf.Eq.(8.4)],that is,the lack of decayof fluctuations.As a result ofsmallperturbations the systemcontinuouslyswitchesto orbitswith different

frequencies,and there will be no average, \"preferred\"orbit.Oscillationsofthis type may only describenoisetype of effects rather than physicallyobservableperiodiceffectscharacterized by sharply definedamplitudes andperiods.

Further consequencesof existenceof a constant of motion are seen by

combiningEqs.(8.6)and (8.9)and by observingthat:

dr

and, therefore,alsothat:

du

~dt~with

j\\ \342\200\224

fi =

dr

d2r

dv

drdu

dv

di~

d2r

dt~du

(8.9)

, , , , (8-10)dv du du dv

We seethat in the variables(u, v) the model representsa conservativesystemadmitting a (regular)first integral r.The latter plays the sameroleas doesenergy in mechanics,as is apparent from Eq. (8.9),which assumesa\" Hamiltonian \" form.

Intuitively, it is clear that a conservative system cannot be structurallystablein the sensedefinedin Section5.4(Andronov, Vit, and Khaikin, 1966).Thisgivesonemore argument against the useofmodelsof this kind asmodelsfor oscillationsobservedin nature. This is to becontrastedwith the behaviorof modelsgiving riseto oscillatorybehavior pastan instability. As discussedat length in the previous chapter, theseoscillationsare endowedby bothasymptotic and structural stability, which ensure a sharply reproduciblebehavior in time.

It is instructive to point out that for certain exceptionalvalues of theparameters,a (nonconservative)systemcapableof undergoing a limit-cycleoscillation can reduce to a conservative system.To seethis, considerasystem involving two variable intermediates near the threshold of anasymptotically orbitally stableperiodicmotion. According to Chapter 6,

164 SomeFurther Aspects ofDissipathe Structures and Self-organization Phenomena

the matrix of coefficientsof the linearized mass-balanceequationsadmitsa pair of purely imaginary roots.Thus, in the notation of Section6.5:

The linearizedbalanceequationstake the following form:

dxItdy

Setting

\342\200\224 = aux + al2y

\342\200\224 = a2lx-

dx dr dy dr(o.lz)

dt dy dt dx

we verify that the existencecondition (8.10)is satisfied:

(8.13a)cy dx dx dy

We concludethat a nonlinear systembehavesasa conservativesystemin thevicinity of the point of marginal stability. The \"energy\" function -fcan becomputedup to an arbitrary constant, and the result is:

x2 y2-r= -fl21y + fli2 y + anxy (8.13b)

Note that this form is not necessarilydefinite unlessal2 >0, a2i <0and a\\t + al2a2i< 0.The latter inequality is a necessarycondition for thereferencesteady state to behave likea center.

From the point of view of thermodynamics the constant of motion (8.13b)does not have a specialsignificance,as it dependson the detailsof thechemicalkineticsvia the linearizedcoefficientsau,a12,a22\342\226\240In contrast, theconstant of motion in the Lotka-Volterramodel [seeEq. (8.8)]is closelyrelated to the excessentropy, 32S[seeEq.D.30)]around the referencestate.Indeed,in an ideal mixture the thermodynamicderivativesappearingin theexpressionD.30)yield:

dXj0 Xo

d/iY\\ kB T

^Yjo=

~Y^

JyH=

(ixjo

8.3. Simple Models Giving Rise to Limit Cycles 165

As a result:

We return briefly to the significanceof the constant of motion in Chapter11.

The occurrenceof sustainedoscillationsin the Lotka-Volterramodel is aconsequenceof having a purely irreversiblesetof reactionsas in Eq.(8.1).Onthe other hand, the samesystem can be analyzed closeto equilibrium by

introducing a rate constant k of back reactions.One finds (Lefeveret al.,1967)that if the overall affinity is lessthan a \"critical\" value,the steady stateis an asymptotically stablenode.It is only beyond this critical value that

dampedoscillationscan occurduring the evolution to the steady state, which

now behaves likea stablefocus.Finally, in the limit where the affinity tendsto infinite (which can be achieved by taking k -* 0) the oscillationsbecomesustained and one recovers the resultsdevelopedin this section.

8.3.SIMPLEMODELSGIVING RISETO LIMITCYCLES

Activation by the Product

In addition to simpleautocatalysis,whoseeffectsare analyzed in Chapter7,a reaction sequencecan give riseto more complexactivation (positivefeed-feedback) processes,whereby oneof the reaction productsstimulates its ownsynthesisor the synthesisof one of its precursors.A biochemicalexampleofthis situation is discussedin Chapter14.

HigginsA964)and Sel'kovA968)workedout mathematical modelsforthis type of process.Sel'kov'sschemesappearin the form:

\302\243\342\200\224/<*.*>

^ = //[/(A\", Y) -07] (8.15)

X, Y may represent,respectively,the substrateand productconcentrationsofone (or of a seriesof) enzymaticreactions,v is the rate of entry of X, /?therate of Y consumption, and n is related to the enzymaticconstants.Finally,/ is an increasing function of Y, at least within a certain range of values.Sel'kovfinds that sustainedoscillationsof the limit cycle type may ariseprovided the degreeof productactivation is higher than one.This impliesthat / dependson Y through a powerlaw of the form f(X, Y1

+\302\243)

with e >0.


End-productInhibition

Activation is a very general processin nature and is a necessaryprerequisitein a number of important reaction chains.Nevertheless,it is fair to say that

negative feedbackprocessesare even more common, especiallyin biology.One of the best-knownexamplesis that in which the productof a chaindiminishesthe rate of synthesis of somedistant precursorof the inhibitorysubstance.Moralesand McKay A967) and Walter A969, 1970)studiedmodelsfor this type of feedback,which may be summarizedas follows:

dXi~~d7~~

(8.16)dXi - k

where X{ are the concentrations and f(Xn+l) is a decreasingfunction ofXn+ j describingthe effectof inhibition. The most widely usedform for / is:

f(Xn+i)=t ,C

VD (8-17)

A more detailedanalysisshowsthat the exponentp is related to the nature ofthe enzyme catalyzing the conversion of the inital product into X,. It canbe shown that this type of system exhibits limit-cyclebehavior providedn > 2 and p >2.In fact,by increasingthe number of intermediatestepsonefinds that the oscillatory behavior occursfor relatively small p. In contrast,if n = 2, the valuesof p necessaryfor periodicsolutionsare as high as p = 9.The existenceof oscillatory solutionsin a large classof negative feedbackcontrol processeshas recently beenestablishedby Tyson A975b)using an

elegant topologicalmethod.An early model treating the effectof inhibition on the oscillatorybehavior

has beenproposedby Spanglerand SnellA961,1967).

End-productInhibition and/or SubstrateInhibition

Sel'kovand co-workers(Sel'kov,1967;Samoilenkoand Sel'kov,1971)andDegnA968)studiedmodelswhere the substratecan have an inhibitory role.The most commonlyenvisagedmechanismis onewherea reversiblereaction

gJ. Simple Models Giving Rise to Limit Cycles 167

betweena substrate,S,and the enzyme-substratecomplex,ES,forming aninactivecomplex,ES2,is addedto a classicalMichaelis-Mentenscheme:

S + E \342\200\224\302\243i-> ES

ES \342\200\224^-> E + P (8.18)

S+ ES <

k*> ES2

where P is the productof the enzymaticreaction.This systemcan behandledby the so-calledquasisteady-stateapproximation, which is illustrated in

somedetail on a concretemodel in Chapter14.Briefly, one assumesthat the

enzymatic forms vary on a fast scaleand adjust instantaneously, to thesubstrateor productvariations. One then obtainsfor the sequence(8.18):

dt 1 +(ki/k2)S+ (k,

where the total enzymeconcentration

Eo = E + ES + ES2= const (8.20)remains constant in time.

Equation (8.19)has to besupplementedwith additional terms expressingthe rates ofsubstrateentry, substrateand productefflux, and influenceof the

producton enzyme activity. Assuming a well-stirred medium with a first-orderdecayof the productand a convectivetransport of the substratefrom anoutsidereservoir,oneobtains equationsof the form (for a singlesubstrate-product reaction):

dJt= h(s0- s) - f(s,P)

(8.21)dP\342\200\224 = f(S,P)-PPP

where/ is a nonincreasing function of both S and P.An exampleof the S-dependenceof this function is provided by Eq.(8.19).Onecan show that if thedegreeof inhibition is higher than or equal to two, then Eq.(8.21)can exhibit

limit-cycleoscillationsas well as multiple steady states.

TemperatureOscillations

Hitherto we have neglectedthermal effectsin nonlinear kineticsby antici-anticipating a very efficient energy transfer through heat conduction,or suitable\"homeostatic\"mechanisms ensuring a constant value of the temperature.


Whenever theseconditionsare not satisfied, temperature can play a veryefficient roleas a generator of oscillations,especiallyif the system releasesheat locally through certain exothermic reactions.The main reasonfor this

is the highly nonlineardependenceofrate constantson the temperature givenby the Arrhenius law:

k(T) ocexp \342\200\224

RT (8.22)

where T is the temperature, R the gasconstant, and E the activation energy.The fact that k(T) is an increasingfunction of T, together with the presenceofexothermicreactions,constitutesan obviousfeedbackwherebya (random)increaseof T acceleratesthe reaction, which subsequently releasesheatfaster and further increasesthe temperature of the mixture.

The first exampleof sustained oscillationsin temperature-dependentsystemshas beenencounteredin chemicalreactor theory, particularly in theso-calledadiabatic stirred tank reactor (Gavalas,1968).Let V be the volumeof a reaction vesselfed by a stream of volumerate q, in which the concentra-concentration of the reactant is Xo and the temperature To (seeFig.8.2).A reactantdisappearsinsideV by an irreversible first-order reaction at a rate k(T)X,where k(T) is given by Eq. (8.22).Adequate stirring maintains the contentsuniform in X and T.Of course,X = X(t), T = T(t),although q is maintainedconstant.The mass-energy-balanceequationsin the tank give:

dA=l(Xo-X)~k(T)X(8.23)

(CPAT1 = qCP(T0- (AH)Vk(T)X

where CP = heat capacity of reactant, C = heat capacity of the walls, andAH = heat of reaction.

Theseequationshave beenstudiedin detail both analytically and numeri-numerically.

It has beenshown that, dependingon the values of the parameters

1x0, To

X, TV

Q

X, T

Figure 8.2. An open reactor subject to a constant flux q.X0,T0:reactant concentration and

temperature at the entry. X, T:reactant concentration and temperature inside reactor.

8.4. Multiple Steady States and All-or-None Transitions 169

q, Xo, To,AH, C,CP,the systemmay exhibit multiple steady states,oscilla-oscillations of the limit-cycletype, or even multiple limit cycles.More recently,Rosen A973)has consideredthe effectof temperature dependencieson thetrimolecular model analyzed in Chapter7 and found also the possibilityoftemperature-induced limit cycles.

The additional effect of spatial inhomogeneities has been analyzedby CohenA972),Keller A974),and Amundson A974),in the context of thetubular chemicalreactor problem.The main differencebetweenthis problemand thosebasedof the reaction-diffusionequationsstudiedin the precedingchapters is the presenceof convection terms in the balance equations,reflecting the convective heat-masstransport along the tube. Turbulencesets up an axial dispensionthat is usually describedby an empirical law

patterned after Fick'sor Fourier'slaws. This confers to the equationsofevolution a parabolic character similar to that of reaction-diffusionequations.

Again one finds multiple solutionswith complexstability properties,including spatiotemporally dependentregimes similar to those found in

the trimolecular model.

8.4. MULTIPLESTEADY STATESAND ALL-OR-NONETRANSITIONS

GeneralComments

In this sectionwe considerspatially uniform systemsand study situationswhere the equations of evolution admit more than onesteady-statesolution.This type of situation leadsto abrupt transitions between simultaneouslystablesteady statesinvolving neither symmetry breakingnor time ordering,and gives rise, therefore, to some interesting thermodynamic problems.Couplingwith both spaceand time-dependent phenomena is discussedin

Section8.10and Chapter 16within the context of morphogenesisanddevelopment.

Hysteresisphenomena associated with multiple steady states wereconjectured long ago by Rashevsky A938) in connection with biologicalsystems.Bierman A954) and Spanglerand SnellA961) discussedmodelsinvolving product inhibition, with two or more stable stationary states.Moreover, for certain values of the parameters,the equationsof chemicalreactor theory [seeEq. (8.23)]admit several steady states.This has someextremelyimportant practicalconsequencesrelated to the yield of the reactor(Gavalas, 1968).

An interesting classof modelsleading to multiple steady states is that

involving simpleautocatalytic steps.EdelsteinA970),Goldbeterand Nicolis


A972)and Dekkerand SpeidelA972)have suggestedmodelsalong this line.Their main interest is that they provide mechanisms of amplification ofsmall effects,which may have played an important roleduring chemicalevolution (wediscussthis aspectin Chapter17).

A still different classof modelsare thosereferring to regulatory processesat the geneticlevel(Cherniavskii,Grigorov,and Poliakova, 1967;Babloyantzand Nicolis,1972;Babloyantzand Sanglier,1972).Mostof thesemodelswereinspiredby the Jacob-Monodscheme(Jacoband Monod,1961)which isknown to apply to bacterial regulatory processes.

In this sectionwe illustrate the main aspectsof multiple steady-statetransitions on a simpleclassof modelsinvolving a single chemical inter-intermediate. Suchmodelswere analyzed by SchloglA971,1972),JanssenA974),McNeiland Walls A974),and Matheson,Walls, and GardinerA975).

A SimpleAutocatalytic Model

Considerthe following nonlinear reaction scheme:

A + 2X <

*'> 3X

(8.24a)X \302\253==\342\231\246 B

The overall reaction is:A < B (8.24b)

and describesthe conversion of the initial reactant A into B via the inter-intermediate X which, in addition, can catalyze its own production.The systemis opento interaction with infinite reservoirsof reactant A and B, so that theconcentrations of A and B are kept constant in the system.Possibleways ofrealizing this situation experimentallywere already discussedin Chapter7.The rate equation is:

^= ~k2X3+ ktAX2 -k3X+ k4B (8.25)at

This equation admits an equilibrium solution, provided the conditionsensuring a simultaneousequilibrium of both reactions(8.24a)are fulfilled:

LAY2 \342\200\224 b Y3

8.4. Multiple Steady States and All-or-None Transitions 171

As the rates kt are constant and characteristicof the mechanism,we interpretEq.(8.26a)as a condition on the ratio A/B:

*\302\253,

= D =t^ (8.26b)

Whenever R # Req,the affinity of the overallreaction(8.24b)is nonvanishing,and the systemoperatesunder nonequilibrium conditions.The steady-statesolutions of Eq.(8.25)obey the equation

X30 -aX2+ kXo~b = 0 (8.27a)

wherewe set

a = ^~, b =^, k = ~ (8.27b)k2 k2 k2

As is well known from elementary algebra,the nature of the roots of this

cubic equation depends on the sign of the expressionq3 + r2, where(Abramowitz and Stegun, 1964):

9=7^2+,^ t, (8-28)r = 2?\" ~ iok + \\b

If q3 + r2 >0, then the equation admits onereal root and onepair ofcomplexconjugateroots.If q3 + r2 <0 the equation admits three real roots.Thesetwo regimesare separatedin the spaceof parameters(q, r) or (a, b, k)by the curve

q3 + r2 = 0 (8.29)along which the equation admits at least two identicalroots:

XV -1/3 i (SI \"X(\\ct\\qI = JL 02 =\342\200\224^ '^ (o.JUaj

whereas the third root is given by

a3X03 = 2^3+\"

(8.30b)

From theserelations oneseesthat the three roots becomeidentical if onecould have r = 0, that is, alsoq = 0.Accordingto Eq.(8.28)this implies

a = C/cI/2


Equation (8.29)can beexpressedin terms of a and b as follows:

b2 + 2(\302\243a3-

\\ak)b + \302\243k3

-^a2k2= 0 (8.31a)If bt,b2 are the two rootsof this equation such that bt <b2,then the existenceof three real solutions of Eq.(8.27a)requires

bi <b<b2 (8.31b)We note that in terms of a and b the equilibrium condition (8.26b)reads:

Onecan verify that this condition can never be fulfilled as long as inequality(8.31b)is satisfied.Thus, multiple steady states in this modelcan only arisefar from equilibrium.

In the physicallyacceptablepart of the (a,b)plane(i.e.,the part a >0,b >0)relations (8.31)are representedgraphically in Fig.8.3a.At a0 = 2kl/2 thecurve b = b(a) has onevanishing root bt = 0 and a secondpositive one,equal to b2 = Tfk312. The curve becomesmeaninglessbelow a

\342\200\224amin,

for which the discriminantof the quadratic (8.3la)vanishes.Thiscorrespondsto the caseof three equal roots of Eq.(8.27a),namely, amin = CkI'2.In theneighborhoodofamin, b has the behavior indicated in the Fig.8.3a,as longas k remains finite. If k -\302\273 0, then b becomesnegative.This limiting caseis,

1 root

b = b(a)

Figure 8.3a. Linear stability diagram for model (8.24)in spaceof parameters b and a, Eq.

(8.27b).Region inside the curve b = b(a) (e.g.,ft, < b < b2 for a = a,) is region of multiple

steady states.

gj. Multiple Steady States and All-or-None Transitions 173

therefore,unphysical and has to be discarded.The behavior of b in termsofk is qualitatively similar.Again for a -> 0, Eq.(8.31a)can neverbesatisfiedfor physicallyacceptablevaluesofb and k. Figure8.3bdepictsthe dependenceof X at the steady state on the parameter b for fixed k and a such that

flmin <a<a0.The analogy with the van der Waals theory of phasetransitions (Landau

and Lifshitz, 1957)is striking. In fact,Fig.8.3has exactly the same structureas the isotherms of a densegase in the coexistenceregion, provided b isinterpreted as the \"pressure\"p and X, the \"specificvolume\" v. The critical

point in the present problemcorrespondsto the regimeof three equal roots,XOi = ^02 = ^03-At this point the \"isotherm\" b = b(X) has a horizontalinflexion point.

Onecan show by linear stability analysis that the stateson branchesOQand PR are stable,whereasthe stateson the branch PQ are unstable.It isalso apparent from this result that a hystereris in X may occuras b varies,as illustrated by the arrows along the lines P'Q,QQ\\Q'P,PP'.Moreover,states on branch P'Qcan evolve to the higher branch PQ'(and vice versa)even before states Q or P are reached,provided the perturbations actingon the referencestate exceedthe values correspondingto the intermediatebranch QP.A system with multiple steady statesis,therefore,endowedwith

an intrinsic excitability. We examine somebiologicalimplications of this

property in Chapter15.Moreover, the possibilityof abrupt transitions to ahigher value of the variable makessuch systemssuitable for the modelingof explosivereactions (Gray, 1974).

Figure 8.3b. Steady-state diagram representing concentration X versus the parameter b.Dotted part PQdenotes unstable branch. In region b, < b < b2 the system displays multiplesteady-states and hysteresis. At Q or P there occur finite jumps of X, respectively, towardbranches PR or QO.


ThermodynamicInterpretation

As we pointed out in the precedingsubsectionthe occurrenceof multiplesteady states in the model (8.24)requiresa finite distancefrom thermo-thermodynamic equilibrium. On the other hand, in contrast to what happensfor

limit-cycle and symmetry-breaking transitions, it is often impossibletounambiguously determinethe thermodynamicbranch ofsolutions(Edelstein,1970;Glansdorffand Prigogine,1971).To seethis, considerthe steady-stateequation (8.27)in connection with Fig.8.3b.In addition to the parameter b,the values of X are alsodetermined by the ratio {ki/k2) = k and by a. It is

quite conceivable,therefore, that different paths can be found in the param-parameter spacespannedby q and r (cf.Eq.(8.28))and leading from a valueof X onthe lower branch to a value of X on the higher branch.Among thesepaths,somecertainly crossthe hysteresisregion;this implies an instability of thethermodynamicbranch, which couldbedefinedasthe branch of lowervaluesof X. Other paths, however,couldavoid this region and lead to the upperbranch smoothly,that is,without an instability. From this point ofview, thereis an analogy with equilibrium phase transitions, where the branch ofequilibrium states is degenerate(Kobatake,1970).This degeneracyreflectsthe fact that the steady statesdepictedon Fig.8.3bbelongto the samebranch,in the sensethat they can bejoinedby a smooth curve OQPR.

The situation is different in systemswhere multistationarity arisesas aresult of coexistenceof different branchesof solutions,as illustrated in

Fig.8.3c.Forbmin < b < b0 the systempresentsthree steady states,onelyingon branch (i) and two lying on branch (ii). At b = bu which is a bifurcationpoint, an exchangeof stability between branches(i) and (ii) may occur.Onecan speakof a single-valuedthermodynamic branch (i) here and of a transi-transition to statesof a new type belonging to the classof dissipativestructures.Simplemodelsgiving rise to this type of steady-statebehavior have beenworked out (Venieratos, 1976)within the context of prebioticpolymersynthesisand competition (seealsoChapter17).

AU-or-noneTransitionsand CatastropheTheory

It is instructive to phrase the results of this section in the language ofcatastrophetheory (introduced in Section6.4).To this end,we first redefinethe variable X in Eq. (8.25)in orderto eliminate the quadratic term in Xin the right-hand side.Setting the following:

Z=X-\\^A (8.32)3k2

Multiple Steady States and AII-or-None Transitions

X

175

Figure 8.3c. Steady-state diagram representing bifurcation of multiple solutions at point ft,.Dotted lines show unstable branches. At ftmin < b < ft, the transition between the two stablebranches requires a finite jump.

we obtain for Z an equation of the standard form (Thorn, 1972):

where

3k3k2 - k\\A2u =

\342\200\224^

= u(a, b, k)

\\\\ 27 k\\ k2

(8.33a)

(8.33b)

t = k2t

The next point to realizeis that a systeminvolving a singlevariable alwaysderivesfrom a potential. Thus, Eq.(8.33a)can be written in the form

dZ~dZ

(8.34)

where t~ is the integral of the right-hand side(up to an arbitrary constant):

i\" = ?l+ u^- + vZ (8.35)4 2


3 sol.

Figure 8.4a. Stability diagram of system described by cubic rate equation involving two

parameters [Eq.(8.33a)]according to catastrophe theory. Oc= line of conflict; 0 = cuspsingularity.

In the u, v plane the region correspondingto a singleminimum of V fromthat correspondingto two minima and one maximum are given by thecondition of coalescenceof two rootsof the cubicC.33a):

4u3 + 21v2= 0 (8.36)This equation, which is to be comparedwith Eq.(8.29),definesa semicubicparabola(a) + (b)representedin Fig.8.4a.As (a)or (b)(Fig.8.4a)are crossedfrom outside,two new steady-statesolutionsemerge.Onceinsidethe domainlimited by the curves (a) and (b\\ which correspondsto the multiple steady-state region ofFig.8.3b,the two minima of V generallyhavedifferent heights,exceptalong a set of pointsdefining curve (c),where, say, V(Zol) = V(Z03),Z02beingthe value of Z at the maximum of V. This curve, which emanatesfrom the cusp O, correspondsto what is called in catastrophetheor.y aconflict,in the sensethat the two statesZ0l and Zo3 are equally dominantattractors.* Therefore,the systemcan evolveto oneor to the other asx -* oo.This idea,which is motivated by Maxwell'srule familiar from first-orderphase transitions (Landau and Lifshitz, 1957)can be substantiated by astochasticanalysisof fluctuations (wediscussthis point in Section12.6).

It is interesting to investigate the behavior of Z along the curve (a) + (b)of Fig.8.4a.In this respect,we first note that in a three-dimensionalspacespannedby Z, u and v, O is necessarilythe projectionof that point O'of the surface Z = Z(u, v) where the cubic(8.32a)admits three coalescentroots.Beyond (a) + (b) all vertical lines emanating from the (u, v) planeintersect the surface Z =

Z(\302\253, v) at a single point. Inside(a) + (b), on theother hand, one has intersection at three points.Thus, as u decreasestowardnegativevalues,the imageO'ofOon the surfacebecomesa point of bifurcation

* As a matter of fact, in the notation of this subsection, the conflict line Ocshould besituated

symmetrically with respect to branches (a) and (b).

g.4. Multiple Steady States and AU-or-None Transitions 177

Figure 8.4b. Visualization of Riemann-Hugoniot cusp catastrophe (seeprojection on u-vplane) asphenomenon of bifurcation of new branches of solutions Z (seesurface). 0'= bifurca-

bifurcation point corresponding to cusp singularity. Dashed areaon curve corresponds to region of

multiple steady-state solutions of equation for Z.

of a new branch of steady state solutions(a')+ (b')as shown on Figure 8.4b(Thompson and Hunt, 1973,1975).The resemblanceof curves O'b'a with

the bifurcationdiagrams derived in Chapter7 is striking. Whenever point O'is accessibleto the system,one has on crossingthis point\342\200\224and indeedtheentirecurveO'c' a situation similar to the formation of a shockwaveorto afirst-orderphasetransition in the region of coexistingphases.In particular,starting from a smooth wave,, characterized by a well-definedvelocity ofpropagation, one would switch to a regime characterized by two distinctpropagation velocities.Thisphenomenonhas beenanalyzedby Riemann andHugoniot and may, therefore,beappropriatelycalledthe Riemann-Hugoniotcatastrophe(Thorn,1972).

In the spaceof the physicochemical parametersa,b,k the behavior isqualitatively similar. The cusp singularity of Fig.8.4occursin the vicinity


ofamin in Fig.8.3a,which again is a bifurcationpoint in the usual senseof theterm. Finally, as stressedrepeatedlythroughout this monograph, in chemicalsystemsinvolving more than onevariableand operating far from equilibrium,it is generallynot possibleto construct a potential generating the equationsof evolution. Hence,the analogiesdrawn in this subsectionwith catastrophetheory are specificto a particular classof systems.

8.5.TWO-DIMENSIONALPROBLEMS

Despitetheir pedagogicalinterest, one-dimensionalproblemsin nonlinearkineticsprovide rather unrealisticmodelsof real systems.This is particularlytrue in biology,wheremany crucialprocessestake placein (two-dimensional)surfaces,such as cell membranes.The purposeof the presentsectionis tooutline some resultsobtained on two-dimensional dissipative structures.We seethat the symmetry features of the spatial domain introduce a highly

interesting interplay between the size of the systemand the form of thepatterns.Moreover,the variety of possiblepatterns is greatly enhanced.As in Chapter7, we illustrate the main ideason the trimolecular model,Eq.G.13).

Linear Stability Analysis

Let 4>i{r), \342\200\224kfbe the eigenfunctions and eigenvalues of the Laplacian

operatorwithin the spatial domain:

V20,. = -kf<j>i'

(8.37)

These quantities are determined uniquely, oncethe size,geometry, andboundary conditionsare specified.We may recall that in the one-dimensionalcase(seeSection7.4)<j>i

were trigonometric functions,whereask( were equalto i2n2/l2.Fora given ku the stability propertiesof the uniform steady-statesolution Xo = A, Yo

= (B/A) [seeEq. G.16)]in the two-dimensionalcaseare determined by the samerelations as in Section7.4.In particular:

\342\200\242 The characteristic equation admits one non-negative and one negativeroot when

B > B, = 1 + /42 \342\200\224 + ~^-j+ Dikf (8.38)

with Bc= min Bt

$.5. Two-Dimensional Problems 179

\342\200\242 The characteristic equation admits onepair of complexconjugate rootswith a nonnegativereal part when

B > B,= 1 + A2 + (Di + D2)kf (8.39)with Bc \342\200\224 min Bt.

The new feature added by the higher dimensionality is that the eigen-eigenfunctions

of the linearizedproblemare now different. Forinstance, for zeroflux boundary conditionsonewould have:

(8.40)nxi,nyi

= 0, 1,2,...for a rectangleof sideslx and ly along the two coordinateaxes,and

(eine \302\261e~ine)Jn(kir) (8.41)t

n = 0, 1,2,...for a circleof radius R, where Jn is the Besselfunction of integer ordern.The permitted valuesof kt in the two casesare:

kf = n2(^+ ?g\\ (8.40a)

and

A.I (k.r\\nk,R) = 0dJ\"{kir)

dr

n = 0, 1,2,...;i= 1,2,...(8.41a)

An additional feature of great importance is the occurrenceof degenerateeigenfunctions.This impliesthat evenat the first instability from the thermo-dynamic branch, multiple (primary) bifurcationscouldbecomepossible.

When n # 0 degeneracyoccursalways for the circle,which presentsforeach value of kt (and, thus, alsoof a}t) two independenteigenfunctions:

0,-,i = ct cosnd Jn(k(r)

(j>U2= c2 sin nd Jn(ktr)

A similar situation is found for the squarewhenevernx! # nyi. Degeneracycan alsoappearfor the rectangle,although this requiressomerather stringent


X X

Figure 8.5. Bifurcation parameter versus the eigenvalue of Laplaceoperator. U denotesunstable uniform steady-state. (A) shows caseof pure degeneracy; (B) shows caseof\"accidental\" degeneracy.

conditionson the lengths of the sides.Still higher degeneraciesbecomepossiblein the caseof a sphere(Hanson,1974b).

It shouldbepointedout that, even if the first bifurcationdoesnot leadto a\"pure\" degeneracyof the type discussedabove, it may happen that twodifferent bifurcationsbecomepossiblefor identical values of the bifurcationparameter.This caseof \"accidental\"degeneracy is illustrated in Fig.8.5(seealsoBoa,1974).

NumericalSimulations of Steady-StateSolutions

The analytical computation of the bifurcating steady-state solutionsin thenondegeneratecaseis straightforward. More interesting from the mathe-mathematical point of view is the situation of a doubleora higher ordereigenvaluesince,accordingto the theorem mentioned in Section6.6,the nature andstability of the bifurcating solutions are not known a priori.This point wasinvestigated by Sattinger and McLeodA973) on a general basis,and byShiffmann A975)on the particular caseof the trimolecularmodel.We do notdescribethe results here but, rather, report on the numerical simulationsof two-dimensional patterns for the trimolecular model (Erneux andHerschkowitz-Kaufman,1975).

Figures8.6to 8.11representthe stationary structures, obtained for theintermediate X when the unstable steady state (Xo, Yo) is slightly perturbed(in somefigures the positionof the initial perturbation appliedto the uniformunstable solution is indicated by an arrow). In each of those figures the

resulting pattern reflects the geometrical propertiesof one of the unstableeigenfunctions<\302\243,(r).

The latter can becharacterized by two integer numbersthat aredefinedfrom the linearstability analysis[seeEqs.(8.40a)and (8.41a)]:

8.5. Two-Dimensional Problems 181

1 //

Figure 8.6. Polar steady-state dissipative structure for trimolecular model in circular domainwith zero-flux boundary conditions. Broken dotted line: unstable uniform steady state. R =0.1,A = 2,B = 4.6,D, = 3.25x 1(T3,D2 = 1.62x 1(T2.

nx, ny for the rectangleand n, i for the circle.Thestationarydissipativestructureobtained can be related to thosemodeswhich have an appreciablepositivereal part of wi.

Morespecifically,Fig.8.6presentsthe circlewith no fluxes at the bound-boundaries; clearly, the stationary structure has the propertiesof the unstableeigenfunctionn = 1,i = 1 with its characteristicpolarity. Noticethat in onedimension polarity correspondsto spontaneousonset of a macroscopicconcentration gradient along the system, resulting from different values ofX or Y at the boundariesr = 0 and r = I.The organizedpattern that appearsin Figure 8.6is not a trivial extension of onedimensional polar structuresto the circle;in fact, we observethat the circular limits permit us to have,in the samestructure, a region of spacethat is greatly organized (around0^0)and another quasiuniform region (around 0 ~ tt/2). In Figure 8.7we consideredthe circlefor fixed concentrations at the boundaries;thestructure is angle independentand correspondsto the eigenfunctionn = 0,' = 2.Figure 8.8showsthe rectangle for the sameboundary conditionsandcorrespondsto the eigenfunctionnx =

ny= 1.This casewas alsoconsidered

for a higher value of B.The resulting pattern (Fig.8.9)doesno longer, as in

Fig. 8.8,totally reflect the symmetry properties of the eigenfunctions4>(nx = 1,ny

= 1)but undergoesa grooving in its central region which


Figure 8.7. Steady-state dissipative structure for trimolecular model in circular domain withfixed concentrations at boundaries. R = 0.2,A = 2,B = 4.6,D, = 1.6x 10~3,D2= 8 x 1CT3.

increaseswith B. This showsthe influence on the bifurcation solution ofsubharmonic terms (seealso Section7.6),which modify the geometricalaspectsof the first approximation 4>{nx = 1,ny

= 1)for increasingvalues ofthe bifurcation parameter B \342\200\224 Bc.

Figures8.10and 8.11present,for a degenerateeigenvalue, the squarewith no fluxes at the boundaries.Twodifferent initial conditionsare imposed.Fig.8.10refers to the casewhere we slightly perturbedthe unstable steadystate at onepoint. This leadsto the simplestructure representedon Fig.8.10

Figure 8.8. Steady-state dissipative structure for trimolecular model in rectangle with fixed

concentrations at boundaries. \\x= 0.231./,.= 0.165.A = 2,B = 3.9,D, = 1.6x 10\023, D2 =

8 x 10~3.

Figure 8.9. Concentration pattern under sameconditions as Fig. 8.8except for a more super-supercritical B = 4.1.

Figure 8.10.Stable steady-state dissipativestructure for trimolecular model in a squaresubject to zero-flux boundary conditions.

lx =ly

= 0.132873,A = 2, B = 4, \302\243>,

=1.6x Kr3,D,= 8 x 10~3.

Figure 8.11.Unstable concentration patternfound under sameconditions as in Fig. 8.10.

183


which can be characterized by nx = 0, ny= 1;Fig.8.11presentsa second

casewherewe have perturbedat another point. The emergingstructure is the

degeneratesolutioncorrespondingto <j>(nx= 0, ny

= 1)+ (j)(nx = 1,ny= 0).

But this structure is not stable,on a slight perturbation it tendseither to thestructure correspondingto the shapeof eigenfunctionscj)(nx

= 0, ny\342\200\224 1)or

to the structure characterized by <j>(nx\342\200\224

\\,ny= 0).

The one-dimensionalstudiesdescribedin Chapter7 show that severalstable solutionscan be obtained from different initial conditions.Thesesolutionsare all related to oneof the unstable eigenfunctions,generallythosecorrespondingto an appreciablereal part for coi.A fortiori, this is also thecaseof two-dimensional systems,where the possibilityof new bifurcationsincreasesrapidly with increasing values of the bifurcation parameter.Forexample,if wecomparea one-dimensionalsystemof length 1,and a rectangleof sides1 and 0.5,both with zero fluxes at the boundaries,we have, for theeigenvaluesk,:

onedimension:kt \342\200\224 nn,

two dimensions:kt = n(nx

n = 0, 1,...An ?I/2, nx = 0, 1,...

ny= 0, 1,...

Supposethe physicochemicalparametersare chosensuch that the values ofkj allowed,for which the uniform steady state is unstable, obey to:

2tt < kt < 3tt

Thenoneobservesthat only the eigenfunction4>(n = 2)satisfiesthis conditionin one dimension,whereasin two dimensions,three different solutions

Figure 8.12.Comparison between the bifurca-

bifurcation diagrams for one-dimensional system (upperpart) and two-dimensional system (lower part).

8.5. Two-Dimensional Problems 185

namely, ^nx = 0, ny= 1),<j>(nx

= 1,ny= 1),and <j>(nx

= 2, ny= 1),are to be

considered.The situation can be illustrated by the general bifurcationdiagram shown in Fig.8.12.At B*, only onebifurcationhas appearedfor onedimension,when one increasesthe value of B from zero.On the contrary, in

two-dimensionalsystems,several bifurcations may occurbefore B*.More-Moreover, as we have already shown, in two dimensionsmultiple solutionscanalso emergefrom the same bifurcation, as a consequenceof degenerateeigenvaluesin highly symmetricalsystems.Computersimulations confirmentirely this striking multiplicity of dissipativestructures in two dimensions.One can really speakhere, following HansonA974b),of a \"quantization\"of the statesof a macroscopicsystem.

Time-periodicand Wave-likeSolutions

In many situations of physicochemical and biologicalinterest, solutionsthat are apparently time-periodicseemto emergeand display,in addition,a nontrivial spacedependence.Among the most striking exampleswe mayquote the occurrenceof rotating spiral wave fronts in the Belousov-Zhabotinskireaction,as well as the appearanceofpulsesof chemicalactivityon cell surfaces,which seemto bear someconnection with the subsequentdeformation of those surfaces.Examplesare discussedin some detail in

Chapters13 and 16.In the presentsectionwe focus on the mathematicalaspectsof the problemand discussthe feasibility of having spatiotemporalpatterns whose symmetry is lesserthan the symmetry of the spatial domain.We deal exclusivelywith systems subject to zero flux conditions,as in

systemswith constant boundary conditionsthere may be an artificial spacedependenceinduced by the boundaries.Similar considerationshold forwave-likesolutions in periodicgeometries,such as the surface of a sphere(Auchmuty and Nicolis,1976b).

We consider systems involving two chemical variables and, moreparticularly, the trimolecular model.Let the reaction spacebe a circleofradius R. According to the analysis of Sections7.4and 7.12,for zero-fluxboundary conditionsthe first bifurcation from the thermodynamic branchnecessarilyleadsto a (uniform) limit cycle.Thus, any spatially dependentsolution is the result of either a subsequentprimary bifurcation from the

thermodynamic branch or a secondarybifurcation from the limit cycle.Inboth cases,it is extremelydifficult to assessthe stability of these solutions.True,we know from the analysisofone-dimensionalsystemsthat, well within

the supercriticalregion,multiple solutionsof the rate equationsare possible,and this is corroboratedby the resultsof computer simulations reportedin

Chapter7.Still, oneshouldbeawareof the fact that sofar there is no rigorousproof of stability of thosepatterns.


Independentlyof stability considerations,one can construct the spatio-temporal solutions using bifurcation theory or other methods. Let usillustrate the situation on the exampleof rotating waves.Let r, 0 be the polarcoordinates.Two attitudes are possible.Either one seeksfor specialsolutionsX(r, 0, t), which are singly periodicin 0 (seeStanshine,1975,for a calculationon the Belousov-Zhabotinskireaction) and for which

X = X(a)ot- <j>(r) \302\261 0) (8.42)

for r sufficiently large.One finds that such solutionsare not defined forr < r0,wherer0 is a certaincriticallength.When they are defined,the relation<j>(r) \302\261

0 = const is the equation for the involute of a circle,observedexperimentallyby Winfree A972).Computersimulationsby the latter author(Winfree, 1974a)on a system undergoing discontinuouskinetics haveproducedqualitatively similar results.

The secondpossibilityis to seekfor bifurcating time-periodicsolutionswithout postulating any specificr or ^-dependencies(Auchmuty and Nicolis,1976b;Erneuxand Herschkowitz-Kaufman, 1976).This ismoregeneral than

the previous method, as in a boundedsystem any stablewave-likesolutionis necessarilytime-periodic(or at least almost periodic).Usingthe methodoutlined in Section7.12,onefinds for the first nonuniform bifurcating solutionfrom the thermodynamic branch:

X(r, 0, t) = (B - Bc)ll2[ricosD>!+ Qf)cos0

+ r2 cos(<j>2+ \302\2432f)sin d^J^k^r)GO

+ (B- Bc) X (am + am cosBQf+ i//J)Jo(kOmr)m=1

GC

+ (B - Bc) X {[\302\253m2+ am2 cosBfit+ <Am2)]cos 20

+ [<+ am2 cosBQf+ i/C2)]sin20}x J2(k2mr) <8-43)

and a similar expressionfor Y. The variouscoefficientsare determinedby thesystem'sparametersand by the normalization conditions.kt, kOm and klmare related to zerosof derivativesof Besselfunctions as follows:

J'o(kOmR) = 0

kl = min(/c,: k, > 0) with J'^/c^) = 0

J'2(k2mR) = 0 (8.44)

Expression(8.43)presents several interesting properties, the mostremarkableof which is that it remainswell-definedat the center of the circle.Thus the concentration front predictedby this expression,although rotating,

1.75t = 0

1.51

(A)

1.50t = 0.14

1.51

1.75

2.50

2.72

Figure 8.13.Equal concentration curves for X in trimolecular model in circle of radius R =0.5861subject to zero-flux boundary conditions. Full and broken lines refer, respectively, toconcentrations larger or smaller than values on (unstable) steady state Xo = 2. A = 2,D, =8 x 1O~3,D2 = 4 x 10~3, B = 5.4.(A) to (F) describe concentration pattern at variousstages ofperiodic solution.

187

1.50t = 0.36

1.67 1.44

(C)

4.00t = 2.45

4.07\342\231\246

*3.75

(D)

Figure 8.13.(Continued)

4.00t = 3.36

3.75

4.25

3.65 14.45

(E)

1.75

t = 3.70

1.5U

2.252.502.75

3.00

(F)


189

4.1 4.7

1.7

(A)

2.0 2.3

t = 0.931.7

2.9

3.5

(B)

Figure 8.14.Rotating solution for trimolecular model arising under same conditions as in

Fig. 8.13but for more supercritical value of bifurcation parameter, B = 5.8.

190

t = 2.525

4.1

1.7

4.1 4.7

2.0k

(D)


191


can never be a true spiral.Moreover, the velocityof rotation is not a trueconstant as it dependson the position.

The resultsof computer simulationsof time-periodicsolutions on a circleprovide some very interesting clues for future theoretical developments(Erneuxand Herschkowitz-Kaufman,1976).Oneof the most striking aspectsrevealedis,again, the multiplicity ofstablesolutions.As an example,considerthe trimolecular model.At the first bifurcation point Bo = A2 + 1,thesituation is relatively simple.A uniform limit cyclesolution emerges,which

remains stablefor values of B significantly higher than Bo.The situation is

entirely different beyond the secondbifurcation point Bj (cf. Section7.4).One finds a four-fold multiplicity of time-periodicand space-dependentpatterns that subsistduring a very largenumber ofperiodsand are,moreover,rapidly restoredwhen subjectto a small perturbation. From the standpointof bifurcation theoretical analysis, this would mean that: (a) the parametersrur2,...,appearingin Eq.(8.43)are multiple valued and (b) although thebranchesemergingat B^ are unstable for B near Bu they are subsequentlystabilized(e.g.,by means of secondarybifurcating branches).

Onecan further show that there exist two groupsof qualitatively differentsolutions,one displaying the same featuresas the time-periodicsolutionsinone dimension(cf.Section7.13),asshown in Fig.8.13,and a secondgroupofrotating solutionsappearingwheneverthe valueof the bifurcationparameterB is sufficiently supercritical.Figure 8.14describessome successivecon-configurations of the isoconcentration curves in the courseof one periodof thephenomenon for one of the two rotating solutions.Which of the two typesofsolutiondominatesdependson the initial conditions.Moreover,one showsthat the concentrations remain well-defined,and presumablynonoscillatory,at the center.Thus, the reaction front is not a true spiral,which is in agreementwith the general theoretical predictionsbasedon Eq.(8.43).

8.6.SYSTEMSINVOLVING MORETHAN TWO CHEMICALVARIABLES

In systems involving three or more chemical intermediates some newpossibilitiesof self-organization arise.Firstly, the Hanusse-Tyson-Lighttheorem no longerrequiresa trimolecularstep.Secondly,the first bifurcationfrom the thermodynamic branch can lead to time and space-dependentstablepatterns, even for zero-fluxboundary conditions.On the other hand,the difficulty in analyzing such systemsincreasesconsiderably,already at thelevelof linear stability analysis.Graphicaland topologicalmethods (Clarke,1974)might prove to be useful in this context.

The situation remainsrelatively tractable in the caseof three intermediates

8.6. Systems Involving More Than Two Chemical Variables 193

(Hanusse,1973).The characteristicequation takesthe form:

co3- To2+ So- A = 0 (8.45)where T and A have the same interpretation as in Eq.F.21)and 8 is the sumof principal minors of rank two of the coefficientmatrix of the linearizedequations.Dependingon the signsof these three parametersone can havesingular points behaving like saddlepoints,stableor unstable nodes,andstable or unstable foci. The latter is the most favorable situation for theexistenceof limit cycles.Regarding symmetry-breakingtransitions leadingto steady states, it has been theorized (Hanusse,1973)that for a systeminvolving three intermediates undergoing uni- and bimolecular steps toexhibit a symmetry-breakingtransition, it is necessarythat at leastoneof the

principal minors of rank 2 benegative.Another interesting result specifictouni- and bimolecular processesis that the diagonal terms of the matrix oflinearizedequations can only benegativeor zero.

The occurrenceof various forms of self-organization has been demon-demonstrated numerically on the model (Hanusse,1973):

Y + Z

X + Z <

4 ' X + B (8.46)ft-4

with/c,- = l,/c_!= /c_2= /c_3= 0.1,k_4 = 0.005,A = B = 1.Let us now lookmore closelyon the nature of the first bifurcating solution

for zero-fluxor periodicboundary conditionsin a systemof three variables.(Auchmuty and Nicolis,1976b).As in the caseof two variables,the bifurcationof a steady-state solution occurswhen the bifurcation parameter X crossesacritical value kc correspondingto the equality

A(AC)= 0 (8.47)

If\342\200\224kf

is the eigenvalueof the Laplaceoperator,then Eq.(8.47)is a cubicequation for kf. Therefore, it generallygivesa nontrivial spacedependencecorrespondingto a k\\ # 0, in much the sameway as in the analysisof Section7.4.


Considernow the first bifurcating time-periodicsolution.ForX = Xc,the characteristic equation admits the roots:

\302\253i,2

=\302\261iy (yreal)a (8.46)

Inserting cou2and W3 into Eq.(8.45)and equating real and imaginary parts,we find:

T(lc,I\302\256

= co3<0

S(XC, ki) = y2 > 0 (8.49)

Moreover, ll/c9= A(lc,/c9 (8.50)

The point is that this equation is cubic for k'.As a result, at Xc the cor-corresponding value k\\ need not be the lowest eigenvalue of the Laplacian,that is,zero for no-flux or for periodicconditions,but may correspondtoone of the other eigenvalues.This is reflectedby a spacedependenceof thesolution whosesymmetry propertiescan now belessthan thoseof the spatialdomain.In particular,a rotating solutioncan now appearasa first bifurcatingsolution, a property that would automatically guarantee its stability.

Onemay comparethe Xc computed from Eq.(8.50)and the critical valueX\302\260 correspondingto the onset of oscillationsin the absenceof diffusion:

T(X\302\260, Q)d(X\302\260, 0) =A(lc\302\260, 0) (8.51)

As neither S nor A are monotonic functions of kt, this relation can wellcorrespondto a value of X\302\260 that is higher than Xc. Thus, a system which isnonoscillatory under homogeneousconditionscan exhibit time-periodic(and space-dependent)solution in the presenceof diffusion.

Even for systemswith two chemical variables,one might still have theseresults when thermal or electrical effects introduce a third (nonchemical)variable.Forinstance,for the thermal effectsone must supplementthe secondEq.(8.23)with the conduction term, XV2 T.The resulting equation wouldhavethe same character as the mass-balanceequations,to which it would becoupledthrough the concentration dependenceof the heat-sourceterm andthe temperature dependenceof the rate constants.

Further examplesof systemsinvolving several variables are given in

Chapter 13on the Belousov-Zhabotinskireaction and in Chapter 14devoted to enzyme-catalyzedreactions.

8.7. Coupled Oscillators 195

8.7. COUPLEDOSCILLATORS

In Section7.13,wementioned that a number ofself-organizationphenomena\342\200\224most notably the formation of spatiotemporaldissipative structures\342\200\224can

be viewed as the result of coupling between nonlinear oscillators.Undercertain conditionsthis coupling, which is due to diffusion, can synchronizethe various oscillatorsand maintain fixed phase relations between them,even if the intrinsic periodsof the individual oscillatorsare different.

Populationsof coupledbiochemicaloscillatorshave often beenused tomodel physiological rhythms (Winfree, 1967,1975;Pavlidis, 1973),spatio-temporal control of various developmental processes,cell division,contactinhibition, and other phenomena in living organisms.(We discusssomeofthesephenomena in detail in Part IV). The purposeof the presentsectionis toprovide a brief introductory reviewon the subject.Generally speaking,onecan distinguish betweenthree broadly different types of coupling:

\342\200\242 Couplingthrough diffusion or through other passiveforms of exchangeofmatter.

\342\200\242 Chemicalcoupling.\342\200\242 Couplingbetween individual oscillatorsand an external time-dependent

field.

In the first class,in addition to the couplingsdiscussedin Section7.13,one may cite Van der Pol oscillatorscoupledthrough Fickian diffusion

(Polyakova and Romanovski, 1971)or oscillatorscoupled via a massexchangeof at least one chemical.The rate of this processis usually taken tobe of the form D(X2 \342\200\224 X{),where X,(i = 1,2)representsthe chemicalvariableexchangedbetweenone coupleofoscillators(Lefever,1968;Landahland Licko, 1973;Torre,1975;Tyson and Kaufman, 1975).A beautiful

experimental setup realizing, by simple means,the couplingbetween twochemical oscillatorsof the Belousov-Zhabotinskitype has beenreportedby Marekand Stuchl A975).In each case,dependingon the relative valuesof the parametersand the couplingcoefficients,one can havesynchronizationby entrainment at the highest frequencyor at a multiple of this frequency,irregularsynchronizationat multiples of the driving frequency,enhancementof the amplitude of the oscillators,or quenching of the o scillatorybehaviorin someor all of the oscillators.When conditionsfor synchronizationare notfulfilled, one can observe the phenomenon of rhythm splitting where,owingto the interaction, the periodof the slower(driven) oscillatoris split into twoparts by the action of the faster oscillator.In somecases,the lifetime of


oscillatorybehaviordependsstrongly on the initial phasedifferencesbetweenoscillators.In others, synchronization may never be achieved, and aregime similar to almost-periodicoscillationscan result. Finally, it is

possiblesometimesto initiate sustainedoscillationsin systems for which

oscillationsare dampedor inexistentwhen uncoupled.In the secondclassofcouplings,describedabove as \"chemical\"couplings,

one may quote the coupling in \"series\"between two identical oscillators,where the initial productof the chemicalmechanisminvolved in the secondoscillatoris a decay productof the first oscillator(Tyson, 1973).One finds,again, the possibilityof synchronization,as well as that of almost-periodicsolutionsand subharmonic resonance.Another situation, (Pavlidis, 1971)is that of a coupling through a common pool for one of the substancesinvolved.Forinstance, in a population of cellsone of the productsof thereactionsinsideeach cell many be returned to the interstitial fluid. Theamounts returned from eachcellare mixed,and their aggregateconcentrationcan affectthe variouscells.Thismodelexhibitsboth entrainment and rhythm

slitting. Finally, Richie and Woomack A966) report a study of Rayleighoscillatorscoupledvia a linearly dependentterm.

A general classof couplingsbetweentwo oscillatorshas been analyzed byRuelle A973).He finds that oscillationswith the periodsmTi and nT2 canarise(Ti, T2 being the intrinsic periodsof the two oscillators),wherem and n

are small integers such that n/m ^ TJT2.A particularly important question,in view of its physiologicalimplications

concernsthe order of magnitude of the overall period comparedto theindividual periods.The answer dependsmost certainly on the intensity ofthe coupling.It is possiblethat for varying coupling parameters one canhave both a decreaseof the period through entrainment (as mentionedearlier in this section)or an increaseof the period if the coupling is\"small\" enough.Moreover,the number ofoscillatorscan alsoplay somerole.Computer simulations of coupledtrimolecular oscillatorsthrough diffusion

(Herschkowitz-Kaufman, 1973)confirm these points,but no analytic workhas beenreportedthus far.

The final point concernsthe influenceof external fields on a populationof nonlinear oscillators.A systematic analytic and numerical study of thebehaviorofa singleVan derPol oscillatorin a periodicexternalfield has beenreportedby Nicolis,Galanos,and ProtonotariosA973).Dependingon the

intensity of the field and on the difference between intrinsic and driving

periods,a number of interestingphenomena are found suchas:(a)frequencyentrainment, (b) phase locking,(c) a bounded phase difference betweenoscillatorand driving field, (d) an oscillation with a free-running phasedifference,or (e)quenching of the oscillatory behavior.

A different approachinvolvesstudy ofa population ofotherwiseuncoupled

8.8- Heterogeneous Catalysis and Localized Transitions 197

oscillatorswith a statisticaldistribution of intrinsic periodsbeingdriven by asingle external frequency (Kreifeldt, 1970).It is found that, under certainconditions,the distribution of entrained frequencieshas a high central peakaround the driving frequencyand a dip on either side.This result may havesome implications on the interpretation of electroencephalographicdata,particularly of that part of the spectrum around 10Hz.

A most remarkablefeatureofnonlinear oscillatorsis the way in which they

couplewith external stimuli in the form of pulses(Pavlidis, 1973;Winfree,1975).Dependingon the time, r0, the stimulus is administrated and on its

strength, D, the oscillatorrespondsby resetting its phase(e.g.,by attainingits maximum) with a variable delay 6 = 9(t0,D).Fora critical combination(t*, D*), the oscillation is almost switched off\342\200\224 9 becomesindeterminate.This singularity is reflected by the fact that the rhythm's amplitude after

perturbation is strongly affectedand subsequentlycan berestarted,but with

an arbitrary phase.An analytic treatment of this fascinatingphenomenon isstill lacking,but numerical analysesof simplemodelshave now reproducedthe phase singularity originally postulated by Winfree A970, 1975)inconnection with biologicalrhythms.

8.8.HETEROGENEOUSCATALYSIS AND LOCALIZEDTRANSITIONS

In many situations of physicochemicaland biologicalinterest, nonequilib-rium reactionsare localizedin a particular regionwithin a bulk medium that

itself is undergoingreactionsand transport processes.Let D,be the diffusion

coefficients,F,the reaction rates in the bulk, andGj\302\260\302\260

the reaction rates that

are taken to bewell localizedat the sites{rx}(which may lie,e.g.,on a catalyticsurface).The mass-balanceequationstake the following form (Ortolevaand Ross,1972):

^ + D, V2Pi + I G?\\{pj})S{r-n(g 52)

Let {p*(r)}be the steady state of the reaction system in the bulk satisfyingthe boundary conditions:

F,{{pf})+ D,.V2pf = 0 (8.53)

If the medium between localized sites contains no essentialnonlinearity,then this state will be asymptotically stable.In order to study the effectof


the heterogeneouscatalyst, we linearize around this state. Let L be thematrix of coefficientsof the bulk terms:

A first approximation in the catalytic strength is:Pi = pf + nt

with

ir= I Lunj + Di y2ni + TG^({pJ}K(r-r\") <8-54)

This set of equations is supplementedby the following physicallyreasonableconditions.Firstly, in an infinite medium we want nt reducedto a space-independent expressionas \\r

\342\200\224

ra\\ -> oo, if pf itself is space-independentinthis limit. Secondly,integrating Eq.(8.54)over an infinitesimal region abouteach site ra we find (sincethe concentration must remain bounded):

-G\302\253({p?})

= 0 (8.55)

This relation expressesthe discontinuity of the flux around the localizationpoints.

Equation (8.54)constitute a set of inhomogeneousdifferential equationswith coefficientsLi}that are generallyspace-dependent.It can betransformedto an equivalent integral equation by introducing the matrix propagatorT(r,r',t) (the Green'sfunction), which satisfiesthe relations:

4- = lt + dv2rdt

T(r,r',t = 0) = Id(r- r')

where / is the unit matrix and D the matrix of diffusion coefficients.Oneobtains:

nir',t) = p?(r*) + 11f dtT,y,r^ t - f')Gj[{ptV,f')}] (8.56)

The steadystate or the oscillatorysolutionsofEq.(8.56)can bestudied,and in

generalone expectsmultiple solutionsas in the caseof dissipativestructuresin bulk media studiedin previoussections.Let us illustrate more specificallythe situation for the caseofa homogeneouszero-orderstate pj(r)= pj.Then,Eq. (8.54)can be inverted directly by introducing the eigenvectors, um,

8.8- Heterogeneous Catalysis and Localized Transitions 199

of the operatorL + DV2.Forsteady-state solutions and a singlecatalyticsite at r\" = 0 in a one-dimensionalmedium, one has:

\302\273.\342\226\240

= Z <m)wmexp(-/cm|x|)m

with

det|L+fc\302\243D|

= 0D 1Lum = -Kum

The expansioncoefficientscj^and the spectrum of km are found by solvingthe linear algebraicequationsarising by substituting Eq. (8.57)into Eqs.(8.54)and (8.55).Assuming the eigenvalueproblem(8.57)has a solution, onecan formally write the solution of the rate equations in terms of Fouriertransformsas:

zn j j_\302\273<(*)

= TI \\dk eikxlL - k2D\\x G/{pf}) (8.58)

We want to investigate the nature of these solutions,assuming that the(homogeneous)steadystate pjof the bulk is stable.Now, this impliesthat the

eigenvalues,zm of the matrix L have negative real parts. As a result, if aspatial structure is to appear,it will beduespecificallyto the diffusion terms.Undersomecircumstances(Ortolevaand Ross,1972)theseterms can inducea true symmetry-breakinginstability leadingto a globaldissipativestructure.However, a much more common type of solution correspondsto an

undulatory (anddamped)pattern around the catalyst,which doesnot involvean instability of the thermodynamic branch.This pattern is realized whenthe following conditionsare satisfied:

Re km >0, Imkm # 0 (8.59)

with Re zm <0.Observationsof spatial patterns in chemicalsystemsmay well correspond

to either dissipativestructures or to local,undulatory structures.The latter

may emanate from glasssurfaces, dust particles,or other heterogeneitiesthat exist in experimental situations, in a manner similar to nucleation in

equilibrium phasetransitions. A careful analysis, involving the size depen-dependence or other qualitative propertiesof the pattern can show whether asymmetry-breakingtransition or a simpledampedspatial oscillation hastaken place.


8.9.SYSTEMSINVOLVING PHOTOCHEMICALSTEPS

The illumination of chemical systemsprovides the possibilityof interestingfeedback mechanisms and hence a great variety of phenomena.*Considerfirst the simplereaction (Nitzan and Ross,1973;Nitzan, Ortoleva,and Ross,1974):

A + hv <> A* <\342\226\240 A + heat

B

We assumethe molecule A to be of sufficient complexitysuch that after theabsorptionof a photon, it undergoesa rapid radiationlesstransition to avibrationally excitedstate, A*, followedby rapidthermal deexcitation.Thisresults in a net heating of the reacting mixture proportionalto the con-concentration of A. Considernow a reacting mixture (A, B) in a closedvessel(atchemical equilibrium) where a steady illumination is imposed.A new non-equilibrium state is achievedafter sometime, which will satisfy the equations:

A A

\342\200\224 = -k,(T)A + k2(T)B

d^= []A-AH~-oi(T-To) (8.60)

Here/?,a representabsorptionand coolingcoefficients,respectively,7^ is theexternal temperature, and AH is related to the enthalpy change arising fromreaction A ?\302\261 B. Theseequations are couplednonlinearly becauseof the

temperature dependenceof the rate coefficients[seeEq.(8.22)].Forendo-thermic reactions one finds a singlesteady state that is stable.No limit cyclebehavior is observedfor the system.Forexothermicreactionsone finds the

possibilityof three steady states,of which two are stableand one is unstable,as in the modelsdiscussedin Section8.4.Couplingwith transport (diffusionand heat conduction)allowsfor the formationofspatialdissipativestructures.

The very interesting feature of thesephenomena is that they may arisein closedsystems,asonly an energyflux is required.Thishaspotential implica-implication in the theory of evolution of prebioticpolymers.

Photochemical reactionsalsoplay a very important role in atmosphericphenomena at and above the stratosphericlevel. For a pure oxygenatmospherethese reactions are parts of the so-calledChapman sequence(Nicolet, 1964).They comprisethe oxygenphotodissociation:

O2 + hv -^-\302\273 2O- (8.61)\342\200\242 For a recent experimental observation, seeYamazaki, Fujita and Baba A976).

8.9. Systems Involving Photochemical Steps 201

and the photolysisof ozone:

O3 + hv \342\200\224A^ O-+ O2 (8.62)Additional stepsin the sequenceinclude a three-bodyrecombination

O-+ O2 + M kAT)> O3 + M + 24 kcal (8.63)

and a bimolecularrecombination:

O-+ O3 K*( >> 2O2 + 94 kcal (8.64)The rate of the latter reactionsdependson temperature, and this introducesan interesting feedbackbetweenthermal and chemicalvariables.Additionalfeedbacksarise from solar heating and infrared cooling,which affect thetemperature and at the same time depend strongly on the chemicalcomposition.

In practice,a pure oxygenatmosphereoperating as a closedsystem is nota soundassumption.On the one side,the extremelyreactiveforms O \342\200\242and O3combine with other gaseouscomponentsof the atmosphereor with thelithosphere.And on the other side,oxygen is producedby plant photo-photosynthesis. Thiscouplingbetweenatmosphere,biosphere,and lithospherewasrecentlyanalyzedby Kozak,Nicolis,Sanglier,and KressA976)in the contextof the evolution of the earth'satmosphere.Assuming the following simpleforms for the rate of O2 production and for the rates of O-and O3 con-consumptions :

A \342\200\224*\342\200\224 O2

O-+(reactants) \342\200\224^\342\200\224> (products) (8.65)

O3 + (reactants) 3\342\200\224> (products)Kozak and colleagueswrite equations of the form:

C-r=v - J2X + J3Z- k2(T)MXY+ 2k3(T)YZat

-f-= 2J2X+ J3Z- k2(T)MXY- k3(T)YZ- d^Ydt

dZ\342\200\224 = -J3Z+ k2{T)XYM- k3{T)YZ-d3Z

dTc\342\200\224 = tx{T0 - T) + AH2k2{T)MXY+ AH3k3(T)YZ+ (radiativeterms)

(8.66)


HereX = [O2],Y = [O],Z = [O3],c is the specificheat of the fluid, a. isNewton'sthermal coefficient,Tois the surfacetemperature, and AH, are theheats of reactions(8.63)and (8.64).

Equations (8.66)have been studiedboth analytically and by computersimulation. Experimentalvalues have beenusedfor J2,J3,k2(T),k3(T),To,and c,whereasv, du d3 and a havebeentreated as variableparameters.Initialconditionssimulating thoseprevailing in an early biotic atmospherewereused.It has beenfound that the O2 concentration tends to a steady-statelevel after exhibiting a pronouncedovershoot to a value exceedingappre-appreciably the steady-state one.The time scaleof evolution is very slow,of theorder of thousandsof years.This behavior can be comparedto one ofthe premisesof the Berkner-Marshalltheory of atmosphericevolution(Berknerand Marshall,1964;Broda,1975),which postulatesthe occurrenceof two \"quantum jumps\" in the O2 levelcorresponding,respectively,to bebeginning of the Cambrian Periodand of the Devonian Period,which sawthe developmentof the first forests.

The development of such modelsof the atmosphereshouldalso be ofinterest in the understanding of large-scalephenomena in contemporaryatmosphericdynamics. To this end a more refined treatment is necessary,incorporating both the influenceof molecular diffusion and of convection.Perturbations of the earth's atmosphere\342\200\224that is, the pollution problem-couldalsobe studiedfrom the same standpoint.

8.10.SOMEFURTHER METHODSOF ANALYSIS OFREACTION-DIFFUSIONEQUATIONS

The analysisof dissipativestructures carriedout thus far has beenbasedonlinear stability analysis,bifurcationtheory, and numerical simulations.Onlythe latter of thesemethodscouldgive information on the behavior (includingstability) of the solutions of the correspondingreaction-diffusion equationsaway from the point of the first bifurcation. In contrast, bifurcation andstability analysesonly enabledus to construct the solutionsin the immediatevicinity of the bifurcationpoints.As a result, the correspondingpatterns wereof small amplitude and did not exhibit any sharptransition betweenregionsofwidely divergentconcentrationsofa certain chemicalsubstance.Similarly,the limit cycle oscillationswere quasisinusoidaland did not give rise toflashesof activity followedby a periodof relativequiescence.

The formationofsurfacesofdiscontinuity separatingqualitatively different

regionsin a reaction space,or the emergenceof burstsof activity, is a verycommon phenomenon in situations of physicochemical and biologicalinterest.Whether one dealswith crystal growth, membrane transport, or cell

8.10.SomeFurther Methods ofAnalysis ofReaction-Diffusion Equations 203

mitosis one is confronted with phenomena involving two time or spacescales\342\200\224a slow one associationwith the bulk phenomenon and a fast oneseparating two successivebulk phenomena.In terms of the differential

equations describingthe system,this situation is characteristic of the highly

supercriticalregion beyond the point of the first bifurcation. New methodsare obviously neededin order to tackle these situations. They are usuallybasedon asymptotic expansionswith respectto an appropriateparameter;hence,they are referredto as asymptotic methods.

A different type of problemariseswhen the number of chemicalvariablespresentcannot be reduceddrastically to, say, within one to three.In this

caseit becomesincreasinglydifficult and awkward to rely entirely on analytic(and especiallyon perturbative) methods. Combinatorial or topologicalmethods can sometimesbe of help, and later we comment briefly on somerecent approachesbasedon thesemethods.

