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Hybrid method for the chemical master equation. (English) Zbl 1126.80010
Summary: The chemical master equation is solved by a hybrid method coupling a macroscopic, deterministic description with a mesoscopic, stochastic model. The molecular species are divided into one subset where the expected values of the number of molecules are computed and one subset with species with a stochastic variation in the number of molecules. The macroscopic equations resemble the reaction rate equations and the probability distribution for the stochastic variables satisfy a master equation. The probability distribution is obtained by the stochastic simulation algorithm due to Gillespie. The equations are coupled via a summation over the mesoscale variables. This summation is approximated by quasi-Monte Carlo methods. The error in the approximations is analyzed. The hybrid method is applied to three chemical systems from molecular cell biology.

MSC:
80M25 Other numerical methods (thermodynamics) (MSC2010)
80A32 Chemically reacting flows
65C05 Monte Carlo methods
65H10 Numerical computation of solutions to systems of equations
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
Software:
Algorithm 823
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References:
[1] Barkai, N.; Leibler, S., Circadian clocks limited by noise, Nature, 403, 267-268, (2000)
[2] Burrage, K.; Hegland, M.; MacNamara, S.; Sidje, R.B., A Krylov-based finite state projection algorithm for solving the chemical master equation arising in the discrete modeling of biological systems, (), 21-38
[3] Caflisch, R.E., Monte Carlo and quasi-Monte Carlo methods, Acta numer., 1-49, (1998) · Zbl 0949.65003
[4] 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) · Zbl 1088.80004
[5] Cao, Y.; Gillespie, D.T.; Petzold, L.R., The slow-scale stochastic simulation algorithm, J. chem. phys., 122, 014116, (2005)
[6] Cormen, T.H.; Leiserson, C.E.; Rivest, R.L.; Stein, C., Introduction to algorithms, (2001), The MIT Press Cambridge, MA · Zbl 1047.68161
[7] E, W.; Liu, D.; Vanden-Eijnden, E., Nested stochastic simulation algorithm for chemical kinetic systems with disparate rates, J. chem. phys., 123, 194107, (2005)
[8] 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) · Zbl 1162.80003
[9] Elf, J.; Paulsson, J.; Berg, O.G.; Ehrenberg, M., Near-critical phenomena in intracellular metabolite pools, Biophys. J., 84, 154-170, (2003)
[10] Engblom, S., Computing the moments of high dimensional solutions of the master equation, Appl. math. comput., 180, 498-515, (2006) · Zbl 1103.65011
[11] Erban, R.; Kevrekidis, I.G.; Adalsteinsson, D.; Elston, T.C., Gene regulatory networks: a coarse-grained equation-free approach to multiscale computation, J. chem. phys., 124, 084106, (2006)
[12] Faure, H., Discrépance de suites associées à un système de numération (en dimension s), Acta aritm., 41, 337-351, (1982) · Zbl 0442.10035
[13] L. Ferm, P. Lötstedt, Numerical method for coupling the macro and meso scales in stochastic chemical kinetics, Technical Report 2006-001, Department of Information Technology, Uppsala University, Uppsala, Sweden, 2006. Available at <http://www.it.uu.se/research/publications/reports/2006-001/>.
