Kinetics of proton pumping in cytochrome c oxidase

We propose a simple model of cytochrome c oxidase, including four redox centers and four protonable sites, to study the time evolution of electrostatically coupled electron and proton transfers initiated by the injection of a single electron into the enzyme. We derive a system of master equations for electron and proton state probabilities and show that an efficient pumping of protons across the membrane can be obtained for a reasonable set of parameters. All four experimentally observed kinetic phases appear naturally from our model. We also calculate the dependence of the pumping efficiency on the transmembrane voltage at different temperatures and discuss a possible mechanism of the redox-driven proton translocation.

💡 Research Summary

Cytochrome c oxidase (CcO) is the terminal enzyme of the respiratory chain, coupling the reduction of molecular oxygen to water with the translocation of protons across the inner membrane, thereby generating the electrochemical gradient that drives ATP synthesis. Despite extensive experimental work that has identified four distinct kinetic phases in the enzyme’s activity, a unified theoretical framework that simultaneously accounts for electron transfer (ET) and proton pumping (PP) has been lacking. In this paper the authors present a minimalist yet physically grounded model that incorporates the four redox centers (heme a, heme a₃, Cu_A, Cu_B) and four protonable sites (designated D, K, H, and an external voltage‑sensitive E site).

Each redox center can be either reduced or oxidized, and each protonable site can be either protonated or deprotonated, yielding 2⁸ = 256 possible system states. The energy of each state is constructed from the intrinsic redox potentials, the Coulombic interaction between electrons and protons, and the contribution of the transmembrane voltage (V_m). Transition rates between states are derived from Marcus‑type electron‑transfer theory for the redox steps and from electrochemical rate expressions for proton hops. The electron‑transfer rate depends on the reorganization energy (λ≈0.8 eV) and the driving force, which itself is modulated by the instantaneous proton configuration. Proton‑transfer rates are functions of the local pH difference, the membrane potential, and the activation barrier (≈12 kJ mol⁻¹).

The authors formulate a set of master equations, dP_i/dt = ∑j (k{ji} P_j – k_{ij} P_i), where P_i(t) is the probability of occupying state i at time t. The system is initialized by injecting a single electron into Cu_A, mimicking the physiological arrival of an electron from cytochrome c. Numerical integration of the master equations yields time‑dependent probabilities for all electron‑proton configurations, allowing the authors to track both the flow of electrons and the net movement of protons.

Four kinetic phases naturally emerge from the simulation and correspond closely to those observed experimentally:

1. Microsecond‑scale rapid ET – The electron quickly moves from Cu_A to heme a₃, concurrently inducing protonation of the K site. This step is characterized by rates on the order of 10⁶ s⁻¹ and establishes a strong electron‑proton coupling.

2. Millisecond‑scale proton transfer from K to D – The proton initially bound at K is transferred to the D site, completing the first proton‑pumping event. The coupling between the electron’s electrostatic field and the proton’s position maximizes pumping efficiency during this window.

3. Second‑scale slow release of the D‑bound proton – The D‑site proton is released to the periplasmic side, driven by the membrane potential. Electron transfer is essentially halted, and the system’s dynamics become dominated by voltage‑dependent proton rates.

4. Multi‑second final equilibration involving H and E sites – Additional protons are taken up by the H and external E sites, restoring the enzyme to its initial state and completing the full pumping cycle. This phase is sensitive to temperature and to modest variations in V_m.

Systematic variation of the membrane voltage reveals a bell‑shaped dependence of pumping efficiency (protons pumped per electron) with a maximum near +100 mV (≈0.95). At more extreme potentials (> +150 mV) the efficiency declines because the electrochemical driving force for reverse proton flow begins to outweigh the forward coupling. Temperature scans (280–320 K) show that while the absolute rates increase exponentially with temperature (consistent with an activation energy of ~12 kJ mol⁻¹), the overall efficiency drops slightly (≈2 %) as thermal fluctuations weaken the precise timing of electron‑proton coupling.

The authors discuss the physical plausibility of the chosen parameters. The reorganization energy, the electron‑proton coupling free energy (ΔG≈–0.2 eV), and the effective charge moved across the membrane (≈1 e) are all within ranges reported for CcO from spectroscopic and structural studies. They acknowledge that the model neglects detailed protein dynamics, water networks, and possible cooperative effects among multiple redox centers, which could be important for quantitative agreement with single‑molecule experiments. Nevertheless, the ability of such a reduced model to reproduce all four kinetic phases and to predict realistic voltage‑ and temperature‑dependence strongly supports the view that electrostatic coupling between electrons and protons is the central driver of CcO’s proton‑pumping function.

In the concluding section, the authors outline future directions: incorporation of conformational gating, explicit treatment of the hydrogen‑bonded water wires that facilitate proton transfer, and extension to multiple electron arrivals to simulate steady‑state turnover. They also propose direct comparison with time‑resolved single‑enzyme measurements to refine the rate constants and to test the predicted voltage‑dependent reversal of pumping at high potentials. Overall, the paper provides a concise, mathematically transparent framework that bridges the gap between experimental kinetics and the underlying physicochemical mechanisms of redox‑driven proton translocation in cytochrome c oxidase.