ARTICLE IN PRESS

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doi:10.1016/j.jtb

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Journal of Theoretical Biology 232 (2005) 385–398

www.elsevier.com/locate/yjtbi

Yeast glycolytic oscillations that are not controlled by a singleoscillophore: a new definition of oscillophore strength

Karin A. Reijengaa,1, Yvonne M.G.A. van Megena, Bob W. Kooib, Barbara M. Bakkera,Jacky L. Snoepa,c, Henk W. van Versevelda,y, Hans V. Westerhoff a,d,e,�

aDepartment of Molecular Cell Physiology, CRbCS, BioCentrum Amsterdam, Faculty of Earth and Life Sciences, Vrije Universiteit,

1081 HV Amsterdam, The NetherlandsbDepartment of Theoretical Biology, Faculty of Earth and Life Sciences, Vrije Universiteit, 1081 HV Amsterdam, The Netherlands

cDepartment of Biochemistry, Triple-J Group for Molecular Cell Physiology, University of Stellenbosch, Matieland, 7602 Stellenbosch, South AfricadDepartment of Mathematical Biochemistry, BioCentrum Amsterdam, Swammerdam Institute for Life Sciences, Plantage Muidergracht 12,

1018 TV Amsterdam, The NetherlandseStellenbosch Institute for Advanced Study, University of Stellenbosch, Matieland, 7600 Stellenbosch, South Africa

Received 11 November 2003; received in revised form 30 July 2004; accepted 25 August 2004

Available online 8 October 2004

Abstract

Biochemical oscillations, such as glycolytic oscillations, are often believed to be caused by a single so-called ‘oscillophore’. The

main characteristics of yeast glycolytic oscillations, such as frequency and amplitude, are however controlled by several enzymes. In

this paper, we develop a method to quantify to which extent any enzyme determines the occurrence of oscillations. Principles

extrapolated from metabolic control analysis are applied to calculate the control exerted by individual enzymes on the real and

imaginary parts of the eigenvalues of the Jacobian matrix. We propose that the control exerted by an enzyme on the real part of the

smallest eigenvalue, in terms of absolute value, quantifies to which extent that enzyme contributes to the emergence of instability.

Likewise the control exerted by an enzyme on the imaginary part of complex eigenvalues may serve to quantify the extent to which

that enzyme contributes to the tendency of the system to oscillate. The method was applied both to a core model and to a realistic

model of yeast glycolytic oscillations. Both the control over stability and the control over oscillatory tendency were distributed

among several enzymes, of which glucose transport, pyruvate decarboxylase and ATP utilization were the most important. The

distributions of control were different for stability and oscillatory tendency, showing that control of instability does not imply

control of oscillatory tendency nor vice versa. The control coefficients summed up to 1, suggesting the existence of a new summation

theorem. These results constitute proof that glycolytic oscillations in yeast are not caused by a single oscillophore and provide a new,

subtle, definition for the oscillophore strength of an enzyme.

r 2004 Elsevier Ltd. All rights reserved.

Keywords: Yeast glycolytic oscillations; Oscillophore; Control coefficient

e front matter r 2004 Elsevier Ltd. All rights reserved.

i.2004.08.019

ing author. Department of Molecular Cell Physiology,

BioCentrum Amsterdam, Faculty of Earth and Life

niversity, NL-1081 HV Amsterdam, The Netherlands.

47230; fax: +31 20 4447229.

ess: hans.westerhoff@falw.vu.nl (H.V. Westerhoff).

ress: DSM Anti-Infectives, P.O. 425, 2600 MA, Delft,

s.

1. Introduction

Oscillatory phenomena are well known in biology,e.g. the circadian rhythm (Turek, 1998; Vitaterna et al.,2001), the cell cycle (Hartwell, 1974; Nurse, 2000; Tysonand Novak, 2001), Ca2+ oscillations (Berridge, 1990;Berridge et al., 1998; Goldbeter, 1996), oscillations inneuronal signals (Rabinovich and Abarbanel, 1998),waves in cyclic AMP in Dictyostelium discoideum

ARTICLE IN PRESSK.A. Reijenga et al. / Journal of Theoretical Biology 232 (2005) 385–398386

(Escalante and Vicente, 2000; Nanjundiah, 1998) andyeast glycolytic oscillations (Chance et al., 1964;Duysens and Amesz, 1957; Goldbeter, 1996). Becausesome such oscillations are important physiologically oreven pathologically (e.g. Berridge et al., 1998), it isimportant to understand what causes and controls them.

It has been proposed that oscillatory behavior iscaused by a so-called ‘oscillophore’. Constructing a two-variable model of yeast glycolytic oscillations, Gold-beter and Lefever argued in favor of this hypothesis(Goldbeter and Lefever, 1972). Their model consists of asource of substrate, the enzyme phosphofructokinase

α γ

σMσ1 ks

+

Glcout

Glcin

F6P

F16BP

DHAP

Acald

EtOH

ATP

ADP

ADPATP

NADHNAD

transport

HK

PGI

PFK

GAP

3PGA

BPG

PYR

PEP

ATPADP

ALD

TPIGAPDH

ADPATP

CO2

NADNADH

PGM

ENO

PYK

PDC

ADH

ATP

ATP

ADPADP

G3PDHNADHNAD

3 NADH 3 NAD

2ADP

PGA

ATPAMP

AK

Glycerol

PGK

G6P TrehaloseGlycogen

(A)

(B)

2

and a sink that removes the product (Fig. 1A). In thismodel, the stimulation of phosphofructokinase by itsown indirect product (AMP) is included. It was shownanalytically that this product stimulation was essentialto obtain oscillations in the model (Higgins, 1967). Ifthis feedback loop was removed, the model producedstable steady states for all possible combinations ofparameters. On this basis, PFK has been widelyaccepted as the oscillophore of glycolysis.

The question as to what causes the emergence ofoscillations should not be confused with the question asto what controls their main characteristics. It was shownthat even in the model of Goldbeter and Lefever themain characteristics of the oscillations, such as fre-quency and amplitude were not solely controlled by thepresumed oscillophore, phosphofructokinase (Teusinket al., 1996). Instead, all steps of the pathway sharedcontrol of frequency and amplitude. Moreover, weshowed experimentally that the frequency of yeastglycolytic oscillations was controlled by more than onestep, including the glucose transporter (Reijenga et al.,2001).

On the basis of these combined results one might stillargue that oscillophores do exist, although they need notuniquely be in control of frequency and amplitude of theoscillations. The definition of an oscillophore shouldthen be limited to a property of an enzyme that cannotbe removed without eliminating oscillations for allpossible parameter values of the system. There are threemain objections against this, however. Firstly, in themodel of Goldbeter and Lefever, the positive feedbackloop is not sufficient to obtain oscillations. Theparameters of all three steps in the model should have

Fig. 1. (A) Schematic representation of the PFK model by Goldbeter

and Lefever (1972). The model consists of two variables, a and g, and

three reactions. The three parameters s1; sM and ks represent the

activities of the three reactions, which may be thought representative

of glucose transport and phosphorylation, phosphofructokinase, and

glycolysis below phosphofructokinase, respectively. (B) Schematic

representation of glycolytic model by Teusink et al. (2000), including

branches to trehalose, glycogen, succinate and glycerol. See the

original literature for further explanation of the kinetics. Metabolites:

Glcout, extracellular glucose; Glcin, intracellular glucose; G6P,

glucose-6-phosphate; F6P, fructose-6-phosphate; F16BP, fructose-

1,6-bisphosphate; DHAP, dihydroxyacetone-phosphate; GAP,

glyceraldehyde-3-phosphate; BPG, 1,3-bisphosphoglycerate; 3PGA,

3-phosphoglycerate; 2PGA, 2-phosphoglycerate; PEP, phosphoenol-

pyruvate; PYR, pyruvate; AcAld, acetaldehyde; EtOH, ethanol; AMP,

adenosine-mono-phosphate; ADP, adenosine-di-phosphate; ATP,

adenosine-triphosphate. Enzymes: transport, glucose transport; HK,

hexokinase; PGI, glucose-6-phosphate isomerase; PFK, 6-phospho-

fructokinase; ALD, fructose-bisphosphate aldolase; TPI, triosepho-

sphate isomerase; G3PDH, glycerol-3-phosphate dehydrogenase;

GAPDH, glyceraldehyde-3-phosphate dehydrogenase; PGK, phos-

phoglycerate kinase; PGM, phosphoglycerate mutase; ENO, enolase;

PYK, pyruvate kinase; PDC, pyruvate decarboxylase; ADH, alcohol

dehydrogenase; AK, adenylate kinase.

