Reflected Diffusions and (Bio)Chemical Reaction Networks

Abstract: 

Joint work with Saul Leite, Federal University of Juiz de Fora, Brazil

David Anderson, U. Wisconsin-Madison

Des Higham, U. Strathclyde  

Continuous-time Markov chain models are often used to describe the  stochastic dynamics of networks of reacting chemical species, especially in the growing field of systems biology. 

Discrete-event stochastic simulation of these models rapidly becomes computationally intensive. Consequently,  more tractable diffusion approximations are commonly used in numerical computation, even for modest-sized networks.

However, existing approximations (e.g., van Kampen and Langevin),  do not respect the constraint that chemical concentrations are never negative.

n this talk, we propose an approximation for such Markov chains, via reflected diffusion processes, that respects the fact that concentrations of chemical species are non-negative. This fixes a difficulty with Langevin approximations that they are frequently only valid until the boundary of the positive orthant is reached.

Our approximation has the added advantage that it can be written down immediately from the chemical reactions. This contrasts with the van Kampen approximation, which involves a two-stage procedure --- first  solving a deterministic ordinary differential equation, followed by a stochastic differential equation for fluctuations around those solutions. 

An invariance principle for reflected diffusions, due to Kang and Williams, is adapted in justifying our approximation under mild assumptions.  Some numerical examples illustrate the advantages of our approximation over direct simulation of the Markov chain or use of the van Kampen approximation.