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Generalized master equation for stochastic gene expression with non-Markovian degradation

Changhong Shi*

Xiyan Yang

Tianshou Zhou† and Jiajun Zhang‡

  • *Contact author: ashi_248@163.com
  • †Contact author: mcszhtsh@mail.sysu.edu.cn
  • ‡Contact author: zhjiajun@mail.sysu.edu.cn
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Phys. Rev. Research 8, 013008 – Published 7 January, 2026

DOI: https://doi.org/10.1103/qbnx-97l4

Abstract

mRNAs and proteins are believed to degrade in non-Markovian fashions, challenging the common assumption of exponential mRNA and protein lifetimes. Here, we use the queue theory to derive a generalized master equation for a three-stage model of gene expression with nonexponential mRNA and protein degradations. Analytical and numerical results show that the protein degradation memory enlarges the bimodal region and increases the protein noise, whereas the mRNA degradation memory enhances bimodality but reduces the protein noise by lowering the translational burst variance. While these results provide insight into the roles of degradation memories in shaping gene-expression patterns, the generalized master equation is a useful tool for studying gene-expression dynamics in complex cellular environments.

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References (53)

  1. A. Raj and A. van Oudenaarden, Nature, nurture, or chance: Stochastic gene expression and its consequences, Cell 135, 216 (2008).
  2. M. B. Elowitz, A. J. Levine, E. D. Siggia, and P. S. Swain, Stochastic gene expression in a single cell, Science 297, 1183 (2002).
  3. W. J. Blake, M. Kærn, C. R. Cantor, and J. J. Collins, Noise in eukaryotic gene expression, Nature (London) 422, 633 (2003).
  4. D. Kulasiri and R. Kosarwal, Chemical Master Equation for Large Biological Networks (Springer, 2021).
  5. L. Ham, M. A. Coomer, K. Öcal, R. Grima, and M. P. Stumpf, A stochastic vs deterministic perspective on the timing of cellular events, Nat. Commun. 15, 5286 (2024).
  6. A. G. Nicoll, J. Szavits-Nossan, M. R. Evans, and R. Grima, Transient power-law behaviour following induction distinguishes between competing models of stochastic gene expression, Nat. Commun. 16, 2833 (2025).
  7. V. Shahrezaei and P. S. Swain, Analytical distributions for stochastic gene expression, Proc. Natl. Acad. Sci. USA 105, 17256 (2008).
  8. J. Zhang, L. Chen, and T. Zhou, Analytical distribution and tunability of noise in a model of promoter progress, Biophys. J. 102, 1247 (2012).
  9. J. Zhang and T. Zhou, Promoter-mediated transcriptional dynamics, Biophys. J. 106, 479 (2014).
  10. S. Luo, Z. Wang, Z. Zhang, T. Zhou, and J. Zhang, Genome-wide inference reveals that feedback regulations constrain promoter-dependent transcriptional burst kinetics, Nucleic Acids Res. 51, 68 (2023).
  11. J. M. Pedraza and J. Paulsson, Effects of molecular memory and bursting on fluctuations in gene expression, Science 319, 339 (2008).
  12. T. Jia and R. V. Kulkarni, Intrinsic noise in stochastic models of gene expression with molecular memory and bursting, Phys. Rev. Lett. 106, 058102 (2011).
  13. J. Zhang and T. Zhou, Markovian approaches to modeling intracellular reaction processes with molecular memory, Proc. Natl. Acad. Sci. USA 116, 23542 (2019).
  14. E. McShane, C. Sin, H. Zauber, J. N. Wells, N. Donnelly, X. Wang, J. Hou, W. Chen, Z. Storchova, J. A. Marsh, et al., Kinetic analysis of protein stability reveals age-dependent degradation, Cell 167, 803 (2016).
