What can be learned about the enzyme ATPase from single-molecule studies of its subunit F1?

Abstract We summarize the different types of single molecule experiments on the F1 component of FOF1-ATP Synthase and what has been learned from them. We also describe results from our recent studies on interpreting the experiments using a chemical-mechanical theory for these biological motors.


Introduction
There is now a substantial body of research on singlemolecule studies of the component F 1 of F O F 1 -ATP Synthase (Adachi et al. 2007(Adachi et al. , 2012Noji et al. 1997;Sielaff et al. 2008;Spetzler et al. 2009;Watanabe et al. 2010;Yasuda et al. 2001) In the physiological operation of the F O F 1 -ATP synthase the proton transfer between two offset ion channels in the F O causes its c-ring to rotate. This c-ring rotation induces rotation of the central γ shaft, which controls the chemistry in the F 1 α 3 β 3 ring (Boyer, 1993;Braig et al. 2000;Junge & Nelson, 2015;Mukherjee & Warshel, 2011Senior, 2007;Walker, 2013;Weber, 2010). In treating this modular structure the system has been studied by a 'divide and conquer' method of separately analyzing the function of the F 1 ring with the embedded shaft. The crystal structure of the F 1 -ATPase (Braig et al. 2000) provides a starting point for these mechanistic studies (Walker, 2003). The kinetic study of the water-soluble F 1 -ATPase, although itself artificial, can play a key role in the understanding of the mechanism of the whole ATP synthase.
In the present paper we summarize (Section 4) what we believe has been learned from the single molecule studies, both experimental and theoretical. When coupled with information extracted from ensemble experiments (Boyer, 1993;Braig et al. 2000), single-molecule experiments in the F 1 -ATPase can uncover the dynamics of the coupling between the γ rotor and the F 1 stator (Senior, 2007;Weber, 2010), e.g., by probing the rotor angle dependence of the individual substeps in the stator. To realize this potential, structurally interpretable information needs to be extracted from single-molecule experiments. One additional item of the present paper is to illustrate the role of an analytical theory that treats the chemo-mechanics, provides a framework for treating and predicting experimental results, and suggests new experiments. The results reveal features linking single-molecule trajectories with past and future and atomistic simulations.
At the outset it should be stated that although much is being learned from the single molecule studies there are some limitations. The F O F 1 -ATP synthase consists of the two units F O and F 1 but in the single molecule studies it is the F 1 part, which has been studied thus far. Since there is then no proton gradient across the F O ion channels to drive, via the c-ring and the γ shaft, the opening and closing of β subunits in the F 1 , almost all of the single molecule studies have been performed for the reverse reaction, namely the hydrolysis of ATP (Kamerlin et al. 2013), catalyzed by the F 1 -ATPase. We note in passing that in some bacteria the F O F 1 -ATP synthase works in reverse, it hydrolyzes ATP to generate ion gradient and in vitro the enzyme can operate in both directions depending on the nucleotide and proton concentration (Pedersen & Carafoli, 1987). With the assumption of the reversible nature of the individual steps in the overall process one then learns details of the mechanism of the ATP synthesis in the system. There are nevertheless some single molecule studies described in Section 2 of ATP synthesis directly using controlled rotation (Adachi et al. 2007(Adachi et al. , 2012. In the absence of F O driven rotation, the F 1 shaft spontaneously rotates in the hydrolysis direction, but in these controlled rotation experiments the rotation and hence the function can be artificially reversed using magnetic tweezers by forcing the γ shaft to rotate in the opposite direction, thus effectively emulating the function of the F O .
A second note is that most of the single molecule studies are of a quasistatic nature, in that the rotation of the γ subunit is controlled in one way or another. Nevertheless, an important quantity, the timescale of the single-molecule events, is in the millisecond to second range, which is typical for the intact F O F 1 -ATP synthase. This value contrasts with the 'free rotation' microsecond range when the concentration of ATP is high and there is no load on the rotor. Novel information on the F 1 -ATPase is being obtained from these experiments on a fine 'scale' not physiologically accessible. In summary, much is being learned about the various dynamic aspects that one can expect to apply in the F O F 1 -ATP synthase.
The single-molecule studies are of various types: stalling, controlled and free rotation, FRET, AFM, and polarized fluorescence. Recently, we have interpreted some of these single-molecule results in terms of a reaction rate theory that combines the chemistry with the mechanical contribution associated with the motor. These studies are complementary to earlier studies largely of ensemble measurements (cf. Boyer, 1993 and references therein). To this end we describe a chemical-mechanical theory (Volkán-Kacsó & Marcus, 2015) that we have formulated to treat several types of singlemolecule experiments on this biomolecular motor (Adachi et al. 2012;Watanabe et al. 2012). The theory draws heavily on a theory of electron and other transfers in solution (Marcus, 1968(Marcus, , 1993Marcus & Sutin, 1985). We show how the torsional elasticity effects (Sielaff et al. 2008) in the motor can be coupled in the theory to various physical and chemical processes, such as nucleotide binding and ATP hydrolysis. We focus in the present review on four types of single-molecule experiments that have been performed on the F 1 component depicted in Fig. 1 of the enzyme, F 1 F O -ATPase. A summary of what we believe has been learned from these single molecule studies is given in Section 4.
The functional form of the chemical reaction rate theory that we employ was originally designed for electron transfer reactions. It has been shown nevertheless that the functional form of the reaction rate, more particularly the free energy of activation in terms of the standard free energy of reaction applies to other types of transfer processes within a range |ΔG 0 | < λ, using a notation introduced later (Lewis & Hu, 1984;Marcus, 1968;Murdoch, 1983).
In this paper we describe in Section 2 various types of single molecule experiments, in Section 3 the theory and its application to single molecule experiments on the F 1 -ATPase of thermophilic bacillus PS3 is described, and in Section 4 what has been learned from these single molecule experiments is summarized. In particular, in Section 4.2 a discussion on the dwell angles and substeps in two kinds of F 1 -ATPase, thermophilic bacillus and mammalian, is included.

