Graphic Requirements for Multistationarity
By Christophe Soulé. ComPlexUs, 2004
268 citations recorded by the source corpus.
We discuss properties which must be satisfied by a genetic network in order for it to allow differentiation. These conditions are expressed as follows in mathematical terms. Let F be a differentiable mapping from a finite dimensional real vector space to itself. The signs of the entries of the Jacobian matrix of F at a given point a define an interaction graph, i.e. a finite oriented finite graph G(a) where each edge is equipped with a sign. René Thomas conjectured 20 years ago that if F has at least two nondegenerate zeroes, there exists a such that G(a) contains a positive circuit. Different authors proved this in special cases, and we give here a general proof of the conjecture. In particular, in this way we get a necessary condition for genetic networks to lead to multistationarity, and therefore to differentiation. We use for our proof the mathematical literature on global univalence, and we show how to derive from it several variants of Thomas rule, some of which had been anticipated by Kaufman and Thomas.
Subfields: Biophysics, Cell Biology, Molecular Biology
View Paper: https://www.semanticscholar.org/paper/3c75d5fc94985f7ad8b1e0fc5ed13a0952199ff7
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Citations in the corpus
- A methodology for the structural and functional analysis of signaling and regulatory networks (2006, background): “Jacobian matrix and interaction graph Several works have highlighted the strong relationships between interaction graphs and the Jacobian matrix J, the latter obtained from a dynamical model of the network under investigation [10,35,39].”