Research article Special Issues

Modeling stochastic gene expression: From Markov to non-Markov models

  • Received: 24 May 2020 Accepted: 12 July 2020 Published: 12 August 2020
  • Gene expression is an inherently noisy process due to low copy numbers of mRNA or protein. The involved reaction events may happen in a Markov fashion but also in a non-Markov manner, depending on waiting-time distributions for the occurrence of reaction events. In recent years, many mechanistic models of stochastic gene expression have been developed to forecast fluctuations in mRNA or protein levels. Here we discus commonalities between these models as well as their extensions from Markov to non-Markov models, focusing on the contributions of noisy sources to the protein level. We derive a useful formula for the protein noise quantified by the ratio of the variance over the squared mean. This formula, expressed in terms of the frequencies of the probabilistic events, can be used in the fast evaluation of fluctuations in the protein abundance. Although the detail of the formula may vary from gene to gene, it highlights sources of the protein noise, which can be decomposed into two parts: spontaneous fluctuations resulting from the birth and death of the protein and forced fluctuations originated from switching between the promoter states.

    Citation: Zhenquan Zhang, Junhao Liang, Zihao Wang, Jiajun Zhang, Tianshou Zhou. Modeling stochastic gene expression: From Markov to non-Markov models[J]. Mathematical Biosciences and Engineering, 2020, 17(5): 5304-5325. doi: 10.3934/mbe.2020287

    Related Papers:

  • Gene expression is an inherently noisy process due to low copy numbers of mRNA or protein. The involved reaction events may happen in a Markov fashion but also in a non-Markov manner, depending on waiting-time distributions for the occurrence of reaction events. In recent years, many mechanistic models of stochastic gene expression have been developed to forecast fluctuations in mRNA or protein levels. Here we discus commonalities between these models as well as their extensions from Markov to non-Markov models, focusing on the contributions of noisy sources to the protein level. We derive a useful formula for the protein noise quantified by the ratio of the variance over the squared mean. This formula, expressed in terms of the frequencies of the probabilistic events, can be used in the fast evaluation of fluctuations in the protein abundance. Although the detail of the formula may vary from gene to gene, it highlights sources of the protein noise, which can be decomposed into two parts: spontaneous fluctuations resulting from the birth and death of the protein and forced fluctuations originated from switching between the promoter states.


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    [1] A. Sanchez, S. Choubey, J. Kondev, Stochastic models of transcription: From single molecules to single cells, Methods, 62 (2013), 13-25.
    [2] J. Zhang, T. Zhou, Promoter-mediated transcriptional dynamics, Biophys. J., 106 (2014), 479-488.
    [3] T. Zhou, J. Zhang, Analytical results for a multistate gene model, SIAM J. Appl. Math., 72 (2012), 789-818.
    [4] L. Cai, N. Friedman, X. S. Xie, Stochastic protein expression in individual cells at the single molecule level, Nature, 440 (2006), 358-362.
    [5] T. Liu, J. Zhang, T. Zhou, Effect of interaction between chromatin loops on cell-to-cell variability in gene expression, PLoS Comput. Biol., 12 (2016), e1004917.
    [6] D. R. Rigney, W. C. Schieve, Stochastic model of linear, continuous protein-synthesis in bacterial populations, J. Theor. Biol., 69 (1977), 761-766.
    [7] O. G. Berg, A model for the statistical fluctuations of protein numbers in a microbial population, J. Theor. Biol., 71 (1978), 587-603.
    [8] P. K. Tapaswi, R. K. Roychoudhury, T. Prasad, A stochastic model of gene activation and RNA synthesis during embryogenesis, Sankhyā Indian J. Statist. Ser. B, 49 (1987), 51-67.
    [9] J. Peccoud, B. Ycart, Markovian modeling of gene-product synthesis, Theor. Popul. Biol., 48 (1995), 222-234.
    [10] D. R. Rigney, Stochastic model of constitutive protein levels in growing and dividing bacterial cells, J. Theor. Biol., 76 (4), 453-480.
    [11] D. R. Rigney, Stochastic models of cellular variability. In Kinetic Logic A Boolean Approach to the Analysis of Complex Regulatory Systems, Springer, Berlin, Heidelberg, 1979.
