Elasticity, friction, and pathway of γ-subunit rotation in F_oF₁-ATP synthase

Significance

F_oF₁-ATP synthase produces the ATP essential for cellular functions from bacteria to humans. Rotation of its central γ-subunit couples proton translocation in the membrane-embedded F_o motor to ATP synthesis in the catalytic F₁ motor. To explain its high efficiency, determine its top speed, and characterize its mechanism, we construct a viscoelastic model of the F₁ rotary motor from molecular dynamics simulation trajectories. We find that the γ-subunit is just flexible enough to compensate for the incommensurate eightfold and threefold rotational symmetries of mammalian F_o and F₁ motors, respectively. The resulting energetic constraints dictate a unique pathway for the coupled rotations of the F_o and F₁ rotary motors, and explain the fine stepping seen in single-molecule experiments.

Abstract

We combine molecular simulations and mechanical modeling to explore the mechanism of energy conversion in the coupled rotary motors of F_oF₁-ATP synthase. A torsional viscoelastic model with frictional dissipation quantitatively reproduces the dynamics and energetics seen in atomistic molecular dynamics simulations of torque-driven γ-subunit rotation in the F₁-ATPase rotary motor. The torsional elastic coefficients determined from the simulations agree with results from independent single-molecule experiments probing different segments of the γ-subunit, which resolves a long-lasting controversy. At steady rotational speeds of ∼1 kHz corresponding to experimental turnover, the calculated frictional dissipation of less than k_BT per rotation is consistent with the high thermodynamic efficiency of the fully reversible motor. Without load, the maximum rotational speed during transitions between dwells is reached at ∼1 MHz. Energetic constraints dictate a unique pathway for the coupled rotations of the F_o and F₁ rotary motors in ATP synthase, and explain the need for the finer stepping of the F₁ motor in the mammalian system, as seen in recent experiments. Compensating for incommensurate eightfold and threefold rotational symmetries in F_o and F₁, respectively, a significant fraction of the external mechanical work is transiently stored as elastic energy in the γ-subunit. The general framework developed here should be applicable to other molecular machines.

F_oF₁-ATP synthase is essential for life. From bacteria to human, this protein synthesizes ATP from ADP and inorganic phosphate P_i in its F₁ domain, powered by an electrochemical proton gradient that drives the rotation of its membrane-embedded F_o domain (1–5). Its two rotary motors, F₁ and F_o, are coupled through the γ-subunit forming their central shaft (2). ATP synthase is a fully reversible motor, in which the rotational direction switches according to different sources of energy (2, 6). In hydrolysis mode, the F₁ motor pumps protons against an electrochemical gradient across the membrane-embedded F_o part, converting ATP to ADP and P_i (7, 8).

F₁ has a symmetric ring structure composed of three αβ-subunits with the asymmetric γ-subunit sitting inside the ring (9, 10). Each αβ-subunit has a catalytic site located at the αβ-domain interface. The F₁ ring has a pseudothreefold symmetry with the three αβ-subunits taking three different conformations, E (empty), TP (ATP-bound), and DP (ADP bound) (9–11). The F_o part is composed of a c ring and an a subunit (3, 12). Driven by protons passing through the interface of the c ring and the a subunit, the c ring rotates together with the γ-subunit (rotor) relative to the a subunit, which is connected to the F₁ ring through the peripheral stalk of the b subunit (stator) (12). Interestingly, in nature, one finds a large variation in the number of subunits in the c ring. In animal mitochondria, one finds c₈ rings, requiring a minimal number of eight proton translocations for the synthesis of three ATP, at least 20% fewer protons than in bacteria and plant chloroplasts with c₁₀–c₁₅ rings (13, 14). The resulting symmetry mismatches between F₁ and F_o (15–17) clearly distinguish the biomolecular motor from macroscopic machines.

Key open questions concern the detailed rotational pathway of the two coupled rotary motors, the impact of the rotational symmetry mismatch between the F_o and F₁ motors on the motor mechanics, the resulting need for transient energy storage, the role of frictional dissipation, and the molecular elements associated with stepping of the F₁ motor (18–24). Here we explore these questions by building a dissipative mechanical model of the F₁ motor on the basis of atomistic molecular dynamics (MD) simulations. Friction and torsional elasticity of the γ-subunit are central to the efficient function of the coupled F_oF₁ nanomotors (15, 25, 26). For γ-subunits cross-linked with the α₃β₃-ring, estimates have been obtained by monitoring thermal angle fluctuations in single-molecule experiments (16, 27) and MD simulations (28). To probe the elastic and frictional properties under mechanical load over broad ranges of rotation angles and angular velocities, we induce torque-driven γ-subunit rotation in MD simulations (20, 29). From the resulting mechanical deformation and energy dissipation, we construct a fully quantitative viscoelastic model. We account for the torsional elasticity and friction by describing the rotational motion of the γ-subunit as overdamped Langevin dynamics on a 2D harmonic free energy surface. The model quantifies the magnitude of transient elastic energy storage compensating for the incommensurate rotational symmetries of the F_o and F₁ motors (30). The resulting energetic constraints allow us to map out a detailed pathway for their coupled rotary motions, and to rationalize the finer stepping of the mammalian F₁ motor seen in recent experiments (31), with only eight c subunits in the corresponding F_o motor. By quantifying the frictional dissipation, we identify a key contributor to the high thermodynamic efficiency of the F₁ motor. The general framework developed here for F₁ should be applicable also to other molecular machines.

Results

Two-Domain Langevin Model of γ-Rotation in F₁-ATPase.

To construct a Langevin diffusion model, the γ-subunit was coarse-grained into a collection of domains that are coupled by torsional springs and rotate in different frictional environments (Fig. 1A). To determine the minimal number of domains, we examined the spatial extent of coupled rotary motions along the γ-subunit axis. Using the flexible rotor method (20), we applied torque on the γ-subunit in atomistic MD simulations. A relatively weak harmonic restraint of $\kappa =\mathrm{2,000}\hspace{0.17em}\text{kcal}\cdot {\text{mol}}^{-1}\cdot {\text{rad}}^{-2}\approx 0.61\hspace{0.17em}\text{kcal}\cdot {\text{mol}}^{-1}\cdot {\text{deg}}^{-2}$ on the average angle at angular velocities of $\omega =1-10°\text{/ns}$ resulted in a gradual buildup of tension and permitted γ-subunit twisting and substantial angle fluctuations. In the torque MD simulations, we probed rotary motions of up to $55°$ in the protruded part of the γ-subunit and $20°$ at its core (Fig. 1A).

Fig. 1.

Langevin diffusion model of γ-rotation in F₁-ATPase. (A) Coarse graining of the γ-subunit into two domains connected by torsional springs and embedded in different viscous environments. (B) Local rotation angles of the γ-subunit along axis in torque MD simulation, scaled by the target angle $\omega t$ for $\omega =1$°/ns. The profiles are shown for times t = 35 ns, 40 ns, and 50 ns, averaged over three independent trajectories. The near-perfect superposition of the profiles is consistent with quasi-equilibrium conditions in a linearly elastic model. Dashed lines show the optimized quasi-equilibrium solution of the two-domain viscoelastic model. Arrows indicate crosslinks in the single-molecule experiments of Sielaff et al. (16) and Okuno et al. (27). Residues γC87 and γA270 in E. coli correspond to γC78 and γA256, respectively, in bovine mitochondria, and residue γR84 in Bacillus PS3 corresponds to γR75.

