The F-, A-, and V-ATPases are unique biological rotary motors, which perform active ion transport by utilizing the energy from ATP hydrolysis (Forgac, 2007). F-ATPase as an ATP synthase functions in the mitochondria, chloroplasts, and oxidative bacteria (Walker, 2013). In archaea, A-ATPase functions as the ATP synthase similar to F-ATP synthase; its structure and subunit composition resemble those of the V-ATPase (Grüber et al., 2014). V-ATPase functions as a proton-transporting ATPase in various organelles, plasma membranes of eukaryotic cells, and bacteria (Kakinuma et al., 1999; Imamura et al., 2005; Forgac, 2007). They consist of a hydrophilic globular catalytic domain (F₁, A₁, or V₁) and a hydrophobic membrane-embedded domain (F_o, A_o, or V_o), which facilitates ion translocation across membranes (Cross and Müller, 2004), as shown in the schematic model of Enterococcus hirae V-ATPase (Figure 1).

E. hirae V-ATPase functions as a primary ion (Na⁺) pump (Murata et al., 1996, 1999, 2000, 2001, 2005a, 2008) to maintain the homeostasis of intra-cellular ionic environment at high salt concentrations outside providing high salt tolerance to this organism (Kakinuma et al., 1999). It is composed of nine subunits with amino-acid sequences that are homologous to those of the corresponding subunits of eukaryotic V-ATPases (Murata et al., 2001, 2005b; Yamato et al., 2016). V₁ is composed of an A₃B₃ hexameric ring functioning as an ATP hydrolyzing rotary motor, with its DF shaft (Saijo et al., 2011; Minagawa et al., 2013) located inside the A₃B₃ ring. A and B are the catalytic and non-catalytic subunits, respectively, which form one catalytic nucleotide-binding A₁B₁ pair and the hexameric ring is composed of three such pairs. Single molecule observations revealed that the DF shaft rotates in three 120° steps in a 360° rotation without apparent sub-steps (Minagawa et al., 2013; Ueno et al., 2014; Iino et al., 2015) and the observed dwells are thought to correspond to the catalytic dwell position (Arai et al., 2013). Its rotation speed (∼100 rps at 100 μM ATP) is comparable to that of bacterial F-ATP synthase (Iino et al., 2015). The DF shaft is connected to the c-rotor ring in the membrane via the d subunit (Saijo et al., 2011). The c decamer rotor ring which binds Na⁺ and the a subunit form the V_o domain (Murata et al., 2003, 2005a). Na⁺ is believed to be translocated across the membrane through the interface between the a subunit and the c-rotor ring, by utilizing the rotation energy of the c-rotor transmitted via the DF shaft through d subunit (Mizutani et al., 2011). Since a subunit is thought to have two half channels that open to either side of the membrane, a Na⁺ needs to rotate with the c-rotor ring in order to migrate from one half channel to the other (Murata et al., 2005a, 2008; Mizutani et al., 2011). Two peripheral EG stalks are believed to connect the V₁ and V_o domains (Yamamoto et al., 2008). Structural, single-molecule, and computational analyses of the V₁ and A₃B₃ complexes have been extensively conducted to elucidate the rotation catalysis mechanism of the V₁ rotary motor (Arai et al., 2013; Minagawa et al., 2013; Ueno et al., 2014; Suzuki et al., 2016; Isaka et al., 2017; Singharoy et al., 2017). Here, we summarize such studies and discuss the updated rotation mechanism.

Various crystal structures of the V₁ and A₃B₃ complexes of E. hirae V₁-ATPase (Figure 2) have been obtained (Arai et al., 2013; Suzuki et al., 2016). The structure of the A₃B₃ complex consists of three domains, namely the N-terminal β-barrel domain, tightly connecting the hexamer, the central domain, and the C-terminal domain, forming three nucleotide binding sites. A nucleotide binding site is formed between the A and B subunits functioning as a pair. To examine the structural differences, the C-terminal domains viewed from the N-terminal side are shown in the surface representations, with and without the nucleotides (Figure 2).

