BIT Numerical Mathematics

, Volume 50, Issue 4, pp 797–822 | Cite as

Efficient simulation of discrete stochastic reaction systems with a splitting method



Stochastic reaction systems with discrete particle numbers are usually described by a continuous-time Markov process. Realizations of this process can be generated with the stochastic simulation algorithm, but simulating highly reactive systems is computationally costly because the computational work scales with the number of reaction events. We present a new approach which avoids this drawback and increases the efficiency considerably at the cost of a small approximation error. The approach is based on the fact that the time-dependent probability distribution associated to the Markov process is explicitly known for monomolecular, autocatalytic and certain catalytic reaction channels. More complicated reaction systems can often be decomposed into several parts some of which can be treated analytically. These subsystems are propagated in an alternating fashion similar to a splitting method for ordinary differential equations. We illustrate this approach by numerical examples and prove an error bound for the splitting error.


Stochastic simulation algorithm Discrete stochastic reaction systems Splitting methods Analytic solution formulas Error bounds Chemical master equation 

Mathematics Subject Classification (2000)

60J75 65L70 47D06 92D25 92C42 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alfonsi, A., Cancès, E., Turinici, G., Ventura, B.D., Huisinga, W.: Adaptive simulation of hybrid stochastic and deterministic models for biochemical systems. In: ESAIM: Proc., vol. 14, pp. 1–13 (2005) Google Scholar
  2. 2.
    Anderson, D.F., Ganguly, A., Kurtz, T.G.: Error analysis of tau-leap simulation methods. Ann. Appl. Probab. (2010, to appear). arXiv:0909.4790v2 [math.PR]
  3. 3.
    Banasiak, J.: A complete description of dynamics generated by birth-and-death problem: a semigroup approach. In: Rudnicki, R. (ed.) Mathematical Modelling of Population Dynamics. Collection of Papers from the Conference, Będlewo, Poland, June 24–28, 2002. Banach Center Publications, vol. 63, pp. 165–176 . Polish Academy of Sciences, Institute of Mathematics, Warsaw (2004) Google Scholar
  4. 4.
    Burrage, K., Tian, T.: Poisson Runge-Kutta methods for chemical reaction systems. In: Sun, Y.L.W., Tang, T. (eds.) Advances in Scientific Computing and Applications, pp. 82–96. Science Press, Beijing (2004) Google Scholar
  5. 5.
    Burrage, K., Tian, T., Burrage, P.: A multi-scaled approach for simulating chemical reaction systems. Prog. Biophys. Mol. Biol. 85, 217–234 (2004) CrossRefGoogle Scholar
  6. 6.
    Cao, Y., Li, H., Petzold, L.: Efficient formulation of the stochastic simulation algorithm for chemically reacting systems. J. Chem. Phys. 121, 4059 (2004) CrossRefGoogle Scholar
  7. 7.
    Cao, Y., Gillespie, D.T., Petzold, L.: Multiscale stochastic simulation algorithm with stochastic partial equilibrium assumption for chemically reacting systems. J. Comput. Phys. 206(2), 395–411 (2005) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Cao, Y., Gillespie, D.T., Petzold, L.R.: Avoiding negative populations in explicit Poisson tau-leaping. J. Chem. Phys. 123, 054104 (2005) CrossRefGoogle Scholar
  9. 9.
    Cao, Y., Gillespie, D.T., Petzold, L.R.: The slow-scale stochastic simulation algorithm. J. Chem. Phys. 122, 014116 (2005) CrossRefGoogle Scholar
  10. 10.
    Cao, Y., Gillespie, D.T., Petzold, L.R.: Efficient step size selection for the tau-leaping simulation method. J. Chem. Phys. 124, 044109 (2006) CrossRefGoogle Scholar
  11. 11.
    Descombes, S., Schatzman, M.: Strang’s formula for holomorphic semi-groups. J. Math. Pures Appl. Sér. IX 81(1), 93–114 (2002) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    E, W., Liu, D., Vanden-Eijnden, E.: Nested stochastic simulation algorithm for chemical kinetic systems with disparate rates. J. Chem. Phys. 123, 194107 (2005) CrossRefGoogle Scholar
  13. 13.
    Engblom, S.: Numerical Solution Methods in Stochastic Chemical Kinetics. PhD thesis, Uppsala University (2008) Google Scholar
  14. 14.
    Faou, E.: Analysis of splitting methods for reaction-diffusion problems using stochastic calculus. Math. Comput. 78, 1467–1483 (2009) MATHMathSciNetGoogle Scholar
  15. 15.
    Ferm, L., Hellander, A., Lötstedt, P.: An adaptive algorithm for simulation of stochastic reaction-diffusion processes. J. Comput. Phys. 229(2), 343–360 (2010) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Gibson, M.A., Bruck, J.: Efficient exact stochastic simulation of chemical systems with many species and many channels. J. Phys. Chem. A 104(9), 1876–1889 (2000) Google Scholar
  17. 17.
    Gillespie, D.T.: A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Comput. Phys. 22(4), 403–434 (1976) CrossRefMathSciNetGoogle Scholar
  18. 18.
    Gillespie, D.T.: Approximate accelerated stochastic simulation of chemically reacting systems. J. Chem. Phys. 115, 1716 (2001) CrossRefGoogle Scholar
  19. 19.
    Gillespie, D.T., Petzold, L.R.: Improved leap-size selection for accelerated stochastic simulation. J. Chem. Phys. 119, 8229 (2003) CrossRefGoogle Scholar
  20. 20.
    Goutsias, J.: Quasiequilibrium approximation of fast reaction kinetics in stochastic biochemical systems. J. Chem. Phys. 122, 184102 (2005) CrossRefGoogle Scholar
  21. 21.
