The Annals of Applied Probability

Error analysis of tau-leap simulation methods

David F. Anderson, Arnab Ganguly, and Thomas G. Kurtz

Full-text: Open access


We perform an error analysis for numerical approximation methods of continuous time Markov chain models commonly found in the chemistry and biochemistry literature. The motivation for the analysis is to be able to compare the accuracy of different approximation methods and, specifically, Euler tau-leaping and midpoint tau-leaping. We perform our analysis under a scaling in which the size of the time discretization is inversely proportional to some (bounded) power of the norm of the state of the system. We argue that this is a more appropriate scaling than that found in previous error analyses in which the size of the time discretization goes to zero independent of the rest of the model. Under the present scaling, we show that midpoint tau-leaping achieves a higher order of accuracy, in both a weak and a strong sense, than Euler tau-leaping; a result that is in contrast to previous analyses. We present examples that demonstrate our findings.

Article information

Ann. Appl. Probab., Volume 21, Number 6 (2011), 2226-2262.

First available in Project Euclid: 23 November 2011

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H35: Computational methods for stochastic equations [See also 65C30] 65C99: None of the above, but in this section
Secondary: 92C40: Biochemistry, molecular biology

Tau-leaping simulation error analysis reaction networks Markov chain chemical master equation


Anderson, David F.; Ganguly, Arnab; Kurtz, Thomas G. Error analysis of tau-leap simulation methods. Ann. Appl. Probab. 21 (2011), no. 6, 2226--2262. doi:10.1214/10-AAP756.

Export citation


  • [1] Anderson, D. F. (2007). A modified next reaction method for simulating chemical systems with time dependent propensities and delays. J. Chem. Phys. 127 214107.
  • [2] Anderson, D. F. (2008). Incorporating postleap checks in tau-leaping. J. Chem. Phys. 128 054103.
  • [3] Cao, Y., Gillespie, D. T. and Petzold, L. R. (2005). Avoiding negative populations in explicit Poisson tau-leaping. J. Chem. Phys. 123 054104.
  • [4] Cao, Y., Gillespie, D. T. and Petzold, L. R. (2006). Efficient step size selection for the tau-leaping simulation method. J. Chem. Phys. 124 044109.
  • [5] Chatterjee, A., Vlachos, D. G. and Katsoulakis, M. A. (2005). Binomial distribution based tau-leap accelerated stochastic simulation. J. Chem. Phys. 122 024112.
  • [6] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • [7] Ethier, S. N. and Kurtz, T. G. (2005). Markov Processes: Characterization and Convergence, 2nd ed. Wiley, New York.
  • [8] Gibson, M. A. and Bruck, J. (2000). Efficient exact stochastic simulation of chemical systems with many species and many channels. J. Phys. Chem. A 105 1876–1889.
  • [9] Gillespie, D. T. (1976). A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Comput. Phys. 22 403–434.
  • [10] Gillespie, D. T. (1977). Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem. 81 2340–2361.
  • [11] Gillespie, D. T. (2001). Approximate accelerated simulation of chemically reaction systems. J. Chem. Phys. 115 1716–1733.
  • [12] Gillespie, D. T. and Petzold, L. R. (2003). Improved leap-size selection for accelerated stochastic simulation. J. Chem. Phys. 119 8229–8234.
  • [13] Kloeden, P. E. and Platen, E. (1992). Numerical Solution of Stochastic Differential Equations. Applications of Mathematics (New York) 23. Springer, Berlin.
  • [14] Kurtz, T. G. (1972). The relationship between stochastic and deterministic models for chemical reactions. J. Chem. Phys. 57 2976–2978.
  • [15] Kurtz, T. G. (1977/78). Strong approximation theorems for density dependent Markov chains. Stochastic Processes Appl. 6 223–240.
  • [16] Kurtz, T. G. (1981). Approximation of Population Processes. CBMS-NSF Regional Conference Series in Applied Mathematics 36. SIAM, Philadelphia, PA.
  • [17] Li, T. (2007). Analysis of explicit tau-leaping schemes for simulating chemically reacting systems. Multiscale Model. Simul. 6 417–436 (electronic).
  • [18] Rathinam, M., Petzold, L. R., Cao, Y. and Gillespie, D. T. (2005). Consistency and stability of tau-leaping schemes for chemical reaction systems. Multiscale Model. Simul. 4 867–895 (electronic).
  • [19] Wilkinson, D. J. (2006). Stochastic Modelling for Systems Biology. Chapman and Hall/CRC, Boca Raton, FL.