Necessary conditions for multistationarity in discrete dynamical systems

Adrien Richard (1, 2) , Jean-Paul Comet (2)

1 IBIS - Modeling, simulation, measurement, and control of bacterial regulatory networks
2 IBISC - Informatique, Biologie Intégrative et Systèmes Complexes

R. Thomas conjectured, 20 years ago, that the presence of a positive circuit in the interaction graph of a dynamical system is a necessary condition for the presence of several stable states. Recently, E. Remy et al. stated and proved the conjecture for Boolean dynamical systems. Using a similar approach, we generalize the result to discrete dynamical systems, and by focusing on the asynchronous dynamics that R. Thomas used in the course of his analysis of genetic networks, we obtain a more general variant of R. Thomas' conjecture. In this way, we get a necessary condition for genetic networks to lead to differentiation.

Mots clés: Discrete dynamical system; Discrete Jacobian matrix; Interaction graph; Multistationarity; Positive circuit

Domaines: Mathématique discrète [cs.DM]

Adrien Richard, Jean-Paul Comet. Necessary conditions for multistationarity in discrete dynamical systems. Discrete Applied Mathematics, 2007, 155 (18), pp.2403--2413. ⟨10.1016/j.dam.2007.04.019⟩. ⟨hal-00967300⟩