Multiscale Models for Chemical Reaction Processes

Dennis Cox
Statistics Department
Rice University

Abstract

            Numerous researchers in the biological sciences have used the Stochastic Simulation Algorithm of Gillespie to simulate the chemical reaction processes associated with cell level modeling of genetic pathways.  Because of the complexity of these reaction systems, it is difficult to perform the detailed computations, and numerous approximations have sprung up, including so-called  "multiscale" approximations.  These involve continuous state approximations to the discrete states of the Gillespie algorithm which are either modeled with ordinary or stochastic differential equation components.  The only mathematical justification we know of for these approximations depends on "expansions of the master equation," which means showing that the forward Kolmogorov equation has a limit of the desired type, and these methods are somewhat problematic.

            Our group has begun to develop a mathematical theory based on strong approximation methods to establish convergence of the solutions of the model equations to the conjectured limits.  A key ingredient that has been missing from the previous analyses is a consistent partitioning of the chemical species and the reactions.  We will give a rigorous description and provide examples that conform to this consistency condition and other examples that cannot be so modeled.  We will present the results on the "quasi-deterministic" type of model  (a mixture of jump processes and deterministic processes described by differential equations in between the jumps) and indicate the extension to include components described by stochastic differential equations.

This is joint work with Yue Wu, Jesse Turner, and Josue Noyola-Martinez