In the biological self-assembly, a gain of the translational, configurational entropy of water ascribed mainly to the mitigation of water crowding (i.e., entropic correlations among water molecules) in the system plays imperative roles (Kinoshita, 2013; Oshima and Kinoshita, 2015; Kinoshita, 2016) (Supplementary Section S1). The most fundamental self-assembly is protein folding (Figure 1). The presence of the backbone or a side chain of a protein generates an excluded volume (EV) to which the centers of water molecules are inaccessible (Harano and Kinoshita, 2005; Yoshidome et al., 2008; Yasuda et al., 2010; Hayashi et al., 2017; Hayashi et al., 2018). The formation of α-helix or β-sheet is accompanied by a reduction of the total EV generated by the backbone and an increase in the total volume available for the translational displacement of water molecules in the entire system, leading to a water-entropy gain. The formation undergoes an energy increase arising from the loss of protein-water hydrogen bonds. However, this energy increase is compensated with an energy decrease by the ensuring of protein intramolecular hydrogen bonds. Therefore, α-helix and β-sheet are very advantageous structural units from both entropic and energetic viewpoints (Hayashi et al., 2017; Hayashi et al., 2018). The close packing of side chains also leads to a reduction of the total EV followed by a water-entropy gain. The geometric properties of side chains play essential roles in the close packing. It was experimentally and theoretically shown that the water-entropy gain upon folding of apoplastocyanin (a protein with 99 residues) is as large as ∼670k _B (k _B is the Boltzmann constant) (Yoshidome et al., 2008). A protein folds so that its backbone and side chains can closely be packed with the formation of as much α-helix and β-sheet as possible (Hayashi et al., 2017; Hayashi et al., 2018). It is often, however, that the overall, sufficiently close packing cannot be achieved, in which case only the portions amenable to close packing are preferentially packed: In a protein, some portions are closely packed whereas the other portions are not. This nonuniform packing is more favorable than the overall uniform packing in terms of the water entropy (Yoshidome et al., 2011; Kinoshita, 2021a).

As rationalized in Supplementary Section S1, the translational, configurational entropy of water is a crucially important factor in the functional expression of F₁-ATPase (Kinoshita, 2021a; Kinoshita, 2021b). Moreover, the nonuniform packing in F₁-ATPase plays an essential role in its rotation mechanism. Hereafter, we use the terms, “packing efficiency (PE)” and “packing structure.” For a protein complex, from a viewpoint of the water entropy, it is required that not only the backbone and side chains of each protein in the complex but also the atoms in the protein interfaces be packed as closely as possible. However, it is often impossible to meet this requirement through uniform packing. The water entropy can become higher for nonuniform packing than for the uniform packing. The nonuniform packing means that only the portions of each protein and protein interfaces amenable to close packing are preferentially packed (Yoshidome et al., 2011; Kinoshita, 2021a). The PE signifies how close the packing is: When a protein in a complex or a protein interface is more closely packed, we state that its PE is higher. The physical meaning of the PE of a subunit or a subcomplex is made more definite in Section 4.1. The packing structure represents the differences in the PE among proteins and protein interfaces in a complex.

In determining which physical factors are more important than the others, it is absolutely necessary to employ an accurate model accounting for all the physical factors and compare their relative magnitudes quantitatively. By meeting this requirement for protein folding/denaturation problems (Inoue et al., 2020a; Inoue et al., 2020b) and protein-ligand binding processes (Yamada et al., 2019), we drew the conclusion that in aqueous environment the EV effect is a pivotal physical factor. Here, the EV effect works for increasing the translational, configurational entropy of water primarily by reducing the total EV. Matubayasi and coworkers (Kamo et al., 2016; Togunaga et al., 2018), using their all-atom MD simulation with explicit water based on a new solution theory in energy representation, showed that a useful modeling of protein folding is possible on the basis of not the electrostatics but the EV effect. (Graziano and coworkers (Merlino et al., 2017) also suggested that the EV effect is a principal driving force of protein folding.) This is because, for example, the decrease in protein intramolecular electrostatic interaction energy upon folding or transition to a more compact structure is accompanied by the penalty of electrostatic component of the energetic dehydration (i.e., the energy increase): The former is quite large but almost cancelled out by the latter that is equally large (Kinoshita, 2016; Kinoshita, 2021a), which can never be reproduced by a dielectric continuum model for water. In general, for a solute in aqueous solution, the EV effect becomes stronger as its size increases.

