The Finite State Projection Approach for the Analysis of Stochastic Noise in Gene Networks

Abstract

In order to capture important subcellular dynamics, researchers in computational biology have begun to turn to mesoscopic models in which molecular interactions at the gene level behave as discrete stochastic events. While the trajectories of such models cannot be described with deterministic expressions, the probability distributions of these trajectories can be described by the set of linear ordinary differential equations known as the chemical master equation (CME). Until recently, it has been believed that the CME could only be solved analytically in the most trivial of problems, and the CME has been analyzed almost exclusively with kinetic Monte Carlo (KMC) algorithms. However, concepts from linear systems theory have enabled the finite state projection (FSP) approach and have significantly enhanced our ability to solve the CME without resorting to KMC simulations. In this paper, we review the FSP approach and introduce a variety of systems-theory-based modifications and enhancements to the FSP algorithm. Notions such as observability, controllability, and minimal realizations enable large reductions and increase efficiency with little to no loss in accuracy. Model reduction techniques based upon linear perturbation theory allow for the systematic projection of multiple time-scale dynamics onto a slowly varying manifold of much smaller dimension. We also present a powerful new reduction approach, in which we perform computations on a small subset of configuration grid points and then interpolate to find the distribution on the full set. The power of the FSP and its various reduction approaches is illustrated on few important models of genetic regulatory networks.Index Terms-Chemical master equation, gene regulatory network, linear systems, Markov process, model reduction, stochastic process, systems biology./ . During his undergraduate and graduate work, he was with the Penn State Rotorcraft Center of Excellence to develop detailed aeroelastic models with which to analyze helicopter noise and vibrations. While at the University of California, Santa Barbara, he has developed a stochastic model of the Pap Pili epigenetic switch in E. coli, and he formulated the finite state projection solution for the chemical master equation. His main research interests include applying systems theory to the modeling and analysis of stochastic gene regulatory networks.

Direct Solution of the Chemical Master Equation Using Quantized Tensor Trains

“…A stochastic model of the toggle switch was considered in [62] and consists of the following four reactions: where and represent the two repressors. Denote the species counts of each by and , respectively.…”

“…The stochastic model admits a bimodal stationary distribution over a wide range of values of the rate constants. We consider the set of parameters from [62] which were selected to test the efficiency of using available numerical algorithms to calculate matrix exponentials to solve low dimensional FSP approximations of the CME. We then scaled the parameters so that a larger set of states would be required to guarantee an FSP truncation with low approximation error.…”

“…Effectively, this reduces the states in the truncation from to , a reduction of about a third. A similar truncation was used for this model in [62].…”

Temperature Control of Fimbriation Circuit Switch in Uropathogenic Escherichia coli: Quantitative Analysis via Automated Model Abstraction

“…One approach toward addressing such challenges is the ongoing development of advanced analytical and numerical approximation methods—whether with respect to time (e.g., tau-leaping [60],[134]), state space (e.g., finite state projection [135],[136]), or other system variable—that are capable of significantly accelerating the analysis of master equation-type models to within a specified level of precision. This potentially makes feasible accurate computational analysis of molecular dynamics behind physiologically-meaningful biological networks that are otherwise too demanding for exact kinetic simulations (as, for example, is the case with bacterial systems grown in rich media or other such initial/external conditions).…”

Biochemical fluctuations, optimisation and the linear noise approximation

“…Both analytic and numerical solutions of this chemical master equation are difficult to obtain for most biologically relevant systems. Even though there exist methods to numerically solve the master equation [6] these are only feasible for relatively simple systems. A popular substitute is to apply Gillespie's stochastic simulation algorithm [7] to calculate single trajectories of the system's dynamics.…”