Relations between gene regulatory networks and cell dynamics in Boolean models

An asynchronous Boolean dynamics to some extent represents the joint evolution of a system of Boolean-discretized variables. In a biological context, these kinds of objects are used to model the evolution of the gene expression levels. With such a dynamics, one can associate a (genetic) regulatory graph summarizing the influence of each variable on the others. The first of Thomas's rules, formally proved in particular in the asynchronous Boolean framework, states that the presence of several stationary states in a dynamics arises only if the corresponding regulatory graph contains a positive feedback loop. In the present work, we first give a necessary condition for the presence of a single stationary state in a dynamics and next derive a necessary condition for multistationarity which is slightly stronger than that required in the first of Thomas's rules. Next, we reverse the approach and study the properties of dynamics corresponding to a particular class of regulatory graphs, that are made up of several circuits sharing a common component. We prove that the corresponding dynamics contains at most two stationary states and give more specific results for when the regulatory graphs contain less than two positive (resp. negative) circuits. Moreover, we show that the behavior of a dynamics cannot be predicted if its regulatory graph contains both at least two positive circuits and two negative circuits (all sharing a common component). In particular, it may contain zero, one or two stationary states.

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