Asymptotic Behaviorof NonlinearOscillators

We illustrate the supercriticalbehaviorof nonlinear oscillatorson the simpleexample of the trimolecular model (Lavenda, Nicolis,and Herschkowitz-Kaufman, 1971;Turner, 1974;Boa, 1974).Considerthe rate equationsG.13)in the absenceof diffusion. According to Section7.4,the conditionfor the formation of a homogeneouslimit cycle is B > Bo = A2 + 1.Con-Consequently, the strongly supercriticalbehavior is to be found in the limit

A fiit (8.67)A

It is conveniant to introduce

Equations (8.13)yield:

dZ

~dt~ +

-> co,

the new

Z =

Y \342\200\224 Z =

A = finite

variable

X + Y

= f(Y, Z)

I) (8.68a)

Thesalientpoint to observein theseequationsis that, becauseof the smallnessof \\/B,dY/dt will bevery large [formally(dY/dt) -\302\273 ooasB -\302\273 oo]everywhereexcepton the curve

g(Y,Z)= 0 (8.68b)


Similarly, the phase-spacetrajectories[cf.Eq.6.15]:dZ _ 1 f(Y,Z)

will be practically horizontal everywhere in the (Z, Y) plane exceptin thecurve (8.68b).

Figure 8.15representsthe curve g = 0 in the phasespaceof the originalvariables (X, Y). From Eq.(8.68a)it may be seenthat this curve consistsoftwo branches,given by:

and

X ^ 0 (path D'E)

XY =* B (path G'H'E) (8.68c)

The correctedphase-spacetrajectory for finite values of B which is alsoplotted in the samefigure, can befound asfollows.We first observethat asthe

x g g

(B+1JY44

Figure 8.15.Motion in X- Y plane for limit-cycle solution of trimolecular model in regime of

relaxation oscillations. Dotted lines represent asymptotic curve (g(X, Y) = 0)corresponding toB -\302\273 x.Correctedphase trajectory for finite values of B is represented by full lines.


system is traveling along part DEof the limit cycle,which is expectedto benear the line DE'[seeEq.(8.68c)],X ~ 0(B~J),and dX/dt almost vanishes:

(A = fl + i,f = 0A)) (8.69)

Introducing the variable\302\243

in the trajectory equation and seekingfor

asymptoticsolutionsof this equation as X -* oo givesan analytic representa-representation of the trajectory along DE,which breaksdown for I, = 2, that is, for7 = X2/4A. Presumably, what happensis that in the regime along EF, Y

reachesits maximum value, X beginsto increase,and\302\243

ceasesto be of 0A).In order to study this part of the motion, the natural variablesare

2A X2X =

\342\200\224{\\+u), Y =\342\200\224(\\+p) (8.70a)

By solving the trajectory equation in thesevariables,one finds that:

X2Ymax

= \342\200\224 + CA113+ o(l) (8.70b)4A

where\302\243

is the first positivezero of the Airy function Ai( \342\200\224 z).The next stage of evolution is along the path FG.This path is closeto the

line

X + y = const= ymax (8.71)

Subsequently,X diminishesfrom its maximum value of 0(A2), while Y

remains smaller than 0(A). To study this regimewe set:

X=y+ ?a*y* (8J2)

It turns out that this seriessolution divergesfor Y = B1'2,correspondingtopoint H on the trajectory. A partial resummation yields a solution of theform [cf.secondEq.(8.68c)]:

X-f-j^... ,,73,


Thereafter, the representativepoint of the system travels along path HD,which is closeto the line

X + Y = const= 2B1/2 (8.74)until curve dX/dt = 0 is crossedagain and the cycleis completed.

In analogy to the solution just described,the periodof the oscillationscan alsobesplit into threeparts:(a) 7; = TDE,(b)T2= TEF,and(c)T3= TGH,as it is easyto seethat when B -> oo,the jumpsfrom F to G and from H to Doccurinstantaneously.

From the rate equations and F.15)one obtains:\"Ye dY _ye~ YD

\342\226\240o(A - \\)X-X2Y~ A

anddX_ Cx\"d_~

}XrA-X-1Xg n \342\200\224

y\\\342\200\224 XX

= -finA-In/I-2In 2+0A) (8.75)Thus, taking into account Eq. (8.70)and the values of YF, YE, and YD, wefind for the periodof the limit cyclethe expression:

_ A2 j_ _2A^ 3

A deeperanalysis would be requiredto ascertain the exact value of thecontribution of o(l).

An important new phenomenon that may arisein the supercriticalregionof time-periodicsolutions is that of secondarybifurcation, analyzed in

Section7.9for the steady-state solutions.To extend this analysis to time-periodicsolutions,it is necessaryto investigate the stability propertiesofthesesolutions.Sucha theory is availablefor ordinary differential equationsand is dueto Floquet (Minorski,1962).An extensionof this theory to partial-differential equations has beenobtained by Sattinger and Joseph(Joseph,1972)in the context of fluid-dynamic stability problems.By applyingthese techniques one finds that the limit cycle for the trimolecular modelcannot undergoa secondarybifurcation,neither in the spatiallyhomogeneouscase(Lefeverand Nicolis,1971)nor in the presenceof diffusion (Erneuxand Herschkowitz-Kaufman,1976).


f<0

/=0

Figure 8.16.Assumed shape of level curve f(X, Y) = 0.Branch Y = h2(X) is unstable, whereas

h0 and h, correspond to stable solutions.

Singular Perturbation and Formation of Surfacesof Discontinuity

Considera finite reactor with two variable reactants, whosediffusion coeffi-coefficients Dl,D2are very different.Moreover,assumethat the chemicalkineticsdefinesa bistableprocess,as in Section8.4.Settinge2 = (/>1/D2)and scalingthe spaceand time variables by the diffusion coefficientD2we can write therate equations(Fife,1976;seealsoOrtoleva and Ross,1975):

~ = f(X, Y)

8Y\342\200\224 -V2Y= g{X, Y)

(8.77)

with c <^ 1.Owing to the bistablecharacter of the kinetics,one of the levelcurves/ = 0 or g = 0 has a sigmoidalshape.Supposingthis to be the casefor the curve f(X, Y) = 0, we obtain the form shown in Figure 8.16.

A simplemodel giving this type of bistablebehavior is:A +Y \342\200\224

X

3XD + 2X

\342\200\224> X + Y

\342\200\224* Y + B\342\200\224\302\273 C\342\200\224* 3X + E (8.78a)

with D, <g D2(Di = Dx,D2 = DY). We want to describethe time evolutionof the systemunder the following boundary and initial conditions:

n-VX = ai(X0- X),X(r, 0) =

</\302\273(r)

Y(t, 0) = <ftr)

(8.78b)


wheren is the normal to the boundary surfaceand</>, \\ji

are smooth functionsof space.Owing to the smallnessof e, the evolution can bedecomposedintothe following two stages.

STAGE 1:DEVELOPMENT OF LARGE GRADIENTS

We neglectcV2X, at least initially. Thus

^ 2(879)

dtV Y '

We first regard Y as time independent:Y = Y(r) =

tfr(r)

We lookat the trajectoriesin the {X, Y) space(Fig.8.17)for a fixed r.As Y istime independent,these trajectoriesare obviously horizontal. Imagine nowa line segment (a')within the spatial domain.This line has an \"image\"(Y, X) on the initial phaseplane.Supposemoreover that steady statesalongthe curve / = 0 have different stability properties,as in the example ofSection8.4.In particular, let the stateson the part of/ = 0 having a negativeslopebe unstable and all others bestable.Owing to these different stability

propertiesof the stateson the levelcurve / = 0, each segmentof {a')movesin a different fashion in the trajectory space(X, Y). Thus, the part of (a')above the horizontal line Y, is attracted to the segment MN of the curve

Figure 8.17.Evolution of various initial conditions on curve (a') in reaction spacetoward

different portions of stable branches h0 and h,.

8.10.SomeFurther Methods ofAnalysis ofReaction\342\200\224Diffusion Equations 209

f = 0,whereasthe part below Y3 tends to the segmentOF.As branch HPKisunstable, the part between Yl and Y2 tendsto GH,and that between Y2 andy3 tends to KL.

If this simplifiedmodel were to remain valid as t -* x, an asymptoticstate would be reachedwhereby X would suffer a seriesof discontinuitiesalong ML,KH,and GF.However,the gradient V Y would remain bounded.Alternatively, within the domain V wewould have surfacesof discontinuity,F, separating two different states.The situation is describedon Fig.8.18,where the line (a')is also drawn. It is reminescent of Thorn's catastrophetheory (seeSection6.4),although the type of equation dealt with in this

sectionis different from thoseencounteredin Thorn's theory. Someanalogieswith first-order phasetransitions shouldalsobepointedout.

Obviously,the picture we just drew is valid until

(8.80)

Theseextremely high gradients act acrossthe surfacesF,.If\342\200\224as generallyhappens\342\200\224 Y is time dependent,the trajectoriesin the phaseplane will nolongerbehorizontal. They may go to infinity, to negativevalues,or becomeperiodic,although for the particular model(8.78)this doesnot apply.Still,the general picture of the evolution describedabove remains valid and thefinal result is the developmentof large gradients acrossFf.

STAGE 2:SLOWMIGRATION OF THE CARRIERS

As | \\X | beginsto build to the value (8.80),the correspondingcarrier beginsto migrate. Supposethat the discontinuity at F leadsfrom Xo to Xx. One

Figure 8.18.Formation of surfaces of discontinuity, F,.within reaction space.


can then do a standard match asymptotic analysis (Fife, 1972),wherebya stretchedvariable n alongthe normal x to F is defined(seeFig.8.18):

(8.81)

, weKeepingdominant terms and noting that Y remains smooth acrossFwrite the equation for X in the following form:

^-^= /(*>\302\253 (8.82)

This providesus with a single nonlinear diffusion equation describingtheevolution of the front surface.The boundary conditionsare:

x^Xo as,-._ooX -* Xx as n -* + oo

This equation admits solutionsof the form shown on Fig.8.19describingasmooth transition between the two states.However, the front surface, in

general,moves with time:X(ri, t) = U(n - ct) (8.84)

Thesesolutionsmay or may not beunique. Note that in the initial variablesthe velocitywould beec.Its valuedependson spacethrough Y = Y(r).

Onecan show that if the rate function/changessign only oncein the courseof the transition, then the velocityof propagation c and the form of the wavefront are unique and stable.Moreover, if the transition takesplacebetweentwo attractive branchesof/(e.g.,GHand MN),then ccouldbezero.

Oncethe surfaceF beginsto migrate it can either go up to the boundary \302\243

of the reaction spaceand merge into a boundary layer,collidewith anothersurfaceand annihilate, or someof its parts can approachto somestationaryconfigurationsFo.The velocityc on each point of Fo is zero.It can beshown(Fife, 1976)that this implies that the two states Xo, X^ are \"equally

Figure 8.19.Solutions of nonlinear diffusion

equation (8.82)describing evolution of front

surface.

8.10.SomeFurther Methods ofAnalysis ofReaction-Diffusion Equations 211dominant\" that is, that a Maxwell rule familiar from first-order phasetransitions is verified (seeFig.8.17):

1 fdX = 0 (8.85)GHPKL

The configuration Fo can be found directly by requiring specialtypes ofsolution of Eq. (8.82).Fortwo or more variables, explicit stability criteriahavebeenobtained for systemsin onespacedimension.Note that the forma-formation of Fo is a typically far-from-equilibriumphenomenon, although noinstability with respectto diffusion needsto be involved,as in the formationof a dissipativestructure.

Combinatorialand TopologicalMethods

Chemical networks\342\200\224especially within the context of biologicalcontrolprocessesat the genetic level\342\200\224often bear strong similarities to discreteswitching networks (Sugita, 1963;Kauffman, 1969,Glass and Kauffman,1973;Thomas, 1973,Thomas and Van Ham,1974;Glass,1975).The reasonis that the rates of synthesis of enzymes, fE, dependusually on the con-concentration of various effectors,X, in a highly nonlinear fashion (seealsoChapter 15):

XX\"h =^^ (8-86)

where X is a production constant, 6 is a \"threshold,\" and n is a parametermeasuring the cooperativity of the process.Forn > 2, Eq.(8.86)representsa sigmoidal function of X. Forsomepurposesit can be substituted by aBooleanfunction B = B(X) of a discontinuousvariable X, such that:

B=l ifl-9 (887)b = o xx<e (8-87)

The rate equationsdescribingthe control processnow take the form:

dX\342\200\224- = AjB^!,X2,...,X{.!,Xl+1,...,Xn)-yiXi (8.88)at

(i =!,...,\302\253)

For n variables there are 2\" Boolean\"state vectors\" (X = Xu...,Xn).As a Booleanfunction assignsa valueof 1 or 0 to each of thesevectors,thereis a choiceof 22\"\"' values for the componentsB, of the \"rate vector\"

B=(B1,...,flI1)inEq.(8.88).


Graphical,stochastic,and other types of analysis have been developedfor studying Eq. (8.88).Phenomena qualitatively similar to thosedescribedin previous sectionshave been found, such as multiple steady states andsustainedoscillations.However,theseresultsgive no detailedinformation onthe stability of the solutions,and practically no information on the quanti-quantitative aspectsof evolution,such as relaxation times or periodsof oscillation.Morework is neededto elucidate the relation betweendiscrete(i.e.,Boolean)and continuous formalism.It appearsresonableto exceptthat the dynamicsof isolatedcellscan in somecasesbe better representedby a discontinuousformalism, especiallyasfar asgeneticregulatoryprocessesinvolving a limitednumber of effector moleculesare concerned.In contrast, the collectivebehaviorof largeassembliesofcellsis to bedescribedby the usualcontinuousformalism of chemicalkineticsusedthroughout this monograph.

8.11.THERMODYNAMIC ASPECTSOF DISSIPATIVESTRUCTURES

The phenomenon of bifurcationaccompanyingthe transition to a dissipativestructure is analyzed in Chapter4 in respectto irreversiblethermodynamics.It is shown that the instability of a steady-state solution on the thermo-dynamic branch implies the appearanceof a negative excessentropy pro-production around the referencestate.The vanishing of this quantity determinesthe \"thermodynamic threshold\" separating the unstable regime from thestableone.Theseresults have been extendedrecently to the analysis of the

stability propertiesof time-dependent statesas well (Nazarea and Nicolis,1975;deSobrino,1975).

In this sectionweare concernedwith a different problem.Assuming that adissipativestructure has been formed,we would like to find a set of thermo-thermodynamic propertiescharacterizing this structure as uniquely as possible.Alternatively, we seeka state function whosepropertiesare indicativeof the

propertiesof the dissipativestructure itself.A general solution to this important problemis not yet available.Instead,

we considerin this sectionthe \"inverse\" question,namely, given a certainstate function such as entropy or entropy production,how this function

behaves on a dissipative structure. We considerthis question successivelyfor spatial structures, uniform systems,and systemsinvolving multiple timescales.

Entropy and Entropy Productionon a SpatialDissipativeStructure

We begin by deriving an expressionfor the total entropy productiondifference,AP, between an arbitrary state and the reference state on the

8.11.Thermodynamic Aspects ofDissipative Structures 213

thermodynamic branch. Accordingto Eqs.D.23)and D.27)we have:

AP =fdVZ(JkXk-J\302\260X\302\260k)J k

= UvY.xksjk+( (8.89a)

As seen in Chapter4, the last term vanishes if we assumethat for eachconstituent\342\200\224including the initial and final productsof the reactions\342\200\224the

concentration is keptfixed at the boundariesor the flux acrossthe boundarysurface is vanishing. Thus, we obtain:

=1J

(8.89b)

where/i, is the chemicalpotential ofspeciesi given by the expressionadoptedthroughout this monograph [seeEq.C.10)]:

Hi = n? + kT log Pi

In the secondterm we perform a partial integration. The surface term is, in

general, nonvanishing, and the remaining terms can be expressedin termsof the excessbalanceequationsD.2).Finally

1 f= -- dLn \342\200\242X Hi Si,

f kT\302\243

log Pi -f1 (8.90)

Considernow the casein which a steady-statespace-dependentdissipativestructure has been formed.By definition, <5p, = p{ \342\200\224 pf is time independentinside the system, both for the variable intermediates and for the initial

and the final products.Thus, the only surviving term in Eq.(8.90)is:

(APH=-If dY. n \342\200\242X Hi Sh (8.91)1 Jz i

where the subscript o indicatesthat AP is evaluated for a steady-statestructure formed around the thermodynamic branch. It is instructive to


evaluate this expressionfor the one-dimensionalmodelsanalyzed in Chapter7.Onehas,for a reaction line of length /:

T(APH= -X j\"i(Q\302\2535/i@ +i i

On the other hand

j = -D-8^

and

Thus

(8.92)

We seethat the entropy-production differenceis entirely determined by thedifferences of the slopesof the concentration profiles between the twoboundaries.Forzero-fluxboundary conditionstheseslopesvanish.Similarly,in the caseof localized dissipative structures onehas pt(l) = p-0)(/);p,@) =piO)@), and thus <5j,@)= <5/,(/) = 0.In both cases,the entropy-productiondifference vanishes (Herschkowitz-Kaufman, 1973).In contrast, for fixedboundary conditionsand for delocalizeddissipativestructures the slopesofthe concentrationsat the boundariesare generallydifferent (seeSection7.6).In this case:

(APH # 0 (fixed boundary conditions) (8.93)The valueand the sign of this differencedependson the detailsof the chemicalkinetics.Under certain conditionson A and B it has been shown that in

the trimolecular model AP can be a positive decreasingfunction of the

wavelength of the structure (Auchmuty and Nicolis,1975).Forthe entropy differencewe have, from Eq.C.5)(seealsoSchlogl,1974;

Ishidaand Matsumoto, 1975):

AS = [dV Asv = - \342\200\224 idV\302\243

L,.dptJ *\342\226\240 J | J

1 f ...v / , c._,(\342\200\242,

. \\= \\dV > nf opj + kT log p:dptT J t \\ J )

8.11-Thermodynamic Aspects ofDissipative Structures 215

or, on integrating between the referencestate {pf}and an arbitrary state:

AS = -kjdV X Pi log^- k

jdV \302\243 6p{\\ogp?- l +\302\2430

(8.94)

Considernow the entropy differenceAS on a time-independent spatialdissipative structure. Becauseof the convexity property of entropy as afunction of the concentrations,we have:

-AS = As/ < ASip-y) (8.95)

where ~p~y is the spaceaverage of the concentrations:

p-y =^\\dV Pi(r)

In somecasesp~y can be related to the values of concentrations on thethermodynamic branch. In the trimolecular model, for fixed boundaryconditions and for an even value of the critical wave number (seeSections7.6and 7.8),onehas from Eq.G.56)that

Xv = Xo + 0[(fl - flc)]Yv =

Yo + 0[(fl - BC)]\" '

On the other hand, becauseof the nonlinear dependenceon pt, AS containsthe effectof the terms of order(B \342\200\224 BcI12determining the dominant contri-contribution to X and Y. In the vicinity of the transition to the dissipativestructurethis yields:

S < S(X0,y0) = So (8.97)This result reflects the idea that symmetry breaking engendersorder,andthereforediminishesthe entropy of the system.

SystemsMaintainedUniform in Space

A spatially uniform system can undergo dissipative instabilities leading tomultiple steady statesor to chemical oscillations.In the secondcase,theinstantaneous-value of AP has no specialsignificance,but oneexpectsthat

the averageAPT over a periodof the oscillationhas a macroscopicmeaning.From Eq. (8.91)one seesthat the contributions coming from the volumeintegral vanish, as (dSpi/dt)T vanishesby definition of a periodictrajectory.Similarly, the volume integral doesnot contribute in a transition between


multiple steady states, as the state variables are time-independent.Weconcludethat

(APH = --or (8.98)

(APf =-1f^n-X^I7In general, thesequantities are nonvanishing, even though in a homo-

homogeneous system onewould tend at first sight to cancel all terms related tofluxes.In reality, an openuniform systemis alwayssubjectto suchfluxes fromthe outsideworld, with the additional requirement that onceinside thesystem,the different constituents must bedistributed almost instantaneouslyin a uniform fashion. This can be achievedeither by extremely high valuesof the diffusion coefficientsof the initial and final productsor by adequatestirring.

As an example,considerthe trimolecularmodelin the homogeneouslimitand supposethat A,B are such that the systemperformssustainedoscillations.We assumethat the systemis bound by walls that are impermeableto X andY but permit diffusion of A and B from the outsideworld.Onceinsidethesystem A and B attain instantaneously uniform compositions,and for this

to bethe caseweassumeDA -* ooand\302\243>B-> oo.Considerfirst the distributionof A. We have [seeEq.G.69)]:

??\342\226\240 A

0 < r < 1 (8.99a)vi vr~

with

The stationary solution of this equation is

(8.99b)

(899c)

dA

dt-A -

A((

^DAd2A

dr2

A(\\) =

0

An

and the diffusion current of A is given by:

In the limit DA -* oo this expressionremains finite at the boundariesandequal to

Ja@)= y, JaQ)=~~ (8-lOOa)

8.U.Thermodynamk Aspects ofDissipative Structures 217

Moreover, the diffusion term in Eq.(8.99a)reducesto

lim DA ~ = Ao =^= const (8.100b)DA^oc drz dt

Thus, the equation for A reads:

*A=-A+*iA- (8.101a)dt dt

Similarly, if B is subjectto fixed boundary conditions,onewould have in

the limit DB-* oo:dl d^ (8.101b)BX +dt dt

Coming now to the entropy-production difference,APT, over a periodofthe oscillation,we obtain:

T APT = ^@) <^@)T+ nB(Q)SjB(O)T- ^A) SjA(l) - /iB(l) SjB(l)

or, thanks to the fixed boundary conditions:

T APT = OiJ + kBT log A0)(djJQ^- djA(l)T)

+ in% + kBT log Bo)(djB@)T- SjB(\\)T) (8.102)

Accordingto Eq.(8.100a)the flux ofA at eachboundary is constant and equalto \302\261A0/2. Thus, ^(O) = SjA(\\) = 0.However,the situation for B is slightlydifferent. From Eq.(8.101b)we obtain, at the steady state for B:

~ = BXT (8.103)dt

Now one can easilyseefrom the rate equationsfor the model,that* XT =Xo = A. It sufficesto add the equations for X and Y:

d(X + Y)~j

= A -Xdt

and integrate over one periodof the oscillation.Thus

= BA = constdt

and

* Note that, in general, YT^ Yo for this model.


We concludethat (Lefeverand Nicolis,1971):

APr = 0 (8.104)The above derivation showsclearly that this result can be extendedto allsystemswhere the average fluxes of the input chemicalsare the sameat thereferencestate and along the new trajectory. In the caseof transition betweenmultiple steady states this is not generallyso,and the entropy productionvaries, although from relation (8.98)one cannot deduceany general trendsaccompanying this variation (wediscussthis point in the next subsection).

Considerthe behavior of the entropy differenceAS. From the convexityproperty invoked in the precedingsubsection,we have, in the caseof limit-

cycleformation:

AST <AS(fcT}) (8.105)In general,ptT dependson the detailsof the kinetics;hence,it is not possibleto deduceany generaltrends.In somecases,however,onecandrawinterestingconclusions.As an example,considerfirst the Lotka-Volterramodel (8.2).Dividing through by X and Y and integrating the rate equations over aperiodwe obtain:

= kiA-k2Y

jt\\ogY)T= k2XT-k3B

or

YT =-^= y0, ZT = -~?= Xo (8.106)

Inequality (8.105)becomes:ST <S(X0,Y0)= S0 (8.107)

Thus, the passageto a periodicmotion is accompaniedby a decreaseofentropy. However, it would be erroneousto concludefrom this result that

the Lotka-Volterramodel exhibits temporal ordering.Indeed,as we sawin Section8.2,a physicalstate cannot beassignedto a well-definedtrajectorybecauseof the densenetwork of trajectoriesaround the steady-state point.

It is curiousto observethat in the trimolecular model,inequality (8.107)is not generallyverified.As wepointed out already, XT = Xo for this model,but YT # y0. As a result, inequality (8.105)is not equivalent to Eq.(8.107).Theseresultsshow the inadequacy of entropy and, to someextent, alsoof

8.11.Thermodynamic Aspects ofDissipative Structures 219

the entropy production in characterizing uniquely the coherent behaviorassociatedwith the formation of a dissipativestructure.

Still, in the immediate vicinity of the transition to a limit cycle onecanobtain more sharp results.As seen in Chapter7, for zero flux boundaryconditions in the trimolecular model,one has correctionsto the thermo-thermodynamic branch of the order(B \342\200\224 BcI/2.If we integrate over a periodthesecorrectionsvanish, and one obtains a result similar to Eq.(8.96):

W =X\302\260

+ \302\260

(8 108y = y0 + o((B- bc))(8-108

For B near Bc this yields:ST<S(X0,Y0)= S0 (8.109)

Finally, the entropy differenceseemsto obeyno general rules in the caseof multiple steady-state transitions, although in some specificexamplesasystematicvariation has been found (McNeiland Walls, 1974).

SystemsEvolving Accordingto TwoTimeScales

The results of the preceedingsubsectionsindicate that we are still lackingan adequate thermodynamic measure of complexity and coherence.Inview of the variability of nonlinear systemsfar from equilibrium, this con-conclusion shouldnot comeas a great surprise.Nevertheless, it appearsthat

certain general statements can be made regarding the propertiesof entropyproduction during someportionsof the evolution of the system\342\200\224sometimes

the most significant ones.This point becomesmore transparent in systemsevolving accordingto two time scales.In thesesystems,oneor severalstepsof the reaction sequencebecomevery rapid comparedto others.Usually,if a chemical instability is compatiblewith this type of kinetics,we have oneof the following situations:

(i) The correspondingdissipative structure belongsto the highly super-supercritical region,asin the caseof the modelsanalyzedin the first two subsectionsof Section8.10.(ii) The initial system undergoesa structural instability (cf. Section5.4),which qualitatively alter its nature. Examplesof this secondtype of instabilityare discussedin Chapters17and 18.

Intuitively, one expectsthat a reaction stepbecomesfast when, for a givenpositiveaffinity dM correspondingto the production ofa certain intermediatesubstanceM,the rate wM becomespositiveand large.A simpleinspectionofEq.(8.89a)and (8.89b)showsthat, as long as such a fast stepis switchedon,


the entropy production differenceAP is bound to be positive and large.If,after a certain periodof time, this step becomesextinguished or attains arate comparableto the others,onemight expectthe entropy productionAP to again followthe propertiesoutlined in the two previous subsections.

As an example,considerthe evolution of the entropy production per unit

volume, A<7, of the relaxation oscillation for the trimolecular model (seebeginning of Section8.10,path FG in Fig.8.15).One has, accordingtoEq.(8.90),the condition X + Y = C = constalong FGand the resultsof the

preceedingsubsection:

~ AaF^G = -\302\243

IX log X + (C- X)\\og(C -X)]+-^-5^-kB dt kBT dt

= ^-l-X log X - (C- X)\\og(C -^ \302\243oX dt kB T

Performing the derivativewith respectto X we find:

Now, C ^ Xmax. Moreover, along FG, dX/dt >0 and X starts to evolvefrom a small value of 0(/l)correspondingto the vicinity of point F.Thus,along the first stagesof the fast build up of X, C > X, X > A, and

A<7f^G>0 (8.111)until X reachesa valueof the orderof the amplitude of the oscillation,C.

Similar conclusionscan be drawn from modelsinvolving multiple steadystates (Goldbeterand Nicolis,1972;Prigogine,Nicolis,and Babloyantz,1972;Hiernaux and Babloyantz, 1976).This point is expandedin moredetail in Chapter17devoted to prebioticevolution.

PartIII

StochasticMethods

Chapter9

GeneralComments

9.1.INTRODUCTION

The analysis outlined in the precedingchaptersis basedon a determinicticcausaldescription,provided by the equationsof chemicalkinetics.However,there exist a number of instanceswhere such a descriptionmay not be ade-adequate. The main reasonis that the very existenceof many degreesoffreedomin macroscopicsystemsimpliesautomatically the appearanceoffluctuations.Let N be the number of degreesof freedom.In a typicalprobleminvolving asystem in a not-too-dilutephase,N is of the orderof 1023.On the other hand,the macroscopicdescriptionof the same system is based on a restrictednumber ofvariables{a^...,an}a.ndn<^ N\342\200\224for instance,chemicalpotentialsor concentrations, temperature, pressure,and so on.A given macroscopicstate is always associatedwith rapid transitions between different atomicstates.As a result, the macroscopicvariablesare subjectto deviationsaroundcertain \"reference\"values correspondingto the results observedin anexperiment involving macroscopicdevices.Thesedeviations appearto theobserver as random events and are preciselythe fluctuations.Oncea mac-macroscopic fluctuation is produced,the system respondsaccordingto definitephenomenologicallaws.

Onecouldexpectthat fluctuations, although measurable,shouldremainsmall comparedto the macroscopicvalues.In statistical mechanics,this canbe shown explicitlyfor systemsin thermodynamicequilibrium, exceptat thepoints of phasetransitions such as the liquid-vaportransition (Landau andLifshitz, 1957).In these casessmall fluctuations (in the presenceof a criticalpoint) or finite fluctuations (in the caseof first-order transitions involvingnucleation and metastability) are amplified,attain a macroscopiclevel,anddrive the systemto a new phase.Thiscorrespondsto a macroscopicevolutioninvolving abrupt changesof certain thermodynamic quantities. Moreover,the initial \"reference\"phasebecomesunstable.In the critical region aroundthe instability the system exhibits a marked coherent behavior, frequentlyaccompaniedby long-range fluctuations and the breakingof spatial sym-symmetries (Brout, 1965).

223

224 General Comments

As seenrepeatedly in the previous chapters,nonlinear systemsdriven farfrom equilibrium can alsoundergo transitionsand instabilities.It is, therefore,essentialto construct a theory of fluctuations around far-from-equilibriumstatesto supplementthe predictionsof the macroscopicdescription,especiallyin the neighborhoodof instabilities.The purposeof the presentchapter andChapters10-12is to assessthe preciserole of fluctuations in the spontaneousemergenceof patterns of the dissipative structure type. We dealexclusivelywith the internal fluctuations generatedby the systemitself.The influenceofexternal noiseon the instabilitiesand the associatedbifurcation phenomenahas been analyzed recentlyby Horsthemkeand Malek-MansourA976).

9.2.STOCHASTICFORMULATION

Fluctuation theory is a branch of statistical mechanics.Let p({rj,{p,},r) bethe N-particle probability distribution function of the system (r,,p, arepositionsand momenta of the N particles).Oncep is known, the microscopicstate of the system is determined.Considernow the ensembleof variables{a,}determining the macroscopicstate of the same system. In general,a,- = a,-({i;},{Pj}).The probability P({a,}){(/a,}that the system be at amacroscopicstate such that all a, take values between {a,}and {a,+ da{)isgiven in terms of p by the relation:

{a,-,a( + da{]' '

If P isexpandedaround somereferencestate{a\302\260}, Eq.(9.1)can provide the

probability of a fluctuation around this referencestate.Fora system at thermodynamicequilibrium the distribution function p is

known, and Eq.(9.1)can serve as the starting point of the theory of fluctua-fluctuations (seeSection9.4).Forsystemsaway from equilibrium, however,relation(9.1)is not very useful in view of the difficulties encounteredin determiningtheform ofp. Forthis reason,we follow here a method \"intermediate\" betweenthe macroscopicdescriptionand the completestatistical mechanical treat-treatment. The basicideaof this theory, known as stochastictheory, is to considerthat the variation of a, valuesdue to a fluctuation is a random, or stochasticprocess,that is, a phenomenon where thesevalues do not dependon the

independentvariable(time) in a well-definedmanner.Thus,an observationofthe different membersofa representativeensembleofsystemsyieldsdifferent

9.2. Stochastic Formulation 225

functions a,{t).Hence,we definesuitable probability distributions,such as:Pi({a},t){da}= probability of finding {a}

within {a},{a + da}at time t.

P2(iai}ti'W2}t2){dai}{da2]= probability of finding {a}within {flj},{a,+ da^}at time rtthen {a}within {a2},{a2+ da2}at time t2

with

lP/{fli}r,:---{flj}rJ-)=lP,>0{aji (9.3)

U= 1.2,...)It is often useful to describeprobability distributions by a few \"typical

values.\" Among these,the \"expectation,\"or \"averagevalue\" is by far themost important. Its definition followsthe customary notion of an arithmeticaverage:

= Y.akPMa},t) (9.4)(a!

By construction, <ak> doesnot dependexplicitlyon the fluctuations.In orderto analyzethe relativeimportance of the latter in respectto the averages,oneintroduces expectationsof quadratic or higher order combinations of thestochasticvariables:

<aka,yt = Y.aliaiPiUa},t)(etc.) (9.5)la!

On expanding ak around the mean <a^>one can switch from Eq.(9.5)to the

variances,namely:

(SahSat}= <(ak - (ah\302\273(a,-

<a,\302\273> (etc.)

In particular, one has:

((dahJ}= (a?y - (aky2

(Sak5aty = (akaty - <aky(aty (etc.) (9.6)

The notion of probability distribution is familiar from statisticsand from the

theory ofgames.It is a very remarkablefact that thereexista few distributionsthat occurin a surprisingly great variety of problems,from geneticsandeconometryto gambling.The threeprincipaldistributions,with ramificationsthroughout probability theory,are the (a) binomial, (b) Poissonian,and (c)thenormal, or Gaussiandistributions (Feller,1957).


Distributions(a) and (b) arise,typically, in problemsinvolving Bernoullitrials, that is,repeatedindependent events (or trials) with only two possibleoutcomesfor each trial having the sameprobabilitiesthroughout the trials.Itis usual to denotethe two probabilitiesby p and q and to usethe time-honoredterminology of the theory of gamesreferring, respectively,to p as \" success\"

and q as \"failure.\" Clearly,p >0, q >0, and p + q = 1.In problemsinvolving the binomial distribution oneis interested only in

the total number of successesproducedin a successionof n Bernouillitrials,but not in their order.Onethen shows(Feller,1957)that the probability that

n trials result in k successesand n \342\200\224 k failures@ < k < n) is:

\\p(\\Pr (9.7a)

The averageand variance of this distribution are, respectively:

= np

<<5/c2> = np(\\- p) (9.7b)

Supposewe fix an arbitrary (time) interval of length t and divide it intosubintervalsof length l/n.If a particular subinterval contains at least oneofour random events,we agreeto call this possibilitya \"success.\"Considerasuccessionof Bernoulli trials with the sameprobability pn of success.Thetotal number of such trials is the integer nearest to nt. We assumethat, asn \342\200\224> oo:

h

Then,oneshows(Feller,1957)that asn -* x the probabilityofhaving exactlyk successesin the trials is given by the Poissoniandistribution:

k!

The averageand variance are, respectively:

<*> = h<<5/c2> = Xt = </c>

The status of the Gaussiandistribution is somewhat different. Typically,this arisesas a limiting caseafter the laws of large numbers are applied.Inparticular, onehas (Feller,1957):

9.2. Stochastic Formulation 227

Lawoflargenumbers. If {ak}is a sequenceof mutually independentrandomvariableswith a commondistribution and if the expectationvalue{ak}= <a>exists,then for everyc >0 as n -* oo.

+ (9.9)

In otherwords,the probability that the nonweightedaverage (av + \342\200\242\342\200\242\342\200\242+ an)/n

differs from the mean by lessthan a prescribede tends to one.

Centrallimit theorem. Suppose,in addition,that the variance<(<5akJ> = a2exists.Then, for every fixed fl and for n -* oo:

j (9.10a)

whereO(/?)is the normal distribution

(9-iob)

The integrand in Eq.(9.10b)is calledthe Gaussiandensity function.

Note that the validity of the central limit theorem imposesautomaticallyan order of magnitude for the variance of the fluctuations relative to themean value. Indeed,let A be an extensivestochasticvariable:

A = ci! + \342\200\242\342\200\242\342\200\242+ an

{A}= n(a}wheren is now related to the sizeof the system.Accordingto Eq.(9.10a)thevariance {SA2}is bound to be of ordera2n, that is,

Alternatively, the orderof magnitude of the most probablefluctuations SA

are related to </l>by

SA oc</l>1/2ocn1'2The relativeimportance offluctuations,therefore,diminishesasthe sizeof thesystem increases:

SA SA I0

Indeed,this is what happensin the caseof the binomial and Poissoniandistributions consideredearlier in this section.


Imagine now a situation wherebydA/(A}tendsto zero more slowly than

n~1/2,remains constant, or even tendsto infinity as n -* oc.Then, the scalesof macroscopicaveragesand of fluctuations are not clearly separated.Thismeans that the averagevalue <a>or {A}is not representativeof the systemor,alternatively, that the \"sampling\" procedureallowed by the law of largenumbers and consistingof identifying, for all practical purposes,the arith-arithmetic mean to the stochasticaverage is not entirely meaningful. Hence,wesay that in this casewehave a breakdown of the laws of large numbers. We seeshortly that this situation is characteristicofphasetransitions at equilibrium,as well as of nonequilibrium instabilities leading to dissipative structures.

Finally, it is important to point out that, while in the Poissoniancasetheaverageand the varianceare linked through Eq.(9.8b),in the caseofa processdescribedby a Gaussiandistribution thereappeartwo independentquantities,the mean and the variance.We discussthis point in the following chapters,asit appears that in stochasticproblemsinvolving chemical reactions thePoissoniandistribution plays a privilegedrole.

9.3.MARKOVIAN PROCESSES

Section9.2is concernedmostly with independent trials such that the jointprobability of a sequenceof events satisfies the multiplicative property.The theory of Markov processesprovides a generalization that consistsof permitting the outcome of any trial to dependon the outcome of pre-preceding trials. To this end one introducesthe conditional probabilities,W2(Wi}t11 {a2}t2)and others, related to the P, values in Eq.(9.2)by:

P2({al}tl:{a2}t2)=W2({al}tl\\ {a2}t2)x P,({<ii},'i)(9.11)

(etc),with [cf.Eq.(9.3)]:Tw2({a1}tl\\{a2}t2)=l,W2>0 (etc.) (9.12)(U2l

A set of numbers satisfying /property (9.12)definesa stochasticmatrix.By definition, a processis Markovian if the conditional probability W2

contains all necessaryinformation on the process.Stated differently, the\"memory\" of the past history of the systemis limited to one(moregenerallyto a small number of steps)backwards.Such processesare widely spreadin

probability theory. One of the simplest illustrations is the problem ofrandom walk. At time 0 a \"particle\" is at its initial positionz and at times1,2,3,....(or At, 2At, where At = ti+i \342\200\224 r,) it movesa unit step in the

9.3. Markovian Processes 229

positive or negative direction,with probabilitiesp and q, respectively,de-depending on whether the correspondingtrial resulted in

\" success\" or \"failure.\"

The random walk can be either unrestricted or terminate when the particlefor the first time reachespoints0 or a, which act asabsorbingbarriers.In thelatter casethe matrix of conditional probabilitiesis:

W-, =

1

q000

00

q

0

p0 p

...o...o...oq....o

00000

000

p1

(9.13)

Alternatively, for eachofthe interior states(i.e.,for pointsdifferent from 0or a)transitions are possibleto the right and the left neighbors:

W2(i, t = n\\i + 1,r = n + 1) = p

W2(i, t = n\\i- \\,t = n + 1) =

<?

However,no transition is possiblefrom either 0 or a to any other point;onceon thesestates,the system stays fixed there forever. Despiteits simplicity,this modeldescribesin a quite realisticway the motion of a heavy particle in

a fluid of light particles,known as Brownian motion (seeSection10.3).In the more picturesquelanguageof betting, the sameproblemconsidersa

gambler who wins or losesone unit of money with probabilitiesp and q,respectively.If his initial capital is z and he plays against an adversary with

initial capital a \342\200\224 z, the game continues until the gambler'scapital iseither reducedto zero or has increasedto a, that is, until one of the two

players is ruined.This classical\"ruin problem\"is equivalent to someprob-problems in population geneticsinvolving extinctionof a mutant or ofa speciesina set of competingpopulations.Similarproblemsarisein chemicalreactionsinvolving autocatalytic steps(Malek-Mansourand Nicolis,1975).

A secondillustration of the notion of Markovian processis providedby theEhrenfesturn model (Kac,1959).Let N objects(e.g.,N balls)bedistributedin two containers A and B.Take t2

\342\200\224

tx = At = 1.At time t = n a ball ischosenat random and transferred from one container to the other. Let thestate of the systembedetermined by the number of ballsin container A. andsupposethat at t = n thereare exactlyk ballsin A {k < N).Then the probabil-probability

for reaching at t = n + 1 the next state, which will be {k \342\200\224 1)or (k + 1)balls in container A, dependsonly on the number of balls in A at t = n.

Thus, the sequenceof events defines a Markovian process.On the otherhand, the transition k -> k - 1 implies that a ball has been removed from


A, whereas the transition k -\302\273 k + 1 impliesthat the ball is removed from B.The correspondingconditional probabilitiesare, clearly:

W2(k,t = n\\k- l,r= n + 1)= -J

W2(fc, t = n\\k + l,t= n + \\)= \\

W2 =0otherwise (9.14)Actually, in the precedingillustrationswehaveconsidereda specialclassof

Markovian processeshaving transition probabilitiesdependingon f i and t2

only through the difference Af = r2\342\200\224 fj. Such processesare known as

stationary Markovian processes.Usingthe definitionsof this sectionone canwrite for such a processan equation of evolution for P2,which is referred toas the Chapman-Kolmogorovequation (Feller,1957):

P2({ai};{a2},r)= X W2({a}\\{a2),At) x P2({ax}-{a},t - Af) (9.15)(a)

for any Af such that 0 < At < t. Forsimplicity, we assumehere and in theremaining of this sectionthat the variables {a}are discrete.In this caseonespeaksof a Markov chain. In the caseof continuous variables (e.g.,becauseofspacedependencies)the sum has to besubstituted by an integral.Note alsothat, owing to the definition (9.11)of W2, Eq.(9.15)is nonlinear in P2.It canalsobe transformed into an equivalent form displaying W2 as an unknown

function.In Eq.(9.15),let Af representthe duration of an elementary event. Then,

W2(At) is the probability that a systemwill make a singletransition from state{a}to state {a2}during Af. As wedeal with processesthat are continuous in

time it is more convenient for our purposesto work with transition prob-probabilities per unit time. To this end we set k = {a},I = {a2}and define:

\"A,\020 At

= (probability/time)for a transition between two different states(k \302\245= I)

(9.16)_ ' ~

W2(k\\k,At)*kk~A,imo a7

= (probability/time) that a transition occursfrom state k (9.17)

Clearlywkl >0 (k / /) (9 ]8jwkk <0

9.3. Markovian Processes 231

Moreover,using Eq.(9.12):5>H= 0 (9.19)

i

Assuming now that the wkl valuesexistand are finite, one can easilyderivean equation ofevolutionfor the probabilitiesPx({a},t) = P(k, f) by summing

Eq.(9.15)over {ax}and by expanding P(k, t - Af) in Taylor seriesaroundP(k, t). One finds straightforwardly:

dP(k,f) ,dt \\

= l[wlkP(l,t)-wklP(k,tn (9.20)l*k

We see that the time variation of P is due to the competition between the

\"gain\" terms, associatedwith the transitions / -\302\273 k, and the \"loss\" termsassociatedwith the transitions k -\302\273 /. Hereafterwereferto this relation as themaster equation. This equation plays a great role in the analyses of the

following chapters.As a simpleillustration, considerfirst the Chapman-Kalmogorovequation

(9.15)for the Ehrenfest model.From (9.14)one has, summing over alltransitions changing the population k of the spheresin one of the containers:

P(k, t + Af) = 1 - ^\342\200\224-i

)P(k -1,0+ ^ir~ pC< + !'f) (9-21a)

fork = 1,...,N- 1,1

P@, t + Af) = \342\200\224 P(l,f)N(9.22)

P(N,t+ At) = ~P(N - l,f)

In the original model Af is taken to be the unity. If the limit Af ^ 0 is takenand the left-hand sideof Eq.(9.21)is expandedin powersof Af, we observethat in order to convert this equation into the master equation (9.20)it is

necessaryto divide by Af and give a meaning to quantities such as(Aty1(k + \\/N) and so on in the limit Af -\302\273 0.This is not possibleunlessadditional prescriptionsare made on the transition probabilities(9.14).Thisproblemisdiscussedin Section10.3using the random walk asillustrative

example.Returning to the general equation (9.20)we concludethat, within the

framework of Markovian processes,the problemof fluctuations amounts toconstructing and solving an equation of this form adaptedeach time to theproblemconsidered.As we see later, the structure of the solution depends


critically on two factors. The first factor is the macroscopicreferencestatearound which fluctuations are studied.In particular, the constraintsmaintain-maintaining

the systemin a state away from equilibrium are shown to deeplyinfluencethe behavior of fluctuations. The secondcrucial factor is the form of thetransition probabilities.In the examplesof random walk and of the urn

model discussedearlier in this section the transition probabilitieswereeither constant or linear functions of the random variables. Someof the\"chemicalgames\"discussedin Chapter10sharethese properties.As a rule,however, they are much more complicatedas they are determined bytransition probabilitiesthat are nonlinearfunctionsofthe stochasticvariables.This nonlinearity leadsto new important features absent in the standardexamplessuch as Brownian motion. In either case,the stochastictreatmentgives additional information with respectto the macroscopicdescription.Although the latter is recoveredunder certain conditionsas the behavior ofthe expectationvalueof the stochasticvariable,one has,in addition, a meansfor calculatingvariancesand other related quantities reflectingexplicitlytheeffect of fluctuations. Interesting examplesof gamesdefined by chemicalreactionsmay be found in Eigen and Winkler A975).

Hitherto we have assumedthat in the stochasticvariables {a,}the processis Markovian. Actually, this dependson the choiceof these variablesand onthe nature of the processesdescribedby the macroscopicequationsofevolution. As an example,considerthe processof diffusion of a distributionof N colloidalparticlesin a volume Keachone ofwhich performsa Brownianmotion. The local density p satisfiesFick'slaw ofdiffusion. Supposethat onenow choosesas variables in the correspondingstochastictreatment, thepositionand the momentum of a representative Brownian particle, or thenumber XAV(i) of Brownianparticlesin a small volume AV having momentabetween p, and p, + dp;.Oneof the classicalresults of probability theory is,then, that the stochasticprocessis a Gaussian-Markovian process.*Thesituation is entirely different if one considersas stochasticvariable thenumber

X\302\261vof particlesin AK whatever their velocities(Kac,1959).One

finds that the correspondingstochasticprocessis non-Markovian.Moreover,it doesnot seempossibleto derive a closedequation for P(X\302\261V, t) owing tothe very complicateddependenceof this function on the probabilitydistribu-distribution of positionsand velocitiesof the individual particles.A similar situationis encounteredin Chapter12.

9.4.THE EQUILIBRIUMLIMIT

Beforewebegin our analysisof nonequilibrium systemswe briefly discussinthis sectionthe propertiesof fluctuations in equilibrium systemsaway from* Such processesarealsoknown in the mathematical literature asdiffusion processes.

9.4. The Equilibrium Limit 233

points of phasetransition. This problemcan be solved in its most generalform, and one can say that it iscurrently a closedsubject.Onereasonfor this

is that the equilibrium state is characterized (seeChapters1 and 2) by anumber of thermodynamic potentials that attain at that point their extremalvalues, namely, entropy for an isolated system, free energy at constanttemperature and volume, and soon.

In order to see how the thermodynamic potentials are related to the

probability of a fluctuation, we first considerthe caseof a singlecompositionvariable denotedby X. We assumethat this variable is extensive*and that

the systemthat ismaintained isothermalcommunicateswith a largeexternalreservoir maintaining the corresponding\"chemicalpotential\" /< constant.According to statistical mechanics the equilibrium state is describedby the

grand-canonical ensemble,which we write here in the following form(Landau and Lifshitz, 1957):

pcp= expp<b{p, p, F)exppQiX- E(X)) (9.23)

where O is the generalizedthermodynamic potential (seealso Section3.5),[i is the reciprocaltemperature (in units of Boltzmann's constant kB), and Eis the internal energy,which dependson X as well as on parameterssuch asthe positions,momenta, and internal energiesof the particles.

From Eq. (9.23)one can deducethe probability distribution for X, by

summing over the position,momenta, and internal statesof the particlesinthe volume V:

pX) fexp - pE(X){dri}{dp,}= A exp0(O+

where A is a normalization factor. One recognizesimmediately that theintegral overcoordinatesand momenta is related to the freeenergy,F(X,/?, V)

correspondingto the value X of the extensivevariable.We concludethat:

Pep(X) = A exp0[0(/I,p, V) + fiX- F(X,p, K)] (9.24)

Note that the free-energy function, F, appearing in thermodynamics isdifferent from the quantity F{X,p, V) introduced in Eq.(9.24)(Callen,1960,1965;Tisza and Quay, 1963).The former is related to the thermodynamicpotential O by the relation:

O = F -fiX (9.25)

* For a reaction-diffusion system X denotes, typically, the number of particles of a reactingspecieswithin the system ofvolume V and temperature T.


whereX is the macroscopicvalue of X observedin the system,and F = F(X).Relation (9.24)becomes:

t X)] (9.26)

Applying formally the well-knownthermodynamic identities to F one has:

(AF)TV= F - F = n AX = n(X - X) = -T(AS)EV

whereAS isevaluatedat constant energyand volume rather than at constantT and V.

Thus, the roleof the term ji{X- X) in Eq. (9.26)is to remove from AF

(or from AS) the first-order terms of the expansionof F (or S)around themacroscopicstate. Calling A;S this \"internal\" entropy difference. Weconcludethat:

PJLX)ocexpfi-(AiS)! (9.27)

The reasonwhy A; Srefersto the part of AS dueto an internal fluctuation,is that a systematicexternal disturbancemodifying X only affectsthe first-order terms.According to the conditionsdevelopedin Section3.5,one hasthat AjS <0.

Relation(9.27)is the celebratedEinstein formula for fluctuations(Einstein,1903,1910).Although here we have been concernedwith fluctuations ofcompositionvariables, the sameexpressionevaluated under more generalconditionscan also generate the fluctuations of other extensive thermo-thermodynamic quantities. We may also point out that instead of consideringanopensystemin contact with a reservoir,wemay equallywell apply the resultsto evaluate the fluctuations within a small volume AV of a large isolatedsystemof volume V. In the latter case,the subsystemin the volume V \342\200\224 AV

surrounding AV actsas a heat-and-matter reservoir.Forsmall fluctuations around equilibrium, one can expandEq.(9.27)and

retain the first nontrivial terms only. Oneobviously finds

^sJ (9.28)

where%S2S)eis the second-orderexcessentropy familiar from Part I.In this

form, the probability function can be extendedstraightforwardly to severalfluctuating variables.RecallingexpressionD.29b)for <52Sone finds:

(929)

9.4. The Equilibrium Limit 235

where X, denotenow the numbers of particlesof constituents 1 to n andbXi = Xt \342\200\224

Xj eq.As we saw repeatedly in Chapters3 and 4, the quadraticfrom appearingin the exponential ispositivedefinite. Thus, the distributionof fluctuations at equilibrium is Gaussian.The maximum value of thisdistribution related to the most probablestate of the system,X1?, aswell as itsaverageare both identical to the mascroscopic(equilibrium) values X,!-eq:

<*\302\273>\302\253,

= XT =XLeq (9.30)Leq

Moreover,a direct calculation where <5X, are treated as continuous variablesshows that the mean-squaredeviation or variance of the fluctuations is

(Landau and Lifshitz, 1957):

<SX,5X^= ^7@) \\(9.31)

where (d/i/dX)' denotes the inverse of the matrix whose elements are

(d/ij/dXj)^.Fora dilute system n can be substituted by expressionC.10)and one obtains:

(dXjOXj)^=<A,->eq d*J = Ai-eq dfj (9.32)

We conludethat the fluctuations exhibit in this casea Poissonianvariance.This can alsobe shown directly from Eqs.(9.23)and (9.24)in the limit of adilute solution.In this case,E(X) = YJ= i e,and (Landau and Lifshitz, 1957):

Pep(X) ex e\342\200\224\\ jdr,dp, expp(n -\302\243,

where the factor X!accountsfor the indistinguishability of the particles.Bydefinition of the chemicalpotential:

Jdr, dp, expftp - e,)=

and

F^X) = /I ^^ (9.33)

The normalization condition gives A = e~<x>;therefore, Eq. (9.23)is aPoissoniandistribution [cf.Eq.9.8)],with a mean value <X>which, accord-according

to Eq.(9.20),is equal to the macroscopicvalueof X correspondingto thestate of equilibrium.

Relations (9.29)to (9.33)can be interpreted in an alternative way asfollows.We first see,from Eq. (9.32),that the quantity


which providesa measureof the relativeimportance of fluctuations,behavesas:

((dXi))eq 1

YX vi/2 (9-J4)

Ai,eq Ai.eq

In a macroscopicsystem X, ^ -> oo, and Eq. (9.34)becomesa very smallquantity. In other words,there is in this casea clearcut distinction betweenmacroscopicdescriptionand fluctuations. This is also reflected in the factthat the maximum of the probability distribution (9.29)is situated at themacroscopicvalue X,eq.Thus, fluctuations around this value lead the

systemto configurations that becomeincreasinglyimprobablewith the sizeof the fluctuation.All ofthesestatementsare, in fact,equivalentto the stabilityof the state of equilibrium wheneverthe matrix of the coefficients(duJdXj)^definesa positive definite quadratic form, that is, to the condition A(S <0.This property fails only when the system is in the vicinity of a transitionleading to a separationbetweendifferent phases(Landau and Lifshitz, 1957).

The theory of fluctuations outlined in this sectionwas basedentirely onequilibrium thermodynamics and statistical mechanics.In principle,oneshouldbeableto reproducetheseresults in the most generalcaseof a systemat equilibrium from the master equation (9.20)provided the stochasticvariables are chosenadequately, and the transition probabilitieswk, areconstructedin such a way that they reflectcorrectly the conditionsprevailingat equilibrium. Among the latter, the condition of detailed balance plays acrucial role (DeGrootand Mazur, 1962).It statesthat each elementarystepin a sequencelike a chemical reaction can occurwith the same probabilityas its \"inverse,\" that is, as the stepobtained from the \"direct\" one on timereversal.*

9.5.FLUCTUATIONSIN NONEQUILIBRIUMSYSTEMS:AN HISTORICALSURVEY

The successof Einstein'stheory of fluctuations around equilibrium suggeststhat the extensionto nonequilibrium situations couldbe basedon similarideas,that is:

\342\200\242 The use of some suitable \"potential\" extending the notion of thermo-dynamic potentials to far-from-equilibriumsituations.

\342\200\242 The useofextensivestochasticvariables referring to the systemasa whole.

\342\200\242We may recall that this condition alsoplays an important role in the derivation ofthe Onsager

reciprocity relations (see'Section3.3).

9.5. Fluctuations in Nonequilibrium Systems: An Historical Survey 237

This would bejustified from the fact that the basicrelations (9.30)to (9.32)dependon the volume of the system in a universal fashion:

<(<5XJ>=aK, a = const

and suggest in this way a remarkable property of similarity betweenvariancesand volumesof different sizes.

We started these investigations with the study of simple examples(Prigogine,1954;Prigogineand Mayer, 1955;Nicolisand Babloyantz,1969)that confirmedat first the possibilityof a theory of nonequilibrium fluctua-fluctuations basedon the excessentropy, S2S.However, the study of nonlinearsituations led to the quite unexpectedconclusionthat these ideascan onlyapply to specialsituations.Considerfirst the possibilityofdenninga potentialthat would determine the probability of a fluctuation. As we saw repeatedlyin the first partsof this monograph, one cannotexpectto find sucha potentialexcept in the caseof systems involving a single variable and in the caseofsystems operating in the linear range of irreversible processes.Forsuchsystemsone can, indeed,show that the probability function takesthe form:

P ocexp(A7r) (9.35)where Axe is a generalized excessfree-energyfunction, also referred to askinetic potential (Prigogine and Mayer, 1955;Graham and Haken,1969;Haken,1975a,b).In the presenceof instabilities An has some remarkablepropertiessimilar to the Landau-Ginzburgexpressiondetermining thefreeenergy in the critical region of equilibrium second-orderphasetransi-transitions (Stanley, 1971).In systemsinvolving more than one variable andoperating far from equilibrium, all of these resultsbreakdown;moreover,no obvious way of extending them appearsto exist.

Let us comenow to the problemof the choiceof stochasticvariables.At

equilibrium (McQuarrie,1967)and for the most general sequenceof uni-molecular reactionsaway from equilibrium (Nicolisand Babloyantz, 1969)it has been shown that fluctuations can be correctly describedby extensivevariables referring to the system as a whole. In contrast, for nonlinearreactions operating far from equilibrium a most striking result has beenshown\342\200\224the variance of fluctuations changesqualitatively as their scaleincreases.Small-scalefluctuations,affecting only the region of dimension ofthe order of the mean free path, can be describedby a straightforwardextensionof Einstein'stheory. The only differencewith equilibrium is thatthe parameters appearingin the probability distribution [seeEq.(9.29)]areevaluated at the nonequilibrium referencestate.This is quite similar to theideasunderlying the notion of local equilibrium in kinetic theory of gases(Chapman and Cowling, 1952).This distribution function representsa


nonequilibrium situation,yet its functional form is the sameas in equilibrium.The differenceliessimply in the fact that the state variablescan evolvein timeand can be inhomogeneousin space,as a result of the nonequilibriumconstraints.

In contrast, large-scalefluctuations affecting macroscopicvolumescomparableto the size of the systembehave in a distinctly nonequilibriumfashion. In the presenceof instabilities they can be amplified and drive the

system to a new regimedifferent from the initial referencestate (NicolisandPrigogine, 1971;Nicolis,1972,1974a;Nicolis,Malek-Mansour,Kitahara,and Van Nypelseer,1974;Nicolis,Malek-Mansour,Van Nypelseer,andKitahara, 1976).We are far from the simple \"universal\" lawsof fluctuationsas describedby a Poissonian-typedistribution.

A correctdescriptionof fluctuations in nonequilibrium systemsmust

necessarilybe a localdescriptioncapableof differentiating between fluctua-fluctuations of variable ranges and coherencelengths (Nicolis,Malek-Mansour,Kitahara, and Van Nypelseer,1974;Malek-Mansourand Nicolis,1975;Gardiner,McNeil,Walls and Matheson,1976;Lemarchand and Nicolis,1976).

Ourprincipalgoal in Chapters10-12is to describethesenew developmentsin detail and to substantiate them on representative, if simple,examples.We first describe,in Chapter10,the basicideasof the formulation basedonthe use of extensivevariables. Chapters11and 12are devoted to the localtheory of fluctuations and to the analysis of the critical behavior of themacrovariables in the presenceof instabilities.

Chapter10

Birth-and-DeathDescriptionof Fluctuations

10.1.MASTER EQUATIONFOR BIRTH-AND-DEATH PROCESSES

As we emphasizedin Chapter 9, in the master equation descriptionoffluctuations one has to assignto the systema set of transition probabilitiesdescribingthe processin the spaceof someappropriatestochasticvariables.Considerfirst the intuitively appealingchoice,which consistsof introducingin the masterequation the variablesthat describethe systemasa wholeon thelevelof the macroscopicrate equations.In a sequenceof chemicalreactionsthese variables correspondto the numbers {X,},i = 1,...,n of particlesofthe n chemical speciespresent.A particular step of the sequenceproducesadditional particlesof a certain set of componentsand consumesparticlesof other components.We thus obtain a \"birth-and-death\"processin thespaceof {X,}values.The transition probabilitieswk, as dennedin Eq.(9.16)then dependon a set of integers rip (rip may be positive,negative,or zero)describingthe changeof the number X, due to reactionp:

wkl=

W(rip)=

w({X,--

rip\\-> {X,}) A0.1)

Equation (9.20)becomes:dP({Xi},t)_ \\^(X})P\302\253X -r }t)

ut p

-^w({Xj-> {X,.+ r,.,,})P({X,.},r) (io.2)p

This equation describesthe time evolution of the probability function

P({Xt},t) in terms of the various chemicalprocesses.However,we have stillto expressw in terms of the X values.Obviously,the w values are related tothe frequencies of reactive collisionsbetween chemical species.Considerfirst a simpleexampleinvolving three speciesX, Y, Z interacting through

X + Y -^ 2Z

239

240 Birth-and-Death Description ofFluctuations

The transition probability w = w(X + \\,Y + 1,Z \342\200\224 2 -\302\273 X, Y, Z) is propor-proportional to the number of binary collisionbetweenX and Y beforethe reactiontakesplace.Thus

w = k(X + i)(y+l)where fc is the rate constant. Note that the term (+ 1)addedto X and Y onthe right-hand sideof this expressionis, in fact, equal to \342\200\224 vx, \342\200\224vY,

wherevx,vY are the stoichiometric coefficientsof X, Y in the reaction.In ourparticular examplethese coefficientsare equal to the jumps, \342\200\224 rx and \342\200\224

rY

undergone by X and Y during the reaction. This result can be extendedtoautocatalytic reactions,where the reactant may appearon both sides.Forinstance for the reaction

A + X \302\261> 2X

the transition for X is X \342\200\224 1 -> X, and w becomes:

w(A + 1,X - 1- A, X) = k(A + \\)(X- 1)

Again, the term(\342\200\224 1) addedto X is equal to \342\200\224 vx, sincethe stoichiometric

coefficientof X in the reaction is now equal to + 1.The jump, rx is alsoequalto + 1.Hereafter, therefore, the jump appearingin expressionA0.1)can beidentified with the stoichiometriccoefficientof i in reaction p.

Theseexpressionscan beeasilygeneralizedto all caseswhere the stoichio-stoichiometric coefficientvip (or the jump rip of Xt) is 0, \342\200\224 1 or + 1,provided thestoichiometric coefficientsof all reactants appearingon the left-hand sideof the reaction are themselves0, \342\200\224 1,or + 1.We obtain:

-rip) rip = 0, -1,+1.A0.3a)

The product is taken over all speciesthat participate in the collisionmechan-mechanism considered.Supposenow that in someof the steps the stoichiometriccoefficientsvip of the reactantson the left-handsideofthe reactionare different

from 0, -1,or + 1.If the jump sufferedby this reactant is rip, then one has to

expressthe number of ways that vip moleculescan be selectedout of the

Xj\342\200\224

rip moleculesi that are present.This gives:

MX,- rip, {X)}-X,, {X)})= k,,(X,

-rip)

\342\226\240\342\226\240\342\226\240(X, - rip-

vip + \\)/vip !(vip

> 0) A0.3b)

As an example,one has for the process

w(X + 2,E-1 -\302\273 X,E)= ^(X + 2)(X+ 1)

10.2.Limitations ofBirth-and-Death Formalism 241

whereasfor the reaction

2X \302\261> E + X

we obtain the different expression

MX + \\,E-1 -> X,E)= ^(X + l)X

It can beshown(Malek-Mansourand Nicolis,1975)that Eqs.A0.3)are the

only forms of w compatiblewith the requirement of having a Poissoniandistribution at equilibrium.As pointed out in Section9.4,a systemdescribedby the Poissoniandistribution gives riseto statistical averages identical tothe macroscopicvalues. Thus, the equilibrium laws of chemical kineticsbecomeexactconsequencesof the master equation A0.2).We have ampleopportunity to illustrate further relations A0.3)in the sequel.

10.2.LIMITATIONSOF BIRTH-AND-DEATH FORMALISM

Becauseof its simplicity and intuitive appeal,birth-and-death formalismhas been widely used (Barucha-Reid, 1960,McQuarrie,1967).It is onlyrecentlythat its limitations wereclearly recognized.Firstly, the fact that thetransition probabilitiesare computed in terms of \"collective\"variablesreferring to the entire system[seeEq.A0.3)]impliesthat only very exceptionalfluctuations are retained in the description.

Moreover, by treating the system as a whole one discards importantaspectsof fluctuations associatedwith such propertiesas the size,rangeover which they extend, and correlation length over which two parts of thesystem can feeleachother. Oneexpectsthat thesepropertiesshouldplay animportant role in the vicinity of criticalphenomena,exactlyas in equilibriumstatistical mechanics.As is well known in this case(Landau and Lifshitz,1957),even in a systemthat is macroscopicallyhomogeneous,the fluctuationsbreaklocallythis homogeneityand generate in this way additional stochasticvariablesthat must be incorporatedin the description.

Finally, becauseof the global character of the birth-and-death method,there is no way to expresssatisfactorily the nature of the macroscopic(nonequilibrium) state around which fluctuations are evaluated in Eq.A0.2).In particular, the question of relativeclosenessof this state to a local equi-equilibrium regime\342\200\224which is a crucialelement in the validity of the localdescrip-description of irreversible processesadopted throughout this monograph\342\200\224is

left open.A natural remedy of this is to apply the birth-and-death formalism to a

subvolume AK of the entire system that is sufficiently small to permit a


birth-and-death-likedescriptionof the processestherein and yet sufficiently

large to remain closeto local equilibrium.Sucha volume would becoupledautomatically to the remainder of the systemby the transport of matter andenergyacrossthe surfaceseparating AV and V \342\200\224 AV.

The purposeof Chapters11and 12is to develop in detail such a localtheory of fluctuations. Before we attempt this, however, we devote the re-remaining of this chapter to the birth-and-death formalism.The reasonfor this

is both pedagogicaland substantial;most of the methodsof analysisdevel-developed

for the solution of the birth-and-death master equationsare applicableto the more complicatedlocal theory as well.Moreover,even in this, morerefined treatment of fluctuations the contribution of chemical reactionswithin the subvolume AV are describedby a modifiedbirth-and-death pro-process. The latter is therefore,a basiccomponentof the dynamicsoffluctuationsin nonequilibrium systems.

