Computation of switch time distributions in stochastic gene regulatory networks

Many gene regulatory networks are modeled at the mesoscopic scale, where chemical populations are assumed to change according a discrete state (jump) Markov process. The chemical master equation (CME) for such a process is typically infinite dimensional and is unlikely to be computationally tractable without further 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. In this paper, we show that this "error" has far more significance than simply the distance between the approximate and exact solutions of the CME. In particular, we show that apart from its use as a measure for the quality of approximation, this error term serves as an exact measure of the rate of first transition from one system region to another. We demonstrate how this term may be used to directly determine the statistical distributions for stochastic switch rates, escape times, trajectory periods, and trajectory bifurcations. We illustrate the benefits of this approach to analyze the switching behavior of a stochastic model of Gardner\´s genetic toggle switch.