The intramolecular and intermolecular electrostatic attractive interactions of biomolecules do not work as a driving force of the biological self-assembly. However, the electrostatics is important in the following respect. When charged groups are buried upon protein folding or protein-ligand binding, the contact of oppositely charged groups (i.e., gain of electrostatic attractive interaction between groups with positive and negative charges) are essential for compensating for the electrostatic energetic dehydration penalty, namely, the loss of electrostatic attractive interactions between the group with a positive charge and water oxygens and between the group with a negative charge and water hydrogens. (Water oxygens and hydrogens carry negative and positive partial charges, respectively.) Thus, it is important to insure the charge complementarity in the biological self-assembly. The charge complementarity achieved in the stabilization of the structure of F₁- or V₁-ATPase is to be maintained during the rotation of the central shaft.

The ATP hydrolysis reaction is extremely slow in bulk aqueous solution in the absence of a catalyst. Since an ATP-driven protein or protein complex works as the catalyst for accelerating the reaction, it is involved in the reaction through the ATP hydrolysis cycle composed of binding of ATP to it, hydrolysis of ATP into ADP and Pi in it, and dissociation of ADP and Pi from it (Kinoshita, 2021a).

In F₁-ATPase, a β subunit works as the catalyst. Let C _X (mol/L) be the concentration of X. Unless otherwise specified, we consider the following aqueous-solution condition: C _ATP is sufficiently higher than C _ADP and C _Pi such that the reaction of ATP+H₂O→ADP+Pi (ATP hydrolysis) occurs much more frequently than that of ATP+H₂O←ADP+Pi (ATP synthesis) and the overall reaction is unidirectional and not ATP synthesis but ATP hydrolysis (Supplementary Section S3); and C _ATP is sufficiently high and C _ADP and C _Pi are sufficiently low such that on the whole, binding of ATP to and dissociation of ADP and Pi from a β subunit take place (Supplementary Section S4). It follows that the ATP hydrolysis cycle occurs spontaneously for lowering the system free energy.

Kinoshita’s physical picture (Kinoshita, 2021a; Kinoshita, 2021b) is distinguished from the others in the following respects.

(1) We must consider that the system comprises not only F₁-ATPase but also the aqueous solution of ATP, ADP, and Pi in which F₁-ATPase is immersed when we refer to the decrease in system free energy upon a state change of the system. The system exchanges energy with the surroundings (i.e., the heat bath) through heat and work. For the system comprising F₁-ATPase and the aqueous solution such as the one employed in a single-molecule experiment, the pressure-volume work performed by the system against the pressure of the surroundings P, PΔV (ΔV is the change in system volume), is the only mechanical work. Since PΔV is negligibly small, the system performs essentially no mechanical work (for ΔV<0, the work is performed by the pressure of the surroundings). This argument is based on the fact that a structural change of a protein (Yoshidome et al., 2008; Inoue et al., 2020a) or protein-ligand binding (Yamada et al., 2019) is accompanied by only small ΔV and P|ΔV|<<k _B T (the absolute temperature T is set at 298 K in this article) at P = 1 atm (Supplementary Section S1.1).

The PE and packing structure related to the water entropy can be analyzed only by a statistical-mechanical approach using a molecular model for water and accounting for the complex structure at the atomic level. The measure of the PE of a subunit or subcomplex with nothing bound, η, is defined as η = “ S 1   o f   a   s u b u n i t   o r   s u b c o m p l e x ” (1a)

Here, S ₁ is the loss of water entropy (its major component is the translational, configurational entropy) upon the creation of a cavity, which geometrically matches the polyatomic structure of the subunit or subcomplex, in a fixed position within water, and S ₁∼S where S is the hydration entropy (Supplementary Section S1.1) (Hikiri et al., 2019). Smaller | η | implies higher PE. The measure of the PE of a subunit or subcomplex with chemical compound Y bound, η _Y , is defined as η Y = “ S 1   o f   a   s u b u n i t   o r   s u b c o m p l e x   w i t h   Y   b o u n d ” − “ S 1   o f   Y ” (1b)

