Dynamics of two-component biochemical systems in interacting

BioSystems 43 (1997) 1–24

Dynamics of two-component biochemical systems in interactingcells; Synchronization and desynchronization of oscillations and

multiple steady states

Jana Wolf, Reinhart Heinrich *

Humboldt-Uni6ersity, Institute of Biology, Theoretical Biophysics, In6alidenstrasse 42, D-10115 Berlin, Germany

Received 9 October 1996

Abstract

Systems of interacting cells containing a metabolic pathway with an autocatalytic reaction are investigated. Theindividual cells are considered to be identical and are described by differential equations proposed for the descriptionof glycolytic oscillations. The coupling is realized by exchange of metabolites across the cell membranes. Noconstraints are introduced concerning the number of interacting systems, that is, the analysis applies also topopulations with a high number of cells. Two versions of the model are considered where either the product or thesubstrate of the autocatalytic reaction represents the coupling metabolite (Model I and II, respectively). Model Iexhibits a unique steady state while model II shows multistationary behaviour where the number of steady statesincreases strongly with the number of cells. The characteristic polynomials used for a local stability analysis arefactorized into polynomials of lower degrees. From the various factors different Hopf bifurcations may result inleading for model I, either to asynchronous oscillations with regular phase shifts or to synchronous oscillations of thecells depending on the strength of the coupling and on the cell density. The multitude of steady states obtained formodel II may be grouped into one class of states which are always unstable and another class of states which mayundergo bifurcations leading to synchronous oscillations within subgroups of cells. From these bifurcations numerousdifferent oscillatory regimes may emerge. Leaving the near neighbourhood of the boundary of stability, secondarybifurcations of the limit cycles occur in both models. By symmetry breaking the resulting oscillations for theindividual cells lose their regular phase shifts. These complex dynamic phenomena are studied in more detail for a lownumber of interacting cells. The theoretical results are discussed in the light of recent experimental data on thesynchronization of oscillations in populations of yeast cells. © 1997 Elsevier Science Ireland Ltd.

Keywords: Cell population; Metabolic oscillation; Synchronization; Stability; Bifurcation

* Corresponding author. Tel.: +49 30 20938698; fax: +49 30 20938813; e-mail: [email protected]

0303-2647/97/$17.00 © 1997 Elsevier Science Ireland Ltd. All rights reserved.

PII S 0 3 0 3 -2647 (97 )01688 -2

J. Wolf, R. Heinrich / BioSystems 43 (1997) 1–242

1. Introduction

Periodic behaviour is an ubiquitous phe-nomenon of biological systems. It is found onnearly all structural levels, particularly withinmetabolic systems. In the last decades a greatvariety of oscillations within cellular systems havebeen observed and mathematically described, suchas glycolytic oscillations in different types of cells.More recently, oscillations of intracellular calciumconcentrations have attracted great interest ofexperimentalists and theoreticans; for a recentreview for cellular oscillations see Goldbeter(1996).

In a pioneering work on oscillating biochemicalreactions Higgins (1967) addressed the problem inwhich way a coupling between individual cellsaffects the resulting dynamics, for example, bysynchronizing their oscillations. Later on it hasbeen shown experimentally that a mixing of twocell populations oscillating out of phase may leadto their rapid synchronization (Pye, 1969; Ghoshet al., 1971). Recent experiments on glycolyticoscillations before and after mixing two out-of-phase populations of yeast cells show a synchro-nization which is rather slow (Richard et al.,1996).

The problem of coupling metabolic oscillatorshas also been analyzed in a great number oftheoretical investigations. Due to the difficulties indescription of high-dimensional systems the theo-retical investigations have often been restricted tothe case of two or three coupled oscillators which,of course, is not sufficient for describing cell pop-ulations. However, various results have beenderived also for systems of many interacting oscil-lators (Othmer and Aldridge, 1978; Alexander,1986). Other investigations concern systems ofweakly coupled oscillators (Kopell and Ermen-trout, 1986). Most theoretical work is performedto the case of direct interactions, i.e., where thecoupling terms of the model equations containonly the differences between concentration vari-ables within neighboured cells (Alexander, 1986;Kopell and Ermentrout, 1986). Such a direct cou-pling necessitates a physical contact between thecells. For its mathematical description special spa-tial arrangements have to be assumed, for exam-

ple, linear chains or rings of coupled cells. Foroscillations in cell suspensions it is probably morerealistic to consider an indirect coupling, by tak-ing into account substances which may diffuseacross the cellular membrane into the extracellu-lar space and may enter the cytoplasm of othercells. Recently such a coupling has been sup-ported experimentally for glycolytic oscillations inpopulations of yeast cells (Richard et al., 1996).There, strong support has been given that ac-etaldehyde which permeates the plasma mem-brane mediates the coupling. In particular, it wasshown that the extracellular concentration of ac-etaldehyde oscillates and that the cells respond toacetaldehyde pulses. The experimental results in-dicate that ethanol which was also considered toplay the role of an intercellular messenger (Aon etal., 1992) does not exert this function.

An intriguing question is whether coupling ofoscillating cells is always accompanied by syn-chronization or whether more complex dynamicphenomena may result. Theoretical work on inter-acting identical oscillators has shown that alsoasynchronous behaviour may be expected. Of par-ticular interest is the symmetric case where allphase shifts are proportional to the reciprocalvalue of the number of cells. As far as directcoupling is considered this type of asynchronousbehaviour may be excluded for interacting twocomponent systems (Alexander, 1986). In thepresent paper we show that such an assertion doesnot hold for cell populations where the coupling isrealized by substances which are extruded into theextracellular medium, i.e., besides synchronousoscillations, asynchronous dynamics is possiblealso for interacting two-component systems. Onemay expect therefore, that desynchronized be-haviour of cells may be a common phenomenon.We will demonstrate that it is characterized bystrong variations in the internal states of theindividual cells but nearly constant external con-centrations of the diffusible metabolites.

Generally, sustained oscillations in biochemicalsystems may only arise if one or more reactionsobey a nonlinear kinetics. A main case is thatautocatalytic processes are involved, as for exam-ple, the phosphofructokinase reaction in glycoly-sis (Higgins, 1967; Sel’kov, 1968; Goldbeter and

J. Wolf, R. Heinrich / BioSystems 43 (1997) 1–24 3

Lefever, 1972; Eschrich et al., 1985; Heinrich andSchuster, 1996) or the calcium-induced-calcium-release (CICR) from the endoplasmic reticulum,for recent models see (Goldbeter et al., 1990;Somogyi and Stucki, 1991). It may be interestingto analyze whether there are differences in thedynamics of the cell populations if the couplingsubstance belongs to the pool of substrates or tothe pool of products of the autocatalytic reaction.In yeast cell populations the latter case seems tobe realized since acetaldehyde is one of the end-products of glycolysis. In the present paper weinvestigate both possibilities. By considering asimple model, proposed originally for the expla-nation of glycolytic oscillations, many of the re-sults may be derived analytically. Our analysis isvalid for an arbitrary number of interacting cells.Furthermore, no restrictions are made concerningthe strength of the coupling, which is expressed bythe rate constant for the exchange of substancesbetween the intracellular and extracellular space.Weak and strong coupling follow as special cases.

The results of this theoretical analysis are dis-cussed in the light of recent experimental datapresented by (Richard et al., 1996). In particular,we demonstrate numerical simulations of oscillat-ing cell suspensions before and after mixing oftwo subpopulations. Furthermore, several conclu-sions are drawn concerning the observability ofoscillations, if the cells are desynchronized.

Besides oscillations, multiplicity of steady statesis a well-known characteristic of nonlinear bio-chemical reaction systems (Eschrich et al., 1990;Schellenberger and Hervagault, 1991). Therefore,part of this work is devoted to the analysis of theinterrelationships between these two fundamentalphenomena in the case of cell populations. Wewill show that in the case of multiple stable andunstable steady states, the whole population maysplit into different subgroups of synchronized cellswhere each subgroup may show different be-haviour.

2. Basic model assumptions

We investigate models for suspensions of inter-acting cells, in which the single cells may show

metabolic oscillations. The dynamics of themetabolites of a cell is described by kinetic equa-tions resulting from a feedback-activation mecha-nism which has been proposed for the explanationof glycolytic oscillations (Higgins, 1964, 1967;Sel’kov, 1968).

