Elasticity, friction, and pathway of γ-subunit rotationin FoF1-ATP synthaseKei-ichi Okazaki and Gerhard Hummer1

Department of Theoretical Biophysics, Max Planck Institute of Biophysics, D-60438 Frankfurt am Main, Germany

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

We combine molecular simulations and mechanical modeling toexplore the mechanism of energy conversion in the coupled rotarymotors of FoF1-ATP synthase. A torsional viscoelastic model withfrictional dissipation quantitatively reproduces the dynamics and en-ergetics seen in atomistic molecular dynamics simulations of torque-driven γ-subunit rotation in the F1-ATPase rotary motor. The torsionalelastic coefficients determined from the simulations agree with re-sults from independent single-molecule experiments probing differ-ent segments of the γ-subunit, which resolves a long-lastingcontroversy. At steady rotational speeds of ∼1 kHz correspondingto experimental turnover, the calculated frictional dissipation of lessthan kBT per rotation is consistent with the high thermodynamicefficiency of the fully reversible motor. Without load, the maximumrotational speed during transitions between dwells is reached at ∼1MHz. Energetic constraints dictate a unique pathway for the cou-pled rotations of the Fo and F1 rotary motors in ATP synthase, andexplain the need for the finer stepping of the F1 motor in the mam-malian system, as seen in recent experiments. Compensating forincommensurate eightfold and threefold rotational symmetries inFo and F1, respectively, a significant fraction of the external mechan-ical work is transiently stored as elastic energy in the γ-subunit. Thegeneral framework developed here should be applicable to othermolecular machines.

bioenergetics | molecular motor | mechanochemical coupling

FoF1-ATP synthase is essential for life. From bacteria to hu-man, this protein synthesizes ATP from ADP and inorganic

phosphate Pi in its F1 domain, powered by an electrochemicalproton gradient that drives the rotation of its membrane-embeddedFo domain (1–5). Its two rotary motors, F1 and Fo, are coupledthrough the γ-subunit forming their central shaft (2). ATP synthaseis a fully reversible motor, in which the rotational direction switchesaccording to different sources of energy (2, 6). In hydrolysis mode,the F1 motor pumps protons against an electrochemical gradientacross the membrane-embedded Fo part, converting ATP to ADPand Pi (7, 8).F1 has a symmetric ring structure composed of three αβ-subunits

with the asymmetric γ-subunit sitting inside the ring (9, 10). Eachαβ-subunit has a catalytic site located at the αβ-domain interface.The F1 ring has a pseudothreefold symmetry with the threeαβ-subunits taking three different conformations, E (empty), TP(ATP-bound), and DP (ADP bound) (9–11). The Fo part iscomposed of a c ring and an a subunit (3, 12). Driven by protonspassing through the interface of the c ring and the a subunit, thec ring rotates together with the γ-subunit (rotor) relative to the asubunit, which is connected to the F1 ring through the peripheralstalk of the b subunit (stator) (12). Interestingly, in nature, onefinds a large variation in the number of subunits in the c ring.In animal mitochondria, one finds c8 rings, requiring a minimalnumber of eight proton translocations for the synthesis of threeATP, at least 20% fewer protons than in bacteria and plant chlo-roplasts with c10–c15 rings (13, 14). The resulting symmetry mis-matches between F1 and Fo (15–17) clearly distinguish the bio-molecular 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 Fo and F1 motors on the motormechanics, the resulting need for transient energy storage, the roleof frictional dissipation, and the molecular elements associatedwith stepping of the F1 motor (18–24). Here we explore thesequestions by building a dissipative mechanical model of the F1

