Importance of water entropy in rotation mechanism of F₁-ATPase

Abstract

We briefly review our theoretical study on the rotation scheme of F₁-ATPase. In the scheme, the key factor is the water entropy which has been shown to drive a variety of self-assembly processes in biological systems. We decompose the crystal structure of F₁-ATPase into three sub-complexes each of which is composed of the γ subunit, one of the β subunits, and two α subunits adjacent to them. The β_E, β_TP, and β_DP subunits are involved in the sub-complexes I, II, and III, respectively. We calculate the hydration entropy of each sub-complex using a hybrid of the integral equation theory for molecular liquids and the morphometric approach. It is found that the absolute value of the hydration entropy follows the order, sub-complex I > sub-complex II > sub-complex III. Moreover, the differences are quite large, which manifests highly asymmetrical packing of F₁-ATPase. In our picture, this asymmetrical packing plays crucially important roles in the rotation of the γ subunit. We discuss how the rotation is induced by the water-entropy effect coupled with such chemical processes as ATP binding, ATP hydrolysis, and release of the products.

F_oF₁-ATP synthase is a molecular motor found in biological membranes. It is composed of a water-soluble part, F₁, and a membrane-embedded part, F_o. Coupled with proton transport across membranes, F₁-part of the synthase produces adenosine triphosphate (ATP) from adenosine diphosphate (ADP) and inorganic phosphate (Pi). F₁-ATPase, F₁-part of the synthase, is a rotary molecular motor induced by such chemical processes as the ATP binding, ATP hydrolysis, and release of products (ADP and Pi). In the present review, we focus on the α₃β₃γ complex in F₁-ATPase, which has widely been studied in single-molecule experiments1–10. According to the atomic-level crystal structures11–14 of the α₃β₃γ complex, the α₃β₃ subunits are arranged hexagonally around the γ subunit as shown in Figure 1. During the cycle of the chemical processes mentioned above, the γ subunit rotates in a counterclockwise direction when it is viewed from the F_o side₁. The γ subunit performs a 120° step rotation during hydrolysis of a single ATP molecule2. This step is further resolved into 80° and 40° substeps. The 80° substep is induced by ATP binding3 and/or release of ADP5. Subsequent ATP hydrolysis occurs in 1ms without the rotation4, which is followed by the 40° rotation accompanying release of Pi6. Release of ADP occurs during the 80° rotation5. The 80° and 40° dwells are referred to as catalytic and ATP-waiting dwells, respectively4.

Figure 1.

Ribbon representation of the α₃β₃γ complex viewed from the F_o side11. The γ subunit, represented by yellow ribbon, rotates in a counterclockwise direction1. This figure is drawn using the DS visualizer 2.5.

According to the crystal structure obtained by Abrahams et al.11 in 1994 (Fig. 1), two of the β subunits named β_TP and β_DP are in closed conformation and the other one named β_E takes open conformation, respectively. AMP-PNP (an analog of ATP), and ADP are bound to the β_TP and β_DP subunits, respectively. Nothing is bound to the β_E subunit. The conformations of the three α subunits, on the other hand, are almost the same. (The three α subunits are named α_TP, α_DP, and α_E, respectively, as shown in the Fig. 111.) Most of the crystal structures which have been reported so far represent essentially the same characteristics though the substrates bound are different12,13. For example, in the crystal structure obtained by Bowler et al.13 AMP-PNP binds to the β_DP sub-unit. (It has been experimentally shown that ATP is hydrolyzed in the β_DP subunit7. Thus, ATP binds to the β_DP subunit as well.) The 120° rotation of the γ subunit accompanies the interconversion of the β-subunit structures (β_DP→β_E, β_TP→β_DP, and β_E→β_TP). The structures of the α₃β₃γ complex before and after the 120° rotation are the same. According to single-molecule experiments7–9, most of the crystal structures are in the catalytic dwell state. This implies that the 40° rotation first occurs from the crystal structure, and the 80° rotation is then induced by the ATP binding and/or release of ADP.

Direct interactions between the γ and β subunits have been proposed as important factors for the rotation14–16. For example, Ma et al.14 have suggested that the electrostatic interaction between Arg and Lys residues on the protruding portion of the γ subunit and negatively charged residues of the β subunit, known as the DELSEED motif, plays essential roles in the rotation. However, Hara et al. have shown that the rotation is not influenced by the mutation of a residue in the DELSEED motif to Ala17, indicating that the electrostatic interaction between the γ and β subunits plays no important roles for the rotation. Furthermore, according to the experimental result by Fruike et al.18, the γ subunit retains its rotation in the correct direction as well as its ATP-hydrolysis activity even when most of its axle including the DELSEED motif is removed. These experimental results imply that the rotation of the γ subunit is induced not by the direct interaction between the γ and β subunits but by another factor.