Using the model for glycolytic oscillations inthe form as specified by Sel’kov (1968) the dy-namics of the metabolite concentrations within asingle cell is governed by the equations

dXdt

=61−k2XY g, (1a)

dYdt

=k2XY g−k3Y. (1b)

These equations describe a system where the com-pound X is supplied by a constant input 61 anddegraded by an autocatalytic reaction. The latterreaction produces the compound Y which in turnis degraded. In the case of glycolysis reaction 1and 2 may represent the reactions catalyzed byhexokinase and phosphofructokinase while theprocesses of the lower part of the pathway arelumped into reaction 3. The coefficient g charac-terizes the strength of the product activation. Ithas been demonstrated that system (1) may showoscillations of limit cycle type as long as g\1. Inthe following we consider the special case g=2.Introducing dimensionless quantities for the con-centrations X/C�X, Y/C�Y, for the rate con-stant k3/k2C2�k, for the input rate 61/k2C3 in,and for time tk2C

2� t, where C represents anarbitrary constant concentration, equation system(1) may be rewritten as

dXdt

=6−XY2, (2a)

dYdt

=XY2−kY. (2b)

The reaction system has a unique steady statewith the concentrations

X( =k2

6, (3a)

Y( = 6k

. (3b)


The steady state is locally asymptotically stableupon the condition that

6\kk. (4)

Crossing the boundary of stability this state be-comes unstable via a Hopf-bifurcation. The fre-quency v of the oscillation near to the bifurcationpoint is approximately given by

v2=D=62

k, (5)

where D denotes the determinant of the Jacobianmatrix of equation system (2a,b).

In the present models we consider suspensionsconsisting of an arbitrary number N of interactingcells. Each cell contains an oscillating reactionsystem corresponding to the mechanism describedby equation system (2). It is supposed that theindividual cells interact via the flux of metabolites,which are produced in all cells and may permeatethrough the cell membranes.

For the development of the models we intro-duce the following simplifications:1. All metabolic oscillators have an identical stoi-

chiometry and identical kinetic parameterswith respect to the biochemical transforma-tions as well as the transport processes.

2. All cells are characterized by the same volumeV1 and the same membrane surface area A.The metabolites are distributed homogeneouslyboth in the cellular solution and in the externalsolution with the volume V2.

3. The coupling is realized by a transmembraneflux of a single uncharged metabolite.

We consider two versions of the model differingin the permeating substance. In model I the cou-pling is realized by the product Y and in model IIby the substrate X of the autocatalytic reaction.The fluxes Ji,Y (Model I) and Ji,X (Model II)between the i-th cell and the external medium arefunctions of the permeabilities PY and PX of themembrane to the substances Y and X, respec-tively, and of the differences in the concentrationsof the coupling substances between the cell andthe external medium. We use the relations

Ji,Y=PY(Yi−Y e), (6a)

Ji,X=PX(Xi−X e) (6b)

where Xi and Yi denote the concentrations in thei-th cell. Y e (Model I) and X e (Model II) are theexternal concentrations of the coupling sub-stances. Let us define effective rate constants k ofthe transmembrane diffusion in the following way

k=APV1

, (7)

where P denotes either PY or PX. We call k thecoupling parameter since an interaction of cellstakes place only for k"0. For k=0 the suspen-sion contains noninteracting cells with two-com-ponent oscillators described by Eqs. (2a) and (2b).Concerning the extracellular volume V2 one maythink about different situations. One may assume,for example, that V2 is constant at varying num-bers of cells, which means that the cell density isproportional to N. We concentrate here on an-other possibility where V2 increases linearly withN, which results in a constant cell density atvariations of N. Denoting by 8 the ratio of intra-cellular and extracellular volume for the case N=1 and using a scaled coupling parameter(k/k2C2�k) the differential equation system forN coupled cells reads for model I

dY e

dt=

k8

N� %

N

j=1

Yj−NY e�, (8a)

dXi

dt=6−XiY2

i , (8b)

dYi

dt=XiY2

i −kYi−k(Yi−Y e). (8c)

Analogously one may derive an equation systemfor model II

dX e

dt=

k8

N� %

N

j=1

Xj−NX e�, (9a)

dXi

dt=6−XiY2

i −k(Xi−X e), (9b)

dYi

dt=XiY2

i −kYi. (9c)

For N=1 equation systems (8) and (9) describeoscillators in single cells exchanging a metabolitewith the external solution. At fixed numbers N\1 of cells variations of 8 correspond to changes ofthe cell density.


3. Dynamics of single cells

The three-component systems resulting withN=1 from model I and model II have uniquesteady states. In both cases the stationary concen-trations X( 1 and Y( 1 are identical to those given inEq. (3a) and Eq. (3b), respectively, for the twocomponent system (2). Furthermore, it followsfrom Eq. (8a) and Eq. (9a) that the steady stateconcentrations of the permeating substances X orY are the same in the external solution and withinthe cell.

Using these steady state concentrations onederives from the Jacobian matrix of system (8) thefollowing characteristic equation for the eigenval-ues l of model I

F(l)

=l3+l2�62

k2+k+k8−k�

+l�62

k+62k8

k2 −kk8+62k

k2

�+62k8

k=0,

(10)

and from system (9) for model II

G(l)

=l3+l2�62

k2+k+k8−k�

+l�62

k+62k8

k2 −kk8−kk�

+62k8

k=0.

(11)

According to the Hurwitz-criterion in a three-component system with the characteristic equa-tion l3+a2l

2+a1l+a0=0 a Hopf-bifurcationtakes place if the following conditions are fulfilled

a1a2−a0=0, (12a)

a0\0, (12b)

a1\0, (12c)

a2\0, (12d)

(m

(p"0. (12e)

In relation Eq. (12e) m denotes the real part of thecomplex eigenvalues and p is any bifurcation

parameter. Relations Eqs. (12a), (12b), (12c) and(12d) ensure that the Jacobian matrix has one pairof pure imaginary eigenvalues. For model I oneobtains with the help of Eq. (10)

a1a2−a0

=k2�62(1+8)2

k2 −k8(1+8)�

+k�64(1+8)

k4 +k28−2628

k�

+64

k3−62 (13)

and for model II with Eq. (11)

a1a2−a0

=k2�628(1+8)k2 −k(1+8)2�

+k�648

k4 +k2(1+8)−2628

k�

+64

k3−62.

(14)

In both models the conditions for Hopf-bifurca-tions may be fulfilled. We present the proof exem-plarily for model I using k as bifurcationparameter.

1. It follows from Eq. (13) that condition (12a)can be fulfilled by real non-negative parametervalues.

2. The relation a0\0, Eq. (12b), is always true,whereas the signs of the coefficients a1 and a2 maychange depending on the kinetic parameters. Forall k-values for which condition (12a) is fulfilledthe relation a0\0 implies

sgn(a1)=sgn(a2), (15a)

a1, a2"0. (15b)

3. With the coefficients of the characteristic Eq.(10) one obtains

k((a1a2−a0)(k

=a0−�62

k+

262k8

k2 +kk8+262k

k2

�a2

−�262

k2 +k�

a1. (16)

Eliminating a0 by condition (12a) and taking intoaccount the definition of the coefficient a1 Eq. (16)may be rewritten as


k((a1a2−a0)(k

= −�62k8

k2 +62k

k2 +2kk8�

a2

−�262

k2 +k�

a1. (17)

Due to relations (15a) and (15b) it follows that((a1a2−a0)/(k"0. From Eq. (13) one obtainsin the limit k�0 that (a1a2−a0)��. There-fore, at the lowest value of k for which condi-tion (12a) is fulfilled the following relation holds

((a1a2−a0)(k

B0. (18)

4. With the relations (15) and (17) it followsimmediately that a1 and a2 are both positive ifrelations (12a) and (18) are fulfilled.

5. Inserting m+ iv for the complex eigenvalueinto the general characteristic equation forthree-variable systems yields independent equa-tions for the real part and imaginary part of thepolynomial. After derivation of these equationswith respect to the bifurcation parameter k oneobtains with m=0

(3v2−a1)(m

(k+2a2v

(v

(k=v2 (a2

(k−(a0

(k, (19a)

2a2

(m

(k−2v

(v

(k= −

(a1

(k. (19b)

Taking into account that m=0 implies a1=v2

the solution of system (19) with respect to (m/(kyields

(m

(k= −

((a1a2−a0)/(k2(a1+a2

2). (20)

Because on that part of the surface a1a2−a0=0where condition (18) is true, the coefficient a1 ispositive one obtains (m/(k\0, that is condition(12e) is fulfilled.