motor on the basis of atomistic molecular dynamics (MD) simu-lations. Friction and torsional elasticity of the γ-subunit are centralto the efficient function of the coupled FoF1 nanomotors (15, 25,26). For γ-subunits cross-linked with the α3β3-ring, estimates havebeen obtained by monitoring thermal angle fluctuations in single-molecule experiments (16, 27) and MD simulations (28). To probethe elastic and frictional properties under mechanical load overbroad ranges of rotation angles and angular velocities, we inducetorque-driven γ-subunit rotation in MD simulations (20, 29). Fromthe resulting mechanical deformation and energy dissipation, weconstruct a fully quantitative viscoelastic model. We account forthe torsional elasticity and friction by describing the rotationalmotion of the γ-subunit as overdamped Langevin dynamics on a2D harmonic free energy surface. The model quantifies the mag-nitude of transient elastic energy storage compensating for theincommensurate rotational symmetries of the Fo and F1 motors(30). The resulting energetic constraints allow us to map out adetailed pathway for their coupled rotary motions, and to ratio-nalize the finer stepping of the mammalian F1 motor seen in recentexperiments (31), with only eight c subunits in the correspondingFo motor. By quantifying the frictional dissipation, we identify a keycontributor to the high thermodynamic efficiency of the F1 motor.The general framework developed here for F1 should be applicablealso to other molecular machines.

Significance

FoF1-ATP synthase produces the ATP essential for cellular func-tions from bacteria to humans. Rotation of its central γ-subunitcouples proton translocation in the membrane-embedded Fomotor to ATP synthesis in the catalytic F1 motor. To explain itshigh efficiency, determine its top speed, and characterize itsmechanism, we construct a viscoelastic model of the F1 rotarymotor from molecular dynamics simulation trajectories. We findthat the γ-subunit is just flexible enough to compensate for theincommensurate eightfold and threefold rotational symmetriesof mammalian Fo and F1 motors, respectively. The resulting en-ergetic constraints dictate a unique pathway for the coupledrotations of the Fo and F1 rotary motors, and explain the finestepping seen in single-molecule experiments.

Author contributions: K.O. and G.H. designed research; K.O. performed research; K.O. andG.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.

Freely available online through the PNAS open access option.1To whom correspondence should be addressed. Email: gerhard.hummer@biophys.mpg.de.

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

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ResultsTwo-Domain Langevin Model of γ-Rotation in F1-ATPase. To constructa Langevin diffusion model, the γ-subunit was coarse-grained intoa collection of domains that are coupled by torsional springs androtate in different frictional environments (Fig. 1A). To determinethe minimal number of domains, we examined the spatial extentof coupled rotary motions along the γ-subunit axis. Using theflexible rotor method (20), we applied torque on the γ-subunit inatomistic MD simulations. A relatively weak harmonic restraintof κ= 2,000  kcal ·mol−1 · rad−2 ≈ 0.61  kcal ·mol−1 · deg−2 on the av-erage angle at angular velocities of ω= 1− 10°=ns resulted in agradual buildup of tension and permitted γ-subunit twisting andsubstantial angle fluctuations. In the torque MD simulations,we probed rotary motions of up to 55° in the protruded part ofthe γ-subunit and 20° at its core (Fig. 1A).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 twomain parts. This division is consistent with the structural featuresof the γ-subunit, which consists of a protruded globular part and acoiled-coil helical domain inside the hexamer ring. Our minimalmodel of the γ-subunit rotation near the catalytic dwell thusconsists of domains 1 and 2 connected by a torsional spring withspring constant k12 (see Eq. 4). Domain 1 is coupled harmonicallywith spring constant k1 to the static α3β3-ring subunits. Externaltorque is exerted by a torsional spring κ rotating at an angularvelocity ω. The different viscous environments of the two domainsare described by rotational diffusion coefficients D1 and D2. Ourtwo-domain model can be seen as a further reduction of the nine-segment model of Czub and Grubmüller (28), which divides theγ-subunit into five segments and the α3β3-ring into four. As shownin 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 theother half of segment 4. According to the γ-subunit twist profile(Fig. 1B and Fig. S1), the elastic deformation is concentratedwithin their segment 4. In our two-domain model, we treat thislinearly twisted region by the single torsional spring k12. By con-trast, our torsional spring k1 accounts for the interactions betweensegments 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 (seg-ment 1), whose elastic response in FoF1-ATP synthase will bedetermined also by its interactions with the c ring and the e- andδ-subunits (Fig. S1).To determine the parameters k1, k12, D1, and D2 of the Langevin

