Efficient simulation of stochastic chemical kinetics with the Stochastic Bulirsch-Stoer extrapolation method

BMC Systems Biology

Volumn 8, Issue 1, 2014, Pages

  Székely Jr , Tamás ^a   Burrage, Kevin ^a,b   Zygalakis, Konstantinos C ^c   Barrio, Manuel ^d  

^a UNIVERSITY OF OXFORD   (United Kingdom)

^b QUEENSLAND UNIVERSITY OF TECHNOLOGY   (Australia)

^c UNIVERSITY OF SOUTHAMPTON   (United Kingdom)

^d UNIVERSITY OF VALLADOLID   (Spain)

Author keywords

Bulirsch Stoer; Discrete stochastic methods; High order methods; Stochastic simulation; leap

Indexed keywords

ENZYME INHIBITOR;

ARTICLE; CHEMICAL MODEL; CHEMISTRY; INFORMATION SCIENCE; KINETICS; METHODOLOGY; STATISTICS;

ENZYME INHIBITORS; INFORMATICS; KINETICS; MODELS, CHEMICAL; STOCHASTIC PROCESSES;

EID: 84902300328     PISSN: None     EISSN: 17520509     Source Type: Journal    
DOI: 10.1186/1752-0509-8-71     Document Type: Article

References (47)

