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 |