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The limiting dynamics of a bistable molecular switch with and without noise

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Abstract

We consider the dynamics of a population of organisms containing two mutually inhibitory gene regulatory networks, that can result in a bistable switch-like behaviour. We completely characterize their local and global dynamics in the absence of any noise, and then go on to consider the effects of either noise coming from bursting (transcription or translation), or Gaussian noise in molecular degradation rates when there is a dominant slow variable in the system. We show analytically how the steady state distribution in the population can range from a single unimodal distribution through a bimodal distribution and give the explicit analytic form for the invariant stationary density which is globally asymptotically stable. Rather remarkably, the behaviour of the stationary density with respect to the parameters characterizing the molecular behaviour of the bistable switch is qualitatively identical in the presence of noise coming from bursting as well as in the presence of Gaussian noise in the degradation rate. This implies that one cannot distinguish between either the dominant source or nature of noise based on the stationary molecular distribution in a population of cells. We finally show that the switch model with bursting but two dominant slow genes has an asymptotically stable stationary density.

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Acknowledgments

This work was supported by the Natural Sciences and Engineering Research Council (NSERC, Canada) and the Polish NCN grant no 2014/13/B/ST1/00224. We are grateful to Marc Roussel (Lethbridge), Romain Yvinec (Tours) and Changjing Zhuge (Tsinghua University, Beijing) for helpful comments on this problem. We are especially indebted to the referees and the Associate Editor for their comments that have materially improved this paper.

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Authors and Affiliations

  1. Departments of Physiology, Physics and Mathematics, Centre for Applied Mathematics in Bioscience and Medicine (CAMBAM), McGill University, 3655 Promenade Sir William Osler, Montreal, QC, H3G 1Y6, Canada

    Michael C. Mackey

  2. Institute of Mathematics, University of Silesia, Bankowa 14, 40-007, Katowice, Poland

    Marta Tyran-Kamińska

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  1. Michael C. Mackey
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Correspondence to Michael C. Mackey.

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Mackey, M.C., Tyran-Kamińska, M. The limiting dynamics of a bistable molecular switch with and without noise. J. Math. Biol. 73, 367–395 (2016). https://doi.org/10.1007/s00285-015-0949-1

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  • DOI: https://doi.org/10.1007/s00285-015-0949-1

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