Cellular environments are in essence stochastic, owing to the random character of the biochemical reaction events in a single cell. Stochastic fluctuations may substantially contribute to the dynamics of systems with small copy numbers of some biochemical species. Then, stochastic models are indispensable for properly portraying the behaviour of the system. Sensitivity analysis is one of the central tools for studying stochastic models of cellular dynamics. Here, we propose some finite-difference strategies for estimating parametric sensitivities of higher-order moments of the system state for stochastic discrete biochemical kinetic models. To reduce the variance of the sensitivity estimator, we employ various coupling techniques. The advantages of the proposed methods are illustrated in several models of biochemical systems of practical relevance.
Elowitz, M.B., et al. (2002) Stochastic Gene Expression in a Single Cell. Science, 297, 1183-1186. https://doi.org/10.1126/science.1070919
Federoff, N. and Fontana, W. (2002) Small Numbers of Big Molecules. Science, 297, 1129-1131. https://doi.org/10.1126/science.1075988
Raser, J.M. and O’Shea, E.K. (2004) Control of Stochasticity in Eukaryotic Gene Expression. Science, 304, 1811-1814. https://doi.org/10.1126/science.1098641
Wilkinson, D.J. (2006) Stochastic Modelling for Systems Biology. Chapman & Hall/CRC, Boca Raton. https://doi.org/10.1201/9781420010664
Gillespie, D.T. (1992) A Rigorous Derivation of the Chemical Master Equation. Physica A, 188, 402-425. https://doi.org/10.1016/0378-4371(92)90283-V
Gillespie, D.T. (1976) A General Method for Numerically Simulating the Stochastic Time Evolution of Coupled Chemical Reactions. Journal of Computational Physics, 22, 403-434. https://doi.org/10.1016/0021-9991(76)90041-3
Gillespie, D.T. (1977) Exact Stochastic Simulation of Coupled Chemical Reactions. Journal of Computational Physics, 81, 2340-2361. https://doi.org/10.1021/j100540a008
Kurtz, T.G. (1982) Representation and Approximation of Counting Processes. In: Advances in Filtering and Optimal Stochastic Control, Lecture Notes in Control and Information Sciences, Vol. 42, Springer, Berlin, 177-191. https://doi.org/10.1007/BFb0004537
Gillespie, D.T. (2001) Approximate Accelerated Stochastic Simulation of Chemically Reacting Systems. Journal of Computational Physics, 115, 1716-1733. https://doi.org/10.1063/1.1378322
Padgett, J.M.A. and Ilie, S. (2016) An Adaptive Tau-Leaping Method for Stochastic Simulations of Reaction-Diffusion Systems. AIP Advances, 6, Article ID: 035217. https://doi.org/10.1063/1.4944952
Higham, D.J. (2008) Modeling and Simulating Chemical Reactions. SIAM Review, 50, 347-368. https://doi.org/10.1137/060666457
Li, T. (2007) Analysis of Explicit Tau-Leaping Schemes for Simulating Chemically Reacting Systems. Multiscale Modeling and Simulation, 6, 417-436. https://doi.org/10.1137/06066792X
Simoni, G., Reali, F., Priami, C. and Marchetti, L. (2019) Stochastic Simulation Algorithms for Computational Systems Biology: Exact, Approximate, and Hybrid Methods. Wiley Interdisciplinary Reviews: Systems Biology and Medicine, 11, e1459. https://doi.org/10.1002/wsbm.1459
Sekiguchi, T., Hamada, H. and Okamoto, M. (2019) Inference of General Mass Action-Based State Equations for Oscillatory Biochemical Reaction Systems Using k-Step Genetic Programming. Applied Mathematics, 10, 627-645. https://doi.org/10.4236/am.2019.108045
Varma, A., Morbidelli, M. and Wu, H. (1999) Parametric Sensitivity in Chemical Systems. Cambridge University Press, Cambridge. https://doi.org/10.1017/CBO9780511721779
Morshed, M., Ingalls, B. and Ilie, S. (2017) An Efficient Finite-Difference Strategy for Sensitivity Analysis of Stochastic Models of Biochemical Systems. Biosystems, 151, 43-52. https://doi.org/10.1016/j.biosystems.2016.11.006
Gholami, S. and Ilie, S. (2021) Reducing Stochastic Discrete Models of Biochemical Networks. Applied Mathematics, 12, 449-469. https://doi.org/10.4236/am.2021.125031
Ilie, S. and Gholami, S. (2013) Simplifying Stochastic Mathematical Models of Biochemical Systems. Applied Mathematics, 4, 248-256. https://doi.org/10.4236/am.2013.41A038
Rathinam, M., Sheppard, P.W. and Khammash, M. (2010) Efficient Computation of Parameter Sensitivities of Discrete Stochastic Chemical Reaction Networks. The Journal of Chemical Physics, 132, Article ID: 034103. https://doi.org/10.1063/1.3280166
Anderson, D.F. (2012) An Efficient Finite Difference Method for Parameter Sensitivities of Continuous Time Markov Chains. SIAM Journal on Numerical Analysis, 50, 2237-2258. https://doi.org/10.1137/110849079
Gibson, M.A. and Bruck, J. (2000) Exact Stochastic Simulation of Chemical Systems with Many Species and Many Channels. Journal of Computational Physics, 105, 1876-1889. https://doi.org/10.1021/jp993732q
Morshed, M., Ingalls, B. and Ilie, S. (2018) An Effective Implicit Finite-Difference Method for Sensitivity Analysis of Stiff Stochastic Discrete Biochemical Systems. IET Systems Biology, 12, 123-130. https://doi.org/10.1049/iet-syb.2017.0048
Thanh, V.H. (2019) A Critical Comparison of Rejection-Based Algorithms for Simulation of Large Biochemical Reaction Networks. Bulletin of Mathematical Biology, 81, 3053-3073. https://doi.org/10.1007/s11538-018-0462-y
Sayyidmousavi, A. and Ilie, S. (2017) An Efficient Hybrid Method for Stochastic Reaction-Diffusion Biochemical Systems with Delay. AIP Advances, 7, Article ID: 125305. https://doi.org/10.1063/1.5001760
Ethier, S.N. and Kurtz, T.G. (1986) Markov Processes: Characterization and Convergence. Wiley, New York. https://doi.org/10.1002/9780470316658
Jahnke, T. (2011) On Reduced Models for the Chemical Master Equation. Multiscale Modeling and Simulation, 9, 1646-1676. https://doi.org/10.1137/110821500