FluctuationsandScalinginBiology

T.Vicsek

editor

2001

Contents

Introduction

2

1

Basicconcepts(T.Vicsek)

3

1.1

Fluctuations................................

5

1.1.1

Noiseversus

uctuations.....................

5

1.1.2

Molecularmotorsdrivenbynoiseand

uctuations.......

6

1.2

Scaling...................................

8

1.2.1

Criticalbehaviour.........................

8

1.2.2

Scalingofeventsizes:Avalanches................

9

1.2.3

Scalingofpatternsandsequences:Fractals...........

11

1.2.4

Scalingingroupmotion:Flocks.................

14

2

Introductiontocomplexpatterns,

uctuationsandscaling

17

2.1

Fractalgeometry(T.Vicsek).......................

18

2.1.1

Fractalsasmathematicalandbiologicalobjects........

19

2.1.2

De�nitions.............................

22

2.1.3

Usefulrules............................

23

2.1.4

Self-similarandself-aÆnefractals................

25

2.1.5

Multifractals

...........................

27

2.1.6

Methodsfordeterminingfractaldimensions

..........

28

2.2

Stochasticprocesses(I.Der�enyi)

....................

31

2.2.1

Thephysicsofmicroscopicobjects

...............

31

2.2.2

KramersformulaandArrheniuslaw

..............

33

2.3

Continuousphasetransitions(Z.Csah�ok)................

35

2.3.1

ThePottsmodel.........................

38

2.3.2

Mean-�eldapproximation

....................

38

Bibliography

..................................

41

1

2

CONTENTS

3

Self-organisedcriticality(SOC)(Z.Csah�ok)

45

3.1

SOCmodel................................

46

3.2

Applicationsinbiology..........................

47

3.2.1

SOCmodelofevolution

.....................

48

3.2.2

SOCinlungin

ation.......................

54

Bibliography

..................................

57

4

Patternsandcorrelations

59

4.1

Bacterialcolonies(A.Czir�ok)......................

59

4.1.1

Introduction............................

59

4.1.2

Bacteriaincolonies........................

60

4.1.3

Compactmorphology.......................

67

4.1.4

Branchingmorphology......................

78

4.1.5

Chiralandrotatingcolonies...................

96

5

Microscopicmechanismsofbiologicalmotion(I.Der�enyi,T.Vicsek)105

5.1

Characterisationofmotorproteins

...................105

5.1.1

Cytoskeleton...........................105

5.1.2

Musclecontraction........................109

5.1.3

Rotarymotors...........................113

5.1.4

Motilityassay...........................115

5.2

Fluctuationdriventransport.......................118

5.2.1

Basicratchetmodels.......................120

5.2.2

Briefoverviewofthemodels...................121

5.2.3

Illustrationofthesecondlawofthermodynamics

.......122

5.3

Realisticmodels..............................123

5.3.1

Kinesin

..............................123

Bibliography

..................................131

6

Collectivemotion

141

6.1

Flocking:collectivemotionofself-propelledparticles(A.Czir�ok,T.

Vicsek)...................................141

6.1.1

Modelsandsimulations......................142

6.1.2

Scalingproperties.........................143

6.1.3

FurthervariantsofSPPmodels.................152

6.1.4

Continuumequationsforthe1dsystem.............163

6.1.5

Hydrodynamicformulationfor2D................165

6.1.6

Theexistenceoflong-rangeorder................170

Bibliography

..................................172

Chapter1

Basicconcepts

Thecomplexityofbiologicalsystemsismanifestedinmanyways.Hereweshall

considerthoseaspectsoflifewhichinvolverandom

uctuationsandahierarchical

underlyingstructureresultinginapowerlawdependenceofthevariousquantities

characterisingthesesystems.Aswillbeshown,thesetwofeatures{

uctuationsand

scaling{arefrequentlyandintimatelyrelated,althoughinsomecasestheyappear

independentlyofeachother.

Are

uctuationsanimportant,inherentingredientoflife?Whatistheirorigin

andimpact?

Theseareimportantquestionswhichhavetobeaddressedbeforewe

leadthereadertothemoreadvancedpartsofthetext.

Itisalmostatrivialstatementthatnearlyallprocessesinbiologyinvolverandom-

nessbeyondanegligiblelevel.Ontheotherhand,exceptsomearti�cialsituations,

allphenomenainnaturehavetheelementsofstochasticityinthem.Thereisno

suchathingasanoiselessorabsolutelydeterministicsystemandthisistruefrom

thebehaviourofgalaxiesdowntoelementaryparticles.Intheintermediaterangethe

presenceoftemperature

uctuations,thevariousnonlinearities/instabilitiesrepresent

themainsourceofrandomness.Thisisknowntobesoandisunderstoodinmany

casesfornon-livingsystems.

Biologicalsystemsarenotexceptionsfrom

thesamerule.Perhapsthemost

typicalfeatureofabiologicalobjectorsignalisthattheysimultaneouslypossess

somespeci�cpatternanddeviationsaroundthesepatterns.Ifwetakeasanexample

ananimal,letussayadog,wecanconsiderthefollowingillustrationsoftheabove

statement:i)notwodogsareexactlythesame,buttheyaresimilarinmanyways,

ii)theyhavetypicalreactions,butneverreactinaperfectlyidenticalway,iii)their

heartbeatsalmostregularly,butacloseranalysisshowsspeci�c

uctuationsaround

theaverageheartbeatrate,iv)ifwestudytheelectricsignals(withthehelpof

anelectrode)oftheirbrainweseeanalmostrandomlylookingseriesofspikes,v)

3

4

CHAPTER

1.

BASIC

CONCEPTS

ifwehappenedtoseethemotionoftheindividualcellsorevenlargemoleculesin

theirbody(e.g.,bloodcells,sperms,RNAmolecules)wewouldobserveanerratic

behaviouraroundanaveragetendency.Thelistofsuchexamplescouldbecontinued

forlong.

Aswillbediscussedlater,inmanycasestheabovementionedstochasticchanges

or

uctuationsarenotcompletelyrandomandcanbeassociatedwithpowerlawsor

scaling.

Inshort,ifasystemismadeofmanyinteractingunits,speci�cstatisticalfeatures

involving

uctuationsandscalingemerge.

Thereareafewimportantaspectsof

uctuationswhentheoriginofanapparently

randombehaviourisrelativelywellunderstood.

I) Microscopicobjectsaresubjecttothesocalledthermal

uctuations.Itisafun-

damentalfeatureofallsystemsthatiftheyhaveawellde�nedtemperatureT,than

eachmicroscopicparticle(atomormolecule)inthemhasinaverageanamountof

kineticenergy

1 2kT

(kistheBoltzmannconstant)foreachdegreeoffreedom(modeof

motion).Amorecomplexmoleculehasmanydegreesoffreedom,andisalsosubject

to"kicks"fromtheneighbouringmoleculesmovingwithavelocitycorrespondingto

theirkineticenergy.Asaresult,anindividualmoleculeinteractingwithmanyothers

followsamoreorlessrandomtrajectoryduetothemanysubsequent,randomlyoccur-

ring"kicks"fromtheirneighbours.Itisabeautifulsubjecttounderstandhowsuch

uctuationsaretamedbythespeci�cprocesses(di�usion,enzymaticinteractions)

insidelivingcellsandresultinanorganised(withsome

uctuations)behaviour.

II)

Whenmanysimilar,butnotnecessarilymicroscopicobjects(biologicalornon-

living)arepresentinasystemtherearefurtherreasonstoconsidertherandomaspect

ofthebehaviour.

Ingeneral,ifmanymovingobjectsinteract,themotionoftheindividualobjects

isboundtoberandom-like(likeintheabovementionedmicroscopiccase),evenif

themotionortheinteractionisdeterministicandweconsidermacroscopicobjects.

Thetrajectoriesaresubjecttomanychangesofvaryingdegreeandinsteadoflooking

atsuchprocessesasdeterministicitisconceptuallymoreusefultoassumethatthe

motionisstochasticanditisthestatisticalfeaturesoftheassemblyofobjectswhich

shouldbedeterminedwhendescribingthesystem.Itisnotonlythedirectionof

motionwhichcanbechangedduringinteraction.Forobjectswithadirectedness

theirdirectioncanalsobemodi�edduetosomedirection-dependentinteraction.

Now,asaresultoftheseinteractions,variousspontaneousprocessesmaytake

inthesystem:forattractiveforces,aggregatesarelikelytoform,orgroupswith

thesamedirectednessoftheirmembersmayappear.Schoolsof�shisacommon

1.1.

FLUCTUATIONS

5

exampleforaggregation.However,duetothecomplexnatureofasystemwithmany

objectsmovingbothrandomly,andorderly(inagroup,butwithsomeperturbations)

anotherkindof

uctuations,arandomdistributionofgroupsizesisproduced.Many

timesthedistributionofgroupsizesfollowsapowerlaw,orinotherwords,ascaling

distribution.

III)

Non-linearitiesareknowntoleadtoaverycomplexbehaviourwhich{especially

inthepresenceofthermal

uctuations{canbeconsideredasrandom.Acommon

spatialexampleistheformationofbranchingpatternsunderconditionsleadingtothe

unstablegrowthofmoreadvancedbranches.Inthesecasesthesmallestperturbations

arelikelytoresultinanew,quicklygrowingside-branchandthestructureattains

thewellknownbranchingmorphologysocommoninbiology(e.g.,treesorblood

vesselnetworks).Althoughsuchnetworkspossesssomespeci�cfeaturestypicalfor

thegivenbiologicalobject,theyarealsoirregular.Inaddition,aswillbeshownthey

haveaspeci�chierarchicalstructurebestdescribedbyfractalgeometry.

InthefollowingpartsofthisintroductorychapterIbrieydiscussthebasiccon-

ceptsrelatedto

uctuationsandscalinginbiologyandsummarisethemostimportant

�ndingsobtainedintherelatedinvestigations.

1.1

Fluctuations

1.1.1

Noise

versus

uctuations

Theoriginof

uctuationscanwidelyvary.Inmostofthecases,however,theyare

duetotheabovementioned"thermalnoise"orerraticmotionofmicroscopicparti-

cles.Typically,noiseisnotcorrelatedwhichmeansthatthevalueofthe

uctuating

quantityF

atthelocationr

attimetdoesnotdependonitsvalueatadi�erent

locationandatanearliermomentoftime.Symbolicallywewritethatthecorrelation

functionhastheformofadeltafunction

c(r)=hF(r;t)F(r

0;t

0)i�hF(r;t)ihF(r;t)i=CÆ(r�r

0;t�t0)

wheretheÆfunctionisequaltozeroforanynon-zerovaluesofitsargumentsand

C

issomeconstant.Theaveraging(denotedbyh:::iismadeoverallvaluesofthe

arguments.Thisexpressionholdsfortheuncorrelatedwhiteorshotnoise,whilefor

correlated

uctuationsc(r)hasmorecomplexforms.

Fluctuationscanbemorecomplexthanjustwhitenoise.Manytimestheyrepre-

sentaninherent,characteristicfeature(reactiontowhitenoise)ofthesystemitself.

6

CHAPTER

1.

BASIC

CONCEPTS

Forexample,duetothethermal(white)noisethe

uctuationsinthetotalmagnetisa-

tionofaferromagnetnearitscriticalpointcanstronglyincreaseandexhibitspeci�c

correlations.

Noiseand

uctuationsplayacentralroleinorderingphenomena.Asystemof

manyinteractingunitswithaninteraction"trying"toforcetheunitstobehaveinthe

samewayinthepresenceofstrongexternal

uctuations(noise)maynotbeableto

order.Ontheotherhand,asthemagnitudeofnoisedecreases(e.g.,thetemperature

islowered),theobjectsinthesystemmayalreadyassumetheircommonorcollective

patternofbehaviour,forexample,theyspontaneouslydevelopacommondirection

ofmotion,or�ndtheirrightplaceforacrystallinestructure.Thesenoisedriven

transitionswillbediscussedbrieyinthescalingpartofthisintroductorychapter.

