Structure of ATP synthase under strain during catalysis

ATP synthases are macromolecular machines consisting of an ATP-hydrolysis-driven F1 motor and a proton-translocation-driven FO motor. The F1 and FO motors oppose each other’s action on a shared rotor subcomplex and are held stationary relative to each other by a peripheral stalk. Structures of resting mitochondrial ATP synthases revealed a left-handed curvature of the peripheral stalk even though rotation of the rotor, driven by either ATP hydrolysis in F1 or proton translocation through FO, would apply a right-handed bending force to the stalk. We used cryoEM to image yeast mitochondrial ATP synthase under strain during ATP-hydrolysis-driven rotary catalysis, revealing a large deformation of the peripheral stalk. The structures show how the peripheral stalk opposes the bending force and suggests that during ATP synthesis proton translocation causes accumulation of strain in the stalk, which relaxes by driving the relative rotation of the rotor through six sub-steps within F1, leading to catalysis.

A TP synthases use a transmembrane electrochemical proton motive force (pmf) to generate adenosine triphosphate (ATP) from adenosine diphosphate (ADP) and inorganic phosphate (Pi). The enzyme complex consists of two molecular motors positioned to oppose each other's action on a shared rotor subcomplex (Fig. 1a, left). The membrane-embedded F O motor is driven by proton translocation across the membrane through two offset half channels 1,2 while the soluble F 1 motor is powered by ATP hydrolysis. In Saccharomyces cerevisiae, the F O region contains subunits a, e, f, g, i/j, k, 8, part of subunit b, and the c 10 -ring of the rotor 3 , while the F 1 region includes a trimer of catalytic subunit αβ pairs and subunits γ, δ, and ε from the rotor 4 . Coupling between F 1 and F O requires that the two motors are held stationary relative to each other by a peripheral stalk subcomplex (Fig. 1a, green structure), which in yeast is formed from subunits b, d, h, and OSCP (the oligomycin sensitivity conferral protein).
During ATP synthesis, proton translocation through F O at the interface of subunit a and the c-ring causes the γδεc 10 rotor (Fig. 1a, outlined in black) to turn. Rotation of subunit γ within F 1 leads each αβ pairs to cycle through open, tight, and loose conformations that result in the formation of ATP. Conversely, sequential ATP hydrolysis at each of the three αβ pairs in F 1 causes the γ subunit to turn in the opposite direction, rotating the proton-carrying c-ring against subunit a in F O and pumping protons across the membrane. Even with the rotor turning at hundreds of revolutions per second 5,6 there is little or no 'slip' 7 and the H + :ATP ratio remains constant. In S. cerevisiae this ratio is 10:3 due to the ten proton-carrying c subunits in F O and three catalytic sites in F 1 8 . With this H + :ATP ratio, when 10× the free energy of proton translocation (4e μ H þ ) is more negative than 3× the free energy of ATP hydrolysis (4G ATP ) the F O motor overpowers the F 1    3 4G ATP is more negative than 10 4e μ H þ , F 1 overpowers F O and the enzyme hydrolyzes ATP to pump protons. Synthesis or hydrolysis of each ATP molecule is associated with a~120°rotation of the γ subunit within F 1 , leading to conformations of the enzyme known as rotational State 1, 2, and 3 Ref. 9 . During ATP hydrolysis, which is better studied than ATP synthesis, this~120°rotation is broken down into a~40°sub-step as the enzyme transitions from a 'catalytic dwell' to a 'binding dwell', and an~80°sub-step as the enzyme transitions to the next catalytic dwell 5,10-12 (Fig. 1b, clockwise). ATP hydrolysis likely occurs during the~40°sub-step while ATP binding likely occurs during the~80°sub-step [12][13][14] . Consequently, the expected sequence of states for a 360°rotation of the rotor during ATP synthesis is State 1 binding → State 1 catalytic → State 2 binding → State 2 catalytic → State 3 binding → State 3 catalytic (Fig. 1b, counter-clockwise). The mismatch between these six sub-steps in F 1 and the ten proton-translocation steps in F O suggests that the enzyme cycles between strained and relaxed conformations during catalysis 15,16 . Early cryoEM noted that the peripheral stalks of mitochondrial ATP synthases have a left-handed curvature 17,18 (Fig. 1a, left). However, torque applied to the rotor following proton translocation through F O (Fig. 1a, right, blue arrows) would tend to rotate the α 3 β 3 hexamer in the same direction as the torque, inducing a right-handed curvature of the peripheral stalk as it resists the rotation. Similarly, the opposite torque applied to the opposite end of the rotor by ATP hydrolysis in F 1 (Fig. 1a, right, red arrows) would tend to rotate the membrane-embedded region of F O along with the c-ring, also inducing a right-handed curvature of the peripheral stalk as it resists the rotation 17 . Previously observed structures were obtained in the absence of a pmf or free ATP 9,17-30 , suggesting that the peripheral stalk may act as a spring that has a left-handed curvature when relaxed but a righthanded curvature under strain during catalysis 17 .

