Concentration Robustness in LP Kinetic Systems

For a reaction network 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 with kinetics $K$ is a positive equilibria LP (PLP) system if its set of positive equilibria is an LP set. Analogously, it is a complex balanced equilibria LP (CLP) system if its set of complex balanced equilibria is an LP set. An LP kinetic system is a PLP or CLP system. This paper studies concentration robustness of a species on subsets of equilibria. We present the "species hyperplane criterion", a necessary and sufficient condition for absolute concentration robustness (ACR) for a species of a PLP system. An analogous criterion holds for balanced concentration robustness (BCR) 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 $X$ in a linkage class have ACR and BCR in $X$, respectively. This leads to a broadening of the "low deficiency building blocks" framework 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.