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Journal of Biotechnology 71 (1999) 67–104

The non-linear analysis of cybernetic models. Guidelines formodel formulation

J. Varner 1, D. Ramkrishna *School of Chemical Engineering, 1283 Chemical Engineering Building, Purdue Uni6ersity, West Lafayette, IN 47907, USA

Received 21 October 1997; accepted 9 November 1998

Abstract

A set of guidelines are formulated, using tools from bifurcation theory, that describe the qualitative characteristicsof the different forms of competition for key cellular resources present within the cybernetic framework. Theseguidelines establish the basis of a modular approach for the construction of abstracted cybernetic models of microbialprocesses. This methodology, employed in the subsequent papers of this series, affords the construction of entireclasses of cybernetic models that are guaranteed to possess desired dynamic features, thus reducing the modelformulation burden and yielding insight into the necessary level of metabolic pathway abstraction. © 1999 ElsevierScience B.V. All rights reserved.

Keywords: Cybernetic models; Non-linear analysis; Mathematical models

1. Introduction

The central theme of mathematical modeling isthe abstraction of physical phenomena into asuitably simplified mathematical formalism. Thereexist a number of perspectives that guide the levelof abstraction and hence the method of formula-tion of acceptable mathematical models. One

school of thought, termed first principles or whitebox modeling, relies upon developing an ab-stracted view of the process in which unnecessaryphysical detail has been neglected. If the intuitionbased abstraction of the physical system is cor-rect, then the appropriate phenomena is predictedby the mathematical model. If, however, the con-ception of the system is erroneous then the modelformulation fails. Clearly, a deep and fundamen-tal understanding of the phenomena is requiredfor a consistent level of first principles modelingsuccess. In the context of biological systems, thisintuitive approach requires the presumption ofwhich metabolic intermediates and pathways arecrucial to system behavior and the specific regula-tory role they play. Given that metabolic path-

* Corresponding author. Tel.: +1-765-494-4066; fax: +1-765-494-0805.

E-mail address: [email protected] (D. Ramkr-ishna)

1 Present address: Institute of Biotechnology, ETH-Hong-gerberg, Zurich CH-8093, Switzerland. Tel.: +41-1-633-3141;fax: +41-1-633-1051; e-mail: [email protected] (J.Varner).

0168-1656/99/$ - see front matter © 1999 Elsevier Science B.V. All rights reserved.

PII: S 0168 -1656 (99 )00016 -4

J. Varner, D. Ramkrishna / Journal of Biotechnology 71 (1999) 67–10468

ways can consist of large numbers of coordinatedreactions, this approach may require a number ofiterations to be successful in the general case.

In the area of process control, another method-ology for the development of mathematical mod-els, termed black box modeling, takes afundamentally different tact. This methodologyrelies upon the fitting of basis functions to experi-mental data. Thus, instead of translating physicalintuition into mathematical structure, black boxapproaches force mathematical structure to obeyexperimental observations. Black box approaches,such as Volterra series methods, or the use ofNeural networks have enjoyed a fair degree ofsuccess in a number of different instances wherefirst principles modeling has proven to be difficult(Doyle et al., 1995; Pearson et al., 1996). Theattractiveness of black box approaches stems fromthe strong advantage that they are guaranteed todescribe the system behavior, at least locally. Incomplicated systems, this negates the need to de-velop a deep intuitive understanding of the pro-cess, potentially yielding a large time and effortsavings.

The cybernetic framework developed byRamkrishna and co-workers has long sought toabstract biological systems into highly idealizedmathematical structures that are capable of pre-dicting complex nutrient uptake and growth profi-les (Kompala et al., 1986; Baloo and Ramkrishna,1991; Straight and Ramkrishna, 1991). However,these past endeavors have suffered from the pit-falls of a purely first principles perspective. Assuch the development and application of theframework has been limited in scope to a fewexample systems. Therefore, the objective of thiswork is to approach the problem of cyberneticmodel formulation from a different perspective.Our vision for the development of cyberneticmodels of microbial systems falls somewhere be-tween the two extreme schools of thought. Weseek to formulate a methodology that produces adegree of biological insight higher than black boxmodeling while simultaneously guaranteeing thedescription of macroscopic biological phenomena,such as growth and nutrient assimilation profiles.Moreover, this methodology does not rely solelyupon physical intuition as is true of white box

approaches. Rather, it draws inspiration from aset of operating postulates that govern the dy-namic features displayed by particular topologicalpathway structures. Thus, upon inspection of thetopological structure of a model framework, thequalitative dynamic characteristics are easily iden-tified. This is somewhat akin, by analogy, toassembling a jigsaw puzzle. Imagine that we pos-sess a library of elementary pieces that can beused to assemble the puzzle. Each piece has atopological structure associated with it. Further-more, as a consequence of the structure, eachelementary piece has a particular function. Forexample, some pieces may be corner pieces,whereas others may only fit in the center of thepuzzle. What ever the case may be, clearly, theshape of the piece reflects its purpose or functionin the puzzle.

We propose something very similar, in spirit,for the formulation of abstracted cybernetic mod-els. We postulate that a global metabolic map canbe decomposed into abstracted elementary com-ponents, similar to pieces in a puzzle. The con-verse of this postulate implies that given anappropriate library of elementary components, wecan assemble model structures from abstractedelementary pieces. Bear in mind that each elemen-tary component has a distinct mathematicalconfiguration, thus, this represents the structureside of the argument, i.e. what is the mathematicalstructure needed to describe a microbial process.At issue, however, is the ability to a priori con-struct model formulations that are guaranteed topossess desired dynamic features. The investiga-tion of this issue is the particular focus of thework at hand.

Our library of elementary components consistsof the four elementary pathways derived byStraight and Ramkrishna (1994) (see Fig. 1).These pathways can be subdivided on the basis oftopology, as well as, the type of competitive struc-ture for key cellular resources required for expres-sion of the enzymatic machinery. It is the latterissue, i.e. the competitive structure, that dictatesthe dynamics and moreover the function of theelementary piece. Thus, by understanding the dy-namics associated with each type of competitivestructure, we gain insight into the dynamics dis-

J. Varner, D. Ramkrishna / Journal of Biotechnology 71 (1999) 67–104 69

played by each elementary member of the modelconstruction library. The next step in the develop-ment must be an understanding of the manner inwhich the competitive structure of individual ele-mentary pathways is influenced upon assembly.This is a key issue given the modular theme of themethodology. These two pieces of informationcan then be employed to formulate a set of postu-lates that guide the construction of mathematicalmodels so that specific desired features areobtained.

1.1. Scope of in6estigation

The objective of this work is the analysis, usingtools from bifurcation theory, of the characteris-tics of substitutable and complementary competi-tion as defined by Straight and Ramkrishna(1994). The linear and convergent elementarypathways experience substitutable competition,whereas, the divergent and cyclic units possess acomplementary competitive structure. The distinc-tion between competitive type rest upon thechoice of the cybernetic objective function andresource constraint rather than the topology ofthe pathway. The hallmark of substitutable com-petition is the objective of maximization of thelevel of an end product subject to a constraintupon available resources. Thus, the objective of a

linear and convergent pathway is the maximiza-tion of the end product or the focus of the path-way. Complementary competition, however, doesnot strive towards this type of objective. Rather,the objective of complementary structures is themaximization of the mathematical product of thepathway end products or ring metabolites (in thecase of a cyclic unit.)

In what follows, we analyze the elementaryconvergent and divergent pathways to determinethe qualitative characteristics of each type of com-petition as a function of system parameters. In thelarger picture, we are examining the qualitativecharacteristics of the elementary pathways thatcompose our formulation library. As per ourmodular theme, after completing the analysis ofindividual elementary pathways, we combine aconvergent and a divergent pathway to determinethe qualitative effects upon system behavior of theinteraction of substitutable and complementarycompetition. However, before we introduce andanalyze the model formulations, we begin by dis-cussing some mathematical preliminaries. In par-ticular, we present an overview ofLyapunov–Schmidt reduction and the objectivesof bifurcation theory. These pieces of mathemati-cal machinery are used, subsequently, to analyzethe qualitative aspects of cybernetic model behav-ior. Those readers who seek an understanding ofthe development solely on an application levelmay move directly to Section 1.4 where the analy-sis objectives are summarized. The detail of thebifurcation analysis is presented as an appendix tothis development thereby limiting unnecessaryconfusion stemming from involved mathematicalmanipulation.

1.2. Mathematical preliminaries:Lyapuno6–Schmidt reduction

Lyapunov–Schmidt reduction is an elegant op-erator technique that has its origins in Singularitytheory (Golubitsky and Schaeffer, 1985). Thistechnique allows for the isolation of the aspect ofa dynamic system that is responsible for qualita-tive changes in system behavior. Once this portionhas been isolated, tools from bifurcation theorycan be applied to examine the exact nature of the

Fig. 1. Four elementary pathways derived by Straight andRamkrishna.


qualitative change. Often, using a reduction tech-nique such as Lyapunov–Schmidt allows thestudy of the qualitative nature of an n-dimen-sional system to be reduced to a q-dimensionalsubsystem, where q is equal to the number ofzero eigenvalues of the system Jacobian evalu-ated around an equilibrium point, therebygreatly simplifying the analysis problem. In amore exact mathematical sense, Lyapunov–Schmidt reduction is centered upon the use ofappropriately defined projection operators to iso-late the regions of system space that are respon-sible for bifurcation phenomena. In thesubsequent development, we leave much of thedetail to the reader in the interest of brevity,however, in what follows we present a generaltreatment of the material so to acquaint thereader with the arguments of the mathematicaldevelopment.

Let the time evolution of an arbitrary dynamicsystem in the state variable x be governed by thedifferential equation set

dxdt

=F=F(x, k), x�Rn F:Rn×Rk�Rn

(1.1)

where k is the k-dimensional vector of systemparameters. Furthermore, denote a steady statesolution of Eq. (1.1) as

F(x*, k*)=0 (1.2)

where * denotes the steady state value. Assumingthat F(x, k) is smooth around (x*, k*), if theJacobian of equation set (1.1) evaluated at(x*, k*), denoted as dF�x*, k* has a non-zero de-terminant in an arbitrary neighborhood of theequilibrium point, then equation set (1.1) con-tains no bifurcation behavior by the implicitfunction theorem. If however, there exist a pointin parameter space, denoted by k° along solutionx* at which det (dF�x*, k $

)=0, equation set (1.1)has the possibility of multiple equilibrium solu-tions. Formally the point (x*, k°) is then termeda critical point. Without loss of generality weassume that (x*, k°) is translated to the originand the critical Jacobian in the translated systemis given by

L dF�0, 0 (1.3)The eigenvalues of the critical operator L form

the basis for the decomposition of system space,and eventually, for the isolation of the aspect ofthe dynamic system that is responsible for thebifurcation behavior. We assume for the sake ofease (although the technique is applicable in anycase) that L is diagonalizable. Note this is almostalways possible given matrix operators becausethe original system basis can be rotated to beequivalent to the eigenvector directions, hence,producing a diagonal Jacobian. The only require-ment of the rotation is a non-defective Jacobianfor all values of the bifurcation parameter, i.e.the algebraic multiplicity is equal to geometricmultiplicity. Assuming this is true, the systemspace can easily be decomposed into the compo-nents

X=KerL�RngL (1.4)

where the null space of the critical Jacobian isspanned by the eigenvector set

zN={{zj}:Lzj=0, Ö j} (1.5)

and the range space is spanned by

zR={{zk}:Lzk=skzk, Ö k} (1.6)

Note that because we have assumed the criticalJacobian is diagonal the eigenvector set zN, zR

form the standard basis on Rn. The projectionoperator E which projects elements of the systemspace onto RngL can then be defined as

E(·) %dim(zR)

k=1

�·,zkR�zk

R (1.7)

This operator is crucial to the decompositionof the dynamical system into regions where bi-furcation is present and absent. More precisely,bifurcation is limited to the aspect of the systemthat lies in the KerL. Using the projection opera-tor E we isolate the range space variables interms of the null space variables, which can besubstituted into the null space system aspect. Thelatter set of equations contain the bifurcationproblem and are termed the reduced set or equa-tion.


1.3. Mathematical preliminaries: bifurcationtheory

Bifurcation theory is the mathematical study ofthe qualitative change in system behavior as afunction of parameters. Thus, given the reducedequation, i.e. the aspect of the system that isresponsible for qualitative changes in behavior,bifurcation theory could be employed to studywhere and how system behavior changes. In whatfollows we present a very simplified overview of thesalient aspects of the theory so that the reader isfamiliar with the operations that follow in theanalysis section of the development. For a com-plete discussion of bifurcation theory the reader isreferred to Iooss and Joseph (1990).

Suppose the reduced equation is given by

dzdt

=G(z, k), z�R1 (1.8)

with the equilibrium solution

G(0, k)=0 (1.9)

where we have assumed the reduced equation to be1-dimensional, i.e. the original n-dimensional sys-tem has only a single zero eigenvalue. We need notmake this assumption, although for most bifurca-tion problems encountered such a simplification isnot restrictive. The study of higher dimensionalbifurcation problems (greater than 2-dimensional)is not a well-defined endeavor above a 2-dimen-sional null space (Mcleod and Sattinger, 1973),accordingly, such situations are beyond the scopeof this introduction in any event. It can be shownthat the equilibrium solutions of Eq. (1.8) corre-spond to equilibrium solutions in the originalsystem. Thus, by studying Eq. (1.8) we are analyz-ing how the original system changes as a functionof the system parameters. Bifurcation theory con-cerns itself with examining the system behavior inregions where the conditions of the implicit func-tion theorem fail. More exactly, if the 1-dimen-sional eigenvalue given by the partial

gz (G(z

)0, k

(1.10)

is non-zero for all arbitrary neighborhoods of theequilibrium point (0, k) then Eq. (1.8) displays no

bifurcation behavior, i.e. the implicit function the-orem is satisfied for all values of k. If however, gz

vanishes at some point, denoted as (z0, k0), thispoint is a possible bifurcation point. The objectiveof bifurcation theory is then to determine thequalitative system properties in the neighborhoodof k0.

