Skip to main content
SpringerLink
Log in

A stochastic exponential Euler scheme for simulation of stiff biochemical reaction systems

  • Published:
BIT Numerical Mathematics Aims and scope Submit manuscript

Abstract

In order to simulate stiff biochemical reaction systems, an explicit exponential Euler scheme is derived for multi-dimensional, non-commutative stochastic differential equations with a semilinear drift term. The scheme is of strong order one half and A-stable in mean square. The combination with this and the projection method shows good performance in numerical experiments dealing with an alternative formulation of the chemical Langevin equation for a human ether a-go-go related gene ion channel model.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. Abdulle, A.: Fourth order Chebyshev methods with recurrence relation. SIAM J. Sci. Comput. 23(6), 2041–2054 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  2. Abdulle, A., Cirilli, S.: S-ROCK: Chebyshev methods for stiff stochastic differential equations. SIAM J. Sci. Comput. 30(2), 997–1014 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  3. Abdulle, A., Li, T.: S-ROCK methods for stiff Itô SDEs. Commun. Math. Sci. 6(4), 845–868 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  4. Abdulle, A., Medovikov, A.A.: Second order Chebyshev methods based on orthogonal polynomials. Numer. Math. 90, 1–18 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  5. Adamu, I.A.: Numerical approximation of SDEs and stochastic Swift-Hohenberg equation. Ph.D. thesis, Heriot-Watt University (2011)

  6. Alfonsi, A.: High order discretization schemes for the CIR process: application to affine term structure and Heston models. Math. Comp. 79(269), 209–237 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  7. Arnold, L.: Stochastic Differential Equations: Theory and Applications. Wiley, New York (1974)

    MATH  Google Scholar 

  8. Bayer, C., Szepessy, A., Tempone, R.: Adaptive weak approximation of reflected and stopped diffusions. Monte Carlo Methods Appl. 16(1), 1–67 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  9. Biscay, R., Jimenez, J.C., Riera, J.J., Valdes, P.A.: Local linearization method for numerical solution of stochastic differential equations. Ann. Inst. Statist. Math. 48(4), 631–644 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  10. Brennan, T., Fink, M., Rodriguez, B.: Multiscale modelling of drug-induced effects on cardiac electrophysiological activity. Eur. J. Pharm. Sci. 36(1), 62–77 (2009)

    Article  Google Scholar 

  11. Butcher, J.C.: Numerical Methods for Ordinary Differential Equations, 2nd edn. Wiley, Chichester (2008)

    Book  MATH  Google Scholar 

  12. Carbonell, F., Jimenez, J.C., Biscay, R.J.: Weak local linear discretizations for stochastic differential equations: convergence and numerical schemes. J. Comput. Appl. Math. 197(2), 578–596 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  13. Chen, Y., Ye, X.: Projection onto a simplex. e-print, ArXiv:1101.6081v2 (2011)

  14. Cruz, dl, Biscay, R.J., Jimenez, J.C., Carbonell, F., Ozaki, T.: High order local linearization methods: an approach for constructing A-stable explicit schemes for stochastic differential equations with additive noise. BIT 50(3), 509–539 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  15. Dangerfield, C., Kay, D., Burrage, K.: Modeling ion channel dynamics through reflected stochastic differential equations. Phys. Rev. E85, 051907 (2012)

    Google Scholar 

  16. Ehle, B.L., Lawson, J.D.: Generalized Runge–Kutta processes for stiff initial-value problems. IMA J. Appl. Math. 16(1), 11–21 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  17. Gillespie, D.T.: The chemical Langevin equation. J. Chem. Phys. 113(1), 297–306 (2000)

    Article  Google Scholar 

  18. Higham, D.J.: A-stability and stochastic mean-square stability. BIT 40(2), 404–409 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  19. Hochbruck, M., Lubich, C., Selhofer, H.: Exponential integrators for large systems of differential equations. SIAM J. Sci. Comput. 19(5), 1552–1574 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  20. Hochbruck, M., Ostermann, A.: Explicit exponential Runge–Kutta methods for semilinear parabolic problems. SIAM J. Numer. Anal. 43(3), 1069–1090 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  21. Hochbruck, M., Ostermann, A.: Exponential integrators. Acta Numer. 19, 209–286 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  22. van der Houwen, P.J., Sommeijer, B.P.: On the internal stability of explicit \(m\)-stage Runge–Kutta methods for large \(m\)-values. Z. Angew. Math. Mech. 60, 479–485 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  23. Ilie, S., Morshed, M.: Automatic simulation of the chemical Langevin equation. Appl. Math. 4(1A), 235–241 (2013)

