American Institute of Mathematical Sciences

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July  2021, 26(7): 3717-3735. doi: 10.3934/dcdsb.2020254

Analysis of non-Markovian effects in generalized birth-death models

Zhenquan Zhang , Meiling Chen , Jiajun Zhang and  Tianshou Zhou ,

Key Laboratory of Computational Mathematics, Guangdong Province and School of, Mathematics, Sun Yat-sen University, Guangzhou, 510275, China

* Corresponding author: Tianshou Zhou

Received  October 2019 Revised  June 2020 Published  July 2021 Early access  August 2020

Birth-death processes are a fundamental reaction module for which we can find its prototypes in many scientific fields. For such a kind of module, if all the reaction events are Markovian, the reaction kinetics is simple. However, experimentally observable quantities are in general consequences of a series of reactions, implying that the synthesis of a macromolecule in general involve multiple middle reaction steps with some reactions that would not be specified by experiments. This multistep process can create molecular memory between reaction events, leading to non-Markovian behavior. Based on the theoretical framework established in a recent paper published in [39], we find that the effect of non-Markovianity is equivalent to the introduction of a feedback, non-Markovianity always amplifies the mean level of the product if the death reaction is non-Markovian but always reduces the mean level if the birth reaction is non-Markovian, and in contrast to Markovianity, non-Markovianity can reduce or amplify the product noise, depending on the details of waiting-time distributions characterizing reaction events. Examples analysis indicates that non-Markovianity, whose effects were neglected in previous studies, can significantly impact gene expression.

Keywords: Birth-death process, non-Markovianity, waiting-time distribution.
Mathematics Subject Classification: Primary: 60G07, 60J27; Secondary: 92B05.
Citation: Zhenquan Zhang, Meiling Chen, Jiajun Zhang, Tianshou Zhou. Analysis of non-Markovian effects in generalized birth-death models. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3717-3735. doi: 10.3934/dcdsb.2020254
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J. M. Pedraza and J. Paulsson, Effects of molecular memory and bursting on fluctuations in gene expression, Science, 319 (2008), 339-343.  doi: 10.1126/science.1144331.  Google Scholar

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A. Raj, C. S. Peskin, D. Tranchina, D. Y. Vargas and S. Tyagi, Stochastic mRNA synthesis in mammalian cells, PLoS Biol., 4 (2006), e309. doi: 10.1371/journal.pbio.0040309.  Google Scholar

[28]

M. Salathé, M. Kazandjieva, J. W. Lee, P. Levis, M. W. Feldman and J. H. Jones, A high-resolution human contact network for infectious disease transmission, Proc. Natl. Acad. Sci. U.S.A., 107 (2010), 22020-22025.   Google Scholar

[29]

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[30]

A. Schwabe, K. N. Rybakova and F. J. Bruggeman, Transcription stochasticity of complex gene regulation models, Biophys. J., 103 (2012), 1152-1161.  doi: 10.1016/j.bpj.2012.07.011.  Google Scholar

[31]

V. Shahrezaei and P. S. Swain, Analytical distributions for stochastic gene expression, Proc. Natl. Acad. Sci. U.S.A., 105 (2008), 17256-17261.  doi: 10.1073/pnas.0803850105.  Google Scholar

[32]

M. Starnini, J. P. Gleeson and M. Boguñá, Equivalence between non-Markovian and Markovian dynamics in epidemic spreading processes, Phys. Rev. Lett., 118 (2017), 128301. doi: 10.1103/PhysRevLett.118.128301.  Google Scholar

[33]

P. S. Stumpf, R. C. Smith, M. Lenz, A. Schuppert, F. J. Müller, A. Babtie, T. E. Chan, M. P. Stumpf, C. P. Please, S. D. Howison, F. Arai and B. D. MacArthur, Stem cell differentiation as a non-Markov stochastic process, Cell Syst., 5 (2017), 268-282.  doi: 10.1016/j.cels.2017.08.009.  Google Scholar

[34]

D. M. Suter, N. Molina, D. Gatfield, K. Schneider, U. Schibler and F. Naef, Mammalian genes are transcribed with widely different bursting kinetics, Science, 332 (2011), 472-474.  doi: 10.1126/science.1198817.  Google Scholar

[35]

P. Thomas, N. Popović and R. Grima, Phenotypic switching in gene regulatory networks, Proc. Natl. Acad. Sci. U. S. A., 111 (2014), 6994-6999.  doi: 10.1073/pnas.1400049111.  Google Scholar

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N. G. Van Kampen, Stochastic Processes in Physics and Chemistry, North-Holland, Amsterdam, 2007. Google Scholar

