  At the scale of the individual cell, protein production is a stochastic
process with multiple time scales, combining quick and slow random steps with
discontinuous and smooth variation. Hybrid stochastic processes, in particular
piecewise-deterministic Markov processes (PDMP), are well adapted for
describing such situations. PDMPs approximate the jump Markov processes
traditionally used as models for stochastic chemical reaction networks.
Although hybrid modelling is now well established in biology, these models
remain computationally challenging. We propose several improved methods for
computing time dependent multivariate probability distributions (MPD) of PDMP
models of gene networks. In these models, the promoter dynamics is described by
a finite state, continuous time Markov process, whereas the mRNA and protein
levels follow ordinary differential equations (ODEs). The Monte-Carlo method
combines direct simulation of the PDMP with analytic solutions of the ODEs. The
push-forward method numerically computes the probability measure advected by
the deterministic ODE flow, through the use of analytic expressions of the
corresponding semigroup. Compared to earlier versions of this method, the
probability of the promoter states sequence is computed beyond the naive mean
field theory and adapted for non-linear regulation functions.
