Evaluation of the Multiplane Method for Efficient Simulations of Reaction Networks

TL;DR

The paper evaluates the multiplane method for simulating reaction networks, demonstrating its accuracy in small domains with strong fluctuations and its convergence to rate equations in large domains, offering an efficient stochastic simulation approach.

Contribution

It tests and validates the multiplane method's effectiveness across different domain sizes, extending its applicability for complex reaction network simulations.

Findings

Accurate in small domains with strong fluctuations.

Converges to rate equation results in large domains.

Provides an efficient alternative to direct master equation integration.

Abstract

Reaction networks in the bulk and on surfaces are widespread in physical, chemical and biological systems. In macroscopic systems, which include large populations of reactive species, stochastic fluctuations are negligible and the reaction rates can be evaluated using rate equations. However, many physical systems are partitioned into microscopic domains, where the number of molecules in each domain is small and fluctuations are strong. Under these conditions, the simulation of reaction networks requires stochastic methods such as direct integration of the master equation. However, direct integration of the master equation is infeasible for complex networks, because the number of equations proliferates as the number of reactive species increases. Recently, the multiplane method, which provides a dramatic reduction in the number of equations, was introduced [A. Lipshtat and O. Biham, Phys. Rev. Lett. 93, 170601 (2004)]. The reduction is achieved by breaking the network into a set of maximal fully connected sub-networks (maximal cliques). Lower-dimensional master equations are constructed for the marginal probability distributions associated with the cliques, with suitable couplings between them. In this paper we test the multiplane method and examine its applicability. We show that the method is accurate in the limit of small domains, where fluctuations are strong. It thus provides an efficient framework for the stochastic simulation of complex reaction networks with strong fluctuations, for which rate equations fail and direct integration of the master equation is infeasible. The method also applies in the case of large domains, where it converges to the rate equation results.