Fig. 1 shows for model I the functions k=k(k) resulting from a1a2−a0=0 as well as froma1=0 and a2=0 for fixed values of the otherparameters. The curve resulting from condition(12a) for model I consists of two branches sepa-rating the regions of positive and negative val-ues of (a1a2−a0). On the branch drawn by asolid line the conditions (12b) to (12e) for Hopf-bifurcations are fulfilled, whereas on the broken

line the conditions (12c) and (12d) are violated.The lines a1=0 and a2=0 do not bound theregion of stability because they are locatedwithin the region of negative values of (a1a2−a0). Therefore, in the region of low k-valueswhere (a1a2−a0) is positive, the steady state isstable.

For both models limit cycle oscillationsemerge at transitions from stable to unstablesteady states. With increasing distance from thebifurcation line, that is, with increasing k-values,folded limit cycles and chaos may occur via pe-riod-doubling cascade. These dynamic phenom-ena of single cells have been studied in detail bycalculating Lyapunov-exponents (Wolf, 1994). Infact, the present model for the single cell hasmuch in common with a three-variable systemstudied in (Herzel and Schulmeister, 1987) whichshows similar dynamical properties. However,the model used in the present paper has a moresimple structure because it involves only one re-action step characterized by a nonlinear kineticequation.

Fig. 1. Regions of stable and unstable steady states for a singlecell for model I. The two branches of the function k=k(k)result from the condition a1a2−a0=0. On the branch drawnby a thick solid line the conditions (12a) to (12e) for Hopfbifurcations are fulfilled, whereas on the broken line theconditions (12c) and (12d) are violated. Lines a1=0 and a2=0(thin solid lines) are located within the region of unstablesteady states. Other parameter values: 6=3.0, 8=0.2 (Thesevalues are also used for all other figures.)


4. Coupling via the product of the autocatalyticreaction (Model I)

4.1. Stability analysis of the steady states

It is easy to see from equation system (8) that thesystem of N coupled cells displays independently ofthe value of N always a unique steady state solutionwhich for every cell is identic to that of the single cell,that is, X( i=X( and Y( i=Y( e=Y( for i=1,…, N withX( and Y( given in Eq. (3a) and Eq. (3b). Afterappropriate numbering of the variables the Jacobianmatrix for this steady state has for any value of N avery regular structure, in that it contains besidesseveral non-zero-elements in the first row and in thefirst column only blocks, with two elements per rowand column, along the main diagonale; cf. AppendixA. Due to these properties the characteristic equa-tion for the eigenvalues may be written as follows

D(l)N−1F(l)=0 (21)

for arbitrary values of N ; for the derivation of thisformula, see Appendix A. The factor F(l) is apolynomial of third degree and identic to thatobtained for the single cell; cf. Eq. (10). For the otherterm one obtains

D(l)=l2− tr ·l+D, (22a)

with

tr=k−k−62

k2 , (22b)

D=62

k�

1+k

k�\0, (22c)

where tr and D denote the trace and determinant,respectively, of the (2×2) submatrix which corre-sponds to the above-mentioned blocks of the Jaco-bian (see Eq. (A5)). In this way the full characteristicpolynomial of the order (2N+1) is splitted into aproduct of (N−1) identical quadratic parts and onecubic part. Consequently, only five different eigen-values may exist which are independent of thenumber of cells. Two eigenvalues follow fromD(l)=0 and the other three from F(l)=0. There-fore, there are two possibilities for Hopf bifurcationsleading to sustained oscillations. One possibilityresults from the cubic part of the characteristicpolynomial, that is, at parameter combinations

Fig. 2. Bifurcation lines for coupled oscillators (Model I).Lines k=ka(k) and k=kb(k) where the cubic part and thequadratic part of the characteristic Eq. (21), respectively,yields one pair of pure imaginary eigenvalues. The solid partsof the lines indicate the boundaries of the region of stability,where Hopf bifurcations lead to synchronous or asynchronousoscillations of the cells. The broken parts of the lines arelocated within the region of unstable steady states. The coordi-nates of the points P1 and P2 are given in (24) and (27),respectively. Region 1: stable steady states, regions 2–4: un-stable steady states.

fullfilling conditions (12a) to (12e) by taking intoaccount Eq. (13). For that bifurcation Eq. (12a)gives with conditions (12b)–(12d) raise to a bifurca-tion line k=ka(k) within the (k, k)-plane as shownin Fig. 2. This curve corresponds to the thick solidline in Fig. 1.

A second bifurcation may result from the (N−1)identic quadratic parts D(l) of the characteristicpolynomial. Due to the positive sign of D thebifurcation points are determined by tr=0, that isby

k=kb(k)=k−62

k2 . (23)

The curves ka and kb shown in Fig. 2 have twocommon points. In the first point P1 with thecoordinates

k (1)=62/3, (24a)

k (1)=0, (24b)

they have the same slope

dka

dk=

dkb

dk=3, (25)

but differ in the second derivatives


d2ka

dk2 = −6(1+38)

k, (26a)

d2kb

dk2 = −6k

. (26b)

The second common point P2 is located at

k (2)=�62�1+

18

��1/3

, (27a)

k (2)=� 62

8(1+8)2

�1/3

. (27b)

Relations (24)–(27) ensure that for any values of6 and 8 the lines ka and kb separate the (k, k)-parameter plane into four regions (see Fig. 2). Inregions 2 and 4 the equation F(l)=0 and inregions 3 and 4 the equation D(l)=0 yield eigen-values with positive real parts. Therefore, forparameter values taken from regions 2, 3 and 4the steady state is unstable. In region 1 all eigen-values resulting from Eq. (21) have negative realparts. From these considerations it follows thatfor k\k (2) the stability region 1 is bounded bythe line ka and for kBk (2) by the line kb.

4.2. Dynamical properties

4.2.1. Synchronous and regular asynchronousoscillations

Crossing the boundary of the stability region,oscillations of limit cycle type emerge. However,different types of oscillations arise from the lineska and kb, more precisely from those parts of thelines which are boundaries of the stability region(solid parts of the lines ka and kb in Fig. 2). Figs.3 and 4 show for N=2 the oscillations forparameters values close to these lines. The bifur-cation resulting from the cubic part F(l) of thecharacteristic Eq. (21) leads to synchronous oscil-lations of the cells, that is, there is no phase shiftbetween the variables of cell 1 and the corre-sponding variables of cell 2 (Fig. 3). In contrast tothat the bifurcation following from the quadraticpart D(l) produces asynchronous oscillations ofthe cells (Fig. 4). In the latter case the phase shiftbetween the oscillations of the two cells is half ofthe oscillation period T. A comparison of Figs. 3and 4 shows that in the case of asynchronousoscillations the external metabolite oscillates with

Fig. 3. Synchronous oscillations of two coupled cells. Thedynamics results from a bifurcation at the line ka(k) (see Fig.2). Parameter values: k=3.2, k=3.84. The initial conditionsfor the concentrations of the two cells where nonidentic (X1=4.91, X2=4.93, Y1=0.77, Y2=0.79, Ye=0.78). The curvesfor Yi and Ye were plotted starting from a time t0, where thelimit cycle has been reached.

a much smaller amplitude. Furthermore, the asyn-chronous oscillations have the characteristic thatthe external metabolite oscillates with the doublefrequency compared to the internal substances.

The occurence of stable limit cycles of twocoupled oscillators 1 and 2 with phase shifts t12=0 or t12=T/2 between the corresponding vari-ables is in accord with results from previousinvestigations of two coupled oscillators (Ruelle,

Fig. 4. Regular asynchronous oscillations of two coupled cells.The dynamics results from a bifurcation at the line kb(k) (seeFig. 2). Phase shift of the oscillating cells: t12=T/2. Parametervalues: k=1.0, k=2.50. The curves for Y1, Y2 and Ye wereplotted starting from a time t0, where the limit cycle has beenreached.


Fig. 5. Regular asynchronous oscillations of five coupled cells.Phase shift of the oscillating cells: t=T/5. The amplitudes ofthe Ye oscillations are very small (DYe:0.0001) in compari-son with those of Yi. Parameter values are the same as given inthe legend of Fig. 4.

states, the regular asynchronous oscillations in-creases strongly with the number of interactingcells. For example, in the numerical integrationsshown in Figs. 4 and 5 for N=2 and N=5,respectively, the initial values were chosen in sucha way that the values of the single cells wherenearly identic. However, the time t0 where thelimit cycle is reached is for N=5 about 100 timeslonger than for N=2. This difference is easilyexplained by the fact that the coordinate state ofregular asynchronous oscillations has to be ac-complished by very small regular fluctuations ofthe external concentration. By similar reasoning itfollows that the transient time to reach syn-chronous oscillations, where the external concen-trations changes considerably, is much shorterthan that for regular asynchronous oscillations.

It follows immediately from Eqs. (27a) and(27b) that the region for the coupling parameter k

where regular asynchronous behaviour occurs in-creases with decreasing cell density. The lower thecell density the higher the coupling constant hasto be for synchronization.

