Annals of Applied Probability

Asymptotic analysis of multiscale approximations to reaction networks

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

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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.

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

First available in Project Euclid: 17 January 2007

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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

Reaction networks chemical reactions cellular processes Markov chains averaging scaling limits


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.

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  • Athreya, K. B. and Ney, P. E. (2004). Branching Processes. Dover, Mineola, NY.
  • Ball, F. and Donelly, P. (1992). Branching process approximation of epidemic models. Theory Probab. Appl. 37 119–121.
  • Cao, Y., Gillespie, D. T. and Petzold, L. R. (2005). The slow-scale stochastic simulation algorithm. J. Chem. Phys. 122 014116.
  • Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. Wiley, New York.
  • Gardiner, C. W. (2004). Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, 3rd ed. Springer, Berlin.
  • Haseltine, E. L. and Rawlings, J. B. (2002). Approximate simulation of coupled fast and slow reactions for stochastic chemical kinetics. J. Chem. Phys. 117 6959–6969.
  • Kurtz, T. G. (1977/78). Strong approximation theorems for density dependent Markov chains. Stochastic Process. Appl. 6 223–240.
  • Kurtz, T. G. (1992). Averaging for martingale problems and stochastic approximation. Applied Stochastic Analysis. Lecture Notes in Control and Inform. Sci. 77 186–209. Springer, Berlin.
  • Rao, C. V. and Arkin, A. P. (2003). Stochastic chemical kinetics and the quasi-steady-state assumption: Application to the Gillespie algorithm. J. Chem. Phys. 118 4999–5010.
  • Srivastava, R., You, L., Summers, J. and Yin, J. (2002). Stochastic vs. deterministic modeling of intracellular viral kinetics. J. Theoret. Biol. 218 309–321.
  • Stiefenhofer, M. (1998). Quasi-steady-state approximation for chemical reaction networks. J. Math. Biol. 36 593–609.
  • Van Kampen, N. G. (1981). Stochastic Processes in Physics and Chemistry. North-Holland, Amsterdam.