For bifurcation to occur, two conditions must besatisfied at the bifurcation point. Firstly, alterna-tive equilibrium solutions must exist in a neighbor-hood around the bifurcation point with theadditional property that at least two solutionbranches intersect at the bifurcation point. Todetermine the existence of alternative solutions wesimply solve Eq. (1.9) for zj=zj(k), j=1, 2, . . . n,i.e. the equilibrium solution branches parameter-ized by the bifurcation parameter k. Note when theclosed form solution of Eq. (1.9) is not practical,approximations of the equilibrium behavior can bedetermined using the tools outlined in Iooss (Ioossand Joseph, 1990). These n branches represent thevarious modes of system behavior that are possiblegiven k. Note the number of solution branches, aswell as, the physical significance of each branchmay change as k is varied, thus not all n branchesmay be accessible at any given k. The secondcomponent of bifurcation is an exchange of stabil-ity at the bifurcation point between a base solutionand a bifurcating, alternative solution. This aspectof bifurcation is described by the system eigenval-ues. In particular, points of possible bifurcation aremarked by zero eigenvalues. The significance ofthese points, in a mathematical sense, was dis-cussed earlier in that they mark points where theconditions of the implicit function theorem areviolated. The points where both properties arepresent mark the location where the system ofequations may assume different qualitative longterm properties.

1.4. Analysis objecti6es

We now, in a formal sense, outline the objectivesof our analysis and discussion1. To determine the qualitative properties of key

enzymes that compete in substitutable andcomplementary environments


Table 1Model parameter set(s) where j=0, 1, 2 and k=a, b

Parameter set: elementary pathways

KejKj 0.01a0.01a

aj0.01a 0.001bKe 0k

0.001bak0 mg

max 0.75c

Rj 0.1b b 0.05c

a Units: g gDw−1.b Units: g gDw−1 h−1.c Units: h−1.

model, and then make the formulation completeby deriving the appropriate forms of the cyber-netic variables that modify the rates of reactionand enzyme synthesis. Once the model systemshave been formulated we turn to the analysisportion of the development and analyze eachelementary competitive structure in turn.

3.1. Elementary con6ergent pathway

The framework of the elementary convergentpathway is shown in Fig. 1. The specific materialflux into the elementary pathway is denoted by

Rk, k=0, 1,…, n (3.1)

and is considered a constant. This is consistentwith the notion of investigating pathway featuresunder balanced growth conditions. The metabo-lites pj, j=0,1,…, n are degraded by key enzymesej at the specific rate

rj=m jmax� ej

e imax

� pj

Kj+pj

, j=0, 1,…, n (3.2)

where m jmax, Kj denotes the rate and saturation

constants governing the formation of pn+1 ande j

max denotes the maximum specific level of keyenzyme ej. The key enzyme ej is assumed to beinduced by the intermediate pj and is expressed atthe specific rate

rej=aj

pj

Kej+pj

j=0, 1,…, n (3.3)

where aj, Kejdenote the rate and saturation con-

stants governing the expression of ej. The interme-diate pn+1 is assumed to be consumed at thespecific rate

rn+1=m jmax pn+1

Kn+1+pn+1

(3.4)

where mn+1max , Kn+1 denote the rate and saturation

constants governing the consumption of pn+1. Weassume en+1�en+1

max .The formulation of the cybernetic regulation

for the elementary convergent pathway was pre-sented by Straight and Ramkrishna (1994) and ispresented here for the sake of completeness. Theobjective of an elementary convergent pathway isassumed to be the maximization of the level of

2. To determine how the qualitative properties ofpurely complementary/purely substitutable ele-mentary pathways are influenced by assembly

3. To hypothesize a resource allocation basis forthe qualitative properties of the various cyber-netic competitive environments

4. To formulate a set of guidelines that can beemployed to rationally construct abstractedcybernetic models of microbial processes.

2. Materials and methods

All model systems discussed in this develop-ment were constructed and simulated within theSimulink environment of Matlab. The modelequations were evaluated using the ODE23s rou-tine within Matlab. Unless otherwise noted allmodel systems were simulated using the parame-ter described by Table 1.

3. Model formulation

In this section we formulate and analyze, usingthe tools of bifurcation theory, the cyberneticmodels of the elementary convergent and diver-gent metabolic pathways shown in Fig. 1. In alarger sense we are analyzing the behavior ofsubstitutable and complementary competition.This allows the results of the analysis to be ex-tended to the linear and cyclic elementary path-ways because these share a common cyberneticstructure. In each case we begin by discussing theprocess equations and kinetics that constitute the


intermediate pn+1. This objective is subject to aconstraint upon the resources available for en-zyme expression. These statements formally be-come the constrained optimization problem

max %n

j=0

pn+1j (Rj) subject to

g=R0+R1+ ···+Rn R (3.5)

where p jn+1 denotes the level of pn+1 produced by

the j th route and Rj denotes the amount of re-source allocated to the j th source of pn+1. Theoptimality condition given by

dpn+10

dR0

=dpn+1

1

dR1

= ···=dpn+1

n

dRn

(3.6)

can be rearranged to yield the matching condition

dpn+1j

dpn+1j + %

n

q=0, j

dpn+1q

=dRj

dRj+ %n

q=0, j

dRq

,

j=0, 1,…, n (3.7)

where the lower summation limit indices denotethe exclusion of the j th element from the set. Eq.(3.7) states the optimum operation of the conver-gent pathway occurs when the fractional returnon investment is equal to the fractional allocationof critical resource. If we assume the allocationpolicy described by Eq. (3.7) is implemented atevery instant in time and that allocation takesplace in time dt, the return on resource investmentcan be measured as the reaction rate. Accord-ingly, the cybernetic variable that governs theallocation of critical resource earmarked for con-vergent pathway operation to the synthesis of keyenzyme ej takes the form

ujs=

rj

rj+ %n

q=0, j

rq

, j=0, 1,…, n (3.8)

where superscript s denotes substitutable. Thederivation of the cybernetic variable that governsthe activity of a convergent key enzyme, denotedas 6 j

s, follows from the cybernetic proportionallaw. This simply states that enzyme activity isproportional to the reaction rate that it catalyzes.Functionally, this implies the relationship

6 js�l rj, j=0, 1,…, n (3.9)

The cybernetic variable 6 js, Ö j is constrained to

obey

056 js51, Ö j (3.10)

which implies the proportionality constant l isgoverned by

05l51rj

, Ö j (3.11)

Eq. (3.11) must be valid Ö j which implies l is ofthe form

l=1

max{r0, r1,…rn}(3.12)

Accordingly, the cybernetic variable that gov-erns the activity of convergent pathway key en-zymes is given by

6 js=

rj

max(r0, r1,…, rn), j=0, 1,…, n (3.13)

We now make the model formulation completeby modifying the rates of reaction and enzymesynthesis with the appropriate cybernetic vari-ables. The intermediate pn+1 is synthesized viakey enzyme ej, j=0, 1,…, n at the modified spe-cific rate

rj 6 js, j=0, 1,…, n (3.14)

where rj denotes the specific rate of pn+1 forma-tion and 6 j

s denotes the cybernetic variable thatgoverns the activity of ej, j=0, 1,…, n. The keyenzyme ej is expressed at the modified specific rate

reju j

s, j=0, 1,…, n (3.15)

where rejdenotes the specific rate of ej expression

and ujs denotes the cybernetic variable that gov-

erns the expression of ej. Consistent with theassumption of maximal en+1 level, we assume thecybernetic variable that governs en+1 activitytakes the value nn+1

s �1.The complete set of mass balances governing

the evolution of the convergent pathway modelthat reflects the input of metabolic regulation isgiven by the set

dpj

dt=Rj−rj 6 j

s−rg pj, j=0, 1,…, n


dpn+1

dt= %

n

j=0

rj 6 js−rn+1 6n+1

s −rg pn+1,

dej

dt=rej

u js− (rg+b)ej+rej

*, j=0, 1,…, n

(3.16)

where rg denotes the specific growth rate and rej*

denotes the rate of constitutive enzyme synthesisof ej. The parameter b denotes the rate constantgoverning the first order decay of ej.

3.2. Elementary di6ergent pathway

The framework for the elementary divergentpathway is shown in Fig. 1. Again, consistentwith the development of Straight and Ramkrishna(1994), we assume the elementary divergent path-way is an isolated local element of a larger globalmetabolic map whose only interaction with therest of the pathway is through material flux con-nections. The metabolic flux into the branch pointis denoted by

R0 (3.17)

and is assumed to be constant. This is consistentwith exploring the system dynamics under condi-tions of balanced growth. The formation ofp1

j, j=1, 2,…, n from p0 is catalyzed by the keyenzyme e0

j, j=1, 2,…, n at the specific rate

r0j =m0

j, max� e0j

e0j, max

� p0

K0j +p0

, j=1, 2,…, n

(3.18)

where m0j, max, K0

j denote the rate constant andsaturation constants that govern the formation ofp1

j and e0j, max denotes the maximum level of key

enzyme e0j. The key enzyme(s) e0

j are assumed tobe induced by p0 and expressed at the specific rate

re 0j =a0

j p0

Ke 0j +p0

, j=1, 2,…, n (3.19)

where a0j, Ke 0

j denote the rate and saturationconstants governing the expression of e0

j. Theintermediates p1

j are assumed to be precursors fordownstream metabolic activity and consumed atthe specific rate

r1j =m1

j, max p1j

K1j +p1

j, j=1, 2,…, n (3.20)

where m1j, max, K1

j denote the rate and saturationconstants governing the utilization of p1

j fordownstream metabolic activity. For the sake ofsimplicity we assume e1

j�e0j, max, j=1, 2,…, n.

This assumption is consistent with the notion ofrestricting our analysis to solely the divergentpathway element.

Straight and Ramkrishna (1994) have postu-lated the objective of a divergent branch point isthe maximization of the mathematical product ofthe branch point metabolites. This objective func-tion is subject to a constraint upon the resourcesavailable for the expression of branch point keyenzymes. These statements formally become theconstrained optimization problem

max�5

n

j=1

p1j(R0

j)�

subject to g= %n

j=1

R0j R

(3.21)

where R0j, j=1, 2,…, n denotes the amount of

resources allocated to the expression of e0j, j=

1, 2,…, n. The optimality condition given by

p12p1

3…p1n dp1

1

dR01=p1

1p13…p1

n dp12

dR02=…

=p11p1

2…p1n−1 dp1

n

dR0n (3.22)

can be rearranged to yield the matching condition

dp1j/p1

j

dp1j/p1

j + %n

q=1, j

dp1q/p1

q

=dR0

j

dR0j + %

n

q=1, j

dR0q

(3.23)

Eq. (3.23) states that optimum divergent path-way operation occurs when the fractional returnon investment is equal to the fractional allocationof resources. If we assume Eq. (3.23) is imple-mented at every instant in time and allocationtakes place on the time scale of dt, the cyberneticvariable that governs the allocation of criticalresources for the expression of e0

j, j=1, 2,…, n isgiven by

u0jc =

r0j/p1

j

r0j/p1

j + %n

q=1, j

r0q/p1

q

, j=1, 2,…, n (3.24)

where superscript c denotes complementary pro-cess. The cybernetic variable that governs the


activity of e0j =1, 2,…, n follows from a restate-

ment of the proportional law. In the case of acomplementary pathway the proportional law isgiven by

60jc �l

r0j

p1j, j=1, 2,…, n (3.25)

Note that the rate of reaction is scaled by itsproduct, this is a feature distinct to complemen-tary processes and is an artifact of the objectivefunction. The cybernetic variable 60j

c is con-strained to obey

0560jc 51, j=1, 2,…, n (3.26)

which implies the proportionality constant isbounded by

05l51

r0j/p1

j, j=1, 2,…, n (3.27)

Eq. (3.27) must hold Ö j, thus, it follows 60jc is of

the form

60jc =

r0j/p1

j

max(r01/p1

1, r02/p1

2,…, r0n/p0

n), j=1, 2,…, n

(3.28)

We make the model system complete by modi-fying the rates of reaction and enzyme expressionby the appropriate cybernetic variables. We thenpresent the complete set of mass balances thatconstitute the elementary divergent pathway. Thebranch metabolite precursor p0 is consumed toform the branch metabolites p1

j via the key en-zyme e0

j at the modified specific rate

r0j 60j

c , j=1, 2,…, n (3.29)

where r0j denotes the specific rate of p1

j formationand 60j

c denotes the cybernetic variable that regu-lates the activity of e0

j. The key enzyme e0j is

induced by p0 and expressed at the modified spe-cific rate

re 0j u0j

c , j=1, 2,…, n (3.30)

where re 0j denotes the specific rate of e0

j expressionand u0j

c denotes the cybernetic variable that regu-lates the expression of e0

j. The complete set ofmass balances governing the time evolution of theelementary divergent pathway which include thecybernetic variables are given by the set:

dp0

dt=R0− %

n

j=1

r0j 60j

c −rg p0,

dp1j

dt=r0

j 60jc −r1

j −rg p1j, j=1, 2,…, n

de0j

dt=re 0

j u0jc − (rg+b)e0

j +re 0j* , j=1, 2,…, n

(3.31)

where rg denotes the specific growth rate and re 0j*

denotes the specific rate of constitutive enzymesynthesis of the j th key enzyme. The parameter b

denotes the rate constant for the first order decayof enzyme e0

j.

4. Analysis and discussion

4.1. Substitutable competition

To begin the analysis of the 2n+2-dimensionalelementary convergent pathway model we rescalewith respect to the following set of dimensionlessvariables and parameters

sj pj

Kj

ej ej d

aj

kj m j

max

d(4.1)

t dt nj r*ej

aj

K( i, j Kei

Kj

(4.2)

where d (rg+b) and j=0, 1,…, n. By assump-tion we are investigating the dynamics of theelementary convergent pathway during balancedgrowth conditions. It follows that rg�mg

max underthis assumption. Furthermore, we assume d�mg

max, i.e. the rate constant governing the firstorder decay of enzyme, denoted as b, is smallcompared with the maximum specific growth rate.Substituting the resealed variables and parametersinto the convergent pathway model under theabove assumptions yields the dimensionless sys-tem in vector form

dxdt

=F(x, k, h), F :R2n+2xRn+1xRn+1�R2n+2

(4.3)

where the state vector x and the rate constantvector k are given by

x {s0, s1,…sn, e0, e1,…, en}T (4.4)


k {k0, k1,…, kn}T, (4.5)

h {h0, h1,…, hn}T (4.6)

The goal of our analysis is to characterize thequalitative features of the convergent pathway asa function of the parameter vector k. To achievesuch a goal we call upon tools from nonlinearanalysis, specifically bifurcation and singularitytheory, to determine the location and nature ofpossible bifurcation points. We then determinethe manner in which the system behavior is al-tered as the parameter values are varied through acritical value. To assure a manageable problemthat still provides a hint of the nature of thegeneral n-dimensional problem, we consider thecase of n=2. The mathematical details of theanalysis are presented in the appendix, we chooseonly to summarize and discuss the results in aqualitative light here.