Inthefollowingwhenonlytheirstochasticnatureisrelevantweusenoise,

uctu-

ationsandrandomperturbationsassynonyms.However,randomchangesappearing

asaninherentbehaviour(response)ofthesystemitselfwillalwaysbecalled

uctu-

ations.

Afurtheraspectof

uctuationsinvolvestransportprocessesinthepresenceof

noise.Interestingly,evenuncorrelated

uctuations(thesearerandomchangeswithout

anytendencies)mayresultinabehaviourwithawellde�nedtendency.Thishappens

inthecaseofmolecular

motors,wherewhitenoiseassiststhemotorproteinsto

proceedalongspeci�cintracellulartracks.Ontheotherhand,whitenoisealone

cannotproducecurrents,thiswouldcontradictthesecondlawofthermodynamics.

Herewementionthebasic�ndingsaboutthisfascinatingnewdirectioninbiological

physics.

1.1.2

Molecularmotorsdrivenbynoiseand

uctuations

Intheinorganicworldtransportalwaystakesplacealongamacroscopicgradientof

apotential(orananalogousquantity).Thingsfalldownduetothegravitational

forcewhichcanbeobtainedfromthederivativeofthegravitationalpotential.Even

theglobaltransportofmicroscopicobjectssuchasmoleculestakesplacealongthe

extendedgradientofthesocalledchemicalpotential.Forexample,particlestendto

di�usefromadenserregiontoalessdenseonemakingamacroscopicoveralldistance

ifthegradientofthedensityextendsoverthatdistance.Electronsmoveinawire

followingthegradientoftheelectricpotentialwhichislargeratoneendofthewire

thanattheotherend.

Thisisnothowtransportisrealizedinmostofthebiologicalsystems.Theabove

mechanismtendstobringasystemintoamotionlessstate:astheobjectsmovealong

thepotentialgradienttheysimultaneouslydecreasethevalueoftheoverallgradient.

Forexample,di�erenceintheconcentration(drivingthedi�usionaltransport)de-

1.1.

FLUCTUATIONS

7

creasesintimeastheparticlesdi�usetothespotsofsmallerconcentrations(and

increasethedensityatthesespots).Instead,lifeisaboutgeneratingdi�erences,

buildingstructuresfromalesspatternedenvironment.

Onemechanismfordoingthisismotionalongperiodic,butlocallyasymmetric

structures.Imagineasawtooth-like(orratchet)potential:itispiece-wiselinear,

withtwodi�erentgradients(slopes).Wealsohaveaparticleinthispotential,most

ofthetime"sitting"inoneoftheminima("valleys").Now,ifwepullthisparticle

periodicallybackandforthinthissawtooth-likepotential,thefollowingcasesare

possible:i)theforceislargeenoughfortheparticletobepulledoutfromaminimum

bothtoleftandrightii)theforceisstrongenoughonlytopullouttheparticleinthe

directionofthesmallergradient(steepness),iii)theparticleisnotpulledoutfrom

thevalleybecausetheforceistooweak.

Obviously,caseii)establishesasituationinwhichatransportispossiblewithout

anyglobalpotentialdi�erence:theparticlemovesinthedirectionofthesmaller

slopealthoughtheforceactingonitiszeroinaverage(actsbackandforth).Inthis

wayparticlescanbecollectedatoneendofatrackwithsuchapotential,thus,a

concentrationdi�erence,i.e.,structurecanbebuiltup.

However,lifeisnotsosimple.Suchprocessesoccuronthemolecularlevel,where

uctuationsareverystrongfortworeasons:a)Theparticle(calledmotorprotein)

iskickedbytheothermoleculesinthesystem

randomly,inanoisy,uncorrelated

mannerallthetime,b)theperiodic,deterministicbackandforthdrivingcannotbe

establishedinamicroscopicenvironmentaswell:instead,theenergysuppliedbythe

socalledATPmoleculesandprovidingtheconformationalchangesofthemolecular

motorresultinginitstendencytomovebackandforthalsoarrivesstochastically(the

ATPmoleculesare"consumed"attimesfollowingaPoissondistribution).

Thepictureemergingisthefollowing:Apossible,simpli�edrepresentationof

biologicalmotionisthatofthemotionofaBrownianparticleinanasymmetric

periodicpotential.Thecorrespondingequationhavetoaccountforthestochastic

natureoftheprocess:bothforthewhitenoisecomingfromtheenvironmentandfor

theirregularnatureoftheenergyinput.TherelatedLangevinequationapproachis

discussedindetailsinchapter5.

Thereisoneinterestingpointhere:inthecaseofmotionalongasymmetric

periodicpotentialsnoisemayplayaroleenhancingthetransport.Thisisabit

paradoxical,wearemoreusedtothenotionthatrandom

perturbationstypically

destroytendencies.Inthecaseofmolecularmotors,however,itmayhappen,that

addingwhitenoiseresultsinstrongercurrent.Theeasiestwaytoshowthisisthe

following:Imaginecaseiii)whentheexternalforcepullingtheparticlebackandforth

isnotstrongenoughfortheparticletoescapeinanyofthetwodirections.Now,if

weaddnoise(arandomlychangingsmallamountofextraforce)insomecasesthe

8

CHAPTER

1.

BASIC

CONCEPTS

overallforcemayexceedthecriticalvaluenecessarytopullouttheparticlefroma

valley.Thiswillhappenmorefrequentlyinthedirectionofthesmallerslope,since

thecriticalamountofforceissmallerinthatdirection,andtherewillbeanoverall

currentinthedirectionofthesmallerslope.Thisiswhysuchsystemsarealsocalled

thermalratchets.

Anotherinterestingvariantofthissituationiswhenweconsideronesingleforce

(insteadofthesumofadeterministicperiodicbackandforthactingforceandan

uncorrelatedwhitenoiseone)changingstochastically.Now,ifthissingle

uctuating

forceiscompletelyuncorrelated,orinotherwords,thermalorwhitenoise,thenno

globaltransportispossible.Ifthe

uctuationsarethermalnoise-like,thanthesystem

isinequilibriumandnotransportispossibleinthermalequilibrium.

Ontheotherhand,ifthe

uctuatingforce(noise)iscorrelated,transportalready

becomespossible.Theprobabilitytoleaveinthelesssteepdirectionwillbestilllarger

thanintheoppositedirection.Inthisway,ourratchet"recti�es"the

uctuations,it

isabletomakeuseofitsnon-whitepart.

1.2

Scaling

AquantityF

scalesasafunctionofitsargumentx

ifchangingtheargumentbya

factor(e.g.,changingxforAx)doesnotchangetheformofthefunctionaldependence

ofF

onx(apartfromaconstantfactor).Thisistriviallysoforafunctionoftheform

ofapowerlaw,butisnot,asarule,trueforotherfunctions.Forexample,F

=x

2

scalesbecauseF

0

=F(Ax)=A

2x

2

=A

2F(x),whileF

=log(x+B)doesnotscale

becauseinthiscaseF

0

=F(Ax+B)cannotbereducedtoaformcontainingF(x)

andaconstantfactoronly.

Thescalingquantitiesweshallconsideraremostlyofstochasticnature.Thus,

thespeci�cfunctionaldependenceswillbevalidfortheaverageofthegivenquan-

titiesasafunctionoftheirarguments.Eachrealizationofsomestochasticprocess

hasa

uctuatingoutcome,butmakinganaverageoverseveralprocesses,orovera

singleprocesshavingsequences

uctuatingaroundanaveragecanprovidetheproper

estimateofthequantityofinterest.

1.2.1

Criticalbehaviour

Perhapsthemosttypicalcollectivephenomenaexhibitedbyanassemblyofmany

interactingparticlesarethesocalledphasetransitions,when,asafunctionofanex-

ternalparameter(liketemperature),theparticlescollectivelychangetheirbehaviour.

1.2.

SCALING

9

Forexample,duringfreezingallofthemoleculesofa

uidmovetoaspeci�cposition

sothattheresultingstructurebecomesacrystalwithregularmicroscopicstructure.

Inthevicinityofsuchtransitionsinterestingspatialandtemporalcorrelations

canbeobservedinthesystemsandthesefeatureswillberelevantfromthepointof

themajorityofthetopicsdiscussedinthisbook.

Inparticular,duringsecondorderphasetransitionsthesocalled"criticalstate"

(orcriticalphenomena)canbeobservedinwhichtheordinarilyexponentialfunc-

tionaldependencesarereplacedbyalgebraic(powerlaw)dependenceoftherelevant

quantitiesontheirparameters.Apowerlawdependenceofthequantityn(s)(e.g.,

thenumberofschoolscontainings�sh)isofthefollowingformn(s)�

s�

�

;where�

expressesproportionality,and�

issomeexponent.Thepowerlawdependenceisvery

special:forexample,apowerlawdecayofthenumberofclusters(schoolsof�sh)as

afunctionoftheirsize(numberof�shinaschool)meansthatverylargeclustersmay

occurwithaprobabilitywhichisnotnegligible(thisprobabilitywouldbeextremely

smallifthenumberofclusterswoulddecreaseexponentiallywithgrowingclustersize,

asitdoesforregularstates).Ifaquantitychangesaccordingtoapowerlawwhenthe

parameteritdependsonisgrowinglinearly,wesayitscales,andthecorresponding

exponentiscalledcriticalexponent.

Whyaresuchstatescalledcritical?Becausetheyareextremelysensitivetosmall

changesorperturbations.Ifahumanbeingisinacriticalstateitmeanshisorher

statecangetworseveryeasily.Inthecaseoflatticemodelsasmallchangeinthe

temperaturemayleadtothequickcollapseorbirthofverylargeclusters.Duringthe

lasttwodecadesstatisticalphysicistshaveworkedoutdelicatetheoriesandmethods

tointerpretthebehaviourofsuchcriticaltransitionsandstatesandinthefollowing

weshallconsidertheapplicationoftherelatedconceptstobiologicalphenomena

involvingmanysimilarunits.

Theimportantpointisthatscalingtypicallyinvolvesuniversality:insteadofpar-

ticleswecanimaginesimilarorganisms.Iftheinteractionamongtheseorganismsis

relativelysimple,andisanalogoustothosewhichproducescalingorphasetransitions

innon-livingsystems,thanwecanexpectthesametypeofbehaviourinsuchsystems

oflivingentitiesaswell.

1.2.2

Scalingofeventsizes:Avalanches

Asmentionedabove,scalingcanbeobservedduringequilibriumphasetransition,but

inthefollowingweshallarguethatapowerlawdependenceofthevariousimportant

quantitiescanemergeinthenon-equilibriumworld(oflife)aswell.Infact,itisinthe

non-equilibrium

statewhenstructurescanemergespontaneouslyfromanoriginally

homogeneousmedium.

10

CHAPTER

1.

BASIC

CONCEPTS

Aparticularandimportantdeparturefromequilibriumiswhenthesystem

is

"slowlydriven"toastationarystate.Slowdrivingmaymeanthegradualaddition

ofsomequantity(energy)toaasystem

whichmayalsoloosethisenergydueto

interactions.Iftheinteractionbetweentwopartsofthesystemissuchthatachange

exceedingacriticalvalueofthegivenquantityinoneunitresultsinasimilarex-

ceedingofthesamecriticalvalueintheneighbouringunit,thanlarge,avalanche-like

seriesofchangesmaytakeplaceinthesystemwhenitisclosetoa(critical,balanced)

state.Asthisstateisbothspontaneouslyemergingandcritical,theassociatedphe-

nomenoniscalledInthisstatethesystemisverysensitiveto

uctuations,sincea

smallperturbationmayleadtoalargeavalanche.Inthissensethesystemisina

criticalstate.Notalloftheavalanchesarelarge,themajorityofthemaresmall,

buttheprobabilityofhavingalargeavalanchedoesnotgotozeroveryquicklywith

theavalanchesize.Inmanysuchslowlydrivensystemsscaling(powerlaw)ofthe

avalanchesizedistributioncanbeobserved.Avalanchesaresometimesverylarge(as

weknowfromthenewsonskiingareas,butinthiscontextanearthquakeisalso

anavalanche)andtheyarethesocalledbigeventsinthetheoryofslowlydriven

systems.

