  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%.