At first sight the occurrence of regular asyn-chronous oscillations seems to be in contradictionto the proposition that in the case of symmetricdiffusion for connected two-component systemsany nonsynchronous periodic solution is unstable(Alexander, 1986). A consideration of the systemsstudied in Alexander (1986) shows however, thatthere the systems are coupled directly to eachother which gives raise to coupling terms of theform Ji,k8 (Yi−Yk) where Ji,k denotes thetransmembrane flux of Y from cell k into cell i. Inthe model proposed here, the coupling betweenthe cells is realized in an indirect way, that is, viaan external compound, which for cell suspensionsis a more realistic assumption. This adds to thedifferential equation system for the concentrationsof the intracellular variables one equation for theexternal coupling substance. For such a systemthe proposition mentioned above is not valid.

The fact, that bifurcations on the lines ka andkb lead to synchronous and asynchronous oscilla-tions, respectively, may be understood as follows.Introducing the mean values

1973; Collins and Stewart, 1993; Reick andMosekilde, 1995).

Since the bifurcation lines shown in Fig. 2 areindependent of the number of coupled cells,analogous results are found for N\2. On thesolid part of the line ka where k\k (2) the bifurca-tions always give rise to synchronous oscillations,whereas on the solid part of the line kb wherekBk (2) asynchronous oscillations emerge withphase shifts tij= ( j− i )T/N, where i, j=1,…, Nand iB j. The broken parts of the lines are locatedin the region of instability and are not of rele-vance to the dynamics. That means that the phaseshifts follow a strong rule although the oscilla-tions are asynchronous. We denote this type ofbehaviour regular asynchronous oscillations. Previ-ously, the terms anti-synchronous and anti-phaseoscillations were proposed for the case of only twocoupled oscillators (Kawato and Suzuki, 1980;Alexander, 1986). However, these terms have nowell defined meaning for a high number of cellswhere the phase shifts may attain any multiple ofT/N.

Fig. 5 shows regular asynchronous oscillationsobtained for N=5 by numerical integration. It isseen that the concentration of the externalmetabolite remains nearly constant despite thefact that the internal metabolite concentrationsvary substantially. It is worth mentioning that thetransient time to reach, from arbitrary initial


X=1N

%N

i=1

Xi, (28a)

Y=1N

%N

i=1

Yi, (28b)

and the differences

ji=Xi+1−Xi, (29a)

hi=Yi+1−Yi i=1,…, N−1 (29b)

as system variables the equation system (8) maybe rewritten as

dXdt

=6−XY2+o(jj, hk), (30a)

dYdt

=XY2−kY−k(Y−Ye)+o(jj, hk), (30b)

dY e

dt=k8(Y−Ye), (30c)

and

dji

dt= − (Y2ji+2XYhi)+o(jj, hk) (30d)

dhi

dt= (Y2ji+2XYhi)− (k+k)hi+o(jj, hk),

(30e)

for i=1,…, N−1. o(jj, hk) denote second orhigher order terms of jj and hk. It is seen that inEq. (30a) and Eq. (30b) only nonlinear terms of jj

and hk appear and that the right hand sides of Eq.(30d) and Eq. (30e) contain only terms which areat least of first order in these variables. Eq. (30c)is independent of the differences jj and hk. There-fore, applying a linear stability analysis to equa-tion system (30a)–(30e) for the stationary state(X( i=X( , Y( i=Y( e=Y( , j( i= hi=0) the lin-earized equation system is splitted into one systemdepending only on the perturbations of the meanvalues X, Y and Y e and into (N−1) decoupledtwo-variable systems depending on the perturba-tions of the differences ji, hi.

The characteristic polynomial of equation sys-tem (30a)–(30c) corresponds to F(l) of a singleoscillator given in Eq. (10). The characteristicpolynomials of all systems resulting from Eq.(30d) Eq. (30e) are identic to D(l) given in Eqs.(22a), (22b) and (22c). Therefore, the line kb in

Fig. 2 separates the regions where for fixed meansteady state values X( and Y( the state ji=hi=0for all cells i is stable (regions 1 and 2) or unstable(regions 3 and 4). From that it follows that oscil-lations which occur at transitions from region 1 toregion 2 are synchronous, that is, they retain thesymmetry of the previous steady state. On thecontrary, at transitions from region 1 to 3 thissymmetry is broken and asynchronous oscillationsemerge.

4.2.2. Mixing of two cell populationsIt has been shown experimentally that after

mixing of two populations of cells, which aresynchronized internally, but oscillate about 180°out of phase, the oscillations are strongly dampedbut reappear after some time span. In the experi-ments on oscillations in yeast cells synchroniza-tion takes place after a time corresponding toabout eight periods (Richard et al., 1996). Resultsof a simulation of such a situation are presentedin Fig. 6. For tB12 the two curves show thesustained oscillations of the external concentra-tion of Y in two independent populations of equalsize. At t=12 the two suspensions are mixed suchthat Y e assumes the mean value of the two con-centrations of the single populations. Immediatelyafter mixing the oscillation amplitude is dampedby a factor of about three and the full synchro-

Fig. 6. Mixing of two oscillating cell populations. tB tm: timedependent changes of the external concentrations Ye,1 and Ye,2

of the populations 1 and 2, respectively. Number of cells ineach population: N=10. t\ tm: oscillations of Ye of themixed population. Parameter values: k=10, k=4.15, time ofmixing: tm=12.


Fig. 7. Perturbation of regular asynchronous oscillations. Mean value Xmean of the concentrations Xi (A) and concentration X1 (B)as functions of time for a system of N=10 coupled cells. For tB t1 the population is within a state of regular asynchronousoscillations. At t= t1=20 the external concentration of the coupling substance Y is decreased by 20% and increased by the samefraction at time t= t2=80.

nized state is attained after about nine cycles. Forthe present simulations the value of the couplingparameter k was chosen such that the time courseafter mixing resembles the experimental curves(Fig. 1 in Richard et al. (1996)). Lower (higher)values of the coupling parameter lead to a slower(faster) synchronization after mixing. The calcula-tions were performed for cell populations eachconsisting of N=10 cells. Simulations with ahigher number of cells indicate that the Y e varia-tions after mixing are nearly independent on theirnumber N.

4.2.3. Perturbation of regular asynchronousoscillations

As shown in Figs. 4 and 5 this type of oscilla-tions is characterized by a nearly constant concen-tration of the coupling substance in the externalmedium. Therefore, it may be difficult to distin-guish experimentally whether the individual cellsare in a steady state or in oscillatory states. Thisholds true also if one tries to detect the oscilla-tions by recording other quantities which resultfrom an averaging over all cells, such as byNADH-fluorescence for yeast glycolysis. Suchhidden oscillations should become transiently visi-ble if the regular phase shifts are disturbed. Fig. 7shows results of simulations where in a suspensionof cells in a state of regular asynchronous oscilla-tions the concentration of the external metabolite

is decreased by 20% at time t= t1 and increasedby the same fraction at time t= t2\ t1. It issupposed that a signal may be detected which isdirectly related to the mean value Xmean of thesubstrate concentrations Xi. Whereas Xmean isnearly constant for tB t1 it is abruptly increasedat t= t1 and shows thereafter damped oscillationsfor tB t2. Obviously these dynamics is due to apartly synchronization of the cells at the time ofperturbation. The damping results from the subse-quent restoration of the regular asynchronousstate. A similar phenomenon is obtained by asudden increase of Y e at t= t2. A comparison ofFig. 7A and B shows that the average dynamics,represented by Xmean(t), differs substantialy fromthe dynamics of the concentrations Xi(t) withinthe single cells. The latter concentrations oscillatewith much higher amplitudes as shown for X1 inFig. 7B. This holds true, in particular, for thosetime intervals, where the oscillations of Xmean arenegligible small. Instead of changing the externalconcentration Y e one could also apply a suddentransient increase and decrease of the input rate 6of all cells for obtaining similar results. The curvein Fig. 7A shows some correspondence to experi-mental data, where the oscillations monitored byNADH fluorescence are induced by adding glu-cose and cyanide (Ghosh et al., 1971; Aldridgeand Pye, 1976; Richard et al., 1996) and reacti-vated by an acetaldehyde pulse (Richard et al.,1996).