Thesimplestexampleisthatofasandpile.Imaginethatweaddgrainsofsand

toagrowingpile:asthegrainsaredropped,mostofthetimethesurfaceofthe

pilebecomesonlyslightlyrearranged,however,timetotimethenewgraintriggers

alongseriesofevents:grainsrollingdowntheslopedragmanymoregrainswith

them.Asimplemodelalongalinesegmentwouldcontaincolumnsofparticles.

Anewparticleisdroppedatarandomposition.Iftheheightdi�erencebetween

twocolumnsbecomeslargerthan2,thanfromthehighercolumntwoparticlesare

removedandaddedtothetwoneighbouringcolumn.Particlesattheedgeofthe

segmentaredroppedoutfromthesystemcompletely.

InthisbooktwoexamplesforbiologicalSOCarediscussed.Thestructureof

thelungissuchthatitcanbebroughtintoanalogywiththesandpilemodel.The

airenteringthelunghastogothroughasequenceofairwayseachopeningifthe

pressureexceedsacriticalvalue.Byforcingtheairgraduallyentertheexperimentally

investigatedlung,largejumpsintheterminalairwayresistancehavebeenobserved.

Thesejumpscorrespondedtoavalanches:tothesubsequentopeningofalargeset

ofairwaysinashorttime.Thedistributionofjumpsfollowedapowerlaw.The

observationofstrongly

uctuatingextinctionratesandthecorrespondingSOCrelated

theoryisalsodiscussedinchapter3.

1.2.

SCALING

11

1.2.3

Scalingofpatternsandsequences:Fractals

Natureisfullofbeautifulcomplexshapeswhicharefarmoreintricatethanthe

idealisedformsproposedbyEuclidmorethantwothousandyearsago.Thisispar-

ticularlytrueforthelivingworldwherecomplicatedstructuresaregeneratedduring

embriogenesis.Manyofthesepatternsarerandombranchingnetworks;examplesin-

cludetrees,thenetworkofbloodvessels,airwaysinthelung,neuralnets,etc.These

highlyhierarchicalpatternscanbebestinterpretedintermsoffractalgeometry.

Fractalsarefascinatinggeometricalobjectscharacterisedbyanon-trivialfrac-

tionaldimension.ImagineagrowingpatternwhosemassM

(thenumberofparticles

itcontains)increasesslowerthanthed-thpowerofitsradiusR,wheredisthedi-

mensionofthespaceinwhichthepatternisdeveloping.Thisisclearlydi�erentfrom

thecaseofhomogeneousstructuresthatweareusedto.ForfractalsM(R)�

RD

;

whereD

iscalledthefractaldimensionsinceinmanycasesitisnotaninteger,buta

fractionalnumberlessthand.Iftheaboverelationistrueforapattern,itisbound

tobeself-similarinthesensethatasmallpartofitlooksthesameasthewhole

structureafteritisexpandedisotropically.Fordeterministicmathematicalfractals

theblownuppiecelooksexactlythesameasthewholeobject.Forrandompatterns

self-similarityissatis�edinastochasticmanner.Thefractaldimensioncanalsobe

de�nedthroughtheexpressionc(r)=

1 N

Pr

0

�(r)�(r+r

0)�

rD

�

d

wherec(r)describes

thedensity-densitycorrelationswithinthepatternand�(r)isequaltounityifthereis

aparticleatthepositionr,anditisequaltozerootherwise.Forisotropicstructures

thecorrelationfunctionc(r)isequivalenttotheprobabilitythatone�ndsaparticle

belongingtotheclusteratadistancer=jr�r0jfroma�xedpointonthecluster.

Inthiscaseanaveragingcanbemadeoverthedirectionsaswell.

Themeaningoftheabovestatementsisthatfractalscanbelookedatasstructures

exhibitingscalinginspacesincetheirmassasthefunctionofsizeortheirdensityas

afunctionofdistancebehaveasapowerlaw.

Self-aÆnestructuresrepresentanothertypeoffractals.Forsuchobjectsasmall

partofthefractalmustbeenlargedinananisotropicwaytomatchtheentirepattern.

Forexample,ifthefractalisembeddedintotwodimensions,forself-aÆnefractals

oneachievesmatchingbyrescalingthesizehorizontallyandverticallybydi�erent

factors.

FractalBacterialColonies

Perhapsthebestde�nedbiologicalsystemsexhibitingfractalgrowtharebacterial

colonies.Byacarefulcontroloftheexperimentalconditionsithasbeenpossibleto

obtainwellreproducibleresultsonthedevelopmentofcomplexbranchingpatterns

madeofmanymillionsofbacteriaastheymultiplyonthesurfaceofanagar(gel)

layerinaPetridish.Therelatedbeautifulresultsarediscussedinchapter4.

12

CHAPTER

1.

BASIC

CONCEPTS

Typically,bacterialcoloniesaregrownonsubstrateswithahighnutrientlevel

andintermediateagarconcentration.Undersuch"friendly"conditions,thecolonies

developsimple(almoststructureless)compactpatternswithrelativelysmoothen-

velope.Thisbehaviour�tswellthecontemporaryviewofthebacterialcolonies

asacollectionofindependentunicellularorganisms.However,innature,bacterial

coloniesregularlymustcopewithhostileenvironmentalconditions.Whathappens

ifwecreatehostileconditionsinaPetridishbyusing,forexample,averylowlevel

ofnutrientsorahardsurface(highconcentrationofagar),orboth?Thebacterial

reproductionrate,whichdeterminesthegrowthrateofthecolony,islimitedbythe

levelofnutrientsconcentrationavailableforthebacteria.Thelatterislimitedbythe

di�usionofnutrientstowardsthecolony.Hence,thegrowthofthecolonyresembles

di�usion-limitedgrowthininorganicsystemsleadingtofractalpatterns.

Di�usion-limitedgrowthleadstorandombranchingpatternsbecauseofthefol-

lowinginstability:ifagivenpartofthegrowingsurfaceisabitmoreadvancedthan

thesurroundingregion,thispartwilladvancefasterthantheneighbouringpartsof

thecolony,becauseitwillbeclosertothesourceofthenutrientdi�usingfromthe

outerregionsofthePetridish.Inturn,partslaggingbehindtendnottogrowany

moresinceinthoseregionsnonutrientwillbeavailableasthenutrientdi�using

towardsthecolonywillbeconsumedbytheadvancedpartsofthecolony.Thisis

positive(negative)feedback:aprotrusiongrowsfaster(andproducesabranch)the

screenedfjordsstoptogrowcompletely.Theresultingpatternhasaradiallygrowing

tree-likestructure.

Inrealitythesituationissomewhatmorecomplexsincethebacteriacancom-

municatethroughchemotaxis.Theyareabletopassoninformationabouttheiren-

vironmentandincrease/decreasethegrowthrateatotherpointsinthecolony.The

communicationenableseachbacteriumtobebothactorandspectator(usingBohr's

expressions)duringthecomplexpatterning.Thebacteriadevelopedaparticle-�eld

duality:eachofthebacteriumisalocalised(moving)particlewhichcanproducea

chemicalandphysical�eldarounditself.Forresearchersinthepatternformation

�eld,theabovecommunicationregulationandcontrolmechanismopensanewclass

oftantalisingcomplexmodelsexhibitingamuchricherspectrumofpatternsthanthe

modelsforinorganicsystems.

Allthiscanbeinvestigatedbyconstructingsuitablecomputermodels.Inthe

correspondingcalculationsanumberoffactorsaretakenintoaccount(seechapter

4):goodagreementwiththeexperimentalobservationscanbeachievedbyassuming

anutrientdependentmultiplicationrate,di�usionalmotionofthebacteriaonthe

agarsurface,chemotacticsignalling,etc.Thesesimulationsaredi�erentfromthe

commonapproachesinphysicsandbiology.Physicistsprefertobuildsimplemodels

ignoringmanyofthedetailsandlookforuniversalbehaviour.Biologistsmostly

1.2.

SCALING

13

usespeci�cmodelsre

ectingthebiologicaldetailsofthesystemunderinvestigation.

Themodelsusedtomimicbacterialcolonydevelopmentinthecomputerinterpolate

betweentheseapproachesandareaimingat�ndinguniversalbehaviourtakinginto

accountmostofthebiologicallyrelevantdetails.

Correlationsinthegeneticcode

Onepossiblerepresentationofthevastinformationstoredintheextremelylong

sequencesofDNAdataisarandomwalkbuilttocorrespondtosuchsequences.In

thisapproachDNAismappedontoaprocesswhichcanberegardedasawalk:each

ofthefour\letters"ofaDNAsequenceisidenti�edwithastepinagivendirection.

Then,thespeci�cfeaturesofthiswalkcanbeanalysedusingmethodsborrowedfrom

statisticalphysics.

Giventhewalkonecanlookforcorrelations.Twoseriesofdata(XandY)are

correlatedifthereisarelationshipbetweenthecorrespondingelementsoftheseries.

Whensearchingforcorrelationswithinasinglesequenceofdatawecanaskhowthe

valueXiisrelatedtothevalueXi+j.Bycomparingthetwovalueswiththeaverage

ofX

onecangetinformationaboutthequestionwhethertwovaluesinthedataset

separatedbyjelementsarecorrelated.

Anordinaryrandomwalkhasnolong-rangecorrelations.Oneofthemostrelevant

questionsonecanraiseinthecontextofDNAsequencesisthelocationofcoding

andnon-codingpartsinthegenome.Inthecasethesetwokindsofsub-sequences

havedi�erentkindsofcorrelationswemaybeabletodi�erentiatebetweencoding

andnon-codingpartwithoutanypriorknowledgeaboutthesequences.Indeed,it

hasbeenshownthattherandomwalkscorrespondingtonon-codingpartshavelong-

rangecorrelationsincontrasttothecodingparts(whichhaveshort-rangecorrelations

only). A

sanalternativetotheDNAwalk,thesymbolsequencecorrespondingtoaDNA

moleculecanberegardedasawrittentextcomposedbyusingfourletters.Sincewe

donotknowthe"language"ofthetextwehavetotoapplymethodsdevelopedfor

analysingwritten(natural)textsofunknownorigin.Inparticular,onecanaskthe

questionwhethertwotextswerewritteninthesamelanguageornot.Weexpect

largercorrelationsbetweentextofthesameorigin(language).Heretwosequences

arecorrelatedifthescalarproductofthetwoappropriatelyde�nedvectors(corre-

spondingtothem)hasavaluedi�erentfromthatitwouldhavefortwouncorrelated

sequences.Inthe�rstapproximationhismethodislanguageinsensitive.

WhenapplyingthevectorspacetechniquetoDNAsequencesinawaywelookat

DNAasanencodedtextwritteninanunknownlanguage,still,weexpecttolocate

correlationsbetweenpartsofthesequencesduetosimilaritiesintheirunderlying

structure.Incasethelanguageofthecodingpartsisdi�erentfromthenon-coding

oneswegetahighervalueforthecorrespondingscalarproduct(asithasbeendemon-

14

CHAPTER

1.

BASIC

CONCEPTS

stratedinsomecases).

1.2.4

Scalingingroupmotion:Flocks

Groupmotion(ocking)isabeautifulphenomenonmanytimescapturingoureyes.

Here

ockingisunderstoodinageneralsenseoftheword,includingherdingof

quadrupeds,schoolingof�sh,etc).Inthelastchapterofthisbookweaddress

thequestionwhethertherearesomeglobal,perhapsuniversaltransitionsin

ocking

whenmanyorganismsareinvolvedandsuchparametersasthelevelofperturbations

orthemeandistanceoftheorganismsischanged.

