Biological Proton Pumping in an Oscillating Electric Field

Time-dependent external perturbations provide powerful probes of the function of molecular machines. Here we study biological proton pumping in an oscillating electric field. The protein cytochrome oxidase is the main energy transducer in aerobic life, converting chemical energy into an electric potential by pumping protons across a membrane. With the help of master-equation descriptions that recover the key thermodynamic and kinetic properties of this biological “fuel cell,” we show that the proton pumping efficiency and the electronic currents in steady state depend significantly on the frequency and amplitude of the applied field, allowing us to distinguish between different microscopic mechanisms of the machine. A spectral analysis reveals dominant reaction steps consistent with an electron-gated pumping mechanism.

Figure 1

Proton pumping machine. (a) Schematic of CcO and ATP synthase function. Electron transfer from cytochrome $c$ via ${\mathrm{Cu}}_A$ and heme $a$ to the binuclear center (heme $a_3$ and ${\mathrm{Cu}}_B$) is indicated in red (medium gray). Light blue (light gray) arrows indicate proton translocation, including uptake of protons from the $N$ side of the membrane and release of pumped protons on the $P$ side. Dark blue (dark gray) arrows indicate the redox chemical reaction. ATP synthase generates an ATP from ADP (adenosine diphosphate) and $P_i$ driven by the proton gradient. (b) Kinetic scheme of the three-site model of CcO. Circles and squares show proton and electron sites, respectively. Arrows indicate proton- (light blue [light gray]) and electron-transfer reactions (red [medium gray]). The dark blue (dark gray) arrow denotes the product formation. (c) Reaction diagram of the three-site model. The vertices of the cube correspond to the kinetic states of the model where “$+$” represents a filled proton site, “$-$” a filled electron site, and “0” an empty site. States are labeled in roman numerals according to the binary code of their site occupancies. Lines between vertices indicate allowed transitions with the same color scheme as in (a).

Figure 2

(a) Pumping efficiency and (b) electron flux versus voltage amplitude, $V_1$, at $\omega ={10}^3{\mathrm{s}}^{-1}$ for model 1 (symbols: numerical integration; lines: perturbation theory).

Figure 3

Pumping efficiency (left) and electron flux (right) as a function of frequency $\omega$ at $V_1=50\mathrm{mV}$ and $V_0=0$, $V_0(\eta =\frac{1}{2})$, and $V_0(\eta =0)$ (top to bottom). Arrows indicate the efficiency and electron flux for a time-independent voltage ($V_1=0$).