Torque-coupled thermodynamic model for $F_o{F}_1$-ATPase

$F_o{F}_1$-ATPase is a motor protein complex that utilizes transmembrane ion flow to drive the synthesis of adenosine triphosphate (ATP) from adenosine diphosphate (ADP) and phosphate (Pi). While many theoretical models have been proposed to account for its rotary activity, most of them focus on the $F_o$ or $F_1$ portions separately rather than the complex as a whole. Here, we propose a simple but new torque-coupled thermodynamic model of $F_o{F}_1$-ATPase. Solving this model at steady state, we find that the monotonic variation of each portion's efficiency becomes much more robust over a wide range of parameters when the $F_o$ and $F_1$ portions are coupled together, as compared to cases when they are considered separately. Furthermore, the coupled model predicts the dependence of each portion's kinetic behavior on the parameters of the other. Specifically, the power and efficiency of the $F_1$ portion are quite sensitive to the proton gradient across the membrane, while those of the $F_o$ portion as well as the related Michaelis constants for proton concentrations respond insensitively to concentration changes in the reactants of ATP synthesis. The physiological proton gradient across the membrane in the $F_o$ portion is also shown to be optimal for the Michaelis constants of ADP and phosphate in the $F_1$ portion during ATP synthesis. Together, our coupled model is able to predict key dynamic and thermodynamic features of the $F_o{F}_1$-ATPase in vivo semiquantitatively, and suggests that such coupling approach could be further applied to other biophysical systems.

Figure 1

Geometric structure and separate models for the $F_o$ and $F_1$ portions. (a) Profile of the $F_o{F}_1$-ATP synthase structure, showing the two counter-rotating portions $F_o$ and $F_1$ combined by an asymmetric shaft $\gamma$. $F_o$ is membrane embedded, and $F_1$ is soluble in the cytoplasm. (b) Upper panel: Cross-sectional structure of $F_o$-ATPase, viewing from the extracellular side with its function illustrated. Filled dark and light circles represent whether protons bond to the carrier or not, respectively; the dashed circle represents one proton channel of the two which cannot be seen from this view. The free state $E$ bonds to a proton from the extracellular side and becomes $E_a$. After clockwise rotation of $F_o$, the conformation $E_a$ turns into $E_b$, which goes to again the initial state $E$ by transferring the proton inside the intracellular side. (b) Lower panel: A discrete-state model of $F_o$, simplified from [27]. $k_H$ is the second-order reaction constant. The sequence of events for protons entering the intracellular side of the membrane is shown by green (light gray) arrows, whereas the black arrows indicate the possibility in bringing out the protons from the intracellular side to the extracellular side. (c) Left panel: Discrete-state model of $F_1$, redrawn from [5, 24]. For each ATP molecule to be synthesized (hydrolyzed), the $\gamma$ shaft, shown as the dark yellow solid arrow, rotates 40 and ${80}^{\circ }$ clockwise (anticlockwise) successively. (c) Right panel: The extraction from the left panel. Transition rates $k_{\alpha },k_{-\alpha }$ are binding or dissociation rates for species $\alpha =D\left(\mathrm{ADP}\right),P\left({\mathrm{P}}_{\mathrm{i}}\right),T\left(\mathrm{ATP}\right)$. The transition between states $E_4$ and $E_5$ includes the ${80}^{\circ }$ rotation with reaction rates $k_h^{\prime \prime \prime },k_s^{\prime \prime \prime }$ and the phosphate binding/releasing with $k_D,k_{-D}$, hence $k_1=k_1(k_D,k_s^{\prime \prime \prime })$ and $k_{-1}=k_{-1}(k_{-D},k_h^{\prime \prime \prime })$ [28]. After the completion of one cycle, the state $E_1$ turns to an equivalent state with the $\gamma$ shaft rotating by ${120}^{\circ }$.

