Kinetic derivation of the Hessian geometric structure in chemical reaction networks

The theory of chemical kinetics forms the basis to describe the dynamics of chemical reaction networks. Owing to physical and thermodynamic constraints, the networks possess various structures, which can be utilized to characterize important properties of the networks. In this work, we reveal the Hessian geometry which underlies chemical reaction networks and demonstrate how it originates from the interplay of stoichiometric and thermodynamic constraints. Our derivation is based on kinetics, we assume the law of mass action and characterize the equilibrium states by the detailed balance condition. The obtained geometric structure is then related to thermodynamics via the Hessian geometry appearing in a pure thermodynamic derivation. We demonstrate, based on the fact that both equilibrium and complex balanced states form toric varieties, how the Hessian geometric framework can be extended to nonequilibrium complex balanced steady states. We conclude that Hessian geometry provides a natural framework to capture the thermodynamic aspects of chemical reaction networks.