Accurate hybrid stochastic simulation of a system of coupled chemical or biochemical reactions

TLDR

Exact solutions of well‑mixed stochastic chemical systems via the Master equation can be obtained with the stochastic simulation algorithm, but the cost grows with reaction frequency, making fast reactions expensive to simulate, and hybrid methods are broadly applicable to systems governed by stochastic differential, ordinary differential, and Master equations. The study proposes a hybrid stochastic method that partitions reactions into fast and slow subsets, approximating fast reactions with a chemical Langevin equation and modeling slow reactions with the Next Reaction algorithm, aiming to efficiently monitor discrete slow events while simulating continuous dynamics. The method accelerates simulation by allowing multiple slow reactions per time step of the chemical Langevin integration, yielding substantial speed gains with minimal loss of accuracy, and is validated on a biological pulse generator and a large‑scale benchmark against exact and prior hybrid approaches. Comparisons of probability distributions and weak errors of the first two moments demonstrate that the hybrid method achieves accurate results while reducing computational cost.

Abstract

The dynamical solution of a well-mixed, nonlinear stochastic chemical kinetic system, described by the Master equation, may be exactly computed using the stochastic simulation algorithm. However, because the computational cost scales with the number of reaction occurrences, systems with one or more “fast” reactions become costly to simulate. This paper describes a hybrid stochastic method that partitions the system into subsets of fast and slow reactions, approximates the fast reactions as a continuous Markov process, using a chemical Langevin equation, and accurately describes the slow dynamics using the integral form of the “Next Reaction” variant of the stochastic simulation algorithm. The key innovation of this method is its mechanism of efficiently monitoring the occurrences of slow, discrete events while simultaneously simulating the dynamics of a continuous, stochastic or deterministic process. In addition, by introducing an approximation in which multiple slow reactions may occur within a time step of the numerical integration of the chemical Langevin equation, the hybrid stochastic method performs much faster with only a marginal decrease in accuracy. Multiple examples, including a biological pulse generator and a large-scale system benchmark, are simulated using the exact and proposed hybrid methods as well as, for comparison, a previous hybrid stochastic method. Probability distributions of the solutions are compared and the weak errors of the first two moments are computed. In general, these hybrid methods may be applied to the simulation of the dynamics of a system described by stochastic differential, ordinary differential, and Master equations.