Dynamics of Two-Component Biochemical Systems in Interacting Cells; Synchronization and Desynchronization of Oscillations and Multiple Steady States

BioSystems 43 (1997) 1–24

Dynamics of two-component biochemical systems in interacting cells; 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. The individual cells are considered to be identical and are described by differential equations proposed for the description of glycolytic oscillations. The coupling is realized by exchange of metabolites across the cell membranes. No constraints are introduced concerning the number of interacting systems, that is, the analysis applies also to populations with a high number of cells. Two versions of the model are considered where either the product or the substrate of the autocatalytic reaction represents the coupling metabolite (Model I and II, respectively). Model I exhibits a unique steady state while model II shows multistationary behaviour where the number of steady states increases strongly with the number of cells. The characteristic polynomials used for a local stability analysis are factorized into polynomials of lower degrees. From the various factors different Hopf bifurcations may result in leading for model I, either to asynchronous oscillations with regular phase shifts or to synchronous oscillations of the cells depending on the strength of the coupling and on the cell density. The multitude of steady states obtained for model II may be grouped into one class of states which are always unstable and another class of states which may undergo bifurcations leading to synchronous oscillations within subgroups of cells. From these bifurcations numerous different oscillatory regimes may emerge. Leaving the near neighbourhood of the boundary of stability, secondary bifurcations of the limit cycles occur in both models. By symmetry breaking the resulting oscillations for the individual cells lose their regular phase shifts. These complex dynamic phenomena are studied in more detail for a low number of interacting cells. The theoretical results are discussed in the light of recent experimental data on the synchronization 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]

1. Introduction ple, linear chains or rings of coupled cells. For oscillations in cell suspensions it is probably more Periodic behaviour is an ubiquitous phe- realistic to consider an indirect coupling, by tak- nomenon of biological systems. It is found on ing into account substances which may diffuse nearly all structural levels, particularly within across the cellular membrane into the extracellu- metabolic systems. In the last decades a great lar space and may enter the cytoplasm of other variety of oscillations within cellular systems have cells. Recently such a coupling has been sup- been observed and mathematically described, such ported experimentally for glycolytic oscillations in as glycolytic oscillations in different types of cells. populations of yeast cells (Richard et al., 1996). More recently, oscillations of intracellular calcium There, strong support has been given that ac- concentrations have attracted great interest of etaldehyde which permeates the plasma mem- experimentalists and theoreticans; for a recent brane mediates the coupling. In particular, it was review for cellular oscillations see Goldbeter shown that the extracellular concentration of ac- (1996). etaldehyde oscillates and that the cells respond to In a pioneering work on oscillating biochemical acetaldehyde pulses. The experimental results in- reactions Higgins (1967) addressed the problem in dicate that ethanol which was also considered to which way a coupling between individual cells play the role of an intercellular messenger (Aon et affects the resulting dynamics, for example, by al., 1992) does not exert this function. synchronizing their oscillations. Later on it has An intriguing question is whether coupling of been shown experimentally that a mixing of two oscillating cells is always accompanied by syn- cell populations oscillating out of phase may lead chronization or whether more complex dynamic to their rapid synchronization (Pye, 1969; Ghosh phenomena may result. Theoretical work on inter- et al., 1971). Recent experiments on glycolytic acting identical oscillators has shown that also oscillations before and after mixing two out-of- asynchronous behaviour may be expected. Of par- phase populations of yeast cells show a synchro- ticular interest is the symmetric case where all nization which is rather slow (Richard et al., phase shifts are proportional to the reciprocal 1996). value of the number of cells. As far as direct The problem of coupling metabolic oscillators coupling is considered this type of asynchronous has also been analyzed in a great number of behaviour may be excluded for interacting two theoretical investigations. Due to the difﬁculties in component systems (Alexander, 1986). In the description of high-dimensional systems the theo- present paper we show that such an assertion does retical investigations have often been restricted to not hold for cell populations where the coupling is the case of two or three coupled oscillators which, realized by substances which are extruded into the of course, is not sufﬁcient for describing cell pop- extracellular medium, i.e., besides synchronous ulations. However, various results have been oscillations, asynchronous dynamics is possible derived also for systems of many interacting oscil- also for interacting two-component systems. One lators (Othmer and Aldridge, 1978; Alexander, may expect therefore, that desynchronized be- 1986). Other investigations concern systems of haviour of cells may be a common phenomenon. weakly coupled oscillators (Kopell and Ermen- We will demonstrate that it is characterized by trout, 1986). Most theoretical work is performed strong variations in the internal states of the to the case of direct interactions, i.e., where the individual cells but nearly constant external con- coupling terms of the model equations contain centrations of the diffusible metabolites. only the differences between concentration vari- Generally, sustained oscillations in biochemical ables within neighboured cells (Alexander, 1986; systems may only arise if one or more reactions Kopell and Ermentrout, 1986). Such a direct cou- obey a nonlinear kinetics. A main case is that pling necessitates a physical contact between the autocatalytic processes are involved, as for exam- cells. For its mathematical description special spa- ple, the phosphofructokinase reaction in glycoly- tial arrangements have to be assumed, for exam- 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 and metabolic oscillations. The dynamics of the Schuster, 1996) or the calcium-induced-calcium- metabolites of a cell is described by kinetic equa- release (CICR) from the endoplasmic reticulum, tions resulting from a feedback-activation mecha- for recent models see (Goldbeter et al., 1990; nism which has been proposed for the explanation Somogyi and Stucki, 1991). It may be interesting of glycolytic oscillations (Higgins, 1964, 1967; to analyze whether there are differences in the Sel’kov, 1968). dynamics of the cell populations if the coupling Using the model for glycolytic oscillations in substance belongs to the pool of substrates or to the form as speciﬁed by Sel’kov (1968) the dy- the pool of products of the autocatalytic reaction. namics of the metabolite concentrations within a In yeast cell populations the latter case seems to single cell is governed by the equations be realized since acetaldehyde is one of the end- dX k products of glycolysis. In the present paper we =61 −k2XY , (1a) investigate both possibilities. By considering a dt simple model, proposed originally for the expla- dY =k XY k −k Y. (1b) nation of glycolytic oscillations, many of the re- dt 2 3 sults may be derived analytically. Our analysis is valid for an arbitrary number of interacting cells. These equations describe a system where the com- Furthermore, no restrictions are made concerning pound X is supplied by a constant input 61 and the strength of the coupling, which is expressed by degraded by an autocatalytic reaction. The latter the rate constant for the exchange of substances reaction produces the compound Y which in turn between the intracellular and extracellular space. is degraded. In the case of glycolysis reaction 1 Weak and strong coupling follow as special cases. and 2 may represent the reactions catalyzed by The results of this theoretical analysis are dis- hexokinase and phosphofructokinase while the cussed in the light of recent experimental data processes of the lower part of the pathway are presented by (Richard et al., 1996). In particular, lumped into reaction 3. The coefﬁcient k charac- we demonstrate numerical simulations of oscillat- terizes the strength of the product activation. It ing cell suspensions before and after mixing of has been demonstrated that system (1) may show two subpopulations. Furthermore, several conclu- oscillations of limit cycle type as long as k\1. In sions are drawn concerning the observability of the following we consider the special case k=2. oscillations, if the cells are desynchronized. Introducing dimensionless quantities for the con- Besides oscillations, multiplicity of steady states centrations X/CX, Y/CY, for the rate con- 2 3 is a well-known characteristic of nonlinear bio- stant k3/k2C k, for the input rate 61/k2C in, 2 chemical reaction systems (Eschrich et al., 1990; and for time tk2C t, where C represents an Schellenberger and Hervagault, 1991). Therefore, arbitrary constant concentration, equation system part of this work is devoted to the analysis of the (1) may be rewritten as interrelationships between these two fundamental dX phenomena in the case of cell populations. We =6−XY 2, (2a) dt will show that in the case of multiple stable and unstable steady states, the whole population may dY =XY 2 −kY. (2b) split into different subgroups of synchronized cells dt where each subgroup may show different behaviour. The reaction system has a unique steady state with the concentrations k 2 2. Basic model assumptions X( = , (3a) 6 We investigate models for suspensions of inter- 6 Y( = . (3b) acting cells, in which the single cells may show k 4 J. Wolf, R. Heinrich / BioSystems 43 (1997) 1–24

The steady state is locally asymptotically stable where Xi and Yi denote the concentrations in the upon the condition that i-th cell. Y e (Model I) and X e (Model II) are the external concentrations of the coupling sub- 6 k k. (4) \ stances. Let us deﬁne effective rate constants s of Crossing the boundary of stability this state be- the transmembrane diffusion in the following way comes unstable via a Hopf-bifurcation. The fre- AP quency of the oscillation near to the bifurcation s= , (7) V point is approximately given by 1 where P denotes either P or P . We call s the 6 2 Y X 2 =Z= , (5) coupling parameter since an interaction of cells k takes place only for s"0. For s=0 the suspen- where Z denotes the determinant of the Jacobian sion contains noninteracting cells with two-com- matrix of equation system (2a,b). ponent oscillators described by Eqs. (2a) and (2b).

