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#### Concentration Robustness in LP Kinetic Systems

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##### Citation

Lao, A. R., Lubeni, P. V. N., Magpantay, D. M., & Mendoza, E. R. (2022). Concentration
Robustness in LP Kinetic Systems.* Match-Communications in Mathematical and in Computer Chemistry,*
*88*(1), 29-66. doi:10.46793/match.88-1.029L.

Cite as: https://hdl.handle.net/21.11116/0000-000A-4A12-F

##### Abstract

For a reaction network N with species set S, a log-parametrized (LP) set is a non-empty set of the form E(P, x*) = {x is an element of R->(S) | log x - log x* is an element of P.} where P (called the LP set's flux subspace) is a subspace of R->(S), x* (called the LP set's reference point) is a given element of R->(S), and P-perpendicular to (called the LP set's parameter subspace) is the orthogonal complement of P. A network N with kinetics K is a positive equilibria LP (PLP) system if its set of positive equilibria is an LP set, i.e., E+(N, K) = E(PE, 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+(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).