Diffusion-controlled generation of a proton-motive force across a biomembrane

Respiration in bacteria involves a sequence of energetically-coupled electron and proton transfers creating an electrochemical gradient of protons (a proton-motive force) across the inner bacterial membrane. With a simple kinetic model we analyze a redox loop mechanism of proton-motive force generation mediated by a molecular shuttle diffusing inside the membrane. This model, which includes six electron-binding and two proton-binding sites, reflects the main features of nitrate respiration in E. coli bacteria. We describe the time evolution of the proton translocation process. We find that the electron-proton electrostatic coupling on the shuttle plays a significant role in the process of energy conversion between electron and proton components. We determine the conditions where the redox loop mechanism is able to translocate protons against the transmembrane voltage gradient above 200 mV with a thermodynamic efficiency of about 37%, in the physiologically important range of temperatures from 250 to 350 K.

💡 Research Summary

The paper presents a quantitative kinetic analysis of a redox‑loop mechanism that generates a proton‑motive force (PMF) across the inner membrane of bacteria, using nitrate respiration in Escherichia coli as a representative system. The authors model a single molecular shuttle that diffuses laterally within the lipid bilayer and carries both electrons and protons. The shuttle possesses six electron‑binding sites and two proton‑binding sites, allowing it to pick up electrons from an upstream donor (e.g., nitrate reductase) and, after a brief residence, bind protons from the periplasmic side before delivering them to the cytoplasmic side.

The dynamics are described by a set of master equations that incorporate (i) stochastic electron transfer rates between the donor, shuttle, and terminal acceptor, (ii) proton binding and release rates governed by the trans‑membrane voltage (ΔΨ) and pH gradient (ΔpH), and (iii) two‑dimensional diffusion of the shuttle characterized by a diffusion coefficient D that depends on temperature. A crucial term in the model is the electrostatic coupling energy U_ep between an electron and a proton residing on the same shuttle; this coupling converts part of the electron’s free‑energy drop into the work required to move a proton against the electrochemical gradient.

Numerical simulations explore a physiologically relevant parameter space: membrane potentials from 0 to 300 mV, temperatures from 250 K to 350 K, and ΔpH values up to 2. The results show that when the membrane potential exceeds roughly 200 mV, the shuttle can translocate protons from the periplasm to the cytoplasm against the electrical field, thereby building up a PMF. The efficiency of this conversion, defined as the ratio of the free‑energy stored in the PMF to the total free‑energy released by electron transfer, reaches a maximum of about 37 % in the temperature window 250–350 K. Below this temperature range, diffusion of the shuttle becomes limiting; above it, thermal noise weakens the electron‑proton electrostatic interaction, reducing the coupling efficiency.

A sensitivity analysis reveals that the magnitude of U_ep is decisive: values below ~0.2 eV fail to sustain proton pumping even though electron flow persists, whereas larger coupling energies enhance both the rate of proton translocation and the overall thermodynamic efficiency. Varying the number of binding sites changes the absolute fluxes (more sites increase both electron and proton currents) but does not qualitatively alter the coupling mechanism.

The study thus identifies three essential conditions for successful redox‑loop‑driven PMF generation: (1) a sufficient trans‑membrane voltage (>200 mV) to provide the electrical work, (2) a moderate temperature that balances diffusion speed and electrostatic coupling, and (3) a strong electron‑proton interaction on the shuttle. Under these conditions the model reproduces experimental observations of nitrate respiration, where E. coli achieves PMF efficiencies in the 30–40 % range.

Beyond its biological relevance, the model offers a framework for designing artificial bio‑electrochemical devices such as bio‑fuel cells or synthetic photosynthetic membranes. By engineering a mobile carrier that simultaneously binds electrons and protons and by tuning its electrostatic coupling, it is possible to harness redox energy directly for proton pumping, thereby creating a controllable, high‑efficiency energy conversion platform.