Stalling experiments
In rotor angle manipulation experiments the angle θ of the rotor was externally controlled by magnetic tweezers, while rotation was optically monitored (Adachi et al. 2012). In these experiments magnetic beads were attached to the rotor (Fig. 2a) and the rotor aligned itself to the direction of a controlling external magnetic field.
An advanced version of these experiments are the stalling experiments (Watanabe et al. 2010). In these experiments the freely rotating shaft was allowed to first reach a dwell angle, then the magnetic tweezers were quickly turned on and the rotor shaft immediately assumed a chosen angle θ. The rotor was stalled at this angle for a predetermined time, then released upon which it either moved back to the original dwell or forward to the next dwell, depending on the state of the underlying process (e.g., for ATP binding either ATP-bound or empty) at the moment of release. We note in passing that earlier experiments (described later) have resolved two dwells, the binding dwell (before an 80°s ubstep, cf. Fig. 1a) and the hydrolysis dwell (before a subsequent 40°step).
The observed relative number of forward and back events as a function of stall time was given a simple two-state kinetic interpretation (Watanabe et al. 2012), which then permitted the determination of forward (k f ) and backward (k b ) rate constants of ATP binding, or other processes. From their ratio, a local thermodynamic property, an equilibrium constant for the reaction step, was also obtained. The angle dependence of the rate constants was explained in terms of the effect of the elastic response of the rotor structure on the free energy of the F 1 -ATPase-nucleotide system (Fig. 2b) in the model of elastic transfer of nucleotides or other groups (Volkán-Kacsó & Marcus, 2015). Later, we give some of the experimental and theoretical results and then show what can be extracted from these single-molecule experiments.

Controlled rotation experiments
In these experiments single-molecule fluorescence microscopy was added to the rotation microscopy and magnetic tweezers (Adachi et al. 2012). The individual binding and release events of fluorescent nucleotides were directly monitored, while the shaft was rotated at a constant rate. To detect site occupancy the binding nucleotide species (ATP or ADP) was modified by linking a fluorescent Cy3 moiety to it via a flexible alkane tether (Adachi et al. 2007). When the nucleotide was bound to the F 1 -ATPase the Cy3 moiety emitted a measurable fluorescence signal but when it was in the solution it was dark. Events whereby the occupancy changed between 0 and 1 were thereby determined and analyzed (Fig. 1c). Using a criterion to assign groups of binding events to specific β subunits, in the analysis of Kinosita and co-workers (Adachi et al. 2012) the number of (0 → 1) and (1 → 0) events over the time spent in a 0 and 1 occupancy state yielded forward and reverse rate constants, respectively. Expressions have recently been obtained permitting the extraction of rate constants where multiple events occur, so completing our treatment of single events (Volkán-Kacsó & Marcus, 2017a).

Free rotation experiments
In the first-generation experiments of this type, the unconstrained rotation of the shaft was observed by conventional microscopy, first under a heavy viscous load via imaging the motion of a microfilament attached to the rotor (Noji et al. 1997). Later, the viscous drag was reduced by attaching submicron-sized beads to the rotor (Sakaki et al. 2005;Yasuda et al. 2001). A counter-clockwise (as defined in the previous section) stepping rotation with dwell angles 120 apart was resolved which later was shown to consist of the 40°and 80°s ubsteps mentioned earlier (Watanabe et al. 2010).

High-speed free rotation experiments
In some later versions of the single-molecule rotation experiments, the size of the imaging bead was further reduced (Yasuda et al. 2001). Subsequently gold nanorods (Spetzler et al. 2009) were attached to the rotor shaft, now permitting the angular velocity during the transition between steps to be observed. In particular, the gold nanorods do not limit the rotation velocity via their friction with the water molecules. Using a μs resolution imaging of the rotor angle by monitoring the polarization direction of the light scattered by the nanorod, a feature-rich velocity profile was timeresolved, which reflected the intrinsic dynamics of the rotation (Sielaff et al. 2016;Spetzler et al. 2009).