At the lowest speed, the profiles of the γ-subunit rotation angle along the axis show two plateaus separated by a distinct step at ∼20 Å (Fig. 1B), indicating that the γ-subunit can be divided into two main parts. This division is consistent with the structural features of the γ-subunit, which consists of a protruded globular part and a coiled-coil helical domain inside the hexamer ring. Our minimal model of the γ-subunit rotation near the catalytic dwell thus consists of domains 1 and 2 connected by a torsional spring with spring constant k₁₂ (see Eq. 4). Domain 1 is coupled harmonically with spring constant k₁ to the static α₃β₃-ring subunits. External torque is exerted by a torsional spring κ rotating at an angular velocity ω. The different viscous environments of the two domains are described by rotational diffusion coefficients D₁ and D₂. Our two-domain model can be seen as a further reduction of the nine-segment model of Czub and Grubmüller (28), which divides the γ-subunit into five segments and the α₃β₃-ring into four. As shown in Fig. S1, our domain 1 covers their segment 5, half of segment 4, and, in addition, the C-terminal helix not included in the nine-segment model. Our domain 2 contains segments 1–3 and the other half of segment 4. According to the γ-subunit twist profile (Fig. 1B and Fig. S1), the elastic deformation is concentrated within their segment 4. In our two-domain model, we treat this linearly twisted region by the single torsional spring k₁₂. By contrast, our torsional spring k₁ accounts for the interactions between segments 4–5 and 6–9. Note that we do not explicitly consider the small helix-turn-helix motif at the very top of the γ-subunit (segment 1), whose elastic response in F_oF₁-ATP synthase will be determined also by its interactions with the c ring and the ε- and δ-subunits (Fig. S1).

Fig. S1.

Comparison with the nine-segment model by Czub and Grubmüller (28). The five segments on the γ-subunit are mapped on the twist profile and the structure.

To determine the parameters k₁, k₁₂, D₁, and D₂ of the Langevin diffusion model, we fitted analytical solutions (see SI Text and Figs. S2 and S3) for the time-dependent average rotation angles $x_1\left(t\right)$ of the core helix domain (residues γ1–26, γ228–273) and $x_2\left(t\right)$ of the protruded domain (residues γ27–227) to corresponding trajectories from atomistic MD simulations (Fig. 2). The MD rotational angles were calculated as distance-weighted averages over the angles of individual residues using separate axes for the two domains (see Methods). In a first step, we quantified the elastic properties at a low angular velocity of $1°\text{/ns}$, at which the time-dependent rotation angles $x_1$ and $x_2$ could be fitted with a quasi-equilibrium model (model A; Fig. 2A). Correspondingly, the twist profiles at different time points, rescaled by the target angle ωt, almost perfectly coincide (see Fig. 1B and Eqs. S1 and S2). At rotational speeds exceeding $2°\text{/ns}$, dissipative effects become significant. The resulting deviations from quasi-equilibrium (difference between orange and black lines in Fig. 2A) allow us to extract rotational diffusion coefficients from fits of model B (assuming quasi-equilibrium for domain 2), and model C (with both domains out of equilibrium) (Table S1). In a second step, we performed a global analysis to determine parameter ranges that fit trajectories over the entire range of angular velocities. Plots of ${\chi }^2$ in Fig. 2B show the nonlinear correlations between the fit parameters, permitting variations in $k_{12}$ and $k_1$ by ∼30% around the optimal values $\left(k_{12},k_1\right)=\left(123,309\right)$ (in kilocalories per mole per square radian) (Fig. 2B). Whereas, at low rotational speeds, $D_1$ and $D_2$ are only bounded from below, data at higher speeds provide also upper limits. The overlap of the four ${\chi }^2$ maps for model C delimits the range of $D_1$ and $D_2$ that fit all trajectories well (Fig. 2B). To test the assumption of quasi-equilibrium at $1°\text{/ns}$, we ran additional simulations at lower rotational speeds of $0.2°\text{/ns\hspace{0.17em}}$ and $0.5°\text{/ns}$. As shown in Fig. 2C, both the time-dependent twisting profiles and the ${\chi }^2$ maps for $k_{12}$ and $k_1$ are fully consistent with the results obtained at the higher speeds.

Fig. S2.

Harmonic approximation of torque potential. Brownian dynamics trajectories are shown with noise-averaged trajectories (smooth lines) for the harmonic torque potential (Left) and sin/cos torque potential (Right). The same optimized parameters were used; $k_{12}=122.9\hspace{0.17em}\text{kcal}\cdot {\text{mol}}^{-1}\cdot {\text{rad}}^{-2}$, $k_1=309.4\hspace{0.17em}\text{kcal}\cdot {\text{mol}}^{-1}\cdot {\text{rad}}^{-2}$, $D_1=0.001\hspace{0.17em}{\text{rad}}^2\text{/ns}$ and $D_2=0.002\hspace{0.17em}{\text{rad}}^2\text{/ns}$.

Fig. S3.

Weight factors for the two-domain Langevin model from atomistic torque simulation.

Fig. 2.

Parameterization of the Langevin diffusion model via all-atom torque MD simulations. (A) Angle trajectories for the core ($x_1$, red) and protruded domain ($x_2$, green) from torque MD simulations at angular velocities of ω = 1–10°/ns (left to right). Fits of the quasi-equilibrium model A and the overall optimal rotational diffusion model C ($k_{12}=94$ and $k_1=223$ kcal/mol/rad², and $D_1=0.001$ and $D_2=0.003$ rad²/ns) are shown in black and orange, respectively. (B) Fit quality χ² of quasi-equilibrium model A as a function of $k_{12}$ and $k_1$ at $\omega =1°\text{/ns}$ (Top Left). The ${\chi }_{\omega }^2$ of model C as a function of $D_1$ and $D_2$ at the different rotational speeds $\omega$, and overall (right; $({\chi }_1^2+{\chi }_2^2+{\chi }_5^2+{\chi }_{10}^2)/4$) are shown in the other panels. The blue and yellow points in B indicate the best local and global fits, respectively, with the latter accounting also for the work profiles in Fig. 3. (C) Validation with slower torque trajectories at rotational speeds of ω = 0.2°/ns and 0.5°/ns. (Top) Angle trajectories of domains 1 and 2 are shown in red and green for ω = 0.2°/ns (bottom scale), and in magenta and cyan for ω = 0.5°/ns (top scale), respectively. (Middle and Bottom) Fit quality χ² of quasi-equilibrium model A as a function of $k_{12}$ and $k_1$ at ω = 0.2°/ns and 0.5°/ns. Yellow squares indicate the global optimum determined at ω ≥ 1°/ns.

Table S1.