    Gradinaru, V.: Strang splitting for the time dependent Schrödinger equation on sparse grids. SIAM J. Numer. Anal. 46, 103–123 (2007) CrossRefMathSciNetGoogle Scholar
  22. 22.
    Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration. Structure-preserving Algorithms for Ordinary Differential Equations, 2nd edn. Springer Series in Computational Mathematics, vol. 31. Springer, Berlin (2006) MATHGoogle Scholar
  23. 23.
    Hansen, E., Ostermann, A.: Exponential splitting for unbounded operators. Math. Comput. 78, 1485–1496 (2009) MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Hansen, E., Ostermann, A.: High order splitting methods for analytic semigroups exist. BIT 49(3), 527–542 (2009) MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Haseltine, E.L., Rawlings, J.B.: Approximate simulation of coupled fast and slow reactions for stochastic chemical kinetics. J. Chem. Phys. 117, 6959 (2002) CrossRefGoogle Scholar
  26. 26.
    Hegland, M.: Approximating the solution of the chemical master equation by aggregation. In: Mercer, G.N., Roberts, A.J. (eds.) Proceedings of the 14th Biennial Computational Techniques and Applications Conference, CTAC-2008, ANZIAM J. 50, C371–C384 (2008) Google Scholar
  27. 27.
    Hethcote, H.W.: The mathematics of infectious diseases. SIAM Rev. 42(4), 599–653 (2000) MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Hundsdorfer, W., Verwer, J.: Numerical Solution of Time-dependent Advection-Diffusion-Reaction Equations. Springer Series in Computational Mathematics, vol. 33. Springer, Berlin (2003) MATHGoogle Scholar
  29. 29.
    Jahnke, T.: Splittingverfahren für Schrödingergleichungen. Wiss. Arbeit für das Staatsexamen, Universität Tübingen, Germany (1999) Google Scholar
  30. 30.
    Jahnke, T.: An adaptive wavelet method for the chemical master equation. SIAM J. Sci. Comput. 31(6), 4373–4394 (2010) CrossRefMathSciNetGoogle Scholar
  31. 31.
    Jahnke, T., Huisinga, W.: Solving the chemical master equation for monomolecular reaction systems analytically. J. Math. Biol. 54(1), 1–26 (2007) CrossRefMathSciNetGoogle Scholar
  32. 32.
    Jahnke, T., Lubich, C.: Error bounds for exponential operator splittings. BIT 40(4), 735–744 (2000) MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Li, T.: Analysis of explicit tau-leaping schemes for simulating chemically reacting systems. Multiscale Model. Simul. 6(2), 417–436 (2007) MATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Lubich, C.: A variational splitting integrator for quantum molecular dynamics. Appl. Numer. Math. 48(3–4), 355–368 (2004) MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Lubich, C.: On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations. Math. Comput. 77, 2141–2153 (2008) MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    McAdams, H.H., Arkin, A.P.: Stochastic mechanisms in gene expression. Proc. Natl. Acad. Sci. 94, 814–819 (1997) CrossRefGoogle Scholar
  37. 37.
    McAdams, H.H., Arkin, A.P.: It’s a noisy business! Genetic regulation at the nanomolar scale. Trends Genet. 15, 65–69 (1999) CrossRefGoogle Scholar
  38. 38.
    McLachlan, R.I., Quispel, G.W.: Splitting methods. Acta Numer. 11, 341–434 (2002) MATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    Neuhauser, C., Thalhammer, M.: On the convergence of splitting methods for linear evolutionary Schrödinger equations involving an unbounded potential. BIT 49(1), 199–215 (2009) MATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    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 (2003) CrossRefGoogle Scholar
  41. 41.
    Rathinam, M., El Samad, H.: Reversible-equivalent-monomolecular tau: a leaping method for “small number and stiff” stochastic chemical systems. J. Comput. Phys. 224(3), 897–923 (2007) MATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    Rathinam, M., Petzold, L.R., Cao, Y., Gillespie, D.T.: Stiffness in stochastic chemically reacting systems: The implicit tau-leaping method. J. Chem. Phys. 119, 12784 (2003) CrossRefGoogle Scholar
  43. 43.
    Rathinam, M., Petzold, L.R., Cao, Y., Gillespie, D.T.: Consistency and stability of tau-leaping schemes for chemical reaction systems. Multiscale Model. Simul. 4(3), 867–895 (2005) MATHCrossRefMathSciNetGoogle Scholar
  44. 44.
    Salis, H., Kaznessis, Y.: Accurate hybrid stochastic simulation of a system of coupled chemical or biochemical reactions. J. Chem. Phys. 122, 054103 (2005) CrossRefGoogle Scholar
  45. 45.
    Solari, H.G., Natiello, M.A.: Stochastic population dynamics: The Poisson approximation. Phys. Rev. E 67, 031918 (2003) CrossRefGoogle Scholar
  46. 46.
    Srivastava, R., You, L., Summers, J., Yin, J.: Stochastic vs. deterministic modeling of intracellular viral kinetics. J. Theor. Biol. 218(3), 309–321 (2002) CrossRefMathSciNetGoogle Scholar
  47. 47.
    Thalhammer, M.: High-order exponential operator splitting methods for time-dependent Schrödinger equations. SIAM J. Numer. Anal. 46(4), 2022–2038 (2008) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2010

Authors and Affiliations

  1. 1.Fakultät für Mathematik, Institut für Angewandte und Numerische MathematikKarlsruhe Institute of Technology (KIT)KarlsruheGermany
  2. 2.Institute of Applied Mathematics, Department of Scientific ComputingMiddle East Technical UniversityAnkaraTurkey
  3. 3.Selçuk UniversityKonyaTurkey

Personalised recommendations