[14] Ferm, L.; Lötstedt, P.; Sjöberg, P., Conservative solution of the fokker – planck equation for stochastic chemical reactions, Bit, 46, S61-S83, (2006) · Zbl 1105.65091
[15] Gardiner, C.W., Handbook of stochastic methods, (2002), Springer Berlin · Zbl 0862.60050
[16] Gillespie, D.T., A general method for numerically simulating the stochastic time evolution of coupled chemical reactions, J. comput. phys., 22, 403-434, (1976)
[17] Gillespie, D.T., Approximate accelerated stochastic simulation of chemically reacting systems, J. chem. phys., 115, 1716-1733, (2001)
[18] Glasserman, P., Monte Carlo methods in financial engineering, (2004), Springer New York · Zbl 1038.91045
[19] Hairer, E.; Nørsett, S.P.; Wanner, G., Solving ordinary differential equations I, nonstiff problems, (1993), Springer-Verlag Berlin · Zbl 0789.65048
[20] Haseltine, E.; Rawlings, J., Approximate simulation of coupled fast and slow reactions for stochastic chemical kinetics, J. chem. phys., 117, 6959-6969, (2002)
[21] 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., 205, 708-724, (2007) · Zbl 1121.65009
[22] Hong, H.S.; Hickernell, F.J., Algorithm 823: implementing scrambled digital sequences, ACM trans. math. softw., 29, 95-109, (2003) · Zbl 1068.11049
[23] Hucka, M.; Finney, A.; Sauro, A.; Bolouri, H.M.; Doyle, H.; Kitano, J.C., The systems biology markup language (SBML): a medium for representation and exchange of biochemical network models, Bioinformatics, 19, 524-531, (2003)
[24] van Kampen, N.G., Stochastic processes in physics and chemistry, (1992), North-Holland Amsterdam · Zbl 0974.60020
[25] Kiehl, T.R.; Mattheyses, R.M.; Simmons, M.K., Hybrid simulation of cellular behaviour, Bioinformatics, 20, 316-322, (2004)
[26] Levchenko, A.; Bruck, J.; Sternberg, P.W., Scaffold proteins may biphasically affect the levels of mitogen-activated protein kinase signaling and reduce its threshold properties, Proc. natl. acad. sci. USA, 97, 5818-5823, (2000)
[27] Lodish, H.; Berk, A.; Matsudaira, P.; Kaiser, C.A.; Krieger, M.; Scott, M.P.; Zipursky, S.L.; Darnell, J., Molecular cell biology, (2004), Freeman New York
[28] Lötstedt, P.; Ferm, L., Dimensional reduction of the fokker – planck equation for stochastic chemical reactions, Multiscale meth. simul., 5, 593-614, (2006) · Zbl 1126.82027
[29] Lötstedt, P.; Söderberg, S.; Ramage, A.; Hemmingsson-Fränden, L., Implicit solution of hyperbolic equations with space – time adaptivity, Bit, 42, 134-158, (2002) · Zbl 0999.65084
[30] S. Mac, K. Burrage, R.B. Sidje, Multiscale modeling of chemical kinetics via the master equation, Multiscale Meth. Simul., in press. · Zbl 1153.60370
[31] McAdams, H.H.; Arkin, A., Stochastic mechanisms in gene expression, Proc. natl. acad. sci. USA, 94, 814-819, (1997)
[32] Moskowitz, B.; Caflisch, R.E., Smoothness and dimension reduction in quasi-Monte Carlo methods, Math. comput. modell., 23, 37-54, (1996) · Zbl 0858.65023
[33] Owen, A.B., Monte Carlo variance of scrambled net quadrature, SIAM J. numer. anal., 34, 1884-1910, (1997) · Zbl 0890.65023
[34] Paulsson, J.; Berg, O.G.; Ehrenberg, M., Stochastic focusing: fluctuation-enhanced sensitivity of intracellular regulation, Proc. natl. acad. sci. USA, 97, 7148-7153, (2000)
[35] Puchalka, J.; Kierzek, A.M., Bridging the gap between stochastic and deterministic regimes in the kinetic simulations of the biochemical reaction networks, Biophys. J., 86, 1357-1372, (2004)
[36] Rao, C.V.; Arkin, A.P., Stochastic chemical kinetics and the quasi-steady-state assumption: application to the Gillespie algorithm, J. chem. phys., 118, 4999-5010, (2003)
[37] Rao, C.V.; Wolf, D.M.; Arkin, A.P., Control exploitation and tolerance of intracellular noise, Nature, 420, 231-237, (2002)
[38] Salis, H.; Kaznessis, Y., Accurate hybrid stochastic simulation of a system of coupled chemical or biochemical reactions, J. chem. phys., 122, 054103, (2005)
[39] Samant, A.; Vlachos, D.G., Overcoming stiffness in stochastic simulation stemming from partial equilibrium: a multiscale Monte Carlo algorithm, J. chem. phys., 123, 144114, (2005)
[40] Santoso, L.; Booth, H.S.; Burden, C.J.; Hegland, M., A stochastic model of gene switches, Anziam j., 46, C530-C543, (2005) · Zbl 1074.92014
[41] P. Sjöberg, P. Lötstedt, J. Elf, Fokker-Planck approximation of the master equation in molecular biology, Technical Report 2005-044, Department of Information Technology, Uppsala University, Uppsala, Sweden, Comput. Vis. Sci., in press, doi:10.1007/s00791-006-0045-6.
[42] Thattai, M.; van Oudenaarden, A., Intrinsic noise in gene regulatory networks, Proc. nat. acad. sci. USA, 98, 8614-8619, (2001)
[43] 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)
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