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P. Van Mieghem and R. Van de Bovenkamp, Non-Markovian infection spread dramatically alters the susceptible-infected-susceptible epidemic threshold in networks, Phys. Rev. Lett., 110 (2013), 108701. doi: 10.1103/PhysRevLett.110.108701.  Google Scholar

[38]

J. J. Zhang and T. S. Zhou, Promoter-mediated transcriptional dynamics, Biophys. J., 106 (2014), 479-488.  doi: 10.1016/j.bpj.2013.12.011.  Google Scholar

[39]

J. J. Zhang and T. S. Zhou, Markovian approaches to modeling intracellular reaction processes with molecular memory, Proc. Natl. Acad. Sci. U. S. A., 116 (2019), 23542-23550.  doi: 10.1073/pnas.1913926116.  Google Scholar

[40]

J. J. Zhang, Q. Nie and T. S. Zhou, A moment-convergence method for stochastic analysis of biochemical reaction networks, J. Chem. Phys., 144 (2016), 194109. doi: 10.1063/1.4950767.  Google Scholar

show all references

References:
[1]

T. Aquino and M. Dentz, Chemical continuous time random walks, Phys. Rev. Lett., 119 (2017), 230601. doi: 10.1103/PhysRevLett.119.230601.  Google Scholar

[2]

A. L. Barabasi, The origin of bursts and heavy tails in human dynamics, Nature, 435 (2005), 207-211.  doi: 10.1038/nature03459.  Google Scholar

[3]

M. Barrio, K. Burrage, A. Leier and T. Tian, Oscillatory regulation of Hes1 discrete stochastic delay modelling and simulation, PLoS Comput. Biol., 2 (2006), e117. Google Scholar

[4]

D. Bratsun, D. Volfson, L. S. Tsimring and J. Hasty, Delay-induced stochastic oscillations in gene regulation, Proc. Natl. Acad. Sci. U.S.A., 102 (2005), 14593-14598.  doi: 10.1073/pnas.0503858102.  Google Scholar

[5]

T. Brett and T. Galla, Stochastic processes with distributed delays: Chemical Langevin equation and linear noise approximation, Phys. Rev. Lett., 110 (2013), 250601. doi: 10.1103/PhysRevLett.110.250601.  Google Scholar

[6]

A. Corral, Long-term clustering, scaling, and universality in the temporal occurrence of earthquakes, Phys. Rev. Lett., 92 (2004), 108501. doi: 10.1103/PhysRevLett.92.108501.  Google Scholar

[7]

I. De Vega and D. Alonso, Dynamics of non-Markovian open quantum systems, Rev. Modern Phys., 89 (2017), 015001. doi: 10.1103/RevModPhys.89.015001.  Google Scholar

[8]

J. Delvenne, R. Lambiotte and L. E. C. Rocha, Diffusion on networked systems is a question of time or structure, Nat. Commun., 6 (2015), 1-10.  doi: 10.1038/ncomms8366.  Google Scholar

[9]

W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 2, John Wiley & Sons, New York, 2008. Google Scholar

[10]

C. W. Gardiner, Stochastic Methods: A Handbook for the Natural and Social Sciences, Springer-Verlag, Berlin, 2009.  Google Scholar

[11]

D. T. Gillespie, A general method for numerically simulating the stochastic time evolution of coupled chemical reactions, J. Comput. Phys., 22 (1976), 403-434.  doi: 10.1016/0021-9991(76)90041-3.  Google Scholar

[12]

J. P. Gleeson, K. P. OŚullivan, R. A. Baños and Y. Moreno, Effects of network structure, competition and memory time on social spreading phenomena, Phys. Rev. X, 6 (2016), 021019. doi: 10.1103/PhysRevX.6.021019.  Google Scholar

[13]

T. Guérin, O. Bénichou and R. Voituriez, Non-Markovian polymer reaction kinetics, Nat. Chem., 4 (2012), 568-573.   Google Scholar

[14]

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. doi: 10.1371/journal.pbio.1000607.  Google Scholar

[15]

H. W. Hethcote and P. v. d. Driessche, A SIS epidemic model with variable population size and a delay, J. Math. Biol., 34 (1995), 177-194.  doi: 10.1007/BF00178772.  Google Scholar

[16]

J. C. Jaeger and G. Newstead, An Introduction to the Laplace Transformation with Engineering Applications, Methuen & Co., Ltd., London, John Wiley & Sons, Inc., New York, NY, 1949.  Google Scholar

[17]

T. Jia and R. V. Kulkarni, Intrinsic noise in stochastic models of gene expression with molecular memory, Phys. Rev. Lett., 106 (2011), 058102. doi: 10.1103/PhysRevLett.106.058102.  Google Scholar