Everyonehasexperiencedhowaninitiallyrandomlydirectedgroupofbirdsfeed-

ingonthegroundisspontaneouslyorderedintoawellorganised

ockwhentheyleave

becauseofsomeexternalperturbation.Thisorderingisahighlynon-trivialquestion

sinceinahuge

ockofseveralhundredorthousandbirdsusuallythereisno"leader"

bird(wedonotconsiderheretheV

shapedorotherstructured

ightofsomelarge

bodiedbirds)andnotevenallbirdscanvisuallyinteract.Still,thewhole

ockselects

awellde�neddirection.Suchorderingisfamiliarfromequilibriumphasetransition

ofmagneticsystemsandthecorresponding�ndingsmayprovidecluestotheunder-

standingofthemorecomplexfar-fromequilibriumorderingofmovingorganisms.

Self-propulsionisanessentialfeatureofmostlivingsystems.Inaddition,the

motionoftheorganismsisusuallycontrolledbyinteractionswithotherorganismsin

theirneighbourhoodandrandomnessplaysanimportantroleaswell.Itispossible

todesignsimplecomputermodelswhichsimulatethecollectivemotionandtakeinto

accountthemostrelevantingredientsofthephenomenon.

Asimplemodelofcollectivemotionconsistsofparticlesmovinginone,twoor

threedimensions.Theparticlesarecharacterisedbytheir(o�-lattice)locationx

iand

velocityv

i

pointinginthedirection#i.Toaccountfortheself-propellednatureof

theparticlesthemagnitudeofthevelocityis�xedtov0.Asimplelocalinteraction

isde�nedinthemodel:ateachtimestepagivenparticleassumestheaverage

directionofmotionoftheparticlesinitslocalneighbourhoodwithsomeuncertainty.

Suchamodelisatransportrelated,non-equilibriumanalogueoftheferromagnetic

models.Theanalogyisasfollows:thefunctiontendingtoalignthespinsinthesame

directioninthecaseofequilibriumferromagnetsisreplacedbytheruleofaligning

thedirectionofmotionofparticles,andtheamplitudeoftherandomperturbations

canbeconsideredproportionaltothetemperature.

Inaddition,collectivemotioncanbedescribedbycontinuumequationsaswell.

Thecollectionof"birds"isthenlookedatasparticlesina

uidsubjectto

uctuations

andsatisfyingtheconditionoftryingtomovewithagivenvelocity.

Boththeoreticalapproachesledtotheconclusionthatthereareinteresting,in

1.2.

SCALING

15

casesunexpected(comparedtoequilibriumsystems)transitionsincollectivemotion.

Forexample,ifthenoise(levelofperturbations,correspondingtotemperatureinthe

caseofferromagnets)isdecreased,theoriginallydisordered

ockbecomesorderedin

analogywithsecondorderphasetransitions.Thelevelofglobalorder,the

uctuations

aroundthisorderandseveralrelatedquantitiesallscale,i.e.,behaveaccordingto

powerlawasafunctionofthedistancefromacriticallevelofperturbations.

Pedestriansimulations

Aspecialkindof

ocksisagroupofpeople.Inthelastchapterinterestingappli-

cationsofpedestriansimulationsarediscussed.Justasinthecaseofotherorganisms,

peoplecanberepresentedbyparticles"dressed"bytheappropriateinteractions.Sim-

ulationsofhumansmovingincon�nedplacesleadstoanumberofinterestinge�ects

reproducingrelatedobservations.

Freezingbyheatingisane�ectobservedwhenaparticlesaredriveninopposite

directions.Therelatedsimulationsdemonstratedthatmorenervousorhecticchanges

(heating)ofthedirectionofmotioncancauseabreakdownofaneÆcientpatternof

cooperativeinteractionsand�nallyproduceadeadlock(freezing).Inparticular,this

mayberelevantforpanickingpedestriansinasmokyenvironment,whotendtobuild

upfatalblockings.Thesystemdescribedinthelastchapterconsistsofamesoscopic

numberofdrivenparticleswithrepulsivehard-coreinteractionsmovingintoopposite

directionsunderthein

uenceof

uctuations.Exampleforsuchsystemsispedestrians

walkinginapassage.

Inshort,\freezingbyheating"meansatransitionfroma

uidstate(withself-

organisedlanesofuniformdirectionofmotiontoasolid,crystallised(\frozen")state

justbyincreasingthenoiseamplitude(\temperature").Thisisincontrastto,for

example,melting(whereincreasingthetemperatureincreasestheenergyandorder

isdestroyed)andtonoise-inducedorderinginglassesorgranularmedia,wherein-

creasingthetemperaturedrivesthesystemfromadisorderedmetastablestate(cor-

respondingtoalocalenergyminimum)toanorderedstablestate(corresponding

totheglobalenergyminimum).Instead,\freezingbyheating"showsanincrease

intheorderatincreasingtemperature,althoughthetotalenergyincreasesatthe

sametime.Thecrystallisedstatecanalsobedestroyedbyongoing

uctuationswith

extremenoiseamplitudesgivingrisetoathird,disordered(\gaseous")statewith

randomlydistributedparticles.Thus,withincreasing\temperature"�,wehavethe

atypicalsequenceoftransitions

uid�!

solid�!

gaseous.

Furthervariantsofpedestriansimulationsallowthequantitativeinvestigationof

trailformation,optimisationofpassagegeometries,etc.

InthischapterIhaveattemptedtopresentinasimpli�edmanneraselectionof

concepts,topicsandresultsdiscussedinmuchmoredetailinthemainbodyofthe

book.Fordetailsnecessaryforadeeperunderstandingoftheconceptsand�ndings

16

CHAPTER

1.

BASIC

CONCEPTS

relatedto

uctuationsandscalinginbiologyIadvisethereadertoreadotherparts

ofourbookaswell(wheretherelatedreferencesarealsogiven).

Chapter2

Introductiontocomplexpatterns,

uctuationsandscaling

Inthisbookwemostlyconsidermodelsofreality,since"reality"inthecaseofbiology,

isfartoocomplextoallow

completetheoreticaltreatment.

Ontheotherhand,

wheneveritispossible,wearetryingtopresentmodelsasrealisticaspossiblein

ordertore

ecttheessentialfeaturesofthespeci�cphenomenonoccurringinnature.

Inmanycasesweareconcernedwithsystemsconsistingofmanysimilarobjectsand

thisfeaturehasparticularimplicationsonthekindsofmodelsweconsider.

Variousmodelsallowingexactornumericaltreatmenthavebeenplayinganim-

portantroleinthestudiesofbiologicalprocesses.Becauseofthecomplexityofthe

phenomenaitisusuallyadiÆculttasktodecidewhichofthemanyfactorsin

uenc-

ingtheprocessesarethemostsigni�cant.Inarealsystemthenumberofallpossible

factorscanbetoolarge;thisnumberisdecreasedtoafew

byappropriatemodel

systems.Thus,theinvestigationofthesemodelsprovidesapossibilitytodetectthe

mostrelevantfactors,anddemonstratetheire�ectsintheabsenceofanydisturbance.

Systemsconsistingofmanysimilarunitscanbesuccessfullydescribedintermsof

particles,wherethewordparticlecanstandforamoleculeaswellasformorecomplex

objects,includingorganisms.Then,theparticularnatureofthemodelisgivenby

thefeaturesoftheindividualparticlesandbythewaystheseparticlesinteractwith

eachother.

Thespatialarrangementoftheseparticlesisfrequentlyofmajorinterest.Struc-

turesconsistingofconnectedparticlesareusuallycalledclustersoraggregates.In

mostofthecasestheparticlesareassumedto"exist"onalatticeforcomputational

convenience,andtwoparticlesareregardedasconnectediftheyoccupynearestneigh-

boursitesofthelattice.However,forstudyinguniversalityandrelatedquestions,

o�-latticeorfurtherneighbourversionsofclusteringprocessescanalsobeinvesti-

17

18

CHAPTER2.PATTERNS,FLUCTUATIONS,ANDSCALING

gated.Alatticesitewithaparticleassignedtoitiscalledoccupiedor�lled.An

importantadditionalfeatureincludedintothemajorityofmodelstobedescribedis

stochasticitywhichistypicalforbiologicalphenomena.

Inthischapterwediscussthebasicfeaturesofthecomplexpattersproduced

byawidevarietyofbiologicalgrowthprocesses.Inmanycasesbiologicalgrowth

leadstorandomfractalstructurescharacterisedbyanon-integerdimensionde�ned

below.Inthesecondsectionofthischaptertheprinciplesbehindthemotioninthe

presenceof

uctuationswillbepresented.Sincebiologicalmotionisproducedby

motormoleculesactingonamicroscopicscale,thermalnoiseandotherstochastic

perturbationsareessential.Finally,wediscusscontinuousphasetransitionswhere

theconceptofscalingplaysacentralrole.Scalingmeansapowerlawdependenceofa

quantityonitsargumentand,aswillbedemonstrated,isafeatureshowingupinan

unexpectedlylargeselectionofbiologicalsystems.Weknowfromtheearlystudiesof

scalinginphysicsthatitisafundamentalcharacteristicsofasystem.Thepowerlaw

dependenceofaquantityusuallyinvolvesasimilarbehaviouroftheotherquantities

inasystem;inaddition,theexponent(correspondingtothepowerlaw)istypically

notsensitivetothedetailsuniversal)oftheprocessesconsidered.

2.1

Fractalgeometry

Duringthelastdecadeithaswidelybeenrecognisedbyresearchersworkingindiverse

areasofsciencethatmanyofthestructurescommonlyobservedpossessarather

specialkindofgeometricalcomplexity.Thisawarenessislargelyduetotheactivity

ofBenoitMandelbrot[1]whocalledattentiontotheparticulargeometricalproperties

ofsuchobjectsastheshoreofcontinents,thebranchesoftrees,orthesurfaceof

clouds.Hecoinedthenamefractalforthesecomplexshapestoexpressthattheycan

becharacterisedbyanon-integer(fractal)dimensionality.Withthedevelopmentof

researchinthisdirectionthelistofexamplesoffractalshasbecomeverylong,and

includesstructuresfrommicroscopicaggregatestotheclustersofgalaxies.Objects

ofbiologicaloriginaremanytimesfractal-like.

Beforestartingamoredetaileddescriptionoffractalgeometryletus�rstcon-

siderasimpleexample.Fig.2.1showsaclusterofparticleswhichcanbeusedfor

demonstratingthemainfeaturesoffractals.Thisobjectwasproposedtodescribe

di�usion-limitedgrowth[2]andhasalooplessbranchingstructurereminiscentof

manyshapesofbiologicalorigin.ImagineconcentriccirclesofradiiRcenteredatthe

middleofthecluster.Forsuchanobjectitcanbeshownthatthenumberofparticles

inacircleofradiusRscalesas

N(R)�

RD

;

(2.1)

2.1.

FRACTAL

GEOMETRY

19

Figure2.1:Atypicalstochasticfractalgeneratedinacomputerusingthedi�usion-

limitedaggregationmodel.

whereD

<

disanon-integernumbercalledthefractaldimension.Naturally,fora

realobjecttheabovescalingholdsonlyforlengthscalesbetweenalowerandanupper

cuto�.Obviously,foraregularobjectembeddedintoaddimensionalEuclideanspace

Eq.2.1wouldhavetheformN(R)�

Rd

expressingthefactthatthevolumeofad

dimensionalobjectgrowswithitslinearsizeR

asR

d.Clustershavinganon-trivialD

aretypicallyself-similar.Thispropertymeansthatalargerpartoftheclusterafter

beingreduced\looksthesame"asasmallerpartoftheclusterbeforereduction.This

remarkablefeatureoffractalscanbevisuallyexaminedonFig.2.1,wherepartsof

di�erentsizes(includedintorectangularboxes)canbecomparedfromthispointof

view.

