Absolute concentration robustness in networks with many conservation laws

Nicolette Meshkat, Anne Shiu, Angélica Torres

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.