Piecewise affine models of regulatory genetic networks: Review and probabilistic interpretation

Abstract

A formalism based on piecewise-affine (PWA) differential equations has been shown to be well-suited to modelling genetic regulatory networks. In this paper, we first review some results concerning the qualitative study of these models: we partition the phase space into domains bounded by the threshold hyperplanes. Inside each domain, the system is affine. To define solutions on the surfaces of discontinuity, we use the approach of Filippov, which extends the vector field to a differential inclusion. We obtain a transition graph, describing qualitatively the possible transitions of solutions between domains. In a second part of the paper, we give a new probabilistic interpretation of these transitions, by computing the proportion of the volume of the domain that crosses to one of its adjacent domains.We apply this idea to the model of the bistable switch and to parameter estimation from experimental transition probabilities. © 2010 Springer-Verlag Berlin Heidelberg.