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Analytic solutions for stochastic hybrid models of gene regulatory networks

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Abstract

Discrete-state stochastic models are a popular approach to describe the inherent stochasticity of gene expression in single cells. The analysis of such models is hindered by the fact that the underlying discrete state space is extremely large. Therefore hybrid models, in which protein counts are replaced by average protein concentrations, have become a popular alternative. The evolution of the corresponding probability density functions is given by a coupled system of hyperbolic PDEs. This system has Markovian nature but its hyperbolic structure makes it difficult to apply standard functional analytical methods. We are able to prove convergence towards the stationary solution and determine such equilibrium explicitly by combining abstract methods from the theory of positive operators and elementary ideas from potential analysis.

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Notes

  1. A function is called strongly positive if it is positive outside a set of measure zero (Reed and Simon 1978).

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Acknowledgements

This work is the outcome of a collaboration between two mathematicians working on evolutionary equations (DM) and partial differential equations (PK) and a computer biologist (VW). It has partially been carried out at the ZiF (Center for Interdisciplinary Research) in Bielefeld in the framework of the Cooperation Group Discrete and continuous models in the theory of networks. The authors are indebted to the ZiF for financial support and hospitality. The work of PK was also partially supported by the Swedish Research Council (Grant D0497301). The work of DM was partially supported by the German Research Foundation (Grant 397230547). The work of VW was partially supported by the German Research Foundation (Grant 391984329). We warmly thank Jochen Glück (Ulm) for suggesting to us the proofs of Lemma 4.5 and Theorem 4.6 and for many helpful discussions. We also thank the anonymous referees for pointing us to relevant references. In particular, we have learned that some of our analytic formulae for stationary distributions have already been obtained in Faggionato et al. (2009), Zeiser et al. (2010).

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Authors and Affiliations

  1. Department of Mathematics, Stockholm University, 106 91, Stockholm, Sweden

    Pavel Kurasov

  2. Chair of Analysis, University of Hagen, 58084, Hagen, Germany

    Delio Mugnolo

  3. Computer Science Department, Saarland University, 66123, Saarbrücken, Germany

    Verena Wolf

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  1. Pavel Kurasov
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  2. Delio Mugnolo
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  3. Verena Wolf
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Correspondence to Delio Mugnolo.

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Kurasov, P., Mugnolo, D. & Wolf, V. Analytic solutions for stochastic hybrid models of gene regulatory networks. J. Math. Biol. 82, 9 (2021). https://doi.org/10.1007/s00285-021-01549-7

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