  15. M. Bazzaro, M. K. Lee, A. Zoso, W. L. Stirling, A. Santillan, I.-M. Shih, and R. B. Roden, Ubiquitin-proteasome system stress sensitizes ovarian cancer to proteasome inhibitor-induced apoptosis, Cancer Res. 66, 3754 (2006).
  16. J. C. Taggart, H. Zauber, M. Selbach, G.-W. Li, and E. McShane, Keeping the proportions of protein complex components in check, Cell Syst. 10, 125 (2020).
  17. C. Deneke, R. Lipowsky, and A. Valleriani, Complex degradation processes lead to non-exponential decay patterns and age-dependent decay rates of messenger RNA, PLoS One 8, e55442 (2013).
  18. J. G. Belasco, All things must pass: Contrasts and commonalities in eukaryotic and bacterial mRNA decay, Nat. Rev. Mol. Cell Biol. 11, 467 (2010).
  19. V. Elgart, T. Jia, and R. Kulkarni, Quantifying mRNA synthesis and decay rates using small RNAs, Biophys. J. 98, 2780 (2010).
  20. E. Huntzinger and E. Izaurralde, Gene silencing by microRNAs: Contributions of translational repression and mRNA decay, Nat. Rev. Genet. 12, 99 (2011).
  21. C. Shi, X. Yang, T. Zhou, and J. Zhang, Nascent RNA kinetics with complex promoter architecture: Analytic results and parameter inference, Phys. Rev. E 110, 034413 (2024).
  22. Q. Jiang, X. Fu, S. Yan, R. Li, W. Du, Z. Cao, F. Qian, and R. Grima, Neural network aided approximation and parameter inference of non-Markovian models of gene expression, Nat. Commun. 12, 2618 (2021).
  23. X. Yang, Z. Wang, C. Shi, T. Zhou, and J. Zhang, Deciphering HIV-1 transcription initiation and elongation from single-molecule imaging data, Research 8, 0645 (2025).
  24. C. Shi, Y. Jiang, and T. Zhou, Queuing models of gene expression: Analytical distributions and beyond, Biophys. J. 119, 1606 (2020).
  25. J. Szavits-Nossan and R. Grima, Solving stochastic gene-expression models using queueing theory: A tutorial review, Biophys. J. 123, 1034 (2024).
  26. Y. Wang, J. Szavits-Nossan, Z. Cao, and R. Grima, Joint distribution of nuclear and cytoplasmic mRNA levels in stochastic models of gene expression: Analytical results and parameter inference, Phys. Rev. Lett. 135, 068401 (2025).
  27. H. Xu, S. O. Skinner, A. M. Sokac, and I. Golding, Stochastic kinetics of nascent RNA, Phys. Rev. Lett. 117, 128101 (2016).
  28. C. Shi, X. Yang, J. Zhang, and T. Zhou, Stochastic modeling of the mRNA life process: A generalized master equation, Biophys. J. 122, 4023 (2023).
  29. Y. Li and C. Jia, Effective Markovian dynamics method for solving non-Markovian dynamics of stochastic gene expression, Phys. Rev. Lett. 135, 048401 (2025).
  30. N. Kumar, A. Singh, and R. V. Kulkarni, Transcriptional bursting in gene expression: Analytical results for general stochastic models, PLoS Comput. Biol. 11, e1004292 (2015).
  31. X. Yang, S. Luo, Z. Zhang, Z. Wang, T. Zhou, and J. Zhang, Silent transcription intervals and translational bursting lead to diverse phenotypic switching, Phys. Chem. Chem. Phys. 24, 26600 (2022).
  32. M. Chen, S. Luo, M. Cao, C. Guo, T. Zhou, and J. Zhang, Exact distributions for stochastic gene expression models with arbitrary promoter architecture and translational bursting, Phys. Rev. E 105, 014405 (2022).
  33. V. Ramaswami and M. F. Neuts, Some explicit formulas and computational methods for infinite-server queues with phase-type arrivals, J. Appl. Probab. 17, 498 (1980).
  34. H. H. McAdams and A. Arkin, Stochastic mechanisms in gene expression, Proc. Natl. Acad. Sci. USA 94, 814 (1997).