In the present models we consider suspensions Concerning the extracellular volume V2 one may consisting of an arbitrary number N of interacting think about different situations. One may assume, cells. Each cell contains an oscillating reaction for example, that V2 is constant at varying num- system corresponding to the mechanism described bers of cells, which means that the cell density is by equation system (2). It is supposed that the proportional to N. We concentrate here on an- individual cells interact via the flux of metabolites, other possibility where V2 increases linearly with which are produced in all cells and may permeate N, which results in a constant cell density at through the cell membranes. variations of N. Denoting by the ratio of intra- For the development of the models we intro- cellular and extracellular volume for the case N= duce the following simplifications: 1 and using a scaled coupling parameter 2 1. All metabolic oscillators have an identical stoi- (s/k2C s) the differential equation system for chiometry and identical kinetic parameters N coupled cells reads for model I with respect to the biochemical transforma- dY e s N tions as well as the transport processes. = Y −NY e , (8a) dt N % j 2. All cells are characterized by the same volume j=1 V and the same membrane surface area A. dX 1 i =6−X Y 2, (8b) The metabolites are distributed homogeneously dt i i both in the cellular solution and in the external dY solution with the volume V . i =X Y 2 −kY −s(Y −Y e). (8c) 2 dt i i i i 3. The coupling is realized by a transmembrane flux of a single uncharged metabolite. Analogously one may derive an equation system We consider two versions of the model differing for model II in the permeating substance. In model I the cou- dX e s N pling is realized by the product Y and in model II = X −NX e , (9a) dt N % j by the substrate X of the autocatalytic reaction. j=1 The fluxes J (Model I) and J (Model II) dX i,Y i,X i =6−X Y 2 −s(X −X e), (9b) between the i-th cell and the external medium are dt i i i functions of the permeabilities P and P of the Y X dY membrane to the substances Y and X, respec- i =X Y 2 −kY . (9c) dt i i i tively, and of the differences in the concentrations of the coupling substances between the cell and For N=1 equation systems (8) and (9) describe the external medium. We use the relations oscillators in single cells exchanging a metabolite with the external solution. At fixed numbers N J =P (Y −Y e), (6a) \ i,Y Y i 1 of cells variations of correspond to changes of e Ji,X =PX(Xi −X ) (6b) the cell density. J. Wolf, R. Heinrich / BioSystems 43 (1997) 1–24 5

3. Dynamics of single cells parameter. Relations Eqs. (12a), (12b), (12c) and (12d) ensure that the Jacobian matrix has one pair The three-component systems resulting with of pure imaginary eigenvalues. For model I one N=1 from model I and model II have unique obtains with the help of Eq. (10) steady states. In both cases the stationary concen- a1a2 −a0 trations X( 1 and Y( 1 are identical to those given in Eq. (3a) and Eq. (3b), respectively, for the two 6 2(1+)2 =s 2 −k(1+) component system (2). Furthermore, it follows k 2 from Eq. (8a) and Eq. (9a) that the steady state 4(1+) 2 2 4 concentrations of the permeating substances X or 6 2 6 6 2 +s 4 +k − + 3 −6 (13) Y are the same in the external solution and within k k k the cell. and for model II with Eq. (11) Using these steady state concentrations one derives from the Jacobian matrix of system (8) the a1a2 −a0 following characteristic equation for the eigenval- 6 2(1+) ues u of model I =s 2 −k(1+)2 k 2 F(u) 4 2 4 6 2 26 6 2 2 +s +k (1+)− + − . 6 4 3 6 =u 3 +u 2 +s+s−k k k k k 2 (14) 2 2 2 2 6 6 s 6 s 6 s In both models the conditions for Hopf-bifurca- +u + 2 −ks+ 2 + =0, k k k k tions may be fulfilled. We present the proof exem- (10) plarily for model I using k as bifurcation and from system (9) for model II parameter. 1. It follows from Eq. (13) that condition (12a) G(u) can be fulfilled by real non-negative parameter 6 2 values. u 3 u 2 s s k = + 2 + + − 2. The relation a 0, Eq. (12b), is always true, k 0 \ 2 2 2 whereas the signs of the coefficients a1 and a2 may 6 6 s 6 s change depending on the kinetic parameters. For +u + 2 −ks−ks + =0. k k k all k-values for which condition (12a) is fulfilled (11) the relation a0 \0 implies According to the Hurwitz-criterion in a three- sgn(a )=sgn(a ), (15a) component system with the characteristic equa- 1 2 tion u 3 +a u 2 +a u+a =0 a Hopf-bifurcation 2 1 0 a1, a2 "0. (15b) takes place if the following conditions are fulfilled 3. With the coefficients of the characteristic Eq. a a a 1 2 − 0 =0, (12a) (10) one obtains a0 \0, (12b) ((a a −a ) 6 2 26 2s 26 2s k 1 2 0 =a − + +ks+ a a 0, (12c) 0 2 2 2 1 \ (k k k k 2 a2 \0, (12d) 26 − +k a . (16) (v k 2 1 "0. (12e) (p Eliminating a0 by condition (12a) and taking into

In relation Eq. (12e) v denotes the real part of the account the deﬁnition of the coefﬁcient a1 Eq. (16) complex eigenvalues and p is any bifurcation may be rewritten as 6 J. Wolf, R. Heinrich / BioSystems 43 (1997) 1–24

line the conditions (12c) and (12d) are violated. ((a a −a ) 62s 6 2s k 1 2 0 =− + +2ks a The lines a =0 and a =0 do not bound the (k k 2 k 2 2 1 2 region of stability because they are located 26 2 − +k a . (17) within the region of negative values of (a1a2 − k 2 1 a0). Therefore, in the region of low k-values Due to relations (15a) and (15b) it follows that where (a1a2 −a0) is positive, the steady state is stable. ((a1a2 −a0)/(k"0. From Eq. (13) one obtains For both models limit cycle oscillations in the limit k0 that (a1a2 −a0) . There- fore, at the lowest value of k for which condi- emerge at transitions from stable to unstable tion (12a) is fulﬁlled the following relation holds steady states. With increasing distance from the bifurcation line, that is, with increasing k-values, ((a1a2 −a0) B0. (18) folded limit cycles and chaos may occur via pe- k ( riod-doubling cascade. These dynamic phenom- 4. With the relations (15) and (17) it follows ena of single cells have been studied in detail by immediately that a1 and a2 are both positive if calculating Lyapunov-exponents (Wolf, 1994). In relations (12a) and (18) are fulﬁlled. fact, the present model for the single cell has 5. Inserting v+i for the complex eigenvalue much in common with a three-variable system into the general characteristic equation for studied in (Herzel and Schulmeister, 1987) which three-variable systems yields independent equa- shows similar dynamical properties. However, tions for the real part and imaginary part of the the model used in the present paper has a more polynomial. After derivation of these equations simple structure because it involves only one re- with respect to the bifurcation parameter k one action step characterized by a nonlinear kinetic v obtains with =0 equation. (v ( (a (a (3 2 −a ) +2a = 2 2 − 0 , (19a) 1 (k 2 (k (k (k (v ( (a 2a −2 =− 1 . (19b) 2 (k (k (k 2 Taking into account that v=0 implies a1 = the solution of system (19) with respect to (v/(k yields

(v ((a1a2−a0)/(k =− 2 . (20) (k 2(a1 +a 2)