Let us assume that the fastest rate of p3 forma-tion is r0. The mathematical purpose of such anassumption is detailed in Appendix A, in short, itallows the cybernetic 6 variable to be decomposedinto a form that is more amenable to analysiswhile still maintaining the qualitative features ofthe cybernetic regulatory description. As is true ofall cybernetic developments, the qualitative por-trait of the system dynamics follows directly fromresource allocation policy implemented by themicroorganism. In the case of the convergentpathway, the key enzymes that catalyze the paral-lel routes to p3 formation compete for cellularresources from a single resource pool. Thus, byassuming r0�r1, r2 the key enzyme e0 is seen as amore attractive resource investment from the per-spective of the microorganism because it satisfiesthe local objective better than the other key en-zymes, namely, the production of p3. It followsthat e0 receives the lion’s share of the criticalresources and the synthesis of the remaining ele-mentary pathway enzymes is repressed. This is thebase solution from which we explore the qualita-tive features of the system behavior. When choos-ing different values for the constant vector k weare in effect altering how the microorganism viewsthe key enzymes in terms of their attractiveness asresource investments.

It is shown in the appendix that the elementaryconvergent pathway in which the fastest rate of p3

formation is r0, i.e. e0 is allocated all the criticalresources while e1 and e2 are repressed undergoesbifurcation (pitchfork) when the condition

K1� k1 c1

(1+c1)r0*−1

�K2� k2 c2

(1+c2)r0*−1

�=0 (4.7)

is satisfied where r0* denotes the steady state rateof p3 formation catalyzed by e0 and cj Rj/d.More precisely, condition (4.7) marks the locationin parameter space in which the microbe views theother enzymes that catalyze p3 formation as at-tractive resource investments. Notice there existsthree distinct parameter conditions that allowcondition (4.7) to be satisfied, i.e.

kj

k0

=�1+cj

cj

�u j=1, 2 (4.8)

or both j=1, 2 where u is given by

u e0* s0*

1+s0*(4.9)

and is approximately independent of the systemparameters in a small neighborhood of the bifur-cation point. In terms of the qualitative featuresof the dynamics, these bifurcation points markwhere ej, j=1, 2 assumes non-zero equilibrium so-lutions. Interestingly, the right hand side of thecondition (4.8) is approximately unity, thus, itfollows that as the maximum rate of two compet-ing routes of p3 formation approaches similarvalues the microbe views each as an attractiveinvestment and allocates critical resources to thesynthesis of the key enzymes that catalyze eachparallel route. Condition (4.7) implies three dis-tinct regions of system behavior all of whichspring forth from the base solution. These regionsare shown by simulation of the original modelsystem in Fig. 2. The first region is the basesolution from which we examine the system bifur-cation behavior, i.e. e0 assumes a non-zero steadystate and the other system enzymes are forced tozero, because of a lack of resource allocation.This follows from k0�k1 and k0�k2, i.e. themaximum rate of p3 formation catalyzed by e0 ismuch greater than all other rates making it themicrobes preferred resource investment. The sec-


ond region marks a situation in which e0 and e1

are considered good resource investments while e2

is forced to zero due to a lack of allocation. Thisregion is parametrically characterized by k0#k1

and k0�k2. By symmetry region four followsfrom k0#k2 and k0�k1. Region three marks theunique situation in which all three key enzymesare viewed, from the perspective of the microor-ganism as good resource investments. Thus, itfollows that all three key enzymes are synthesizedin this region. Parametrically, this allocation pat-tern follows from k0#k1 and k0�k2.

From this analysis we see one of the features ofsubstitutable competition is the presence of anenzymatic zero solution. This follows from a re-source allocation argument that stipulates thatenzyme repression is a consequence of a lack ofresource allocation. The question at hand is howthis feature is effected by the competitive structure

as well as other factors such as constitutive en-zyme synthesis. We reserve comment on the influ-ence of constitutive enzyme synthesis until thediscussion in Section 4.3 and consider the compet-itive structure issue presently by examining thequalitative features complementary competition.

4.2. Complementary competition

To ease the algebraic complexity of the analysisand guide the understanding of the qualitativeinteraction present within the equation set, wepropose the following set of dimensionless vari-ables and parameters

r0 p0

K01, rj

p1j

K1j, ej

d

aj

e0j,

kj m0

j, max

d, k̄j

m̄1j, max

d, Gq, j

K0q

K0j

Fig. 2. Simulation results for the elementary convergent pathway in the four regions of system behavior.


K( q, j Ke 0

q

K0j , c0

R0

d, hk

rek*d

where d (rg+b), t dt and j=1, 2,…, n. As-suming balanced growth and slow enzymatic de-cay, substitution of the resealed variables andparameters into the elementary divergent systemmodel yields the resealed system

K01 dr0

dt=c0− %

n

j=1

rj 6 jc−K0

1 r0

K1q drq

dt=rq 6q

c − r̄q−K1q rq, q=1, 2,…, n

dek

dt=rok

ukc −ek+hk, k=1, 2,…, n (4.10)

where the resealed rates take the form(s)

rq=kq eq� r0

Gq, 1+r0

�, q=1,…, n

r̄k= k̄k� rk

1+rk

�, k=1,2,…, n (4.11)

and

rej=

r0

K( j, 1+r0

, j=1,…, n (4.12)

The parameter c0 denotes the resealed specificrate of material influx into p0, and by assumptionis considered as a constant.

It is shown in Appendix A by analysis of theresealed divergent pathway model that key en-zymes experiencing complementary competitionassume non-zero equilibrium levels for all valuesof the system parameters. From a resource alloca-tion perspective, this implies that no key enzymecan dominate over the others under this type ofcompetitive environment. This interesting featurefollows as a consequence of the class of objectivefunctions that characterize complementary com-petition. The class of objective functions thatdefines substitutable competition consist of maxi-mizing the level of a particular intermediate orend product of an isolated pathway, whereas, theobjective of complementary competition is themaximization of the mathematical product of endproducts. This implies, in the case of complemen-tary objectives, that all routes for the productionof end products, for example p1

1, p12,…, p1

n, must

be operating at all times to ensure a non-zeroobjective. Thus, although the level of any one keyenzyme is influenced by the others through re-source sharing, it can not be driven to zero as wastrue of substitutable competition.

4.3. Combination of elementary pathways

Using a modular approach, we seek to con-struct lumped abstracted model frameworks ofmicrobial processes. However, the problem thathas plagued modelers since the beginning of timeis at work in this instance as well. Namely, whatlevel of abstraction is sufficient to suitably de-scribe physical reality? This is an especially ardu-ous question in the modeling of microbialprocesses because of the complexity of biologicalsystems. The approach that we have proposedrelies upon the construction of model frameworksfrom elementary components in such a way as toguarantee the description of desired qualitativebehavioral features. The requirement of the ap-proach is a fundamental understanding of thedynamics displayed by the elementary compo-nents in isolation coupled with awareness of theimpact upon system dynamics of the interactionstemming from assembly. In other words, wemust understand the characteristics of each of theelementary pieces, and moreover, how these char-acteristics interact when the elementary compo-nents are assembled to form the model. Thepurpose of the previous section was the explo-ration of the former, accordingly, we now focusour attention upon the ramifications to systembehavior of assembling elementary units.

The model structure that we consider is shownin Fig. 3. This is an overlapping combination ofan elementary convergent and divergent pathway.Notice that key enzyme e1

a is a member of bothelementary pathways, i.e. is a point of overlapbetween the two elementary components. As suchwe expect, intuitively, the allocation to e1

a to beinfluenced by e0 as well as e1

b competition. How-ever, the competitive structure experienced by e1

a,as well as, the other system enzymes is neithercomplementary nor substitutable. Rather it issome hybrid combination of the two. We explorethe deeper significance of these statements subse-


Fig. 3. Hybrid network composed of an overlapping conver-gent and divergent pathway.

and e1a, max denotes the maximum specific level of

key enzyme e1a. The key enzyme e1

a is assumed tobe induced by the presence of p1 and is expressedat the specific rate

re 1a=a1

a p1

Ke 1a+p1

(4.17)

where a1a, Ke 1

a denote the rate and saturationconstants that govern the expression of e1

a. Theintermediate p1 can also be consumed via keyenzyme e1

b to produce the intermediate p3 at thespecific rate

r1b=m1

b, max� e1b

e1b, max

� p1

K1b+p1

(4.18)

where m1b, max, K1

b denote the rate and saturationconstants that govern the formation of p3 ande1

b, max denotes the maximum specific level of keyenzyme e1

b. The key enzyme e1b is assumed to be

induced by the presence of p1 and is expressed atthe specific rate

re 1b=a1

b p1

Ke 1

b +p1

(4.19)

where a1a, Ke 1

b denote the rate and saturationconstants that govern the expression of e1

b. Theintermediates p2 and p3 are assumed to be precur-sors for downstream metabolites and are con-sumed at the specific rate

rj=m jmax pj

Kj+pj

, j=2, 3 (4.20)

We assume for the sake of simplicity that thekey enzymes which mediate the consumption of p2

and p3 are near their maximum levels, i.e. ej�e j

max, j=2, 3 and can be considered as constant.To make the model formulation complete we

must derive the appropriate forms of the cyber-netic variables that regulate the expression andactivity of key enzymes within the network. Thederivation of the elementary cybernetic variablesthat regulate the convergent and divergent ele-mentary pathways was shown in the previoussection(s) and by Straight and Ramkrishna(1994), so we neglect its reintroduction. We con-struct the cybernetic regulation for the modularnetwork using the tenets of the modular approachdescribed by Varner and Ramkrishna (1998). To

quently, however, in the meantime we formulatethe model equations that constitute theframework.

The material flux into intermediates p0 and p1 isassumed to be constant and denoted as

Rj, j=0, 1 (4.13)

The intermediate p0 is consumed via the keyenzyme e0 to produce the intermediate p2 at thespecific rate

r0=m0max� e0

e0max

� p0

K0+p0

(4.14)

where m0max, K0 denote the rate and saturation

constants that govern p2 formation and e0max de-

notes the maximum level of key enzyme e0. Thekey enzyme e0 is assumed to be induced by thepresence of p0 and is expressed at the specific rate

re 0=a0

p0

Ke 0+p0

(4.15)

where a0, Ke 0denote the rate and saturation

constants that govern the expression of e0. Theintermediate p1 can be consumed by two routes.Firstly, the consumption of p2 via key enzyme e1

a

to produce the intermediate p2 occurs at the spe-cific rate

r1a=m1

a, max� e1a

e1a, max

� p1

K1a+p1

(4.16)

where m1a, max, K1

a denote the rate and saturationconstants that govern the formation of p2 from p1


review, this formulation assumes that metabolicregulation can be decomposed into a regulatoryhierarchy consisting of three interacting layers,namely, elementary, local and global. This type offormulation was developed to provide a system-atic description of cybernetic regulation for mod-els systems formulated from elementarycomponents. The elementary regulatory layer con-trols the allocation of critical resources from theresource pools associated with the elementarypathways to the synthesis of key enzymes compet-ing for resources within a particular elementarypathway. These cybernetic variables are formu-lated following standard cybernetic doctrine, i.e.we postulate an objective function that is subjectto a resource constraint and solve the resultingconstrained optimization problem to yield theforms of the cybernetic variables. The local regu-latory layer describes the interaction of the ele-mentary regulatory components, i.e. overlappingkey enzymes such as e1

a. Because e1a is a member

of two elementary pathways, it can receive criticalresources from each elementary resource pool.Thus, the local layer of regulatory action de-scribes how the multiple elementary allocationsources interact. Lastly, the highest postulatedlevel of metabolic control is termed global con-trol. This regulatory layer controls local metabolicactivity. In a larger sense, this layer of regulatoryaction imparts the higher understanding of themicroorganism to local metabolic activity. Forexample, we postulate the regulatory signals thatcontrol nutritional state dependent metabolic ac-tivity such as maintenance function or storageproduct synthesis are global signals. These signalstranslate the regulatory significance of the nutri-tional state into control action. The cyberneticvariable that reflects all three regulatory levels istermed the complete cybernetic variable. We dis-pense with further introduction of the approachbecause this is not the main focus of the presentdevelopment, however, interested readers arestrongly encouraged to consult the abovereference.

Within the present context, we assume no keyenzyme is subject to global regulation, which im-plies the global cybernetic regulatory componentin this case is unity. The derivation of the local

cybernetic variables follows from the pathwaytopology and the type of intersection of elemen-tary pathways.

