Reaction networks as systems for resource allocation: A variational principle for their non-equilibrium steady states

Within a fully microscopic setting, we derive a variational principle for the non-equilibrium steady states of chemical reaction networks, valid for time-scales over which chemical potentials can be taken to be slowly varying: at stationarity the system minimizes a global function of the reaction fluxes with the form of a Hopfield Hamiltonian with Hebbian couplings, that is explicitly seen to correspond to the rate of decay of entropy production over time. Guided by this analogy, we show that reaction networks can be formally re-cast as systems of interacting reactions that optimize the use of the available compounds by competing for substrates, akin to agents competing for a limited resource in an optimal allocation problem. As an illustration, we analyze the scenario that emerges in two simple cases: that of toy (random) reaction networks and that of a metabolic network model of the human red blood cell.

💡 Research Summary

The paper presents a rigorous variational principle governing the non‑equilibrium steady states (NESS) of chemical reaction networks. Starting from a fully microscopic description, the authors assume that chemical potentials vary slowly compared to the characteristic reaction times, so that reaction fluxes can be treated as quasi‑steady over the observation window. Under this assumption they derive an expression for the time derivative of the entropy production rate, showing that it is proportional to a global quadratic form of the fluxes. This quadratic form has exactly the structure of a Hopfield Hamiltonian:

\