Non-linear aspects of dynamic pattern in cellular networks

Abstract

Succession of one process-generated pattern by another is governed by non-linear transformation and transport processes, exemplified by chemical reactions within well-mixed cells connected by permeable walls into a regular network. Earlier work on the onset of instability is extended to encompass finite-amplitude disturbances of a uniform stationary state. Multiple stationary states of prescribed symmetries in a sixteen-cell network are found and their stability with respect to symmetric disturbances is examined by means of elementary concentration patterns—a “symmetry-adapted basis”—and their interaction properties, as described by direct products of the irreducible representations of the symmetry group of the network. Group theoretic symmetry considerations ease analysis of emerging dynamic pattern after changing reaction and diffusion parameters turn a uniform state unstable, as a simple example indicates. The paper is a step in the direction of mapping the topography of the physico-chemical state space of a developing system, a space corresponding to Waddington's concept of an epigenetic landscape.