A network dynamics approach to chemical reaction networks

TL;DR

This paper explores chemical reaction networks through the lens of nonlinear network dynamics, revealing how complex-balanced assumptions simplify analysis and relate to thermodynamics, stability, and consensus dynamics.

Contribution

It introduces a novel formulation connecting reaction network theory with consensus dynamics and thermodynamics, offering new insights into stability and equilibrium characterization.

Findings

Reaction dynamics can be reformulated using a balanced Laplacian matrix.

The approach provides a new perspective on complex-balancedness via Kirchhoff's Matrix Tree theorem.

Complete stability analysis for reaction networks with inflows and outflows is achieved.

Abstract

A crisp survey is given of chemical reaction networks from the perspective of general nonlinear network dynamics, in particular of consensus dynamics. It is shown how by starting from the complex-balanced assumption the reaction dynamics governed by mass action kinetics can be rewritten into a form which allows for a very simple derivation of a number of key results in chemical reaction network theory, and which directly relates to the thermodynamics of the system. Central in this formulation is the definition of a balanced Laplacian matrix on the graph of chemical complexes together with a resulting fundamental inequality. This directly leads to the characterization of the set of equilibria and their stability. Both the form of the dynamics and the deduced dynamical behavior are very similar to consensus dynamics. The assumption of complex-balancedness is revisited from the point of view of Kirchhoff's Matrix Tree theorem, providing a new perspective. Finally, using the classical idea of extending the graph of chemical complexes by an extra 'zero' complex, a complete steady-state stability analysis of mass action kinetics reaction networks with constant inflows and mass action outflows is given.