The nucleotide-free A₃B₃ (eA₃B₃) shows a unique asymmetrical structure (Arai et al., 2013); three of the A₁B₁ pairs are all in different conformations, with different nucleotide binding affinities, i.e., the ‘empty’ (ATP-unbound form incapable of nucleotide binding; see also the section of ‘free-energy calculations of EhV₁’), the ‘bindable’ (ATP-accessible form), and the ‘bound’ (ATP-bound form) forms. The asymmetrical structure of eA₃B₃ is altered upon binding of the non-hydrolysable ATP analog, adenosine 5’-(β, γ-imino)triphosphate (AMP-PNP). Binding of AMP-PNP to the ‘bound’ and ‘bindable’ forms triggers a conformational change of the ‘bindable’ form to become the ‘bound’ form (2_ATPA₃B₃). These structures suggest that the A₃B₃ complex changes its conformation from one asymmetrical structure to another, through the binding and dissociation of the nucleotides in one direction, which determines the order of ATP hydrolysis and in turn should correspond to the rotational direction of the DF shaft.

The structures of eA₃B₃ and 2_ATPA₃B₃ change in response to the binding of the DF shaft inside the ring, which results in the formation of a more tightly packed ‘tight’ form, compared to the ‘bound’ form in the A₃B₃ complex, with or without the bound AMP-PNP (2_ATPV₁ and eV₁, respectively). The DF shaft binding induced the ‘tight’ form, therefore, the ‘tight’ form is thought to be the major binding form of the DF shaft (Arai et al., 2013). The ‘tight’ form is presumably waiting for ATP hydrolysis to occur during the ‘catalytic dwell’ in the rotary cycles. In this form, the R350 of arginine-finger in the B subunit, believed to catalyze the hydrolysis of ATP, approaches closer to the γ-phosphate of ATP. The other two AB pairs of V₁ adopt either an ‘empty’ or ‘bound’ form; no ‘bindable’ form is observed, indicating that V₁ can only bind two AMP-PNP, but not three.

For the hydrolysis reaction to continue, a number of structural changes in the ‘tight’ form need to be induced via the conversion of ATP into ADP and Pi. Crystal structures of the 2ADP-bound (2_ADPV₁) and the 3ADP-bound (3_ADPV₁) V₁ complexes have recently been obtained in the presence of 20 μM and 2 mM ADP, respectively (Suzuki et al., 2016). In 2_ADPV₁, the ‘tight’ form changes to the ‘ADP-bound’ form, which cooperatively induces a conformational change from the ‘empty’ to the ‘bindable-like’ form; the ‘bindable-like’ form can bind a nucleotide, while the ‘empty’ form cannot. Since electron density of Pi is not observed, even in the presence of 200 μM Pi in the crystallization solution, Pi must have been released soon after ATP hydrolysis at ‘tight’ form, which changes the conformation of 2_ATPV₁ to that of 2_ADPV₁; 2_ADPV₁ is believed to be in the ‘ATP-binding dwell’ state, waiting for ATP to bind. This early release of Pi, in good contrast to the late release in F-ATP synthase as reported (Rees et al., 2012), may be related to their functional differences; F-ATP synthase works as both ATP synthase and ATPase but V-ATPase works specifically as ATP hydrolyzing enzyme. The DF shaft in the 2_ADPV₁ does not rotate significantly, but is tilted toward the ‘ADP-bound’ form owing to the conformational changes induced by the binding of ADP to the ‘tight’ form of eV₁. Such a tilt of the DF shaft without apparent rotation would be difficult to be recognized as an additional sub-step based on single-molecule observations.

The structural differences between the 2ADP-bound (2_ADPV₁) and 3ADP-bound (3_ADPV₁) V₁ complexes, which are considered to be induced by ADP binding to the ‘bindable-like’ form of 2_ADPV₁, were analyzed. The ‘bindable-like’ form changes to an ‘half-closed’ form. A strong electron-density peak for SO₄^2- (a Pi analog) is observed at the nucleotide binding site with ADP:Mg²⁺ in the 3_ADPV₁ complex. The DF shaft and the ‘ADP-bound’ form are slightly attracted to the ‘half-closed’ form; thus, the shifted ‘ADP-bound’ form is rather similar to the ‘tight’ conformation. The nucleotide-binding site is also more similar to that of the ‘tight’ form than that of the ‘ADP-bound’ form. This shifted ‘ADP-bound’ form of 3_ADPV₁ is coined the ‘tight-like’ form. The distances between the β-phosphate of ADP and the interacting residues in the ‘tight-like’ form are slightly longer than those in the ‘ADP-bound’ form, suggesting that the binding affinity for ADP of the ‘tight-like’ form must be lower than that of the ‘ADP-bound’ form. Consequently, an ADP molecule will be easily released from the binding site. The structure of 3_ADPV₁ is, therefore, believed to correspond to the state of waiting for ADP release (‘ADP-release dwell’) in the rotation. However, since the 3_ADPV₁ structure was obtained at an unusually high concentration of ADP (2 mM) for an E. hirae cell, the ‘ADP-release dwell’ state might be a minor intermediate state, which might exist in the catalytic cycle with high ADP and low ATP concentrations (Suzuki et al., 2016; Ueno et al., 2018).

