The Annals of Applied Probability

Separation of time-scales and model reduction for stochastic reaction networks

Hye-Won Kang and Thomas G. Kurtz

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A stochastic model for a chemical reaction network is embedded in a one-parameter family of models with species numbers and rate constants scaled by powers of the parameter. A systematic approach is developed for determining appropriate choices of the exponents that can be applied to large complex networks. When the scaling implies subnetworks have different time-scales, the subnetworks can be approximated separately, providing insight into the behavior of the full network through the analysis of these lower-dimensional approximations.

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Ann. Appl. Probab., Volume 23, Number 2 (2013), 529-583.

First available in Project Euclid: 12 February 2013

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

Reaction networks chemical reactions cellular processes multiple time scales Markov chains averaging scaling limits quasi-steady state assumption


Kang, Hye-Won; Kurtz, Thomas G. Separation of time-scales and model reduction for stochastic reaction networks. Ann. Appl. Probab. 23 (2013), no. 2, 529--583. doi:10.1214/12-AAP841.

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  • Ball, K., Kurtz, T. G., Popovic, L. and Rempala, G. (2006). Asymptotic analysis of multiscale approximations to reaction networks. Ann. Appl. Probab. 16 1925–1961.
  • Cao, Y., Gillespie, D. T. and Petzold, L. R. (2005). The slow-scale stochastic simulation algorithm. J. Chem. Phys. 122 014116. Available at
  • Crudu, A., Debussche, A. and Radulescu, O. (2009). Hybrid stochastic simplifications for multiscale gene networks. BMC Syst. Biol. 3 89.
  • Darden, T. (1979). A pseudo-steady state approximation for stochastic chemical kinetics. Rocky Mountain J. Math. 9 51–71.
  • Darden, T. A. (1982). Enzyme kinetics: Stochastic vs. deterministic models. In Instabilities, Bifurcations, and Fluctuations in Chemical Systems (Austin, Tex., 1980) 248–272. Univ. Texas Press, Austin, TX.
  • Davis, M. H. A. (1993). Markov Models and Optimization. Monographs on Statistics and Applied Probability 49. Chapman & Hall, London.
  • E, W., Liu, D. and Vanden-Eijnden, E. (2005). Nested stochastic simulation algorithm for chemical kinetic systems with disparate rates. J. Chem. Phys. 123 194107. Available at
  • E, W., Liu, D. and Vanden-Eijnden, E. (2007). Nested stochastic simulation algorithms for chemical kinetic systems with multiple time scales. J. Comput. Phys. 221 158–180.
  • Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • Gillespie, D. T. (1977). Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem. 81 2340–2361.
  • Goutsias, J. (2005). Quasiequilibrium approximation of fast reaction kinetics in stochastic biochemical systems. J. Chem. Phys. 122 184102. Available at
  • 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.
  • Hensel, S. C., Rawlings, J. B. and Yin, J. (2009). Stochastic kinetic modeling of vesicular stomatitis virus intracellular growth. Bull. Math. Biol. 71 1671–1692.
  • Jakubowski, A. (1997). A non-Skorohod topology on the Skorohod space. Electron. J. Probab. 2 21 pp. (electronic).
  • Kabanov, Y. M., Liptser, R. S. and Shiryaev, A. N. (1984). Weak and strong convergence of distributions of counting processes. Theory Probab. Appl. 28 303–336.
  • Kang, H.-W. (2011). A multiscale approximation in a heat shock response model of E. coli. Unpublished manuscript. Mathematical Biosciences Institute, Ohio State Univ.
  • Kang, H.-W., Kurtz, T. G. and Popovic, L. (2012). Diffusion approximations for multiscale chemical reaction models. Unpublished manuscript. School of Mathematics, Univ. Minnesota, Dept. Mathematics and Statistics, Univ. Wisconsin, Madison and Dept. Mathematics and Statistics, Concordia Univ.
  • Khas’minskiĭ, R. Z. (1966a). On stochastic processes defined by differential equations with a small parameter. Theory Probab. Appl. 11 211–228.
  • Khas’minskiĭ, R. Z. (1966b). A limit theorem for the solutions of differential equations with random right-hand sides. Theory Probab. Appl. 11 390–406.
  • Kurtz, T. G. (1972). The relationship between stochastic and deterministic models for chemical reactions. J. Chem. Phys. 57 2976–2978.
  • Kurtz, T. G. (1977/78). Strong approximation theorems for density dependent Markov chains. Stochastic Process. Appl. 6 223–240.
  • Kurtz, T. G. (1980). Representations of Markov processes as multiparameter time changes. Ann. Probab. 8 682–715.
  • Kurtz, T. G. (1991). Random time changes and convergence in distribution under the Meyer–Zheng conditions. Ann. Probab. 19 1010–1034.
  • Kurtz, T. G. (1992). Averaging for martingale problems and stochastic approximation. In Applied Stochastic Analysis (New Brunswick, NJ, 1991). Lecture Notes in Control and Information Sciences 177 186–209. Springer, Berlin.
  • Macnamara, S., Burrage, K. and Sidje, R. B. (2007). Multiscale modeling of chemical kinetics via the master equation. Multiscale Model. Simul. 6 1146–1168.
  • Mastny, E. A., Haseltine, E. L. and Rawlings, J. B. (2007). Two classes of quasi-steady-state model reductions for stochastic kinetics. J. Chem. Phys. 127 094106. Available at
  • Meyer, P. A. (1971). Démonstration simplifiée d’un théorème de Knight. In Séminaire de Probabilités, V (Univ. Strasbourg, Année Universitaire 19691970). Lecture Notes in Math. 191 191–195. Springer, Berlin.
  • Meyer, P. A. and Zheng, W. A. (1984). Tightness criteria for laws of semimartingales. Ann. Inst. Henri Poincaré Probab. Stat. 20 353–372.
  • 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.
  • Segel, L. A. and Slemrod, M. (1989). The quasi-steady-state assumption: A case study in perturbation. SIAM Rev. 31 446–477.
  • Srivastava, R., Peterson, M. S. and Bentley, W. E. (2001). Stochastic kinetic analysis of Escherichia coli stress circuit using sigma(32)-targeted antisense. Biotechnol. Bioeng. 75 120–129.
  • Zeiser, S., Franz, U. and Liebscher, V. (2010). Autocatalytic genetic networks modeled by piecewise-deterministic Markov processes. J. Math. Biol. 60 207–246.