The structure of the moduli space of toric dynamical systems of a reaction network

TL;DR

This paper investigates the topological structure of the moduli space of toric dynamical systems derived from reaction networks, revealing its product structure, contractibility, and invariance properties.

Contribution

It proves that the toric locus is homeomorphic to a product of flux vectors and an affine polyhedron, and is a contractible manifold with invariance under affine transformations.

Findings

Toric locus is homeomorphic to a product of flux vectors and an affine polyhedron.

The toric locus is a contractible manifold.

The toric locus is invariant under bijective affine transformations.

Abstract

We consider toric dynamical systems, which are also called complex-balanced mass-action systems. These are remarkably stable polynomial dynamical systems that arise from the analysis of mathematical models of reaction networks when, under the assumption of mass-action kinetics, they can give rise to complex-balanced equilibria. Given a reaction network, we study the moduli space of toric dynamical systems generated by this network, also called the toric locus of the network. The toric locus is an algebraic variety, and we are especially interested in its topological properties. We show that complex-balanced equilibria depend continuously on the parameter values in the toric locus, and, using this result, we prove that the toric locus has a remarkable product structure: it is homeomorphic to the product of the set of complex-balanced flux vectors and the affine invariant polyhedron of the network. In particular, it follows that the toric locus is a contractible manifold. Finally, we show that the toric locus is invariant with respect to bijective affine transformations of the generating reaction network.