Elasticity, friction, and pathway of γsubunit rotation in FoF1ATP ...
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Elasticity, friction, and pathway of subunit rotationin FoF1ATP synthaseKeiichi Okazaki and Gerhard Hummer1
Department of Theoretical Biophysics, Max Planck Institute of Biophysics, D60438 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 FoF1ATP synthase. A torsional viscoelastic model withfrictional dissipation quantitatively reproduces the dynamics and energetics seen in atomistic molecular dynamics simulations of torquedriven subunit rotation in the F1ATPase rotary motor. The torsionalelastic coefficients determined from the simulations agree with results from independent singlemolecule experiments probing different segments of the subunit, which resolves a longlastingcontroversy. 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 coupled rotations of the Fo and F1 rotary motors in ATP synthase, andexplain the need for the finer stepping of the F1 motor in the mammalian system, as seen in recent experiments. Compensating forincommensurate eightfold and threefold rotational symmetries inFo and F1, respectively, a significant fraction of the external mechanical work is transiently stored as elastic energy in the subunit. Thegeneral framework developed here should be applicable to othermolecular machines.
bioenergetics  molecular motor  mechanochemical coupling
FoF1ATP synthase is essential for life. From bacteria to human, this protein synthesizes ATP from ADP and inorganicphosphate Pi in its F1 domain, powered by an electrochemicalproton gradient that drives the rotation of its membraneembeddedFo domain (15). Its two rotary motors, F1 and Fo, are coupledthrough the 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 membraneembedded Fo part, converting ATP to ADPand Pi (7, 8).F1 has a symmetric ring structure composed of three subunits
with the asymmetric subunit sitting inside the ring (9, 10). Eachsubunit has a catalytic site located at the domain interface.The F1 ring has a pseudothreefold symmetry with the threesubunits taking three different conformations, E (empty), TP(ATPbound), and DP (ADP bound) (911). 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 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 chloroplasts with c10c15 rings (13, 14). The resulting symmetry mismatches between F1 and Fo (1517) 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 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 (1824). Here we explore thesequestions by building a dissipative mechanical model of the F1motor on the basis of atomistic molecular dynamics (MD) simulations. Friction and torsional elasticity of the subunit are centralto the efficient function of the coupled FoF1 nanomotors (15, 25,26). For subunits crosslinked with the 33ring, estimates havebeen obtained by monitoring thermal angle fluctuations in singlemolecule 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 inducetorquedriven 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 subunit as overdamped Langevin dynamics on a2D harmonic free energy surface. The model quantifies the magnitude 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 rationalize 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
FoF1ATP synthase produces the ATP essential for cellular functions from bacteria to humans. Rotation of its central subunitcouples proton translocation in the membraneembedded 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 subunit is just flexible enough to compensate for theincommensurate eightfold and threefold rotational symmetriesof mammalian Fo and F1 motors, respectively. The resulting energetic constraints dictate a unique pathway for the coupledrotations of the Fo and F1 rotary motors, and explain the finestepping seen in singlemolecule 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: [email protected]
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1500691112//DCSupplemental.
1072010725  PNAS  August 25, 2015  vol. 112  no. 34 www.pnas.org/cgi/doi/10.1073/pnas.1500691112
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ResultsTwoDomain Langevin Model of Rotation in F1ATPase. To constructa Langevin diffusion model, the subunit was coarsegrained 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 subunit axis. Using theflexible rotor method (20), we applied torque on the subunit inatomistic MD simulations. A relatively weak harmonic restraintof = 2,000 kcal mol1 rad2 0.61 kcal mol1 deg2 on the average angle at angular velocities of = 1 10=ns resulted in agradual buildup of tension and permitted subunit twisting andsubstantial angle fluctuations. In the torque MD simulations,we probed rotary motions of up to 55 in the protruded part ofthe subunit and 20 at its core (Fig. 1A).At the lowest speed, the profiles of the subunit rotation angle
