Piecewise-Linear Models of Genetic Regulatory Networks: Theory and Example

International audienceThe experimental study of genetic regulatory networks has made tremendous progress in recent years resulting in a huge amount of data on the molecular interactions in model organisms. It is therefore not possible anymore to intuitively understand how the genes and interactions together influence the behavior of the system. In order to answer such questions, a rigorous modeling and analysis approach is necessary. In this chapter, we present a family of such models and analysis methods enabling us to better understand the dynam-ics of genetic regulatory networks. We apply such methods to the network that underlies the nutritional stress response of the bacterium E. coli. The functioning and development of living organisms is controlled by large and complex networks of genes, proteins, small molecules, and their interactions, so-called genetic regulatory networks. The study of these networks has recently taken a qualitative leap through the use of modern genomic techniques that allow for the simultaneous measurement of the expression levels of all genes of an organism. This has resulted in an ever growing description of the interactions in the studied genetic regulatory networks. However, it is necessary to go beyond the simple description of the interactions in order to understand the behavior of these networks and their relation with the actual functioning of the organism. Since the networks under study are usually very large, an intuitive approach for their understanding is out of ques-tion. In order to support this work, mathematical and computer tools are necessary: the unambiguous description of the phenomena that mathematical models provide allows for a detailed analysis of the behaviors at play, though they might not exactly represent the exact behavior of the networks. In this chapter, we will be mostly interested in the modeling of the genetic reg-ulatory networks by means of differential equations. This classical approach allows precise numerical predictions of deterministic dynamic properties of genetic regu-latory networks to be made. However, for most networks of biological interest the application of differential equations is far from straightforward. First, the biochemi-cal reaction mechanisms underlying the interactions are usually not or incompletel