It is remarkablethat near equilibrium or in systemsof unimolecularreactionsthe necessityof a local descriptionof fluctuations is not apparent.In a sense,the entire systemor different small componentsof it behave in anidentical fashion and are both described(if the mixture is ideal) by aPoissoniandistribution at the steady state.As soonas the system deviatesfrom these limiting regimes,the nonlinearity and the nonequilibrium con-constraints introduce a coupling between neighboring spatial elements.As aresult, a global descriptionof fluctuations becomesinadequate.

Throughout the remainder of this chapter and in Chapters11and 12welimit ourselvesto reactionsunder conditionsof ideal mixtures. Thus, weneglectthe explicitinfluenceof intermolecularforces into the rate equationsand the transition probabilities.

'

10.3.SOMEMETHODSOF ANALYSIS OF BIRTH-AND-DEATHMASTER EQUATIONS

Herewe are interested primarily in solutionsof the master equation A0.2)|which are asymptotic in one of two ways:(a) either the limit of long timeS||f -\302\273 oo is taken and/or(b) the volume V and the total number of particlesfl|assignedto the system are macroscopic.This is usually expressedby

tjMfij

thermodynamic limit:'\342\200\242$

AT :'\\F-oo,AT - oo, p = \342\200\224= finite A0.4|

Suchsolutions,whenever they exist,are expectedto describeadequatelythe situations encounteredin Part II,where the evolution of the systeminvolved alwayssomelarge-scalemacroscopicelement.

10.3-SomeMethods ofAnalysis ofBirth-and-Death Master Equations 243

We now compilesomemethods for analyzing the asymptotic solutionsof the birth-and-death master equations.The presentation is rather brief, asmost of these methods are illustrated later on by simpleexamples.

Fokker-PlanckEquation

In a number of problemsinvolving Markov chains the \"spacing\" betweenstatesis small.One of the bestknown examplesis Brownian motion (Wax,1954),that is, the motion of a heavy particle of massM in a fluid of light

particlesof massm <| M. In the limit as m/M -\302\273 0, the momentum transferarising from the collisionsbetween Brownian particle and the particlesofthe medium is small with respectto the instantaneous momentum of theBrownian particle.In a system involving chemical reactions,a somewhatanalogous situation can arise in that the step appearingin the birth-and-death master equation A0.2)isexpectedto besmall comparedto the instan-instantaneous valuesof the numbers of particles.

To handle this type of situation we considera smooth function Q({X^),which goeswith sufficient rapidity to zero as X( -\302\273 oo and is otherwise arbi-arbitrary. We multiply both sidesof Eq.A0.2)by Q and integrate over {Xt} con-considering them as continuous variables. Notice that this implies that bothP and w have to be interpreted as probability densities.We obtain:

8t

i~

rip\\

- f },f) A0.5a)

We now expandin the first term on the right-hand side,Q({Xt})in a Taylorseriesaround Q({Xt \342\200\224 rip}).Usingcondition (9.11)weobtain, after switchingto new integration variables X\\ = Xt \342\200\224

rip:

A0.5b)


This expressionis known as the Kramers-Moyalexpansion.In order tounderstand better this structure, we define the transition momentsai.ixh~in-by the relation:

a,:il i,({X,})= -X M{Xi]- {Xt + rip}) f| rikP A0.6)

In general, all a, values are different from zero.A Gaussian-Markovianprocessis characterized by the property:

= 0 for / > 2 A0.7)

Equation A0.5)now becomes:

Integrating partially in the right-hand side and taking into account that

Q({Xt})is an arbitrary function, we concludethat:

,0 A0.8)ij MiMj

Thisequation is known as the Fokker-Planckequation (Wax, 1954).Beforegoing further, it is interesting to comment on the meaning of this

equation in the light of the ideasdevelopedin Sections9.2and 9.3.Let usconsiderthe problemof random walk. Usingthe form (9.13)of the transitionprobabilitiesand taking the time-spacestepsto be Af and Ax, respectively,rather than unity, we write the Chapman-Kolmogorovequation (9.15)inthe form:

F(x,f + Af) = pP(x- Ax, f) + qP(x+ Ax, f) A0.9)

We now considerthe limit of a fast jump, Af -> 0, between neighboringpoints,Ax ->0.Equation A0.9)can then be expandedin powersof Af andAx. The first few terms yield:

.dP dP Ax2d2P q - p A 3 d3PAt \342\200\224 = (q - p Ax \342\200\224 +

\342\200\224-\342\200\2243+ ^\342\200\224- Ax3 j-j + \342\226\240\342\226\240\342\226\240

ct ex 2 ex 6 ox

If we divide through by Af and take the limit Af -> 0,wemust give a meaningto expressionssuch as (Af)\" x(q - p) Ax and Ax2/2 Af. This problem,which

was already mentioned within the context of Eq. (9.21)for the Ehrenfest

10.3-SomeMethods ofAnalysis ofBirth-and-Death Master Equations 245

model, can be solved in simple situations such as Brownian motion (Wax,1954).Indeed,in this caseone showsthat:

q - p = 0 (asq = p =%)

and that Ax2/2 Af existsfor all Ax and Af (in particular for Ax -\302\273 0, Af -\302\273 0)and definesthe diffusion coefficientof the Brownianparticle.It is, therefore,natural to admit that we couldset

Ax2\342\200\224- = D = finite2Af

q - p = --Ax, C = finite.

Thus, all terms containing powersof Ax higher than two cancelin the limitAf \342\200\224\342\226\2720, Ax \342\200\224\342\226\2720, and we have:

A0.10)ot ox ox

which isof the form A0.8)with a constant \"drift coefficient\"\302\243

related to thedifferencebetween the two transition probabilitiesp and q, and with a con-constant \"diffusion coefficient\"D.Thus, random walk provides a model for aGaussianMarkovian process.

The situation is different when chemical reactions are considered.In thiscaseone deals,typically, with processesthat are continuous in time (Af -> 0)but discretein the spaceof the stochasticvariables {X,}.Moreover, thetransition probabilitiesper unit time w are perfectly well defined at theoutset in the limit Af -\302\273 0 [seeEq. A0.3)].Thus, one can write a masterequation directly for dP/dt without making additional prescriptionsas in

the caseof the random walk. As a result, even for the simplest reactionsconditionA0.7)can neverbesatisfiedexactly.*Forinstance,for the reaction

thejumpsr*,= +l,rX2= -l,andw(X-> X + 1)= ktA,w(X -\302\273 X - 1)=k2X.Thus [seeEq.A0.6)]:

Obviously, this expressioncan never be set exactly equal to zero.At best,therefore,Eq.A0.8)is an approximation to the exact solution of the master

In this respect we may mention a theorem (Pawula, 1967)according to which if we assumeat = 0 for someeven /, we areactually assuming a, to bezero for all / > 3.

246 Birtk-and-Death Description ofFluctuations

equation.Even so,van Kampen A961,1969)pointed out that in a non-nonlinear system,by retaining the fully nonlinear contributions in the transitionprobabilitiesax and a2 one keepsterms of the same orderof magnitude asthe higher transition moments. Subsequently,he suggesteda method for

expandingsystematicallythe master equation, which leads to a Fokker-Planckequation whosecoefficientsare this time linearizedaround the valuescorrespondingto the macroscopicevolution.

The main point of van Kampen'smethod is to assumethat relations(9.19)to (9.22)extend to nonequilibrium situations, in the sensethat:

\342\200\242 P({X},t) is Gaussianin the limit of a macroscopicsystem,that is V -\302\273 oo.\342\200\242 {Xt} are distributedaround their most probable values,* {X\"(t)} with a

variance of order V:

, - XT(t)J},oc V

with X?(t)oc V

Thus, any deviation from XJ\" is of the order V1/z. We introduce a newvariable, ^,which is independent of V by setting (for simplicity we considerhereafter and until the end of Section10.3a singlestochasticvariable):

X = Xm(t) + V1/2\302\243 A0.11a)or,in terms of scaledvariables

X=Vx, Xm(t)=Vy(t)x = m+v-*\302\273t

mUb)

We may now substitute this into the master equation A0.2)and focuson the^-dependenceof the probability function P:

P(lt)= P(y(t) + V-1'2t;,t) A0.12)The time derivativedP/dt of P contains a contribution reflectingan explicittime dependence,as well as the fact that y(t) dependsgenerallyon t:

_SP ll28Pdyddt dt

<3\302\243dt

As before, the right-hand-side of the master equation is expandedin a

Kramers-Moyalseries[seeEqs.A0.5b)and A0.8)].Moreover,the transition

* Note that, in general, <A\",> # Xf. Themost probable value is expected to correspond to the

results of macroscopicobservations. In general, <A\"> provides a good approximation to this,unless P is a multihumped distribution.

JO .3- SomeMethods ofAnalysis ofBirth-and-Death Master Equations 247

moments are expandedin powersof F~1/2<i;.Equating equal powersofV\021/2 on both sideswe find:

To orderV1'2:

dy _Tt~y

and to order V\302\260:

where the transition moments are evaluated at the most probablevalue,y(t). The latter evolves in time accordingto Eq. A0.13)which, as we seeshortly, has the same structure as the macroscopicequationsof evolution.

Fora singlestochasticvariable Eq.A0.14)givesa Gaussiandistributionas a steady-state solution, contrary to the equation A0.8) with nonlinearcoefficients.In general, therefore, van Kampen'smethod cannot handlestraightforwardly situations involving instabilities that, as we seelater,involve deviations from the Gaussian behavior. Moreover,the methodbecomesawkward to apply around time-dependent states. In this case,accordingto definition A0.11),the reducedstochasticvariable

\302\243

shoulditself be time dependentto secure the time independencyof the initial

stochasticvariable X.Despitethese limitations, van Kampen'sexpansionconstitutes a powerful

tool of analysis that permits sorting out of somegeneral features of fluctua-fluctuations in nonlinear systems.

Gaussian-Markovianprocessesleading to a Fokker-Planckequationwith nonlinear coefficientsas in Eq. A0.8) can also be representedby aLangevin equation (Wax, 1954).The ideais to extendthe macroscopicequationof evolution for the macrovariable y(t) to a stochasticdifferential equation(Arnold, 1973).In physical applicationsthe stochasticcharacter is usually

expressedthrough a random force term, F(t):

^ = fll(M)+ F{t) A0.15)

The validity of the macroscopicdescriptioncan now be ascertainedonly in

an average senseand this requires,accordingto the classicaltheory:

<F(f)> = 0 A0.16a)Moreover, the Gaussian-Markovianassumption is equivalent to the ab-absence of any correlation betweenvaluesof F at two different times:

<F(fl)F(f2)>= 2BS(t,- t2) A0.16b)


In Eq.A0.16)the bracketsdenotea statistical average over the distributionof F values.

The useof Langevin'smethod in nonlinear systemsfar from equilibriumcan becriticizedon the samegroundsas that of the Fokker-PlanckequationA0.8)(seealsoChapters11and 12).

Hamilton-JacobiEquation Approach

This method, which has beendevelopedby Kubo, Matsuo,and KitaharaA973) (seealso Kitahara, 1974),bearssomeresemblanceto van Kampen'smethod.It is, however, more general in that it can handle situations nearinstabilities in a more straightforward manner.

One again considersa scaledvariable x correspondingto an extensivequantity X:

X = Vx

and expressesthe master equation A0.2) in terms of x.To this end, oneintroduces:

\342\200\242 A new probability distribution:

P\\x,t) = P(X,t) A0.17a)

\342\200\242 A new transition probability:

Vw{x, r) = w(X-^X+ r) A0.17b)

\342\200\242 A \"Hamiltonian,\" H, definedby:

H(x,p)= \302\243A

-e\"'\"p)w(x,r,,) A0.17c)

p

Then, the finite displacementoperation involved in the master equation canbe expressedin terms of an operator form of H, whereby p is substitutedby the derivatived/dx.Oneobtains,instead of Eq.A0.2):

V 8t \\ V8x

This is analogousto Schrodinger'sequation of quantum mechanics, but

instead of h/i one now has the factor \\/V, which is expectedto be small in a

macroscopicsystem.In analogy with quantum mechanics,one can seekfor

10-3-SomeMethods ofAnalysis ofBirth-and-Death Master Equations 249

asymptotic solutions of Eq.A0.18)using WKB-typemethods(Courant andHilbert, 1962).Thesesolutionsare of the form:

P oc exp(F<\302\243(x, t)) A0.19)

Their existencehas beenestablishedby Kubo,Matsuo,and Kitahara A973)and by Kitahara A974) using Feynman's path integral method.Substituting

Eq.A0.19)into Eq.A0.18)one obtains,to the dominant order in V:

0 A0.20)dt V 'dx

Thisequation has the sameform as the Hamilton-Jacobiequation (Courantand Hilbert, 1962).It can be solved by the well-establishedmethod ofcharacteristicequations.The topologicalstructure of the trajectoriesof theseequations in the correspondingphasespacegives information about theuniquenessand stability of their solutions,as well as about the shapeof theprobabilitydistribution.The method becomesincreasinglydifficult to applyfor two or more stochasticvariables.Forsystemsinvolving a singlevariable,however,someremarkable resultshave beenobtained,especiallyin the caseof transitions between multiple steady states.The probability distributionbecomesthen double-humped(or multiple-humped),and the phenomenaaccompanying the instability can be expressedin a language similar toequilibrium first-orderphasetransitions.

Generating-functionMethodand Cumulant Expansion

A still different method, usedwidely in the sequelof the presentmonograph,consistsin expressingthe master equation A0.2) in the generating functionspace.To this end we definethe generatingfunction F({.s,},t) by (McQuarrie,1967;Barucha-Reid,1960):

F({s,},t)=t Y\\s?'P({Xilt) d\302\260-21)

It is easy to see that F is related in a simpleway to the average values ofvariouspowersof Xt. Thus, taking Eq.(9.3)into account and defining

X1XJP({Xk},t)(etc.) A0.22)


wefind:

P({X{},t) = 1

= <*;>

= <(SXjJ>

{Si}=11}

'^^ ^ \302\253=<\302\253,\302\253.> A0.23)

In order to construct the equation of evolution for F we multiply bothsidesof the master equation A0.2)by YJs?'and sum over {Xt}.We obtain:

\302\245

= i-11Uti'MW,}- {x>+ njmx,},t) A0.24)

{Xi\\ p i

In the first term we switch to the new variables X\\ = Xt - rip. We alsotakeinto account the explicitform A0.3)of the transition probabilitiesand noticethat the product

can alsobeexpressedin the form:

1 - d^'\"-\342\200\224-

s|'\+r\")") \342\200\224j- sf'x (contribution of the other constituents)Vjp

. CSj'''

We recall that rip refersto the jumpsufferedby constituent j in the reaction p,and vip to the stoichiometriccoefficientof the reactants i on the left-handsideof the reaction (vip

> 0).On the other hand, the secondterm in Eq.A0.24)introducescontributions of the form:

- {Xt + rip\\)

which can beexpressedas:

VIP'sj'p \342\200\224^ sf'x (contribution of other constituents)

10.3\342\226\240 SomeMethods ofAnalysis ofBirth-and-Death Master Equations 251

Theseexpressionswill be written symbolicallyas:

and

nsH^};{vv)rwj A0-25>

Substituting backinto Eq.A0.24)we obtain:

n *--n***)* \302\273({|};(v>)f,m,o 00.26){ ( ) ({|} )Forinstance, for the reaction

one has:

Va = 1, vx = 0, r.4 = -1, rx = +1, w = L4

and

Equation A0.26)is a linear partial differential equation with variable coef-coefficients. In general, its solution is an arduoustask.Still, in somesituationsone can envisagesystematic expansionproceduresin a way similar to theprevious subsection.Thus, in the limit where the volume consideredis ofmacroscopicdimensionswe set(Prigogineand Nicolis,1971):

F = expNi//({Si},t) A0.27)

where, accordingto definitions A0.23), \\j/is expectedto be of order unity.

Substituting backinto Eq.A0.26)and retaining dominant terms in N one canconvert the equation into a nonlinear differential equation of the first orderthat is somewhat similar to the Hamilton-Jacobiequation A0.20).We seekfor solutions of this equation in the form

\342\200\242A

= ax(t)^+ \342\200\242\342\200\242\342\200\242+ an(tKn + ftn@^ + ...+ Mf)y+ b12(t)\302\2431\302\2432

+b13(t)\302\243i\302\2433

+ --- A0.28)wherewe defined the new variables

fi = s,- - 1 A0.29)


On substituting into Eq.A0.26)and identifying equalpowersof\302\243,\342\200\242

oneobtainsa setofnonlinear ordinarydifferential equationsfor the expansioncoefficientsa,-,bjj, and so on.Theseequationsconstitute an infinite hierarchy and canusually be solved only if a truncation to somefinite order is possible.Ex-Examples are given in the sequel.

A final remark concernsthe physical interpretation of the expansioncoefficients.From Eq.A0.23)and the definition A0.27)of

i/>we find:

bu=i (SX,5Xj)=1[<X;^.>- <X,-><X,>](etc.) A0.30)

This showsthat the expansioncoefficientsare related to the cumulants, orsemiinvariants of the probability distribution (seeKubo, 1962).Another

interestingfeature is that bit and bu are related directlyto the deviation of thedistribution from the Poissonianform, for which relations (9.32)are iden-identically satisfied.

10.4.THE MOMENTEQUATIONS

In this sectionwe investigatemore specificallythe relation between masterequationsand the macroscopicequations of evolution. We have alreadypointedout that the propertiesof the probability distribution are determinedby the entire setof its moments [cf.Eq.A0.22)].Among these,the first and thesecondmoments are of particular interest.Indeed,in severalcaseswhere thedistribution is peakedaround a well-definedvalue, the first moment\342\200\224the

mean\342\200\224is directly related to the maxima of the probability distribution

<*\342\226\240,->=s XT A0.31)

Both <X,>and XT can then beexpectedto describecorrectly the evolutionof macroscopicquantities. The situation is different when multiple humpeddistributions are involved; in this caseone expectsthat the results of amacroscopicobservation are related to the most probablevalues XT and tothe dispersionaround thosevalues,rather than to <X,>,<<5A\",2> and so on.Unfortunately, thesequantitiesare difficult to identify on the masterequation.It is much more natural to obtain, from the latter, resultsreferring to theevolution of the moments of the probability distribution.

10.4.The Moment Equations 253

In this sectionwe derive the exact equationsof evolution for the first few

moments, starting from the master equation A0.2)and discussconditionsunder which theseequations can be truncated to provide direct informationon the behavior of the first and secondmoments. An equivalent methodconsistingof working in the generating function representation is illustratedon the examplesanalyzed in the subsequentsections.

We first considerthe equation for <X,>.We multiply both sidesof Eq.A0.2)by X{ and sum over all valuesof Xt. Moreover, in the first term of the

right-hand sidewe perform the samechange of variable as in the transfor-transformations following Eq.A0.24).We obtain in this way:

^T j},t)at p {Xj)

Accordingto relations A0.6)and A0.22)the right-hand sideis the statisticalaverageof the first transition moment alti:

The main feature of this equation is that it is not a closedequation;owingto the nonlinearity of alti onehas, in general,

Now, aUl{{(Xj)}yis identical to the right-hand side of the macroscopicequations of evolution.*We see,therefore, that the average laws differ fromthe phenomenologicalonesobeyedby the macroscopicvariables by termsrelatedto the variancesor to higher ordermoments.A closeinspectionof theexplicitform A0.3)of the transition probabilitiesin connectionwith definitionA0.6)of the transition moments revealsone further interestingfeature,which

we first illustrate in the two simpleexamples:

X + Y \342\200\224-^\342\200\224\342\226\272 E

and

We have, for the first example: - k^SXSY)* In fact, the equations of chemical kinetics refer to local quantities such asdensities and molefractions. The moment equations reduce to such local expressions if one takes into account

Properly the volume dependence of the rate constants.


Similarly, in the secondcase:

Moregenerally,one has:

i-<x,->= a, i({<Xj>})+ terms containing (<5X,SXj>;(SXf>-<*,.>;\342\200\242\342\200\242

\342\200\242)

at

A0.34)

In other words,the differencebetween (alt,)and the macroscopiclaw isrelated to the differencebetween actual values of variances and Poissonianvariances. Fora Poissoniandistribution this differencevanishes and onerecoversexactly the macroscopicequations(Malek-Mansourand Nicolis,1975).Moreover,as is shown in the examplesof the subsequentsections,even when it is nonvanishing, this differenceis weighted by a factor l/Nwith respectto the macroscopiccontributions, where AT is a suitable sizeparameter.This in turn introducesin fluctuation theory the problemofchosingappropriately this sizeparameter, which can only be solved satis-satisfactorily by a local approachto fluctuations.Suchan approachis developedin Chapters11and 12.

The results obtained sofar on the first moment equation are exact,as theyaccount for the influence of the higher moments on the evolution of theaverage. Quite frequently it is necessaryto introduce someapproximationsin order to obtain information on the evolution of thosehigher momentsthat will directly influencethe averages.

Let us focus on the secondmoment <A\"jX,->. We multiply Eq. A0.2) by

XjXj, sum over X{ and Xj, and perform the sametransformations as forEq.A0.32).We obtain in this way:

Tt<Xi*j>=I E (*\302\253\302\273\342\200\242;,

+ xJ'iP+ r'Pr]P)dt P {Xk)

On the left-handsidewe switch from {X,Xj}to the variances

<8Xt8Xj>= <XtXj> - <

Taking into account onceagain the definition A0.6)of the transition momentsas well as Eq.A0.32)we obtain:

10.4.The Moment Equations 255

The terms of the type {Xialjy can beexpandedaround the averages <X,>,(flij)as follows:

8X,)x UaiJ>+ ife) SX

* \\CAk/<.xy

Thus, the equation for the variancestakesthe following form:

yExisxjy=YJat k

fe) <SXk 5Xj}\\+ <a2Jj({Xk})>+ 0K6X,SXk SX,))khx> J

A0.35)

A still more compactform can be obtained by introducing the followingmatrices:

<a2>:\302\253a2\302\273u= O2.u>

(^ =Kij\302\253X\302\273 A0.36a)

<6X 8A\"> :<\302\247X bXyu = <8X,5Xj}

We note that they all dependon the averages of {X,}and are,therefore,independent of the fluctuations.

Denoting by KT the transposeof the matrix K we may finally write Eq.A0.33)in the following form (Lax,1960):

j($\\hxy=(K-<^X8A-\302\273T + K-<\302\247X8A-> + <*2y+ty.(&XidXkSXly)

A0.36b)

By invoking the sameargumentsasthoseleadingto Eq.A0.34)wecan further

transform Eq.A0.36b)into a form displayingthe deviationof the higher ordervariances from the Poissonianform:

^<5X 8X>= (K \342\226\240<,5X 8X\302\273T + K \342\200\242

<<5X 8X>at+ a2(K-Xj>})+ terms containing

ky,<8Xf>- 3<\302\2535X,2>

?>- <*,\302\273;\342\200\242\342\200\242

\342\200\242) A0.37)


Again, this is not a closedequation.However,insofaras the terms containinghigher ordervariancescan beneglected,one has the closedset of equationsA0.34)and A0.37)from which the mean value vector (X) and the variancematrix <<5X hX~) could be evaluated. Whenever the system has a singleasymptotically stablemacroscopicsteady state, this truncation procedureis legitimate.The solution of the moment equationsthen representsa station-stationary probability distribution, which may or may not have a Poissonianform.The situation isdifferent in the absenceofasymptoticstability.Formarginallystable systems (e.g.,the Lotka-Volterramodel) we expectthe solution toalways remain time dependent(seealso Section10.6).Forsystems under-undergoing bifurcation,on the other hand, after a transient periodof evolution weexpecta new asymptotic solution to beestablishedthat is representativeofthe new regimebeyond instability. Onecan really speakin this secondcaseof \"orderthrough fluctuations.\"

Clearly, in the absenceof asymptotically stablemacroscopicstates,thehigher order moments becomeincreasinglyimportant. Thus, the results ofthe calculationbasedon the truncation of the hierarchyof the moment equa-equations couldonly be regardedas short-time approximationsof the system'sbehavior, starting from an initial distribution of the Poissonianform. Thisis sufficient to provide a rudimentary ideaof the behavior of the system in

the vicinity of the macroscopicsteady state.Oneof the featuresof the localtheory offluctuations developedin Chapters

11and 12is to introduce a new parameter in the moment equations,relatedto the ratio between the diffusion coefficientand a suitable combination ofthe rate constants.This parameter weights the relative contribution of thePoissonianand non-Poissonianterms in the momentequationsand thus pro-provides a meansfor systematicexpansionproceduresfor solving theseequations.

Finally, Eqs.A0.34)and A0.37) illustrate in a striking way the roleoffluctuations in the onset of instabilities.Supposefor a moment that thetruncated forms of these equationsare valid to a good approximation.Let

{X\302\260j}be a steady-state solution.By varying slowly the bifurcation

parameter denotedby A in Part IIwedrive the systemto the threshold for aninstability. Obviously, if the descriptionbasedon the truncated equationsremains valid, the systemis not goingto evolvespontaneously to a new state,but is going to remain on the unstable branch of solutions.According tothe completeform of Eq.A0.34),however,we seethat d(Xj}/dtis going tobuild up to a finite value\342\200\224meaning that the system will be able to evolvespontaneouslyfrom the unstable solution\342\200\224provided the fluctuations mayincreasein time and take appreciablynon-Poissonianvalues.

Theseconsiderationsshed some light on the very meaning of stability,which is one of the central notions of Parts I and II.In essense,the purposeof the macroscopicanalysiscarriedout in thesepartswas to test the response

10.5-Simple Examples 257

of the systemto an externaldisturbance,which, although weak,was assumedto be present initially in the system with a macroscopicallyobservableamplitude.The view adoptedin fluctuation theory,on the other hand, is that

the system has to generate spontaneously this deviation from the averageregime,which couldthen trigger further evolution to a dissipativestructureprovided it contains a non-Poissoniancontribution. In essence,a \"pertur-\"perturbation\" is identified here to the non-Poissonianpart of a fluctuation (seeChapter11.)10.5.SIMPLEEXAMPLES

We now illustrate the ideasand techniques developedthus far on simplemodelsreferring,successively,to linear and to nonlinear sequencesof chemi-chemical reactions.Actually, it is necessaryto distinguish here between two typesof nonlinearity that one may encounter in a physicochemicalproblem:(a) the kinetic nonlinearity, related to the dependenceof the rate lawson themacrovariables and (b) the phenomenologicalnonlinearity, related to thedistancefrom equilibrium (i.e.,to the deviation from the linear range ofirreversibleprocesses).This type of nonlinearity stems,therefore, from thedependenceof the rates on the affinities and can even arisein a sequenceofreactionswhere the rate laws themselvesare linear.

In Section9.3weseethat at equilibrium the fluctuations of the numbers ofparticlesin a dilute mixture obeyto the Poissoniandistribution.The purposeof the presentsectionis to investigatepossibledeviations from this regimedue to the influenceof the kineticsand/orthe distancefrom equilibrium.

UnimolecularReactions

We considerthe following sequence:A-23

A0.38)*21 *32

Theoverall reaction

A < E A0.39a)

has an affinity

A0.39b)cwith the equilibrium constant

K = kl2k23 A0.39c)k2lk32


The system {X}is supposedlyin contact with two large reservoirsof matter

containing A and E, such that the content of the reaction volume in A and Eis renewedinstantaneously,thus maintaining a fixed concentration of A andE therein. Standardchemicalkineticswould describethe system by

A, E = const

dX\342\200\224

dX -\342\200\224 = (k12A + k32E)- (k21 + k2i)X A0.40)

where the bar indicatesthat wedealwith a \"phenomenological\"value of Xdetermined by the macroscopicequations of evolution. Equation A0.40)predictsthat the system tends,as t -> oo,to a unique asymptoticallystablesteady state

In the standard descriptionfluctuations are neglected.In orderto study theireffectwe introduce the probability distribution P(A, X, E,t) and assumethat

the reaction under considerationdefines a Markovian birth-and-deathprocess.Equation A0.2)now takesthe following form:

dP\342\200\224 = kl2(A + l)P(A + \\,X-\\,E,t)~k12AP

+ k21(X+ \\)P(A - \\,X + \\,E,t)- k21XP+ k23(X + \\)P(A, X + 1,E - 1,f) - k23XP+ k32(E + \\)P(A, X - 1,E + 1,f) - k32 EP. A0.42)

This is a finite-differenceequation with linearcoefficients.The finite-differenceproperty is a result of the discretenessof the stochasticvariables\342\200\224the

numbers of particles\342\200\224while the linearity of the coefficientsis a consequenceof the unimolecularcharacter of the reactions A0.38).

The study of this equation is performedmostconvenientlyin the generatingfunction representation.Defining[seeEq.A0.21)]

F(sA,sx,sE,t)=t sis$s%P(A,X,E,t) A0.43)A,X,E=0

we seethat Eq.A0.26)takesthe following form:

dF dF dF^ =

k12(sx-sA)\342\200\224 + k21(sA-sx)\342\200\224

dF dF+ k23(sE -S]C)\342\200\224 + k32(sx- sE)^- A0.44)

10.5.Simple Examples 259

If onesolvesthis equation subjectto the normalization condition, onefinds

that as f -> oo the only steady-statesolution is given by the equilibriumdistribution of Section9.3.This is natural, as the system {A + X + E}hasbeentreated thus far as a closedsystem.Eventually, sucha systemis bound toreach a singleequilibrium state.However,our problemis to study the evolu-evolution of an opensystem {X}in contact with large external reservoirs {A}and{E}.To this end we introduce the reducedprobability distribution

and

P(X,t)=\302\243 P(A,X,E,t)

A,E=0

f(s,t)= X sxP(A,X,E,t)X,A.E =0

A0.45)

Equation A0.44)reducesto:df *

<3f'2

vh n A,E=0EP

A,E=0

~ A0.46)

This equation for/isclosedonly when Y,a,e=oAP, Y,a.e=oEP 1Sexpressedin terms of/ To achieve such a closurewe appealto physical assumptionsthat are expectedto be quite realistic for the type of system considered(Nicolisand Babloyantz, 1969).What we want to describehere is a non-equilibrium steady state;under theseconditionswe have to ensurethat thestate of the reservoirs {A} and {E}varies in a much slowerscalethan thestate of the system {X}.This scaleseparationnow permits the assumptionthat

\302\243\",\302\243=0

AP (etc.),which are conditional averages,do not dependon thestate of {X}.In other terms, the statistical correlation between {A} and

{\302\243}

and the \"small\" system {X}is negligible:

P=(A}P(X,t) A0.47)A,E=0 A,E=0

Suppressingthe average value symbol for the initial and final productparticle numbers, which appearnow as simpleparameters,we may reduceEq.A0.46)to:

^ = A - s)\\(k23 )^- (k12A + k32E)f A0.48)


This equation can be solved straightforwardly at the steady state.TakingconditionsA0.23) into account we find a unique, properly normalizedsolution:

U ^ + ^1 A0.49a)

This solution predictsa steady-stateaverage [seeEq.A0.23)]:

Thus

f(s) = exp[(s- 1)XO]= exP^- 1)<*>] A0.49c)

This identity between statistical average and macroscopicvalue is a con-consequence of the linearity of the scheme.It can alsobe inferred from the first

moment equation A0.32),wherea^X) is now a linear function of X obeyingthe property:

<\302\253,(*)>=

\302\253,\302\253*\302\273

= (ki2A + k32E)- (k21 + /c23)<X> A0.50)

Returning to physicalvariablesand using Eq.A0.45)weobtain a Poissoniandistribution for P(X)(Nicolisand Babloyantz,1969):

It is well known (Landau and Lifshitz, 1957)that in the limit of small fluctua-fluctuations :

SX _ X - Xoa o ^ o

Equation A0.51)reducesto a Gaussiandistribution

P(X)= BnX0)-112exp|~-\342\204\2421 A0.52)L ZAoJ

On the other hand, from Eqs.D.29)and (8.14)we see that ~5X2/2XOis

equal to the second-orderexcessentropy around the steady nonequilibriumstate Xo.Equation A0.52),therefore,reducesto

P ocexpQ-(S2SH\\ A0.53)

We obtain a formula of the samestructure as the Einstein formula (9.27).The differenceis that here the excessentropy is computed around a non-


equilibrium referencestate.It is only in the limit ofzero affinity that Eq.A0.41)gives

^0 = ^eq = -, A

and the nonequilibrium probability function becomesidentical to theEinstein formula. Nevertheless, even far from equilibrium the effect offluctuations remains small and doesnot influencethe macroscopicbehavior,by virtue of relation (9.34),which extendsto the nonequilibriumsituation.

The conclusionthat the theory of fluctuations around nonequilibriumstates can be formulated\342\200\224at least for our simpleunimolecular model\342\200\224in

terms of thermodynamic functions, is intuitively very appealing.Indeed,the local equilibrium assumption, which has beenthe starting point for theextension of thermodynamics to nonequilibrium situations, suggeststhat

such a reduction shouldbe possible,at least within a certain well-definedrange of phenomena.

A Nonlinear Model

We now considerthe simplebimolecular scheme(Nicolisand Prigogine,1971;Nicolis,1972):

*'\302\273 X + M

A0.5*)

Thesystem{X}is again assumedto beopento large reservoirsof A, M,E, D.This time, however, inverse reaction rates are neglected,and the systemoperatesautomatically far from thermodynamic equilibrium.Onecan showthat by taking backreactions into account in the vicinity of equilibrium, themaster equation admits the Poissoniandistribution as long-time solution.In this subsection,however, we are interested exclusively in the non-equilibrium propertiesof the model.

The overall reaction is:2A > E + D

and the macroscopicrate equationsread,for an ideal mixture:

A Y

-r=klAM-k2X2 A0.55a)at

A, M = const


The systemadmits a singlesteady stateI \302\273,f\\l/2

A0.55b)

which is asymptoticallystablewith respectto arbitrary perturbations.In order to study the effect of fluctuations we introduce a probability

distribution P(A, M,X, D, E, t) and assume that Eq. A0.54) defines aMarkovian birth-and-death processin the spaceof thesestochasticvariables.We recall that, accordingto the discussionin Section10.2,this assumptionmight well be unrealistic.Nevertheless,wepursuehere our analysiswith theprimary aim of illustrating the basicideasand techniques of the stochasticmethod.

The next stepis to reducethe master equation A0.2)for P(A, M,X, D,E, t)to a closedequation for the reducedprobability distribution P(X,t) byinvoking the same physical assumptionsas in the precedingsubsection.We obtain in this way the following equation, which now definesa birth-and-death processin the subspaceof the singlestochasticvariable X:

= kxAMP(X -l,f)-kxAMP(X, t)at

y (X + l)(X+ 2)P(X+ 2, f)

-y(X - \\)XP(X,t) A0.56)

We switch to the generating function f(s,t) and find the equation [seealsoEq.A0.26)]

\342\200\224 = kxAM{s \342\200\224

\\)f + \342\200\224 A \342\200\224 s2)-r-j A0.57)

The nonlinearity of the coefficients,aswell as the fact that wehave a second-orderpartial differential equation for/is a consequenceof the bimolecular(i.e.,nonlinear) character of the reaction.

Equation A0.57)can besolvedexactlyat the steady state asfollows(Mazo,1975).Dividing through by A \342\200\224 s)and taking Eq.A0.55b)into account weobtain:

{l+s)dl-2X2f= 0 A0.58)ds2 \302\260'

This is a second-orderequation and requirestwo boundary conditions.Accordingto the definition A0.21)ofgeneratingfunctions,|s(| must not exceed


unity, in order to ensureconvergenceof the seriesexpressionfor F({s,},f).Thus, we seekhere for boundary conditionsat s = 1 and at s = \342\200\224 1.Oneof these is obvious and follows from the first relation A0.23):/(I)= 1.In orderto find the secondcondition weconsiderthe quantity

= I P(X)- I P(X)

On the other hand, one can seeby writing the master equation A0.56)successivelyfor dP(O)/dt,dP(\\)/dt (etc.)that there are equal probabilitiesfor

having an even orodd number of X particlesin the systemas X -> oo.We

concludethat /(-1)= 0.We may now introduce the new variable z = ^s + 1) and write the

equation for g(z) = f(s):

with

3@)= 0, 0A) = 1 A0.59)

The solution to Eq.A0.59)satisfying the boundary conditionsis (MorseandFeshbach,1953)

LM (ia60)

where Ix is the Besselfunction of imaginary argument. Usingthe followingproperty of Besselfunctions

TUz)+ v/v(z) = z/v_1(z)dz

one has:

<*>=7,DX0)

In a macroscopicsystemXo is a very large number. Thus, we may usein theabove relation the asymptoticexpansionsof 70 and /,.This yields:

A0.61)


This result showsthat the macroscopicdescriptionleading to Eq. A0.55)is valid up to terms of orderone.In order to compute the variance of thefluctuations one may evaluate the secondderivative of / from the exactsolution A0.60).However, it is just as well to considerthe first moment

equation A0.32)for the model.At the steady state one has:

<aiW>= o

or

k,AM - k2(X(X- 1)>= 0

Usingagain Eq.A0.55b),we find:

<X2>-X20= (Xy A0.62)

Hence,on substitution of Eq.A0.61)in Eq.A0.62):

EX2)= {X2}-= ^\302\260 + 0A) =^ + 0A) A0.63)

4 4

Clearly, Eq. A0.63) is incompatible with a Poissoniandistribution and,therefore,alsowith the Einstein-likeform A0.53).The factorf in front of<X>is model-dependent.The fluctuations in many other nonlinear modelsinvolving a single stochasticvariable have been tested (Nicolis,1972;Gardiner,McNeil,Walls and Matheson,1976;Van Kampen, 1976).Theanswer is a non-Poissoniandistribution with mean quadratic fluctuationsdependingon the individual kinetic steps.Thus, no general statement com-comparable to Eq.A0.53)seemsto bepossiblefor nonlinear systemsoperating farfrom equilibrium, provideda descriptionbasedon the birth-and-deathmasterequationsis adopted.We discussthis most important point further in thesequel.

10.6.SYSTEMSINVOLVING TWO STOCHASTICVARIABLES: THELOTKA-VOLTERRAMODEL

The next type of examplewe analyze refers to systemsinvolving more than

onevariable. The main difficulty here is that the master equations can nolonger be solvedexactly.Thus, in orderto analyze the effectof fluctuationswe must appealto the asymptotic methods reviewedin Section10.3.

The model we choosefor illustration of these techniques is the Lotka-Volterra model,discussedfrom the macroscopicviewpoint in Section8.2.Themotivation for this choiceis twofold.Firstly, the importanceof this model

10.6. Systems Involving TwoStochastic Variables: The Lotka-Volterra Model 265

in problems involving interacting populations is considerable(seealsoChapter18).And secondly,as wesaw in Section8.2,the steady state and thetime-periodicsolutionsof the rate equations for this model lackthe propertyof asymptotic stability. In this respectthe Lotka-Volterrasystem may beconsideredasprototype ofany systemin the vicinity ofan unstabletransition.

As in the precedingsection,weassumethat the chemicalsteps

A + X *' ' 2X

A0.64)

define a birth-and-death processin the number of particlesspace.In thereducedform wherea summation over the initial and final productvariables{A, B, E} is performed,the master equation A0.2)becomes*:

dP(X Y t),' = A(X - \\)P(X - \\,Y,t)-AXP(X, Y, t)

at

+ (X + \\)(Y- \\)P(X + 1,Y - 1,f) - XYP(X,Y, t)

+ B(Y + \\)P(X. Y + 1,f) - BYP(X,Y, t)

A0.65)

Forsimplicity, all reaction rates have beenset equal to unity.In the generating function representation

F(sx,sY,t)= t sxxsYYP(X,Y,t)Y.X =0

the master equation reads

dF d2F dF dF( + l)fa f) +^+\\)AB= (n + l)fa f) r +^+\\)Ar]B

Ct CQCt] CQ Or]

wherewe used,instead of sx,sY, the variables:

A0.66)

* = Sx~ {

A0.67)n = sY

- l

An important point to bestressedat the outset is that Eqs.A0.65)or A0.66)cannot admit factorizablesolutions.Even if one starts initially with a factor-ized probability distribution, the term containing (X + \\)(Y\342\200\224 1)(or the

* We here formulate the analysis of fluctuations for the chemical analog of the Lotka-Volterramodel.The results can beapplied straightforwardly to the population dynamical version of themodel, which is discussed in Chapter 18.


secondderivative in\302\243,

andr\\

in the generating function representation)intro-introduces correlationsbetweenX and Y. This point is at first sight reasonableinview ofthe X \342\200\224 Y couplingappearingin the chemicalschemeitself.Neverthe-Nevertheless this is surprisingin terms of both kinetic and thermodynamic theoryof nonequilibrium phenomena.The very basis of the local formulationofirreversiblephenomenaadoptedthroughout this monograph is the conceptof local Maxwellianequilibrium, which is assumedto prevail within smallvolume elements of the system.Now, a Maxwellianequilibrium in an idealsystem impliesnecessarilythe factorization of the probability distributionsof the individual particles!

The resolution of this difficulty amounts to realizing that the birth-and-death descriptionneglectsthe local aspectsof a phenomenon and dealsexclusivelywith fluctuations in largevolumes.Becauseof this, the individual

degreesof freedom,which locally are statistically independent,are lumpedat the benefitof the macrovariablessuch as X and Y that refer to the entiresystemand, therefore,neednot yield factorizabledistributions.

Bearing this point in mind we now investigatethe asymptotic solution ofEq.A0.66)by introducing the auxiliary variable i/>, Eq.A0.27),and the limitN -* oo.The equation for the generating function becomes:

1 d2i

Nd\302\243

<

We have set:

A = olN

B = $N A0.69)t = Nt

Clearlya, /? = 0A). In Eq.A0.68)the secondderivativeterm is multiplied by\\/N and, thus, may be consideredas a small quantity for N large.We seeksolutions of this equation in the form A0.28):

A0.70)

wherethe expansioncoefficientsare related to the momentsof the probabilitydistribution by Eq.A0.30).

10.6.Systems Involving Two Stochastic Variables: The Lotka-Volterra Model 267

Substituting into Eq. A0.68) and identifying equal powersof I, and rj

we obtain a nonlinear system of differential equationsfor the expansioncoefficients:

da^ b12~ = aal-a1a2--Yd^bf A0.71)

and, using a matrix notation for compactness:'20Mi\\

\\2ala2j/2(<x.-a2) -2a!

A0.72)

Hadweneglectedthe term containing l/N in Eq.A0.68),the first two relationswould bea closedsetofequationsindependentof the variancebi2.Moreover,

the matrix multiplying I b12I in Eq. A0.72)would simplify. On the other

W/hand, higher orderterms in the expansionA0.70)of

i/>would have introduced

new contributions in the equationsfor the variance, reflectingthe influenceof triple or higher moments.Thesecontributions would all be multiplied byl/N in the equationsand would, therefore,benegligiblein the limit N \342\200\224\342\226\272 oc.

Let us carry out further the consequencesof the limit N -> oo bydroppingall terms proportionalto l/N.The first two equationsbecomethenidentical to the macroscopicrate equationsfor the model [seeEq. (8.2)].Theyadmit the steady-state solution

ax = p, a2 = a A0.73)

Substituting into the equationsfor the varianceswe find the simplified form:

0 -2^ 0\\/ftnN

a 0-\302\243)

U12I A0.74)0 2a 0/U,


The important point is now that these equations do not admit a time-independentsolution.To seethis, we add the equations for bn and b22.We obtain:

jt (\302\253ft,, + pb22)= 2ajS(a+ p) A0.75a)

On the other hand, by differentiating oncemore the equation for bl2we obtain:

d2b12_ db^ db22dx2

~adx

P dx

By eliminating the derivativesofbl x,b22from the first and the third equationsA0.74)weobtain a singleclosedequation for b12:

d2b12 + 4ajSb12= 2aj?(a- ft A0.75b)dx

Tosolve the differential equationswe must specify the initial conditions.Assume that at f = 0 the systemwas describedby a factorizable Poissoniandistribution for X and Y. This means that:

(SXSY),=o = 0 A0.76a)

Accordingto Eq.A0.30)theseconditionsimply for the bu values:

bu{t = 0) = 0 A0.76b)

The solution of Eq.A0.75)satisfying these initial conditionsis

o*u+ Pb22=2a\302\243(a

+ P)x A0.77a)

and

bi2 = a^[1- cosB(ajSI/2T)]- liapI12sinB(ajSI/2-r) A0.77b)

We seethat, despitethe initial Poissoniandistribution and the fact that,

macroscopicallyspeaking,the system is at a steady state, the variances<<5A\022> and <<5V2>increasein time, deviate immediately from the Poissonianregime,and cannot reach a new steady state.Stochastically,therefore, the

steady state A0.73) is meaningless,even in the limit of small fluctuationscorrespondingto the truncation performedon the moment equationsearlierin this sectionby neglecting the 1/JV terms. The system exhibits abnormalfluctuations that increaselinearly in time with a periodic\"background

10.6. Systems Involving Two Stochastic Variables: The Lotka-Volterra Model 269

noise\"whosefrequency2(a/?I/2is the doubleof the frequencyof the macro-macroscopic motion. Eventually, thesefluctuations alter the orderof magnitude ofthe \\/N terms that couldno more beneglectedin the moment equations.As

a result, the averagevalues are driven by the fluctuations to a time-dependentregime far from the steady state (Nicolisand Prigogine, 1971).This is alsoconfirmed by a numerical analysis of the completemaster equation A0.65)(Malek-Mansour,1973;Langlois,Van Nypelseer,and Walker, 1976).Thelatter suggests(seeFig.10.1)that the trajectoriesleave systematically the

steady-statesolution and tend, after performing,a few spirals,either to com-complete

extinction or to a state where Y is absent(seehowever Turner, 1977).Analytically also,onecan show(Reddy,1975)that the master equation can-cannot admit a steady-state solution centeredaround nontrivial values of XandK

This \"chaotic\"behavior is strongly reminiscent of the well-knownphe-phenomenon of turbulence in fluid dynamics,where the fluctuations alsoplay a

55

54

53

52

51

50

49

48

47

46

45

44

43

42

I T T

0.03200.0348 4 0.0256

0.04480.0512

0.0576

0.0640^

\\

o

0.0704

0.0768

0.0832\342\200\242-*

I

0.0912

0.0960

\\ Z \342\200\2420.0192

_Deterministic_steady state

I

o.oooo-Time

44 45 46 47 48 49 50 51 52 53 54 55

Figure 10.1.Phase-spacetrajectory for stochastic averages of X and Y in Lotka-Volterramodel.Initially the system is Poissonian distributed around deterministic steady state. Arrowsindicate values of time corresponding to different points along trajectory.


decisiveroleby altering qualitatively the predictionof the macroscopicanalysis.We refer to this situation as generalized turbulence (NicolisandPrigogine,1971).

The possibilityof spontaneousdeviations from the regime of Poissonianfluctuations provides a striking illustration of breakdownof the lawsof large numbers discussedin Section9.2.Although we return to this pointin Chapters11and 12,we may already point out that this entirely newsituation is a consequenceof the coupling inducedby the chemicalreactions,as a result of which the transitions undergone by the stochasticvariablesarenot statistically independent events, even in the limit of a large systemN -\302\273 oo.

These results may well be comparedwith the analysis of the momentequationsperformed in Section10.4.Indeed,the structure of the system ofequationsA0.72)for bti is identicalto the structureofthe generalsystemA0.37)for the variances.On the other hand, the matrix K appearingin this equationis, in fact, identical to the matrix of coefficientsof the linearized system ofmacroscopicequations[cf. Eq.F.6)].Let A^, u^ be the eigenvaluesand the(orthogonalized)eigenvectorsof K:

where the superscriptj in the definition of the scalarproductrefers to thechemical speciesj.We expand5X and <5X 5X> in Eq.A0.37) in terms ofu^.We obtain:

and

<5X 5X> =\302\243 <a/iav>u/iuv s

\302\243 (T^u^u,.

Notethat the indices[i,v are not to beconfusedwith the indicesi,jidentifyingchemical speciesin the equationsof Section10.4.Performing the matrix

product K \342\226\240<8X 5X> and utilizing the orthogonality of um we obtain, atthe steady state, the following form for Eq.A0.37):

10.6.Systems Involving Two Stochastic Variables: The Lotka-Volerra Model 271

Thus solvingfor a^.we find:

and

<5X 5X> = -X <fl2.,v>u,uv y-^j A0.78)

As we saw repeatedly in Chapters6 and 7, at a state of marginal stabilitythe matrix K has onezero eigenvalueor a pair of eigenvalueswith vanishingrealpart.Let the eigenvaluesbereal, and supposek^ is the criticaleigenvaluevanishing at the point of marginal stability. Comparingwith Eq.A0.78)weconcludethat the variancematrix containsat leastonediverging contribution,namely, the onecorrespondingto the term \\i = v in the sum. This reflectsthe fact found earlier in this section,that the evolution equationsfor thevariances admit no physically reasonablesteady state (Mazo,1970).Theanalysiscan berepeatedif the eigenvalueis complex.

If the referencestate is an equilibrium or a near-equilibriumsteady state,then for a suitable choiceof variables, the matrix K is symmetric and any

macroscopicperturbation decaysmonotonously to this state. However,in a general nonequilibrium situation the symmetry of K is not guaranteed.The regressionto the referencestate can then either be oscillatory or notoccur at all. The argument just developedshowsthat at the same time thebehaviorof the variance matrix tendsto bepathological.

Actually, whenever the macroscopicrate equationshave an oscillatorymode, the fluctuations are themselvesrotating, as one can seeby a closeranalysis of the equation for the variances. Tomita and Tomita A974)(seealsoTomita, Ohta, and Tomita 1974)suggestedthat the rotation of fluctua-fluctuations can be characterized by the angular momentum-likequantity

Mi} = (SXfSXj)- (SXtdXj} A0.79)

where the averages are evaluated at the referencestate, and SX is formallydefinedby the right-hand sidesof the macroscopicrate equationsexpandedaround the referencestate.Theyshow that M is related to the variancematrix

by

M = (K \342\200\242

<r)r - (K \342\200\242

<t) A0.80)

The physical meaning of these relations can be seenbetter by switching tonew setsof variablesdisplaying the \"radial\" and the \"angular\" parts of thefluctuations (Kitahara, 1974).

272 Binh-and-Death Description ofFluctuations


As pointedout in Section10.6,a system in the vicinity of an instability

leading to a dissipative structure behaves in much the same way as theLotka-Volterramodel.Firstly, the linearizedrate equationsacquire a modewith a vanishing real part of the correspondingeigenvalueof the coefficientmatrix K. This results in an increasein the life time of this mode,which

reminds strongly the phenomenon of \"criticalslowingdown\" familiar from

equilibrium phasetransitions (Stanley,1971).Secondly,the evolution of thevariance matrix is alsodominated by this long-living mode.This results in

abnormal fluctuations that increasein time and systematically drive theprobability distribution away from the Poissonianregime.

Theseconjecturescan be verified explicitly on examplesinvolving in-instabilities leading to spatial patterns or to limit cycles(Nicolis,Malek-Mansour,Kitahara, and Van Nypelseer,1974;Nicolis,Malek-Mansour,Van Nypelseer,and Kitahara, 1976;Nicolis,1974a)and to multiple steadystates (McNeiland Walls, 1974;Matheson, Walls, and Gardiner, 1975;Janssen,1974;J.S.Turner, 1975).Nevertheless,aswepointed out repeatedly,the analysisof instabilitiesbasedon the birth-and-death master equations issubjectto seriouslimitations;hencewe discussthe problemof instabilitiesin

further detail in Chapters11and 12,after having developeda basisfor a localdescriptionof fluctuations in nonequilibrium systems.

Chapter11

Effectof Diffusion:Phase-spaceDescriptionand MultivariateMasterEquation

11.1.NECESSITYFOR A LOCALDESCRIPTIONOF FLUCTUATIONS

Let us try to deducesomemore quantitative conclusionsfrom the argumentsdevelopedin Section10.2on the limitations of the birth-and-death formalism.

To understand the onset of a chemicalinstability* we analyzethe behaviorof fluctuations in macroscopicsystemsinvolving a largenumber ofparticles,N -* oo in a macroscopicvolume, V -\302\273 oo (N/V

\342\200\224

finite) and subjecttomacroscopicconstraints.We want, moreover, to be able to handle smalland intermediate fluctuations, which are expectedto be the most efficientones in driving the system away from the unstable state. Supposefor amoment that, in spiteof the fluctuations, the systemremains homogeneousin space.Below the critical point of instability, the probability distributiontakesa Gaussianform:

wherex is the intensivevariablex = (X/V) correspondingto the number ofparticlesX of a certain constituent. The variance

cr\302\253x\302\273 dependson theaverage of x and is independent of the size of the system. From Eq.A1.1)we seethat

dxaz V~m A1.2)

Realizing that V standshere for the size of the entire system, which is ofmacroscopicdimensions,we concludethat a fluctuation obeying Eq.A1.2)* The same arguments should hold for fluctuations in fluid dynamics and, indeed, for all

systems subject to macroscopicconstraints.

273

274 Effect ofDiffusion: Phase-spaceDescription and Multivariate Master Equation

is exceedingly small and cannot possiblyinfluence the behavior of thesystem.

Conversely,if a fluctuation

ExocO(l) A1.3)is to occurwith a finite probability, it is necessarilylocal, in other words,refersto a smallpart of the big systemhaving a volume A V, such that (A V)~112is of 0A). But oneshouldhave to account for the couplingbetweenthis smallsubvolumeand the remaining part of the big system,arisingfrom, for instancethe exchange of matter (through diffusion) and energy (through heat con-conduction) acrossthe surface separating AV and V \342\200\224 AV. In different terms,becauseof the density fluctuations, the system becomeslocally inhomo-geneousand the descriptionbasedon the birth-and-death master equationA0.2)breaksdown.

This argument alsoshowsthat if one wants to maintain the picture of ahomogeneouslarge system fluctuating appreciablyin a coherent way, onemust introduce large-scalefluctuations of small probability. In linearsystemsor in systemscloseto equilibrium both ways of treating fluctuationsare equivalent. This remarkable\"similarity\" property, by virtue of which

questionsof size\342\200\224and thus of homogeneity\342\200\224do not matter, is a directconsequenceof the absenceof correlationsbetweenthe individual stochasticvariables.*Thesefeatures are no longer found in nonlinear, nonequilibriumsystems.

11.2.PHASE-SPACEDESCRIPTIONOF FLUCTUATIONS

Having recognizedthe necessityof a local descriptionof fluctuations far|from equilibrium we first adopt a phase-spacedescriptionwhere this local|character is manifestly preserved.In other words,we chooseas stochastic*variables the number of particles,Xa, of speciesX (for simplicity, we deal;!in this sectionwith a singlevariable intermediate) in a phase-spacevolume\"

AFj, around the state a.This state includesthe positioncoordinater, the:momentum p and all the internal degreesof freedom {e,}that may play a i

rolein the problem.For simplicity, we first argue as if a could only take \\

discretevalues,although the passageto the continuum presentsno special'jjdifficulty. In terms of the velocity-distributionfunction fa of kinetic theory

!j

(Chapman and Cowling,1952),Xx can begiven by: ]

Xx = /.Ar Ap{A\302\243i} (H.4)j* We recall that we deal here with \"ideal\" mixtures where the intermolecular interactions do 'not modify explicitly the thermodynamic relations between state variables.

11.2.Phase-spaceDescription ofFluctuations

f(v)

275

Figure 11.1.Instantaneous velocity distribution f{v).

wherefx is the instantaneous valueof the distribution in velocityand positionspace(seeFig.11.1).

A random fluctuation of the macrovariable X is expressedas

3X = A1.5)

These fluctuations perturb continuously the reference macroscopicstatevariables X,,. The latter obey a kinetic equation of the Boltzmann type(Prigogine,1949a;Rossand Mazur, 1961;Eu, 1974),which we write in the

compactform:

dtA1.6)

Here,Jr describesthe effectof reactive collisions.The elasticcollisionterm

(dXJdt)elmay be set approximately to minus the contribution arising fromthe freemotion of the particles,{dXJdt)now, if onedealswith systemsnear alocalequilibrium regime.

We assumenow that in the spaceof the variables {Xx} the fluctuationsdefine a Markovian stochasticprocessof the birth-and-death type. Thevalidity of this conjecture for the Boltzmann equation has beenjustifiedby SiegertA952)and KacA959).Intuitively, this appearsto bequite reason-reasonable, as one now dealswith small coupledsystems for which the objectionraisedin Sections10.2and 11.1do not apply.The correspondinggeneralizedmaster equation has the form

dpgxj,t)dt A({Xfi} },t) A1.7)


where the transition probabilitiesper unit time A must satisfy the samerelations as in Eqs.(9.8)to (9.11).In particular,

l->{Xa})= 0 A1.8a)

A({Xf} -\302\273 {Xx})>0 ({X,,}* {Xa})

->{Xa})\302\2430 A1.8b)

Moreover, on averaging Eq. A1.7)over Xa one shouldobtain, by makingassumptionssimilar to those in Section10.4,the Boltzmann-likeequationA1.6)as a first approximation:

+ terms containing ((SXX SXpy (etc.) A1.9)A final averaging over a has to producethe macroscopicrate equationsofchemicalkinetics.

11.3.A SIMPLEMODEL

In this sectionwe analyze the solution of the phase-spacemaster equationson a simplebut representativemodel and comparewith the results basedonthe birth-and-death descriptionof fluctuations.

We again considerthe nonlinear model A0.54).Contrary to the birth-and-death analysis,where the only stochasticvariable was the total numberof particles,within the framework of phasespacedescriptionwe have toderive a master equation for the probability function P({XX},t).

There are two basically different processespresent. Firstly, there areelasticcollisions,which do not change the total number of particlesofconstituent X but tend to redistribute the initial Xx values (seeFig.11.1)accordingto the Maxwell-Boltzmann law (seeFig.11.2).Secondly,there arereactive collisions,which change the number of particlesof X presentand

modify the velocitydistribution. A central problemis,therefore,to accountfor the combinedeffectsof thesetwo processes.Severalcasesare conceivable,dependingon the comparative strengths of the elastic and reactive effects.We can delineate three different limiting regions.

ReactiveCollisionsAre Dominant

This correspondsto such problemsas strongly exothermic reactions,orreactions in a rarefied mixture. The relevant equation is that of a stochasticprocessin phaseand number space,but onewhere the velocitydistribution

11.3.A Simple Model

f(v)

277

Figure 11.2.Maxwellian distribution for stochastic variable X,, in the various regions ofvelocity space.

is in no way centeredon a Maxwelliandistribution. Essentially,the variousvelocity regions,and hence also the populationsin the various internalstates and positions,are coupledvery loosely.We are not interestedin this

regime,as it is remote from the range of validity of the local descriptionadoptedthroughout this monograph.

ElasticCollisionsAre Significant

In this casethe stochasticprocessis centeredaround Maxwellianequilibrium.The Boltzmann-likeequation A1.6)appropriatefor A0.54)is:

UA* _ y

with

-X S^klXaXj+

j,Mk = const

A1.10)

T,j_kl and Su-,k,are the transition probabilitiesfor scattering between twomoleculesin states(i,j)into two moleculesin states(k, /) for the reactionscorrespondingto the two stepsin the sequenceA0.54).The systemis assumed


to be macroscopicallyhomogeneousand to behave as a dilute mixture. Inthe sequelit is convenient to define the following coefficients;

kl

We may also note that in a scattering processone has the followingconditions:

*;,\342\226\240.\342\200\236

= o; sH.u= o(]U2)

Theseconditionsexpressthat two particlesin exactly the samestate cannotundergo a collision.

We may now write the master equation correspondingto Eq. A1.10),noting that the reaction sequenceA0.54)has to be interpreted as

x +x -^ f +n AU3)

Using the form of the birth-and-death transition probabilitiesfor thevariables {X^}we have

, t)^ W{X _ u {xrf) _

dt\342\226\240<?\"

^ ii r. \"-'a/fLV'1* ' ll\\\"-jl \"^ 1)* (^i

) +() A1.14)dt Jei \\ dt yrlow

where the prime indicatesthoseparticleswhosestate has not beenaffectedby the reactivecollisions,and the factor 1/2accountsfor the indistinguish-ability of the collidingX-molecules.

In principleonehas to solvethis equation by taking into account explicitlythe effectof elasticcollisions.Herewe adopt an alternative procedure.Wefirst recall that elasticeffectsserve essentially to establish,approximately,a Maxwelldistribution;therefore,we neglect their explicit influenceon themaster equation (Nicolisand Prigogine, 1971;Nicolis,1972).Also, the flow

11.3.A Simple Model 279

term servesto establisha uniform distribution and is henceforthnot includedexplicitly.Equation A1.14)becomes:

dP({x},0dt i

^ i},t)

- XlX'fiP(Xl,X%, {X'e},r)] A1.15)The superscriptedenotesthat the variousmolecularpopulationsare centeredaround a Maxwellian(seeFig.11.2).Noticethat small fluctuations aroundthe Maxwellianare taking placecontinuously, and are in fact essentialtokeep all of the stochasticvariables {X}independent.This equation isdiscussedin detail in Section11.4.ElasticCollisionsAre Dominant

In this limit thermal equilibrium is always rigorously maintained, that is,one neglectsfluctuations around the Maxwellian distribution. Thus, achemical reaction can only provokechangesbetween spatially uniform

equilibrium stateswith different total numbers of particles.As an example,the first step of the sequenceA1.13)producesone particle of X in somevelocitystate vh but the velocitydistribution immediatelyrelaxesfrom

Xv2 exp(-mv2/2kBT)Z~o

to

(X + \\)v2 exp(-mv2/2kBT)__

where Zo is a normalization constant.Thus, the wholeprocessis insensitiveto which velocitystate in fact receivedthe particle.As a result, the stochasticprocessin phasespacedegeneratesto a processinvolving a singlevariable,the number of particlesof constituent X (Kuramoto, 1973;Nitzan and Ross,1974;Nicolis,Allen, and Van Nypelseer, 1974).The master equationbecomes:

u u '= I UP({x+ i}M, t) - P({xM},0]dt

\\f(X + 2)MP({X+ 2}M,r)

- \\XU(X- \\)MP({X}M, f)] A1.16)


where, for example,

(X + 1) Av2,---> A1.17a)z J

We now definethe quantities

kxAM = ^Ta

L ZKB l Jk2 =!S.,4 vl\302\273l exP -^f {v\302\260

+ vfU Ay* Avp A U7b)a/i ^0 L ZKB l J

Then, Eq.A1.16)becomesidentical to the birth-and-death master equationA0.56).This implies* the validity of the non-Poissonianmean-squaredeviation A0.63).

We seethat the birth-and-death result is obtainedby \"freezing\" the velocitydistribution to Maxwellianequilibrium. We expectthis type of behavior tobe representativeof large-scalefluctuations,which evolvemuch more slowlythan the processesrelated to the relaxation of the velocitydistribution. Incontrast, local fluctuations evolvetoo rapidly for this type of behavior to beapplicable.In a sense,therefore,the usual birth-and-death descriptionis notrepresentativeof the dynamicsofa chemicalsystem,but only of the evolutionof the large-scalefluctuations. The latter are expectedto be exceptionalevents,as long as the system is far from an instability.

11.4.APPROXIMATE SOLUTIONOF MASTER EQUATION

We now placeourselves in the secondcaseof Section11.3(where elasticcollisionsare significant), which is expectedto correspondto the behaviorofsmall local fluctuations.Ourpurposeisto analyzethe solutionsof the masterequation A1.15)in a certain well-definedlimit consistentwith the premisesinvolved in the phase-spacedescription.

The difficulty in solvingthe phase-Spacemaster equations arisesfrom theinfinite number of coupledterms contained in the sums over stateson theright-hand side.On the other hand, this very complexityenablesus to expandsystematicallysuch equationsfor small fluctuations; in this casethe limitof small deviations from the macroscopicdistribution function Xa makes

* Note the existence ofvolume factors appearing implicitly in Eq.C.16)through the normaliza-normalization constant Zo.Thesefactors ensure the correct form of the macroscopicrate equations ob-obtained on averaging over the microscopic states.

11.4.Approximate Solution ofMaster Equation 281

sense,as we deal with the evolution of the individual degreesof freedom.Setting

Xk = Xk + 3Xk = Xk + Exk,E< 1 A1.18)

we may expandboth P and the coefficientsin Eq. A1.15)following themethods outlined in the first part of Section10.3.Keepingdominant termsin \302\243 we obtain (Nicolisand Prigogine, 1971;Nicolis,1972):

8P({x},;dt

- dxkP + Xk-~x,P

- dP -

wherewehave taken into account the fact that Xk satisfies,to this order,theBoltzmann-likeequation A1.10).