along the axis show two plateaus separated by a distinct step at 20 (Fig. 1B), indicating that the subunit can be divided into twomain parts. This division is consistent with the structural featuresof the subunit, which consists of a protruded globular part and acoiledcoil helical domain inside the hexamer ring. Our minimalmodel of the 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 33ring subunits. Externaltorque is exerted by a torsional spring rotating at an angularvelocity . The different viscous environments of the two domainsare described by rotational diffusion coefficients D1 and D2. Ourtwodomain model can be seen as a further reduction of the ninesegment model of Czub and Grubmller (28), which divides thesubunit into five segments and the 33ring into four. As shownin Fig. S1, our domain 1 covers their segment 5, half of segment 4,and, in addition, the Cterminal helix not included in the ninesegment model. Our domain 2 contains segments 13 and theother half of segment 4. According to the subunit twist profile(Fig. 1B and Fig. S1), the elastic deformation is concentratedwithin their segment 4. In our twodomain model, we treat thislinearly twisted region by the single torsional spring k12. By contrast, our torsional spring k1 accounts for the interactions betweensegments 45 and 69. Note that we do not explicitly consider the
small helixturnhelix motif at the very top of the subunit (segment 1), whose elastic response in FoF1ATP synthase will bedetermined also by its interactions with the c ring and the e andsubunits (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 timedependent average rotation anglesx1t of the core helix domain (residues 126, 228273) andx2t of the protruded domain (residues 27227) to corresponding trajectories from atomistic MD simulations (Fig. 2).The MD rotational angles were calculated as distanceweightedaverages 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 timedependent rotation angles x1 and x2 could befitted with a quasiequilibrium model (model A; Fig. 2A). Correspondingly, the twist profiles at different time points, rescaled bythe target angle t, almost perfectly coincide (see Fig. 1B and Eqs.S1 and S2). At rotational speeds exceeding 2=ns, dissipative effects become significant. The resulting deviations from quasiequilibrium (difference between orange and black lines in Fig. 2A)allow us to extract rotational diffusion coefficients from fits ofmodel B (assuming quasiequilibrium 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 velocities. Plots of 2 in Fig. 2B show the nonlinear correlationsbetween the fit parameters, permitting variations in k12 and k1 by30% around the optimal values k12, k1= 123, 309 (in kilocalories 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 2maps for model C delimits the range of D1 and D2 that fit alltrajectories well (Fig. 2B). To test the assumption of quasiequilibrium at 1=ns, we ran additional simulations at lower rotational speeds of 0.2=ns and 0.5=ns. As shown in Fig. 2C, boththe timedependent twisting profiles and the 2 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 torquedriven rotationsprovides us with largely independent information that we can usenot only to validate the Langevin diffusion model and finetuneits parameters, but also to quantify the energetics of the rotation.The timedependent 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 rotational speeds of = 1 and 2=ns show little variation (blue andcyan points in Fig. 3), indicating that, at these speeds, rotationindeed occurs at nearequilibrium conditions. However, the accumulated work at a given target angle t increases as the rotational 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 t = 50 is overestimated by 2030%(Fig. 3).This small discrepancy in the work profiles can be corrected
by finetuning the parameters in the Langevin model to fit bothwork and anglevs.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 simulations (Fig. 3). The optimized parameters k12 = 94 and k1 = 223kcalmol1rad2 and D1 = 0.001 and D2 = 0.003 rad2/ns capture
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Fig. 1. Langevin diffusion model of rotation in F1ATPase. (A) Coarsegraining of the subunit into two domains connected by torsional springs andembedded in different viscous environments. (B) Local rotation angles of thesubunit along axis in torque MD simulation, scaled by the target angle t for= 1/ns. The profiles are shown for times t = 35 ns, 40 ns, and 50 ns, averagedover three independent trajectories. The nearperfect superposition of theprofiles is consistent with quasiequilibrium conditions in a linearly elasticmodel. Dashed lines show the optimized quasiequilibrium solution of thetwodomain viscoelastic model. Arrows indicate crosslinks in the singlemolecule experiments of Sielaff et al. (16) and Okuno et al. (27). Residues C87 andA270 in E. coli correspond to C78 and A256, respectively, in bovine mitochondria, and residue R84 in Bacillus PS3 corresponds to R75.
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all trajectories and work profiles well (orange lines in Fig. 2A;solid lines in Fig. 3).With this finetuned Langevin diffusion model, we decompose
the work performed during torquedriven rotation into recoverable 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 rotational speed of 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 (green line in Fig. S4) and the elastic potentialacting on x1 (red line) as a function of t remains almost independent 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.