Single-molecule FRET and polarization experiments
Another single-molecule fluorescence experiment is a more elaborate version of previous imaging techniques. An example is the single-molecule Förster resonance energy transfer (smFRET) method (Zimmermann et al. 2005), which was used in conjunction with rotation imaging to simultaneously monitor subunit conformation changes and rotation (Sugawa et al. 2016;Zimmermann et al. 2005). Another example is the time-resolved detection of the direction of a bimolecular fluorescence species attached to the β subunit, when the latter undergoes conformational changes (Masaike et al. 2008). In the fluorescent probe polarized emission occurs due to a strong dipole moment matrix element coupling the electromagnetic field and the electronic states of the fluorophore along the z axis.

Chemo-mechanical group transfer theory
The theory will be described (Volkán-Kacsó & Marcus, 2015, 2017a) by providing (1) an equation for the  (Braig et al. 2000) viewed in the plane of the α 3 β 3 ring (right). The left subfigure was reproduced with journal permission from Weber (Weber, 2010). dependence of the rate constant of a substep on the standard free energy reaction, and (2) introducing into the thermodynamic free energy equation a mechanical elastic coupling term arising from the interaction of the subunits with the γ subunit of the F 1 -ATPase. The theory and its application to predict controlled rotation data, summarized in this section to make the present review self-contained, is described in more detail in (Volkán-Kacsó & Marcus, 2017b).

Rate constants and free energy relations
We first note that the equations of the 'weak-overlap' theory of electron transfer reactions were adapted to treat 'strong overlap' reactions, such as the transfer of an atom B in the atom transfer reaction A + BC → AB + C (Marcus, 1968) and used for the latter (e.g., Chidsey, 1991;Lewis & Hu, 1984;Murdoch, 1983;Sutin, 1966). In this case the topology is such that there is in effect only one relevant potential energy surface (the lower adiabatic surface) instead of two diabatic surfaces. For such transfer reactions in solution (Marcus, 1968(Marcus, , 1993Marcus & Sutin, 1985) the free energy barrier for the reaction, ΔG*, is given by an equation, which has been approximated in the literature by a ( where, for simplicity of presentation in this paper, we have omitted the 'work terms' that occur in transitioning from a pair of reactants (r) or products (p) in solution to form a collision complex with the F 1 -ATPase. In Eq.
(1), ΔG 0 denotes the standard free energy reaction of the transfer reaction and λ denotes the 'reorganization energy' for the reaction. The rate constant k is given by where A depends on the nature of the process, e.g., whether the reaction is first or second order, and on the details of the transition state of the process. A particular example would be to use a collision theory picture (Volkán-Kacsó & Marcus, 2015) of ATP binding to the weak binding site of an empty β subunit in which the pre-exponential term A is a collision frequency Z, and a term related to an ATP binding outside of the β subunit. For the binding and release of ATP there are extensive data obtained by different methods (Adachi et al. 2012;Braig et al. 2000;Spetzler et al. 2009;Watanabe et al. 2010Watanabe et al. , 2012Yasuda et al. 2001). In the ATP binding an ATP first forms a collision complex (Oster & Wang, 2000) with the F 1 -ATPase, followed by an ATP binding as the next step (an 80°step) in the overall hydrolysis.
The simplest adaptation of Eq. (1) is to use the quadratic relationship, but, because of the different topology of the surfaces for weak and strong interaction, to set ΔG* = 0 when − ΔG 0 ⩾ λ, and ΔG* = λ when ΔG 0 ⩾ λ between the reactants. Whereas the ET reaction rate decreases at highly negative ΔG 0 the 'strong overlap' reaction rate reaches some limiting maximum value, a result clear from the topology of the adiabatic surface, instead of experiencing the 'Marcus inverted effect'. When, instead, ΔG 0 is very positive (⩾ λ) we use for this limiting case ΔG* = ΔG 0 . Accordingly, in passing we note that the relation between this model and a bond energy-bond order model (BEBO) has been discussed (Marcus, 1968 and reference cited therein). In the BEBO model the sum of the bond orders in a reaction AB + C → A + BC is assumed constant throughout the reaction step, so for an atom or group transfer a cooperative effect is seen. Here, the activation energy when B is transferred from A to C is roughly 10% of the AB bond energy for a thermoneutral reaction. A somewhat more detailed account of the application of the BEBO idea for a substep in the F1-ATPase is found elsewhere (Volkán-Kacsó & Marcus, 2017b). The relevance to processes such as ATP binding is that when an ATP enters a channel in an empty β subunit and finally locks into place in the subunit some hydrogen bonds are broken while new hydrogen bonds are formed, perhaps with an approximately conserving the total number of hydrogen bonds.

