Time dependent stochastic mRNA and protein synthesis in piecewise-deterministic models of gene networks

We discuss piecewise-deterministic approximations of gene networks dynamics. These approximations capture in a simple way the stochasticity of gene expression and the propagation of expression noise in networks and circuits. By using partial omega expansions, piecewise deterministic approximations can be formally derived from the more commonly used Markov pure jump processes (chemical master equation). We are interested in time dependent multivariate distributions that describe the stochastic dynamics of the gene networks. This problem is difficult even in the simplified framework of piecewise-deterministic processes. We consider three methods to compute these distributions: the direct Monte-Carlo; the numerical integration of the Liouville-master equation; and the push-forward method. This approach is applied to multivariate fluctuations of gene expression, generated by gene circuits. We find that stochastic fluctuations of the proteome and, much less, those of the transcriptome can discriminate between various circuit topologies.