Galerkin Spectral method applied to the chemical master equation

Communications in Computational Physics

Volumn 5, Issue 5, 2009, Pages 871-896

  Engblom, Stefan ^a  

^a UPPSALA UNIVERSITY   (Sweden)

Author keywords

Chemical reactions; High dimensional problem; Master equation; Moving basis; Spectral Galerkin method

Indexed keywords

EID: 67649603987     PISSN: 18152406     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (50)

1. M. Abramovitz and LA. Stegun. Handbook of Mathematical Functions. Dover, New York, 1970.
2. D. F. Anderson. Incorporating postleap checks in tau-leaping. J. Chem. Phys, 128(5):054103054111, 2008. doi:10.1063/1.2819665.
3. W. J. Anderson. Continuous-Time Markov Chains. Springer Series in Statistics. Springer-Verlag, New York, 1991.
4. B. C. Berndt. Ramanujan's Notebooks, Part I. Springer-Verlag, New York, 1985.
5. K. Burrage, M. Hegland, F. Macnamara, and B. Sidje. A Krylov-based finite state projection algorithm for solving the chemical master equation arising in the discrete modelling of biological systems. In A. N. Langville and W. J. Stewart, editors, Markov Anniversary Meeting: An international conference to celebrate the 150th anniversary of the birth of A. A. Markov, pages 21-38. Boson Books, Raleigh, NC, USA, 2006.
6. Y. Cao, D. Gillespie, and L. Petzold. Multiscale stochastic simulation algorithm with stochastic partial equilibrium assumption for chemically reacting systems. J. Comput. Phys., 206:395-411, 2005. doi:10.1016/j.jcp. 2004.12.014.
7. Y. Cao, L. R. Petzold, M. Rathinam, and D. T. Gillespie. The numerical stability of leaping methods for stochastic simulation of chemically reacting systems. J. Chem. Phys., 121(24):12169-12178, 2004. doi:10.1063/1.1823412.
8. P. Deuflhard, W. Huisinga, T. Jahnke, and M. Wulkow. Adaptive discrete Galerkin methods applied to the chemical master equation. Technical Report 07-04, Zuse Institute Berlin, 2007. http://www.zib.de/bib/pub.
9. P. Deuflhard and M. Wulkow. Computational treatment of polyreaction kinetics by orthogonal polynomials of a discrete variable. IMPACT Comp. Sci. Eng., 1(3):269-301, 1989. doi:10.1016/0899-8248(89)90013-X.
10. T. M. Dunster. Uniform asymptotic expansions for Charlier polynomials. J. Approx. Theory, 112:93-133, 2001. doi:10.1006/jath.2001.3595.
11. E. B. Dynkin. Markov processes, volume I. Academic Press, 1965.
12. W. E, D. Liu, and E. Vanden-Eijnden. Nested stochastic simulation algorithm for chemical kinetic systems with disparate rates. J. Chem. Phys., 123(19), 2005. doi:10.1063/1.2109987.
13. J. Elf and M. Ehrenberg. Spontaneous separation of bi-stable biochemical systems into sparial domains of opposite phases. Syst. Biol., 1(2):230-236, 2004. doi:10.1049/sb:20045021.
14. J. Elf, P. Lötstedt, and P. Sjöberg. Problems of high dimension in molecular biology. In W. Hackbusch, editor, Proceedings of the 19th GAMM-Seminar in Leipzig "High dimensional problems - Numerical Treatement and Applications", pages 21-30, 2003.
15. S. Engblom. Computing the moments of high dimensional solutions of the master equation. Appl. Math. Comput, 180(2):498-515, 2006. doi:10.1016/j.amc. 2005.12.032.
16. S. Engblom. Gaussian quadratures with respect to discrete measures. Technical Report 2006-007, Dept of Information Technology, Uppsala University, Uppsala, Sweden, 2006. http://www.it.uu.se/research.
17. S. Engblom. A discrete spectral method for the chemical master equation. Technical Report 2008-005, Dept of Information Technology, Uppsala University, Uppsala, Sweden, 2008. http://www.it.uu.se/research.
18. L. Perm and P. Lötstedt. Adaptive solution of the master equation in low dimensions. Appl. Numer. Math, In Press, Corrected Proof, 2008.
19. L. Ferm, P. Lotstedt, and P. Sjöberg. Conservative solution of the Fokker-Planck equation for stochastic chemical reactions. BIT, 46:S61-S83, 2006. doi:10.1007/s10543-006-0082-z.
20. C. W. Gardiner. Handbook of Stochastic Methods. Springer Series in Synergetics. Springer-Verlag, Berlin, 3rd edition, 2004.
21. T. S. Gardner, C. R. Cantor, and J. J. Collins. Construction of a genetic toggle switch in Escherichia coli. Nature, 403:339-342, 2000. doi:10.1038/35002131.
22. B. Gaveau, M. Moreau, and J. Toth. Master equation and Fokker-Planck equation: Comparison of entropy and of rate constants. Letters in Math. Phys., 40(2):101-115, 1997.
23. D. T. Gillespie. A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Comput. Phys., 22(4):403-434, 1976. doi:10.1016/0021-9991(76)90041-3.
24. D. T. Gillespie. A rigorous derivation of the chemical master equation. Physica A, 188:404-425, 1992. doi:10.1016/0378-4371(92)90283-V.
