Computing Landscape, Velocity and Flux of Probability Mass in Gene Regulatory Networks

Abstract

Gene regulatory networks are collections of molecules which regulate gene expression, namely, the levels of messenger RNAs (mRNAs) and their protein products. Gene regulatory networks control essential cellular processes, including cellular differentiation, cell fate, signal transduction, and metabolic regulations. Modeling of gene regulatory networks is very important. Many computational models have been developed for modeling gene regulatory networks. Deterministic models study the processes in gene regulatory networks under assumptions of the higher species concentrations. However, when the copy numbers of molecules involved are small and the differences in reaction rates are large, stochasticity plays an important role in cellular processes governed by these networks. The discrete Chemical Master Equation (dCME) provides a fundamental framework for studying the underlying stochastic processes of biological networks. However, directly solving the dCME is challenging due to the enormous size of the state space. Therefore it is necessary to truncate the state space, thus, the accuracy of such truncation needs to be assessed. Here a new technique for effective state space enumeration, based on the introduction of multiple buffer queues for molecular species is proposed. It allows a priori evaluation of the truncation error for each of these buffer queues and computing the error bound of the solution. The method for accurate solution of dCME allows to model behavior of gene regulatory networks in highly stochastic regime. Particularly, stochastic modeling suggests the appearance of phenotypic switch from slow promoter binding, which can lead to distinct expression levels with considerable lifetime. Due to the challenges with solving of dCME, these issues are still not well understood. In this work the multistability in the feed-forward loop (FFL), where feedback and cooperativities are absent, is studied in detail. The extensive exploration of the the full parameter space of reaction rates, including regulations intensities, and the number of genes involved, provides global phase diagrams of the multistable properties of FFL. Furthermore, the studies of the sensitivities of the regulation intensities in FFL suggest that the parameter sensitivities depend on the number of peaks in this network. While the time-evolving probability landscape or equivalently reaction trajectories define the overall stochastic behavior of the systems, vector fields of probability flux and probability velocity can further characterize time-varying and steady-state stochastic properties of the systems, including the degree of departure from the detailed balance at the steady-state for non-equilibrium systems, high probability paths, barriers, and checkpoints between different stable regions in multistable systems, as well as mechanisms of dynamic switching among different cellular states. Conventional probability fluxes on continuous space are ill-defined and are problematic when at boundaries of the state space or when copy…