Skip to main content

Advertisement

SpringerLink
Log in

Extending the Multi-level Method for the Simulation of Stochastic Biological Systems

  • Original Article
  • Published:
Bulletin of Mathematical Biology Aims and scope Submit manuscript

Abstract

The multi-level method for discrete-state systems, first introduced by Anderson and Higham (SIAM Multiscale Model Simul 10(1):146–179, 2012), is a highly efficient simulation technique that can be used to elucidate statistical characteristics of biochemical reaction networks. A single point estimator is produced in a cost-effective manner by combining a number of estimators of differing accuracy in a telescoping sum, and, as such, the method has the potential to revolutionise the field of stochastic simulation. In this paper, we present several refinements of the multi-level method which render it easier to understand and implement, and also more efficient. Given the substantial and complex nature of the multi-level method, the first part of this work reviews existing literature, with the aim of providing a practical guide to the use of the multi-level method. The second part provides the means for a deft implementation of the technique and concludes with a discussion of a number of open problems.

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.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Notes

  1. Note that here, and throughout the rest of this work, we implicitly include the final exact coupling level in our summations, where appropriate.

References

  • Anderson D (2007) A modified next reaction method for simulating chemical systems with time-dependent propensities and delays. J Chem Phys 127(21):214107

    Article  Google Scholar 

  • Anderson D, Higham D (2012) Multi-level Monte Carlo for continuous time Markov chains, with applications in biochemical kinetics. SIAM Multiscale Model Simul 10(1):146–179

    Article  MathSciNet  MATH  Google Scholar 

  • Anderson D, Ganguly A, Kurtz T (2011) Error analysis of tau-leap simulation methods. Ann Appl Probab 21(6):2226–2262

    Article  MathSciNet  MATH  Google Scholar 

  • Arkin A, Ross J, McAdams H (1998) Stochastic kinetic analysis of developmental pathway bifurcation in phage \(\lambda \)-infected escherichia coli cells. Genetics 149(4):1633–1648

    Google Scholar 

  • Auger A, Chatelain P, Koumoutsakos P (2006) R-leaping: accelerating the stochastic simulation algorithm by reaction leaps. J Chem Phys 125(084):103

    Google Scholar 

  • Barrio M, Burrage K, Leier A, Tian T (2006) Oscillatory regulation of hes1: discrete stochastic delay modelling and simulation. PLoS Comput Biol 2(9):e117

    Article  Google Scholar 

  • Bayer C, Hoel H, Von Schwerin E, Tempone R (2012) On non-asymptotic optimal stopping criteria in Monte Carlo simulations. SIAM J Sci Comput 2(36):A869–A885

    MATH  Google Scholar 

  • Cao Y, Li H, Petzold L (2004) Efficient formulation of the stochastic simulation algorithm for chemically reacting systems. J Chem Phys 121(9):4059–4067

    Article  Google Scholar 

  • Cao Y, Gillespie D, Petzold L (2006) Efficient step size selection for the tau-leaping simulation method. J Chem Phys 124(4):044109

    Article  Google Scholar 

  • Cao Y, Gillespie D, Petzold L (2007) Adaptive explicit-implicit tau-leaping method with automatic tau selection. J Chem Phys 126(22):224101

    Article  Google Scholar 

  • El Samad H, Khammash M, Petzold L, Gillespie D (2005) Stochastic modelling of gene regulatory networks. Int J Robust Nonlinear Control 15(15):691–711

    Article  MathSciNet  MATH  Google Scholar 

  • Elowitz M, Levine A, Siggia E, Swain P (2002) Stochastic gene expression in a single cell. Sci Signal 297(5584):1183

    Google Scholar 

  • Engblom S (2009) Spectral approximation of solutions to the chemical master equation. J Comput Appl Math 229(1):208–221

    Article  MathSciNet  MATH  Google Scholar 

  • Erban R, Chapman S, Kevrekidis I, Vejchodskỳ T (2009) Analysis of a stochastic chemical system close to a sniper bifurcation of its mean-field model. SIAM J Appl Math 70(3):984–1016

    Article  MathSciNet  MATH  Google Scholar 

  • Fedoroff N, Fontana W (2002) Small numbers of big molecules. Science 297(5584):1129–1131

    Article  Google Scholar 

  • Gibson M, Bruck J (2000) Efficient exact stochastic simulation of chemical systems with many species and many channels. J Phys Chem A 104(9):1876–1889

    Article  Google Scholar 

  • Giles M (2008) Multilevel Monte Carlo path simulation. Oper Res 56(3):607–617

    Article  MathSciNet  MATH  Google Scholar 

  • Gillespie D (1976) A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J Comput Phys 22(4):403–434