diffusion model, we fitted analytical solutions (see SI Text andFigs. S2 and S3) for the time-dependent average rotation anglesx1ðtÞ of the core helix domain (residues γ1–26, γ228–273) andx2ðtÞ of the protruded domain (residues γ27–227) to corre-sponding trajectories from atomistic MD simulations (Fig. 2).The MD rotational angles were calculated as distance-weightedaverages over the angles of individual residues using separateaxes for the two domains (see Methods). In a first step, wequantified the elastic properties at a low angular velocity of 1°=ns,at which the time-dependent rotation angles x1 and x2 could befitted with a quasi-equilibrium model (model A; Fig. 2A). Corre-spondingly, the twist profiles at different time points, rescaled bythe target angle ωt, almost perfectly coincide (see Fig. 1B and Eqs.S1 and S2). At rotational speeds exceeding 2°=ns, dissipative ef-fects 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 ofmodel B (assuming quasi-equilibrium for domain 2), and model C(with both domains out of equilibrium) (Table S1). In a secondstep, we performed a global analysis to determine parameterranges that fit trajectories over the entire range of angular ve-locities. Plots of χ2 in Fig. 2B show the nonlinear correlationsbetween the fit parameters, permitting variations in k12 and k1 by∼30% around the optimal values ðk12, k1Þ= ð123, 309Þ (in kilo-calories per mole per square radian) (Fig. 2B). Whereas, at lowrotational speeds,D1 andD2 are only bounded from below, data athigher speeds provide also upper limits. The overlap of the four χ2maps for model C delimits the range of D1 and D2 that fit alltrajectories well (Fig. 2B). To test the assumption of quasi-equi-librium at 1°=ns, we ran additional simulations at lower rota-tional speeds of 0.2°=ns  and 0.5°=ns. As shown in Fig. 2C, boththe time-dependent twisting profiles and the χ2 maps for k12and k1 are fully consistent with the results obtained at thehigher speeds.

Work Profile as Additional Information. The combined elastic anddissipative work performed during the torque-driven rotationsprovides us with largely independent information that we can usenot only to validate the Langevin diffusion model and fine-tuneits parameters, but also to quantify the energetics of the γ-rotation.The time-dependent work performed by the external torque iscalculated as in Eq. 5. Fig. 3 shows the work profiles as a functionof the reference rotation angle obtained from actual atomistictorque MD simulations. The work profiles at the lowest rota-tional speeds of ω= 1  and 2°=ns show little variation (blue andcyan points in Fig. 3), indicating that, at these speeds, rotationindeed occurs at near-equilibrium conditions. However, the ac-cumulated work at a given target angle ωt increases as the ro-tational speed becomes faster and the system is driven out ofequilibrium. For comparison, dashed lines in Fig. 3 show thework profiles calculated for the optimized Langevin diffusionmodel. Whereas the overall tendency of the work profiles of thediffusion model is consistent with the results of the atomisticsimulations, 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 bothwork and angle-vs.-time trajectories. From a global analysis,we find that a small increase in D2 and small decreases of k12and k1, well within the tolerance of the fits in Fig. 2, produceagreement with the entire set of data from atomistic simula-tions (Fig. 3). The optimized parameters k12 = 94 and k1 = 223kcal·mol−1·rad−2 and D1 = 0.001 and D2 = 0.003 rad2/ns capture

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Fig. 1. Langevin diffusion model of γ-rotation in F1-ATPase. (A) Coarsegraining of the γ-subunit into two domains connected by torsional springs andembedded in different viscous environments. (B) Local rotation angles of theγ-subunit along axis in torque MD simulation, scaled by the target angle ωt forω= 1°/ns. The profiles are shown for times t = 35 ns, 40 ns, and 50 ns, averagedover three independent trajectories. The near-perfect superposition of theprofiles is consistent with quasi-equilibrium conditions in a linearly elasticmodel. Dashed lines show the optimized quasi-equilibrium solution of thetwo-domain viscoelastic model. Arrows indicate crosslinks in the single-mole-cule experiments of Sielaff et al. (16) and Okuno et al. (27). Residues γC87 andγA270 in E. coli correspond to γC78 and γA256, respectively, in bovine mito-chondria, and residue γR84 in Bacillus PS3 corresponds to γR75.