This simplified equation is now integrated over all x values but one.Defining the reducedprobability functions

Pu...,s(xi,...,xs,t)=jdxs+1\342\226\240\342\226\240\342\226\240P({x},t) A1.20)

and taking into account relation A1.12)we obtain:

8Pi(xa,t) 1 [V 82Pi(xa,t) y F y d2PAxx, Ql

A1.21)

We seethat Eq.A1.19)givesriseto a hierarchy of equations relating dPJdtto Pi2 and, similarly, 3P12/dtto Pi23 (etc.).Supposenow that wechooseaninitial condition correspondingto factorizedreducedprobability functions:

sPi S({XJ,0) = []pi(xh0) s = finite A1.22a)

i= 1

282 Effect ofDiffusion: Phase-spaceDescription and Multhariate Master Equation

Becauseof the dominance of the flow terms over the reactivecollisionterms,we exceptthe factorization condition to be maintained to a goodapproxi-approximation, as long as we dealwith small fluctuations.This is also supportedby the fact that the operatorsin Eq.A1.21)have the same structure as theBoltzmann operator in that they connect two states without spatial cor-correlations between particles.We conclude,therefore, that within terms ofordere:

Pi...J{X.},t)=f[P,(*,,r) A1.22b);= i

This, together with definition A1.18)of the fluctuations, implies that thelast term in A1.21)vanishes. The coefficients of the second derivatived2Pl/dxlof this equation may be further simplifiedby taking into accountagain that Xx satisfies the Boltzmann-like equation A1.10).One finally

obtains:

(?sx) xp*xl)+ V sxx) ^- A L23)

This equation has the same structure as the linearized Fokker-Planckequation A0.14).At the steady state it reducesto the form

as the transition probabilities\302\243,- SajXjcancel in Eq.A1.23).By integration

we obtain

X \022 expf- pj A1.24a)

This givesriseto a variance compatiblewith a Poissoniandistribution (seealsoKlimontovich, 1958):

<xl>= Xa A1.24b)

Similarly, for the higher orderdistribution function:

Now,accordingto the expansionofentropycarriedout repeatedlyin Chapter4 and also in Sections10.5and 9.3,the exponent is just the second-orderexcessentropy evaluated around the referencestate.Thus

P. s({*})ocexp(^) A1.25)

11.5.Molecular Dynamics Studies ofFluctuations 283

We seethat in the nonlinear,nonequilibriummodeldiscussedin this section,small fluctuations behave in agreement with the generalizedEinsteinformula provided the system remains near a local equilibrium regime andcan be treated as a dilute mixture. Moreover, the correlationsbetweenfluctuations in different regionsof spaceand for different velocitiesvanish.This is quite different from the result basedon the birth-and-death masterequations,which predicta non-Poissonianvariance of the fluctuations in

large volumes(seeSection11.6).

11.5.MOLECULARDYNAMICS STUDIESOF FLUCTUATIONS

A computer simulation of the dynamicsof a reacting mixture is,perforce,alocal descriptionof the reactions.It should,therefore, be able to illustratethe difference between phase-spaceand birth-and-death type theories.Moreover,in a machinecalculation,the fluctuations are predominantlysmallin extent, and hencethe resultsof the calculation decidedirectly on the

question of Poissonianbehavior.Portnow A974,1975)conductedcomputer \"experiments\"for the reaction

schemeA1.13),with chemical speciesrepresentedas hard spheres(withAr mass and radius)and the experimentscarriedout at T = 273\302\260K andat a density of p = 0.0017837g/cm3.Collisionswith the walls are specularreflections.The particlesare given identitiescorrespondingto the chemicalspeciesinvolved.Whenever two particlescollide,their identitiesare modifiedin accordancewith the reaction scheme.Thus, all collisionsbetweenreactingspeciesare reactive collisions;there is no activation energy for any of thereactions,and all rate constantsare equal.Sincemany reactionsproceedwithout an activation energy (e.g.,ion-moleculereactions),the model isnot without reality.

As the reaction proceedsthe number of particlesof X is determined atdifferent times. Sufficient time is allowed to elapsebetween measurementsfor memory effects to be unimportant. Moreover,it is verified that the

velocity distribution remains closeto the Maxwellian distribution. Themean and the variance of X are deducedand a test for goodnessof fit to aPoissoniandistribution is made on the observedparticle number distri-distributions.

For model A1.13),25 X particles,25 M particles,and 25 A particlesweretaken initially. The experiment ran for 3.146x 10~8s,and there were1941reactive and 4067 unreactive collisions.After 30 observations themeanand the variancewere,respectively,25.2and 23.3.The expectedvaluesfor both quantities, if the distribution were to be Poissonian,were 24.5,whereas if the birth-and-death descriptionwere valid, the variance should


have been f x B4.5)= 18.3.Thus, the molecular dynamics results arecompellingevidence for a Poissoniandescription,and they underscoretheimportance ofa localdescriptionof fluctuations in nonlinear,nonequilibriumsystems.

Computerexperimentsare currently in progressto investigatethe growthof fluctuations in the vicinity of instabilities. Someresults have alreadybeenreportedby Ortoleva and RossA974b).

11.6.DISCUSSION

From the procedurefollowedin Section11.4it shouldalready be clear that

the Einstein-likeresult A1.25)concerningthe behavior of small local fluctua-fluctuations is quite general and is,in fact,independentof the chemicalmechanism.This can be verified explicitly,even for systemslacking asymptotic stabilitysuch as the Lotka-Volterra*ortrimolecularmodels(Nicolisand Prigogine,1971;Nicolis,1972).We conclude,therefore, that a system in the vicinityof a point of marginal stability cannot evolve by a mechanism of smalllocal fluctuations.

On the other hand, experimentshowsthat there exist systemsthat presentspontaneouslyinstabilitiesleadingto dissipativestructures.Forsuch systemsfluctuations ofa certain type should,therefore,presentan abnormal behaviorsimilarto that predictedby the birth-and-death description,assummarizedin

Section10.6.The results of Section11.3suggestthat this should be thecaseof large-scalefluctuations\342\200\224in a sensebetter defined below\342\200\224that canonly \"see\"the averagestate of the systemand can, therefore,beamplifiedif

the macroscopicequations ofevolution predicta point of marginal stability.We concludethat the onset of a self-organizationprocessesimplies that

there existsa volume element within the systemof dimensionsmuch largerthan the characteristicmolecular dimensionsbut smaller than the totalvolumeof the system,within which fluctuations behave coherently,and that

they add up to a sizableresult and subsequently modify the macroscopicbehavior.

* We may recall, from the analysis in Section 8.2,that in the Lotka-Volterra model (&2SHisrelated to the constant of motion t or, more precisely, to its expansion for small deviationsaround the steady state. Thus, relation A0.25)becomes:

Pacexpffii') A1.25a)

where fl is a parameter. In this limit our result agrees with the statistical analysis of Kernpr

A957, 1959)based on the derivation of a \" Liouville-like\" equation for conservative systemsinvolving a large but even number of variables. For finite fluctuations Kerner's results differ

qualitatively from the predictions based on the stochastic master equations. For recent critical

surveys ofKerner's approach, we refer to Nicolis A972) and May A973).

11.7.Reduction to a Multivariate Master Equation in Concentration Space 285

The passagethrough thermal fluctuations to this macroscopiccoherencevolume may bedescribedasa specificallynonequilibriumnucleation phenom-phenomenon of a new kind. In principle,by solvingthe phase-spacemasterequationsfor finite fluctuations one should be able to display this phenomenonexplicitly.The molecular dynamics methods shouldalsoprovide a meansfor calculating the critical dimensionsof the coherencevolume.

We now adopt a different strategy.Becauseof the complexityof the phase-spacedescription,we try to defineconditionsunder which the phase-spaceequations can reduceto simpler ones,by retaining only those variablesdirectlyaccessibleby macroscopicmeasurement.This enablesus to estimatethe dimensionsof the coherencevolume and to relate it to the system'sparameters.

11.7.REDUCTIONTO A MULTIVARIATE MASTER EQUATIONIN

CONCENTRATIONSPACE

A macroscopicsystemcan alwaysbeviewedas a set of cellscommunicatingbetween themselves via a transport of energy and matter. Let X be thecharacteristic length of each cell,and for simplicity let us assumethat wedeal with transport along a single spatial dimension.We want to set up astochasticdescriptionwherein the variables are the numbers of particleswithin the various cells,independently of their momenta or their internalstates.This will constitute an \"intermediate\" level between the birth anddeath description,where all cellsare lumped into one,and the phase-spacedescriptionwhere the cellsare of microscopicdimensions.We imposeon X

the following conditions:

In eachcellthe averagenumber ofparticlesis significantly larger than one.This implies that X is at least of orderof ctr = lr, where c is the thermal

speedand tr is the relaxation time of the system. Hereafter, the length

lr = ctr is referred to as meanfreepath.

The individual cellsare not experiencingthe macroscopicinhomogeneitiesthat may appearin the system.This implies that X <? tchc, where fch is thetime betweentwo successivereactivecollisions.Thisenablesus to go,in thesequel,to the limit of a continuous system in a macroscopicsense,eventhough X cannot be strictly zero.Combining theseconditionswe find

that the descriptionwe attempt here breaksdown unlessrch > tr. Thisis the caseof all the chemicalreactionswhich can be treated by the mac-macroscopic balanceequations usedthroughout this monograph.


In orderto advance from the phase-spacedescriptionto the reduceddescriptionin terms of cells,we may start from an equation of the typeA1.15)and sum over all values of the occupationnumbers {X,}within theindividual cells.We definethe multivariate probability

p{X1,...,Xn,t)= X PHX,},t) A1.26)jX,J:(Ecell

where Xt now denote the numbers of particleswithin the cells1,2,...,n,

irrespectiveof their velocities,and the operationincluded in the sum has the

following meaning:

I =\342\226\240 I

{.Yj}: tecell {X{r, p,...)}such that: -x < px. py, px < + oc rx, ry, rz: within cell 1,....nIn other words,we sum over all occupationnumbers in velocityspaceandover those occupationnumbers in ordinary space whoseCartesianco-coordinates rx, ry, rz are includedwithin oneof the cells1 to n. The generaliza-generalization to an arbitrary number of reacting chemicalsis straightforward. Byperforming the operation indicated in Eq.A1.26)on the phase-spacemasterequation, the left-hand side gives rise straightforwardly to a term dp/dt.In the righ-hand side we analyze, successively,the structure of the flow

and reactive terms.

Flow Terms

Originally,theseterms describethe effectof the freeflight ofparticlesbetweenelasticcollisionson the probability function P. In a dilute mixture, andprovided ?.satisfiesthe two conditionsenunciated at the beginning of this

section,this term gives,on summation over X(r, p,...),an additive contri-contribution over cells.Fora given cell,this contribution expressesthe frequencyof changeof the cell'scontent in X dueto freemotion. It is well known (Wax,1954;Kac, 1959)that this motion has all the characteristicsof a randomwalk, provided the state of the system doesnot deviate significantly fromthe Maxwellian.Hence,we may write:

Ft =Yj fl\302\260w term

{Xi}: Ic cell,.

x p(Xl,...,Xi-i,Xi-l,Xi+1+ 1,...)+ w(X( - \\,Xi_l+ 1 -+Xi,X,_1)x p(Xl,...,Xi.l+ l,X,-\\,XI+u...)- w(X( -\302\273 Xi - \\)p(Xu ...,*,-_!,X,,Xi+l,...) A1.27)

11.7.Reduction to a Multhariate Master Equation in Concentration SpaceX X X X X

287

i-1 ;+ 1

Figure 11.3.Particle transport between adjacent cells,i, in configuration space.X stands forcharacteristic length of the cells, and dotted line indicates axis along which transport takes

place.

In orderto calculate the transition probabilitiesper unit time, w, werecallthat:

w(Xf -> Xt- 1)= 2 Cdu (dl.xvfx\\i,v, t) A1.28a)

whereZx is the boundary surfacebetween two cells,and/ is the momentumdistribution function in the ith cell,such that

dv fx\\i,-v,r) = -iX, A1.28b)

v is the component of the velocity along the direction of transport (seeFig.11.3).

Invoking onceagain the conditionsimposedon the dimension k we takef*(i) to be nearly Maxwellian,but allow for the number of particlesX tofluctuate and vary from cell to cell.*In a sense,contrary to the birth-and-death description,we assume that fluctuations remain in Maxwellianequilibrium in velocity spacebut allow for spontaneousspatial inhomog-eneities causedby the random thermal motion of the particles.RelationA1.28a)becomes

w(Xt t- 1) = 2 f dv \\dJ.xv~Xi<My, t)

Jo J *\342\226\240

where/ has beennormalized within each cell,which is taken to bea cubeofsideX. Finally, invoking homogeneitywithin each cellwe obtain:

{ -\\)= 2dXtwhere

d = - dv vo

A1.29a)

A1.29b)

* In general, the internal energy and the momentum ofthe center ofmassofcelli alsofluctuate.Tosimplify the analysis we consider here exclusively the problem offluctuations ofcomposition.


By construction,d is independentof the labelof the cell.Moreover,because4>

is nearly Maxwellian,the kinetictheoryofgasesprovidesus with the relation*(Chapman and Cowling,1952)

Joa a. Davvcp =* \342\200\224

K

where D is Fick'sdiffusion coefficientand lr the mean free path. Thus

d

Now, k itself is of the orderof a few mean free paths.Moreover, k = (l/n),where / is the total length of the systemand n the number of cells.Thus

d D^--2= finite A1.30)

Similar expressionshold for the other transition probabilitiesappearingin Eq.A1.27).

ReactiveTerms

Coming now to the reactive terms we obtain, on summation of a masterequation of the type A1.15)over X(r, p,...),an expressionthat in the caseof a bimolecular reaction is of the following form:

*\342\226\240\342\226\240= I T.SafiXaXllP{Xa,Xfi,{X'},t) A1.31)tXi}:l\302\243

celli xp

The difficulty with this term is that the summation refers to a contributionthat is nonlinear in the occupation-number distributions.Moreover,thesedistributions appear explicitly within the probability function P. Again,however,by invoking our assumptionson the sizeof the cellswe expectthat

* Kinetic theory ofgasesisinvoked here on the basis ofour assumption that the reacting mixture

is ideal. Naturally, Z>, lr, and soon arevery complicated functions of the system's parameters if

the reaction takes placein a dense solvent.

11.8.Multhariate Master Equation in a Model System 289

Xa, Xp can beapproximatedby a Maxwellianwith a fluctuating number ofparticles.Taking definition A1.17b)into account we then have:

Rt = k2 I I X(ri)X(r2)IX,}: (rx,ry, rz) within cell i r,.T2

x X P(Xa,Xfi,{X'},t) A1-32)IXi): - oc <px, py, pT< +oc

where k2 is the macroscopicrate constant and the sum over {X,}has beensplit into two parts referring, respectively,to occupationsin velocity andordinary space.Taking into account definition A1.26)and the restrictionsimposedon tu r2 in Eq.A1.32),we finally obtain:

R, = k2Xfp(Xu...,Xi,...,Xn,t) A1.33)

Combining now expressionsA1.27)and A1.33)and summing over cells,we obtain a multivariate masterequation for the reducedprobability functionp(Xi,...,Xn,t). This equation can be written in a system that is macro-scopicallyhomogeneousor inhomogeneous,provided the dimension of thecellssatisfies the requirements discussedin the beginning of this section.Starting from this equation onecan calculate such quantities as the meansor variances within a cell as well as the variances betweencells.The latterprovides information about the spatial correlationsin the system.Thiscalculation is carriedout explicitlyin Section11.8for a model system.Ourmain purposeis to analyze the behavior of the fluctuations as the system is

preparing to approachthe instability point from the pretransitionalregion.

11.8.MULTIVARIATE MASTER EQUATIONIN A MODELSYSTEM

Here and in Sections11.9and 11.10we want to investigate closelytherelationship between stability and fluctuations. Starting from an asymp-asymptotically stable situation, we first obtain information about the origin ofthe non-Poissonianbehavior of fluctuations.We next considerthe limit asthe criticalpoint of instability is approached.We show that in this limit thenon-Poissonianbehaviorgivesriseto somenew and unexpectedphenomena,such as the building up of long-rangecorrelations.

The model we chooseto illustrate thesephenomena is the trimolecularmodeldiscussedextensivelyin Chapter7. In the notation of Section11.7,themacroscopicsolution on the thermodynamic branch readsfor this model:

Xi,0= A YL0 = ^ A1.34)


assuming fixed boundary conditionsof the type G.14)or no-flux boundaryconditions.Hereafter we concentrate on fixed conditionsonly.

Much of our analysis is devoted to the behavior of the solutionsof themultivariate master equation appropriatefor our model around the macro-macroscopic state A1.34).Forreferencehowever,we first compilethe results of abirth-and-death analysisof the trimolecular model,where it is assumedthat

fluctuations remain uniform in space.

HomogeneousLimit

Let p(X, Y, t) be the probability function for the two-variable intermediates.According to the resultsof Chapter 10,the master equation for p(X, Y, t)reads:

l)(X- 2)(Y + \\)p(X - 1,Y + 1,t)- \\)Yp{X, Y,t)

+ B(X + \\)p(X + 1,Y - 1,0-BXp(X,X t)

+ (X + l)p(X+ 1,X f) - Xp(X,K t) A1-35)

Thisequation can behandledby the cumulant expansiontechniquedescribedin Section10.6on the birth-and-death analysisof the Volterra-Lotkamodel.This givesus a set of equationsfor the average values of X and Y and a setof equations for the variancesthat will dependnonlinearlyon <X> and <Y>.As long as the variancesremain of the same orderof magnitude as the meanvalues (fluctuations not far from the Poissonianregime)and the latter arelarge enough (X, Y > 1),the equationsof evolution for the averagesreduceto the macroscopicequations.*Becauseof this, the equations for thevariancesof the secondorderreduceto a linear set of equationsof the form

A0.35)provided, of course,the variances of the third and higher orderareneglected.Furthermore, if we want to describethe early stages of the

spontaneousevolution of the variances starting from the homogeneoussteady state, we can fix (X), <Y> to the values given by Eq. A1.34).Thesystem of equationsfor the variances then reducesto a linear system with

constant coefficientsof the form A0.74).* As we seein subsequent sections, the system is bound to remain in the vicinity ofa Poissonian

regime if the diffusion dominates over the chemical reactions. By neglecting the effects of dif-

diffusion, asone doesin the birth-and-death treatment, one can no longer dispose ofa systematicway ofmeasuring deviations from the Poissonian distribution.

11.8.Multivariate Master Equation in a Model System 291

Defininga \"matrix of variances\"

I) AL36)

we obtain in this way the following set of equations(Lemarchand andNicolis,1976):

~=T+CM A1.37)at

which is reminiscentof Eqs.A0.35)or A0.74)derivedpreviouslyin a different

context.The meaning of the various symbolsis as follows:

T isa constant matrix equal to

\342\200\242 The objectC has the structure of a tensorof fourth orderand may bewritten formally as

C = rx/ + /xF A1.39a)

where F is the matrix of linearizedcoefficientsof the deterministicsystemofequationsin the absenceof diffusion [seeEq.G.20)]:

(B- 1 A2

~{-B -Aand / is the unit matrix

/1 n\\

A1.39c)

The-operation representedin Eq.A1.39a)is calledthe exteriorproduct (ortensorproduct)and converts a 2 x 2 matrix operatedon by C into another2x2matrix. In an explicitrepresentation,Chas four indicesand its elementsare given by:

A1.39d)

where 5*rq is the Kroneckerdelta.

292 Effect ofDiffusion: Phase-spaceDescription and Muttivariate Master Equation

Note that for Poissonianfluctuations one would have:

o

However,as seenin Section10.6,a systemof equationssuch as Eq.A1.37)for the variances generally predicts macroscopicdeviations from thePoissonianregime.At the steady state these deviations are given by thefollowing equation:

PI

or, taking into account Eqs.A1.36)and A1.39)as well as the symmetryof the matrix Mpq:

2(B-\\)MXX + (A2 -

B)MXY + (B - 1 - A2)MYY = -2A(B+ 1)A2MXX + A2MYY = 2AB

(B- 1 - A2)MXX + (A2 - B)Mxy - 2A2MYy = -2ABA1.41)

It iseasilyseenthat this setofequationscannot admit Poissonianfluctuations[Eq.A1.40)].Moreover,as the point of marginal stability B = A2 + 1

correspondingto the bifurcationof the limit cycleis approached(seeSection7.4),the matrix of the coefficientsof Eq.A1.41)becomes:

2A2

A2

0

_ j0

_ J

0A2

-2A2

This matrix is singular. In other words,the steady-state variances tend to

divergeas the critical point is approachedfrom below.We return to this

important point in the subsequentsections.

Effectof Diffusion on MasterEquation

We now adopt the more realisticpicture of a birth-and-death processin the

spaceof the variables Xu Yu ...,Xn, Yn referringto the n number of cellsin

which the systemhas beendivided.In orderto expressthe constraint of fixed

boundary conditionsweconsiderthat the systemis interactingat its endswith

two cells,0 and n + 1,where the concentrations of X and Yare imposedand


equal to Eq. A1.34).The master equation for the probability function

p(X0,y0, Xu Yu...,Xn+uYn + i,t) reads(Lemarchand and Nicolis,1976):|={iV.\"!)-(\302\253+2)p]

+ I ^^.-i)(A-,-2)(yfi =0 Z

n+1 |i \342\200\224 O -^

\\\302\243 (X,

Li=0B\\ \302\243 (X, + 1)P(X,+ 1,YtL

i=0 ,=0

dJ(X0+ \\)p(X0 +\\,Xl-\\)+ (Xn + 1 + \\)p(Xn-

(Xo + Xn+1)p + t (X,i= 1

+ i,y, - i) + (yn+1 + i)P(yn - 1,yB+1

-(yo+yn+,)p+ t(y,

We havenoted explicitlyonly thoseargumentsofpwhosevaluesare differentfrom Xo, Yo,...,Xn+l, Yn+1. Similar multivariate equations have beendevelopedby Gardiner,McNeil,Walls, and MathesonA976) for linearsystems and for systems undergoing transitions between multiple steadystates and by Kuramoto A974) for the bimolecular model A0.59)discussedextensivelyhere and in Chapter10.

Equation A1.42)can be handled in a manner similar to the homogeneousbirth-and-death equation A1.35),in order to get a closedset of equationsfor the variances around the macroscopicstate A1.34).The calculationsleading to these equationsare long and are not reproducedin detail.We


only outline the analysis of the terms arising from the transport of particlesacrosscells.The completefinal result is obtained by combining thesecontributions and those arising from the chemical reactions.Within eachcell,the latter are identical to the expressionsderived in the first subsectionhere (homogeneouslimit).

Contribution of Flow terms to Averages

Multiplying the last two termsofEq.(l1.42)by Xt(i # 0, n + l)andsummingover celloccupationswe obtain:

=d>t H {Xi(Xj + VWXj-1- 1,Xj + 1)flow j=l {Xj}

+ p(Xj+ \\,Xj+1-l)-]-2XiXjP}In the first term we may switch to the new summation variables

X'j= Xj + 1, X'j-i= Xj-i ~ 1' X'J+l = Xj+l \342\200\224 1

If j,j \342\200\224 1,or j + 1 is different from i, the two contributions in the curlybracketwill sum to zero.Thus, the only surviving terms are:

\"f /flow X,

+ (X, + l)Xi+l + X{Xi+l+ (Xt

-2XtXi+l-2XiX,._l}A1.43)

On the right-hand sidewemultiply and divideby the squareof the length ofacell,keepingin mind that k = l/n. We obtain:

dt I -> > ^ ' AL44)\"r /flow

The secondfactor on the right-hand sideis the well-knownfinite-differenceapproximation of the secondderivative operator (Krank, 1970).The first

term is related, accordingto Eq. A1.30),to the Fick diffusion coefficient,Duof the productX providedX ^ lr. Thus, in the limit ofa continuous systemX/l -* 0, we recover Fick'ssecondlaw describingthe rate of the transport ofmatter in the vicinity of a point in space.This is in agreement with both the

macroscopicmodelsanalyzedin Part IIand the remarksin Section11.7aboutthe passagefrom phasespaceto multivariate master equations.


Contribution of Flow terms to Variances

The contribution of the flow terms to the variances is handled in a similarway. We multiply both sidesof Eq.A1.42)by XtXj, Xt Y,-,...,and sum overcelloccupations.We define [cf.Eq.A1.36)]:

MXlX}= <(Xi - <Xi\302\273{Xj

-<*,\302\273> A1.45)

and similarly for MX.Y. (etc.).We obtain, for instance, for MXiXjwith i # j:

XjXj\\ _ , J y y y V (V i\\\\

dt /flow l(X,l k= 1

_!- l,Xk + l) + p(Xk+ \\,Xk+i - 1)]

(Xi( k=i ) \\ Ul /flow

A1.46)'flow

By the same arguments as in the previous subsection,we see that the first

bracket vanishesunlessif k, k \342\200\224 1,or k + 1 is equal to i orto j.This yields,rememberingthat i # j:

i- 1)- 2XfXj + (Xt + \\)Xt+1Xj

flow XiXj

+ XtXi+iXj + (Xi + VXt-iXj + XtXt-iXj

- 2XiXjXi+1- 2XiXi-1XJ+ symmetric terms}

\\ /flow \\ at yflow

or, taking Eq.A1.43)into account:

flow

A1.47)

25*6 Effect ofDiffusion: Phase-spaceDescription and Multivariate Master Equation

In the limit of a continuous system this expressionreducesto a form similarto Fick'slaw, differing from the macroscopiclaw A1.44)by a factor of two.Note that this result involves no approximation related to higher ordervariances,which can only arisefrom the chemicalterms.

Equationsfor Variances

Taking into account the explicitresultsderived in the previous subsectionswe may write the system of equationsfor the variances in the followingcompact form (for details,seeLemarchand and Nicolis,1976)

iMu = Eu + ZKijklMk, i,j=l,...,n A1.48)ax u

In Eq.A1.48)the definition of the variancematrix is the obviousextensionof M[Eq.A1.36)]:

My yl 1 * J

A1-49)

The objectsEtJ and Kijkl are, respectively,2x2matrices having the samestructure as F, T, or S, and B x 2) x B x 2) tensors having the samestructure as C.They differ from T and C defined in Eqs.A1.38)and A1.39)by terms related to diffusion. Moreexplicitly,one finds for constant boundaryconditionsthat (Lemarchandand Nicolis,1976):

(i) The only nonvanishing elementsof the matrix of E are:

where A is the matrix of the coefficientsdt:

o ;j(ii) The only nonvanishing elementsof K are:

Kihij= C - 2(A x / + / x A)

Ki+ulij= KijJ + 1J= Ax I A1.51)Ki.j+l.ij= f^ij.i.j+1= / X A

Note that these propertiesof K confer to the contribution arising from the

transport of matter in Eq.A1.48)the familiar structure appearingin Fick'sequation for diffusion, in agreement with the remarksmade in the precedingsubsection.

11.9.Spatial Correlations in the Trimotecular Model 297

11.9.SPATIAL CORRELATIONSIN THE TRIMOLECULARMODEL

Relation A1.48)is a linear set of equationsthat can be solved straight-straightforwardly. To this end, one must determine (seeChapter 7 for a similardiscussion)the eigenfunctions and eigenvalues of the operator K which,as we just pointed out, has the structure of a diffusion operator in a finite

differencerepresentation.Considerfirst an operatorL,such that

ISj= fJ+x+fj-x-2fj A1.52)Let \342\200\224 kk, uf] be the eigenfunctionsand eigenvalues:

Luf = -Xkuf A1.53a)We take fixed boundary conditionsand seekfor solutionsof the form:

uf = c sin

We find, taking the definition A1.52)into account:

Luf = c{sinnk(j + 1)+ sin /ik(j - 1)- 2 sin /ikj}= 2c{cosfxk sin \\ikj

- sin nkj)= 2(cosjufc- \\)uf

Thus, Eq.A1.53a)is satisfied.In order to satisfy the boundary conditionsas well, we need to have /xk

= (kn/n + 1).We concludethat

(*) \342\200\242kn

\342\226\240

uf = c sin /1 n + \\

= l(\\ - cos-kn-)>0 A1.53b)

If instead of L we have an operatorhaving a matrix structure becauseof the

presenceof more than one chemical,then the operatorexpressingthe effectof diffusion on the equations of evolution is of the form (seeSection7.4for a similar problem)

0

and its \"eigenvaluematrix\" is [cf.definition A1.50b)]:0 d,LY

298 Effect ofDiffusion: Phase-spaceDescription and Multhariate Master Equation

If, in addition, one has the contribution Y [seeEq.A1.40b)]coming fromthe linearized chemical kinetic equations, then the \"eigenvaluematrix\"of the operatorappearingon the right-hand sideof the differential equationsis

Ak = Y-kkA A1.54)

We may now return to systemA1.48).To solvefor Mu wehave to invert the

operatorK whose structure is that of a linearizedchemicalkinetic operatorplusa diffusion operator.

To this end,we seekfor a linear operatorP transforming Eq.A1.48)tothe form:

ivi iii \342\200\224

<*kl ' ^ kl kl V^ ^

\342\200\242-)-))

with

Mij =Y.pUkiM'ki A1.56)

Moreover,we require that P possessan inverse. Substituting Eq. A1.56)into Eq. A1.48)and comparing with Eq. A1.55),we deducethe followingrelation betweenP and i\302\243:

/ . ^rsmn'mnpq ^rspq~& pq {ll.Jlfmn

Keepingin mind the structure A1.51)of K we seekto determine P when

JSfp, has the following form:

i\302\243pq

= C -Xp

A x / - kql x A A1.58)

where kp is the eigenvalue[Eq.A1.53b)]of the diffusion operatorin finite-differencerepresentation.Forfixed p and q relation A1.57)now reads:

{A - 5r0)Pr_!.,.\342\200\236,+ (kp

- 2 + dr0 + dr,n+1)Prspq

+ {A \" 5,o)P,.,-l.M+ (A,\" 2 + <5s0+ Kn+l)PrSpq

+ A -<5s.n + 1)Pr.s+1.M}/x A = 0 . (U 59)

The operation describedinsideeachbracketis identicalto the finite-differenceoperationA1.52)and refers to the indicesr and s successively.This enablesus to seekfor solutionsof Eq.A1.59)in the following factorizedform:

p =i p p*\342\226\240 rspq r rpr sq

11.9.Spatial Correlations in the Trimolecular Model 299

Thus, the equation satisfiedby Prs has a structure similar to the eigenvalueproblemA1.53a).We have, therefore, for fixed boundary conditionsthefollowing expressionfor F:

rpn . sqn\342\200\224J\342\200\224-sin\342\200\224\342\200\224-

n + 1 n + 1Prspq= c sin -^sin-^ A1.60a)

where c is an arbitrary constant. The elements of the inverse operatorP l

are easilyfound to be:--1- l -:-JHLs[nJVL A i.6ob)

[2(n + l)]2c w + 1 w + 1

Knowing F\"'we can easilycompute A'kl in Eq.A1.55)using the expressionA1.50a)for E. We find:

G+4 AS) b*/- A1.61)+ l)c \302\253 + 1

We are now in positionto solve Eq.A1.55)for M'kl. Knowing M'u we cannext determine Mkl from Eqs.A1.56)and A1.60b).We are first interestedin

steady-statesolutions.After a few straightforward computations we find:

A1.62a)The diagonal element

5\302\243kkof the operator <gu, [Eq.A1.58)]is given by the

expression[cf.Eqs.A1.54)and A1.40a)]:kk

= Ak x I + I x Ak A1.62b)It has, therefore, the structure of a B x 2) x B x 2) tensorand representsthe combinedinfluenceof chemical reactionsand diffusion. The operatorS\\ on the other hand, is the B x 2) matrix:

s = r + rs+ srT (ii.62c)Tr being the transposeof the matrix F.It dependssolelyon chemicalkineticcontributions and is,therefore,independent of the labeling i of the cell.

Note that the inverse operation i\302\243k~k may be expressedin terms of Akand of its scalareigenvaluesa>kl, a>k2 that correspondto the eigenvaluesofthe linearizedproblemdealt with in Section7.4.Onefinds (LemarchandandNicolis,1976):

x {a)kla)k2I x I + [Ak -(Mi, + co^)/]

x [A -(<utl + <otl)/]} A1.63)


Let us now discusssomequalitative propertiesof the steady state solutionA1.62a).We see that Mfp contains one contribution correspondingto aPoissonianvariance and one additional term expressingspatial correlationsbetween cells.In order to analyze the relative importance of these twocontributions we display explicitly the size of each cell and then go to thelimit of a continuous system.Let / be the total length of the system.The sizeof each subsystemis obviouslyequal to l/n = A. Forthe chemical reactionoccurring within each cell,this sizeappearsexplicitlyin the kineticequationsfor the numbers of particles.Forthe trimolecular model one has:

jy\\ y2v oydt )ch A2 X

A1.64)dY\\ _BX X2Y

The coefficientsI/A, and I/A2 do not change the structure of the equationsfor the variances,but the three matrices F, 7, and S defined in Section11.8must be redefinedas follows.The same expressionsas before hold, but eachsymbol now representsa concentration instead of a number of particles.Thus, these matrices are independent of the size of each cell.Then, thesolution of Eq.A1.48)is obtained by writing AS and XT in placeof Sand Tin expressionA1.62a).This implies that & [seedefinition A1.62c)]is to bereplacedby XS.Relation A1.62a)now takesthe following form:

x {(F-Xk A) x / + / x (F -XkA)}-l\302\243 A1.65)

Dividing through both sidesby X2 and introducing

/(X,- <*,\342\226\240\302\273 (Xj - <Xj-\302\273=\\\342\200\224x ~~x-^~

A1.66)

where <x>now denotesthe concentration of X, and similarly for m*,^,we obtain:

<y8f;+-lGxiix

A1.67a)

11.9.Spatial Correlations in the Trimolecular Model 301

with

x {(F-^A)x /+ / x (r-^A)}\021^ A1.67b)

The function Gtj dependson n, but it may beshown that it remainsconvergentwhen n tends to infinity, that is, when the limit of a continuous system istaken, providedthe coefficientsdx and d2 remain finite. It appears,therefore,that in this limit the systemis dominated within each cellby Poissonian-typefluctuations, in completeagreement with the results of the phase-spacedescriptionof fluctuations.

Onecan obtain a more explicitform in the limit as n -* oo by taking intoaccount the relation A1.30)between the transport frequenciesd,,d2 andFick'sdiffusion coefficients

\302\243>,, D2.Introducing the matrix

I)we may write the factor F \342\200\224

Xk A appearingin Goas:

On the other hand, in the limit n -* oo one has, far from the boundaries:

. k2n2

k\"(n+IJor

^-ji^-p- 01.39)

This expressionis to be comparedwith the eigenvalue of the second-derivativeoperatoranalyzed in Section7.4.

The final transformation we need is that of the term n dff in expressionA1.67a).In the limit n ->\342\226\240 oo this tends to / d(rl \342\200\224 r2), where 5 is the Diracfunction. As a result, calling ru r2 the coordinatesof cellsUj as their sizerelative to the length / goesto zero:

m**(n.r2) = <x}8(ri- r2) + -Gxx(ri,r2) A l.70a)


where

\"u ri) \342\200\224 Z cos~^~1\342\200\224~

kn \342\200\224 cos-^\342\200\224-\342\200\224- kn

S A1.70b)

As we have pointedout, the influenceof spatial correlationson a givencell is small comparedto the effect of the Poissonian-likecontribution,if the size of the cell is sufficiently small.However,the situation can changecompletelyif the diffusion rate can beneglectedascomparedto the chemicalterm. In this case,the expressionfor GtJ reducesto:

A f kn knG, = > cos(i-(-j) \342\200\224 cos(;\342\200\224 j) :

= -(n+ l)df/(r x / + / x T)-lS A1.71)ExpressionA1.70a)contains,then, two termsof the sameorderofmagnitude:

T\"

88f?Ux> ^ [(F x / + / x D\"Wxx} A1-72)

This is the result obtained from the birth-and-death description,where thesystem is lumped into a singlecell and diffusion is neglected.

In the oppositelimit when the diffusion is very fast the spatial correlationterm A1.70b)becomesstill smaller relative to the Poissoniancontribution,as it is inverselyproportionalto the magnitude of the diffusion coefficients.It is striking to see from this discussionthat the birth-and-death and

phase-spacedescriptionsappear to be two facets of the same problem,correspondingsimply to different limiting situations.

11.10.CRITICALBEHAVIOR

We want to investigate the behavior of the correlation function Gxx(ru r2)as the systemapproachesthe state of marginal stability correspondingto theonset of a steady-state spatial dissipative structure or of a time-periodicsolution.In both caseswe assume that the critical point is approachedfrom below.This enablesus to pose the macroscopicvalues of the con-concentrations equal to the uniform steady-state solution A1.34).

The transition to a dissipative structure in the trimolecular model is

analyzedat length in Chapter7 from the standpoint of a purely macroscopicdescription.As in the beginning of Section11.9,we outline here briefly the

principal modificationsarising in a descriptionin terms of discretecells.

11.10.Critical Behavior 303

Firstly, we recallfrom Part IIthat linear stability is determined by the signof the real partsof the eigenvaluesof the operatorappearingin the linearizedequationsofevolution.In the trimolecularmodeland for a continuous systemthis operator,L is given by Eq. G.20),and its propertiesare discussedin

detail in Section7.4.In the discrete representation the correspondingoperatorAk is defined in Eq.A1.54)or,more explicitly:

f ) A1.73)-A2-AkdJwherekk is given by Eq.A1.53b).The eigenvaluesa)ki, cok2of this matrix canbe constructed explicitlyand are given by expressionssimilar to Eq.G.28).In view of the explicit form A1.63)of the quantity 5\302\243lk appearingin thecorrelation function, we are only interested here in the product and thesum of these roots.We find straightforwardly:

wkl + o)k2= trace of Ak = B- 1 - A2 -

kk(dx + d2)

cokiwk2 = determinant of Ak = A + kk dt)(A + Xk d2) - kkB d2 A1-74)

The macroscopicequations of evolution in the discrete representationinvolving cells1, n presentan instability wheneverthe sum or the productof the two rootscrossesthe value zero.KeepingA, du d2 fixed,we seefrom

Eq.A1.74)that the condition leadingto an instability is written in terms of Bas follows:

B>1 + A2^ + Xk dl +~ =Bk A1.75a)d2 /* d2

for bifurcation of a steady-state structure and

B > 1 + A2 + kk(dl + d2) = Bk A1.75b)for bifurcation of a time-periodicstructure. Theserelations are representedgraphically in Fig.11.4,which replacesFigs.7.1and 7.2of the continuousformalism.

Concerningmore specificallyrelation A1.75a),we seefrom Fig.11.4that

for an instability to lead to a stationary spatial structure it is necessarythat

Bk belarger than the minimum of the marginal stability curve.The valuesofK and Bk correspondingto this minimum are given by expressionssimilar toEq.G.35):

A1.76)A

L =-(d, d2)1/2


1 +A2

Figure 11.4.Marginal stability curves for trimolecular model corresponding to onset of aspatial structure (Bk) and onset ofoscillations (Bk) in representation in terms ofdiscrete cells.

Finally, in drawing Fig.11.4we have supposedthat the minimum of Bkwaslower than that ofBk.This is satisfiedwhen a relation similar to Eq.G.36)holds:

A1.77)

and implies that as B increases,the transition to a steady-state dissipativestructure occursbeforethe transition to a time-periodicpattern.

Coming now to the behavior of the correlation function G*x we first

analyze the dependenceon k of the coefficientykkl<$ in the trigonometricseriesrepresentation of G*x.This quantity plays the role of a \"form factor\"and would be identical to the Fourier transform of the correlation function

if the system were taken to be infinite (Landau and Lifshitz, 1957).Sub-Substituting the explicit expressionsA1.74)to A1.77)into the definitions of5\302\243kk

and S\" given in Section11.9one finds for a discretesystem:

5\302\243kkl&= 2AB{'Al

~'

-A2dl - d2Bd2

+

2 B

A2 dl + (A2 + 1 - B)d2-A2dl-d2

+ di d2 -dt-d,d2 0

x IB- 1 - A2 -Wx+d2)Yxx [A2 + lk(A2 d, - (B - \\)d2) + X2 d,


From this expressionand from Eq. A1.75)it becomesobvious that\302\243\302\243^3

presentsa maximum that becomessharperas B approachesBc.At B = Bcthe form factor diverges.This impliesaccordingto expressionA1.67b)for Gtj that the range of correlationsis increasing as one approachesthecriticalstate.Figure 11.5describesthe shapeof the form factor for a systemof 100cellsand for various values of B up to the value correspondingto thestate of marginal stability.

The increaseof the range of correlationsin the critical region is alsoconfirmedby a direct analysisof the correlation function G*x.The numericalevaluation of Eq.A1.67b)for 100cellsgives the results displayedon Figs.1\\.6a,b and 11.7.ForB well belowcritical,the correlationsare maximum fori =j and then decreasesharply beyond a small distance.As B increasesthere appearsan undulatory pattern which, however, is dampedquickly.Finally, in the immediatevicinity of the criticalpoint the correlationsfeaturean oscillatory distribution that is dampedonly linearly with distance,asshown in Fig.11.7.

The correlation function can also be computed analytically in the limitof a continuous system [seeEq.A1.70)].According to expressionA1.70b),

50

40

30

20

10

B = 4.0

0 50 K

Figure 11.5.Dependence of the form factor if^'S'on wave number in trimolecular model for\342\226\2404

= 2,d, = \\,d, = 4.


Bi

1 +A2

Figure 11.4.Marginal stability curves for trimolecular model corresponding to onset of aspatial structure (Bk) and onset ofoscillations (Bk) in representation in terms ofdiscrete cells.

Finally, in drawing Fig.11.4we have supposedthat the minimum of Bkwaslower than that of Bk. This is satisfiedwhen a relation similarto Eq.G.36)holds:

A1.77)

and implies that as B increases,the transition to a steady-state dissipativestructure occursbeforethe transition to a time-periodicpattern.

Coming now to the behavior of the correlation function G*x we first

analyze the dependenceon k of the coefficient!\302\243kk$ in the trigonometricseriesrepresentation of G*x.This quantity plays the role of a \"form factor\"and would be identical to the Fourier transform of the correlation function

if the system were taken to be infinite (Landau and Lifshitz, 1957).Sub-Substituting the explicit expressionsA1.74)to A1.77)into the definitions ofykk and & given in Section11.9one finds for a discretesystem:,..,.!A2 -A2'

'\342\226\240kk

= 2ABB

f (A1\\2-A2di-d2

dx i

~dxd2

- B)d2 -A2di- d2Bd2

d\\ + dx d2 -dxd20

x IB- 1 - A2 -Xk{dx+d2)T1x \\_A2 + Xk(A2d1 -(B-\\)d2) + X2 dx A1.78)


From this expressionand from Eq.A1.75)it becomesobvious that\302\243\302\243kk$

presentsa maximum that becomessharperas B approachesBc.At B = Bcthe form factor diverges.This implies accordingto expressionA1.67b)for Gu that the range of correlationsis increasing as one approachesthecriticalstate.Figure 11.5describesthe shapeof the form factor for a systemof 100cellsand for various valuesof B up to the value correspondingto thestate of marginal stability.

The increaseof the range of correlationsin the critical region is alsoconfirmedby a direct analysisof the correlation function G?jX. The numericalevaluation of Eq.A1.67b)for 100cellsgives the resultsdisplayedon Figs.l\\.6a,band 11.7.ForB well belowcritical,the correlationsare maximum fori =j and then decreasesharply beyond a small distance.As B increasesthere appearsan undulatory pattern which, however, is dampedquickly.Finally, in the immediatevicinity of the criticalpoint the correlationsfeaturean oscillatory distribution that is dampedonly linearly with distance,asshown in Fig.11.7.

The correlation function can also be computed analytically in the limit

of a continuous system [seeEq.A1.70)].According to expressionA1.70b),

B = 4.0

Rgure 11.5.Dependence of the form factor\342\226\2404 = 2,</, = \\,d2 = 4.

50 K

& on wave number in trimolecular model for

B=3

500

10

Figure 11.6a.Distance dependence of spatialcorrelation function G** well below critical

point. Values of A, dt, d2 areas in Fig. 11.5.

B = 3.5

103

10 20

Figure 11.6b.As bifurcation parameter ap-approaches critical value, range of G** increases'

slightly in respect to behavior shown in Figure

11.6a.

306


XX'

G1j

10\"

5.103

0

A

-\342\200\242

-

V

V

Critical stateB = 4

!\\ A A

V

V

V *

0 50 100

Figure 11.7.Critical behavior of spatial correlation function G\342\204\242 for samevalues of parametersas in Fig. 11.6a.Correlation function displays both linear damping with distance and spatialoscillations with wavelength equal to that of macroscopicconcentration pattern.

G(r1?r2) is a trigonometric serieswhose coefficientsare rational fractions ofk2. Thesefractionscan be split into simpleelementsof the form H/k2+ K2,whereK is generallya complexnumber and each subseriesmay besummed.The calculations are straightforward, and one finally finds for the XXcomponent of G(ru r2) (Lemarchandand Nicolis,1976):

i r2) = + A1.79)

where G*rx is a short-range part exhibiting exponential decay with distance,and Gfrx a long-range term. This term has the form (keepingrx fixed and


displayingonly the r2-dependence):

+ b(K)Kn(\\ -y\\x>sKn(\\ -j)+ c(K) A1.80)

where the coefficientsa, b, c dependon K as well ason A, B and the diffusion

coefficientsD,,D2.In the critical state the wave number K is given by theexpression

where/?, a are the concentrationsof A, Brelated to the correspondingparticlenumbers by P = (B/Q),a = (A/Q),Q being the size of each cell [seeEq.A1.64)].Taking into account the approximate expressionA1.76)for fic wefind:

Comparingwith the macroscopicanalysisof the trimolecular model carriedout in Section7.4[seeespeciallyEq.G.35)]weconcludethat the wavelengthof the correlation pattern is identical to the wavelength of the macroscopicconcentration pattern itself.

If the system is slightly below the critical state then K will have an

imaginary part describingan exponential decay of the correlationswith asmall damping factor. One finds in this case:

== K' + iK\" A1.83)

At the critical state, expressionA1.80)predictsa linear damping of thecorrelationsexpressedby the factor K(\\

\342\200\224 (r2//)) in front of the cosineterm,in agreementwith the numerical results reportedon Fig.11.7.

The results given in Eqs.A1.80)to A1.83)can be regardedfrom a still

different viewpoint, which is frequently adoptedin the analysisofequilibriumcritical phenomena (Stanley,1971;Wilson, 1973).Firstly, we seethat in the

vicinity of the critical point all spatial coordinatesappearingin the long-range part of the correlation function are scaledby the parameter K/L

//.//.Concluding Remarks 309

Accordingto Eq.A1.83),as b approachesthe threshold valuebc,this ratio or,alternatively the equivalent lengths

r =i' r = -L*i L \\ ,171 A1.84)K /C |b \342\200\224

bc\\

are intrinsic parametersof the system, independentof the size /, the size ofthe cellsQ, the boundary conditions,or the microscopicparametersin theatomic scale.Forb -\302\273 bt. these lengths behave as follows:

and

A\"-+oo A1.85)where A,, is the wavelength of the macroscopicconcentration pattern. Forfixed ri the correlation function A1.80)can, therefore,be written as:

Gr(r,,r2) =g\\j\\9n[j,\\

+ const A1.86)

This relation is strongly reminiscent of the scalinghypothesis familiar from

equilibrium phase transitions. In essence,we have demonstratedhere forour model a scaling law for the correlation of fluctuations around a non-equilibrium steady state. Moreover,we have identifiedtwo intrinsic lengths,which can beappropriately referredto as correlationlengths:

\342\200\242 The length A', which takesa finite valueA,,at the criticalpoint and expressesthe structure of the correlation pattern in space.This pattern is macro-macroscopic, as At. itselfis a macroscopiclength.

\342\200\242 The length A\", which diverges at the critical point, thus expressingthefact that the correlationsare not damped in spacebut attain, instead,a self-maintainedregime(wediscussfurther the significanceoftheselengthsin Section12.5).

U.ll.CONCLUDINGREMARKS

In this chapter we show that in a system initially at a homogeneousstate,\342\200\242ong-range correlationsbetween macroscopicfluctuations emerge in thevicinity of and below the critical point of a nonequilibrium instability andconferto the systema distinctly non-Poissonianbehavior.We are witnessing,therefore,a striking breakdownof the \"laws of largenumbers\" ofprobability


theory, correspondingto the fact that different macroscopicregionsin thesystem do not evolve independently but instead,they becomecoupledviatheselong-rangecorrelations.

In contrast to this, if one evaluates the fluctuations within a small boxwhosesize A/ is much smaller than the characteristic length kc, one finds

that the mean-squaredeviation is still describedby a Poissoniandistributionto a goodapproximation (Gardiner,McNeil,Walls, and Matheson1976).Toseethis, weconsiderthe expressionfor the cumulants in the continuous limit,Eqs.A1.70)and A1.80),and compute the quantity:

1= dr^ dr2mxx{rur2)

= A/<x> + - drx dr2Gxx(rur2)J&i

Keepingonly the longrangecontribution ofGxx{rur2),Eq.A1.80),weobtainin the vicinity of and below the critical point:

<SX2)AI = (X\\,(\\ + 0(A/2)) A1.87)Thus, as long as the boxis sufficiently small, it doesnot \"feel\" the long-rangecorrelationsand remains in a Poissonian-likeregime,provided the steady-state expressionsfor M or G usedthroughout the last two sectionsremainmeaningful. Inasmuch one is interestedin the pretransitional phenomenaaccompanying the instability, a steady-stateapproachis certainlyjustified.

On the other hand, when the critical point for the macroscopicequationsin the cell representation [seerelation A1.75)]is crossed,the correlationfunction behavespathologicallybecauseof the divergenceof the form factor[Eq.A1.78)].At this stage,therefore,time-dependentsolutionsof Eq.A1.55)have to beenvisagedto describethe growth of the criticalmodein the vicinityof and above the instability. As this growth will continue, the deviation ofMu from the Poissoniandistribution value, S 5ff becomesincreasinglymarked.Thus, a new nonnegligiblecontribution appearsin the equationsof evolution for the averagevalues of X and Y:

A/ V \\\342\200\224-\342\200\224 = (macroscopicexpression)+ (deviationfrom Poissonian) A1.88)

At the early stagesof growth, the influenceof this new term on <X>,< Y>remainssmallbecauseof the factor

l/(\302\253+ 1)(or 1//in the limit ofcontinuous

system)weighting the correlation function which is responsiblefor the non-Poissoniancontributions.Eventually, after the lapseof a macroscopictime

interval, thesecontributions take overand drive the averageto a new macro-macroscopic regime.The situation is somewhat reminiscent of nucleation theory

11.11.Concluding Remarks 311familiar from the liquid-vaportransition at equilibrium. We return to this

point in Chapter 12,where a simplified formalism to treat the effects ofdiffusion isdeveloped.

A striking Monte-Carlosimulation of the kinetics of fluctuations in the

postcriticalregionhas beenreportedrecentlyby HanusseA976).Thisauthorconsidersan autocatalytic modelpresentingmultiple steady-statetransitions(seealsoSection12.6)and investigatesthe evolution of an initial probabilitydistribution peakedaround one of the (macroscopically)stablebranches.

(a)

100 200 300 (b)Time

Figure 11.8.Nucleation in model system involving multiple steady states, (b) shows time

evolution of the mean, (a)shows time evolution of the variance. Latter begins to increase sharplyas soon assufficiently large \"germ\" appears in the system through fluctuations [points A, B oncurve (b)].


When the values of the diffusion coefficientsrelative to the chemical ratesare small,one observesa rapidtransition to a secondstablebranch, providedthe transition point correspondingto the Maxwell rule (seeSections8.4and 12.6)has beenexceeded.When, on the other hand, the values of thediffusion coefficientsare large, the initial state displaysa finite life time. After

a variable time interval one or several domains appear,where the con-concentration of the chemicalshas a value closeto that correspondingto theother branch.Subsequently,these \"germs\" grow and entrain, by diffusion,the remaining of the systemto the new branch of solutions.

Figure 11.8representsthe time evolution of the mean-squaredeviationand the average value in such a numerical \"experiment.\" At point A (onthe plot for the average)a first germ emerges,and a secondone followsatpoint B. At point C the two germs attain almost simultaneously theboundariesof the system,whereupon two convergingwave fronts are formedthat finally disappearwhen state D (correspondingto a valueof the mean onthe other branch) has been reached.

Finally, when diffusion is very fast the initial state is maintained as longas desired.This correspondsto the property of metastability familiar fromequilibrium phasetransitions.*

We believethat it should bepossibleto test directly the resultsconcerningthe emergenceof long-range correlationsby experiment, in particular by

appropriatescattering measurements involving volumes of variable sizeswithin the system.To our knowledge,no experimentsof this type have beenundertaken so far for physicochemicalor biochemical systems undergoinginstabilitiessuch as those analyzed in Part IV.

* This nucleation phenomenon has been recorded on a film available at the Centre desRecherchesPaul Pascal.Talence.France.

Chapter12

A \"Mean-field\"Descriptionof Fluctations:NonlinearMasterEquation

12.1.INTRODUCTION

Thedescriptionbasedon a multivariate master equation correctlyrepresentsthe behavior of a macroscopicsystem if the number of cellsis large.In this

case,the number of stochasticvariables is alsovery large, and the solutionof the master equationsgoverningthe evolution of the probabilitiesin termsof these variables is greatly complicated.

Hence,we develop in this chapter, a simplified formalism by treating thesystemas a set of two interacting subsystems:(a)a small volume AV and (b)the rest of the system,V \342\200\224 AV as illustrated in Fig.12.1.This,ofcourse,doesnot imply that the particular AV chosenin such a decompositionplays aprivilegedrole.In fact, one can imagine that the entire spaceis filled with

such subvolumesAV, which undergo simultaneouslyfluctuations of variouskinds.In subsequentsections,however,we only analyzeexplicitlythe situa-situation prevailing in one of these volumes by averaging over the rest of thesystem. In other words,we are interestedin a reduced rather than globaldescriptionof fluctuations.

An analysis basedon this picture is especiallyadapted to the study oflocalized fluctuations having a well-defined range. The latter could beidentified to the sizeof AV and be in this way incorporatedexplicitly in thetheory.From the analysisof Sections11.7to 11.10werecall that the range is,in fact,determined by the propertiesof the correlation function G(ru r2).Inparticular, in the vicinity of an instability the range\342\200\224and, therefore, alsoAV\342\200\224is expectedto becomemuch larger than the dimensionsofthe individualcellsconsideredin the multivariate formalism of Chapter11.

5/5

314 A \"Mean-field\" Description ofFluctations: Nonlinear Master Equation

Figure 12.1.Transport of particles acrosssurfaceAE separating a small subvolume AV within amacroscopic system, n: outward normal to AI.v: velocity ofa particle directed toward A V. Dashed

region indicates an area of width of order of mean

path, lr, surrounding surface AZ.

12.2.DERIVATION OF NONLINEAR MASTER EQUATION

We refer again to Fig.12.1and assumethat the system remains macro-scopicallyhomogeneous,although the density and compositionfluctuationscontinuouslybreak this homogeneitylocally,within smallsubvolumessuchasAV. If the referencestate is inhomogeneous,then the analysis we outline in

this sectionmust be modified by suitably redefining the various transitionprobabilities.Thisextendedformalism can then beusedto study fluctuationsin fluid-dynamic systemsas well (Malek-Mansour,Brenigand Horsthemke,1977).

As A V isfinite, our treatment of fluctuations, although local,is formulatedin terms of discretevariables, namely, the number of particlesXin, Xout ofthe various constituents within AV and V \342\200\224 AV, respectively.The cor-corresponding intensivevariablesare definedas:

whereas

We expectthat

limAK->0

AV

= finite

A2.1)

A2.2)

but that the fluctuation Spx.n =pXtn

\342\200\224 <pxin) be, in general, a nonsmoothfunction of AV.

12.2.Derivation ofNonlinear Master Equation 315

Let P\\v(X{n, t) be the probability distribution* within AV andP(Xin, Xout, t) the probability within the entire system.We denote by Rthe contribution of the chemical reactionsto the evolution of P, and by Fthe contribution of the transport processesacrossthe surface AT.. As in

Chapter11,we take into account only the transport of matter by assumingafast thermal relaxation to a constant temperature distribution.The extensionto the nonisothermal casehas beencarriedout by FrancksonA975).Themaster equation takesthe following form:

-rr = Âv(în) + FAv.v-Av(Xin, Xout) A2.3)

In writing this relation we have introducedonceagain the Markovianassumption accordingto which the contribution of chemical reactionsin

AV dependsonly on the stochasticvariables insideAV. As a matter of fact,in what follows we assumethat the stochasticprocesswithin AV is of thebirth-and-death type, although this is not generally the casefor the sto-stochastic processwithin the entire system V.

The transport term F,which isresponsiblefor the coupling betweenAV andV \342\200\224 AV, can easilybededucedfrom the analysisof Eq.A1.27).We have:

F&V.V-AV \342\200\224 Z, {wout(în\342\200\224 h Xout + 1 -\342\226\272 Xin, Xout)

Xout

x P(Xin-hXoul+ l,f)+ win(Xin + 1,XBut -l^Xin,X0Ut)P(Xin + 1,Xout

- 1,0- [w0JXin,X0J+ win(Xin, Xout)-]P(Xin, XBut, t)} A2.4)

The transition probabilitiesper unit time wou, and win describethe frequencyof passageof X particlesacrossAS and are given by expressionssimilar toEq.A1.28a).We simply have to take into account here the possibilityof athree-dimensionaltransport.We have (seeFig.12.1):

f fwout = -

<f>

dS \\ dw n/oul(r, v, f)

win =<f

dS f dw n/in(r, v, f) A2.5)

*Throughout this section we adopt a compactnotation and denote by X the entire vector ofall

chemical constituents present: X = (Xx, . . . , Xn).


The requirementofmacroscopichomogeneity,combinedwith the argumentson the validity of the localequilibrium descriptiondevelopedin Section11.7,permits us to express/inthe form

/out =y 2\\y 0out(V, t)

A2.6)

./in =-^0in(V, f)

with

Then, expressionA2.4)reducesto

7av.v-av = 1I--77\342\200\224l^T <Pds dvvn0out.'vi<0

x P(Xin-\\,Xout+ l, rdS \\ dvv n0inJvi>0

x P(Xin + 1,Xoul- 1,f) - inverseterms V A2.8)

On the right-hand sidewe introduce the decomposition

P(ATin, Xmt, t) = PAV(Xin, t)Pv^AV(X0Ut, t) + G(Xln, Xout, f) A2.9)

where G is the correlation function between AV and V \342\200\224 AK Accordingto Section11.9,G is generallynonvanishing, but its importance relative tothe contributions coming from the local processesinsideAV dependsonthe sizeof the celland the relativemagnitude ofdiffusion and chemicalrates.Following this analysis we assumehere as a first approximation that thecontribution of G remainssmall,that is,that AV and V \342\200\224 AV are statisticallyindependent.It must be emphasizedthat this assumption can only hold if

(AVIV) is sufficiently small and the correlationsare sufficiently short-ranged.In particular, in the limit (AV/V) \342\200\224\342\231\2460 it would bea rigorousconsequenceofthe phase-spacemaster equations.On the other hand, assoonasA V becomesan appreciablepart of V the assumption breaksdown completely(seealsoGardiner,McNeil,Walls, and Matheson,1976).

12.3.Further Properties and Moment Equations 317

Bearing this in mind and utilizing relations A2.7) and A2.8) we maytransform Eq. A2.3) into a closed-form equation (Malek-MansourandNicolis,1975;Prigogine,Nicolis,Herman and Lam, 1975):

= Rav{X) + @<a:>[pak(x- i,r) - pav(x,r)]

\\)pAV(x + l,f) - a:pak(a:,r)] A2.10)

where we have droppedthe subscript\"in\" from X and introduced the co-coefficient :

= _L i dS (*AF JAE Jv.n>0dvvn0(v,f) A2.11)

This coefficient,which is to be comparedwith the coefficientintroducedin Section11.7[seeespeciallyEq.A1.29b)],plays the roleof an \"effectivediffusion frequency\" ofpassageofparticlesacrossAS.In this respectwe maynote that the distribution /out usedin the evaluation of the transition prob-probabilities describesthe motion of particlesdirectedtoward AV and comingfrom a layer of a width lr, that is, of the order of the mean free path of X

specieswithin the reaction medium (Chapman and Cowling, 1952).Thispoint becomesmore obvious if Eq.A2.11)is evaluated using the Maxwellianform for 0.One then obtains [seerelation A1.29c)]:

where / is a characteristicdimensionofA V. As wepointedout in Section12.1,this quantity reflectsthe range, or coherencelength of the fluctuations.

Equation A2.10)provides a generalizationof the birth-and-death descrip-description that displaysthe qualitative features of the more completetheoryoutlined in Chapter11and at the same time is freeof the complexitiesof themultivariate master equation.

12.3.FURTHER PROPERTIESAND MOMENTEQUATIONS

The most characteristic property of the master equation A2.10)is its non-linearity, arising through the factor <X> on the right-hand side.This is dueto the passagefrom a globaldescriptionto a local one.In this respect,it hasthe same origin as the nonlinear terms of the kinetic equationsof statisticalmechanicssuch as the Boltzmann or the Vlassovequations(Balescu,1963).For this reasonwemay referto the descriptionbasedon the nonlinear masterequation as a mean field descriptionof fluctuations.


An alternative way to understand the nonlinearity is to point out that theexternal environment of AV has been taken into account in an averagefashion and that the macroscopichomogeneitycondition A2.7)has relatedthis average to an average over the small system itself. As we see in thesubsequentsectionsthis \"competition\" between subsystemand externalenvironment is responsiblefor the propagation of fluctuations leading to an

instability. The similarity between this descriptionand the Prigogine-Herman theory of vehicular traffic (Prigogineand Herman, 1971)shouldbepointedout.

Independentlyof any physicalcontext, it would beof interest to character-characterize more preciselythe type of stochasticprocessdescribedby the masterequation A2.10).It is likely that this equation belongsto the classofsituationsreferred to by McKeanas \"nonlinearMarkov processes\"(McKean,1969).Even if the processremains Markovian, it certainly loosesits stationarycharacter.Indeed,one of the transition probabilitiesnow dependson theaverage value (X~),which is an explicit function of time.

The final point we want to develop in this sectionconcernsthe momentequationsgeneratedby the nonlinear master equation.Multiplying bothsidesby X and summing over all values of X we find, in the notation ofSection10.4:

^ = <a>+ \302\256<*> t *C>V(X- 1,0-P(X,f)]

+ 2 X [X(x+ 1)Pav(X+1,0-X2PAV(X, t)]A-=0

Performing the by now familiar change of variables within the sums over Xwe find:

d 2 t XPAV(X, t)itin other words,

d<xy,dt

A2.13)

Thus the first moment equation is independent of 2,in agreement with the

requirement of macroscopichomogeneity.To find the equation for the secondmoment we multiply both sidesof the

master equation by X2 and sum over X.We find by a similar calculation:

AV = (a2x)AV + (Xalx}AV -2\302\256[<SX2)AV

- <X>AK] A2.14)at

12.4.Onset ofa Limit Cycle 319

where (SX2}AV is the mean-squaredeviation. We seethat diffusion contri-contributes explicitlyto the evolution of fluctuations through a term expressingthe deviation of the probability distribution function from the Poissonianregime.In the limit 9) -* oo the diffusion term dominates in Eq.A2.14).Thesystemevolvesthen to a steadystate characterized by a Poissonianvariance,since(SX2}AV = (X}AV. This is in qualitative agreementwith the results ofChapter11.Note,however,that in the multivariate descriptionoffluctuations,Eq.A1.43),the evolution ofvariancesMu to the steadystate is not, in general,given by a simpleexponentialdecay,becauseof the finite differencestructureof the operatorKijk, on the right-hand sideof this equation.In contrast, thecontribution of diffusion to the evolution of the mean-squaredeviationswithin AV [seeEq.A2.14)]always tendsto induce an exponential decay in

time (Gardiner,McNeil,Walls, and Matheson,1976).

12.4.ONSETOF A LIMITCYCLE

We now return to the onset of instabilitiesand analyze the nonlinear masterequation for a simple system giving riseto dissipative structures.We againwork on the trimolecular model,whosecritical behavior is analyzed in

Section11.10from the standpoint of the multivariate master equation.Ourpurposenow isto discussthe emergenceofa limit-cyclebehavior,postponinguntil Section12.5the caseof steady-state space-dependentstructures.

As wesaw in Sections7.4to 7.14,the nature of the solution that bifurcatesthe first from the thermodynamicbranch dependson the diffusion coefficientsDu D2 and on the boundary conditions.In particular, for zero flux or forperiodicboundary conditionsthe first bifuractionnecessarilyleadsto a limit

cycle if D, = D2.If, on the other hand, D\\ is sufficiently smaller from D2,asymmetry-breakinginstability becomespossible.

In principle,some of these features could be altered by the stochastictreatment, which goesbeyond the macroscopicanalysis carried out in

Chapter7. Hereafter,however,we are interested in asymptotic solutions ofthe master equation which, as discussedin Chapter10,are well adaptedtodiscussthe earlystagesof instability. Thisenablesus to truncate the hierarchyofmoment equationsto thosefor the second-ordervariances.Becauseof this,the equations for the variancesinvolve the sametime and length scalesasthosefor the averages.The latter reduceto the macroscopicsolutions if averageandmostprobablevaluearecloseenough.This is certainlythe caseofall problemsinvolving \"soft\" transitions, that is, the appearanceof patterns whoseamplitude vanishes as one approachesthe bifurcation point from above. A

striking confirmation of these ideasis found in Section10.6,where the vari-variances in the Lotka-Volterramodel vary at twice the frequency of the


macroscopicmotion, and in Section11.10,where the range of the correlationfunction was determined by the parametersappearingin the macroscopicequationsof evolution.