DiscussionSubunit Elasticity. We determined the torsional elasticity androtational Langevin diffusion coefficients by optimizing a twodomain rotational Langevin diffusion model against the atomistic
torque trajectories at different rotational speeds and the corresponding nonequilibrium work profiles. To assess the elasticityparameters of the rotation, we compared the results from ourviscoelastic model to the values from singlemolecule experiments.Sielaff et al. (16) determined an effective elastic coefficient of about750 pNnm/rad2 for the Escherichia coli subunit between twocrosslinks at residues A270 and A87, 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 constant of k12 660 pNnm/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 pNnm/rad2 obtained in a previous simulation study for a similarsegment of a subunit covalently attached to the hexamer ring (28).Using a similar approach, Okuno et al. (27) estimated an elasticcoefficient of 223 141 pNnm/rad2 for the Bacillus PS3 subunit upto a single crosslink at a position closely matching the full twist x2 ofour model, and moving with the protruding part of the subunit(Fig. 1B). The corresponding effective spring constant in our modelis keff = 1=k1 + 1=k121 460 pNnm/rad2, deviating from themeasurement by less than the combined ( 2 140 pNnm/rad2)
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Fig. 2. Parameterization of the Langevin diffusion model via allatom torque MD simulations. (A) Angle trajectories for the core (x1, red) and protruded domain(x2, green) from torqueMD simulations at angular velocities of = 110/ns (left to right). Fits of the quasiequilibriummodel A and the overall optimal rotationaldiffusion model C (k12 = 94 and k1 = 223 kcal/mol/rad
2, and D1 = 0.001 and D2 = 0.003 rad2/ns) are shown in black and orange, respectively. (B) Fit quality 2 of
quasiequilibriummodel A as a function of k12 and k1 at = 1=ns (Top Left). The 2 of model C as a function ofD1 and D2 at the different rotational speeds , andoverall (right; 21 + 22 + 25 + 210=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 = 0.2/ns and 0.5/ns. (Top) Angletrajectories of domains 1 and 2 are shown in red and green for = 0.2/ns (bottom scale), and inmagenta and cyan for = 0.5/ns (top scale), respectively. (Middleand Bottom) Fit quality 2 of quasiequilibrium model A as a function of k12 and k1 at = 0.2/ns and 0.5/ns. Yellow squares indicate the global optimumdetermined at 1/ns.
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SEs of model and experiment. In effect, we could thus independently validate the two spring constants k1 and k12 in ourmodel, because they were probed in separate measurements.Moreover, our results resolve the longstanding controversy concerning 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 rotational diffusion coefficient by approximating the core part of thesubunit as a cylinder (32),
D1,hydro = kBT4La2 [1]
where = 7 104 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. Thesubunit 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. Approximated as a sphere of radius a 2 nm, we use (32, 33)
D2,hydro = kBT8a3 [2]
to estimate a hydrodynamic rotational diffusion of D2, hydro 0.03 rad2=ns . D2 is thus 10 times smaller than the hydrodynamic limit, which is reasonable because the protruded part isslowed down by interactions, mainly salt bridges, with the tips ofthe subunits.
Mechanical Consequence of FoF1 Symmetry Mismatch. The calculated 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 singlemolecule 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
GFo,F1= Fo=45Gin + F1=120Gout +keff2Fo F12
[3]
where Fo and F1 are the rotation angles of the Fo and F1 subunitsin the ATP synthesis direction, respectively. Gin =200mV4.6 kcal=mol is the input energy per one proton translocationand Gout 12.3 kcal=mol is the output energy per one ATP synthesis (6) at maximum load conditions, 8Gin = 3Gout, for the mammalian ATP synthase. Note that GFo,F1 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 FoF1 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 57 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 FoF1 plane. Here weaccounted for the unknown relative phase Fo 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 Fo =Fo + n 360=8 with n integer). To quantify the effects of subunit asymmetry on the kinetics of ATP synthesis, we developed amaster equation model of the coupled FoF1 rotation on the basisof the angledependent energy function Eq. 3 (see SI Text for details). The resulting mean times for full rotation of the two motorsdepend strongly on the number of F1 substeps. Assuming an elementary rate of k0 1/(100 s) 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 estimates 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 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 theFoF1 plane. Second, the additional substeps seen in recentsinglemolecule experiments on the mammalian F1 motor arecritical for its operation. With the 40+ 80= 120 steps of thebacterial motor, an eightmembered 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 s (see SI Text). Assuming the same elasticity ofthe subunit (keff = 66.0 kcalmol1rad2), F1 motors operatingwith 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 s1 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 phosphate release (31, 36). Alternatively, the bacterial subunit or itsinterface with the c ring could be softer, especially for the thermophilic Bacillus PS3 F1, in which only two substeps have beenfound after extensive singlemolecule experiments. With the stiffness reduced by half in our model (keff = 33.0 kcalmol
1rad2),two and threesubstep 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 timedependent 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 t during the torque MD simulations. Work profiles from the elastic model with globally optimal set ofparameters are shown as solid lines.