Mechanical elastic coupling
We have used the above ideas (Volkán-Kacsó & Marcus, 2015, 2017a in applying Eqs. (1) and (2) to the binding and release of ATP in single-molecule stalling and controlled rotation experiments. The dwell angles 0°, 80°, 120°, 200°, 240°, 320°and 360°are angles of rotation of the rotor with respect to the stator and are the angles of local stability.
A thermodynamic cycle (Scheme 1) was introduced by Volkán-Kacsó & Marcus (2015) to provide a basis for a relationship between the free energies of a ligand binding in free rotation, ΔG 0 0 , and the binding free energy ΔG 0 (θ) at a constant rotor angle θ. We introduced into the expressions for the free energy of the reactants and of the products an elastic coupling (Sielaff et al. 2008) between the rotor and the stator (Volkán-Kacsó & Marcus, 2015) involved in the step. In particular, to the two-state model originally used to derive Eq. (1) we added to each parabolic free energy curve an elastic term. The elastic term in the free energy equations for the reactants and products of an individual step in a subunit are given as a function of θ in Eq. (3), so yielding Eqs. (4) and (5) for the free-energy barrier as a function of θ.
where we have now included a work term W r for attaching the ATP from solution to the exterior of the F 1 -ATPase, and where the standard free energy of reaction ΔG 0 (θ) is given as a function of θ in Eq. (5).
Here, ΔG 0 0 is energy of the final dwell state minus the free energy of the initial dwell state for that step, as in Scheme 1 given earlier.

Application of theory to the single-molecule experiments
Stalling and controlled rotation experiments provide the θ-dependent rate constants and equilibrium constants for the steps in the overall process. The ΔG 0 (θ) is related to the θ-dependent equilibrium constant K(θ) for that step by ΔG 0 (θ) = −kT ln K(θ). The equation for the ln K(θ) for the ATP binding step is seen in Eq. (5) to be linear in θ, Scheme 1. Thermodynamic cycle for a substep showing reactant and product states both when the rotor is stalled at a given angle and when the system is not subject to stalling by the magnetic tweezers. 5 in agreement with the experimental data in Fig. 1. With experimental data from ensemble (Boyer, 1993;Weber & Senior, 1997) and free rotation (Adachi et al. 2012;Spetzler et al. 2009;Yasuda et al. 2001) experiments Eqs.
(4) and (5) were used to predict (Volkán-Kacsó & Marcus, 2015) for the stalling experiments the Bronsted slope ∂ΔG*/∂ΔG 0 of 0·47 which compares with the value of 0·48 in the stalling experiment for the rate of ATP binding and release over the θ range studied. For the binding rate (k f ) the slope of ln k f (θ) versus θ was also predicted for these nucleotides from the Bronsted slope and the ln K(θ) versus θ data, and found to be in reasonable agreement (∼within 10%) with experiment (Volkán-Kacsó & Marcus, 2015). For the spring constant κ of the rotor that appears in Eq. (4) the value of κ = 16 pN nm rad −2 was obtained from the stalling experiments. It was then used in the prediction of the absolute value of the binding and release rate constants in Fig. 3 for the nucleotides in controlled rotation experiments. With no adjustable parameters, good agreement was found between these experiments and our calculations. In the prediction we also used the binding rate constant obtained from the free rotation experiment for the fluorescent ATP species (Spetzler et al. 2009). The agreement between experimental and theoretical rate constants is seen in Fig. 3 where the points (symbols) are experimental and the solid curves theoretical.
The controlled rotation experiments are complementary to the other forms of single-molecule experiments in several respects. There is a region of angles θ where the controlled rotation and stalling experiments overlap, so that from the stalling data, supplemented by ensemble data it was possible, as above, to predict results of the controlled rotation experiments using no adjustable parameters. The comparison is given in Fig. 3, covering the θ-range of ∼−40°to ∼+40°. We note that for these rate and equilibrium constant calculations in Volkán-Kacsó & Marcus (2016) we did not use the Bronsted slope equation introduced in the previous study (Volkán-Kacsó & Marcus, 2015), the equation now being simpler than the one involving the Bronsted slope (one equation instead of two). In the controlled rotation experiments there are bound fluorescent nucleotides that become non-fluorescent upon release into the solution and so they are used to observe individual binding and release events, but there will also be some missed events when the rate of release is very fast. It was possible to take into account these missed events using the theory given in the previous equations. The missed events are an example of artifacts that arise in single-molecule trajectories, and theory has been used as an instrument to account for such artifacts (see e.g., Cossio et al. 2015 and references therein). The results, after correcting for these events, are given in Fig. 3. The observations of the rates without this correction for the missed events is given by the dashed lines in that figure.
Recently, the theory of elastic group was further extended in several respects (Volkán-Kacsó & Marcus, 2017a). A method was described for predicting the experimentally observed lifetime distribution of long binding events in the controlled rotation. These events involve a combination of two or more processes. Using these distributions the long binding events in the experiments can be treated and the rate constants for the hydrolysis and synthesis reactions occurring during these events extracted. As discussed in a later section a near symmetry of the data about the angle of −40°and a 'turnover' in the binding rate data versus rotor angle for angles greater than ∼40°was interpreted using diffusion-reaction kinetics.

What has been learned from single-molecule experiments
4.1 The depositing of the F 1 -ATPase on the slide does not influence the rotation kinetics A first question is the relevance of single-molecule studies to the behavior of the actual F 1 -ATPase system when in the former the F 1 -ATP lies on some surface rather than diffused in solution. In Fig. 4 single-molecule and ensemble kinetic data are plotted as a function of ATP concentration: the rate of the F 1 -ATPase rotation in single-molecule experiments and the number of hydrolyzed ATP molecules per 120°rotation in the ensemble experiments.
These Michaelis-Menten curves from single-molecule rotation experiments are practically identical with the corresponding data from ensemble measurements. This important property illustrates that F 1 -ATPase molecules attached to a microscope slide and imaged using a filament or a micro bead behave in this respect as if they were simply suspended in solution. Hence, on that basis, the kinetic and dynamic information extracted from single-molecule rotation experiments are relevant to the physiological functionality of the F 1 -ATPase.