2.1.1

Fractalsasmathematicalandbiologicalobjects

Inadditiontoself-similaritymentionedabove,acharacteristicpropertyoffractals

isrelatedtotheirvolumewithrespecttotheirlinearsize.Todemonstratethiswe

�rstneedtointroduceafewnotions.WecallembeddingdimensiontheEuclidean

dimensiond

ofthespacethefractalcanbeembeddedin.Furthermore,d

hasto

bethesmallestsuchdimension.Obviously,thevolumeofafractal(oranyobject),

20

CHAPTER2.PATTERNS,FLUCTUATIONS,ANDSCALING

V(l),canbemeasuredbycoveringitwithddimensionalballsofradiusl.Thenthe

expression

V(l)=N(l)l

d

(2.2)

givesanestimateofthevolume,whereN(l)isthenumberofballsneededtocoverthe

objectcompletelyandlismuchsmallerthanthelinearsizeLofthewholestructure.

Thestructureisregardedtobecoverediftheregionoccupiedbytheballsincludesit

entirely.Thephrase\numberofballsneededtocover"correspondstotherequirement

thatN(l)shouldbethesmallestnumberofballswithwhichthecoveringcanbe

achieved.ForordinaryobjectsV(l)quicklyattainsaconstantvalue,whileforfractals

typicallyV(l)!

0asl!

0.Ontheotherhand,thesurfaceoffractalsmaybe

anomalouslylargewithrespecttoL.

ThereisanalternativewaytodetermineN(l)whichisequivalenttothede�nition

givenabove.Considerad-dimensionalhypercubiclatticeoflatticespacinglwhich

occupiesthesameregionofspacewheretheobjectislocated.Thenthenumberof

boxes(meshunits)ofvolumeldwhichoverlapwiththestructurecanbeusedasa

de�nitionforN(l)aswell.Thisapproachiscalledboxcounting.

ReturningtotheclustershowninFig.2.1wecansaythatitcanbeembedded

intoaplane(d=2).Measuringthetotallengthofitsbranches(correspondingto

thesurfaceinatwo-dimensionalspace)wewould�ndthatittendstogrowalmost

inde�nitelywiththedecreasinglengthlofthemeasuringsticks.Atthesametime,

themeasured\area"ofthecluster(volumeind=2)goestozeroifwedetermine

itbyusingdiscsofdecreasingradius.Thereasonforthisisrootedintheextremely

complicated,self-similarcharacterofthecluster.Therefore,suchacollectionof

branchestobede�nitelymuch"longer"thanalinebuthavingin�nitelysmallarea:

itisneitheraone-noratwo-dimensionalobject.

Thus,thevolumeofa�nitegeometricalstructuremeasuredaccordingtoEq.2.2

maygotozerowiththedecreasingsizeofthecoveringballswhile,simultaneously,its

measuredsurfacemaydivergefollowingapowerlawinsteadofthebetterbehaving

exponentialconvergence.Ingeneral,wecallaphysicalobjectfractal,ifmeasuringits

volume,surfaceorlengthwithd,d�

1etc.dimensionalhyperballsitisnotpossible

toobtainawellconverging�nitemeasureforthesequantitieswhenchanginglover

severalordersofmagnitude.

Itispossibletoconstructmathematicalobjectswhichsatisfythecriterionof

self-similarityexactly,andtheirmeasuredvolumedependsonleveniflor(l=L)

becomessmallerthanany�nitevalue.Fig.2.2givesexampleshowonecanconstruct

suchfractalsusinganiterationprocedure.Usuallyonestartswithasimpleinitial

con�gurationofunits(Fig.2.2a)orwithageometricalobject(Fig.2.2b).Then,in

thegrowingcasethissimpleseedcon�guration(Fig.2.2a,k=2)isrepeatedlyadded

2.1.

FRACTAL

GEOMETRY

21

k=3

k=0

k=1

k=2

Figure2.2:Fig.2.2ademonstrateshowonecangenerateagrowingfractalusingan

iterationprocedure.InFig.2.2bananalogousstructureisconstructedbysubsequent

divisionsoftheoriginalsquare.Bothproceduresleadtofractalsfork!

1

withthe

samedimensionD

'

1:465[3].

toitselfinsuchawaythattheseedcon�gurationisregardedasaunitandinthe

newstructuretheseunitsarearrangedwithrespecttoeachotheraccordingtothe

samesymmetryastheoriginalunitsintheseedcon�guration.Inthenextstage

thepreviouscon�gurationisalwayslookedatastheseed.TheconstructionofFig.

2.2bisbasedondivisionoftheoriginalobjectanditcanbewellfollowedhowthe

subsequentreplacementofthesquareswith�vesmallersquaresleadstoaself-similar,

scaleinvariantstructure.

Onecangeneratemanypossiblepatternsbythistechnique;thefractalshown

inFig.2.2waschosenjustbecauseithasanopenbranchingstructureanalogousto

manyobservedbiologicalfractals[3].Onlythe�rstcoupleofsteps(uptok=3)of

theconstructionareshown.Mathematicalfractalsareproducedafterin�nitenumber

ofsuchiterations.Inthisk

!

1

limitthefractaldisplayedinFig.2.2abecomes

in�nitelylarge,whilethedetailsofFig.2.2bbecomeso�nethatthepictureseems

to\evaporate"andcannotbeseenanymore.Ourexampleshowsaconnected

construction,butdisconnectedobjectsdistributedinanontrivialwayinspacecan

alsoformafractal.

Inanyrealsystemthereisalwaysalowercuto�ofthelengthscale;inourcase

thisisrepresentedbythesizeoftheparticles.Inaddition,arealobjecthasa�nite

22

CHAPTER2.PATTERNS,FLUCTUATIONS,ANDSCALING

linearsizewhichinevitablyintroducesanuppercuto�ofthescaleonwhichfractal

scalingcanbeobserved.Thisleadsustotheconclusionthat,incontrasttothe

mathematicalfractals,forfractalsobservedinnaturalphenomena(includingbiology)

theanomalousscalingofthevolumecanbeobservedonlybetweentwowellde�ned

lengthscales.

Then,apossiblede�nitionforabiologicalfractalcanbebasedontherequirement

thatapowerlawscalingofN(l)hastoholdoveratleasttwoordersofmagnitude.

2.1.2

De�nitions

BecauseofthetwomaintypesoffractalsdemonstratedinFig.2.2,tode�neand

determinethefractaldimensionDonetypicallyusestworelatedapproaches.

Forfractalshaving�xedLanddetailsonverysmalllengthscaleDisde�ned

throughthescalingofN(l)asafunctionofdecreasingl,whereN(l)isthenumber

ofddimensionalballsofdiameterlneededtocoverthestructure.

Inthecaseofgrowingfractals,wherethereexistsasmallesttypicalsizea,one

cutsoutd-dimensionalregionsoflinearsizeLfromtheobjectandthevolume,V(L),

ofthefractalwithintheseregionsisconsideredasafunctionofthelinearsizeLof

theobject.WhendeterminingV(L),thestructureiscoveredbyballsorboxesof

unitvolume(l=a=1isusuallyassumed),thereforeV(L)=N(L),whereN(L)is

thenumberofsuchballs.

ThefactthatanobjectisamathematicalfractalthenmeansthatN(l)diverges

asl!

0orL!

1

,respectively,accordingtoanon-integerexponent.

Correspondingly,forfractalshavinga�nitesizeandin�nitelysmallrami�cations

wehave

N(l)�

l�D

(2.3)

with

D=liml!

0

lnN(l)

ln(1=l)

;

(2.4)

while

N(L)�

LD

(2.5)

and

D=

limL!

1

lnN(L)

ln(L)

:

(2.6)

forthegrowingcase,wherel=1.Here,aswellasinthefollowingexpressionsthe

symbol�

meansthattheproportionalityfactor,notwrittenoutin2.3,isindependent

ofl.

2.1.

FRACTAL

GEOMETRY

23

Obviously,theabovede�nitionsfornon-fractalobjectsgiveatrivialvaluefor

D

coincidingwiththeembeddingEuclideandimensiond.Forexample,thearea

(correspondingtothevolumeV(L)ind=2)ofacirclegrowsasitssquaredradius

whichaccordingto2.6resultsinD=2.

Nowweareinthepositiontocalculatethedimensionoftheobjectsshownin

Fig.2.2.Itisevidentfromthe�gurethatforthegrowingcase

N(L)=5k

with

L=3k;

(2.7)

wherekisthenumberofiterationscompleted.Fromhereusing2.6wegetthevalue

D

=ln5=ln3=1:465:::whichisanumberbetweend=1andd=2justaswe

expected.

2.1.3

Usefulrules

Inthissectionwementionafewruleswhichcanbeusefulinpredictingvarious

propertiesrelatedtothefractalstructureofanobject.Ofcourse,becauseofthe

greatvarietyofself-similargeometriesthenumberofpossibleexceptionsisnotsmall

andtheruleslistedbelowshouldberegarded,atleastinpart,asstartingpointsfor

moreaccurateconclusions.

a)Manytimesitistheprojectionofafractalwhichisofinterestorcanbeexper-

imentallystudied(e.g.apictureofafractalembeddedintod=3).Ingeneral,

projectingaD<dsdimensionalfractalontoadsdimensionalsurfaceresultsin

astructurewiththesamefractaldimensionDp=D.ForD�

dstheprojection

�llsthesurface,Dp=ds.

b)Itfollowsfroma)thatforD<dsthedensitycorrelationsc(r)(seenextsection)

withintheprojectedimagedecayasapowerlawwithanexponentds�Dinstead

ofd�

Dwhichistheexponentcharacterisingthealgebraicdecayofc(r)ind.

c)Cuttingoutads

dimensionalslice(cross-section)ofaD

dimensionalfractal

embeddedintoaddimensionalspaceusuallyleadstoaD+ds�ddimensional

object.Thisseemstobetrueforself-aÆnefractals(nextsection)aswell,with

Dbeingtheirlocaldimension

d)ConsidertwosetsAandBhavingfractaldimensionsDA

andDB,respectively.

MultiplyingthemtogetherresultsinafractalwithD=DA

+DB.Asasimple

example,imagineafractalwhichismadeofparallelsticksarrangedinsucha

waythatitscross-sectionisthefractalshowninFig.2.2b.Thedimensionof

thisobjectisD=1+ln5=ln3.

24

CHAPTER2.PATTERNS,FLUCTUATIONS,ANDSCALING

e)TheunionoftwofractalsetsAandB

withDA

>

DB

hasthedimension

D=DA.

f)ThefractaldimensionoftheintersectionoftwofractalswithDA

andDB

isgiven

byDA\B

=DA

+DB

�

d.Toseethis,consideraboxoflinearsizeLwithin

theoverlappingregionoftwogrowingstochasticfractals.ThedensityofAand

BparticlesisrespectivelyproportionaltoLDA

=Ld

andLDB

=Ld.Thenumber

ofoverlappingsitesN

�

LDA

\

B

isproportionaltothesedensitiesandtothe

volumeoftheboxwhichleadstotheabovegivenrelation.Theruleconcerning

intersectionsoffractalswithsmoothhypersurfaces(rulec)isaspecialcaseof

thepresentone.

g)Thedistributionofemptyregions(holes)inafractalofdimensionDscalesas

afunctionoftheirlinearsizewithanexponent�

D�

1.

Self-similaritycanbedirectlycheckedforadeterministicfractalconstructedby

iteration,butinthecaseofrandomstructuresoneneedsothermethodstodetectthe

fractalcharacterofagivenobject.Infact,random

fractalsareself-similaronlyina

statisticalsense(notexactly)andtodescribethemitismoreappropriatetousethe

termscaleinvariancethanself-similarity.Naturally,fordemonstratingthepresence

offractalscalingonecanusethede�nitionbasedoncoveringthegivenstructurewith

ballsofvaryingradii,however,thiswouldbearathertroublesomeprocedure.Itis

moree�ectivetocalculatethesocalleddensity-densityorpaircorrelationfunction

c(~ r)=

1 VX ~ r

0

�(~ r+~ r0)�(~ r

0)

(2.8)

whichistheexpectationvalueoftheeventthattwopointsseparatedby~ rbelongto

thestructure.ForgrowingfractalsthevolumeoftheobjectisV=N,whereNisthe

numberofparticlesinthecluster,and2.9givestheprobabilityof�ndingaparticle

attheposition~ r+~ r0,ifthereisoneat~ r0.In2.9�isthelocaldensity,i.e.,�(~ r)=1if

thepoint~ rbelongstotheobject,otherwiseitisequaltozero.Ordinaryfractalsare

typicallyisotropic(thecorrelationsarenotdependentonthedirection)whichmeans

thatthedensitycorrelationsdependonlyonthedistancersothatc(~ r)=c(r).

