Single molecule thermodynamics of ATP synthesis by F$_1$-ATPase

F$_\mathrm{o}$F$_1$-ATP synthase is a factory for synthesizing ATP in virtually all cells. Its core machinery is the subcomplex F$_1$-motor (F$_1$-ATPase) and performs the reversible mechanochemical coupling. Isolated F$_1$-motor hydrolyzes ATP, which is accompanied by unidirectional rotation of its central $\gamma$-shaft. When a strong opposing torque is imposed, the $\gamma$-shaft rotates in the opposite direction and drives the F$_1$-motor to synthesize ATP. This mechanical-to-chemical free-energy transduction is the final and central step of the multistep cellular ATP-synthetic pathway. Here, we determined the amount of mechanical work exploited by the F$_1$-motor to synthesize an ATP molecule during forced rotations using methodology combining a nonequilibrium theory and single molecule measurements of responses to external torque. We found that the internal dissipation of the motor is negligible even during rotations far from a quasistatic process.

FoF1-ATP synthase is a factory for synthesizing ATP in virtually all cells. Its core machinery is the subcomplex F1-motor (F1-ATPase) and performs the reversible mechanochemical coupling. Isolated F1-motor hydrolyzes ATP, which is accompanied by unidirectional rotation of its central γ-shaft. When a strong opposing torque is imposed, the γ-shaft rotates in the opposite direction and drives the F1-motor to synthesize ATP. This mechanical-to-chemical free-energy transduction is the final and central step of the multistep cellular ATP-synthetic pathway. Here, we determined the amount of mechanical work exploited by the F1-motor to synthesize an ATP molecule during forced rotations using methodology combining a nonequilibrium theory and single molecule measurements of responses to external torque. We found that the work exploited by the motor amounts only to that is thermodynamically required for the ATP synthesis. Specifically, F1-motor converts mechanical work to chemical free energy at quite a high efficiency with negligible dissipation inside the motor even during rotations far from a quasistatic process.
F o F 1 -ATP synthase comprises an Fo-motor embedded in a membrane (inner membrane of the mitochondria in the eukaryotic cells) and an F 1 -motor protruding from the membrane (Fig. 1a). Proton translocation through the Fo-motor driven by the transmembrane electrochemical potential unidirectionally rotates the c-ring of the F omotor. Because the c-ring is connected to the γ-shaft of F 1 -motor, the c-ring imposes torque on the γ-shaft causing it to rotate. The forced rotation of the γ-shaft induces the F 1 -motor's stator α 3 β 3 γ ring to synthesize ATP from ADP and P i (inorganic phosphate). The F 1 -motor converts the mechanical work transferred from the c-ring to the chemical free energy of ATP, ∆µ [1,2]. Approximately 95% of cellular ATP is synthesized by this rotary mechanochemical transduction. In contrast, the isolated F 1 -motor hydrolyzes ATP to ADP and phosphate and rotates the γ-shaft by converting ∆µ to mechanical motion under physiological conditions [3,4]. However, the rotational direction is opposite to that of the ATP-synthetic rotation. The γ-shaft rotates 120 • per ATP hydrolysis [2,4]. Thus, the F 1 -motor reversibly transduces freeenergy between mechanical work and ∆µ [3][4][5][6].
The detailed kinetics and reaction scheme of the F 1motor's rotations have been studied intensively. In contrast, knowledge regarding the energetics is limited [4,[7][8][9][10][11] due to the difficulty in studying the energetics of such nanosized engines working at an energy scale comparable to thermal energy. An experimental methodology was recently established by combining measurements of single molecule responses with nonequilibrium theory [8][9][10][11]. Previous research reveals the high efficiency of the ATP-hydrolytic rotations by the isolated F 1 -motor as follows: (i) The maximum work performed by the F 1 -motor during a 120 • rotation is equal to ∆µ within experimental error [9]. Specifically, the F 1 -motor achieves a unity thermodynamic efficiency at the stalled state where the mean rotational rate vanishes because of a hindering external torque. (ii) The F 1 -motor converts ∆µ to mechanical work with negligible internal heat dissipation even during rotations far from quasistatic process [8]. (iii) The motor achieves such a high efficiency by switching its chemical state depending on the angular position of the γ-shaft [10].
