Modeling chemical reaction networks on the Pontryagin bundle with the Hamilton-Pontryagin approach

Abstract

The Lagrange-d'Alembert-Pontryagin principle is a versatile approach to model dynamical systems including resistive forces from the Lagrangian view. We show in this work, how this method can be applied to open stoichiometric reaction networks. We define a degenerate Lagrangian as the sum of the chemical potential of the different chemical species, and define the Lagrangian forces as connection between chemical affinities and the reaction fluxes. We used this formulation to apply a variational approach including only one kinetic parameter per chemical reaction. Moreover, the stoichiometry of the network is used to define a Dirac structure, which includes all stoichiometric constraints and is defined as the Whitney sum of a distribution and its annihilator. By these means we derive the equations for mass-action kinetics as implicit Euler-Lagrange equations. This results in possible access to variational optimization methods and a direct access to variational integrators for biochemical reaction networks.