Nested stochastic simulation algorithm for chemical kinetic systems with disparate rates

Abstract

An efficient simulation algorithm for chemical kinetic systems with disparate rates is proposed. This new algorithm is quite general, and it amounts to a simple and seamless modification of the classical stochastic simulation algorithm (SSA), also known as the Gillespie [J. Comput. Phys. 22, 403 (1976); J. Phys. Chem. 81, 2340 (1977)] algorithm. The basic idea is to use an outer SSA to simulate the slow processes with rates computed from an inner SSA which simulates the fast reactions. Averaging theorems for Markov processes can be used to identify the fast and slow variables in the system as well as the effective dynamics over the slow time scale, even though the algorithm itself does not rely on such information. This nested SSA can be easily generalized to systems with more than two separated time scales. Convergence and efficiency of the algorithm are discussed using the established error estimates and illustrated through examples.

A cutoff phenomenon in accelerated stochastic simulations of chemical kinetics via flow averaging (FLAVOR-SSA)

“…Peles et al 12 have formulated an efficient algorithm for the direct solution of the master equation with multiple time-scales. In a separate line of work, Weinan et al 13,14 have used drift averaging techniques 15 to accelerate stochastic simulations, while Gear and Kevrekidis 16 and Kevrekidis et al 17 The extension of FLAVORS to stochastic systems results in a simple and non-intrusive algorithm for the effective simulation of a stiff set of reactions. The objective of this article is to provide a description of a novel method, based on the concept of flow averaging, for the simulation of stiff discrete-space, continuous-time Markov processes.…”

Parallel in Time Simulation of Multiscale Stochastic Chemical Kinetics

“…homogenization can be put in contrast to other multiscale methods [7,11]. Typically one computes a solution to a homogenized equation obtained by determining some kind of averaged coefficients.…”

“…Several different model reduction techniques have been proposed for this situation [7,11,26]. Also, an implicit version of the tau-leap method has been developed [41], but this method converges in a very weak sense only [8,32,42].…”

Adaptive deployment of model reductions for tau-leaping simulation

“…However, neither of the two algorithms is efficient for stiff systems, where vastly different time scales are involved. 7,8 Thus, numerous model reductions have been proposed to accelerate both algorithms, including the slow-scale SSA (ssSSA), 7 the nested SSA, 9,10 the stochastic quasi-steady-state approximation (sQSSA), [11][12][13] the stochastic Michaelis-Menten model reduction (M-M), 11,12,14,15 and the time-dependent solution method. 16 Among these methods, the ssSSA 7 eliminates the need to simulate many fast reactions by approximating the fast subsystems that reach stochastic partial equilibrium very quickly between two consecutive slow reactions by their partial equilibrium states.…”