Discrete time piecewise models of genetic regulatory networks: single cell and population dynamics.

We introduce simple models of genetic regulatory networks and we proceed to the mathematical analysis of their dynamics. The models are discrete time dynamical systems generated by piecewise and contracting mappings whose variables represent gene expression levels. When compared to other models of regulatory networks, these models have an additional parameter which is identified as quantifying interaction delays. In spite of their simplicity, their dynamics presents a rich variety of behaviours. This phenomenology is not limited to piecewise models but extends to smooth nonlinear discrete time models of regulatory networks. After a short review of the general properties of the dynamics of such type of models we shall concentrate on a specific biological question: the allelic exclusion in the recombination mechanism of the Tbeta chains during the maturation of the T cells of the immune system. This example shows the relevance of considering a model of the histories of single cells followed by the analysis of the dynamics of populations of such cells.