Approximate Stochastic Models for Chemically Reacting Systems

In the stochastic formulation of chemical reactions, the time evolution of the first M order statistical moments for the number of molecules of the different species involved is generally not closed, in the sense that they depend on moments of order higher than M. For analysis purposes, the time evolution of the first M order moments is often made to be closed by approximating the higher order moments as nonlinear functions of moments up to order M, which we refer to as the moment closure functions. Recent work has introduced the technique of derivativematching, where the moment closure functions are obtained by matching time derivatives of the exact (not closed) moment equations with that of the approximate (closed) equations for some initial time and set of initial conditions. However, for multispecies reactions these results have been restricted to a second order of truncation, i.e. M= 2. This paper extends these results by providing explicit formulas to construct moment closure functions for any arbitrary order of truncation M. Striking features of these moment closure functions are that they are independent of the reaction parameters (reaction rates and stoichiometry). Moreover, by increasing M, the closed moment equations provide more accurate approximations to the exact moment equations. We demonstrate the usefulness of our result by applying it to an example of stochastic focusing motivated from a gene cascade network, where the stochastic mean differs from the chemical rate equations. Moment estimates from the closed moment equations are compared with those obtained from a large number of Monte Carlo simulations.