Ju l 2 00 5 Asymptotic analysis of multiscale approximations to reaction networks ∗

A reaction network is a chemical system involving multiple reactions and chemical species. Stochastic models of such networks treat the system as a continuous time Markov chain on the number of molecules of each species with reactions as possible transitions of the chain. In many cases of biological interest some of the chemical species in the network are present in much greater abundance than others and reaction rate constants can vary over several orders of magnitude. We consider approaches to approximation of such models that take the multiscale nature of the system into account. Our primary example is a model of a cell’s viral infection for which we apply a combination of averaging and law of large number arguments to show that the “slow” component of the model can be approximated by a deterministic equation and to characterize the asymptotic distribution of the “fast” components. The main goal is to illustrate techniques that can be used to reduce the dimensionality of much more complex models. MSC 2000 subject classification. 60J27, 60J80, 60F17, 92C45, 80A30 Reaction networks, chemical reactions, cellular processes, Markov chains, averaging, scaling limits Research supported by VIGRE Grant, University of Indiana Research supported by NSF Grant DMS-0503983 Research supported in part by CGeMM Intramural Grant, University of Louisville