Fast adaptive uniformization of the chemical master equation

HiBi09 - 2009 International Workshop on High Performance Computational Systems Biology

Volumn , Issue , 2009, Pages 118-127

  Didier, Frédéric ^a   Henzinger, Thomas A ^a,c   Mateescu, Maria ^a   Wolf, Verena ^b  

^a EPFL   (Switzerland)

^b SAARLAND UNIVERSITY   (Germany)

^c INSTITUTE OF SCIENCE AND TECHNOLOGY AUSTRIA   (Austria)

Author keywords

Biochemical reactions; Chemical master equation; Markov chains; Transient analysis; Uniformization

Indexed keywords

BIOCHEMICAL REACTIONS; CHEMICAL MASTER EQUATION; MARKOV CHAIN; MARKOV CHAINS; UNIFORMIZATION;

APPROXIMATION ALGORITHMS; BIOLOGY; CHEMICAL ANALYSIS; COMPUTATIONAL EFFICIENCY; MARKOV PROCESSES; STOCHASTIC SYSTEMS;

TRANSIENT ANALYSIS;

EID: 73449119937     PISSN: None     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.1109/HiBi.2009.23     Document Type: Conference Paper

References (47)

1. A. Arkin, J. Ross, and H. H. McAdams. Stochastic kinetic analysis of developmental pathway bifurcation in phage λ-infected E. coli cells. Genetics, 149:1633 - 1648, 1998.
2. N. Barkai and S. Leibler. Biological rhythms: Circadian clocks limited by noise. Nature, 403:267 - 268, 2000.
3. 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, Proceedings of the Markov 150th Anniversary Conference, pages 21 - 38. Boson Books, 2006.
4. H. Busch, W. Sandmann, and V. Wolf. A numerical aggregation algorithm for the enzyme-catalyzed substrate conversion. In Proc, of CMSB, volume 4210 of LNCS, pages 298 - 311. Springer, 2006.
5. E. de Souza e Silva and R. Gail. Transient solutions for Markov chains. In Computational Probability, chapter 3, pages 43 - 79. Kluwer Academic Publishers, 2000.
6. F. Didier, T. A. Henzinger, M. Mateescu, and V. Wolf. Approximation of event probabilities in noisy cellular processes. In Proc. of CMSB, 2009. To appear.
7. J. D. Diener and W. H. Sanders. Empirical comparison of uniformization methods for continuous-time Markov chains. In Computations with Markov Chains, pages 547 - 570. Kluwer Academic Publishers, Boston, 1995.
8. M. B. Elowitz, M. J. Levine, E. D. Siggia, and P. S. Swain. Stochastic gene expression in a single cell. Science, 297:1183 - 1186, 2002.
9. S. Engblom. Galerkin spectral method applied to the chemical master equation. Comm. Comput. Phys., 5:871 - 896, 2009.
10. N. Fedoroff and W. Fontana. Small numbers of big molecules. Science, 297:1129 - 1131, 2002.
11. W. Feller. An Introduction to Probability Theory and Its Applications, Volume 1. Wiley, January 1968.
12. B. L. Fox and P. W. Glynn. Computing Poisson probabilities. Communications of the ACM, 31(4):440 - 445, 1988.
13. D. Gonze, J. Halloy, and A. Goldbeter. Robustness of circadian rhythms with respect to molecular noise. PNAS, USA, 99(2):673 - 678, 2002.
14. J. Goutsias. Quasiequilibrium approximation of fast reaction kinetics in stochastic biochemical systems. J. Chem. Phys., 122(18):184102, 2005.
15. W. K. Grassmann, editor. Computational Probability. Kluwer Academic Publishers, 2000.
16. D. Gross and D. Miller. The randomization technique as a modeling tool and solution procedure for transient Markov processes. Operations Research, 32(2):926 - 944, 1984.
17. E. M. Hahn, H. Hermanns, B. Wachter, and L. Zhang. INFAMY: An infinite-state Markov model checker. In Proc. CAV, LNCS. Springer, 2009. To appear.
18. E. L. Haseltine and J. B. Rawlings. Approximate simulation of coupled fast and slow reactions for chemical kinetics. J. Chem. Phys., 117:6959 - 6969, 2002.
19. 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., 205:708 - 724, 2007.
20. A. Hellander. Efficient computation of transient solutions of the chemical master equation based on uniformization and quasi-Monte carlo. J. Chem. Phys., 128(15):154109, 2008.
21. D. A. Henderson, R. J. Boys, C. J. Proctor, and D. J. Wilkinson. Linking systems biology models to data: a stochastic kinetic model of p53 oscillations. In A. O'Hagan and M. West, editors, Handbook of Applied Bayesian Analysis. Oxford University Press, 2009.