We concludethat if we are interestedin the early stagesof formation of alimit cyclein a small subvolume, AK of a large (infinite) system,we may takeD, = D2.This can be translated in the stochasticformalism by the equality

2i = 92= 9 A2.15)where

\302\243?,

= 9X, 92= 9Y.The equations for the varianceshavea form similar to Eq.A1.37),but they

are now supplementedby diffusion terms, in agreement with Eq.A2.14).Itis alsoconvenient to switch to reducedvariables:

A B

A2.16)X Yx = ~~~, y = \342\200\224

whereNo is of the orderof the total number ofparticleswithin AK. Moreover,we introduce instead of the variance vector M = {(SX2},(SXSY},<<5Y2\302\273

the vector b = {biUb12,b22)expressingthe deviations from the Poissonianregime,as in Eq.A0.30):

b22=\342\200\224 l(SY2}- <Y>] A2.17)No

We obtain in this way the following set ofequations,which for concisenesswewrite in matrix form as in Section11.26(Nicolis,Malek-Mansour,Van

Nypelseer,and Kitahara, 1976):

'2(P- 1)-29 2a2 0

/?- a2 - 1 -29 a2

-2/J -2a2-29/

A2.18)


where t is a reducedtime variable.It shouldbeevidentby now that the prop-properties of this system are determined by the eigenvaluesof the matrix of thecoefficients.Working out the characteristic equation, we find:

Wl =\342\200\236 a,

~~ l ~ Wr^ A2A9)

co\302\261

= P \342\200\224 a \342\200\224 1 \342\200\224 29\302\261 sjM

where

M = (P - a2 - 1 -29J- 4fj3>2 + (a2 + 1 -P)9 + a2]= (p - a2 - IJ - 4a2 A2.20)

Equation A2.18)admits a stable steady-statesolutionsif coi <0, Reco\302\261 <0.This happens:

(i) For any 3>, if

0 < p < a2 + 1 A2.21)

(ii) For

9 > \342\200\224~

\" ~\\ if a2 + 1 <P < (a + IJ A2.22)

(iii) For

9 > \342\200\224 it p > (a + 1) A/./3)

In these regions,a local fluctuation doesnot increase.Thus, the referencestate

<*>~a,(y)^- A2.24)a

remains unaffectedduring a macroscopicperiodof time.

We may now combinetheserelations with Eq.A2.12)relating 9 to thecoherencelength of the fluctuation. We define a critical length, \\c, by therelation

inn \\

A2.25)

where3icis given by Eqs.A2.22)and A2.23)when the equality sign is satisfied.Figure 12.2representsthe relation between lc and the chemicalparameter pappearingin the expressionfor 3>c.The curve Ic

\342\200\224 IC(P)has an asymptote at


a+ 1 <\302\253+ 1)

Figure 12.2.Coherence length of fluctuation at critical point versus chemical parameter fl in

trimolecular model under macroscopically homogeneous conditions. HereL, oc (/I- a2 -

I)\021

and L2 oc {/?- a2 - 1 + [(/?- sr2 - IJ-4a2]\022)-'.

ft = a2 + 1.At )8 = (a + IJ the two branches(a) and (b) given respectivelyby:

'r 2(D//r)- a2

and

2(D//r)- a2 - 1 + - a2 - IJ - 4a2

A2.26)

A2.27)

merge,and this resultsin an angular point with the right derivativetending toinfinity.

It is instructive to phrase these results in the language of equilibriumcritical phenomena (Stanley, 1971).We first observe that the transition toinstability is reflectedby the appearanceof a limit cyclehaving a nonvanish-ing radius,r. Thus, it is natural to define r as the order parameter in thesystem.Now,accordingto Section7.12,r varies in terms of /?as:

rcc(p-&I/2 for p > pcr = 0 for p < pc

A2.28)

wherepc is the point where the first instability is occuring\342\200\224the analog of thecritical point of equilibrium phasetransitions:

pc = a2 + 1 A2.29)

Relation A2.26)becomes:

lc x (p - pe) A2.30)


We have, therefore,found a critical exponentof 1/2for the orderparameterand of -1for the quantity lc that plays the roleof the correlation length in thetheory. The correspondingexponentsin any theory of equilibrium criticalphenomena based on Landau ideas-knownas a self-consistenttheory ofcritical phenomena\342\200\224would be, respectively, 1/2and

\342\200\2241/2. Thus, despitethe self-consistentfield character of the masterequation A2.10),thereemergesa \"nonclassical\"exponent,namely, \342\200\224 1.An inspection of Eq.A2.12)showsthat this nonclassicalbehavior stemsfrom the appearanceof the relaxationlength in the equation relating 3) and /. This, in turn, is related to the non-equilibrium character of the phenomenon, which introducesinto the theorythe effectof excitationssuch as soundwaveswhose velocityis related to themean free path lr and whose wavelengthsare \"intermediate\" between theshort wavelength scalerelated to atomic dimensionsand the macroscopicscalerelated to the coherencelength of the fluctuations. It is instructive torealize the analogy between this picture and the ideasunderlying Wilson'stheory of equilibrium phasetransitions (Wilson, 1973).Analogies betweenchemical instabilities and phasetransitions have also beenpointedout by

Nitzan, Ortoleva, Deutch and RossA974) and by Kuramoto and TsuzukiA975).

Bearingthesepoints in mind one may now interpret the diagramofFig.12.2as follows.Supposea fluctuation appearsat the vicinity of a point insideAV. If the range of this fluctuation, that is,the length overwhich it preservesacoherent character, is in the dashed region of the figure, then the decayprocessesof this local fluctuation (roughly measuredby S>~')take over the

amplificationmechanisms,and the fluctuation diesout. Forp < a2 + 1evenif an infinite coherencelength is imposedinitially, the fluctuation decays.But for p > a2 + 1 those distrubanceswhose range exceedslc are amplifiedand spreadthroughout the system.The latter remain stable with respecttofluctuations with I < lc. Thus, diffusion plays a stabilizing role for certainfluctuations, even in the supercritical region.Forp > (a + IJ the range ofdecayingfluctuations decreasesrapidly and finally, far in the unstable region(P > a2 + 1),lc -> 0 and the evolution away from the reference state is

spontaneous.This picture is reminescentof the phenomenon of nucleationfamiliar from equilibrium phasetransitions (Zettlemoyer,1969).

We closethis sectionby a remark concerningthe compositionfluctuationsthemselveswithin AV. We first observe from Eq.A2.18)that the variancesdiverge in the neighborhoodof the criticalpoint, as the eigenvalueco^ of thecoefficientmatrix vanishesat this point.Thus

<<5A:2>oc\342\200\224 A2.31)

324 A'\342\226\240'\342\226\240Mean-field'\" Description ofFluctations: Nonlinear Master Equation

and similarly for <<5Y2>(etc.).Now, from Fig.12.2we seethat at the criticalpoint \\c

\342\200\224\342\226\272 x,in other words,3> \342\200\224\342\226\2720.Thus

A232)

We seethat the critical exponentexpressingthe divergenceof (SX2}is equalto twice the value of the critical exponent expressingthe way the orderparameter varies in the supercriticalregion.Again, this relation is analogousto thoseprevailing in equilibrium phasetransitions.

12.5.ONSETOF A SPATIAL DISSIPATIVE STRUCTURE

We considernow the casewhere X and Y in the trimolecular model arecharacterized by two different coefficients3>i and 3>2-As recalledin theSection12.4,in this casethe macroscopicbalance equationsadmit solutionsdisplayinga regular spatial structure. Herewe want to understand the onsetof these patterns through fluctuations, starting from an initially uniform

systemat the steady state A2.24).Following the sameprocedureas in Section12.4and truncating the hier-

hierarchy ofmoment equationsto the levelofsecond-ordervariances,one obtainsa differential systemfor mean quadratic fluctuations that has the same formas in Eq.A2.18),exceptthat the matrix of the coefficientsis now given by:

A2.33)2@-1)-2S

-00

0-a22a2-1-0,-20 \342\200\224 2'

0a2

a2 -

The eigenvaluesof this matrix are

to, = p - a2 - 1 -(\302\251,

+ 22)oi+ = P - a2 - 1 -@, + 92) \302\261 ^/M A2.34)

with

M = IP- a2 - 1 -@, + \302\2562)]2

2- /?0j + S>2 + a2\302\256, + a2) A2.35)

The secondmoment equation admits a nonoscillatory instability of the

steady-statesolution if one of the rootsco\302\261, a>i vanishes.This happensunder

12.5.Onset ofa Spatial Dissipative Structure 325

the following conditions,which have the same structure as relations en-encountered in Section7.4:

67)

P < pc = 1 + a2 + 9, + 92Introducing the coherencelength / through the relation A2.12):

we finally obtain:

A2.36)

A2.37)

A3.38)

A2.39a)

A\302\253l+a!+^l+ri (,2.39b)

Figure 12.3representsthe criticalcurves lc = lc{P)correspondingto relationsA2.39a)and A2.39b)in the (/, /?) plane for a = 2 and (D,//ri)= ^\302\243>2/U-

The important new point is that in the region /?\302\260

<P < P] the instabilityof the referencestate involvesfluctuations offinite coherencelength. In this

same region, the imposition of large-scalefluctuations (/ -> oo) stabilizesthe steadystate.Beyondp = p],however,that is, beyondthe asymptoteof thecurve p =

p~c, the behavior becomessimilar to that outlined in Section12.4.

Unstable

e\302\260c el el-2 e

Figure 12.3.Coherence length of fluctuation at critical point versus chemical parameter ji in

trimolecular model, a = 2,and ratio DJD2=|is compatible with emergence of steady-statedissipative structures asdeduced from deterministic analysis.


In contrast, the \"orderparameter\"behavesin exactlythe sameway as in thelimit-cyclecase.Indeed,aswesaw in Sections7.6and 7.7,the amplitudeof theemergingdissipativestructure beyond the first instability at

/?\302\260variesas:

||x||ac.(p- P\302\260V2 A2.40)

At first sight, the occurrenceof a finite coherencelength and continuousorderparameter seemsparadoxicaland difficult to fit in the framework of themore familiar equilibrium criticalphenomena.A closerinspection,however,revealsthe existenceofa secondcharacteristiccoherencelength* that divergesas one moves to the critical point /?\302\260

from the pretransition region.In this

way the problemunder considerationresemblescloselythe crystallizationproblem(Brout, 1965).In the latter, in the orderedphase,there existsafinite characteristic length (the lattice parameter),whereas in the disorderedphasethere appearlong-range correlationsup to the solidificationpoint.

To seethe appearanceof the secondcoherencelength in our problemweproceedas in the end of Section11.10.We considerthe marginal stability

equation A2.39a)in the equivalent form:

+ -= 0 A2.41)

where

'ri 'r2

From the results ofChapter11weexpectthe quantity 1//to scalethe spatialcoordinater2 \342\200\224 r, appearingin the correlationfunction G(AVU AV2) betweentwo spatial regionst:

U AV2) = g(\"-^A A2.43)

If 1//is real, the function G then describesa self-maintainedcorrelation ofmacroscopicrange betweenspatial regions.In contrast, a complexvalue of1//impliesspatial damping in the form of evanescentwavesemanating fromAK The point is that Eq.A2.41)gives riseto complex/ values in the \"dis-\"disordered

\"region/? <

/?\302\260.Onefinds, in exactlythe sameway asin Section11.10:|\302\2614

* A similar remark has been made for the Benard problem by Zaitsev and Shliomis A971).Seealsothe work by Lekkerkerker and Boon A975).t Note the change in notation between this and Chapter 11.Thus, in Section 11.10/ denotesthe length of the system, which in the present chapter is taken to be infinite. Thecoherence

length denoted by ).in Section 11.10is now denoted by /.

12.6.Multiple Steady-state Transitions and Metastability 327

where1 _ _ 1 / 2 0, \\

1 1\342\200\224 = \342\200\224

I/J\342\200\224

/?\302\260|1/2(/?\342\200\224 /?_I/2 A2.45b)' #1

and

A2.46)

If p is slightly subcritical, then

P_ <P <P\302\260c A2.47)

and /\" is,indeed,a real quantity. As the critical point p\302\260is approached,the

length /\" divergesas [seealsoEqs.A1.80)and A1.81)]:

(P-p-)\"iir rc'that is,accordingto a \"classicalexponent\"equal to \\, which isthe sameasthe

exponentappearingin the orderparameter equation A2.40).Thus, the wavesemanatingfrom a regionA Fareno longerevanescentbut give rise,instead,toa sustainedregimecharacterizedby the length /'.A simplecalculationidenticalto the one reportedin Section11.10showsthat /' is identicalto /\302\260 at the criticalpoint (seeFig.12.3).

12.6.MULTIPLESTEADY-STATE TRANSITIONSAND METASTABILITY

We illustrate the basicideasdirectly on a simplemodel.Considera latticesurfaceelementwith N binding sites.The systemis in contact with a reservoirofa constituent Xewhereconcentrationsare heldfixed.Xe may combinewith

a freesite,and beadsorbedby the lattice. We shall call X the correspondingcompositionvariable. On the other hand, X can undergo a desorptionpro-process and again becomeXe.Thesephenomena can berepresentedas:

X

X \342\200\224\302\243*-\302\273 XP+F A2.49)


where F denotesthe freesites.The rate equationsare:

dK-kFX-kX--dFdt dt

where X denotesthe macroscopicallyobservedvalue of X.Now,

X + F = const = N

and

dX__ ,dt

~ iy

Introducing the absorption-desorptionequilibrium constant k = k2/k1Xe,and setting k^Xe \342\200\224 a, we have:

A Y^

\342\200\224 = a[N - A + k)X~\\ A2.50a)dt

Furthermore, we assumethat the adsorption-desorptionprocessis cooper-cooperative, that is, that k is influencedby the number of X already adsorbed.Wechoosefor illustrative purposethe relation

k = k(X) = k exp -~- A2.50b)

where K and n are constants.Note, however, that this expressionis notarbitrary. It can beattached to a mean-fieldtype of treatment of the lattice,asis frequently done in the context of equilibrium phase transitions (Brout,1965).

EquationsA2.50)can bestudiednumerically at the steady state.Onefinds(Prigogine,Lefever,J.S.Turner, and J.W. Turner, 1975;Lefever,1975)that

when n >4 there alwaysexistsan interval Kt < K < K2 where,for each K,the equation admits three equilibrium solutions(seecontinuous S curvein Fig.12.4).Two of them are stable,the third one(intermediatebranch in

Fig.12.4)beingunstable.Notethat the instabilitiesfound here do not requirenecessarilynonequilibrium constraints.Thus, in this sectionwe dealwith aproblemthat is not specificto nonequilibrium situations, in contrast to theproblemsstudiedin Sections12.4and 12.5.

Systemsof this type have been studied extensively from the stochasticviewpoint, using the birth-and-death formalism (Landauer,1962;Janssen,1974;Matheson,Walls, and Gardiner,1975).Onefinds that at a certain valueof the parameter determining the stability propertiesof the macroscopicequations,the systemdisplaysa bimodal distribution with two equal heights

12.6.Multiple Steady-state Transitions and Metastability 329

2.55In K, 2.75

N= 200

2.95In Kc 3.15lnK

3.35inK7

Figure 12.4.Bottom: branch of macroscopic steady states calculated from Eq. A2.52) asfunction of In K for N = 100and i\\

= 6 (continuous curve); dashed line: branch of averagevalues calculated from master equation without diffusion. Top:In !2)casfunction of K for N = 50,100,200.Dashed areasindicate domains where value of<X}/N lies when 3)>@c.

correspondingto a simultaneousoccupation of the two stablebranches.Forthe model of this sectionthis critical value is

A2.51)

The analogy with the van der Waals theory of first-order phasetransitions(Brout, 1965)is striking and has been repeatedly emphasized.ForK # Kcthe distribution evolvesto a singlebranch, although the time ofevolutiongets


longer with larger N. Although this is somewhat reminiscentof the pheno-phenomenon of metastability, the correspondenceis difficult to establish.

Prigogine,Lefever,J.S.Turner, and T. W. Turner A975)have investigatedthis phenomenon by a local approachto fluctuations basedon a masterequation of the type A2.10).Intuitively, it is clearthat only by this treatment,which introducesthe conceptof the range of a fluctuation, couldone tacklethe problemof metastability.The master equation reads:

, t) ,{N x + 1)p(x l t) _ (N _ X)P(X,t)dt

/ (x + n\\ - 1,0- XKri

-1,0-P(X,t)]\\)P{X +1,0-XP{X,0] A2.52)

Herethe expressionA2.12)for 3) has to beadaptedto the fact that we dealwith a surfaceAS rather than with a subvolumeAV. Consequently:

3*^ A2.53)

where AP is the perimeter of the surfaceelement AS.In dealing with Eq.A2.52)onemust be aware of the fact that all of the

techniquesbasedon asymptotic expansionsused in the precedingsectionsbreakdown completely.This is dueto the fact that the transitions associatedwith a multiple steady-stateproblemare \"

hard,\" in the sensethat the distancebetween final and referencestates is finite. In other words,the stochasticaverage (X) (which appearsin the master equation) and the most probablevalue X (which appears in the macrosocopicrate equation) differ by amacroscopicamount. Thus,onehas to solveEq.A2.52)asa wholeand avoidmoment truncations. Fortunately, thanks to the validity of the detailedbalancecondition, the master equation for this model can be solvedexactlyat the steady state by iteration. One finds (Prigogine,Lefever,J.S.Turner,and J.W. Turner, 1975):

\302\243A2.54)

For3> = 0onerecoversthe birth-and-deathdescription,and in particular oneshowsthat the first moment <X> is always a single-valuedfunction of K(Fig.12.4,dashedcurve).In contrast, if 3) is greater than somecritical value\302\256c,Eq. A2.54)yieldstwo equilibrium solutions in the interval Ki < K < K2-In addition to the distribution correspondingto the stablebranch found in

12.7.Asymptotic Solutions ofNonlinear Master Equation 331

the birth-and-death description,onefinds a new distribution with an averagenear the valueof X on the other branch, which henceappearsasa metastablebranch in the usual senseof the term. Onecan show that 3)cdependson K.On the other hand, accordingto Eq.A2.53)3>dependsalsoon the sizeN, asAP/AS ocN~112.Thus, the value of 3) at the critical state where the newbranch appearsgives information about the sizeof the fluctuations beyondwhich the transition from the metastableto the stablebranchbecomespossible.This correspondsto the critical nucleusof germination.

Theseideascouldbe interesting to a broadclassof phenomena in which

metastablestatesare involved, suchasthe formation of fractures in metalsorthe ignition of explosivereactions.

12.7.ASYMPTOTICSOLUTIONSOF NONLINEAR MASTER EQUATION

Having now beenacquainted with the practiceand predictionsof the mean-field theory of fluctuations,we return to the nonlinear master equation andanalyze somegeneral features of its solutions.As we saw in Section12.4to12.6,an exactsolutionof this equation isgenerallyan arduoustask.Therefore,it would behighly desirableto develop approximate but systematicschemesfor calculating thesesolutions.

The resemblenceof Eq. A2.10)to the equationsencounteredin kinetictheory suggeststhat onecouldseekfor asymptoticsolutionsof a kind similarto the Chapman-Enskogsolution of the Boltzmannequation (Chapman andCowling, 1952).These,known also as normal solutions,are based on the

presenceofprocessesevolving accordingto two time scalesin the Boltzmannequation:(a)one associatedwith the freemotion of the particlesand (b) theother associatedwith collisions.Now, this is exactly what happensin thenonlinear master equation A2.10)and, indeed,in the multivariate masterequationaswell,exceptthat the roleofthe reactionand diffusion contributionsare somewhat the \"inverse\" of the collisionand freeparticle contributions in

the Boltzmann equation.Indeed,in our casethe reactivecollisionstend to

destroy the Poissoniandistribution\342\200\224the analog of local equilibrium in

kinetic theory\342\200\224whereas diffusion tendsto reestablishthis distribution. Inso-Insofar as

RAV(X) cc rjh1

and

fchS> oc 'f-> 1


where rch is the characteristicscaleof(macroscopic)evolutionofcompositionvariables due to the chemical reactions,onecan introduce in Eq.A2.10)adimensionlesssmallnessparameter

e = -^\342\200\224 = {-^-) = ly<\\ A2.55)t9 \\DtJ 'The master equation takesthe form [setting t = (r/rch)]

^f \\\302\253y[AV(X - 1,0- Pav(X,01

+ (X + \\)PAV(X + 1,r) - XPAV(X, t)} A2.56)

and we want to investigate the existenceof asymptotic expansionswith

respectto this parameter:P = P@)(X,t) + eF^X, t)+ \342\226\240\342\226\240\342\226\240 A2.57)

where we drop from now on the subscriptAV. A similar procedurecan beenvisagedfor the multivariate master equation.

As is frequently the casewith asymptoticexpansions,it is very difficult todetermine the range of validity of Eq.A2.57).In particular, onewould liketo know whether a power-seriessolution in ehas a finite radiusofconvergence,or if not, whether an asymptotic (and generallynonconvergent)expansionin

powersofeexists,orfinally, whethereis not agoodexpansionparameter at all.Hereafter,we regard Eq.A2.57)as aformal seriesand try to determine the

successivecoefficients PA) from the master equation. Substituting into

Eq.A2.56)and taking into account that R dependslinearly on P(X)and that

it alwaysremains finite, we have (Horsthemke,Malek-Mansourand Hayez,1977):

To ordere^1:-i,r)-p@)(x,r)]

+ (X + \\)Pm(X +l,f)-XP@)(X, 0= 0 A2.58)To order e\302\260:

-^\342\200\224 = r(p{0))+ Kxyo)[p(i)(x- i,o- p{1\\x,o]at

+ (X + \\)pA\\x + i,o- xp{i)(x,t)+ (Xyl)\\_P@\\X

- 1,0- P@)(X,0]} (etc.) A2.59)

Herewe have alsoexpandedformally <X> as:+ \342\200\242\342\200\242\342\200\242 A2.60)

72.7. Asymptotic Solutions ofNonlinear Master Equation 333

The point is now that the equation at the dominant order, Eq. A2.58)admits a Poissoniandistribution as the only solution,with an as yet undeter-undetermined mean <X>@):

p@) =e-<x><\302\260>

_^A/ ' A2.61)

In order to compute (X)*01we go to the next order,Eq. A2.59),multiplyboth sidesby X, and sum over X. We find, in the notation of Section12.3:

sincethe effectof diffusion cancels.Moreover,as the average is taken over aPoissoniandistribution onecan, accordingto the resultsof Section10.4,identify the right-hand side(a1A-><0)to the macroscopickinetic law:

@)

) A2.62)

Thus, <X>@) and P<0)are now entirely determined, thanks to the macro-macroscopic equationsof evolution.The situation isstrongly reminiscentof kinetictheory of gases,where the parametersappearingin the local Maxwelliandistribution\342\200\224temperature, density,and convectionvelocity\342\200\224are determinedby the equationsof fluid dynamics.

The next step is to determine PA) from Eq.A2.59),keepingin mind that

both P@> and <X>@) are known. We have:- l,o- pa\\x,0]+ {x+ \\)P{l\\x + i,r) - xpv\\x,t)

dPi0)[P@\302\273(X

- 1,r) - P@)(X,tj] - R(P{0))+ \342\200\224r- A2.62a)dt

The finite-differenceoperator appearingon the right-hand side is alwaysinvertible.Forinstance, in the generating function representation it wouldgive riseto the extremelysimpleexpression

= M(F@), <X>A)) A2.63)

<X>@)FA1 = right-hand side

where

F^\\s,t)= Y,sxP{1)(X,t)(etc.)x

Thus,FA1 can becomputedstraightforwardly provided (X}A)can bedeter-determined. This is achievedby working out the equation to ordere,which yields


a differential equation for (X>A).As PUIis,in general,different from Poisson-ian, the equation for <X>A1 contains correctionsfrom the macroscopiclaw:

+ corrections A2.63a)at

This schemecan becontinued to get as many correctionterms toand Pi0)(X,t) as desired.Having P onecan then compute the variances,forinstance:

CX2)= <<5X2><\302\260> + eEX2y^ + \342\226\240\342\226\240\342\226\240

= (XyO)+IyCX2y^ + --- A2.64)The remarkablepoint that appearshere is that the propertiesof fluctuationsin a volume A V dependexplicitlyon the sizeof this volume.When the lattergoesto zero one recovers a Poissonianvariance. Otherwiseone obtainscorrectionterms.This illustrates in a very transparent way the passagefromshort-range fluctuations, describableby the Fokker-Plancklimit ofthe phase-spacemasterequations,to large-scalefluctuations.A similar resultisobtainedin Section11.11from the multivariate master equation by integrating thecorrelation function G(rl,r2)over a finite volume(seealsoGardiner,McNeil,Walls, and Matheson,1976).This extremelystriking featureof fluctuations in

nonequilibrium systemsshould bedirectlytestableby scatteringexperimentsfrom a reacting fluid.

A final remark concerning systemsgiving riseto instabilities is in order.Within the framework of the e-expansion,[Eqs.A2.57)-(12.60)],instabilitymanifestsitselfexclusivelythrough the equationsfor the successiveapproxi-approximations (X}@\\...to the averagevalue<X}.The correspondingprobabilitydistributions PA) merely

\"adjust\" to the evolution of the averages,although

they are at the sametime deviating from the Poissoniandistribution. Onecan expect,however, that in the vicinity of the instability the successivecorrectionsbecomeincreasinglyimportant, partly becausee itselfwould nolonger bea small quantity, owing to the long-rangecharacter of the correla-correlations. Eventually the expansionbreaksdown and the dependenceof thesolution on / becomesmore complicatedand not necessarilyanalytic.


The theory of fluctuations in nonequilibrium systemsoutlined in this andprecedingchaptershas enabledus to draw somedefinite and unexpectedconclusions.Firstly, we have seen that the propertiesof fluctuations in afinite volume dependexplicitlyon the size of this volume. Secondly,in the

vicinity of an instability the systemdevelopslong-rangespatial correlations,

12.8.Concluding Remarks 335

even if macroscopicallyit remainshomogeneousin space.Thesecorrelationsare characterized by a coherencelength that is an intrinsic parameter of thesystem and that divergesas oneapproachesthe critical point. Thirdly, theonset of nonequilibrium instabilities in macroscopicsystemsis due to thedeviation of the fluctuations from the Poissoniandistribution.

Despitetheseadvances,oneshould beaware of the fact that many impor-important problemsremain unsolved.The main reasonfor this is the difficulty tosolve the completemaster equation without truncating the hierarchy ofmoments, save for a few particular casesinvolving a single variable. As aresult,onecannot yet estimatesatisfactorilythe a priori probability ofhavinga fluctuation of a certain sizeand range appearingspontaneously in the

system,nor the rate of growth of such a fluctuation beyond instability.

Similarly, a number of quantitative predictionsbased on asymptoticsolutionssuch as truncated moment expansionscouldconceivablybemodi-modified by a more refined treatment. We have in mind, in particular, the oc-occurrence of \"classical\"exponentsarising in the analyses of Sections11.10,12.4,and 12.5.This couldwell be the consequenceof having neglectedtriplecorrelationsin the multivariate master-equation treatment of Section11.10,and of having a mean field theory of fluctuations in the analysisof the presentchapter.A more satisfactory treatment of correlationsmight well producenonclassicalexponentsin a way similar to Wilson's extension of Landau'smean field theory of equilibrium critical phenomena.

Part IV

ControlMechanismsinChemicaland BiologicalSystems

We comenow to the main objectof our study. We want to show how themethodsand techniquesdevelopedso far and illustrated on modelsmay beappliedin the analysis of concreteproblemsin a wide variety of subjects.In this part we first treat various topics related to self-organization in

chemistry and biology.Chapter13is devoted to the analysis of dissipativestructures in singlereaction chains.In Chapters14and 15we are concernedwith biochemicalreactionsat the subcellularand cellular levels.Finally, in

Chapter16we considermore complexsituations at the supercellularlevelinvolving the \"collective\"behaviorofpopulationsofcoupledcells.In Part V

the conceptof self-organization is extendedto evolutionary problemsin-involving competition.In Chapter17wediscussthe evolutionofinteractingandreplicating macromolecules.Our purposeis to investigate whether such apopulation can attain statescharacterized by an ordersimilar to that of the

geneticcode.This is the problemof the origin of life.Finally, in Chapter18we analyze self-organizationphenomena at the level of interacting popula-populations in an ecosystem.

Chapter13

Self-organizationinChemicalReactions

13.1.INTRODUCTION

We limit ourselvesto a discussionof reactionsin homogeneousphase,where the breakingof spatial homogeneity and the emergenceof spatio-temporal order are most striking. The earliest reportedperiodicchemicalreaction in homogeneoussolution is the catalyticdecompositionofhydrogenperoxideby the iodicacid-iodineoxidation couple(Bray, 1921).Despitethe

many experimental studiesof this reaction, the detailedmechanism of theoscillation itself remained a matter of controversy until recently (Nicolisand Portnow, 1973;Noyesand Field,1974;Sharma and Noyes,1976).

BelousovA958)reporteda secondcaseof an oscillatorychemicalreactionin homogeneoussolution, which is the main subjectof the presentchapter,specifically,the oxidation of citric acid by potassiumbromate catalyzed bythe ceric-cerousion couple.ZhabotinskiA964) pursuedthe study of thereaction and demonstrated that the cerium catalyst couldbe replacedby

manganeseor ferroinand that the citricacidreducingagentcouldbereplacedby a variety of organic compounds,either possessinga methylene groupor easily forming such a group during oxidation (e.g.,malonic or bromo-malonic acid).Usual experimental studies involve a reaction mixture atabout 25\302\260 consistingof potassiumbromate, malonic or bromomalonicacid,and eeriesulfate (or someequivalent compound)dissolvedin sulfuric acid.Dependingon the conditionsimposedon the system,the different phenomenaobservedare compiledin Section13.2.

13.2.BELOUSOV-ZHABOTINSKIREACTION:EXPERIMENTAL FACTS

SpatiallyUniform Mixture

When the reaction occursin a well-stirred medium, sustainedoscillationsin the chemicalconcentrations appearfor certain rangesof initial composi-composition of the mixture. Figure 13.1describesthe temporal evolution of the

339

340 Self-organization in Chemical Reactions

15 17.0 2410Minutes

\302\273-

Figure 13.1.Potentiometric traces at 25\302\260 of log[Br~]and log[Ce4+ /Ce3+ ] versus time duringBelousov reaction: [malonic acid]= 0.032M.[KBrO3]= 0.063M,[Ce(NH4J(NO3M]0=0.01M,[H2SO4]= 0.8M,and [KBr]0= 1.5x 10\025M [from R. J.Field A972), J.Chem.Educ. 49,308].

concentrationsof Br\" and of Ce4+/Ce3+ . The periodis of the orderof theminute, whereas the life time of the phenomenon is of the orderof the hour.Eventually the oscillationsdie out, as the system remains closedto masstransfer and the raw materials necessaryfor the reaction are exhausted.Thus, although the initial mixture may be removed very far from thermo-dynamic equilibrium, it finally tends to the state of equilibrium whereoscillatorybehavior is ruled out. Recently,however,Marekand Stuchl A975)and Marekand Svobodova(1975)reportedoscillationsfor this reactionunderopen-systemconditions.Note that with bromomalonic acid presentinitially,oscillationsin the ceric-cerousconcentration begin immediately,whereaswith malonic acidthere is an induction period.The sharpness,stability, andreproducibilityof the oscillationsand the existenceofa threshold ofchemicalconcentrations necessaryfor oscillatory behavior prove that the chemicaloscillationsshown in Fig.13.1belongto the classof temporal dissipativestructures.This is substantiated by the mathematical model of the reactionanalyzed in Section13.4.

Spatially InhomogeneousMixture in OneDimension

When the reaction is carriedout in a thin, long vertical tube (Busse,1969;Herschkowitz-Kaufman,1970)there appearhorizontal bandscorrespondingto alternatively high concentration regionsof the chemicals(seeFig.13.2).The analogy with the spatial dissipativestructures displayedin Chapter7 is

tempting. Recently,however,Kopelland HowardA973b)and ThoenesA973)pointed out that under usual experimental conditionsthese patterns arelikely to becausedby external temperature and density gradientsrather than

by a symmetry breakinginducedby diffusion (seealso Beckand Varadi,1972).Nevertheless,it should be mentioned that Marek and SvobodovaA975)have recently reportedresultsindicating the existenceof steady-state,space-dependentsolutions in a flow reactor undergoing the Belousov-Zhabotinskireaction.

Figure 13.2.Horizontal bands in the Belousov-Zhabotinski reaction. Light strips correspond to

regions where oxidation steps are predominant(after M. Herschkowitz-Kaufman, 1970).

SpatiallyInhomogeneousMixture in Twoand Three Dimensions

Finally, when the reaction takesplacein an unstirred shallowlayer(e.g.,in apetri dish)or in a usual container, various forms of wavelike activity areobserved(Zaikin and Zhabotinski,1970;Winfree, 1972,1974b).In the thin

layer the most common ones are concentration waves with cylindricalsymmetry or rotating spiralwaves. Figure 13.3showsthe successivecon-concentration patterns in the latter case.It is noteworthy that the appearanceof


Figure 13.3. Spiral wave activity in Belousov-Zhabotinski reaction carried out in petri dish

(after A. M. Zhabotinski, 1974).

thesewavesdoesnot necessarilyrequire a locallyoscillatingreaction.On theother hand, a leadingcenter area exceedinga critical dimension seemsto bealwaysinvolved.In the three-dimensionalcasethe spiralstake the form of an

unfolding scroll.Sometimesthe scrollaxiscan closeinto itself.The reactionremains oscillatoryeverywherein spaceexceptalong the axis.It is generallyacceptedthat the emergenceof thesepatterns is likely to requirean instability

13.3.Mechanism 343

causedby diffusion. We have, therefore,a striking experimental illustrationof a spatiotemporaldissipativestructure. Equally striking is the resemblanceof the patterns of Fig.13.3to the patterns deducedfrom the theoreticalmodelanalyzed in Chapter7 on the one hand, and on the other hand, to thetype of activity onetendsto associatewith biologicalsystems.

13.3.MECHANISM

In order to interpret these fascinating and extremely spectacularphe-phenomena* on a theoreticalbasis,it is necessaryto have the detailedmechanismof the reaction.Recently,Noyes,Field,Koros,and co-workers(seeNoyesandField,1974for a recent survey) completedan extensiveseriesof experimentson the malonic acid-bromatereaction and proposeda detailedkineticmechanism comprisingnot lessthan 11steps.Fortunately, they were ableto simplify their more detailed mechanism and interpret the oscillationsinhomogeneoussolution in terms of the propertiesof three key substances:(a) HBrO2,which seemsto play the roleof a switch intermediate, (b) Br\",which seemsto play the roleof a control intermediate, and (c)Ce4+,which

can be regardedas a regenerationintermediate in the sensethat it is rapidlyproducedwhen the systemis switchedin onedirectionand permits thereafterthe formation of the control intermediateBr~.Moreprecisely,the followingreactionsseemto dominate,accordingto the abundanceof Br\" :\342\200\242 With enough Br

~:(i) BrO3-+ Br + 2H+ kl

> HBrO2+ HOBr

(ii) HBrO2+ Br~ + H+ \342\200\224^-> 2HOBr\342\200\242 With small quantities of Br~ left, Ce3+ is oxidizedaccordingto:

(iii) BrO3 + HBrO2+ H+ k'> 2BrO2-+ H2O

(iv) BrO2-+ Ce3++ H+ ki> HBrO2+ Ce4+

(v) 2HBrO2 \342\200\224^-> BrO3\" + HOBr+ H +

The first step is rate limiting, whereas HOBrdisappearsquickly by com-combining with malonicacid.From (i) and (ii), a quasisteadystate is reachedwith

a concentration

wherekjk2~ 10\"9.\342\200\242A nice feature of this reaction is that it can be carried out extremely easily. The interestedreader is urged to try to apply oneofthe recipesgiven by Winfree or by Herschkowitz-Kaufman.


Step(iii) is rate limiting. Steps(iii) and (iv) taken together are equivalentto an autocatalyticgenerationof HBrO2\342\200\242A new quasisteadystate is reachedwith

where

r^ ^ 10\024, k3 = l0*M~h~l.

It isgratifying to see,in this concretesystem,the importance of autocatalyticprocesseswhich wasanticipated in the analysisof the mathematicalmodelofChapter7.

Now from (ii) and (iii) it appearsthat Br~ and BrOjcompetefor HBrO2.Autocalytic productionof the latter will beimpossibleas long as

/c2[Br\"]> /^[BrOj] A3.3)

Thus, at the critical concentration value

[Br-]f=~ [BrOj] =* 5 x lO^BrC^] A3.4)k2

the reaction switchesfrom pathway (i)-(ii)to pathway (iii)-(v).As [HBrO2]increases,then from (ii) Br\" is consumed,and [Br~]dropsbelow the criticalvalue A3.4).On the other hand, the producedCe4+regeneratesBr^ accord-according

to the global reaction:

(vi) 4Ce4++ BrCH(COOHJ+ H2O+ HOBr k\">

2Br\"+4Ce3++ 3CO2+ 6H+

Subsequently,[Br\"]exceedsthe threshold value A3.4)and [HBrO2]comesbackto the levelgiven by Eq.A3.1).In this way the occurrenceofoscillationsis explainedqualitatively. A quantitative study of the model basedon themechanism just outlined confirms the existenceof oscillationssimilar tothoseshown on Fig.13.1,as shown in Section13.4.

This successfulanalysis of the mechanism of the Belousov-Zhabotinskireaction has motivated a number of further investigationson nonbiologicalreactionsgiving rise to dissipative structures.Thus, recently Pacault, deKepper,and HanusseA974, 1975)have conductedexperimentson the

Briggs-Rauscheroscillating reaction (Briggsand Rauscher, 1973).Thisreaction involves hydrogen peroxide,malonic acid, KIO3,MnSO4,andHC1O4and can becarriedout under open-systemconditions.In addition tosustained oscillationsone observesabrupt transitions in the chemical

13.4.The\"Oregonator\" 345

concentrations, indicating the existenceof multiple steadystatesasdiscussedbriefly in Section13.7.

Finally, Janjic and colleagues(Janjic and Stroot,1974;Stroot and Janjic,1975)report a seriesof studieson oscillating reactionswhereby a numberof ketoniccompoundsgenerate oscillationsin Belousov-Zhabotinski-typereactionsinvolving bromate as well as a catalytic agent.

13.4.THE \"OREGONATOR\"

We dealexclusivelywith modelsbasedon the reaction mechanismoutlinedin Section13.3.A different classof modelsbasedon a mechanismsuggestedby Zhabotinskihas beenworkedout by Zhabotinskiand co-workers(seeZhabotinski,1974for a recent survey), ClarkeA973),Tomita and KitaharaA975),and Othmer A975).

The model discussedbelow has been introduced by Field and NoyesA974,1975)and isfrequently referredto in the literature asthe \"Oregonator.\"Let

X = [HBrO2]Y = [Br~] A3.5)

Z = 2[Ce4+]be the concentrations of the three key substanceswhose behavior wasdescribedin Section13.3.Moreover,we set

A = B = [BrOj]P,Q= waste-productconcentration A3.6)

Then, weseefrom Section13.3that reaction (i)describesthe conversionof Y

to X, reaction (ii) the simultaneous inactivation of X and Y, reactions(iii)and (iv) the autocatalytic generation of X, reaction (v) the bimoleculardecompositionof X, and the globalreaction (vi) the transformation of Zinto Y. Hence,wewrite the following steps:

A + Y \342\200\224^-\302\273 X

X + Y

A3.7a)


34Here/ is a suitable stoichiometriccoefficient,*the rate constantskY to kcontain the effectof the concentration of H + , and k6 contains the effectofbromomalonic acid.The overall reaction is a linear combination of (Noyes,1976b):

/A + 2B * /P + Q A3.7b)for any/

(/ - 1)A + 2B \342\226\272 (/ + 1)P for / > 1

2B \302\273 2/P+ A - /)Q for / < 1

B \342\226\272 P for / = 1

Note that all reactionsare taken to beirreversible.The reversibleOregona-tor was recently studiedby Field A975).A comparisonwith the detailedmechanismsuggeststhe following valuesfor the rate constants:

k1 = 1.34M-Vk2 = 1.6x 109M-V

Ic5=4x107M\"VThe valuesof k6 and/areusedas parameters.Finally, in the region studiedexperimentally A = B = [BrO3~]= 0.06M,whereas[H+]= 0.8M.Here-Hereafter thesevaluesare taken constant, and this amounts to treating the systemas open.Becauseof this, of the (quadratic) nonlinearity of three of the stepsand the large distance from equilibrium, the system has the necessaryingredients for presentingcooperative behavior. We alsonote that such abehavior would be perfectly compatiblewith the Hanusse-Tyson-Lighttheorem, as the model comprisesthree variable intermediates.

Considerfirst a spatiallyuniform mixture. The rate equationsare:_ V A V V W _l_ b BV \"I b Y2~\342\200\224 ftjjii K2J\\. I T\" K34.D-A 1.K 5 j\\

at

\342\200\224 = -kyAY - k2XY + fkbZdt

~ = k3ABX-k(,Z A3.9)at

Theseequationsdisplaywidely different constantswith ordersranging from1 to 109.Thus, we expectthe systemto evolveaccordingto two time scales

*Actually, the regeneration step for Br^ described by A3.7a)might not bethe most appropriate

one.SeeFaraday Symposium on Oscillatory Phenomena, 1974.

13.5.Oscillatory Behavior 347

and perform, in particular, relaxation oscillationsin a way similar to the

systemtreated in the first subsectionof Section8.10.To seethis, wecast thedifferential equationsin terms of the dimensionlessvariablesx,y, z and t andthe dimensionlessparametersq, s,w definedasfollows:

k

A3.10)

and

rr4k2k34B V

The balance equationsA3.9)become:

dx l 2^\342\200\224 = s{y- xy + x-

qxA)

dl=X-(-y-xy+ fz) A3.12)

dzJx

= w(x - z)

Forthe estimates given above, one has q = 8.375x 10~6,s = 77.27andw = 0.1610kb. We seeimmediately that the system involves indeedtwotime scales,sincethe evolutionofxis determinedby the inversetime constants, which is large,whereasthat of y and z by inversetime constantsoforderofor lessthan unity.

13.5.OSCILLATORYBEHAVIOR

Equations A3.12)admit a trivial solution x0 = j0= z0 = 0 which is

always unstable, and a single positive steady-statesolution.This solutioncan beevaluated explicitlyfrom Eq.A3.12):

z0 = x0

y0 = t~~ - i[U+ /) - <7*ol A3.13a)


where

We want now to investigatethe possibilityof having chemicaloscillationsof the limit-cycletype. An indication of the existenceof such solutionsis if

(x0,)>o>zo)is an unstable criticalpoint.A linear analysisabout this point canbecarriedout explicitly,by the methods developedin Chapter6 and appliedon the exampleof Chapter7.Setting

x = x0 + xeM (etc.)one finds the characteristic equation (Fieldand Noyes,1974;Murray, 1974;Hastingsand Murray, 1975):

a>3 - Ta>2+ dot - A = 0 A3.14)where T,S,A dependon x0,y0, z0 and on the parameterss,w,f, q of which w

and/aretaken to vary whereass and q are kept fixed.We want at least one of the necessaryand sufficient conditionsfor the

rootsof Eq.A3.14)to have negativereal parts be violated.Theseconditionsare (Abramowitz and Stegun, 1964):

T<0, A <0, A - T3 >0 A3.15)Onecan seethat for all realistic values of the parametersthe first two con-conditions are satisfied.After somealgebra,the instability condition which isthe inverseof the third inequality can becast in the form

0 < w < wc{f) A3.16)where the stability curve wc(f) is given by

wc(/)= -^ [\302\2432+ /d -

*\342\200\236)]

^ 2 + /(I - xo)Y - 4E2[2qx20+ xo(q- 1)+ /]}1/2

A3.17)with

E = sy0 + l~ + 2qs\\x0 -\\ sVs / s

This relation is meaningful only if the right-hand sideis positive and this

requires:

2qx2 + xo(q- 1)+ / <0 A3.18)

J3.5. Oscillatory Behavior 349

With x0 as a function of/(seeA3.13b))and for the numerical valueschosenfor q, s and A the last inequality gives:

fu <f <fic A3.19)where

flc^0.500; /2c~2-412

Figure 13.4representsgraphically inequalities A3.16)to A3.19),where k6hasbeenusedasparameter insteadofw. It is worth noting the steepnessof this

stability diagram, ascomparedto thosediscussedin Chapter7.Forvaluesofk6 (or w) and/within the unstable region,sinceby Eq.A3.15)

the product of the rootsis alwaysnegativethere is at least one root, say col,which is negative.Thereexist,therefore,exceptionalpaths (seealsodiscussionin Section6.5)along which the perturbations decay back to the steady state

Coming now to the behavior of the systempast the instability, one shouldfirst quotea general result due to Hastingsand Murray A975)who proved,using topologicalmethods,that the modelequationspossessat least onefinite-amplitudeperiodictrajectory. The periodicsolutionshave beencon-constructed analytically using bifurcation theory (Erneux, 1977;Tyson, 1976)

480.0

360.0-

240.0

120.0-

0.0

- m1=1

i

@.9935, 450.05)

Stable

3.00.0 1.0 2.0

Figure 13.4. Linear stability diagram for Oregonator for q = 8.375 x 10~6,s = 77.27,iv = 0.1610**.


7.0-6.0

5.0

4.0

3.0

2.0

1.0

0.0

\342\226\2401.0

6 4.0 \342\200\224

\342\200\224

-

~l 1 1

\\i

35.8'

1

B

\\\\

Xrfvk\\

I

W3N?*o

|

\302\273

V>V\\

268

1 1

50

|

A

| |-3

1 1

-2 -1 |0

\\ny

(B)

|1

|2

|3-4.0-3.0 -2.0-1.00.0 1.0 2.0 3.0 4.0

log [Br~] + 6.5228

(A)

Figure 13.5.(A) Projection of limit-cycle solution for Oregonator in spaceof logarithmicvariables X and Y. (B)Characteristic curve in the X-Y plane obtained by applying the quasi-steady-state relation, Eq. A3.20).

or asymptotic methods(Stanshine, 1975).One finds that in the vicinity offu and/2c the singular point behaveslike a focus,unstable (if/lc </ <f2c)orstable (if/</lcor/>/2c).Within the unstable region the asymptoticanalysis yields a limit cycle of a finite amplitude, whereas for / <fu onefinds a small-amplitudeunstable limit cycle surroundedby a stableone oflarge amplitude. A similar situation is encounteredin Chapter 15.It ispossiblethat both limit cyclesin this region belongto subcritical branchesbifurcating from/lc (for the unstable cycle)and from/2c(for the stablecycle).

The results of computer simulations (Fieldand Noyes,1974)confirm thetheoretical predictionsand illustrate the almost discontinuouscharacter ofthe oscillationsarising from the occurrenceof two time scales.Figure 13.5/4representsthe projectionof the limit cycle in the X-Y plane.This shape,together with Eq.A3.12)suggeststhat a reasonableapproximation would beto assumethat x adjusts instantaneously to the concentration of y. Thevalidity of this \"quasisteady-stateassumption\"is basedon the fact that thefirst equation A3.12)can be written as:

1 dx-\342\200\224

s ax2

xy + x \342\200\224 qx

13.6.Spatial Patterns 351

Inx

5.0

3.0

1.0

1 0

I

\342\200\224

\342\200\224

i

I

I I

\342\200\224

i i

0.0 150.0 300.0r

450.0 600.0

Figure 13.6. Logarithmic plot of In x following a 6.5% decreaseof y from steady-state solution.Perturbation applied at t = 153.

Sinces ^> 1 the left-handsidecan besetequal to zero,and xcan beexpressedin terms of y by the algebraicrelation

x = {A - y) + [(i - yJ +2<z

A3.20)

which is representedin Figure 13.5B.One can show numerically that the

projectionsof the limit cyclein the Y-Z plane for the completemodeland forthe model where assumption A3.20)has beenmadeare quite similar.

Finally, the numerical analysis showsthat perturbations around a stablesteady state decay backto this state after displayinga flash of activity which

can last a very limited periodof time, as shown on Fig.13.6.

13.6SPATIAL PATTERNS

When diffusion is taken into account, the rate equations for the Oregonatorbecomemore complexto analyze.StanshineA975) performedan asymptoticanalysis of these equations,as describedbriefly in Section8.5.He obtainssolutions in the form of spiralwaves emanating from a \"leading center.\"It would be very desirableto completethese considerationswith linearstability- analysis to determine the role of diffusion in the initiation of the

spatial patterns describedin Section13.2.This would settle the point raisedby someauthors (e.g.,Thones,1973;Smoesand Dreitlein, 1973)according

i52 Self-organization in Chemical Reactions

to which all spatial patterns\342\200\224including spiral waves\342\200\224can be interpretedin terms ofthe dependenceof the frequencyoflocaloscillatorson temperatureand concentration. In this respect,however,we may note that the analysisof Chapter 7 and Section8.6suggeststhat spatiotemporalpatterns canappear in chemical systems,notably systems involving more than twovariables,asa result ofa symmetry-breakinginstability causedby diffusion.

A general analysis of the equationsgoverning the wave front in the

Oregonatorhas recently beenreportedby Murray A976).Wave solutions,which correspondto moving bandsof concentration,are shown to exist,andupper bounds for the wave speed are provided.Numerical results with

parameter values obtained from experiment, comparefavorably with theobservations on the Belousov-Zhabotinskireagent.

13.7.BRIGGS-RAUSCHERREACTION

Pacault and co-workers(Pacault,deKepper,and Hanusse,1975;deKepper,Pacault, and Rossi1976)have realizedan open homogeneousreactor forthe study ofthe Briggs-Rauscherreactionmentionedat the endofSection13.3.In addition to parameterssuch as the rate of stirring, the temperature,pressure,and rate of influx within the reactor, they control the concentrationsof a number of \"initial\" chemicalspeciesoncethe latter are injected within

Figure 13.7.Experimental stability diagram for Briggs-Rauscher reaction in spaceofchemical

constraints.

13.7.Briggs\342\200\224Rauscher Reaction

E

353

t

[CH,

0

[KI03],(COOHJ]O

[MnS04]o

5

= 0,17mol. 1

= 0,058mol.= 0,004mol.

10irii

-,_I\021,

I-1,[H2O2]O

[HC1OJ,,

= 1,2mol.I\021;

=0,019mol. I\"';T= 25-C.

Figure 13.8. Double periodicities arising in Briggs-Rauscher reaction.

the reactor and are suitably homogenized.These\"chemicalconstraints\"are the concentrations of (H2O2J,CH2(CO2HJ,and KIO3.The responseof the system to theseconstraints is determined by measuring the instan-instantaneous values of the temperature, redox potential, and concentration ofiodine,which is formed during the reaction.

Dependingon the values of the various constraints, one finds oscillatoryor nonoscillatorystates.Figure 13.7depictsthe state diagram of the systemin

the spaceof the three chemical constraints referred to above. Inside thesurfaceone has oscillatorybehavior,whereasoutsidethe systemevolvesto asteady state.

A very interesting point concernsthe transitions between states. Oncrossingthe boundariesof the state diagram one can, for certain additionalslight variations of the parameters,switch betweenan oscillatorystate and asteady state correspondingto a different (average)concentration level,andviceversa.Thesediscontinuoustransitions may or may not beaccompaniedby hysteresis.

A different possibilityis illustrated in Fig.13.8.Insteadof performing asingle jump to a new state, the system switches periodically between two

oscillatory steps having different periods.Such \"doubleperiodicities\"arisefrequently in systemsinvolving coupledoscillators(cf.Section8.7)andcouldprovideuseful indications in the elaboration ofplausiblemodelsfor thereaction.

Chapter14

RegulatoryProcessesat theSubcellularLevel

14.1.METABOLICOSCILLATIONS

Oscillatoryand other kinds of self-organizationphenomena are observedatall levelsof biologicalorganization, from the molecular to the supercellular,or even to the socialone,with periodsranging from secondsto days or evenyears.In this chapter we are interestedin biochemicaloscillationsat thesubcellularlevel and the comparisonof some theoretical models with

experimentaldata.The most characteristicfeature of this type of oscillationis that they involve enzyme regulation at a certain stage of the reactionsequence.This may be due to one of two causes:(a) periodicity in enzymesynthesis due to control mechanisms at the genetic level (seeChapter 15)or (b) periodicity in enzyme activity due to regulation at the level of theenzyme itself; these so-calledmetabolic oscillationsare the main subjectof the presentchapter.While the first type of oscillation, known also asepigeneticoscillations,have a periodof the orderof the hour, the periodofmetabolic oscillationsis of the orderof the minute.

Severalreviewsof biochemicaloscillationsare available in the literature(Hessand Boiteux, 1971;Goldbeterand Caplan,1976).In this chapter wefocus on a few representativeexamplesof oscillatoryenzymesgiving risetometabolic oscillationsand related self-organizationphenomena.We treat,successively,the phosphofructokinasereaction in connection with the

glycolytic oscillations,the adenyl cyclasereaction, which catalyzes the

periodicsynthesisofcyclicAMP in someorganisms,and reactions involvingmembrane-boundenzymes.

14.2.THE GLYCOLYTICCYCLE

Glycolyticoscillations,which have beenstudiedfor more than 10years,arethe best-knownexampleof metabolic oscillations.They have beenobserved

354

14.2.The Glycolytic Cycle

GLU

355

TPI

GAP

3-PG

IPGM

2-PG

PPH 1Lii .PEP I

-ADP I

i

\342\204\242

Figure 14.1.Glycolytic pathway. Framed parts

represent steps involving allosteric enzymes PFK and

PYK.

in yeast cells,yeast cell-freeextracts, beef-heartextracts,rat skeletal-muscleextracts,and tumor cells.Finally, a reconstituted glycolyticsystemhas beenstudiedin great detail and depth by Hessand co-workers(seeHessand

Boiteux,1971,for a survey).Glycolysisis a phenomenon of the greatest importance for the energetics

of living cells.It consistsin the degradation of one moleculeof glucoseandthe overallproductionoftwo moleculesofATP,by meansofa linearsequenceof enzyme-catalyzed reactions summarized in Figure 14.1.The globalreaction reads:

glucose+ 2ADP +2P,\342\226\240

+ 2NAD

2(pyruvate) + 2ATP + 2NADH A4.1)

556 Regulatory Processesat the Subcellular Level

At severalpoints along the chain a number ofbranchingstake place,couplingglycolysisto other energetic processesin the cell,such as the Krebscycleand the respiratory chain. A coupling with the biosynthetic pathways alsooccurs,via the adenine nucleotides.

Let us now summarize the principal experimental facts for the recon-reconstituted glycolytic system, which is the basis for the theoretical modelpresentedin Section14.3.

Characteristicsof Oscillations

(i) Oscillationsin the concentrations of all metabolites of the chain areobservedfor certain (weak)rates of glycolyticsubstrateinjection.(ii) The periodof the phenomenon isof the orderof the minute and dependson the temperature. The concentrations of the metabolites vary periodicallyin a range between 10\025and 10~3M.(iii) All glycolytic intermediates oscillatewith the same period but with

different phases.They can be classifiedin two groupsdiffering by a phaseangle Aa, which is an increasing function of the substrate injection rate.

395nm

Yeast extractE7.4mg/ml)

Glucose input 100mM/hr

= 1-99= 5s

Figure 14.2.Experimental plots for NADH oscillations recorded by absorbance measurementsat 395nm. 355nm. and 355-395nm.

14.2.The Glycolytic Cycle $57

Within each of the two groupsthe constituents oscillatein phaseor with aphaselag of 180\302\260. Figure 14.2representsthe time variation of NADHduringseveralperiodsof the oscillation.(iv) The shape,amplitude, and periodof oscillation dependon the rate ofentry of the substrate.Belowa critical rate of injection the systempresentsdampedoscillationsor evolvesmonotonously to a steady state characterizedby a high level of NADH.When the rate of entry exceedsa certain criticalvalue the oscillationsare again damped,and one observesa steady statecorrespondingto a low levelof NADH.Thus, glycolyticoscillationsariseina bounded domain of rate of influx of the substrate.This fact, reminiscentofthe stability diagram of Fig.13.4for the Belousov-Zhabotinskireaction,plays a crucial role in the construction of the theoretical model reportedinSection14.3.

Roleof Phosphofructokinase(PFK)

By plotting the phaseof the glycolyticintermediates as a function of theirpositionalong the chain a crossoverdiagram may be realized indicating a

180\302\260 phaseshift at the levelof phosphofructokinase(seeFig.14.1).A similarshift is found for pyruvate kinase(PYK), whereas a smaller shift can beattributed to glyceraldehyde-phosphatedehydrogenase(GAPDH).Thisindicatesthe basicrole of PFKand PYK as principal pointsof control ofthe oscillation.

The important role of PFKis further confirmed by the observation that

fructose-6-phosphate(F6P),the substrate of PFK, seemsto be the lastglycolytic substrate capable of generating oscillatory behavior. Thus,addition of fructose-1,6-diphosphate(FDP)which \"by-passes\"the PFKstepdoesnot result in time periodicitiesin the pathway.

Now, accordingto Fig.14.1,PFKcatalyzes the quasiirreversiblereaction

ATP + F6P > ADP + FDP A4.2)

Moreover,the allostericnature of this enzyme*is revealedby its cooperativekineticswith respectto its substrates,products,and effectors (Mansour,1972).In particular, PFKis directlyactivated by its product,ADP, aswell asby AMP.This givesriseto an autocatalytic processreminiscentof the modelmechanismsdiscussedin Chapter7.3,which is rather uncommon in enzymeregulation. As we see shortly, this autocatalysis providesthe molecularbasisof glycolyticperiodicities.Note also that the secondproduct of theenzyme, FDP,does not play any regulatory role in the periodicitiesin

* We assume that the reader is familiar with the basic notions of allosteric proteins. A goodreview with illustrations on several regulatory enzymes is given by Bernhard A968).


yeast.Finally, the substrate ATP plays an inhibitory role which disappearsunder the action of the secondsubstrate,F6P,which is an activator of the

enzyme.As a result of theseregulations PFK presentsa periodic\"on-off\"per-

performance in the courseof oscillations,passingfrom 1% to 80% of its maxi-maximum activity. This suggeststhat the enzyme conformation itself oscillatesbetween an \"inactive\" and an \"active\" form. The control by the variousadenine nucleotidesthat as we saw gives rise to the periodicoperation ofPFK,isalsoresponsiblefor the propagation of the periodicpulseof enzymeactivity to the lower part of the pathway.

Similar experimental observations have been made on intact cellsandcell extracts.

14.3.ALLOSTERICMODELFOR GLYCOLYTICOSCILLATIONS

Introduction

The first theoretical model of the glycolytic pathway which predictedoscillatory behavior was workedout by HigginsA964). Shortly after the

discovery of the privilegedrole of the PFK in the glycolyticchain, Sel'kovA968) proposed an elegant theoretical model for glycolytic oscillationsbasedsolelyon the PFKcatalyzed reaction.The model involvesa phenom-enologicalAnsatz concerning the activation of PFK by its product,andgives a first picture of the observedbehavior. Subsequently,GoldbeterandLefever A972) workedout a more elaboratemodel taking into accountexplicitly the allostericnature of PFK.

The assumptionsunderlying the Goldbeter-Lefevermodel may be sum-summarized as follows(seeFig.14.3):(i) Structural hypotheses.We considera monosubstrateallostericenzymeconsistingof two protomers.The protomersexist under two conformations,R and T, which may differ by their catalytic activity and by their affinity

toward the substrate.The reversible transition between these two confor-conformations isfully concerted,in agreement with the allostericmodel of Monod,Wyman, and Changeux A965).The complexesobtained on binding of thesubstrateto the R and T forms decomposeirreversiblyto yield the product.The latter is a positive effectorof the enzymeand bindsexclusivelyto the Rconformation, which has a larger affinity toward the substrateand/or alarger turnover number than the T state.(ii) Environmental hypotheses.The system is open;the substrate is suppliedat a constant rate, whereas the product is removed proportionally to itsconcentration.

o nd\" a O

O d ^\"\"\"a O

o+ o o

o\342\200\242

o

o\342\200\242

o o\342\200\242

o\342\200\242

o\342\200\242

o

o\342\200\242

d'

m

+ o

d'

+ o

Figure 14.3. Allosteric dimer enzyme model activated by reaction product. Substrate (#)is supplied at a constant rate. Product (O)binds exclusively

w to active R state ofenzyme and is removed from system proportionally to its concentration, a,-,d, are association and dissociation constants of the\302\253 enzymatic complexes;k, k' are irreversible decomposition rate constants of forms R and T.

Although the precedingassumptionsfollow from the regulatorypropertiesof phosphofructokinaseoutlined in Section14.2,several simplificationshave beenmade.A detailedmodel should take into account the two sub-substrates ofPFK,namely ATP and F6P.As discussedin Section14.2,glycolyticoscillationsare controlledby the coupleATP/ADP rather than by the coupleF6P/FDP,which strongly suggestsa two-variable monosubstrateenzymemodel.This assumption simplifiesconsiderablythe analytical treatment ofthe system.The two variables consideredare the substrate ATP and the

productADP.The fact that F6Pis the substrateutilized experimentally in

the control ofglycolyticoscillations,asATP is beingproducedby the glycoly-glycolytic system itself, doesnot affect this hypothesis;the important point isthat the combination of F6Pand ATP producesfinally the activator ADP,the causeof the instability leading to oscillations.We can neglectthe secondproduct,FDP,which has no regulatory control (Betzand Sel'kov,1969)anddoesnot induce any phaseshift (Hessand Boiteux, 1968a)under oscillatoryconditionsin yeast.Moreover, the accumulation of FDPhas no effectonglycolyticperiodicities(Hessand Boiteux,personalcommunication).

Phosphofructokinaseis often a tetramer (Mansour,1972)as in E. coli(Blangy,Buc,and Monod,1968).Association-dissociationphenomena mayalsoplay a role in the regulation of the enzyme(Mansour,1972).Here,weconsidera singlemolecular enzyme speciesand restrict our analysis to thesimplestand minimal caseof a dimer.It shouldbe stressedthat the maincharacteristicsof allostericproteins,namely, cooperative interactions, arealready presentwhen the enzyme consistsof two protomers.Extension ofthe model to n protomersshowsno major differencewith the dimer case(Plesser,1975).

The model consideredis a mixed K-V system in the terminology ofMonod and co-workersA965):Indeed,the substrateexhibits differentialaffinity toward two enzymeconformations (K effect),while at the same timethe statesR and T differ in their catalyticproperties(V effect).The reasonforconsideringthe generalcaseof a K-Vsystemis twofold.The expressionsforperfect K or V systemsare easily derived as limiting casesof the generalequations.On the other hand, the roleof enzymeinhibition by the substratein the mechanismof instability can be readily analyzed in such a system.As

pointedout by Monodand co-workersA965),such an inhibition is observedin the concertedK-V system provided the substratebindswith the highestaffinity to the less active state of the protein. In the initial publicationGoldbeterand LefeverA972) treated the casein which the T state is totallyinactive.Herewepresent,therefore,a generalizationof theseresults(seealsoGoldbeter1973;Goldbeterand Nicolis,1976).The last hypothesisconcern-concerning

the enzymeregardsthe catalytic step,which is treated as irreversible in

agreement with the experimental observations (Hess,Boiteux,and Kruger,1969),on the far from equilibrium operation of the PFKreaction.

14.3.Allosteric Model for Glycolytic Oscillations j/j/

As to the environmentalconstraints, the hypothesisof a constant input ofsubstrateis only an approximation in vivo. It corresponds,nevertheless,tothe conditionsfor the experiments with yeast cellsand extracts.The effectof a periodicinput of substrateand the caseof a random substrateinjectionrate are consideredbriefly in Section14.4.The latter situation is likely torepresenta more realisticdescriptionof the system in vivo.

The last hypothesisconcernsthe sink of the product.We considera non-saturated sink for the product,taking into account the large concentrationof enzymes following PFK in the glycolyticsequence(Hess,Boiteux, andKruger, 1969).In such a way, the only nonlinearity in the systemarisesfromthe regulation and cooperativity of the allostericenzymeconsidered.

From a thermodynamicpoint of view, the requirement that the limit cyclebehavior only arises in open systems operating far from equilibrium iscertainly satisfied,as the sourceof substrate,the sink of the product,and thecatalytic stepsare purely irreversible.

The model correspondingto the precedingassumptionsis representedin

Fig.14.3.We denoteby a, a' the kinetic constants for the binding of thesubstrateto the R and T states,respectively;the correspondingdissociationconstantsare denotedd and d'.Similarly, binding of the productto the Rstate and dissociationof the resulting complexescorrespondto constantsa2,d2.We considerthe simplecasewherea2 = a and d2 = d.The constantsk,,k2 are related to the interconversionofRo into To;k and k!are the catalyticconstantsrelated to the irreversibledecompositionof the enzyme-substratecomplexesin the R and T states,respectively(usually k! will be zero).Theconstant input rate of the substrateis denotedvx, whereas ^s is the kineticconstant related to the outflow of the product.