Based on the various structures of A₃B₃ and V₁ obtained with or without the nucleotides, we propose a chronology of the main events occurring during one ATP hydrolysis and 120° rotation (Figure 3, model 1 and model 2) as follows:

In regard to the question of how the shaft rotates, several hypotheses have been put forth, such as the typical push–pull mechanism (Kinosita et al., 2004) and a type of thermal ratchet mechanism (Yamato et al., 2016, 2017). In E. hirae V₁, the DF shaft rotates 120° in one step and the traveling distance of the amino acids of the shaft to interact with the motor ring subunits during such 120° rotation in one step appears too long to dissociate from the previous motor subunits and reach the next ones through a push–pull mechanism. A thermal ratchet mechanism, therefore, appears to be utilized by V₁ instead of a push–pull one.

To directly investigate the large-scale and long-time dynamics, such as the DF shaft rotation coupled to the motions of the A₃B₃ ring in a straightforward fashion, multiscale molecular dynamics (MD) simulations were conducted using an approach combining a coarse-grained (CG) model with all-atom MD simulations (Isaka et al., 2017). A CG model of V₁-ATPase was built from the catalytic-dwell crystal structure [2_ATPV₁, PDB ID: 3VR6 (Arai et al., 2013)], and one amino-acid residue was represented by one bead located on its Cα atom (Figure 4A). Nucleotides and solvent molecules were not explicitly included in the CG model. We employed the Gō potential [the atomic interaction based CG model (Li et al., 2011)], in which the nucleotide-binding states were implicitly taken into account through the subunit structures. To optimize the CG parameters, the fluctuation of CG residues around a minimum of the Gō potential used in CG-MD simulations was matched to those of all-atom MD simulations at equilibrium near the 2_ATPV₁ crystal structure, using the fluctuation-matching methodology (Isaka et al., 2017). Tuning of the CG parameter in terms of protein fluctuation is important to simulate conformational changes induced by ligand binding, because, according to the picture rendered by linear-response theory (Ikeguchi et al., 2005), structural changes upon ligand binding occur as a response to the equilibrium fluctuation of the ligand-free state. Using the tuned CG parameter, the shaft rotation was examined using a multiple-Gō model (Okazaki et al., 2006; Okazaki and Takada, 2008; Yao et al., 2010; Kenzaki et al., 2011), in which two minima were set at the structures, one before and one after ATP hydrolysis, corresponding to the 2_ATPV₁ crystal structure and its 120°-rotated structure, respectively.

CG-MD simulations revealed structurally essential features underlying the DF shaft rotation at the CG residue resolution. Several CG-MD simulations have produced a successful 120° shaft rotation with no sub-step (panel g in Figure 4B), consistent with that observed in single-molecule experiments (Iino et al., 2015). A 120° rotation in ∼100 × 10³ steps may approximately correspond to a sub-millisecond regime, because a 120° rotation is completed within 0.2 ms according to single-molecule experiments (Iino et al., 2015). Typical conformations during shaft rotation are illustrated in panels a–f of Figure 4B, alongside the time-evolution of the rotation angle. Here, the three AB subunits are termed A_IB_I, A_IIB_II, and A_IIIB_III (panel a in Figure 4B), and they adopt the ‘empty,’ ‘bound,’ and ‘tight’ structures before the rotation, respectively. From the intermediate structures and conformational changes during rotation, two key structural features were identified: The first is that the A_IB_I pair spontaneously adopts the ‘bindable-like’ structure observed in the 2_ADPV₁ crystal structure just before the beginning of the shaft rotation (panel c in Figure 4B). Because the ‘bindable-like’ structure was not used as the input structures, this spontaneous emergence is not trivial. The formation of the ‘bindable-like’ structure was observed in all examined simulations; however, the emergence of the ‘bindable-like’ structure is not the only requirement for successful shaft rotation. In several simulations, shaft rotation did not occur spontaneously due to steric hindrance between the shaft and the B_I subunit. Said differently, the B_I subunit acts as a gate, and the shaft can pass through the gate by the creation of space between the A_III and the B_I subunits, i.e., the ‘open-gate’ conformation is achieved. The second structural feature is that the separation of the A_III and the B_I subunits from each other avoids the steric hindrance during rotation (panels d and e in Figure 4B). The maximal width of the gate was ∼28 Å, and such large openings are not observed in the crystal structures.

