2021/10/26 by Lao, Angelyn R., Lubenia, Patrick Vincent N., Magpantay, Daryl M. +1 · 1 citation
#Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2110.13845
For a reaction network with species set \mathscrS, a log-parametrized (LP) set is a non-empty set of the form E(P, x^*) = \x ∈ ℝ^\mathscrS_> | log x - log x^* ∈ P^⊥\ where P (called the LP set's flux subspace) is a subspace of ℝ^\mathscrS, x^* (called the LP set's reference point) is a given element of ℝ^\mathscrS_>, and P^⊥ (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.