## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 16, Number 4 (2006), 1925-1961.

### Asymptotic analysis of multiscale approximations to reaction networks

Karen Ball, Thomas G. Kurtz, Lea Popovic, and Greg Rempala

#### Abstract

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.

#### Article information

**Source**

Ann. Appl. Probab., Volume 16, Number 4 (2006), 1925-1961.

**Dates**

First available in Project Euclid: 17 January 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1169065212

**Digital Object Identifier**

doi:10.1214/105051606000000420

**Mathematical Reviews number (MathSciNet)**

MR2288709

**Zentralblatt MATH identifier**

1118.92031

**Subjects**

Primary: 92C45: Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.) [See also 80A30] 80A30: Chemical kinetics [See also 76V05, 92C45, 92E20] 60J27: Continuous-time Markov processes on discrete state spaces 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60F17: Functional limit theorems; invariance principles

**Keywords**

Reaction networks chemical reactions cellular processes Markov chains averaging scaling limits

#### Citation

Ball, Karen; Kurtz, Thomas G.; Popovic, Lea; Rempala, Greg. Asymptotic analysis of multiscale approximations to reaction networks. Ann. Appl. Probab. 16 (2006), no. 4, 1925--1961. doi:10.1214/105051606000000420. https://projecteuclid.org/euclid.aoap/1169065212