4.2
There is a concerted kinetics of binding, hydrolysis and release events in the 3 active subunits of the F 1 A scheme shown in Table 1 has been established by a series of single-molecule experiments, in which ATP binding and ADP release (from a different β subunit) are concerted (Beke-Somfai et al. 2012) events (80°substep) and ATP hydrolysis and Pi release (again, from a different β subunit) occur during a 40°substep (Watanabe et al. 2010).
The single-molecule findings leading to the coupling scheme are the following: 1. ATP binding occurs during the 80°step: The dwell before an 80°substep is ATP concentration dependent and the dependence is linear at low concentrations (Michaelis-Menten curve), identifying it as a binding dwell. ATP binding is considered the first step in the kinetic scheme, so the binding dwell is set to 0° (Yasuda et al. 2001). 2. ATP hydrolysis occurs during a 40°step much later in the scheme, namely during the 200°to 240°step: Mutations at the catalytic pocket have been shown not to affect the 80°s tep but to reduce the length of the dwell before a 40°s tep by 2 orders of magnitude, indicating that the hydrolysis occurs during the latter. Using a mixture of ATP and Cy3-ATP, it was shown in fluorescent trajectories that a Cy3-ATP bound at 0°will undergo hydrolysis during the 200°to 240°step (Adachi et al. 2007). 3. ADP release occurs during an 80°step, namely from 240°t o 320°: Using a mixture of ATP and Cy3-ATP, fluorescent Cy3-ATP was seen to undergo hydrolysis (the Cy3-ATP in solution, like ATP, induced the spontaneous rotation in F 1 -ATPase) and the resulting fluorescent Cy3-ADP was released during 240°to 320°step (Adachi et al. 2007). 4. Pi release occurs during the last 40°step, from 320°to 360°: Increasing the concentration of Pi retards the rate of the 40°step, without affecting the 80°step (Adachi et al. 2007). In single-molecule experiments this finding is explained by the rebinding of Pi at high Pi concentrations, which will effectively lengthen the dwell before the 40°step. Accordingly, Pi release is associated with a 40°s tep (Adachi et al. 2012), but which of the three? The hydrolysis occurs at 200°to 240°, so Pi release cannot occur before 200°. Stalling experiments showed that the hydrolysis step 200°to 240°is reversible at low Pi concentrations, so Pi remains in the pocket until the last 40°step, namely the 320°to 360°step (Watanabe et al. 2010). Accordingly, the release of the P i is preceded by the ADP release.
A crystallographic method of studying the chemo-mechanics of rotation is to use substitutions in the binding species and thus prepare various stable states that can be resolved in X-ray structure (Bason et al. 2014). Using this approach in the case of mammalian (human and bovine) F 1 -ATPase Walker and coworkers (Bason et al. 2014) found three such states about 65°, 25°and 30°apart. A somewhat similar 80°, 10°and 30°division of the 120°for the dwells of the thermophilic Bacillus ATPase studied in current single-molecule experiments has been suggested (Adachi et al. 2007) based on a mutant that retards the rate and permits an improved time-resolution. Kinosita and coworkers (Adachi et al. 2007) suggested that the combined 10°and 30°substeps give rise to the 40°substep resolved in previous single-molecule experiments (Nishizaka et al. 2004) and associated with hydrolysis and Pi release in a different subunit (Watanabe & Noji, 2014;Watanabe et al. 2010). The 10°is presumably the hydrolysis step, a similar small 15°hydrolysis step is supported by a very approximate independent analysis (Volkán-Kacsó & Marcus, 2017b), but the latter can be improved when more accurate kinetic data in the stalling experiments become available. If the highly time-resolved experiments, resolved on the microsecond scale (Sielaff et al. 2016), become available at lower ATP concentrations they too can provide relevant data on these divisions of the 120°. So, a further study of the dwell angles is of interest in relating results from these different approaches.

Unidirectional concerted stepping rotation occurs in the presence of fluorescent ATP and in mutant F1-ATPase species
The kinetics in the F 1 -ATPase, when only the fluorescent Cy3-ATP was in the buffer, (there was no ATP), was shown by Kinosita and co-workers (Nishizaka et al. 2004) to be practically identical with the kinetics when the rotation was driven by unlabeled ATP. While the chemo-mechanical coupling and the stepping were identical, the average rate of rotation was lower with Cy3-ATP than with ATP, owing to the lower rate of Cy3-ATP binding.
In single-molecule experiments, such as stalling, one of three β subunits was mutated to reduce the hydrolysis and catalysis rates, which was needed to properly resolve and study the catalytic step (Adachi et al. 2012). The rates were further reduced by using a slowly hydrolyzing mutant nucleotide analog ATPγS instead of the ATP itself. Again, the stepping, directional rotation and so the coupling scheme was not altered by these changes, only smaller rate constants were seen in the free rotation trajectories (Nishizaka et al. 2004).

