Exact Computation of Velocity Field and Probability Flux of Time-Evolving Probability Landscape of Stochastic Networks

Abstract

Stochastic networks play important roles in cellular processes such as gene regulation, maintenance of epigenetic states, and decision of cellular fate. The discrete chemical master equation (dCME) provides a general framework for studying stochasticity in cellular mesoscopic systems. The time-evolving probabilistic landscape governed by dCME gives a comprehensive picture of the stochastic behavior of the network. Probability flux and entropy production of stochastic network have been previously studied using Fokker-Plank equation based on Gaussian processes. Here we formulate the time-dependent evolution of the probability landscape as a general diffusion process [1], and construct the velocity field and the probability flux based on exactly computed time-evolving probabilistic landscape. We dispense with Gaussian approximations and the truncation of the jump operator necessary when using the Fokker-Plank equation. We computed the dynamics of probability fluxes and vector fields over the full state space across different time scale. We showed details of the time-evolving probability fluxes and vector fields for the ubiquitous birth-death process, the bistable Schlogl process, the oscillatory Schnakenberg model, and the toggle switch model at different time points using the exact ACME algorithm [2]. Our results identify important cellular states, characterize high probability paths and trajectories along which the probability mass of the network state evolve. Furthermore, we provide exact assessment of dissipative entropy production associated with the complex diffusion process.[1] Eisenberg, Bob et al. The Journal of Chemical Physics 133.10 (2010): 104104.[2] Cao, Youfang et al. BMC Systems Biology 2.1 (2008): 30.