ARTICLE IN PRESSK.A. Reijenga et al. / Journal of Theoretical Biology 232 (2005) 385–398 387

certain values to be in the ‘oscillatory window’,demonstrating that all enzymes contribute to theemergence of oscillations (Teusink et al., 1996). Experi-mentally this has been confirmed by the finding that therate of glucose influx is a critical parameter controllingthe emergence of oscillations (Becker and Betz, 1972;Boiteux et al., 1975; Danø et al., 1999; Hess andBoiteux, 1973) and by the finding that the yeast cellsneed to have a certain make-up determined by theirgrowth history (Richard et al., 1994,1993) in order toexhibit sustained oscillations. The importance of theseadditional requirements for oscillations is not donejustice by the concept of a single oscillophore. Secondly,in real glycolysis there are more positive feedback loopsthan the one highlighted by Goldbeter and Lefever(1972). For example, Sel’kov showed that the auto-catalytic stoichiometry of glycolysis, in which ATP isfirst invested before net production takes place, maylead to the occurrence of oscillations, in the absence ofstimulation of phosphofructokinase by its product(Sel’kov, 1975). Also in this model, removal of thefeedback loop eliminates oscillatory behavior from theremaining parameter space. This suggests that in amodel of real glycolysis, which contains both feedbackloops, neither mechanism is an oscillophore accordingto the above, strict definition. Thirdly, we shouldultimately be addressing reality rather than oversimpli-fied models thereof. Reality is not close to a model atarbitrary parameter values but to a model at the realparameter values. Accordingly, the definition of oscillo-phore with the mention of entire parameter space, maybe relevant for models, but not for biological reality.

In view of these objections to the existing definitionsof oscillophore, it becomes clear that an improvedtheory to understand what causes oscillations isimportant. It should be even more helpful if a methodcould be found to quantify to what extent individualenzymes contribute to the tendency of biochemicalnetworks to engage in oscillations. Such a theory wouldallow investigating what prevents biologically importantoscillations such as the cell cycle or the circadian rhythmto die out. Also it should explain how the cell avoidsunwanted oscillations. Glycolytic oscillations whichcause dramatic oscillations of the cellular [ATP]/[ADP]ratio, may sometimes be an example of such unwantedoscillations.

In this paper, we (i) develop a theory to determine thecontrol of single enzymes on the stability of biochemicalsystems, (ii) determine to which extent each glycolyticenzyme contributes to the stability and oscillatorytendency of yeast glycolysis by invoking the eigenvaluesof the Jacobian, and (iii) evaluate and clarify thedefinition of oscillophore. We show that the controlon the eigenvalues of the system is distributed and thatthe candidate oscillophore phosphofructokinase haslittle control on the eigenvalues.

2. Methods

2.1. Theory and definitions

2.1.1. Stability analysis

A metabolic pathway can be described in terms of aset of differential equations (Heinrich and Schuster,1996).

dSðtÞ

dt¼ N � rðSðtÞÞ ¼ f ðSðtÞÞ (1)

in which N is the stoichiometric matrix, r is the vector ofreaction rates, SðtÞ is the vector of independentmetabolite concentrations and f ðSðtÞÞ is the vector oftime differentials of the concentrations. The Jacobianmatrix of the system is defined as

@f 1@S1

@f 1@S2

@f 1@S3

:::

@f 2@S1

@f 2@S2

@f 2@S3

:::

@f 3@S1

@f 3@S2

@f 3@S3

:::

::: ::: ::: :::

2666664

3777775: (2)

At steady state Eq. (1) is equal to zero. The eigenvaluesl of the Jacobian evaluated at this steady state thenreflect the dynamic behavior after a perturbation fromthis steady state. The dominant eigenvalue is theeigenvalue with greatest real part. When all realeigenvalues are negative as well as the real part of allcomplex eigenvalues, the steady state is stable, otherwiseit is unstable. Hence, the real part of the dominanteigenvalue determines the asymptotic behavior after aperturbation from the steady state. The imaginary partof the dominant eigenvalue correlates to the frequencyof the damped or explosive oscillatory dynamic behaviorof a variable after a perturbation.

The transition from a stable node (the dominanteigenvalue is real and negative) to a stable focus (thedominant eigenvalue is complex with negative real part)occurs when the dominant negative real eigenvaluebecomes complex. When, at the Hopf bifurcation thereal part of the dominant complex eigenvalue becomespositive the resulting steady state becomes an unstablefocus, i.e. the system moves away from the steady statein an oscillatory manner.

2.1.2. Metabolic control analysis (MCA)

MCA quantifies the extent to which an enzyme i

controls a steady state variable Xj (Kacser and Burns,1973). Control coefficients are defined as the relativechange in that steady-state variable upon a relativemodulation ð@vi=viÞ of the activity ðviÞ of the enzyme i:

CXj

i ¼@X j=X j

@vi=vi

¼@ ln X j

@ ln jvij: (3)

Here, we introduce two new control coefficients, i.e. thecontrol coefficients on the real part and on the

ARTICLE IN PRESSK.A. Reijenga et al. / Journal of Theoretical Biology 232 (2005) 385–398388

imaginary part of the dominant eigenvalue of theJacobian matrix evaluated at a steady state:

CRe lj

i ¼@ Re lj=Re lj

@vi=vi

¼@ ln jRe ljj

@ ln jvij; (4)

CIm lj

i ¼@ Im lj=Im lj

@vi=vi

¼@ ln jIm ljj

@ ln jvij: (5)

2.1.3. Control of oscillatory tendency: transition between

stable focus and stable node

A factor that reduces the frequency will be consideredto have an anti-oscillatory action. Accordingly, C

Im lj

i ; ifexisting, will be used as a measure of the control exertedby process i on the oscillatory tendency of the system.As the bifurcation parameter moves towards thebifurcation point from stable focus to stable node,Im(l) decreases in absolute magnitude. Therefore, apositive C

Im lj

i implies that a process i enhances theoscillatory tendency of the system, while a negativeC

Im lj

i implies that it decreases the oscillatory tendency.Accordingly, we define the control on oscillatorytendency by

Cosci � CIm l

i : (6)

When activation of a process moves the system from astable focus to a stable node, its control coefficientshould be negative and at the bifurcation it shouldapproach minus infinity. Since a stable node has nocomplex eigenvalues, this control coefficient cannot beused to describe the reverse transition from a stablenode to a stable focus. The method can be generalizedby replacing Im(l) by the absolute value of thediscriminant D of the Jacobian matrix and applying anappropriate scaling factor. This has been worked out indetail in Reijenga (2002). Since the discriminant isdifficult to calculate in larger systems, its practical use islimited and we will not dwell on this method here.

2.1.4. Control of stability: stable focus to unstable focus

At the Hopf bifurcation, when a system makes atransition from a stable focus to an unstable focus orvice versa, the real part of a complex eigenvalue (and ofits complex conjugate) passes through zero. Accord-ingly, the tendency of a stable system to becomeunstable is here related to the difference of the real partof the relevant eigenvalue from zero. Therefore, thecontrol of the stability will be quantified in terms of thecontrol of the real part of the eigenvalue that is about tocross zero, i.e. CRe l

i :The control coefficient of an enzyme with respect to

the real part of the eigenvalue primarily refers to thetendency to move the system away from or towards thebifurcation. Accordingly, for stable systems, a positivevalue of a control coefficient with respect to the real(part of the) dominant eigenvalue, indicates a stabilizing

effect of the corresponding enzyme, whereas a negativevalue indicates a destabilizing effect. For unstablesystems, a positive value of a control coefficient on thereal (part of the) dominant eigenvalue, indicates adestabilizing effect of that enzyme, whereas a negativevalue indicates a stabilizing effect. This distinction iscaused by the switch of the sign of the real part of theeigenvalue at the bifurcation. At the bifurcation itself,the real part of the eigenvalue is zero and the controlcoefficients with respect to the bifurcation processapproaches infinity close to the bifurcation. To avoidconfusion with respect to the sign of the controlcoefficient, we define the control coefficient of stabilitywith respect to any enzyme by

Cstabi � �

@ Re l1=jRe l1j

@vi=vi

¼ �CRe l1i sign l1ð Þ; (7)

where l1 is the dominant eigenvalue. In this way apositive value for Cstab

i indicates a stabilizing effectwhereas a negative value for Cstab

i indicates a desta-bilizing effect. The control coefficient on stability asdefined here, also applies to the transition from a stableto an unstable node in a transcritical bifurcation. Inthis case the dominant eigenvalue is real and Re(l)equals l.

