We consider some mathematical issues raised by the modelling of gene
networks. The expression of genes is governed by a complex set of regulations,
which is often described symbolically by interaction graphs. Once such a graph
has been established, there remains the difficult task to decide which
dynamical properties of the gene network can be inferred from it, in the
absence of precise quantitative data about their regulation. In this paper we
discuss a rule proposed by R.Thomas according to which the possibility for the
network to have several stationary states implies the existence of a positive
circuit in the corresponding interaction graph. We prove that, when properly
formulated in rigorous terms, this rule becomes a theorem valid for several
different types of formal models of gene networks. This result is already known
for models of differential or boolean type. We show here that a stronger
version of it holds in the differential setup when the decay of protein
concentrations is taken into account. This allows us to verify also the
validity of Thomas' rule in the context of piecewise-linear models and the
corresponding discrete models. We discuss open problems as well.Comment: To appear in Notes Comptes-Rendus Acad. Sc. Paris, Biologi