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elasticity, the twosubstep F1 could thus achieve a reasonable ATPsynthesis rate. However, the interface between subunit and cring cannot be much softer because their angledependent interaction must have 10fold symmetry for a c10 ring. If we assumethat an elastic potential with an interfacial spring constant kcholds in each of the n minima all of the way to a cuspshapedbarrier, the nfold periodic free energy surface will have barriers ofheight kc=n2/2. 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 subunit on a millisecond timescale, theinterfacial spring constant kc should exceed 100 kcalmol1rad2.Putting this torsional spring in series with the springs k1 and k12determined here, one would find an effective sprint constant ofkeff = k11 + k112 + k1c 1 > 40 kcalmol1rad2, at least half of ourvalue for the subunit and 33ring alone.It will also be interesting to compare the rotor stiffness of F and
Vtype ATPases (37), also to examine if the ATP hydrolysisdrivenproton pump function of the latter imposes different requirementson the mechanochemical coupling.
Dissipation at Average Rotational Speed. From the rotational diffusion coefficients D1 and D2, we can estimate the frictional dissipation per full cycle. Here we implicitly assume, first, that therotational speed is uniform and, second, that the calculated friction applies uniformly to the entire rotational range, ignoringadditional dissipation, e.g., during barrier crossing events. Although not exact, the estimate should nevertheless serve as auseful reference. Under these two assumptions, the overall diffusion coefficient (obtained by adding the respective frictioncoefficients) is Deff = 1=D1 + 1=D21 0.00075 rad2=ns. Thefrictional dissipation in a full turn at an angular velocity of= 1 kHz corresponding to experimental turnover rates is then2kBT=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 = kBT=D,define a fundamental limit on the rotation, the maximum rotational speed between the dwells without load. This limiting speedhas so far not been determined experimentally. The reason isthat, in singlemolecule experiments, a relatively big probe has tobe attached to visualize the rotations. When the amount of dissipation for a 120=2=3 rotation W = 2=3kBT=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 max. For the estimateddiffusion coefficient Deff , we find max 0.4 =ns, corresponding to300 ns per 2=3 rotational transition. Note that this speed iscomparable to that of the torque MD simulations, and that functionally relevant conformational changes of the subunits havebeen observed at such rotary speeds (20, 29). This prediction canbe tested by further reducing the size of the probe used in thesinglemolecule 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 rotation inF1ATPase from atomistic MD trajectories, we deduced thetorsional elasticity and rotational friction coefficients governingthe subunit rotation. The estimated torsional elastic coefficientsagree with singlemolecule experiments probing different segments of the subunit (16, 27), thus resolving a controversy intheir interpretation. The overall elastic coefficient is also consistent with the results of earlier equilibrium simulations of a crosslinked subunit (28).The subunit appears to be barely soft enough to absorb the
required elastic energy in the coupled rotation of the symmetrymismatched Fo and F1 motors of ATP synthase. This relative stiffness 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,
B C
Fo [deg]
A
F1
FoC8
()3
Mitochondrialinner membrane
D
0
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
F1 [
deg]
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 FoF1 rotary motion under reversible conditions of 200mV proton motive force (pmf) in mitochondria. (A) Schematicof coupled FoF1 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 subunit modeled with the effective spring constant keff = 66 kcal mol
1 rad2 determined herefrom the MD simulations as a function of the rotation angles Fo and F1 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 F1 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 Fo and F1 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 220230 mV for mitochondria. With only a single 40step, highenergy intermediates (>10 kcal/mol) arise, inconsistent with efficient operation.
10724  www.pnas.org/cgi/doi/10.1073/pnas.1500691112 Okazaki and Hummer
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the mismatch is so large that additional substeps are required, thusproviding a rationale for the recent singlemolecule observation oftwo intermediate steps instead of one (31).The estimated rotational diffusion coefficient for the coiled
coil core part of the subunit is 70 times smaller than thehydrodynamic limit, which is likely due to tight interactionswith the subunit bearing. For the protruded part, the diffusion coefficient is 10 times smaller than the hydrodynamiclimit, possibly due to salt bridges with the tips of the subunits.From these estimated rotational diffusion coefficients, we predicta maximum rotational speed of max 1MHz between the dwellswithout load, and estimate an energy dissipation of < 1 kBT perrotation at an average speed of 1 kHz. The F1 rotary motorthus operates at very small dissipation levels, which is essential forthe high overall thermodynamic efficiency of FoF1ATP 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 C atomfrom the axis.