The key enzyme e0 is a member of only a singleelementary pathway (convergent). Thus, it com-petes for key cellular resources only from the poolearmarked for convergent pathway operation. Itfollows that the local regulatory component con-sists solely of the elementary substitutable regula-tion, i.e.

u0l =u0

s =r0

r0+r1a (4.21)

where the superscript l denotes local control. Be-cause e0 is not sensitive to any global regulatorysignals, the complete cybernetic variable that reg-ulates the expression of e0 is given functionally as

u0=u0l (4.22)

The complete cybernetic variable that governsthe activity of e0 follows by analogy and is givenfunctionally as

60=60l =

r0

max(r0, r1a)

(4.23)

The key enzyme e1a is a member of both the

elementary convergent, as well as, divergent path-way(s). As such, its synthesis machinery competesfor key cellular resources from both elementaryresource pools. The elementary cybernetic vari-able that governs the allocation of critical re-sources from the pool earmarked for convergentpathway operation, denoted as u1a

s , was derivedpreviously and is given functionally as

u1as =

r1a

r1a+r0

(4.24)

where superscript s denotes substitutable. The keyenzyme e1

a is also a member of the elementarydivergent pathway. The elementary cyberneticvariable that governs the allocation of criticalresources from the resource pool earmarked foroperation of the elementary divergent pathway,denoted as u1a

c , was derived previously and isgiven functionally as

u1ac =

r1a/p2

r1a/p2+r1

b/p3

(4.25)


where superscript c denotes complementary. Thelocal cybernetic variable that governs the expres-sion of e1

a, denoted by u1al must reflect both

possible allocation sources because e1a is sensitive

to the objectives of both elementary pathways.Therefore, we postulate the local cybernetic vari-able consists of the product of the elementaryregulatory components, i.e.

u1al =u1a

c u1as (4.26)

where the multiplication follows as a consequenceof the multiple allocation sources. Because e1

a isnot subject to any global regulatory signals, thecomplete cybernetic variable that governs e1

a syn-thesis consists solely of the local component andtakes the form

u1a=u1al (4.27)

The functional form of the complete cyberneticvariable that governs the activity of e1

a follows byanalogy and is given functionally as

61a=61al =61a

c 61as (4.28)

where the elementary cybernetic variables take theform

61ac =

r1a/p2

max(r1a/p2, r1

b/p3)61a

s =r1

a

max(r1a, r0)

(4.29)

Lastly, the key enzyme e1b is only a member of

the elementary divergent pathway which impliesthat it can only receive resources from the poolearmarked for divergent pathway operation. Thecybernetic variable that governs the allocation ofcritical resources for the expression of e1

b, denotedas u1b

c , was derived previously and takes the form

u1bc =

r1b/p3

r1a/p2+r1

b/p3

(4.30)

Because the key enzyme e1b is a member of only

a single elementary pathway, the local cyberneticregulatory component that regulates its expressionis given solely by the elementary component, i.e.

u1bl =u1b

c (4.31)

Moreover, e1b expression is not sensitive to

global regulatory concerns. Accordingly, the com-plete cybernetic variable that governs the synthe-sis of e1

b is given by

u1b=u1bl (4.32)

The functional form of the cybernetic variablethat regulates the activity of e1

b follows by analogyand is given by

61b=61bl =

r1b/p3

max(r1a/p2, r1

b/p3)(4.33)

We now make the model formulation completeby modifying the rates of reaction and enzymesynthesis by the appropriate cybernetic variables.

The production of p2 from p0 is catalyzed bythe key enzyme e0 at the modified specific rate

r0 60 (4.34)

where r0 denotes the specific rate of p2 formationfrom p0 and 60 denotes the complete cyberneticvariable that governs the activity of e0. The keyenzyme e0 is induced in the presence of p0 and isexpressed at the modified specific rate

re 0u0 (4.35)

where re 0denotes the specific rate of expression of

e0 and u0 denotes the complete cybernetic variablethat governs the synthesis of e0.

The intermediate p1 is consumed to produce p2

via key enzyme e1a at the modified specific rate

r1a 61a (4.36)

where r1a denotes the specific rate of p2 formation

from p1 and 61a denotes the complete cyberneticvariable that governs the activity of e1

a. The keyenzyme e1

a is assumed to be induced in the pres-ence of p1 and is expressed at the modified specificrate

re 1a u1a (4.37)

where re 1a denotes the specific rate of e1

a expressionand u1a denotes the complete cybernetic variablegoverning the synthesis of e1

a.The intermediate p1 can also be consumed to

produce p3 via key enzyme e1b at the modified

specific rate

r1b 61b (4.38)

where r1b denotes the specific rate of p3 formation

and 61b denotes the complete cybernetic variablethat governs the activity of e1

b. The key enzyme e1b


is assumed to be expressed at the modified specificrate

re 1b u1b (4.39)

where re 1b denotes the specific rate of e1

b expressionand u1b denotes the complete cybernetic variablethat governs the synthesis of e1

b.The model framework, including the influence

of metabolic regulation, is given by the set ofdifferential equations

dp0

dt=R0−r0 60−rg p0

dp1

dt=R1−r1

a 61a−r1b 61b−rg p1

dp2

dt=r0 60+r1

a 61a−r2−rg p2

dp3

dt=r1

b 61b−r3−rg p3

de0

dt=re 0

u0− (rg+b)e0+re 0*

de1j

dt=re 1

j u1j− (rg+b)e1j +re 1

j* , j=a, b (4.40)

where rg denotes the specific growth rate and b

denotes the rate constant governing the first orderdecay of the k th key enzyme.

As alluded to earlier, the hypothetical frame-work shown in Fig. 3 is a combination of aconvergent and a divergent elementary pathway.More exactly, the key enzyme e1

a is a member ofboth the convergent and divergent elementarycomponents. From a cybernetic perspective, thisimplies that e1

a competes for key cellular resourcesin both a substitutable, as well as, a complemen-tary environment. In the analysis section weshowed that substitutable and complementarycompetition are marked by qualitatively differenttypes of dynamic behavior. Specifically, in a sub-stitutable environment, it is possible for key en-zymes to assume zero steady states, dependingupon the system parameters, whereas, a comple-mentary environment is characterized by a lack ofbifurcation behavior. Bear in mind, that thesebehavioral traits are inherent to the isolated ele-mentary pathways. Intuitively, we expect e0 to

behave much like a purely substitutable enzyme,and e1

b as a complementary enzyme, however,what behavior should we expect from e1

a? In otherwords, does e1

a behave like a substitutable enzyme,i.e. it has a zero solution or does it behave in acomplementary fashion by only assuming non-zero steady states? The answer to this questioncan be obtained by understanding the resourceallocation structure and the properties of elemen-tary pathways. To this end we postulate thefollowing:

Definition 4.1. A key enzyme that is a memberof both a substitutable and a complementary ele-mentary pathway competes for key cellular re-sources in both competitive environments. Suchan enzyme is termed partially substitutable/par-tially complementary.

Postulate 4.1. A partially substitutable/partiallycomplementary key enzyme, because of the re-source allocation structure, has the possibility of azero solution whose stability is a function ofsystem parameters.

Postulate 4.1 is proven in Appendix A so wedispense with a formal discussion here. Rather wefocus upon the qualitative aspects of the systembehavior and moreover the manner in which thisbehavior follows straightaway from the resourceallocation structure. We have seen in the preced-ing sections that substitutable competition affordsthe possibility of enzymatic zero solutions,whereas, complementary competition yields nosuch behavior. From the modular formulation ofthe cybernetic variables described above and for-mulated in Varner and Ramkrishna (1998) thelocal level of regulation stems directly from solu-tion of the isolated constrained optimizationproblems. In particular, key enzyme e1

a competesfor cellular resources in two distinct competitiveenvironments, i.e. it experiences partially comple-mentary/partially substitutable competition forkey cellular resources. The cybernetic variablethat governs the synthesis of e1

a because of thepossibility of joint allocation is given by

u1a=u1as u1a

c (4.41)

where the functional forms of the elementarycybernetic variables u1a

j , j=s, c follow from thesolution of the isolated constrained optimization


problem. We postulate the presence of the substi-tutable regulatory component affords the possibil-ity of a zero solution for e1

a.The equilibrium behavior of e1

a (characterizedin a rigorous sense in Appendix A) can be borneout through qualitative resource allocation argu-ments. The substitutable aspect of the allocationof critical resources to e1

a competes with e0 toproduce the intermediate p2, whereas, the comple-mentary aspect of the cybernetic regulation com-petes to maximize the mathematical product of p2

and p3. Given this, one could imagine havingalready characterized the qualitative nature ofsubstitutable competition that if r0�r1

a the inter-mediate p2 is produced solely by e0, i.e. e0 is abetter resource investment from the microorgan-ism viewpoint for the synthesis of p2 and conse-quently e1

a levels are driven to zero. The

simulation results for this case are shown in Fig.4. Some remarks are in order before we proceed.

Remark 4.1. Note the complementary objectiveis satisfied under this condition as well as thesubstitutable.

Remark 4.2. Bear in mind e1b always assumes a

non-zero equilibrium solution because it competesin only a complementary environment.

Remark 4.3. Note that investigation of the caser0�r1

a yields no bifurcation behavior. Thus, inthis allocation regime e1

a is always viewed as a badresource investment for the production of p2 rela-tive to e0.

However, e1a does not compete solely in a com-

plementary environment. Rather it feels the influ-ence of both types of competitive structures. Incontrast to the case above, imagine the situationin which r0�r1

a. This case is marked by e1a assum-

Fig. 4. Parallel network evolution when r1a�r0, Ö t. The key enzyme e1

a evolves toward a zero steady state.


Fig. 5. Simulation of hybrid network for r1a�r0. Note that e0 can assume both a zero and non-zero steady state. Steady state values;

Top plate: (e0/e0max, ea

1/ea, max)= (0.2167, 0.1194). Bottom plate: (e0/e0

max, ea1/e1

a, max)= (0, 0.2205).

ing a non-zero equilibrium solution and e0 levelsbeing driven to zero. Following the same logicas the substitutable analysis, as e0 becomes amore attractive resource investment from the mi-croorganisms viewpoint it assumes a non-zeroequilibrium solution. In fact it is shown in Ap-pendix A that the base zero solution (e0=0,e1

a"0) bifurcates to a non-zero level of e0 whenthe condition

k0

k1a=U (4.42)

is satisfied where U is the compound parameter

U k2 k3

k3

(4.43)

Notice that condition (4.42) is very similar instructure to the bifurcation condition determined

in the pure substitutable case with the exceptionof U. The compound parameter U can beshown to be the influence stemming from thecomplementary aspect of the system regulation.Clearly, this is the case because U reflects theregulatory influence stemming from the metabo-lite feedback loops that are a characteristic ofcomplementary competition. In particular, thecomplementary regulatory aspect influences e0

expression (via e1a synthesis) depending upon the

rate of removal of the intermediates p2 and p3.If U is small, the maximum rate of formation ofthe intermediate p2 via key enzyme e1

a must belarge when compared with k0 for e0 to be anattractive resource investment. The converse be-ing true if U is large. This case is simulatedusing the original model system and the resultsare shown in Fig. 5.


Thus, a partially substitutable/partially com-plementary key enzyme can assume both a non-zero and a zero equilibrium solution dependingupon the system parameters. In particular, ifr0�r1

a then e0 is seen as an attractive resourceinvestment and e1

a levels are driven to zero. Inthis case the intermediate p2 is produced in asubstitutable manner. However, if r1

a�r0 usingthe same arguments, e0 levels can be driven tozero because e1

a represents a better resource in-vestment from the perspective of the microor-ganism. It follows in this case p2 is produced ina complementary fashion. Lastly, as k0\Uk1 e0

and e1 are both synthesized and p2 is producedvia both routes. This behavioral switch followsas a consequence of the substitutable aspect ofthe system regulation.

4.4. Effect of constituti6e enzyme synthesis

The presence of constitutive enzyme synthesisdoes nothing to qualitatively alter the comple-mentary aspect of the system regulation, how-ever, as alluded to previously it can significantlyalter the characteristics of substitutable environ-ments. More precisely, it destroys the possibilityof substitutable zero solutions. Intuitively, onewould expect such a phenomena because thoseenzymes that are seen as poor resource invest-ments can not be driven to zero because someexpression is taking place, i.e. constitutive ex-pression, independently of the inducible portion.From a mathematical perspective, the presenceof constitutive enzyme synthesis is formallytreated as a system imperfection or breakingproblem as outlined by Iooss and Joseph (1990).The details of the analysis are presented in Ap-pendix A. Within the present discussion, theparametric location and the physical significanceof the bifurcation are altered somewhat by thepresence of constitutive enzyme synthesis.More precisely, bifurcation in this context isbetween two non-zero equilibrium solutionsrather than between a zero and non-zero equi-librium behavior. Note this statement assumesall enzymes possess a constitutive synthesis ele-ment.

5. Topological abstraction guidelines

The analysis of the model framework(s) shownabove shed light upon the type of behavior thatcan be expected from particular competitive struc-tures. The objective of this section is to translatethis understanding into a set of guidelines thatcan be used to formulate cybernetic model sys-tems. In reality, these guidelines are nothing morethan a restatement of the previous results in con-densed form. We choose to state the guidelines inthe form of postulates and illustrate the applica-tion of these postulates in subsequent papers.

Postulate 5.1. The qualitative dynamics of acybernetic model can be determined by inspectionof the topological structure. Specifically, the dy-namics follow from the competitive structure thata key enzyme is experiencing. The competitivestructure is a non-unique function of the topol-ogy. Convergent and linear fragments are substi-tutable, whereas, divergent and cyclic units arecomplementary.

Remark 5.1. This assumes that the abstractedmodel framework is constructed from the elemen-tary pieces as derived by Straight and Ramkr-ishna. In a more general sense, the behavior stemsfrom the structure of the competition for cellularresources and in particular the form of the objec-tive function and resource constraint.

Postulate 5.2. Key enzymes that compete forcellular resources in a substitutable manner havethe possibility of both non-zero, as well as, zeroequilibrium solutions depending upon the systemparameters. As a rule of thumb, in the absence ofconstitutive enzyme synthesis, the fastest processin a substitutable competitive environment domi-nates the allocation process, thus, driving the levelof key enzyme(s) that catalyze the slower pro-cesses to zero.

Postulate 5.3. Key enzymes that compete forcellular resources in a complementary mannerevolve only toward non-zero steady states for allvalues of the system parameters.

Postulate 5.4. Overlapping enzymes that lie inmultiple allocation regimes can receive key cellu-lar resources from each of the elementary resourcepools. Thus, because the optimization for eachelementary allocation problem is performed in


isolation, overlapping key enzymes possess thecharacteristics of the totality of competitive struc-tures in which it competes.

Postulate 5.5. Key enzymes that compete inpartially complementary/partially substitutableenvironments assume the qualitative properties ofthe substitutable environment, i.e. have the possi-bility of assuming a zero solution in the face ofsevere competition for key cellular resources.

Remark 5.2. In the case of partially substi-tutable/partially complementary competition, thecomplementary competitive aspect alters the loca-tion of bifurcation of the substitutable competi-tion. The communication between elementarypathways is a consequence of having a single keyenzyme compete for resources from both elemen-tary pools.

