Hamiltonian Perspective on Compartmental Reaction-Diffusion Networks

TL;DR

This paper presents a geometric Hamiltonian framework for reaction-diffusion networks, providing a discretization method and stability analysis that rules out spatial pattern persistence in compartmental models.

Contribution

It introduces a novel geometric formulation of reaction-diffusion systems using port-Hamiltonian structures and develops a systematic discretization approach for compartmental modeling.

Findings

Reaction-diffusion systems are modeled as port-Hamiltonian systems.

A discretization method recovers compartmental ODE models from PDEs.

Spatial patterns do not persist in the analyzed compartmental networks.

Abstract

Inspired by the recent developments in modeling and analysis of reaction networks, we provide a geometric formulation of the reversible reaction networks under the influence of diffusion. Using the graph knowledge of the underlying reaction network, the obtained reaction-diffusion system is a distributed-parameter port-Hamiltonian system on a compact spatial domain. Motivated by the need for computer based design, we offer a spatially consistent discretization of the PDE system and, in a systematic manner, recover a compartmental ODE model on a simplicial triangulation of the spatial domain. Exploring the properties of a balanced weighted Laplacian matrix of the reaction network and the Laplacian of the simplicial complex, we characterize the space of equilibrium points and provide a simple stability analysis on the state space modulo the space of equilibrium points. The paper rules out the possibility of the persistence of spatial patterns for the compartmental balanced reaction-diffusion networks.