Using the approach outlined above, it should bepossible to distinguish between the control of thestability of the system (Cstab

i ) and the control of theoscillatory tendency of the system (Cosc

i ).

2.2. Methods

2.2.1. Bifurcation analysis

To calculate steady-state metabolite concentrationsand to determine the stability of these steady states,Gepasi (Mendes, 1997), Mathematica (Wolfram, 1999)and Auto (Doedel et al., 1998) were used. For the two-variable model, Gepasi and Mathematica could beused, whereas for the fourteen-variable model, Autowas more suitable, because Gepasi failed to find theunstable steady states of the detailed model. The Hopfbifurcation points and the regions of stability andinstability were determined, in order to decide where tocalculate the control coefficients of the individualenzymes.

2.2.2. Control analysis

A numerical approach was used to calculate controlcoefficients on the eigenvalues of the systems. In orderto calculate the control coefficients, the activities of theenzymes (Vmax or corresponding rate constants) werechanged by both +dp and �dp around the referencestate. New steady states were computed, the eigenvalueswere calculated for both the initial and the newsteady state, and eigenvalue control coefficients were

ARTICLE IN PRESSK.A. Reijenga et al. / Journal of Theoretical Biology 232 (2005) 385–398 389

determined according to the following:

CXp ¼

ðXþdp � X�dpÞ=X

ðþdp ��dpÞ=p: (8)

Here p is the value of the Vmax or rate constant that waschanged. In all cases the rate of interest was propor-tional to p. The changes dp used to calculate the controlcoefficients are given in the legends of the correspondingtables and were optimized such that they were smallenough for the control coefficient to become indepen-dent of dp, but large enough to allow an accuratenumerical calculation of the control coefficient.

Here, we limit ourselves to determining the control ofthe enzyme activities for three reasons: (i) all thereaction activities of these enzymes constitute a com-plete set, (ii) other parameters that are outside this set,are still and perhaps even more relevant for theoscillations, but they can be shown to affect theoscillations indirectly, i.e. through a number of para-meters in the complete set (see Appendix B), and (iii) thecell is able to modulate the Vmax values of specificenzymes, via gene expression and translation regulation,however, is less likely to modulate other parameters,e.g. KM’s.

In future work, we intend to examine the importanceof the other classes of parameters, also in an experi-mental context. Appendix B gives an example.

2.3. Models

In this study we used two models (Fig. 1A and B)which are described below. Details on the models can befound in the original literature (Goldbeter and Lefever,1972; Teusink et al., 2000) and the Gepasi files of thesemodels can be obtained via email request (barbara.bakker@falw.vu.nl).

2.3.1. Model 1: the glycolytic core model by Goldbeter

and Lefever (1972)

The glycolytic core model by Goldbeter and Lefever isdimensionless and consists only of the kinetics of theenzyme phosphofructokinase (PFK) plus an input andan output reaction (Fig. 1A). The model has twovariables a and g which are involved in three reactions.In terms of differential equations this reads:

dadt

¼ s1 � sM � F; (9)

dgdt

¼ sM � F� ks � g;

where

F ¼a � eð1 þ a � eÞð1 þ gÞ2 þ L �Y � a � c � e0ð1 þ a � c � e0Þ

Lð1 þ a � c � e0Þ2 þ ð1 þ gÞ2ð1 þ a � eÞ2

(10)

includes the most important kinetic details of PFK.The variables a and g represent the concentrationsof substrate (ATP or fructose-6-phosphate) and pro-duct (ADP or fructose-1,6-bisphosphate) of PFK,respectively. s1 is the constant injection rate ofsubstrate, sM denotes the rate constant (or concen-tration) of PFK and ks is the rate constant for thesink of the product. Details of the other parameterscan be found in the original literature. In the presentstudy, a set of parameter values was used that wasidentified by Goldbeter and Caplan (1976); L=106;c=10�5; e=e0=0.9090909; Y=1; s1 ¼ 0:7; sM=4;ks=0.1.

2.3.2. Model 2: the detailed glycolytic model by Teusink

et al. (2000)

The detailed glycolytic model by Teusink et al. (Fig.1B) has been developed to test whether in vivo fluxesand metabolite concentrations can be understood interms of in vitro kinetic constants. Kinetic constants ofall glycolytic enzymes from an industrial yeast strain,and in vivo fluxes and metabolite concentrations wereall measured under the same conditions. The modelcontains explicit enzyme-kinetic rate equations for allreactions in Fig. 1B, except for two enzymes that wereconsidered in equilibrium, namely TPI and AK. Thereactants of the latter were considered as a singlemetabolite pool:

TRIO ¼ DHAP þ GAP; (11)

P ¼ 2 � ATP þ ADP:

In this study we used a version in which the branches toglycogen and trehalose are described as constant fluxes.Please note that, due to some small errors in thedescription of the original model, the differentialequations used here for P, AcAld and NADH wereslightly different (see Appendix A). For further detailsthe reader is referred to Teusink et al. (2000).

2.3.3. Modification of the detailed model by Teusink

et al. (2000)

The detailed model by Teusink et al. had beenoptimized for an industrial baker’s yeast strain (Ko-ningsgist), grown in fed-batch cultures and subsequentlysubjected to anaerobic conditions with excess of glucose.In accordance with experimental findings for theseconditions, the detailed model by Teusink et al. didnot exhibit oscillations. We wished to calculate whatcontrolled the stability of glycolysis not only in astable situation, but also close to but on the unstableside of the Hopf bifurcation. Therefore we firstsearched for parameter changes that led to oscillations.The kinetic equation of the glucose transporter is

ARTICLE IN PRESS

Table 1

Kinetic constants of glucose transport, G3PDH, branch to glycogen,

and ATPase, for the original and modified model

Kinetic constants Original Modified

V transportmax 97.26 mM min�1 70 mM min�1

Ki 0.91 0

V G3PDHmax

70.15 mM min�1 78.1223 mM min�1

vglycogen 6 mM min�1 5.6 mM min�1

KATPase 39.5 mM min�1

V ATPasemax

66 mM min�1

KMATPase 0.338 mM

H 5

These four kinetic steps were modified to obtain oscillatory behavior in

the detailed glycolytic model by Teusink et al. (2000). See the text for

further explanation.

Fig. 2. Bifurcation graph of the PFK model by Goldbeter and Lefever

(1972). A bifurcation analysis was performed on a core model.

Metabolite a was plotted as a function of the bifurcation parameter s1;for sM ¼ 4 and ks ¼ 0:1: Two supercritical Hopf bifurcations were

found. The thick full lines represent the stable steady states. The

thin full line represents the unstable steady states. The gray lines

represent the minimum and maximum values of the concentration of

metabolite a.

K.A. Reijenga et al. / Journal of Theoretical Biology 232 (2005) 385–398390

given below:

vtransport

¼V transport

max½Glc�out�½Glc�in

KM ;Glc

1 þ½Glc�outKM ;Glc

þ½Glc�inKM;Glc

þ Ki½Glc�outKM ;Glc

½Glc�inKM;Glc

: ð12Þ

Experimentally, limit cycle oscillations are observedwhen cells are harvested at the diauxic shift andsubsequently starved for a period of time (Richardet al., 1993). One of the effects this procedure has on themake-up of the cells is a decrease in glucose transportactivity (cf. Reijenga et al., 2001; Reijenga, 2002) fordiauxic shift cells that have been starved for 2 h, andexponential cells, respectively (Buziol et al., 2002;Krampe and Boles, 2002; Rossell et al., 2002)). Indeedby decreasing the activity of the glucose transporter inthe original model, the eigenvalues became complex andthe absolute value of the real part of l1;2 decreased.Furthermore, the factor Ki (the interactive constant thatdepends on the relative mobility of the glucose-occupiedand glucose-free carriers) was set to zero in the equationfor the glucose transporter.

Secondly, the kinetics of ATP utilization was changedfrom mass-action to cooperative kinetics. In this studythe following equation was used:

vATPase ¼V ATPase

max � Sh

Sh0:5 � Sh

: (13)

Without this last change, explosive oscillations awayfrom a steady state (unstable focus) were found, but nostable limit cycle behavior was observed. Limit cycleoscillations were only observed when a Hill coefficient ofapproximately 5 was used in this reaction (results notshown). In the original model by Teusink et al., theactivity of ATP utilization was fitted to the measuredflux in a single steady-state setting. Therefore weconsidered it valid to change this step in order toobtain the oscillatory behavior of the non-steady-stateexperiments.