Nowwecanusethepaircorrelationfunctionintroducedaboveasacriterionfor

fractalgeometry.Anobjectisnon-triviallyscaleinvariantifitscorrelationfunction

determinedaccordingto2.9isunchangeduptoaconstantunderrescalingoflengths

byanarbitraryfactorb:

c(br)�

b��c(r)

(2.9)

2.1.

FRACTAL

GEOMETRY

25

with�anon-integernumberlargerthanzeroandlessthand.Itcanbeshownthat

theonlyfunctionwhichsatis�es2.9isthepowerlawdependenceofc(r)onr

c(r)�

r�

�

(2.10)

correspondingtoanalgebraicdecayofthelocaldensitywithinarandom

fractal,

sincethepaircorrelationfunctionisproportionaltothedensitydistributionaround

agivenpoint.LetuscalculatethenumberofparticlesN(L)withinasphereofradius

Lfromtheirdensitydistribution

N(L)�

Z L 0c(r)ddr�

Ld�

�;

(2.11)

wherethesummationin2.8hasbeenreplacedbyintegration.Comparing2.11with

2.5wearriveattherelation

D

=d��

(2.12)

whichisaresultwidelyusedforthedeterminationofD

fromthedensitycorrelations

withinarandomfractal.

2.1.4

Self-sim

ilarand

self-aÆ

ne

fractals

Therearethreemajortypesoffractalsasconcerningtheirscalingbehaviour.Self-

similarfractalsareinvariantunderisotropicrescalingofthecoordinates,whilefor

self-aÆnefractalsscaleinvarianceholdsforaÆne(anisotropic)transformation.Until

thispointmainlytheformercasehasbeendiscussed.

Therandom

motionofaparticlerepresentsaparticularlysimpleexampleof

stochasticprocessesleadingtogrowingfractalstructures.Awidelystudiedcaseis

whentheparticleundergoesarandomwalk(Brownianordi�usionalmotion)making

stepsoflengthdistributedaccordingtoaGaussianinrandomlyselecteddirections.

SuchprocessescanbedescribedintermsofthemeansquareddistanceR

2

=hR

2(t)i

madebytheparticlesduringagiventimeintervalt.Forrandom

walksR

2

�

t

independentlyofdwhichmeansthattheBrowniantrajectoryisarandomfractal

inspaceswithd>

2.Indeed,measuringthevolumeofthetrajectorybythetotal

numberofplacesvisitedbytheparticlemakingtsteps,(N(R)�

t),theabove

expressionisequivalentto

N(R)�R

2

(2.13)

andcomparing2.13with2.5weconcludethatforrandomwalksD

=2<d

ifd>2.

Inthiscase,ratherunusually,thefractaldimensionisanintegernumber.However,

thefactthatitisde�nitelysmallerthantheembeddingdimensionindicatesthatthe

objectmustbenon-triviallyscaleinvariant.

26

CHAPTER2.PATTERNS,FLUCTUATIONS,ANDSCALING

Inmanyphysicallyrelevantcasesthestructureoftheobjectsissuchthatitis

invariantunderdilationtransformationonlyifthelengthsarerescaledbydirection

dependentfactors.Theseanisotropicfractalsarecalledself-aÆne[4,5,6].

Single-valued,nowhere-di�erentiablefunctionsrepresentasimpleandtypical

forminwhichself-aÆnefractalsappear.IfsuchafunctionF(x)hastheproperty

F(x)'b�H

F(bx)

(2.14)

itisself-aÆne,whereH>0issomeexponent.Eq.2.14expressesthefactthatthe

functionisinvariantunderthefollowingrescaling:shrinkingalongthexaxisbya

factor1=b,followedbyrescalingofvaluesofthefunction(measuredinadirection

perpendiculartothedirectioninwhichtheargumentumischanged)byadi�erent

factorequaltob�H

.Inotherwords,byshrinkingthefunctionusingtheappro-

priatedirection-dependentfactors,itisrescaledontoitself.Forsomedeterministic

self-aÆnefunctionsthiscanbedoneexactly,whileforrandomfunctionstheabove

considerationsarevalidinastochasticsense(expressedbyusingthesign').

Ade�nitionofself-aÆnityequivalentto2.14isgivenbytheexpressionforthe

heightcorrelationfunctionc(�x)

c(�x)=h[F(x+�x)�F(x)]

2i��x

2H

(2.15)

whichcanbeeasilyusedforthedeterminationoftheexponentH.Inadditionto

functionssatisfying2.14and2.15,therearealsoself-aÆnefractalsdi�erentfrom

single-valuedfunctions.

Letus�rstconstructadeterministicself-aÆnemodel,inordertohaveanobject

whichcanbetreatedexactly.

Anactualconstructionofsuchaboundedself-aÆnefunctionontheunitinterval

isdemonstratedinFig.2.3.Theobjectisgeneratedbyarecursiveprocedureby

replacingtheintervalsofthepreviouscon�gurationwiththegeneratorhavingthe

formofanasymmetricletterzmadeoffourintervals.However,thereplacement

thistimeshouldbedoneinamannerdi�erentfromtheearlierpractice.Hereevery

intervalisregardedasadiagonalofarectanglebecomingincreasinglyelongated

duringtheiteration.Thebasisoftherectangleisdividedintofourequalpartsand

thez-shapedgeneratorreplacesthediagonalinsuchawaythatitsturnoversare

alwaysatanalogouspositions(atthe�rstquarterandthemiddleofthebasis).The

functionbecomesself-aÆneinthek!

1

limit.

Sucharandomfunctionis,forexample,theplotofthedistancesX(t)measured

fromtheoriginasafunctionoftimet,ofaBrownianparticledi�usinginonedimen-

sion.ItisobviousthatasocalledfractionalBrownianplotforwhichhX

2 H

(t)i�t2H

satis�es2.15.

2.1.

FRACTAL

GEOMETRY

27

k=1

k=2

k=3

Figure2.3:Self-aÆnefunctionscanbegeneratedbyiterationprocedures.ThisThe

single-valuedcharacterofthefunctionispreservedbyanappropriatedistortionof

thez-shapedgenerator(k=1)ofthestructure[3].

NextwegiveafurtherbasicfeatureoffractionalBrownianmotion.

CalculatingtheFourierspectrumofafractionalBrownianfunctionone�ndsthat

thecoeÆcientsoftheseries,A(f),areindependentGaussianrandomvariablesand

theirabsolutevaluescaleswiththefrequencyfaccordingtoapowerlaw

jA(f)j�f�

H

�

1 2

:

(2.16)

2.1.5

M

ultifractals

Intheprevioussectionscomplexgeometricalstructureswerediscussedwhichcould

beinterpretedintermsofasinglefractaldimension.Thepresentsectionismainly

concernedwiththedevelopmentofaformalismforthedescriptionofthesituation

whenasingulardistributionisde�nedonafractal[7,8].

Itistypicalforalargeclassofphenomenainnaturethatthebehaviourofasystem

isdeterminedbythespatialdistributionofascalarquantity,e.g.,concentration,

electricpotential,probability,etc..Forsimplergeometriesthisdistributionfunction

anditsderivativesarerelativelysmooth,andtheyusuallycontainonlyafew(ornone)

singularities,wherethewordsingularcorrespondstoalocalpowerlawbehaviourof

thefunction.(Inotherwords,wecallafunctionsingularintheregionsurrounding

point~ xifitslocalintegraldivergesorvanisheswithanon-integerexponentwhen

28

CHAPTER2.PATTERNS,FLUCTUATIONS,ANDSCALING

theregionofintegrationgoestozero).Inthecaseoffractalsthesituationisquite

di�erent:aprocessinnatureinvolvingafractalmayleadtoaspatialdistributionof

therelevantquantitieswhichpossessesin�nitelymanysingularities.

Asanexample,consideranisolated,chargedobject.Ifthisobjecthassharptips,

theelectric�eldaroundthesetipsbecomesverylargeinaccordwiththebehaviour

ofthesolutionoftheLaplaceequationforthepotential.Inthecaseofchargingthe

branchingfractalsproducedinthek!

1

limitofconstructionsshowninFig.2.1or

2.2onehasin�nitenumberoftipsandcorrespondingsingularitiesoftheelectric�eld.

Moreover,tipsbeingatdi�erentpositions,ingeneralhavedi�erentlocalenvironments

(con�gurationoftheobjectintheregionsurroundingthegiventip)whicha�ectthe

strengthofsingularityassociatedwiththatposition.

Theabovediscussedtimeindependentdistributionsde�nedonafractalsubstrate

arecalledfractalmeasures.Ingeneral,afractalmeasurepossessesanin�nitenumber

ofsingularitiesofin�nitelymanytypes.Theterm

\multifractality"expressesthe

factthatpointscorrespondingtoagiventypeofsingularitytypicallyformafractal

subsetwhosedimensiondependsonthetypeofsingularity.Thedescriptionofthe

multifractalformalismgoesbeyondthescopeofthepresentsection,butcanbefound,

forexample,inRef.3.

2.1.6

Methodsfordeterminingfractaldimensions

Whenonetriestodeterminethefractaldimensionofbiologicalstructuresinpractice,

itusuallyturnsoutthatthedirectapplicationofde�nitionsforD

givenintheprevious

sectionsisine�ectiveorcannotbeaccomplished.Instead,oneisledtomeasureor

calculatequantitieswhichcanbeshowntoberelatedtothefractaldimensionofthe

objects.Threemainapproachesareusedforthedeterminationofthesequantities:

experimental,computerandtheoretical.

Experimentalmethodsformeasuringfractaldimensions

Anumberofexperimentaltechniqueshavebeenusedtomeasurethefractaldimension

ofscaleinvariantstructuresgrowninvariousexperiments.Themostwidelyapplied

methodscanbedividedintothefollowingcategories:(a)digitalimageprocessingof

two-dimensionalpictures,(b)scatteringexperimentsand(c)directmeasurementof

dimension-dependentphysicalproperties.

(a)Digitisingtheimageofafractalobjectisastandardwayofobtainingquan-

titativedataaboutgeometricalshapes.Theinformationispickedupbyascanneror

anordinaryvideocameraandtransmittedintothememoryofacomputer(typicallya

2.1.

FRACTAL

GEOMETRY

29

PC).Thedataarestoredintheformofatwo-dimensionalarrayofpixelswhosenon-

zero(equaltozero)elementscorrespondtoregionsoccupied(notoccupied)bythe

image.Oncetheyareinthecomputer,thedatacanbeevaluatedusingthemethods

describedinthenextsection,wherecalculationofD

forcomputergeneratedclusters

isdiscussed.

Theonlyprincipalquestionrelatedtoprocessingofpicturesarisesiftwo-dimensional

imagesofobjectsembeddedintothreedimensionsareconsidered.Ithasalreadybeen

mentionedthatthefractaldimensionoftheprojectionofanobjectontoa(d�

m)-

dimensionalplaneisthesameasitsoriginalfractaldimension,ifD

<

d�

m.

(b)Scatteringexperimentsrepresentapowerfulmethodtomeasurethefractal

dimensionofstructures.Dependingonthecharacteristiclengthscalesassociated

withtheobjecttobestudied,light,X-rayorneutronscatteringcanbeusedto

revealfractalproperties.Thereareanumberofpossibilitiestocarryoutascattering

experiment.Onecaninvestigatei)thestructurefactorofasinglefractalobject,ii)

scatteringbymanyclustersgrowingintime,iii)thescatteredbeamfromafractal

surface,etc.