KineticEquations

We considerhere the limit of a homogeneoussystem.The effectof diffusion

is briefly analyzed in Section14.5.Let S and P denotethe concentration of the substrateand the product,

respectively;Ri} representsthe concentration of the enzymespeciesin the Rstate with i moleculesof Sand j moleculesof P bound.Ro and To are theconcentrations of the two enzymeconformations free of ligands;Tt and T2are the two enzyme-substratecomplexesin the T state.We set:

^\342\226\2401

= ^0 + \0201 + ^02

\302\2433

= ^20 + ^21+ ^22


The system is describedby the following set of kinetic equations:\"I

\342\200\224 = -k^o+ k2 To- 2aPR0 + dR0l - 2aSR0+ (d + k)R10dt

= 2aPR0 - dR01- aPR01 + 2dRO2- 2aSR01+ (ddR01

dt

dR\302\2602 = aPROi- 2dRO2- 2aSRO2+ (d + k)Rl2dt ' = 2aSR0- (d + k)R10- aSR10+ 2(d + k)R20

dRl0dt

p..= 2aSR01- (d + k)Ru - aSRu + 2{d + k)R21dt

dR12dt

dR20dt

dR2idt

dR22dt

0 I ri 1 T* ^ t \302\243~* T* i 111 i 1 I-\342\200\224\342\200\224 = /c^o ~ k2T0 \342\200\224 2aST0+ (a + k

= 2aSRO2-(d+ k)Ri2-aSRl2+ 2(d + k)R22

= aSRl0- 2(d + k)R20

= aSRn- 2(d + k)R21 A4.4)

= aSRi2- 2(d + k)R22

~ = 2a'ST0- (</' + k')Ti - a'S^+ 2(d'+ k')T2

IT,a'ST,- 2(d' + k')T2dt

dSdt

12 2 3 0

+ rfi1\342\200\224a b li + 2a 12

\342\200\224 = -2aFi?0+ (d - aP)/?oi + 2rf/?O2 + ^2

+ Jt'7\\ + 2k'T2- ksP

14.3.Allosteric Model for Glycolytic Oscillations 363

together with the conservation relation

2 2I Ru + YT; = D0 = const A4.5)i.j-0 i =0

where

\302\243Ry

=\302\243i+r2+\302\2433 A4.6)

ij

The study of Eq.A4.4)is considerablysimplified if one observesthat in thecaseS $> Do and P $> Do, they involve two separatetime scales.Thus, onecan make the usual assumption of a quasistationary state for the enzymaticforms (Heineken,Tsuchiya, and Aris, 1967).The evolution equationsforthese forms then reduceto algebraicrelations that permit expressionof theconcentration of the various enzymecomplexesin terms of the metaboliteconcentrations.

It is convenient here to normalize the metabolite concentrations bydivision through the dissociationconstant of their complexeswith theenzyme in the R state (Monod,Wyman, and Changeux, 1965):

where

KR(S)=

KR(P)= -=KR A4.8)a

The dissociationconstant of the enzyme-substratecomplexesin the T stateis given by

KT(S) =- A4.9)

FollowingMonodand co-workersA965)wealsointroduce the parameters

t _ *i KR(S)

The allosteric constant, L, equal to the equilibrium ratio To/Ro,gives therelative amounts of protein in the T and R statesin the absenceof ligands.


The nonexclusivebinding coefficient,c,givesa measureof the relative affinityof the substratefor both enzymeconformations.Both parametersL and carecloselyrelated to the cooperativity of the enzymekinetics.

Finally writing

k , k<14\302\273

and using the quasisteady-stateassumption for the enzyme,one obtains:

PoO + yJZl \" ' ac^\"+(l+7)e'+ 1/ v \"\\ e+1

e'+ 1/ v \"V e+1

ac Vc'+ \\J \\ \302\243+1

InsertingrelationA4.12)into the evolutionequationsofa and y, oneobtainsthe equations:

da r,

dy A4-13)ft=f(a,y)-k,y

where

7e'+ 1/ V \302\243+1.

A4.14)

14.3.Allosteric Model for Glycolytic Oscillations 365

Let us denote by 9 the ratio of the turnover numbers of the T and Rstates:

0 = ^<1 A4.15)

We note that a perfect K system is defined by 9 = 1,c < 1,whereas both 9and c are smaller than 1 in a K-V system.A perfect V system is dennedby9 < 1,c = 1 (Monod,Wyman, and Changeux, 1965).

On the other hand, the term 2kD0appearingin Eqs. A4.13)\342\200\224A4.14) is

equal to the maximum rate of the enzymereaction, vM. Let us write:

With these notations, the evolutionequationsofa and y take the followingform:

\342\200\224 = a1-/(a,y)

A4.17)f{*y)ky

with

e'+ 1/ v

\"V e+ 1.A4.18)

All quantities appearingin theseequationsare dimensionless,exceptthe

parametersa1,aM, ks, whose dimension is that of an inverse of time. It is

gratifying to point out that, in the limit of very large values of the allostericconstant L ^> (ct, + ksJ/k2,of e(e^> 1),and for the parameter c = 0, the rateequationsreduce to a form similar to the trimolecular model [seeEq.G.13)].This limiting version of the allostericmodel has been analyzedrecently by LefeverA975).

Stationary Statesand Stability Analysis

At the steady state the normalized concentration of the product is simply:

7o = ~ A4.19)


The correspondingsubstrate concentration a0 is given by the followingrelation:

a0 Bax-

aA

+ a,(L'+ 1)= 0 A4.20)

with L = L/(l + y0J.Oneseesimmediatelythat the systemcannot reach any physicalstationary

state when the rate of entrance of the substrate,vu exceedsthe maximumreaction rate vM. Otherwise,the systemadmits a singlestationary state, <x0,

given by the largest of the rootsof Eq.A4.20).The existenceof multiple stationary statescannot be ruled out when the

number of protomersconstituting the enzymeis larger than two. It has beenshown (Wyman, 1969)that allostericsystems may admit three stationarystates when the number of protomerstends to infinity as in membranesystems(seealso Blumenthal, Changeux, and Lefever,1970).The couplingof cooperativity to positive feedbackin the model consideredcouldbringabout a similar effectwith a finite number of enzymesubunits.However,theanalysis of such modelsindicatesthe existenceof a single physical steadystate, as in the dimer case(Plesser,1975).

The next stepin the analysisis to determine the stability propertiesof the

stationary state. Thesepropertiesare closelyrelated to the dynamic be-behavior of the system,as shown in Chapters6 and 7.

Let us investigatethe responseof the systemto infinitesimal perturbationsda, dy from the singular point (<x0,y0) by means of a normal modeanalysis.The linearization of Eqs.A4.17)and A4.18)leadsto the following system:

pAC8*BC6yatA4.21)

^ = AC da + (BC- ks)dyat

Theseequationsadmit solutions of the form

da = aea\"c * o* A4.22a)dy = y emt

provided a> satisfiesthe characteristicequation:

co2+ [C{A- B) + kjco+ ksAC = 0 A4.22b)

14.3.Allosterk Model for Glycolytic Oscillations 367

We haveset:

1 +-

\302\243+1

e+ 1/ Ve'+ l/\\ e'+1,2

e+ 1A e'+ 1/ Ve+ 1 A e'+ 1

e'+ 1A e+l

+ (l + To)A4.23)

A study of the characteristicequation enablesus to determine the nature ofthe singular point (a0,y0) in the phaseplane (a,y). Now, the quantity C is

alwayspositive.In the caseof a perfectK system@ = 1),this isalsotrue forA. In fact, the latter quantity is alwayspositive for any value of 6, providedc and c are lessthan one, which is generallyensured.Thus, the steady statecan only becomeunstable (i.e.,Re co>0) for

C(A - B) + ks <0 A4.24a)

The critical point of marginal stability isdennedby

C(A - B) + ks = 0 A4.24b)

In the neighborhoodof and above this critical situation, the singular pointenters the unstable region as a focus.The steady state is then enclosedby alimit cycle (Fig.14.4)correspondingto sustainedoscillationsof the con-concentrations a and y in time. Far from the critical region the singular pointbecomesan unstable node,but remainsenclosedby a limit cycleas shown bynumerical integration.

Insertion of the value

ks = C(B- A)


20

15

10

Figure 14.4. Evolution to limit-cycle solution in a-y phase plane obtained by integration of

Eq. A4.17)on analog computer. Values of a, y at unstable steady state are a0 = 16,y0 = 2.Parameter values are a, =0.2s , ks = 0.1s5 x 10\022

145s.

= 103s~',Do = 5 x 10~4mM, KR =mM, L = 7.5 x 106,c = 10~2, \302\243

= 0.1,e' = 6 = 0. Period of the oscillations:

in the expressionfor to at marginal stability yields the linear periodof theoscillations

'T InC[A(B- A4.25)

This relation is in goodagreement with numerical resultswhen the systemhas just entered the unstable domain.

As shown in Fig.14.4,the limit cycleis stablesinceit can bereachedfromeither inside(e.g.,starting near the unstablesteadystate)oroutside.Numericalintegration with various initial conditionsalways leadsto a unique asymp-asymptotic trajectory. Thus, transitions between multiple limit cyclesseemto beexcludedin this system.

By evaluating condition A4.24b)on a digital computer, a set of stability

diagrams can be constructed similar to those establishedin Chapter 7.

14.4.Limit-Cycle Oscillations 369

Thesediagramsdescribethe behaviorof the systemasa function of the main

parametersa1,aM,L,9,c,and ks.A direct examination of these diagrams allows a number of qualitative

observations (Goldbeterand Lefever,1972):

\342\200\242 Sustainedoscillationsare observedin K as well as in K-V systems.\342\200\242 The unstabledomain generallycorrespondsto largevaluesof the allosteric

constant, L,of the orderof thoseobtained experimentallyfor a number ofallostericproteins.

\342\200\242 Substrateinhibition of the enzyme is not a prerequisitefor limit cyclebe-behavior sincethe systemmay oscillatefor c = 0, 9 = 0.

In Section14.4weanalyze in somedetail the characteristicsof the periodicsolution in the caseof a constant inflow of substrate.Where possible,thetheoretical predictionsare comparedto the experimental facts concerningglycolyticoscillations.

14.4. LIMIT-CYCLEOSCILLATIONS

The evolution of the system to a limit cyclehas been followednumericallyfor a wide range of values of the various parameters.The oscillationscorre-correspond well to the behavior observedon cell extracts,for both the orderofmagnitude of the periodA min) and the concentrations (between 10~4and

Particular attention was given to the effectof the variation of the rate ofsubstrateentry a^ on the characteristicsof the oscillations.Note that autogether with the total enzyme concentration Do and the rate of removalks, are the principal parametersthat can becontrolledin \"in vitro\" experi-experiments.

Figure 14.5describesthe influenceofax on the amplitudeof the metabolitey and on the period of the oscillations.We observethat the amplitudedisplaysa maximum and that the curve y = y(ox) is slightly asymmetricand extendsbetween two finite values of ax, which correspondto statesofmarginal stability. The period,on the other hand, displaysan inflexionpoint correspondingto the maximum of the amplitude and decreaseswithin most of the interval between the two extremevalues of ax.

The numerical simulationsshow that the form of the oscillationscan varyconsiderablybetween almost sinusoidaland distinctly nonsinusoidal.Aperiodicvariation of the enzymaticactivity is alsoobservedas a function of

r(s)

8 10 12 14

Figure 14.5. Period and amplitude of oscillations as function of substrate injection rate forks= 1 s~',L = 5 x 106,and other parameters (except ct,) as in Fig. 14.4.

ff,. The activation factor is found to behigher for low valuesof the injectionvelocity,in agreement with experiment.

The influence of other parametersrelated to environmental conditions(e.g.,Do and ks) is similar; oscillationsare observedfor a restrictedintervalof values of these parameters.On the other hand, the oscillatory behaviorsubsistsfor a wide range of variation of the structural parametersof the

enzyme (e.g.,L, 9, c,and k). Finally, the value of the periodseemsto beremarkably insensitiveto the valuesof the parameters.

The role of the cooperativity in the oscillationsis reflected by the fact,that limit cyclescan only be observedfor large values of the allostericcon-constant. The roleof the activation by the product is equally important. Indeed,the numericalstudy of the characteristicequation showsthat the steadystatebecomesan unstable focus only when the normalized concentration of the

product,y, is higher than unity. In other words,the instability occursassoonas the product of the reaction can induce an appreciabledisplacementof the

enzymaticequilibrium toward the active forms.

14.5.Effect ofExternal Disturbances on Limit-Cycle Oscillation 371

14.5.EFFECTOF EXTERNAL DISTURBANCESON LIMIT-CYCLEOSCILLATION

A substrateinflow fluctuating in a given range is likely to representthe in

vivo situation more realistically than a constant sourceconsideredin theprevious sections.Recently,the responseof the allostericenzymeoscillatorto a periodicand to a stochasticinput of substratewas investigated bothexperimentally (Boiteux, Goldbeter,and Hess, 1975)and by numericalsimulations(Goldbeterand Nicolis,1976).Herewe only discussthe effectofdiscontinuousadditionsof ADP or ATP (Goldbeter,1976).Experimentally,one observesthat such pulsesof ATP or ADP phaseshift the oscillations,whereas no such effect is observedwith F6Por FDP.This illustrates theimportant roleof adenylate control in the mechanism of glycolyticoscil-oscillations.

In a typical experiment(Pye, 1969),0.7mM of ADP causesa delayof morethan one minute in a yeast extract oscillating with a periodof 5 min, whenaddedat the NADHminimum. The resultsof the correspondingsimulationon the allostericmodel for the PFKare shown in Fig.14.6.In Fig.14.6,4,Hthe unperturbed oscillationsof the productand the substrate are represented,and the numerical values of the parametersare such that the periodis of300s.The effectof addition of a pulseof 14units of y [correspondingto0.7mM ADP for KR(P) = 5 x 10\022mM;seeEq.A4.7)]during 5 s isstudiedfor different timesover the period.As indicated in Fig.14.6B,C,F,G such anaddition of the reaction producthas no effectover most of the period.Whenaddedat an ADP minimum, an immediate responseis observedin the formof a small peak of the product as indicated in Fig.14.6\302\243. This phaseshift

results in a delay of more than one minute in the next peaks,which areidenticalto thoseobtainedin Fig.\\4.6A. Earlier addition ofADP causesonlya minor phaseshift, as shown in Fig.14.6D.The reasonfor this is that thesubstrate has to accumulate to a critical level before being consumedonaddition of the positiveeffector(i.e.,of ADP).This condition is not satisfiedin Fig.14.6D.

The precedingresults are in qualitative and quantitative agreement with

experiment.They are also strongly reminiscent of the general behavior ofnonlinear oscillatorsto external pulsesoutlined in Chapter8.This analogycan be pushed further by constructing the phase-shift curve 9 = 9(t0) re-referred to in Section8.7.Figure 14.7representsthis curve,for a fixed durationof the stimulus and for variable intensities.

The fact that the phaseshift is limited to a restrictedportion of the periodprovides a molecular basisfor the refractory period observedin a numberof biochemical oscillators,which is thought to play an important role in

supercellularphenomena related to development and morphogenesis.Wereturn to this point in Section14.7.

O -^a;

P

8

J I

Ei-

og

Og

os

J I

62

31

500 1000 1500Time (s)

(G)

150

75

I I I

500 1000 1500Time (s)

(H)

150

75

500 1000Time (s)

1500

Figure 14.6. Phase shift of oscillations on titration by reaction product. Arrows denote time of addition of product (amplitude. 14units of -/ and dura-

duration ofapplication of pulse, 5 s).Curves established for <r, = 0.5s\021,^ = 0.1s\" ',<xM= 8 s\" \\ L = 5 x IO6,c= 10\025, r. = <;' = IO'\\9=1.

3/47^r 3i i

40302010

(S)

+250 \342\200\224

+200 \342\200\224

+150 \342\200\224

+100 \342\200\224

-50 \342\200\224

-100\342\200\224

-150\342\200\224

Figure 14.7.Phase-response curve for PFK model on titration by reaction product. Phaseshift Arj>, corresponding to delay (A0 > 0) or advance (A0 < 0) of oscillations, is given asfunction of addition time

<j>

over period T = 312s.Phase 4>= 0 corresponds to maximum in

product concentration, 7 =y\302\260M

= 42.4.In upper inset, first maximum in y, denoted >'M, induced

by product addition is given; phase shift is considered asphase advance when yM exceedsvalue

JyJJ,. The two curves correspond to addition of 14units of y within period of 2s@-00),andof 6units of y within 2s

(\342\200\242-\342\200\242-\342\200\242).

Both curves are obtained by integration of Eq. A4.17)for a, \342\200\224

0.5s-',fes= 0.1s-',(TM= 8s\"',c= e'= lO\023, c = 1(T5,0 = \\,L = 5 x 106.

374

14.6.Patterns ofSpatiotemporal Organization in the Allosteric Enzyme Model 375

14.6.PATTERNS OF SPATIOTEMPORALORGANIZATION IN THEALLOSTERICENZYME MODEL

In our precedinganalysis the effectof diffusion has beenentirely neglected.The situation of spatial homogeneity certainly correspondsto the experi-experimental conditionsunder which the glycolyticsystem isolatedfrom yeast ormuscle cellsis studiedin a continuously stirred medium. Analytically, theassumption of instantaneous homogenization allows the descriptionof theevolution of the model system by a set of ordinary differential equations.

In vivo, or in the absenceof stirring in experiments with cell extracts,diffusion processes,aswell as the influenceof the geometryof the systemandof the boundary conditions,ceaseto benegligible.From the generalanalysisof Chapter7 we know that diffusion processesenlarge the possibilitiesofstructurization in openfar-from-equilibriumsystemsby allowing the spon-spontaneous formation of new patterns of spatiotemporal organization. TheBelousov-Zhabotinskireaction provides a striking example of existenceofsuch patterns, particularly in the form of propagating concentration waves.The study ofmodels,suchas the trimolecularscheme,leadsto similar results.

In this sectionwe intend to determine the conditionsunder which theallostericenzyme model consideredin the presentchapter exhibits period-periodicities in time and space.This can lead to a better understanding of themechanismsprevailing in the generation of coherent behavior from cellularmetabolism.Theeffectofdiffusion in the systemis studiedunder the followingsimplifying assumptions:(a) one-dimensionaldiffusion of the substrateandthe productand (b) spatially uniform distribution and negligiblediffusion ofthe enzymaticforms.*

Under these conditions,adopting a quasistationary-state hypothesis forthe enzyme as in the homogeneouscase,the evolution of the metabolitesin time and spaceis given by the partial differential equations,which are tobecomparedwith Eqs.A4.13)and A4.14):

da d2<x

dt1 1 dr2

dy _ d2y

with

e+1/ A4.26)*This condition is realized experimentally in artificial membranes containing immobilized

enzymes.


The spatial coordinateis denotedby r\\ Dx and D2 denote,respectively,thediffusion coefficientsof the substrateand the product.All other parametershave the same significanceas in Section14.3.

We want to investigate the stability of the homogeneousstationary state(a0,y0) given by Eqs.A4.19)and A4.20)with respectto space-dependentperturbations.Considerthe caseof fixed concentrations at the boundaries,equal to the steady-state values (ao,yo).The linearized equationsA4.26)around this state admit solutionsof the form:

\302\243y

= y e\302\260* sin \342\200\224\342\200\224

m=l,2,..., A4.27)and the characteristic equation becomes(Goldbeterand Nicolis,1976):

+ ksCA = 0 A4.28)

The coefficientsA, B,C have the same interpretation as in Section14.3.A study of Eq.A4.28)showsthat for realisticvaluesof the various param-

parameters no instability of the thermodynamic branch leading to a steady-statespatial dissipativestructure is possible.In contrast, spatiotemporalpatternscanappearbeyond instability (Goldbeter,1973).Dependingon the dimensionof the systemthese patterns can beeither standing!or propagating wavesor,finally, quasiuniform limit-cycleoscillations(see,in this respect,the classifica-classification of time-periodicdissipativestructures developedin Chapter7). Figure14.8describesa regime of propagating wave.The resemblencewith the be-behavior of the trimoleculardescribedin Chapter7 is striking.

A particularly fascinatingphenomenon is that the regimesdominated bydiffusion and by the oscillatorychemicalreaction are separatedby a criticallength of the system, /*. This point has beenanalyzed recently by NazareaA974),who showedthat for / < /* a reaction-diffusion system similar toEq.A4.26)can only admit a unique solution on the thermodynamic branch,as briefly discussedin Section7.13.An approximate expressionfor this

t These\"standing waves\" should not be confused'with the steady-state patterns showing aspaceperiodicity, which may alsobe solutions of reaction-diffusion equations under certainconditions.

14.6.Patterns ofSpatiotemporal Organization in the AHosteric Enzyme Model

Space (cm) Space (cm)0.3

Figure 14.8.Spatial distribution of the product concentration during regime of propagatingconcentration wave in allosteric model. Length / = 0.3cm, <t, = 5/<s

= 0.5s\" ',aM = 102s~'.L = 5 x 106,c = \\02, e =0.1,e' = 6 = 0, \302\243>,

= D2 = 10\"cm2 s1.Period of phenomenonis T= 202.8s.

critical length is obtained easily as follows.Sincethe sign of the derivativedct/dt changeson the limit cycle, the sourceterm at in Eq.A4.26)is of theorderof the nonlinear contribution from the enzymatic reaction.Sincethekinetic term is of the orderof au one is left with the approximate relation

jl^ - \342\200\224 A4.29)

subject to the boundary conditionsa = a0 in r = 0 and r = /. The solution

^=-w/+wxr

+\302\253\302\260 A430)

passesthrough a maximum am at r \342\200\224 A/2), wherethe following relationholds:

\302\253m= j | A4.31)


The ratio (xM/a0 gives a measure of the relative importance of chemicalreactionsand diffusion; the distribution oc(r) is uniform for infinite diffusion

or for small /, and becomesparabolicas the kineticcontribution increases.Ifthe critical length is dennedas that for which the two terms on the right-handsideof Eq.A4.31)are of the sameorder,one obtains the expression

/\342\200\242

* ^^ A4.32)

The dependenceof the critical length on the squareroot of the ratiokinetic term) agreeswith dimensionalanalysisand with the simulation study.

The influenceof the length of the systemon the stability propertiesof therate equations,which wasalsopointed out in the analysisof the trimolecularmodel (seeSection7.8),illustrates the role of diffusion or,more precisely,ofdiffusion coefficient/(lengthJ as a coupling parameter between neighboringspatial regions.When this coupling is very strong, the systemremains on thethermodynamicbranch.When it is loose,the chemicalkineticstakesoverandcan give riseto sustainedoscillations.Betweenthese two extremesone canhave \"intermediate\" forms of organization in the form of concentrationwaves.

What are the spacescalesfor the various structures in the glycolyticexample?The effectof diffusion has beenstudied with the same values ofparametersfor which the limit-cycle behavior in the homogeneouscasematches the oscillationsobservedin yeast and muscle with respectto theperiodand amplitude. Inserting thesevalues in the model in the presenceofdiffusion, one finds that the dimensionsover which space-dependentstruc-structures ariseare supracellularand vary between10\"2and 1cm for the wavelikesolutions.Although the equation derived for /* is a simpleestimate, it yieldsgood agreement with the simulation results.For the parametersused in

Fig.14.8,relation A4.32)givesa critical length of the orderof2.5x 10~2cm;a regime of propagating waves is achieved accordingly for / = 0.3cm, asshown in Fig.14.8.Thus, the spatiotemporal behavior clearlymanifestsitselfat a supercellularlevel. On the basis of simulations, the behavior of thesystemasa function of dimension/ in the caseDx = D2can besummarizedin

the following way:

0.1<(///*) < 1 :time-independentregimeon the thermodynamicbranch1.0<(///*) < 10:standing waves

/ ^ 10/*:propagating waves/ ^> 10/*:oscillationsquasihomogeneousin space

Recentexperimentsby Hessand co-workers(Hess,privatecommunication)confirm the existenceof spatiotemporalpatterns in glycolysis,with char-characteristic lengths in the millimeter range and with periodscomparabletothoseof the homogeneousoscillation.

14.7.Periodic Synthesis ofcAMP 379

14.7.PERIODICSYNTHESISOF cAMP

SignalsofcyclicAMP emitted by living cellsseemto control a number of keyvital processesat various levelsof complexity.One of the most fascinatingphenomena is the processof aggregation of the amoebaDictyostelium dis-coideumleading from isolatedcells to multicellular differentiated bodies(Sussman,1964).This processinvolvesan oriented responseof the cellsto agradient of cyclicAMP in the extracellular medium, known as chemotaxis.It has long beenknown that amoebaeof the abovementionedspeciesaggre-aggregate in a wavelike manner around a number of centers.Pulsesof cyclicAMPemitted by the centerswith a periodof severalminutes propagateoutward toother cellsthrough a relay mechanism that allows the formation of largeaggregationterritories (Gerisch,1968).In connection with this phenomenon,metabolicoscillationscontrolled by cAMP havebeenobservedin suspensioncultures of aggregation-competent D. discoideumcells(Gerischand Hess,1974)and shown to arise independently of the glycolyticoscillations.

In this sectionwe describea plausiblemechanism for the oscillationsofcAMP,basedon the regulation of enzymesdirectly involved in the synthesisof this metabolite (Goldbeter,1975).The two regulatory enzymesconsideredin the model,ATP pyrophosphohydrolase(EJ and adenyl cyclase(E2),transform ATP into 5'AMPand cAMP,respectively,as indicated in Fig.14.9.In D. discoideum,this system is positively controlled(RossomandoandSussman,1973);adenylcyclaseisactivatedby 5'AMP,whereasATPpyrophos-phorydrolaseis activated by cAMP.In both casesthe interactions of the

-ATP

Figure 14.9. Mechanism for oscillatory syn-synthesis of cAMP in slime mold Dictyosteliumdiscoideum.


enzymes with the effectorsare highly cooperative, suggestingthe allostericnature of the enzymes. Finally, the two reaction productsare coupledbyphosphodiesterase,which transforms cAMP into 5'AMP.The latter leavesthe system via another enzymatic step involving 5'nucleotidase.*

The rate equationsfor the modelcan bedevelopedin excatlythe samewayas in Section14.3for the PFKreaction.The differenceis that one dealsherewith three coupledvariables, namely, ATP, 5'AMP,and cAMP, and tworegulatory enzymes.Denoting by Lt, aMi (i = 1,2) the allostericand the nor-normalized maximum activity of enzyme \302\243,, by ax the normalized sourcetermof the substrate,by k x, k2 the rate constants for the sink of5'AMPand cAMP,and by a, /J, y the normalizedconcentrationsof ATP, 5'AMPand cAMP,oneobtains:

^ = _ f{a y)_ /\342\200\242 (a $

dt1

j = f2(*J)~ k2y A4.33)

where

<tm1ocA + a)(l + yJJ1 Li + A + aJ(l + yJ

h = zf+^T^lTTW <R34)

Theseequationsare to becomparedto Eqs.A4.13)and A4.14).They containthe additional simplificationthat the nonexclusivebinding coefficientof thesubstratehas beenset equal to zero for the two enzymes.

The stability analysisaround the steady-statesolution ofEq.A4.33)showsthat the latter can becomean unstable focus.Within the unstable domain the

system performs sustained oscillationsof the limit-cycle type shown onFig.14.10.The periodof the phenomenon, which is well within the 3-5-minrange observedexperimentally, is largely independentof the enzymaticconstants and of the substrate injectionrate. As in the PFKreaction, the roleofenzymecooperativityin the onsetof the oscillationsis illustratedby the factthat the oscillatory domain correspondsto large values of the allosteric

* A model for a circadian rhythm involving cAMP and based on negative feedback has been

reported by Cummings A975).

14.7.Periodic Synthesis ofcAMP 381

21

18

8 15

r 3

100 200 300 400 500 600Time (s)

700 800 900 1000

Figure 14.10.Periodic variation of normalized cAMP concentration obtained from numerical

integration of Eq. A4.33).Numerical values of parameters are<tm, = 0.4s\021, aM2 = 10s~l,kl = k2 = 0.1s\"\\ a, = 0.5s\021, Z-! = L2 = 106.

constants,ofthe orderof106.Thepulsatorynature ofthe oscillationsdisplayedin Fig.14.10and the stability of the periodcorrespondwell to the releaseofpulsesof cAMP by the aggregation centersas observedexperimentally(Gerisch,1968).Morerecently,the oscillatorydynamicsof adenylcyclasehasbeen directly demonstrated in suspensionsof D. discoideumcells(Gerischand Wick, 1975).

Further simulationson this system(Goldbeterand Segel,1977)suggestthat

the mechanismgiving riseto the cAMP oscillationsmay alsoberesponsiblefor the relay of the cAMP signal taking placeduring aggregation.The mainpoint is that, outsidethe oscillatory domain, the system describedby Eq.A4.33)can amplify an external cAMP pulse if the latter activates adenylcyclasevia binding to the membranecAMP receptor.Undertheseconditions,a new amount of cAMP is synthesizedin the form of a pulse,thus relayingthe chemotacticsignal.

Finally, thanks to the formation of 5'AMP from cAMP in the phosphodie-sterasereaction, sustainedoscillationscan still arisein the absenceof ATPpyrophosphohydrolase.In contrast, adenylcyclaseseemsto beindispensablefor oscillatorybehavior (Goldbeter,1975).


14.8.REACTIONSINVOLVING MEMBRANE-BOUND ENZYMES

Within the living cell,the majority of enzymesare attached to membranestructures or contained in cell organelles.In recent years the binding ofenzymesinto synthetic membraneswasachieved,enabling study of the inter-interaction between diffusion and enzyme kinetics under biologicallyplausibleconditions.Ofparticular interest is the possibilityof achievinghomogeneousdistribution of enzyme molecules(seeThomas, 1975 for a recent survey).Moreover,it appearsthat immobilization increasesdrastically the life timeof the activeenzymaticforms.

Generally speaking,the phenomena observedin such systemscan beclassifiedinto two categories.

Effectof MembraneStructure on Enzyme Behavior

The binding of enzymesinto an insolublephaseallows the creation of struc-structural modesthat include diffusion constraints.Onepossibleconsequenceisan asymmetricaldistribution of activesitesin the structure.This can result in

phenomena similar to active transport, in the sensethat a certain metaboliteis transportedacrossthe membrane against gradient (Broun,Thomas, andSelegny,1972).

Effectof LocalConcentrationDistribution of Reactants on EnzymeBehavior

The local environment within a membrane undergoing enzymatic reactionsis the result of a balancebetween the flow of matter and the reactions.As aresult, the substrateand product concentrations in the membrane may varyin time or differ from point to point acrossthe membrane aswell as from theouter solution.Forinstance, sustainedoscillationshave beenobservedby

Caplan, Naparstek,and Zabusky A973) in the hydrolysis of benzoyl-L-arginin ethyl ester by a membrane containing bound papain.Computersimulation of the model rate equationsreproducethis behavior. A strikingexampleof spatial organization is reported in a study involving glucose-oxidaseand urease,in the presenceof their substrates.Theglucose-oxidasereaction increasesthe pH,whereas the ureasereaction decreasesthe pH.Asthe enzyme activity dependsstrongly on the pH, the reaction sequenceiseffectively autocatalytic (Thomas, Goldbeter,and Lefever, 1977).FinallyChanu and Delmotte (Delmotte and Chanu, 1971;Delmotte, 1975)haveconductedextensiveexperimentson the ionic transport propertiesof mem-membranes containing immobilizedenzymesimmersedin an opensystemreactor

14.8.Reactions Involving Membrane-bound Enzymes 383

and observedcooperative phenomena in the form of simultaneouslystablemultiple regimesof transport.

The mathematical analysis of these phenomena followsthe lines adoptedthroughout this monograph.The main point to be retained at the outset isthat the usual expressionsof the activity of the various enzymes (whetherallostericor simply Michaelian)have to beunderstoodin a localsense.Thus,the time evolution of, for example,a substrateconcentration S, takes theform (assuminga one-dimensionalsystemwith diffusion along the thicknessof the membrane)

PC /JC-Ds^ A4.35)

where F(S)describesthe effect of the enzyme reaction.This equation isgenerallycoupledto those for the productsand for H+ orOH\".

The boundary conditions,especiallyfor the substrateconcentration,can besymmetrical,fixed,or variable.Oneof the boundariescan bean impermeablewall, such as in the casewhere an enzymecoating is attached to a sensitiveglasselectrode.

We have already insisted,in Chapter7 and in the precedingsectionsof the

presentchapter, on the variability of situations encounteredin systemsobey-obeying

reaction-diffusionequations.To illustrate this for the caseofimmobilizedenzymes,considerfirst a urate-oxidasemembrane separating two compart-compartments containing a substrate solution at a concentration So.Supposethe

enzymekinetics showsan inhibition by substrateexcess.The rate equationfor Sreadsat the steady state (Duban,Kernevez,and Thomas, 1976)

d2s

with

and

= s0,(\302\243)

= 0 A4.36)

(R37)

The equationsare written in dimensionlessvariables.The length unit is themembrane thickness,the concentration unit is the Michaelisconstant of the

enzyrfle, KM, and a and k are positiveconstantsrelated to the diffusion con-constant, thicknessof the membrane, and characteristicsof the enzyme.

A study of Eq. A4.36)showsthat the system can admit multiple steadystatesand hysteresisphenomena similar to thosedescribedin Section8.4.


This can be best seenby approximating the diffusion flux through the ex-expression

So-S

where / is the thicknessof the membrane, which is now taken to be inactiveand separating a reservoir from a compartment of thickness/ containing ahomogeneousenzymesolution.

When a cofactor is present and plays a rate-limiting role,the system isdescribedby the equations(Duban,Kernevez,and Thomas, 1976)

dsJt~da

<raF{s) -

aaF(s)-

d2s1

dx2

d2a\"\302\253d?

A4.38)

where a is the reducedcofactor concentration, and the boundary conditionsas well as F(s) are given by similar expressionsas before.

Numerical and analytical studiesshow that this system of equationsadmits oscillatory solutions of the limit-cycletype in a range of parametervalues. Again, this can be best seen by simplifying the partial differentialequationsby meansofan approximateexpressionfor the diffusion flux, which

leadsto a systemof ordinary differential equations.A more completetreat-treatment taking spatial derivatives into account predicts the possibilityofspatial patterns and/or time periodicities.

14.9.THE PHYSIOLOGICALSIGNIFICANCEOF METABOLICOSCILLATIONS

In this chapter wehave thus far shown that cellularmetabolismcan generatecoherentbehavior in the form ofsustainedoscillationsand/orspatial patternsextendingover macroscopicdistances.The occurrenceof regulatoryenzymesand/or feedback processesin the great majority of biochemical reactionssuggestthat this type of behavior is far from exceptional in living cells.

Independentlyof their specificmetaboliccontext, coherent patterns of the

dissipativestructure type are endowedwith powerful regulatory properties,someofwhich are pointed out in Chapter7.Further remarkson thesegeneralaspectsare to be found in recent review papers on chemical oscillations(Nicolisand Portnow, 1973;Noyesand Field,1974;Goldbeterand Nicolis,1976;Goldbeterand Caplan,1976).

In this sectionwe insist more specificallyon the role of metabolic oscilla-oscillations at the subcellularlevel.We review briefly three important regulatory

14.9. The Physiological Significance ofMetabolic Oscillations 385

phenomena where these oscillationsseem to play a prominent role:(a)modulation of the adenylic energy charge, (b) onset of developmental pro-processes,and (c)problemof circadian rhythms.

Modulationof Adenylic Energy Charge

As wesaw in the first sectionsof this chapter, the PFKreaction takesplaceatthe upperpart of the glycolyticcycle,whose primary role is to synthesizeATP. Thanks to the regulation via the oscillations,one can say whether thesystem is functioning accordingto whether the ATP levelis low (high levelsof ADP and AMP), orhigh (low levelsof ADPand AMP) or,equivalently,accordingto the energetic needsof the cell.Atkinson A965)has quantifiedthis ideaby introducing the notion of adenylicenergycharge,which dependson the various adenylatesas follows

AEC = (ATP) + j(ADP)(ATP) + (ADP) + (AMP)

0 <AEC < 1 A4.39)

He suggestedthat this quantity controlscellular metabolism, in the sensethat a high valueofAEC inhibits the sequencesofmetabolismthat regenerateATP and at the same time activates the sequencesutilizing ATP.

The allostericmodel of glycolyticoscillationsenablesus to analyze theeffectofoscillationson the adenylicenergycharge.Figure 14.11representsthetime evolutionof this quantity, which is seento beperiodic,asa result of the

1.0

0.8

0.6

0.4

0.2

y> ,

1/

/

, s

I(

f/

\342\200\224

I I I I ,150 300 450

Time (s)600 750

Figure 14.11.Periodic variation of energy charge calculated from definition A4.39) in con-conjunction with Eq. A4.17).


\"entrainment\" by the glycolyticoscillations.The point is that these oscilla-oscillations lead to a drasticloweringof the levelof energy charge during an ap-appreciable periodof time.This suggeststhat one of the rolesof the glycolyticoscillatormight be to induce an alternative switch on and off of reactionsutilizing ATP and of thoseproducingATP (Goldbeter,1974).

CellAggregation and Pattern Formation

Goodwinand CohenA969)(seealsoRobertsonand Cohen,1972;RobertsonDrage,and Cohen,1972)have conjecturedthat chemicalwavesprovide thebasisof the control of a large classof developmentalprocesses.The resultsof Section14.6concerningthe emergenceof spatiotemporal periodicitiesofsupercellulardimensionsin the PFKreaction supportthese ideas.

Although there is no universal agreement on the role of spatiotemporalcontrol in higher organisms,it iscertain that periodicphenomena do play arole in the early stagesof aggregation and in the differentiation of certainamoebaebelongingto the family of slimemolds.

In this respect,the periodicsynthesisofcAMP in theseorganismsanalyzedin Section14.7fulfills an obvious physiologicalrole.Thus, Darrnon, Brachet,and da SilvaA975) have shown that the absenceof aggregationand morpho-morphogenesis in certain mutants of D.discoideumcan be overcome by subjectingthe amoebaeto periodicpulsesof cAMP.In contrast, continuous flow ofcAMPfails to promote differentiation.Similar resultshave beenobtained byGerischand co-workersA975) with D.discoideumcells,in which the appear-appearance of cell-contact sitescharacteristic of the aggregation stage has beenblocked.This morphogenetic block,however, can be bridgedby periodiccAMPpulses.

CircadianRhythms

Many organismsdisplayan innate low-frequencybehaviorwith periodsof theorderof one day.The basicpropertiesof these oscillations,such as stabilityof the period,phaseresetting,or entrainment by an external stimulus, enablethe organism to adapt in a flexibleway to external conditionsand to keep,through suitable phase relations, an identical successionof various vital

processes.As suggestedby Winfree A967) and PavlidisA973),a circadian rhythm may

originate from a coupling of a number of biochemical oscillatorswhoseperiodis of the orderof the minute. The periodstability, phaseresetting,andentrainment observedfor the PFKreaction in both model and experimentsaddcredenceto this view (seealsoWinfree, 1975).Nevertheless,the mechan-mechanisms by which the periodcan be lengthenedremains a matter of speculation.

Chapter15

RegulatoryProcessesatCellularLevel

15.1.INTRODUCTION

We now turn to the secondgeneral type of control mechanismin living cells,namely, the control of the synthesis of proteins.We know that one of themost important roles of proteins is catalysis of various reactions.Now,proteinsare generally stable molecules(life times of the order of severalminutes), whereas catalysis is a very fast process.Thus, it is not unusual tohave a situation where the protein level in the cell tendsto be too high. In

responseto this, the organismusessomesubstancesto repressthe synthesisofthe macromolecules.In other circumstancesit may happen that the organismis subjectto a violent changeof the environment.New typesofmacromoleculeare neededto catalyze the reactionsinvolving the new substratesavailable.The ingenious trick employed by living cellsis that a new substratecanitself induce the synthesisof these macromolecules.*

In the first few sectionsof this chapter we show how thesenegative andpositivefeedbackprocessesand their combination may give riseto coherentphenomena in the form of all-or-nonetransitions or of sustainedoscillationsof the limit-cycletype.The latter belongto the classof epigeneticoscillations,accordingto the terminologyof Section14.1,with periodscomparableto theduration of the cellular cycle itself. These phenomena are illustrated in

regulation of the lac operon in the bacterium Escherichiacoli(\302\243. coli);the

bacterial regulatory mechanismsare now well establishedsincethe pioneer-pioneering

work of Jacoband MonodA961).In contrast, regulation in eukaryoticcellsis, to a large extent, still an open problem(seehowever Britten andDavidson,1969,for someinterestingcontributions in modellingthis phenom-phenomenon).

* We assume that the reader is familiar with the main steps of the biosynthesis of proteins, aswell aswith the basic concepts ofthe genetic code.An excellent survey is found in the monographby Watson A965).

387

388 Regulatory Processesat Cellular Level

In the last sectionsof the presentchapter we are concernedwith theregulation of the cellularcycle.After reviewing evidenceof the existenceof amitotic oscillator we briefly present the work of Kauffman, A974a,b;Kauffman and Wille, 1975),which provides a beautiful illustration of therole of sustainedoscillationsin the regulation of vital activities.

15.2.LAC OPERON

We first analyze the utilization of a disaccharide,lactose,as a carbonandenergysourceby the bacteriumE.coli.Although E.colicontains the enzymesnecessaryfor the metabolism of glucoseunder all conditions,cellsgrown inthe absenceof lactoseare unable to metabolize this substanceimmediatelywhen they are suddenly exposedto a lactoseenvironment. After a certainlag,however,they beginto assimilatethis sugarat a high rate. Lactose-growncells,on the other hand, can metabolize this sugar at almost the same rateas they metabolizeglucose.

A comparisonof the enzymaticcontent ofglucose-and lactose-growncellsshowsthe presenceat high levelsof certain enzymes in lactose-growncells,which are barely detectablein glucose-grownones.Theseinclude:(a) fi-galactosidase,which hydrolyzeslactoseto its constituent monosaccharides,glucoseand galactose,(b) galactosidepermease,which mediates the entryof lactoseinto the cells,and (c)a sequenceof enzymesnot essentialfor thelactosemetabolism,suchasthiogalactosidetransacetylaseaswell asenzymesparticipating in the galactoseconversionto a glucosederivative.

It is now establishedthat the synthesisof these enzymesis regulated by apart of the bacterial genome, the lac operon (Beckwith and Zipser,1970)which is shown in Figure 15.1.A repressor,R, is synthesizedby the gene i

along the operon.In its active form it combineswith the operator,o, andinhibits the transcription of the structural genesz, y, a synthesizing, respec-respectively, /?-galactosidase(E), permease(M),and thiogalactosidetransacetylase(A). An inducer, I, whose entry is facilitated by permease,liberatestheoperator,and thus the synthesisof proteinscan begin.Allolactose,a lactose

'i

pi

\302\260

i iy

iQ

i ij j~

1 i'

! >

1I I II y |y

8BR \342\200\242! E MAFigure 15.1.Schematic representation of lacoperon.

15.3.Mathematical Model for Induction ofji-galactosidase 389

isomerformed by a reaction converting the 1-4bondof lactoseinto a 1-6bond, is the natural inducer of the lac operon in E. coli cellsgrown in alactose-richmedium.However,in many experimentsthe enzymaticsynthesisis inducedby the so-calledgratuituous inducersthat cannot bedecomposedby /?-galactosidase,such as TMG (methyl-/?-D-thiogalacto-pyranoside),TDG (/?-D-galacto-i-s-thio-/?-D-galactoside)or IPTG(isopropyl-/?-D-thio-galactoside).

Thus far wehave beenconcernedwith the positivefeedbackrelated to theaction of the inducer.When lactose(or, more usually, a precursorof thelactosecatabolism)isusedinsteadof a gratuituous inducer,the control of thelac operon is subjectto two adverse processes.As before, the inducer de-repressesthe operator,thus exerting a positive feedbackaction.However,glucose,which, aswe sawearlier in this section,is a productof the galactosemetabolism, is a highly effectiverepressorof the lac operon and exerts,therefore, a negative feedback action. Actually, this action, known as cata-bolite repression,takesplacevia a loweringof intracellular concentration ofcAMP,which normally stimulates protein synthesis (Pastan and Perlman,1970;Zubay, Schwartz,and Beckwith,1970).

In Sections15.3and 15.4we analyze the consequenceof the inductionprocess,whereasin Section15.5we examine the combinedeffect of thepositive and negativefeedbackson the regulation of the lac operon.

15.3.MATHEMATICAL MODELFOR INDUCTIONOFj3-GALACTOSIDASE

The investigationsofNovickand Weiner A957,1959)and Cohnand HoribataA959)have shown that the induction of /?-galactosidasein E.coli givesriseto all-or-nonephenomena.Thus, in a genotypicallyhomogeneouspopula-population,

one finds two distinct levels of induction, namely, noninduced andmaximally induced.Theselevelscan be maintained during several genera-generations. Moreprecisely,below a certain critical concentration of the inducerthe rate of synthesis of /?-galactosidase(or of permease)is almost zero,whereas beyond this value it rapidly attains its maximum value. Everything

happensas if the bacteria couldundergo an abrupt transition between twolevelsof enzymesynthesis,in the vicinity of the critical concentration.

The modelshown in Fig.15.2(Babloyantzand Sanglier,1972)incorporatesthe most important experimentaldata on the inducer'saction and the sub-subsequent enzymesynthesis.We have simply addedto the schemein Fig.15.1the entry of external inducer, Ie,into the cell via the action of permease,and the subsequentinactivation of the active repressorR by the internal


Ip

I\302\260

I*

I

R' -- ~> R

Figure 15.2. Regulation of lacoperon circuit.

inducer I(.The correspondingchemicalstepsare:

(i) R' . *,'> R

A:,

A1)

(iii)

(iv)

(v)

(vi)

(vii)

R +

R +

n +

M

O+

O+

M

E

< t\342\200\224

k3

k4

k5 t

kb

kj

o

F,

M

F2

F3

M

A5.1)

Step(i) describesthe equilibrium between the active repressorR and theinactive form R' synthesized by the regulatory gene i. R bounds to theoperatorO+ and blocksit by forming the complexRO = O~ in step (ii).In the presenceof n, moleculesof inducer Ij,part of the repressorforms Fiand, therefore, the operatoris liberated in step (iii). Step (iv) describesthetranscriptionof the structural genesoncethe operatorO+has beenliberated.\\] representsthe pool of amino acids.Entry of the inducer into the cell isrepresentedby step (v), where it is understoodthat k5 $> k'$ to allow for anaccumulation of I( into the cell.Steps(vi) and (vii) representthe dilution ofthe enzymeswithin the medium.

It must berealized that in the modeldescribedby Eq.A5.1)severalsimpli-simplifications have beenmade;the repressoris, in fact, an allostericprotein with

four subunits.Thus, steps(i) and (iii) give only a global picture of its kinetics.

15.4.AU-or-None Transitions ^gj

Moreover, the transport of inducer is a facilitated processinvolving manysteps.Both aspectshave beenanalyzed in more detailed models(Sanglier,1976),and no major qualitative change with respectto the resultsof thesimplified model A5.1)has been found. Finally, the processof enzymesynthesisisalsoa complexphenomenon involving severalsteps.

We are now in a positionto write the rate equationsdescribingthe evolu-evolution of the various reactants in the scheme15.1.To this end,we considerthat r\\, R', and F^ are in excessand hence may be treated as constants.Moreover,we neglectthe influenceof diffusion in the equationsof evolutionon the basisof both the limited dimensionsof bacterial cellsand the homo-homogeneity of the extracellular medium. The latter condition may be compro-compromised in the presenceof attractants in the nutrient medium, resulting in achemotactic (oriented)responseof the cells.

We may now write the following equations:

^= k,R'- k\\R- k3RI^'+ k'3Fx - k2RO+ + k'2O~ A5.2)

^ A5.3)

A5.4)

A5.5)

,k3RI^'+ n^Fy A5.6)

Theseequationsare to besupplementedby the condition expressingthat

the total amount of operator,O+ + O~,is conservedwithin the cell:O+ + O~ =

%= const A5.7)

15.4.ALL-OR-NONETRANSITIONS

EquationsA5.2)to A5.6)have beensolved numerically,both at the steadystate and for the time-dependent behavior of the variables using Bairtow'smethod.Wherever possible,the values of the various constantshave beenchosenin agreement with the experimentalresults(Sanglier,1976;Sanglierand Nicolis,1976).This fixes the value n, \342\200\224 2 for the stoichiometric co-coefficient in step(iii) of scheme15.1,aswell as the valuesfor R, Fj = [/?/],/,

dt~~

dETt~

dt~

-k2RO++ k'2O

rjk^O* \342\200\224 k-i E

k5IeM- kslxM


1.5 \342\200\224

0.5 \342\200\224

-56

y*

I

I

0.30 1.17log (le)

(A)

2.04

Figure 15.3.Theoretical curves of stationary-state solutions for E in lacoperon model obtained

by using experimental rate constants and k, = 0.08,k\\\342\200\224 10.

rj, k2,k'2, k3,k'3, k5,k'5. The remaining constants ku k\\, Ie,/c4,k6,k-, havebeenusedasparametersin the computer simulationsand varied over a widerange of valuesas seenin the captionsof the subsequentfigures.

Let us first discussthe propertiesof the stationary states.Figure 15.3representsthe valuesof the enzymeconcentrationasa function of the inducerconcentration in the external medium. We see that the system displaysmultiple steady statesfor a limited range of values of /e.The stateson the

upper and lower branchesare stable,where as those of the intermediatebranch are unstable.This is in agreementwith the experimentalresultson theall-or-nonecharacter of the induction phenomena.Moreover,one finds bothinduced as well as noninduced bacteria for the same concentration of theexternal sugar. The numerical values compare also favorably with the

experimental results.Thus, from Figure 15.3onefinds, for example,for Eat the noninduced state, 2 x 10\023fiM comparedto the experimentalvalueof 3.3 x 1CT3nM, whereas for E at the inducedstate the theoretical andexperimentalvalues are, respectively,2 /iM and 3.3/iM.

This behavior is due specificallyto the action of the permease.A numericalsimulation of the so-calledcryptic strain y~, which lackspermease,givesasigmoidaldependenceof the enzymeconcentration on /e instead of multiple

steady states.It is also of interest to simulate the kineticsof the induction processas

measuredby the temporal evolution of the enzymeconcentration after fixing

15.4.AH-or-None Transitions 393

100 50Bacterial mass (/ug X ml\"')

100

Figure 15.4. Kinetics of /?-galactosidase induction for normal and cryptic type cells:experi-experimental curves.

a certain inducer concentration initially. This behavior iseasilyaccessibletoexperiment. Figure 15.4describesthe experimental enzyme concentrationrelative to the increaseof bacterial mass (rather than in terms of time).In this way one can identify the differential velocityof synthesis:

dE/dt _ dEdB/dt

~dB A5.8)

where B = bacterial mass.Again, for bacteriacapableof synthesizing permease,for an intermediate

concentration of the external inducer the responseof the population is

\"inhomogeneous\"in the sensethat one finds both inducedand noninducedbacteria.This is reflectedby an accelerationof the rate of synthesisbeforeaconstant value is attained, as shown in Fig.15.4.

Thesequalitative featuresare accountedfor satisfactorilyin the numericalsimulation of the time-dependentbehavior of the theoretical model.Figure15.5representsthe time variation of the enzymeconcentration for a bacterialpopulation synthesizing permease.In addition to the general shapeof thecurves, the critical time Tc correspondingto the duration of the lag phaseis of the sameorderof magnitude in the theoretical and experimentalcurves.Note,however,that by construction the model doesnot account for the lagarising from the delay of the processof protein synthesisitself.


7.2

7.15

7.1

7.05

0 Tc 20 40t(min)

Figure 15.5.Theoretical curves for /?-galactosidase induction in normal-type cells.

Finally, the model can easily be adapted to account for the peculiarbehavior of the mutant X86 whoseinduction curve displaysa depression(correspondingto a regime of noninduced bacteria) for intermediate valuesof the inducer concentration in the external medium (Sanglier,1976).

15.5.CATABOLITEREPRESSION:SUSTAINEDOSCILLATIONSAND

THRESHOLDPHENOMENA

The following control circuit (Fig.15.6)can reasonably be expectedtodescribethe salient featuresof the regulation of the lac operonactivity in the

presenceof the negative feedbackarising from catabolite repression.In

15.5.Catabolite Repression: Sustained Oscillations and Threshold Phenomena

ip

io

I*

Iy

i

395

Figure 15.6. Regulation of lacoperon in presence of catabolite repression.

addition to the seven stepsalready encounteredin schemeA5.1),one nowhas the following processes:

A5.9)

A5.10)

Thesereactionsdescribethe stimulating effect of G, the product of theinternal inducer'scatabolism,in the formation of active repressor.Thisactivationstepis again taken to becooperative,as in step(iii) ofschemeA5.1)in order to account for the allostericcharacter of the repressor.Owing tothe similarity between R and R' we subsequently take nG ^ nI = 2.Un-Unfortunately the rate constantsk8,k8, and kg are not known experimentallyand are consideredas variable parametersin the computer simulationsreportedin the sequel.*

Onemay now write the rate equationsfor the variablesR,0+,E,M,G,I{,under the conditionsdiscussedin Section15.3.As a matter of fact, we givehere only the contracted form of these equationsobtained by invoking aquasisteady-stateapproximation similar to that discussedfor the allostericmodel of Chapter 14.More precisely,we take all genetic and enzymaticvariables to adjust instantaneously to the metabolite concentrations Gand /j.The validity of this approximation restson the usually small con-concentrations of macromolecules,as well as on the occurrenceof fast stepsin

the decompositionof the intermediate enzymaticcomplexes.* As pointed out in Section 15.2,G acts on the genome via the cAMP. Thus step A5.9) shouldberegarded asa global representation of this complex phenomenon.


Setting

we obtain the reducedset:dG = -2ksR'G2

k's = k'8DH = k\\+ k'2

t = kiR'+ F

2k'8(k8R'G2+ t)

(k8R'G2 2F:- (k9

where

A5.11)

D(G,I,)

D(G,1.)

D(G,I,)= /c7{fc2(/c8/?'G2+ t) + k'2(k3lf + n)}

A5.12)

A5.13)Theseequationshave been solved numerically on a CDCcomputer usingBairtow'smethod (Sanglierand Nicolis,1976).The behavior in the vicinity

6 8

Figure 15.7. Periodic trajectory in phase plane spanned by /, and G.Unstable limit cycle isindicated by dotted lines. Numerical values: k, = 0.2min~ '.k\\

= 0.008min\021, R' = 10~2/(M,ks = 0.03min\"

F, = 10\023 uMk9 = 5 x 10\023

3 x 10\026 min\"

1

\\xM 2, k's =k2 = 4 x 105 min\"

min /iM = 0.2min\" 2, k'3 = 60min\"

\\ k'2 = 0.03min\021, r\\ki= 5 x 10

1nM~\\ k5 = 0.6min\021 iiM'\\ k's = 0.006min\021 \\iM~\\ k6 = k7 =

2.002x 10\023 ixM. Crossindicates position of(stable) steady state.

15.5.Catabolite Repression: Sustained Oscillations and Threshold Phenomena 397

8.24-

6.38-

4.52-\\

2.66-

0.81-

1 \\

1

11

11t1

LU366. 732. 1098. 1464. 1830.

Figure 15.8.Time evolution of concentration with same kinetic constants as in Fig. 15.7.Initial values: If = 5.5pM, G\302\260

=

of the steady states\342\200\224including stability\342\200\224has been studiedvia Merson'smethod.Experimental values have been used for the constantsspecifiedin Section15.4.

Fora wide range of values of the different constantsand for k\\ >0.1one finds for the systemofequationsA5.12)and A5.13)a singlestablesteadystate that behaves like a focus.Keepingall other constants fixed but taking0.000248< k\\ <0.1,one finds a stable focus surroundedby two limit

cycles(Fig.15.7),an unstable one,with a period around 110min, and astableone with a periodof 300min. The time evolutionof the concentrationsis shown in Figs.15.8and 15.9.We seethat in the immediate vicinity of andinside the unstable cycle the system spiresduring several periodsbeforebeingattracted to the stablesteadystate.On the other hand, initial conditionscloseto but outsidethe unstable cycle are attracted to the outer (stable)limit cycle.

Finally, for k\\ <0.000248one finds multiple steady states,and in partic-particular: (a) a stablefocus surroundedby an unstable limit cycleof a periodofabout 110min, (b) a saddlepoint, and (c)a stablenode.

The most natural way to topologically accommodatethese types ofbehavior is shown in Figs.15.10and 15.11.In addition to an unstable limit

6.49H

5.86-I

5.22-

4.57

3.93-\\\\1

1 ^228. 457. 685.

t

914. 1142.

Figure 15.9.Time evolution of concentration with same kinetic constants as in Fig. 15.7.Initial values: 1\302\260

= 5.5/iM, G\302\260= 16.2pM.

Figure 15.10.Phase-spacerepresentation of evolution of concentrations /, and G using samekinetic constants as in Fig. 15.7,except k'8 = 10~5min~' y.M~l. Unstable limit cycle is in-

indicated by dotted lines.

398

15.5.Catabolite Repression: Sustained Oscillations and Threshold Phenomena 399

03

0.25

0.2

0.15

0.1

0.05

n

-

SNC

1-\302\253 1 .

SP -

-j r* 1\342\200\224

0.05 0.1 0.15 0.2 0.25

Figure 15.11.Magnified phase-spacediagram around node and saddle point with samekinetic constants as in Fig. 15.10.

cycle,the systemnow features a saddle-nodeseparatrix loop.Owing to thisclosed(but unstable) trajectory, for certain initial conditionsthe systemevolvesfor quite a long time before being attracted by the stablenode,asindicated clearly in Fig.15.10.

The value k\\ ^ 0.00024792min~ * has beendetermined approximatelyas

the threshold value where the saddlepoint and the nodemerge to give riseto a nonsimple singular point of the saddle-nodetype. This thresholdseparatesthe multiple limit-cycle from the multiple steady-state regime.

Forother sets of values of the parametersthe system features a singleasymptoticallystablelimit cycleof a periodof about 190min surroundingan unstable focus.The various possibilitiesfound on exploringthe param-parameters ku k\\, k8, k9,Ie,Fy are compiledin Table 15.1.

Unstablelimit cyclescorrespondingto subcritical branchesbifurcatingfrom steady stateshave beenfound in chemicalreactor theory,asmentioned

Table15.1

lmin\021 min tiM

kgmin 1fiM

1Kj

Steady states\"

0.2 6 x 10\023 0.03

0.2 6 x 0.003

0.02 6 x 10\024 0.00030.002 6 x 10\025 0.003

91100

51100

9910099100 .

5000

500

5005000

0.10.0970.0080.0002480.000247

10\025

2.00.10.0080.0002470.000248

510

702

SF no LC

\\ SF+ ULC{T~ 110min)J + SLC(r ~ 300min)

i SF + ULC

SPSNSF no LC

1\\

UF + SLC(T~1500min

)

) UF

[ SP) SN

UF + SLC(r~190min)

SF no LC

F:N:SP:LC:S:U:

focusnodesaddlepointlimit cyclestableunstable

75.5. Catabolite Repression: Sustained Oscillations and Threshold Phenomena 401

in Section8.3.Their existenceis due to the exponentialnonlinearity arisingfrom the temperature dependenceof the rate constants.The resultsof thepresentsectionestablishthe existenceof unstable limit cyclesas well as ofmultiple ones in isothermal systemsundergoing mass-actionkinetics,that

is, in systemssubjectto polynomial nonlinearities.*From a general standpoint, multiple limit-cyclebehavior has someinter-

interesting implications.Forinstance,a systemat a stablesteadystate surroundedby two limit cyclesis endowedby an intrinsic excitability,in the sensethat

perturbations from this state exceedinga certain threshold evolveto a stableperiodicregime instead of decayingbackto the referencestate.Excitabilityand threshold phenomena are widely spreadin biology.It seems,therefore,reasonableto expectthat multiple limit cycles,which together with multiplesteady statesare the two alternatives for explaining this type of behavior,play an increasinglyimportant role in biochemicalkinetics.Next, multiplelimit cyclesprovide an efficient regulatory mechanismfor varying the periodof an oscillatory motion over a wide range of values.Of equal importancemay be the overshoot behavior exhibited by the concentrations during theevolutiondepictedin Figure 15.10.

Let us comenow to the more specificproblemof the lacoperonregulation.The periodicsynthesis of /?-galactosidasein E. coli has beenobservedbyboth Goodwin A969) in synchronous cultures and Knorre A968) in

asynchronouscultures.Mastersand DonachieA966)found that the periodicsynthesisof various enzymesin synchronouspopulationsof Bacillussubtilis

arisesfrom variation in the repressorconcentration.Thus, it appearsthat the

oscillatoryenzymesynthesisis not a directconsequenceof DNA replication.Rather, it reflects the intrinsic regulatory mechanisms involved in the syn-synthesis. However,the periodof the oscillation is quite closeto the generationtime.This is in goodqualitative agreementwith our model,which predictsaperiodof 110min on the unstable limit cycleand of 190min on the (unique)stable limit cycle correspondingto the third set of parameter values in

Table 15.1.Another point of agreement is the increaseof the frequencyofoscillationsas the external lactoseconcentration increases.

In spiteof the endogenouscharacter of the oscillationspredictedby themodel,it is quite possiblethat theseoscillationscan be influencedby the cellcycle.This should be especiallyso in synchronous cultures, whereas in

asynchronous ones damped oscillationsarising from destructive inter-interference of the individual control circuits shouldbe expected.Again, this is

*Quite recently, in his doctoral dissertation, J.A. Stanshine (M.I.T.,Mathematics Department,

1975)points out the existence of one unstable and one stable limit cycle in the Field-Noyesmodel for the Belousov-Zhabotinski reaction. SeealsoBruns, Bailey and Luss A973).


in agreementwith simulationsof a more detailedschemetaking into accountexplicitlygrowth and celldivision(Sanglier,1976).

15.6.CONTROLOF CELLULARDIVISION

Having seen that the simplestgenetic regulatory processcan already giveriseto coherent phenomena,we turn to the problemof the regulation of thecellular cycleproper.

Mitosisis an eventof short duration relativeto the cellularcycle.Owingtothe sequentialcharacter of the various processesthat must occurduring this

cycle,such as DNA replication or the doubling of enzymesand organelles,it has beenassumedthat this cycleis a sequenceofdiscretestates,eachcausingthe next.

In contrast, in the particular caseof the amoebaPhysarum polycephalumin the plasmodialstage,experiment suggeststhat the \"mitotic clock\"is acontinuous biochemical oscillator that commands mitosis and DNAreplication, but that the latter processesare not themselvesat the origin ofthe oscillator.

Physarum polycephalum belongsto the group of mycetozoaand is charac-characterized by the fact that growth occurswithout cellular division. It consistsof a cytoplasmic mass containing up to 108 nuclei. As the cytoplasmiccontent increases,the nucleiundergodivisionin virtually completesynchronyevery 10-12hr.

Let us now review the most important data suggestingthe existenceof acontinuous biochemical oscillator(Kauffman and Wille, 1975;Kauffman,1974b).

Fusion Experiments

Plasmodiaat two different pointsin the life cycle(we shall say that they areat two different \"phases\")are fused.The nuclei and the cytoplasm mix well

during 2-6hr, and the mitosisof the resulting pair is studied.The followingresultsare found:

(i) Synchronization.Let the initial phases(j)A, (j)B of two fused plasmodiabesuch that <pA < (j)B.Then, if (j)B is sufficiently far from mitosisand (j>B

\342\200\224

(j>A

is sufficiently small, the resulting pair undergoesmitosis at a phase inter-intermediate between (j)A and (j)B and such that the mitosis of A is accelerated,whereas that of B is sloweddown.

15.6.Control ofCellular Division 403

This result suggeststhe existenceof one or severalcytoplasmicmitogenicsubstancescapableof initiating nuclear division when a critical value isexceeded.Subsequently,these substancesare consumed,and their con-concentration drops and reaccumulates during the next cycle.Moreover, assynchronization is a ubiquitous property of nonlinear oscillators(seeChapter7 and Section8.7),this type of experimentstrongly suggeststhat themitogensshouldthemselvesundergo a periodicnonlinear oscillation.

(ii) Mitotic block.If (j)B is closeto mitosis and cj>B\342\200\224

<pA is between \\\\ and6 hr, there is no acceleration in the mitosisof A, but one observes,instead, adelay of 2-7hr.

(iii) Abortive prophase.A almost arrives up to mitosis but then \"misses\"it and presentsan aberrant form of nucleolar reconstruction leading to aninterphase morphology.(iv) Arc discontinuity. Let the phasesof A, B be plotted along the cir-circumference of a circle(Fig.15.12).For reference, we considera third

plasmodiumCwhosephase(j)cison the other sideof A on the circumference.We assumethat, oncefused, two plasmodiamix well and fairly rapidly.Let (j)AB be the phaseof the mixed pair after mixing. We expectthat, as <pB

moves away from (j)A to (j)Bl, (j)B2,...,(j)AB moves as indicated in Figure15.12.Beyond a phase (j)B* of B, however, synchronization leadsthe fusedpair to a phase(j)AB\\ying on the samearc as C.Thus, at B* the phaseof thefused pair undergoesa discontinuity. Sucha discontinuity is indeedfound

experimentally.The ingenious ideaof Kauffman has beenthat, by studying the behavior

under fusion experiments,one can obtain direct information on the nature of

B2

Figure 15.12.Illustration of arc discontinuity

arising on fusion of two plasmodia of Physarum

polycephalum.


the mitotic oscillatoritself.In fact, this ideashouldappearquite natural tothe readerfamiliar with the analysisofChapter7 of this monograph.Indeed,let us begin with the Ansatz, suggestedby the synchronizationexperiments,that mitosisis directedby a nonlinear oscillatorwhich, by its very stability,must undergo a limit cycletype of motion. In a fusion experiment,two suchoscillatorsare coupledvia a diffusion-likemass transfer (seealso Section7.12).Obviously,then, the behavior of the final pair can give information onthe behavior of the initial limit cycle.