1. Arkin A, Ross J, McAdams HH. Stochastic kinetic analysis of developmental pathway bifurcation in phage λ-infected Escherichia coli cells. Genetics 1998, 149:1633-1648.
2. Kaern M, Elston T, Blake W, Collins J. Stochasticity in gene expression: from theories to phenotypes. Nat Rev Genet 2005, 6:451-464.
3. Munsky B, Neuert G, van Oudenaarden A. Using gene expression noise to understand gene regulation. Science 2012, 336:183.
4. Fedoroff N, Fontana W. Small numbers of big molecules. Science 2002, 297:1129-1131.
5. Elowitz M, Levine A, Siggia E, Swain P. Stochastic gene expression in a single cell. Science 2002, 297:1183-1186.
6. Gillespie DT, Mangel M. Conditioned averages in chemical kinetics. J Chem Phys 1981, 75:704-709.
7. Goutsias J. Classical versus stochastic kinetics modeling of biochemical reaction systems. Biophys J 2007, 92:2350-2365.
8. Wilkinson DJ. Stochastic modelling for quantitative description of heterogeneous biological systems. Nat Rev Genet 2009, 10:122-133.
9. Kurtz TG. Limit theorems for sequences of jump Markov processes approximating ordinary differential processes. J Appl Probab 1971, 8:344-356.
10. Gillespie DT. Exact stochastic simulation of coupled chemical reactions. J Phys Chem 1977, 81:2340-2361.
11. Gillespie DT. Stochastic simulation of chemical kinetics. Annu Rev Phys Chem 2007, 58:35-55.
12. Gillespie DT. Approximate accelerated stochastic simulation of chemically reacting systems. J Chem Phys 2001, 115:1716-1733.
13. Gillespie DT, Petzold LR. Improved leap-size selection for accelerated stochastic simulation. J Chem Phys 2003, 119:8229-8234.
14. Cao Y, Gillespie DT, Petzold LR. Efficient step size selection for the tau-leaping simulation method. J Chem Phys 2006, 124:044109.
15. Tian TH, Burrage K. Binomial leap methods for simulating stochastic chemical kinetics. J Chem Phys 2004, 121:10356-10364.
16. Chatterjee A, Vlachos DG, Katsoulakis MA. Binomial distribution based tau-leap accelerated stochastic simulation. J Chem Phys 2005, 122:024112.
17. Peng X, Zhou W, Wang Y. Efficient binomial leap method for simulating chemical kinetics. J Chem Phys 2007, 126:224109.
18. Pettigrew MF, Resat H. Multinomial tau-leaping method for stochastic kinetic simulations. J Chem Phys 2007, 126:084101.
19. Xu Z, Cai X. Unbiased tau-leap methods for stochastic simulation of chemically reacting systems. J Chem Phys 2008, 128:154112.
20. Hu Y, Li T. Highly accurate tau-leaping methods with random corrections. J Chem Phys 2009, 130:124109.
21. Hu Y, Li T, Min B. A weak second order tau-leaping method for chemical kinetic systems. J Chem Phys 2011, 135:024113.
22. Székely T, Burrage K, Erban R, Zygalakis KC. A higher-order numerical framework for stochastic simulation of chemical reaction systems. BMC Syst Biol 2012, 6:85.
23. Richardson LF. The approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam. Phil Trans Roy Soc Lond 1910, 210:307-357.
24. Hairer E, Nørsett SP, Wanner G. Solving Ordinary Differential Equations: Nonstiff Problems. 2nd edition 1993, Berlin: Springer-Verlag.
25. Bulirsch R, Stoer J. Numerical treatment of ordinary differential equations by extrapolation methods. Numerische Mathematik 1966, 8:1-13.
26. Deuflhard P. Recent progress in extrapolation methods for ordinary differential equations. SIAM Rev 1985, 27(4):505-535.
27. Gragg WB. On extrapolation algorithms for ordinary initial value problems. SIAM J Numer Anal 1965, 2:384-403.
28. Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical Recipes in C: The Art of Scientific Computing. 2nd edition 1992, Cambridge: Cambridge University Press.
29. Kurtz TG. Strong approximation theorems for density dependent Markov chains. Stochastic Processes Appl 1978, 6:223-240.
30. van Kampen NG. Stochastic Processes in Physics and Chemistry. 3rd edition 2007, Amsterdam: Elsevier.
31. Goutsias J, Jenkinson G. Markovian dynamics on complex reaction networks. Phys Rep 2013, 529:199-264.
32. Kurtz TG. Representations of Markov processes as multiparameter time changes. Ann Probab 1980, 8:682-715.
33. Marquez-Lago T, Burrage K. Binomial tau-leap spatial stochastic simulation algorithm for applications in chemical kinetics. J Chem Phys 2007, 127:104101.
34. Elf J, Ehrenberg M. Spontaneous separation of bi-stable biochemical systems into spatial domains of opposite phases. Syst Biol 2004, 1:230-236.
35. Rathinam M, Petzold LR, Cao Y, Gillespie DT. Consistency and stability of tau-leaping schemes for chemical reaction systems. Multiscale Model Simul 2005, 4(3):867-895.
36. Li T. Analysis of explicit tau-leaping schemes for simulating chemically reacting systems. Multiscale Model Simul 2007, 6(2):417-436.
37. Burrage K, Tian T. Poisson Runge-Kutta methods for chemical reaction systems. Advances in Scientific Computing and Applications 2004, 82-96. Beijing/New York: Science Press, Sun YLW, Tang T.
38. Anderson DF, Ganguly A, Kurtz TG. Error analysis of tau-leap simulation methods. Ann Appl Probab 2011, 21(6):2226-2262.
39. Hu Y, Li T, Min B. The weak convergence analysis of tau-leaping methods: revisited. Comm Math Sci 2011, 9:965-996.
40. Burrage K, Burrage PM. Order conditions of stochastic Runge-Kutta methods by B-Series. SIAM J Numer Anal 2000, 38(5):1626-1646.
41. Milstein GN, Tretyakov MV. Numerical methods in the weak sense for stochastic differential equations with small noise. SIAM J Numer Anal 1997, 34:2142-2167.
42. Milstein GN, Tretyakov MV. Mean-square numerical methods for stochastic differential equations with small noises. SIAM J Sci Comput 1997, 18:1067-1087.
43. Buckwar E, Rößler A, Winkler R. Stochastic Runge-Kutta methods for Ito SODEs with small noise. SIAM J Sci Comput 2010, 32:1789-1808.
44. Rué P, Villa-Freixà J, Burrage K. Simulation methods with extended stability for stiff biochemical kinetics. BMC Syst Biol 2010, 4:110-123.
45. Cao Y, Gillespie DT, Petzold LR. The adaptive explicit-implicit tau-leaping method with automatic tau selection. J Chem Phys 2007, 126:224101.
46. Goutsias J. Quasiequilibrium approximation of fast reaction kinetics in stochastic biochemical systems. J Chem Phys 2005, 122:184102.
47. MacNamara S, Burrage K, Sidje R. Multiscale modeling of chemical kinetics via the master equation. Multiscale Model Simul 2008, 6(4):1146-1168.