[18]

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 (2014), 011041. doi: 10.1103/PhysRevX.4.011041.  Google Scholar

[19]

I. Z. Kiss, G. Röst and Z. Vizi, Generalization of pairwise models to non-Markovian epidemics on networks, Phys. Rev. Lett., 115 (2015), 078701. doi: 10.1103/PhysRevLett.115.078701.  Google Scholar

[20]

N. Kumar, A. Singh and R. V. Kulkarni, Transcriptional bursting in gene expression: Analytical results for general stochastic models, PLoS Comput. Biol., 11 (2015), e1004292. doi: 10.1371/journal.pcbi.1004292.  Google Scholar

[21]

D. R. Larson, What do expression dynamics tell us about the mechanism of transcription?, Curr. Opin. Genet. Dev., 21 (2011), 591-599.  doi: 10.1016/j.gde.2011.07.010.  Google Scholar

[22]

N. Masuda, M. A. Porter and R. Lambiotte, Random walks and diffusion on networks, Phys. Rep., 716 (2017), 1-58.  doi: 10.1016/j.physrep.2017.07.007.  Google Scholar

[23]

A. S. Novozhilov, G. P. Karev and E. V. Koonin, Biological applications of the theory of birth-and-death processes, Briefings in Bioinformatics, 7 (2006), 70-85.  doi: 10.1093/bib/bbk006.  Google Scholar

[24]

E. Pardoux, Markov Processes and Applications: Algorithms, Networks, Genome and Finance, Vol. 796, John Wiley & Sons, New York, 2008. doi: 10.1002/9780470721872.  Google Scholar

[25]

J. Peccoud and B. Ycart, Markovian modeling of gene product synthesis, Theor. Popul. Biol., 48 (1995), 222-234.  doi: 10.1006/tpbi.1995.1027.  Google Scholar

[26]

J. M. Pedraza and J. Paulsson, Effects of molecular memory and bursting on fluctuations in gene expression, Science, 319 (2008), 339-343.  doi: 10.1126/science.1144331.  Google Scholar

[27]

A. Raj, C. S. Peskin, D. Tranchina, D. Y. Vargas and S. Tyagi, Stochastic mRNA synthesis in mammalian cells, PLoS Biol., 4 (2006), e309. doi: 10.1371/journal.pbio.0040309.  Google Scholar

[28]

M. Salathé, M. Kazandjieva, J. W. Lee, P. Levis, M. W. Feldman and J. H. Jones, A high-resolution human contact network for infectious disease transmission, Proc. Natl. Acad. Sci. U.S.A., 107 (2010), 22020-22025.   Google Scholar

[29]

I. Scholtes, N. Wider, R. Pfitzner, A. Garas, C. J. Tessone and F. Schweitzer, Causality-driven slow-down and speed-up of diffusion in non-Markovian temporal networks, Nat. Commun., 5 (2014), 1-9.  doi: 10.1038/ncomms6024.  Google Scholar

[30]

A. Schwabe, K. N. Rybakova and F. J. Bruggeman, Transcription stochasticity of complex gene regulation models, Biophys. J., 103 (2012), 1152-1161.  doi: 10.1016/j.bpj.2012.07.011.  Google Scholar

[31]

V. Shahrezaei and P. S. Swain, Analytical distributions for stochastic gene expression, Proc. Natl. Acad. Sci. U.S.A., 105 (2008), 17256-17261.  doi: 10.1073/pnas.0803850105.  Google Scholar

[32]

M. Starnini, J. P. Gleeson and M. Boguñá, Equivalence between non-Markovian and Markovian dynamics in epidemic spreading processes, Phys. Rev. Lett., 118 (2017), 128301. doi: 10.1103/PhysRevLett.118.128301.  Google Scholar

[33]

P. S. Stumpf, R. C. Smith, M. Lenz, A. Schuppert, F. J. Müller, A. Babtie, T. E. Chan, M. P. Stumpf, C. P. Please, S. D. Howison, F. Arai and B. D. MacArthur, Stem cell differentiation as a non-Markov stochastic process, Cell Syst., 5 (2017), 268-282.  doi: 10.1016/j.cels.2017.08.009.  Google Scholar

[34]

D. M. Suter, N. Molina, D. Gatfield, K. Schneider, U. Schibler and F. Naef, Mammalian genes are transcribed with widely different bursting kinetics, Science, 332 (2011), 472-474.  doi: 10.1126/science.1198817.  Google Scholar

[35]

P. Thomas, N. Popović and R. Grima, Phenotypic switching in gene regulatory networks, Proc. Natl. Acad. Sci. U. S. A., 111 (2014), 6994-6999.  doi: 10.1073/pnas.1400049111.  Google Scholar