A possible mechanism underlying the 120° shaft rotation was proposed on the basis of the CG-MD simulations (Figure 4B). Although nucleotides were not included in the CG model, their binding states could be estimated from the conformational changes of the three AB subunits. ATP hydrolysis and Pi release occurs in the A_IIIB_III pair, and the A_III subunit moves outward (panels a and b in Figure 4B). The A_III subunit interacts with both the B_I subunit and the shaft, and then the A_III subunit pulls them outward, inducing the separation of the A_IB_I pair and a tilt of the shaft (panel b in Figure 4B). Owing to the outside movement of the B_I subunit, the A_IB_I pair undergoes a conformational change to the ‘bindable-like’ structure (panel c in Figure 4B). The emergence of the ‘bindable-like’ structure is reasonable because the incoming ATP can bind to the ‘bindable-like’ structure, which has a more open interface than the ‘empty’ structure as described in the previous section. When ATP binds to the ‘bindable-like’ structure, a closing motion of the A_IB_I pair from the ‘bindable-like’ structure to the ‘bound’ structure is induced. In addition, when ADP release occurs at the A_IIIB_III pair, the A_III subunit moves further outward. Consequently, both the outward movement of the A_III subunit and the closing motion of the A_IB_I pair cause a separation of the A_III and the B_I subunits from each other (panels c–e in Figure 4B). Owing to their movements in opposite directions, the separation becomes large, resulting in the creation of a space, i.e., opening the gate enough to avoid steric hindrance between the shaft and the B_I subunit. Finally, the shaft passes through the gate (panels e and f in Figure 4B), and then the gate closes, coupled with the closing motion of the A_IB_I pair. The closed gate prevents reverse rotation. In this mechanism, the cooperatively rearranging motions among the AB subunits play a crucial role.

Analysis of interaction patterns among the AB subunits and the shaft provide insight into the mechanism by which the dynamical rearrangements of the AB subunits propagate the shaft (Figure 4C). The region of the shaft enclosed in the A₃B₃ ring consists of two helices of the N- and C-terminus, denoted D_N and D_C, respectively. At the initial state, only the A_I subunit is in contact with D_N. After the emergence of the ‘bindable-like’ structure, the shaft is tilted, coupled with the outward movement of the A_III subunit. The tilt of the shaft triggers the rotation, and causes an inward shift of the interface between the A_I subunit and the shaft. Coupling with the closing motion of the A_I subunit itself, the A_I subunit approaches the D_C and then comes into contact with both the D_N and D_C, just before the open-gate conformation. Such a contact pattern is similar to that between the half-closed structure and the shaft observed in the 3_ADPV₁ crystal structure (Suzuki et al., 2016). After the open-gate conformation, the shaft enters into the space between the A_III and the B_I subunits and rotates, coupling to the further outward motion of the A_III subunit and the closing motion of the A_I subunit. At the final state, the A_I subunit is in contact with the D_C only. In summary, three events cooperatively contribute to the shaft rotation: (i) the closing motion of the A_I subunit pushes the shaft, (ii) the outward motion of the A_III subunit pulls the shaft, and (iii) the open-gate conformation allows the shaft to rotate.

An important difference between this mechanism and the rotation model inferred from a series of crystal structures (model 1 in Figure 3) is the A_IIIB_III pair, which is the adjacent pair of the ‘half-closed’ structure. As described above, the A_III subunit should move outward in order to avoid the steric hindrance. Therefore, the A_IIIB_III pair should adopt the open (‘empty’) structure, implying that the conformations of the three AB pairs in the 3_ADPV₁ crystal structure do not emerge together during a successful 120° rotation.