Heat-ShockExperiments

In a second group of experimentsone acts directly on the individual

plasmodiaby applying short (of the orderof 30min) heat shocksat a higher-than-ambient temperature. One finds that the plasmodiamay missthe first,or even the secondmitosis, which would happen under normal conditions.Subsequently,however, they again undergo nuclear division in completesynchrony. Everything happensas if the \"chemicalclock\"commandingmitosis can still measure the time correctly, but that under the effectof theperturbation generated by the shock it needsa certain interval to adjustbefore resetting its phase.

15.7.QUANTITATIVE MODEL

We want now to interpret the data compiledin Section15.6on the basisof aquantitative model.The stability requirement suggestsa limit cycle-typeoscillator,and the simplestsuch oscillatorcan be constructedby couplingtwo biochemicalvariables. Let X be the bifurcation parameter.By varyingX from Xc, where a small-amplitude limit cycle first bifurcates from thethermodynamic branch (cf.Section7.12)up to the limit X -> oo (cf.Section8.10,first subsection),one can account for various forms of oscillatorybehavior, from sinusoidalto discontinuous.By comparing with experimentone can have indicationsboth on the reasonablenessof a two-variablemodeland on how far within the relaxation oscillation regime one must look forthe functioning of the limit cycle.Rather than remain general,let us considera specificexample(Tysonand Kauffman, 1975).

Let X representa stableprotein and Y an active form of this protein.The conversionof X to Y can alsobecatalyzedby a sequenceofenzymes,andlet the overall rate of this conversionbeproportionalto Y2.OnceYexceedsa critical value Yc it boundswith the chromosomesand initiates mitosis.Thus Y is not really consumedbut merely leaves the cytoplasmicmedium.

15.7.

We

Quantitative Model

obtain in this way the following

X -X -Y -

model

A\342\226\272

B\342\226\272

cY2

d>

X

Y

Y

405

The rate equationsdescribingthe evolution of X and Y in a single plas-modium are (setting c = d = 1):

d4~ = A -BX-XY2dt

^-= BX + XY2 - Y A5.14)dt

Comparingwith Eq.G.13)in the absenceof diffusion, we realize that themodel has somestriking similarities with the trimolecular model.In fact,the differenceis that the rolesofX and Y are inverted in the autocatalyticstepand that the sink term for X is replacedhere by a sink term for Y. Exceptfor this difference,the analysis of Eq.A5.14)proceedsalong the same linesas in Sections7.3and 7.4.Onefinds that there isa singlesteady-statesolution:

*\302\260 irr^' Y\302\260

A A5-15)D + A

This solution becomesan unstable focus when the following inequality issatisfied:

2A2 > B + A2 + 1 A5.16)B + A

A limit cyclethen emergesfrom the steadystate that becomesmore and moreof the relaxation type as B decreasesto zero.

In order to discussfusion experimentson the basis of this model,weassumethat the cytoplasmicand nucleartransfer within the pair is equivalentto a diffusion betweentwo identicalcells1 and 2, all of whosewallswith the

exceptionof the one separating them are impermeableto the flux ofmitogens.The mitogen flux, Jxacrossa unit area of this separating surface is given byFick'sfirst law:


where Ar is the length of the cell,with a similar expressionfor Y. In ordertofind from Eq. A5.17)the total rate of change of X within, say, cell 1,wemultiply by the surface area S and divide by the volume V of the cell.Weobtain:

=1TtAX2-X1)= S1(X2-X1) A5.18)w v

Combiningwith Eq. A5.14)we obtain four coupledequationsdescribingthe evolution of X and Y in the two cells:

dt

dX

-= A -BXl - XXY\\ + 6^X2- Xt)

= A - BX2- X2Y\\ + d^Xi -X2)

'^= BXl +XlY2l-Yl+S2(Y2-

= BX2+ X2 Y22-Y2+S2(Yl - y2) A5.19)

A two-celldiscreteversionof the trimolecular model similar to Eq.A5.19)was analyzed by Prigogine and LefeverA968) well before the bifurcationtheoretical analysisadoptedin this monograph wasdevelopedand has givenrise, in fact, to the first unambiguous exampleof symmetry breakingin

chemicalkinetics.Onemay now study the behavior of the two diffusion-coupledoscillators

describedby Eq.A5.19)and comparewith the resultsof the fusion experi-experiments. As it turns out, Eqs.A5.19)have many types of solution (TysonandKauffman, 1975).Here,however, we are interested only in the followingtwo possibilities.

First, the phasesof two individual oscillatorsare initially nearly identical,and then diffusion is switchedon.One then finds by numerical simulationof Eq.A5.19)that the homogeneouslimit cycle,which is still a solution ofthese equationsas long as inequality A5.16)is satisfied,becomesunstable.It splitsinto two local limit cyclesprevailing in the two cells(seeFig.15.13).However, their differencewith the homogeneouslimit cycle is small. Thustheir amplitudes exceedboth the critical value Yc and as a result mitosistakesplacein synchrony.

Second,the initial phase difference is appreciable.After switching ondiffusion one finds an inhomogeneouslimit cycleoscillation leading to onelimit cyclewith amplitude almost identicalto that of the homogeneouslimit

cycle,whereasthe amplitude of the secondcycle is strongly reduced(see

15.7.Quantitative Model

Y

407

IHLC

Figure 15.13.Inhomogeneous limit-cycle solution (IHLC)arising in model A5.19)wheninitial phase difference between individual oscillators is small.

Fig.15.14).As a result, if Yc is higher than this secondamplitude, there isasymmetricinhibition ofmitosisin one copy.This can beeither a permanent(such as in Fig.15.14)or transient phenomenon.One copy,say A, maymiss the first mitosis but eventually synchronize with the secondcopy Bwith an appreciabledelay.

We concludethat, despiteits simplicity, this model providesa qualitativeexplanation of most of the fusion experiments,including the phenomenon ofmitotic block.By adjusting the numerical values of the parameters<5i, <52,A, B one can alsoobtain a semiquantitative agreement.As it turns out, the

Figure 15.14.Strongly inhomogeneous limit-cycle solution arising when initial phase difference

is appreciable.


most favorable range of the chemicalparametersis that correspondingto amoderate relaxation oscillator, for example, A = \\, B = ^o, whereas fordiffusion coefficientsa good choiceis <5X

= 0.15,52 = 0.10,and for thethreshold,Yc

= 1.7.A qualitative failure of the moderelatesto the arc discontinuity. Experi-

Experiments show that the dependenceof</>B\302\253

on <j>A (seeFig.15.12)is given by acurve comprisinga region of negativeslope,contrary to the model that givessolely curves with a positive slope.It has beenargued (Kauffman, 1974b;Kauffman and Wille, 1975)that this property of the arc discontinuity isrelated to the topology of the isochrones,that is,of thosecurveson the phaseplanethat separateportionsof trajectoriesdescribedat equal time intervals.The notion of isochroneis also useful in interpreting the heat-shockexperimentsdescribedin Section15.6.For further details,the reader isreferred to the original publicationscited therein.

Chapter16

CellularDifferentiationandPatternFormation

16.1.INTRODUCTORYREMARKS

In this Chapter we are interested in large-scalephenomena involvingassembliesof interacting cells.Our aim is to seehow cellular communica-communications can generate regular spatial patterns at the supercellularlevel andat the same time control the timing and localizationof the variousprocessesinvolved in the formation, as well as in the maintenance of thesepatterns.In biologyoneof the most striking illustrations of this phenomenon occursduring embryonicdevelopment.

A higher organism is constituted of a number of differentiatedcellsthat

can be characterized by their function and form. The molecular basis ofdifferentiation\342\200\224that is, the control of the transcription and translation ofgenetic information\342\200\224is not consideredhere in detail (seeBritten andDavidson,1969;Davidson and Britten, 1973).Rather,wefocuson the follow-following problem.An advancedorganism,suchas a vertebrate,contains 100or sodiscretecellular types.Differentorganismsmanifest a variety of shapesandforms that is quantitatively certainly much greater than the differencesin

the constituting cells.The understanding of this variety, specifically,of the

spatial organization of different cellular types, is the problemof patternformation,ormorphogenesis(Bonner,1974;Wolpert, 1975).

The complexityof the problemcan beappreciatedif one realizesthat eventhe most advanced organism issuesfrom a single cell,namely, the zygote.Ontogenesisis, therefore, a spatiotemporalprocessimplying subcellularmechanismsthat are finally reflectedby morphologicalmodificationsat thesupercellularlevel.Obviously, this passagerequiresa sharp regulation of avariety of processessuch as:(a)cellular differentiation,growth, and mitosis,(b) cellular movements (chemotaxis,cell sorting), (c) formation of cellularjunctions,and (d) the collectivemovements at the supercellularlevel(e.g.,gastrulation).

409

410 Cellular Differentiation and Pattern Formation

A detaileddescriptionaccounting in a quantitative way for both themolecular and globalaspectsof the problemis certainly premature at thepresenttime. It is,therefore,interestingto study morphogenesisby means ofsimplified but nevertheless representative models based on those ex-experimentally establishedelements that continuously recur in embryonicdevelopment and are largely independent of the detailsof the subcellularphenomena.This analysis enablesus to correlatethe molecular and themacroscopicaspectsof morphogenesisand to sort out the essentialfeaturesdetermining pattern formation.

The viewpoint we adopt is motivated by the considerationsdevelopedthroughout this monograph.In other words,we considerthat different

macroscopicpatterns correspondto different stable solutions of a set ofnonlinear differential equations describingthe evolution of the chemicalsubstancesinvolved. Theseequationsare developedin Sections16.4to16.6and appliedto concretebiologicalsituations in Section16.7.Sections16.2and 16.3are devoted to an account of the mechanisms determiningcellular interactions during development.

16.2.POSITIONALINFORMATION

The analysis of pattern formation becomesmore transparent when viewedat the levelofa morphogeneticfield.The latter can bedefinedas an assemblyof functionally coupledcellswhosedevelopment is under the control of acommon regulatory process(Robertsonand Cohen,1972).Typically,such afield comprisesof 10to 100cells.Forinstance,in the earlysea-urchingastrulaonefinds 30 cellsover a length of 0.2mm. We want to understand how aninitial assemblyof identicalcellswithin the field can give riseto differentiatedcellsthat are spatiallyorganized.We would alsolike to obtain information onthe stability of this spatial pattern.

It has long beenknown in embroyology(seeBrachet, 1974,for an excellentsurvey) that there exist physiologicalgradients within eggsduring develop-development. ChildA941)has insistedheavily on the relationsbetweenthesegradientsand subcellularregulation by stating that

\" If a gradient extendsover morethan a singlecell,cellsalong its courserepresentdifferent levels,and thesedifferencesprovide a basisfor differencein geneaction, certainlyan essentialfactor in differentiation.\" Such observations led gradually to the conceptofpositionalinformation, an idea first proposedby Drieschin 1895and which

is now widely acceptedin embryology.Wolpert A969, 1971,1975)developedfurther the idea of positional

information and appliedit to a variety ofdevelopmentalproblems.Hepointsout that positionalinformation implies the existenceof a coordinatesystem

16.2.Positional Information

Blue

M

411White Red

Figure 16.1.\"French flag problem\" illustrating positional differentiation.

within which cellpositionis specified.This resultsin a spatial differentiationthat precedes,and is independent of, molecular differentiation at the sub-cellular level.The latter takesplacein a secondstage,during which positionalinformation is interpreted by the individual cells.Wolpert postulatestheuniversality of this mechanism.

A very instructive illustration ofWolpert's ideasis provided by the so-called\"French flag problem.\"Considera one-dimensionalfield. The positionofcellscan be defined with respectto one of the extremal points (unipolarsystem)or with respectto both (bipolarsystem).Supposethat the resultingpositional information is such that in the first third of the field a bluepigmentis produced,in the secondone no pigment is produced,and in the final onea red pigment is produced.This gives riseto a \" French flag,\" as shown onFig.16.1.Supposefurther that positionalinformation is relatedto the gradientof a substanceM,the morphogeru which diffusesbetweena sourceand a sink.Assuming diffusion is describedby Fick'slaw and that the morphogen is notinteracting with the cells,the evolution of the concentration M of M is givenby:

dM d2M'llr2'

The steady-state solution of this equation (DM is taken constant) is:M,-M2M = r + M2

A6.1)

A6.2)

whereMx, M2 are the concentrations of M at the sourceand at the sink and /the length of the system,respectively.

An important property related to the linearprofileofMgiven by Eq.A6.2)is size in variance; supposethe length of the system is varied from / to /'.We say that there is size invariance, if the concentration of morphogen at


point r of the initial system is equal to the concentration at point r' of themodifiedsystem,where r and r' are connectedby

In a linear gradient this property is trivially satisfiedprovided the boundaryconditions(valuesofMat the sourceand sink) are reestablishedin the modifiedfield.

A qualitative behaviorcorrespondingto sizeinvariancecorrespondsto the

property known in embryologyas morphallaxis.A different kind of patternregeneration is known as epimorphosis;the positional information within

the new field /' remains the same as in the initial one,and restoration of the

pattern is achievedby growth (seeFig.16.2).

B jw Ft

(A)

(B)

Epimorphosis

1

1

I

Figure 16.2.Regeneration of French flag pattern. Intact system is asin {A) and comprises three

equal regions of blue (B),white (W), and red (R).Positional information is represented by linear

gradient with boundary values M, and M2. If flag is cut at level X-X corresponding to /cth

cell and left-hand pieceis removed, then morphallactic regeneration occurs as in (B).Value of

positional information at cut now becomesM, and new gradient is established (dashed line).

In epimorphic regeneration, (C),positional information at cut remains sameand restoration of

pattern is achieved by growth.

16.3.Mechanisms Involved in Positional Information 413

Accordingto Wolpert, the existenceof a sourceand a sink postulatedsofar* are justified in the sensethat an organizerhas been shown to exist in agreat variety of fields,which inducesdifferentiation in the neighboringcells.This is the caseof hypostome in Hydra, of micromeresin seaurchin, or ofHensen'snodein chick(seeRobertsonand Cohen,1972for a survey). Oneofthe principal goalsof this chapter (Section16.4and 16.5)is to analyze the

origin of positional information, that is,the establishment and maintenanceof the boundary conditionsgiving rise to a sourceand a sink within the

morphogeneticfield.

16.3.MECHANISMS INVOLVED IN POSITIONALINFORMATION

The diffusion of a morphogen betweena sourceand a sink consideredin theillustration of Wolpert's ideas(Section16.2),is only one of the numerouscandidates potentially responsiblefor the establishment of positionalinformation and for the subsequentpattern formation. As shown by CrickA970), the time necessaryfor a morphogen to diffuse through a field oftypical dimensionsis compatiblewith experimental data. The diffusion

coefficientappearsto beof the orderof0.8x 10~6cm2 s~1,which is aboutthe value of the diffusion coefficient in solution of low-molecular-weightsubstancessuch as cyclicAMP.

Forreasonsexplainedin Section16.4,most of the ideasof this chapter areillustrated by assuming that diffusion is, indeed,responsiblefor positionalinformation. Before this, however, we compilebriefly in this sectionsomeother processeslikely to influencepattern formation.

Active Transport

A morphogen can propagatealong a field by either facilitated or activetransport.This can be the result of an asymmetry in the cellular propertiesgiving rise to polarizedphenomena.As a result, a gradient can build upalong the field that is oppositeto the direction of the morphogen'stransport(Lawrence,1966;Wilby and Webster, 1970).Although no sourceand sinkneed to beconsideredexplicitly,the cellson at least one of the boundariesplay a privileged role.An alternative model involving nonpolarized cellsand assuming the existenceof a sourceand a sink has been elaboratedby Babloyantz and Hiernaux A974).

* The reader has certainly noticed by now that the assumption ofa source and a sink amountsto a boundary-value problem whereby fixed concentrations are imposed on the boundaries.


Cellular Contacts

Cell communication can be establishedwithout having necessarilythepropagationofa substancealong the field.Considera one-dimensionalsystemof N polarizedcellsand supposethat the state of differentiation of each cellis determinedby the concentration ofa morphogenM.It may happen that thelevelMofM in the ith cellis determinedby cell i \342\200\224 1 via a seriesofactivationand/or inhibition mechanisms involving membrane-bound enzymes andmetabolites (MacMahon,1973;Babloyantz, 1977;Garay,1977).Typically,this givesriseto a steady-stateprofileof M describedby the following equa-equation:

Under certain conditionsthis expressioncan give rise to differentiatedpatterns.The latter dependsolelyon the number of cellsand hence cannotdisplaysizeinvariance.

The situation is different in the caseof bipolarfields,where the state of acell is determined by two morphogensand is influencedby the state of both

neighboring cells.Patterns displaying size invariance can be generated(Garay, 1977).Thereexistsalsoa possibilityofbifurcationsleadingto multiplesolutions,one of which correspondsto a uniform distribution of the mor-morphogens, aswell as of wavelikebehavior (Babloyantz,1977).

Roleof Oscillations

Goodwinand CohenA969)suggestedthat positional information can arisefrom the formation of a phasegradient. They assumethat each cell in thefield ispotentially an autonomous oscillatorof periodT. During this periodthe cell evolves through three states:(a) the (brief)emissionof a signal,say at times 0+ , T + , 2T + , (b) a long refractory period, TR, duringwhich the cell is insensitive,and (c)a time interval T \342\200\224

TR, during which thecellis sensitive,although it cannot emit.When a cellemits a signalSreachinga neighboringcell,the latter in turn emits,after a delay of Ats, provided it issensitive.This gives rise to an organizing wave.Positional information canarise if there existsa gradient of frequenciesof the autonomous oscillatorsalong the field. Under certain conditionsthe oscillatorhaving the highestfrequency can entrain the others,and this constitutes an organizer in the

biologicalsenseof the term. Note that this implies an initially polarizedfield.

Goodwinand CohenA969) postulatethat after signal S in released,asecondpulseP related to S is produced.The latter propagatesthrough the

16.4.Dissipative Structures and the Onset ofPolarity 415

field and establishesa phasegradient giving riseto a more accuratepositionalinformation.

The incidenceof these phenomena in pattern formation is not well-established.Nevertheless,the aggregation of cellular slime moldsdiscussedin Chapter 14provides a striking illustration showing the ubiquitous roleof oscillationsin initiating the morphogeneticsignal, in amplifying andrelaying it along the field,and in regulating the sizeand other propertiesofthe resulting multicellular body.

On the cellular level,the role of concentration wavespropagating alongmembranes during the early stagesof development has been emphasizedby GoodwinA975), Kauffman A974a), and Blumenthal A975).Electricalsignalshavealsobeendiscoveredin Acetabularia(Novak and Bentrup, 1972;Novak, 1975)and in Fucus (Nuccitelliand Jaffe,1974).

CellSorting

The formation ofa morphogeneticpattern can alsoarisefrom the occupationof specificpositionsby differentiated cellsas a result of a \"cell sorting.\"This mechanismsupposesthat cellscan respondto specificsignalsby movingautonomously within the field. This movement can be oriented toward aspecificattractant emitted by another cell of the same species(chemotaxis;seealsoSection14.7),or it can bethe result of characteristicadhesiveproper-properties. Both suchphenomenaappearto bepresentin different stagesofdevelop-development of the slime mold Dictyostelium discoideum(Kellerand Segel,1970;Garrod,1974).

16.4.DISSIPATIVESTRUCTURESAND THE ONSETOF POLARITY

A system of coupledcellscapableof undergoing differentiation can alsodisplaypattern formation for various reasons.Firstly, a certain asymmetrycan be built into the individual cellsat the outset (seefirst and third sub-subsections in Section16.3).Secondly,the cellsat boundariescan be subjecttospecificconditionsgiving rise to a sourceand a sink\342\200\224that is, to a polarityalong the field (seesecondsubsectionin Section16.3).A final possibilityisthat a homogeneoustissueof identical and unpolarized cellsbecomesin-

homogeneousspontaneously.This can only happen if a symmetry-breakinginstability is involved.As mentioned repeatedlythroughout this monograph,diffusion is one of the very general mechanismscapableof inducing such an

instability. Morerecently,the role of cellularcontactsin symmetry-breakinginstabilitieshas beenrecognized(Garay, 1977;Babloyantz, 1977).Neverthe-Nevertheless,

from now on we choosethe mechanism of cellular communication


through diffusion to illustrate the ideasunderlying pattern formation. Wefirst deal,in this section,with the onsetofpolarity in a previouslyunpolarizedfield and with the stability of the resulting gradient. The role of this primarypattern in inducing a secondarypattern of differentiatedcellsis examined in

Section16.6.The fact that we are concernedwith a symmetry-breaking instability

inducedby diffusion implies that the primary pattern is necessarilya spatialdissipativestructure, We want, moreover, to ensureits stability with respectto smallperturbations;otherwise,the pattern would not havea physiologicalmeaning.From the analysisofSection6.6and ofChapter7 we then concludethat:

(i) Nonlinear interactions between at least two morphogenetic substancesmust be involved.

(ii) It is sufficient that the pattern arisesvia a supercriticalbifurcationfrom theuniform steady state on the thermodynamic branch involving an eigenvalueproblemof odd multiplicity.

(iii) The rate equationsmust admit positiveand boundedsolutions.

The first of these ideaswas introduced in the context ofmorphogenesisby

Turing A952).The particular modelshe used to illustrate his ideaswererecentlychallenged(Bardand Lauder,1974;Cooke,1975)on the groundsthat

they couldnot yield regulativepatterns.A bifurcation analysis carriedoutrecently on one of Turing's models(Erneux, Hiernaux, and Nicolis1977)revealed that this model displayspeculiarbehavior becauseproperties(ii)and (iii) are not satisfied.Asidefrom this, the ideasof Turing and the notionof dissipative structure remain perfectlycompatiblewith the phenomenonof pattern formation in morphogenesis.

Giererand Meinhardt A972, 1974)have achieved further clarificationofthe mechanismof pattern formation by pointing out the role of the couplingbetweena short-rangedactivationphenomena and a long-ranged inhibition.Forinstance, in the caseof Hydra the head couldbeformedwhen the activa-activator concentration exceedsa critical value. Onceformed, the head couldexert an inhibition extending over a long distance,not permitting further

formation of heads.Theseideas instigated them to suggest a number ofmodelsfor pattern formation.Oneofthesemodelsis taken up and analyzedin

Section16.5.

16.5.A QUANTITATIVE MODEL

We want to understand how two coupledmorphogens,say X and Y, can giveriseto a spontaneousgradient along a field.Onceestablished,this gradient

16.5.A Quantitative Model 417

canprovide the positional information necessaryfor the formation ofpatternsof differentiatedcells.

The rate equations have the following general form

= MX, Y) + Dt V2 Xdt

dY-^= f2(X, Y) + D2V2Y A6.4)

As we are interested in the spontaneousonset of the gradient, the boundaryconditionsto be consideredare zero fluxes at the extremities of the field.Fora one-dimensionalarray of cellsthis gives:

dX dY = 0 A6.5)

where / is the length of the field.Let (Xo,y0) be the uniform steady-state solution of Eq.A6.4).We study

the stability toward inhomogeneous perturbations (x, y) which to a first

approximation satisfy the linearizedequations(seeSection7.4):

dy

dt 4hdX A6.6)

BecauseofEq.A6.5)weseekfor solutionsof this linearizedsystemin the form

1\\em-' cosmnr

m = 0, 1,... A6.7)

In particular, we are interested in the onset of a stationary polar pattern,ojm

= 0.From Eqs.A6.6)and A6.7)one obtains a relation betweenm, /, DuD2,and the chemicalparameters involved infi and/2 through the character-characteristic equation (seeSection7.4 for a similar analysis of the trimolecularmodel):

A sm2n2U- \"' T

AdX

AdY

AdY

Dm2n2

I2

= 0 A6.8)

Let us evaluate Eq.A6.8)on a specificexample.The latter must beregardedas a minimal model usedfor illustrative purposes,rather than asa limitationon the validity of the general arguments advanced in this section.


Supposethat morphogen X\342\200\224 the activator\342\200\224is producedat a constant ratek^A as well as via an autocatalytic step involving a third molecule P. X

catalyzes the formation of Y from a precursorC at a rate k2CX2.Moreover,P transforms Y to X. Finally, either X and Y have a finite life time and yieldwaste productsF and G, or they bound with the genome of the cells.Ineither case,we assumethat they leave the reaction spacevia unimolecularsteps.Thesestepscan be representedas follows(Babloyantzand Hiernaux,1975):

Y

B + X

+

+

C

X

Y

P

P

k2x2^

k* ,

'kh

k7

Y

F

G

X

P + 2X A6.9)

Using a quasisteady-stateassumption for P one can derive straight-straightforwardly the form of the rate functions/!and f2 appearingin Eq.A6.4):

f k A 4. h

2f2 = k2CX2- k4Y A6.10)

(Giererand Meinhardt, 1972).The steady-state solution is:

= k1A0 k

_

k3 k2k3k5C

k\\k2A2C ikik-jktAB k4k26k2B2+ +k\\k5 k2k\\k\\C A6.11)

and the characteristic equation A6.8)takesthe explicitform:

\342\200\236\342\226\240 ... m2n2/ , 2aX0D2. , , ,*,,~> ^A = DiD7-^r-+ \342\200\224,- ( k3D2 + kAD, ^\342\200\224 ) + k3kA = 0 A6.12a)

16.5.A Quantitative Model

with

419

a. = k6k7BA6.12b)

The study of A in terms of the length / for various m values permits one todetermine the points of marginal stability for which relation A = 0 issatisfied,(seeFig.16.3).

Form = 1,A vanishesat /},indicatinga possibilityfor onset of instability.If the total length of the system is lessthan /} the homogeneousstate of thesystem remains stable,and there is no possibilityfor sourceand sink for-formation. If the length is increasedbeyond /},there is possibilityfor spon-spontaneous pattern formation dueto the instability of the homogeneousstate ofthe system.Moreover,weseefrom Fig.16.3that, up to the length l\\, one mayonly have a structure with one high and one low concentration point.If, on the other hand, we take a systemof length betweenl\\ and

\\\\and lessthan

/3, two possibilitiesarise,due to the overlapping of various m values for agiven length of the system;structures with one high-concentrationregionandstructures with two high-concentration regionsmay appear,Thus, the form

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0.1

\342\200\224

I I II\\i V- E In co

E - r

4i 'k^r\342\200\224 \\ y\\ 3^^\342\200\224

\342\200\224

-

\342\200\224

--

\342\200\224^.\342\200\224^\342\200\224^-^

\\-~-\342\200\224' 6

\342\200\224

Figure 16.3. Dependence of characteristic determinant A [seeEqs. A6.8) and A6.12a)]on

length / for different values of wave number m, showing wavelengths that can be fitted within

given length.


49 \342\200\224

15Cell position

Figure 16.4.Concentration profile of morphogen Y in model A6.9) for length / = 0.9and for

M = 0.0005.B=\\,C=\\.k2= 5.ki=0.1,kA= 0.7,k5 = 1,kb

= 1,k7 = 2, D, = 0.002,

of the final pattern is a function of the magnitude of the initial perturbationsand their location in the system.

As / increasesthe systemmay becomeunstable for higher valuesofm. Thus,a high number ofwavelengthscan befitted into the systemand morecomplexpatterns may appear.Theseconjecturescan be verified by computer simula-simulations. Considera homogeneoussystemwith parameter valuescorrespondingto an unstable steady state leadingto spatial structure formation. At t = 0 aperturbation is introduced into the system. If the length / of the system isbetween l\\ and l\\, one obtains a structure with one peakas seenin Fig.16.4.This structure displaysa gradient appearingspontaneouslyalong the system.The form of the gradient is unique, but there are two possiblebifurcatingsolutions that are symmetrical, as discussedin Sections7.5-7.7.Fromthe biologicalpoint of view, this multiplicity is immaterial; the notion ofsymmetry becomesmeaningful only after the first gradient has beenestab-established from an initially homogeneoussituation. Similar results are obtainedin two dimensions,although the patterns are much more spectacularin this

case,(e.g.,trimolecular model discussedin Section8.5).In addition to thenumericalsimulations,a bifurcationtheoretical analysisofschemeA6.9)hasbeencarriedout along the lines of Sections7.5-7.7.(Hiernaux, 1976).Usingthe length / as bifurcation parameter, the possibilityof supercriticalbifurca-bifurcation at a simpleeigenvalueleading to two spatially symmetric solutionshasbeenestablished.This guarantees the stability of the patterns, in agreementwith the results of the numerical calculations.

The stability of the morphogeneticpatterns derived in this sectionimplies^.regulationwithin the field, at least for lengths within the interval/

\\<l<l\\.

In particular, we seethat after removal of a part of the field, the remainingreestablishesa stablepolarpattern as long as / remains larger than /}.This

16.6.Positional Differentiation 421

property is lessstringent than the size invariance invoked in Section16.2.Nevertheless,it can account for the maintenance of a certain characteristicstructure, as observedin experimentsinvolving Hydra regeneration. In thelatter, one couldrelate the regeneration of the head to the attainment of acertain critical threshold, rather than to a very narrow interval of concen-concentration values as postulatedby other investigators (Hicklin,Hornbruch,Wolpert, and Clarke,1973).

16.6.POSITIONALDIFFERENTIATION

Supposenow that, by a mechanism of the type discussedin the previoussections,a polarpattern hasdevelopedwithin a field.In addition to polarity,this pattern endowsthe field with well-defined boundary values of the

morphogensand, more generally, with positionalinformation for the in-individual cells.We want to analyzehow this information is interpreted by theindividual cellsand inducesa secondarypattern of differentiation.

We assumethat one (or several)morphogen(s)give rise in a certain way,not specifiedhere, to a substanceS that can diffuse along the field betweenasourceand a sink. The latter are specifiedin a unique fashiononcethe primarypattern of the morphogenshas been establishedand may, therefore, beconsideredto be \"imposed\"at the outset.This assumption can be relaxed(Babloyantz and Hiernaux, 1975)but is maintained here for simplicity.

In each cell,it isassumedthat Sactsat the geneticlevel*and regulates the

synthesisof a protein E. We require that the E S relationship be sigmoidalor S-shaped,as in the modelsanalyzed in Sections8.4and 15.4.In particular,it is assumedthat thereexistsa thresholdvalueScofSfor which a considerableenhancement occursin the synthesis of E. This kind of \"all-or-none\"be-behavior, which wastacitly assumedin the analysisofthe \" French flag problem''of Section16.2,may beachievedthrough repression,activation,or combina-combination of these mechanismsknown to operatein bacteria and probablyalsoin

eukaryotes(Jacoband Monod,1961).Onecan construct quantitative modelsexhibiting all-or-nonebehavior with elements proposed by Britten andDavidson A969) and Georgiev A969) for genetic regulation in higher or-organisms (Hiernaux, 1976).However, this would introduce unnecessaryassumptionsand a great number of variables.Instead,weconsidera model,

* Thus, while our model indicates how the morphogens may becoupled to the genome, it doesdoesnot specify how they areproduced genetically.


based on the bacterial induction mechanisms of the Jacob-Monodtype(Babloyantzand Nicolis,1972;Babloyantzand Sanglier,1972).

R

R

n

R

+ o+

+ 21

+ o+

S + E

E

'k2

*3

k.ks'k6

k1t

ks ,'k9

*10

R

O\"

Ft

E + O

1 + E

F A6.13)

A repressorR is synthesizedfrom a precursorR' and can repressthe activeoperatorO+ into O\",thus blockingthe synthesisof the enzymeE.The lattercatalyzesa reaction whosesubstrateis the morphogen Sand whoseproductIacts as an inducer by inactivating the repressorR. /ct,...,fcu are the rateconstants,F is the decayproductof the enzymeE, r\\ representsthe precursorsof enzymeE and F!representsthe repressor-inducercomplex.

Forthe entire morphogeneticfield of N cells,the variablesassociatedwith

the model are Of,Of,Rh S,,and /, (for i = 1,N).As before,we take thevariations of /?,-,Of,Of at the quasisteadystate.Then, the time evolution of\302\243,, /j,and S, is given by (Babloyantzand Hiernaux, 1975):

~T~=-.\342\200\224]\342\200\224^r, ]\342\200\224,\342\200\224r )\342\200\224; ;\342\200\224;\342\200\224ff

\342\200\224 Kio-t; + Ku^fit u u R -4- u u r -L- U w -\\- w u /

^ = _fc I^Rll+J^FJl+ k p +k ES _ k EI

^ =-/c8\302\243,S,. + kgEJi+ -^(Si+,+ S,_!- 2Sf) i = 2,...,N - 1

at Ar

A6.14)

We haveassumedthat only the morphogen Sdiffusesthroughout the field.Diffusion has beenapproximatedby Fick'slaw, which has beenwritten in afinite-differenceapproximation suitable for numerical calculations.D is the

(constant)diffusion coefficientof S and Ar is the dimensionof a cell.

16.6.Positional Differentiation 423

15Cell position

Figure 16.5.Gradient of diffusion of morphogen S in source-sink field of 30cells.

We assignto the boundary cells1and N valuesofE,I,and Scorrespondingto the pointson the lowerand on the upperbranch of the E-Scharacteristiccurve. This establishesa sourceand a sink in the morphogeneticfield.

The steady-state solutionsof the Eqs.A6.14)give the stationary spatialdistribution of E,I,and S.Theseequationshave been solvednumerically bythe Runge-Kutta method on a CDCcomputer.Figure 16.5presentsthestationary gradient ofthe morphogenSbetweensourceand sink.Thepositionof the sourcedeterminesthe polarity of the field.Figure 16.6describesthe

spatial distribution of the enzymeE correspondingto the above mentionedgradient.It is seen that a smooth variation in the gradient of morphogenmay generate a discontinuousresponsecurve for E. This property is dueto the nonlinear character of the E-Srelationship and the existenceof athreshold concentration Sc for the synthesis of E. In a morphogeneticfield of N cells,due to the processof diffusion, the concentration of S is notidentical in all cells.The cellsexposedto a concentration S> Scsynthesizeappreciablequantities of E and are differentiated,while thosewith S < Scremain undifferentiated.Thus, the field is separatedinto two distinct regionsof high and low concentrations of E.

15Cell position

Figure 16.6. Pattern of celldifferentiation corresponding to gradient in Fig. 16.5.


1 30Cell position

Figure 16.7. Pattern of cell differentiation in source field of 60cells.

Fora given dimensionof the field, the proportionof inducedcellsis deter-determined by the value of the concentration of Sat the sourceand at the sink andby the magnitude of diffusion coefficientD.

If the concentration of the substancessuddenlychangesin one or severalcells,the perturbedpattern later returns to its originalform. Thus,the patternsof Figs.16.5and 16.6are stable.Earlier analysesof mechanismssomewhatsimilar to Eq.A6.14)by EdelsteinA972) had producedunstable solutionsand hencecouldnot account for the patterns associatedwith morphogenesis.

If one imposesthe value of the sourceat both boundariesof the morpho-genetic field, one obtains the differentiatedpatterns of Fig.16.7.This cor-corresponds to the duplication of the pattern of Fig.16.6and displaysdiffer-differentiated cellsnear the two boundariesof the field.Forgiven valuesofSand D,this property holdsonly if the dimension of the field is not smaller than acertain criticalvalue.Theseresultscan easilybeextendedto two-dimensionalmorphogeneticfields(Babloyantzand Hiernaux, 1975).

In conclusion,the main interest of the modelspresentedthus far in this

chapter liesin their relativesimplicity and in the fact that the resultsobtainedare largely independentof the detailsof the regulatory processesat the sub-cellular level.We may say, therefore,that we have a few \"generalprinciples\"underlying pattern formation.The purposeof Section16.7is to illustrate theuse of these principlesin concretesituations involving the development ofsimpleorganisms.

16.7.APPLICATIONS

We do not attempt to presenta quantitative analysis, which at this time ismeaninglessin view of the largenumber ofvariablesinvolved in any biologic-biologically

realistic situation. Rather, we focus on the general but quite concretepredictionthat in the courseof developmenta critical sizemust beexceededbeforepositional information\342\200\224and hencepattern formation\342\200\224can takeplace.

16.7.Applications 425

We refer to two extremelydifferent organisms:(a)Hydra, which in the adultstageis a highly differentiatedanimal involving 17different cellulartypesand(b) Acetabularia, which is a unicellular organism capableof undergoingmorphological changesgiving riseto the appearanceof a cap in the apicalregion.In both cases,becauseof the morphology of the organisms,one canspeakof a one-dimensionalmorphogenetic field.

Considerfirst the caseof Hydra.Schallerand GiererA973)have demon-demonstrated the existenceof the gradient ofa substanceextendingalong the animalthat activates the formation of the head (Schaller,1975).This substancehasbeenisolatedand is a peptideof low molecular weight. Moreover, BerkingA974) has isolatedan inhibitory substancethat is again distributedalong theanimal and inhibits the formation of the head region,consistingofhypostomeand tentacles.Thesediscoveriesaddcredenceto modelsof the type discussedin Section16.5involving nonlinear interactions betweenan activator and aninhibitor. In the analysisin Section16.5wehaveshown that the length ofsucha systemmust exceeda criticalvaluebeforea region of high concentration ofthe activator appears.If one acceptsto identify this with the head region ofHydra, then one obtains an interpretation of a fact well-knownto zoologists,namely that Hydra must exceeda critical size before acquiring its morpho-morphological

characteristics.The situation is fairly analogous in the caseofAcetabularia.A gradientofan

inhibitory substance,I,producedby the nucleus(which is in the rhizoidregionoppositeto the cap)has beensuggestedby certain authors (e.g.,Bonotto,Lurquin, Baugnet-Mahieu,et al.,1972).Thissubstanceblocksthe action ofan

agent unmasking the ribonucleoproteins,RNP. This agent, Do, can be aproteolytic enzyme(Brachet,1975).Moreover, a gradient of RNP extendingfrom the apical to the basalregionhasbeenfound (Brachet,1964;Wall, 1973),which codesfor the synthesisof enzymesinitiating cap formation.

Taking only thosegradients into account a minimal modelhas beendevel-developed (Rommelaereand Hiernaux,1975),which can besummarizedasfollows:

(mRNA)M + vDfl

r] + mRNA

E k6

> mRNA 4

\342\226\272 (mRNA)'

\342\226\272 E*

\342\226\240M +

+ sE

vD,

+ P

A6.15)


D,denotesthe inactive form of the unmasking substanceDo,which is takento be a cooperative enzyme as in the modelsanalyzed in Chapter 15.Theactive form Do unmasksthe messengerRNA, which in turn mediatesthe

synthesis of the enzyme E relating to the cap formation. Both I and E aredegradedto yield inactive forms I* and E*.

The rate equationsdescribingthis model have beensolved numericallyunder the following conditions:

(i) A linear gradient of (mRNA)M is supposedto exist at the outset,

(ii) The inhibitor I diffusesalong the systemand obeysmixedconditionsatthe boundaries;it is held constant in the region of the nucleus from which it

emanates and cannot flow outward in the apical region.

The numerical simulations yield a critical length, beyond which the syn-synthesis of enzyme E becomesappreciable.One is tempted to concludefromthis that the cap formation cannot take placebefore a critical elongation isattained. This is in agreement with the experimentalobservations.

Throughout this and the precedingsectionsof the presentchapter wehaveargued in terms of nonlocalizedreactions,typically with reactionsoccurringwithin the cytoplasm.It is plausiblethat membrane-bound substancesplayan important role in morphogenesis.A striking illustration is provided by thediscoveryofelectricalsignalsaccompanyingthe regenerationofAcetabularia(Novak and Bentrup, 1972;Novak, 1975),which are related to the existenceof a transmembrane potential. A more completemodel of morphogenesisshould, therefore, contain both cytoplasmic and membrane elements.Nevertheless,the general ideasoutlined in this chapter would still apply, asthey rest only on the qualitative propertiesof the reaction sequencessuch asstability and symmetry breaking rather than on the quantitative details.

Part V

Evolutionand PopulationDynamics

Chapter17

Thermodynamicsof Evolution

17.1THE NOTIONOF COMPETITION

We haveseenthat the formation and maintenanceof self-organizingsystemsare the results of nonequilibrium constraints and appropriatenonlinearcouplingscharacterizingregulatory processesat the molecular level.In this

final part ofthe monographwewant to analyzethe influenceofa third elementnot explicitlypresentin our precedingconsiderations,namely, the competitionbetween the entities constituting the system.

Competition is a very general phenomenon in nature. It occurswheneverthe resourcesor raw materials necessaryfor synthesis,growth or survival,are limited or becomescarce.The interactions between the elementspresenttake then the form of a struggle,whoseoutcome may betwofold.Either someof the entities are eliminated from the system,ora sort of \"dynamical equi-equilibrium\" is realized,enabling the coexistenceof widely different entities. Inthis chapter we discussprimarily the influenceof the first type of process,which is usedas the basisof a theory of prebioticevolution of matter. Theproblemof coexistenceof competingpopulationsis discussedin Chapter18.In both caseswe insist on thosegeneral aspectsof the phenomena that arelargely independent of the mechanisticdetailsof the processesat the mole-molecular level.

17.2.PREBIOTICEVOLUTION:GENERAL PRESENTATION

Present-daytheoryofevolution is a subtlecombinationof the resultsaccumu-accumulated in molecular biology since 1952and of Darwin'soriginal ideas.Evolution is consideredto be the result of random mutations arising from

\"errors\" in the replication of the geneticmaterial. The errorsare inevitable,in view of the complexity of the various biochemical processesand of the

diversity of the external perturbations.Obviously, in the absenceof further

constraints there is no competition, and the errors propagateindefinitely.

Darwin'sfundamental discoverywas to realize that natural selectiondirects

429

the mechanism by which an organism can survive and increasein com-complexity. This mechanismoperatesassoonasthe environment cannot supporta population exceedinga criticalsize.It selectsthosespeciesorgenotypesthat

produce the highest number of descendantsunder the existing externalconditions.In this way, natural selectionprevents the accumulation ofmistakesand at the same time allows improvements(through mutations) totake place.

This picture of selectionthrough \"survival of the fittest,\" which is stronglyreminiscent of the gambler'sruin problemor the related phenomenon ofextinction (seeChapter9), already implies the existenceof self-maintainingand self-reproducingsystems.Strictly speaking,therefore, it is not a theoryof prebioticevolution. This latter problemmay be formulated in successivestepsas follows.

(i) Under primitive earth conditions(probablyprevailing three or fourbillionyears ago),smallorganiccompoundssuchasacids,bases,and sugars,could be synthesized at an appreciablerate. This view is widely acceptedsince Miller'shistorical experiment (Miller, 1953;seeFoxand Harada,1971for a recent survey).(ii) Next, thesemoleculesmust join into polymers having a new type ofactivity. The concentration of thesemoleculeswould have been very lowunder prebioticconditions;therefore,somemechanismmust have operatedin orderto concentrate this dilute mixture in preferential places.Thesepoints are discussedin Section17.3.(iii) The following stepis especiallycrucial,namely, it is possibleto conceiveof a type of selectionpressure,compatiblewith the interactions betweentheseactive polymers, that would direct the systemto increasingcomplexityandorganization?At a certainstagealong this road,the systemmust havebecomecapableof accumulating information from pastexperience,in sometype ofprimitive geneticcode.What are the rules that haveprevailedin the formationof this code,whoseexistenceis now a well-establishedfact in biology?Wediscussthis problemin somelength in Section17.4and 17.5starting froman account of Eigen'simportant contribution.

17.3.PREBIOTICPOLYMERFORMATION

The most common mechanismof polymerformation is linearchain propaga-propagation. However, the free energy of binding of the monomers that constitutethe actual biopolymers\342\200\224nucleic acidsand proteins\342\200\224is

such that the yield ofthis processwould beextremelysmall.Hence,we assumethat a biopolymerhaving a direct evolutionary significancemust be a part of an autocatalyticcycle that enhancesthe rate of synthesisof the polymers involved.

In this sectionwe illustrate the roleof autocatalysis on simplemodels

17.3.Prebiotk Polymer Formation 431

describing the synthesis of low-molecular-weight homopolymer units,such as poly-U(polyuridylic acid)orpoly-A (polyadenylicacid).Our mainmotivation for introducing this simplification is to allow for an explicitkinetic and thermodynamicanalysisof the rate equations.The pricewehaveto pay is that the characteristicpropertiesof certain polymerchains to act astemplatesfor their own synthesisor for the synthesisof other polymersis notbuilt in the models.This difficulty is resolvedby introducing reaction mech-mechanisms involving suitable autocatalytic stepsgiving riseto the sametype ofkineticequationsasthe templateprocesses.A much moresophisticatedmodeltaking into account known data on polynucleotide interactions has beenanalyzed by BabloyantzA972).

The general picture is as follows.A precursorI of an active monomer X

enters into the system. Onceformed, X gives rise to a dimeric moleculeX2 in either of the following three ways (Goldbeterand Nicolis,1972):

SimpleCondensationof TwoMonomersX (ModelC)

The overall reaction is:

X2 Isa \"Template\"for Its OwnFormation (ModelT)

A7.1a)

A7.1b)

X

\342\226\240 condensation

2X,

template

A7.2a)

432 Thermodynamics ofEvolution

The overall reaction is now

41 2F A7.2b)

X2 IsSynthesizedon a Template,with an Additional Autocatalytic Effect(ModelTC)

condensation

template + autocatalysis

X

The overall reaction remainsas in Eq.A7.2b):41 . 2F

A7.3a)

A7.3b)

Assuming a spatially homogeneousmixture, one can derive straight-straightforwardly the rate equations for the three models.At the steady state, onecan seethat theseequations reduceto algebraicequations for X2 that are,respectively,of the fourth degree(model TC),third degree(modelT), andseconddegree(model C).Theseequations have been solved numericallyand the results are as follows(Goldbeterand Nicolis,1972).

Firstly, when the overall affinity vanishes (system maintained at equi-equilibrium), the three modelsyield exactly the sameresult:

A7.4)

This correspondsto a low-yield polymer synthesis,wherein the auto-autocatalytic or template stepsare completelyswitchedoff.

Let us now cometo the nonequilibrium behavior. It turns out that thelatter dependscritically on the magnitude of the parameter a2, which

determinesthe relative importance of direct polymer synthesis and auto-autocatalytic processes.Fora2 large, schemesT and TC operateon the same

17.3.Prebiotic Polymer Formation 433

branch asC.Thisbranch correspondsto a maximum utilization ofmonomersby C and is referred to hereafteras the \"optimal\" branch.

Fora2 relatively small the competition between the two types of mechan-mechanism becomesapparent.The synthesison the \"template\" becomespredomi-predominant,

and the mechanismsT and TCgive riseto far-from-equilibriumabrupttransitions of the sigmoidaltype (modelT)or to transitions betweenmultiplesteady states(model TC).The situation for this latter caseis illustrated in

Fig.17.1,where / is usedas parameter measuring the deviation from equi-equilibrium. The resemblancewith the simplemodel discussedin Section8.4is

striking. Thesystemevolvesfrom a state of low polymerconcentration,which

correspondsto direct synthesis(C)at low valuesof a2 and may bereferredto

3

2

i

o

-1-2-3-4

-5-6-

-7

-8I I I I I

-10-9 -8 -7 -6 -5 -4 -3 -2 -1log/

Figure 17.1.Multiple steady states as function of / for model TC.Solid lines refer to steadystates of system TCfor a2 = l(T4,a4= a5 = 104,a,= 102,a3= 10~2,a6= 2,F = 5 x 10\"9,Kt = 102, K2 = 2 x 106, Ky = 1,Kt = K5 = 103, Kb = 2, K being equilibrium constants ofreactions. (#,+) and (O, x) refer to minimal and optimal branches, respectively; latter

corresponds to steady-state operation of model Cfor a2 = 10~4and for a2 = 106,respectively.


asminimal branch, to an optimal branch characterizedby a polymerconcentra-concentration that may behigher by severalordersof magnitude.At the sametime the(polymer/monomer) ratio in the system is enhanced.Both low and upperbranchesare shown to be stable,whereas the states on the intermediatebranch are unstable.

ModelsT and TCprovidealsoa very instructive illustration of the thermo-dynamic arguments advanced in Section8.11.The numerical evaluation ofthe total entropy production,P as well as of the molar entropy production,P/n, showsthat the transition to the optimal branchesin both systemscorre-corresponds to a jump in thesequantities.A closeanalysisof the rates and affinities

of the various reactionsshowsthat this jump resultsin from a manifoldincreaseof the rate of the template and/or catalytic reactions,whereas therate ofdirectcondensationdiminishesstill further. In other words,the systemswitchesto a fast reaction pathway and at the same time it enhancesits rateof dissipation.This is in completeagreement with the considerationsde-developed in the third part of Section8.11.

Finally, while Pcontinues to increasewith / beyond the transition region,the molar entropy production P/n beginsto decrease.This occurssimul-simultaneously with a decreaseof rates and affinities for the template and catalyticprocesses,owing to the fact that the inversestepsbegin to build up along thereaction sequence.

This behaviorof the dissipationfunction perunit massis strikingly similarto the existing thermal data on specificoxygenconsumption during embry-embryonic development(Zotin and Zotina, 1967).It appearsthat during a fractionof early embryonic life this quantity (which is expectedto be an increasingfunction of entropy production per unit mass)is sharply increasing,whereassubsequentlyit steadily decreases.This latter tendency continues duringadult life, with a few notable exceptionsassociatedwith such phenomena astissueregeneration or malignant growth.

One is tempted to argue that only after synthesis of the key substancesnecessaryfor its survival (which implies an increasein dissipation)doesan

organism tend to adjust its entropy productionto a low value compatiblewith the externalconstraints.At this point,contact can probablybemadewith

Darwin'sideaof the \" survival of the fittest,\" becausea low rate ofdissipationis likely to give to an organism a selectiveadvantage.

17.4.BIOPOLYMERCOMPETITIONAND HYPERCYCLES

We are now concernedwith the next stageofevolution,namely, the behaviorof interacting populationsof biopolymers(Eigen,1971).The problemisformulated by first assuming that by somemechanism(e.g.,the oneoutlined

17.4.Biopolymer Competition and Hypercycles 435

in Section17.3)it has been possibleto produceappreciableamounts ofpolymers having the following properties:

(i) Under the maintenance of a finite energy and matter flow (e.g.,relatedto the entry of monomers in the system as in the previous section)thesesubstancescanmetabolizein the sensethat they may undergotransformationsfrom energy-richto energy-deficientcompounds.(ii) The substanceshave certain autocatalytic properties arising from

template action. In particular, the overall rate of productionof constituenti is a least proportionalto its concentration. This is to be comparedwith

Eq.(8.2)for the Lotka-Volterramodel,where autocatalysis gives riseto asimilar property.(iii) The replication of i is subjectto errors.This implies that there is apossibilityfor producingfrom i a set of other substances;A/7= 1,...,n).

A general form of rate equationssatisfying theseconditionsis (assumingagain a spatiallyhomogeneousmixture):

-^= (AtQ, - Dt)Xt + I <t>uXj-

</>OiX, i = 1,...,n A7.5)at j=!U*>)

At and D, represent,respectively,the rate of formation of substancei asdirectedby the template i and the rate of decompositionof i. Qt measuresthe \"quality\" of the templateaction, that is,its ability to reproducei faithfully.

Clearly, 0 < Qx < 1.<pOi representsthe rate of dilution of i and reflects thefact that as the systemgrows,the relativepolymercontent within the reactionvolume is constantlymodified.Finally, <\302\243,7 representthe rates ofspontaneousproductionof i arising from errors in the replication of j species.A givenreplication processat template k produceseither substancek at a rateAkQkXk>or substances)/ k at a total rate Ak( 1 \342\200\224 Qk)Xk. We concludethat:

We must now specifythe constraints acting on the system.In addition tothe nonequilibrium conditionsrelated to the monomer (or the energy)flow into the system,onehas to imposea constraint implying somesort ofselection.One of the conditionsproposedby Eigen for this purposeis theconservation relation:

\302\243X,-

= const A7.7)


That this condition may lead to selectionis obvious from the fact that anyincreasein the concentration ofoneof the i constituents necessarilyimpliesadecreasein the concentration of the other substances.

Summing over i in Eq. A7.5)and taking relations A7.6)and A7.7) intoaccount we find that the Ah D,,and (poi must satisfy the following condition:

Xi A7.8)

Thiscondition implieseithera recyclingof the wasteproductsinto the system,or a suitableadjustment of the dilution rate. Forinstance,taking all (f)Oi to beidentical we find:

*.-*.-SMJ.OixDefiningexcessproductivity as

\302\243,

= Ai -D, A7.10a)and mean productivity as

E =^^=<t>0 A7.10b)

we may now transform Eq.A7.5)into a form with conditionsA7.6)to A7.9)built in. Substituting 4>Oi from Eq.A7.9)we find

dX \342\200\224

\"

\342\200\224i = (Wf - E)X, +\302\243 frjXj i=l,...,\302\253 A7.11)at j-!

where we dennedthe selectivevalue as

Wf = A,Qt -D, A7.12)The nature of the competitiondescribedby Eq.A7.11)is bestillustratedon

the exampleof \"self-reproductivecatalytic hypercycles\"(Eigen, 1971),depictedin Fig.17.2.Hypercyclesreflectthe fact that nucleicacids(denotedby I,- in Fig.17.2)are a necessaryprerequisitefor self-organizationbecausethey possessthe ability to act as templates.However,they require a catalyticfactor that couplesthe different template mechanisms.This fact is providedby the presenceof protein chains, denotedby E, in Fig.17.2.The wholehypercycleis assumedclosed,that is,the final protein En feedsbackon I,.

Supposenow that, as a result of errors,the hypercyclegives rise to sidebranches.As a consequenceof the conservation condition A7.7) thesenew

hypercyclescompetewith the original one.Under certain conditionsthis

competition may lead to selectionand, becauseof the nonlinearities,selection

17.4.Biopolymer Competition and Hypercycles 437

Figure 17.2. An n-membered self-reproductive hypercycle of the type developed by Eigen.Polynucleotides I, are information carriers capable of self-reproduction. In addition, poly-eptides E, serve as catalysts for the production of polynucleotides (Ef_, catalyzes productionof I,) and, moreover, I, provide information for synthesis of E,. Hypercycle is closed,that is,En catalyzes the production of I,.Such autocatalytic systems may be advantageous duringevolution.

is very sharp.Thus, among many competingsystemsonly onecan survive in

an appreciablequantity. In the examplestreated in Eigen'spaper (Eigen,1971;Jones,Enns,and Rangnekar, 1976),the survivor is characterizedby the

highest value function [seedefinition A7.12)].Now, a system respondsto an errorand evolves further only if it is not

sufficiently stableinitially. Hence,oneis tempted to think that the ultimate

goal ofprebioticevolutionwould beto direct the systemto a state ofoptimalstability toward its own errors(or fluctuations,as errorsmay be thought ofas a new kind of fluctuation). This searchfor optimal stability shouldthengive a more precisemeaning to the Darwinian \"survival of the fittest\"

principle.The final state will bea systempossessinga means for minimizingerrors.This devicecouldbe thought of as the precursorof the geneticcode.


In the considerationsdevelopedin this sectionand in Section17.3it isassumedthat the polymerscouldbepresentin macroscopicallyappreciablequantities,otherwisethe variousinteractionswould not beeffective.Mechan-Mechanisms able to localize thesereactionswithin specificcompartments should,therefore,haveplayedan important rolein prebioticevolution.In this respectwe may point out that the possibilityof creating, in a limited region of space,concentrations that may be much higher than in the homogeneousmixtureis built into the very conceptof spatial dissipative structure. In particular,the examplestreated in Chapters7 and 8illustratehow a systemcan reachandmaintain new configurations where the probability for certain reactionsisenhancedand thereby triggers further evolution. The problemof compart-compartment formation in the context of prebioticevolution has beenconsideredalong theselines by Tyson A975a)and Hiernaux and BabloyantzA976).

17.5.EVOLUTIONVIEWED AS A PROBLEMOF STABILITY

We want now to derivesomegeneral thermodynamicand kineticcriteria forstability and evolution. Supposethat we have initially a set of interactingpolymer speciesX, (i = 1,...,n) that all exist in relatively abundant quan-quantities. The time evolution of the set {X{}is describedby the following rateequations:

\342\200\224^

= F? + F^Xj})= O.-dXj}) i=l n A7.13)

HereFet is a supply term related to the exchanges(e.g.,of monomers)with theexternal environment, and F, describesthe chemical processesinsidethesystem.It is assumedthat the system is maintained uniform and that thereexists at least one asymptotically stable steady-state (or time-periodic)solution of Eq. A7.13).This means that all compositionfluctuations dXj

regressin time, that is, that all n roots w{ of the characteristic equation (seeSection6.5)have negativereal parts.

In addition to compositionfluctuations the system may be subjecttofluctuations tending to alter the average (deterministic)laws of evolution.For instance, a hitherto absent \"mutant\" or \"error copy\" of a polymermay suddenly appear in the system. Subsequently,new reaction pathwaysmay be openedeven if the new copy is present in very small quantities;small parameters that would normally play no rolein the macroscopicevolution become,therefore,essentialas a result of these fluctuations.

How are we to introduce the influenceof such structural fluctuations intothe description?Naturally, a spontaneouserroror mutation cannot initially

17.5.Evolution Viewed asa Problem ofStability 439

figure in our deterministic Eq.A7.13),which is a statistical average over alarger number of elements.Rather, it constitutes a stochasticprocessthat

couldbe studiedindependently of Eq. A7.13)by the methods outlined in

Chapters10-12of this monograph (seealsoAllen, 1975;Eigenand Winkler,1975).Supposenow that the conditionsfor spreadingof the errorover asufficient volume or surfacearea are fulfilled. Then, the new copyconstitutesa macroscopicfluctuation of the previously prevailing equations.CallingYj (j = 1,...,m) the new copies,we obtain a perturbedset of differentialequationswhoseorderhas increasedby m comparedto the orderof A7.13).Theseequationsread:

dY,

dt

A7.14)= G,({Xj},{Yk},c)i = 1 n /=l,...,w

Herec is a parameter characteristicof the propertiesof the enlarged systemand such that in the limit c -\302\273 0 the original equationsare recovered.*Thismeans that there existsa value Yk({Xj}) given byj0)= 0 A7.15a)

such that

)},0)= Fi A7.15b)

Naturally, we expectYk given by Eq.A7.15a)to be practically zero.Forcdifferent from zero but small, the stability of Eq.A7.14)will be determinedby a characteristicequation of degreen + m. This equation will have n + mrootsand n of thesemust havevaluescloseto thoseof the initial characteristicequation and, in particular, have negative real parts. The stability of thereferencestate of the initial systemor,alternatively, the stability of the state{Xf},{Yk} of the enlarged systemA7.14),can only be threatened by the newroots a>n+k @ < k <m). To illustrate this, take m = 1.The only way torecover the characteristic equation of the initial system in the limit e -\302\273 0 isthen to have a correctionterm of order \302\243 multiplying either the (n + l)stpowerof the characteristicroot, or the term of the characteristicequation that

is independent of the characteristicroot:cco\"+'+ an(c)co\" + \342\226\240\342\226\240\342\226\240+ a,(fi)co+ ao(c)= 0 A7.16a)

* Note that in the equations for Y, there is no flow ofmass term included. This reflects the fact

that {Y} is produced systematically solely via autocatalysis and is, therefore, essentially a closedsystem.


or

co\"+1 + an{r)co\" + \342\226\240\342\226\240\342\226\240+ a{(E)co+ mo(c)= 0 A7.16b)

Forc small the new root con+ , dependson c in oneof two ways:

\342\226\240l--~ A7-17a)

if we are in caseA7.16a),or

A7.17b)

if we are in caseA7.16b).Dependingon the valuesof the parametersand onthe kinetics,the coefficientin the right-hand sideof either Eq.A7.17)may bepositive.Thus wereach the important conclusionthat by taking into accounta small and, at first sight, \"spurious\" parameter, the stability propertiesmaychange.In other words,onedeals here with a problemof the structuralstability (seeSection5.4)of the reducedsystemE.13)with respectto perturba-perturbations that introduce sidereactionsincreasing the order of the differentialsystem.Note that in caseA7.17a)the departurefrom the unstable regime isvery fast for e -\302\273 0, whereas for caseA7.17b)the unstable modeevolvesveryslowly.This evolution may lead,for example,to a new state of high Y con-concentration, which can evenbedominated by the new substances.

One can derive a somewhat explicit kinetic criterion for evolution bylinearizing the secondequation A7.14)around the referencestate {Xj},{%}correspondingto c = 0.We obtain, setting Y,

= Y, + SY,:

'=\342\226\240 \302\273

The condition for evolving away from the values % of Y is, then, that thecharacteristicdeterminant of this m x m system:

det = 0 A7.19)

shouldlead to at least oneroot having a positive real part.The remarkablefeature of this criterion is that it doesnot dependexplicitlyon the propertiesof the original system.Fora single additional copy (m = 1)the criterion

17.6.Evolutionary Feedback 44j

yieldssimply

>0 A7.20)

in other words,that the presenceof Y should enhance the rate of its synthesis.This statement is equivalent to that of an autocatalytic action of Y.

Theseconsiderationshavebeenillustratedon evolutionmodelsfor systemsin which the new substancesintroduce a drasticchange in the functionalproperties(Prigogine,Nicolis,and Babloyantz,1972).Typically,c.is then theinverseof somerate constant k characterizing the new function and the newtime scaleimposedon the system. The essentialproperty that instabilityoccursfor e -\302\273 0, that is, for k j> 1,means that the new copieslead to anincreasein the interactions within the system and/or between the systemand environment.

17.6.EVOLUTIONARY FEEDBACK

When inequality A7.20)or its generalization to several new substancesissatisfied,then the systemof Y valuesswitchesfrom the referencestate {Yk] to aregimecharacterizedby new reactionshaving a positivevelocity[seeSection8.11,especiallyEq.(8.89b)].Sincethesereactionswereabsentinitially, theiraffinity is boundto bealsopositive.Thus, during the first stagesof evolutionaway from the referencestate the system increasesits rate of dissipationasmeasuredby the entropy production.If the instability correspondsto Eq.A7.17a)this increaseis very sharp,as discussedin Section8.11(third sub-subsection). This result has beenillustrated on models(Prigogine,Nicolis,andBabloyantz, 1972;Goldbeterand Nicolis,1972;Hiernaux and Babloyantz,1976)which alsoindicatea similar behaviorfor the specificrate ofdissipation,that is,for the entropy production per unit mass(or permole).

The point is that the instability, which is triggered by nonequilibriumenvironmentalconditions[i.e.,by Fc- / 0 and large in Eq.A7.13)]maintain-maintaining

acontinuous energydissipation,further increasesthe levelof dissipationand therebycreatesconditionsfavorableto the appearanceofnew instabilities.In other words,someirreversibleprocessestaking placeinsidethe systemarefunctioning more intenselyand have, accordingly,increasedtheir departurefrom the equilibrium state.Hence,the probability that there existsa classoffluctuations,with respectto which theseprocessesare in turn unstable, hasbecomehigher.On the contrary, if the resultof the instability wereto decreasethe levelof dissipation,one would get closerto the propertiesof an isolatedsystemat equilibrium, that is, closerto a regime in which all fluctuations are


damped.Thismechanismofevolution through a successionof transitions hasbeencalledthe evolutionary feedbackand can be visualizedas follows:

Nonequilibrium Threshold(Eq. 17.20)

Instability through structuralfluctuation (Eq. 17.17)

Increaseddissipation <

(Eq. 8.89)

We are led,therefore, to regard energy dissipationas the driving force ofevolution. In this respectit is pertinent to point out (de Duve, 1974)that,

despitethe increasein organization and complexityof living systems,therehas been an accelerationof biologicalevolution in the courseof time. Itseemsas though each new step increasing functional organization has in

itselfthe germs for further evolution.The purposeof Section17.7is to relate the level of dissipationwith the

functional organization of simplereaction networks.

17.7.ENERGY DISSIPATIONIN SIMPLEREACTION NETWORKS

SimpleLinear Networks

Let us first considerthe transformation of someinitial substrateA to a final

productE through JV intermediate productsX, (i = 1,...,JV). We want tostudy the behavior of the steady-stateentropy productionper unit mass(orpermole)<rm, asa function of the topologyof the reaction scheme,that is,asafunction of the reaction \"connections\"between the chemical species.Weassumethe reactionsfollow linearkineticsand that all forward and backwardrate constants are equal to k. We have:

z\\2 ' * ' ^ \"^N ^

A7.21)

17.7.Energy Dissipation in Simple Reaction Networks 44s

The steady-statesolution of the rate equationsis easily found to be:(N+1-QA+ iE

X N+\\ A7-22)

and the total entropy productionis,assuming a uniform systemand takingdefinitionsC.14)and C.9)into account:

P = k(A- X,)ln~ + k{Xx-

X2)\\n ^ + \342\200\242\342\200\242\342\200\242+ k(XN- E)ln ^

A7.23)where A, E, and Xt denotethe total mass(or number of moles)of the cor-corresponding constituents within the reaction volume.

