Graphic requirements for multistability and attractive cycles in a Boolean dynamical framework |
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Authors: | lisabeth Remy Paul Ruet Denis Thieffry |
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Institution: | aCNRS, Institut de Mathématiques de Luminy, UMR 6206, Campus de Luminy, Case 907, 13288 Marseille Cedex 9, France;bTechnologies Avancées pour le Génome et la Clinique, UMR 628, INSERM, Université de la Méditerranée, Campus de Luminy, Case 928, 13288 Marseille Cedex 9, France |
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Abstract: | To each Boolean function 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. |
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Keywords: | Discrete dynamical systems Boolean networks Regulatory networks Genetic regulation Differentiation Homeostasis Thomas' rules Discrete Jacobian matrix Jacobian conjecture |
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