4.2.4. The case of strong couplingOne may conclude from Fig. 2, that a strong

coupling between the cells allows only syn-chronous oscillations. It may be interesting toconsider the limiting case k��, where Xi=Xand Yi=Y. The parameter k may be eliminatedfrom Eq. (8c) by adding Eq. (8a) multiplied by1/8 which results in

dYdt

+18

dY e

dt=XY2−kY. (31)

Application of the quasi-equilibrium approxima-tion to the equation odY e/dt=Y−Y e with o=1/k yields Y=Y e for o�0. In this way thedynamics of the whole system may be describedby the equations

dXdt

=6−XY2, (32a)

dYdt

=8

1+8(XY2−kY). (32b)

This equation system differs from that of thedecoupled oscillators (see equation system (8)with k=0) only in a volume dependent factor inthe equation for the variable Y. A linear stabilityanalysis of this system reveals that the steady stateis stable upon the condition that

62�1+18

�\k3. (33)

Surprisingly, the k-value where the steady statebecomes unstable equals k (2), where the curves ka

and kb intersect (see Eq. (27a) and Fig. 2). Thestability condition (33) for strongly coupled cellsis volume dependent. A decrease of the cell den-sity, that is a lowering of 8 has a stabilizing effect.This corresponds to the experimental fact, thatsustained synchronous oscillations of yeast cellsare preferentially found at high cell densities (Aonet al., 1992).

4.2.5. Frequency of oscillationsDetermining the frequency of the coupled oscil-

lators near the boundary of the stability regionfrom the determinant of the Jacobian matrix ofEq. (32a) and Eq. (32b) and comparing the resultwith that obtained for k=0 yields

v �k��=� 8

1+8

�2/3

v �k=0=� V1

V1+V2

�2/3

v �k=0.

(34)

We may conclude that the frequency of the oscil-lations decreases at the transition from decoupledto strongly coupled cells. The decrease is thestronger the larger the extracellular volume com-pared with the cell volume. It can be understoodby the fact that for large extracellular volume thetotal amount of the coupling substance which hasto be exchanged must be very high. This is relatedto the deposition effect proposed by Sel’kov to beresponsible for the increase of the oscillation pe-riod by taking into account the reversible ex-change of glycolytic compound with a pool ofpolysaccharides (Sel’kov, 1980).

Using for the frequency of oscillations near theboundary of the stability region the relationsv2=D for two-component systems and v2=a1

for three-component systems one may calculatethe frequency v for all sets of parameter values.Numerical evaluation of D at tr=0 (see Eqs.(22a), (22b) and (22c)) and of a1 from Eq. (10) ata1a2−a0=0 with Eq. (13) yields v(k) as shownin Fig. 8. Only the solid parts of the lines are of

Fig. 8. Frequency of synchronous and regular asynchronousoscillations. The frequencies are calculated as functions of theparameter k at k-values corresponding to the bifurcation lineska and kb. The solid lines correspond to those regions ofparameter values where the synchronous and asynchronousoscillations, respectively, are stable. Point P1: uncoupled case(k=0); Point P2: transition between stable synchronous andstable asynchronous oscillations; Point P3: strong coupling(k��). The curves are independent of the number N ofinteracting cells.


Fig. 9. Bifurcation diagram of two coupled oscillators. kBkH:steady state concentrations X( i ; kHBkBkS: maxima and min-ima of regular asynchronous oscillations; k\kS: maxima andminima of nonregular asynchronous oscillations. Inset A: de-tailed representation of the symmetry breaking of the minimaof regular asynchronous oscillations. Inset B: Phase shifts ofthe regular asynchronouos oscillations (t12=T/2) and nonreg-ular oscillations (t12BT/2). Parameter values: N=2, k=1.0.

diagram shows the steady state concentrations X( 1

and X( 2. Beyond the bifurcation point k\kH themaxima and minima of the oscillating variablesX1 and X2 are plotted. There is a region kHBkBkS, where the lines for the two cells coincidedespite a phase shift of t12=T/2 between thevariables. In this region regular asynchronous be-haviour is found. It is seen that an increase of thebifurcation parameter k leads to a further symme-try breaking. For k\kS the shapes of the oscilla-tions of X1 and X2 differ which is reflected bydifferences in the maxima and minima of thesevariables. Furthermore, the phase shift betweenthe variables of the two cells is no longer T/2, butdecreases monotonically with increasing k asshown in the inset B to Fig. 9. That means thatfor k\kS nonregular asynchronous oscillationsoccur. It is worth mentioning, that at the pointk=kS a single limit cycle bifurcates into two limitcycles, which may be transformed into each otherby exchanging the numbers of the cells. Examplesfor a single limit cycle with phase shift T/2 forkBkS and for two limit cycles for k\kS areshown in Fig. 10A and B, respectively. In bothcases only the trajectories of the variables X1 andX2 are shown.

For higher values of k, where the region ofstability is bound by the line ka a similar sec-ondary bifurcation may occur by symmetrybreaking. At increasing values of the bifurcationparameter k the oscillations remain first syn-chronous. Beyond a critical value kS the limitcycle bifurcates into two cycles. Each of themrepresents nonregular asynchronous oscillations.Starting with a phase shift t12=0 for k=kS thephase shift increases with increasing values of k.Only for a very strong coupling no symmetrybreaking bifurcations occur.

5. Coupling via the substrate of the autocatalyticreaction (Model II)

5.1. Steady states

Whereas in model I the stationary state isunique and independent of the number of coupledcells, model II shows generally multiple steady

physical meaning since the region of stability isdetermined either by the cubic or by the quadraticpart of the characteristic polynomial (Eq. (21)).The points P1 and P3 correspond to the limitingcases k=0 and k��. Point P2 corresponds tothe intersection point of the lines ka and kb, wherea transition between synchronous and regularasynchronous oscillations occurs. As seen fromFig. 8 this transition is accompanied by a discon-tinuity in the parameter dependency of the fre-quency. Generally, the frequency forasynchronous oscillations of internal metabolitesis higher than that for the synchronous oscilla-tions.

4.2.6. General aspectsLeaving the near neighbourhood of the bifurca-

tion points by increasing the bifurcation parame-ter k more complex dynamic phenomena mayoccur. Fig. 9 shows for a system of two interact-ing cells a bifurcation diagram of the variables X1

and X2 with varying parameter k.The value of the coupling parameter k was

chosen from the intervall where the line kb(k)bounds the stability region. In this case the pointof a Hopf-bifurcation k=kH is determined by Eq.(22b) with tr=0. For low k-values (kBkH), the


Fig. 10. Asynchronous oscillations shown in the (X1, X2) phase plane. (A) Regular asynchronous oscillations; (B) Nonregularasynchronous oscillations. The limit cycle 2 follows from limit cycle 1 by renumbering of the cells. The two limit cycles are reachedfrom different initial conditions. Parameter values: N=2, k=1.0, k=2.50 (A), k=3.48 (B).

states. Using Eq. (9a) the external metabolite hasthe steady state concentration

X( e=1N

%N

j=1

X( j. (35)

Eq. (9c) can be fulfilled by either

Y( i=kX( i

or Y( i=0. (36a,b)

We number the cells so that Y( i"0 for 15 i5rand Y( i=0 for r+15 i5N. The case r=0 isexcluded since there exists no stationary state as itfollows immediately from equation system (9).Introducing X( e from Eq. (35) and the Y( i valuesfrom Eqs. (36a,b) into Eq. (9b) leads to

0=6−k2

X( i

−kX( i+k

N%N

j=1

X( j, for 15 i5r

(37a)

and

0=6−kX( i+k

N%N

j=1

X( j, for r+15 i5N,

(37b)

respectively. Subtracting Eq. (37a) for i\1 fromthat for i=1 one arrives at an equation which hastwo solutions for X( i as a function of X( 1. By anappropriate numbering of the first r cells thisleads to

X( i=X( 1 for 15 i5q (38a)

and

X( i=k2

kX( 1

for q+15 i5r. (38b)

Subtracting Eq. (37b) from Eq. (37a) for i=1 oneobtains

X( i=X( 1+k2

kX( 1

for r+15 i5N. (39)

5.1.1. Case 15q=r5NIntroducing the two possible solutions for X( i,

Eq. (38a) and Eq. (39), into Eq. (37a) for i=1gives a linear equation for X( 1. Calculating thenthe X( i-values by using again Eq. (38a) and Eq.(39) and the corresponding Y( i-values yields

X( i=rk2

N6, (40a)

Y( i=N6rk

, 15 i5r, (40b)

X( i=N6rk

+rk2

N6, (40c)

Y( i=0, r+15 i5N. (40d)

With the help of Eq. (40a) Eq. (40c) and Eq. (35)one obtains for the steady state concentration ofthe external metabolite


X( e=rk2

N6+

(N−r)6rk

. (40e)

Since r may attain any integer value from 1 to Nequation system (40) gives N different solutionsfor the steady state concentrations of the metabo-lites. They result from a situation, where r cellshave Y( -values unequal to zero, whereas (N−r) ofN cells are in a state, characterized by vanishingvalues for Y( . In the special case r=N the steadystates in all cells are the same and identic to thatfor N=1. In the following this stationary state iscalled main steady state.

