Generalized mass action systems: Complex balancing equilibria and sign vectors of the stoichiometric and kinetic-order subspaces

TL;DR

This paper extends chemical reaction network theory to generalized mass action systems with arbitrary nonnegative power-law rate functions, establishing conditions for the existence and uniqueness of complex balancing equilibria.

Contribution

It introduces the concept of kinetic complexes and subspaces, and shows that many classical results extend to these generalized systems, including conditions for equilibria.

Findings

Sign vector coincidence ensures unique positive complex balancing equilibrium.

Multiple equilibria can occur in the same compatibility class in generalized systems.

Classical results are extended to more realistic intracellular reaction models.

Abstract

Mass action systems capture chemical reaction networks in homogeneous and dilute solutions. We suggest a notion of generalized mass action systems that admits arbitrary nonnegative power-law rate functions and serves as a more realistic model for reaction networks in intracellular environments. In addition to the chemical complexes and the related stoichiometric subspace, we introduce corresponding kinetic complexes, which represent the nonnegative exponents in the rate functions and determine the kinetic-order subspace. We show that several results of Chemical Reaction Network Theory carry over to the case of generalized mass action kinetics. Our main result essentially states that, if the sign vectors of the stoichiometric and the kinetic-order subspace coincide, there exists a unique positive complex balancing equilibrium in every stoichiometric compatibility class. However, in contrast to classical mass action systems, multiple complex balancing equilibria in one stoichiometric compatibility class are possible in general.