Analysis of the multiplane method for stochastic simulations of reaction networks with fluctuations

Chemical reaction networks are commonly modeled by rate equations, which are systems of ordinary differential equations describing the evolution of species concentrations. Such models break down at low concentrations, where stochastic effects become dominant. Instead, one has to solve the master equation that governs the multidimensional probability distribution of particle populations. For large networks such an approach is often computationally prohibitive due to the exponential dependence of the number of states on the number of components. The multiplane method is a dimension reduction technique that exploits the structure of the network to derive approximate dynamics for the marginal distributions of pairs of coreacting species. This method was introduced in [A. Lipshtat and O. Biham, Phys. Rev. Lett., 93 (2004), 170601] as an uncontrolled approximation for specific examples in the context of interstellar chemical reactions. In this paper we formalize the method and prove that it is asymptotically exact in the two extreme limits of small and large population sizes. Our analysis concentrates on steady-state conditions, although numerical simulations indicate that the method is equally well applicable to time-dependent solutions. This analysis partially explains the surprisingly high accuracy of the method.