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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 recover-able elastic energy and dissipation (Fig. S4). At a rotational speedof 1°=ns, nearly the entire exerted work is stored as elastic energy.This rotational speed is 10,000 times faster than the average ro-tational speed of ω≈ 100°/ms at turnover that is dominated by thedwell times at intermediate states. At speeds exceeding 2°/ns,dissipation becomes relevant. Remarkably, the sum of the intrinsicelastic energy of γ (green line in Fig. S4) and the elastic potentialacting on x1 (red line) as a function of ωt remains almost in-dependent of the rotational speed. However, their relativecontributions differ as the rotational speed increases. At higherspeeds, x1 lags behind due to its higher friction. As a result, thespring k12 becomes increasingly stretched, absorbing relativelymore of the elastic energy than k1.

Discussionγ-Subunit Elasticity. We determined the torsional elasticity androtational Langevin diffusion coefficients by optimizing a two-domain rotational Langevin diffusion model against the atomistic

torque trajectories at different rotational speeds and the cor-responding nonequilibrium work profiles. To assess the elasticityparameters of the γ-rotation, we compared the results from ourviscoelastic model to the values from single-molecule experiments.Sielaff et al. (16) determined an effective elastic coefficient of about750 pN·nm/rad2 for the Escherichia coli γ-subunit between twocrosslinks at residues γA270 and γA87, albeit with a significantuncertainty because of their difference measurement. As indicatedin Fig. 1B, this segment closely matches the x2 − x1 torsional twist inour model, for which we have determined an effective spring con-stant of k12 ≈ 660 pN·nm/rad2. The agreement between our modeland the experiment is thus excellent, considering our estimated fituncertainties of about 30%. Our estimate for k12 also agrees with the620 pN·nm/rad2 obtained in a previous simulation study for a similarsegment of a γ-subunit covalently attached to the hexamer ring (28).Using a similar approach, Okuno et al. (27) estimated an elasticcoefficient of 223± 141 pN·nm/rad2 for the Bacillus PS3 γ-subunit upto a single cross-link at a position closely matching the full twist x2 ofour model, and moving with the protruding part of the γ-subunit(Fig. 1B). The corresponding effective spring constant in our modelis keff = ð1=k1 + 1=k12Þ−1 ≈ 460 pN·nm/rad2, deviating from themeasurement by less than the combined (≈ 2× 140 pN·nm/rad2)

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Fig. 2. Parameterization of the Langevin diffusion model via all-atom torque MD simulations. (A) Angle trajectories for the core (x1, red) and protruded domain(x2, green) from torqueMD simulations at angular velocities of ω = 1–10°/ns (left to right). Fits of the quasi-equilibriummodel A and the overall optimal rotationaldiffusion model C (k12 = 94 and k1 = 223 kcal/mol/rad2, and D1 = 0.001 and D2 = 0.003 rad2/ns) are shown in black and orange, respectively. (B) Fit quality χ2 ofquasi-equilibriummodel A as a function of k12 and k1 at ω= 1°=ns (Top Left). The χ2ω of model C as a function ofD1 and D2 at the different rotational speeds ω, andoverall (right; ðχ21 + χ22 + χ25 + χ210Þ=4) are shown in the other panels. The blue and yellow points in B indicate the best local and global fits, respectively, with thelatter 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) Angletrajectories of domains 1 and 2 are shown in red and green for ω = 0.2°/ns (bottom scale), and inmagenta and cyan for ω = 0.5°/ns (top scale), respectively. (Middleand Bottom) Fit quality χ2 of quasi-equilibrium model A as a function of k12 and k1 at ω = 0.2°/ns and 0.5°/ns. Yellow squares indicate the global optimumdetermined at ω ≥ 1°/ns.