We have recently proposed the rotation scheme of the γ subunit based on the water-entropy effect19. In the present review, we briefly discuss how the water-entropy effect plays essential roles in the rotation of the γ subunit. We point out that the asymmetric packing found in the α₃β₃γ complex and the water-entropy effect provide important clues to the mechanism.

Water-entropy Effect in Biological Systems

Importance of water-entropy effect

Our recent theoretical analyses based on statistical thermodynamics of fluids have shown that the water entropy is the key quantity in elucidating the mechanism of such processes as protein folding/unfolding, molecular recognition between guest ligands and host enzymes, and aggregation of protein molecules like the amyloid-fibril formation20,21. As an illustration of the water-entropy effect, we consider a tight packing of the three side chains shown in Figure 2. The tight packing is induced by the excluded volume (EV) effect. Here the EV is defined as the volume which the centers of water molecules cannot enter22. In the left-hand side of Figure 2, for example, the EV corresponds to the molecular volume of the side chains plus the volume shown in gray. When a tight packing is formed, the EVs overlap, leading to a reduction of the EV. This decrease provides a corresponding increase in the total volume available to the translational displacement of the coexisting water molecules and in the number of accessible configurations of the water. Thus, tight packing leads to a gain in the water entropy. The native structure is the structure with almost the maximum water entropy20,21. Note that the interaction between protein atoms arising from the water-entropy effect is an indirect interaction via water molecules. According to the usual view23, the water adjacent to a nonpolar group is entropically unstable, and protein folding is driven by the release of such unfavorable water to the bulk through the burial of nonpolar groups. However, we have shown that the entropic gain originating from this view is too small to reproduce the experimental result of apoplastocyanin (apoPC) folding24. Our recent studies on solvation thermodynamics have shown that “hydrophobic interaction” is driven primarily by the water entropy gain originating from the EV effect20.

Figure 2.

Close packing of three side chains. The excluded volume generated by a side chain is the volume occupied by the side chain itself plus the volume shown in gray. When side chains are closely packed, the excluded volumes overlap. The total volume available to the translational displacement of water molecules increases by the overlapped volume shown in black, leading to a water-entropy gain.

The above explanation is made under isochoric (constant-volume) condition while experiments are performed under isobaric (constant-pressure) one. However, it is experimentally known that the volume change upon folding ΔV is almost 025,26 (V is the partial molar volume of a protein, which is defined by the volume change upon insertion of a protein). Thus, protein folds under constant-volume and constant-pressure conditions. The above explanation is also valid under isobaric condition. The reason why ΔV is almost 0 is explained as follows. As discussed in ref 27 and in Supporting Information, V can be expressed as

$$V\approx V_{\text{ex}}-\zeta A,$$ (1)

where V_ex is the EV, A is the water-accessible surface area, and ζ (>0) represents the average number density of water near the protein surface. When protein folds, both of V_ex and A reduce. The change of the volume upon folding becomes nearly 0 because the reduction of V_ex is almost cancelled out by the second term of Eq. (1). In the later subsection (“Hydration entropies of three sub-complexes”), we validate that the rotation mechanism of F₁-ATPase can also be discussed under isochoric condition.

The validity of our picture of protein folding has been shown in experiments. Here we discuss the experimental results by Terazima and coworkers. They developed a novel experimental technique which enables us to directly measure the enthalpic change, ΔH, upon protein folding at 298K26. They showed that folding of apoPC accompanies an enthalpic loss of 870 kJ/mol that is very large. This indicates that the folding leads to a great gain of the water entropy which surpasses the enthalpic loss and a loss of the protein conformational entropy. The water-entropy gain upon folding can be estimated as follows20,24. The free-energy change upon folding, ΔG, can be expressed as ΔG=ΔH−T(ΔS_C+ΔS) where ΔS_C is the conformational-entropy loss and ΔS is the water-entropy gain. The loss −TΔS_C can be estimated to be in the range, 305 kJ/mol < −TΔS_C < 956 kJ/mol24. It is assumed that the free-energy gain takes the most probable value shared by a number of proteins, −50 kJ/mol. Using ΔH=870 kJ/mol, the water-entropy gain is estimated to be in the range, −1876 kJ/mol < −TΔS < −1225 kJ/mol. It follows that quite a large water-entropy gain occurs during the folding of apoPC. This large gain can never be explained by the contribution from the water molecules near the protein surface alone20,21.

