Geometry of nonequilibrium reaction networks

Sara Dal Cengio (1) , Vivien Lecomte (1) , Matteo Polettini (2)

1 LIPhy - Laboratoire Interdisciplinaire de Physique [Saint Martin d’Hères]
2 Complex Systems and Statistical Mechanics

Building on Kirchhoff's treatment of electrical circuits, Hill and Schnakenberg - among others - proposed a celebrated theory for the thermodynamics of Markov processes and linear biochemical networks that exploited tools from graph theory to build fundamental nonequilibrium observables. However, such simple geometrical interpretation does not carry through for arbitrary chemical reaction networks because reactions can be many-to-many and are thus represented by a hypergraph, rather than a graph. Here we generalize some of the geometric intuitions behind the Hill-Schnakenberg approach to arbitrary reaction networks. In particular, we give simple procedures to build bases of cycles (encoding stationary nonequilibrium behavior) and cocycles (encoding relaxation), to interpret them in terms of circulations and gradients, and to use them to properly project nonequilibrium observables onto the relevant subspaces. We develop the theory for chemical reaction networks endowed with mass-action kinetics and enrich the description with insights from the corresponding stochastic models. Finally, basing on the linear regime assumption, we deploy the formalism to propose a reconstruction algorithm for metabolic networks consistent with Kirchhoff's Voltage and Current Laws.

Domaines: Mécanique statistique [cond-mat.stat-mech]; Physique mathématique [math-ph]

Sara Dal Cengio, Vivien Lecomte, Matteo Polettini. Geometry of nonequilibrium reaction networks. 2022. ⟨hal-03764894⟩