Results and discussion
The peripheral stalk shows pronounced bending under strain during ATP hydrolysis. We purified S. cerevisiae ATP synthase with the detergent n-Dodecyl-β-D-Maltopyranoside (DDM), which results in a monomeric preparation of the enzyme 30,31 , and determined its structure by cryoEM (Supplementary Figs. 1 and 2, Supplementary Tables 1 and 2). A high-resolution map of the intact complex was generated by combining multiple maps from focused refinements ( Fig. 1c and Supplementary Fig. 1c). In this map, the peripheral stalk shows the left-handed curvature seen previously. Three-dimensional (3D) classification allowed particle images to be separated into six rotor positions, corresponding to the catalytic and binding dwells for each of the three main rotational states. These conformations resemble recent catalytic and binding dwell structures for an isolated bacterial F 1 subcomplex imaged during ATP hydrolysis, where the absence of the peripheral stalk resulted in all catalytic dwell structures being identical and all binding dwell structures being identical 32 . For yeast ATP synthase imaged without ATP, the catalytic dwell structures show αβ tight either in the open conformation lacking nucleotide or in a closed conformation with weak nucleotide density, and the binding dwell structures show αβ tight only in an open conformation without nucleotide ( Supplementary Fig. 3). The existence of αβ tight in an open conformation without visible nucleotide density is likely an artifact from loss of ATP during the purification of the enzyme. Further classification of the State 1 catalytic conformation resulted in classes distinguished by variability in the position of the peripheral stalk and a slight rotation of the rotor relative to subunit a. These classes were designated as State 1 catalytic (a) (Fig. 1d, light blue) to State 1 catalytic (d) (Fig. 1d, dark blue) in order of increasing straightening of the peripheral stalk (Supplementary Movie 1, 'no ATP' condition). As these structures were determined in the absence of free ATP, they likely represent energetically similar conformations that can be reached by thermal fluctuation of the enzyme structure 9,26 .
To test the hypothesis that the peripheral stalk of ATP synthase deforms under strain, we next added ATP to the preparation and froze cryoEM specimens. The presence of ATP in the cryoEM sample buffer introduces a concentration-dependant background noise and loss of contrast in images of DDM-solubilized ATP synthase ( Supplementary Fig. 5A). Therefore, a delicate balance is required to provide sufficient ATP in solution to ensure that the enzyme is active as grids are frozen, while simultaneously maintaining sufficient contrast and image quality for image analysis. Initially, a small dataset of images was collected with a screening electron microscope where 10 mM ATP was added to the preparation at 4°C and cryoEM specimens were frozen within 10 s. Analysis of this dataset revealed conformations of the enzyme not seen in the absence of free ATP ( Supplementary  Fig. 4). Therefore, a large dataset was collected for the specimen with a high-resolution microscope ( Supplementary Fig. 5). Classification of the resulting dataset yielded maps showing six different F 1 states, corresponding to the catalytic and binding dwell structures from each of the three main rotational states. Subclassification of these populations separated each catalytic and each binding state into conformations with increasing rotation of the rotor relative to subunit a, and increasingly strained peripheral stalks, designated as 'a', 'b', 'c', etc. Overall, 27 unique conformations were identified: State 1 binding (a to d), State 1 catalytic (a to h), State 2 binding (a to b), State 2 catalytic (a to e), State 3 binding (a to c), and State 3 catalytic (a to e). Overlaying the eight State 1 catalytic (a to h) structures reveals that during ATP hydrolysis the peripheral stalk exhibits a large bending motion, transitioning from a left-handed curvature (Fig. 1e, light purple) to the predicted right-handed curvature (Fig. 1e, dark purple; Supplementary Movie 1, 'During ATP hydrolysis' condition). Without ATP and during ATP hydrolysis, the left-handed curvature of the peripheral stalk remains the most highly populated conformation of the enzyme (Fig. 1f).