Because on that part of the surface a1a2 −a0 =0 where condition (18) is true, the coefficient a1 is positive one obtains (v/(k\0, that is condition (12e) is fulfilled. Fig. 1 shows for model I the functions s= s(k) resulting from a1a2 −a0 =0 as well as from Fig. 1. Regions of stable and unstable steady states for a single a1 =0 and a2 =0 for fixed values of the other cell for model I. The two branches of the function s=s(k) parameters. The curve resulting from condition result from the condition a1a2 −a0 =0. On the branch drawn (12a) for model I consists of two branches sepa- by a thick solid line the conditions (12a) to (12e) for Hopf rating the regions of positive and negative val- bifurcations are fulfilled, whereas on the broken line the conditions (12c) and (12d) are violated. Lines a1 =0 and a2 =0 ues of (a1a2 −a0). On the branch drawn by a (thin solid lines) are located within the region of unstable solid line the conditions (12b) to (12e) for Hopf- steady states. Other parameter values: 6=3.0, =0.2 (These bifurcations are fulfilled, whereas on the broken values are also used for all other figures.) J. Wolf, R. Heinrich / BioSystems 43 (1997) 1–24 7

4. Coupling via the product of the autocatalytic reaction (Model I) 4.1. Stability analysis of the steady states

It is easy to see from equation system (8) that the system of N coupled cells displays independently of the value of N always a unique steady state solution which for every cell is identic to that of the single cell, e that is, X( i =X( and Y( i =Y( =Y( for i=1,…, N with X( and Y( given in Eq. (3a) and Eq. (3b). After appropriate numbering of the variables the Jacobian matrix for this steady state has for any value of N a very regular structure, in that it contains besides Fig. 2. Bifurcation lines for coupled oscillators (Model I). several non-zero-elements in the first row and in the Lines s=sa(k) and s=sb(k) where the cubic part and the first column only blocks, with two elements per row quadratic part of the characteristic Eq. (21), respectively, and column, along the main diagonale; cf. Appendix yields one pair of pure imaginary eigenvalues. The solid parts A. Due to these properties the characteristic equa- of the lines indicate the boundaries of the region of stability, tion for the eigenvalues may be written as follows where Hopf bifurcations lead to synchronous or asynchronous oscillations of the cells. The broken parts of the lines are D(u)N−1F(u)=0 (21) located within the region of unstable steady states. The coordinates of the points P1 and P2 are given in (24) and (27), for arbitrary values of N; for the derivation of this respectively. Region 1: stable steady states, regions 2–4: un- formula, see Appendix A. The factor F(u)isa stable steady states. polynomial of third degree and identic to that obtained for the single cell; cf. Eq. (10). For the other fullfilling conditions (12a) to (12e) by taking into term one obtains account Eq. (13). For that bifurcation Eq. (12a) gives with conditions (12b)–(12d) raise to a bifurca- D(u)=u 2 −tr ·u+Z, (22a) tion line s=sa(k) within the (k, s)-plane as shown with in Fig. 2. This curve corresponds to the thick solid line in Fig. 1. 6 2 tr=k−s− , (22b) A second bifurcation may result from the (N−1) k 2 identic quadratic parts D(u) of the characteristic 6 2 s polynomial. Due to the positive sign of Z the Z= 1+ \0, (22c) k k bifurcation points are determined by tr=0, that is by where tr and Z denote the trace and determinant, 2 respectively, of the (2×2) submatrix which corre- 6 s=sb(k)=k− 2 . (23) sponds to the above-mentioned blocks of the Jaco- k bian (see Eq. (A5)). In this way the full characteristic The curves sa and sb shown in Fig. 2 have two polynomial of the order (2N+1) is splitted into a common points. In the first point P1 with the product of (N−1) identical quadratic parts and one coordinates cubic part. Consequently, only five different eigen- k (1) =6 2/3, (24a) values may exist which are independent of the number of cells. Two eigenvalues follow from s (1) =0, (24b) D(u)=0 and the other three from F(u)=0. There- they have the same slope fore, there are two possibilities for Hopf bifurcations ds ds leading to sustained oscillations. One possibility a = b =3, (25) results from the cubic part of the characteristic dk dk polynomial, that is, at parameter combinations but differ in the second derivatives 8 J. Wolf, R. Heinrich / BioSystems 43 (1997) 1–24 d2s 6(1+3) a =− , (26a) dk 2 k d2s 6 b =− . (26b) dk 2 k

The second common point P2 is located at 1 1/3 k (2) = 6 2 1+ , (27a) 6 2 1/3 s (2) = . (27b) (1+)2 Relations (24)–(27) ensure that for any values of 6 and the lines s and s separate the (k, s)- a b Fig. 3. Synchronous oscillations of two coupled cells. The parameter plane into four regions (see Fig. 2). In dynamics results from a bifurcation at the line sa(k) (see Fig. regions 2 and 4 the equation F(u)=0 and in 2). Parameter values: s=3.2, k=3.84. The initial conditions regions 3 and 4 the equation D(u)=0 yield eigen- for the concentrations of the two cells where nonidentic (X1 = e values with positive real parts. Therefore, for 4.91, X2 =4.93, Y1 =0.77, Y2 =0.79, Y =0.78). The curves for Y and Ye were plotted starting from a time t , where the parameter values taken from regions 2, 3 and 4 i 0 limit cycle has been reached. the steady state is unstable. In region 1 all eigenvalues resulting from Eq. (21) have negative real a much smaller amplitude. Furthermore, the asyn- parts. From these considerations it follows that chronous oscillations have the characteristic that for s\s (2) the stability region 1 is bounded by (2) the external metabolite oscillates with the double the line sa and for sBs by the line sb. frequency compared to the internal substances. 4.2. Dynamical properties The occurence of stable limit cycles of two coupled oscillators 1 and 2 with phase shifts ~12 = 4.2.1. Synchronous and regular asynchronous 0or~12 =T/2 between the corresponding vari- oscillations ables is in accord with results from previous Crossing the boundary of the stability region, investigations of two coupled oscillators (Ruelle, oscillations of limit cycle type emerge. However, different types of oscillations arise from the lines sa and sb, more precisely from those parts of the lines which are boundaries of the stability region

(solid parts of the lines sa and sb in Fig. 2). Figs. 3 and 4 show for N=2 the oscillations for parameters values close to these lines. The bifurcation resulting from the cubic part F(u)ofthe characteristic Eq. (21) leads to synchronous oscillations of the cells, that is, there is no phase shift between the variables of cell 1 and the corresponding variables of cell 2 (Fig. 3). In contrast to that the bifurcation following from the quadratic part D(u) produces asynchronous oscillations of the cells (Fig. 4). In the latter case the phase shift Fig. 4. Regular asynchronous oscillations of two coupled cells. The dynamics results from a bifurcation at the line s (k) (see between the oscillations of the two cells is half of b Fig. 2). Phase shift of the oscillating cells: ~12 =T/2. Parameter the oscillation period T. A comparison of Figs. 3 values: s=1.0, k=2.50. The curves for Y , Y and Ye were and 4 shows that in the case of asynchronous 1 2 plotted starting from a time t0, where the limit cycle has been oscillations the external metabolite oscillates with reached. J. Wolf, R. Heinrich / BioSystems 43 (1997) 1–24 9

states, the regular asynchronous oscillations increases strongly with the number of interacting cells. For example, in the numerical integrations shown in Figs. 4 and 5 for N=2 and N=5, respectively, the initial values were chosen in such a way that the values of the single cells where

