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

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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] 0303-2647/97/$17.00 © 1997 Elsevier Science Ireland Ltd. All rights reserved. PII S0303-2647(97)01688-2 2 J. Wolf, R. Heinrich / BioSystems 43 (1997) 1–24 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 g 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 g −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 g charac- we demonstrate numerical simulations of oscillat- terizes the strength of the product activation. It ing cell suspensions
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