The flexible peripheral stalk accommodates rigid rotation of the rotor during ATP hydrolysis. To facilitate comparison of the ATP synthase conformations that occur during ATP hydrolysis, backbone models of the protein structure were fit flexibly into each of the 27 maps (Fig. 2a). Remarkably, the α 3 β 3 γδεc 10 models from all 18 catalytic dwell conformations could be overlaid with high-fidelity (Fig. 2b, left), as could the nine α 3 β 3 γδεc 10 models from binding dwell conformations (Fig. 2b, right), with some limited flexibility at the interface between F 1 and the c-ring. This observation shows that, other than being in a catalytic or binding dwell conformation, the differences between the structures are mostly due to deformation of the peripheral stalk subunits and the rotation of the c-ring relative to subunit a in F O . Comparison of the eight State 1 catalytic models shows that the α 3 β 3 γδεc 10 rotor can turn~80°against subunit a in F O , or more than one-fifth of a complete revolution, before transition to the next binding dwell conformation (Fig. 2c). Bending of the peripheral stalk and not the central rotor of the complex supports suggestions that the peripheral stalk is the most compliant part of the enzyme and stores energy during rotary catalysis 9,21,22,26,28,33,34 .
The peripheral stalk bends by deformation of subunits d, f, and h. The peripheral stalk of yeast ATP synthase contains subunits b, d, h, and OSCP (Fig. 3a). Although atomic models for subunits b, d, and OSCP have been constructed from previous cryoEM of ATP synthase 30 , model quality for the 92-residue subunit h in earlier structures was low due to flexibility in both the peripheral stalk overall and subunit h specifically. Focused refinement of the peripheral stalk in the current structure resulted in continuous density for most of subunit h, allowing for construction of an atomic model for residues 1 to 62 based on predictions from AlphaFold 35 (Fig. 3a, blue; Supplementary Fig. 6). Interestingly, despite density immediately C-terminal of His62 in subunit h appearing disordered, an additional density that interacts with subunits a, d, f, and 8 indicates that the C terminus of the protein reaches the membrane surface, as suggested previously 36 (Fig. 3a, dashed box). Therefore, subunit h spans the entire distance from F 1 to F O , a role usually attributed only to subunit b, and different from subunit F 6 , the shorter mammalian homolog of subunit h 29 .
