Multivariate Normal Approximation for the Stochastic Simulation Algorithm: Limit Theorem and Applications

Stochastic approaches in systems biology are being used increasingly to model the heterogeneity and the intrinsic stochasticity of living systems, especially at the single-cell level. The stochastic simulation algorithm – also known as the Gillespie algorithm – is currently the most widely used method to simulate the time course of a system of bio-chemical reactions in a stochastic way.In this article, we present a central limit theorem for the Gillespie stochastic trajectories when the living system has reached a steady-state, that is when the internal bio-molecules concentrations are assumed to be at equilibrium. It appears that the stochastic behavior in steady-state is entirely characterized by the stoichiometry matrix of the system and a single vector of reaction probabilities.We propose several applications of this result such as deriving multivariate confidence regions for the time course of the system and a constraints-based approach which extends the flux balance analysis framework to the stochastic case.