The enthalpic loss upon folding of apoPC can be explained as follows20,21. Protein folding leads to a lowering of the intramolecular energy by the formation of hydrogen bonds and salt bridges. We note, however, that the interaction between protein atoms and water molecules (e.g., protein-water hydrogen bonding) is lost upon the folding, leading to an increase in the protein-water interaction energy, the so-called dehydration penalty. The dehydration penalty is largely compensated if the intramolecular hydrogen bonds and/or salt bridges are formed. However, the formation is not always reached, giving rise to the dehydration penalty. The experimental result of apoPC indicates that the dehydration penalty is dominant and the enthalpy change is positive. We believe that the enthalpic loss upon folding occurs not only for apoPC but also for the other proteins. Although it has been reported for many proteins that the folding accompanies negative changes in the system enthalpy and entropy at room temperature28, we have argued that it is an artifact arising from the experimental and estimation procedures for the changes employed. Details are described in ref 20.

Calculation of the hydration entropy based on the statistical thermodynamics of fluids

The quantity we focus on hereafter is the hydration entropy (HE), S, which represents the water-entropy loss upon the insertion of a solute with a prescribed structure. A larger absolute value of S, |S|, means a larger magnitude of the loss. The water-entropy change upon the conformational change of a solute from structure A to structure B is equal to the HE of structure B minus that of structure A.

The HE can be calculated by the angle-dependent integral equation theory29–32, an elaborate theory of statistical-thermodynamics for molecular liquids. This theory allows us to calculate the thermodynamic quantities of hydration of a solute, as well as the density and orientational structures of water near the solute. In the theory, water is not a dielectric continuum but an ensemble of water molecules. The number of water molecules is infinitely large. The high reliability of the angle-dependent integral equation theory has been verified in a number of applications. For example, the hydration free energies of small nonpolar solutes calculated by the theory combined with the multipolar water model29,30 are in perfect agreement with those from Monte Carlo simulations with the SPC/E and TIP4P water models31. The dielectric constant for bulk water, which is determined from the water-water orientational correlation functions, is in good agreement with the experimental data31. The theory is also capable of elucidating the hydrophilic hydration experimentally known32.

Despite these successful results, the application of the angle-dependent integral equation theory to complex solute molecules like proteins is rather difficult due to the mathematical complexity. This problem can be overcome by combining it with the morphometric approach33. We calculate the HE by a hybrid of the angle-dependent integral equation theory combined with the multipolar water model29,30 and the morphometric approach. Details of our hybrid method are described in refs 19, 24. In the calculation of HE, the (x, y, z)-coordinates of solute atoms are used as part of the input data to characterize each structure. The polyatomic structure, which is crucially important for the HE, is accounted for on the atomic level: All of the atoms constructing the protein complex and ATP (i.e., hydrogen, oxygen, carbon, nitrogen, sulfur, and phosphorus) are incorporated in the structures. It has been shown that by our hybrid method, very fast and accurate computation of the HE is possible. For example, we have calculated the water-entropy gain upon folding of apoPC24. In the calculation, we have assumed that the unfolded state comprises a set of randomcoil structures. The HE of the unfolded state has been taken to be the average of the values for those structures. The water-entropy gain thus obtained is −TΔS=−1658 kJ/mol, which is certainly in the range estimated above, implying that the quantitative reproduction of the experimentally measured changes in thermodynamic quantities is successfully performed. Further, although in our hybrid method the polyatomic structure is accounted for on the atomic level, the calculation for α₃β₃γ complex with ~48000 atoms is finished only in a few minutes on the Xeon workstation. The calculation of the HE is performed under isochoric condition due to theoretical convenience. The validity of employing isochoric condition is described in the subsections “Importance of water-entropy effect” and “Hydration entropies of three sub-complexes.