Optimized parameters from angle trajectory fits at different rotational speeds (ω) for the three models A, B, and C

ω	Model	$k_{12}$	$k_1$	$D_1$	$D_2$	${\chi }^2$
1°/ns	A	123	309			1.00
	C	94	223	0.0029	0.0025	1.00
	C	94	223	0.0010	0.0030	1.05
2°/ns	B	123	309	0.0016		1.00
	C	123	309	0.0016	0.016	1.00
	C	94	223	0.0010	0.0022	1.00
	C	94	223	0.0010	0.0030	1.01
5°/ns	B	123	309	0.0008		1.00
	C	123	309	0.0007	0.0023	1.00
	C	94	223	0.0007	0.0016	1.00
	C	94	223	0.0010	0.0030	1.18
10°/ns	B	123	309	0.0018		1.00
	C	123	309	0.0011	0.0022	1.00
	C	94	223	0.0011	0.0019	1.00
	C	94	223	0.0010	0.0030	1.33

Torsional spring constants are in kilocalories per mole per square radian, and rotational diffusion coefficients are in square radians per nanosecond.

Work Profile as Additional Information.

The combined elastic and dissipative work performed during the torque-driven rotations provides us with largely independent information that we can use not only to validate the Langevin diffusion model and fine-tune its parameters, but also to quantify the energetics of the γ-rotation. The time-dependent work performed by the external torque is calculated as in Eq. 5. Fig. 3 shows the work profiles as a function of the reference rotation angle obtained from actual atomistic torque MD simulations. The work profiles at the lowest rotational speeds of $\omega =1\hspace{0.17em}$ and $2°\text{/ns}$ show little variation (blue and cyan points in Fig. 3), indicating that, at these speeds, rotation indeed occurs at near-equilibrium conditions. However, the accumulated work at a given target angle ωt increases as the rotational speed becomes faster and the system is driven out of equilibrium. For comparison, dashed lines in Fig. 3 show the work profiles calculated for the optimized Langevin diffusion model. Whereas the overall tendency of the work profiles of the diffusion model is consistent with the results of the atomistic simulations, the work at ωt = 50° is overestimated by 20–30% (Fig. 3).

Fig. 3.

Nonequilibrium work profiles $w_t$ from torque MD simulations (see Methods). The mechanical work performed by the external torque in MD simulations at various rotational speeds (points with different colors for the different speeds; see Inset showing time-dependent work) is compared to the corresponding work calculated from the elastic model with optimized parameters (dashed lines with the same coloring scheme). The work is plotted as a function of the target angle $\omega t$ during the torque MD simulations. Work profiles from the elastic model with globally optimal set of parameters are shown as solid lines.

This small discrepancy in the work profiles can be corrected by fine-tuning the parameters in the Langevin model to fit both work and angle-vs.-time trajectories. From a global analysis, we find that a small increase in $D_2$ and small decreases of $k_{12}$ and $k_1$, well within the tolerance of the fits in Fig. 2, produce agreement with the entire set of data from atomistic simulations (Fig. 3). The optimized parameters $k_{12}=94$ and $k_1=223$ kcal⋅mol−¹⋅rad⁻² and $D_1=0.001$ and $D_2=0.003$ rad²/ns capture all trajectories and work profiles well (orange lines in Fig. 2A; solid lines in Fig. 3).

With this fine-tuned Langevin diffusion model, we decompose the work performed during torque-driven rotation into recoverable elastic energy and dissipation (Fig. S4). At a rotational speed of $1°\text{/ns}$, nearly the entire exerted work is stored as elastic energy. This rotational speed is 10,000 times faster than the average rotational speed of $\omega \approx 100$°/ms at turnover that is dominated by the dwell times at intermediate states. At speeds exceeding $2°$/ns, dissipation becomes relevant. Remarkably, the sum of the intrinsic elastic energy of γ (green line in Fig. S4) and the elastic potential acting on $x_1$ (red line) as a function of ωt remains almost independent of the rotational speed. However, their relative contributions differ as the rotational speed increases. At higher speeds, $x_1$ lags behind due to its higher friction. As a result, the spring k₁₂ becomes increasingly stretched, absorbing relatively more of the elastic energy than k₁.

Fig. S4.

Work profiles decomposed into the elastic energy stored in the γ-subunit and the dissipated work; ω is (A) 1°/ns, (B) 2°/ns, (C) 5°/ns, and (D) 10°/ns. The green and red lines represent the torsional elastic energy between the two domains of the γ-subunit and the torsional elastic energy between the γ-subunit and the ring subunits, respectively. The remainder, indicated in gray shading, is the dissipated work.

Discussion

γ-Subunit Elasticity.

We determined the torsional elasticity and rotational Langevin diffusion coefficients by optimizing a two-domain rotational Langevin diffusion model against the atomistic torque trajectories at different rotational speeds and the corresponding nonequilibrium work profiles. To assess the elasticity parameters of the γ-rotation, we compared the results from our viscoelastic model to the values from single-molecule experiments. Sielaff et al. (16) determined an effective elastic coefficient of about $750$ pN⋅nm/rad² for the Escherichia coli γ-subunit between two crosslinks at residues γA270 and γA87, albeit with a significant uncertainty because of their difference measurement. As indicated in Fig. 1B, this segment closely matches the $x_2-x_1$ torsional twist in our model, for which we have determined an effective spring constant of $k_{12}\approx 660$ pN⋅nm/rad². The agreement between our model and the experiment is thus excellent, considering our estimated fit uncertainties of about 30%. Our estimate for k₁₂ also agrees with the 620 pN⋅nm/rad² obtained in a previous simulation study for a similar segment of a γ-subunit covalently attached to the hexamer ring (28). Using a similar approach, Okuno et al. (27) estimated an elastic coefficient of $223\pm 141$ pN⋅nm/rad² for the Bacillus PS3 γ-subunit up to a single cross-link at a position closely matching the full twist $x_2$ of our model, and moving with the protruding part of the γ-subunit (Fig. 1B). The corresponding effective spring constant in our model is $k_{eff}={\left(1/k_1+1/k_{12}\right)}^{-1}\approx 460$ pN⋅nm/rad², deviating from the measurement by less than the combined ($\approx 2\times 140$ pN·nm/rad²) SEs of model and experiment. In effect, we could thus independently validate the two spring constants k₁ and k₁₂ in our model, because they were probed in separate measurements. Moreover, our results resolve the long-standing controversy concerning the disagreement between the two measurements, with one probing k₁₂ and the other probing $k_{eff}$.

Hydrodynamics and Rotational Friction.

