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Markov-Chain Perturbation and Approximation Bounds in Stochastic Biochemical Kinetics

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  • Alexander Y. Mitrophanov

    (Frederick National Laboratory for Cancer Research, National Institutes of Health, Frederick, MD 21702, USA)

Abstract

Markov chain perturbation theory is a rapidly developing subfield of the theory of stochastic processes. This review outlines emerging applications of this theory in the analysis of stochastic models of chemical reactions, with a particular focus on biochemistry and molecular biology. We begin by discussing the general problem of approximate modeling in stochastic chemical kinetics. We then briefly review some essential mathematical results pertaining to perturbation bounds for continuous-time Markov chains, emphasizing the relationship between robustness under perturbations and the rate of exponential convergence to the stationary distribution. We illustrate the use of these results to analyze stochastic models of biochemical reactions by providing concrete examples. Particular attention is given to fundamental problems related to approximation accuracy in model reduction. These include the partial thermodynamic limit, the irreversible-reaction limit, parametric uncertainty analysis, and model reduction for linear reaction networks. We conclude by discussing generalizations and future developments of these methodologies, such as the need for time-inhomogeneous Markov models.

Suggested Citation

  • Alexander Y. Mitrophanov, 2025. "Markov-Chain Perturbation and Approximation Bounds in Stochastic Biochemical Kinetics," Mathematics, MDPI, vol. 13(13), pages 1-20, June.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:13:p:2059-:d:1684382
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    References listed on IDEAS

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    1. Chris Sherlock, 2021. "Direct statistical inference for finite Markov jump processes via the matrix exponential," Computational Statistics, Springer, vol. 36(4), pages 2863-2887, December.
    2. Alexander Y. Mitrophanov, 2024. "The Arsenal of Perturbation Bounds for Finite Continuous-Time Markov Chains: A Perspective," Mathematics, MDPI, vol. 12(11), pages 1-15, May.
    3. Yann Vestring & Javad Tavakoli, 2024. "Confidence Regions for Steady-State Probabilities and Additive Functionals Based on a Single Sample Path of an Ergodic Markov Chain," Mathematics, MDPI, vol. 12(23), pages 1-11, November.
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    5. Yacov Satin & Rostislav Razumchik & Ilya Usov & Alexander Zeifman, 2023. "Numerical Computation of Distributions in Finite-State Inhomogeneous Continuous Time Markov Chains, Based on Ergodicity Bounds and Piecewise Constant Approximation," Mathematics, MDPI, vol. 11(20), pages 1-12, October.
    6. Alexander Zeifman & Victor Korolev & Yacov Satin, 2020. "Two Approaches to the Construction of Perturbation Bounds for Continuous-Time Markov Chains," Mathematics, MDPI, vol. 8(2), pages 1-25, February.
    7. Zeifman, A.I., 1995. "Upper and lower bounds on the rate of convergence for nonhomogeneous birth and death processes," Stochastic Processes and their Applications, Elsevier, vol. 59(1), pages 157-173, September.
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