Evaluationofnumericaldata

Throughoutthissectionweassumethattheinformationaboutthestochasticstruc-

turesisstoredintheformofd-dimensionalarrayswhichcorrespondtothevalues

ofafunctiongivenatthenodes(orsites)ofsomeunderlyinglattice.Inthecaseof

studyinggeometricalscalingonly,thevalueofthefunctionattributedtoapointwith

givencoordinates(thepointbeingde�nedthroughtheindexesofthearray)iseither

1(thepointbelongstothefractal)or0(thesiteisempty).Whenmultifractalprop-

ertiesareinvestigatedthesitefunctiontakesonarbitraryvalues.Ingeneral,such

discretesetsofnumbersareobtainedbytwomainmethods:i)bydigitisingpictures

takenfromobjectsproducedinexperiments,ii)bynumericalproceduresusedforthe

simulationofvariousbiologicalstructures.Forconvenience,inthefollowingweshall

frequentlyusetheterminology\particle"foralatticesitewhichbelongstothefractal

(is�lled)andclusterfortheobjectsmadeofconnectedparticles.

BelowwediscusshowtomeasureD

forasingleobject.Tomaketheestimates

moreaccurateoneusuallycalculatesthefractaldimensionformanyclustersand

averagesovertheresults.

PerhapsthemostpracticalmethodistodeterminethenumberofparticlesN(R)=

RD

withinaregionoflinearsizeR

andobtainthefractaldimensionD

fromtheslopes

oftheplotslnN(R)versuslnR.IfthecentersoftheregionsofradiusR

arethepar-

ticlesofthecluster,thanN(R)isequivalenttotheintegralofthedensitycorrelation

function.Inpracticeonechoosesasubsetofrandomlyselectedparticlesofthefractal

30

CHAPTER2.PATTERNS,FLUCTUATIONS,ANDSCALING

[N(R

)]ln

Rln

R=

1

Rm

ax

Slop

e=D

Figure2.4:Schematiclog-logplotofthenumericallydeterminednumberofparticles

N(R)belongingtoafractalandbeingwithinasphereofradiusR.IfR

issmaller

thantheparticlesizeorlargerthanthelinearsizeofthestructureatrivialbehaviour

isobserved.Thefractaldimensionisobtainedby�ttingastraightlinetothedata

inthescalingregion[3].

(asmanyasneededforareasonablestatistics)anddetermineshN(R)iforasequence

ofgrowingR(orcountsthenumberofparticlesinboxesoflinearsizeL).Inorderto

avoidundesirablee�ectscausedbyanomalouscontributionsappearingattheedgeof

theclusteroneshouldnotchooseparticlesascentresclosetotheboundaryregion.

ThesituationisshowninFig.2.4.Typicallythereisadeviationfromscalingfor

smallandlargescales.

TheroughnessexponentH

correspondingtoself-aÆnefractalsisusuallydeter-

minedfromthede�nition2.15.Analternativemethodistoinvestigatethescalingof

thestandarddeviation�(l)=[hF

2(x)ix�hF(x)i

2 x]1=2

oftheself-aÆnefunctionF

h�(l)i�lH;

(2.17)

wherethelefthandsideistheaverageofthestandarddeviationofthefunctionF

calculatedforregionsoflinearsizel.TheroughnessexponentH

canbecalculated

bydetermining�(l)forpartsoftheinterfacesforvariousl.Anaveragingshould

bemadeoverthesegmentsofthesamelengthandtheresultsplottedonadouble

2.2.

STOCHASTIC

PROCESSES

31

logarithmicplotasafunctionofl.

2.2

Stochasticprocesses

2.2.1

Thephysicsofmicroscopicobjects

Everybiologicalprocesseventuallytakesplaceatthemolecularlevel.Thephysicsof

thismicroscopicrealmisfundamentallydi�erentfromthephysicsofourmacroscopic

world,andrequiresacompletelydi�erentdescription.Firstofall,asthelength-scale

andvelocity-scalegodowntomolecularscales,theReynoldsnumbergoesdowntoo,

andweapproachtheoverdampedregimeinwhichinertiaplaysnoroleanymore[9]

andwherethevelocity(andnottheacceleration)oftheobjectsisproportionalto

theforcesactingonthem.Secondly,thereisBrownianmotion.Microscopicobjects

arebeingrandomlykickedaroundbymoleculesofthesurroundingmedium,andthe

processeshaveaninherentlystochasticnature.

Thetimescaleofmacroscopicprocessesissetbythevelocityandaccelerationof

massiveobjects,thethermal

uctuationsarenegligible,andwhenwedesignamacro-

scopicdevicewetrytosuppressanystochasticelementasmuchaspossible.Onthe

otherhand,formicroscopicobjectseverydegreeoffreedomhasinevitablyasignif-

icant

1 2kBTthermalenergyonaverage(whereTdenotestheabsolutetemperature

andkB

istheBoltzmanncoeÆcient),thetimingoftheprocessesissetbythermally

assistedevents(suchasdi�usionoractivatedtransitionsoverenergybarriers),and

forthedesignofmicroscopicdevicesthermal

uctuationsshouldbeexploitedrather

thansuppressed.

Ingeneral,themotionofanyobjectinathermalenvironmentcanbedescribed

bytheLangevinequation[10]:

mx(t)=�

_x(t)+

p 2D�(t)+F(x;t);

(2.18)

wherex,m,

,andD

denotetheposition,mass,viscousfrictioncoeÆcient,and

di�usioncoeÆcientoftheobject,respectively.Thethreeforcetermsontheright

handsideoftheequationaretheviscousfrictionbythemedium,

_x(t);thethermal

noisecomingfromthemoleculesofthemedium,

p 2D�(t);andalltheotherforces

unrelatedtothemedium,F(x;t).Sincethethermalnoisetermisastochasticfunction

theLangevinequationisreferredtoasastochasticdi�erentialequation.Thenoise

factor,�(t),isusuallymodelledbyaGaussianwhitenoisewithzerotimeaverage,

h�(t)i=0;

(2.19)

32

CHAPTER2.PATTERNS,FLUCTUATIONS,ANDSCALING

andautocorrelationfunction

h�(t)�(t

0)i=Æ(t�t0):

(2.20)

Boththeviscousfrictionandthethermalnoiseareexertedbythemedium,and

arenotindependent.Theirmagnitudesareconnectedbythe

uctuation-dissipation

theorem(orEinsteinrelation):

D=

kBT

:

(2.21)

ForbiomoleculesinwatersolutiontheLangevinequationcanbesimpli�ed.The

ratio� relax

=

m=isacharacteristictimescaleoftheLangevinequation(2.18),

andtellsushowlongittakesforaparticletolooseitsinitialvelocityviaviscous

frictionifthethermalnoiseandFareturnedo�.Multiplyingthisbytheparticle's

characteristicvelocityv,wegetthecharacteristicdistance�relax

=vm=onwhich

theparticlecomestoahalt.Comparingthisdistancetothecharacteristicsizeof

theparticlea,wegetsomeinformationaboutthestrengthoftheviscousdamping:

if�relax=a�

1thedampingisstrong,becausetheparticlestopsonamuchshorter

distancethanitssize;andif�relax=a�

1thedampingisweak.Supposingthatm

isproportionaltoa

3�and

isproportionaltoa�(cf.Stokeslaw),where�and�

arethedensityanddynamicviscosityofthemediumrespectively,�relax=abecomes

proportionalto

R=

va

�=�

=va �;

(2.22)

whichiscalledtheReynoldsnumber(�=�=�isthekinematicviscosity).Thus,itis

theReynoldsnumberthatcharacterisesthestrengthofthedamping.LowReynolds

numbermeansstrongdamping.

LetusnowestimatetheReynoldsnumberforbiologicalmolecules.Thetypical

sizeofaproteinisintheorderofnanometers(a�1nm),thedensityanddynamic

viscosityofwaterare��10

3

kg/m

3

and��10�

3

kg/s/m.Themaximalforcesacting

onaproteinareintheorderofpiconewtons(afewkBToverafewnanometers),thus

thecharacteristicvelocityofaproteincannotbemuchlargerthanv�1pN=(a�)�

1m/s.ThisshowsthattheReynoldsnumberforbiomoleculesisintheorderof10�

3

orevensmaller,i.e.,weareinthestronglydampedoroverdampedregime.R=10�

3

isasomewhatshockingresult.Itmeansthattheviscousfrictioncanstopaprotein

onadistance(�10

�

3

nm)muchshorterthanthesizeoftheatoms.

Inthisoverdampedregimewhentheforceschange,thevelocityofaparticle

relaxessoquickly(during� relax)andonsuchasmalldistance(�relax

�

a)thatthe

accelerationterm(thederivativeofthevelocitywithrespecttotime)onthelefthand

2.2.

STOCHASTIC

PROCESSES

33

sideoftheLangevinequation(2.18)canbeneglected:

_x(t)=F(x;t)=

+

p2D�(t):

(2.23)

Thiskindofreductioniscalledadiabaticeliminationofthefastvariables[11,10].

SincethemotionofbiomoleculescanbewelldescribedbytheoverdampedLangevin

equation,fromnowonwewilluseonlythisversionoftheequation,andalsotheterm

Langevinequationwillalwaysrefertoitsoverdampedversion(2.23).

Fromthisstochasticordinarydi�erentialequationonecanderiveadeterministic

partialdi�erentialequation,theFokker-Planckequation(orSmoluchowskiequation)

[10],whichdescribesthetimeevolutionoftheprobabilitydensityP(x;t)ofthepo-

sitionoftheparticle:

@tP(x;t)=�@xJ(x;t);

(2.24)

where

J(x;t)=

F(x;t)

P(x;t)�

kBT

@xP(x;t)

(2.25)

istheprobabilitycurrentoftheparticle.Iftheforce�eldF(x;t)isthenegative

gradientofapotential:F(x;t)=�@xV(x;t),theprobabilitycurrentcanbewritten

intheform

J(x;t)=�

kBT e�

V(x;t)=kB

T@x

� eV(x;t)=kB

TP(x;t)�:

(2.26)

2.2.2

Kram

ersform

ula

and

Arrheniuslaw

InmanysystemsBrownianparticlesarewigglingindeeppotentialwells(compared

tokBT)forlongperiodsoftime,rarelyinterruptedbyquickjumpsintooneofthe

neighbouringwells.Ifthepotentialisstaticorchangesinamuchlongertime-scale

thanthedurationofthesejumps(whichisusuallythecase),akineticapproachcan

beusedtodescribethemotionoftheparticles,withtransitionrateconstantsbetween

discretestates.

Thedisciplineofratetheory(forreviewseeRef.[12])wascreatedwhenArrhenius

[13]extensivelydiscussedvariousreaction-ratedataandshowedthattheyvaryona

logarithmicscalelinearlytotheinversetemperatureT�

1.Inotherwords,theescape

(orjumping)rateconstants,k,followtheArrheniuslaw

k=�e�

�E=kB

T;

(2.27)

where�Edenotesthethresholdenergyforactivationand�isafrequencyprefactor.

UsingKramers'method[14,10],theArrheniuslawcanbeeasilyderivedfrom

theFokker-PlanckequationforanoverdampedBrownianparticlemovinginaone-

dimensionalpotentialV(x)(depictedinFig.2.5),ifthepotentialwellfromwhich

34

CHAPTER2.PATTERNS,FLUCTUATIONS,ANDSCALING

theparticleistryingtoescapeismuchdeeperthankBT.Inthiscasetheprobability

densitynearthebottomcanbewellapproximatedbyitsequilibriumvalue

Peq(x)=P0e�

V(x)=kB

T;

(2.28)

whichcanbederivedfromEq.(2.26)bysettingitsright-handsidetozero.The

normalisationfactorisapproximately

P0=

1

Z c be�

V(x)=kB

Tdx

;

(2.29)

becausethevastmajorityoftheprobabilityfallsintotheinterval[b,c]wherethe

potentialdi�erencefromthebottomofthepotentialisnotlargerthanafew(�5)

kBT.Kramers'approachisbasedontheassumptionthattheprobabilitycurrentover

thepotentialbarrierbetweenBandCisaconstantJ.Indeed,thisconditionholds,

becausetheinterval[B,C]containsonlyaverysmallfractionofthetotalprobability.