Postulate 5.6. Constitutive enzyme synthesis de-stroys the possibility of zero solutions in a substi-tutable competitive environment. It has noqualitative effect upon the dynamics of comple-mentary structures.

These postulates represent a set of rules thatcan be used to guide the construction of ab-stracted cybernetic models. In the general case,given an arbitrary model structure, they can beemployed to determine the qualitative dynamicsthat a particular model system displays. We illus-trate this approach in subsequent papers by con-sidering the construction and analysis of modelsystems capable of describing complex uptakepatterns.

6. Conclusions

In this work we have investigated the qualita-tive nature of cybernetic models using tools frombifurcation theory. Specifically, we have deter-mined the qualitative properties of substitutableand complementary competition for key cellularresources as a function of the system parameters.The hallmark of substitutable competition is thepossibility of an enzymatic zero solution. In cy-bernetic terms this implies the key enzyme whichcatalyzes the fastest rate dominates the allocationof critical resources, leaving little resource for theremaining enzymes. Moreover, as a function of

system parameters substitutable competition wasshown to undergo bifurcation between states ofequitable sharing of resources and complete dom-inance by specific key enzymes. Complementarycompetition, on the other hand, is marked by theabsence of an enzymatic zero solution. In fact, nobifurcation behavior was observed for all valuesof the system parameters. The combination ofsubstitutable and complementary competition re-sulted in a regulatory structure whose dynamicfeatures bear a striking resemblance to purelysubstitutable competition. The partially substi-tutable partially complementary structure sharesthe qualitative features of elementary substi-tutable competition, namely, the existence of anenzymatic zero solution. However, the parametriclocation of bifurcation is influenced by the com-plementary aspects of the system.

In a larger sense this work serves as an intro-duction to a conceptual methodology that can beemployed to guide the abstraction of metabolicpathways. In particular, we have solved the cyber-netic structure function relationship. In otherwords, we now have an understanding of thequalitative features that can be expected fromsubstitutable and complementary competition.This is the function side of the equation. Thestructural aspect, i.e. the abstraction of metabolicpathways, stems from a modular constructionconcept. This methodology follows from the pos-tulate that an arbitrary metabolic network can bedecomposed and abstracted into elementary com-ponents, similar to pieces in a jigsaw puzzle. Byunderstanding the function associated with a par-ticular piece, and moreover, how this function isaltered by interaction with other elementary com-ponents we gain insight into the qualitative natureof the dynamics of the abstracted pathway model.This affords the ability to approach the problemof formulating a cybernetic model with a desiredset of dynamical features from a grey box model-ing perspective. In other words, using this ap-proach, we can produce entire classes of modelsystems with desired behavioral traits, simply byassembling the elementary pieces in a particularfashion.

This paper is the first of a series to be publishedelsewhere in the literature. The objective of the


remaining papers is to illustrate these concepts byconsidering a number of example systems. Firstly,we consider the mineralization of Aniline in thepresence/absence of secondary carbon and nitro-gen sources. For this system we present two de-generate cybernetic model formulations, both ofwhich are capable of describing experimental nu-tritional uptake patterns. The last two examplesare purely illustrative in scope and are presentedin a more general spirit. The first formulation is acybernetic model that is capable of describing anarbitrary nutritional uptake pattern given threesubstrates. This model is a generalization of themodel framework recently published by Ramakr-ishna et al. (1997). We then having analyzed thisstructure, generalize it to the n-dimensional mix-ture problem and present a model framework thatis capable of predicting uptake patterns for ncarbon sources.

Appendix A. Bifurcation analysis of competitiveenvironments

A.1. Bifurcation analysis: substitutablecompetition

To begin the analysis of the 2n+2-dimensionalelementary convergent pathway model we rescalewith respect to the following set of dimensionlessvariables and parameters

sj pj

Kj

ej ej d

aj

kj m j

max

d(A.1)

t dt hj rej*

aj

K( i, j Kei

Kj

(A.2)

where d (rg+b) and j=0, 1, …, n. By assump-tion we are investigating the dynamics of theelementary convergent pathway during balancedgrowth conditions. It follows that rg�mg

max underthis assumption. Furthermore, we assume d�mg

max, i.e. the rate constant governing the firstorder decay of enzyme, denoted as b, is smallcompared with the maximum specific growth rate.Substituting the recealed variables and parametersinto the convergent pathway model under theabove assumptions yields the dimensionless sys-tem in vector form

dxdt

=F(x, k, h), F :R2n+2×Rn+1�R2n+2

(A.3)

where the state vector x and the rate constantvector k are given by

x {s0, s1,…, sn, e0, e1,…, en}T (A.4)

k {k0, k1,…, kn}T, (A.5)

h {h0, h1,…, hn}T (A.6)

The goal of our analysis is to characterize thequalitative features of the convergent pathway asa function of the parameter vector k. To achievesuch a goal we call upon tools from non-linearanalysis, specifically bifurcation and singularitytheory, to determine the location and nature ofpossible bifurcation points. We then determinethe manner in which the system behavior is al-tered as the parameter values are varied through acritical value. To assure a manageable problemthat still provides a hint of the nature of thegeneral n-dimensional problem, we consider thecase of n=2. Before we can begin the bifurcationanalysis, however, we address a technical issuerelated to the form of the cybernetic 6 variable.

For the j th substitutable process, the elementarycybernetic variable that governs the activity of ej,denoted as 6 j

s, takes the form

6 js=

rj

max(r1, r2,…, rj,…, rn), Ö j (A.7)

The max function in the denominator of 6 js

makes the system very difficult to deal with ana-lytically. Accordingly, we employ a methodologythat has been developed to combat this concern,yet still maintains the essential information thatthe 6 variable imparts to the system. Let us sup-pose that the i th rate of pn+1 formation is thefastest rate. Dividing the numerator and the de-nominator of (A.7) by ri yields

6 js=

gj, i

max(g1, i, g2, i,…, gj, i,…, gn, i), Ö j,

gj, i rj

ri

(A.8)

Following the same logic the elementary substi-tutable cybernetic variable that governs the ex-pression of ej, denoted as uj

s, given by


ujs=

rj

rj+ %n

k=1, j

rk

, Ö j (A.9)

can be rewritten as

ujs=

gj, i

gj, i+ %n

q=1, j

gq, i

, Ö j (A.10)

where we divided the numerator and denominatorby the fastest rate of pn+1 formation. We arbitrar-ily assign ri r0. This does not have to be so,however, it eases the algebraic complexity whilenot restricting the generality of analysis. Accord-ingly, we rewrite Eqs. (A.8) and (A.10) with n=2to yield

6 js=

gj, 0

maxq(gq, 0),

ujs=

gj, 0

gj, 0+ %2

q=0, j

gq, 0

, j=0, 1, 2 (A.11)

When j=0 we arrive at the control variable set

60s =

1max(1, g1, 0, g2, 0)

, u1s =

1

1+ %2

q=1

gq, 0

(A.12)

Because we have assumed the r0 is the fastestrate, for some period of time t*� [t1, t2] 60

s =1.During t* the other 6 s variables, i.e. 61

s, 62s are

given by

61s =g1, 0, 62

s =g2, 0 (A.13)

Following the same logic, the cybernetic vari-able u0

s can be rewritten as

u0s =

1

1+ %2

q=1

gq, 0

(A.14)

If 1� (g1, 0=g2, 0) while t*� [t1, t2], then u0s#1

and

u1s#g1, 0, u2

s#g2, 0, Ö t* (A.15)

This breakdown has the effect of reducing ouroriginal system model to two distinct sets of equa-tions which we term the unregulated and theregulated set, respectively. The unregulated set is

composed of the mass balances for the quantitiesthat take part in the fastest process. Because wehave assumed r0 to be the fastest, the unregulatedset is made up of the s0 and e0 balances

K0

ds0

dt#c0−k0 e0 s̄0−K0 s0, s̄0

s0

1+s0

(A.16)

de0

dt#re 0

−e0+h0 (A.17)

where 60s =1, u0

s#1 and cj Rj/d denotes therescaled material input flux towards p0. By con-trast, the regulated set consists of the remainderof the system mass balances, i.e.

Kj

dsj

dt#cj−kj ej s̄j gj, 0−Kj sj, j=1, 2 (A.18)

dei

dt#

si

K( i, j+sj

gi, 0−ei+hi, i=1, 2 (A.19)

These two sets of equations (regulated and un-regulated) define what is termed the region set andcan be recast in vector form as

dxRS

dt#H(xRS,k, h),

xRS {xUR, xR}T (A.20)

where xUR, xR are the state variable vectors of theunregulated and regulated set, respectively andH(xRS, k, h) is defined as

H(xRS, k, h) {f(xUR, k, h), g(xRS, k, h)}T

(A.21)

where f(xUR, k, h), g(xRS, k, h) are the vectorscorresponding to the right hand sides of the un-regulated and regulated sets, respectively. Theequation set (A.20) defines a region in the solutionstate space in which the cybernetic regulatorymachinery can be broken down as shown. Clearly,this decomposition hinges upon the rate r0. If r0

remains the fastest rate for Ö t then this regionbecomes invariant with respect to time. However,if r0 remains the fastest rate for only some finiteperiod of time, i.e. there exist non-infiniteboundaries for the interval [t1, t2] then the regula-tory system can be broken down as shown only aslong as t� t*. This distinction has a profound


effect upon system behavior. If the regulatoryregion is invariant, then the long time behavior ofthe original system is the stable equilibrium be-havior of equation set (A.20). Conversely, if weenter and/or leave the regulatory region in somefinite period of time, the behavior of equation set(A.20) will appear only as a transient in theoverall long time behavior of the full systemmodel. Moreover, it can be shown that there willalways exist, if the cybernetic system possesses astable steady state, an invariant regulatory region.Thus, those regions which appear as transientssimply represent the manner in which the regula-tory picture changes with respect to time as wemove towards a stable steady state. In otherwords, we move through the transient regions onour way towards the invariant region as timeprogresses.

The dynamics displayed by the unregulated setare approximately independent of the rest of thesystem, thus, as t�� xUR can be approximatedby the long time equilibrium behavior of equationset (A.16). To determine this behavior we mustsolve the system

f(xUR, k, h)=0 (A.22)

i.e. determine the equilibrium behavior of theunregulated system. If we assume K( 0, 0�s0, Ö t(this assumption implies enzyme expression is sat-urated with respect to inducer metabolite) then(A.22) has the solution set (e*0, s*0 )T

e0*=1+h0,

K0(s0*)2+ (K0+k0(1+h0)−c0)s0*−c0=0(A.23)

which can be shown to be a stable equilibriumpoint for all values of the system parameters. Theregulated set given by

dxR

dt#g(xR, k, 0) (A.24)

has an equilibrium solution set x*R=(s*1 , e*1 , s*2 , e*2 )T

xR*={c1, 0, c2, 0}T (A.25)

Notice that we have defined our equilibrium setas the solutions of the system g(xR, k, 0), h=0.

This is done so as to consider h (which denotesthe scaled specific rate of constitutive enzymesynthesis) as an imperfection parameter. Solutionset (A.25) denotes the system zero solution forenzymes ej, j=1, 2. Thus, even at this early stage,we see that substitutable competition has an allo-cation structure that allows for some enzyme lev-els to be driven to zero. From a resourceallocation viewpoint, this implies that one keyenzyme, if competitive enough, can dominate overthe others. Introducing the new variable y�R4

given by

xR=xR*+y (A.26)

translates equilibrium set (A.25) to the origin.This solution set forms the base set from whichwe investigate any possible bifurcation that mayarise in the system. A point of possible bifurcationis marked by a zero eigenvalue, i.e. the Jacobianmatrix becomes singular at this point. In a moreprecise sense we seek the parameter value k0 suchthat Det{J(k0)}=0. Physically, by examining bi-furcation behavior with respect to the systemparameters, specifically the process rate constants,we are exploring the ramifications of differentlevels of competitive vigor amongst the enzymeexpression systems. The translated system givenby

dydt#g(y, k, 0) (A.27)

has the Det (J) at the origin, where ji, k (gi

(yk

,

given by

K1� k1 c1

(1+c1)r0*−1

�K2� k2 c2

(1+c2)r0*−1

�(A.28)

where r0* is the rate of the fastest process evalu-ated at the equilibrium point of the unregulatedsystem. If we define

r0* k0 u, u e0* s0*

1+s0*(A.29)

then the critical value of the system parametersi.e. points of possible bifurcation are given by

bj=kj

k0

� cj

(1+cj)u�

−1, j=1, 2 (A.30)


Examination of Eq. (A.30) clearly indicatesthree distinct bifurcation possibilities exist forthe regulated system. Firstly, b2 could indepen-dently cross through a critical value i.e. b2=0,b3"0 which implies

k1

k0

=�1+c1

c1

�u (A.31)

Similarly b3 could cross through zero indepen-dently (b2"0, b3=0) which implies

k2

k0

=�1+c2

c2

�u (A.32)

Both of the previous cases mark simple cross-ings with a 1-dimensional Jacobian null-space.The last case, however, involves b2=b3=0simultaneously which implies a 2-dimensionalnull-space and is a slightly more involvedproblem.

The physical significance of the parametric re-lationship amongst the critical condition(s) hasan interesting interpretation in terms of resourceallocation. If the material flux into the elemen-tary convergent pathway is large, i.e. cj�1 thenthe right hand side of the critical condition isapproximately unity. This implies that bifurca-tion occurs when the ratio of the rate constantsis unity. From a resource allocation viewpoint,this condition marks the point where the invest-ment attractiveness of competing processes isapproximately equal.

We begin the bifurcation analysis with thefirst two cases, then comment upon the 2-dimen-sional null-space problem.

A.1.1. Case a, b: 1-dimensional null-spaceThe Jacobian matrix for the regulated bal-

ances when evaluated on the base solution caneasily be shown to be of the form

J=�J1

00J2

�(A.33)

where 0 denotes a 2×2 null block and theblocks Ji are given by

Ji=�−Ki

00b i

�, i=1, 2 (A.34)

Two observations are useful at this point.Firstly note that the Jacobian of the regulatedsystem is diagonal. This will be of great usesubsequently when we employ a reduction tech-nique to isolate the bifurcation problem. Sec-ondly, note that the balances for the second andthird process are not coupled and are identical.The latter feature follows as a consequence ofthe regulatory region decomposition. We takeadvantage of these properties because they implywe have two degenerate bifurcation problems.We need only to study one to gain informationabout both systems.

