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Asymptotic analysis of multiscale approximations to reaction networks. (English) Zbl 1118.92031
Summary: 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 a 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.

92C45 Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.)
60J27 Continuous-time Markov processes on discrete state spaces
92C37 Cell biology
80A30 Chemical kinetics in thermodynamics and heat transfer
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60F17 Functional limit theorems; invariance principles
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