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# Elasticity, friction, and pathway of γ-subunit rotation in F_{o}F_{1}-ATP synthase

Edited by Ken A. Dill, Stony Brook University, Stony Brook, NY, and approved July 16, 2015 (received for review January 12, 2015)

## Significance

F_{o}F_{1}-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 F_{1} motor. To explain its high efficiency, determine its top speed, and characterize its mechanism, we construct a viscoelastic model of the F_{1} 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 F_{1} motors, respectively. The resulting energetic constraints dictate a unique pathway for the coupled rotations of the F_{o} and F_{1} 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_{o}F_{1}-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 F_{1}-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*_{B}*T* 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 F_{1} rotary motors in ATP synthase, and explain the need for the finer stepping of the F_{1} motor in the mammalian system, as seen in recent experiments. Compensating for incommensurate eightfold and threefold rotational symmetries in F_{o} and F_{1}, 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_{o}F_{1}-ATP synthase is essential for life. From bacteria to human, this protein synthesizes ATP from ADP and inorganic phosphate P_{i} in its F_{1} domain, powered by an electrochemical proton gradient that drives the rotation of its membrane-embedded F_{o} domain (1⇓⇓⇓–5). Its two rotary motors, F_{1} 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 F_{1} motor pumps protons against an electrochemical gradient across the membrane-embedded F_{o} part, converting ATP to ADP and P_{i} (7, 8).

F_{1} 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 F_{1} 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 F_{1} 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 c_{8} 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 c_{10}–c_{15} rings (13, 14). The resulting symmetry mismatches between F_{1} 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 F_{1} 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 F_{1} motor (18⇓⇓⇓⇓⇓–24). Here we explore these questions by building a dissipative mechanical model of the F_{1} 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_{o}F_{1} nanomotors (15, 25, 26). For γ-subunits cross-linked with the α_{3}β_{3}-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 F_{1} 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 F_{1} 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 F_{1} motor. The general framework developed here for F_{1} should be applicable also to other molecular machines.

## Results

### Two-Domain Langevin Model of γ-Rotation in F_{1}-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. 1*A*). 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 *A*).

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. 1*B*), 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 *k*_{12} (see Eq. **4**). Domain 1 is coupled harmonically with spring constant *k*_{1} to the static α_{3}β_{3}-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 *D*_{1} and *D*_{2}. 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 α_{3}β_{3}-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. 1*B* 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 *k*_{12}. By contrast, our torsional spring *k*_{1} 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_{o}F_{1}-ATP synthase will be determined also by its interactions with the c ring and the ε- and δ-subunits (Fig. S1).

To determine the parameters *k*_{1}, *k*_{12}, *D*_{1}, and *D*_{2} of the Langevin diffusion model, we fitted analytical solutions (see *SI Text* and Figs. S2 and S3) for the time-dependent average rotation angles *Methods*). In a first step, we quantified the elastic properties at a low angular velocity of *A*). Correspondingly, the twist profiles at different time points, rescaled by the target angle *ω*t, almost perfectly coincide (see Fig. 1*B* and Eqs. **S1** and **S2**). At rotational speeds exceeding *A*) 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 *B* show the nonlinear correlations between the fit parameters, permitting variations in *B*). Whereas, at low rotational speeds, *B*). To test the assumption of quasi-equilibrium at *C*, both the time-dependent twisting profiles and the

### 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 *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).

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 ^{1}⋅rad^{−2} and ^{2}/ns capture all trajectories and work profiles well (orange lines in Fig. 2*A*; 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 *ωt* remains almost independent of the rotational speed. However, their relative contributions differ as the rotational speed increases. At higher speeds, *k*_{12} becomes increasingly stretched, absorbing relatively more of the elastic energy than *k*_{1}.

## 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 ^{2} 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. 1*B*, this segment closely matches the ^{2}. The agreement between our model and the experiment is thus excellent, considering our estimated fit uncertainties of about 30%. Our estimate for *k*_{12} also agrees with the 620 pN⋅nm/rad^{2} 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 ^{2} for the *Bacillus PS3* γ-subunit up to a single cross-link at a position closely matching the full twist *B*). The corresponding effective spring constant in our model is ^{2}, deviating from the measurement by less than the combined (^{2}) SEs of model and experiment. In effect, we could thus independently validate the two spring constants *k*_{1} and *k*_{12} 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 *k*_{12} and the other probing

### Hydrodynamics and Rotational Friction.

For the diffusion coefficients of the core part, ^{2}/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}. The γ-subunit rotating inside F_{1} 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} 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}–F_{1} Symmetry Mismatch.

