Spatial Differentiation and Positive Circuits in a Discrete Framework

  • Authors:

  • Anne Crumière;Paul Ruet

  • Affiliations:

  • CNRS - Institut de Mathématiques de Luminy, Campus de Luminy, Case 907, 13288 Marseille Cedex 9 (France);CNRS - Institut de Mathématiques de Luminy, Campus de Luminy, Case 907, 13288 Marseille Cedex 9 (France)

  • Venue:
  • Electronic Notes in Theoretical Computer Science (ENTCS)
  • Year:
  • 2008

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Abstract

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.