For the diffusion coefficients of the core part, $D_1$, we obtained ∼0.001 rad²/ns from the angle trajectories at different rotational speeds and the work profiles. This value has the same order of magnitude as previous estimates from autocorrelation functions of MD trajectories (28). To get a sense of the relevant scale, we estimated a hydrodynamic rotational diffusion coefficient by approximating the core part of the γ-subunit as a cylinder (32),

$$D_{1,hydro}=k_{B}T/4\pi \eta L{a}^2$$ [1]

where $\eta =7\times {10}^{-4}\hspace{0.17em}\text{kg}/\text{m}\cdot \text{s}$ is the viscosity of water at 310 K, $L\approx 7\hspace{0.17em}\text{nm}$ is the height of the cylinder, and $a\approx 1\hspace{0.17em}\text{nm}$ is the radius of the cylinder. The resulting $D_{1,\hspace{0.17em}hydro}\approx 0.07{\hspace{0.17em}\text{rad}}^2\text{/ns}$ is ∼70 D₁. The γ-subunit rotating inside F₁ thus experiences significantly higher friction than in solution, reflecting the tight hydrophobic and steric interactions with the ring subunits. For the diffusion coefficient of the protruded part, $D_2$, we obtained $\sim 0.003\hspace{0.17em}{\text{rad}}^2\text{/ns}$. Approximated as a sphere of radius $a\approx 2\hspace{0.17em}\text{nm}$, we use (32, 33)

$$D_{2,hydro}=k_{B}T/8\pi \eta a^3$$ [2]

to estimate a hydrodynamic rotational diffusion of $D_{2,\hspace{0.17em}hydro}\approx 0.03\hspace{0.17em}{\text{rad}}^2\text{/ns\hspace{0.17em}}$. D₂ is thus ∼10 times smaller than the hydrodynamic limit, which is reasonable because the protruded part is slowed down by interactions, mainly salt bridges, with the tips of the αβ-subunits.

Mechanical Consequence of F_o–F₁ Symmetry Mismatch.

The calculated elastic properties allow us to construct a detailed pathway for the coupled rotary motion of the F_o and F₁ motors in mammalian ATP synthase. A step size of $360°/8=45°$ is assumed for F_o, and substeps of $\sim 25°+\sim 30°+\sim 65°$ = 120$°$ are assumed for F₁ on the basis of recent single-molecule experiments on human mitochondrial F₁ (31), with the cost of ATP spread proportionally among them. The free energy of the system for the discrete (sub)steps then becomes

$$G\left({\varphi }_{Fo},{\varphi }_{F1}\right)=\left({\varphi }_{Fo}/45°\right)G_{in}+\left({\varphi }_{F1}/120°\right)G_{out}+\frac{k_{eff}}{2}{\left({\varphi }_{Fo}-{\varphi }_{F1}\right)}^2$$ [3]

where ${\varphi }_{Fo}$ and ${\varphi }_{F1}$ are the rotation angles of the F_o and F₁ subunits in the ATP synthesis direction, respectively. $G_{in}=-200\hspace{0.17em}\text{mV}\approx -4.6\hspace{0.17em}\text{kcal/mol}$ is the input energy per one proton translocation and $G_{out}\approx 12.3\hspace{0.17em}\text{kcal/mol}$ is the output energy per one ATP synthesis (6) at maximum load conditions, $8G_{in}=3G_{out}$, for the mammalian ATP synthase. Note that $G\left({\varphi }_{Fo},{\varphi }_{F1}\right)$ accounts only for the energy of metastable intermediate states, not the energy barriers between them.

This free energy function severely restricts the possible pathways of the coupled F_o–F₁ rotary motion. To ensure significant activity, we require that, during a full cycle, the free energy difference between the least and most populated intermediates should not exceed 5–7 kcal/mol, an upper limit supported by the kinetic analysis discussed below. Fig. 4B indicates all intermediates that exceed this limit shaded in red, which leaves only a single pathway for the coupled rotary motion through the ${\varphi }_{Fo}$-${\varphi }_{F1}$ plane. Here we accounted for the unknown relative phase $\mathrm{\Delta }{\varphi }_{Fo}$ of the F_o and F₁ motors (Fig. 4B and Fig. S5A), and for uncertainties of $\pm 3°$ in the F₁ substep sizes (Fig. 4D and Fig. S5B; F_o steps as ${\varphi }_{Fo}=\mathrm{\Delta }{\varphi }_{Fo}+n\times 360°/8$ with n integer). To quantify the effects of subunit asymmetry on the kinetics of ATP synthesis, we developed a master equation model of the coupled F_o–F₁ rotation on the basis of the angle-dependent energy function Eq. 3 (see SI Text for details). The resulting mean times for full rotation of the two motors depend strongly on the number of F₁ substeps. Assuming an elementary rate of k₀ ≈ 1/(100 μs) set by proton binding to F_o, ATP would be produced at a rate of ∼0.6/s and 30/s with two and three substeps, respectively (Fig. S6 and Table S2). Although these estimates are clearly rough, they nonetheless imply that having three substeps ensures an efficient operation with the small c₈ ring of mammalian F₁, using only eight protons per three ATP.

Fig. 4.

Pathway and energetics of coupled F_o–F₁ rotary motion under reversible conditions of ∼200-mV proton motive force (pmf) in mitochondria. (A) Schematic of coupled F_o–F₁ rotary motion in animal mitochondria. Pathway of coupled motion with (B) three substeps of 25°, 30°, and 65°; (C) two substeps of 40° and 80°; and (D) three substeps of 28°, 33°, and 59°. Results are shown for a γ-subunit modeled with the effective spring constant $k_{eff}=66\hspace{0.17em}\text{kcal}\cdot {\text{mol}}^{-1}\cdot {\text{rad}}^{-2}$ determined here from the MD simulations as a function of the rotation angles ${\varphi }_{Fo}$ and ${\varphi }_{F1}$ of the F_o and F₁ subunits, respectively. Numbers in blue indicate the free energy of the respective conformations. Here we assume that k₁ and k₁₂ apply consistently to all microstates. The pmf corresponds to an input free energy of $-4.6\hspace{0.17em}\text{kcal/mol}$ per single proton translocation and an output free energy of $4.6\times 8/3\approx 12.3\hspace{0.17em}\text{kcal/mol}$ per single ATP synthesis. For simplicity, the latter energy cost is assumed to be accumulated evenly as a function of the ${\varphi }_{F1}$ angle. The minimum energy paths are shown as blue lines for three (B) or two (C) substeps of the F₁ motor. An adjustment in the relative phases of ${\varphi }_{Fo}$ and ${\varphi }_{F1}$ was made to lower the energy of the microstates along the path, as indicated by the slight shifts in angles (Fig. S5A). Red crosses and shading indicate conformations with free energies larger than 7 kcal/mol, where the minimum free energy is set to zero. For the three substeps of F₁ (B and D), all microstates along the path are reachable at ∼200 mV pmf, well below the maximum pmf of ∼220–230 mV for mitochondria. With only a single 40° step, high-energy intermediates (>10 kcal/mol) arise, inconsistent with efficient operation.

Fig. S5.

Pathways and energetics of coupled F_o–F₁ rotary motion with a c₈ ring under near-reversible conditions of ∼200-mV pmf in mitochondria as functions of F_o and F₁ rotation angles ϕ_Fo and ϕ_F1, respectively, in ATP synthesis direction. The difference between maximum and minimum free energies along the minimum free energy paths is shown as a function of the relative phase $\mathrm{\Delta }{\varphi }_{Fo}$ of ϕ_Fo and ϕ_F1 for (Left) three and (Right) two F₁ substeps (A) with fixed step sizes and (B) with step size variations of up to $\pm 3°$. Red circles represent the values used for plotting the pathways in Fig. 4 and in C. (C) Optimal pathway at maximum load conditions with minimally adjusted F₁ step sizes and two F₁ substeps (blue line and squares). Red crosses indicate conformations with free energies exceeding 7 kcal/mol relative to the minimum along the path.