Comparison of single occupancy experiments with multiple occupancy experiments and comparison of corresponding rates
The θ-dependent rate of ATP binding is not influenced by the occupancy of the neighboring sites and thus by the total occupancy (Sakaki et al. 2005). In the stalling experiments (Watanabe et al. 2012) used to study ATP binding, when the rotor angle was stalled at angles ranging from about −45°to +45°, the events of ATP binding and release as a function of stalling time followed a two-state kinetics. During longer stall times multiple binding and release events occurred. When an ATP is first bound, the binding induces a fast release of ADP from the counter clockwise neighbor β subunit (Adachi et al. 2007). Given the low ADP concentration maintained in the experiments (to prevent the so-called 'ADP inhibited' state from forming) any subsequent ATP binding and release events occurred with only a Pi in the latter β subunit. Accordingly, whereas the first ATP binding event occurred in the presence of ADP in the second subunit, any ATP release and subsequent ATP binding events occurred without the ADP in the second subunit. Yet, when the stall times for this ATP binding/release step were increased, the statistics of forward and backward steps and hence the θ-dependent rate constant of ATP binding were not altered.
In fluorescent Cy3-ATP free rotation (Adachi et al. 2012) and controlled rotation experiments (Watanabe et al. 2012) there was no difference in the rate constant of Cy3-ATP binding, even though the site occupancy was 2-3 in the former and only 0 and 1 in the latter. In other experimental studies that accessed very low nM ATP concentration regimes it was also concluded that the rate constant of binding of ATP was not influenced by the nucleotide occupancy in the neighboring subunits (Sakaki et al. 2005).
In contrast, the release rate of ADP is determined by whether ATP is bound or not to a neighboring site. In the chemo-mechanical scheme given in Table 1 the ADP release during the 80°step is faster than 1 s −1 . This upper limit is due to the rate limiting hydrolysis step seen in the highresolution single-molecule rotation experiments (Yasuda et al. 2001). Meanwhile, the spontaneous release rate of ADP, i.e., when it is not induced by the binding of an ATP to the clockwise neighbor, is, very slow, on the order of tens of seconds (Adachi et al. 2012). Consequently, the binding of ATP accelerates the release of ADP in the counter clockwise subunit by at least 4 orders of magnitude, reflecting the role of conformational changes induced by the binding.
The dwell angle increases in the counter clockwise direction. The species occupying the pockets of ring β subunits 1, 2, and 3 are represented in the binding and catalytic dwells.

8
The above observations have been used in our theoretical treatment of ATP binding and release rates in controlled rotation and stalling experiments (Volkán-Kacsó & Marcus, 2015, 2017a. They establish the condition that the 80°step is controlled by the ATP binding/release events in a single β subunit: these events induce opening and closing conformational changes in that subunit while any changes in the other two subunits would 'tag along'. Using a single reaction coordinate associated with the first β subunit, the changes in the latter were taken into account (Volkán-Kacsó & Marcus, 2015).

4.5
The concerted kinetics in the ring subunits drives a unidirectional rotation kinetics even in the absence of a rotor shaft High-speed AFM measurements (Uchihashi et al. 2011) in a rotorless F 1 -ATPase demonstrated that the opening-closing kinetics of the β subunits follow a unidirectional counterclockwise rotation pattern. The result is consistent with the kinetic scheme in the counter-clockwise rotation observed in the complete F 1 , albeit the individual steps in the rotorless F 1 -ATPase are slower. Using the rotation pattern, Noji and coworkers (Uchihashi et al. 2011) extracted the rotation rate constant versus ATP concentration reproduced in Fig. 5.
The strong dependence of binding and release rate constants on the rotor angle could suggest, as previously proposed, a so-called 'γ-dictator' model (Uchihashi et al. 2011), whereby this concerted mechanism in the hydrolysis and presumably in the synthesis cycles is due to the interaction between rotor and ring subunits, alone. Indeed, our present treatment of controlled rotation experiments indicates that the single-site activity can be described by such a model. However, when considering higher nucleotide concentrations, in particular in rotation experiments for F 1 -ATPase under physiologically relevant conditions, giving rise to a multi-site activity, events of binding/release and hydrolysis/ synthesis occurring in pockets of different β subunits are directly coupled. These inter-subunit interactions are strong enough to determine the directional rotation kinetics even without the γ shaft (Uchihashi et al. 2011), or a truncated shaft (Furuike et al. 2008). So while the angular position θ of the latter does affect the rates of the processes in the β subunits, it is not essential in mediating the interactions between the β subunits that lead to the concerted kinetics.

