This paper studies chemical reaction networks with poly-PL kinetics, i.e. positive linear combinations of power law kinetics. The analysis of such systems is motivated by the study of veloz et al., that proposed to analyze the dynamics of Evolutionary Game Theory models using Chemical Reaction Network Theory (CRNT) in the form of polynomial kinetics (POK). Our approach is based on the fact that poly-PL kinetics generate power law dynamical systems, which via a method recently introduced by G. Craciun can be mapped to EMAK systems. These are the analogue of mass action kinetics on Euclidean embedded graphs (E-graph). Our main structural results show the coincidence of the stoichiometric subspaces of the original network and its associated E-graph as well as the conservation of the positive dependency, which is a necessary condition for the existence of positive equilibria. However, our overall analysis shows Craciun’s method is only of limited use for studying PYK systems since only very special kinetics reflect the structural properties of the original chemical kinetic system.
Keywords: Chemical Reaction Networks, E-graph, Poly-PL Kinetics, power law kinetics[1] Arceo CPP, Jose EC, Marin-Sanguino A, Mendoza ER (2015) Chemical reaction network approaches to biochemical systems theory. Mathematical biosciences, 269:135–152.
[2] Arceo CPP, Jose EC, Lao AR, Mendoza E.R (2017) Reaction networks and kinetics of biochemical systems. Mathematical biosciences, 283:13–29.
[3] Arceo CP, Jose EC, Lao AR, Mendoza ER. (2018) Reactant subspaces and kinetics of chemical reaction networks. Journal of Mathematical Chemistry, 56(2):395–422.
[4] Arceo CP, Jose EC, Lao AR, Mendoza ER. (2019) Chemical reaction networks: Filipino contributions to their theory and its applications. Phil. J. Science 148 (2): 249-261.
[5] Boros, B On the positive steady states of deficiency of one mass action system. PhD thesis, Eotvos Lorand University.
[6] Boros, B. (2018) Existence of positive steady states for weakly reversible mass action systems. Journal on Mathematical Analysis.
[7] Brunner J, Craciun G. (2018) Robust persistence and permanence of polynomial and power law dynamical systems. SIAM Journal on Applied Mathematics, 78(2):801-825.
[8] Cortez MJ, Nazareno AN, Mendoza ER. (2018) A computational approach to linear conjugacy in a class of power law kinetic systems. J. Math. Chem. 56 2: 336-357.
[9] Craciun G. (2018) Polynomial Dynamical Systems, Reaction Networks, and Toric Differential Inclusions. SIAM J. Appl. Algebra Geometry, 3(1), 87–106.
[10] Cressman R, Tao Y. (2014) The replicator equation and other game dynamics. Pro-
ceedings of theNational Academy of Sciences , 111 (2014), 10810–10817.
[11] Farinas HF, Mendoza ER, Lao AR. (2019) Species subsets and embedded networks of S-systems. submitted
[12] Feinberg M. (1979) Lectures on chemical reaction networks. notes of lectures given
at the mathematics research center of the university of wisconsin 1979 16
[13] Feinberg M, Horn FJM (1977) Chemical mechanism structure and the coincidence of the stoichiometric and kinetic subspaces. Arch. Rational Mech. Anal. 66:83-97.
[14] Feinberg M. (1987) Chemical reaction network structure and the stability of complex isothermal reactors: I.The deficiency zero and deficiency one theorems. Chemical Engineering Science 42 2229-
2268.
[15] Fortun NT, Mendoza ER, Razon LF, Lao AR. (2019) Robustness in power law kinetic systems with reactant-determined interactions. Proceedings of the Japan Conference on Geometry, Graphs and
Games 2018, Lecture Notes in Computer Science (in press).
[16] Fortun N, Lao A, Razon E, Mendoza E. (2019) A deficiency zero theorem for a class of power law kinetic systems with in dependent decompositions. MATCH Commun. Math. Comput. Chem. 81 621-638
[17] Fortun NT, Mendoza ER, Razon LF, Lao AR. (2018) A Deficiency One Algorithm for Power in Law Kinetic Systems with Reactant-Determined Interactions. J. Math.Chem. 56 (10): 2929-2962.
[18] Fortun NT, Lao AR, Razon LF, Mendoza ER. (2018) Multistationarity in Earth ́s Pre-industrial Carbon Cycle Models. Manila J. Science 11, 81-96
[19] Gross E, Harrington H, Meshkat N, Shiu AJ. (2018) Joining and decomposing reaction networks. submitted
[20] Guberman M, Altschuler S, Wu L (2003) Mass action reaction networks and the deficiency zero theorem. Undergraduate Thesis (Bachelor of Arts with honors), Harvard University
[21] Hernandez B, Mendoza E, de los Reyes A (2019) A computational approach to multi-stationarity in power law kinetic systems. submitted
[22] Horn F, Jackson R. (1972) General mass action kinetics. Arch. Rational Mech. Anal, 47:187–194.
[23] Johnston MD, Siegel D, Szederkenyi G (2012) Dynamic Equivalence and Linear Conjugacy of Chemical Reaction Networks: New Results and Methods. MATCH Commun.Math. Comput. Chem.
68: 443-468.
[24] Johnston MD (2014) Translated Chemical Reaction Networks. Bull.Math.Biol. 76 1081-1116 .
[25] Mendoza ER, Talabis DASJ, Jose EC. (2018) Positive equilibria of weakly reversible power law kinetic systems with linear independent interactions. J. Math Chem: 10910-018-0909-2
[26] Generalized Mass Action Systems and Positive Solutions of Polynomial Equations with Real and Symbolic Exponents. Proceedings of CASC 2014, (eds. V.P. Gerdt, W. Koepf, W.M. Seiler, E.H.
Vorozhtsov), Lecture Notes in Comput. Sci. 8660, pp. 302-323
[27] Nazareno A, Eclarin RP, Mendoza E, Lao A (2019) Linear conjugacy of chemical kinetic systems. submitted
[28] Shinar G, Feinberg M. (2011) Design principles for robust biochemical reaction networks: what works, what cannot work and what might almost work. Mathematical biosciences, 231 1:39–48.
[29] Talabis DASJ, Arceo CPP, Mendoza ER. (2018) Positive equilibria for a class of power-law kinetics. J. Math Chem 56(2), 358–394
[30] Talabis DASJ, Mendoza ER, Jose EC. (2019) Complex balanced equilibria of weakly reversible power law kinetic systems with linear independent interactions. submitted
[31] Talabis DASJ, Magpantay D, Mendoza ER, Nocon EG, Jose EC. (2019) A Weak 17
Reversibility Theorem for poly-PL kinetics and the replicator equation. submitted
[32] Veloz T, Razeto-Barry P, Dittrich P, Fajardo A. (2014) Reaction networks and evolutionary game theory. J.Math. Biol. 68: 181-206.
[33] Wiuf C, Feliu E. (2013) Power-law kinetics and determinant criteria for the preclusion of multistationarity in networks of interacting species. I AM J ApplDyn Syst 12: 1685-1721