Transient analysis of stochastic switches and trajectories with applications to gene regulatory networks

Many gene regulatory networks are modelled at the mesoscopic scale, where chemical populations change according to a discrete state (jump) Markov process. The chemical master equation (CME) for such a process is typically infinite dimensional and unlikely to be computationally tractable without reduction. The recently proposed finite state projection (FSP) technique allows for a bulk reduction of the CME while explicitly keeping track of its own approximation error. In previous work, this error has been reduced in order to obtain more accurate CME solutions for many biological examples. Here, it is shown that this 'error' has far more significance than simply the distance between the approximate and exact solutions of the CME. In particular, the original FSP error term serves as an exact measure of the rate of first transition from one system region to another. As such, this term enables one to (i) directly determine the statistical distributions for stochastic switch rates, escape times, trajectory periods and trajectory bifurcations, and (ii) evaluate how likely it is that a system will express certain behaviours during certain intervals of time. This article also presents two systems-theory based FSP model reduction approaches that are particularly useful in such studies. The benefits of these approaches are illustrated in the analysis of the stochastic switching behaviour of Gardner's genetic toggle switch.