The weak convergence analysis of Tau-Leaping methods: Revisited

Communications in Mathematical Sciences

Volumn 9, Issue 4, 2011, Pages 965-996

  Hu, Yucheng ^a   Li, Tiejun ^a   Min, Bin ^a  

^a Peking University   (China)

Author keywords

Chemical reaction kinetics; Convergence analysis; Large volume scaling; Rooted tree theory

Indexed keywords

EID: 79961222409     PISSN: 15396746     EISSN: None     Source Type: Journal    
DOI: 10.4310/CMS.2011.v9.n4.a2     Document Type: Article

References (32)

1. D. Anderson, A. Ganguly, and T. Kurtz, Error analysis of tau-leap simulation methods, Ann. App. Prob., accepted, 2010.
2. A. Arkin, J. Ross, and H. McAdams, Stochastic kinetic analysis of developmental pathway bifurcation in phage λ-infected Escherichia coli cells, Genetics, 149, 1633-1648, 1998.
3. A. Bortz, M. Kalos, and J. Lebowitz, A new algorithm for Monte Carlo simulation of Ising spin systems, J. Comp. Phys., 17, 10-18, 1975.
4. K. Burrage and B.P.M., Conditions of Stochatic Runge-Kutta Methods by B-Series, SIAM J. Numer. Anal., 38, 1626-1646, 2001.
5. K. Burrage and T. Tian, Poisson Runge-Kutta methods for chemical reaction systems, in Third InternationalWorkshop on Scientific Computing and Applications, City University of Hong Kong, 2003.
6. J. Butcher, Coefficients for the study of Runge-Kutta integration processes, J. Austral. Math. Soc., 3, 185-201, 1963.
7. Y. Cao, D. Gillespie, and L. Petzold, Avoiding negative populations in explicit Poisson tauleaping, J. Chem. Phys., 123, 054104, 2005.
8. Y. Cao, D. Gillespie, and L. Petzold, Efficient stepsize selection for the tau-leaping method, J. Chem. Phys., 124, 44109, 2006.
9. Y. Cao and L. Petzold, Trapezoidal tau-leaping formula for the stochastic simulation of biochemical systems, in Proceedings of the FOSBE 2005, University of California Santa Barbara, 2005.
10. Y. Cao, L. Petzold, and D. Gillespie, The slow-scale stochastic simulation algorithm, J. Chem. Phys., 122, 14116, 2005.
11. W. E, D. Liu, and E. Vanden-Eijnden, Nested stochastic simulation algorithm for chemical kinetic systems with disparate rates, J. Chem. Phys., 123, 194107, 2005.
12. M. Elowitz, A. Levine, E. Siggia, and P. Swain, Stochastic gene expression in a single cell, Science, 297, 1183-1186, 2002.
13. D. Endy and R. Brent, Modelling cellular behavior, Nature, 409, 391-395, 2001.
14. M. Gibson and J. Bruck, Efficient exact stochastic simulation of chemical systems with many species and many channels, J. Phys. Chem. A, 104, 1876-1889, 2000.
15. D. Gillespie, Stochastic simulation of chemical processes, J. Comput. Phys., 22, 403, 1976.
16. D. Gillespie, Exact stochastic simulation of coupled chemical reactions, J. Phys. Chem., 81, 2340-2361, 1977.
17. D. Gillespie, Approximate accelerated stochastic simulation of chemically reacting systems, J. Chem. Phys., 115, 1716, 2001.
18. G. Gong and M. Qian, Lectures on Applied Stochastic Processes and the Stochastic Modeling in Algorithms and Intelligent Computing, Tsinghua University Press (in Chinese), Beijing, 2004.
19. J. Gross and J. Yellen, Graph Theory and its Applications, CRC Press, Boca Raton, Florida, 2006.
20. E. Hairer, S. Norsett, and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, Springer, Berlin and Heidelberg, 1993.
21. E. Haseltine and J. Rawlings, Approximate simulation of coupled fast and slow reactions for stochastic chemical kinetics, J. Chem. Phys., 117, 6959, 2002.
22. Y. Hu and T. Li, Highly accurate tau-leaping methods with random corrections, J. Chem. Phys., 130, 124109, 2009.
23. Y. Hu, T. Li, and B. Min, Local truncation error analysis of tau-leaping methods: Revisited, technical report 016, School of Mathematical Sciences, Peking University, Beijing, 2009.
24. T. Kurtz, The relationship between stochastic and deterministic models for chemical reactions, J. Chem. Phys., 57, 2976-2978, 1972.
25. T. Kurtz, Strong approximation theorems for density dependent Markov chains, Stoch. Proc. Appl., 6, 223-240, 1977/78.
26. T. Li, Analysis of explicit tau-leaping schemes for simulating chemically reacting systems, Multi. Model. Simul., 6, 417, 2007.
27. H. McAdams and A. Arkin, Stochastic mechanisms in gene expression, Proc. Natl. Acad. Sci. USA, 94, 814-819, 1997.
28. S. Plyasunov, On hybrid simulation schemes for stochastic reaction dynamics, unpublished, 2005.
29. M. Rathinam, L. Petzold, Y. Cao, and D. Gillespie, Stiffness in stochastic chemically reacting systems: The implicit tau-leaping method, J. Chem. Phys., 119, 12784, 2003.
30. M. Rathinam, L. Petzold, Y. Cao, and D. Gillespie, Consistency and stability of tau-leaping schemes for chemical reaction systems, Multi. Model. Simul., 4, 867, 2005.
31. A. Rössler, Stochastic Taylor expansions for the expectation of functionals of diffusion processes, Stoch. Anal. Appl., 22, 1553, 2004.
32. A. Slepoy, A. Thompson, and S. Plimpton, A constant-time kinetic Monte Carlo algorithm for simulation of large biochemical reaction networks, J. Chem. Phys., 128, 205101, 2008.