Journal of Scientific Computing

, Volume 34, Issue 2, pp 127–151 | Cite as

A Hierarchy of Approximations of the Master Equation Scaled by a Size Parameter

  • Lars Ferm
  • Per Lötstedt
  • Andreas Hellander


Solutions of the master equation are approximated using a hierarchy of models based on the solution of ordinary differential equations: the macroscopic equations, the linear noise approximation and the moment equations. The advantage with the approximations is that the computational work with deterministic algorithms grows as a polynomial in the number of species instead of an exponential growth with conventional methods for the master equation. The relation between the approximations is investigated theoretically and in numerical examples. The solutions converge to the macroscopic equations when a parameter measuring the size of the system grows. A computational criterion is suggested for estimating the accuracy of the approximations. The numerical examples are models for the migration of people, in population dynamics and in molecular biology.


Master equation Reaction rate equations Linear noise approximation Moment equations 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Barkai, N., Leibler, S.: Circadian clocks limited by noise. Nature 403, 267–268 (2000) Google Scholar
  2. 2.
    Caflisch, R.E.: Monte Carlo and quasi-Monte Carlo methods. Acta Numer. 7, 1–49 (1998) MathSciNetGoogle Scholar
  3. 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) zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Chen, W.-Y., Bokka, A.: Stochastic modeling of nonlinear epidemiology. J. Theor. Biol. 234, 455–470 (2005) CrossRefMathSciNetGoogle Scholar
  5. 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) CrossRefGoogle Scholar
  6. 6.
    Dieudonné, J.: Foundations of Modern Analysis. Academic Press, New York (1969) zbMATHGoogle Scholar
  7. 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) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Elf, J., Ehrenberg, M.: Fast evaluation of fluctuations in biochemical networks with the linear noise approximation. Genome Res. 13, 2475–2484 (2003) CrossRefGoogle Scholar
  9. 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 (2003)
  10. 10.
    Elf, J., Paulsson, J., Berg, O.G., Ehrenberg, M.: Near-critical phenomena in intracellular metabolite pools. Biophys. J. 84, 154–170 (2003) CrossRefGoogle Scholar
  11. 11.
    Engblom, S.: Computing the moments of high dimensional solutions of the master equation. Appl. Math. Comput. 180, 498–515 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Engquist, B., Runborg, O.: Computational high frequency wave propagation. Acta Numer. 3, 181–266 (2003) CrossRefMathSciNetGoogle Scholar
  13. 13.
    Escudera, C., Buceta, J., de la Rubia, F.J., Lindenberg, K.: Extinction in population dynamics. Phys. Rev. E 69, 021908 (2004) CrossRefMathSciNetGoogle Scholar
  14. 14.
    Ethier, S.N., Kurtz, T.G.: Markov Processes, Characterization and Convergence. Wiley, New York (1986) zbMATHGoogle Scholar
  15. 15.
    Faure, H.: Discrépance de suites associées à un système de numération (en dimension s). Acta Aritm. 41, 337–351 (1982) zbMATHMathSciNetGoogle Scholar
  16. 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 (2007)
  17. 17.
    Ferm, L., Lötstedt, P., Sjöberg, P.: Conservative solution of the Fokker-Planck equation for stochastic chemical reactions. BIT 46, S61–S83 (2006) zbMATHCrossRefGoogle Scholar
  18. 18.
    Fox, R.F., Keizer, J.: Amplification of intrinsic fluctuations by chaotic dynamics in physical systems. Phys. Rev. A 43, 1709–1720 (1991) CrossRefMathSciNetGoogle Scholar
  19. 19.
    Gardiner, C.W.: Handbook of Stochastic Methods, 3rd edn. Springer, Berlin (2004) zbMATHGoogle Scholar
  20. 20.
    Giles, M.B., Süli, E.: Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality. Acta Numer. 11, 145–236 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  21. 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) CrossRefMathSciNetGoogle Scholar
  22. 22.
    Givon, D., Kupferman, R., Stewart, A.: Extracting macroscopic dynamics: model problems and algorithms. Nonlinearity 17, R55–R127 (2004) zbMATHCrossRefGoogle Scholar
  23. 23.
    Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I. Nonstiff Problems, 2nd edn. Springer, Berlin (1993) zbMATHGoogle Scholar
  24. 24.
    Hong, H.S., Hickernell, F.J.: Algorithm 823: implementing scrambled digital sequences. ACM Trans. Math. Softw. 29, 95–109 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Kurtz, T.G.: Solutions of ordinary differential equations as limits of pure jump Markov processes. J. Appl. Probab. 7, 49–58 (1970) zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Kurtz, T.G.: Limit theorems for sequences of jump Markov processes approximating ordinary differential processes. J. Appl. Probab. 7, 344–356 (1971) CrossRefMathSciNetGoogle Scholar
  27. 27.
    Lötstedt, P., Ferm, L.: Dimensional reduction of the Fokker-Planck equation for stochastic chemical reactions. Multiscale Methods Simul. 5, 593–614 (2006) zbMATHCrossRefGoogle Scholar
  28. 28.
    McAdams, H.H., Arkin, A.: Stochastic mechanisms in gene expression. Proc. Nat. Acad. Sci. USA 94, 814–819 (1997) CrossRefGoogle Scholar
  29. 29.
    McAdams, H.H., Arkin, A.: It’s a noisy business. Genetic regulation at the nanomolar scale. Trends Genet. 15, 65–69 (1999) CrossRefGoogle Scholar
  30. 30.
    McKane, A.J., Newman, T.J.: Stochastic models in population biology and their deterministic analogs. Phys. Rev. E 70, 041902 (2004) CrossRefMathSciNetGoogle Scholar
  31. 31.
    Murray, J.D.: Mathematical Biology I. An Introduction, 3rd edn. Springer, New York (2002) Google Scholar
  32. 32.
    Oden, J.T., Prudhomme, S.: Estimation of modeling error in computational mechanics. J. Comput. Phys. 182, 496–515 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Owen, A.B.: Monte Carlo variance of scrambled net quadrature. SIAM J. Numer. Anal. 34, 1884–1910 (1997) zbMATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Risken, H.: The Fokker-Planck Equation, 2nd edn. Springer, Berlin (1996) zbMATHGoogle Scholar
  35. 35.
    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 Google Scholar
  36. 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) CrossRefGoogle Scholar
  37. 37.
    Strogatz, S.H.: Nonlinear Dynamics and Chaos. Perseus Books, Cambridge (1994) Google Scholar
  38. 38.
    Succi, S.: The Lattice Boltzmann Equation for Fluid Dynamics and Beyond. Clarendon, Oxford (2001) zbMATHGoogle Scholar
  39. 39.
    van Kampen, N.G.: The expansion of the master equation. Adv. Chem. Phys. 34, 245–309 (1976) CrossRefGoogle Scholar
  40. 40.
    van Kampen, N.G.: Stochastic Processes in Physics and Chemistry. North-Holland, Amsterdam (1992) Google Scholar
  41. 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) CrossRefGoogle Scholar
  42. 42.
    Weidlich, W.: Sociodynamics. A Systematic Approach to Mathematical Modelling in the Social Sciences. Taylor and Francis, London (2000) zbMATHGoogle Scholar
  43. 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) zbMATHMathSciNetGoogle Scholar
  44. 44.
    Wilcox, D.C.: Turbulence modeling for CFD. DCW Industries, La Cañada, CA (1994) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Division of Scientific Computing, Department of Information TechnologyUppsala UniversityUppsalaSweden

Personalised recommendations