Absolute Concentration Robustness in Networks with Low-Dimensional Stoichiometric Subspace

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 xi = 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.