Smaller | η _Y | implies higher PE. For the comparison between a subunit or subcomplex with nothing bound and that with Y bound in terms of the PE, Eqs 1a, 1b can be replaced by Eqs 2a, 2b, respectively: η ’ = “ S 1   o f   a   s u b u n i t   o r   s u b c o m p l e x ” + “ S 1   o f   Y ” (2a) η ’ Y = “ S 1   o f   a   s u b u n i t   o r   s u b c o m p l e x   w i t h   Y   b o u n d ” (2b)

It is unequivocal that the following equation holds: η − η Y = η ’ − η ’ Y (3)

Figure 2 illustrates the state stabilized in aqueous solution of nucleotides, which was experimentally observed (Bowler et al., 2007). Two nucleotides are bound to two of the three β subunits, respectively. AMP-PNP is an analogue of ATP and not hydrolyzed. AMP-PNP, which can thus be regarded as ATP whose hydrolysis does not occur, is used when the solution of a crystal structure is undertaken. States where the number of nucleotides bound to the three β subunits is zero, one, and three should be less stable (Supplementary Section S5), which is also the case for V₁-ATPase. (Hereafter, unless otherwise mentioned, ATP and AMP-PNP are not distinguished.) It was observed that β_DP, β_TP, and β_E take closed, closed, and open structures, respectively (Bowler et al., 2007). Namely, the backbone and side chains in a β subunit with a nucleotide bound is more closely packed than those in a β subunit with no nucleotides bound. Let β(Y) denote the PE of a β subunit to which chemical compound Y is bound and β denote the PE of a β subunit with nothing bound. It is experimentally definite that the following inequalities hold: β A T P > β ;   β A D P > β (4)

The state illustrated in Figure 2A is stabilized so that the water entropy can be almost maximized.

In aqueous solution of ATP, ADP, and Pi, even without the γ subunit, the structural rotation in the counterclockwise direction, “β[open]-β[closed]-β[closed] shown in Figure 2A” → “β[closed]-β[open]-β[closed] after 120° rotation” → “β[closed]-β[closed]-β[open] after 240° rotation” → ⋅⋅⋅, occurs in the α₃β₃ complex (Uchihashi et al., 2011). Here, β[open], for instance, denotes a β subunit possessing open structure. This experimental result indicates that the rotation mechanism is programmed in the α₃β₃ complex.

For F₁-ATPase, we then define three subcomplexes regardless of the presence of the γ subunit as follows (Figure 3A): S u b c o m p l e x   1 : β E , α E , a n d   α T P , S u b c o m p l e x   2 : β T P , α T P , a n d   α D P , S u b c o m p l e x   3 : β D P , α D P , a n d   α E .

The three subcomplexes are named in respect of their positions. Hence, after the γ subunit rotates by 120° (Figure 3B), subcomplex 3, for instance, now comprises β_E, α_E, and α_TP. β_DP, β_TP, and β_E take closed, closed, and open structures, respectively: The backbone and side chains are more closely packed in β_DP or β_TP than in β_E.

It is the best to define the three subcomplexes for V₁-ATPase here. They are defined as follows regardless of the presence of the DF subunit (Figure 3C): S u b c o m p l e x   1 : A E , B E , a n d   B T P , S u b c o m p l e x   2 : A T P , B T P , a n d   B D P , S u b c o m p l e x   3 : A D P , B D P , a n d   B E .

The three subcomplexes are named in respect of their positions. Hence, after the DF subunit rotates by 120° (Figure 3D), subcomplex 3, for instance, now comprises A_E, B_E, and B_TP. In this study, V₁-ATPase is viewed from the V_o side and the central shaft, the DF subunit, rotates in the counterclockwise direction. The structure of the A₃B₃DF complex stabilized in aqueous solution of AMP-PNP and ADP is depicted in Figure 4. The naming for the six subunits of V₁-ATPase adopted is the same as that for the six subunits of F₁-ATPase. Hence, a detailed comparison between these two rotary molecular motors can be made. (In Supplementary Table S1, we summarize the differences between previous studies by Murata and coworkers (Arai et al., 2013; Suzuki et al., 2016) and this study in naming the A-B subunit pairs.) The subscript “DP” or “TP” for “A” in the naming does not necessarily imply that ADP or ATP is bound to the A subunit.

Kinoshita’s physical picture was constructed on the basis of observations in single-molecule experiments by Noji and coworkers (Yasuda et al., 1998; Yasuda et al., 2001; Adachi et al., 2007; Watanabe et al., 2010), those by Yoshida and coworkers (Shimabukuro et al., 2003), and theoretical analyses performed by Kinoshita and coworkers (Yoshidome et al., 2011; Yoshidome et al., 2012). The basal concepts hypothesized in this physical picture can be summarized as follows.

