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International audienceR. 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

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René Thomas’ discrete modelling of gene regulatory networks (GRN) is a well-known approach to study the dynamics resulting from a set of interacting genes. It deals with some parameters which reflect the possible targets of trajectories. Those parameters are a priori unknown, but they may generally be deduced from a well-chosen set of biologically observed trajectories. Besides, it neglects the time delays for a gene to pass from one level of expression to another one. The purpose of this paper is to show that we can account for time delays of increasing or decreasing expression levels of genes in a GRN, while preserving powerful enough computer-aided reasoning capabilities. We designed a more accurate abstraction of GRN where delays are now supposed to be non-null unknown new parameters. We show that such models, together with hybrid model-checking algorithms, make it possible to obtain some results about the behaviour of a network of interacting genes, since dynamics depend on the respective values of the parameters. The characteristic of our approach is that, among possible execution trajectories in the model, we can automatically find out both viability cycles and absorption in capture basins. As a running example, we show that we are able to discriminate between various possible dynamics of mucus production in the bacterium Pseudomonas aeruginosa.