Anotherassumptionisthattheprobabilitydistributionforx�

C

iszero,because

theparticlehasbasicallynochancetogetbacktothewellandtheescapecanbe

consideredtobecompleted.Thus,afterrearrangingandintegratingEq.(2.26)from

BtoCweget

J

kBT

Z C BeV

(x)=kB

Tdx=�

� eV(x)=kB

TPeq(x)� C x=B

;

(2.30)

kT

~5BkT

~5B

B

V

ba

cA

Cx

JFigure2.5:PotentialV(x)withadeepwellandabarrieroverwhichanoverdamped

Brownianparticletriestoescape.

2.3.

CONTINUOUSPHASE

TRANSITIONS

35

wheretheexpressionbetweenthesquarebracketsisP0

forx=Bandzeroforx=C.

From

thisthecurrentJoverthebarrier(whichisequivalenttotheescaperate

constantk)canbeexpressedas

J�

k=

D

Z c be�

V(x)=kB

Tdx

Z C BeV

(x)=kB

Tdx

=

De�

[V(A)�V(a)]=kB

T

Z c be�

[V(x)�V(a)]=kB

Tdx

Z C Be�

[V(A)�V(x)]=kB

Tdx

:

ThisexpressionhasindeedthesameformasthatoftheArrheniuslaw(2.27).Here

theactivationenergy�EistheheightofthebarrierV(A)�

V(a),andthefrequency

prefactor�dependsonlyontheshapeofthepotentialnearthebottomofthewell

andthetopofthebarrier.

ThisderivationholdsevenifthepotentialV(x)changesintimebutmuchslower

thentheintrawellrelaxationtimeoftheparticle[15].Inthiscasetheescaperate

constantbecomesalsotimedependent.

Mostchemicalreactionscanalsobedescribedintermsofkineticrateconstants,

soiftheyarepresent,theyrepresentanothersourceofstochasticityinmolecular

processesinadditiontothethermalnoise.Thewaitingtimeforanyescapeprocess

orchemicalreactioncharacterisedbyarateconstantkhasanexponentialdistribution

withmeanvalue1=k.

2.3

Continuousphasetransitions

Inthefollowingchapterwediscusstherelevanceofthesocalledself-organised

critically(SOC)forbiology.However,beforedescribingthismoreadvanced,non-

equilibriumconceptwegiveashortintroductiontothecloselyrelatedprecursor,the

secondorder(orcontinuous)phasetransitionoccurringinequilibrium.

Phasetransitionscanbeeasilyunderstoodonasimplethermodynamiclevel.Let

usconsiderasubstancewhichcanexistintwodi�erentphaseslikewater(liquidand

ice)oriron(paramagneticandferromagnetic).Usuallyoneofthephasesisdisordered

whiletheotherisordered.Thedistinctionisbasedonthesymmetryofthestate:

thesymmetric(orisotropic)stateisthedisorderedone.Tocharacterisethestrength

oforderingatgivenvaluesofthermodynamicparameters(likethetemperature)we

introduceanorderparameter.Itmeasureshowwellorderedthesubstanceis.By

conventioniftheorderparameteriszerowespeakofacompletelydisorderedstate

36

CHAPTER2.PATTERNS,FLUCTUATIONS,ANDSCALING

0.0

0.2

0.4

0.6

0.8

1.0

m

-10123

F(T,m)

T>

Tc

T=

Tc

T<

Tc

Tc

T

0.0

0.5

1.0

m

(a)

0.0

0.2

0.4

0.6

0.8

1.0

m

-10123

F(T,m)

T>

Tc

T=

Tc

T<

Tc

Tc

T

0.0

0.5

1.0

m

(b)

Figure2.6:Tworoutestoaphasetransition:(a)�rstorderand(b)secondorder.

Theinsetsshowtheorderparameterasthefunctionoftemperature.

(e.g.,liquidwater)andifitisnon-zerothenthesubstanceisinanorderedstate

(e.g.,icecrystal).Asitisknownfromthermodynamics,atconstanttemperaturethe

phasewiththelowestfreeenergyisstable.Sincethefreeenergygenerallycanalso

beafunctionoftheorderparameter,thisenergyminimumrequirementwillselect

whichphasecanbeobservedatagiventemperatureT.Inthiscasethetemperature

playstheroleofcontrolparameterwhichallowsfortuningthesystemtothephase

transition.

InFig.2.6wesketchtwopossiblewaysleadingtoaphasetransition.Whenthe

temperatureisabovethecriticaltemperatureTc

(wherethetransitionoccurs)theonly

globallystablesolutiontotheenergyminimumcriterionisthephasewithzeroorder

parameter(m=0).ThismeansthatforT

>

Tc

thesystemisinitsdisorderedstate.

Fig.2.6(a)showsacasewhenloweringthetemperature,atT

=Tc

�rstanon-trivial

(m>

0)minimumappearsandthenitshiftstolargervaluesasthetemperatureis

lowered(seeinsetinFig.2.6(a)).Thisisthescenarioofa�rstorderphasetransition.

Thecharacteristicfeatureofthistypeoftransitionisajumpdiscontinuityinthe

orderparameter(andcertainotherquantities).Incontrary,Fig.2.6(b)demonstrates

anothertypeoftransitionwheretheorderparameterchangescontinuouslyasthe

temperatureisloweredbelowTc.Thistypeoftransitionisreferredasasecondorder

phasetransitionandithasreceivedmuchlargerattentioninthepastdecades.A

motivationforthisinterestliesinthespecialpropertiesofsuchtransitions.

Withoutlossofgeneralityletusconsideramorespeci�cexampleforasecond

ordertransition:amagneticmaterialattemperatureT

andmagnetic�eldH

.If

thissystemshowsaparamagnetic{ferromagneticsecondordertransitionatTc

(for

2.3.

CONTINUOUSPHASE

TRANSITIONS

37

H

=0)thenitisconvenienttousethereducedtemperature

t=

T�

Tc

Tc

(2.31)

ascontrolparameterinsteadofT.

Measuringthephysicalpropertiesofthesamplerevealsdivergencesofvarious

physicalobservablesasthecriticaltemperatureisapproached.Theorderparameter

inthiscaseisthezero-�eld(H=0)magnetisationM0

sinceitiszeroifT>Tc,and

non-zerobelowTc.InthevicinityofTc

itbehavesas

M0(t)�jtj�;

(2.32)

where�isthecriticalexponentofthemagnetisation.Similarly,forthespeci�cheat

(whichgivesthechangeofenergyforasmallchangeoftemperature)

CH=0(t)�jtj��;

(2.33)

andthesusceptibility(whichisthesensitivityofthemagnetisationwithrespectto

theexternal�eldH)scalesas

�(T)�jtj�

:

(2.34)

Bothofthesequantitiesdescribearesponseofthesystemtosomeexternalperturba-

tion.ClosetoTc

theydivergeshowingthattherethesystemisextremelysensitive:

itisinacriticalstate.

Theabovede�nedthreecriticalexponents(�;�;)arenotindependentofeach

other.Amoredetailedanalysisshowsthattheexponentrelation

�=

2���

2

(2.35)

holds. N

earthetransitionpointthespontaneous

uctuationsinthesystem

become

largeduethehighsusceptibilities.Forthecaseofa

uidthesestrong

uctuations

areobservableasthedecreaseoflighttransmittance(criticalopalescence).Sincethe

lengthscale�associatedwiththese

uctuations,i.e.,thetypicalsizeof

uiddroplets,

alsohaspowerlawdivergence

��t��;

(2.36)

atthecriticalpointtherewillbenotypicallengthscaleexceptthetriviallower

(atomicsize)andupper(systemsize)scales.Thisfactismanifestedviathefractal[3]

structureofthe

uctuationsinthesystemanditiscloselyconnectedtootherpower

38

CHAPTER2.PATTERNS,FLUCTUATIONS,ANDSCALING

lawdivergencespresentatTc.Thefractalityfromtheexperimenter'spointofview

meansthatthe

uctuationsarestatisticallyinvariantunderthetransformation

(x;y;:::)7�!

(�x;�y;:::);

(2.37)

i.e.,notypicallengthscalecanbeidenti�ed.

2.3.1

ThePottsmodel

Usingthethermodynamicapproachitispossibletoderivethecriticalexponents

onlyheuristically,basedonsymmetryarguments,supposingsomeformofthe(coarse

grained)freeenergynearthecriticalpoint.ThismethodisusedbytheLandau

theoryofcriticalphenomena.Toovercomethelimitationsofthisapproachonehas

tointroducemodelswhichincludemoredetailsabouttheinteractionsleadingtothe

phasetransition.Anumberofdi�erentmodelscanbeconstructeddependingon

thelevelofabstractionatwhichinteractionsarehandled.Herewediscussarather

generallatticemodelintroducedbyPottsin1952[16].

Theq-statePottsmodelconsistsofasetof\spins"(orparticles)fsigeachof

whichmayhaveintegervaluessi

=0;1;:::(q�1).Thesespinssitonalatticeand

theHamiltonian(theenergyfunction)isde�nedas

H[fsig]=�X

JÆ Kr(si;sj);

(2.38)

whereÆ KristheKroneckerdeltafunctionandthesummationgoesovernearestneigh-

boursonly(shortrangeinteraction).ThemeaningoftheenergyfunctionEq.(2.38)

isthatonlyparticleswithsi=sj\like"eachother,onlysuchcombinationslowerthe

energyofthesystemasillustratedinFig.2.7.

Ifthetemperatureishighthenalltheqstateswillbeequallypopulated,so

anyquantitycanserveasanorderparameterwhichmeasuresthedi�erenceofthe

distributionofspinstatesfromuniform.Atlowtemperaturesthesystemorganises

itselfintoacon�gurationwheremostspinsareinarandomlyselectedstatewhilethe

otherstatesareweaklypopulated.

ThePottsmodelisrelatedtomanyotherlatticemodelsinstatisticalphysics[17].

Forq=2(twostates:`up'and`down')itisequivalenttothewellknownIsingmodel.

Theq=1limitreproducesthepercolationproblem,whileq=0canbemappedto

theresistornetworkproblem.

2.3.2

Mean-�eldapproximation

Althoughsomeofthespecialcasesmentionedabovecanbesolvedexactly,nogeneral

solutionexiststothePottsmodelitself.Herewepresentthesimplestapproachto

2.3.

CONTINUOUSPHASE

TRANSITIONS

39

��������

��������

����������

����������

��������

����������

����������

���

���

���

���

���

���

������

������

��������

��������

���

���

���

���

������

������ ���

���

���

���

���

���

��������

���

���

���

���

H 0 -J

Figure2.7:InteractionenergyofparticlesinthePottsmodel.

determinetheexponents[17].Considerthefollowingslightlymodi�edHamiltonian

H[fsig]=�

zJ N

X i<j

Æ Kr(si;sj);

(2.39)

whereziscoordinationnumberofthelattice(numberofneighbours,i.e.,z=4fora

planarsquarelattice)andN

isthetotalnumberofspins.IncontrasttoEq.(2.38)

thisHamiltonianallowslongrangeinteractionssinceinteractionofeveryspin-pair

contributestothetotalenergy.Inotherwords,aspinisa�ectednotonlybyits

neighbours,butratherbythemeanstateofthewholesystem.Forthisreasonsuch

anapproximationiscalledamean-�elddescription.Thisapproachhasseveralde�-

ciencies:byintroducinglongrangeinteractionitneglectsthe

uctuationswhichare

essentialpartsofphasetransitions,andthereforedoesnotgivecorrectresultsfor

thecriticaltemperatureandtheexponents.Nevertheless,sincegivesaqualitatively

correctpictureitisworthexamining.

Insteadofaccountingforthestateofeverys