    [12] T. B. Kepler, T. C. Elston, Stochasticity in transcriptional regulation: origins, consequences, and mathematical representations, Biophys. J., 81 (2001), 3116-3136.
    [13] M. Thattai, A. Van Oudenaarden, Intrinsic noise in gene regulatory networks, Proc. Natl. Acad. Sci. U.S.A., 98 (2001), 8614-8619.
    [14] P. S. Swain, M. B. Elowitz, E. D. Siggia, Intrinsic and extrinsic contributions to stochasticity in gene expression, Proc. Natl. Acad. Sci. U.S.A., 99 (2002), 12795-12800.
    [15] M. Sasai, P. G. Wolynes, Stochastic gene expression as a many-body problem, Proc. Natl. Acad. Sci. U.S.A., 100 (2003), 2374-2379.
    [16] T. Jia, R. V. Kulkarni, Intrinsic noise in stochastic models of gene expression with molecular memory and bursting, Phys. Rev. Lett., 106 (2011), 058102.
    [17] Z. Cao, R. Grima, Linear mapping approximation of gene regulatory networks with stochastic dynamics, Nat. Commun., 9 (2018), 1-15.
    [18] C. V. Harper, B. Finkenstädt, D. J. Woodcock, S. Friedrichsen, S. Semprini, L. Ashall, et al., Dynamic analysis of stochastic transcription cycles, PLoS Biol., 9 (2011), e1000607.
    [19] M. R. Green, Eukaryotic transcription activation: Right on target, Mol. Cell, 18 (2005), 399-402.
    [20] J. Paulsson, Models of stochastic gene expression, Phys. Life Rev., 2 (2005), 157-175.
    [21] G. Hornung, R. Bar-Ziv, D. Rosin, N. Tokuriki, D. S. Tawfik, M. Oren, et al., Noise-mean relationship in mutated promoters, Genom. Res., 22 (2012), 2409-2417.
    [22] Q. Li, G. Barkess, H. Qian, Chromatin looping and the probability of transcription, Trend. Genet., 22 (2006), 197-202.
    [23] C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, Springer, Berlin, Heidelberg, 2004.
    [24] J. D. Jordan, E. M. Landau, R. Iyengar, Signaling networks: The origins of cellular multitasking, Cell, 103 (2000), 193-200.
    [25] L. Bintu, J. Yong, Y. E. Antebi, K. McCue, Y. Kazuki, N. Uno, et al., Dynamics of epigenetic regulation at the single-cell level, Science, 351 (2016), 720-724.
    [26] C. W. Gardiner, Stochastic Methods: a handbook for the natural and social sciences, Springer, New York, 2009.
    [27] N. G. Van Kampen, Stochastic Processes in Physics and Chemistry, North-Holland, Amsterdam, 2007.
    [28] E. Pardoux, Markov Processes and Applications: Algorithms, Networks, Genome and Finance, vol 796, John Wiley & Sons, New York, 2008.
    [29] H. Andersson, T. Britton, Stochastic epidemic models and their statistical analysis, vol. 151, Springer Science & Business Media, 2012.
    [30] M. Salathé, M. Kazandjieva, J. W. Lee, P. Levis, M. W. Feldman, J. H. Jones, A high-resolution human contact network for infectious disease transmission, Proc. Natl. Acad. Sci. U.S.A., 107 (2010), 22020-22025.
    [31] A. Corral, Long-term clustering, scaling, and universality in the temporal occurrence of earthquakes, Phys. Rev. Lett., 92 (2004), 108501.
    [32] P. S. Stumpf, R. C. Smith, M. Lenz, A. Schuppert, F. J. Müller, A. Babtie, et al., Stem cell differentiation as a non-Markov stochastic process, Cell Syst., 5 (2017), 268-282.
    [33] D. M. Suter, N. Molina, D. Gatfield, K. Schneider, U. Schibler, F. Naef, Mammalian genes are transcribed with widely different bursting kinetics, Science, 332 (2011), 472-474.
    [34] T. Guérin, O. Bénichou, R. Voituriez, Non-Markovian polymer reaction kinetics, Nat. Chem., 4 (2012), 568-573.