Rotation Scheme of the γ Subunit based on the Water-entropy Effect

Asymmetric packing in the α₃β₃γ complex

It has been pointed out that the packing of the α₃β₃γ complex in crystal structures is highly asymmetrical. For example, using an all-atom molecular dynamics (MD) simulation, Ito and Ikeguchi34 have analyzed packing at the interfaces between adjacent subunits in terms of the number of stable contacts. The stable contacts are defined as the inter-subunit residue pairs maintaining their inter-atomic distances less than 4.5Å for 98% of snapshots in the MD trajectory. The number of stable contacts can be a measure of tightness of packing at the interfaces between adjacent subunits. Ito and Ikeguchi have found that the packing at the interfaces between β_DP, adjacent α subunits, and γ subunit is especially tight, implying that the four subunits are strongly interacted (See Fig. 3(a)). Experimental studies have shown that in the structure illustrated in Figures 1 and 3(a), the rotation of the γ subunit occurs upon a structural change of β_DP, the catalytically active subunit7,9. It is likely that perturbation of the tight packing by the structural change of β_DP induces the movement of the γ subunit.

Figure 3.

Summary of experimental results using schematic representation of the α₃β₃γ complex viewed from the F_o side. (a) Schematic representations of the crystal structure shown in Figure 1. The yellow lines represent that the packing in two adjacent subunits is especially tight34. Three circular arcs denote sub-complexes I (green), II (red), and III (blue), respectively. The definition of each sub-complex is described in “Hydration entropies of three sub-complexes”. The other figures represent (b) the overall conformation after the first 40° rotation of the γ subunit and (c) the overall conformation after the second 80° rotation of the γ subunit. Primes are added to the subunits in (b) because their conformations should be different from those in (a). The definition of the arrow at the center of the γ subunit is shown in the original paper19.

We have considered that the asymmetric packing is driven primarily by the water-entropy effect and that it should be analyzed in terms of the water entropy. In our study, we use the same crystal structure of the α₃β₃γ complex as that employed by Ito and Ikeguchi (the α₃β₃γ complex of the crystal structure from Bovine Heart Mitochondria (PDBID: 2JDI)13). In the crystal structure, all subunits except for the β_E subunit have AMP-PMP and Mg²⁺. Neither of the nucleotide nor Mg²⁺ is bound to the β_E subunit. We replace AMP-PMP by ATP because the results by single-molecule experiments have indicated that most of the crystal structures are in the catalytic dwell state7,9.

Measures of tight packing between adjacent subunits

As a first step, we investigate whether the results of the MD simulation by Ito and Ikeguchi can be reproduced in terms of the water-entropy effect. As a measure of tight packing between adjacent subunits, we calculate ΔS_ij≡S_ij–S_i–S_j where S_ij is the HE of subunit pair i-j and S_i is the HE of subunit i. (i-j=α_TP-β_E, α_E-β_E, α_E-β_DP, α_DP-β_TP, α_TP-β_TP, α_DP-β_DP, α_DP-γ, α_TP-γ, α_E-γ, β_E-γ, β_TP-γ, and β_DP-γ.) The calculation procedure is as follows: (i) we first take an arbitrary subunit pair i-j from the crystal structure; (ii) the HE of subunit pair i-j is calculated; (iii) we calculate the HE of each subunit (subunit i or j); and (iv) ΔS_ij is calculated. ΔS_ij represents the water-entropy gain upon formation of subunit pair i-j. It becomes larger as the tightness of interface packing between subunit pair i-j increases because the tight packing between adjacent subunits leads to the large reduction of the EV. Thus, ΔS_ij is a measure of the tightness like the number of stable contacts calculated in the MD simulation34.

As shown in Figure 4, ΔS_ij/k_B is highly correlated with the number of stable contacts obtained in the MD simulation by Ito and Ikeguchi34 (the correlation coefficient is 0.96). Thus, the framework of the results of the MD simulation is successfully reproduced using our hybrid method focused on the water-entropy effect. In the MD simulation, quite a long computation is inevitable. On the other hand, the calculation of the HE for one subunit pair is finished only in a few seconds by our hybrid method. Nevertheless, the packing characteristics estimated by the MD simulation can beautifully be reproduced, which is remarkable.

Figure 4.

Comparison between the numbers of stable contacts estimated by the MD simulation with all-atom potentials34 and the water-entropy gain upon the formation of subunit pair i–j, S_ij/k_B. Each value is shown in the original paper19 and in the supporting information of the paper of Ito and Ikeguchi34.

Hydration entropies of three sub-complexes

The 120° rotation of the γ subunit is induced by overall conformational changes of the β subunits as well as changes in the interface packing between subunits. For this reason, we define the following three sub-complexes as shown Figure 3(a):

• Sub-complex I: γ, β_E, α_E, and α_TP,

• Sub-complex II: γ, β_TP, α_TP, and α_DP,

• Sub-complex III: γ, β_DP, α_DP, and α_E.