Central to the investigation of how the V₁-motor operates is the underlying free-energy change that characterizes, on the one hand, the energy source, i.e., ATP, and, on the other hand, the conformational transition, i.e., the motor action. Based on the studies of Boyer on the so-called binding-change model for the rotational catalysis in F-type ATP synthase (Hutton and Boyer, 1979), which was demonstrated by Gao et al. (2003), employing molecular simulations of the F₁ domain, a similar approach was taken for V₁. In particular, the binding affinities of the nucleotides (ATP or ADP.Pi) have been determined employing the alchemical free-energy perturbation (FEP) methodology between the ‘tight,’ ‘bound,’ and ‘empty’ pockets at the AB interface. The binding affinities of ATP to the ‘tight,’ ‘bound,’ and ‘empty’ sites are 11.6 kcal/mol, 8.9 kcal/mol, and 4.1 kcal/mol, respectively, and that for ADP.Pi is 8.9 kcal/mol at the ‘tight’ site and 4.3 kcal/mol at the ‘empty’ site (Singharoy et al., 2017). Thus, the chemical energy in terms of these binding-affinity differences to be utilized by the AB protein subunits to undergo conformational transitions and the central DF shaft to rotate is estimated to be 11.6 kcal/mol (ATP affinity to the ‘tight’ state) – 4.3 kcal/mol (ADP.Pi affinity to the ‘empty’ state) = 7.3 kcal/mol. Importantly, in the ‘empty’ site, there is minimal interaction between the R262 residue of the A subunit and the terminal phosphate of the ATP, since the conformation of the R350 residue on the B subunit prevents entry of ATP into the pocket (Figure 5). Consequently, ATP-affinity to the ‘empty’ site is the least.

The chemical energy produced due to ATP hydrolysis in an aqueous solution is readily dissipated in the environment. However, the same event occurring at protein–protein interfaces induces binding-affinity changes due to side-chain reorganization of the binding pockets, a process that occurs at a much slower timescale. Thus, the binding-affinity changes resulting from ATP hydrolysis potentially serve as a design principle that ATPase employs to prevent dissipation and to channel the chemical energy into mechanical work. Complementing the aforementioned FEP calculations, the binding affinity changes derived from single-molecule experiments, as a function of shaft rotation, reveal that a binding pocket undergoes a cycle of the tight → empty → bound transition during the 120° rotation of the shaft, during which the pocket experiences energy changes of 5–6 kcal/mol over the millisecond timescale (Adachi et al., 2012), much slower than the picosecond scale of energy dissipation of ATP hydrolysis in an aqueous solution.

Conversion of the chemical work into mechanical work is captured by employing a second type of free-energy methodology, namely geometric transformation schemes (Chipot and Pohorille, 2007). The mechanical changes within a chemically charged V₁ following the hydrolysis step is probed using string simulations with swarms of trajectories (Pan et al., 2008). Combination of the FEP and the string methodologies offers a general theoretical framework for capturing a nanoscale motor in action (Singharoy and Chipot, 2017; Benson et al., 2018). Energy changes along the most probable conformational transition pathway in V₁, underlying the rotation of the central shaft as a mechanical response to ATP hydrolysis, product (ADP.Pi) release, and binding of a new reactant ATP was found to be approximately 6 kcal/mol (Singharoy et al., 2017).

Qualitative examination of the simulations performed employing the string method with swarms of trajectories of the entire V₁ (Singharoy and Chipot, 2017; Singharoy et al., 2017) did not reveal any significant difference in the conformation of the motor structures determined by crystallography (Arai et al., 2013). Analysis of the sequence of events characterizing the conformational transition in V₁, however, unveiled additional, subtle, albeit key milestones, absent in the structural studies.

Similar to the crystallographic structures (Figure 6), first, the ‘tight’ conformation transforms into the ‘empty’ form, which prefaces further opening of the adjacent ‘empty’ interface, transforming the latter into a ‘bindable’ site. The newly formed ‘bindable’ site provides more open conformations than the ‘empty’ site, facilitating access of a loosely bound ATP to the ATP-binding residues (binding affinity 4.3 kcal/mol). The stated tight → empty transformation also weakens the AB-DF interface, allowing the bent shaft to straighten. Second, upon ATP association to the ‘bindable’ site of the A subunit, the corresponding interface closes through a hinge-bending motion to a ‘bound’ state. This bindable → bound transformation induces a wringing deformation on the straightened central shaft at the locus, where it interacts with the newly formed ATP-bound A subunit. Third, the wrung shaft rotates by 120°, featuring two ‘bound’ and one ‘empty’ state. Lastly, the ‘bound’ site already occupied by ATP evolves into the ‘tight’ form completing the rotatory catalysis mechanism in V-type ATPase. The bound → tight transformation induces a bend on the 120°-rotated shaft, reestablishing its adhesive contact with the ‘tight’ interface.