Taking Eq.A7.22)into account we find that, at the steady state

A -A\"?

=X\302\260i

-X\302\2602

= \342\200\242\342\200\242\342\200\242= X%- E A7.24)

Thus,

Po = M - Xi)ln4

-\"TiTT\"! AZ25)

Dividing through by the total mass(or number of moles)of the system:N

m = A + E + Y^Xi1=1

and taking again Eq.A7.22)into account we arrive at an expressionfor <r\302\260

solely in terms of the ratio A/E-that is, in terms of the overall affinity of thereaction- and of the number of intermediates:

k (N + \\)(N + 2)[_(A/E) + 1] E

Thus, for a given overallaffinity, the reaction chain seesits specificentropyproduction decreasewhen its length increases,that is, when more inter-intermediate stepsintervenebeforeE is produced.

Let us now regard Eq.A7.21)as a polymerization reaction. Each stepi ofthe chain describesthe addition of a monomer A, to the intermediateX,-_ , in

such a way that k = ltAj = const,where /, is the rate constant for the ith

addition.The final polymerE is producedat a rate proportionalto the steady-state value of XN, that is, from Eq.A7.22):

fc^o kA+ NE

A7 27)


This is an increasingfunction of N as long as E/A > 1,otherwise the rate ofsynthesisof E decreaseswith the length. In particular, this is the casewhenthe overall affinity is triggeredto infinity, that is,if A/E -* oo at constant A. At

the sametime, the rate of depletionof monomers A becomes:

dt N A7.28)

that is,it diminisheswith the length of the chain.In conclusion,we find that for a large and positive overall affinity the

decreaseofspecificrate ofdissipationwith the length of the chain is associatedwith a poorfunctional performance as measuredby the yield in polymers.This result (Prigogine and Lefever,1975)can alsobe expressedas follows.From Eq.A7.22)onecan seethat for

A/E-* co, A> X{>X2>\342\226\240\342\226\240\342\226\240> XN> E.

Alternatively, there is a continuous degradation of the chemicalpotential:

Ha > J\"x, > \342\200\242\342\200\242\342\200\242> HxN >He A7.29)

ForN -* oo the individual differencesfix. \342\200\224

jUXj+1 tend to zero, and as aresult the overall rate of transformation of A into E is very slow.

The situation is different when the initial and final productsA and E canbe connectedby more than onepathway. Forinstance, considera seriesof\"parallel\" connectionsinvolving each a singleintermediate product:

A7.30)

Following the sameprocedureas before onefinds that when /c, = k~--\342\226\240\342\226\240\342\226\240 = k, <r\302\260 is given by (Prigogineand Lefever,1975):

N[tjA/E)-l-\\ A

(N + 2MA/E)+1] EA7.31)

17.7.Energy Dissipation in Simple Reaction Networks 445

In contrast to Eq.A7.26),this value increaseswith the number of loopsin thenetwork and tendstoward a finite constant value for N \342\200\224> oo.At the sametime, the functional \"efficiency\" of the network is enhanced as E is producedat a higher rate.

An alternative interpretation of this result is as follows.Starting with alinear chain such as Eq. A7.21),an evolutionary processtransforming it

progressivelyinto the form A7.30) is associatedwith increasein both thespecificrate of dissipationand the functional efficiency.Fora given value ofthe constraint A/E onemay say that the systemhas managedto interact morestrongly with the externalworld in such a way that it is traversed by a higherflux of matter.

SimpleCatalytic Networks

The next step is to considernonlinear reactions involving catalytic steps,which are more realisticfrom the biochemicalpoint ofview. A typicalschemeis the following:

A7.32)

where the catalyst M is formeddueto the action of the end constituent of thechain and, subsequently,catalyzes some(or all) of the reactions.The com-computations show (Prigogine,1965)that if A/E -> oo and the strength of the

catalyticeffectis sufficiently large,the rate ofentropy productionis enhancedenormously.Significantly, onefinds that at the sametime

but

Pe1 A7.33)

In other words,the effectof catalysis is to propagatethe chemicalpotentialwithout degradation along the chain and to sustain betweenXN and the final

productE a very large affinity of reaction. We find here oneof the key ideasbehind the notion of evolutionary feedback;it is by keepingsomeof thereactionsvery far from equilibrium that a systembecomesmore efficient.

446 Thermodynamics oj Evolution

17.8.A BIOCHEMICALILLUSTRATION

When, by addition of a suitable substrate,a chemicalpotential is appliedtoan enzymaticprocesssuch as the glycolyticpathway (seeChapter14),fluxesaregeneratedand the variouscomponentprocessespassinto anonequilibriumstate.Consider,for instance, the caseof a steady-state functioning. Onecancompute the minimum energy requiredto maintain a steady-state flux for areaction within a nonequilibrium chain (Hess,1963,1975)asfollows.Considerthe following reaction:

A ,'

> E A7.34)k2

If 01is the affinity of this reaction at the steady state:

a=kBT\\nk^then the amount ofenergyperunit time requiredto maintain the steadystateis:

d-^-= (kiA0 ~k2E0)kBT\\nk^ A7.35)dt k2 Eo

= entropy-production rate

Experimental values for the ratio (Ao/Eo) have been reportedby HessA963) in the caseof the glycolyticreactions in tumor cells.On the basisofthesevalues,glycolyticreactions can be classifiedinto two groupsshowingdifferent steady-statebehavior.

(i) Reactionswith near-equilibrium enzymes.In these reactions,for instance,the glucosephosphateisomerasecatalyzing the reversible isomerization ofglucose6-phosphateinto fructose6-phosphate,the net flux is low comparedto the maximal activity of the enzyme.(ii) Reactions with enzymes operating quasi-irreversibly,with a reversefluxnegligibly small comparedto the net flux, and having regulatory functions. Inthis case,the freeenergyof the reaction is mostly usedfor control.Examplesof this secondclassinclude pyruvate kinaseand phosphofructokinase.Forinstance, in pyruvate kinaseonefinds that the deviation from equilibrium in

the glycolyticsystemas indicated by Ct is about 5 kcalmole\"'.

Usingthesedata one can concludethat along the glycolyticpathway thechemicalpotential differenceis practicallyconcentrated in the vicinity of thetwo principal regulatoryenzymes,phosphofructokinaseand pyruvate kinase,

17.8.A Biochemical Illustration 447

whereasin the part ofthe chainbetweenthesetwo enzymesthere is a \"plateau \"

of chemicalpotential. The situation is strikingly similar to that encounteredin the model of the secondsubsectionin Section17.7.

It is tempting to point out the analogy betweenthe evolutionaryfeedbackand the conclusionthat the energy cost for control is associatedwith the

complexityof the metabolicpathway to becontrolled.Needlessto say,wearestill at the very start in the understanding of the origin of bioenergeticpathways!

Chapter18

Ecosystems

18.1.INTRODUCTION

In Chapter 17 we have seenthat, starting from an initial chaotic situation,the competition between units endowedwith autocatalytic propertiescangive rise to a functional organization via a successionof nonequilibriuminstabilities.Now, the lawsdescribingthe growth, decay,and interaction ofbiologicalpopulationsand socialsystemsare very closelyanalogousto thoseof chemical kineticsin general and to thoseof competing biopolymersinparticular.

The purposeof this chapter is to exploretheseanalogiesand undertakean analysis of interacting populationsin an ecosystemusing the methodsand ideasunderlying the theory of dissipativestructures.As in Chapter17,we try here to sort out somegeneral features and trends that are largelyindependent of the details of the phenomena at the level of the singleindividual.

We first review,in Sections18.2and 18.3,the basicequationsofpopulationdynamics and illustrate the type of orderedbehavior that can be predictedfrom these equations.Sections18.4and 18.5are devoted to ecologicalevolution. We first outline a general formulation of this problemand then

apply it on examplesinvolving coloniesof socialinsects.Section18.6isdevoted to the questionof stability of complexsystemsand to the time-honoredproblemof \"the limits of complexity.\"

18.2.BASIC EQUATIONS

As in the previouschapters,in orderto set up the appropriaterate equationsdescribingthe processwe must specifythe interactions betweenthe:(a)com-components of the system and (b) system and the externalworld.Mathematically,the latter influences the evolution equations through either boundaryconditionsor systematicconstraints intervening explicitlywithin the equa-equations.

448

18.2.Bask Equations 449

Let us first discussthe dynamical processesinside the system. As a rule,an assemblyof populationsis subjectto the following kinds of processes.

(i) Processesof genetic origin. As the entities constituting the populationsare living beings,they reproducethemselvesat a frequency /c, and die at afrequencydt. Moreover, they are subjectto \"mutations\" that from time totime may change the nature of the system in an unpredictableway, as in thecaseof prebioticevolution examined in Section17.5.(ii) Processesinvolving competition, intra- or inter-specific.Quite frequently,theseprocessesarisefrom the fact that in a medium where the amount ofresourcesis limited, the growth of an organism eventually takes placeatthe expenseof the others.As it has long beenknown from early studiesonhuman or animal populations,this resultsin a saturation in the growth. Inother cases,competitionmay involve direct interactionsbetweenindividuals,castes,orspecies,suchaspredation and aggression.By definition, all kinds ofcompetition give riseto nonlinear contributions in the rate equations,(iii) Regulatoryprocesses.Theseprocessesensurethe coordination of theactivities of the populationsin spaceand time. They give rise to feedbacksthat is, to nonlinear interactions, in the sensethat they favor\342\200\224directly orindirectly\342\200\224the growth of a certain part of the population which is necessaryfor the survival of the entire population.The formation of soldiersin socialinsectsis a characteristicexample.(iv) Communication processes.The above three processesare local in thesensethat they take placein any smallarea or volumeelement in the system.In addition to theseprocessespopulationscommunicate betweenneighboringor even distant areas.Spatial dispersion (Pielou,1969)or migration(Maynard-Smith, 1974)are examplesof means for communication. Sensingdevices and/or chemical agents such as pheromonesin socialinsectsmayalsobe involved.

Coming now to the constraints acting on the system we remark that an

ecosystemcommunicates with the external world through a separatingsurface, Z. In general, the conditionsprevailing in the external world arenot identicalwith thoseinsideE.In particular, the numbers of individuals ofspeciesi insideand outsideare different: Xt # X]. Similarly for the energyper unit volume or per unit area, ev # e\\. Thesedifferencesare felt by the

systemas constraints inducing a flow of matter and energy within L, in thesameway as in the caseof physicochemicalsystemsdescribedin Chapter5.Thus,an ecosystemis generallyan opensystem.It is easyto think ofexamples,such as the biosphereas a whole being subjectto a flux of solarenergyor arelatively advanced societybeing sustainedthanks to an exchangeof energyand information with the surroundings.

4S0 Ecosystems

We want now to write the explicit form of the rate equationsdescribingtheseprocesses.Let us first illustrate the procedurein the simplecaseof asinglespecies.In the presenceof food, A, the individuals, X, multiply at arate that to first approximation can be set equal to*:

(d\302\261\\= kAX A8.1a)

\\ at /birth

Similarly, the rate of mortality is:\342\200\224 = -dX A8.1b)dt /dealh

Note that Eqs.A8.1)are isomorphicto the rate ofan autocatalyticproductionof a chemicalX and to that of a dilution or an inactivation of X:

A + X \342\200\224*\342\200\224\302\273 2X

X \342\200\224d-^ A8.2)

If A were unlimited, then Eq. A8.1)would predicta population explosionthrough an exponential growth of X,f provided kA > d, and an exponentialdecayto zero provided kA <d. In fact,A is generallylimited and as a resultits rate of consumption must be taken into account explicitly.This rate isequal to

(^1) = -kAX A8.3a)\\ dt /cons

On the other hand, A is reproducedin the sameway as X or, if it is a simpleorganic matter, it arisesfrom the decay processesof other living organisms.This processis quite complexand involves the so-calleddecomposers.A

simplelimiting caseis that where A is recycledentirely into the systemthrough the death of X:

(d4) = dX A8.3b)V dt /prod

From Eqs.A8.1)and A8.3)we concludethat in this limiting casethere is aconversationcondition for the total amount oforganic matter present(closedsystem):

A + X = N = const A8.4)

* Expression A8.1a)implies the presenceof a single phenotype. For an extension to several

phenotypes, we refer to Allen A976).t This is usually referred to asMalthusian growth.

18.2.BasicEquations 451

Substituting A from this relation we obtain the following equation of evolu-evolution for X:

A V\342\200\224 = kX(N - X) - dX A8.5)dt

This is the celebratedVerhulst equation for logisticgrowth, which has beenwidely usedin the study of population trends (Goel,Maitra, and Montroll,1971).As mentioned before, the term N \342\200\224 X introducesa saturation of thepopulation to a finite steady-state level:

X0 = N-~ A8.6)k

It can be shown straightforwardly that this state is asymptotically stablewheneverit exists(i.e.,Xo >0),whereasthe trivial state Xo = 0 correspond-corresponding

to the extinctionof the population is unstable.*In the caseof several interacting populationsthe saturation term takes

the form (May, 1973)r 1

A8.7)

and the general equation of evolution taking into account processes(i) to(iv) cited above is:

+ Fc({Xj})

+ FR({Xj})+ FM({Xj},{X\302\260}). A8.8)

The nonlinear functions Fc,FR describe,respectively,the rate of competitionother than that implied by Eq.A8.7)and the rate of regulation. FM standsfor migration, movement, and so on, and dependson both internal andexternal valuesof Xt.

As a secondexampleof Eq. A8.8),considera system involving a singleprey, X and its predator,Y. We assumethat the prey has an unlimitedreservoir of food (Nprey

-> oo) and that the predator feeds entirely on the

prey.The latter diesentirely becauseof predation.Theseassumptionsimplythat Pij ->0,FR= FM = 0, and dprey

= 0 in Eq.A8.28).Moreover, we takefor Fc the simplestpossibleform describingthe frequency of binary en-encounters betweenX and Y:

\\Fc(X,Y)\\ = sXY A8.9)

* Note that the existence of two steady states is not in contradiction with the assumption ofaclosedsystem, Eq. A8.4). Indeed, we deal here with a system displaced from the regime ofdetailed balance.

452 Ecosystems

Again, it is instructive to point out that Eq.A8.9)has the sameform as therate of the autocatalytic reaction:

X + Y \342\200\224^\342\200\224\342\226\272 2Y A8.10)

Keepingtheseremarksin mind we may reduceEq.A8.8)to the followingsimpleform:

^ = kAX - sXYdt

^ sXYdY A8.11)at

Theseare identical to the equationsof the Lotka-Volterramodel analyzedin Section16.2(seealsoGoel,Maitra, and Montroll, 1971).

Sofar Eq.A8.8)have beenregardedas deterministic,In fact, theseequa-equations define a stochasticprocessin the spaceof {X{}values for the samereasonsas chemicalreactions.Thus, following the methods of Part III,onecan derive master equations to completethe macroscopicpredictions.Moreover,mutations introduce structural fluctuations that confer to thesystem an additional stochasticelement. Finally, an ecosystemis subjectto environmental fluctuations that continuously influencethe values of thecoefficients kt, Nt (etc.)in Eq. A8.8).In this chapter we are concernedprimarily with the effect of structural fluctuations. A detailed analysis ofenvironmental fluctuations can be found in May'smonograph (May, 1973).

18.3.EXAMPLE OF ORDEREDBEHAVIOR: ORGANIZATION IN

INSECTSOCIETIES

The closeresemblanceof the rate equationsto thoseof chemical systems,and the existenceof nonequilibrium constraints, promptsus to ask whether

dissipativestructures occurat this levelof descriptionof biologicalpopula-populations, causingpatterns oforganization similar to thosefound on the chemicalreactionsanalyzed throughout this monograph.

In recent years, limit-cyclesolutionsof model equationsof populationdynamics have been discovered(May, 1973).The possibilityof spatialorderinghas also been pointed out (Segeland Jackson,1972).Herewedevelop a different examplereferring to the rapidly growing field of socio-biology.Although the type of behavior predictedby this model is probablyquite general, in our formulation we refer specifically to social insects(Deneubourg,1976).

18.3.Example ofOrdered Behavior: Organization in Insect Societies 453

Among insectssocialorganization attains a maximum complexity withthe hymenopteraand termites(Wilson, 1971),and the survival of the individu-individual is practically impossibleoutsidethe group.The regulation of castes,nestconstruction, formation of paths, and the transport of material or of preyare different examplesof collectivebehavior extending beyond the scaleofthe single individual.

Considerthe problemof the early stagesof construction of a termite'snest.It seemsthat this occursin two phases(Grasse,1959).Firstly, there is anuncoordinated phase characterized by a random depositionof buildingmaterial.However,when by chanceoneof thesedepositsbecomessufficiently

large a secondphasebegins,in that the termites depositmaterial on that

aggregation preferentially.A pillar or wall growsdependingon the initial

compositionof the deposit.If theseunits are isolated,construction stops,otherwisean arch may result.

The following simplemathematical model (Deneubourg,1976)showshowsuch \"orderthrough fluctuations\" appears.Let C be the concentration ofinsectsdepositingthe material,whoseconcentration isP.The basicassump-assumption we adopt is that a chemical substancethat termites mix with thebuilding material is emitted. Let H be its concentration. The substancedif-diffuses freely in spaceand its odor attracts termites toward the regionsofhigh H, that is, toward the regionswhere there is an excessof alreadydepositedmaterial. This orientation mechanism in a gradient of odor maynot be the only onepresent,but it is the only oneretained in the model.

The main stepsin the processcan be summarizedas follows:

C '\302\273 P 2 ' H + P*

+

A8.12)where the loopdescribesthe indirect autocatalytic effectof H on P via theattraction of C \342\200\242P* representsthe \"inactive\" material, that is,the materialat a state where no odoringsubstancecan beemitted.

The rate equationsare:dP _\342\200\224 = ktC-k2PdH

]r D ]r* is.2* <*4= k2Pk4H+ DHV2Hct

~ = F* -ktC+ DV2C +y\\-(C\\H) 08-13)

In the last equation Fe representsthe flux of insectsbringing material intothe spacewherethe work is performed,D the coefficientofrandom dispersion

4S4 Ecosystems

Figure T8.T. Spatial pattern of density P obtained by numerically solving Eq. A8.13)in regionof instability of uniform steady state. Numerical values of parameters: F'= 3,fc,

= k2 = k4 =0.8888,D = 0.01,DH

= 6.25x 10~4,y = -0.4629x 10\022.

(D >0),and\342\200\224k{C

the rate of disappearanceof \"active insects\"in the sensethat oncethey depositthe material the termites do not participate in thework for a while. Finally, the last term representsthe attraction by the odor.It has been set proportional to a (constant) attraction coefficienty(y <0)and to the divergenceofa \"

field.\" The latter isproportional to the localdensityof insectsand to the gradient of odor.A similar term has been used in the

modeling of the chemotactic motion of cellular slime molds during the

aggregation phase(seeChapter15and Kellerand Segel,1970).Under certain conditions,Eq.A8.13)may admit two steady states.The

first is the uniform state:

c -p\302\260~^7

Ho = \342\200\224 A8.14)kA

A linear stability analysis following the lines of Section7.4showsthat thisstate may becomeunstable.The system then evolves to a structured con-configuration depictedon Fig.18.1.This configurationfeaturesa regular patternof material depositionwhich is expectedto simulate the regularity of the

pillar successionobservedin actual termite nests.A detailedanalysisof the characteristicequation showsthat only when the

density of the depositedmaterial* reachesa certain value doesthe uniform

* According to Eq. A8.14)the density of deposited material is directly proportional to the

density of insects.

18.4.Evolution ofEcosystems 455

distribution becomean untenable solution of the rate equations.This is in

agreement with the experimentalobservations.Moreover, it is seenthat thewavelength of the pattern isa function of the dispersioncoefficientsD and DHand the sizeof the system.In particular, structuration becomesimpossibleifthe dimensionsare below somecritical value. This is quite similar to theconclusionreachedin Section16.5devoted to the onset of polarity in amorphogeneticfield.

It is striking to see,on this simpleexample,that the performanceofcomplextasks leading to regular structures does not necessarilyrequiredistancemeasuring or other similar devices.Rather, it appearsto be an inevitableconsequenceof the nonlinear interactions within the system and of theconstraints exertedby the external environment.

18.4.EVOLUTIONOF ECOSYSTEMS

Thus far we have analyzed some aspectsof the behavior of an alreadyestablishedecosystem.We want now to understand how the interactionsand other aspectsof the functioning of the systemcame about as a result ofevolution. As in Chapter17,to study evolution we have to take into accountthree factors: (a) reproduction,(b) selectionthrough competition, and(c)variation through \"mutations.\"

The first two aspectsare already incorporatedinto the basicequationsA8.8).In orderto account for the effectof \"mutations\" we adopt the samepicture as in Section17.5.We obtain in this way (Allen, 1976;seealsoCzaplewski,1973)a criterion for the spreadof a structural fluctuation in theform A7.19),namely, that the characteristicdeterminant

dG, 1 - cotdet/o

= 0 A8.15)

shouldlead to at least oneroot having a positive real part. HereG, denotethe ratesof evolution of the mutants {Yk} (k, I = 1,...,m):

^ = G,({Yk},{Xj}) k,l=l,...,m;j=l,...,nA8.16)

and {Xj}denote the initially presentspeciesor phenotypes.Finally, thederivative in Eq. A8.15)is evaluated around a reference state where the\"mutants\" are absent:

{Xj}= {XJ}; {Yk} = {0} A8.17)Note that in the precedingargument the exact mechanism of structuralfluctuations is left open.Darwinism supposesan origin of fluctuations

456 Ecosystems

basedon random genetic variations. This is certainly appropriatefor manyaspectsof biologicalevolution. Socialand ethic evolution, on the otherhand, includeslearning mechanisms, innovations, and inventions of theindividuals trying to adapt to the environment. Note, however, that the

precisemechanism would play an essentialroleif we wanted to calculatethe time scaleof the evolution.

Let us examinein detail how the evolutionaryprocessoccurs,in the simplecaseof a singlespeciesin a medium of limited resources(Allen, 1976).Thebehavior of the reference system is describedby Eqs. A8.5) and A8.6).Now, supposethat by some\"accident\"oneof the individuals born at acertain moment is different. Let X2 be the number of individuals of this newtype, and supposethat after sometime there is a sufficient number of X2present to enabledescriptionof their evolution by an equation such as[seeEq.A8.8)]:

^ kX(NX2-pXi)-d2X2 A8.18)

X2 may have different values of k, N, and d and in addition may exploitdifferent food resources.This is expressedby the factor /?, where 0 < /? < 1.If P = 1,X2 usesexactlythe sameresourcesas X u while for /? = 0 they haveno common resources.Partial overlap is expressedby /? between zero andunity.

The new equationsfor the whole systemare, instead of Eq.A8.5):A V

\302\261 fcX^JV X pX) dX

dX7 - A. 2 \342\200\224 pA i) \342\200\224

(X2 A. 2 \\\\o.\\y)dt

and, the state existingat the moment the mutant appearsis:

X\\ = JV, - jp X5 = 0 A8.20)

The responseof the initial systemto a small quantity of mutant is determin-deterministic and can be calculated using Eq.A8.15)to establishthe condition for aroot of the characteristic equation evaluated around state A8.20),to have apositive real part.This is, then, the condition for growth and for an evolu-evolutionary stepto occur.Onefinds that, if

18.4.Evolution ofEcosystems 457

Figure T8.2. Niche occupied successively by speciesof increasing effectiveness.

then the mutant grows to some finite value and occupiesa \"niche\" in

the system.What will happen to X{ ? Severalcasesare possible.If the speciesX2 is a mutant occupyingexactly the sameniche as Xu then /? = 1 and wefind that X2 grows if

N2-^>Nt-^ A8.22)

and completelyreplacesXt; the total systemmovesto the stablesteady stateX\302\260

= 0, X2 = N2 \342\200\224 d2/k2,and X{ has becomeextinct.Successivemutationswithin the sameniche are rejectedfor values of N \342\200\224 (d/k) lessthan the pre-preexisting one and replacethat type if N \342\200\224 (d/k) is larger.We conclude,therefore, that evolution leadsto the steadily growing exploitation of eachniche and that the population carriedby each band of resourcesincreases.Thus, evolution appearsas in Fig.18.2.

Another possibleevolution is that the speciesX2 may differ from Xx in itschoiceof resources.In the caseof the exploitation of an entirely different

niche /? = 0, and the condition for the growth of X2 is

JV, -^ A8.23)

Thus, if X2 is viable in this niche it growsto a steady population X2 =N2 \342\200\224 {d2jk2)and coexistswith X\\ = N{ -djkx.Again evolution leadsto

458 Ecosystems

a fuller exploitation of the medium. If we considerthe intermediate rangewhen there is an overlap of resources,we find two cases.In addition to

if we have

then X2 replacesXx and the final population isagain greater than the initial

oneNx\342\200\224 (di/ki).The secondcaseariseswhen the conditions

*,-\302\243,18.26)

are fulfilled. Then X2 grows but coexistswith Xt.The final state is:

X\302\2602

=IN2-y~p(n{-j^jf/O- P2) 08.27)

Again, the total populationa a 1 / a

;i + p) > N, -~ A8.28)^Summarizing, then, we find that if there isa certain plasticityof the \"genetic\"matter, it can only result in the greater exploitation of the environment.This increasingeffectivenessof the exploitation of each resourceprovidesabasisfor understanding the notion of \" survival of the fittest\" familiar fromDarwin'stheory. In particular, our arguments show that a definition of the\"fittest\" can only be obtained through a suitable analysis of the evolutionequationsdescribingthe ecosystem.

Usingthe above ideasonecan alsostudy the evolution of a predator-preyecosystem(Allen, 1975).Onefinds that the prey evolvesso as to exploit theavailable resourcesmore efficiently and to avoid capture by the predator.The predator,on the other hand, evolyesso as to increasethe frequencyofcapture of prey and to decreaseits death rate. The result of this \"arms race\"is that the ratio of the biomassof predator to prey increasesslowly with

evolution.

18.5.Structural Instabilities and Increase ofComplexity: Division ofLabor 459

18.5.STRUCTURALINSTABILITIESAND INCREASE OFCOMPLEXITY:DIVISIONOF LABOR

As has been briefly discussedat the beginning of Section18.4,the sourcesof\" innovation \" are not exclusivelygenetic,but can referto changesin behavior.In this sectionwe analyze one modeof evolution, namely, the formation ofcooperativegroups.Within these may evolvedivisionsof labor,hierarchicalrelationshipsand castes,mechanisms of population regulation, and evenaltruism. Herewe are interested more specificallyin the division of labor(Deneubourgand Allen, 1976).

We start from the basicequationsA8.8) and let X{ correspondto thenumber of ants in a colony i, whereas /?0- correspondsto the fraction ofoverlap of the territoriesexploitedby coloniesi and).We take, for simplicity,JVj

= N = const,

Pu =Pfi, Pu = 1, and 0 < pu < 1

Supposethat initially all ant coloniesare of the sametype and that theindividuals that make up this type of colony are all identical. Then ptj

=Pij

\342\200\224 1.In the caseof two coloniesEq.A8.8)would read:dX,-\342\200\224- = kX{(N- X, - X2) - dXx

= kX2(N - Xl - X2) - dX2 A8.29)dX2

dt

Theseequationsadmit the trivial solution

X, = a 2 = 0 (lo.JUaja solution correspondingto the extinction of oneof the colonies

X\302\260{

= 0, X% = N--; Xi = 0, X\302\260i

= N -- A8.30b)K K

and a nontrivial solution given by

Ai +A2 = iV (lo.JUCJk

This latter solution is degenerate,and this is translated by the fact that alinearized stability analysis around this state showsmarginal stability. Inother words,in the competition describedby Eq.A8.29)the \"winner\" canbeeither X!or X2,dependingon the initial state of the system.

460 Ecosystems

Supposenow that oneof the two colonies,say X2, has \"invented\" thedivision of labor.Insteadof being composedof identical individuals, it nowcomprisesof: (a) \"Workers,\" Y, who are responsiblefor the growth of thecolonyand (b)\"soldiers,\"Z, whoseonly role is to aggresscolonyX t (hereafterdenotedby X). We assumethat the colony has alsodevelopeda mechanismfor regulating the relative populationsof Z and Y.

We want to seeunder what conditionsthe division of labor is imposedon the ecosystem.This problemis viewedhere as a problemof competitionbetweencolonies(X) and (Y + Z) or,alternatively, asa problemof evolutionof two identical colonies(X), (Y) under the influence of small structuralfluctuations (Z).

The equationsbecome:A *V

\342\200\224 = kX(N - X - Y - Z) -dX-pXZat

^ = kY(N - X - Y -Z)-dY-F(Y, Z)at

~r = F(Y,Z)-dZ'

A8.31)at

where pXZ is the aggressionterm; X and Y are assumedto bemorphologic-morphologically

identical and are thus given the samevalues of k and d\\ finally,F(Y, Z)is the regulation function that is representedby a Verhulst-like form [seeEq.A8.5)]:

F(Y,Z)= a}YZ-a2Z2 A8.32)

One can establish straightforwardly that for kN \342\200\224 d > 0 the trivial

solution Xo = Yo = Zo = 0 of Eq. A8.31)and the nonzero solutionXo Y0Z0 jL 0 are both unstable.Thus, we are interested solely in the prop-properties of the state

Xo #0, Yo # 0, Zo = 0 A8.33a)

or of the state

Xo = 0, Yo^0, Zo#0 A8.33b)

As regardsthe first state, we already pointed out [seediscussionfollowingEq.A8.29)]that for fluctuations in the (X, Y) plane oneobtains a property ofmarginal stability. In orderto study the effectof structural fluctuationsof Z,it sufficesto apply the criterion, Eq.A8.15),for /= 1:

-k -?)\\ >0 A8.34)

18.5.Structural Instabilities and Increase ofComplexity: Division ofLabor 461

or,introducing the last equation A8.31)as well as expressionA8.32)for F:alYo-d>0 A8.35)

Now, from Eq.A8.30c)we know that

with

kx + ky=

Substituting into Eq.A8.35)we find:

-(^+ l)d>0 A8.36)

The conclusionto be drawn from this relation is that the spreadof thestructural fluctuation engendering division of labor is favored by N large,that is, by the wealth of the environment exploited.It is also favored by o^large, that is, by a regulation enhancing the population of Z values within

the colony,oncethey appearthrough a structural fluctuation.The propertiesof the steadystate A8.33b)can bestudiedfollowing exactly

the sameprocedure.This time, the colony (Y + Z) is assumedto have beenestablished,and we are lookingfor its responseto the appearanceof acolony of nonspecialists(X).Working out condition A8.15)for the first

equation A8.31)onefinds (Deneubourgand Allen, 1976):

k(N -Yo- Zo) - d - pZ0>0 A8.37)

This condition is not satisfied if p or Yo + Zo are appreciable.In otherwords, if the differentiated colony is wealthy and aggressive,it will not

permit the developmentof X. Even if X can build up, however, it can onlylead the system to state A8.33a):colony (Y + Z) is not eliminated from themilieu. Rather, it modifies its structure to better adapt to the external con-conditions.

A number of observations(Wilson, 1971,1975)seemto confirm the con-conclusions reachedfrom the analysisof the model,especiallythoseconcerningthe dimensionsof the colony.Thus, it appears that polymorphism in ant

speciesliving in temperate climatesis much lesspronouncedthan in tropicalregionswhere the mean dimension of a colony is larger.Similarly, in largebee coloniesthe degreeof polyethism of workers is highly developed.Finally, in small beecoloniesthe morphologicaldifferencebetween\"queen,\"and \"workers\" is lesspronouncedthan in large ones.

462 Ecosystems

18.6.STABILITYAND COMPLEXITY

We have seenthat evolution toward increasingcomplexityand organizationis the result of structural fluctuations\342\200\224mutations or innovations\342\200\224that canappearsuddenly in a previouslystablesystem and drive it subsequentlyto anew regime.Obviously,there isno limit to the type of fluctuation that maybeconsidered,and no ecologicalequationscan beclaimedto be structurallystable to all possibleinnovations. It can, therefore, be legitimately askedwhether there is a limit to complexity.The very existenceofcomplexsystemssuch as a tropical forest (May, 1973)ora modern society illustrate well theneedfor answeringsuch a question.

In this sectionwe want to stressthat a complexsystem is subjectto twoadverse tendencies.On the oneside,the more complexthe system\342\200\224that is,the more variables there are in interaction\342\200\224the higher the degreeof thecharacteristic equation [seeEq.A8.15)]determining its stability, the greaterthe chancesthere are, therefore, of having at least onepositive root andhence instability of the referencestate (May, 1973).

On the other hand, the argument basedon the characteristic equation isvalid only after the fluctuation has occurredand extendedup to the macro-macroscopic range.Therefore,to have a more completeview of the stability versuscomplexityproblem,we must specify:

(i) The a priori probability for having a certain fluctuation in a complexsystem.(ii) The probability that this fluctuation spreadsand attains a macroscopicamplitude and range.

Both problemsbelongto stochastictheory and can, in principle,be tackledby the methodsdevelopedin Chapters11and 12.The study of the simplechemicalmodelsconsideredin thesechaptershas shown that the evolutionof a fluctuation dependson the competition between growth and dampingthrough diffusion or,more generally, through surface effects.The latter,which in the nonlinear master equation formulation of Chapter 12aredescribedby the coefficient3),measurethe degreeof coupling between asystemand its surroundings and are stronger the more complexthe system.Consequently,the tendency to damp the fluctuations is accentuated.Inother words,steps (i) and (ii) referring to the purely stochasticstage ofevolution are more difficult to realize in complexsystemsin that they requirethe nucleation of fluctuations exceedingsome critical range (Prigogine,Nicolis,Herman, and Lam, 1975).

18.6.Stability and Complexity 463

Therefore, we reach the conclusionthat a sufficiently complexsystem isgenerally in a metastable state.The value of the threshold for metastabilitydepends,in a complicatedfashion, on the system'sparametersand theexternal conditions.The question of the stability of complexsystemsseems,therefore,to have a lessclearcut answer than thoseconsideredup to present.

Obviously,it would be interesting to comeup with quantitative measure-measurements of the factors that determinethe growth of instabilitiesin ecosystemsorsocieties.It seemsfascinating to us that these difficult questionscan beconsideredevenapproximately by means of mathematical modeling.

PerspectivesandConcludingRemarks

1. INTRODUCTION

In this final chapter we want to reviewa few problemsof basicinterest that

are still in an exploratory phase.This may give us an indication about thedirection in which the methods studiedin this monograph may find new

applicationsin the near future. Section2 is devoted to the mathematicalaspectsof self-organization.In Section2 we reviewsomeaspectsof fluctua-fluctuation theory when appliedto chain reactionsand explosions.In Section3 wereviewsomerecent work on neural and immune networks.In Section4 wediscussin somedetail a most interestingmodel of the immune surveillanceagainst cancer.This involvesa transition from what can be appropriatelycalledmicrocancerto macrocancer.The Garay-Lefevermodel uses manyof the tools describedin this monograph, such as a study of ecologicalequationsin Part V (appliedhere to the growth ofcancercells)and the theoryof multiple steady-state transitions including fluctuations as discussedin

Chapter12.It seemsfacinating to observethat there may exista closeanalogybetweensomeclassesofdiseasesand nonequilibrium transitions, in contrastwith the classicalteachings of Cl.Bernard,where onealways considersdiseaseas a continuous perturbation of the normal state (Cl.Bernard,1855,1856).

We closethis chapter with a few remarkson the sociologicaland epistemo-logicalconsequencesof our approach.

2. FLUCTUATIONCHEMISTRY

Oneof the principal pointsof focus of the presentmonograph has been theideathat the onsetof a self-organizationprocessisdictated by the behaviorof the fluctuations. The main reason is that a self-organizingsystem is

464

2. Fluctuation Chemistry **-

necessarilyundergoing instabilities, and hence is capable of amplifyingcertain disturbancesincluding someof its own fluctuations.

Despitethe generality of feedbackphenomena giving riseto instabilitiesand, subsequently,to dissipativestructures,onecan still argue that in everydaylaboratory chemistryself-organizationis the exception,not the rule.In thissectionwe want to suggestthat chain reactions,of the utmost importance in

chemicalkinetics,are likely to beinfluencedby fluctuations.We developourarguments primarily for explosivereactions.

Let us first considera chain-branching explosionand assumethat thermaleffectscan be neglected.As well known (Dainton, 1966)the simplest suchprocesscan be decomposedinto an initiation step, producinga certainpopulation of free radicalsX, a chain-branching step leading to an (auto-catalytic) amplification of this population, and a recombination and/ortermination step,where the free radicalsare inactivated.Schematically

I{X)> X initiation

2X chain branching

P + M termination

X + A recombination A)

A, P,M, are concentrations of precursorsand of final productssupposedlyin excessand are hencetaken to beconstant.

The rate equation for X is

_ = I(X)+ (B - T)X- RX2 B)at

where we set

B = k2A

T=k3AM Ca)R = kA

The initiation term I(X)is,(in the absenceof external stimulus,) determinedby an equilibrium reaction and ishencea small contribution, especiallywhenthe concentration of X beginsto attain appreciablevalues.Fora first-orderinitiation reaction, a typical expressionfor I(X)would be:

l(X) = Io- XX Cb)

Therefore,onecaneasilyverify that Eq.B)admitsa singlepositivesteady-statesolution that remainsasymptoticallystablefor all nonvanishing valuesof the

466 Perspectives and Concluding Remarks

parameters.As B increasesand crossesthe value B s T, the steady-stateconcentration Xo of X increasessharply, although it remains continuous.Hencethe relation

B ~ T D)

definesthe explosionlimit.

Supposenow onestarts initially with a small population of free radicals.Beforethe steady-statelevel Xo is reached,an induction time x is necessary.Now, becauseof the smallnessof X for t <f t, the instantaneous values ofcompositionwithin various small volumesAVt is likely to be quite different

from the (macroscopic)valueX determinedby solving Eq.B).In other words,thefluctuations ofcompositionare expectedto play a role,suchasby modifyingthe induction time necessaryfor explosion.This ideais alsosupportedby theobservation that in the limit / -> 0 Eq.B)doesgive rise to an instability,

leading to a transition between two steady-state levelsXol = 0 and Xo2 =(B \342\200\224 T)/R. This transition, reminiscent of the problemstreated in Section12.6,has been analyzed by McNeiland Walls A974)in a somewhat different

context. They found that in the vicinity of B = T fluctuations are indeedimportant. Hence,for / finite and small one can expectsomeof theseresultsto still apply.

Further work is necessaryto substantiate theseideasand to extend theinvestigationsto more realistic explosionmodels.The inclusion of thermaleffectsis certainly necessary.In the framework of a macroscopictreatment,Gray (seeGray, 1974for a recent survey) has workedout simplemodelsfor such thermal explosionswhich, typically, giveriseto transitions betweenmultiple steady states.As stressedrepeatedly in Chapters11and 12,in this

casefluctuations are certainlyof the utmost importance and, in fact,dictatethe conditionsfor the stability or metastability of the various branches.This, in turn, determinesthe possibilityof a spontaneoustransition betweendifferent states.

3. NEURAL AND IMMUNE NETWORKS

We haveseenthat populationsofcoupledcellsmay give riseto \"coherentbe-behavior,\" extending beyond the level of the single cell.Thesephenomenareflect the numerous regulatory and coordination processesexertedin the

system,and for this reasonthey are associatedwith the \"network behavior\"of the system. This is to be contrasted with the situations encountered in

traditional problemsin physical chemistry or in molecular biology, wheresmall-and large-scalephenomena are linked in a simplemanner.

In this sectionwe want to comment briefly on two further instanceswhere

3. Neural and Immune Networks 467

the network behavior of the systemseemsto play a prominent role,namely,the activity of the central nervous system and the immune response.

Activity of Central NervousSystem

Throughout this monograph we have seenthat the dynamics of a spatiallydistributedsystemmay give riseto oscillations,traveling waves,and stationarystructures.Now, rythmic activity has long beenknown to be a ubiquitousfeature of the brain.The patterns of activity associatedwith EEGprovidethe best-knownillustration. Moreover, it has beenshown (Freeman, 1968)that the mammalian brain may respondto external excitations by eitherdamped or sustainedoscillations.Hence,it is tempting to undertake amodeling of neural nets basedon the samekind of considerationsas thoseunderlying the conceptof dissipativestructure.

Wilson and Cowan (seeWilson, 1974for a recent survey) and Freeman(seeFreeman, 1975for a recent survey) developedelegant modelsof neuraldynamics basedon the useof a sigmoidal expressionfor the input-outputfunction of a group of neurons, This procedureis justified as a necessaryconsequenceof a statistical distribution of thresholdsamong individualneurons.A limit-cyclesolution generated by modelsof this type has beenimplicated in the handling of information in the olfactory bulb (Freeman,1975).On the other hand, multiple stead-statesolutions have been invoked(Wilson, 1974)to simulateexperimentsofhysteresisin binocular vision.

A different type of model has beendevelopedby Kaczmarek A976) that

considersexplicitlythe patterns of activity oisingleneurons.Functions suchas frequency of firing vs membrane potential and the ionic equilibriumpotentials of the neuronal membranes are incorporatedinto the theory.In addition to \"normal\" patterns of activity, the model gives rise to cellfiring patterns and membranepotential variations similar to thoseobservedduring the progressionofepilepticseizures(seealsoKaczmarekand Babloy-antz, 1977).

Mostof the work carriedout sofar in the modelingof the nervoussystemisconcernedwith the electricalactivity of neural masses,for example,with therates of firing of populationsof excitatory and inhibitory neurons.Despiteits intuitive appeal,this approachis not particularly convenientfor analyzingthe effectsof external (e.g.,electric)fieldson the brain.

Recently, Nazarea A977) developeda complementary approachto thislatter problem.Instead of regarding the brain as an electrical networkmediated by chemical effectors\342\200\224the neurotransmitters and the various ionspresentin the neural membranes\342\200\224he focuseson the spatiotemporalproper-properties of the chemical effectors themselves. As most of these are chargedspecies,the chemicalrates couplewith the electric field in a straightforward


way. This createsthe possibilityof theoretical interpretation of such phe-phenomena as the direct-current polarization of the cerebralcortexor theasymmetricfrequencyshift when the polarity of the electrodesis changed.

Needlessto say, numerous challenging problemsremain unresolved in

this field.Among these,the study of modelssimulating the geometry of theneural array, its size,the boundary conditions,and the influence of ap-appropriate sensoryinputs during development on the neural configurationsisof specialinterest.

The Immune Response

A living organism has the ability to defend himself against the invasion offoreign substancessuch as tissuesor microorganisms,among others,which

\"Foreign\"stimulus

Antigen

Epitope

Recognizingand potentiallyresponding set

Unspedific parallelset

Bufferingsets

Figure 1.The immune system contains a set /)( of combining sites (paratopes) on immuno-

globulin molecules and on cellreceptors that recognize a given epitope (E)of an antigen. Thisrecognizing set includes the potentially responding lymphocytes. The molecules of set p, alsodisplay a set i2 of idiotopes. Apart from recognizing the foreign epitope, the set />, likewise

recognizes a set i2 of idiotopes which thus constitutes, within the immune system, a kind ofinternal image of the foreign epitope. This set i2 occurs in molecular association with a set p2 of

paratopes. Likewise, the set i, is recognized, within the immune system, by a set p2 of paratopesthat represent antiidiotypic antibodies. Besidethe recognizing set pli1, there is a parallel set pj,of immunoglobulins and cellreceptors that display idiotopes of th set it in molecular associationwith combining sites that do not fit the foreign epitope. As a first approximation, the arrowsindicate a stimulatory effect when idiotopes arerecognized by paratopes on cell receptors and asuppressive effect when paratopes recognize idiotopes on cellreceptors. Successive groups ofever larger sets encompass the entire network of the immune system (after Jerne, 1974).

4. Immune Surveillance Against Cancer 469

are usually referred to as antigens.This is achieved by the immune system,which isconstituted by specializedcells,the lymphocytes(which can undergomany stagesof differentiation),and by the moleculesproducedby them.Among thesethe antibodiesplay an essentialrole, becauseof their abilityto combine with the antigens. This property is related to the particularstructure of the antibodies,which disposeof specific binding sites\342\200\224the

paratopes\342\200\224recognizing specific types of antigens. On the other hand,antibodieshave sites presenting antigenic properties\342\200\224the idiotopes\342\200\224and

are hence susceptibleof binding with the paratopesof other antibodies.Thisdual roleofantibodies,togetherwith their very pronouncedvariability,

led JerneA973, 1974)to postulate the existenceof an immune network (seeFig.1)consideredresponsiblefor the observedbehavior of the immune

system.In other words,the claim is that such phenomena as the immune

response,the memory, the tolerance,and the immune surveillanceconstitutenot the random superpositionof events as the molecular level but, rather,reflect the collective behavior of large assembliesof coupledlymphocytes(seealsoUrbain,1976).

Elegant modelsdesignedto substantiate theseideashave beendevelopedby Richter A975) and by Hoffman A975) in the context of the immune

response.The stability ofsimpleimmune networkshas recentlybeenanalyzedby Hiernaux A977).

It is the presentauthors'opinion that the modelingofimmune networkswill

be an increasingly important subject of theoretical biology in the nextfew years.

4. IMMUNE SURVEILLANCE AGAINST CANCER

The functioning of the immune system is also closelyrelated to certainpathological states of living organisms,among which canceroccupiesaunique position.

Our present understanding of the cancerproblem suggeststhat thecanceroustransformation of a normal tissuecouldbe describedas follows(seeCairns,1975and Calvin, 1975for recent surveys):

(i) In normal tissuesthe pattern of supercellularorganization is determinedby a complexinterplay betweenshort- and long-range interactions betweenthe cells.Theseinteractions control, among other processes,the mechanismof cellular replication,(ii) Variousproteinsmediating these interactions are geneticallycoded.


(iii) Any disturbancein this geneticcontrol, either \"spontaneous\"(somaticmutations) or dueto the action of environmentalfactors (pyhsical,chemical,or biological),couldyield cellswith an abnormal pattern of proteinscoded.Subsequently,thesecellsmay exhibit a different capacity of responseto theinteractionswith normalcells.In the caseofthe malignant transformation,thecellsare characterized by a high proliferativeadvantage,(iv) The normal mechanismsof defenseof the organism tend to destroytheabnormal cellsor to slowdown their rate of proliferation. It seemsthat oneof the most important mechanismsof defenseinvolves the immune system.The immune cellsrecognizethe abnormal proteinsand destroythe cellsthat

bear them. This means that the immune system exertssomesort of \"surveil-\"surveillance\" on the very small population of transformed cells(\"microcancerpopulation\.

The exact nature and propertiesof the processesmentioned in points(i)-(iii)is still largelyhypothetical,mainly becausetheseprocessescorrespondto molecular changestaking place at the level of the single cell.On thecontrary, in respectto point (iv), even if the underlying molecularevents arenot known in detail, at least there existsexcellentdata on the kineticsoftumoral growth and on the competition of this processwith the immunesurveillance.The purposeof the pesentsection is to show that this informa-

information can beusedto formulate a mathematical modelfor the phenomenon ofimmune surveillanceagainst cancer,which may elucidatesomevery generalphysicochemical aspects of carcinogenesis.The model is based on thefollowing minimum setof data, all of which are compatiblewith present-dayideason the cancerproblem.

There existsa \"source\"producingmalignant cells.Its intensity dependsstrongly on environmental factors and is usually very small (in fact, hardlydetectable)in normal tissues.There existsin the organism a population ofcytotoxiccells(Hellstromand Hellstrom,1974),generallyformedby differentcellular species,which are capableof recognizingand destroying the can-cancerous cells.This impliesthat, in addition to the freecytotoxiccells,there alsoexist configurations where cancerouscellsare bound to the cytotoxic cellsurface.Subsequently,cytotoxiccellslike T lymphocytesmay emit chemicalsignalscalled lymphokines, which further enhance the cytotoxic activity.Moreover,it appearsthat lymphokines inhibit the diffusion of macrophagesand other cellular species.In other words,the motility of the cytotoxiccellsis reducedin the regionswhere malignant cellsare present.

One may now construct a \"minimal\" model compatiblewith thesedata.Let AV be the volume of the tissue,B and X the density of normal andmalignant cells therein, and Mo and M, the density of free and boundcytotoxiccells.The interaction betweencellularspeciescan berepresentedby

4. Immune Surveillance Against Cancer

the following scheme:B

Mo\342\200\224g\342\200\224

M,\342\200\224^

Mo + P E)

HereP standsfor the productsof degradationof X. The spontaneouscon-conversion of B to X (in fact, B = N -X, N being the total density of cells)isassumedto occurat a fixed rate a. Finally, the kinetic constants kx, k2characterize the binding and destruction of the malignant cellsby the cyto-toxic cells.

Considerfirst the caseof a uniform distribution of cytotoxiccells.SchemeE) then gives rise to the following kinetic equations(Garay and Lefever,1977;Lefeverand Garay, 1977):

i y = (N -X)(a+ IX) -kxM0X F)at

= -kxM0X+ k2Ml G)dt

where the term XX addedto a accountsfor cellular multiplication.EquationsF)-(8)have been solved using numerical values for N, a, X,

ki, and k2, drawn from experimental data. The main result is that thesystem presents a transition between two simultaneously stable steadystates,one correspondingto a low value of X, which couldbeappropriatelyreferred to as a \"microcancer,\"and one where X is high and which corre-corresponds, therefore, to a \"cancer.\"Moreover,one showsthat the transitionregion is a region of metastability rather than of completestability of themicrocancerstate.This impliesthat there existsa critical \"nucleus,\" that is,acritical microtumor size that must bereachedbefore malignant growth candevelop.This can happen either as the result of a spontaneousstatisticalfluctuation or by an externalagent (in much the sameway asa seedprovokescrystallization in a supercooledliquid). Experimentconfirmsthe existenceofsuch a threshold for cancerdevelopment (Baldwin, 1976;Carnaud andGaray,1977).

The above considerationsstressthe importance of stochasticelements in

canceronset.This has been repeatedly recognizedin the literature (e.g.,Huebneret al.,1970)and can beattributed to the fact that the emergenceof

canceris a rare and random event that in most casesseemsto arisefrom asinglecell.

Garay and Lefevercarriedout an extensivestochasticanalysis of modelE)basedon the considerationsdevelopedin Chapters10and 12.Thisanalysis,which alsotakesinto account the mobility of the cytotoxiccells,has providedquantitative estimates for such quantities as the width of the metastableregion and the critical value of N (or volume element AV) beyond which

malignant growth becomespossible.The main point to be retained from this short account is that a theoretical

descriptionof canceronset is possibleand that such a descriptionhas toincorporateexplicitly the usually neglectedeffectsof fluctuations. Similarideasare likely to apply to a host of other problemsinvolving the develop-development ofpathologicalbehavior in living organisms.As mentioned in Section1,a striking idea suggestedby thesetheoretical developments is that patho-pathological behavior is not a mere quantitative modificationof normal behavior(Cl.Bernard,1855,1856).Rather, it is separatedfrom the latter by a dis-discontinuity that can bebridgedonly if the system'sparametersand the initial

conditionssatisfy someappropriaterelations.

5. SOCIALSYSTEMSAND EPISTEMOLOGICALASPECTS

The variousproblemsand applicationsdiscussedin this monograph illustratethe central importance of the problemsof self-organization.One of thestriking features emerging is the existenceof various levels of description.We observearound us a great variety of behavior.Onecouldevenstate that

weobservea basicdichotomy; on one sidewehave the basiclawsofclassicaland quantum dynamics,and on the other we have natural phenomena that

may well berepresentedby \"games,\" in the terminologyofEigenand Winkler

A975). Moreover, in a given system we may observevarious levels of be-behavior. We now begin to seehow wecan switch from one type ofdescriptionto another. As is well known, all interactions of interest,notably in biologicalmedia, are of short range (e.g.,valency and van der Waals's interactions).These interactions lead to the formation of \"conservative\"structuresand determine alsothe valuesof characteristic rate and diffusion constants.Yet the theory of dissipative structures showsthat under someconditions(involving among othersa sufficient distancefrom equilibrium) the equa-equations describingthe chemical kineticstogether with diffusion may leadto a new long-rangeorder.Thecoherenceintroducedby dissipativestructuresis always characterized,as emphasizedseveral times, by a supermolecularscale that leads to a modification of the \"space-timestructure\" in which

the moleculesare embedded.Similarly, the equationsof ecologystudied

5. SocialSystems and Epistemological Aspects 473

in Part V contain parametersthat may well be determined by the geneticmaterial. Yet the statesdescribedby theseequationsrefer to a \"supergenetic\"aspectof evolution. Again we obtain a feedbackbetween the structures andthe units that form the system and the evolution of the system as a whole.

It is, of course,very tempting to apply these considerationsand speciallystructural stability theory to problemsof socioculturalevolution.

The main difficulty is to determine the relevant variables. In somecases,such as problemsconcerning vehicular traffic flow, this is relatively simple(Prigogineand Herman, 1971).In other problems,however,oneneedsto in-introduce such elusive variables as \"quality of life\" which are much moredifficult to handle in a quantitative manner. Let us indicate briefly two inter-interesting examples.

The first refersto the progressiveurbanization of a geographicalarea.It isgenerallyacceptedthat, in the long run, local populationsgrow or decreaseaccordingto employmentopportunities,which in turn dependon the marketavailable for goods and services.This introducesa nonlinear feedbackmechanism in the growth of population density correspondingto the so-called\"urban multiplier.\"

Thisproblemwasrecentlystudied(Allen, Deneubourg,and Sanglier,1977)starting from a logisticequation like Eq.A8.5).The carrying capacityof anyparticular localitycan beincreasedif it is the seat ofeconomicfunctions (e.g.,the point of productionof somegoodsor services).The model studiesthe

impact of successiveinnovations on an initially homogeneouspopulation.When various economicfunctions are launched, the nonlinear interactionmechanisms between population density and employment opportunitiesresult in the formation of spatial structures, and an urban hierarchy evolves.

Oncemore, both stochasticand deterministic elements comeinto play.We have a random element in the time and location of lauchingofeconomicfunctions, whereastheir survival and growth is governedby the deterministiceconomicconstraints of the availablemarket.

Similarmethodshavebeenappliedto the analysisof the remarkable socialorganization of the Katchin tribesof highland Burma (Leach,1961).In this

societywe find two forms of organization and a well-definedmechanismofinstability that leads from one to another. Moreover,the mechanism ofinstability is related to the rules of parental relations studiedby L6vi-StraussA958).In a sense,theserules play, in this anthropological context, the sameroleas the catalytic reactionsin biochemicalcycles.

In the mathematical model of the organization change (DeneubourgandPahaut, 1977),a stochasticelement is introducedto simulate the diversity ofhuman behavior.At each moment the structural stability of the deterministicmodel describingthe \"

average\" behavior is testedby injectingfluctuations,

which eventually trigger the transition between the two forms of socialorganization.


It is worthwhile to notice with Leachthat \"changeisno longer somethingthat isdoneto us by nature but somethingwe can chooseto do to nature\342\200\224

and to ourselves.\"Clearly, in this perspectiveone of the major objectivesofscienceis to elucidate the dynamicsofchange.The methodsdescribedin this

monograph might be one step in this direction.Although much work remains to bedone,it already clearly appearsthat

self-organizationisan emergingparadigmofscience(Jantsch,1975)emphasiz-emphasizing macroscopiccoordination processesat many levels,in which nonlinearprocessesand nonequilibrium conditionsplay a significant role.*

* The epistemological aspectsof the ideas summarized above are presented in a forthcomingmonograph by one of the authors (I.Prigogine) and I.Stengers.

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AddendumMathematicalProblems

Throughout this monograph, most of the analytical resultsobtained onpattern formation were basedon approximate solutions of reaction-diffu-reaction-diffusion equations using suitable perturbative expansions.Clearly, exact resultswould be highly desirable,as they would enable us to tackle problemsinvolving arbitrary deviations from the first bifurcation point.

Oneof the most challengingquestionsin this direction is the possibilityto explore fully the bifurcation diagram of a reaction-diffusion system,including information on stability of the branchesand on secondaryandhigher bifurcations. Recently, Lefever,Herschkowitz-Kaufmanand TurnerA977) were able to carry out this program in a particular limiting caseofthe trimolecular model.

Considerthe familiar schemeG.10)where both the entry term A and thedecay term X are deleted.In the absenceof diffusion such a system isclosedto mass transfer with respect to X, Y. However, inclusion ofdiffusion can model appropriately an open system, if the boundary condi-conditions are compatiblewith nonvanishing fluxes of X and Y. The rateequations become(we take a one-dimensionalmedium of length /= 1):

) 2 9/2where

f(X,Y) =X2Y-BXand

Y(O)=Y(l)=B/\302\243 B)

Introducing the new variables

C)

487

488 Addendum Mathematical Problems

we may lump the equations at the steady state into the single relation:

K-F(W) D)

where K is an integration constant, and

Solutionsof eq.D) can be constructedin terms of elliptic functions.They feature bifurcation propertiesand space dependenciessimilar tothose describedin Section7.6.The novel point is, of course,that the

amplitude and period of the solutions and the shape of the bifurcationdiagram canbe calculatedwithout using perturbation theory. Moreover, in

the particular caseDX=D2 the eigenvalues of the linearized equations

around the space-dependentsolutions of eq.D) can be computed exactly,as they are given by a Lame equation.

An additional feature of the steady-state solutions is worth pointing out.We first write relations A) as

dr2

d2Ydr2 F)

In this form, the equations have a structure similar to those of classicaldynamics, provided time is replacedby the spatial variable r. Suchphe-phenomena as the formation of spatial dissipativestructures may, therefore, beviewed as the analogsof the formation of periodicorbits,which, as is wellknown, characterize the motion of certain classesof dynamical systems.

Finally, it has beenstressedrepeatedly(Lorenz,1963;Nitecki, 1971)that in systemsof nonlinear differential equations involving three or morevariables new types of bifurcations leadingto \"strange attractors\"becomepossible.The resulting random, or \"chaotic,\" behavior might well prove tobe a genericproperty of complexreaction networks.In this respect thenonlinear laws that can lead to order far from equilibrium couldrefer to asmall\342\200\224although extremely important\342\200\224class of systems!The situation issomewhat reminiscent of the status of integrable systems of classicalmechanics(Moser1973)which, although of the utmost importance, refernevertheless to rather exceptionalpatterns of dynamical behavior.

Index

Acetabularia, 425Activation, 165,357,418Active transport, 413Adenyl cyclase,379ADP, 355,360

oscillatory behavior, 369,379Affinity, 33,213,444Allosteric constant, 363cAMP, 379

and catabolite repression, 389oscillatory behavior, in Dictyostelium,

379,380and relay, 381

Arc discontinuity, 403ATP, 355,360,379

and adenylic energy charge, 385oscillatory behavior, 369,379

Belousov-Zhabotinski reaction, 339mathematical model, 345mechanism, 343

Bendixson negative criterion, 88/3-galactosidase, seeInduction, of/3-galac-

tosidaseBifurcation, 71,83

diagram, 111,113,121,124eigenvalue, at simple, 86secondary, 120,192of steady-state solutions, 106,109,113,

120,180of time-periodic solutions, 140of waves, 153,185

Booleannets, 211Boundary conditions, 29,64Briggs-Rauscher reaction, 352Brusselator, 93

bifurcation analysis, 106,140,153asclosedsystem, 156stability analysis, 96

Catalysis, 90,93,165,169heterogeneous, 197

Catastrophe theory, 74and all-or-none transitions, 174

Cellsorting, 415Cellular contacts, 414Center, 82Central limit theorem, 227Chain reactions, 465Chapman-Kolmogorov equation, 230Chapman sequence, seeOzoneCharacteristic equation, 77

for Brusselator, 98Chemical potential, 33Chemical reactors, 168,382Chemotaxis, 379Circadian rhythms, 195,386Competition, 429,449Conflict, 176Conservation equations, 27,63Constant of motion, 161,284Correlations, 297

.long range, 305,307Critical exponents, 115,309,323,

326Curie symmetry principle, 40Cusp, 176

Detailed balance, 41,330Diffusion, 46

and Fick's law, 48and positional information, 410,415

Dissipative structures, 60bifurcation analysis of, 106,140,153localized, 131multiplicity of, 127and polarity, onset of, 415qualitative properties of, 115thermodynamic properties of, 212in two dimensions, 178

Distribution, binomial, 226Gaussian, 227,235Poisson, 226,235,

260489

490 Index

Ecosystems, basicequations, 449evolution of,455

Eigenvalue, 84of Laplaceoperator, 98,178multiple, 85,121-

Elliptic functions, 465Entrainment, 195,414Entropy, 22,32,55,212

balance equation, 33excess,56flow, 24,34production, 24,34,212,443

Enzyme, allosteric, 357Michaelian, 167

Escherichia coli, 387Evolution, prebiotic, 429

and structural stability, 438Evolutionary feedback,441,445Evolution criterion, 50

and kinetic potential, 51Excitability, 173,401

Floquet theory, 206Fluctuations, 223

in Brusselator, 289,319circulation of, 271and Einstein formula, 234around equilibrium, 232in finite volumes, 310in Lotka-Volterra model, 264molecular dynamics study of, 283in nonlinear model, 261in unimolecular reactions, 257

Fluxes, generalized, 35Focus, 82Fokker-Planck equation, 243Forces,generalized, 35Fredholm alternative, 108,143Fusion, cellular, 403

Generating function, cumulant-, 251,266,320

moment-, 249Gibbs formula, 33Glucose-oxidase,382Glycolysis, 354

and adenylic energy charge, 385allosteric model of, 358external disturbances, effect of, 371oscillatory behavior, 356,369

spatio-temporal patterns, 375

Hamilton-Jacobi equation, 248Heat-shock, 404Hopf bifurcation, 141Hydra, 425Hypercycles, 436

Immune networks, 468Immune surveillance, 469Induction, of/3-galactosidase, 388

all-or-none transitions, 389oscillatory behavior, 395

Inhibition, 166,418Insects, division of labor in, 459

social organization in, 453Instability, seeDissipative structures; Stabil-

Stability; and Symmetry-breaking

K-V system, 360

Lacoperon, 388Landau-Ginzburg potential, 117Langevin equation, 247Large numbers, law of, 227Length, critical, 151

in allosteric model, for glycolysis, 378Leray-Schauder theorem, 85Limit cycles,87

in Belousov-Zhabotinski reaction, 340in Brusselator, 140in glycolysis, 369in lacoperon regulation, 394in mitosis, 404models of, 165multiple, 350,401

Lotka-Volterra model, 160

Markov process,229Master equation, 231

for birth and death process,239multivariate, 285nonlinear, 317in phase space,275

Maxwell rule, 176,211Membrane-bound enzymes, 382Metastability, 327,462Minimum entropy production, 42Mitotic oscillator, 402

mathematical model of,405

Index 491

Moment, seeProbability distribution

Morphogenesis, 409

Neural networks, 467Node, 79Nonexclusive binding coefficient, 364Normal mode, 78

in Brusselator, 103Nucleation, 312,323,331,462

Onsager reciprocity relations, 40Opensystems, 24,26Order parameter, 116,322,326Oregonator, 345Organizer, embryonic, 413Oscillations, conservative, 160

epigenetic, 387past an instability, 87metabolic, 354periodic, almost, 196in trimolecular model, 140seealsoLimit cycles

Oscillators, 195asymptotic behavior of, 203coupled through diffusion, 147

Ozone,201

Papain, 382Pattern, seeMorphogenesis; Symmetry-

breakingPermease,388Phase-shift, 371,415Phase-singularity, 197Phenomenological relations, 36,39,49Phosphodiesterase, 381Phosphofructokinase, 357,446

mathematical model, of the reaction, 358Photochemical reactions, 200Physarum polycephalum, 402Polarity, 104

in morphogenesis, 415Polymer, prebiotic, 430,434Positional differentiation, 421Positional information, 410,413Potential, kinetic, 51,75

thermodynamic, 43,233,236Probability distribution, 224

moments of, 225,252Pyrophosphohydrolase, 379Pyruvate kinase, 357,446

Reaction-diffusion equations, 27,64Refractory period, 371,414Repression, 388,394,422Riemann-Hugoniot catastrophe, 177

Saddle point, 81Separatrix, 82,88,399Singular points, multiple, 83

simple, 79Sizeinvariance, 411,421Solvability condition, seeFredholm alterna-

alternative

Stability, 46,55,63asymptotic, 65and complexity, 462linearized, 71Lyapounov, 65marginal, 79,86orbital, 66structural, 68thermodynamic criterion of, 59

Steady states, 41multiple, 169,391

Stochastic differential equations, 247Stoichiometric coefficient, 28,240Symmetry-breaking, 115,415Synchronization, 147,195,402

Thermodynamic branch, 57Transition probabilities, 228,230

for birth and death process,240for diffusion, 287,315

Trimolecular model,seeBrusselator

Turning points, 134Turnover numbers, 364

Urban growth, 473Urease, 382Van derWaals theory, 173,329Variance, 225

divergence of, 271,292,324

Wave, in allosteric model, ofglycolysis, 376in Belousov-Zhabotinski reaction, 341,

351in Brusselator, 149,153,185and formation, of surfaces of discontinu-

discontinuity,207

kinematic, 147rotating, 186trigger, 147

Wave number, 98

self organization in nonequilibrrium systems - prigogine

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