  35. B. Munsky and M. Khammash, The finite state projection algorithm for the solution of the chemical master equation, J. Chem. Phys. 124, 044104 (2006).
  36. S. Kullback and R. A. Leibler, On information and sufficiency, Ann. Math. Stat. 22, 79 (1951).
  37. D. Fraser and M. Kaern, A chance at survival: Gene expression noise and phenotypic diversification strategies, Mol. Microbiol. 71, 1333 (2009).
  38. M. Acar, J. T. Mettetal, and A. van Oudenaarden, Stochastic switching as a survival strategy in fluctuating environments, Nat. Genet. 40, 471 (2008).
  39. F. Jiao, Q. Sun, M. Tang, J. Yu, and B. Zheng, Distribution modes and their corresponding parameter regions in stochastic gene transcription, SIAM J. Appl. Math. 75, 2396 (2015).
  40. B. Barzel and O. Biham, Binomial moment equations for stochastic reaction systems, Phys. Rev. Lett. 106, 150602 (2011).
  41. J. Zhang, Q. Nie, and T. Zhou, A moment-convergence method for stochastic analysis of biochemical reaction networks, J. Chem. Phys. 144, 194106 (2016).
  42. A. Valleriani, Z. Ignatova, A. Nagar, and R. Lipowsky, Turnover of messenger RNA: Polysome statistics beyond the steady state, Europhys. Lett. 89, 58003 (2010).
  43. C. Chevalier, J. Dorignac, Y. Ibrahim, A. Choquet, A. David, J. Ripoll, E. Rivals, F. Geniet, N.-O. Walliser, J. Palmeri, et al., Physical modeling of ribosomes along messenger RNA: Estimating kinetic parameters from ribosome profiling experiments using a ballistic model, PLoS Comput. Biol. 19, e1011522 (2023).
  44. H. Yin, X. Wen, and H. Wang, Stochastic modeling of the mRNA kinetic process: A delay chemical master-equation model based on a survival probability, Phys. Rev. E 112, 044405 (2025).
  45. M. Dobrzyński and F. J. Bruggeman, Elongation dynamics shape bursty transcription and translation, Proc. Natl. Acad. Sci. USA 106, 2583 (2009).
  46. N. A. M. Monk, Oscillatory expression of Hes1, p53, and NF-κB driven by transcriptional time delays, Curr. Biol. 13, 1409 (2003).
  47. D. Bratsun, D. Volfson, L. S. Tsimring, and J. Hasty, Delay-induced stochastic oscillations in gene regulation, Proc. Natl. Acad. Sci. USA 102, 14593 (2005).
  48. T. Galla, Intrinsic fluctuations in stochastic delay systems: Theoretical description and application to a simple model of gene regulation, Phys. Rev. E 80, 021909 (2009).
  49. J. Holehouse, Z. Cao, and R. Grima, Stochastic modeling of autoregulatory genetic feedback loops: A review and comparative study, Biophys. J. 118, 1517 (2020).
  50. C. Jia and R. Grima, Holimap: An accurate and efficient method for solving stochastic gene network dynamics, Nat. Commun. 15, 6557 (2024).
  51. R. D. Dar, B. S. Razooky, A. Singh, T. V. Trimeloni, J. M. McCollum, C. D. Cox, M. L. Simpson, and L. S. Weinberger, Transcriptional burst frequency and burst size are equally modulated across the human genome, Proc. Natl. Acad. Sci. USA 109, 17454 (2012).
  52. J. Breda, M. Zavolan, and E. van Nimwegen, Bayesian inference of gene expression states from single-cell RNA-seq data, Nat. Biotechnol. 39, 1008 (2021).
  53. B. Munsky, G. Li, Z. R. Fox, D. P. Shepherd, and G. Neuert, Distribution shapes govern the discovery of predictive models for gene regulation, Proc. Natl. Acad. Sci. USA 115, 7533 (2018).

Outline

Information

Phys. Rev. Research 8, 013008– Published 7 January, 2026
  • Received 20 August 2025
  • Accepted 17 December 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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