Aligning the eight structures corresponding to State 1 catalytic by their F O regions reveals that the dramatic bending of the peripheral stalk is facilitated mainly by deformation of subunits b, d, and h ( Fig. 3b; Supplementary Movie 1, 'During ATP hydrolysis' condition). In conformations that show only slight bending of the peripheral stalk, such as State 1 catalytic (b to d), deviation from the relaxed State 1 catalytic (a) conformation is mediated primarily by a pivot point in subunits b and d close to the membrane surface (Fig. 3c, red arrow). In the more strained conformations like State 1 catalytic (h), a second pivot point in subunit b at the top of subunit d is apparent (Fig. 3c, yellow arrow). The two pivot points are located at either end of subunit d, indicating that subunit d controls where the peripheral stalk bends and likely acts to oppose the bending force, inducing the left-handed curvature of the peripheral stalk when it is not under strain. The structure of subunit d, with an α-helical hairpin that allows it to push against subunit b, is ideal for its role of applying a force that attempts to restore the relaxed conformation of the peripheral stalk during ATP hydrolysis or synthesis ( Fig. 3d; Supplementary Movie 1, orange subunit). Subunit h contains two disordered regions close to the two pivot points defined by subunit d, which allows it to withstand the large conformational changes that occurs around the pivot points (Fig. 3d,  Overall rotation cycle of yeast ATP synthase. Despite the presence of a high concentration of ATP in the buffer used for freezing specimens during ATP hydrolysis, State 1 catalytic (a), the least strained of the State 1 catalytic conformations, appears to have MgADP bound in its αβ tight site (Fig. 4a, left). In contrast, refinement of the F 1 region with particle images combined from State 1 catalytic (e to h), the four most strained of the State 1 catalytic conformations, resulted in a structure similar to State 1 catalytic (a) but with what appears to be MgATP bound to αβ tight (Fig. 4a,  right). In the presence of free ATP, ATP hydrolysis occurs at the αβ tight site and MgADP within the site is expected to inhibit this hydrolytic activity. Therefore, the presence of MgADP in αβ tight of the non-strained conformation suggests that many of the complexes in this conformation are in the well-known MgADP inhibited state 4,37 . Inactive complexes have been detected previously even in the presence of free ATP 38,39 . In contrast, the structures that show the more strained peripheral stalks appear to be calculated from images of active enzyme particles. Density for the binding dwell conformations suggests that they contain MgADP with Pi in the αβ tight site ( Supplementary Fig. 7a), as was seen in the bacterial F 1 region during ATP hydrolysis 32 .
To place the 27 conformations of ATP synthase observed during ATP hydrolysis into a rotational sequence, the positions of subunit γ relative to α 3 β 3 in F 1 (Fig. 4b) and of the c-ring relative to subunit a in F O (Fig. 4c) were measured and plotted on circles that represent a 360°rotation. As described above, the α 3 β 3 γδεc 10 subcomplex is found in three catalytic dwell conformations and three binding dwell conformations, resulting in only six unique positions of subunit γ relative to α 3 β 3 in F 1 (Fig. 2b). Consistent with the isolated bacterial F 1 region 10,32 , ATP hydrolysis in αβ tight of the yeast catalytic dwell conformation appears to induce a slightly more open conformation of the αβ pair and a~36°r otation of the rotor (Supplementary Fig. 7b), leaving the enzyme in a binding dwell. MgADP and Pi are then released from the αβ tight site and ATP binding to the αβ open site drives an~84°r otation of the rotor to the next catalytic dwell conformation. Repetition of this process two more times completes the 360°r otation cycle for ATP hydrolysis (Fig. 4b, clockwise arrows), while for ATP synthesis the reverse reaction is driven by rotation of the rotor in the opposite direction (Fig. 4b, counter-clockwise arrows).
In contrast to the six unique positions of subunit γ relative to α 3 β 3 in F 1 , there are 27 unique positions of the c-ring relative to subunit a in F O . Plotting the angle of the c-ring relative to subunit a in F O produces a series of arcs that show the range of rotation of the ring within each catalytic or binding dwell state (Fig. 4c, black, blue, yellow, green, red, and cyan arcs). These arcs reveal that as the c-ring rotates in the ATP hydrolysis direction, each state exhibits a decreasing strain on the peripheral stalk (Fig. 4c, clockwise arrow). For example, for State 1 catalytic (Fig. 4c, black  arc), rotation of the c-ring in the ATP hydrolysis direction occurs during the transition from State 1 catalytic (h) → State 1 catalytic (a). As ATP hydrolysis in a catalytic αβ tight site causes the transition from a catalytic dwell to a binding dwell, the order of states indicates that the power stroke of ATP hydrolysis forces the peripheral stalk into a more strained conformation (e.g., State 1 catalytic (a) → State 1 binding (d)). This strain subsequently relaxes as the c-ring continues to turn in the ATP hydrolysis direction (e.g., State 1 binding (d) → State 1 binding (a)). Conversely, rotation of the c-ring in the direction driven by proton translocation during ATP synthesis (Fig. 4c, counter-clockwise arrow) leads to increasing strain on the peripheral stalk (e.g., State 1 binding (a) → State 1 binding (d)), which relaxes as ATP is formed in the catalytic site and the enzyme transitions from a binding dwell conformation to a catalytic dwell conformation (e.g., State 1 binding (d) → State 1 catalytic (a)).
