Stochastic modeling of biochemical reactions taking place at the cellular level has become the subject of intense research in recent years. Molecular interactions in a single cell exhibit random fluctuations. These fluctuations may be significant when small populations of some reacting species are present and then a stochastic description of the cellular dynamics is required. Often, the biochemically reacting systems encountered in applications consist of many species interacting through many reaction channels. Also, the dynamics of such systems is typically non-linear and presents multiple time-scales. Consequently, the stochastic mathematical models of biochemical systems can be quite complex and their analysis challenging. In this paper, we present a method to reduce a stochastic continuous model of well-stirred biochemical systems, the Chemical Langevin Equation, while preserving the overall behavior of the system. Several tests of our method on models of practical interest gave excellent results.
H. de Jong, “Modeling and Simulation of Genetic Regulatory Systems,” Journal of Computational Biology, Vol. 9, No. 1, 2002, pp. 67-103. doi:10.1089/10665270252833208
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
J. M. Raser and E. K. O’Shea, “Control of Stochasticity in Eukaryotic Gene Expression,” Science, Vol. 304, No. 5678, 2004, pp. 1811-1814. doi:10.1126/science.1098641
D. J. Wilkinson, “Stochastic Modelling for Systems Biology,” Chapman & Hall/CRC, Leiden, 2006.
S. Ilie and A. Teslya, “An Adaptive Stepsize Method for the Chemical Langevin Equation,” The Journal of Chemical Physics, Vol. 136, No. 18, 2012, pp. 184101-184115. doi:10.1063/1.4711143
M. S. Okino and M. L. Mavrovouniotis, “Simplification of Mathematical Models of Chemical Reactions,” Chemical Reviews, Vol. 98, No. 2, 1998, pp. 391-408. doi:10.1021/cr950223l
L. R. Petzold and W. J. Zhu, “Model Reduction for Chemical Kinetics: An Optimization Approach,” AICHE Journal, Vol. 44, No. 4, 1999, pp. 869-886. doi:10.1002/aic.690450418
H. Huang, M. Fairweather, J. F. Griffiths, A. S. Tomlin and R. B. Brad, “A Systematic Lumping Approach for the Reduction of Comprehensive Kinetic Models,” Proceedings of the Combustion Institute, Vol. 30, No.1, 2004, pp. 1309-1316. doi:10.1016/j.proci.2004.08.001
G. Li and H. Rabitz, “A General Analysis of Exact Lumping in Chemical Kinetics,” Chemical Engineering Science, Vol. 44, No. 6, 1989, pp. 1413-1430.
A. Kumar, P. D. Christofides and P. Daoutidis, “Singular Perturbation Modeling of Non-Linear Processes with Non-Explicit Time-Scale Separation,” Chemical Engineering Science, Vol. 53, No. 8, 1998, pp. 1491-1504. doi:10.1016/S0009-2509(98)00006-2
N. Vora and P. Daoutidis, “Nonlinear Model Reduction of Chemical Reaction System,” AICHE Journal, Vol. 47, No. 10, 2001, pp. 2320-2332. doi:10.1002/aic.690471016
Y. Cao, D. T. Gillespie and L. Petzold, “The Slow-Scale Stochastic Simulation Algorithm,” The Journal of Chemical Physics, Vol. 122, No. 1, 2005, pp. 014116-014134. doi:10.1063/1.1824902
C. V. Rao and A. P. Arkin, “Stochastic Chemical Kinetics and the Quasi-Steady-State Assumption: Application to the Gillespie Algorithm,” The Journal of Chemical Physics, Vol. 118, No. 11, 2003, pp. 4999-5010. doi:10.1063/1.1545446
A. Samant and D. Vlachos, “Overcoming Stiffness in Stochastic Simulation Stemming from Partial Equilibrium: A Multiscale Monte-Carlo Algorithm,” The Journal of Chemical Physics, Vol. 123, No. 14, 2005, pp. 144114-144122. doi:10.1063/1.2046628
D. T. Gillespie, “The Chemical Langevin Equations,” The Journal of Chemical Physics, Vol. 113, No. 1, 2000, pp. 297-306. doi:10.1063/1.481811
D. T. Gillespie, “Exact Stochastic Simulation of Coupled Chemical Reactions,” The Journal of Physical Chemistry, Vol. 81, No. 25, 1977, pp. 2340-2361. doi:10.1021/j100540a008
D. T. Gillespie, “A General Method for Numerically Simulating the Stochastic Time Evolution of Coupled Chemical Reactions”, The Journal of Computational Physics, Vol. 22, No. 4, 1976, pp. 403-434. doi:10.1016/0021-9991(76)90041-3
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
N. G. van Kampen, “Stochastic Processes in Physics and Chemistry,” Elsevier, Amsterdam, 2007.
A. Varma, M. Morbidelli and H. Wu, “Parametric Sensitivity in Chemical Systems,” Cambridge University Press, Cambridge, 1999. doi:10.1017/CBO9780511721779
M. Rathinam, P. W. Sheppard and M. Khammash, “Efficient Computation of Parameter Sensitivities of Discrete Stochastic Chemical Reaction Networks,” The Journal of Chemical Physics, Vol. 132, No. 3, 2010, pp. 034103-034116. doi:10.1063/1.3280166
P. W. Sheppard, M. Rathinam and M. Khammash, “A Pathwise Derivative Approach to the Computation of Parameter Sensitivities in Discrete Stochastic Chemical Systems,” The Journal of Chemical Physics, Vol. 136, No. 3, 2012, pp. 034115-034128. doi:10.1063/1.3677230
Y. C. Ho and X. R. Cao, “Optimization and Perturbation Analysis and Queueing Networks,” Journal of Optimization Theory and Applications, Vol. 40, No. 4, 1983, pp. 559-582. doi:10.1007/BF00933971
R. Suri and M. Zazanis, “Perturbation Analysis Gives Strongly Consistent Sensitivity Estimates for M/G/1 Queue,” Management Science, Vol. 34, No. 1, 1988, pp. 39-64. doi:10.1287/mnsc.34.1.39
V. Sotiropoulos, M. N. Contou-Carrere, P. Daoutidis and Y. N. Kaznessis, “Model Reduction of Multiscale Chemical Langevin Equations: A Numerical Case Study,” IEEE/ACM Transactions on Computational Biology and Bioinformatics, Vol. 6, No. 3, 2009, pp. 470-482. doi:10.1109/TCBB.2009.23
T. Jahnke, “On Reduced Models for Chemical Master Equation,” Multiscale Modeling and Simulation, Vol. 9, No. 4, 2011, pp. 1646-1676.
S. L. Cotter, K. C. Zygalakis, I. G. Kevrekidis and R. Erban, “A Constrained Approach to Multiscale Stochastic Simulation of Chemically Reacting Systems,” The Journal of Chemical Physics, Vol. 135, No. 9, 2011, pp. 094102-094114. doi:10.1063/1.3624333