Nonequilibrium Thermodynamics of Chemical Reaction Networks: Wisdom from Stochastic Thermodynamics

We build a rigorous nonequilibrium thermodynamic description for open chemical reaction networks of elementary reactions. Their dynamics is described by deterministic rate equations with mass action kinetics. Our most general framework considers open networks driven by time-dependent chemostats. The energy and entropy balances are established and a nonequilibrium Gibbs free energy is introduced. The difference between this latter and its equilibrium form represents the minimal work done by the chemostats to bring the network to its nonequilibrium state. It is minimized in nondriven detailed-balanced networks (i.e., networks that relax to equilibrium states) and has an interesting information-theoretic interpretation. We further show that the entropy production of complex-balanced networks (i.e., networks that relax to special kinds of nonequilibrium steady states) splits into two non-negative contributions: one characterizing the dissipation of the nonequilibrium steady state and the other the transients due to relaxation and driving. Our theory lays the path to study time-dependent energy and information transduction in biochemical networks.

Popular Summary

Open chemical networks (OCNs) are large sets of coupled chemical reactions fueled by fluxes of chemicals from the outside world, and metabolic networks of living cells are one key example. The dynamics and topology of OCNs have been extensively studied by both mathematicians and biochemists with the goal of characterizing dynamical behavior and model metabolism on genome-wide scales. Despite the fact that OCNs are archetypes of thermodynamical machines and that metabolic modelers are in search of thermodynamical constrains for their models, the thermodynamics of OCNs remain poorly understood, particularly for nonequilibrium and beyond-steady-state regimes. Here, we build a rigorous nonequilibrium thermodynamical theory of OCNs based on their topological properties.

We consider open networks of elementary reactions driven by time-varying chemical concentrations. Focusing on both transient and equilibrium properties, we identify the energy and entropy of OCNs and establish the corresponding first and second laws of thermodynamics. We also introduce the nonequilibrium Gibbs free energy of OCNs, show how it quantifies their information content, and find that it represents the minimal chemical work necessary to bring a network from its equilibrium state to its nonequilibrium state. In the absence of fueling forces from the outside world, the nonequilibrium Gibbs free energy becomes a potential that is minimized during relaxation to equilibrium.

Our work paves the way for systematic studies of energy and information processing in biochemical networks such as kinetic proofreading.

Figure 1

Representation of a closed CRN. The chemical species are $\{X_{\mathrm{a}},\dots ,X_{\mathrm{e}}\}$. The two reaction pathways are labeled by 1 and 2. The nonzero stoichiometric coefficients are $-{\nabla }_{+1}^{\mathrm{a}}=-1$, ${\nabla }_{-1}^{\mathrm{b}}=2$, and ${\nabla }_{-1}^{\mathrm{c}}=1$ for the first forward reaction and $-{\nabla }_{+2}^{\mathrm{c}}=-1$, $-{\nabla }_{+2}^{\mathrm{d}}=-1$, and ${\nabla }_{-2}^{\mathrm{e}}=1$ for the second one. Since the network is closed, no chemical species is exchanged with the environment.

Figure 2

Representation of an open CRN. With respect to the CRN in Fig. 1, the species $X_{\mathrm{a}}$ and $X_{\mathrm{e}}$ are chemostatted, hence, represented as $Y_{\mathrm{a}}$ and $Y_{\mathrm{e}}$. The green boxes on the sides represent the reservoirs of chemostatted species.

Figure 3

Specific implementation of the CRN in Fig. 2.

Figure 4

Pictorial representation of the transformation between two nonequilibrium concentration distributions. The nonequilibrium transformation (blue line) is compared with the equilibrium one (green line). The equilibrium transformation depends on the equilibrium states corresponding to the initial and final concentration distributions. In Sec. 3e3, for an arbitrary CRN, these equilibrium states are obtained by first closing the network and then letting it relax to equilibrium. Instead, in Sec. 5, for a detailed balance CRN, the equilibrium states are obtained by simply stopping the time-dependent driving and letting the system spontaneously relax to equilibrium.