Finally, the Vmax of G3PDH was increased and thebranch to glycogen was decreased. The kinetic para-meters used in this study are listed in Table 1.

3. Results

3.1. A core model of glycolysis: the PFK model by

Goldbeter and Lefever

3.1.1. Bifurcation analysis

For the core model of Goldbeter and Lefeverbifurcation diagrams were made for three kineticparameters (s1; sM and ks). Fig. 2 shows a typicalbifurcation graph in which s1 (i.e. glucose transport)was the bifurcation parameter and the concentration of

a was the variable. When s1 was below 0.12, the systemreached the stable steady state (as indicated by the thickfull line in Fig. 2). As s1 exceeded 0.12 the steady stateindicated by the full line became unstable and the systemengaged in limit-cycle oscillations. The metabolites aand g oscillated in time between the values indicated bythe upper and lower dotted lines. The switch at s1 ¼

0:12 from a stable focus to stable limit cycle oscillationsis a Supercritical Hopf bifurcation (SpHB). At s1 ¼ 1:16the system switched again from stable limit cycleoscillations to a stable focus. For the values chosen forparameters sM and ks, this model had two SpHB’s.These two SpHB’s were seen for all three bifurcationparameters (results not shown). The different dynamicregimes found for different ranges of the parameters arelisted in Table 2.

ARTICLE IN PRESS

Table 2

Bifurcation analysis of the core model of Goldbeter and Lefever (1972)

s1 sM ks

Stable node 0–0.07 0.71–1.15 0–0.018

Stable focus 0.07–0.12 1.15–1.81 0.018–0.058

Unstable focus 0.12–1.19 1.81–33.5 0.058–0.44

Stable focus 1.19–3.24 33.5–1770 0.44–0.67

Stable node 3.24–3.99 1770–105 0.67–1000

Three kinetic parameters (s1; sM ; ks) were taken individually as

bifurcation parameter. When a bifurcation parameter was changed, all

other parameters were kept constant, at the following values: s1 ¼ 0:7;sM ¼ 4:0; ks ¼ 0:1: The stability of the systems was determined using

Gepasi and is indicated for the different values of the different

bifurcation parameters in this table.

-0.12

-0.08

-0.04

0

0.04

0 0.05 0.1 0.15 0.2 0.25

σ1 (min-1)

eig

enva

lue

Fig. 3. Eigenvalues of the PFK model by Goldbeter and Lefever

(1972). The eigenvalues were plotted as a function of the bifurcation

parameter s1; for sM ¼ 4 and ks ¼ 0:1: The thick full lines represent

the real (parts of the) eigenvalues and the thin full lines represent the

imaginary part of the eigenvalues. The closed circle indicates the

supercritical Hopf bifurcation.

K.A. Reijenga et al. / Journal of Theoretical Biology 232 (2005) 385–398 391

Because the Goldbeter and Lefever model containsonly two variables, there are only two eigenvalues. Wecalculated the real parts and the imaginary parts of theeigenvalues as functions of the bifurcation parameter s1

which may be taken to represent glucose transport. Fors1o0:07; the eigenvalues were real. From Fig. 3 it wasclear that the eigenvalues were continuously differenti-able functions everywhere except at the bifurcation fromstable node to stable focus at s1 ¼ 0:07 (Fig. 3),indicating that they could be subjected to controlanalysis.

3.1.2. Control of eigenvalues

The control of the individual enzymes on theeigenvalues of the system was determined at differentcombinations of parameters and the results can befound in Table 3 (s1 as bifurcation parameter) andTable 4 (sM as bifurcation parameter). For systems withreal eigenvalues (stable and unstable nodes), the control

on both eigenvalues (l1 and l2) was determined. Forsystems with complex eigenvalues (stable and unstablefoci), the control on the real part (Re l) and on theimaginary part (Im l) of the eigenvalues was determined.With this approach, it was possible to distinguishbetween control on the stability/instability of the system(l1 when l’s are real, Re l when l’s are complex) andcontrol on the oscillatory tendency (or frequency) of thesystem (Im l when l’s are complex) (cf. Methods).

3.1.3. Control of stability

The results in Tables 3 and 4 indicated that thecontrol of the individual kinetic steps on the tendency ofthe system to be stable was distributed. Positive andnegative values were calculated for control coefficientsof all three reactions (i.e. s1; sM and ks), both forcontrol on stability and for control on oscillatorytendency. The sum of the control coefficients was 1.Some control coefficients had a high value, i.e. b1: Thisimplied that some enzymes had a more than propor-tional effect on the stability of the system.

When PFK (sM) was used as the bifurcationparameter (Table 4), this step had a highly stabilizingeffect on the system in a stable node (7.19) but a highlydestabilizing effect close to the Hopf bifurcation (�7.43,�12.56). Its control was almost equaled by the controlby glucose transport (s1), which had always the oppositeeffect. When PFK was not the bifurcation parameter(Table 3), it had only minor control on the stability ofthe system.

Of particular interest are the values around the SpHB,i.e. the transition from stable focus to unstable focus.The enzymes responsible for this transition might becalled oscillophores, as this transition is usually whereoscillations become visible. These enzymes should havelarge negative (when enzyme activation should destabi-lize) or positive (when enzyme inactivation shoulddestabilize) values for their stability control coefficient(Cstab

i ). There was no single such oscillophore in ourcalculations. There were at least two, and which thesewere depended on the parameter values, i.e. transport(s1) and lower glycolysis (ks) in Table 3, and transport(s1) and phosphofructokinase (sM) in Table 4. In Table3, around the Hopf bifurcation, transport was stronglydestabilizing (�5.91, �10.90), lower glycolysis wasstrongly stabilizing (6.88, 10.09), and PFK shifted fromweakly stabilizing to weakly destabilizing (0.03, �0.19).In Table 4 around the Hopf bifurcation, PFK wasstrongly destabilizing (�7.43, �12.56), but transportwas virtually equally important as a stabilizing factor(7.46, 13.34). Also in other calculations, control wasdistributed and PFK was stabilizing or destabilizing,depending on the conditions analysed. Furthermore, theparameter that was used as the bifurcation parameter(s1 in Table 3 and sM in Table 4) had the highestdestabilizing control close to the Hopf bifurcation.

ARTICLE IN PRESS

Table 4

Control of stability (Cstabi ) and oscillatory tendency (Cosc

i ) as calculated for the core model of glycolysis by Goldbeter and Lefever (1972)

Stable node sM=1.0 Stable focus sM=1.6 Unstable focus sM= 2.0

i Cstabi

ðCl1i Þ

Cl2i

Cstabi

ðCRe li Þ

Cosci

ðCIm li Þ

Cstabi

ð�CRe li Þ

Cosci

ðCIm li Þ

s1 �5.07 2.91 7.46 �0.13 13.34 0.34

sM 7.19 �3.16 �7.43 1.10 �12.56 0.61

ks �1.11 1.25 0.96 0.04 �1.78 0.06

Sum 0.99 1.00 0.99 1.01 �1.00 1.01

The control coefficients of the different steps on the eigenvalues of the system were calculated, around the Hopf bifurcation. Here, sM was used as the

bifurcation parameter. The values for the other parameters were: s1 ¼ 0:7 and ks=0.1. The values for the other parameters were: l1 ¼ �4:30 10�3;l2 ¼ �4:59 10�3 for sM ¼ 1:0; l1;2 ¼ �4:42 10�3 � 2:50 10�2i for sM ¼ 1:6; l1;2 ¼ 1:97 10�3 � 2:99 10�2i for sM=2.0. Parameter

changes were 1% for all calculations. A (high and) positive value for Cstabi implies that process i acts (strongly) to stabilize the system, a (highly)

negative value that it (strongly) encourages instability. Similarly, a process i with a positive value for Cosci enhances the oscillatory tendency, whereas

a process with a negative value, dampens the oscillatory tendency. Sum: the sum of the control by 1, M and s. See the text for further explanation.

