Solving the chemical master equation using sliding windows

BMC Systems Biology

Volumn 4, Issue , 2010, Pages

  Wolf, Verena ^a   Goel, Rushil ^b   Mateescu, Maria ^c   Henzinger, Thomas A ^d  

^a SAARLAND UNIVERSITY   (Germany)

^b INDIAN INSTITUTE OF TECHNOLOGY BOMBAY   (India)

^c EPFL   (Switzerland)

^d INSTITUTE OF SCIENCE AND TECHNOLOGY AUSTRIA   (Austria)

Author keywords

[No Author keywords available]

Indexed keywords

ENZYME;

ALGORITHM; ARTICLE; CHEMICAL MODEL; CHEMISTRY; COMPUTER PROGRAM; COMPUTER SIMULATION; GENE EXPRESSION PROFILING; GENE EXPRESSION REGULATION; PROBABILITY; REPRODUCIBILITY; STATISTICAL MODEL; STATISTICS; SYSTEMS BIOLOGY; THEORETICAL MODEL;

ALGORITHMS; COMPUTER SIMULATION; ENZYMES; GENE EXPRESSION PROFILING; GENE EXPRESSION REGULATION; MODELS, CHEMICAL; MODELS, STATISTICAL; MODELS, THEORETICAL; PROBABILITY; REPRODUCIBILITY OF RESULTS; SOFTWARE; STOCHASTIC PROCESSES; SYSTEMS BIOLOGY;

EID: 77950644387     PISSN: None     EISSN: 17520509     Source Type: Journal    
DOI: 10.1186/1752-0509-4-42     Document Type: Article

References (54)

