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Parametric Toricity of Steady State Varieties of Reaction Networks

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Computer Algebra in Scientific Computing (CASC 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12865))

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Abstract

We study real steady state varieties of the dynamics of chemical reaction networks. The dynamics are derived using mass action kinetics with parametric reaction rates. The models studied are not inherently parametric in nature. Rather, our interest in parameters is motivated by parameter uncertainty, as reaction rates are typically either measured with limited precision or estimated. We aim at detecting toricity and shifted toricity, using a framework that has been recently introduced and studied for the non-parametric case over both the real and the complex numbers. While toricity requires that the variety specifies a subgroup of the direct power of the multiplicative group of the underlying field, shifted toricity requires only a coset. In the non-parametric case these requirements establish real decision problems. In the presence of parameters we must go further and derive necessary and sufficient conditions in the parameters for toricity or shifted toricity to hold. Technically, we use real quantifier elimination methods. Our computations on biological networks here once more confirm shifted toricity as a relevant concept, while toricity holds only for degenerate parameter choices.

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Notes

  1. 1.

    Alternatively, one could temporarily introduce constants \(\mathbf {k}\) and state equivalence in an extended theory of real closed fields: \({\text {Th}}(\mathbb {R}) \cup \{\mathbf {k}>0\} \,\models \, \varphi \longleftrightarrow \varphi '\). This point of view is common in algebraic model theory and has been taken in [14].

  2. 2.

    https://sourceforge.net/projects/reduce-algebra/.

  3. 3.

    https://www.ebi.ac.uk/biomodels/BIOMD0000000101.

  4. 4.

    i.e., the ones with “\(\emptyset \)” on their left hand side or right hand side, respectively.

  5. 5.

    https://www.ebi.ac.uk/biomodels/BIOMD0000000001.

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Acknowledgments

This work has been supported by the interdisciplinary bilateral project ANR-17-CE40-0036/DFG-391322026 SYMBIONT [3, 4]. We are grateful to our project partner Ovidiu Radulescu for helping us understand part of the biological background.

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Authors and Affiliations

  1. CNRS, Inria, and the University of Lorraine, Nancy, France

    Thomas Sturm

  2. MPI Informatics, Saarland Informatics Campus, Saarbrücken, Germany

    Hamid Rahkooy & Thomas Sturm

  3. Saarland University, Saarland Informatics Campus, Saarbrücken, Germany

    Thomas Sturm

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  1. Hamid Rahkooy
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  2. Thomas Sturm
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Correspondence to Thomas Sturm .

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Editors and Affiliations

  1. Université de Lille, Villeneuve d’Ascq, France

    François Boulier

  2. Coventry University, Coventry, UK

    Matthew England

  3. Plekhanov Russian University of Economics, Moscow, Russia

    Timur M. Sadykov

  4. Institute of Theoretical and Applied Mechanics, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

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Rahkooy, H., Sturm, T. (2021). Parametric Toricity of Steady State Varieties of Reaction Networks. In: Boulier, F., England, M., Sadykov, T.M., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2021. Lecture Notes in Computer Science(), vol 12865. Springer, Cham. https://doi.org/10.1007/978-3-030-85165-1_18

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