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  • Letter
  • Open Access

Non-Markovian gene expression

Ohad Vilk1,2,3, Ralf Metzler4,5, and Michael Assaf1

  • 1Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
  • 2Movement Ecology Lab, Department of Ecology, Evolution and Behavior, Alexander Silberman Institute of Life Sciences, Faculty of Science, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
  • 3Minerva Center for Movement Ecology, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
  • 4Institute of Physics and Astronomy, University of Potsdam, Potsdam 14476, Germany
  • 5Asia Pacific Centre for Theoretical Physics, Pohang 37673, Republic of Korea

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Phys. Rev. Research 6, L022026 – Published 2 May, 2024

DOI: https://doi.org/10.1103/PhysRevResearch.6.L022026

Abstract

We study two non-Markovian gene-expression models in which protein production is a stochastic process with a fat-tailed nonexponential waiting time distribution (WTD). For both models, we find two distinct scaling regimes separated by an exponentially long time, proportional to the mean first passage time (MFPT) to a ground state (with zero proteins) of the dynamics, from which the system can only exit via a nonexponential reaction. At times shorter than the MFPT the dynamics are stationary and ergodic, entailing similarity across different realizations of the same process, with an increased Fano factor of the protein distribution, even when the WTD has a finite cutoff. Notably, at times longer than the MFPT the dynamics are nonstationary and nonergodic, entailing significant variability across different realizations. The MFPT to the ground state is shown to directly affect the average population sizes and we postulate that the transition to nonergodicity is universal in such non-Markovian models.

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References (62)