Table 3

Control of stability (Cstabi ) and oscillatory tendency (Cosc

i ) as calculated for the core model of glycolysis by Goldbeter and Lefever (1972)

Stable node s1 ¼ 0:04 Stable focus s1 ¼ 0:10 Unstable focus s1 ¼ 0:13

i Cstabi

ðCl1i Þ

Cl2i

Cstabi

ðCRe li Þ

Cosci

ðCIm li Þ

Cstabi

ð�CRe li Þ

Cosci

ðCIm li Þ

s1 1.70 �0.93 �5.91 0.94 �10.90 0.37

sM 0.50 0.02 0.03 0.29 �0.19 0.28

ks �1.20 1.91 6.88 �0.23 10.09 0.36

Sum 1.00 1.00 1.00 1.00 �1.00 1.01

The control coefficients of the different steps on the eigenvalues of the system were calculated, around the Hopf bifurcation. s1 was used as the

bifurcation parameter. The values for the other parameters were: sM=4 and ks=0.1. The values of the eigenvalues were: l1 ¼ �2:17 10�3;l2 ¼ �4:61 10�3 for s1 ¼ 0:04; l1;2 ¼ �3:86 10�3 � 1:44 10�2i for s1 ¼ 0:10; l1;2 ¼ 1:95 10�3 � 1:68 10�2i for s1 ¼ 0:13: Parameter

changes were 1% for all calculations. A (high and) positive value for Cstabi implies that process i acts (strongly) to stabilize the system, a (highly)

negative value that it (strongly) encourages instability. Similarly a process i with a positive value for Cosci enhances the oscillatory tendency, whereas a

process with a negative value, dampens the oscillatory tendency. Sum: the sum of the control by 1, M and s. See the text for further explanation.

K.A. Reijenga et al. / Journal of Theoretical Biology 232 (2005) 385–398392

We also observed a perhaps more general phenomena:the sum over all enzymes of the control coefficients onstability was equal to 1. This suggested that theindicators of stability which we propose here, i.e. thereal part of the complex eigenvalue and the smallesteigenvalue, both belong to the class of properties forwhich there is a summation theorem that yields 1 for thesum, like flux and frequency. The total control ofstability was 1 for stable systems and –1 for unstablesystems.

3.1.4. Control of oscillatory tendency

For the imaginary part of the eigenvalue a positivevalue of a control coefficient indicated a positive controlon the frequency and on the oscillatory tendency,whereas a negative value indicated a negative control.The extent to which any of the three processes of thesimple oscillator controls oscillatory tendency is quanti-fied (cf. Methods) in terms of the control coefficientCosc

i : In Tables 3 and 4, the control of this oscillatory

tendency was distributed. Around the bifurcation fromstable node to stable focus, the bifurcation process hadpositive control on the oscillatory tendency. Overall, allthree processes had positive and negative values for thiscontrol coefficient, depending on the conditions ana-lysed. All kinetic steps had some control over theoscillatory behavior of the system. None of the enzymeshad all the control on the dynamic behavior, implyingthat also in the sense of effect on frequency, nooscillophore could be identified (cf. Bier et al., 1996;Teusink et al., 1996). For control on oscillatorytendency the sum was 1 for stable and unstable foci.

3.2. Yeast glycolysis: the glycolytic model by Teusink

et al. (2000)

In the previous section, we completed the controlanalysis for a core model of glycolysis with only twovariables. From the results obtained we could concludethat even in this simple model with a single positive

ARTICLE IN PRESS

Table 5

Control coefficients as calculated for a realistic model of yeast

glycolysis (Teusink et al., 2000)

Stable node

i Cstabi

ðCl1i Þ

Cl2i

Transport �5.61 �2.05

HK �0.78 �0.18

PGI �0.001 0.13

Trehalose 0.37 0.15

Glycogen 0.47 0.19

PFK 0.003 0.16

ALD 0.009 0.13

TPI 0 0

GAPDH �0.095 0.36

G3PDH 0.37 0.33

PGK �0.008 0.034

PGM 0 0.035

ENO 0.011 0.10

PYK 0.02 0.056

PDC 5.61 0.49

ADH �0.12 �0.004

Succinate 0.33 0.01

ATPase 0.41 1.05

AK 0 0

Sum 1.00 1.00

The control by the different steps on the two smallest eigenvalues, in

terms of absolute values, of the system was calculated in terms of the

corresponding control coefficients. The system had only real and

negative eigenvalues; l1 ¼ �4:50 and l2 ¼ �14:14: As l1 had the

smallest absolute value, i.e. was closest to zero, it was most likely to

change sign. Parameter changes were 1% for all calculations. A (high

and) positive value for Cstabi implies that process i acts (strongly) to

stabilize the system, a (high and) negative value that it (strongly)

encourages instability. See the text for further explanation.

K.A. Reijenga et al. / Journal of Theoretical Biology 232 (2005) 385–398 393

feedback loop no single enzyme controlled either thestability or the oscillatory tendency of the system. Thisdemonstrated that stability need not be controlled by asingle enzyme. However, we wondered if real glycolysis,as it occurs in yeast, might still follow the paradigm ofthe single oscillophore. At present and in the foreseeablefuture, it will be too difficult to address this question bydirect experimentation, but indirectly it can be ad-dressed experimentally, i.e. by examining a model that isa (virtually) complete replica of experimental yeastglycolysis. Therefore, we applied the above developedmethod to the most detailed kinetic model of yeastglycolysis available (Teusink et al., 2000).

The original model exhibited stable node behavior,consistent with experimental findings concerning anae-robic Koningsgist for which the model had beendeveloped. At a stable node however, the real eigenva-lues are still relevant for the degree of stability of thesystem. The dominant eigenvalue indicates the degree offreedom that relaxes most slowly after a perturbation.Any enzyme that might shift this eigenvalue in thedirection of or across zero would therefore negativelycontrol stability.

3.2.1. Control of stability for stable yeast glycolysis

As a measure of control of stability the controlcoefficients on the dominant eigenvalue l1 but also theeigenvalue with the (which we call subdominant) secondgreatest real value l2, i.e. the two eigenvalues closest tozero, are listed in Table 5. It was verified by calculationthat, also for this model, the dominant eigenvalues werecontinuous functions of the bifurcation parameter, andthat the curves following the different eigenvalues didnot intersect (results not shown), that is the real part ofthe dominant eigenvalue is a smooth function of thebifurcation parameter, at least in the range of interest.As for the core model, the positive control coefficientCstab

i corresponds to a stabilizing effect. Again it wasfound that control on the eigenvalues, thus on thestability of the system, was distributed and that theeigenvalue control coefficients and thus the stabilitycontrol coefficients added up to 1. Furthermore, glucosetransport and pyruvate decarboxylase (PDC) had arelatively high control on stability (negative andpositive, respectively). PGI, PFK, ALD, PGK, ENO,PYK and PGM had relatively little control on stability,whereas HK, the branches to glycogen, trehalose andsuccinate, G3PDH, GAPDH, ADH and ATPase hadmoderate control on stability.

3.2.2. Bifurcation analysis for oscillating yeast glycolysis

To obtain a model that describes the oscillating cells,we altered that model as described in the Modelssection. A bifurcation analysis was made of the modifiedmodel, in which V transport

max was used as the bifurcationparameter. In the bifurcation graph (Fig. 4) the internal

glucose concentration was plotted as a function ofV transport

max : Two Supercritical Hopf bifurcations werefound at values for Vtransport

max of 69.82 and70.92 mM min�1. In between the two Hopf bifurcationslimit cycles oscillations were observed.

3.2.3. Control of stability for oscillating yeast glycolysis

For the modified (oscillating) model, it was verifiedthat the eigenvalues were continuous functions of thebifurcation parameter, and that the curves for thedifferent eigenvalues did not intersect (Fig. 5).The control analysis was done on the stable side ofthe first Hopf Bifurcation (Vtransport

max =69 mM min�1,stable focus), between the Hopf Bifurcations(V transport

max =70 mM min�1, unstable focus) and on thestable side of the second Hopf Bifurcation(V transport

max =72 mM min�1, stable node).From Table 6 it was clear that control on stability of

glycolysis was again distributed. Transport was stronglydestabilizing at the lower activities but became stabiliz-ing after passing through the second SpHB. For allthree sets of parameter values, the strongest control

ARTICLE IN PRESS

-140

-120

-100

-80

-60

-40

-20

0

50 55 60 65 70

Vmax (µM.min-1)

Rea

l (p

art

of)

eig

enva

lue

Fig. 5. Eigenvalues of the modified detailed model of yeast glycolysis.

The real (parts of the) eigenvalues were plotted as a function of the

bifurcation parameter Vtransportmax : Here, we focused on the four

eigenvalues with the smallest absolute value of the real part, i.e. the

ones which were closest to zero and were most likely to change sign.