5.1.2. Case qBr5N.Introducing the three possible solutions for X( i

(Eq. (38a), Eq. (38b) and Eq. (39)) into Eq. (37a)for i=1 gives a quadratic equation for X( 1. Fromthat one obtains

X( i=N6

2k(r−q)9S, for 15 i5q. (41a)

with

S='� N6

2k(r−q)�2

−qk2

k(r−q). (41b)

Introducing Eqs. (41a) and (41b) for i=1 intoEq. (38b) and Eq. (39) yields

X( i=�r−q

q�� N6

2(r−q)k�S

�, q+15 i5r,

(41c)

and

X( i=Nr6

2q(r−q)k9

(2q−r)Sq

, r+15 i5N,

(41d)

respectively. The corresponding Y( i values followfrom Eq. (36a,b). For the concentration of theexternal metabolite one derives from Eq. (35) andEq. (41a) Eq. (41b) Eq. (41c) Eq. (41d)

X( e=6

2k

�Nq

+N

r−q−2

�9

(2q−r)Sq

. (41e)

It is easy to see that the two solutions for X( i

differing in the sign of the square roots S lead tophysically indistinguishable steady states whichonly differ in the numbering of the cells. There-fore, the lower signs may be omitted.

Considering all possible r and q values withqBr at a given number of cells equation system(41) yields N(N−1)/2 different solutions. To-gether with the N solutions for r=q given inequation system (40) the number of different solu-tions amounts to N(N+1)/2. This number ofsteady states corresponds to a special arbitrarynumbering of the cells. Further steady state resultfrom a mere renumbering of the cells. Accord-ingly, the total number of steady states may beevaluated as follows

Z(N)= %N

r=1

%r

q=1

N !(N−r)!(r−q)!q !

=3N−2N.

(42)

The solutions given in Eq. (41a) Eq. (41b) Eq.(41c) Eq. (41d) Eq. (41e) may become complex forcertain sets of kinetic parameters. Therefore, thetotal number of real steady state may be smallerthan that given in Eq. (42).

The possibility of generating different steadystates by coupling of cells, which have onlyunique steady states in the uncoupled case, is aninteresting kinetic phenomenon. It means, more-over, that the various cells of the population differin their dynamical properties. In the presentmodel, these differences are rather extreme sincefor some cells the metabolism is drastically re-duced. (For all cells where Y( i=0 the steady stateactivities of the second and third reaction in sys-tem (1) are equal to zero, that is, the intermediateX is not metabolized via the main pathway, buttransported out of the cell.)

A high number of stationary states character-ized by Y( i"0 is also obtained if one uses for therate equation of the autocatalytic reaction inequation system (2), instead of 62=XY2, the ex-pression

62= (a+Y2)X, (43)

where a represents a normalized first order rateconstant characterizing the activity of that reac-tion in the absence of Y (see (Heinrich et al.,1977) for the case of single cells and (Ashkenaziand Othmer, 1978) for the case of two interactingcells). For a"0 the calculation of the steadystates for an arbitrary number of coupled cells isalso possible and presented in Appendix B.


Fig. 11. Steady states for model II for the case for nonvanishing ground activity of the autocatalytic reaction. The steady stateconcentration Y( of one cell is plotted versus the kinetic parameter k for N=2 (A) and N=3 (B). Curves a: main steady state,Curves b, b1, b2 and c: nonsymmetric steady states (see text). Parameter values: a=0.05, k=1.0.

Fig. 11A and 11B show for N=2 and N=3,respectively, the concentrations of Y within onecell as functions of the bifurcation parameter k.For N=2 there are two possible types of steadystates. Curve a in Fig. 11A represents the symmet-ric main steady state, where Y( 1=Y( 2 and curve bnonsymmetric steady states with Y( 1"Y( 2. ForN=3 (Fig. 11B) the number of possible cases forstationary states increases. Besides the full sym-metric main steady state (curve a) there exist twopossibilities for steady states with lower sym-metries. On curves b1 and b2 the concentrationswithin the two cells are always identical, whereasthe concentrations of the third cell differ fromthese. The curve c represents the nonsymmetriccase, where the steady state concentration of thethree cells are different. It is easy to see that thelower branches of the curves, b, b1 and b2, tendfor a�0 to Y( =0, that is, they represent steadystates of model II for rBN. The advantage ofconsidering the case a=0 is that the stabilityanalysis may be carried out in an explicit way.

5.2. Stability analysis and dynamical properties

5.2.1. Case 15qBr5NAccording to equation system (41) real steady

state solutions only exist if

k2BN262

4 q(r−q)k. (44)

In Appendix C it is shown that all these steadystates are unstable.

5.2.2. Case 15q=r5NFor a given number rBN the cells may be

subdivided into two groups with the steady statesolutions (40a,b) and (40c,d), respectively. Thecharacteristic polynomial for this state may becalculated in a similar way as shown in AppendixA for model I. One obtains for rBN

E1(l)r−1E2(l)N−r−1G1(l)=0, (45)

where

E1(l)=l2− tr ·l+D

=l2+l�N262

r2k2 +k−k�

+N262

r2k−kk,

(46a)

E2(l)= (l+k)(l+k), (46b)

and

G1(l)= (l+k)�

E1(l)

((1−r/N)k28− (l+k)(l+k8))

+rk28

N(l−k)(l+k)

n

. (46c)

In these equations r/N denotes the fraction ofcells, where Y( i"0.


There are 2(r−1) eigenvalues following fromE1(l)=0 and, for rBN, 2(N−r−1) negativeeigenvalues from E2(l)=0. G1(l)=0 representsan equation of fifth order in l. One of its solutionis l= −k. The other four solutions are the zerosof the fourth-order-polynomial in square brack-ets.

In the case r=N, that is for the main steadystate, the polynomial G1(l) becomes proportionalto (l+k)(l+k) such that E2(l) drops out fromthe characteristic polynomial (45). Therefore, thecharacteristic equation simplifies to

E(l)N−1G(l)=0 (47)

where E(l)=E1(l) for r=N. G(l) is the polyno-mial of third order derived for the case of a singlecell (cf. Eq. (11)).

5.2.3. Bifurcations of the main steady stateBifurcations may result from the eigenvalues of

the cubic or quadratic part of Eq. (47). Concern-ing the first case condition (12a) gives togetherwith Eq. (11) a relation between the parameters kand k for fixed values of the other parameters(line ka in Fig. 12). Consideration of the quadraticpart shows that two types of bifurcations may

occur namely at tr=0 with D\0, or at D=0.The corresponding functions k=kb(k) and k=kc(k), respectively, result from Eq. (46a) withr=N.

The line which would result from the functionkb(k) is not relevant for the boundary of stability,which may be seen as follows. The functions ka(k)and kb(k) have two intersection points. One point(P1) is located on the k-axis and has the samecoordinates as the point P1 in model I (see Eq.(24a) and Eq. (24b)). The coordinates of thesecond point P2 are determined by

k3=628

(1+8), (48a)

k3= −62

82(1+8). (48b)

P2 is located outside the region of physical mean-ingful values of the parameter k. At point P1 bothlines have the same slope (dka/dk=dkb/dk=3).They differ in the second derivatives, which are

d2ka

dk2 =6(−1+38)

k, (49a)

d2kb

dk2 = −6k

. (49b)

Since 8\0 the curvature of kb at k=0 is there-fore always smaller than that of ka. From theslope and the curvatures of the functions ka andkb it follows that in the vicinity of the point P1 forall given values of k the corresponding k-valueson the line kb are larger than those on the line ka.From that and from the fact that these lines haveno intersection point for positive parameter valuesit follows that the line kb cannot be a boundary ofthe stability region.

The lines kc and ka intersect in the point P3

which has the coordinates

k (3)=62/3�8

2+2−

'�8

2+1

�2

+1n

1/3

\0,

k (3)=� 6

k (3)

�2

\0. (50)

The region of stability is bounded either by theline ka for kBk (3) or by the line kc for k\k (3)

(solid parts of the corresponding curves in Fig.12).

Fig. 12. Bifurcation lines of Model II for N=3. The lines ka

and kc following from a1a2−a0=0 (Eq. (14)) and D=0 (Eq.(46a)), respectively, represent bifurcation points of the mainsteady state. The lines kd and kf represent bifurcation pointsresulting from Eq. (51a) for rBN. ke follows from D=0 (Eq.(46a)) with r=2. The solid parts of the lines indicateboundaries of stability region for the various steady states.The broken parts of the lines are located within the regions ofunstable steady states.


