Coupled Oscillator Models for a Set of Communicating Cells

Abstract

We investigate how the structure of cell to cell interaction may affect the periodic behavior of a system of coupled oscillatory gene networks. Using the cellular circadian clock model of Goldbeter ’95 as a cellular model, we propose a general model for coupling these cellular models via transfer of substrate from cell to cell.

We proposes a system of coupled oscillators whose dynamics are determined by multiple independent cellular models and a network of cell-cell communication links acting as perturbation forces on neighboring cell states. We will assume that no production or consumption occurs other than through the mechanisms of each individual cell; hence our model is a closed system without external stimuli. We are interested in the how the cell-cell communication network may contribute to the stability of the composite system. Under the assumption of conserved concentration levels found in the system, the nature of cell-cell interactions may be summarized by geometric properties of stochastic matrices.

The assumption of conservation of substrate allows us to describe an affine condition on the coupling model; a subclass of the affine condition is connected to stochastic matrices and finite state Markov Chains and suggests how the transient states of the Markov Chain act as pumps and work to entrain the system.

In the paper we demonstrate the diversity of dynamics which the general model for coupled oscillator system is capable.