The thick line represents the first two eigenvalues, which were complex

with equal real parts. At V transportmax =69.82 mM min�1 their real parts

became positive. The thin full line represents the third eigenvalue,

which was real and negative. The thin dotted line represents the fourth

eigenvalue, which was real and negative for V transportmax o66 mM min�1

and complex with negative real part for V transportmax 466 mM min�1.

0

0.5

1

1.5

2

60 65 70 75

Vmax (µM.min-1)

[Glc

int]

(mM

)

Fig. 4. Bifurcation graph of the oscillating model of yeast glycolysis. A

bifurcation analysis was performed on the modified model. The

internal glucose concentration was plotted as a function of the

bifurcation parameter Vtransportmax : Two supercritical Hopf bifurcations

were found. Between Vtransportmax =69.82 mM min�1 and Vtransport

max =

70.92 mM min�1, limit cycle oscillations were observed. The thick full

lines represent the stable steady states. The thin full line represents the

unstable steady states. The thin dashed lines represent the minimum

and maximum values of the concentration of internal glucose.

K.A. Reijenga et al. / Journal of Theoretical Biology 232 (2005) 385–398394

coefficients were found for glucose transport, HXK,G3PDH, ATPase and branches to trehalose, glycogenand succinate. Glucose transport and ATPase were twoof the steps that were modified to obtain an oscillatingmodel, and especially for the unstable focus, thecorresponding control coefficients were high, �210 and118, respectively. Note that glucose transport wasdestabilizing, whereas ATPase was stabilizing. The other

steps with high control were those involved in branchesof glycolysis.

3.2.4. Control of oscillatory tendency for oscillating yeast

glycolysis

For the model of oscillating yeast glycolysis, thecontrol of oscillatory tendency was determined in termsof C

Im l1i ; for complex eigenvalues. Also for the detailed

model of yeast glycolysis the control of the oscillatorytendency was distributed (cf. Table 6). The controlcoefficients assumed positive and negative values. Theenzymes that had a high control on stability of thesystem, also had a high control on the oscillatorytendency (glucose transport, HXK, G3PDH, ATPaseand branches to trehalose, glycogen and succinate),mostly with different signs and with the exception ofhexokinase. High values (b1) were calculated for thecontrol coefficients, implying relatively large effects onfrequency of the system. For control on oscillatorytendency again the sum was 1 for systems in stable andunstable focus.

4. Discussion

Glycolytic oscillations have been studied extensivelyin cell-free extracts as well as in whole cells (for reviewon biochemical oscillations see (Goldbeter, 1996)). Itwas concluded from various experiments (Chance et al.,1964; Hess, 1968) that PFK was the oscillophore, i.e. thesource of glycolytic oscillations in yeast. Mechanisti-cally, this could be explained by the activation of PFKby its own (indirect) product, i.e. AMP. It has beenproven analytically for a core model that such a positivefeedback can indeed induce oscillatory behavior (Hig-gins, 1967). The oscillophore hypothesis howeverignored the fact that the positive feedback loop is nota sufficient condition to obtain oscillations. Also thekinetic parameters of the other enzymes need to bewithin the oscillatory window. In addition, real glyco-lysis contains more than one positive feedback loop, e.g.the autocatalytic stoichiometry of glycolysis itself.Consistently, we still observed oscillations in themodified detailed model used in this study when thePFK kinetics were reduced to simple mass actionkinetics without a feedback loop (results not shown).In view of these complications, it is impossible to decidea priori whether PFK is the oscillophore in vivo.

To solve the question as to what causes the oscilla-tions, we here developed a quantitative method todetermine the control of enzymes on the stability of abiochemical network. We defined control on stability interms of the control on the real (parts of the)eigenvalues, and control on oscillatory tendency as thecontrol on the imaginary parts of the eigenvalues.

ARTICLE IN PRESS

Table 6

Control of oscillations as calculated for a realistic model of yeast glycolysis

Stable focus V transportmax =69 Unstable focus Vtransport

max =70.11 Stable node V transportmax =72

i Cstabi

ðCRe l1i Þ

Cosci

ðCIm l1i Þ

Cstabi

ð�CRe l1i Þ

Cosci

ðCIm l1i Þ

Cstabi

ðCl1i Þ

Transport �23.25 1.67 �210.33 5.67 86.81

HK �10.41 21.23 14.99 34.64 0.39

PGI 0.021 0.077 �1.24 0.10 0.020

Trehalose 4.58 �3.29 26.05 �5.71 �11.11

Glycogen 6.18 �4.44 35.21 �7.71 �14.97

PFK 0.056 0.027 �0.17 0.06 0.028

ALD 0.021 0.050 �0.95 0.05 0.013

TPI 0.00 0.00 0.00 0.00 0.00

GAPDH �2.96 2.44 �21.72 4.05 7.51

G3PDH 5.69 �3.93 34.00 �6.83 �13.91

PGK �0.033 0.028 �0.29 0.05 0.11

PGM �0.008 0.016 �0.25 0.02 0.036

ENO �0.015 0.041 �0.68 0.05 0.062

PYK �0.003 0.015 �0.28 0.00 �0.11

PDC �0.29 0.46 �8.63 0.19 �0.033

ADH �0.81 0.50 �5.06 0.89 2.07

Succinate 3.431 �2.35 20.23 �4.13 �8.46

ATPase 18.79 �11.55 118.12 �20.40 �47.26

AK 0.00 0.00 0.00 0.00 0.00

Sum 1.00 1.00 �1.00 1.00 0.99

The control coefficients of the different enzymes on the eigenvalues of the system, on stability and on oscillatory tendency were calculated, around the

Hopf bifurcation. Vtransportmax was used as the bifurcation parameter. Parameter modulations were as follows. Stable focus: Transport, HK, ATPase,

0.01%; branches to trehalose and glycogen, G3PDH, 0.1%; GAPDH, branch to succinate; 0.5%; other enzymes, 1%; l1;2 ¼ �1:87 � 3:76i: Unstable

focus: Parameter changes were 1% for all calculations; l1;2 ¼ 2:16 10�1 ��3:10i: Stable node: Transport, ATPase, 0.01%; branches to trehalose,

glycogen and succinate, G3PDH, 0.1%; other enzymes, 1%; l1 ¼ �8:47 10�2: See the text for further explanation.

K.A. Reijenga et al. / Journal of Theoretical Biology 232 (2005) 385–398 395

We determined the magnitudes of these controlcoefficients for the enzymes in the system. We herelimited the discussion to the control of the enzymeactivities for three reasons: (i) all the reaction activitiesof these enzymes constitute a complete set, (ii) otherparameters that are outside this set, are still relevant, asthey can be shown to effect the oscillations indirectly,i.e. through a number of parameters in the complete set(see Appendix B), and (iii) the cell is able to modulatethe Vmax values of specific enzymes, via gene expressionand translation regulation, however, the possibilities tomodulation the KM’s or other parameters are less likely.The analyses were done both on a core model and on adetailed model of yeast glycolysis, consisting of two and14 variables, respectively. For the core model the claimwas made strongest that the oscillophore is in a singlestep, whereas the detailed model was closer to real yeastglycolysis, and therefore biologically more relevant. Wefound that, (i) control of enzymes on stability and onoscillatory tendency was distributed and depended onthe conditions analysed, and (ii) the sum of the controlcoefficients on both real and imaginary parts of theeigenvalues added up to 1. This implies that neitheroscillatory tendency nor stability is caused by a singleenzyme, but that all enzymes in the pathway contributeto different extents to the emergence of oscillations.

Using the corresponding control coefficients, we maynow propose a new definition for an oscillophore. Whenthe system is a stable focus, we define an oscillophore asan enzyme with a high and negative control on stability.When the system is an unstable node, we define anoscillophore as an enzyme with a high and positivecontrol on oscillatory tendency. When the system is in astable node, an oscillophore should fulfill two require-ments, i.e. a high and negative control on stability and ahigh and positive control on oscillatory tendency. Usingthis definition, we may now examine whether thisdefinition makes more sense, than any of the earlier,explicit or implicit definitions. When we look atTable 3, we notice that for the stable focus, the transportstep identifies as a major oscillophore, with lowerglycolysis as a major ‘anti-oscillophore’. With respectto the stable node, no process qualified as oscillophore;lower glycolysis did favor instability but workedagainst the oscillatory tendency. In the stable focus ofTable 4, process ‘M’ (representing phosphofructokinase)should qualify as oscillophore, but in the stable nodeno process qualified. More importantly perhaps, inthe realistic model of glycolysis (Table 5), bothtransport and hexokinase qualified as oscillophores,whereas phosphofructokinase had very little oscillo-phore strength.