Fit of Langevin Diffusion Model. To fit the Langevin diffusionmodel to the torqueinduced angle trajectories, we minimized 2 = 1=2Nt
Pt xfit1 t x1t=12 +
xfit2 t x2t=22, whereNt is number of data points in a particular time course;i is defined as the SD from the best fit, i =
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=Nt
Pt xoptfiti t xit2
q.
The energy function of the Langevin diffusion model is
U=k122x1 x22 + k12 x
21 +
2x1 + 1 x2 t2 [4]
where x1 and x2 represent the rotation angles of the core and the protrudedparts of the subunit, respectively, and = 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 timedependent torque potential,respectively. The analytical solutions for the timedependent average anglesx1t and x2t are fitted to the corresponding MD trajectories (see SI Text)for models A (assuming quasiequilibrium for both domains), B (assumingquasiequilibrium for the protruding domain 2), and C (full nonequilibrium).
The timedependent nonequilibrium work performed by the externaltorque is calculated from (38)
wt =Zt
0
Ux1t, x2t, tt
dt [5]
where U is the timedependent 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.).
1. Mitchell P (1961) Coupling of phosphorylation to electron and hydrogen transfer by achemiosmotic type of mechanism. Nature 191(4784):144148.
2. Junge W, Sielaff H, Engelbrecht S (2009) Torque generation and elastic power transmission in the rotary FOF1ATPase. Nature 459(7245):364370.
3. Stewart AG, Laming EM, Sobti M, Stock D (2014) Rotary ATPasesDynamic molecularmachines. Curr Opin Struct Biol 25:4048.
4. Watanabe R, et al. (2013) Biased Brownian stepping rotation of FOF1ATP synthasedriven by proton motive force. Nat Commun 4(3):1631.
5. Pogoryelov D, et al. (2010) Microscopic rotary mechanism of ion translocation in theFo complex of ATP synthases. Nat Chem Biol 6(12):891899.
6. Kinosita K, Jr, Adachi K, Itoh H (2004) Rotation of F1ATPase: How an ATPdrivenmolecular machine may work. Annu Rev Biophys Biomol Struct 33:245268.
7. Noji H, Yasuda R, Yoshida M, Kinosita K, Jr (1997) Direct observation of the rotationof F1ATPase. Nature 386(6622):299302.
8. Yasuda R, Noji H, Kinosita K, Jr, Yoshida M (1998) F1ATPase is a highly efficientmolecular motor that rotates with discrete 120 degree steps. Cell 93(7):11171124.
9. Abrahams JP, Leslie AG, Lutter R, Walker JE (1994) Structure at 2.8 resolution ofF1ATPase from bovine heart mitochondria. Nature 370(6491):621628.
10. Bowler MW, Montgomery MG, Leslie AG, Walker JE (2007) Ground state structure ofF1ATPase from bovine heart mitochondria at 1.9 resolution. J Biol Chem 282(19):1423814242.
11. Okazaki K, Takada S (2011) Structural comparison of F1ATPase: Interplay amongenzyme structures, catalysis, and rotations. Structure 19(4):588598.
12. Stock D, Leslie AGW, Walker JE (1999) Molecular architecture of the rotary motor inATP synthase. Science 286(5445):17001705.
13. Watt IN, Montgomery MG, Runswick MJ, Leslie AGW, Walker JE (2010) Bioenergeticcost of making an adenosine triphosphate molecule in animal mitochondria. ProcNatl Acad Sci USA 107(39):1682316827.
14. Stewart AG, Sobti M, Harvey RP, Stock D (2013) Rotary ATPases: Models, machineelements and technical specifications. BioArchitecture 3(1):212.
15. Mitome N, Suzuki T, Hayashi S, Yoshida M (2004) Thermophilic ATP synthase has adecamer cring: Indication of noninteger 10:3 H+/ATP ratio and permissive elasticcoupling. Proc Natl Acad Sci USA 101(33):1215912164.
16. Sielaff H, et al. (2008) Domain compliance and elastic power transmission in rotaryFOF1ATPase. Proc Natl Acad Sci USA 105(46):1776017765.
