On the Clausius formulation of the second law in stationary chemical networks through the theorems of the alternative

TL;DR

This paper applies Gordan's theorem to chemical reaction networks at steady state, linking thermodynamic potentials and reaction loops, and proposes dual computational methods for energy analysis in metabolic networks.

Contribution

It introduces a theoretical equivalence between the Clausius second law formulation and Gordan's theorem in steady-state chemical networks, enabling new computational approaches.

Findings

Exclusion of reaction loops correlates with thermodynamic potential definition.

Calculating free energy and detecting infeasible loops are dual problems.

The approach improves efficiency in cellular metabolism energy analysis.

Abstract

In this article the Gordan theorem is applied to the thermodynamics of a chemical reaction network at steady state. From a theoretical viewpoint it is equivalent to the Clausius formulation of the second law for the out of equilibrium steady states of chemical networks, i.e. it states that the exclusion (presence) of closed reactions loops makes possible (impossible) the definition of a thermodynamic potential and vice versa. On the computational side, it reveals that calculating reactions free energy and searching infeasible loops in flux states are dual problems whose solutions are alternatively inconsistent. The relevance of this result for applications is discussed with an example in the field of constraints-based modeling of cellular metabolism where it leads to efficient and scalable methods to afford the energy balance analysis.