However, the primary physiological role played by the F 1 -motor is mechanochemical ATP synthesis driven by the forced-rotations of the γ-shaft and not hydrolysis. This raises the fundamental unanswered question regarding the free-energy transduction as follows: How much work is required to drive the F 1 -motor to synthesize an ATP molecule? To address this question, we improved available methodology [8] and investigated the energetics of the forced ATP-synthetic rotations by the isolated F 1 -motor.

Experiment
An isolated F 1 -motor molecule was adhered to an upper glass surface (Fig. 1b) [3,4]. A submicron probe particle was attached to the γ-shaft with an elastic protein linker. We observed the probe's rotations with a conventional optical microscope, and we imposed a torque on the probe using an electrorotation method [8,9,[12][13][14]. The electrorotation method uses a highfrequency alternating-current electric field generated by quadrupolar electrodes and imposes a torque with a controlled magnitude (see Methods). Thus, we can measure the rotational response of a single motor molecule as a function of external torque. This system mimics the mechanochemical transduction inside the F o F 1 -ATP synthase; the upper glass surface, the probe and the torque imposed correspond to the stator, the c-ring and the proton-motive force that drives the c-ring, respectively.
Consider the energetics of this system as follows (Fig. 1c): Under sufficiently strong torque in the ATPsynthetic direction, the probe-shaft complex rotates in that direction [9]. The amount of the torque multiplied by 120 • with a sign depending on the rotational direction is the work, W ext , performed on the probe per ATP cycle. A portion of W ext , W , is transferred to the motor by rotating the γ-shaft with the elastic linker and the remainder, Q probe , dissipates through the viscous friction of the probe.
The motor receives W , converts it to the chemical free energy change ∆µ, and synthesizes an ATP molecule at its α 3 β 3 γ ring. The remainder, Q motor ≡ W − ∆µ, dissipates irreversibly from the motor. Because ∆µ is the thermodynamic limit for the work necessary to synthesize an ATP molecule, ∆µ/W is bound by 1 and can be treated as the efficiency of the mechanical-to-chemical free-energy transduction by the motor. These statements are valid when the probe and the γshaft are thermally insulated in that their time-scales of motions are well separated and ∆µ is not small. If these time scales are similar and ∆µ is small, the probe and the γ-shaft move in tandem to some extent and the heat dissipated from the system cannot be separated well into Q motor and Q probe ; for example, a part of the energy absorbed by the motor to overcome an energy barrier for a chemical reaction may dissipate through the probe's motion as a part of Q probe . Then, W evaluated by (1) is no longer work that is transferred to the motor, and ∆µ/W can be greater than 1 [15][16][17]. The present experiment should satisfy the thermally insulated condition as follows: the time-scale difference between the probe and shaft is determined by the rotational frictional coefficient, which is proportional to the cubic of the rotational diameter; the diameters of the probe and the γ-shaft were 300 nm and 2 nm, respectively. In addition, ∆µ is about 18 k B T , which is supposed to be sufficiently high for the above condition to be satisfied.
We can evaluate the work transferred to the nanosized motor W from the thermodynamic quantities regarding the probe W ext and Q probe using the energy balance (1). W ext is easily calculated as noted, whereas it is usually difficult to measure heat Q probe in a microscopic system subjected to thermal fluctuations. This is because, although heat can be defined as the microscopic energy exchange with the solvent [18], the thermal fluctuating force is inaccessible using experimental manipulations. However, a nonequilibrium equality derived by Harada and Sasa [19][20][21] enables us to calculate Q probe from quantities that can be derived experimentally: the fluctuation and the response to a small external torque perturbation of the rotational rate (see Materials and Methods for details). Fluctuation and response are related by the fluctuation response relation (FRR) around the equilibrium state [22]. The FRR is generally violated far from equilibrium. Therefore, the extent of the FRR violation indicates how far the system is driven to nonequilibrium. The Harada and Sasa's relation relates the extent of the FRR violation to the heat dissipation at a nonequilibrium steady state.