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SEs of model and experiment. In effect, we could thus inde-pendently validate the two spring constants k1 and k12 in ourmodel, because they were probed in separate measurements.Moreover, our results resolve the long-standing controversy con-cerning the disagreement between the two measurements, with oneprobing k12 and the other probing keff .

Hydrodynamics and Rotational Friction. For the diffusion coefficientsof the core part, D1, we obtained ∼0.001 rad2/ns from the angletrajectories at different rotational speeds and the work profiles.This value has the same order of magnitude as previous estimatesfrom autocorrelation functions of MD trajectories (28). To get asense of the relevant scale, we estimated a hydrodynamic rota-tional diffusion coefficient by approximating the core part of theγ-subunit as a cylinder (32),

D1,hydro = kBT�4πηLa2 [1]

where η= 7× 10�4   kg=m · s is the viscosity of water at 310 K,L≈ 7  nm is the height of the cylinder, and a≈ 1  nm is the radiusof the cylinder. The resultingD1,  hydro ≈ 0.07  rad2=ns is ∼70D1. Theγ-subunit rotating inside F1 thus experiences significantly higherfriction than in solution, reflecting the tight hydrophobic and stericinteractions with the ring subunits. For the diffusion coefficientof the protruded part, D2, we obtained ∼ 0.003  rad2=ns. Approxi-mated as a sphere of radius a≈ 2  nm, we use (32, 33)

D2,hydro = kBT�8πηa3 [2]

to estimate a hydrodynamic rotational diffusion of D2,  hydro ≈0.03  rad2=ns  . D2 is thus ∼10 times smaller than the hydrody-namic limit, which is reasonable because the protruded part isslowed down by interactions, mainly salt bridges, with the tips ofthe αβ-subunits.

Mechanical Consequence of Fo–F1 Symmetry Mismatch. The calcu-lated elastic properties allow us to construct a detailed pathway forthe coupled rotary motion of the Fo and F1 motors in mammalianATP synthase. A step size of 360°=8= 45° is assumed for Fo, andsubsteps of ∼ 25°+ ∼ 30°+ ∼ 65°= 120° are assumed for F1 on thebasis of recent single-molecule experiments on human mito-chondrial F1 (31), with the cost of ATP spread proportionally

among them. The free energy of the system for the discrete (sub)steps then becomes

GðϕFo,ϕF1Þ= ðϕFo=45°ÞGin + ðϕF1=120°ÞGout +keff2ðϕFo −ϕF1Þ2

[3]

where ϕFo and ϕF1 are the rotation angles of the Fo and F1 subunitsin the ATP synthesis direction, respectively. Gin =−200 mV≈−4.6  kcal=mol is the input energy per one proton translocationand Gout ≈ 12.3  kcal=mol is the output energy per one ATP syn-thesis (6) at maximum load conditions, 8Gin = 3Gout, for the mam-malian ATP synthase. Note that GðϕFo,ϕF1Þ accounts only for theenergy of metastable intermediate states, not the energy barriersbetween them.This free energy function severely restricts the possible pathways