(1) Irrespective of the presence of nucleotides bound and the incorporation of the γ subunit, the PE of each subcomplex is determined by that of the β subunit in it. For instance, when the PE for a β subunit follows the order, “β subunit in subcomplex 3”>“β subunit in subcomplex 2”>“β subunit in subcomplex 1”, the PE for a subcomplex follows the order, “subcomplex 3”>“subcomplex 2”>“subcomplex 1”.

In the construction of the physical picture detailed in Section 7.1.2, it was assumed that the following inequality holds: β A T P ‐ H 2 O > β A T P A T P ‐ H 2 O > β A T P ;   β A T P ‐ H 2 O > β A D P + P i > β P i ; β A T P > β > β P i . (7)

Comparing Order (7) with Orders (5) and (6), we notice that the inequality, β(ATP-H₂O)>β(ATP(ATP-H₂O))>β(ATP), is to be corroborated in further studies.

(3) The chemical compound bound to each β subunit is sequentially changed by the ATP binding to it, progress of ATP hydrolysis into ADP and Pi in it, and dissociation of ADP and Pi from it. As a result, the PEs of each β subunit and the subcomplex in which it is incorporated are also sequentially changed, which leads to sequential changes in packing structure of the α₃β₃ complex.

Some of the basal concepts described above are to be examined in further studies, and they are outlined in Table 1 (the others have already been justified). For F₁-ATPase, however, crystal structures are experimentally known only for the α₃β₃γ complexes with no nucleotides bound and with two AMP-PNPs bound (Kinoshita, 2021a). A crystal structure of the α₃β₃γ complex with one AMP-PNP and one ADP bound was also solved, but the ADP binding was made by a bound azide ion (Bowler et al., 2007). There is not a crystal structure where ADP is bound to a β subunit without the azide ion. Hence, the difference in the PE between two β subunits to which ATP or AMP-PNP and ADP are bound, respectively, cannot be examined. Since crystal structures without the γ subunit (i.e., the α₃β₃ complex) are not found in the literature, effects due to the incorporation of the γ subunit cannot be analyzed. These problems can be solved by treating V₁-ATPase instead.

Crystal structures were solved for the A₃B₃ complexes with no nucleotides bound (the PDB Code is 3VR2 and the resolution R is 2.8 Å) and two AMP-PNPs bound (3VR3 and R = 3.4 Å) and for the A₃B₃DF complexes with no nucleotides bound (3VR4 and R = 2.2 Å) and two AMP-PNPs bound (3VR6 and R = 2.7 Å) (Arai et al., 2013). A crystal structure is also available for the A₃B₃DF complex with two ADPs bound (5KNB and R = 3.3 Å): It is a state stabilized in aqueous solution of ADP in the absence of azide ions (Suzuki et al., 2016). Unlike for F₁-ATPase, we can analyze not only the separate effects due to the binding of nucleotides and the incorporation of the DF subunit in the A₃B₃ complex but also the difference between two A subunits to which ATP and ADP are bound, respectively, in the PE. See Supplementary Section S6 for more information.

In this study, the hydration entropy is calculated by adopting fixed structures. We believe, however, that the basal aspects of our conclusions are not likely to be altered by accounting for the structural fluctuation.

For calculating the hydration entropy S ₁ (S ₁∼S: S ₁ can be referred to as the hydration entropy) of a solute (i.e., a chemical compound, protein, or protein complex) with a fixed structure, we employ our hybrid (Hikiri et al., 2019) of the angle-dependent integral equation theory (ADIET) combined with a multipolar model for water (Kusalik and Patey, 1988a; Kusalik and Patey, 1988b; Kinoshita, 2008) and the morphometric approach (MA) (Roth et al., 2006; Oshima and Kinoshita, 2015; Hayashi et al., 2018), the ADIET-MA method. Since the ADIET is employed, not only the translational contribution but also the orientational one is included in S ₁ though the latter is much smaller (Supplementary Section S1.1). More detailed descriptions of the ADIET and the MA are given in Supplementary Section S7. The high accuracy of the ADIET-MA method was corroborated in our earlier works (Hikiri et al., 2019; Yamada et al., 2019).

