Stochastic models for chemically reacting systems using polynomial stochastic hybrid systems

Abstract

Abstract. A stochastic model for chemical reactions is presented, which represents the population of various species involved in a chemical reaction as the continuous state of a polynomial Stochastic Hybrid System (pSHS). pSHSs correspond to stochastic hybrid systems with polynomial continuous vector fields, reset maps, and transition intensities. We show that for pSHSs, the dynamics of the statistical moments of its continuous states, evolves according to infinite-dimensional linear ordinary differential equations (ODEs), which can be approximated by finite-dimensional nonlinear ODEs with arbitrary precision. Based on this result, a procedure to build this types of approximation is provided.This procedure is used to construct approximate stochastic models for a variety of chemical reactions that have appeared in literature. These reactions include a simple bimolecular reaction, for which one can solve the master equation; a decaying-dimerizing reaction set which exhibits two distinct time scales; a reaction for which the chemical rate equations have a continuum of equilibrium points; and the bistable Schögl reaction. The accuracy of the approximate models is investigated by comparing with Monte Carlo simulations or the solution to the Master equation, when available.

A Derivative Matching Approach to Moment Closure for the Stochastic Logistic Model

“…Another direction for future research is the extension of the results in this paper to multi-species birth-death Markov processes. Primary results on this topic can be found in Hespanha and Singh (2005).…”

Symmetry-Based Model Reduction for Approximate Stochastic Analysis

“…In practice, this corresponds to building a new set of differential equations, one for each moment, and can be performed in several ways. For example, when the rate functions associated with chemical reactions are polynomial, moment equations may be calculated using the moment-generating function [10], or equivalently adopting a generator operator [18] or by a probability generating function [33]. Extensions to more general cases may require more sophisticated methods, e.g.…”

“…Traditionally they were derived manually. Recently, methods for automatic MC derivation were proposed [10,18,33] for biochemical reaction systems.…”

Stochastic hybrid systems for studying biochemical processes

“…However, equation (2.1) can be modified to allow the continuous state x to evolve according to a stochastic differential equation (SDE) rather than an ODE (Hespanha & Singh 2005). Another straightforward extension of the above SHS is to include deterministic transitions between discrete states, where a transition is triggered when a certain 'guard condition' is satisfied (Hespanha 2005).…”

“…Recent work has proposed a novel moment closure technique based on derivative matching, where the moment closure is done by matching time derivatives of the exact (not closed) moment equations with that of the approximate (closed) moment equations at some initial time t 0 and set of initial conditions (Hespanha & Singh 2005). In particular, this derivative-matching approach attempts to determine nonlinear functions 4 for which…”