1. Elowitz MB, Levine MJ, Siggia ED, Swain PS. Stochastic Gene Expression in a Single Cell. Science 2002, 297:1183-1186. 10.1126/science.1070919, 12183631.
2. Ozbudak EM, Thattai M, Kurtser I, Grossman AD, van Oudenaarden A. Regulation of Noise in the Expression of a Single Gene. Nature Genetics 2002, 31:69-73. 10.1038/ng869, 11967532.
3. Blake WJ, Kaern M, Cantor CR, Collins JJ. Noise in Eukaryotic Gene Expression. Nature 2003, 422:633-637. 10.1038/nature01546, 12687005.
4. Fedoroff N, Fontana W. Small Numbers of Big Molecules. Science 2002, 297:1129-1131. 10.1126/science.1075988, 12183614.
5. Kierzek A, Zaim J, Zielenkiewicz P. The Effect of Transcription and Translation Initiation Frequencies on the Stochastic Fluctuations in Prokaryotic Gene Expression. J Biol Chem 2001, 276(11):8165-8172. 10.1074/jbc.M006264200, 11062240.
6. Paulsson J. Summing up the noise in gene networks. Nature 2004, 427(6973):415-418. 10.1038/nature02257, 14749823.
7. Swain PS, Elowitz MB, Siggia ED. Intrinsic and extrinsic contributions to stochasticity in gene expression. PNAS, USA 2002, 99(20):12795-12800.
8. Turner TE, Schnell S, Burrage K. Stochastic approaches for modelling in vivo reactions. Comp Biol Chem 2004, 28:165-178.
9. Wilkinson DJ. Stochastic Modelling for Systems Biology 2006, Chapman & Hall.
10. McAdams HH, Arkin A. It's a noisy business!. Trends in Genetics 1999, 15(2):65-69. 10.1016/S0168-9525(98)01659-X, 10098409.
11. Rao C, Wolf D, Arkin A. Control, exploitation and tolerance of intracellular noise. Nature 2002, 420(6912):231-237. 10.1038/nature01258, 12432408.
12. Thattai M, van Oudenaarden A. Intrinsic Noise in Gene Regulatory Networks. PNAS, USA 2001, 98(15):8614-8619.
13. McAdams HH, Arkin A. Stochastic mechanisms in gene expression. PNAS, USA 1997, 94:814-819.
14. 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. 1460268, 9691025.
15. Srivastava R, You L, Summers J, Yin J. Stochastic vs. Deterministic Modeling of Intracellular Viral Kinetics. J Theor Biol 2002, 218:309-321. 10.1006/jtbi.2002.3078, 12381432.
16. Gillespie DT. Markov Processes 1992, Academic Press.
17. Gillespie DT. Exact Stochastic Simulation of Coupled Chemical Reactions. J Phys Chem 1977, 81(25):2340-2361.
18. Gillespie DT. Approximate Accelerated Stochastic Simulation of Chemically Reacting Systems. J Chem Phys 2001, 115(4):1716-1732.
19. Stewart WJ. Introduction to the Numerical Solution of Markov Chains 1995, Princeton University Press.
20. Munsky B, Khammash M. The finite state projection algorithm for the solution of the chemical master equation. J Chem Phys 2006, 124:044144.
21. Burrage K, Hegland M, Macnamara F, Sidje B. A Krylov-based Finite State Projection algorithm for solving the chemical master equation arising in the discrete modelling of biological systems. Proc of the Markov 150th Anniversary Conference 2006, 21-38. Boson Books, Langville AN, Stewart WJ.
22. Sjöberg P. Numerical Methods for Stochastic Modeling of Genes and Proteins. Phd thesis 2007, Uppsala University, Sweden.
23. Sjöberg P, Lötstedt P, Elf J. Fokker-Planck approximation of the master equation in molecular biology. Computing and Visualization in Science 2009, 12:37-50.
24. Hegland M, Burden C, Santoso L, Macnamara S, Booth H. A solver for the stochastic master equation applied to gene regulatory networks. J Comput Appl Math 2007, 205:708-724.
25. Engblom S. Galerkin spectral method applied to the chemical master equation. Comm Comput Phys 2009, 5:871-896.
26. Busch H, Sandmann W, Wolf V. A Numerical Aggregation Algorithm for the Enzyme-Catalyzed Substrate Conversion. Proc of CMSB 2006, 4210:298-311. LNCS, Springer.
27. Peles S, Munsky B, Khammash M. Reduction and Solution of the chemical master equation using time scale separation and finite state projection. J Chem Phys 2006, 125:204104. 10.1063/1.2397685, 17144687.
28. Cao Y, Gillespie D, Petzold L. Efficient step size selection for the tau-leaping simulation method. J Chem Phys 2006, 124(4):044109-11. 10.1063/1.2159468, 16460151.
29. Kampen N. Stochastic Processes in Physics and Chemistry 2007, Elsevier, 3.
30. Çinlar E. Introduction to Stochastic Processes 1975, Prentice-Hall.
31. Moler CB, Van Loan CF. Nineteen Dubious Ways to Compute the Exponential of a Matrix. SIAM Review 1978, 20(4):801-836.
32. Grassmann WK, . (Ed) Computational Probability 2000, Kluwer Academic Publishers, (Ed).
33. Sidje R, Stewart W. A survey of methods for computing large sparse matrix exponentials arising in Markov chains. Markov Chains, Computational Statistics and Data Analysis 29 1996, 345-368.
34. Jensen A. Markoff chains as an aid in the study of Markoff processes. Skandinavisk Aktuarietidskrift 1953, 36:87-91.
35. Gross D, Miller D. The randomization technique as a modeling tool and solution procedure for transient Markov processes. Operations Research 1984, 32(2):926-944.
36. Philippe B, Sidje R. Transient solutions of Markov processes by Krylov subspaces. Proc of the 2nd Int Workshop on the Numerical Solution of Markov Chains 1995, 95-119. Kluwer Academic Publishers.
37. Hairer E, Norsett S, Wanner G. Solving Ordinary Differential Equations I: Nonstiff Problems 2008, Springer.
38. Hairer E, Wanner G. Solving Ordinary Differential Equations II 2004, Stiff and Differential-Algebraic Problems Springer.
39. Reibman A, Trivedi K. Numerical transient analysis of Markov models. Comput Oper Res 1988, 15:19-36.
40. Zhang J, Watson LT, Cao Y. A Modified Uniformization Method for the Solution of the Chemical Master Equation. Tech. Rep. TR-07-31, Computer Science, Virginia Tech 2007,
41. Hellander A. Efficient computation of transient solutions of the chemical master equation based on uniformization and quasi-Monte Carlo. J Chem Phys 2008, 128(15):154109. 10.1063/1.2897976, 18433192.
42. Sidje R, Burrage K, MacNamara S. Inexact Uniformization Method for Computing Transient Distributions of Markov Chains. SIAM J Sci Comput 2007, 29(6):2562-2580.
43. de Souza e Silva E, Gail R. Transient Solutions for Markov Chains. Computational Probability 2000, 43-79. Kluwer Academic Publishers, Grassmann WK.
44. Fox BL, Glynn PW. Computing Poisson probabilities. Communications of the ACM 1988, 31(4):440-445.
45. Dunkel J, Stahl H. On the transient analysis of stiff Markov chains. Proc of the 3rd IFIP Working Conference on Dependable Computing for Critical Applications 1993, 137-160.
46. Gallopoulos E, Saad Y. On the parallel solution of parabolic equations. Proc. ACM SIGARCH-89 1989, 17-28. ACM press.
47. Saad Y. Analysis of some Krylov subspace approximations to the matrix exponential operator. SIAM J Numer Anal 1992, 29:209-228.
48. Sidje RB. EXPOKIT: Software Package for Computing Matrix Exponentials. ACM Transactions on Mathematical Software 1998, 24:130-156.
49. Baker G. The essentials of Padé approximants 1975, Academic Press, New York.
50. Goutsias J. Quasiequilibrium Approximation of Fast Reaction Kinetics in Stochastic Biochemical Systems. J Chem Phys 2005, 122(18):184102. 10.1063/1.1889434, 15918689.
51. Gardner T, Cantor C, Collins J. Construction of a genetic toggle switch in Escherichia coli. Nature 2000, 403:339-342. 10.1038/35002131, 10659857.
52. Didier F, Henzinger TA, Mateescu M, Wolf V. Approximation of Event Probabilities in Noisy Cellular Processes. Proc. of CMSB 2009, 5688:173. LNBI.
53. Ferm L, Lötstedt P, Hellander A. A Hierarchy of Approximations of the Master Equation Scaled by a Size Parameter. Journal of Scientific Computing 2008, 34:127-151.
54. Henzinger T, Mateescu M, Wolf V. Sliding Window Abstraction for Infinite Markov Chains. Proc CAV 2009, 5643:337-352. LNCS, Springer.