Invertible Matrix Method for Power Law Systems: New Deficiency Results for Power Law Reactant-Determined Kinetic Systems

Source abstract

A Power Law Reactant-Determined Kinetics is Tˆ−rank maximal (to type PL-TIK) if the associated Tˆ−matrix has a maximal column rank, that is, Power Law Tˆ−independent Kinetic (PL-TIK) systems have Tˆ−matrices with linearly independent column vectors per linkage class. The concept of PL-TIK systems, introduced by Talabis et al. in 2018, is important to characterize complex balanced dynamical systems with power law kinetics. This study proposes an invertible matrix-based approach to construct these systems instead of relying to the linear independence of column vectors. Moreover, this approach allows us to reformulate known results associated to PL-TIK systems, such as the Zero Kinetic Reactant Deficiency Theorem (ZKRDT) and the Weak Reversibility Theorem for PL-RDK systems based on PL-TIK Decompositions. Lastly, we apply this new construct to qualitatively describe the long term behavior of some known complex dynamical systems, such as the Glucose and Ketone Body Metabolism with Medium-Chain Triglycerides model studied by Espina et al. (2022), and the Human Dihydrofolate Reductase (DHFR) Catalysis model studied by Craciun et al. (2006).