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A geometric analysis of fast-slow models for stochastic gene expression

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Abstract

Stochastic models for gene expression frequently exhibit dynamics on several different scales. One potential time-scale separation is caused by significant differences in the lifetimes of mRNA and protein; the ratio of the two degradation rates gives a natural small parameter in the resulting chemical master equation, allowing for the application of perturbation techniques. Here, we develop a framework for the analysis of a family of ‘fast-slow’ models for gene expression that is based on geometric singular perturbation theory. We illustrate our approach by giving a complete characterisation of a standard two-stage model which assumes transcription, translation, and degradation to be first-order reactions. In particular, we present a systematic expansion procedure for the probability-generating function that can in principle be taken to any order in the perturbation parameter, allowing for an approximation of the corresponding propagator probabilities to that same order. For illustrative purposes, we perform this expansion explicitly to first order, both on the fast and the slow time-scales; then, we combine the resulting asymptotics into a composite fast-slow expansion that is uniformly valid in time. In the process, we extend, and prove rigorously, results previously obtained by Shahrezaei and Swain (Proc Natl Acad Sci USA 105(45):17256–17261, 2008) and Bokes et al. (J Math Biol 64(5):829–854, 2012; J Math Biol 65(3):493–520, 2012). We verify our asymptotics by numerical simulation, and we explore its practical applicability and the effects of a variation in the system parameters and the time-scale separation. Focussing on biologically relevant parameter regimes that induce translational bursting, as well as those in which mRNA is frequently transcribed, we find that the first-order correction can significantly improve the steady-state probability distribution. Similarly, in the time-dependent scenario, inclusion of the first-order fast asymptotics results in a uniform approximation for the propagator probabilities that is superior to the slow dynamics alone. Finally, we discuss the generalisation of our geometric framework to models for regulated gene expression that involve additional stages.

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Authors and Affiliations

  1. School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, James Clerk Maxwell Building, King’s Buildings, Mayfield Road, Edinburgh, EH9 3JZ, UK

    Nikola Popović

  2. Institute of Computational Biology, Helmholtz Zentrum München, German Research Center for Environmental Health, Ingolstädter Landstraße 1, 85764, Neuherberg, Germany

    Carsten Marr

  3. SynthSys-Systems and Synthetic Biology, University of Edinburgh, CH Waddington Building, King’s Buildings, Mayfield Road, Edinburgh, EH9 3JD, UK

    Peter S. Swain

Authors
  1. Nikola Popović
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  2. Carsten Marr
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  3. Peter S. Swain
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Correspondence to Nikola Popović.

Additional information

Nikola Popović is grateful to Peter De Maesschalck and Ramon Grima for their careful reading of drafts of the manuscript and numerous helpful suggestions, as well as to Peter Szmolyan for stimulating discussions. Moreover, the authors acknowledge grant support from the Moray Endowment Fund, as well as from MAXIMATHS, an initiative by the School of Mathematics at the University of Edinburgh aimed at maximising the impact of mathematics in science and engineering. Finally, the authors thank three anonymous referees for valuable comments which greatly improved the original manuscript.

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Popović, N., Marr, C. & Swain, P.S. A geometric analysis of fast-slow models for stochastic gene expression. J. Math. Biol. 72, 87–122 (2016). https://doi.org/10.1007/s00285-015-0876-1

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  • DOI: https://doi.org/10.1007/s00285-015-0876-1

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