Kinetic models of redox-coupled proton pumping

Abstract

Cytochrome c oxidase, the terminal enzyme of the respiratory chain, pumps protons across the inner mitochondrial membrane against an opposing electrochemical gradient by reducing oxygen to water. To explore the fundamental mechanisms of such redox-coupled proton pumps, we develop kinetic models at the single-molecule level consistent with basic physical principles. We demonstrate that pumping against potentials >150 mV can be achieved purely through electrostatic couplings, given an asymmetric arrangement of charge centers; however, nonlinear gates are essential for highly efficient real enzymes. The fundamental requirements for proton pumping identified here highlight a possible evolutionary origin of cytochrome c oxidase pumping. The general design principles are relevant also for other molecular machines and suggest future applications in biology-inspired fuel cells.

Fig. 1.

Schematic of CcO function. (A) Flux diagram of a Lundegårdh–Mitchell loop in which an electric potential across a protonically sealed membrane is created by taking up protons and electrons from opposite sides. (B) Flux diagram of a true proton pump. J_el is the electron flux, J_up is the proton uptake flux from the N side, J_pump is the net proton pumping flux to the P side, and J_prod is the product flux. Flux conservation requires that J_prod = J_el = J_up − J_pump. (C) CcO proton pump. Electron transfer from cytochrome c via Cu_A and heme a to the binuclear center (heme a₃ and Cu_B) is indicated in red. Light blue arrows indicate proton translocation, including uptake of both chemical and pumped protons from the negatively charged N side and release of pumped protons on the positively charged P side. Blue arrows indicate uptake of dioxygen and release of water. (D) Kinetic scheme of the three-site model. Circles and squares show proton and electron sites, respectively. Arrows indicate proton (light blue) and electron (red) transfer reactions with intrinsic rate coefficients κ_μν. The blue arrow denotes the irreversible product formation.

Fig. 2.

Energy landscape representation of electron-coupled proton pumping. Electron reduction induces “ratcheting” between left-leaning energy landscapes for protons (black) by lowering the barriers for proton uptake from the N side (states A, B, and D) and right-leaning landscapes favoring proton release to the P side (states C, E, and F). Proton and electron free energy surfaces (red; shifted vertically) are drawn to scale for model b of Table 1 (including barriers for an attempt frequency of 10⁹ s⁻¹). (A) Initial proton uptake from the N side. (B) Proton translocation to the pump site 2. (C) Electron transfer. (D) Uptake of the second proton from the N side. (E) Release of the pumped proton to the P side. (F) Formation of product “1/2H₂O.” Thick black arrows indicate the dominant steady state flux. The dashed blue arrow indicates “slip,” i.e., product formation without pumping. The dashed black arrow between F and A indicates that the reaction is reversible in principle, halting the pump eventually in a closed system together with depletion of substrates and buildup of product water. Only the six states with significant population are shown. With a “water gate,” internal proton transfer is accelerated, as indicated by the dashed barrier in B, thus increasing the pumping efficiency.

Fig. 3.

Pumping efficiency η as a function of membrane potential V_m for a three-site model (dashed line), and different four-site models (solid lines and Inset) that were optimized in different V_m regimes. The voltage drops linearly across the membrane. To mimic CcO, the net thermodynamic driving force of the redox reaction was set to 0.5 V by making product formation reversible. The thermodynamic efficiency of the model shown in purple, (1 + η)V_m/(0.5 V), reaches ≈40% at ≈150 mV.