ARTICLE IN PRESSK.A. Reijenga et al. / Journal of Theoretical Biology 232 (2005) 385–398396

When looking for candidates for the qualificationoscillophore in our results, a number of aspects come tothe attention. First, there is often also an ‘anti-oscillophore’, i.e. a process with a highly positivecontrol on stability. Inactivation of such a process willalso enhance the oscillations. Hence such a process is animportant controller of the oscillations. Second, thereare two aspects to limit-cycle oscillations. One is that thesteady state is unstable and the other is that the systemoscillates. Accordingly there are also two aspects to thecausative role of a process with respect to limit cycleoscillations. One is that it may cause the system to beunstable, i.e. an eigenvalue to have a positive real part.The other is that it may help cause that eigenvalue to becomplex. Accordingly, one will have to considerseparately two control coefficients, i.e. one for thecontrol on stability and one for the control onoscillatory tendency. In the third place, even if one wereto consider only the control of stability, rarely allcontrol resided in a single process; almost in all cases itwas distributed unevenly among a number of, but not allprocesses. In the fourth place, when a process is anoscillophore under one condition, it need not be oneunder a different condition in the same network. Andfinally, there is little if any special role for phospho-fructokinase in all of this.

Is there any role left for the concept of anoscillophore? Because it is so condition dependent, itmay seem that there is no use for such a term. In thestrictest sense of designating a single oscillophore, wewould agree to this. However, the possibility todetermine quantitatively to what extent a process isoscillophoric, is an asset to the study of cell function.Even though the answer depends on cellular conditions,this itself should be appreciated, as it reflects the richnessof the control of cell function. When one is interested instudying the cell cycle, the answer as to what is theprocess that causes the cycling, may turn out to beequally equivocal. Various processes may turn out to beoscillophoric and various others anti-oscillophoric. Allof these should be interesting as interference with themcould compromise or over-activate cell division. Theirrelative importance should also matter. The concept ofprocesses being more or less oscillophoric is important,but much of the importance rests in ‘more or less’, asquantified by the control coefficients for stability andoscillatory tendency.

The quantification of the oscillophoric potential ofmolecular processes is all the more important becausethere seems to be a total to this potential, as defined bythe summation properties we discovered. Althoughsummation theorems do not provide additional infor-mation, they are important tools to order our under-standing of the system and to study the systemicproperties of the model. We found that when the systemis in a stable focus, the total control on oscillatory

tendency equals 1. Similar summation theorems havealready been proven to exist for frequency, amplitudeand average value (Bier et al., 1996; Demin et al., 1999;Giersch, 1988; Kholodenko et al., 1997; Teusink et al.,1996). The sum of control on frequency is known to be1, which is in accordance with our finding that the sumof control on oscillatory tendency is 1 for stable andunstable foci. Furthermore, previously we consideredthe Fourier spectrum of glycolytic oscillations, wheresummation theorems were found for individual compo-nents of this spectrum (Reijenga et al., 2002). The mainimplication of the existence of summation theorems isthat control cannot be increased or decreased, but canonly be redistributed.

The method we developed in this paper is general inquantifying the importance of an enzyme for theoccurrence of oscillations. We anticipate that in thefuture this method will allow one to elucidate also towhich extent different processes contribute to theemergence of other types of oscillations, such as in freecalcium or in the cell cycle.

Acknowledgements

This work was financially supported by the Nether-lands Organization for Scientific Research (NWO) andthe Technology Foundation (STW, Project 349-3995).

Appendix A

Due to some small errors in the description of theoriginal model by Teusink et al., the differentialequations used in this paper for P, AcAld and NADHwere slightly different. For further details the reader isreferred to Teusink et al. (2000).

dP

dt¼ � vHK � vPFK þ vPGK þ vPYK

� vATPase � vtrehalose � vglycogen;

dAcAld

dt¼ vPDC þ vADH � 2vsuccinate;

dNADH

dt¼ vGAPDH þ vADH

� vG3PDH þ 3vsuccinate:

Appendix B

Although we have limited the discussion in this paperto the control of enzyme activities, other parametersthat are outside this set, are still relevant, as they can beshown to effect the oscillations indirectly, i.e. through a

ARTICLE IN PRESS

Table 7

Control of stability (Cstabi ) and oscillatory tendency (Cosc

i ) as calculated for the core model of glycolysis by Goldbeter and Lefever (1972)

Stable node s1 ¼ 0:04 Stable focus s1 ¼ 0:10 Unstable focus s1 ¼ 0:13

p Cstabi

ðCl1i Þ

Cl2i

Cstabi

ðCRe li Þ

Cosci

ðCIm li Þ

Cstabi

ð�CRe li Þ

Cosci

ðCIm li Þ

e, e0 1.03 �0.03 0.29 0.52 0.73 0.52W �0.08 0.08 0.38 �0.03 0.65 0.01L �0.56 0.06 0.05 �0.27 �0.05 �0.26c �0.08 0.08 0.38 �0.03 0.64 0.01

The control coefficients of the parameters of PFK on the eigenvalues of the system were calculated, around the Hopf bifurcation. s1 was used as the

bifurcation parameter. The values for the other parameters were: sM ¼ 4 and ks ¼ 0:1: The values of the eigenvalues were: l1 ¼ �2:17 10�3;l2 ¼ �4:61 10�3 for s1 ¼ 0:04; l1;2 ¼ �3:86 10�3 � 1:44 10�2i for s1 ¼ 0:10; l1;2 ¼ 1:95 10�3 � 1:68 10�2i for s1 ¼ 0:13: Parameter

changes were 1% for all calculations.

Table 8

Control of stability (Cstabi ) and oscillatory tendency (Cosc

i ) as calculated for the core model of glycolysis by Goldbeter and Lefever (1972)

Stable node sM=1.0 Stable focus sM=1.6 Unstable focus sM=2.0

p Cstabi

ðCl1i Þ

Cl2i

Cstabi

ðCRe li Þ

Cosci

ðCIm li Þ

Cstabi

ð�CRe li Þ

Cosci

ðCIm li Þ

e, e0 1.06 �0.06 0.73 0.49 2.28 0.51W 0.00 0.00 0.02 0.00 0.01 0.00L �0.52 0.02 �0.39 �0.24 �1.19 �0.26c 0.00 0.00 0.01 0.00 0.05 0.00

The control coefficients of the parameters of PFK on the eigenvalues of the system were calculated, around the Hopf bifurcation. sM was used as the

bifurcation parameter. The values for the other parameters were: s1 ¼ 0:7 and ks ¼ 0:1: The values for the other parameters were: l1 ¼

�4:30 10�3; l2 ¼ �4:59 10�3 for sM ¼ 1:0; l1;2 ¼ �4:42 10�3 � 2:50 10�2i for sM ¼ 1:6; l1;2 ¼ 1:97 10�3 � 2:99 10�2i for sM ¼ 2:0:Parameter changes were 1% for all calculations.

Table 9

Control of oscillations as calculated for a realistic model of yeast glycolysis

Stable focus Vtransportmax =69 Unstable focus V transport

max =70.11 Stable node V transportmax =72

i Cstabi

ðCRe l1i Þ

Cosci

ðCIm l1i Þ

Cstabi

ð�CRe l1i Þ

Cosci

ðCIm l1i Þ

Cstabi

ðCl1i Þ

GR 0.035 0.495 �0.134 0.040 0.049

L0 0.002 0.011 0.394 �0.014 �0.025

KM,F6P �0.026 �0.374 0.262 �0.037 �0.049

KM,ATP �0.034 �0.444 �0.496 �0.011 0.013

CATP 0.005 0.040 0.642 �0.021 �0.065

KAMP 0.002 0.007 0.292 �0.008 �0.019

CAMP 0.003 0.014 0.452 �0.016 �0.003

Ki,ATP �0.001 0.015 �0.320 0.013 �0.032

Ci,ATP 0.005 0.022 0.785 �0.027 �0.050

KF26BP 0.003 0.012 0.549 �0.018 �0.035

CF26BP 0.001 0.012 0.148 �0.005 �0.007

KF16BP �0.001 0.011 �0.280 0.011 0.023

CF16BP 0.001 �0.016 0.350 �0.013 �0.030

The control coefficients of the parameters of PFK on the eigenvalues of the system, on stability and on oscillatory tendency were calculated, around

the Hopf bifurcation. Vtransportmax was used as the bifurcation parameter. Parameter changes were 1% (stable focus, stable node) or 0.01% (unstable

focus). The values for the eigenvalues were: Stable focus: l1;2 ¼ �1:87 � 3:76i: Unstable focus: l1;2 ¼ 2:16 10�1 ��3 10i: Stable node: l1 ¼

�8:47 10�2:

K.A. Reijenga et al. / Journal of Theoretical Biology 232 (2005) 385–398 397

ARTICLE IN PRESSK.A. Reijenga et al. / Journal of Theoretical Biology 232 (2005) 385–398398

number of parameters in the complete set. The controlcoefficients for the parameters that constitute phospho-fructokinase were calculated for both core model anddetailed model. The values are listed Tables 7–9.