nearly identic. However, the time t0 where the limit cycle is reached is for N=5 about 100 times longer than for N=2. This difference is easily explained by the fact that the coordinate state of regular asynchronous oscillations has to be accomplished by very small regular fluctuations of Fig. 5. Regular asynchronous oscillations of five coupled cells. the external concentration. By similar reasoning it Phase shift of the oscillating cells: ~=T/5. The amplitudes of follows that the transient time to reach syn- e e the Y oscillations are very small (DY :0.0001) in compari- chronous oscillations, where the external concen- son with those of Yi. Parameter values are the same as given in the legend of Fig. 4. trations changes considerably, is much shorter than that for regular asynchronous oscillations. 1973; Collins and Stewart, 1993; Reick and It follows immediately from Eqs. (27a) and Mosekilde, 1995). (27b) that the region for the coupling parameter s Since the bifurcation lines shown in Fig. 2 are where regular asynchronous behaviour occurs in- independent of the number of coupled cells, creases with decreasing cell density. The lower the analogous results are found for N\2. On the cell density the higher the coupling constant has (2) solid part of the line sa where s\s the bifurca- to be for synchronization. tions always give rise to synchronous oscillations, At first sight the occurrence of regular asyn- whereas on the solid part of the line sb where chronous oscillations seems to be in contradiction sBs (2) asynchronous oscillations emerge with to the proposition that in the case of symmetric phase shifts ~ij =( j−i )T/N, where i, j=1,…, N diffusion for connected two-component systems and iBj. The broken parts of the lines are located any nonsynchronous periodic solution is unstable in the region of instability and are not of rele- (Alexander, 1986). A consideration of the systems vance to the dynamics. That means that the phase studied in Alexander (1986) shows however, that shifts follow a strong rule although the oscilla- there the systems are coupled directly to each tions are asynchronous. We denote this type of other which gives raise to coupling terms of the behaviour regular asynchronous oscillations. Previ- form Ji,k 8(Yi −Yk) where Ji,k denotes the ously, the terms anti-synchronous and anti-phase transmembrane flux of Y from cell k into cell i.In oscillations were proposed for the case of only two the model proposed here, the coupling between coupled oscillators (Kawato and Suzuki, 1980; the cells is realized in an indirect way, that is, via Alexander, 1986). However, these terms have no an external compound, which for cell suspensions well defined meaning for a high number of cells is a more realistic assumption. This adds to the where the phase shifts may attain any multiple of T/N. differential equation system for the concentrations Fig. 5 shows regular asynchronous oscillations of the intracellular variables one equation for the obtained for N=5 by numerical integration. It is external coupling substance. For such a system seen that the concentration of the external the proposition mentioned above is not valid. metabolite remains nearly constant despite the The fact, that bifurcations on the lines sa and fact that the internal metabolite concentrations sb lead to synchronous and asynchronous oscilla- vary substantially. It is worth mentioning that the tions, respectively, may be understood as follows. transient time to reach, from arbitrary initial Introducing the mean values 10 J. Wolf, R. Heinrich / BioSystems 43 (1997) 1–24 1 N X= Xi, (28a) Fig. 2 separates the regions where for fixed mean N i=%1 steady state values X( and Y( the state xi =pi =0 1 N for all cells i is stable (regions 1 and 2) or unstable Y= Yi, (28b) N i=%1 (regions 3 and 4). From that it follows that oscillations which occur at transitions from region 1 to and the differences region 2 are synchronous, that is, they retain the xi =Xi+1 −Xi, (29a) symmetry of the previous steady state. On the contrary, at transitions from region 1 to 3 this pi =Yi+1 −Yi i=1,…, N−1 (29b) symmetry is broken and asynchronous oscillations as system variables the equation system (8) may emerge. be rewritten as 4.2.2. Mixing of two cell populations dX 2 It has been shown experimentally that after =6−XY +o(xj, pk), (30a) dt mixing of two populations of cells, which are dY synchronized internally, but oscillate about 180° =XY 2 −kY−s(Y−Y e)+o(x , p ), (30b) dt j k out of phase, the oscillations are strongly damped but reappear after some time span. In the experi- dY e =s(Y−Y e), (30c) ments on oscillations in yeast cells synchroniza- dt tion takes place after a time corresponding to and about eight periods (Richard et al., 1996). Results of a simulation of such a situation are presented dx i =−(Y2x+2XYp )+o(x , p ) (30d) in Fig. 6. For tB12 the two curves show the dt i i j k sustained oscillations of the external concentration of Y in two independent populations of equal dpi 2 =(Y xi +2XYpi)−(k+s)pi +o(xj, pk), size. At t=12 the two suspensions are mixed such dt e (30e) that Y assumes the mean value of the two concentrations of the single populations. Immediately for i=1,…, N−1. o(xj, pk) denote second or after mixing the oscillation amplitude is damped higher order terms of xj and pk. It is seen that in by a factor of about three and the full synchro- Eq. (30a) and Eq. (30b) only nonlinear terms of xj and pk appear and that the right hand sides of Eq. (30d) and Eq. (30e) contain only terms which are at least of first order in these variables. Eq. (30c) is independent of the differences xj and pk. There- fore, applying a linear stability analysis to equation system (30a)–(30e) for the stationary state e (X( i =X( , Y( i =Y( =Y( , x( i =p¯ i =0) the lin- earized equation system is splitted into one system depending only on the perturbations of the mean values X, Y and Y e and into (N−1) decoupled two-variable systems depending on the perturbations of the differences xi, pi. The characteristic polynomial of equation system (30a)–(30c) corresponds to F(u) of a single Fig. 6. Mixing of two oscillating cell populations. tBtm: time e,1 e,2 oscillator given in Eq. (10). The characteristic dependent changes of the external concentrations Y and Y of the populations 1 and 2, respectively. Number of cells in polynomials of all systems resulting from Eq. e each population: N=10. t\tm: oscillations of Y of the (30d) Eq. (30e) are identic to D(u) given in Eqs. mixed population. Parameter values: s=10, k=4.15, time of

(22a), (22b) and (22c). Therefore, the line sb in mixing: tm =12. J. Wolf, R. Heinrich / BioSystems 43 (1997) 1–24 11

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 tBt1 the population is within a state of regular asynchronous oscillations. At t=t1 =20 the external concentration of the coupling substance Y is decreased by 20% and increased by the same fraction at time t=t2 =80.

nized state is attained after about nine cycles. For is decreased by 20% at time t=t1 and increased the present simulations the value of the coupling by the same fraction at time t=t2 \t1.Itis parameter s was chosen such that the time course supposed that a signal may be detected which is after mixing resembles the experimental curves directly related to the mean value Xmean of the (Fig. 1 in Richard et al. (1996)). Lower (higher) substrate concentrations Xi. Whereas Xmean is values of the coupling parameter lead to a slower nearly constant for tBt1 it is abruptly increased (faster) synchronization after mixing. The calcula- at t=t1 and shows thereafter damped oscillations tions were performed for cell populations each for tBt2. Obviously these dynamics is due to a consisting of N=10 cells. Simulations with a partly synchronization of the cells at the time of higher number of cells indicate that the Y e varia- perturbation. The damping results from the subse- tions after mixing are nearly independent on their quent restoration of the regular asynchronous number N. state. A similar phenomenon is obtained by a e sudden increase of Y at t=t2. A comparison of Fig. 7A and B shows that the average dynamics, 4.2.3. Perturbation of regular asynchronous represented by Xmean(t), differs substantialy from oscillations the dynamics of the concentrations Xi(t) within As shown in Figs. 4 and 5 this type of oscilla- the single cells. The latter concentrations oscillate tions is characterized by a nearly constant concen- with much higher amplitudes as shown for X1 in tration of the coupling substance in the external Fig. 7B. This holds true, in particular, for those medium. Therefore, it may be difficult to distin- time intervals, where the oscillations of Xmean are guish experimentally whether the individual cells negligible small. Instead of changing the external are in a steady state or in oscillatory states. This concentration Y e one could also apply a sudden holds true also if one tries to detect the oscilla- transient increase and decrease of the input rate 6 tions by recording other quantities which result of all cells for obtaining similar results. The curve from an averaging over all cells, such as by in Fig. 7A shows some correspondence to experi- NADH-fluorescence for yeast glycolysis. Such mental data, where the oscillations monitored by hidden oscillations should become transiently visi- NADH fluorescence are induced by adding glu- ble if the regular phase shifts are disturbed. Fig. 7 cose and cyanide (Ghosh et al., 1971; Aldridge shows results of simulations where in a suspension and Pye, 1976; Richard et al., 1996) and reacti- of cells in a state of regular asynchronous oscilla- vated by an acetaldehyde pulse (Richard et al., tions the concentration of the external metabolite 1996). 12 J. Wolf, R. Heinrich / BioSystems 43 (1997) 1–24

