Nonnegative linear elimination for chemical reaction networks

TL;DR

This paper develops conditions for the nonnegative and unique solutions of linear steady state equations in chemical reaction networks, using graph-theoretic methods involving spanning forests of a multidigraph.

Contribution

It introduces a novel graph-based framework to guarantee nonnegativity and uniqueness of solutions in linear elimination problems for reaction networks.

Findings

Conditions for nonnegativity of solutions are established.

Spanning forests of a multidigraph characterize solution properties.

Results are applicable beyond systems biology in applied sciences.

Abstract

We consider linear elimination of variables in steady state equations of a chemical reaction network. Particular subsets of variables corresponding to sets of so-called reactant-noninteracting species, are introduced. The steady state equations for the variables in such a set, taken together with potential linear conservation laws in the variables, define a linear system of equations. We give conditions that guarantee that the solution to this system is nonnegative, provided it is unique. The results are framed in terms of spanning forests of a particular multidigraph derived from the reaction network and thereby conditions for uniqueness and nonnegativity of a solution are derived by means of the multidigraph. Though our motivation comes from applications in systems biology, the results have general applicability in applied sciences.