The bifurcations which occur on the solid partof the line ka lead to synchronous oscillations ofthe cells. In the present case asynchronous oscilla-tions may be excluded because the line tr=0 ofthe quadratic part of the characteristic polynomial(47) is not a boundary of the stability region (cf.corresponding discussion for model I in Section4.2).

For k\k (3) the main steady state becomesunstable at the line kc since among the 2(N−1)eigenvalues of the quadratic part of the character-istic polynomial N-1 real eigenvalues become pos-itive.

5.2.4. Bifurcation of steady states with q=rBNBifurcations may result from the solutions of

either E1(l)=0 or G1(l)=0. As for the case ofthe main steady state the bifurcation line derivedfrom tr=0 of the quadratic part is no boundaryof the stability region. The other type of bifurca-tions which results from D=0 may exist as longas r\1. The corresponding bifurcation line forr=2 and N=3 is shown in Fig. 12 (line ke).

Since the term in brackets in G1(l) is a polyno-mial of fourth order the conditions for the oc-curence of Hopf-bifurcations are

a1a2a3−a0a23−a2

1=0, (51a)

a0\0, (51b)

a1\0, (51c)

a2\0 (51d)

a3\0 (51e)

where the ai denote the coefficients of the polyno-mial (Bautin, 1949). The curves kd and kf result-ing from condition (51a) for r=q=2 andr=q=1, respectively, are shown in Fig. 12.

The dynamical properties of model II which arereflected by the bifurcation lines shown in Fig. 12for N=3 may be summarized as follows.

Low values of the coupling parameter k : Forlow values of the bifurcation parameter k allsteady states resulting from equation system (40)are stable. Crossing the line ka the main steadystate undergoes a bifurcation and limit cyclescorresponding to synchronous oscillations of allthree cells occur near the the region of stability.

At the line kd the steady state where one of thethree cells have no metabolism via the main path-way shows a bifurcation giving rise to syn-chronous oscillations of the two other cells. Afurther bifurcation takes place at the line kf wherethe steady state characterized by two degeneratedcells becomes unstable. All oscillations in degener-ated cells have the property that only the concen-trations of the metabolites Xi oscillate whereas theconcentrations Yi remain zero.

High values of the coupling parameter k : Inthis case there exists the possibility that the stateswith r\1 become unstable not via Hopf-bifurca-tion but due to a transition from D\0 to DB0.These transitions are characterized by a symmetrybreaking. For example, at the line kc one changesfrom the main steady state with three identicalcells to one of the steady states where only twocells are identical.

6. Discussion

The systems analyzed in the present paper maybe considered as minimal models for the descrip-tion of coupled oscillators in cell suspensions.They are based on a two-component system,which includes a positive feedback mechanism.The nonlinearity of the equations which, from themathematical point of view, is necessary for theemergence of oscillations is provided by a tri-molecular term. Recently, a more simple chemicaloscillatory mechanism has been proposed whichcontains only mono- and bimolecular reactions(Wilhelm and Heinrich, 1995). However, this sys-tem includes three compounds. In fact, it has beenshown that any chemical oscillator with onlymass-action kinetics has to include at least threevariables (Hanusse, 1972). While it is still unclearwhether a three component oscillator with onlyone bimolecular term has any biochemical mean-ing the model used in the present paper reflectsvarious properties of the glycolytic oscillator.

Despite its simple structure the presented modelallows the investigation of various problems,which are of relevance for the interpretation ofexperimental data. The main results may be sum-marized as follows:


1. Depending on the kinetic parameters coupledbiochemical oscillators may show synchronousand asynchronous oscillations.

2. There are two different possibilities for asyn-chronous oscillations. (a) For parameter valuesnear to the boundary of stability regular asyn-chronous behaviour occur, where the phase shiftsbetween the individual cells are integer multiplesof the reciprocal value of the number of cells. Theoccurrence of these regular asynchronous oscilla-tions depends on the type of the coupling. In thepresent model it is found only for the case wherethe coupling substance belongs to the pool ofproducts of the autocatalytic reaction (Model I).For small values of the coupling strength or forlow cell densities regular asynchronous oscilla-tions occur in a large range of parameter values.(b) Leaving the near neighbourhood of the pointsof Hopf-bifurcations nonregular asynchronousbehaviour may arise by secondary symmetrybreaking bifurcations, either from the branch ofsynchronous oscillations or from the branch ofregular asynchronous oscillations. The mode ofnonregular asynchronous behaviour is character-ized by nonidentic oscillations in different cellsand by nonregular phase shifts.

3. Regular asynchronous oscillations in suspen-sions of a large number of cells are characterizedby a nearly constant concentration of the cou-pling metabolite. It may lead to the phenomenonof hidden oscillations, where strong variations inthe internal metabolite concentrations are notreflected by changes in the extracellular mediumor the mean values of internal variables. This hasconsequences for the experimental observabilityof the oscillations. As shown in Section 4.2 hiddenoscillations may come into view for a certain timespan after perturbation of the system. A nearlyconstant external concentration has to be distin-guished from a fixed concentration of the cou-pling substance. In the latter case there is nocommunication between the oscillating cells andthe phase shifts are entirely determined by theinitial conditions and may change after perturba-tions of the variables. In contrast to that thephase shifts of regular asynchronous oscillationsare stable against perturbations.

4. Complex dynamic phenomena may occur ifthe coupling opens the possibility of multiplesteady states as shown in model II. Then thewhole population may be within different stablesteady states or different oscillations states. This isaccompanied by distinct steady states or distinctoscillatory states of the single cells. The conclu-sion is that cells which would behave identically ifuncoupled, may show a distinct behaviour in thecase of coupling. In a more general context thisreasoning may have consequences for processes ofcell differentiation in multicellular complexes.

In our model coupling of the cells is performedby a metabolite, which is produced in all cells andwhich permeates their membranes. This type ofinteraction has been proposed for many systems.For yeast cell populations acetaldehyde was dis-cussed to mediate the coupling. Sometimes, ac-etaldehyde has been reported to be adesynchronizer of the oscillations (Ghosh et al.,1971), whereas other authors call this substance asynchronizer (Richard et al., 1996). Without dis-cussing the problem whether or not acetaldehydemediates the interaction, we want to emphasizethat any substance may play both roles. As shownin Section 4.2 it depends on the kinetic parameterswhether synchronous or asynchronous oscillationsoccur, although in both cases the coupling isrealized by the same substance, that is by theproduct of the autocatalytic reaction.

In some sense the models I and II show comple-mentary properties with respect to symmetrybreaking. In model I there is a unique steady statewhich may bifurcate into a full symmetric limitcycle corresponding to synchronous oscillationsor into a limit cycle, characterized by regularasynchronous oscillations. In model II regularasynchronous oscillations are excluded, but a highnumber of nonsymmetric steady states arises.

We want to stress that the models presented inthis paper are applicable to any number of inter-acting cells, but there are fundamental differencesin the dynamics of model I and II concerningtheir dependence on the number of cells. Thedynamical properties of model I are nearly inde-pendent of N as long as steady states and limitcycles following from the first Hopf bifurcationare considered. The difference in the cell number


only affects the phase shifts for regular asyn-chronous oscillations and the dynamics of theexternal metabolites. In contrast to that thereis a dramatic increase of possible steady stateswith increasing cell number in model II.

A crucial simplification of the present mod-els is that the diffusion processes within theexternal volume are considered to be very fastcompared to the transmembrane exchange ofthe coupling substance and to the characteris-tic times of the intracellular reactions. Inclu-sion of slow extracellular diffusion processesmay be an interesting subject if the cell havea definite spatial arrangement, like in tissues.Therefore, the model applies to oscillations incell suspensions where the cells have no fixedpositions, for example due to stirring.

Throughout the paper it was assumed thatall cells of the populations are identic. Con-cerning the basic oscillatory mechanism this iscertainly justified if all cells of the populationbelong to the same type. There may be ofcourse differences in the kinetic constants andin morphological parameters, but one may ex-pect that there are only slight deviations fromthe mean values. (For typical distributions ofthe cellular volume and of the membranearea, see (Svetina, 1982)).

