A hierarchy of approximations of the master equation scaled by a size parameter

Journal of Scientific Computing

Volumn 34, Issue 2, 2008, Pages 127-151

  Ferm, Lars ^a   Lötstedt, Per ^a   Hellander, Andreas ^a  

^a UPPSALA UNIVERSITY   (Sweden)

Author keywords

Linear noise approximation; Master equation; Moment equations; Reaction rate equations

Indexed keywords

LINEAR NOISE APPROXIMATION; MASTER EQUATION; MOMENT EQUATIONS; REACTION RATE EQUATIONS;

APPROXIMATION ALGORITHMS; LINEAR EQUATIONS; MOLECULAR BIOLOGY; ORDINARY DIFFERENTIAL EQUATIONS; PARAMETER ESTIMATION; POLYNOMIALS;

HIERARCHICAL SYSTEMS;

EID: 38349133325     PISSN: 08857474     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10915-007-9179-z     Document Type: Article

References (44)

1. Barkai, N., Leibler, S.: Circadian clocks limited by noise. Nature 403, 267-268 (2000)
2. Caflisch, R.E.: Monte Carlo and quasi-Monte Carlo methods. Acta Numer. 7, 1-49 (1998)
3. Cao, Y., Gillespie, D., Petzold, L.: Multiscale stochastic simulation algorithm with stochastic partial equilibrium assumption for chemically reacting systems. J. Comput. Phys. 206, 395-411 (2005)
4. Chen, W.-Y., Bokka, A.: Stochastic modeling of nonlinear epidemiology. J. Theor. Biol. 234, 455-470 (2005)
5. Dieckmann, U., Marrow, P., Law, R.: Evolutionary cycling in predator-prey interactions: population dynamics and the Red Queen. J. Theor. Biol. 176, 91-102 (1995)
6. Dieudonné, J.: Foundations of Modern Analysis. Academic Press, New York (1969)
7. E, W., Liu, D., Vanden-Eijnden, E.: Nested stochastic simulation algorithm for chemical kinetic systems with multiple time scales. J. Comput. Phys. 221, 158-180 (2007)
8. Elf, J., Ehrenberg, M.: Fast evaluation of fluctuations in biochemical networks with the linear noise approximation. Genome Res. 13, 2475-2484 (2003)
9. Elf, J., Lötstedt, P., Sjöberg, P.: Problems of high dimension in molecular biology. In: Hackbusch, W. (ed.) High-Dimensional Problems-Numerical Treatment and Applications. Proceedings of the 19th GAMM-Seminar, Leipzig, 2003, pp. 21-30; available at http://www.mis.mpg.de/ conferences/gamm/2003/ (2003)
10. Elf, J., Paulsson, J., Berg, O.G., Ehrenberg, M.: Near-critical phenomena in intracellular metabolite pools. Biophys. J. 84, 154-170 (2003)
11. Engblom, S.: Computing the moments of high dimensional solutions of the master equation. Appl. Math. Comput. 180, 498-515 (2006)
12. Engquist, B., Runborg, O.: Computational high frequency wave propagation. Acta Numer. 3, 181-266 (2003)
13. Escudera, C., Buceta, J., de la Rubia, F.J., Lindenberg, K.: Extinction in population dynamics. Phys. Rev. E 69, 021908 (2004)
14. Ethier, S.N., Kurtz, T.G.: Markov Processes, Characterization and Convergence. Wiley, New York (1986)
15. Faure, H.: Discrépance de suites associées à un système de numération (en dimension s). Acta Aritm. 41, 337-351 (1982)
16. Ferm, L., Lötstedt, P.: Adaptive solution of the master equation in low dimensions. Technical report 2007-023, Department of Information Technology, Uppsala University, Uppsala, Sweden; available at http://www.it.uu.se/research/ publications/reports/2007-023/ (2007)
17. Ferm, L., Lötstedt, P., Sjöberg, P.: Conservative solution of the Fokker-Planck equation for stochastic chemical reactions. BIT 46, S61-S83 (2006)
18. Fox, R.F., Keizer, J.: Amplification of intrinsic fluctuations by chaotic dynamics in physical systems. Phys. Rev. A 43, 1709-1720 (1991)
