## Journal of Applied Mathematics

### Adaptive Time-Stepping Using Control Theory for the Chemical Langevin Equation

#### Abstract

Stochastic modeling of biochemical systems has been the subject of intense research in recent years due to the large number of important applications of these systems. A critical stochastic model of well-stirred biochemical systems in the regime of relatively large molecular numbers, far from the thermodynamic limit, is the chemical Langevin equation. This model is represented as a system of stochastic differential equations, with multiplicative and noncommutative noise. Often biochemical systems in applications evolve on multiple time-scales; examples include slow transcription and fast dimerization reactions. The existence of multiple time-scales leads to mathematical stiffness, which is a major challenge for the numerical simulation. Consequently, there is a demand for efficient and accurate numerical methods to approximate the solution of these models. In this paper, we design an adaptive time-stepping method, based on control theory, for the numerical solution of the chemical Langevin equation. The underlying approximation method is the Milstein scheme. The adaptive strategy is tested on several models of interest and is shown to have improved efficiency and accuracy compared with the existing variable and constant-step methods.

#### Article information

Source
J. Appl. Math., Volume 2015 (2015), Article ID 567275, 10 pages.

Dates
First available in Project Euclid: 15 April 2015

https://projecteuclid.org/euclid.jam/1429105037

Digital Object Identifier
doi:10.1155/2015/567275

Mathematical Reviews number (MathSciNet)
MR3314971

Zentralblatt MATH identifier
1347.92024

#### Citation

Ilie, Silvana; Morshed, Monjur. Adaptive Time-Stepping Using Control Theory for the Chemical Langevin Equation. J. Appl. Math. 2015 (2015), Article ID 567275, 10 pages. doi:10.1155/2015/567275. https://projecteuclid.org/euclid.jam/1429105037

#### References

• A. Arkin, J. Ross, and H. H. McAdams, “Stochastic kinetic analysis of developmental pathway bifurcation in phage $\lambda$-infected Escherichia coli cells,” Genetics, vol. 149, no. 4, pp. 1633–1648, 1998.
• W. J. Blake, M. Kærn, C. R. Cantor, and J. J. Collins, “Noise in eukaryotic gene expression,” Nature, vol. 422, no. 6932, pp. 633–637, 2003.
• M. B. Elowitz, A. J. Levine, E. D. Siggia, and P. S. Swain, “Stochastic gene expression in a single cell,” Science, vol. 297, no. 5584, pp. 1183–1186, 2002.
• D. T. Gillespie, “A rigorous derivation of the chemical master equation,” Physica A: Statistical Mechanics and its Applications, vol. 188, no. 1–3, pp. 404–425, 1992.
• D. T. Gillespie, “A general method for numerically simulating the stochastic time evolution of coupled chemical reactions,” Journal of Computational Physics, vol. 22, no. 4, pp. 403–434, 1976.
• D. T. Gillespie, “Exact stochastic simulation of coupled chemical reactions,” Journal of Physical Chemistry, vol. 81, no. 25, pp. 2340–2361, 1977.
• Y. Cao, D. T. Gillespie, and L. R. Petzold, “The slow-scale stochastic simulation algorithm,” The Journal of Chemical Physics, vol. 122, no. 1, Article ID 014116, 2005.
• D. T. Gillespie, “Approximate accelerated stochastic simulation of chemically reacting systems,” Journal of Chemical Physics, vol. 115, no. 4, pp. 1716–1733, 2001.
• C. V. Rao and A. P. Arkin, “Stochastic chemical kinetics and the quasi-steady-state assumption: application to the Gillespie algorithm,” Journal of Chemical Physics, vol. 118, no. 11, pp. 4999–5010, 2003.
• A. Samant and D. G. Vlachos, “Overcoming stiffness in stochastic simulation stemming from partial equilibrium: a multiscale Monte Carlo algorithm,” Journal of Chemical Physics, vol. 123, no. 14, Article ID 144114, 2005.
• D. T. Gillespie, “The Chemical Langevin equation,” Journal of Chemical Physics, vol. 113, no. 1, pp. 297–306, 2000.
• H. Salis, V. Sotiropoulos, and Y. N. Kaznessis, “Multiscale Hy3S: hybrid stochastic simulation for supercomputers,” BMC Bioinformatics, vol. 7, article 93, 2006.
• S. Ilie, “Variable time-stepping in the pathwise numerical solution of the Chemical Langevin equation,” Journal of Chemical Physics, vol. 137, no. 23, Article ID 234110, 2012.
• P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, vol. 23 of Applications of Mathematics (New York), Springer, Berlin, Germany, 1999.
• J. G. Gaines and T. J. Lyons, “Variable step size control in the numerical solution of stochastic differential equations,” SIAM Journal on Applied Mathematics, vol. 57, no. 5, pp. 1455–1484, 1997.
• A. Szepessy, R. Tempone, and G.. Zouraris, “Adaptive weak approximation of stochastic differential equations,” Communications on Pure and Applied Mathematics, vol. 54, no. 10, pp. 1169–1214, 2001.
• P. M. Burrage, R. Herdiana, and K. Burrage, “Adaptive stepsize based on control theory for stochastic differential equations,” Journal of Computational and Applied Mathematics, vol. 170, no. 2, pp. 317–336, 2004.
• K. Burrage and P. M. Burrage, “A variable stepsize implementation for stochastic differential equations,” SIAM Journal on Scientific Computing, vol. 24, no. 3, pp. 848–864, 2002.
• K. Burrage, P. M. Burrage, and T. Tian, “Numerical methods for strong solutions of stochastic differential equations: an overview,” Proceedings of The Royal Society of London Series A, vol. 460, no. 2041, pp. 373–402, 2004.
• V. Sotiropoulos and Y. N. Kaznessis, “An adaptive time step scheme for a system of stochastic differential equations with multiple multiplicative noise: chemical Langevin equation, a proof of concept,” Journal of Chemical Physics, vol. 128, no. 1, Article ID 014103, 2008.
• S. Ilie and A. Teslya, “An adaptive stepsize method for the chemical Langevin equation,” Journal of Chemical Physics, vol. 136, no. 18, Article ID 184101, 2012.
• G. Söderlind, “Automatic control and adaptive time-stepping,” Numerical Algorithms, vol. 31, no. 1–4, pp. 281–310, 2002.
• G. Söderlind, “Digital filters in adaptive time-stepping,” ACM Transactions on Mathematical Software, vol. 29, no. 1, pp. 1–26, 2003.
• T. G. Kurtz, “Strong approximation theorems for density dependent Markov chains,” Stochastic Processes and Their Applications, vol. 6, no. 3, pp. 223–240, 1978.
• E. Hairer, G. Wanner, and S. P. Nørsett, Solving Ordinary Differential Equations I, Springer, Berlin, Germany, 2nd edition, 2009.
• Y. Cao, L. R. Petzold, M. Rathinam, and D. T. Gillespie, “The numerical stability of leaping methods for stochastic simulation of chemically reacting systems,” Journal of Chemical Physics, vol. 121, no. 24, pp. 12169–12178, 2004. \endinput