Spatial Differentiation and Positive Circuits in a Discrete Framework

The biologist R. Thomas has enounced a rule relating multistationarity in a system of genes interacting in a single cell to the existence of a positive circuit in the regulatory graph of the system. In this paper, we address the question of a similar rule for spatial differentiation. We consider the interactions of genes in several biological cells located on a 1-dimensional infinite grid, and we assume that the expression levels of genes are Boolean. We show that the existence of a positive circuit is a necessary condition for a specific form of multistationarity, which naturally corresponds to spatial differentiation.