Figure 2

Efficiency profiles for $F_o$ and $F_1$ portions in coupled and uncoupled conditions. The profiles for the coupled situations are shown in solid lines; the green solid circle stands for the physiological condition. (a) Efficiency of $F_o$ vs the proton gradient, with intracellular ${\mathrm{pH}}_b=8.4$. In the uncoupled condition with fixed different values of the average torque $M$, the efficiency decreases with the increasing of the proton gradient; the coupled case leads to much slower decreasing, even becoming increasing at certain parameter ranges. The physiological condition $\left(\mathrm{\Delta }\mathrm{pH}=1.4\right)$ gives the $F_o$ portion's efficiency around $84.7\%.$ (b), (c) Efficiency of $F_1$ vs phosphate concentration (with $\left[T\right]=4\mathrm{mM},\left[D\right]=0.3\mathrm{mM})$ and the fraction of ADP over ADP plus ATP (with $\left[P\right]=6\mathrm{mM})$, as the total concentration of ADP and ATP is fixed. The uncoupled condition gives a monotonically decreasing profile for the efficiency; in the coupled case, the decreasing of efficiency becomes much slower, even increasing at a certain parameter range. The physiological condition $\left(p=0.07\right)$ gives the $F_1$ portion's efficiency around $66\%.$

Figure 3

The sensitivity of a separate portion's power and efficiency in the coupled model. In each figure the red (gray) and indigo (dashed light gray) lines stand for the power and efficiency profile, respectively, and the blue line embedded in each figure gives the torque on the shaft. (a)–(c) Power and efficiency of the $F_o$ portion changing with the total concentration of ADP and ATP (with $p=0.07,\left[P\right]=6\mathrm{mM})$, the phosphate concentration (with $\left[T\right]=4\mathrm{mM},\left[D\right]=0.3\mathrm{mM})$, as well as the ratio of ADP concentration (with $\left[T\right]+\left[D\right]=4.3\mathrm{mM},\left[P\right]=6\mathrm{mM})$. The power and efficiency and the resulted average torque response stably when the tuned parameters are around their physiological values. A zooming-in subfigure for the profile with $c_o$ ranging in [0, 0.01 mM] is inserted in (a). (d) Power and efficiency of the $F_1$ portion changing with the proton gradient, with $\left[T\right]=4\mathrm{mM},\left[D\right]=0.3\mathrm{mM},\left[P\right]=6\mathrm{mM}$. The profiles of power, efficiency, and average torque are all $S$ shaped with maximal slope at $\mathrm{\Delta }\mathrm{pH}=1.2.$

Figure 4

Michaelis constants of the reactants for the rotary flux of the whole enzyme. (a), (b) $K_m\left(\left[{\mathrm{H}}^{+}\right]\right)$ depends on the phosphate concentration (with $\left[D\right]=0.3\mathrm{mM},\left[T\right]=4\mathrm{mM})$ or the total concentration of ATP and ADP (with $\left[P\right]=6\mathrm{mM},p=0.07)$. In (a), the two constants, $K_m\left({\left[H^{+}\right]}_a\right)$ and $K_m\left({\left[H^{+}\right]}_b\right)$, reach a plateau even when $\left[P\right]$ is quite low, and remain stable at a certain range of $\left[P\right]$. In (b), the Michaelis constants are rather stable and there is a maximal value for both at $c_0\sim 1.8\mathrm{mM}.$ A zooming-in graph of the profile with $c_o$ ranging in [0, 0.01 mM] is inserted. (c), (d) $K_m\left(\left[P\right]\right)$ and $K_m\left(\left[D\right]\right)$ depend on the proton gradient across the membrane with $\left[T\right]=4\mathrm{mM}$ and ${\mathrm{pH}}_b=8.4$. The two curves shows their minimal value at around $\mathrm{\Delta }\mathrm{pH}\sim 1.4.$ When the proton gradient is high, the Michaelis constant can even increase.