Direct information on the opening and closing of the subunits
Several single-molecule experiments have provided methods to monitor the opening-closing kinetics.
1. Optical imaging of F 1 -ATPase rotation was used in tandem with single-molecule imaging of polarized light emission from fluorophores attached to a β subunit (Masaike et al. 2008). With this arrangement the directional changes of the subunits in the plane of the F 1 ring were monitored. It was demonstrated that in addition to hinge bending conformational changes, these subunits also undergo a twisting motion during the chemo-mechanical cycle. 2. In combined rotation and FRET imaging experiments the subunit conformational changes were monitored while the F 1 underwent typical stepping rotation (Sugawa et al. 2016). The FRET signal was analyzed using statistical theoretical methods and showed that in the active F1-ATPase the empty subunit is in an open conformation while the nucleotide-bound subunits are in closed conformations. 3. As described earlier, high-speed AFM measurements were used to monitor the rotary kinetics of opening and closing of the β subunits (Uchihashi et al. 2011). As in item 2 above, in these experiments the empty subunits awaiting the binding of an ATP were in an open state, while the other two subunits, presumably with nucleotides in their catalytic pockets, were in a closed state.
4.7 Exponential rate and equilibrium constant versus rotor angle dependence of binding/release and catalysis As described in Section 3 stalling and controlled rotation experiments revealed exponential-like forward and back rate constants for ATP binding in the region of angular overlap in these experiments (Adachi et al. 2012;Watanabe et al. 2012). Such exponential functions apply to the hydrolysis, Pi release, and various nucleotide binding rates in regions where these events occur in spontaneous rotation. The forward and back rate constants have opposite angledependent trends, and the equilibrium constant is strictly exponential, as predicted by theory (Volkán-Kacsó & Marcus, 2016).

4.8
There is a turnover in the ATP binding rate around 80°T he turnover in the ATP binding rate constants versus rotor angle plot seen in Fig. 6a indicates that the closing of the binding channel continues as the angle is rotated beyond 80°. The turnover is consistent with structural data and the idea of a continuum of open-to-close states between −40°a nd 140°.
We suggested (Volkán-Kacsó & Marcus, 2016) that a diffusion process while the ATP is entering the channel in the subunit after the initial binding of the ATP to the outside of the F 1 -ATPase. The latter then serves as the reactant state in the transfer process. The binding channel leading to the binding pocket (Fig. 6c), which closes as the rotor is moved in the positive direction (Fig. 6b) retards the diffusion process to the extent that it becomes the bottleneck during binding for angles beyond 50°or so. Conversely, in this standard diffusion-reaction scheme (Marcus, 1960;Noyes, 1961), the transfer reaction is the bottleneck in the θ-region from about −45°to +45°where stalling and controlled rotation experiment overlap.
4.9 Near symmetry of the kinetic data about −40°and about +140°A s seen in Fig. 6c the β subunit is the most open at −40°and the most closed presumably at +140°(0°is defined as the binding dwell angle at which the β subunit in question is empty and is about to undergo ATP binding associated with a subsequent 80°substep). The minimum at −40°d oes not correspond to a specific dwell angle. The structural changes of opening and closing are consistent with the trends of binding/release versus rotor angle data and with spontaneous free rotation experiments. In particular: 1. Moving the rotor to the left or right starting from −40°in both cases induces a closing of the β subunit which accelerates binding. This behavior is consistent with the spontaneous forward rotation induced by binding of ATP in free rotation experiments occurring in a θ-range to the right of −40°(from 0°to 80°). 2. Rotating the γ rotor towards −40°from the left or right in both cases induces an opening of the β subunit, which accelerates release. This result, in turn, is consistent with the spontaneous forward rotation induced by the release of ADP in free rotation experiments occurring in a θ-range to the left of −40°(from of −120°to −40°).
A degree of asymmetry is also seen in the binding/release rate versus rotor angle data relative to the angle from −40°i n both directions: Notably, in the presence of Pi in the solution, ATP binding to the right in Fig. 6a occurs at much higher rates than ATP binding to the left. Thus, Pi binding hinders ATP binding in the synthesis direction, but does not prevent ADP binding, a behavior that is consistent with the hydrolysis scheme being the inverse of the synthesis scheme established by single-molecule experiments (Watanabe et al. 2010;Yasuda et al. 2001). (In the hydrolysis direction Pi is the last to be released during the final 320°-360°substep.) 4.10 In the active F 1 -ATPase the Pi release occurs after ADP release In the chemo-mechanical scheme from Table 1 established by the single-molecule experiments, Pi is the last to be released during the final 320°-360°substep. A corresponding structure, i.e., with ATP in one subunit, ATP (or ADP and Pi) in another subunit and Pi only in the third subunit has yet to be found. This observation raises the possibility that the 'ground state' structure reported in X-ray crystallography (Braig et al. 2000) is a stable structure that is not on the kinetic pathway of F 1 -ATPase rotation. Based on recent stalling experiments it was suggested that the crystallography structure, in which one subunit appears to contain ADP without Pi, is the so-called ADP inhibited state (Watanabe & Noji, 2014). This result is a reminder that an X-ray structure may sometimes represent the most stable structure and not always the kinetically active structure. It is clear that the structural information is an essential component for understanding the mechanism and that the kinetic data can provide useful additional data for determining the detailed mechanism.
4.11 Angle-dependent rate constants can be extracted from the controlled systems such as stalling or controlled rotation experiments As seen in Fig. 2, the rate constant versus rotor angle are quasi-exponential and in the range of overlap between the two experiments the forward rate constants and the backward rate constant have opposite slopes. The elastic group transfer theory was used to interpret and predict these angle-dependent kinetic (Volkán-Kacsó & Marcus, 2016). Furthermore, we recently devised a method to use the lifetime distribution function of long binding events from the controlled rotation experiments to extract the rate constants for hydrolysis and synthesis (Volkán-Kacsó & Marcus, 2017a). We note that due to the angle dependence of the binding, chemical reaction and release rate constants, these waiting time distributions do not necessarily follow simple exponential dependence, and careful treatment is needed to use the data for extraction of the kinetic properties from these distributions.