4.2.4. The case of strong coupling 2/3 V 2/3 = = 1 . One may conclude from Fig. 2, that a strong k s=0 s=0 1+ V1+V2 coupling between the cells allows only syn- (34) chronous oscillations. It may be interesting to consider the limiting case s , where Xi =X We may conclude that the frequency of the oscil- and Yi =Y. The parameter s may be eliminated lations decreases at the transition from decoupled from Eq. (8c) by adding Eq. (8a) multiplied by to strongly coupled cells. The decrease is the 1/ which results in stronger the larger the extracellular volume compared with the cell volume. It can be understood dY 1 dY e + =XY 2 −kY. (31) by the fact that for large extracellular volume the dt dt total amount of the coupling substance which has Application of the quasi-equilibrium approxima- to be exchanged must be very high. This is related tion to the equation mdY e/dt=Y−Y e with m=1/ to the deposition effect proposed by Sel’kov to be s yields Y=Y e for m0. In this way the responsible for the increase of the oscillation pe- dynamics of the whole system may be described riod by taking into account the reversible ex- by the equations change of glycolytic compound with a pool of polysaccharides (Sel’kov, 1980). dX =6−XY 2, (32a) Using for the frequency of oscillations near the dt boundary of the stability region the relations 2 2 dY =Z for two-component systems and =a1 = (XY 2 −kY). (32b) for three-component systems one may calculate dt 1+ the frequency for all sets of parameter values. This equation system differs from that of the Numerical evaluation of Z at tr=0 (see Eqs. decoupled oscillators (see equation system (8) (22a), (22b) and (22c)) and of a1 from Eq. (10) at with s=0) only in a volume dependent factor in a1a2 −a0 =0 with Eq. (13) yields (k) as shown the equation for the variable Y. A linear stability in Fig. 8. Only the solid parts of the lines are of analysis of this system reveals that the steady state is stable upon the condition that 1 6 2 1+ \k 3. (33) Surprisingly, the k-value where the steady state (2) becomes unstable equals k , where the curves sa and sb intersect (see Eq. (27a) and Fig. 2). The stability condition (33) for strongly coupled cells is volume dependent. A decrease of the cell density, that is a lowering of has a stabilizing effect. This corresponds to the experimental fact, that sustained synchronous oscillations of yeast cells are preferentially found at high cell densities (Aon et al., 1992). Fig. 8. Frequency of synchronous and regular asynchronous oscillations. The frequencies are calculated as functions of the parameter k at s-values corresponding to the bifurcation lines

4.2.5. Frequency of oscillations sa and sb. The solid lines correspond to those regions of Determining the frequency of the coupled oscil- parameter values where the synchronous and asynchronous oscillations, respectively, are stable. Point P : uncoupled case lators near the boundary of the stability region 1 (s=0); Point P : transition between stable synchronous and from the determinant of the Jacobian matrix of 2 stable asynchronous oscillations; Point P3: strong coupling Eq. (32a) and Eq. (32b) and comparing the result (s ). The curves are independent of the number N of with that obtained for s=0 yields interacting cells. J. Wolf, R. Heinrich / BioSystems 43 (1997) 1–24 13

diagram shows the steady state concentrations X( 1 H and X( 2. Beyond the bifurcation point k\k the maxima and minima of the oscillating variables H X1 and X2 are plotted. There is a region k BkB k S, where the lines for the two cells coincide

despite a phase shift of ~12 =T/2 between the variables. In this region regular asynchronous behaviour is found. It is seen that an increase of the bifurcation parameter k leads to a further symmetry breaking. For k\k S the shapes of the oscillations of X1 and X2 differ which is reﬂected by differences in the maxima and minima of these variables. Furthermore, the phase shift between Fig. 9. Bifurcation diagram of two coupled oscillators. kBkH: the variables of the two cells is no longer T/2, but steady state concentrations X ; kH k kS: maxima and min- ( i B B decreases monotonically with increasing k as ima of regular asynchronous oscillations; k\kS: maxima and minima of nonregular asynchronous oscillations. Inset A: de- shown in the inset B to Fig. 9. That means that S tailed representation of the symmetry breaking of the minima for k\k nonregular asynchronous oscillations of regular asynchronous oscillations. Inset B: Phase shifts of occur. It is worth mentioning, that at the point S the regular asynchronouos oscillations (~12 =T/2) and nonreg- k=k a single limit cycle bifurcates into two limit ular oscillations (~ T 2). Parameter values: N 2, s 1.0. 12 B / = = cycles, which may be transformed into each other by exchanging the numbers of the cells. Examples physical meaning since the region of stability is for a single limit cycle with phase shift T/2 for determined either by the cubic or by the quadratic S S kBk and for two limit cycles for k\k are part of the characteristic polynomial (Eq. (21)). shown in Fig. 10A and B, respectively. In both The points P1 and P3 correspond to the limiting cases only the trajectories of the variables X1 and cases s=0 and s . Point P2 corresponds to X2 are shown. the intersection point of the lines sa and sb, where For higher values of s, where the region of a transition between synchronous and regular stability is bound by the line sa a similar sec- asynchronous oscillations occurs. As seen from ondary bifurcation may occur by symmetry Fig. 8 this transition is accompanied by a discon- breaking. At increasing values of the bifurcation tinuity in the parameter dependency of the fre- parameter k the oscillations remain ﬁrst syn- quency. Generally, the frequency for chronous. Beyond a critical value k S the limit asynchronous oscillations of internal metabolites cycle bifurcates into two cycles. Each of them is higher than that for the synchronous oscilla- represents nonregular asynchronous oscillations. tions. S Starting with a phase shift ~12 =0 for k=k the phase shift increases with increasing values of k. 4.2.6. General aspects Only for a very strong coupling no symmetry Leaving the near neighbourhood of the bifurca- breaking bifurcations occur. tion points by increasing the bifurcation parameter k more complex dynamic phenomena may occur. Fig. 9 shows for a system of two interact- 5. Coupling via the substrate of the autocatalytic ing cells a bifurcation diagram of the variables X1 reaction (Model II) and X2 with varying parameter k. The value of the coupling parameter s was 5.1. Steady states chosen from the intervall where the line sb(k) bounds the stability region. In this case the point Whereas in model I the stationary state is of a Hopf-bifurcation k=k H is determined by Eq. unique and independent of the number of coupled (22b) with tr=0. For low k-values (kBk H), the cells, model II shows generally multiple steady 14 J. Wolf, R. Heinrich / BioSystems 43 (1997) 1–24