17. Wchter A, et al. (2011) Two rotary motors in FATP synthase are elastically coupledby a flexible rotor and a stiff stator stalk. Proc Natl Acad Sci USA 108(10):39243929.
18. Koga N, Takada S (2006) Foldingbased molecular simulations reveal mechanisms ofthe rotary motor F1ATPase. Proc Natl Acad Sci USA 103(14):53675372.
19. Pu J, Karplus M (2008) How subunit coupling produces the gammasubunit rotarymotion in F1ATPase. Proc Natl Acad Sci USA 105(4):11921197.
20. Okazaki K, Hummer G (2013) Phosphate release coupled to rotary motion of F1ATPase. Proc Natl Acad Sci USA 110(41):1646816473.
21. Nam K, Pu J, Karplus M (2014) Trapping the ATP binding state leads to a detailed understanding of the F1ATPase mechanism. Proc Natl Acad Sci USA 111(50):1785117856.
22. Mukherjee S, Warshel A (2015) Dissecting the role of the subunit in the rotarychemical coupling and torque generation of F1ATPase. Proc Natl Acad Sci USA 112(9):27462751.
23. Dittrich M, Hayashi S, Schulten K (2003) On the mechanism of ATP hydrolysis inF1ATPase. Biophys J 85(4):22532266.
24. Mukherjee S, Warshel A (2012) Realistic simulations of the coupling between theprotomotive force and the mechanical rotation of the F0ATPase. Proc Natl Acad SciUSA 109(37):1487614881.
25. Wang H, Oster G (1998) Energy transduction in the F1 motor of ATP synthase. Nature396(6708):279282.
26. Sun SX, Wang H, Oster G (2004) Asymmetry in the F1ATPase and its implications forthe rotational cycle. Biophys J 86(3):13731384.
27. Okuno D, Iino R, Noji H (2010) Stiffness of gamma subunit of F1ATPase. Eur Biophys J39(12):15891596.
28. Czub J, Grubmller H (2011) Torsional elasticity and energetics of F1ATPase. Proc NatlAcad Sci USA 108(18):74087413.
29. Czub J, Grubmller H (2014) Rotation triggers nucleotideindependent conformationaltransition of the empty subunit of FATPase. J Am Chem Soc 136(19):69606968.
30. Pnke O, Cherepanov DA, Gumbiowski K, Engelbrecht S, Junge W (2001) Viscoelasticdynamics of actin filaments coupled to rotary FATPase: Angular torque profile of theenzyme. Biophys J 81(3):12201233.
31. Suzuki T, Tanaka K, Wakabayashi C, Saita E, Yoshida M (2014) Chemomechanicalcoupling of human mitochondrial F1ATPase motor. Nat Chem Biol 10(11):930936.
32. Howard J (2001) Mechanics of Motor Proteins and the Cytoskeleton (Sinauer Assoc,Sunderland, MA).
33. Berg HC (1993) Random Walks in Biology (Princeton Univ Press, Princeton, NJ).34. Bilyard T, et al. (2013) Highresolution singlemolecule characterization of the enzy
matic states in Escherichia coli F1ATPase. Philos Trans R Soc Lond B Biol Sci 368(1611):20120023.
35. Martin JL, Ishmukhametov R, Hornung T, Ahmad Z, Frasch WD (2014) Anatomy ofF1ATPase powered rotation. Proc Natl Acad Sci USA 111(10):37153720.
36. Bason JV, Montgomery MG, Leslie AG, Walker JE (2015) How release of phosphatefrom mammalian F1ATPase generates a rotary substep. Proc Natl Acad Sci USA112(19):60096014.
37. Zhao J, Benlekbir S, Rubinstein JL (2015) Electron cryomicroscopy observation of rotational states in a eukaryotic VATPase. Nature 521(7551):241245.
38. Hummer G, Szabo A (2001) Free energy reconstruction from nonequilibrium singlemolecule pulling experiments. Proc Natl Acad Sci USA 98(7):36583661.
39. Zwanzig R, Szabo A, Bagchi B (1992) Levinthals paradox. Proc Natl Acad Sci USA 89(1):2022.
40. Elston T, WangH, Oster G (1998) Energy transduction in ATP synthase. Nature 391(6666):510513.
41. Shoup D, Lipari G, Szabo A (1981) Diffusioncontrolled bimolecular reaction rates. Theeffect of rotational diffusion and orientation constraints. Biophys J 36(3):697714.
Okazaki and Hummer PNAS  August 25, 2015  vol. 112  no. 34  10725
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