We analyze the PEs of A_DP, A_TP, and A_E and those of the three subcomplexes. The key quantity is S ₁, and smaller | S ₁ | implies that the protein or subcomplex is more closely packed and its PE is higher. We use the crystal structures of the A₃B₃ complexes with no nucleotides bound and two AMP-PNPs bound (A₃B₃⋅2AMP-PNP) and those of the A₃B₃DF complexes with no nucleotides bound, two AMP-PNPs bound (A₃B₃DF⋅2AMP-PNP), and two ADPs bound (A₃B₃DF⋅2ADP). In A₃B₃⋅2AMP-PNP and A₃B₃DF⋅2AMP-PNP, AMP-PNP and Mg²⁺ are bound to A_DP and A_TP but nothing is bound to A_E. In the calculation of S ₁, AMP-PNP is replaced by ATP. Since A_E has fewer atoms than the other two A subunits, S ₁ of ATP-Mg²⁺ is added to S ₁ of any protein, protein pair, or protein complex including A_E for impartial comparison among their PEs (i.e., Eqs 2a, 2b are adopted). A similar treatment is made for A₃B₃DF⋅2ADP. For all the crystal structures, the slight overlaps of the protein atoms are removed using a standard energy-minimization technique (Yoshidome et al., 2011).

The water-entropy gain upon contact of subunits I and J, ΔS _IJ , is given by Δ S I J = “ S 1   o f   s u b u n i t   p a i r   I − J ” − “ S 1   o f   s u b u n i t   I ” + “ S 1   o f   s u b u n i t   J ” (8)

Subunit pair I-J is taken from a complex, and subunits I and J are obtained by simply separating the pair. A-B, A-DF, and B-DF are considered as the subunit pairs. Larger ΔS _IJ represents that the interface between subunits I and J are more closely packed, i.e., the PE of the interface is higher. For the interface between an A subunit with ATP bound and a B subunit, ΔS _IJ is calculated for A⋅ATP+B→A⋅ATP-B (I = A⋅ATP and J = B where A⋅ATP is an A subunit with ATP bound).

We calculate the water-entropy gain ΔS ₁ (∼ΔS) upon association of A_DP and B_DP or A_TP and B_TP for the three cases where nothing, ATP, and ADP bind to A_DP or A_TP, respectively. ΔS ₁ upon A_DP+ATP+B_DP→A_DP ⋅ATP-B_DP, for example, is “S ₁ of A_DP ⋅ATP-B_DP”−(“S ₁ of A_DP”+“S ₁ of ATP”+“S ₁ of B_DP”). A_DP, A_TP, and B_DP are obtained by simply separating A_DP ⋅ATP-B_DP in the calculation. Larger ΔS ₁ implies higher affinity between the A and B subunits and closer packing of their interface.

The packing structure of A₃B₃DF⋅2ATP, one of the most stable states, is illustrated in Figure 5A. The hydration entropies of subcomplexes 1, 2, and 3 in this complex, which provide information on their PEs, are given in Table 2. (See Supplementary Section S9 emphasizing that the calculation results are reliable.) We find the following: “ P E   o f   s u b c o m p l e x   2 ” > “ P E   o f   s u b c o m p l e x   3 ” > “ P E   o f   s u b c o m p l e x   1 ” (9) “ P E   o f   A T P ” > “ P E   o f   A D P ” > “ P E   o f   A E ” (10)

Thus, the PE of each subcomplex is determined by that of the A subunit in it, as in the case of α₃β₃γ⋅2ATP. Among the six A-B interfaces, A_DP-B_DP, A_TP-B_TP, and A_E-B_E are relatively more closely packed (A_DP-B_DP is the most closely packed). Among the six A-DF and B-DF interfaces, B_E-DF and A_TP-DF are relatively more closely packed (B_E-DF is the most closely packed).

The packing structure of V₁-ATPase (A₃B₃DF⋅2ATP) (Figure 5A) can be compared with that of F₁-ATPase (α₃β₃γ⋅2ATP) (Figure 5E). The packing structure of F₁-ATPase was unveiled in Ref. (Yoshidome et al., 2011). In F₁-ATPase, “PE of subcomplex 3”>“PE of subcomplex 2”>“PE of subcomplex 1” and “PE of β_DP”>“PE of β_TP”>“PE of β_E”.