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

states. Using Eq. (9a) the external metabolite has X( i =X( 1 for 15i5q (38a) the steady state concentration and N e 1 2 X( = X( j. (35) k N j=%1 X( i = for q+15i5r. (38b) sX( 1 Eq. (9c) can be fulfilled by either Subtracting Eq. (37b) from Eq. (37a) for i=1 one k obtains Y( i = or Y( i =0. (36a,b) 2 X( i k X( i =X( 1 + for r+15i5N. (39) sX( We number the cells so that Y( i "0 for 15i5r 1 and Y( i =0 for r+15i5N. The case r=0is excluded since there exists no stationary state as it 5.1.1. Case 15q=r5N follows immediately from equation system (9). Introducing the two possible solutions for X( i, e Eq. (38a) and Eq. (39), into Eq. (37a) for i=1 Introducing X( from Eq. (35) and the Y( i values from Eqs. (36a,b) into Eq. (9b) leads to gives a linear equation for X( 1. Calculating then the X( i-values by using again Eq. (38a) and Eq. k 2 s N (39) and the corresponding Y( i-values yields 0=6− −sX( i + X( j, for 15i5r X( N % i j=1 rk 2 (37a) X( = , (40a) i N6 and N6 Y( i = ,15i5r, (40b) s N rk 0= −sX( + X( , for r+1 i N, 6 i % j 5 5 N j=1 N6 rk 2 (37b) X( = + , (40c) i rs N6 respectively. Subtracting Eq. (37a) for i\1 from Y( i =0, r+15i5N. (40d) that for i=1 one arrives at an equation which has two solutions for X( i as a function of X( 1.Byan With the help of Eq. (40a) Eq. (40c) and Eq. (35) appropriate numbering of the first r cells this one obtains for the steady state concentration of leads to the external metabolite J. Wolf, R. Heinrich / BioSystems 43 (1997) 1–24 15 rk 2 (N−r)6 X( e = + . (40e) Considering all possible r and q values with N6 rs qBr at a given number of cells equation system Since r may attain any integer value from 1 to N (41) yields N(N−1)/2 different solutions. To- equation system (40) gives N different solutions gether with the N solutions for r=q given in for the steady state concentrations of the metabo- equation system (40) the number of different solu- lites. They result from a situation, where r cells tions amounts to N(N+1)/2. This number of have Y( -values unequal to zero, whereas (N−r)of steady states corresponds to a special arbitrary Ncells are in a state, characterized by vanishing numbering of the cells. Further steady state result values for Y( . In the special case r=N the steady from a mere renumbering of the cells. Accord- states in all cells are the same and identic to that ingly, the total number of steady states may be for N=1. In the following this stationary state is evaluated as follows main steady state called . N r N! Z(N)= =3N −2N. % % (N r)!(r q)!q! 5.1.2. Case q r N. r=1 q=1 − − B 5 (42) Introducing the three possible solutions for X( i (Eq. (38a), Eq. (38b) and Eq. (39)) into Eq. (37a) The solutions given in Eq. (41a) Eq. (41b) Eq. for i=1 gives a quadratic equation for X( 1. From (41c) Eq. (41d) Eq. (41e) may become complex for that one obtains certain sets of kinetic parameters. Therefore, the total number of real steady state may be smaller N6 X( i = 9S, for 15i5q. (41a) than that given in Eq. (42). 2s(r q) − The possibility of generating different steady with states by coupling of cells, which have only unique steady states in the uncoupled case, is an N6 2 qk 2 S= − . (41b) interesting kinetic phenomenon. It means, more- 2s(r q) s(r q) ' − − over, that the various cells of the population differ Introducing Eqs. (41a) and (41b) for i=1 into in their dynamical properties. In the present Eq. (38b) and Eq. (39) yields model, these differences are rather extreme since for some cells the metabolism is drastically re- r−q N6 X( i = S , q+15i5r, duced. (For all cells where Y( i =0 the steady state q 2(r q)s − activities of the second and third reaction in sys- (41c) tem (1) are equal to zero, that is, the intermediate and X is not metabolized via the main pathway, but transported out of the cell.) Nr6 (2q−r)S X( i = 9 , r+15i5N, A high number of stationary states character- 2q(r−q)s q ized by Y( 0 is also obtained if one uses for the (41d) i " rate equation of the autocatalytic reaction in 2 respectively. The corresponding Y( i values follow equation system (2), instead of 62 =XY , the ex- from Eq. (36a,b). For the concentration of the pression external metabolite one derives from Eq. (35) and 6 =(h+Y 2)X, (43) Eq. (41a) Eq. (41b) Eq. (41c) Eq. (41d) 2 where h represents a normalized first order rate 6 N N (2q−r)S X( e = + −2 9 . (41e) constant characterizing the activity of that reac- s q r q q 2 − tion in the absence of Y (see (Heinrich et al.,

It is easy to see that the two solutions for X( i 1977) for the case of single cells and (Ashkenazi differing in the sign of the square roots S lead to and Othmer, 1978) for the case of two interacting physically indistinguishable steady states which cells). For h"0 the calculation of the steady only differ in the numbering of the cells. There- states for an arbitrary number of coupled cells is fore, the lower signs may be omitted. also possible and presented in Appendix B. 16 J. Wolf, R. Heinrich / BioSystems 43 (1997) 1–24

Fig. 11. Steady states for model II for the case for nonvanishing ground activity of the autocatalytic reaction. The steady state concentration 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: h=0.05, s=1.0.

Fig. 11A and 11B show for N=2 and N=3, In Appendix C it is shown that all these steady respectively, the concentrations of Y within one states are unstable. cell as functions of the bifurcation parameter k. For N=2 there are two possible types of steady 5.2.2. Case 15q=r5N states. Curve a in Fig. 11A represents the symmet- For a given number rBN the cells may be ric main steady state, where Y( 1 =Y( 2 and curve b subdivided into two groups with the steady state nonsymmetric steady states with Y( 1 "Y( 2. For solutions (40a,b) and (40c,d), respectively. The N=3 (Fig. 11B) the number of possible cases for characteristic polynomial for this state may be stationary states increases. Besides the full sym- calculated in a similar way as shown in Appendix metric main steady state (curve a) there exist two A for model I. One obtains for rBN possibilities for steady states with lower sym- r−1 N−r−1 metries. On curves b1 and b2 the concentrations E1(u) E2(u) G1(u)=0, (45) within the two cells are always identical, whereas where the concentrations of the third cell differ from 2 these. The curve c represents the nonsymmetric E1(u)=u −tr ·u+Z case, where the steady state concentration of the 2 2 2 2 2 N 6 N 6 three cells are different. It is easy to see that the =u +u 2 2 +s−k + 2 −sk, lower branches of the curves, b, b1 and b2, tend r k r k (46a) for h0toY(=0, that is, they represent steady states of model II for r N. The advantage of B E2(u)=(u+k)(u+s), (46b) considering the case h=0 is that the stability analysis may be carried out in an explicit way. and

G (u) (u k) E (u) 5.2. Stability analysis and dynamical properties 1 = + 1 ((1−r/N)s 2−(u+s)(u+s)) 5.2.1. Case 15qBr5N According to equation system (41) real steady rs 2 + (u−k)(u+s) . (46c) state solutions only exist if N n N 26 2 In these equations r/N denotes the fraction of k 2 B . (44) 4 q(r−q)s cells, where Y( i "0. J. Wolf, R. Heinrich / BioSystems 43 (1997) 1–24 17

There are 2(r−1) eigenvalues following from occur namely at tr=0 with Z\0, or at Z=0. E1(u)=0 and, for rBN,2(N−r−1) negative The corresponding functions s=sb(k) and s= eigenvalues from E2(u)=0. G1(u)=0 represents sc(k), respectively, result from Eq. (46a) with an equation of fifth order in u. One of its solution r=N. is u=−k. The other four solutions are the zeros The line which would result from the function of the fourth-order-polynomial in square brack- sb(k) is not relevant for the boundary of stability, ets. which may be seen as follows. The functions sa(k) In the case r=N, that is for the main steady and sb(k) have two intersection points. One point state, the polynomial G1(u) becomes proportional (P1) is located on the k-axis and has the same to (u+k)(u+s) such that E2(u) drops out from coordinates as the point P1 in model I (see Eq. the characteristic polynomial (45). Therefore, the (24a) and Eq. (24b)). The coordinates of the characteristic equation simplifies to second point P2 are determined by 2 N−1 6 E(u) G(u)=0 (47) k 3 = , (48a) (1+) where E(u) E (u) for r N. G(u) is the polyno- 2 = 1 = 6 mial of third order derived for the case of a single s 3 =− . (48b) 2(1+) cell (cf. Eq. (11)). P2 is located outside the region of physical mean- 5.2.3. Bifurcations of the main steady state ingful values of the parameter s. At point P1 both Bifurcations may result from the eigenvalues of lines have the same slope (dsa/dk=dsb/dk=3). the cubic or quadratic part of Eq. (47). Concern- They differ in the second derivatives, which are ing the first case condition (12a) gives together d2s 6(−1+3) a = , (49a) with Eq. (11) a relation between the parameters k dk 2 k and s for fixed values of the other parameters d2s 6 (line s in Fig. 12). Consideration of the quadratic b a 2 =− . (49b) part shows that two types of bifurcations may dk k

Since \0 the curvature of sb at s=0 is therefore always smaller than that of sa. From the slope and the curvatures of the functions sa and sb it follows that in the vicinity of the point P1 for all given values of s the corresponding k-values

on the line sb are larger than those on the line sa. From that and from the fact that these lines have no intersection point for positive parameter values

it follows that the line sb cannot be a boundary of the stability region.