of the coupled Fo–F1 rotary motion. To ensure significant activity,we require that, during a full cycle, the free energy difference be-tween the least and most populated intermediates should not ex-ceed 5–7 kcal/mol, an upper limit supported by the kinetic analysisdiscussed below. Fig. 4B indicates all intermediates that exceed thislimit shaded in red, which leaves only a single pathway for thecoupled rotary motion through the ϕFo-ϕF1 plane. Here weaccounted for the unknown relative phase ΔϕFo of the Fo and F1motors (Fig. 4B and Fig. S5A), and for uncertainties of ±3° inthe F1 substep sizes (Fig. 4D and Fig. S5B; Fo steps as ϕFo =ΔϕFo + n× 360°=8 with n integer). To quantify the effects of sub-unit asymmetry on the kinetics of ATP synthesis, we developed amaster equation model of the coupled Fo–F1 rotation on the basisof the angle-dependent energy function Eq. 3 (see SI Text for de-tails). The resulting mean times for full rotation of the two motorsdepend strongly on the number of F1 substeps. Assuming an ele-mentary rate of k0 ≈ 1/(100 μs) set by proton binding to Fo, ATPwould be produced at a rate of ∼0.6/s and 30/s with two and threesubsteps, respectively (Fig. S6 and Table S2). Although these es-timates are clearly rough, they nonetheless imply that having threesubsteps ensures an efficient operation with the small c8 ring ofmammalian F1, using only eight protons per three ATP.The symmetry mismatch between the mammalian Fo and F1

rotary motors thus has important consequences. First, a significantamount of energy must be stored in the γ-subunit (30), whichappears to be just soft enough to absorb sufficient elastic energy,but not too soft to allow for slipping and alternate pathways in theϕFo–ϕF1 plane. Second, the additional substeps seen in recentsingle-molecule experiments on the mammalian F1 motor arecritical for its operation. With the ∼ 40°+ ∼ 80°= 120° steps of thebacterial 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). Inthis case, the Fo step size is 360°=10= 36°. Under the conditionsof E. coli FoF1 experiments (4), we estimate a faster elementarystep of 1/k0 ≈ 30 μs (see SI Text). Assuming the same elasticity ofthe γ-subunit (keff = 66.0 kcal·mol−1·rad−2), F1 motors operatingwith two and three substeps require ∼20 s and ∼40 ms for full ro-tation, respectively (Table S2). The time scale with the three sub-steps (∼40 ms) agrees well with the experimental values (∼50 ms;∼60 s−1 for a 120° step) at a maximum load condition (4). Thisresult suggests a possible third substep, yet unresolved, for theE. coli F1 (34, 35). This third step could be associated with phos-phate release (31, 36). Alternatively, the bacterial γ-subunit or itsinterface with the c ring could be softer, especially for the ther-mophilic Bacillus PS3 F1, in which only two substeps have beenfound after extensive single-molecule experiments. With the stiff-ness reduced by half in our model (keff = 33.0 kcal·mol−1·rad−2),two- and three-substep F1 motors would operate on time scalesof ∼80 ms and ∼4 ms, respectively (Table S2). With this softer

Fig. 3. Nonequilibrium work profiles wt from torque MD simulations (seeMethods). The mechanical work performed by the external torque in MDsimulations at various rotational speeds (points with different colors for thedifferent speeds; see Inset showing time-dependent work) is compared tothe corresponding work calculated from the elastic model with optimizedparameters (dashed lines with the same coloring scheme). The work isplotted as a function of the target angle ωt during the torque MD simula-tions. Work profiles from the elastic model with globally optimal set ofparameters are shown as solid lines.

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elasticity, the two-substep F1 could thus achieve a reasonable ATPsynthesis rate. However, the interface between γ-subunit and cring cannot be much softer because their angle-dependent in-teraction must have 10-fold symmetry for a c10 ring. If we assumethat an elastic potential with an interfacial spring constant kγcholds in each of the n minima all of the way to a cusp-shapedbarrier, the n-fold periodic free energy surface will have barriers ofheight kγcðπ=nÞ2/2. For n = 10 of the bacterial c10 rings and aminimum barrier height of at least 5 kcal/mol to prevent slippingof the c ring against the γ-subunit on a millisecond timescale, theinterfacial spring constant kγc should exceed 100 kcal·mol−1·rad−2.Putting this torsional spring in series with the springs k1 and k12determined here, one would find an effective sprint constant ofkeff = ðk−11 + k−112 + k−1γc Þ−1 > 40 kcal·mol−1·rad−2, at least half of ourvalue for the γ-subunit and α3β3-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-drivenproton pump function of the latter imposes different requirementson the mechanochemical coupling.