We now discuss the values of ΔS _IJ /k _B for the DF-A and DF-B interfaces. ΔS _IJ /k _B is largest (60.9) for the DF-B_E interface and smallest (21.2) for the DF-B_TP interface (Figure 5A), and the difference between these two values, ∼40, is significantly large. The orientation of the DF subunit is determined so that the sum of the six values of ΔS _IJ /k _B can be maximized. The sum is ∼242. If the DF subunit took an unfavorable orientation and, for example, ΔS _IJ /k _B became 21.2 for all the six interfaces, the sum would be ∼127, giving rise to quite a large water-entropy loss of ∼115k _B. Thus, the water entropy is strongly dependent on the orientation of the DF subunit for a packing structure of the A₃B₃DF complex given.

The packing structure of A₃B₃ is illustrated in Figure 5B. The hydration entropies of subcomplexes 1, 2, and 3 in this complex are included in Table 2. We find the following: “ P E   o f   s u b c o m p l e x   1 ” > “ P E   o f   s u b c o m p l e x   2 ” > “ P E   o f   s u b c o m p l e x   3 ” (11) “ P E   o f   A E ” > “ P E   o f   A T P ” > “ P E   o f   A D P ” (12)

The PE of each subcomplex is determined by that of the A subunit in it. However, Orders (11) and (12) are different from Orders (9) and (10), respectively. Unlike in A₃B₃DF⋅2ATP, A_E-B_E is the most closely packed among the six A-B interfaces.

Even the structure of A₃B₃, the complex with no nucleotides bound and without the DF subunit, is asymmetric in the sense that the structures of the three A subunits (and the three B subunits) are different from one another. This result is in line with the feature of the crystal structure solved by Murata and coworkers (Arai et al., 2013). For F₁-ATPase, on the other hand, it was observed using the high-speed atomic force microscopy that the structure of the α₃β₃ complex with no nucleotides bound and without the γ subunit is symmetric (Uchihashi et al., 2011).

Figure 5C illustrates the packing structure of A₃B₃ ⋅2ATP. The hydration entropies of subcomplexes 1, 2, and 3 in this complex are included in Table 2. We find the following: “ P E   o f   s u b c o m p l e x   3 ” > “ P E   o f   s u b c o m p l e x   2 ” > “ P E   o f   s u b c o m p l e x   1 ” (13) “ P E   o f   A D P ” > “ P E   o f   A T P ” > “ P E   o f   A E ” (14)

The PE of each subcomplex is determined by that of the A subunit in it. Comparing the packing structures in Figures 5B, C, we notice that by the ATP binding to A_DP and A_TP, A_DP-B_DP and A_TP-B_TP become much more closely packed and A_E, which was the most closely packed among the three A subunits, becomes the least closely packed (in other words, A_DP and A_TP become considerably more closely packed than A_E). Among the six A-B interfaces, A_DP-B_DP, A_TP-B_TP, and A_E-B_E are relatively more closely packed as in the case of A₃B₃DF⋅2ATP (compare Figures 5A, C). The ATP binding to an A subunit results in not only higher PE of this A subunit but also closer packing of the interface between this A subunit and one of its adjacent B subunits.

The packing structure of A₃B₃DF is illustrated in Figure 5D. The hydration entropies of subcomplexes 1, 2, and 3 in this complex are included in Table 2. We find the following: “ P E   o f   s u b c o m p l e x   2 ” > “ P E   o f   s u b c o m p l e x   3 ” > “ P E   o f   s u b c o m p l e x   1 ” (15) “ P E   o f   A T P ” > “ P E   o f   A D P ” > “ P E   o f   A E ” (16)

The PE of each subcomplex is determined by that of the A subunit in it. Among the six A-B interfaces, A_E-B_E is the most closely packed. Among the six A-DF and B-DF interfaces, B_E-DF and A_TP-DF are relatively more closely packed. Comparing the packing structures in Figures 5B, D, we notice that by the incorporation of the DF subunit, A_DP-B_DP and A_TP-B_TP become more closely packed and A_E, which was the most closely packed among the three A subunits, becomes the least closely packed.