Notably, the most strained conformation of some of the states show less rotation of the c-ring in the ATP hydrolysis direction than less strained conformations of the preceding state (Fig. 4c,  shaded areas). For example, the transition from State 1 catalytic (a) to State 1 binding (d) during ATP hydrolysis would involve the c-ring rotating 23°in the ATP synthesis direction. The same apparent 'backstepping' can be seen at the transition from State 3 catalytic → State 3 binding , and State 2 binding → State 1 catalytic . This backstepping of the c-ring would bend the peripheral stalk in the opposite direction of the applied force and is physically unlikely. Therefore, the unstrained conformations appear to show inactive complexes that are not part of the rotary sequence during substrate turnover. By extension, these data suggest that during rotary catalysis the peripheral stalk becomes strained and does not relax fully until catalysis stops. Construction of a movie showing rotation in the hydrolysis direction based on the most strained conformation of the enzyme illustrates the amount of deformation that can occur during ATP hydrolysis (Supplementary Movie 2, 'ATP hydrolysis' cycle). Similarly, a video can be constructed showing rotation in the ATP synthesis direction based on the most strained conformations (Supplementary Movie 2, 'ATP synthesis' cycle). Together, these data illustrate how in active ATP synthase the peripheral stalk can serve as a buffer for energy that deforms under strain. ATP synthase c-rings from different species can have between eight and 17 c subunits 40 and the flexibility of the peripheral likely allows this variability. The storage of energy during rotation would also be expected to smooth the transmission of power between the F 1 and F O motors despite symmetry mismatch, which is thought to be essential for the high turnover rate of the enzyme 41,42 . In the fully active enzyme, the peripheral stalk likely remains deformed as the enzyme runs, with the degree of bending dependent on the rate of turnover, and with the enzyme only becoming fully relaxed in the absence of ATP or a proton motive force. At present cryoEM appears uniquely capable of providing high-resolution insight into changes in enzyme conformation that occur during catalysis. Experiments to quantify strain within ATP synthase in solution conditions, such as single-molecule fluorescence resonance energy transfer 43 , could complement the findings reported here and reveal how strain changes within individual molecules during each rotation cycle.

Methods
Yeast growth and ATP synthase purification. Yeast strain USY006 containing a 6 × His tag at the N terminus of the β subunits was grown in YPGD media (1% [w/v] yeast extract, 2% [w/v] peptone, 3% [v/v] glycerol, 0.2% [w/v] glucose) with a 11 L fermenter (New Brunswick Scientific) for~48 h at 30°C until saturation. All purification steps were performed at 4°C. Yeast cell walls were broken with bead beating, and cell debris was removed by centrifugation at 5000 × g for 30 min. Mitochondria were collected by centrifugation at 25,000 × g for 30 min, before being washed with phosphate buffer (50 mM sodium phosphate pH 9.0, 5 mM 6-aminocaproic acid, 5 mM benzamidine, 1 mM PMSF) for 30 min. Washed mitochondria were collected by centrifugation at 184,000 × g for 30 min, before being resuspended in buffer MgCl 2 ). Fractions containing ATP synthase were pooled, and the protein was concentrated to~15 mg/ml prior to cryoEM grid freezing or storage at −80°C.
CryoEM specimen preparation. Glycerol in the ATP synthase preparation was removed with a Zeba Spin desalting column (Thermo Fisher Scientific [TFS]) before freezing cryoEM specimens. Holey gold films with~2 µm holes were nanofabricated 44 on 300 mesh Maxtaform copper-rhodium grids (Electron Microscopy Sciences). Specimens with 10 mM ATP were prepared by first applying 0.4 µL of 50 mM ATP in buffer (70 mM Tris-HCl pH 7.4, 0.05% [w/w] DDM, 100 mM NaCl, 55 mM MgCl 2 ) onto a grid that had been glow-discharged in air for 2 min. Freshly prepared ATP synthase (1.6 µL) was mixed quickly with the ATP solution on the grid before blotting for 1 s in an EM GP2 grid freezing device (Leica) at 4°C and 100% humidity and plunge frozen in liquid ethane. Specimens without ATP were prepared the same way except that the mixing step was omitted.