We name the sub-complexes in terms of their positions in the crystal structure. For example, when the γ subunit rotates by 120° (see Fig. 3(c)), the arrangement changes and sub-complex III now comprises γ, β_E, α_E, and α_TP. We first generate the three sub-complexes from the crystal structure (see Fig. 5) and then calculate the HE of each sub-complex. To make the number of the atoms in each sub-complex impartial, we add the HE of ATP-Mg²⁺ to the HE of sub-complex I. The values of the HE of the three sub-complexes are shown in Table 1. It follows that the value of |S|/k_B (k_B is the Boltzmann constant) is in the order, sub-complex III < sub-complex II < sub-complex I. Therefore, the water-entropy loss upon the insertion of sub-complex III is the smallest.

Figure 5.

Schematic representation of the generation of the three sub-complexes from the crystal structure.

Table 1.

Hydration entropy, S/k_B, for each sub-complex.

	I	II	III
S/kB	−61245	−61028	−60619

The definition of each sub-complex is described in “Hydration entropies of three sub-complexes”.

Since the conformations of the complex before and after the 120° rotation of the γ subunit are the same, there is no water-entropy change upon the 120° rotation. However, the free energy of the system reduces by the free-energy decrease by ATP hydrolysis. It is interesting to note that as observed in Table 1 the rotation leads to the water-entropy gains of 217k_B and 409k_B (540 kJ/mol and 1010 kJ/mol at 298 K, respectively) in sub-complex I and sub-complex II, respectively, and to the water-entropy loss of 626k_B (1550 kJ/mol at 298 K) in sub-complex III. This result indicates that the rotation leads to tighter packing in sub-complexes I and II, and looser packing in sub-complex III, respectively. It should be noted that the gains and loss are quite large. Any loss caused in one of the sub-complexes during the rotation is compensated by the gains occurring in the other sub-complexes. Without this compensation, the system would undergo a significantly large free-energy barrier hindering a smooth rotation of the γ subunit.

As described in “Calculation of the hydration entropy based on the statistical thermodynamics of fluids”, the calculation of the HE is performed under isochoric condition. This can be validated by a discussion which is similar that given in “Importance of water-entropy effect”. As packing becomes tighter in sub-complexes I and II upon rotation, both of V_ex and A reduce. Since the reduction of V_ex is almost cancelled out by the second term of Eq. (1), change in V of sub-complexes I and II upon rotation is quite small. This is also true for sub-complex III: a looser packing upon rotation leads to an increase in V_ex and A. Due to the cancelation of an increase in V_ex by the second term of Eq. (1)V of subcomplex III is also almost unchanged upon rotation. The rotation mechanism can reasonably be discussed under isochoric condition.

Rotation scheme predicted from the analysis on the water-entropy effect

We first summarize the experimental results that have been reported.

1. Figure 3(a) is the crystal structure. Both of β_TP and β_DP are in closed conformations with ATP binding while β_E adopts an open conformation11–13.

2. From Figure 3(a), the γ subunit rotates by 40° (Fig. 3(a) → Fig. 3(b))6. During the rotation, the hydrolysis of ATP in β_DP4 and release of Pi6 occur, with the result that β_DP changes its conformation into a half-open one9. The β subunit with the changed conformation is denoted by β_DP^HO. The α₃β₃γ complex now takes the overall conformation shown in Figure 3(b). Primes are added to the subunits in Figure 3(b) because their conformations should be different from those in Figure 3(a). Although the crystal structure corresponding to Figure 3(b) has not been determined yet, it has been shown by single-molecule experiments9 that β_E’, β_DP^HO, and β_TP’ are in open, half-open, and closed conformations, respectively. Here we comment on the time of Pi release. There are two probable pictures6: Pi produced from ATP hydrolysis is immediately released35, or the Pi release is suspended for the next 120° rotation10. The release occurs from the β_DP subunit in the former picture while it occurs from the β_E subunit in the latter one. Although the time of the Pi release is still an open question, our rotation mechanism discussed below is not influenced by this uncertainty.

3. From Figure 3(b), the γ subunit rotates by 80° (Fig. 3(b) → Fig. 3(c)). During the rotation, the conformational changes9, β_E’→β_TP and β_DP^HO→β_E, occur due to the ATP binding3 and release of ADP6, respectively. The α³β³γ complex now takes the overall conformation shown in Figure 3(c), which is the same as that of the crystal structure (see Fig. 3(a)). However, the free energy of the system G_System in Figure 3(c) is lower than that in Figure 3(a) by the free-energy decrease arising from ATP hydrolysis.