    Article  MathSciNet  Google Scholar 

  • Gillespie D (1977) Exact stochastic simulation of coupled chemical reactions. J Phys Chem 81(25):2340–2361

    Article  Google Scholar 

  • Gillespie D (2001) Approximate accelerated stochastic simulation of chemically reacting systems. J Chem Phys 115(4):1716–1733

    Article  Google Scholar 

  • Gillespie D (2005) Stochastic chemical kinetics. In: Yip S (ed) Handbook of materials modeling. Springer, Netherlands, pp 1735–1752

    Chapter  Google Scholar 

  • Higham D (2008) Modeling and simulating chemical reactions. SIAM Rev 50(2):347–368

    Article  MathSciNet  MATH  Google Scholar 

  • Hou Z, Xin H (2003) Internal noise stochastic resonance in a circadian clock system. J Chem Phys 119:11508

    Article  Google Scholar 

  • Huang C, Ferrell J (1996) Ultrasensitivity in the mitogen-activated protein kinase cascade. Proc Natl Acad Sci 93(19):10078–10083

    Article  Google Scholar 

  • Jahnke T (2011) On reduced models for the chemical master equation. Multiscale Model Simul 9(4):1646–1676

    Article  MathSciNet  MATH  Google Scholar 

  • Jahnke T, Huisinga W (2007) Solving the chemical master equation for monomolecular reaction systems analytically. J Math Biol 54(1):1–26

    Article  MathSciNet  MATH  Google Scholar 

  • Jahnke T, Huisinga W (2008) A dynamical low-rank approach to the chemical master equation. Bull Math Biol 70(8):2283–2302

    Article  MathSciNet  MATH  Google Scholar 

  • Jahnke T, Udrescu T (2010) Solving chemical master equations by adaptive wavelet compression. J Comput Phys 229(16):5724–5741

    Article  MathSciNet  MATH  Google Scholar 

  • Kavehrad M, Joseph M (1986) Maximum entropy and the method of moments in performance evaluation of digital communications systems. IEEE Trans Commun 34(12):1183–1189

    Article  Google Scholar 

  • Kurtz T (1980) Representations of Markov processes as multiparameter time changes. Ann Probab 8(4):682–715

  • Li H, Petzold L (2006) Logarithmic direct method for discrete stochastic simulation of chemically reacting systems. Technical Report, Department of Computer Science, University of California, Santa Barbara

  • Li T (2007) Analysis of explicit tau-leaping schemes for simulating chemically reacting systems. SIAM Multiscale Model Simul 6(2):417–436

    Article  MathSciNet  MATH  Google Scholar 

  • MacNamara S, Bersani A, Burrage K, Sidje R (2008) Stochastic chemical kinetics and the total quasi-steady-state assumption: application to the stochastic simulation algorithm and chemical master equation. J Chem Phys 129(9):095–105

    Article  Google Scholar 

  • McCollum J, Peterson G, Cox C, Simpson M, Samatova N (2006) The sorting direct method for stochastic simulation of biochemical systems with varying reaction execution behavior. Comput Biol Chem 30(1):39–49

    Article  MATH  Google Scholar 

  • Norris J (1998) Markov chains. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Paulsson J, Berg O, Ehrenberg M (2000) Stochastic focusing: fluctuation-enhanced sensitivity of intracellular regulation. Proc Natl Acad Sci USA 97(13):7148–7153

    Article  Google Scholar 

  • Székely T, Burrage K, Erban R, Zygalakis K (2012) A higher-order numerical framework for stochastic simulation of chemical reaction systems. BMC Syst Biol 6(1):85

    Article  Google Scholar 

  • Vellela M, Qian H (2009) Stochastic dynamics and non-equilibrium thermodynamics of a bistable chemical system: the schlögl model revisited. J R Soc Interface 6(39):925–940

    Article  Google Scholar 

  • Yates C, Burrage K (2011) Look before you leap: a confidence-based method for selecting species criticality while avoiding negative populations in \(\tau \)-leaping. J Chem Phys 134(8):084–109

    Article  Google Scholar 

  • Yates C, Klingbeil G (2013) Recycling random numbers in the stochastic simulation algorithm. J Chem Phys 138(9):094–103

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematical Institute, University of Oxford, Woodstock Road, Oxford, OX2 6GG, UK

    Christopher Lester, Ruth E. Baker & Michael B. Giles

  2. Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, UK

    Christian A. Yates

Authors
  1. Christopher Lester
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ruth E. Baker
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Michael B. Giles
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Christian A. Yates
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christopher Lester.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lester, C., Baker, R.E., Giles, M.B. et al. Extending the Multi-level Method for the Simulation of Stochastic Biological Systems. Bull Math Biol 78, 1640–1677 (2016). https://doi.org/10.1007/s11538-016-0178-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11538-016-0178-9

Keywords

Search

Navigation