attractive cycles in a.pdf

Graphic requirements for multistability and attractive cycles in a Boolean dynamical framework élisabeth Remy?, Paul Ruet?, Denis Thieffry? 1 ?CNRS - Institut de Mathématiques de Luminy, UMR 6206 Campus de Luminy, Case 907, 13288 Marseille Cedex 9 (France). Email: {remy,ruet}@iml.univ-mrs.fr ?Technologies Avancées pour le Génome et la Clinique, UMR 628, INSERM, Université de la Méditerranée, Campus de Luminy, Case 928, 13288 Marseille Cedex 9 (France). Email: thieffry@tagc.univ-mrs.fr Abstract To each Boolean function f : {0, 1}n → {0, 1}n and each x ∈ {0, 1}n, we associate a signed directed graph G(x), and we show that the existence of a positive circuit in G(x) for some x is a necessary condition for the existence of several fixed points in the dynamics (the sign of a circuit being defined as the product of the signs of its edges), and that the existence of a negative circuit is a necessary condition for the existence of an attractive cycle. These two results are inspired by rules for discrete models of genetic regulatory networks proposed by the biologist R. Thomas. The proof of the first result is modelled after a recent proof of the discrete Jacobian conjecture. Key words: Discrete dynamical systems, Boolean networks, Regulatory networks, Genetic regulation, Differentiation, Homeostasis, Thomas’ rules, Discrete Jacobian matrix, Jacobian conjecture PACS: 37N25, 39A12, 92C40, 92D10 1 Research partly supported by the French Ministry of Research (ACI IMPbio) and the Agence Nationale de la Recherche (ANR-05-JCJC-0126-01). A preliminary ver- sion of this article appears as preprint IML 2005-08 of the Institut de Mathématiques de Luminy. Preprint submitted to Elsevier 20 December 2007 1 Introduction This article deals with properties of Boolean dynamical systems raised in the context of biology. Biologists often represent the results of their genetic and molecular investiga- tions in terms of graphs. General information on molecular biology may be found in [2] or in t