Absolute concentration robustness in networks with low-dimensional stoichiometric subspace

TL;DR

This paper provides simple criteria to determine absolute concentration robustness (ACR) in reaction networks with low-dimensional stoichiometric subspaces, especially those with many conservation laws or few reactions.

Contribution

It introduces straightforward methods for assessing ACR in simple networks, including networks with one or two species, and characterizes all two-reaction, two-species networks with ACR.

Findings

ACR can be easily checked by inspecting the network structure.

Networks with many conservation laws are often ACR.

Only three families of two-reaction, two-species networks exhibit ACR.

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.