In the following experiment, we evaluated Q probe by Harada-Sasa relation, obtained W from (1), and compared it with ∆µ to evaluate the energetics of mechanicalto-chemical free energy transduction. We also discuss the energetics of the chemical-to-mechanical transduction in the ATP-hydrolytic rotations.

Single-molecule response measurement and violation of the fluctuation-response relation
The γ-shaft's rotations were assessed using a large dimeric probe (diameter = 300 nm) at a relatively high ATP concentration (10 µM ATP, 10 µM ADP, 1 mM P i ). Under this condition, the probe rotated smoothly without clear steps in the absence of external torque (Fig.  2a). When we applied external torque to the probe using the electrorotation method in the direction opposite to the rotations, the rotational rate decreased ( Fig. 2a and  b). At torques greater than ∆µ/120 • , the rotation was reversed, and the γ-shaft rotated in the ATP-synthetic direction. This is consistent with a previous result showing that the maximum work that the F 1 -motor can exert per ATP cycle is similar to ∆µ [9]. Specifically, the thermodynamic efficiency of the F 1 -motor is nearly 1 at the stalled state where the mean rotational rate vanishes.
The FRR is violated at low frequencies (Fig. 2c) where the motor drives the probe to a nonequilibrium state [8], whereas the FRR is held at high frequencies where the probe's motion is expected to behave as a freely rotating Brownian particle at equilibrium. The extent of the FRR violation, or twice that of the shaded area in Fig. 2c, corresponds to the integral in Harada-Sasa relation (see Materials and Methods). When we increased the magnitude of the load from zero, the extent of the FRR violation decreased in the ATP-hydrolytic rotations, nearly vanished at the stalled state, and increased again in the ATP-synthetic rotations.
Energetics of mechanical-to-chemical free energy transduction In Fig. 3a, Q probe evaluated by Harada-Sasa equality and W ext (= ±(torque) × 120 • ) are shown. We found that Q probe linearly decreased as we increased the load and vanished around the stalled state. This indicates that, at the stalled state, the probe behaved like a rotational Brownian motion at equilibrium in a 120 • -spacing periodic potential. In the ATP-synthetic rotations, the value of Q probe increased as the torque increased further, mainly because the rotational rate increased with torque. The difference W = W ext −Q probe is the work received by the motor (Fig. 3b). In the ATP-synthetic rotations, we found that W = ∆µ over a broad range of torque, where ∆µ is the thermodynamic limit for the work necessary to synthesize an ATP molecule. This implies that the motor's internal dissipation Q motor = W − ∆µ is negligible and the F 1 -motor can convert most of W to ∆µ even during rotations at a high rotational rate such as 50Hz. Note that the rate of the proton-driven rotations of the thermophilic F o F 1 -ATP synthase is 3-4 Hz at saturated ADP and Pi concentrations at a room temperature [23].
When the motor rotates in the ATP-hydrolytic direction under weak external torque, −W corresponds to the work performed by the motor transferred to the probe. A part of −W is used to increase the potential of the probe against load −W ext , and the remainder, Q probe , dissipates. In Fig. 3b, we show that W is similar to −∆µ in good agreement with a previous result [8] that the motor can consume ∆µ in the rotational degree of freedom with negligible internal irreversible heat production even during rotations. However, we observed a small deviation of W from −∆µ at a small torque magnitude (Fig.  3c inset), which was not observed in the previous experiment. W increased with torque and reached ∆µ around the stalled state.