The lines sc and sa intersect in the point P3 which has the coordinates 2 1/3 k (3) =6 2/3 +2− +1 +1 \0, 2 2 Fig. 12. Bifurcation lines of Model II for N=3. The lines sa ' n and s following from a a a 0 (Eq. (14)) and Z 0 (Eq. 2 c 1 2 − 0 = = 6 (46a)), respectively, represent bifurcation points of the main s (3) = \0. (50) k (3) steady state. The lines sd and sf represent bifurcation points resulting from Eq. (51a) for r N. s follows from Z 0 (Eq. B e = The region of stability is bounded either by the (46a)) with r=2. The solid parts of the lines indicate (3) (3) boundaries of stability region for the various steady states. line sa for sBs or by the line sc for s\s The broken parts of the lines are located within the regions of (solid parts of the corresponding curves in Fig. unstable steady states. 12). 18 J. Wolf, R. Heinrich / BioSystems 43 (1997) 1–24

The bifurcations which occur on the solid part At the line sd the steady state where one of the of the line sa lead to synchronous oscillations of three cells have no metabolism via the main path- the cells. In the present case asynchronous oscilla- way shows a bifurcation giving rise to syn- tions may be excluded because the line tr=0of chronous oscillations of the two other cells. A the quadratic part of the characteristic polynomial further bifurcation takes place at the line sf where (47) is not a boundary of the stability region (cf. the steady state characterized by two degenerated corresponding discussion for model I in Section cells becomes unstable. All oscillations in degener- 4.2). ated cells have the property that only the concen- (3) For s\s the main steady state becomes trations of the metabolites Xi oscillate whereas the unstable at the line sc since among the 2(N−1) concentrations Yi remain zero. eigenvalues of the quadratic part of the character- High values of the coupling parameter s:In istic polynomial N-1 real eigenvalues become pos- this case there exists the possibility that the states itive. with r\1 become unstable not via Hopf-bifurcation but due to a transition from Z\0toZB0. 5.2.4. Bifurcation of steady states with q=rBN These transitions are characterized by a symmetry Bifurcations may result from the solutions of breaking. For example, at the line sc one changes either E1(u)=0orG1(u)=0. As for the case of from the main steady state with three identical the main steady state the bifurcation line derived cells to one of the steady states where only two from tr=0 of the quadratic part is no boundary cells are identical. of the stability region. The other type of bifurcations which results from Z=0 may exist as long as r\1. The corresponding bifurcation line for 6. Discussion r=2 and N=3 is shown in Fig. 12 (line se). Since the term in brackets in G1(u) is a polyno- The systems analyzed in the present paper may mial of fourth order the conditions for the oc- be considered as minimal models for the descrip- curence of Hopf-bifurcations are tion of coupled oscillators in cell suspensions. They are based on a two-component system, a a a −a a 2 −a 2 =0, (51a) 1 2 3 0 3 1 which includes a positive feedback mechanism. a0 \0, (51b) The nonlinearity of the equations which, from the mathematical point of view, is necessary for the a 0, (51c) 1 \ emergence of oscillations is provided by a tri- a2 \0 (51d) molecular term. Recently, a more simple chemical oscillatory mechanism has been proposed which a 0 (51e) 3 \ contains only mono- and bimolecular reactions where the ai denote the coefficients of the polyno- (Wilhelm and Heinrich, 1995). However, this sys- mial (Bautin, 1949). The curves sd and sf result- tem includes three compounds. In fact, it has been ing from condition (51a) for r=q=2 and shown that any chemical oscillator with only r=q=1, respectively, are shown in Fig. 12. mass-action kinetics has to include at least three The dynamical properties of model II which are variables (Hanusse, 1972). While it is still unclear reflected by the bifurcation lines shown in Fig. 12 whether a three component oscillator with only for N=3 may be summarized as follows. one bimolecular term has any biochemical mean- Low values of the coupling parameter s: For ing the model used in the present paper reflects low values of the bifurcation parameter k all various properties of the glycolytic oscillator. steady states resulting from equation system (40) Despite its simple structure the presented model are stable. Crossing the line sa the main steady allows the investigation of various problems, state undergoes a bifurcation and limit cycles which are of relevance for the interpretation of corresponding to synchronous oscillations of all experimental data. The main results may be sum- three cells occur near the the region of stability. marized as follows: J. Wolf, R. Heinrich / BioSystems 43 (1997) 1–24 19

1. Depending on the kinetic parameters coupled 4. Complex dynamic phenomena may occur if biochemical oscillators may show synchronous the coupling opens the possibility of multiple and asynchronous oscillations. steady states as shown in model II. Then the 2. There are two different possibilities for asyn- whole population may be within different stable chronous oscillations. (a) For parameter values steady states or different oscillations states. This is near to the boundary of stability regular asyn- accompanied by distinct steady states or distinct chronous behaviour occur, where the phase shifts oscillatory states of the single cells. The conclu- between the individual cells are integer multiples sion is that cells which would behave identically if of the reciprocal value of the number of cells. The uncoupled, may show a distinct behaviour in the occurrence of these regular asynchronous oscilla- case of coupling. In a more general context this tions depends on the type of the coupling. In the reasoning may have consequences for processes of present model it is found only for the case where cell differentiation in multicellular complexes. the coupling substance belongs to the pool of In our model coupling of the cells is performed products of the autocatalytic reaction (Model I). by a metabolite, which is produced in all cells and For small values of the coupling strength or for which permeates their membranes. This type of low cell densities regular asynchronous oscilla- interaction has been proposed for many systems. tions occur in a large range of parameter values. For yeast cell populations acetaldehyde was dis- (b) Leaving the near neighbourhood of the points cussed to mediate the coupling. Sometimes, ac- of Hopf-bifurcations nonregular asynchronous etaldehyde has been reported to be a behaviour may arise by secondary symmetry desynchronizer of the oscillations (Ghosh et al., 1971), whereas other authors call this substance a breaking bifurcations, either from the branch of synchronizer (Richard et al., 1996). Without dis- synchronous oscillations or from the branch of cussing the problem whether or not acetaldehyde regular asynchronous oscillations. The mode of mediates the interaction, we want to emphasize nonregular asynchronous behaviour is character- that any substance may play both roles. As shown ized by nonidentic oscillations in different cells in Section 4.2 it depends on the kinetic parameters and by nonregular phase shifts. whether synchronous or asynchronous oscillations 3. Regular asynchronous oscillations in suspen- occur, although in both cases the coupling is sions of a large number of cells are characterized realized by the same substance, that is by the by a nearly constant concentration of the cou- product of the autocatalytic reaction. pling metabolite. It may lead to the phenomenon In some sense the models I and II show comple- of hidden oscillations, where strong variations in mentary properties with respect to symmetry the internal metabolite concentrations are not breaking. In model I there is a unique steady state reflected by changes in the extracellular medium which may bifurcate into a full symmetric limit or the mean values of internal variables. This has cycle corresponding to synchronous oscillations consequences for the experimental observability or into a limit cycle, characterized by regular of the oscillations. As shown in Section 4.2 hidden asynchronous oscillations. In model II regular oscillations may come into view for a certain time asynchronous oscillations are excluded, but a high span after perturbation of the system. A nearly number of nonsymmetric steady states arises. constant external concentration has to be distin- We want to stress that the models presented in guished from a fixed concentration of the cou- this paper are applicable to any number of inter- pling substance. In the latter case there is no acting cells, but there are fundamental differences communication between the oscillating cells and in the dynamics of model I and II concerning the phase shifts are entirely determined by the their dependence on the number of cells. The initial conditions and may change after perturba- dynamical properties of model I are nearly inde- tions of the variables. In contrast to that the pendent of N as long as steady states and limit phase shifts of regular asynchronous oscillations cycles following from the first Hopf bifurcation are stable against perturbations. are considered. The difference in the cell number 20 J. Wolf, R. Heinrich / BioSystems 43 (1997) 1–24 only affects the phase shifts for regular asyn- With a realistic metabolic model which con- chronous oscillations and the dynamics of the tain more than two metabolites one could an- external metabolites. In contrast to that there alyze the potential effects of different coupling is a dramatic increase of possible steady states substances, including those of permeable inor- with increasing cell number in model II. ganic ions if the model takes into account the A crucial simplification of the present mod- corresponding membrane transport processes. els is that the diffusion processes within the It would be interesting to investigate whether external volume are considered to be very fast our result that asynchronous oscillations only compared to the transmembrane exchange of occur if the coupling substance belongs to the the coupling substance and to the characteris- pool of products of the autocatalytic reaction tic times of the intracellular reactions. Inclu- may be generalized to more complex oscillat- sion of slow extracellular diffusion processes ing systems. However, many metabolic inter- may be an interesting subject if the cell have mediates of glycolysis may be excluded to be a definite spatial arrangement, like in tissues. intercellular messengers since they are phos- Therefore, the model applies to oscillations in phorylated and cannot leave the cell. Most of cell suspensions where the cells have no fixed the substances which have been considered as positions, for example due to stirring. candidates for signallers, especially pyruvate, Throughout the paper it was assumed that ethanol and acetaldehyde are produced in the all cells of the populations are identic. Con- lower part of glycolysis. Since the kinetic cerning the basic oscillatory mechanism this is properties of phosphofructokinase which is lo- certainly justified if all cells of the population cated in the upper part of glycolysis are es- belong to the same type. There may be of sential for the emergence of oscillations these course differences in the kinetic constants and endproducts can only coordinate the behaviour in morphological parameters, but one may ex- of the cells if the the pathway contains effi- pect that there are only slight deviations from cient feedback mechanisms. These could be the mean values. (For typical distributions of based on stoichiometric interactions, per- the cellular volume and of the membrane formed, for example, by ATP or NADH, or area, see (Svetina, 1982)). on regulatory couplings. Depending on the Despite its simplicity the models reveals cell sort there are for the latter type of inter- many interesting dynamic phenomena. At the actions many possibilities such as inhibition of present state a comparison with experimental phosphofructokinase by ATP or phospho- data could be accomplished only in a semi- enolpyruvate and inhibition of hexokinase by quantitative manner. It would be an intriguing glucose 6-phosphate. task to use recent models of anaerobic yeast cell metabolism for a more extended theoretical approach for oscillations in cell suspen- Acknowledgements sions. The rather elaborated model presented in (Galazzo and Bailey, 1990) may serve as a This work was supported by a grant of the base. It includes the main processes of glycol- Deutsche Forschungsgemeinschaft (Ref. He ysis and associated reactions and uses realistic 2049/1-2). kinetic equations for the various enzymes. It is restricted to the description of stationary states but its extension to time dependent Appendix A. The characteristic polynomial for states would be straightforward. Very detailed the eigenvalues for N coupled oscillators (Model models of glycolysis exist also for other cell I) types (Rapoport et al., 1976; Werner and Heinrich, 1985; Joshi and Palsson, 1989; The characteristic equation of model I for 1990). N interacting cells reads J. Wolf, R. Heinrich / BioSystems 43 (1997) 1–24 21