    Article  Google Scholar 

  24. Jentzen, A., Kloeden, E.: Overcoming the order barrier in the numerical approximation of stochastic partial differential equations with additive space-time noise. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 465, 649–667 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  25. Jimenez, J.C.: A simple algebraic expression to evaluate the local linearization schemes for stochastic differential equations. Appl. Math. Lett. 15(6), 775–780 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  26. Jimenez, J.C., de la Cruz Cancino, H.: Convergence rate of strong local linearization schemes for stochastic differential equations with additive noise. BIT 52(2), 357–382 (2012)

  27. Jimenez, J.C., Shoji, I., Ozaki, T.: Simulation of stochastic differential equations through the local linearization method. a comparative study. J. Stat. Phys. 94(3–4), 587–602 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  28. Kiehn, J., Lacerda, A.E., Brown, A.M.: Pathways of HERG inactivation. Am. J. Physiol. Heart Circ. Physiol. 277(1), 199–210 (1999)

    Google Scholar 

  29. Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations. Springer, New York (1999). Corrected Third Printing.

  30. Komori, Y., Burrage, K.: Weak second order S-ROCK methods for Stratonovich stochastic differential equations. J. Comput. Appl. Math. 236(11), 2895–2908 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  31. Komori, Y., Burrage, K.: Strong first order S-ROCK methods for stochastic differential equations. J. Comput. Appl. Math. 242, 261–274 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  32. Lawson, J.D.: Generalized Runge–Kutta processes for stable systems with large Lipschitz constants. SIAM J. Numer. Anal. 4(3), 372–380 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  33. Mélykúti, B., Burrage, K., Zygalakis, K.C.: Fast stochastic simulation of biochemical reaction systems by alternative formulations of the chemical Langevin equation. J. Chem. Phys. 132(16), 164109 (2010)

    Article  Google Scholar 

  34. Mora, C.M.: Weak exponential schemes for stochastic differential equations with additive noise. IMA J. Numer. Anal. 25(3), 486–506 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  35. Pettersson, R.: Approximations for stochastic differential equations with reflecting convex boundaries. Stochastic Process Appl. 59(2), 295–308 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  36. Pope, D.: An exponential method of numerical integration of ordinary differential equations. Comm. ACM 6(8), 491–493 (1963)

    Article  MATH  MathSciNet  Google Scholar 

  37. Rößler, A.: Runge–Kutta methods for the strong approximation of solutions of stochastic differential equations. SIAM J. Numer. Anal. 48(3), 922–952 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  38. Shi, C., Xiao, Y., Zhang, C.: The convergence and MS stability of exponential Euler method for semilinear stochastic differential equations. Abstr. Appl. Anal. 2012, 35040701 (2012)

    MathSciNet  Google Scholar 

  39. Shoji, I.: A note on convergence rate of a linearization method for the discretization of stochastic differential equations. Commun. Nonlinear Sci. Numer. Simul. 16(7), 2667–2671 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  40. Skorohod, A.V.: Stochastic equations for diffusion processes with a boundary. Theory Probab. Appl. 6, 287–298 (1961)

    MathSciNet  Google Scholar 

  41. Skorohod, A.V.: Stochastic equations for diffusion processes with boundaries II. Theory Probab. Appl. 7, 5–25 (1962)

    MathSciNet  Google Scholar 

  42. Tanaka, H.: Stochastic differential equations with reflecting boundary condition in convex regions. Hiroshima Math. J. 9(1), 163–177 (1979)

    MATH  MathSciNet  Google Scholar 

  43. Wiktorsson, M.: Joint characteristic function and simultaneous simulation of iterated Itô integrals for multiple independent Brownian motions. Ann. Prob. 11(2), 470–487 (2001)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors would like to thank the referees for their valuable and careful comments which helped them to improve the earlier versions of this paper.

Author information

Authors and Affiliations

  1. Department of Systems Design and Informatics, Kyushu Institute of Technology, Iizuka, 820-8502, Japan

    Yoshio Komori

  2. Department of Computer Science, University of Oxford, Wolfson Building Parks Road, Oxford, OX1 3QD, UK

    Kevin Burrage

  3. School of Mathematics, Queensland University of Technology, Brisbane, QLD, 4001, Australia

    Kevin Burrage

Authors
  1. Yoshio Komori
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kevin Burrage
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yoshio Komori.

Additional information

Communicated by Raul Tempone.

An initial presentation relating to this work was given in the 19th IMACS World Congress. This work was partially supported by JSPS Grant-in-Aid for Scientific Research No. 23540143. It was also partially supported by overseas study program of faculty at Kyushu Institute of Technology.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Komori, Y., Burrage, K. A stochastic exponential Euler scheme for simulation of stiff biochemical reaction systems. Bit Numer Math 54, 1067–1085 (2014). https://doi.org/10.1007/s10543-014-0485-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10543-014-0485-1

Keywords

Mathematics Subject Classfication (2010)

Search

Navigation