We also measured W of a mutant F 1 -motor molecule with mutations around the nucleotide binding site (βT165S and βY341W) and the γ-subunit's hinge region (βG181A) [7] (Fig. 3c and d). This mutant produced a smaller maximum work than the wild-type, supposedly caused by weak binding of ATP [9]. Figure 3d shows that |W | is significantly less than |∆µ|; the mutant is less efficient than the wild type and produces a finite amount of internal dissipation.

Discussion
Here we evaluated the energetics of the free-energy conversion by the F 1 -motor during rotations far from a quasistatic process in mechanical-to-chemical and chemicalto-mechanical energy transduction. This was achieved by combining a single molecule response measurement mimicking the mechanical coupling of F o F 1 -ATP synthase with a nonequilibrium theory. Our main finding is that, in the ATP-synthetic rotations, the F 1 -motor exploits mechanical work for the ATP synthesis at quite a high efficiency with negligible internal dissipation even during rotations far from a quasistatic process. In the ATP-hydrolytic rotations, the motor exploits most of ∆µ for the mechanical work in agreement with a previous result [8]. Moreover, we observed a small deviation between ∆µ and the converted mechanical work, which was not observed in the previous experiment. Such a tendency was predicted by a numerical simulation [17]. As noted, the probe and the γ-shaft are insulated in that the time scales of their motions are well separated and ∆µ is large, and therefore, we were able to evaluate the amount of work passed to the motor. The control of ∆µ, the linker's spring constant, and the probe's frictional resistance is intriguing. It not only facilitates exploration of the potential of this measurement system for studying energetics, but also helps us to understand the mechanical coupling between F 1 and F o -motors of ATP synthase.
In the present study, we assumed tight coupling between ATP synthesis/hydrolysis and the 120 • rotation and compared W with ∆µ to assess the efficiency. Although, the 120 • rotational step and the ATP hydrolysis are tightly coupled [2,4], this has yet to be firmly established for ATP synthesis. The results of a pioneering experiment on the forced rotations of the γ shaft that were induced using magnetic tweezers imply that less than one ATP molecule is synthesized in each 120 • synthetic-direction rotation [2]. In that study, the tweezers were rotated at a high rate of 10 Hz. Magnetic tweezers impose a trapping force, in contrast to the constant torque imposed by the electrorotation method. At a high manipulation rate the angular positions of the probe and tweezers can be apart. Then, magnetic tweezers can impose a very strong torque, which may induce slippage of the mechanochemical coupling. The results of a more recent experiment using magnetic tweezers manipulated at a low rate of 0.05-0.8 Hz suggests that attachment and detachment of nucleotides and the angular position of the γ shaft are tightly coupled [24]. Further, the previous result showing that the maximum work of the F 1 -motor is similar to ∆µ over a broad range of ∆µ, suggests tight coupling [9]. The mutant's low efficiency implies that it fails to completely couple mechanical rotation and ATP hydrolysis and synthesis. Some mechanical steps may not accompany chemical reactions.
Macroscopic engines operated at a finite rate inevitably generates turbulence, and additional energy dissipates through microscopic degrees of freedom as irreversible heat. In contrast, the F 1 -motor is itself microscopic and possibly utilizes thermal fluctuations. Therefore, some of the microscopic degrees of freedom might be not hidden and are accessible to the F 1 -motor. In the ATP-hydrolytic rotations, previous studies suggest that the F 1 -motor shifts the mechanical potentials discontinuously depending on the γ-shaft's angular position [10,25]. Only by nanosized machines can perform such an operation; they minimize irreversible heat and achieve a highly efficient free-energy conversion within a finite-time. This highlights the remarkable property of nanosized engines. Further studies will be required to elucidate the molecular mechanism of the highly efficient mechanochemical couplings of ATP-synthetic rotations.