  1. M. B. Elowitz, A. J. Levine, E. D. Siggia, and P. S. Swain, Stochastic gene expression in a single cell, Science 297, 1183 (2002).
  2. W. J. Blake, M. Kærn, C. R. Cantor, and J. J. Collins, Noise in eukaryotic gene expression, Nature (London) 422, 633 (2003).
  3. M. Kaern, T. C. Elston, W. J. Blake, and J. J. Collins, Stochasticity in gene expression: from theories to phenotypes, Nat. Rev. Gen. 6, 451 (2005).
  4. V. Shahrezaei and P. S. Swain, Analytical distributions for stochastic gene expression, Proc. Natl. Acad. Sci. USA 105, 17256 (2008).
  5. A. Bar-Even, J. Paulsson, N. Maheshri, M. Carmi, E. O'Shea, Y. Pilpel, and N. Barkai, Noise in protein expression scales with natural protein abundance, Nat. Genet. 38, 636 (2006).
  6. A. Raj and A. Van Oudenaarden, Nature, nurture, or chance: stochastic gene expression and its consequences, Cell 135, 216 (2008).
  7. J. M. Raser and E. K. O'shea, Noise in gene expression: origins, consequences, and control, Science 309, 2010 (2005).
  8. B. Munsky, G. Neuert, and A. Van Oudenaarden, Using gene expression noise to understand gene regulation, Science 336, 183 (2012).
  9. J. M. Raser and E. K. O'Shea, Control of stochasticity in eukaryotic gene expression, Science 304, 1811 (2004).
  10. G. K. Ackers, A. D. Johnson, and M. A. Shea, Quantitative model for gene regulation by lambda phage repressor., Proc. Natl. Acad. Sci. USA 79, 1129 (1982).
  11. C. Gardiner, Stochastic Methods (Springer, Berlin, 2009), Vol. 4.
  12. J. E. M. Hornos, D. Schultz, G. C. P. Innocentini, J. Wang, A. M. Walczak, J. N. Onuchic, and P. G. Wolynes, Self-regulating gene: an exact solution, Phys. Rev. E 72, 051907 (2005).
  13. M. J. Morelli, S. Tănase-Nicola, R. J. Allen, and P. R. Ten Wolde, Reaction coordinates for the flipping of genetic switches, Biophys. J. 94, 3413 (2008).
  14. M. Assaf, E. Roberts, and Z. Luthey-Schulten, Determining the stability of genetic switches: Explicitly accounting for MRNA noise, Phys. Rev. Lett. 106, 248102 (2011).
  15. M. Assaf, E. Roberts, Z. Luthey-Schulten, and N. Goldenfeld, Extrinsic noise driven phenotype switching in a self-regulating gene, Phys. Rev. Lett. 111, 058102 (2013).
  16. E. Roberts, S. Be'er, C. Bohrer, R. Sharma, and M. Assaf, Dynamics of simple gene-network motifs subject to extrinsic fluctuations, Phys. Rev. E 92, 062717 (2015).
  17. H. Yin, S. Liu, and X. Wen, Optimal transition paths of phenotypic switching in a non-Markovian self-regulation gene expression, Phys. Rev. E 103, 022409 (2021).
  18. E. Aurell and K. Sneppen, Epigenetics as a first exit problem, Phys. Rev. Lett. 88, 048101 (2002).
  19. I. Golding and A. Amir, Gene expression in growing cells: A biophysical primer, arXiv:2311.12143.
  20. H.-H. Jo, J. I. Perotti, K. Kaski, and J. Kertész, Analytically solvable model of spreading dynamics with non-Poissonian processes, Phys. Rev. X 4, 011041 (2014).
  21. Z. Zhang, M. Chen, J. Zhang, and T. Zhou, Analysis of non-Markovian effects in generalized birth-death models, Discrete Contin. Dyn. Syst. B 26, 6377 (2021).
  22. 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).
  23. T. Brett and T. Galla, Stochastic processes with distributed delays: Chemical Langevin equation and linear-noise approximation, Phys. Rev. Lett. 110, 250601 (2013).
  24. A. Schwabe, K. N. Rybakova, and F. J. Bruggeman, Transcription stochasticity of complex gene regulation models, Biophys. J. 103, 1152 (2012).
  25. 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).
  26. S. Baghram, F. Nikakhtar, M. R. R. Tabar, S. Rahvar, R. K. Sheth, K. Lehnertz, and M. Sahimi, Exact enumeration approach to first-passage time distribution of non-Markov random walks, Phys. Rev. E 99, 062101 (2019).
  27. A. Godec and R. Metzler, Universal proximity effect in target search kinetics in the few-encounter limit, Phys. Rev. X 6, 041037 (2016).
  28. D. S. Grebenkov, R. Metzler, and G. Oshanin, Strong defocusing of molecular reaction times results from an interplay of geometry and reaction control, Commun. Chem. 1, 96 (2018).
  29. D. S. Grebenkov, R. Metzler, and G. Oshanin, From single-particle stochastic kinetics to macroscopic reaction rates: fastest first-passage time of N random walkers, New J. Phys. 22, 103004 (2020).
  30. F. C. Collins and G. E. Kimball, Diffusion-controlled reaction rates, J. Coll. Sci. 4, 425 (1949).
  31. M. Coppey, O. Bénichou, R. Voituriez, and M. Moreau, Kinetics of target site localization of a protein on DNA: a stochastic approach, Biophys. J. 87, 1640 (2004).
  32. S. Condamin, O. Bénichou, V. Tejedor, R. Voituriez, and J. Klafter, First-passage times in complex scale-invariant media, Nature (London) 450, 77 (2007).
  33. O. Bénichou, C. Chevalier, J. Klafter, B. Meyer, and R. Voituriez, Geometry-controlled kinetics, Nat. Chem. 2, 472 (2010).
