Dynamics of two-component biochemical systems in interacting cells; synchronization and desynchronization of oscillations and multiple steady states.

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.