Comparison among Figures 5B–D leads us to the conclusion that the change in packing structure caused by the incorporation of the DF subunit is similar to that caused by the ATP bindings to A_DP and A_TP. Namely, the basal characteristics of packing structure shown in Figure 5A are constructed by the interplay of the two factors, the ATP bindings to two of the A subunits (factor 1) and the incorporation of the DF subunit (factor 2). The feature that all of A_DP-B_DP, A_TP-B_TP, and A_E-B_E are quite closely packed, which is found only in Figures 5A, C, originates primarily from factor 1. Factor 2 makes a larger contribution to the establishment of Orders (9) and (10) than factor 1. A_DP-B_DP becomes the most closely packed by factors 1 and 2. Another significant finding is that | S ₁/k _B | of the A₃B₃ complex decreases by ∼1,670 upon the incorporation of the DF subunit, indicating that the DF subunit largely stabilizes the A₃B₃ complex.

The packing structure and the hydration entropies of subcomplexes 1, 2, and 3 of A₃B₃DF⋅2ADP can be compared to those of A₃B₃DF⋅2ATP in Figures 5A, F and in Table 2. Orders (9) and (10) hold in both of the two complexes. However, the packing structure of A₃B₃DF⋅2ADP possesses the following characteristics in comparison with that of A₃B₃DF⋅2ATP: The interfaces of the A_TP-B_TP and A_DP-B_DP pairs are looser; and the PEs of the three A subunits and the complex are considerably lower. It is important to look at the values of ΔS _IJ /k _B for the DF-A and DF-B interfaces. The sum of the six values of ΔS _IJ /k _B in A₃B₃DF⋅2ATP is ∼242 whereas that in A₃B₃DF⋅2ADP is only ∼122. This result is indicative that the A₃B₃ complex is much more stabilized in terms of the water entropy by the incorporation of the DF subunit in the presence of two ATPs bound than in that of two ADPs. On the basis of these results and the discussion on Figure 2A given above, the stability of the A₃B₃DF complex follows the order, A₃B₃DF⋅2ATP>A₃B₃DF⋅ATP⋅ADP>A₃B₃DF⋅2ADP, and the stability difference is quite large.

Table 3 presents the water-entropy gains upon association of A_DP and B_DP or A_TP and B_TP for the three cases where nothing, ATP, and ADP bind to A_DP or A_TP, respectively. In the Table, A_DP, A_DP-B_DP, and A_DP ⋅ATP-B_DP, for example, denote isolated A_DP, complex of A_DP and B_DP, and complex of A_DP with ATP bound and B_DP, respectively. We find that the participation of ATP or ADP in the association increases the affinity between the A and B subunits. The increase is larger for ATP than for ADP.

Significant results are as follows: Upon binding of a nucleotide, the PE of an A subunit becomes higher. A single-molecule experiment for V₁-ATPase suggested that after ATP is hydrolyzed into ADP and Pi, Pi dissociates first (Iida et al., 2019). Therefore, the chemical compounds of interest are ATP, ADP+Pi, ADP, ATP-H₂O, and ATP(ATP-H₂O). As for the PE of an A subunit to which a chemical compound is bound, “A(ATP)>A(ADP)>A” is shown in this study (“β(ATP)>β(ADP)>β” should be applicable to F₁-ATPase as well). We assume that the order for V₁-ATPase corresponding to Order (7) for F₁-ATPase is the following: A A T P > A A T P A T P ‐ H 2 O > A A T P ‐ H 2 O ; A A T P ‐ H 2 O > A A D P + P i > A A D P > A ; A A T P > A (17)

Here, A(Y) denotes the PE of an A subunit to which chemical compound Y is bound, and A denotes the PE of an A subunit with nothing bound. When the chemical compound binding to an A subunit changes from ATP to ADP, for example, the PE of this A subunit becomes lower (i.e., its packing becomes looser).

In Supplementary Section S10, we briefly discuss a case where protein side chains within the protein-ligand or protein-protein binding interface are significantly flexible (Rosenzweig and Kay, 2014; Wand and Sharp, 2018), possibly more flexible than those in an isolated protein, leading to an increase in the total conformational entropy upon binding (such a case is not applicable to F₁- and V₁-ATPases).

First, we review the physical picture recently developed by Kinoshita (Kinoshita, 2021a; Kinoshita, 2021b).

Kinoshita (Kinoshita, 2021a; Kinoshita, 2021b) claimed that water plays leading roles in the functional expression of F₁-ATPase. In what follows, we refine his claim and describe it in detail. The statements are applicable to V₁-ATPase when “α”, “β”, and “γ” are replaced by “B”, “A”, and “DF”, respectively.