Despite its simplicity the models revealsmany interesting dynamic phenomena. At thepresent state a comparison with experimentaldata could be accomplished only in a semi-quantitative manner. It would be an intriguingtask to use recent models of anaerobic yeastcell metabolism for a more extended theoreti-cal approach for oscillations in cell suspen-sions. The rather elaborated model presentedin (Galazzo and Bailey, 1990) may serve as abase. It includes the main processes of glycol-ysis and associated reactions and uses realistickinetic equations for the various enzymes. Itis restricted to the description of stationarystates but its extension to time dependentstates would be straightforward. Very detailedmodels of glycolysis exist also for other celltypes (Rapoport et al., 1976; Werner andHeinrich, 1985; Joshi and Palsson, 1989;1990).

With a realistic metabolic model which con-tain more than two metabolites one could an-alyze the potential effects of different couplingsubstances, including those of permeable inor-ganic ions if the model takes into account thecorresponding membrane transport processes.It would be interesting to investigate whetherour result that asynchronous oscillations onlyoccur if the coupling substance belongs to thepool of products of the autocatalytic reactionmay be generalized to more complex oscillat-ing systems. However, many metabolic inter-mediates of glycolysis may be excluded to beintercellular messengers since they are phos-phorylated and cannot leave the cell. Most ofthe substances which have been considered ascandidates for signallers, especially pyruvate,ethanol and acetaldehyde are produced in thelower part of glycolysis. Since the kineticproperties of phosphofructokinase which is lo-cated in the upper part of glycolysis are es-sential for the emergence of oscillations theseendproducts can only coordinate the behaviourof the cells if the the pathway contains effi-cient feedback mechanisms. These could bebased on stoichiometric interactions, per-formed, for example, by ATP or NADH, oron regulatory couplings. Depending on thecell sort there are for the latter type of inter-actions many possibilities such as inhibition ofphosphofructokinase by ATP or phospho-enolpyruvate and inhibition of hexokinase byglucose 6-phosphate.

Acknowledgements

This work was supported by a grant of theDeutsche Forschungsgemeinschaft (Ref. He2049/1-2).

Appendix A. The characteristic polynomial forthe eigenvalues for N coupled oscillators (ModelI)

The characteristic equation of model I forN interacting cells reads


sequently to the determinants of lower orderwhere J denotes the Jacobian of the equationsystem (8) and I the identity matrix. Eq. (A1)takes into account that the kinetic parameters andthe steady state concentrations for all cells are thesame. J has been arranged in such a way, that theelements of the (2i )-th and (2i+1)-th row orcolumn correspond to the variables of the i-thcell, Xi and Yi, respectively. The first row and thefirst column refers to the variable Y e. For theelements aij one derives

a11= −k8, a12=k8

N, a21=k, (A2)

a22= −62

k2, a23= −2k, a32=62

k2 ,

a33=k−k. (A3)

Expanding the determinant in products of minorsof the full determinant Det(N) and the threenonzero elements of the last column one arrives atthe following formula

Det(N)=AN−1B+Det(N−1)A (A4)

where Det(N−1) denotes the determinant of amatrix which follows from that of Eq. (A1) bydeleting the last two rows and columns. Further-more, one derives

A=l2−l(a22+a33)+a22a33−a23a32

=l2−l�

k−k−62

k2

�+62

k�

1+k

k�

, (A5)

B= −a12a21(a22−l)=k28

N�62

k2+l�

. (A6)

Repeating the expansion for Det(N−1) and sub-sequently to the determinants of lower order

Det(N) can eventually be expressed as

Det(N)=AN−1((N−1)B+Det(1)) (A7)

In this equation Det(1) denotes the determinantof a matrix containing the elements from the firstthree rows and columns of the full matrix. Fromthat it follows immediately that the factor (N−1)B+Det(1) is identic to the characteristic poly-nomial F(l) of a single cell given in Eq. (10). Thisleads to the characteristic polynomial of N inter-acting cells given in Eq. (21).

Appendix B. Steady states of model II for thecase of nonzero ground activity of theautocatalytic reaction

Using Eq. (43) the steady state conditions forthe variables Xi and Yi may be transformed into

0=6−kY( i−k(X( i−X( e) (B1a)

and

X( i=kY( i

a+Y( 2i

for i=1,…, N. (B1b)

Inserting Eq. (B1b) into Eq. (B1a) for i=1 yieldswith X( e from Eq. (35)

0=6−kY( 1−kkN

�(N−1)Y( 1

a+Y( 21

− %j"1

Y( j

a+Y( 2j

�.

(B2)

Subtracting Eq. (B1a) for i\1 from that for i=1one obtains by consideration of Eq. (B1b)

Y( 1−Y( i= −k� Y( 1

a+Y( 21

−Y( i

a+Y( 2i

�. (B3)


With hi=Y( i−Y( 1 one derives from Eq. (B3)

h3i (Y( 2

1+a)+h2i (2Y( 3

1+2aY( 1−kY( 1)

+hi(Y( 41+2aY( 2

1−kY( 21+a2+ak)=0 (B4)

which has the solutions hi=0, hi=h+ and hi=h−. Denoting by u, 6 and w the numbers of cellswith these three solutions, respectively, onederives from Eq. (B2) with u+6+w=N−1

k=6�

Y( 1

−k

N�(N−1−u)Y( 1

a+Y( 21

−6(Y( 1+h+)

a+ (Y( 1+h+)2

−w(Y( 1+h−)

a+ (Y( 1+h−)2

�n−1

. (B5)

This equation may be used to calculate for allpossible values of u, 6 and w the rate constant k asa function of Y( 1. This yields for N=2 and N=3the curves given in Fig. 11A,B.

Appendix C. Stability analysis of steady statescharacterized by qBr (Model II)

We prove that the steady states given in Eq.(41a) Eq. (41b) Eq. (41c) Eq. (41d) Eq. (41e) for15qBr are always unstable. The characteristicequation for these steady states may be derived bya similar method as shown for model I in Ap-pendix A. For rBN one obtains

E1,a(l)q−1E1,b(l)r−q−1E2(l)N−r−1G2(l)=0,(C1)

with

E1,a(l)=l2+l�k2

X( 21

+k−k�

+k�k2

X( 21

−k�

,

(C2)

E1,b(l)=l2+l�k2X( 2

1

k2 +k−k�

+k�k2X( 2

1

k2 −k�

,

(C3)

and E2(l) which has the same form as given inEq. (46b). The function G2(l) is here a polyno-mial of seventh degree. It turns out that forstability analysis only the term a0 is relevant. Itreads

a0=k38k

N�

rkk2− (r−q)k2X( 21−

qk4

X( 21

�. (C4)

In the case r=N the polynomial reads

E1,a(l)q−1E1,b(l)r−q−1G3(l)=0, (C5)

where G3(l) is a polynomial of fifth degree with

a0=k28

N�

Nkk2− (N−q)k2X( 21 −

qk4

X( 21

�. (C6)

One may distinguish four different cases: (A) q\1 and r−q\1, (B) q=1 and r−q\1, (C) q\1and r−q=1, (D) q=1 and r−q=1.

Case (A): There are four eigenvalues resultingfrom the two quadratic equations E1,a(l)=0 andE1,b(l)=0. For the real parts of these eigenvaluesto be negative it is necessary that

k2

X( 21

−k\0 (C7a)

and

k2X( 21

k2 −k\0. (C7b)

It is obvious that these conditions can not befulfilled simultaneously.

Case (B): There are two eigenvalues resultingfrom the quadratic equation E1,b(l)=0 and someeigenvalues resulting from G2(l)=0 (seven eigen-values for rBN and five eigenvalues for r=N).For the latter eigenvalues to have negative realparts it is necessary that a0\0 with a0 given inEq. (C4) and Eq. (C6), respectively. For r5N thecondition a0\0 is fulfilled if

k2

(r−1)kBX( 2

1Bk2

k. (C8)

This is in contradiction to condition (C7b) fornegative real parts of the eigenvalues resultingfrom E1,b(l)=0.

Case (C): There are two eigenvalues resultingfrom the quadratic equation E1,a(l)=0 and someeigenvalues resulting from G2(l)=0 (seven eigen-values for rBN and five eigenvalues for r=N).For the latter eigenvalues to have negative realparts it is necessary that a0\0 with a0 given inEq. (C4) and Eq. (C6), respectively. For r5N thecondition a0\0 is fulfilled if


k2

kBX( 2

1Bqk2

k. (C9)

This is in contradiction to condition (C7a) fornegative real parts of the eigenvalues resultingfrom E1,a(l)=0.

Case (D): In this case the two polynomialsE1,a(l) and E1,b(l) do not appear in the character-istic equation. For q=1 and r−q=1 the coeffi-cient a0 may be written as

a0= −k38k

N�

kX( 1−k2

X( 1

�2

(C10)

which means that the necessary condition forstability of the steady states, a0\0, can not befulfilled in this case.

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