The toric locus of a reaction network is a smooth manifold

TL;DR

This paper proves that the toric locus of a reaction network forms a smooth manifold, showing its geometric structure and how equilibria depend smoothly on parameters and initial conditions.

Contribution

It establishes that the toric locus is a smoothly embedded submanifold and characterizes its structure as a product space, revealing smooth dependence of equilibria.

Findings

Toric locus is a smoothly embedded submanifold.

Toric locus is diffeomorphic to a product space.

Complex-balanced equilibrium depends smoothly on parameters and initial conditions.

Abstract

We show that the toric locus of a reaction network is a smoothly embedded submanifold of the Euclidean space. More precisely, we prove that the toric locus of a reaction network is the image of an embedding and it is diffeomorphic to the product space between the affine invariant polyhedron of the network and its set of complex-balanced flux vectors. Moreover, we prove that within each affine invariant polyhedron, the complex-balanced equilibrium depends smoothly on the parameters (i.e., reaction rate constants). We also show that the complex-balanced equilibrium depends smoothly on the initial conditions.