  34. A. V. Weigel, B. Simon, M. M. Tamkun, and D. Krapf, Ergodic and nonergodic processes coexist in the plasma membrane as observed by single-molecule tracking, Proc. Natl. Acad. Sci. USA 108, 6438 (2011).
  35. G. Bel and E. Barkai, Weak ergodicity breaking in the continuous-time random walk, Phys. Rev. Lett. 94, 240602 (2005).
  36. S. Burov, R. Metzler, and E. Barkai, Aging and nonergodicity beyond the Khinchin theorem, Proc. Natl. Acad. Sci. USA 107, 13228 (2010).
  37. E. Barkai, Y. Garini, and R. Metzler, Strange kinetics of single molecules in living cells, Phys. Today 65(8), 29 (2012).
  38. R. Metzler, J.-H. Jeon, A. G. Cherstvy, and E. Barkai, Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking, Phys. Chem. Chem. Phys. 16, 24128 (2014).
  39. J.-H. Jeon, V. Tejedor, S. Burov, E. Barkai, C. Selhuber-Unkel, K. Berg-Sørensen, L. Oddershede, and R. Metzler, In vivo anomalous diffusion and weak ergodicity breaking of lipid granules, Phys. Rev. Lett. 106, 048103 (2011).
  40. S. A. Tabei, S. Burov, H. Y. Kim, A. Kuznetsov, T. Huynh, J. Jureller, L. H. Philipson, A. R. Dinner, and N. F. Scherer, Intracellular transport of insulin granules is a subordinated random walk, Proc. Natl. Acad. Sci. USA 110, 4911 (2013).
  41. Z. R. Fox, E. Barkai, and D. Krapf, Aging power spectrum of membrane protein transport and other subordinated random walks, Nat. Commun. 12, 6162 (2021).
  42. O. Vilk, Y. Orchan, M. Charter, N. Ganot, S. Toledo, R. Nathan, and M. Assaf, Ergodicity breaking in area-restricted search of avian predators, Phys. Rev. X 12, 031005 (2022).
  43. O. Vilk, E. Aghion, T. Avgar, C. Beta, O. Nagel, A. Sabri, R. Sarfati, D. K. Schwartz, M. Weiss, D. Krapf et al., Unravelling the origins of anomalous diffusion: From molecules to migrating storks, Phys. Rev. Res. 4, 033055 (2022).
  44. R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep. 339, 1 (2000).
  45. T. Aquino and M. Dentz, Chemical continuous time random walks, Phys. Rev. Lett. 119, 230601 (2017).
  46. A. Leier and T. T. Marquez-Lago, Delay chemical master equation: direct and closed-form solutions, Proc. Math. Phys. Eng. Sci. 471, 20150049 (2015).
  47. 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).
  48. A. A. Kurilovich, V. N. Mantsevich, K. J. Stevenson, A. V. Chechkin, and V. V. Palyulin, Complex diffusion-based kinetics of photoluminescence in semiconductor nanoplatelets, Phys. Chem. Chem. Phys. 22, 24686 (2020).
  49. T. J. Doerries, A. V. Chechkin, R. Schumer, and R. Metzler, Rate equations, spatial moments, and concentration profiles for mobile-immobile models with power-law and mixed waiting time distributions, Phys. Rev. E 105, 014105 (2022).
  50. A. J. Bray, S. N. Majumdar, and G. Schehr, Persistence and first-passage properties in nonequilibrium systems, Adv. Phys. 62, 225 (2013).
  51. H. Chen, K. Shiroguchi, H. Ge, and X. S. Xie, Genome-wide study of MRNA degradation and transcript elongation in Escherichia coli, Mol. Syst. Biol. 11, 781 (2015).
  52. D. F. Anderson, A modified next reaction method for simulating chemical systems with time dependent propensities and delays, J. Chem. Phys. 127, 214107 (2007).
  53. M. Boguñá, L. F. Lafuerza, R. Toral, and M. Á. Serrano, Simulating non-Markovian stochastic processes, Phys. Rev. E 90, 042108 (2014).
  54. N. Masuda and L. E. Rocha, A Gillespie algorithm for non-Markovian stochastic processes, SIAM Rev. 60, 95 (2018).
  55. M. I. Dykman, E. Mori, J. Ross, and P. Hunt, Large fluctuations and optimal paths in chemical kinetics, J. Chem. Phys. 100, 5735 (1994).
  56. M. Assaf and B. Meerson, Extinction of metastable stochastic populations, Phys. Rev. E 81, 021116 (2010).
  57. M. Assaf and B. Meerson, WKB theory of large deviations in stochastic populations, J. Phys. A: Math. Theor. 50, 263001 (2017).
  58. M. Thattai and A. Van Oudenaarden, Intrinsic noise in gene regulatory networks, Proc. Natl. Acad. Sci. USA 98, 8614 (2001).
  59. P. J. Choi, L. Cai, K. Frieda, and X. S. Xie, A stochastic single-molecule event triggers phenotype switching of a bacterial cell, Science 322, 442 (2008).
  60. Y. He, S. Burov, R. Metzler, and E. Barkai, Random time-scale invariant diffusion and transport coefficients, Phys. Rev. Lett. 101, 058101 (2008).
  61. F. Thiel and I. M. Sokolov, Weak ergodicity breaking in an anomalous diffusion process of mixed origins, Phys. Rev. E 89, 012136 (2014).
  62. C. N. Angstmann, B. I. Henry, and A. V. McGann, A fractional-order infectivity SIR model, Physica A 452, 86 (2016).

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Phys. Rev. Research 6, L022026– Published 2 May, 2024
  • Received 7 August 2023
  • Accepted 15 March 2024
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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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