Electrostatic models of electron-driven proton transfer across a lipid membrane

We present two models for electron-driven uphill proton transport across lipid membranes, with the electron energy converted to the proton gradient via the electrostatic interaction. In the first model, associated with the cytochrome c oxidase complex in the inner mitochondria membranes, the electrostatic coupling to the site occupied by an electron lowers the energy level of the proton-binding site, making the proton transfer possible. In the second model, roughly describing the redox loop in a nitrate respiration of E. coli bacteria, an electron displaces a proton from the negative side of the membrane to a shuttle, which subsequently diffuses across the membrane and unloads the proton to its positive side. We show that both models can be described by the same approach, which can be significantly simplified if the system is separated into several clusters, with strong Coulomb interaction inside each cluster and weak transfer couplings between them. We derive and solve the equations of motion for the electron and proton creation/annihilation operators, taking into account the appropriate Coulomb terms, tunnel couplings, and the interaction with the environment. For the second model, these equations of motion are solved jointly with a Langevin-type equation for the shuttle position. We obtain expressions for the electron and proton currents and determine their dependence on the electron and proton voltage build-ups, on-site charging energies, reorganization energies, temperature, and other system parameters. We show that the quantum yield in our models can be up to 100% and the power-conversion efficiency can reach 35%.

💡 Research Summary

The paper develops and analyzes two distinct electrostatic models that describe how the energy released by electron transfer can be harnessed to drive uphill proton transport across a lipid membrane. The first model is inspired by the cytochrome c oxidase complex in the inner mitochondrial membrane. In this representation the membrane region is divided into two strongly interacting clusters: one containing an electron‑binding site and the other a proton‑binding site. When an electron occupies its site, the Coulomb attraction lowers the energy of the neighboring proton site, effectively reducing the proton binding barrier and allowing a proton to move against its electrochemical gradient. The authors formulate the dynamics using creation and annihilation operators for electrons and protons, include on‑site charging energies (U), inter‑cluster tunnelling amplitudes (V), and a harmonic bath that provides reorganization energy (λ) and thermal relaxation. By treating the intra‑cluster Coulomb interaction as strong and the inter‑cluster transfer as weak, they derive a set of master‑equation‑like operator equations that can be solved analytically or numerically. The resulting electron and proton currents (J_e, J_p) are expressed as functions of the electron and proton voltage differences (Δμ_e, Δμ_p), temperature, and the aforementioned parameters. Under appropriate bias the proton current matches the electron current, yielding a quantum yield η_Q = |J_p|/|J_e| approaching unity.

The second model abstracts the redox loop observed in nitrate respiration of Escherichia coli. Here an electron first reduces a mobile shuttle molecule on the negative side of the membrane, displacing a bound proton. The shuttle then diffuses across the membrane, described by a Langevin‑type equation for its coordinate x(t), and releases the proton on the positive side. The electron‑shuttle and proton‑shuttle couplings depend on x, leading to position‑dependent tunnelling matrices T(x). The authors couple the same operator equations for electrons and protons to the stochastic Langevin dynamics, producing a set of coupled differential‑stochastic equations. Solving these jointly yields explicit expressions for J_e and J_p that incorporate both quantum tunnelling and classical diffusion of the shuttle.

Both models share a common theoretical framework: strong Coulomb interaction within each cluster, weak inter‑cluster transfer, and a bath that supplies reorganization energy. Parameter sweeps reveal that the efficiency of energy conversion, defined as η = (Δμ_p · J_p)/(Δμ_e · J_e), can reach up to 35 % when the electron‑proton coupling U, the reorganization energy λ, and the tunnelling rates Γ are optimally tuned. The quantum yield can be as high as 100 %, indicating that each transferred electron can drive one proton against the gradient. Temperature influences the results through thermal activation of the bath; higher temperatures increase dissipation and reduce η, but the effect can be mitigated by adjusting λ and Γ.

In summary, the work provides a unified quantum‑mechanical description of two biologically relevant proton‑pumping mechanisms, demonstrates how electrostatic coupling can convert electron energy into a proton gradient with high yield, and offers analytical tools that could be applied to the design of synthetic nano‑devices mimicking biological energy transduction.