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Parametric Toricity of Steady State Varieties of Reaction Networks


Rahkooy,  Hamid
Automation of Logic, MPI for Informatics, Max Planck Society;


Sturm,  Thomas       
Automation of Logic, MPI for Informatics, Max Planck Society;

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Rahkooy, H., & Sturm, T. (2021). Parametric Toricity of Steady State Varieties of Reaction Networks. Retrieved from https://arxiv.org/abs/2105.10853.

Cite as: https://hdl.handle.net/21.11116/0000-0009-4B11-0
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.