Fokker-Planck approximation of the master equation in molecular biology

Computing and Visualization in Science

Volumn 12, Issue 1, 2009, Pages 37-50

  Sjöberg, Paul ^a   Lötstedt, Per ^a   Elf, Johan ^a,b  

^a UPPSALA UNIVERSITY   (Sweden)

^b HARVARD UNIVERSITY   (United States)

Author keywords

Fokker Planck equation; Master equation; Numerical solution; Stochastic simulation algorithm

Indexed keywords

BIOCHEMISTRY; CHEMICAL REACTIONS; DIFFERENCE EQUATIONS; GRAFTING (CHEMICAL); MATHEMATICAL MODELS; MOLECULAR BIOLOGY; NUMERICAL ANALYSIS; RECONNAISSANCE AIRCRAFT; STOCHASTIC CONTROL SYSTEMS; SYNTHESIS (CHEMICAL); SYSTEMS ANALYSIS; TIMING JITTER;

CELL PROCESSES; FINITE VOLUMES; HIGH ACCURACIES; LOW DIMENSIONAL; MASTER EQUATION; NUMERICAL SOLUTION; STATE SPACES; STEADY STATES; STOCHASTIC SIMULATION ALGORITHM; TIME DEPENDENTS;

FOKKER PLANCK EQUATION;

EID: 58149480366     PISSN: 14329360     EISSN: 14330369     Source Type: Journal    
DOI: 10.1007/s00791-006-0045-6     Document Type: Article

References (30)

1. D.G. Aronson J. Serrin 1967 Local behavior of solutions of quasilinear parabolic equations Arch. Rat. Mech. Anal. 25 81 122
2. S. Benzer 1953 Induced synthesis of enzymes in bacteria analyzed at the cellular level Biochim. Biophys. Acta 11 383 395
3. O.G. Berg 1978 A model for the statistical fluctuations of protein numbers in a microbial population J. Theor. Biol. 71 587 603
4. O.G. Berg 1978 On diffusion-controlled dissociation J. Chem. Phys. 31 47 57
5. Y. Cao D. Gillespie L. Petzold 2005 Multiscale stochastic simulation algorithm with stochastic partial equilibrium assumption for chemically reacting systems J. Comput. Phys. 206 395 411
6. Dieudonné J. (1969). Foundations of modern analysis, Academic, New York
7. J. Elf O.G. Berg M. Ehrenberg 2001 Comparison of repressor and transcriptional attenuator systems for control of amino acid biosynthetic operons J. Mol. Biol. 313 941 954
8. J. Elf J. Paulsson O.G. Berg M. Ehrenberg 2003 Near-critical phenomena in intracellular metabolite pools Biophys. J. 84 154 170
9. M.B. Elowitz M.G. Surette P.-E. Wolf J.B. Stock S. Leibler 1999 Protein mobility in the cytoplasm of Escherichia coli J. Bacteriol. 181 197 203
10. Érdi P., Tóth J. (1988) Mathematical models of chemical reactions. Princeton University Press, Princeton
11. Ferm, L., Lötstedt, P., Sjöberg, P.: Conservative solution of the Fokker-Planck equation for stochastic chemical reactions. BIT (in press, 2006)
12. Fersht A. (1998) Structure and mechanism in protein science: a guide to enzyme catalysis and protein folding. W. H. Freeman & Co, New York
13. Gardiner C.W. (1985) Handbook of stochastic methods. Springer, Berlin Heidelberg New York
14. T.S. Gardner C.R. Cantor J.J. Collins 2000 Construction of a genetic toggle switch in Escherichia coli Nature 403 339 342
15. Gilbarg D., Trudinger N.S. (1977) Elliptic partial differential equations of second order. Springer, Berlin Heidelberg New York
16. D.T. Gillespie 1976 A general method for numerically simulating the stochastic time evolution of coupled chemical reactions J. Comput. Phys. 22 403 434
17. Greenbaum A. (1997) Iterative methods for solving linear systems. SIAM, Philadelphia
18. Hairer E., Nørsett S.P., Wanner G. (1993) Solving ordinary differential equations, nonstiff problems. 2nd edn., Springer, Berlin Heidelberg New York
19. E.L. Haseltine J.B. Rawlings 2002 Approximate simulation of coupled fast and slow reactions for stochastic chemical kinetics J. Chem. Phys. 117 6959 6969
20. John F. (1980) Partial differential equations. 3rd edn., Springer, Berlin Heidelberg New York
21. M. Kærn T.C. Elston W.J. Blake J.J. Collins 2005 Stochasticity in gene expression: from theories to phenotypes Nat. Rev. Genet. 6 451 464
22. van Kampen N.G. (1992) Stochastic processes in physics and chemistry. Elsevier, Amsterdam
23. Larsen R.J., Marx M.L. (1986) An introduction to mathematical statistics and its applications. 2nd edn., Prentice-Hall, Englewood Cliffs
24. Lehoucq R.B., Sorensen D.C., Yang C. (1998) ARPACK users' guide: solution of large-scale eigenvalue problems with implicitly restarted Arnoldi methods. SIAM, Philadelphia
25. P. Lötstedt S. Söderberg A. Ramage L. Hemmingsson- Frändén 2002 Implicit solution of hyperbolic equations with space-time adaptivity BIT 42 134 158
26. MATLAB: The MathWorks, Inc., Natick, MA, USA, http://www.mathworks.com
27. A. Novick M. Weiner 1957 Enzyme induction as an all-or-none phenomenon Proc. Natl. Acad. Sci. USA 43 553 566
28. G.L.G. Sleijpen H.A. van der Vorst 1996 A Jacobi-Davidson iteration method for linear eigenvalue problems SIAM J. Matrix Anal. Appl. 17 401 425
29. H.A. van der Vorst 1992 Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems SIAM J. Sci. Stat. Comput. 13 631 644
30. X.S. Xie 2002 Single-molecule approach to dispersed kinetics and dynamic disorder: Probing conformational fluctuation and enzymatic dynamics J. Chem. Phys. 117 11024 11032