22. T. Henzinger, M. Mateescu, and V. Wolf. Sliding window abstraction for infinite Markov chains. In Proc. CAV, volume 5643 of LNCS, pages 337 - 352. Springer, 2009.
23. T. A. Henzinger, B. Jobstmann, and V. Wolf. Formalisms for specifying Markovian population models. In Proc. LIX Colloquium Reachability Problems, volume 5797 of LNCS. Springer, 2009.
24. A. Jensen. Markoff chains as an aid in the study of Markoff processes. Skandinavisk Aktuarietidskrift, 36:87 - 91, 1953.
25. N. G. v. Kampen. Stochastic Processes in Physics and Chemistry. Elsevier, 3rd edition, 2007.
26. H. Maamar, A. Raj, and D. Dubnau. Noise in gene expression determines cell fate in Bacillus subtilis. Science, 317(5837):526 - 529, 2007.
27. R. A. Marie, A. L. Reibman, and K. S. Trivedi. Transient analysis of acyclic Markov chains. Perform. Eval., 7(3):175 - 194, 1987.
28. H. H. McAdams and A. Arkin. Stochastic mechanisms in gene expression. PNAS, USA, 94:814 - 819, 1997.
29. H. H. McAdams and A. Arkin. It's a noisy business! Trends in Genetics, 15(2):65 - 69, 1999.
30. C. B. Moler and C. F. Van Loan. Nineteen dubious ways to compute the exponential of a matrix. SIAM Review, 20(4):801 - 836, 1978.
31. B. Munsky and M. Khammash. The finite state projection algorithm for the solution of the chemical master equation. J. Chem. Phys., 124:044144, 2006.
32. P. Patel, B. Arcangioli, S. Baker, A. Bensimon, and N. Rhind. DNA replication origins fire stochastically in fission yeast. Mol. Biol. Cell, 17:308 - 316, 2006.
33. J. Paulsson. Summing up the noise in gene networks. Nature, 427(6973):415 - 418, 2004.
34. S. Peles, B. Munsky, and M. Khammash. Reduction and solution of the chemical master equation using time scale separation and finite state projection. J. Chem. Phys., 125:204104, 2006.
35. B. Philippe and R. Sidje. Transient solutions of Markov processes by Krylov subspaces. In Proc. of the 2nd International Workshop on the Numerical Solution of Markov Chains, pages 95 - 119. Kluwer Academic Publishers, 1995.
36. C. Rao, D. Wolf, and A. Arkin. Control, exploitation and tolerance of intracellular noise. Nature, 420(6912):231 - 237, 2002.
37. A. Reibman and K. Trivedi. Numerical transient analysis of Markov models. Comput. Oper. Res., 15(1):19 - 36, 1988.
38. R. Sidje, K. Burrage, and S. MacNamara. Inexact uniformization method for computing transient distributions of Markov chains. SIAM J. Sci. Comput., 29(6):2562 - 2580, 2007.
39. R. Sidje and W. Stewart. A survey of methods for computing large sparse matrix exponentials arising in Markov chains. In Markov Chains, Computational Statistics and Data Analysis 29, pages 345 - 368, 1996.
40. P. Sj̈oberg. Numerical Methods for Stochastic Modeling of Genes and Proteins. Phd thesis, Uppsala University, Sweden, 2007.
41. P. Sj̈oberg, P. L̈ostedt, and J. Elf. Fokker-Planck approximation of the master equation in molecular biology. CComputing and Visualization in Science, 12:37 - 50, 2009.
42. W. J. Stewart. Introduction to the Numerical Solution of Markov Chains. Princeton University Press, 1995.
43. M. Thattai and A. van Oudenaarden. Intrinsic noise in gene regulatory networks. PNAS, USA, 98(15):8614 - 8619, July 2001.
44. A. van Moorsel and W. Sanders. Adaptive uniformization. ORSA Communications in Statistics: Stochastic Models, 10(3):619 - 648, 1994.
45. A. Warmflash and A. Dinner. Signatures of combinatorial regulation in intrinsic biological noise. PNAS, 105(45):17262 - 17267, 2008.
46. J. Zhang, L. T. Watson, and Y. Cao. A modified uniformization method for the solution of the chemical master equation. Technical Report TR-07-31, Computer Science, Virginia Tech., 2007.
47. L. Zhang, H. Hermanns, E. M. Hahn, and B. Wachter. Time-bounded model checking of infinite-state continuous-time Markov chains. In ACSD, 2008. China.