# Fundamental Decompositions and Multistationarity of Power-Law Kinetic Systems

## TL;DR

This paper investigates the properties of the fundamental decomposition of chemical reaction networks, particularly its independence and incidence-independence, and applies these findings to enhance the multistationarity analysis of power-law kinetic systems.

## Contribution

It provides necessary and sufficient conditions for the properties of the $\\mathscr{F}$-decomposition and improves the multistationarity algorithm for power-law systems by removing the need for certain transformations.

## Key findings

- Derived conditions for independence and incidence-independence of the $\\mathscr{F}$-decomposition.
- Identified network classes where the $\\mathscr{F}$-decomposition aligns with other decompositions.
- Improved the multistationarity algorithm by eliminating the transformation step for certain systems.

## Abstract

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 network's stoichiometric subspace is the direct sum of the subnetworks' stoichiometric subspaces) and its incidence-independence (i.e., the image of the network's 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 Schmitz's carbon cycle model recently analyzed by Fortun et al.