Dissipation at Average Rotational Speed. From the rotational diffu-sion coefficients D1 and D2, we can estimate the frictional dissi-pation per full cycle. Here we implicitly assume, first, that therotational speed is uniform and, second, that the calculated fric-tion applies uniformly to the entire rotational range, ignoringadditional dissipation, e.g., during barrier crossing events. Al-though not exact, the estimate should nevertheless serve as auseful reference. Under these two assumptions, the overall diffu-sion coefficient (obtained by adding the respective frictioncoefficients) is Deff = ð1=D1 + 1=D2Þ−1 ≈ 0.00075  rad2=ns. Thefrictional dissipation in a full turn at an angular velocity ofω= 1 kHz corresponding to experimental turnover rates is then2πωkBT=Deff ≈ 0.05  kBT. At this rotational speed, dissipationcould thus effectively be ignored. However, as discussed in ref.20, the actual rotational motions are likely much faster than theaverage, because much of the time is spent waiting in metastablestates (in particular, for the arrival of substrates and the escapeof product). To account for short transition path times, wecalculate in the following also the maximum rotational speedenergetically permitted for these transitions.

Maximum Rotational Speed. The estimated rotational diffusioncoefficients, or friction coefficients from the relation ξ= kBT=D,define a fundamental limit on the rotation, the maximum rota-tional speed between the dwells without load. This limiting speedhas so far not been determined experimentally. The reason isthat, in single-molecule experiments, a relatively big probe has tobe attached to visualize the rotations. When the amount of dissi-pation for a 120°=2π=3 rotation W = ð2π=3ÞωkBT=Deff is equatedwith the ATP hydrolysis energy of ∼ 20kBT at typical cellularconditions (33), assuming an energy conversion efficiency of 100%from the ATP hydrolysis reaction to mechanical rotation, it shouldgive the maximum rotational speed ωmax. For the estimateddiffusion coefficient Deff , we find ωmax ≈ 0.4  °=ns, corresponding to∼300 ns per 2π=3 rotational transition. Note that this speed iscomparable to that of the torque MD simulations, and that func-tionally relevant conformational changes of the αβ-subunits havebeen observed at such rotary speeds (20, 29). This prediction canbe tested by further reducing the size of the probe used in thesingle-molecule experiments. Based on our estimates, if the loadcan be reduced to a diameter of ∼20 nm, then friction will bedominated by the protein, not by the load.

ConclusionsWe developed a multiscale framework to deduce mechanical andfrictional properties of biomolecular machines from atomisticMD simulations, and applied it to the F1 motor of ATP synthase.By constructing a Langevin diffusion model of the γ-rotation inF1-ATPase from atomistic MD trajectories, we deduced thetorsional elasticity and rotational friction coefficients governingthe γ-subunit rotation. The estimated torsional elastic coefficientsagree with single-molecule experiments probing different seg-ments of the γ-subunit (16, 27), thus resolving a controversy intheir interpretation. The overall elastic coefficient is also consis-tent 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 Fo and F1 motors of ATP synthase. This relative stiff-ness ensures that the two motors are tightly coupled, forcing themonto a unique rotational pathway. For ATP synthase in mammalianmitochondria, which use only eight protons to synthesize three ATP,

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80

120

200

240

320

360

8 143 188 233 278 323 368

0.0

10.3 1.6

3.2

2.9

1.4 10.80.0

0.0

0

6292

120

182212240

302332360

-1 134 179 224 269 314 359

0.05.0

0.3

0.4 0.3

0.1

4.15.4 0.1

4.5

0.0

0 00 0

φ F1 [d

eg]

