Graphic Requirements for Multistationarity

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. Ren\'e Thomas conjectured twenty years ago that, if $F$ has at least two non degenerate 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, we get this way 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.