Abstract
For a reaction network \(\mathscr{N}\) with species set \(\mathscr{S}\) , a log-parametrized (LP) set is a non-empty set of the form \(E(P, x^*) = \{x \in \mathbb{R}^\mathscr{S}_> \mid \log x - \log x^* \in P^\perp\}\) where \(P\) (called the LP set's
flux subspace) is a subspace of \( \mathbb{R}^\mathscr{S}\), \(x^*\) (called the LP set's reference point) is a given element of \(\mathbb{R}^\mathscr{S}_{>}\) and \(P^\perp\) (called the LP set’s parameter subspace) is the orthogonal
complement of \(P\). A network \(\mathscr{N}\) with kinetics \(K\) is a positive equilibria LP (PLP) system if its set of positive equilibria is an LP set, i.e., \(E_+(\mathscr{N}, K) = E(P_E, x^∗)\) where \(P_E\) is the flux subspace
and \(x^∗\) is a given positive equilibrium. Analogously, it is a complex balanced equilibria LP (CLP) system if its set of complex balanced equilibria is an LP set, i.e., \(Z_+(\mathscr{N}, K) = E(P_Z, x^∗)\) where \(P_Z\) is the flux
subspace and \(x^∗\) is a given complex balanced equilibrium. An LP kinetic system is a PLP or CLP system. This paper studies concentration robustness of a species on subsets of equilibria, i.e., the invariance of the species concentration
at all equilibria in the subset. We present the "species hyperplane criterion", a necessary and sufficient condition for absolute concentration robustness (ACR), i.e., invariance at all positive equilibria, for a species of a PLP system.
An analogous criterion holds for balanced concentration robustness (BCR), i.e., invariance at all complex balanced equilibria, for species of a CLP system. These criteria also lead to interesting necessary properties of LP systems with
concentration robustness. Furthermore, we show that PLP and CLP power law systems with Shinar-Feinberg reaction pairs in species \(X\), i.e., their rows in the kinetic order matrix differ only in \(X\), in a linkage class have ACR and
BCR in \(X\), respectively. This leads to a broadening of the "low deficiency building blocks" framework introduced by Fortun and Mendoza (2020) to include LP systems of Shinar-Feinberg type with arbitrary deficiency. Finally, we apply
our results to species concentration robustness in LP systems with poly-PL kinetics, i.e., sums of power law kinetics, including a refinement of a result on evolutionary games with poly-PL payoff functions and replicator dynamics by Talabis
et al (2020).
