Craciun’s E-MAK Method: A Dynamics-focused Approach to Poly-PL Kinetics
pp 1-21 (Vol 13 2020)
D.M. Magpantay 1,2*,
E.R. Mendoza 1,3,4,5, and
E.G. Nocon 1
1 Mathematics and Statistics Department, De La Salle University, Manila, Philippines
2 College of Arts and Sciences, Batangas State University, Batangas City, Philippines
3 Institute of Mathematical Sciences and Physics, University of the Philippines Los Baños, Laguna, Philippines
4 Max Planck Institute of Biochemistry, Martinsried, Munich, Germany
5 LMU Faculty of Physics, Geschwister-Scholl-Platz 1, Munich 80539, Germany
*Corresponding author: [email protected]
ABSTRACT
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.