A second notable observation in agreement with the crystallographic data is that, at any point across the pathway, the nucleotide (ATP or ADP.Pi) binding ability of the binding pocket in the A subunit is correlated with the A-shaft interaction: the ‘empty,’ the ‘bound’ and the ‘tight’ sites with the lowest, intermediate and highest ATP-binding affinity, respectively, belong to domains that characterize minimal, primarily electrostatic, and combined electrostatic and hydrophobic interactions with the shaft.

Finally, a critical comparison of the simulations incorporating and devoid of the central shaft confirms that in the absence of the latter, the A₃B₃ ring is looser and more prone to energy dissipation, albeit still capable of catalytic activity, thus reinforcing the idea that the DF domain improves dramatically energy-conversion efficiency. Such loose AB interfaces have also been observed in the crystal structures of the isolated A₃B₃ bereft of the shaft, furnishing a third point of agreement between the computational and crystallographic findings.

A key, yet justifiable discrepancy between the experimental and the computational models of rotational catalysis in V-type ATPases stems from the dynamical property of the central shaft. Given that the shaft is always devoid of deformation in the various crystal structures, which represent the local free-energy minima of the conformational landscape of ATPase motor-action, it is quite intuitive to assume that the shaft rotates as a rigid-body (Arai et al., 2013). Refining this idea, the simulations reveal that the central shaft within V₁ first undergoes a wringing transformation, followed by the rotation of the deformed shaft, and finally restores its configuration in the rotated state. Indeed, at either end of the rotation, the simulations predicted, in line with the structural models, that the shaft remains bereft of deformation. However, the pathway revealed deformability of the shaft—a finding that is consistent with single-molecule experiments, which evince the possibility of energy storage within a shaft due to its inherent elasticity (Sielaff et al., 2008; Martin et al., 2018).

Deformability of the central shaft inculcates a key design principle by virtue of which the overall time of the rotation step occurs over a millisecond timescale. Because the shaft rotates in layers and not as a whole, a larger barrier of rotation is split into smaller barriers, which can be overcome in a more biologically relevant timescale. A cumulative transition time of 1.09 ms is estimated for the ATP-binding and 120°rotation (Figure 6). This time should be corrected by the diffusive ATP hydrolysis-product, i.e., ADP.Pi, release time, which was estimated in an independent study to be ∼2.6 ms (Okazaki and Hummer, 2013). Thus, one complete 360° rotation is expected to take 3 × (2.6 + 1.09) = 11.07 ms. These rate estimates add up a rotation speed of about 90.3 s^-1, which is in good agreement with the single-molecule measurement of an average rotation rate of 89–115 s^-1 (Ueno et al., 2014).

A combined rotational mechanism based on the crystal structures (Figure 3) and the computer simulation studies (Figure 4, 6, 7) is summarized as described below.

Static snapshots and dynamical simulation pictures of the V₁ rotary motor from E. hirae are presented and compared in this review article. Simulation studies provide a complementary view of the rotation and ATP hydrolysis, by connecting the static intermediate structures during rotation. Most of them are consistent and complementary: After the ‘empty’ form changes to the ‘bindable’ form, new ATP is bound to induce further conformational changes to drive the shaft rotation, which appears to undergo a wringing movement during rotation. However, certain parts are inconsistent, perhaps due to the insufficient structural information, or suboptimal simulation strategy. The unique asymmetry of the A₃B₃ ring with three identical A₁B₁ pairs is in line with its meta-stable structure. The mechanism of forming such asymmetrical meta-stable structure will be elucidated in the near future. Further biochemical, crystallographic, and long-time atomic-scale simulation studies will clarify the basic principles of the chemo-mechanical coupling mechanism of the rotary motor, transitioning between such meta-stable structures.

All authors discussed findings, analyzed literature, and wrote the manuscript.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.