4.12
There is a relation between the rate constants in stalling, controlled and free rotation experiments The angle-dependent rate constants in Figs 3 and 6 have been used (Volkán-Kacsó & Marcus, 2017b) to calculate the rate constants that can be extracted from the distribution of dwell times in free rotation trajectories. To do so, the fluctuations of the rotor angle about the initial dwell angle were used to calculate the average rate constants (Volkán-Kacsó & Marcus, 2017b). In particular, for the ATP binding step, the initial angle fluctuates about θ i = 0°for the forward process or it fluctuates about θ i = 80°for the reverse process. The idea of using angle-dependent rate constants, k f (θ) and k b (θ), arises from an approximation, which is valid on the ms to s timescales resolved in free rotation experiments.
In the latter are used microbeads, in which the stepping in the rotation appears as a discrete process of jumping between subsequent dwells, 0°, 80°, 120°, … In this case, transition state theory and hence a rate description is applicable. A quasi-static θ applies whenever a microbead is used with a slow response time due in part to the friction with the water molecules in the buffer solution.
The rotation rates reported for ATP synthesis in bacterial, mitochondrial and chloroplast enzymes under physiological conditions are slow, about 100 s −1 (Cross et al. 1982;Etzold et al. 1997;Junesch & Gräber, 1985), and are similar to singlemolecule studies on F 1 -ATPase. The latter probes ms to s timescales, relevant to the physiologically relevant function. The viscosity-loaded conditions imposed by the use of the microbeads, are also physiologically relevant, as demonstrated by Junge and coworkers (Panke et al. 2001), who argued that a compliant rotor running under high load (and hence elastically twisted) is necessary for an efficient coupling between the F O and F 1 , given that their subunit counts are incommensurate.

There is a harmonic behavior of the restoring force
The Gaussian distribution of the fluctuations about the dwell angle seen in single-molecule experiments (Adachi et al. 2012;Sielaff et al. 2008) correspond to a harmonic behavior. The elastic behavior is due to the twisting and bending of structural elements in the ATPase, such as the rotor and the β lever arm, described by a spring constant κ (Volkán-Kacsó & Marcus, 2015). The linearity of the plot of the logarithm of the equilibrium constant versus θ supports the assumption that the same κ can be used in the two terms in Eq. (3).
4.14 Using the rate constant versus rotor angle data one can predict the dwell angles and the step size in the free rotation The theory has provided simple mathematical relations between the dwell angles, the spring constant and log k versus θ slopes (Volkán-Kacsó & Marcus, 2016, 2017a When the logarithm of the ratio of the forward and reverse rate constants is linear in θ the difference of dwell angles bracketing the step can be predicted (Volkán-Kacsó & Marcus, 2017b). To do so an independent estimate of the spring constant is necessary, e.g., from fluctuation experiments (Sielaff et al. 2008). Then using the angle-dependent rate constant data from stalling experiments, the difference of dwell angles bracketing the hydrolysis step were predicted from the free rotation trajectories (Volkán-Kacsó & Marcus, 2017b). We note that structurally the hydrolysis step is coupled to structural changes of the ring subunits and thus the γ shaft (Beke-Somfai et al. 2013). The linearity condition was not fulfilled for the ATP binding step but was roughly fulfilled for the hydrolysis although the present data are relatively sparse (Fig. 3c in Watanabe et al. 2012).

What is not known from current single-molecule experiments
a. The single molecule studies have been performed in the F 1 -ATPase and not on the F O F 1 ATP synthase; b. Detailed correlation at the atomic level of the shaft rotation in relation to the opening and closing and other conformational changes of the β subunits are not clarified; c. Improved resolution of the dwell angles.

Concluding summary
In summary, single-molecule experiments have revealed a number of new features on the mechanism of F1-ATPase, also relevant for the understanding of the behavior of the complete ATP synthase. Among these features are the following: a. Similar behavior is found for F1-ATPase on a microscope slide and F1-ATPase dispersed in solution; b. Concerted kinetics of binding, hydrolysis and release events in the 3 subunits during a rotation with 40°and 80°substeps; c. In F 1 -ATPase rotation hydrolysis occurs 200°after ATP binding and Pi release follows ADP release; d. The ring subunits drive a unidirectional counterclockwise rotation even in the absence of a rotor shaft; e. Opening and closing of the β subunits during nucleotide binding and release drive shaft rotation; f. There are quantitative relations between the rate constants in stalling, controlled and free rotation experiments;