Fig. S6.

Mean first passage time (MFPT) to the end state in one full rotation of F_oF₁. The states along the x axis are on the minimum energy pathway in Fig. 4. States 1 and N − 1 (the rightmost state of each curve) correspond to the initial state $\left({\varphi }_{o1}=0°,{\varphi }_{11}=0°\right)$ and one state before the end state $\left({\varphi }_{oN}=360°,{\varphi }_{1N}=360°\right)$, respectively.

Table S2.

MFPT for full rotatory cycles of both F_o and F₁ computed from the kinetic model

Cn	keff	nstepf1	Step size	$\mathrm{\Delta }{\varphi }_o$	MFPT
8	66.0	2	80, 40	8	5.0 × 104
8	66.0	3	65, 30, 25	0	1.2 × 103
8	66.0	3	62, 30, 28	−1	9.5 × 102
10	66.0	2	85, 35	−5	6.6 × 105
10	66.0	3	60, 25, 35	−6	1.3 × 103
10	33.0	2	85, 35	1	2.6 × 103
10	33.0	3	60, 25, 35	1	1.3 × 102

C_n is number of c-ring subunits, k_eff is the effective γ-subunit elastic coefficient in kilocalories per mole per square radian, nstepf1 and step size are the number of substeps in 120° rotation of F₁ and the step sizes in degrees, respectively, and $\mathrm{\Delta }{\varphi }_o$ is the relative phase of F_o and F₁ (in degrees). The mean first passage times for the 360° rotations of both F_o and F₁ are given in units of $1/k_0$.

The symmetry mismatch between the mammalian F_o and F₁ rotary motors thus has important consequences. First, a significant amount of energy must be stored in the γ-subunit (30), which appears to be just soft enough to absorb sufficient elastic energy, but not too soft to allow for slipping and alternate pathways in the ${\varphi }_{Fo}$–${\varphi }_{F1}$ plane. Second, the additional substeps seen in recent single-molecule experiments on the mammalian F₁ motor are critical for its operation. With the $\sim 40°+\sim 80°=120°$ steps of the bacterial motor, an eight-membered c ring would not be feasible (Fig. 4C and Fig. S5C).

We also extended the model to bacterial systems with c₁₀ rings, such as E. coli (4) or thermophilic Bacillus PS3 (15) (Fig. S7). In this case, the F_o step size is $360°/10=36°$. Under the conditions of E. coli F_oF₁ experiments (4), we estimate a faster elementary step of 1/k₀ ≈ 30 μs (see SI Text). Assuming the same elasticity of the γ-subunit (k_eff = 66.0 kcal⋅mol⁻¹⋅rad⁻²), F₁ motors operating with two and three substeps require ∼20 s and ∼40 ms for full rotation, respectively (Table S2). The time scale with the three substeps (∼40 ms) agrees well with the experimental values (∼50 ms; ∼60 s⁻¹ for a 120° step) at a maximum load condition (4). This result suggests a possible third substep, yet unresolved, for the E. coli F₁ (34, 35). This third step could be associated with phosphate release (31, 36). Alternatively, the bacterial γ-subunit or its interface with the c ring could be softer, especially for the thermophilic Bacillus PS3 F₁, in which only two substeps have been found after extensive single-molecule experiments. With the stiffness reduced by half in our model (k_eff = 33.0 kcal⋅mol⁻¹⋅rad⁻²), two- and three-substep F₁ motors would operate on time scales of ∼80 ms and ∼4 ms, respectively (Table S2). With this softer elasticity, the two-substep F₁ could thus achieve a reasonable ATP synthesis rate. However, the interface between γ-subunit and c ring cannot be much softer because their angle-dependent interaction must have 10-fold symmetry for a c₁₀ ring. If we assume that an elastic potential with an interfacial spring constant k_γc holds in each of the n minima all of the way to a cusp-shaped barrier, the n-fold periodic free energy surface will have barriers of height $k_{\gamma c}{(\pi /n)}^2$/2. For n = 10 of the bacterial c₁₀ rings and a minimum barrier height of at least 5 kcal/mol to prevent slipping of the c ring against the γ-subunit on a millisecond timescale, the interfacial spring constant $k_{\gamma c}$ should exceed 100 kcal⋅mol⁻¹⋅rad⁻². Putting this torsional spring in series with the springs k₁ and k₁₂ determined here, one would find an effective sprint constant of $k_{eff}={(k_1^{-1}+k_{12}^{-1}+k_{\gamma c}^{-1})}^{-1}>40$ kcal⋅mol⁻¹⋅rad⁻², at least half of our value for the γ-subunit and α₃β₃-ring alone.

Fig. S7.

Pathways and energetics of coupled F_o–F₁ rotary motion with a c₁₀ ring under near-reversible conditions of ∼200-mV pmf as functions of F_o and F₁ rotation angles ϕ_Fo and ϕ_F1, respectively, in ATP synthesis direction. The difference between maximum and minimum free energies along the minimum free energy paths is shown as a function of the relative phase $\mathrm{\Delta }{\varphi }_{Fo}$ of ϕ_Fo and ϕ_F1 for (Left) three and (Right) two F₁ substeps with k_eff = 66 kcal⋅mol⁻¹⋅rad⁻² (A) and k_eff = 33 kcal⋅mol⁻¹⋅rad⁻² (B). Red arrows indicate $\mathrm{\Delta }{\varphi }_{Fo}$ used for further analysis. (C) Optimal pathway at maximum load conditions for k_eff = 66 kcal⋅mol⁻¹⋅rad⁻² with three F₁ substeps is shown as blue line and squares. (D) Optimal pathway at maximum load conditions for k_eff = 33 kcal⋅mol⁻¹⋅rad⁻² with two F₁ substeps. Red crosses indicate conformations with free energies exceeding 7 kcal/mol relative to the minimum along the path.

It will also be interesting to compare the rotor stiffness of F- and V-type ATPases (37), also to examine if the ATP hydrolysis-driven proton pump function of the latter imposes different requirements on the mechanochemical coupling.

Dissipation at Average Rotational Speed.

