Node Balanced Steady States: Unifying and Generalizing Complex and Detailed Balanced Steady States

TL;DR

This paper introduces a unified framework for steady states in chemical reaction networks, generalizing existing concepts by using reaction graphs with multiple nodes per complex, and explores their properties and conditions for existence.

Contribution

It generalizes the concept of balanced steady states through reaction graphs, unifies complex and detailed balance, and introduces the deficiency concept for these states.

Findings

Node balanced steady states generalize complex and detailed balanced states.

Properties like uniqueness and stability are preserved for node balanced states.

The deficiency relates to the existence conditions of positive steady states.

Abstract

We introduce a unifying and generalizing framework for complex and detailed balanced steady states in chemical reaction network theory. To this end, we generalize the graph commonly used to represent a reaction network. Specifically, we introduce a graph, called a reaction graph, that has one edge for each reaction but potentially multiple nodes for each complex. A special class of steady states, called node balanced steady states, is naturally associated with such a reaction graph. We show that complex and detailed balanced steady states are special cases of node balanced steady states by choosing appropriate reaction graphs. Further, we show that node balanced steady states have properties analogous to complex balanced steady states, such as uniqueness and asymptotical stability in each stoichiometric compatibility class. Moreover, we associate an integer, called the deficiency, to a reaction graph that gives the number of independent relations in the reaction rate constants that need to be satisfied for a positive node balanced steady state to exist. The set of reaction graphs (modulo isomorphism) is equipped with a partial order that has the complex balanced reaction graph as minimal element. We relate this order to the deficiency and to the set of reaction rate constants for which a positive node balanced steady state exists.