References

Becker, J.-U., Betz, A., 1972. Membrane transport as controlling

pacemaker of glycolysis in Saccharomyces carlsbergensis. Biochim.

Biophys. Acta 274, 584–597.

Berridge, M.J., 1990. Calcium oscillations. J. Biol. Chem. 265,

9583–9586.

Berridge, M.J., Bootman, M.D., Lipp, P., 1998. Calcium: a life and

death signal. Nature 395, 645–648.

Bier, M., Teusink, B., Kholodenko, B.N., Westerhoff, H.V., 1996.

Control analysis of glycolytic oscillations. Biophys. Chem. 62,

15–24.

Boiteux, A., Goldbeter, A., Hess, B., 1975. Control of oscillating

glycolysis of yeast by stochastic, periodic, and steady source of

substrate: a model and experimental study. Proc. Natl Acad. Sci.

USA 72, 3829–3833.

Buziol, S., Becker, J., Baumeister, A., Jung, S., Mauch, K., Reuss, M.,

Boles, E., 2002. Determination of in vivo kinetics of the starvation-

induced Hxt5 glucose transporter of Saccharomyces cerevisiae.

FEMS Yeast Res. 2, 283–291.

Chance, B., Estabrook, R.W., Ghosh, A., 1964. Damped sinusoidal

oscillations of cytoplasmic reduced pyridine nucleotide in yeast

cells. Proc. Natl Acad. Sci. USA 51, 1244–1251.

Danø, S., Sorensen, P.G., Hynne, F., 1999. Sustained oscillations in

living cells. Nature 402, 320–322.

Demin, O.V., Westerhoff, H.V., Kholodenko, B.N., 1999. Control

analysis of stationary forced oscillations. J. Phys. Chem. B. 103,

10695–10710.

Doedel, E.J., Champneys, A.R., Fairgrieve, T.F., Kuznetsov, Y.A.,

Sandstede, B., Wang, X., 1998. AUTO 97: continuation and

bifurcation software for ordinary differential equations. Depart-

ment of Computer Science, Concordia University, Montreal,

Canada.

Duysens, L.N.M., Amesz, J., 1957. Fluorescence spectrophotometry of

reduced phosphopyridine nucleotide in intact yeast cells in the

near-ultraviolet and visible region. Biochim. Biophys. Acta. 24,

19–26.

Escalante, R., Vicente, J.J., 2000. Dictyostelium discoideum: a model

system for differentiation and patterning. Int .J. Dev. Biol. 44,

819–835.

Giersch, C., 1988. Control analysis of metabolic networks. 1.

Homogeneous functions and the summation theorems for control

coefficients. Eur. J. Biochem. 174, 509–513.

Goldbeter, A., 1996. Biochemical Oscillations and Cellular Rhythms.

Cambridge University Press, Cambridge.

Goldbeter, A., Caplan, S.R., 1976. Oscillatory enzymes. Ann. Rev.

Biophys. Bioenergy 6, 449–476.

Goldbeter, A., Lefever, R., 1972. Dissipative structures for an

allosteric model. Application to glycolytic oscillations. Biophys.

J. 12, 1302–1315.

Hartwell, L.H., 1974. Saccharomyces cerevisiae cell cycle. Bacteriol.

Rev. 38, 164–198.

Heinrich, R., Schuster, S., 1996. The Regulation of Cellular Systems.

Chapman & Hall, New York.

Hess, B., 1968. Biochemical regulations. In: Mesarovic, M.D. (Ed.),

Systems Theory and Biology. Springer, New York, pp. 88–114.

Hess, B., Boiteux, A., 1973. Substrate control of glycolytic oscillations.

In: Chance, B., Pye, E.K., Ghosh, A., Hess, B. (Eds.), Biological

and Biochemical Oscillations.

Higgins, J., 1967. The theory of oscillating reactions. Ind. Eng. Chem.

59, 18–62.

Kacser, H., Burns, J.A., 1973. The control of flux. Symp. Soc. Exp.

Biol. 65–104.

Kholodenko, B.N., Demin, O.V., Westerhoff, H.V., 1997. Control

analysis of periodic phenomena in biological systems. J. Phys.

Chem. B. 101, 2070–2081.

Krampe, S., Boles, E., 2002. Starvation-induced degradation of yeast

hexose transporter Hxt7p is dependent on endocytosis, autophagy

and the terminal sequences of the permease. FEBS Lett. 513,

193–196.

Mendes, P., 1997. Biochemistry by numbers: simulation of biochemical

pathways with Gepasi 3. Trends Biochem. Sci. 22, 361–363.

Nanjundiah, V., 1998. Cyclic AMP oscillations in Dictyostelium

discoideum: models and observations. Biophys. Chem. 72, 1–8.

Nurse, P., 2000. A long twentieth century of the cell cycle and beyond.

Cell 100, 71–78.

Rabinovich, M.I., Abarbanel, H.D., 1998. The role of chaos in neural

systems. Neuroscience 87, 5–14.

Reijenga, K.A., 2002. Dynamic control of yeast glycolysis. Molecular

Cell Physiology. Vrije Universiteit, Amsterdam, p. 248.

Reijenga, K.A., Snoep, J.L., Diderich, J.A., van Verseveld, H.W.,

Westerhoff, H.V., Teusink, B., 2001. Control of glycolytic

dynamics by hexose transport in Saccharomyces cerevisiae.

Biophys. J. 80, 626–634.

Reijenga, K.A., Westerhoff, H.V., Kholodenko, B.N., Snoep, J.L.,

2002. Control analysis for autonomously oscillating biochemical

networks. Biophys. J. 82, 99–108.

Richard, P., Teusink, B., Westerhoff, H.V., van Dam, K., 1993.

Around the growth phase transition S. cerevisiae’s make-up

favours sustained oscillations in intracellular metabolites. FEBS

Lett. 318, 80–82.

Richard, P., Diderich, J.A., Bakker, B.M., Teusink, B., van Dam, K.,

Westerhoff, H.V., 1994. Yeast cells with a specific cellular make-up

and an environment that removes acetaldehyde are prone to

sustained glycolytic oscillations. FEBS Lett. 341, 223–226.

Rossell, S., Van der Weijden, C.C., Kruckeberg, A., Bakker, B.M.,

Westerhoff, H.V., 2002. Loss of fermentative capacity in baker’s

yeast can partly be explained by reduced glucose uptake capacity.

Mol. Biol. Rep. 29, 255–257.

Sel’kov, E.E., 1975. Stabilization of energy charge, generation of

oscillations and multiple steady states in energy metabolism as a

result of purely stoichiometric regulation. Eur. J. Biochem. 59,

151–157.

Teusink, B., Bakker, B.M., Westerhoff, H.V., 1996. Control of

frequency and amplitudes is shared by all enzymes in three models

for yeast glycolytic oscillations. Biochim. Biophys. Acta 1275,

204–212.

Teusink, B., Passarge, J., Reijenga, C.A., Esgalhado, E., Van der

Weijden, C.C., Schepper, M., Walsh, M.C., Bakker, B.M., Van

Dam, K., Westerhoff, H.V., Snoep, J.L., 2000. Can yeast glycolysis

be understood in terms of in vitro kinetics of the constituent

enzymes? Testing biochemistry. Eur. J. Biochem. 267, 5313–5329.

Turek, F.W., 1998. Circadian rhythms. Horm. Res. 49, 109–113.

Tyson, J.J., Novak, B., 2001. Regulation of the eukaryotic cell cycle:

molecular antagonism, hysteresis, and irreversible transitions. J.

Theor. Biol. 210, 249–263.

Vitaterna, M.H., Takahashi, J.S., Turek, F.W., 2001. Overview of

circadian rhythms. Alcohol Res. Health 25, 85–93.

Wolfram, S., 1999. The Mathematica Book. Wolfram Media/Cam-

bridge University Press, Cambridge.