Hybrid models of genetic networks: Mathematical challenges and biological relevance

We review results concerning dynamics in a class of hybrid ordinary differential equations which incorporates logical control to yield piecewise linear equations. These equations relate qualitative features of the structure of networks to qualitative properties of the dynamics. Because of their simple structure, they have been studied using techniques from discrete mathematics and nonlinear dynamics. Initially developed as a qualitataive description of gene regulatory networks, many generalizations of the basic approach have been developed. In particular, we show how this qualitative approach may be adapted to switching biochemical systems without degradation, illustrated by an example of a motif in which two branches of a pathway may be regulated differently when the thresholds for the two pathways are separated.