As is standard practice, we utilize a reductiontechnique to isolate the portion of the systemthat is responsible for the bifurcation. In ourcase, because the Jacobian is diagonal and hencepossesses a complete orthonormal family of ei-genvectors we utilize Lyapunov–Schmidt reduc-tion (Golubitsky and Schaeffer, 1985) to obtainthe reduced system or equation. Furthermore,because our bifurcation problem is 1-dimen-sional, the reduced system will consist of a sin-gle equation whose behavior will control thequalitative aspects of the system dynamics. Inthis development we skip much of the technicaldetail related to the reduction procedure andpresent only the reduced equation. However,firstly, to make the algebra more tractable, weapproximate the system by a Taylor seriesaround the origin retaining up to cubic terms.Note, because of the degeneracy of the systemwe need only expand and study one set of bal-ances. Accordingly, to keep the analysis general,we expand and study the balances around thei th process i.e. ri where i=1, 2. More preciselyif we define zk {yk, yk+1}T and further de-fine gk {gk, gk+1}T where k=1 or 3 thenthe degenerate problem can be caste in vectorform

dzk

dt=gk(z, k, h), z�R2 (A.35)

(see Eq. (A.27)). Expanding Eq. (A.35) in aTaylor series about the origin yields


where G2k(zk, b, h) denotes the vector of second

order terms. In a more practical sense, the sym-metry of J implies that for a 1-dimensional cross-ing one bifurcation parameter is held constantwhile the other is varied. Accordingly, we assumeb1 is held constant while we vary b2, i.e. we varythe parametric nature of e2 consumption. Apply-ing Lyapunov–Schmidt reduction to equation set(A.36) and relaxing the requirement that hj=0yields the reduced equation

dz

dt=b2 z−a(b2)z3+h2,

a(b2)=�u(1+b2)3k1 K2

c2(K2+c2)�

(A.37)

where z is the reduced variable. Note in theabsence of constitutive enzyme synthesis Eq.(A.37) is the pitchfork bifurcation normal form.Thus, as b2 is varied through the critical value weexpect z to bifurcate from the zero solution to anon-zero equilibrium solution. As a matter ofconvenience, we choose to eliminate the parame-ter dependence of the coefficient of z3. Accord-ingly, we introduce the scaled reduced variable

z u

f(b2)(A.38)

where f(b2)"0, Ö b2. Substituting this rescalinginto Eq. (A.37) yields

dudt

=b2 u−u3+ h̄2(b2) (A.39)

f(b2) a(b2) (A.40)

h̄2(b2) h2(a(b2)) (A.41)

The solutions of the reduced Eq. (A.39) controlthe qualitative nature of the dynamics of the fullsystem. More exactly, these solutions correspond,qualitatively, with the equilibrium behavior of e2.Thus, by examining Eq. (A.39) we gain insight asto how the dynamics change as a function ofsystem parameters. In our case we are using thecompound parameter b2 as the bifurcationparameter. Physically, this parameter reflects the

dzk

dt=Jk

!k=1k=3

J1(b1)J2(b2)

zk+G2k(zk, b, h)+O(3), k=1 or 3

influence of the rescaled reaction rate constants aswell as the rescaled specific input flux and satura-tion constants. In the absence of constitutive en-zyme synthesis, Eq. (A.39) has the equilibriumsolutions

u1=0 u29=9b (A.42)

As expected in the neighborhood of the bifurca-tion point two (physically realistic) solutions arepossible for the reduced variable j. In biologicalterms, this implies that two solutions are possiblefor the key enzyme e2 as a function of the systemparameters. The solution u1 corresponds directlywith the zero solution of (A.25). However, u29

corresponds to a non-zero level of e2. Whichsolution is observed is a function of the stabilityproperties. The stability of the solution set (A.42)to small disturbances is governed by the sign ofthe eigenvalue associated with the j th solutionbranch. The eigenvalue in 1-dimension is reallynothing more than the linear coefficient of thetaylor series expansion of the reduced equationwith respect to the reduced variable u around thebifurcation point, denoted as (b0, u0

j), as a func-tion of b. More precisely, the 1-dimensional ei-genvalue of the general reduced equation h(b, u)is given by

lj=(h(u)b, u j

=hu (A.43)

where lj denotes the eigenvalue along the j th

solution branch. It is easily shown that hu van-ishes at the bifurcation point for double pointbifurcation, yielding no stability information. Ac-cordingly, we must appeal to the stability infor-mation contained in the higher order terms of theeigenvalue expansion. Expanding hu around thebifurcation point with respect to the parameter b

and retaining the linear terms yields

lb, j=#�(hu

(b+(hu

dudb

�db+O(�db �2) (A.44)


where db b−b0 and where lb, j denotes theeigenvalue expansion along the j th solution branch.Given the form of Eq. (A.39), lb, j reduces to theset

lb, 1=db lb, 29= −2d b (A.45)

This indicates that solution 1, i.e. the zerosolution is stable when b\b0 and unstable whenbBb0, the converse being true for solution 2. Inthe present case, b0=0 implies the zero solutionstrictly exchanges stability (pitchfork bifurcation)with the bifurcating solution. The stability analysisresults are summarized in Table 2 where systembehavior is summarized as a function of (b1, b2).The analysis prediction are simulated and theresults presented in Fig. 2.

Let us return to the discussion of bifurcationbehavior as a consequence of the system resourceallocation structure. We have shown that bifurca-tion (in the absence of constitutive enzyme synthe-sis) occurs when bj=0. This condition physically,as alluded to previously, marks a location in whichthe resource investment attractiveness of one pro-cess versus another is approximately equal. Inparticular we have shown that as bj moves throughzero from the left the enzymatic zero solutionexchanges stability with a non-zero enzyme level.Recast in terms of resource allocation, bj=0 marksthe boundary at which ej can be driven to zero. Inother words, as bj is varied through the criticalvalue, the attractiveness of the j th process as aresource investment increases and finally becomessignificant enough to warrant investment by themicroorganism.

A.2. Two-dimensional null space

Investigation of the double degenerate case isactually a more general development than the1-dimensional null space analysis and yields similarinformation with one notable exception. In theprevious section we, in effect, analyzed the crossingon the bj axis at any point other than through theorigin. Analysis of the double degenerate systemaffords this possibility. However, given the previ-ous results, it is easily seen from Fig. 2 that acrossing through the origin of the (b1, b2) planeresults in the symmetric allocation reversal betweene1 and e2. In other words, in quadrant 2 of the(b1, b2) plane e1 is synthesized and e2 assumes a zerosolution. As we move through the origin intoquadrant 4, this steady state profile is symmetricallyreversed, i.e. e2 becomes a more attractive resourceinvestment than e1 and the allocation decision toexpress e1 over e2 is reversed. Thus, from a quali-tative perspective no new information is gained bypursuing the double degenerate analysis. Accord-ingly, we neglect the consideration of this case withthe understanding that such an endeavor is readilyavailable given the tools presented by Iooss (Ioossand Joseph, 1990) and Mcleod (Mcleod and Sat-tinger, 1973).

A.3. Constituti6e enzyme synthesis

In the previous analysis we have neglected con-stitutive enzyme synthesis, however, from the ap-pearance of hj in Eq. (A.39) its presence mayqualitative alter the dynamics of the system. In fact,upon closer examination of Eq. (A.39) an impor-tant feature can be borne out, namely, the destruc-tion of the system zero solution. Presently, weneglect the consideration of this case and choose torevisit the issue subsequently in Section A.5.

A.4. Bifurcation analysis: elementary di6ergentpathway

To ease the algebraic complexity of the analysisand guide the understanding of the qualitativeinteraction present within the equation set, wepropose the following set of dimensionless vari-ables and parameters

Table 2System behavior as a function of (b1, b2)a

Parameter region e0 e1 e2

+b1\0, b2\0 +++ 0+b1\0, b2B0

+ 0 0b1B0, b2B0+b1B0, b2\0 0+

a The + symbols indicate non-zero enzymatic steady stateswhereas 0 indicates stable zero solutions. Note results obtainedunder the assumption of no constitutive enzyme synthesis.


r0 p0

K01, rj

p1j

K1j, ej

d

aj

e0j,

kj m0

j, max

d, k̄j

m̄1j, max

d, q, j

K0q

K0j

K( q, j Ke 0

q

K0j c0

R0

dhk

rek*d

where d (rg+b), t dt and j=1, 2,…, n. As-suming balanced growth and slow enzymatic de-cay, substitution of the rescaled variables andparameters into the elementary divergent systemmodel yields the rescaled system

K01 dr0

dt=c0− %

n

j=1

rj 6 jc−K0

1 r0

K1q drq

dt=rq 6q

c − r̄q−K1qrq, q=1, 2,…, n

dek

dt=rek

ukc −ek+hk, k=1, 2,…, n (A.46)

where the rescaled rates take the form(s)

rq=kq eq� r0

Gq, 1+r0

�, q=1,…, n

r̄k= k̄k� rk

1+rk

�, k=1, 2,…, n (A.47)

and

rej=

r0

K( j, 1+r0

, j=1,…, n (A.48)

As was true in the convergent pathway, wemust employ the regulatory region concept tosimplify the structure of the cybernetic regulation.For a general elementary complementary process,the cybernetic variable that governs the activity ofthe j th key enzyme, denoted as 6 j

c, is given func-tionally as

6 jc=

r0j/p1

j

max(r01/p1

1, r02/p1

2,…, r1n/p1

n), j=1, 2,…,n

(A.49)

Dividing the numerator and denominator bythe largest scaled rate

r0k

p1k (A.50)

yields the rescaled 6 variable

6 jc=

r0j

p1j

�p1k

r0k

�max

��r01

r0k

��p1k

p11

�,…,

�r0j

r0k

��p1k

p1j

�,…,

�r0n

r0k

��p1k

p1n

��j=1, 2,…, n (A.51)

If we define the quantity z( h, j as

z( h, j �r0

h

r0j

��p1j

p1h

�(A.52)

Eq. (A.51) can be recast in the form

6 jc=

z( j, k

max({z( q, k}q=1n )

, j=1, 2,…, n (A.53)

The cybernetic variable that regulates the ex-pression of the j th enzyme belonging to an ele-mentary complementary pathway, denoted as uj

c,is given functionally as

ujc=

r0j/p1

j

%n

q=1

r0q/p1

q

, j=1, 2,…, n (A.54)

Division of the numerator and denominator byEq. (A.50) and substituting Eq. (A.52) yields thecontrol variable

ujc=

z( j, k

%n

q=1

z( q, k

, j=1, 2,…, n, Ö k (A.55)

Our intention is to invoke the regulatory regionconcept developed in the analysis of the elemen-tary convergent process to decompose the regula-tion in this case. However, because of thestructure of the system of equations, we cannotdecompose the model into unregulated and regu-lated portions while still maintaining the inherentmathematical structure of the cybernetic regula-tion. Accordingly, we turn to time scale argu-ments and the properties of the cyberneticvariables to decompose the system regulation. Toease the analysis we consider the n=2 case.

Because we are analyzing an elementary pro-cess, i.e. one in which all enzymes present canonly receive resources from the pool earmarkedfor the operation of the k th elementary pathwaythe cybernetic u variables have the property


%n

j=1

ujc=1 (A.56)

Using this property we are able to rewrite uqc as

uqc =1− %

n

j=1, q

u jc (A.57)

If we assume, for the sake of argument, thatr0

1/p11�r0

j/p1j, Ö j for some time t* on the interval

t*� [t0, t1], t1] t0 then Eq. (A.55) can be recast asthe set

u1c�1, u2

c�z2, 1 (A.58)

Using Eqs. (A.58) and (A.57) the elementarydivergent enzyme balances can be rewritten as

de1

dt#re 1

(1−z2, 1)−e1+h1, (A.59)

de2

dt#re 2

z2, 1−e2+h2 (A.60)

Following the same logic, the cybernetic 6 vari-ables can be decomposed as

61c�1, 62

c�z2, 1 (A.61)

which implies the balances on the dimensionlessmetabolites rj, j=1, 2,…, n take the form

K01 dr0

dt#c0−r0

1−r02 z2, 1−K0

1 r0

K11 dr1

dt#r0

1−r11−K1

2 r1

K12 dr2

dt#r0

2 z2, 1−r12−K1

2 r1 (A.62)

Before we proceed, we make some observationsabout the decomposed enzyme and metabolitebalances. Notice, as an artifact of the originalrescaling, the j th time derivative term of themetabolite balances is multiplied by the ( j )th satu-ration constant. If we define the relationship

K01=g1, j K0

j, j=1, 2,…, n (A.63)

(g1,1=1) then the metabolite balances can be re-defined in vector form as

K01 dr

dt#GF(r, e, K) (A.64)

where r (r0, r1, r2)T, {gj, k=0, gj, j"0}�G andF(r, e, K) denotes the vector of the right hand

sides of the decomposed metabolite balances. In asimilar way, we rewrite the enzyme balances invector form

de

dt#G(r, e, K0) (A.65)

where e (e0, e1) and G(r, e, K0) denotes the vec-tor of right hand sides of the decomposed enzymebalances. The system

K01 dr

dt#GF(r, e, K)

de

dt#G(r, e, K0) (A.66)

clearly operates on multiple time scales, i.e. theparameter K0

1 can act as a singular perturbationparameter that can be employed to separate thetime evolution of the system into slow and fastmoving components. More exactly, if we considerthe limiting case of ��K0

1��1 the metabolite bal-ances are approximately constant and the onlydynamic movement is with respect to enzymelevel, i.e.