CryoEM data collection. Preliminary cryoEM data was collected with FEI Tecnai F20 electron microscope operated at 200 kV and equipped with a Gatan K2 Summit camera. Images with this microscope were acquired as movies with 30 fractions at 5 e/ pixel/s and a calibrated pixel size of 1.45 Å/pixel. CryoEM movies for high-resolution analysis were collected with a Titan Krios G3 microscope operated at 300 kV and equipped with a Falcon 4 camera (TFS). Automated data collection was performed with EPU. For the dataset including ATP, 10,037 movies, each consisting of 30 fractions, were collected at a nominal magnification of ×59,000, corresponding to a calibrated pixel size of 1.348 Å. The exposure rate and the total exposure of the specimen were 6.1 e − /pixel/s and~40 e − /Å 2 , respectively. For the ATP-free dataset, 8817 30-fraction movies were collected at a nominal magnification of ×75,000, corresponding to a calibrated pixel size of 1.046 Å. The exposure rate and the total exposure for this specimen were 4.2 e/pixel/second and~39 e/Å 2 , respectively.
Image analysis. Data collection was monitored with cryoSPARC Live 45 to screen and select high-quality micrographs. All other image analysis steps were performed with cryoSPARC except where mentioned. Movie fractions were aligned with patch-based motion correction and contrast transfer function (CTF) parameters were estimated with patch-based CTF estimation. After removing movies with undesirable motion or CTF fit, 7474 and 4059 movies from the dataset including ATP and the ATP-free dataset were selected for further processing, respectively. Movie fractions were aligned with MotionCor 46 with a 7 × 7 grid and averaged micrographs from the aligned movies were subjected to patch-based CTF estimation. For the ATP-free dataset, particle selection was performed with Topaz 47 . For the dataset including ATP, templates for particle selection were generated from 2D classification of manually selected particle images. After particle selection, 2,534,488 particle images were extracted for the dataset with ATP and 442,025 particle images were extracted for the ATP-free dataset. Low-quality particle images were removed with two rounds of 2D classifications, yielding 1,109,677 and 422,765 particle images for the dataset including ATP and the ATP-free dataset, respectively. Further cleaning with ab initio 3D classification and heterogeneous refinement reduced dataset sizes to 915,825 and 379,817 particle images, respectively. The remaining particle images were classified into three classes, corresponding to the three main rotational states of the enzyme, and each class was refined with non-uniform refinement 48 . For the dataset including ATP, local refinement was performed with all particle images with a mask including α 3 β 3 γδε from the F 1 region. CTF parameters of individual particle images were re-estimated with local CTF refinement, and masked refinement was performed again with updated CTF parameters. Image alignment parameters were then converted to Relion 49 .star file format with pyem (https://doi.org/10.5281/zenodo.3576630) and individual particle motion was corrected with Bayesian polishing 50 . For the ATP-free dataset, Bayesian polishing was performed with an intact ATP synthase map reconstructed with all particle images and particle images were down-sampled to a pixel size of 1.308 Å. Motion corrected images were imported back to cryoSPARC, refined, and CTF parameters re-estimated. For the dataset including ATP, particle images were initially classified into four classes. Iterative ab initio classification and heterogeneous refinement of each of the four classes yielded 27 unique rotational states of ATP synthase. The 27 structures were named State 1 binding (a to d), State 1 catalytic (a to h), State 2 binding (a to b), State 2 catalytic (a to e), State 3 binding (a to c), and State 3 catalytic (a to e), and had resolutions ranging from 4.4 to 7.8 Å after refinement. Masked local refinement of the F 1 region of State 1 catalytic (a) yielded a 3.5 Å resolution map representing the MgADP inhibited state. Local refinement of the F 1 region of combined State 1 catalytic (e to h) and State 1 binding (a to d) yielded two 4.0 Å resolution maps. To better visualize the nucleotide density in maps, density modification 51 of locally-refined maps of State 1 catalytic (a), State 1 catalytic (e to