From the rotational diffusion coefficients $D_1$ and $D_2$, we can estimate the frictional dissipation per full cycle. Here we implicitly assume, first, that the rotational speed is uniform and, second, that the calculated friction applies uniformly to the entire rotational range, ignoring additional dissipation, e.g., during barrier crossing events. Although not exact, the estimate should nevertheless serve as a useful reference. Under these two assumptions, the overall diffusion coefficient (obtained by adding the respective friction coefficients) is $D_{eff}={\left(1/D_1+1/D_2\right)}^{-1}\approx 0.00075\hspace{0.17em}{\text{rad}}^2\text{/ns}$. The frictional dissipation in a full turn at an angular velocity of $\omega =1$ kHz corresponding to experimental turnover rates is then $2\pi \omega k_{B}T/D_{eff}\approx 0.05\hspace{0.17em}{k}_{B}T$. At this rotational speed, dissipation could thus effectively be ignored. However, as discussed in ref. 20, the actual rotational motions are likely much faster than the average, because much of the time is spent waiting in metastable states (in particular, for the arrival of substrates and the escape of product). To account for short transition path times, we calculate in the following also the maximum rotational speed energetically permitted for these transitions.

Maximum Rotational Speed.

The estimated rotational diffusion coefficients, or friction coefficients from the relation $\xi =k_{B}T/D$, define a fundamental limit on the rotation, the maximum rotational speed between the dwells without load. This limiting speed has so far not been determined experimentally. The reason is that, in single-molecule experiments, a relatively big probe has to be attached to visualize the rotations. When the amount of dissipation for a 120°=$2\pi /3$ rotation $W=(2\pi /3)\omega k_{B}T/D_{eff}$ is equated with the ATP hydrolysis energy of $\sim 20k_{B}T$ at typical cellular conditions (33), assuming an energy conversion efficiency of 100% from the ATP hydrolysis reaction to mechanical rotation, it should give the maximum rotational speed ${\omega }_{max}$. For the estimated diffusion coefficient $D_{eff}$, we find ${\omega }_{max}\approx 0.4\hspace{0.17em}°\text{/ns}$, corresponding to ∼300 ns per $2\pi /3$ rotational transition. Note that this speed is comparable to that of the torque MD simulations, and that functionally relevant conformational changes of the αβ-subunits have been observed at such rotary speeds (20, 29). This prediction can be tested by further reducing the size of the probe used in the single-molecule experiments. Based on our estimates, if the load can be reduced to a diameter of ∼20 nm, then friction will be dominated by the protein, not by the load.

Conclusions

We developed a multiscale framework to deduce mechanical and frictional properties of biomolecular machines from atomistic MD simulations, and applied it to the F₁ motor of ATP synthase. By constructing a Langevin diffusion model of the γ-rotation in F₁-ATPase from atomistic MD trajectories, we deduced the torsional elasticity and rotational friction coefficients governing the γ-subunit rotation. The estimated torsional elastic coefficients agree with single-molecule experiments probing different segments of the γ-subunit (16, 27), thus resolving a controversy in their interpretation. The overall elastic coefficient is also consistent with the results of earlier equilibrium simulations of a cross-linked γ-subunit (28).

The γ-subunit appears to be barely soft enough to absorb the required elastic energy in the coupled rotation of the symmetry-mismatched F_o and F₁ motors of ATP synthase. This relative stiffness ensures that the two motors are tightly coupled, forcing them onto a unique rotational pathway. For ATP synthase in mammalian mitochondria, which use only eight protons to synthesize three ATP, the mismatch is so large that additional substeps are required, thus providing a rationale for the recent single-molecule observation of two intermediate steps instead of one (31).

The estimated rotational diffusion coefficient for the coiled-coil core part of the γ-subunit is ∼70 times smaller than the hydrodynamic limit, which is likely due to tight interactions with the αβ-subunit bearing. For the protruded part, the diffusion coefficient is ∼10 times smaller than the hydrodynamic limit, possibly due to salt bridges with the tips of the αβ-subunits. From these estimated rotational diffusion coefficients, we predict a maximum rotational speed of ${\omega }_{max}\approx 1$ MHz between the dwells without load, and estimate an energy dissipation of < 1 k_BT per rotation at an average speed of $\omega \approx 1$ kHz. The F₁ rotary motor thus operates at very small dissipation levels, which is essential for the high overall thermodynamic efficiency of F_oF₁-ATP synthase.

Methods

MD Simulations.

The MD simulations were performed as in ref. 20. However, here the rotation angles of each residue were calculated with separate local axes for the two domains passing through their respective centers of mass and orientated parallel to the symmetry axis of F₁. The angles were then averaged for each domain, weighted by the normal distance of the Cα atom from the axis.

Fit of Langevin Diffusion Model.

To fit the Langevin diffusion model to the torque-induced angle trajectories, we minimized ${\chi }^2=\left(1/2N_t\right){{\displaystyle \sum }}_t\left[{\left(\left[x_1^{fit}\left(t\right)-x_1\left(t\right)\right]/{\sigma }_1\right)}^2+{\left(\left[x_2^{fit}\left(t\right)-x_2\left(t\right)\right]/{\sigma }_2\right)}^2\right]$, where $N_t$ is number of data points in a particular time course; ${\sigma }_i$ is defined as the SD from the best fit, ${\sigma }_i=\sqrt{\left(1/N_t\right){\displaystyle \sum }_t{\left(x_i^{optfit}\left(t\right)-x_i\left(t\right)\right)}^2}$.

The energy function of the Langevin diffusion model is

$$U=\frac{k_{12}}{2}{\left(x_1-x_2\right)}^2+\frac{k_1}{2}{x}_1^2+\frac{\kappa }{2}{\left[\rho x_1+\left(1-\rho \right)x_2-\omega t\right]}^2$$ [4]

where $x_1$ and $x_2$ represent the rotation angles of the core and the protruded parts of the γ-subunit, respectively, and $\rho =0.2$ is weight factor of the two domains used to determine the average angle (Fig. S3). Note that the driving torque (last term) is a harmonic approximation of the periodic potential used in our atomistic simulations. The three terms represent the harmonic potential between the two domains, the harmonic potential describing the interaction with ring subunits, and the time-dependent torque potential, respectively. The analytical solutions for the time-dependent average angles $\overline{x_1\left(t\right)}$ and $\overline{x_2\left(t\right)}$ are fitted to the corresponding MD trajectories (see SI Text) for models A (assuming quasi-equilibrium for both domains), B (assuming quasi-equilibrium for the protruding domain 2), and C (full nonequilibrium).

The time-dependent nonequilibrium work performed by the external torque is calculated from (38)

$$w_t=\underset{0}{\overset{t}{{\displaystyle \int }}}\frac{\partial U\left(x_1\left(t\prime \right),x_2\left(t\prime \right),t\prime \right)}{\partial t\prime }dt\prime$$ [5]

where U is the time-dependent biasing potential in Eq. 4, through which the torque is exerted. An analogous expression was used to determine the work in the torque MD simulations.

SI Text

Model A: Quasi-Equilibrium Model.

