Abstract
Studies about the set of positive equilibria (\(E_+\)) of kinetic systems have been focused on mass action, and not that much on power law kinetic (PLK) systems, even for PL-RDK systems (PLK systems where two reactions with identical reactant complexes have the same kinetic order vectors). For mass action, reactions with different reactants have different kinetic order rows. A PL-RDK system satisfying this property is called factor span surjective (PL-FSK). In this work, we show that a cycle terminal PL-FSK system with \(E_+\ne \varnothing \) and has independent linkage classes (ILC) is a poly-PLP system, i.e., \(E_+\) is the disjoint union of log-parametrized sets. The key insight for the extension is that factor span surjectivity induces an isomorphic digraph structure on the kinetic complexes. The result also completes, for ILC networks, the structural analysis of the original complex balanced generalized mass action systems (GMAS) by Müller and Regensburger. We also identify a large set of PL-RDK systems where non-emptiness of \(E_+\) is a necessary and sufficient condition for non-emptiness of each set of positive equilibria for each linkage class. These results extend those of Boros on mass action systems with ILC. We conclude this paper with two applications of our results. Firstly, we consider absolute complex balancing (ACB), i.e., the property that each positive equilibrium is complex balanced, in poly-PLP systems. Finally, we use the new results to study absolute concentration robustness (ACR) in these systems. In particular, we obtain a species hyperplane containment criterion to determine ACR in the system species.
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Abbreviations
- CLP:
-
Complex balanced equilibria log parametrized
- CRN:
-
Chemical reaction network
- GMAS:
-
Generalized mass action system
- ILC:
-
Independent linkage classes
- PL-FSK:
-
Power law with factor span surjective kinetics
- PLK:
-
Power law kinetics
- PL-NDK:
-
Power law with non-reactant-determined kinetics
- PL-NIK:
-
Power law with non-inhibitory kinetics
- PLP:
-
Positive equilibria log parametrized
- PL-RDK:
-
Power law with reactant-determined kinetics
- PL-RLK:
-
Power law with reactant set linear independent kinetics
- PL-TIK:
-
Power law with \(\widehat{T}\)-rank maximal kinetics
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Acknowledgements
This work was supported by the Institute for Basic Science in Daejeon, Republic of Korea with funding information IBS-R029-C3.
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Hernandez, B.S., Mendoza, E.R. Positive equilibria of power law kinetics on networks with independent linkage classes. J Math Chem 61, 630–651 (2023). https://doi.org/10.1007/s10910-022-01432-w
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DOI: https://doi.org/10.1007/s10910-022-01432-w