Single molecule response measurement
The experimental setup is essentially the same as that in the previous studies [8][9][10]14]. F1 molecules derived from a thermophilic Bacillus PS3 with mutations for the rotation assay (His 6 -αC193S/W463F, His 10 -β, γS107C/I210C, denoted by wild type) [2] or a mutant with βT165S, βY341W, and βG181A were adhered on a cover slip functionalized by Ni 2+ -NTA. Rotations of the γ shaft were probed by streptavidin-coated dimeric polystyrene particles (diameter = 300 nm, Seradyn) attached to the biotylated γ shaft in a buffer containing 5 mM MOPS/KOH, 10 µM MgATP, 10 µM MgADP, 1 mM Pi, and 1 mM MgCl 2 (pH 6.9). Observation was performed on a phase-contrast upright microscope (Olympus) with a 100 objective and a high-speed camera (Basler) at 2,000 Hz. The data including a long pause presumably due to the MgADP-inhibited state are excluded from the analysis. We applied torque on the probe by using rotating electric field at 15 MHz generated with the quadrupolar electrodes patterned on the glass surface of the chamber [8,9,12,13,26]. The torque magnitude was controlled by controlling the electrodes voltage.
where · N is the ensemble average under a sufficiently small probe torque N (t). Because of the causality, R(t) = 0 if t < 0. Around an equilibrium state, C(t) and R(t) are related by the fluctuation response relation (FRR): C(t) = k B T R(t) [22]. The FRR is generally violated far from equilibrium. The equality by Harada and Sasa relates the extent of the FRR violation to heat dissipation at a nonequilibrium steady state [19][20][21]. In the frequency space, the equality is expressed as whereC(f ) andR(f ) are the Fourier transforms of C(t) and R(t), respectively, at frequency f , andR ′ (f ) is the real part ofR(f ). The FRR becomesC(f ) = 2k B TR ′ (f ) in the frequency space, Γ is the rotational frictional coefficient, and 3v s is the mean stepping rate that corresponds to the rate of ATP hydrolysis or synthesis.C(f ) was calculated from the rotational trajectories by a fast Fourier transform method and Wiener-Khintchine theorem. Γ was obtained by taking the average ofC(f ) around 300 Hz since the FRR, lim f →∞C (f ) = 2k B T /Γ, is supposed to hold in such a high frequency region [8,9]. Γ was 0.075±0.012 k B Ts/rad 2 (mean±SD, N=70).
For evaluatingR(f ), we added a small torque N (t) = N 0 i sin(2πf i t) , where f i = 1, 4, 10, 20, 40, 80, 160, 250, and 400 Hz, in addition to the constant load. N 0 is unknown a priori. We measured v(t) N , performed a Fourier transform, and obtained R(f i ) N 0 at multiple frequencies f i . Then, we obtained N 0 by comparingC(f ) andR(f )N 0 around at high frequency regions because of the FRR [8,9]. The frequency regions for calibration were determined by eye (around 300Hz typically). Finally, we obtained R(f i ). N 0 was 0.94±0.19 k B T/rad (mean±SD, N=70). We calculated the integration in (2) between -400 Hz and 400 Hz. The contribution from the outside of this region is supposed to be negligible because of the FRR at high frequencies. We omitted data without apparent convergence ofC(f ) andR ′ (f ) at the high frequency region (8 out of 78). When torque multiplied by 120 • was greater than 80 k B T, the FRR was violated at high frequencies even around 300 Hz, where the torque magnitude was calibrated according to the FRR.
We thank valuable discussions with Kyogo Kawaguchi, Takahiro Sagawa, Masaki Sano, Shin-ichi Sasa, and Hiroshi Ueno. This work was supported by Japan Science and Technology Agency (JST) and Grant-in-Aid for Scientific Research on Priority Areas. S.T. was supported by Alexander von Humboldt foundation. , where x is the torque multiplied by 120 • . The dashed line in d was fitted by eye. We excluded 1 point, which was aberrant because of the vanishingly small rotational rate, from the graphs displayed in panels a and b (torque × 120 • , Q probe , and W are 20.0 kBT , -40.5 kBT , and -60.5 kBT , respectively).