GF(r, e, 0)#0 (A.67)

de

dt#G(r, e, 0) (A.68)

However, if we define a new time scale l givenby

l t

K01 (A.69)

Eq. (A.66) can be recast as

dr

dl#GF(r, e, K) (A.70)

de

dl#K0

1 G(r, e, K01) (A.71)

In the limiting case of K01�0 the l time scale is

infinite and the system displays no dynamic move-ment with respect to enzyme level, however, themetabolite balances are free to evolve, i.e. thesystem evolution as K0

1�0 is governed by

dr

dl#GF(r, e, K)


de

dl#0 (A.72)

These two time scales represent the limitingbehavior of the system. In physical terms, the t

time scale is the short time scale in which en-zyme synthesis is occurring. On this scale, thelevel of metabolites is approximately constant.The dynamic evolution of metabolite level on thel time scale is a much longer time scale andreflects the physical argument that enzymatic cat-alyst is required to effect changes in the metabo-lite level. Of course, ��K0

1��1 need not be true inwhich case some mixture of the behavior inher-ent to the individual time scales prevails.Presently, the time scale of enzyme synthesis rep-resents the aspect of system in which bifurcationcan occur, i.e. the qualitative nature of enzymelevel drives the qualitative dynamics of metabo-lite level because of the requirement for enzy-matic catalyst. Accordingly, to determine thequalitative dynamics of the system with respectto changes in parameters we need only to investi-gate the dynamics of the enzyme balances. Thisimplies in a formal sense, we need only investi-gate the qualitative dynamics of the system

GF(r, e, 0)#0 (A.73)

de

dt#G(r, e, 0) (A.74)

with respect to changes in the system parameters.To determine a closed form solution of the alge-braic equation F(r, e, 0)=0 certain assumptionsmust be made to simplify the structure of thesystem. Firstly, we assume that all metabolicfunction is saturated with respect to the branchpoint precursor p0 (consistent with the assump-tion of ��K0

1��1), this includes both catalytic, aswell as, enzyme expression function. This allowsthe balances for rj, j=1, 2 to be decoupled fromthe r0 balance. Secondly, we assume that rj�1, j=1, 2, i.e. the consumption of the branchmetabolites rj, j=1, 2 is subsaturated with re-spect to metabolite level. Under these assump-tions, the algebraic equation F(r, e, 0)=0 has asolution set of the form

rq�� kq

k( q k( 1�

eq, q=1, 2,…, n (A.75)

Substituting Eq. (A.75) into the decomposedsystem regulation i.e. z2, 1 yields

z2, 1=k̄1 k̄2

k̄1

(A.76)

which implies the decomposed enzyme balancestake the functional form

de1

dt#�

1−k̄1 k̄2

k̄1

�−e1+h1+O(z2, 1)

de2

dt#

k̄1 k̄2

k̄1

−e2+h2+O(z2, 1) (A.77)

Equation set (A.77) controls the qualitativepicture of enzyme expression in the limit of smallK0

1. There are some interesting observations thatcome forth upon examination of this set. Firstly,the enzyme balances are coupled only throughthe kinetic nature of branch metabolite consump-tion, i.e. the parameters k̄j, j=1, 2. This is a con-sequence of the feedback repression/inhibition ofthe branch leg key enzyme by its product asdescribed by the cybernetic variables. Secondly,even in the absence of constitutive enzyme syn-thesis, no zero solution is possible for the en-zyme balances. This is in stark contrast to thegeneral convergent elementary pathway in whichboth zero and non-zero steady state enzyme lev-els are possible.

Even at this level of approximation, the quali-tative features of the elementary divergent path-way are evident. In particular, there exists arather complex interplay between the repressionof the j th branch point key enzyme by its corre-sponding product and the activation of the op-posite leg by the respective branch metabolites.This regulatory feature is evident from the sensi-tivity of steady state enzyme level to the parame-ter(s) k̄j, j=1, 2. The steady state level of keyenzyme e1 is positively sensitive to the kineticparameter k̄1 and negatively sensitive to k̄2.From Eq. (A.75), as k̄1 increases, the level ofbranch metabolite p1 decreases, i.e. the branchmetabolite is being consumed for downstreammetabolite formation at a greater rate resultingin a lower steady state level. In turn, the loweredlevel of p1 eases the repressive effect felt by the


e1 expression machinery and e1 synthesis pro-ceeds at a faster rate resulting in a higher steadystate enzyme level. As k̄2 is increased, the pro-motion of e1 expression (as described by 1−z2, 1) decreases. Physically, this implies that asdownstream r2 consumption is increased, thesteady state level of e1 falls and vice-versa. Ac-cordingly, the expression of e0 is positively sen-sitive to r2 levels. The same type of argumentscan be developed to explain the parameter de-pendence of the steady state level of e2. Note,however, that e2 level is positively sensitive to k̄2

and negatively sensitive to the parameter k̄1.This is an artifact of the assumption of r0

1/r1�r0

2/r2 and can be shown to be a symmetric ef-fect. In a deeper sense this effect is the result ofthe cybernetic objective function. The objectiveof maximizing the mathematical product of thebranch metabolite level implies the optimum so-lution occurs when the branch metabolite levelsare equal. The assumption of r0

1/r1�r02/r2 (as-

suming k1�k2) indicates a disparity in the rela-tive level of branch point metabolite.Specifically, this assumption imparts the condi-tion r2�r1. It follows that e2 expression in thisinstance is severely repressed because of the highlevels of branch metabolite. If the downstreamconsumption of the metabolite r2 is increased,the repression force upon the e2 expression ma-chinery is eased and e2 synthesis can proceed atan increased rate. However, because the level ofr2 drops the expression of e1 suffers and r1

decreases in turn reducing e2 expression and soforth. Thus, the joint positive effector relation-ship between the opposite branch metabolitesand the opposite branch enzymatic machinerycoupled with the repressive effects of productinhibition/repression act to lock the branch in-termediate level.

The qualitative significance of the findings forcomplementary competition are best understoodwhen contrasted with the substitutable counter-part. In particular, we have shown that no bi-furcation behavior is possible, and moreover, noenzymatic zero solution for an elementary struc-ture experiencing complementary competition.Thus, the possibility of zero solutions is reservedfor substitutable environments only.

A.5. Bifurcation analysis: partiallysubstitutable/partially complementaryen6ironments

We choose to prove postulate 4.1 by construc-tion using the same non-linear analysis techniquesemployed in the previous section. Specifically, forpostulate 4.1 to hold, e1

a must have a zero solutionwhose stability is a function of system parameters.Accordingly, we must firstly show that e1

a=0 is apossible equilibrium solution, and secondly, thate1

a=0 can undergo bifurcation as a function ofsystem parameters.

To simplify the algebraic aspects of the analysisand bring forth clearly the qualitative interactionsof the system equations we rescale with respect tothe following set of dimensionless variables andparameters:

r0 p0

K0

, r1 p1

K1a, r2

p2

K2

, r3 p3

K3

e0 �d

a0

�e0, e1

a � d

a1a

�e1

a, e1b

� d

a1b

�e1

b,

cj Rj

d, j=0, 1

kj m j

max

d, j=0, 2, 3 k1

j m1

j, max

d, j=a, b

Gq, j K1

q

K1j, Kq, j

Keq

Kej

K( q, j Ke 1

q

Kj

, t dt, h0 re 0*d

,

h1j

re 1j*

d, j=a, b

where d (rg+b). We assume, consistent withthe assumption of investigation of network behav-ior during balanced growth, that rg�mg

max andenzyme decay is slow. If we substitute the dimen-sionless variables and parameters into the modelequations we arrive at the rescaled set

K0

dr0

dt=c0−k0 e0 r̄0 60−K0 r0

K1a dr1

dt=c1−k1

a e1a r̄1

a 61a−k1b e1

b r̄1b 61b−K1

a r1a


K2

dr2

dt=k0 e0 r̄0+k1

a e1ar̄1

a 61a−r2−K2 r2

K3

dr3

dt=k1

b e1b r̄1

b 61b−r3−K3 r3

de0

dt=u0−e0+h0

de1j

dt=u1j−e1

j +h1j j=a, b (A.78)

where

r̄0 r0

1+r0

r̄1a

r1

1+r1

r̄1b

r1

Gb,a+r1

(A.79)

and the rescaled rates r2 and r3 take the form

r2=k2

r2

1+r2

r3=k3

r3

1+r3

(A.80)

We assume for the sake of simplicity that allenzyme expression is saturated with respect toinducer metabolite. The rescaled equation set, aswas true previously, clearly operates on multipletime scales. Enzyme expression operates on theshort time scale, and the metabolite balances areapproximately constant in the limit of ��K0��BB1where we have assumed the relationship

K0=g0, j Kj, j=0, 2, 3

K0=g0, 1a K1a g0, j�Ö j (A.81)

Using arguments similar to the elementary di-vergent pathway analysis, the metabolite balanceswritten with respect to the time scale of enzymeexpression are given by the set of algebraicequations

c0−k0 e0 60�0

c1−k1a e1

a 61a−k1b e1

b 61b�0

k0 e0 60+k1a e1

a 61a−k2 r2�0

k1b e1

b 61b−k3 r3�0 (A.82)

where we have assumed saturation with respect torj, j=0, 1 and subsaturation with respect torj, j=2, 3, i.e.

rj\\1, j=0, 1 rjBB1, j=2, 3(A.83)

The hallmark of substitutable competition isthe existence of enzymatic zero solutions (assum-ing no constitutive enzyme expression.) More ex-actly, we have shown that the fastest rate insubstitutable set dominates the allocation of criti-cal resources. Using this as a guide, we decom-pose the cybernetic regulation into two regions.Firstly, we assume the production of p2 from p0

catalyzed by key enzyme e0 is much faster thatany other route. In this region we expect e0 toreceive the lions share of substitutable resources.The second region we consider is the converse inwhich the production of p2 via p0 is much slowerthan any other route. In each case we assumer1

a/r2�r1b/r3 and K2�K3.

A.5.1. r0�r1a

In this region the cybernetic variables that regu-late enzyme expression can be decomposed intothe set

u0�1−r1

a

r0

u1a��r1

a

r0

��r1a r3

r1b r2

�u1b�1−

r1a r3

r1b r2

where we make use of the k th elementary pathwayproperty

%j

u jk=1, Ö k (A.84)

The rates of reaction, because of the assump-tion of saturation with respect to p0 and p1 takethe form

r0=k0 e0 r1j =k1

j e1j, j=a, b (A.85)

The cybernetic variables that regulate enzymeactivity can be decomposed as

60�1 61a��r1

a

r0

��r1a r3

r1b r2

�61b�1 (A.86)

where the rates of reaction are given above. Themetabolite balances, written with respect to thetime scale of enzyme synthesis, are approximatelyconstant. However, the balances can not be dis-carded straightaway because of the explicitmetabolite dependence stemming from the com-plementary regulatory component. Accordingly,


we substitute the regulatory decomposition shownabove into the metabolite balances and explicitlysolve for the level of r2 and r3 as a function ofthe enzyme level and the system parameters.These functions can then be substituted into theenzyme balances so as to decouple them from theremaining portion of state space. Solving themetabolite balances for r2 and r3 yields thesolutions

(k2 k0 e0)r22− (k0 e0)2r2−

(k1a e1

a)3

k3

#0 (A.87)

r3#k1

b e1b

k3

(A.88)

Note the quadratic equation in r2 is sensitive toboth modes of r2 synthesis. As per our originalassumption regarding the relative magnitudes ofreaction rates given the substitutable nature of e1

a

expression, we assume that e1a�0 as time evolves.

This implies that r2 during enzyme expression ison the order

r2�k0 e0

k2

(A.89)

Substituting r2 and r3 into the enzyme balancesyields the dynamic set

de0

dt#�

1−k1

a e1a

k0 e0

�−e0+h0 (A.90)

de1a

dt#�k1

a e1a

k0 e0

�(D)−e1

a+h1a (A.91)

de1b

dt#�

1−k1

a e1a

k0 e0

�−e1

b+h1b (A.92)

where the compound parameter D is given by theratio

D k2

k3

(A.93)

Notice that e1b is a coupled to the expression of

the remaining enzymes as a slave variable. Inother words, it is driven by the movement of e1

a,however, exerts no influence upon the remainingsystem variables. Accordingly, as time evolves e1

b

moves towards

e1b#

�1−

k1a e1

a

k0 e0

�+h1

b (A.94)

irrespective of the dynamics of e0 and e1a. This

implies the qualitative nature of enzyme expres-sion (in the limit of small e1

a) is controlled solelyby the e1

a and e0 balances which can be recast inthe vector form

de

dt#G(e, k, h) (A.95)

where the state vector e {e0, e1a}T and k, h de-

note the parameter and rescaled constitutive en-zyme synthesis vector(s), respectively. Theequilibrium behavior of equation set (A.95) isdetermined from the solutions of the algebraicsystem

G(e0, k0, 0)=0 (A.96)

where e0, k0 denote the steady state rescaledenzyme and parameter vector(s), respectively.Also note that we have assumed h=0 at equi-librium. This assumption allows for the investiga-tion of the effects of constitutive enzyme synthesisas a breaking parameter, or an imperfection. Thisis similar to the arguments presented during theanalysis of the elementary convergent pathway,and will be touched on subsequently. A possiblesolution of Eq. (A.113) is given by the set

e0=1 e1a=0 (A.97)

Notice solution (A.97), when coupled with theequilibrium solution for e1

b, is indicative of com-pletely parallel operation. In other words, themetabolites r2 and r3 are produced only from r0

and r1, respectively. We use solution (A.97) as abase from which we explore the parameter depen-dence of the equilibrium behavior. More exactly,we ‘ride’ along solution (A.97) as a function ofthe system parameters and search for possiblepoints of bifurcation which are marked by zeroeigenvalues of the system Jacobian. To ease thealgebraic burden, we propose the translation

e=e0+x x�R2 (A.98)

which simply slides the equilibrium point to theorigin of the new state space variable x. Weapproximate vector Eq. (A.95) in the new statevariable x around the origin as the Taylor seriesretaining up to cubic terms


dxdt

=Jx+F2(x, k)+O(x3) (A.99)

where F2(x, k) denotes the vector of second orderterms and J denotes the Jacobian matrix. TheJacobian matrix for the translated enzyme systemevaluated at the origin is given functionally as

J=ÃÃ

Ã

Á

Ä

−1

0

−k1

a

k0

−1

ÃÃ

Ã

Â

Å

(A.100)

The eigenvalues of the Jacobian are −1, −1which is indicative of a globally stable solution,i.e. no possibility of bifurcation. Thus, in thepresent regulatory region, the parallel operationof the system is the only type of behavior that ispossible. If the regulatory decomposition stem-ming from the assumption r1

a�r0 is invariant, thesystem trajectories once entering the region re-main for all time. It follows that, under theseconditions, the long time behavior of the originalset of equations evolved towards parallel networkoperation. The simulation of this case is shown inFig. 4.

