Reactant subspaces and kinetics of chemical reaction networks

Abstract

This paper studies a chemical reaction network's (CRN) reactant subspace, i.e. the linear subspace generated by its reactant complexes, to elucidate its role in the system's kinetic behaviour. We introduce concepts such as reactant rank and reactant deficiency and compare them with their analogues currently used in chemical reaction network theory. We construct a classification of CRNs based on the type of intersection between the reactant subspace R and the stoichiometric subspace S and identify the subnetwork of S-complexes, i.e. complexes which, when viewed as vectors, are contained in S, as a tool to study the network classes, which play a key role in the kinetic behaviour. Our main results on new connections between reactant subspaces and kinetic properties are (1) determination of kinetic characteristics of CRNs with zero reactant deficiency by considering the difference between (network) deficiency and reactant deficiency, (2) resolution of the coincidence problem between the reactant and kinetic subspaces for complex factorizable kinetics via an analogue of the generalized Feinberg-Horn theorem, and (3) construction of an appropriate subspace for the parametrization and uniqueness of positive equilibria for complex factorizable power law kinetics, extending the work of Müller and Regensburger.

Positive equilibria of a class of power-law kinetics

“…As shown in [1], for a PL-RDK on a cycle-terminal network, the kinetic flux subspace coincides with the kinetic order subspaceS M R introduced by Müller and Regensburger in [15]. Moreover, for a MAK system on an RSS network, it coincides with the stoichiometric subspace.Recall also from [1] that a PL-RDK kinetics is called Y-kernel aligned (YKA) if ker Y res ⊂ ker T .…”

“…Arceo et al [1] conducted an initial systematic study of the reactant subspace R's connection to kinetic behavior and identified the set of RSS networks as particularly interesting.…”

“…A characterization of RSS networks is easily found in [1]: In other words, in an RSS network, every complex is a linear combination of reactant complexes. RSS networks will be studied in more detail in Sect.…”

“…Arceo et al [1] introduced the kinetic flux subspaceS of a PL-RDK system on an RSS network. It is defined as follows:…”

“…Moreover, for a MAK system on an RSS network, it coincides with the stoichiometric subspace.Recall also from [1] that a PL-RDK kinetics is called Y-kernel aligned (YKA) if ker Y res ⊂ ker T .…”