The calculated elastic properties allow us to construct a detailed pathway for the coupled rotary motion of the F_{o} and F_{1} motors in mammalian ATP synthase. A step size of _{o}, and substeps of _{1} on the basis of recent single-molecule experiments on human mitochondrial F_{1} (31), with the cost of ATP spread proportionally among them. The free energy of the system for the discrete (sub)steps then becomes_{o} and F_{1} subunits in the ATP synthesis direction, respectively.

This free energy function severely restricts the possible pathways of the coupled F_{o}–F_{1} 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. 4*B* indicates all intermediates that exceed this limit shaded in red, which leaves only a single pathway for the coupled rotary motion through the _{o} and F_{1} motors (Fig. 4*B* and Fig. S5*A*), and for uncertainties of _{1} substep sizes (Fig. 4*D* and Fig. S5*B*; F_{o} steps as *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}–F_{1} 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 F_{1} substeps. Assuming an elementary rate of *k*_{0} ≈ 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 c_{8} ring of mammalian F_{1}, using only eight protons per three ATP.

The symmetry mismatch between the mammalian F_{o} and F_{1} 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 _{1} motor are critical for its operation. With the *C* and Fig. S5*C*).

We also extended the model to bacterial systems with c_{10} rings, such as *E. coli* (4) or thermophilic *Bacillus PS3* (15) (Fig. S7). In this case, the F_{o} step size is *E. coli* F_{o}F_{1} experiments (4), we estimate a faster elementary step of *1/k*_{0} ≈ 30 μs (see *SI Text*). Assuming the same elasticity of the γ-subunit (k_{eff} = 66.0 kcal⋅mol^{−1}⋅rad^{−2}), F_{1} 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* F_{1} (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* F_{1}, 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 F_{1} motors would operate on time scales of ∼80 ms and ∼4 ms, respectively (Table S2). With this softer elasticity, the two-substep F_{1} 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 c_{10} 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 *n* = 10 of the bacterial c_{10} 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 ^{−1}⋅rad^{−2}. Putting this torsional spring in series with the springs *k*_{1} and *k*_{12} determined here, one would find an effective sprint constant of ^{−1}⋅rad^{−2}, at least half of our value for the γ-subunit and α_{3}β_{3}-ring alone.

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

### Maximum Rotational Speed.

The estimated rotational diffusion coefficients, or friction coefficients from the relation

## Conclusions

We developed a multiscale framework to deduce mechanical and frictional properties of biomolecular machines from atomistic MD simulations, and applied it to the F_{1} motor of ATP synthase. By constructing a Langevin diffusion model of the γ-rotation in F_{1}-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 F_{1} 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 *k*_{B}*T* per rotation at an average speed of _{1} rotary motor thus operates at very small dissipation levels, which is essential for the high overall thermodynamic efficiency of F_{o}F_{1}-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 F_{1}. 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

The energy function of the Langevin diffusion model is

where *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)

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 *ω*t, are independent of time *t*, as shown in Fig. 1*B*, 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 (

### Model C: General Rotational Diffusion Model in Matrix Form.

If both *x*_{1} and *x*_{2} are out of equilibrium, the noise-averaged diffusion equations can be cast in matrix form. The diffusion equations of

With new parameters

### 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,

### Overdamped Langevin Dynamics.

The two rotation angles are assumed to evolve according to overdamped Langevin dynamics,*h* is the time step, and the

### 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_{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

### Kinetic Model for Coupled F_{o}F_{1} 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 *K* was constructed in the following way. Only transitions between states within one rotatory step of either F_{o} or F_{1} are considered, including also simultaneous steps of both F_{o} and F_{1}. We define states *i* by their discrete rotatory angles, *i* to state *j* as *i* as given by Eq. **3**. Thus, the elements of the rate matrix *K* become*N* with rotatory angles *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

To estimate the time scale of the prefactor (*1/k*_{0}), 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 *k*_{0} 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*_{0}∼1/*k*_{in}∼100 μs ∼100 μs. For the comparison with the single-molecule experiments of *E. coli* F_{o}F_{1}, 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.

## Acknowledgments

We thank Profs. Wolfgang Junge, Martin Karplus, John Walker, Arieh Warshel, and Rikiya Watanabe for discussions. This work was supported by the Max Planck Society (K.O. and G.H.) and a postdoctoral fellowship for research abroad from the Japan Society for the Promotion of Science (K.O.).

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: gerhard.hummer{at}biophys.mpg.de.

Author contributions: K.O. and G.H. designed research; K.O. performed research; K.O. and G.H. analyzed data; and K.O. and G.H. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1500691112/-/DCSupplemental.

Freely available online through the PNAS open access option.

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