With respect to the behavior postulate, we haveshown that e1

a does possess a zero solution inaccordance with the substitutable component ofthe local regulation. Let us now consider a secondcase in which we assume r1

a�r0.

A.5.2. r1a�r0

In this region the cybernetic variables that regu-late enzyme expression decompose into the set

u0�r0

r1a u1a�

�1−

r0

r1a

��r1ar3

r1br2

�u1b�

�1−

r1a r3

r1b r2

�(A.101)

where, again, we make use of the elementarypathway property

%q

uqk=1, Ö k (A.102)

The cybernetic variables that govern the en-zyme activity decomposes into the set

60�r0

r1a 61a�

�r1a r3

r1b r2

�61b�1 (A.103)

Because the metabolite balances operate on theslow time scale the level of the various metaboliteappears to be approximately constant on the timescale of enzyme synthesis. Therefore, we can solveequation set (A.82) for the metabolite level as afunction of enzyme concentration and systemparameters. Notice, because of the assumption ofsaturation with respect to r0, the regulatory de-composition is not coupled to the r0 balance.However, the complementary regulatory compo-nent does contain r2 and r3, thus, we solve the r2

and r3 balances written with respect to the timescale of enzyme synthesis to yield the equations

k2(k1a e1

a)r22− (k0 e0)2r2−

(k1a e1

a)3

k3

�0 (A.104)

r3�k1

b e1b

k3

(A.105)

Interestingly, the quadratic in r2 indicates thetwo possible modes of formation of r2. Given theassumption of r1

a�r0, we assume e0�0 as timeevolves which implies that the r2 is on the orderof

r2�� k1

a

k2 k3

�e1

a (A.106)

Note Eq. (A.106) is identical in form to thebranch metabolite level shown in the elementarydivergent pathway analysis. This is to be expectedbecause with e0�0, the production of r2 is purelycomplementary. Substituting r2 and r3 into theenzyme balances yields the enzyme expression set

de0

dt#�k0 e0

k1a e1

a

�−e0+h0

de1a

dt#�

1−k0 e0

k1a e1

a

�U−e1

a+h1a (A.107)

de1b

dt# (1−U)−e1

b+h1b (A.108)

where U is given as

U k2 k3

k3

(A.109)


Equation set (A.107) controls the qualitativeproperties of the system in the limit of small e0.Notice the similarities of this equation set withthose derived during the analysis of the individualelementary pathways. The influence of each of theindividual types of elementary competition isclearly visible from the e1

a balance. In particular,the regulatory decomposition is composed of thetwo distinct elementary components. The term�

1−k0 e0

k1a e1

a

�(A.110)

corresponds to the substitutable influence of thee1

a regulation. In particular, note the dependenceof the substitutable regulatory input upon e0. Asthe e0 level increases, the share of resources allo-cated to e1

a expression must decrease because ofthe limited amount of critical resources. Thisreflects the give and take relationship inherent tosubstitutable competition. The term U is the regu-latory influence stemming from the elementarycomplementary pathway. As was true in thepurely complementary case, this compoundparameter reflects the complex interaction be-tween metabolite inhibition/repression and theopposite branch positive effector relationship.Thus, the promotion of e1

a expression is sensitiveto both types of local resource competition.

The e1b balance is not dynamically coupled to

the remaining set, however, parametrically ea1 and

eb1 are linked. This implies that as time evolves,

the key enzyme eb1 assumes the steady state

e1b# (1−U)+h1

b (A.111)

independently of the dynamics of ea1 and e0. Ac-

cordingly, the qualitative dynamics of the hybridnetwork are controlled solely by the e0 and ea

1

balances recast in vector form as

de

dt#G(e, k, h) (A.112)

where the enzymatic state vector e is given bye= (e0, e1

a)T and G(e, k) denotes the vector ofright hand sides of the e0 and e1

a balances. Theequilibrium behavior of system (A.112) can bedetermined by solving the set of algebraic equa-tions given by

G(e0, k0, 0)=0 (A.113)

where e0, k0 denote the equilibrium rescaled en-zyme level and parameter vector, respectively. Apossible solution of equation set (A.113) is givenby

e00=0 e1

a, 0=U (A.114)

Note the above solution corresponds to purelycomplementary operation of the network. Equi-librium solution (A.114) serves as the base fromwhich we explore the qualitative nature of thesystem. Accordingly, we propose the translation(translates the equilibrium point to the origin)

e=e0+x, x�R2 (A.115)

where x denotes the translated state vector. Forsimplicity, we approximate system (A.112) as aTaylor series (retaining up to cubic terms) aroundthe origin

de

dt=Jx+F2(x, k)+ O(x3) (A.116)

where F2(x, k) denotes the vector of second orderterms and J denotes the Jacobian matrix. Ourintention is to utilize Lyapunov–Schmidt reduc-tion to abstract the portion of the system that isresponsible for the qualitative changes in systemdynamics. This task is greatly simplified if theJacobian is self-adjoint with respect to the usualinner product on Rn, i.e. the Jacobian is symmet-ric. Accordingly, we propose the coordinatetransformation

y=Mx, y�R2 (A.117)

which simply rotates the coordinate axis such thatthe principle directions are equivalent to the ei-genvector directions. The 2×2 modal matrix Mconsists of the system eigenvectors and is givenfunctionally as

M=ÃÃ

Ã

Á

Ä

0

1

−1U

1

ÃÃ

Ã

Â

Å

(A.118)

Note the modal matrix M exists for all valuesof the system parameters as long as r2 and r3 are


precursors for downstream metabolic activity.Substituting transformation (A.117) into equationset (A.116) yields the rotated system

dydt

=M−1G(y, k)=By+H2(y, k)+O(3)

(A.119)

where

B M−1JM H2(y, k) M−1F2(y, k)(A.120)

The rotated Jacobian, denoted by B, is a 2×2diagonal matrix given functionally as

B=ÃÃ

Ã

Á

Ä

−1

0

0

k0

k1a U

−1ÃÃ

Ã

Â

Å

(A.121)

Clearly, there exists a possibility of bifurcation,i.e. a zero eigenvalue when the system parametersobey the relationship

k0

k1a U

=1 (A.122)

Note that the critical condition is very similarto the purely substitutable case, with the excep-tion of U. More exactly, it was shown, that purelysubstitutable structures bifurcate when the ratioof the rate constants of competing processes are inthe neighborhood of unity. This structure isclearly visible in the form of the critical conditionshown above, with the addition of the comple-mentary regulatory input described by the com-pound parameter U. The complementaryregulatory input alters the parametric location ofthe bifurcation of the substitutable system, how-ever, does not destroy the existence of bifurcation.Let us examine this in more detail.

For the sake of algebraic simplicity, the criticalcondition is translated to the origin of parameterspace by the transformation

l= −1+k0

k1 U(A.123)

which implies l=0 is the new point of possiblebifurcation. Application of Lyapunov–Schmidt

reduction to the rotated system, relaxing the con-ditions upon h=0, yields the reduced system

dx

dt=g=l x−

1+l

Ux(U h0+h1

a+x)−U h0

(A.124)

where x denotes projected null space state vari-able. Eq. (A.124) controls the qualitative aspectsof enzyme evolution. Specifically, the equilibriumbehavior of Eq. (A.124) corresponds to equi-librium solutions e0.

The analysis of the reduced equation is centeredupon two broad cases, namely, the presence andabsence of constitutive enzyme synthesis. We be-gin by considering the latter case. The reducedequation in this instance becomes

dx

dt=g=l x−

(1+l)U

x2 (A.125)

where l denotes the bifurcation parameter. Toeliminate the parameter dependence of thequadratic term coefficient, we propose therescaling

z x

f(l), f(0)"0 (A.126)

which after substitution yields the rescaled re-duced system

dx

dt=g=l z−z2 (A.127)

Eq. (A.127) is the transcritical bifurcation nor-mal form. The rescaled reduced equation possessa zero solution (which can be shown to be equiva-lent to (A.114)) whose stability is governed by thesign of the bifurcation parameter l. Additionally,a second solution z=l is also possible. Thissolution can be shown to correspond to the casein which both e1

a, as well as, e0 are present (weexpand this discussion subsequently.) The stabilityof the system, i.e. which equilibrium solution isobserved, is dependent upon the sign of theparameter l. In the general case, the bifurcationparameter can be parameterized as

l=l(z) (A.128)

along the equilibrium trajectory, i.e.


g(l(z), z)=0 (A.129)

which implies

dgdz

=(g(l

dl

dz+(g(z

(A.130)

The sign of dgdz evaluated along the j th solution

branch determines the stability. If dgdzB0 when

evaluated along the j th equilibrium solution, thenthis solution is stable, the converse being true forunstable solutions. For our system, dg

dz takes theform

dgdz

=l−2z (A.131)

It follows straightaway that z=0 is stable whenlB0, otherwise, z=l is the stable system behav-ior. At the point of bifurcation, i.e. l=0 Eq.(A.131) vanishes yielding no stability informationas to the exchange of stability. Expanding gz withrespect to l around the bifurcation point andretaining up to linear terms yields

gz, l=�(gz

(l+(gz

(z

�dz

dl

��dl+Ol2 (A.132)

where dl l−l0. Given Eq. (A.131), the eigen-value expansion reduces to

gz, l=!

1−2�dz

dl

)j

�"dl+Ol2 (A.133)

The zero solution is stable when lB0, whereas,the bifurcating solution assumes stability as l isvaried through the origin. The system undergoestranscritical bifurcation at l=0.

In terms of the original system, the solution(s)z=0 and z=l correspond to

e0=0 e0=1−k0

k1a U

(A.134)

e1a=U e1

a=k0

k1a, (A.135)

respectively. Accordingly, the original model sys-tem assumes a solution in the neighborhood ofthe zero solution when the system parametersobey the inequality

k0\k1a U (A.136)

otherwise the bifurcating solution is the stablesolution and the system evolves towards

e0=1−k0

k1a U

e1a=

k0

k1a (A.137)

These results are illustrated by simulation of theoriginal system model in Fig. 6.

The inclusion of constitutive enzyme synthesisbreaks the symmetric nature of bifurcation andresults in a qualitatively different dynamics. If werelax the assumption h=0 the original reducedequation in x becomes

g=l x−1+l

b(x(b h0+h1

a+x))−b h0 (A.138)

Proceeding in much the same manner as theprevious case, let us rescale the reduced variableby some function of the bifurcation parameter l,i.e.

z x

f(l), f(0)"0 (A.139)

which after substitution in Eq. (A.138) yields

g=�

l−1+l

bh*�

z−z2−h0(1+l) (A.140)

where h* and f(l) take the form

h* (b h0+h1a) f(l)

b

1+l(A.141)

Eq. (A.140) has several features that distinguishit from the previous case. Firstly, note that nomathematical zero solution is possible in the hj"0, Ö j case. Thus, the presence of constitutive en-zyme synthesis, as discussed during thesubstitutable analysis section, acts to disrupt thecompetition between competing enzyme systemsand affords a non-zero minimum level. In practi-cal terms this implies no matter how fierce thecompetition to produce p2, e0 and ea

1 are presentat non-zero levels for all values of the systemparameters. The exact level, however, can bifur-cate between two possible equilibrium solutionswhich exchange stability at�

l−1+l

bh*�

=0 (A.142)


Fig. 6. Simulation of hybrid overlapping network in the absence of constitutive enzyme synthesis. Notice e1a can assume both a zero

and a non-zero steady state solution depending upon system parameters. Steady state values; Top plate: (e0/e0max, ea

1/e1a, max)=

(0.2167, 0.1194). Bottom plate: (e0/e0max, ea

1/e1a, max)= (0.2526, 0).

Because our goal is only to determine the qual-itative aspects of this regulatory region, the con-clusion that constitutive enzyme synthesis nolonger allows the existence of a zero solution issufficient. Thus, we dispense with a formal analy-sis of the bifurcation behavior of reduced Eq.(A.140) with the understanding that the bifurca-tion that Eq. (A.140) undergoes is between twonon-zero levels of enzyme e1

a. In fact, guidelineshave been developed by Iooss and Joseph thatgovern how the bifurcation profile of an unper-turbed system behaves under perturbation, weneglect such considerations leaving the interestedreader to further pursue the subject.

References

Baloo, S., Ramkrishna, D., 1991. Metabolic regulation inbacterial continuous cultures I. Biotechnol. Bioeng. 20,1337–1352.

Doyle, F., Ogunnaike, B., Pearson, R., 1995. Non-linearmodel based control using second-order volterra models.Automatica 31, 697–714.

Golubitsky, M., Schaeffer, D., 1985. Singularities and groupsin bifurcation theory, vol. 1. Springer-Verlag, Berlin.

Iooss, G., Joseph, D., 1990. Elementary stability and bifurca-tion theory. Springer-Verlag, New York.

Kompala, D., Jansen, N., Tsao, G., Ramkrishna, D., 1986.Investigation of bacterial growth on mixed substrates: Ex-perimental evaluation of cybernetic models. Biotechnol.Bioeng. 28, 1337–1352.


Mcleod, J., Sattinger, D., 1973. Loss of stability and bifurca-tion at a double eigenvalue. J. Funct. Anal. 14, 62.

Pearson, R., Ogunnaike, B., Doyle, F., 1996. Identification ofstructurally constrained second-order volterra models.IEEE Trans. Signal Process. 44, 2837–2846.

Ramakrishna, R., Ramkrishna, D., Konopka, A., 1997. Cy-bernetic modeling of mixed substitutable substrate environ-ments. Preferential and simultaneous utilization.Biotechnol. Bioeng. 52, 144–154.

Straight, J., Ramkrishna, D., 1991. Complex growth dynamicsin batch cultures: Experiments and cybernetic models. Bio-technol. Biotech. 37, 895–909.

Straight, J., Ramkrishna, D., 1994. Cybernetic modeling andregulation of metabolic pathways. Growth on complemen-tary nutrients. Biotechnol. Prog. 10, 574–587.

Varner, J., Ramkrishna, D., 1998. Metabolic engineering froma cybernetic perspective I. Theoretical preliminaries.Metab. Eng. (submitted Jan 1998).

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