At a relatively slow rotational speed in our torque simulations, angle trajectories are nearly straight lines that pass through the origin and are almost indistinguishable at ω = 1°/ns and 2°/ns. At 1°/ns, rotation is thus slow enough to achieve local equilibrium despite the time-dependent bias on the rotation angle. With the local equilibrium assumption, the average angles of the two domains $\overline{x_1}$, $\overline{x_2}$ satisfy

$${\frac{\partial U}{\partial x_1}|}_{\overline{x_2},t}=0$$

$${\frac{\partial U}{\partial x_2}|}_{\overline{x_1},t}=0.$$

By solving these equations, we get

$$\overline{x_1\left(t\right)}=\frac{k_{12}\kappa }{k_{12}\kappa +k_1\left[k_{12}+\kappa {\left(1-\rho \right)}^2\right]}\omega t$$ [S1]

$$\overline{x_2\left(t\right)}=\frac{\kappa \left[k_{12}+k_1\left(1-\rho \right)\right]}{k_{12}\kappa +k_1\left[k_{12}+\kappa {\left(1-\rho \right)}^2\right]}\omega t.$$ [S2]

Note that these solutions coincide with the linear terms in the solutions of model B. Note further that the twist profiles, scaled by ωt, are independent of time t, as shown in Fig. 1B, which is also consistent with ideal elastic behavior at quasi-equilibrium conditions.

Model B: Rotational Langevin Diffusion Model with Fast Relaxation of Protruded Part.

In model B, we assume that only the protruded part ($x_2$) quickly relaxes, remaining in local equilibrium, whereas the core part ($x_1$) evolves with a finite diffusion coefficient. The noise-averaged diffusion equations for $x_1$ and $x_2$ then become

$$\frac{d\overline{x_1}}{dt}=-D_1{\frac{\partial U}{\partial x_1}|}_{\overline{x_2}}$$

and

$${\frac{\partial U}{\partial x_2}|}_{\overline{x_1}}=0$$

where $U$ is in $k_{B}T$ units, and ${\overline{x}}_i$ represents the noise-averaged rotation angle of the $i$ th domain of the γ-subunit. These two equations lead to a first-order linear differential equation $\left(d\overline{x_1}/dt\right)=a\overline{x_1}+bt$ that can be generally solved as (with $\overline{x_1\left(0\right)}=\overline{x_2\left(0\right)}=0)$

$$\overline{x_1\left(t\right)}=\frac{b}{a^2}\left[e^{at}-at-1\right]$$

and

$$\overline{x_2\left(t\right)}=c\overline{x_1\left(t\right)}+dt$$

where

$$a=-D_1\left[\frac{k_{12}\kappa }{k_{12}+\kappa {\left(1-\rho \right)}^2}+k_1\right],b=D_1\omega \frac{k_{12}\kappa }{k_{12}+\kappa {\left(1-\rho \right)}^2},c=\frac{k_{12}-\left(1-\rho \right)\rho \kappa }{k_{12}+\kappa {\left(1-\rho \right)}^2},d=\frac{\left(1-\rho \right)\kappa \omega }{k_{12}+\kappa {\left(1-\rho \right)}^2}.$$

Model C: General Rotational Diffusion Model in Matrix Form.

If both x₁ and x₂ are out of equilibrium, the noise-averaged diffusion equations can be cast in matrix form. The diffusion equations of $\overline{x_1}$ and $\overline{x_2}$ become

$$\frac{d\overline{x_1}}{dt}=-D_1{\frac{\partial U}{\partial x_1}|}_{\overline{x_2}}$$

$$\frac{d\overline{x_2}}{dt}=-D_2{\frac{\partial U}{\partial x_2}|}_{\overline{x_1}}$$

or, in matrix form,

$$\frac{d\overline{X}}{dt}=A\cdot \overline{X}+B\left(t\right)$$

where $\overline{X}={\left(\begin{array}{cc}\overline{x_1}& \overline{x_2}\end{array}\right)}^T$. The matrix $A$ and vector $B$ are determined by partial derivatives of the harmonic energy function $U$,

$$A=\left(\begin{array}{cc}-D_1\left(k_{12}+k_1+\kappa {\rho }^2\right)& -D_1\left(-k_{12}+\kappa \rho \left(1-\rho \right)\right)\\ -D_2\left(-k_{12}+\kappa \rho \left(1-\rho \right)\right)& -D_2\left(k_{12}+\kappa {\left(1-\rho \right)}^2\right)\end{array}\right)$$

$$B\left(t\right)=\left(\begin{array}{c}{D}_1\kappa \rho \omega t\\ D_2\kappa \left(1-\rho \right)\omega t\end{array}\right).$$

The solution can be written formally as

$$\overline{X\left(t\right)}=e^{At}\cdot \overline{X\left(0\right)}+{\displaystyle \underset{0}{\overset{t}{\int }}ds\hspace{0.17em}{e}^{A\left(t-s\right)}\cdot B\left(s\right)}.$$

With ${\lambda }_{\pm }$ the eigenvalues of matrix $A$,

$${\lambda }_{\pm }=\frac{A_{11}+A_{22}\pm {\left({\left(A_{11}-A_{22}\right)}^2+4A_{12}{A}_{21}\right)}^{1/2}}{2},$$

the matrix exponential can be written as

$$e^{A\left(t-s\right)}=\frac{1}{{\lambda }_{-}-{\lambda }_{+}}\left(\begin{array}{cc}\left({\lambda }_{-}-A_{11}\right)e^{{\lambda }_{+}\left(t-s\right)}-\left({\lambda }_{+}-A_{11}\right)e^{{\lambda }_{-}\left(t-s\right)}& -A_{12}\left(e^{{\lambda }_{+}\left(t-s\right)}-e^{{\lambda }_{-}\left(t-s\right)}\right)\\ -A_{21}\left(e^{{\lambda }_{+}\left(t-s\right)}-e^{{\lambda }_{-}\left(t-s\right)}\right)& -\left({\lambda }_{+}-A_{11}\right)e^{{\lambda }_{+}\left(t-s\right)}+\left({\lambda }_{-}-A_{11}\right)e^{{\lambda }_{-}\left(t-s\right)}\end{array}\right).$$

With new parameters $P_{ij}$ defined through

$$e^{A\left(t-s\right)}\cdot B\left(s\right)=\left(\begin{array}{c}{P}_{11}{e}^{{\lambda }_{+}\left(t-s\right)}s+P_{12}{e}^{{\lambda }_{-}\left(t-s\right)}s\\ P_{21}{e}^{{\lambda }_{+}\left(t-s\right)}s+P_{22}{e}^{{\lambda }_{-}\left(t-s\right)}s\end{array}\right),$$

we obtain

$$\overline{X\left(t\right)}=\left(\begin{array}{c}\frac{P_{11}}{{\lambda }_{+}^2}\left(e^{{\lambda }_{+}t}-{\lambda }_{+}t-1\right)+\frac{P_{12}}{{\lambda }_{-}^2}\left(e^{{\lambda }_{-}t}-{\lambda }_{-}t-1\right)\\ \frac{P_{21}}{{\lambda }_{+}^2}\left(e^{{\lambda }_{+}t}-{\lambda }_{+}t-1\right)+\frac{P_{22}}{{\lambda }_{-}^2}\left(e^{{\lambda }_{-}t}-{\lambda }_{-}t-1\right)\end{array}\right).$$

Validation of Harmonic Approximation of Torque Potential.