    [35] A. L. Barabasi, The origin of bursts and heavy tails in human dynamics, Nature, 435 (2005), 207-211.
    [36] J. M. Pedraza, J. Paulsson, Effects of molecular memory and bursting on fluctuations in gene expression, Science, 319 (2008), 339-343.
    [37] G. Srinivasan, D. M. Tartakovsky, B. A. Robinson, A. B. Aceves, Quantification of uncertainty in geochemical reactions, Water Resour. Res., 43 (2007), W12415.
    [38] S. Condamin, O. Bénichou, V. Tejedor, R. Voituriez, J. Klafter, First-passage times in complex scale-invariant media, Nature, 450 (2007), 77-80.
    [39] G. Guigas, M. Weiss, Sampling the cell with anomalous diffusion—the discovery of slowness, Biophys. J., 94 (2008), 90-94.
    [40] Y. Meroz, I. M. Sokolov, J. Klafter, Distribution of first-passage times to specific targets on compactly explored fractal structures, Phys. Rev. E, 83 (2011), 020104.
    [41] M. Dentz, A. Russian, P. Gouze, Self-averaging and ergodicity of subdiffusion in quenched random media, Phys. Rev. E, 93 (2016), 010101.
    [42] A. A. Ovchinnikov, Y. B. Zeldovich, Role of density fluctuations in bimolecular reaction kinetics, Chem. Phys., 28 (1978), 215-218.
    [43] M. Dobrzyński, F. J. Bruggeman, Elongation dynamics shape bursty transcription and translation, Proc. Natl. Acad. Sci. U.S.A., 106 (2009), 2583-2588.
    [44] D. R. Larson, D. Zenklusen, B. Wu, J. A. Chao, R. H. Singer, Real-time observation of transcription initiation and elongation on an endogenous yeast gene, Science, 332 (2011), 475-478.
    [45] S. Yunger, L. Rosenfeld, Y. Garini, Y. Shav-Tal, Single-allele analysis of transcription kinetics in living mammalian cells, Nat. Methods, 7 (2010), 631-633.
    [46] I. Golding, J. Paulsson, S. M. Zawilski, E. C. Cox, Real-time kinetics of gene activity in individual bacteria, Cell, 123 (2005), 1025-1036.
    [47] T. Muramoto, D. Cannon, M. Gierliński, A. Corrigan, G. J. Barton, J. R. Chubb, Live imaging of nascent RNA dynamics reveals distinct types of transcriptional pulse regulation, Proc. Natl. Acad. Sci. U.S.A., 109 (2012), 7350-7355.
    [48] A. Raj, C. S. Peskin, D. Tranchina, D. Y. Vargas, S. Tyagi, Stochastic mRNA synthesis in mammalian cells, PLoS Biol., 4 (2006), e309.
    [49] D. G. Spiller, C. D. Wood, D. A. Rand, M. R. White, Measurement of single-cell dynamics, Nature, 465 (2010), 736-745.
    [50] A. Eldar, M. B. Elowitz, Functional roles for noise in genetic circuits, Nature, 467 (2010), 167-173.
    [51] B. Zoller, D. Nicolas, N. Molina, F. Naef, Structure of silent transcription intervals and noise characteristics of mammalian genes, Mol. Syst. Biol., 11 (2015), 823.
    [52] T. R. Sokolowski, T. Erdmann, P. R. Ten Wolde, Mutual repression enhances the steepness and precision of gene expression boundaries, PLoS Comput. Biol., 8 (2012), e1002654.
    [53] J. Paulsson, Summing up the noise in gene networks, Nature, 427 (2001), 415-418.
    [54] D. R. Larson, What do expression dynamics tell us about the mechanism of transcription?, Curr. Opin. Gen. Dev., 21 (2011), 591-599.
    [55] V. Shahrezaei, P. S. Swain, Analytical distributions for stochastic gene expression, Proc. Natl. Acad. Sci. U.S.A., 105 (2008), 17256-17261.
    [56] N. Kumar, A. Singh, R. V. Kulkarni, Transcriptional bursting in gene expression: analytical results for general stochastic models, PLoS Comput. Biol., 11 (2015), e1004292.