25. D. T. Gillespie. Approximate accelerated stochastic simulation of chemically reacting systems. J. Chem. Phys., 115(4):1716-1733, 2001. doi:10.1063/1.1378322.
26. P. Guptasarama. Does replication-induced transcription regulate synthesis of the myriad low copy number proteins of Escherichia coli? Bioessays, 17(11):987-997, 1995. doi: 10.1002/bies.950171112.
27. E. L. Haseltine and J. B. Rawlings. Approximate simulation of coupled fast and slow reactions for stochastic chemical kinetics. J. Chem. Phys., 117(15), 2002. doi:10.1063/1.1505860.
28. M. Haviv. Aggregation/disaggregation method for computing the stationary distribution of a Markov chain. SIAM J. Numer. Anal., 24(4):952-966, 1987. doi:10.1137/0724062.
29. M. Hegland, C. Burden, L. Santoso, S. MacNamara, and H. Booth. A solver for the stochastic master equation applied to gene regulatory networks. J. Comput. Appl. Math., 205(2):708-724, 2007. doi:10.1016/j.cam.2006.02.053.
30. J. S. Hesthaven, S. Gottlieb, and D. Gottlieb. Spectral Methods for Time-Dependent Problems. Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge, 2007.
31. N. G. van Kampen. Stochastic Processes in Physics and Chemistry. Elsevier, Amsterdam, 5th edition, 2004.
32. R. Koekoek and R. F. Swarttouw. The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue. Technical Report 98-17, Dept. of Technical Mathematics and Informatics, Delft University of Technology, 1998. http://aw.twi.tudelft.nl/~koekoek/askey.html.
33. T. Li. Analysis of explicit tau-leaping schemes for simulating chemically reacting systems. Multiscale Model. Simul., 6(2):417-436, 2007. doi:10.1137/06066792X.
34. T. Li, A. Abdulle, and W. E. Effectiveness of implicit methods for stiff stochastic differential equations. Commun. Comput. Phys., 3(2):295-307, 2008.
35. P. Lötstedt and L. Ferm. Dimensional reduction of the Fokker-Planck equation for stochastic chemical reactions. Multiscale Model. Simul., 5:593-614, 2006.
36. S. MacNamara, K. Burrage, and R. B. Sidje. Multiscale modeling of chemical kinetics via the master equation. Multiscale Model. Simul., 6(4):1146-1168, 2008. doi:10.1137/060678154.
37. H. G. Matthies and A. Keese. Galerkin methods for linear and nonlinear stochastic partial differential equations. Comput. Methods Appl. Mech. Engrg., 194(12-16):1295-1331, 2005. doi:10.1016/j.cma.2004.05.027.
38. B. Munsky and M. Khammash. The finite state projection algorithm for the solution of the chemical master equation. J. Chem. Phys., 124(4):044104, 2006. doi:10.1063/1.2145882.
39. F. W. J. Olver. Asymptotics and Special Functions. Academic Press, New York, 1974.
40. M. Rathinam, L. Petzold, Y. Cao, and D. T. Gillespie. Stiffness in stochastic chemically reacting systems: The implicit tau-leaping method. J. Chem. Phys., 119(24):12784-12794, 2003. doi:10.1063/1.1627296.
41. M. Rathinam, L. R. Petzold, Y. Cao, and D. T. Gillespie. Consistency and stability of tauleaping schemes for chemical reaction systems. Multiscale Model. Simul., 4(3):867-895, 2005. doi:10.1137/040603206.
42. M. S. Samoilov and A. P. Arkin. Deviant effects in molecular reaction pathways. Nature Biotech., 24:1235-1240, 2006. doi:10.1038/nbt1253.
43. J. Shen. Stable and efficient spectral methods in unbounded domains using Laguerre functions. SIAM J. Numer. Anal., 38(4):1113-1133, 2000.
44. P. Sjöberg, P. Lötstedt, and J. Elf. Fokker-Planck approximation of the master equation in molecular biology. Comp. Vis. Sci., To appear, 2008. doi:10.1007/s00791-006-0045-6.
45. Y. Togashi and K. Kaneko. Molecular discreteness in reaction-diffusion systems yields steady states not seen in the continuum limit. Phys. Rev. E, 70(2):020901-l-020901-4, 2004. doi:10.1103/PhysRevE.70.020901.
46. J. M. G. Vilar, H. Y. Kueh, N. Barkai, and S. Leibler. Mechanism of noise-resistance in genetic oscillators. Proc. Nat. Acad. Sci., 99:5988-5992, 2002. doi:10.1073/pnas.092133899.
47. M. Wulkow. Adaptive treatment of polyreactions in weighted sequence spaces. IMPACT Comp. Sci. Eng., 4(2):153-193, 1992. doi:10.1016/0899-8248(92) 90020-9.
48. M. Wulkow. The simulation of molecular weight distributions in polyreaction kinetics by discrete Galerkin methods. Macromol. Theory. Simul., 5(3):393-416, 1996. doi:10.1002/mats.1996.040050303.
49. D. Xiu and J. S. Hesthaven. High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput., 27(3):1118-1139, 2005. doi:10.1137/040615201.
50. D. Xiu and G. E. Karniadakis. The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput, 24(2):619-644, 2002. doi:10.1137/S1064827501387826.