Stochastic Modeling of Biochemical Reactions

The most common theoretical approach to model the interactions in a biochemical process is through chemical reactions. Often for these reactions, the dynamics of the first M-order statistical moments of the species populations do not form a closed system of differential equations, in the sense that the time-derivatives of first M-order moments generally depend on moments of order higher than M. However, for analysis purposes, these dynamics are often made to be closed by approximating the needed derivatives of the first M-order moments by nonlinear functions of the same moments. These functions are called the moment closure functions. This paper presents a systematic procedure to construct these moment closure functions. This is done by first assuming that they exhibit a certain separable form, and then 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. Using these results a stochastic model for gene expression is investigated. We show that in gene expression mechanisms, in which a protein inhibits its own transcription, the resulting negative feedback reduces stochastic variations in the protein populations.