h), and State 1 binding (a to d) was performed in Phenix 52 . For the ATP-free dataset, a similar 3D classification strategy yielded nine F 1 states, namely State 1, 2, and 3 catalytic with αβ tight in a closed conformation, State 1, 2, and 3 catalytic with αβ tight in an open conformation, and State 1, 2, and 3 binding with αβ tight in an open conformation (Supplementary Fig. 1 and 3). These states included 56,739,52,468,24,879,31,559,47,065,65,651,19,922,57,468, and 23,622 particle images, respectively. Local refinement of the F 1 region with these images yielded maps at 3.4, 3.4, 3.6, 3.5, 3.5, 3.4, 3.7, 3.4, and 3.7 Å resolution, respectively. When the three rotational states are combined, the F 1 regions of State catalytic with β tight closed, State catalytic with β tight open, and State catalytic with β binding open reached 3.2, 3.2, and 3.3 Å resolution, respectively. Classification of particle images contributing to State 1 catalytic from the ATP-free dataset yielded 4 classes with different c-ring positions relative to subunit a, which demonstrates the flexibility of the peripheral stalk in the absence of free ATP (Fig. 1d, Supplementary Fig. 1). Maps of these states were calculated from 41,506, 8810, 10,550, and 1984 particle images and reached 3.8, 4.4, 4.4, and 7.1 Å resolutions after refinement, respectively. A similar classification strategy was employed with the other two catalytic states, and particle images of the most relaxed State 1, 2, 3 catalytic (a) (191,939 particle images) were used to calculate locally-refined maps of OSCP with its contact site on F 1 , the remainder of the peripheral stalk, and the F O region. These maps were combined with the map from local refinement of the F 1 region of the State 1 catalytic with αβ tight closed using the 'vop maximum' function in USCF Chimera to generate a composite map of the entire complex.
Model building and refinement. To build atomic models of the F 1 region, the crystal structure of yeast F 1 (PDB 2HLD) 53 was fitted as a rigid body into locallyrefined maps of F 1 in UCSF Chimera 54 . Models were manually adjusted in Coot 55 before being imported into ISOLDE 56 within ChimeraX 57 to improve dihedral angles and rotamer fitting. A final round of refinement was performed with Phenix and the resulting models were evaluated with Molprobity 58 and EMRinger 59 . To build backbone models of the full complex, a mosaic model was first assembled by rigid-body fitting of a yeast F 1 crystal structure (PDB 2HLD), subunits abc 10 dfi from a yeast F O cryoEM structure (PDB 6B2Z) 60 , the peripheral stalk region without subunit h from a yeast monomer cryoEM structure (PDB 6CP3) 30 , and domains of a subunit h atomic model predicted with AlphaFold 35 into the unsharpened maps. Molecular Dynamics Flexible Fitting 61 was then performed for the 27 rotational states of the dataset including ATP to generate corresponding backbone models. Figures and movies were generated with ChimeraX and UCSF Chimera.
Reporting summary. Further information on research design is available in the Nature Research Reporting Summary linked to this article.

Data availability
The data that support this study are available from the corresponding author upon reasonable request. Cryo-EM maps generated in this study have been deposited in the Received: 11 March 2022; Accepted: 5 April 2022; Fig. 4 Sequence of conformations in the ATP hydrolysis and ATP synthesis cycles. a The unstrained conformation of State 1 catalytic shows density consistent with MgADP in the αβ tight catalytic pair (left, yellow), suggesting an ADP-inhibited state, while the strained conformations show density consistent with MgATP (right, purple) suggesting an active state. b Measurement of the rotation of the rotor subunit γ within the F 1 region shows 36°and 84°sub-steps between catalytic and binding dwell conformations. c Measurement of the rotation of the c-ring relative to subunit a in F O . Within each binding or catalytic dwell conformation the peripheral stalk becomes increasingly strained as the c-ring rotates in the ATP synthesis direction and increasingly relaxed as the c-ring rotates in the ATP hydrolysis direction. Transition between catalytic and binding dwell conformations would require backstepping of the c-ring (shaded areas) if every conformation occurred during the rotary cycle. Colors in (b) and (c) are defined as in Fig. 1b