Analytic Methods for Modeling Stochastic Regulatory Networks

Recent single-cell experiments have revived interest in the unavoidable or intrinsic noise in biochemical and genetic networks arising from the small number of molecules of the participating species. That is, rather than modeling regulatory networks in terms of the deterministic dynamics of concentrations, we model the dynamics of the probability of a given copy number of the reactants in single cells. Most of the modeling activity of the last decade has centered on stochastic simulation, i.e., Monte Carlo methods for generating stochastic time series. Here we review the mathematical description in terms of probability distributions, introducing the relevant derivations and illustrating several cases for which analytic progress can be made either instead of or before turning to numerical computation. Analytic progress can be useful both for suggesting more efficient numerical methods and for obviating the computational expense of, for example, exploring parametric dependence.