sequently to the determinants of lower order Det(N) can eventually be expressed as where J denotes the Jacobian of the equation N−1 system (8) and I the identity matrix. Eq. (A1) Det(N)=A ((N−1)B+Det(1)) (A7) takes into account that the kinetic parameters and In this equation Det(1) denotes the determinant the steady state concentrations for all cells are the of a matrix containing the elements from the first same. J has been arranged in such a way, that the three rows and columns of the full matrix. From elements of the (2i )-th and (2i+1)-th row or that it follows immediately that the factor (N− column correspond to the variables of the i-th 1)B+Det(1) is identic to the characteristic poly- cell, Xi and Yi, respectively. The first row and the e nomial F(u) of a single cell given in Eq. (10). This first column refers to the variable Y . For the leads to the characteristic polynomial of N inter- elements aij one derives acting cells given in Eq. (21). s a =−s, a = , a =s, (A2) 11 12 N 21 Appendix B. Steady states of model II for the 2 2 6 6 case of nonzero ground activity of the a22 =− , a23 =−2k, a32 = , k2 k 2 autocatalytic reaction

a33 =k−s. (A3) Using Eq. (43) the steady state conditions for Expanding the determinant in products of minors the variables Xi and Yi may be transformed into of the full determinant Det(N) and the three e nonzero elements of the last column one arrives at 0=6−kY( i −s(X( i −X( ) (B1a) the following formula and Det(N)=A N−1B+Det(N−1)A (A4) kY( X = i for i=1,…, N. (B1b) where Det(N−1) denotes the determinant of a ( i 2 h+Y( i matrix which follows from that of Eq. (A1) by deleting the last two rows and columns. Further- Inserting Eq. (B1b) into Eq. (B1a) for i=1 yields more, one derives with X( e from Eq. (35)

2 A=u −u(a22 +a33)+a22a33 −a23a32 sk (N−1)Y( 1 Y( j 0=6−kY( 1 − 2 − 2 . 2 2 N h+Y % h+Y 6 6 s ( 1 j"1 ( j =u 2 −u k−s− + 1+ , (A5) (B2) k 2 k k s 2 6 2 Subtracting Eq. (B1a) for i\1 from that for i=1 B=−a12a21(a22 −u)= +u . (A6) one obtains by consideration of Eq. (B1b) N k 2 Repeating the expansion for Det(N−1) and sub- Y( 1 Y( i Y( 1 −Y( i =−s 2− 2 . (B3) sequently to the determinants of lower order h+Y( 1 h+Y( i 22 J. Wolf, R. Heinrich / BioSystems 43 (1997) 1–24

3 4 With pi =Y( i −Y( 1 one derives from Eq. (B3) s k 2 2 2 qk a0 = rsk −(r−q)s X( 1 − 2 . (C4) 3 2 2 3 N X( 1 p i (Y( 1 +h)+p i (2Y( 1 +2hY( 1 −sY( 1)

4 2 2 2 In the case r=N the polynomial reads +pi(Y( 1 +2hY( 1 −sY( 1 +h +hs)=0 (B4) q−1 r−q−1 + E1,a(u) E1,b(u) G3(u)=0, (C5) which has the solutions pi =0, pi =p and pi = − u w p . Denoting by , 6 and the numbers of cells where G3(u) is a polynomial of ﬁfth degree with with these three solutions, respectively, one s 2 qk 4 derives from Eq. (B2) with u+6+w=N−1 2 2 2 a0 = Nsk −(N−q)s X( 1 − 2 . (C6) N X( 1 k=6 Y( 1 One may distinguish four different cases: (A) q\ 1 and r−q 1, (B) q=1 and r−q 1, (C) q 1 + \ \ \ s (N−1−u)Y( 1 6(Y( 1 +p ) and r−q=1, (D) q=1 and r−q=1. − 2 − + 2 N h+Y( 1 h+(Y( 1 +p ) Case (A): There are four eigenvalues resulting − −1 w(Y( +p ) from the two quadratic equations E1,a(u)=0 and − 1 . (B5) h+(Y( +p−)2 E1,b(u)=0. For the real parts of these eigenvalues 1 n to be negative it is necessary that This equation may be used to calculate for all 2 possible values of u, 6 and w the rate constant k as k 2 −s\0 (C7a) a function of Y( 1. This yields for N=2 and N=3 X( 1 the curves given in Fig. 11A,B. and

2 2 s X( 1 −s\0. (C7b) Appendix C. Stability analysis of steady states k 2 characterized by qBr (Model II) It is obvious that these conditions can not be fulfilled simultaneously. We prove that the steady states given in Eq. Case (B): There are two eigenvalues resulting (41a) Eq. (41b) Eq. (41c) Eq. (41d) Eq. (41e) for from the quadratic equation E (u)=0 and some 1 q r are always unstable. The characteristic 1,b 5 B eigenvalues resulting from G (u)=0 (seven eigen- equation for these steady states may be derived by 2 values for rBN and five eigenvalues for r=N). a similar method as shown for model I in Ap- For the latter eigenvalues to have negative real pendix A. For rBN one obtains parts it is necessary that a0 \0 with a0 given in q−1 r−q−1 N−r−1 E1,a(u) E1,b(u) E2(u) G2(u)=0, Eq. (C4) and Eq. (C6), respectively. For r5N the (C1) condition a0 \0 is fulfilled if 2 2 with k 2 k BX( 1 B . (C8) k 2 k 2 (r−1)s s E (u)=u 2 +u +s−k +k −s , 1,a 2 2 This is in contradiction to condition (C7b) for X( 1 X( 1 (C2) negative real parts of the eigenvalues resulting from E (u)=0. s 2X( 2 s 2X( 2 1,b E (u)=u 2 +u 1 +s−k +k 1 −s , Case (C): There are two eigenvalues resulting 1,b k 2 k 2 from the quadratic equation E (u)=0 and some (C3) 1,a eigenvalues resulting from G2(u)=0 (seven eigen- and E2(u) which has the same form as given in values for rBN and five eigenvalues for r=N). Eq. (46b). The function G2(u) is here a polyno- For the latter eigenvalues to have negative real mial of seventh degree. It turns out that for parts it is necessary that a0 \0 with a0 given in stability analysis only the term a0 is relevant. It Eq. (C4) and Eq. (C6), respectively. For r5N the reads condition a0 \0 is fulfilled if J. Wolf, R. Heinrich / BioSystems 43 (1997) 1–24 23

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