    [57] Z. Wang, Z. Zhang, T. Zhou, Exact distributions for stochastic models of gene expression with arbitrary regulation, Sci. China Math., 63 (2020), 485-500.
    [58] P. Liu, Z. Yuan, L. Huang, T. Zhou, Roles of factorial noise in inducing bimodal gene expression, Phys. Rev. E, 91 (2015), 062706.
    [59] J. Zhang, Q. Nie, T. Zhou, A moment-convergence method for stochastic analysis of biochemical reaction networks, J. Chem. Phys., 144 (2016), 194109.
    [60] A. B. O. Daalhuis, Confluent hypergeometric functions, NIST Handb. Math. Funct., 2010.
    [61] T. Aquino, M. Dentz, Chemical continuous time random walks, Phys. Rev. Lett., 119 (2017), 230601.
    [62] N. Masuda, M. A. Porter, R. Lambiotte, Random walks and diffusion on networks, Phys. Rep., 716 (2017), 1-58.
    [63] R. Kutner, J. Masoliver, The continuous time random walk, still trendy: fifty-year history, state of art and outlook, Eur. Phys. J. B, 90 (2017), 50.
    [64] L. Liu, B. R. K. Kashyap, J. G. C. Templeton, On the GIX/G/∞ system, J. Appl. Prob., 27 (1990), 671-683.
    [65] A. R. Stinchcombe, C. S. Peskin, D. Tranchina, Population density approach for discrete mRNA distributions in generalized switching models for stochastic gene expression, Phys. Rev. E, 85 (2012), 061919.
    [66] N. Masuda, L. E. Rocha, A Gillespie algorithm for non-Markovian stochastic processes, SIAM Rev., 60 (2018), 95-115.
    [67] C. Deneke, R. Lipowsky, A. Valleriani, Complex degradation processes lead to non-exponential decay patterns and age-dependent decay rates of messenger RNA, PloS One, 8 (2013), e55442.
    [68] B. C. Arnold, Majorization: Here, there and everywhere, Statist. Sci., 22 (2007), 407-413.
    [69] A. David, S. Larry, The least variable phase type distribution is Erlang, Stochastic Models, 3 (1987), 467-473.
    [70] J. Zhang, T. Zhou, Markovian approaches to modeling intracellular reaction processes with molecular memory, Proc. Natl. Acad. Sci. U.S.A., 116 (2019), 23542-23550.
    [71] H. Qiu, B. Zhang, T. Zhou, Analytical results for a generalized model of bursty gene expression with molecular memory, Phys. Rev. E, 100 (2019), 012128.
    [72] A. Coulon, C. C. Chow, R. H. Singer, D. R. Larson, Eukaryotic transcriptional dynamics: From single molecules to cell populations, Nat. Rev. Genet., 14 (2013), 572-584.
    [73] W. J. Blake, M. Kærn, C. R. Cantor, J. J. Collins, Noise in eukaryotic gene expression, Nature, 422 (2003), 633-637.
    [74] J. M. Raser, E. K. O'Shea, Control of stochasticity in eukaryotic gene expression, Science, 304 (2004), 1811-1814.
    [75] N. Friedman, L. Cai, X. S. Xie, Linking stochastic dynamics to population distribution: an analytical framework of gene expression, Phys. Rev. Lett., 97 (2006), 168302.
    [76] A. M. Kringstein, F. M. Rossi, A. Hofmann, H. M. Blau, Graded transcriptional response to different concentrations of a single transactivator, Proc. Natl. Acad. Sci. U.S.A., 95 (1998), 13670-13675.
    [77] J. Stewart-Ornstein, C. Nelson, J. DeRisi, J. S. Weissman, H. El-Samad, Msn2 coordinates a stoichiometric gene expression program, Curr. Biol., 23 (2013), 2336-2345.
    [78] J. Paulsson, M. Ehrenberg, Noise in a minimal regulatory network: plasmid copy number control, Quart. Rev. Biophys., 34 (2001), 1-59.
    [79] M. B. Elowitz, A. J. Levine, E. D. Siggia, P. S. Swain, Stochastic gene expression in a single cell, Science, 297 (2002), 1183-1186.
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