Discrete and Continuous Models of Probability Flux on Switching Dynamics: A Case Study of the Toggle-Switch System

Abstract

Stochasticity plays important roles in gene regulatory networks. While the discrete Chemical Master Equation (dCME) provides a fundamental framework for studying stochastic gene regulatory network systems, its continuous approximations such as Fokker-Planck and Langevin models are widely used. While deterministic flux is generally not applicable for stochastic networks, we introduce here a new deterministic approach called the Liouville flux model. Based on the law of mass action, it allows the probability flux to be computed at every state using pre-specified probability distribution. With these models, a challenging but important task is to assess the limitations and applicability of these approximations in a consistent manner. Here we report results in the analysis of the switching dynamics of the toggle-switch system using three classes of flux models: the Liouville flux model, the Fokker-Planck flux model constructed from the Kramers-Moyal expansion of the dCME, and the discrete flux model we developed that enables construction of the global time-evolving and steady state flow maps of probability fluxes in all directions at every microstate. We discuss the details of the differences using these flux models.