The structure of the moduli spaces of toric dynamical systems

TL;DR

This paper investigates the topological structure of the parameter space (toric locus) of complex-balanced mass-action systems, showing it is connected, has a product structure, and is invariant under certain transformations.

Contribution

It proves the connectedness of the toric locus, describes its product structure, and demonstrates invariance under affine transformations, advancing understanding of the parameter space topology.

Findings

The complex-balanced equilibria depend continuously on parameters.

The toric locus is connected for any toric dynamical system.

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

Abstract

We consider complex-balanced mass-action systems, or toric dynamical systems. They are remarkably stable polynomial dynamical systems arising from reaction networks seen as Euclidean embedded graphs. We study the moduli spaces of toric dynamical systems, called the toric locus: given a reaction network, we are interested in the topological structure of the set of parameters giving rise to toric dynamical systems. First we show that the complex-balanced equilibria depend continuously on the parameter values. Using this result, we prove that the toric locus of any toric dynamical system is connected. In particular, we emphasize its product structure: it is homeomorphic to the product of the set of complex-balanced flux vectors and the affine invariant polyhedron. Finally, we show that the toric locus is invariant with respect to bijective affine transformations of the generating reaction network.