The fundamental decomposition of a chemical reaction network (also called its $mathscr{F}$-decomposition) is the set of subnetworks generated by the partition of its set of reactions into the fundamental classes introduced by Ji and Feinberg in 2011 as the basis of their higher deficiency algorithm for mass action systems. The first part of this paper studies the properties of the $mathscr{F}$-decomposition, in particular, its independence (i.e., the networks stoichiometric subspace is the direct sum of the subnetworks stoichiometric subspaces) and its incidence-independence (i.e., the image of the networks incidence map is the direct sum of the incidence maps images of the subnetworks). We derive necessary and sufficient conditions for these properties and identify network classes where the $mathscr{F}$-decomposition coincides with other known decompositions. The second part of the paper applies the above-mentioned results to improve the Multistationarity Algorithm for power-law kinetic systems (MSA), a general computational approach that we introduced in previous work. We show that for systems with non-reactant determined interactions but with an independent $mathscr{F}$-decomposition, the transformation to a dynamically equivalent system with reactant-determined interactions -- required in the original MSA -- is not necessary. We illustrate this improvement with the subnetwork of Schmitzs carbon cycle model recently analyzed by Fortun et al.
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