[36]

N. G. Van Kampen, Stochastic Processes in Physics and Chemistry, North-Holland, Amsterdam, 2007. Google Scholar

[37]

P. Van Mieghem and R. Van de Bovenkamp, Non-Markovian infection spread dramatically alters the susceptible-infected-susceptible epidemic threshold in networks, Phys. Rev. Lett., 110 (2013), 108701. doi: 10.1103/PhysRevLett.110.108701.  Google Scholar

[38]

J. J. Zhang and T. S. Zhou, Promoter-mediated transcriptional dynamics, Biophys. J., 106 (2014), 479-488.  doi: 10.1016/j.bpj.2013.12.011.  Google Scholar

[39]

J. J. Zhang and T. S. Zhou, Markovian approaches to modeling intracellular reaction processes with molecular memory, Proc. Natl. Acad. Sci. U. S. A., 116 (2019), 23542-23550.  doi: 10.1073/pnas.1913926116.  Google Scholar

[40]

J. J. Zhang, Q. Nie and T. S. Zhou, A moment-convergence method for stochastic analysis of biochemical reaction networks, J. Chem. Phys., 144 (2016), 194109. doi: 10.1063/1.4950767.  Google Scholar

Figure 1.  Schematic diagram for birth-death reaction process (A), where $ \psi_{1} \left(t\right) $ and $ \psi_{2} \left(t;n\right) $ are interreaction waiting-time distributions, which may be exponential (B) or non-exponential (C). Here $ n $ represents the number of molecules of reactive species $ X $
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Figure 2.  Difference between dynamic probability distributions obtained in the original non-Markovian model (curves with empty circles) and in the constructed Markovian model (colored curves), both being used in the modeling of a gene transcription process. Parameter values are set as $ k_{1} = 3,\lambda_{1} = 90,\lambda_{2} = 1 $, and the initial conditions are set as the same in two cases. In the diagram, '$ t = \inf $' means that the distributions after $ t>10 $ are approximately the same, implying that the stationary distribution exists
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11]. Parameter values are set as: (B) $ k_{d} = 1,\lambda_{d} = 1 $ and (C) $ k_{b} = 1,\lambda_{b} = 10 $. In (B) and (C), we use 1000 realizations to obtain numerical results and error bars to indicate the error ranges of the numerical results">Figure 3.  Characteristics of stationary protein distribution in a generalized model of constitutive gene expression (A), where solid lines represent theoretical predictions and empty circles represent numerical results obtained by the Gillespie algorithm [11]. Parameter values are set as: (B) $ k_{d} = 1,\lambda_{d} = 1 $ and (C) $ k_{b} = 1,\lambda_{b} = 10 $. In (B) and (C), we use 1000 realizations to obtain numerical results and error bars to indicate the error ranges of the numerical results
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11]. Parameter values are set as: (B) $ \mu = 10,\delta = 1 $, $ b = 2,k_{on} = 1,\lambda_{on} = 0.5,k_{off} = 1,\lambda_{off} = 0.1 $ for Markovianity, whereas $ k_{on} = 4,\lambda_{on} = 4 $ and the other parameter values are kept unchanged for non-Markovianity; (C)$ \mu = 10,\delta = 1,b = 2 $, $ k_{on} = 1,\lambda_{on} = 0.5 $, $ k_{off} = 1,\lambda_{off} = 0.1 $ for Markovianity, whereas $ k_{off} = 4,\lambda_{off} = 0.4 $ and the other parameter values are kept unchanged for non-Markovianity. In (B) and (C), we use 1000 realizations to obtain numerical results and error bars to indicate the error ranges of the numerical results">Figure 4.  Characteristics of stationary protein distribution in a generalized model of bursty gene expression (A), where solid lines represent theoretical predictions and empty circles represent numerical results obtained by the Gillespie algorithm [11]. Parameter values are set as: (B) $ \mu = 10,\delta = 1 $, $ b = 2,k_{on} = 1,\lambda_{on} = 0.5,k_{off} = 1,\lambda_{off} = 0.1 $ for Markovianity, whereas $ k_{on} = 4,\lambda_{on} = 4 $ and the other parameter values are kept unchanged for non-Markovianity; (C)$ \mu = 10,\delta = 1,b = 2 $, $ k_{on} = 1,\lambda_{on} = 0.5 $, $ k_{off} = 1,\lambda_{off} = 0.1 $ for Markovianity, whereas $ k_{off} = 4,\lambda_{off} = 0.4 $ and the other parameter values are kept unchanged for non-Markovianity. In (B) and (C), we use 1000 realizations to obtain numerical results and error bars to indicate the error ranges of the numerical results
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