The Linear Noise Approximation for Spatially Dependent Biochemical Networks

An algorithm for computing the linear noise approximation (LNA) of the reaction–diffusion master equation (RDME) is developed and tested. The RDME is often used as a model for biochemical reaction networks. The LNA is derived for a general discretization of the spatial domain of the problem. If M is the number of chemical species in the network and N is the number of nodes in the discretization in space, then the computational work to determine approximations of the mean and the covariances of the probability distributions is proportional to M2N2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M^2N^2$$\end{document} in a straightforward implementation. In our LNA algorithm, the work is proportional to M2N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M^2N$$\end{document}. Since N usually is larger than M, this is a significant reduction. The accuracy of the approximation in the algorithm is estimated analytically and evaluated in numerical experiments.