0

6595

120

185215240

305335360

0 135 180 225 270 315 360

0.06.1

0.8

0.7 0.8

0.0 3.8 6.0

0.0

3.7

45 90

0.0

53 98 44 89

Fig. 4. Pathway and energetics of coupled Fo–F1 rotary motion under reversible conditions of ∼200-mV proton motive force (pmf) in mitochondria. (A) Schematicof coupled Fo–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 keff = 66  kcal ·mol−1 · rad−2 determined herefrom the MD simulations as a function of the rotation angles ϕFo and ϕF1 of the Fo and F1 subunits, respectively. Numbers in blue indicate the free energy of therespective conformations. Here we assume that k1 and k12 apply consistently to all microstates. The pmf corresponds to an input free energy of −4.6  kcal=mol persingle proton translocation and an output free energy of 4.6× 8=3≈ 12.3  kcal=mol per single ATP synthesis. For simplicity, the latter energy cost is assumed to beaccumulated evenly as a function of the ϕF1 angle. The minimum energy paths are shown as blue lines for three (B) or two (C) substeps of the F1 motor. Anadjustment in the relative phases of ϕFo and ϕF1 wasmade to lower the energy of themicrostates 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.

10724 | www.pnas.org/cgi/doi/10.1073/pnas.1500691112 Okazaki and Hummer

the mismatch is so large that additional substeps are required, thusproviding a rationale for the recent single-molecule observation oftwo 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 thehydrodynamic limit, which is likely due to tight interactionswith the αβ-subunit bearing. For the protruded part, the dif-fusion coefficient is ∼10 times smaller than the hydrodynamiclimit, possibly due to salt bridges with the tips of the αβ-subunits.From these estimated rotational diffusion coefficients, we predicta maximum rotational speed of ωmax ≈ 1MHz between the dwellswithout load, and estimate an energy dissipation of < 1 kBT perrotation at an average speed of ω≈ 1 kHz. The F1 rotary motorthus operates at very small dissipation levels, which is essential forthe high overall thermodynamic efficiency of FoF1-ATP synthase.

MethodsMD Simulations. The MD simulations were performed as in ref. 20. However,here the rotation angles of each residue were calculated with separate localaxes for the two domains passing through their respective centers of massand orientated parallel to the symmetry axis of F1. The angles were thenaveraged for each domain, weighted by the normal distance of the Cα atomfrom the axis.

Fit of Langevin Diffusion Model. To fit the Langevin diffusionmodel to the torque-induced angle trajectories, we minimized χ2 = ð1=2NtÞ

Pt ½ð½xfit1 ðtÞ− x1ðtÞ�=σ1Þ2 +

ð½xfit2 ðtÞ− x2ðtÞ�=σ2Þ2�, whereNt is number of data points in a particular time course;

σi is defined as the SD from the best fit, σi =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1=NtÞ

Pt ðxoptfiti ðtÞ− xiðtÞÞ2

q.

The energy function of the Langevin diffusion model is

U=k122ðx1 − x2Þ2 + k1

2x21 +

κ

2½ρx1 + ð1− ρÞx2 −ωt�2 [4]

where x1 and x2 represent the rotation angles of the core and the protrudedparts of the γ-subunit, respectively, and ρ= 0.2 is weight factor of the twodomains used to determine the average angle (Fig. S3). Note that the drivingtorque (last term) is a harmonic approximation of the periodic potentialused in our atomistic simulations. The three terms represent the harmonicpotential between the two domains, the harmonic potential describing theinteraction with ring subunits, and the time-dependent torque potential,respectively. The analytical solutions for the time-dependent average anglesx1ðtÞ and x2ðtÞ are fitted to the corresponding MD trajectories (see SI Text)for models A (assuming quasi-equilibrium for both domains), B (assumingquasi-equilibrium for the protruding domain 2), and C (full nonequilibrium).

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

wt =Zt

0

∂Uðx1ðt′Þ, x2ðt′Þ, t′Þ∂t′

dt′ [5]

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

ACKNOWLEDGMENTS. We thank Profs. Wolfgang Junge, Martin Karplus, JohnWalker, Arieh Warshel, and Rikiya Watanabe for discussions. This work wassupported by theMax Planck Society (K.O. andG.H.) and a postdoctoral fellowshipfor research abroad from the Japan Society for the Promotion of Science (K.O.).

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