We obtained analytic solutions of the Langevin diffusion model by assuming that all potentials are harmonic. However, in the atomistic torque MD simulations, a fully periodic sine and cosine-based torque potential was used to drive the rotation. To check if the harmonic approximation is justified, we generated trajectories of the Langevin diffusion model by numerically solving noise-averaged ordinary differential equations with harmonic and sine and cosine torque potentials, as well as Brownian dynamics simulations described below in Overdamped Langevin Dynamics. We find that the trajectories of the harmonic and anharmonic models are nearly indistinguishable, as shown in Fig. S1. The effective spring constant κ of the harmonic model can be derived directly from the anharmonic model through Taylor expansion of the time-dependent torque potential,

$$\begin{array}{l}U\left(x_1,x_2,t\right)=\frac{\kappa }{2}{\left(\rho \cos x_1+\left(1-\rho \right)\cos x_2-\cos \omega t\right)}^2+\frac{\kappa }{2}{\left(\rho \sin x_1+\left(1-\rho \right)\sin x_2-\sin \omega t\right)}^2\\ =\frac{\kappa }{2}\left[{\left\{\rho \cos \left(x_1-\omega t\right)+\left(1-\rho \right)\cos \left(x_2-\omega t\right)-1\right\}}^2+{\left\{\rho \sin \left(x_1-\omega t\right)+\left(1-\rho \right)\sin \left(x_2-\omega t\right)\right\}}^2\right]\\ \approx \frac{\kappa }{2}{\left[\rho x_1+\left(1-\rho \right)x_2-\omega t\right]}^2.\end{array}$$

Overdamped Langevin Dynamics.

The two rotation angles are assumed to evolve according to overdamped Langevin dynamics,

$$\frac{d{x}_i}{dt}=-\frac{D_i}{k_{B}T}\frac{\partial U}{\partial x_i}+R_i$$

$$\langle R_i\left(t^{\prime }\right)R_j\left(t\right)\rangle =2D_i{\delta }_{ij}\delta \left(t-t^{\prime }\right),$$

where $R_i$ is the random force. For the harmonic case, the corresponding Green’s functions can be solved analytically. The noise-averaged equations for the mean positions $\overline{x_i\left(t\right)}$ are then ordinary differential equations of the same form but without the random force. For the anharmonic potentials, trajectories can be generated numerically by applying the forward-Euler integrator,

$$x_i\left(t+h\right)=x_i\left(t\right)-\frac{D_i}{k_{B}T}\frac{\partial U}{\partial x_i}h+d_i\left(t\right),$$

where h is the time step, and the $d_i\left(t\right)$ are Gaussian random numbers with zero mean and SD $\sqrt{2D_{i}h}$.

Weight Factor of the Domains for Averaging Angle.

In the torque simulations, we bias the average angle of the γ-subunit (or its sine and cosine). The averages are calculated with weights determined by the residue normal distances from the respective axis. To determine the weight factor ρ in the Langevin diffusion model, the weights for the core and protruded parts of the γ-subunit were calculated for the atomistic torque trajectories as

$$\rho \left(x_i\right)=\frac{{{\displaystyle \sum }}_{j\in x_i}{r}_j}{{{\displaystyle \sum }}_{j\in all}{r}_j}$$

where $r_j$ is distance of each residue (C_α atom) in the γ-subunit from the respective rotation axis. As shown in Fig. S2, these exact weights are basically constant during the trajectory. We thus take the average $\rho =\rho \left(x_1\right)=0.2$.

Kinetic Model for Coupled F_oF₁ Rotation.

We developed a kinetic model from the energetic description of the coupled rotation on the basis of the angle-dependent energy function Eq. 3. For the master equation $\left(d/dt\right)\overrightarrow{p}\left(t\right)=K\overrightarrow{p}\left(t\right)$, the rate matrix K was constructed in the following way. Only transitions between states within one rotatory step of either F_o or F₁ are considered, including also simultaneous steps of both F_o and F₁. We define states i by their discrete rotatory angles, $({\varphi }_{oi},{\varphi }_{1i})$. To ensure detailed balance, we use a linear free energy relation to define the transition rate from state i to state j as $k_{i\to j}=k_0\text{exp}\left[-\left(G_j-G_i\right)/2k_{B}T\right]$, where $k_0$ is a preexponential factor that sets the intrinsic transition rate in the absence of any driving force, and $G_i$ is the free energy of rotatory state i as given by Eq. 3. Thus, the elements of the rate matrix K become

$$K_{ij}=\{\begin{array}{c}{k}_{j\to i}\hspace{0.17em}\left(i\ne j\right)\\ -{\displaystyle {\sum }_{l\ne j}{k}_{j\to l}\left(i=j\right)}\end{array}.$$

For this rate matrix, we computed the mean first passage times from any state to the end state N with rotatory angles $\left({\varphi }_{oN}=360°,{\varphi }_{1N}=360°\right)$ by solving the following equation (39):

$$K{\prime }^T\left(\begin{array}{c}{\tau }_1\\ ⋮\\ {\tau }_{N-1}\end{array}\right)=\left(\begin{array}{c}-1\\ ⋮\\ -1\end{array}\right)$$

where $K\text{'}$ is an (N − 1) × (N − 1) matrix constructed by deleting row N and column N of the rate matrix K. In Table S2, we report the mean first passage times for a full cycle, starting from $\left({\varphi }_{o1}=0°,{\varphi }_{11}=0°\right)$ and ending at $\left({\varphi }_{oN}=360°,{\varphi }_{1N}=360°\right)$.

To estimate the time scale of the prefactor (1/k₀), we assume that proton binding to the conserved Glu/Asp in the c subunit is the rate-limiting elementary step (40). For simplicity, we use a uniform k₀ for all transitions. By assuming a diffusion-limited reaction and accounting for surface charge effects (40), we estimate the rate for proton binding from the acidic side of the membrane as $k_{in}=k_D\left[H^{+}\right]\cdot \exp \left(\mathrm{\Delta }\varphi /k_{B}T\right)$. With parameters taken from earlier studies (40, 41), $k_D=4rD=4\hspace{0.17em}\times \hspace{0.17em}0.5\hspace{0.17em}\text{nm}\hspace{0.17em}\times \hspace{0.17em}9.3\times {10}^9{\hspace{0.17em}\text{nm}}^2\text{/s}\hspace{0.17em}=1.86\times {10}^{10}\hspace{0.17em}{\text{nm}}^3\text{/s}$, $\left[H^{+}\right]={10}^{-pH}\cdot 0.6\hspace{0.17em}{\text{nm}}^{-3}$, $\mathrm{\Delta }\varphi =2.3k_{B}T$, and pH 7 for the intermembrane space of mitochondria, the rate becomes $k_{in}=1.1\times {10}^4\hspace{0.17em}{\text{s}}^{-1}$, which gives a time scale of 1/k₀∼1/k_in∼100 μs ∼100 μs. For the comparison with the single-molecule experiments of E. coli F_oF₁, we use the above formula without the surface charge effect, because the fluorescent dye used to determine pH preferentially localizes on the bilayer surface, reporting the surface pH directly (4). Thus, with pH 5.5 on the acidic side of membrane, it gives a time scale of proton binding of 1/k_in∼30 μs.