Biochemical systems have important practical applications, in particular to understanding critical intra-cellular processes. Often biochemical kinetic models represent cellular processes as systems of chemical reactions, traditionally modeled by the deterministic reaction rate equations. In the cellular environment, many biological processes are inherently stochastic. The stochastic fluctuations due to the presence of some low molecular populations may have a great impact on the biochemical system behavior. Then, stochastic models are required for an accurate description of the system dynamics. An important stochastic model of biochemical kinetics is the Chemical Langevin Equation. In this work, we provide a numerical method for approximating the solution of the Chemical Langevin Equation, namely the derivative-free Milstein scheme. The method is compared with the widely used strategy for this class of problems, the Milstein method. As opposed to the Milstein scheme, the proposed strategy has the advantage that it does not require the calculation of exact derivatives, while having the same strong order of accuracy as the Milstein scheme. Therefore it may be used for an automatic simulation of the numerical solution of the Chemical Langevin Equation. The tests on several models of practical interest show that our method performs very well.
Stochastic Biochemical Kinetics; Chemical Langevin Equation; Derivative-Free Milstein Method
S. Ilie and M. Morshed, "Automatic Simulation of the Chemical Langevin Equation," Applied Mathematics, Vol. 4 No. 1, 2013, pp. 235-241. doi: 10.4236/am.2013.41A036.
[1] H. Kitano, “Computational Systems Biology,” Nature, Vol. 420, No. 6912, 2002, pp. 206-210. doi:10.1038/nature01254
[2] M. B. Elowitz and S. Leibler, “A Synthetic Oscillatory Network of Transcriptional Regulators,” Nature, Vol. 403, No. 6767, 2000, pp. 335-338. doi:10.1038/35002125
[3] A. P. Arkin, J. Ross and H. H. McAdams, “Stochastic Kinetic Analysis of Developmental Pathway Bifurcation in Phage-Infected Escherichia coli Cells,” Genetics, Vol. 149, 1998, pp. 1633-1648.
[4] W. J. Blake, M. Kaern, C. R. Cantor and J. J. Collins, “Noise in Eukaryotic Gene Expression,” Nature, Vol. 422, No. 6932, 2003, pp. 633-637. doi:10.1038/nature01546
[5] N. Federoff and W. Fontana, “Small Numbers of Big Molecules,” Science, Vol. 297, No. 5584, 2002, pp. 1129-1131. doi:10.1126/science.1075988
[6] 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, 2002, pp. 1183-1186. doi:10.1126/science.1070919
[7] D. T. Gillespie, “A Rigorous Derivation of the Chemical Master Equation,” Physica A, Vol. 188, No. 1-3, 1992, pp. 402-425. doi:10.1016/0378-4371(92)90283-V
[8] 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, 1976, pp. 403-434. doi:10.1016/0021-9991(76)90041-3
[9] D. T. Gillespie, “Exact Stochastic Simulation of Coupled Chemical Reactions,” Journal of Physical Chemistry, Vol. 81, No. 25, 1977, pp. 2340-2361. doi:10.1021/j100540a008
[10] Y. Cao, D. T. Gillespie and L. Petzold, “The Slow-Scale Stochastic Simulation Algorithm,” Journal of Computational Physics, Vol. 122, No. 1, 2005, pp. 01411601-01411618. doi:10.1063/1.1824902
[11] D. T. Gillespie, “Approximate Accelerated Stochastic Simulation of Chemically Reacting Systems,” Journal of Chemical Physics, Vol. 115, No. 4, 2001, pp. 1716-1733. doi:10.1063/1.1378322
[12] T. Li, “Analysis of Explicit Tau-Leaping Schemes for Simulating Chemically Reacting Systems,” SIAM Multiscale Modeling & Simulation, Vol. 6, No. 2, 2007, pp. 417-436.
[13] 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, 2003, pp. 4999-5010. doi:10.1063/1.1545446
[14] A. Samant and D. Vlachos, “Overcoming Stiffness in Stochastic Simulation Stemming from Partial Equilibrium: A Multiscale Monte-Carlo Algorithm,” Journal of Chemical Physics, Vol. 123, No. 14, 2005, pp. 144114-144122. doi:10.1063/1.2046628
[15] D. T. Gillespie, “The Chemical Langevin Equation,” Journal of Chemical Physics, Vol. 113, No. 1, 2000, pp. 297-306. doi:10.1063/1.481811
[16] S. Ilie and A. Teslya, “An Adaptive Stepsize Method for the Chemical Langevin Equation,” Journal of Chemical Physics, Vol. 136, No. 18, 2012, pp. 184101-184115. doi:10.1063/1.4711143
[17] S. Ilie, “Variable Time-Stepping in the Pathwise Numerical Solution of the Chemical Langevin Equation,” Journal of Chemical Physics, Vol. 137, No. 23, 2012, pp. 234110-234119. doi:10.1063/1.4771660
[18] S. Ilie, W. H. Enright and K. R. Jackson, “Numerical Solution of Stochastic Models of Biochemical Kinetics,” Canadian Applied Mathematics Quarterly, Vol. 17, No. 3, 2009, pp. 523-554.
[19] C. W. Gardiner, “Stochastic Methods: A Handbook for the Natural and Social Sciences,” Springer, Berlin, 2009.
[20] P. E. Kloeden and E. Platen, “Numerical Solution of Stochastic Differential Equations,” Springer-Verlag, Berlin, 1992.
[21] K. Burrage, P. M. Burrage and T. Tian, “Numerical Methods for Strong Solutions of Stochastic Differential Equations: An Overview,” Proceedings of the Royal Society A, Vol. 460, No. 2041, 2004, pp. 373-402. doi:10.1098/rspa.2003.1247
[22] MATLAB, “The Language of Technical Computing,” 2009. www.mathworks.com.
[23] D. J. Wilkinson, “Stochastic Modelling for Systems Biology,” Chapman & Hall/CRC, London, 2006.