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arXiv · 2105.00109

Absolute concentration robustness in networks with low-dimensional stoichiometric subspace

Nicolette Meshkat·Anne Shiu·Angélica Torres
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Abstract

A reaction system exhibits "absolute concentration robustness" (ACR) in some species if the positive steady-state value of that species does not depend on initial conditions. Mathematically, this means that the positive part of the variety of the steady-state ideal lies entirely in a hyperplane of the form $x_i=c$, for some $c>0$. Deciding whether a given reaction system -- or those arising from some reaction network -- exhibits ACR is difficult in general, but here we show that for many simple networks, assessing ACR is straightforward. Indeed, our criteria for ACR can be performed by simply inspecting a network or its standard embedding into Euclidean space. Our main results pertain to networks with many conservation laws, so that all reactions are parallel to one other. Such "one-dimensional" networks include those networks having only one species. We also consider networks with only two reactions, and show that ACR is characterized by a well-known criterion of Shinar and Feinberg. Finally, up to some natural ACR-preserving operations -- relabeling species, lengthening a reaction, and so on -- only three families of networks with two reactions and two species have ACR. Our results are proven using algebraic and combinatorial techniques.

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Nicolette Meshkat, Anne Shiu, Angélica Torres. 2021-04-30. Absolute concentration robustness in networks with low-dimensional stoichiometric subspace. https://arxiv.org/abs/2105.00109

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