A rigorous framework for multiscale simulation of stochastic cellular networks.

Noise and stochasticity are fundamental to biology and derive from the very nature of biochemical reactions where thermal motion of molecules translates into randomness in the sequence and timing of reactions. This randomness leads to cell-cell variability even in clonal populations. Stochastic biochemical networks are modeled as continuous time discrete state Markov processes whose probability density functions evolve according to a chemical master equation (CME). The CME is not solvable but for the simplest cases, and one has to resort to kinetic Monte Carlo techniques to simulate the stochastic trajectories of the biochemical network under study. A commonly used such algorithm is the stochastic simulation algorithm (SSA). Because it tracks every biochemical reaction that occurs in a given system, the SSA presents computational difficulties especially when there is a vast disparity in the timescales of the reactions or in the number of molecules involved in these reactions. This is common in cellular networks, and many approximation algorithms have evolved to alleviate the computational burdens of the SSA. Here, we present a rigorously derived modified CME framework based on the partition of a biochemically reacting system into restricted and unrestricted reactions. Although this modified CME decomposition is as analytically difficult as the original CME, it can be naturally used to generate a hierarchy of approximations at different levels of accuracy. Most importantly, some previously derived algorithms are demonstrated to be limiting cases of our formulation. We apply our methods to biologically relevant test systems to demonstrate their accuracy and efficiency.