19. Gardiner, C.W.: Handbook of Stochastic Methods, 3rd edn. Springer, Berlin (2004)
20. Giles, M.B., Süli, E.: Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality. Acta Numer. 11, 145-236 (2002)
21. Gillespie, D.T.: A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Comput. Phys. 22, 403-434 (1976)
22. Givon, D., Kupferman, R., Stewart, A.: Extracting macroscopic dynamics: model problems and algorithms. Nonlinearity 17, R55-R127 (2004)
23. Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I. Nonstiff Problems, 2nd edn. Springer, Berlin (1993)
24. Hong, H.S., Hickernell, F.J.: Algorithm 823: implementing scrambled digital sequences. ACM Trans. Math. Softw. 29, 95-109 (2003)
25. Kurtz, T.G.: Solutions of ordinary differential equations as limits of pure jump Markov processes. J. Appl. Probab. 7, 49-58 (1970)
26. Kurtz, T.G.: Limit theorems for sequences of jump Markov processes approximating ordinary differential processes. J. Appl. Probab. 7, 344-356 (1971)
27. Lötstedt, P., Ferm, L.: Dimensional reduction of the Fokker-Planck equation for stochastic chemical reactions. Multiscale Methods Simul. 5, 593-614 (2006)
28. McAdams, H.H., Arkin, A.: Stochastic mechanisms in gene expression. Proc. Nat. Acad. Sci. USA 94, 814-819 (1997)
29. McAdams, H.H., Arkin, A.: It's a noisy business. Genetic regulation at the nanomolar scale. Trends Genet. 15, 65-69 (1999)
30. McKane, A.J., Newman, T.J.: Stochastic models in population biology and their deterministic analogs. Phys. Rev. E 70, 041902 (2004)
31. Murray, J.D.: Mathematical Biology I. An Introduction, 3rd edn. Springer, New York (2002)
32. Oden, J.T., Prudhomme, S.: Estimation of modeling error in computational mechanics. J. Comput. Phys. 182, 496-515 (2002)
33. Owen, A.B.: Monte Carlo variance of scrambled net quadrature. SIAM J. Numer. Anal. 34, 1884-1910 (1997)
34. Risken, H.: The Fokker-Planck Equation, 2nd edn. Springer, Berlin (1996)
35. P. Sjöberg P. Lötstedt J. Elf 2007 Fokker-Planck approximation of the master equation in molecular biology Comput. Vis. Sci. 10.1007/s00791-006-0045-6 Sjöberg, P., Lötstedt, P., Elf, J.: Fokker-Planck approximation of the master equation in molecular biology. Comput. Vis. Sci. (2007). doi: 10.1007/s00791-006-0045-6
36. Stollenwerk, N., Jensen, V.A.A.: Meningitis, pathogenicity near criticality: the epidemiology of meningococcal disease as a model for accidental pathogens. J. Theor. Biol. 222, 347-359 (2003)
37. Strogatz, S.H.: Nonlinear Dynamics and Chaos. Perseus Books, Cambridge (1994)
38. Succi, S.: The Lattice Boltzmann Equation for Fluid Dynamics and Beyond. Clarendon, Oxford (2001)
39. van Kampen, N.G.: The expansion of the master equation. Adv. Chem. Phys. 34, 245-309 (1976)
40. van Kampen, N.G.: Stochastic Processes in Physics and Chemistry. North-Holland, Amsterdam (1992)
41. Vilar, J.M.G., Kueh, H.Y., Barkai, N., Leibler, S.: Mechanisms of noise-resistance in genetic oscillators. Proc. Nat. Acad. Sci. USA 99, 5988-5992 (2002)
42. Weidlich, W.: Sociodynamics. A Systematic Approach to Mathematical Modelling in the Social Sciences. Taylor and Francis, London (2000)
43. Weidlich, W.: Thirty years of sociodynamics. An integrated strategy of modelling in the social sciences: applications to migration and urban evolution. Chaos Solit. Fract. 24, 45-56 (2005)
44. Wilcox, D.C.: Turbulence modeling for CFD. DCW Industries, La Cañada, CA (1994)