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Multistationarity in the Space of Total Concentrations for Systems that Admit a Monomial Parametrization

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Abstract

We apply tools from real algebraic geometry to the problem of multistationarity of chemical reaction networks. A particular focus is on the case of reaction networks whose steady states admit a monomial parametrization. For such systems, we show that in the space of total concentrations multistationarity is scale invariant: If there is multistationarity for some value of the total concentrations, then there is multistationarity on the entire ray containing this value (possibly for different rate constants)—and vice versa. Moreover, for these networks it is possible to decide about multistationarity independent of the rate constants by formulating semi-algebraic conditions that involve only concentration variables. These conditions can easily be extended to include total concentrations. Hence, quantifier elimination may give new insights into multistationarity regions in the space of total concentrations. To demonstrate this, we show that for the distributive phosphorylation of a protein at two binding sites multistationarity is only possible if the total concentration of the substrate is larger than either the total concentration of the kinase or the total concentration of the phosphatase. This result is enabled by the chamber decomposition of the space of total concentrations from polyhedral geometry. Together with the corresponding sufficiency result of Bihan et al., this yields a characterization of multistationarity up to lower-dimensional regions.

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Acknowledgements

This project is funded by the Deutsche Forschungsgemeinschaft, 284057449. Alexandru Iosif and Thomas Kahle are also partially supported by the DFG-RTG “MathCore,” 314838170. We thank the anonymous reviewers for their valuable comments. One reviewer helped to improve the paper by providing a simpler proof of Lemma 3.10, clarifying the statements of Theorems 3.15 and 3.18, and pointing us to Frédéric et al. (2018, Theorem 4.1) which yields Corollary 4.13.

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Authors and Affiliations

  1. Hochschule für Technik und Wirtschaft, Berlin, Germany

    Carsten Conradi

  2. Joint Research Center for Computational Biomedicine, Aachen, Germany

    Alexandru Iosif

  3. Otto-von-Guericke Universität, Magdeburg, Germany

    Thomas Kahle

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  1. Carsten Conradi
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  2. Alexandru Iosif
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Correspondence to Carsten Conradi.

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Conradi, C., Iosif, A. & Kahle, T. Multistationarity in the Space of Total Concentrations for Systems that Admit a Monomial Parametrization. Bull Math Biol 81, 4174–4209 (2019). https://doi.org/10.1007/s11538-019-00639-4

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