The experimental results summarized above can be explained in terms of the water entropy as follows. (We consider the situation where the ATP concentration is much higher than the ADP and Pi concentrations.)

1. Figure 3(a): Two of the β subunits are in closed conformation and the other one takes an open conformation. In this situation, the α₃β₃γ complex is packed with high asymmetry so that the water entropy in the presence of the α₃β₃γ complex can be made as high as possible. The tightest packing is formed in sub-complex III. The reasons why such an asymmetric packing is formed are due to an asymmetric nature of the γ subunit and to nonuniform binding of nucleotides to the three β subunits. Here we discuss why such asymmetric packing is formed by the water-entropy effect. As explained in Figure 2, it is desired for a protein or a complex of proteins that the backbones and side chains be tightly packed as a three-dimensional jigsaw puzzle. However, this is not always possible. When the overall, impartial tight packing is not achievable in a protein, the portions which can tightly be packed are chosen for the preferential tight packing. This action is taken by the water-entropy effect. For example, the native structure of yeast frataxin36 has a large valley and a tail as shown in Figure 6. Nevertheless, |S| of the native structure is almost minimized37 because the other portions are tightly packed. If an impartial packing was undertaken, the valley and/or the tail could be removed but the resultant packing would become rather loose, causing a larger value of |S|.

2. Figure 3(a) → Figure 3(b): The conformational change of β_DP into β_DP^HO gives rise to a looser packing in sub-complex III. This conformational change would cause a decrease in the water entropy of sub-complex III. For suppressing the decrease in the water entropy, the 40° rotation of the γ subunit in a counterclockwise direction is required (the angle 40° comes from the experimental observation). As a consequence, the packing in sub-complex II becomes tighter, leading to a water-entropy gain. The water-entropy loss in sub-complex III is thus almost compensated by this water-entropy gain, leading to a smooth rotation of the γ subunit. Further, since Pi is highly charged, its release leads to a large decrease in the hydration energy. Thus, G_System in Figure 3(b) becomes lower than that in Figure 3(a).

3. Figure 3(b) → Figure 3(c): The conformational changes of β_E’ and β_DP^HO (β_E’→β_TP and β_DP^HO→β_E) are induced by the ATP binding and ADP release, respectively.

Figure 6.

Solvent-surface representation of the native structure of yeast frataxin (PDBID: 2ga5)36 drawn by the DS visualizer 2.5. This structure has the large valley marked by the blue ellipse and the tail pointed by the red arrow.

The latter one would lead to a water-entropy loss in subcomplex III. On the other hand, a water-entropy gain occurs in sub-complex I by the former conformational change. Further, the packing in sub-complex II would become tighter by the 80° rotation of the γ subunit, leading to a water-entropy gain. The γ subunit would rotate smoothly by the compensation between the water-entropy loss in sub-complex III and these water-entropy gains. By the 80° rotation, the overall conformation of the α₃β₃γ complex becomes that as shown in Figure 3(c). Since two of the β subunits are now in closed conformations and the other takes an open conformation, the conformation stabilized as shown in Figure 3(a) is recovered. Sub-complex II in Figure 3(c) forms the tightest packing which is the same as that in sub-complex III in Figure 3(a). The absolute value of the HE of each sub-complex in Figure 3(c) is in the order, sub-complex II < sub-complex I < sub-complex III. The water entropy of the α₃β₃γ complex remains the same upon the overall-conformational change, Figure 3(a) → Figure 3(c), but G_System is lowered by the free-energy decrease due to ATP hydrolysis. We remark that G_System is in the order, Figure 3(a) > Figure 3(b) > Figure 3(c).

Concluding Remarks and Perspective

We have reviewed our rotation scheme of the γ subunit19. In our scheme, the asymmetric packing of the α₃β₃γ complex and the water-entropy effect arising from the translational displacement of water molecules induce the rotation during the chemical processes of the ATP binding, ATP hydrolysis, and release of products. Here we give perspectives to construct the complete rotation mechanism in the following two paragraphs.

In the present review, we have constructed the rotation scheme by analyzing the HEs of the three sub-complexes in the crystal structure corresponding to that shown in Figure 3(a). In order to validate the rotation scheme we have proposed and to construct the complete rotation mechanism, the conformations which appear between the conformations before and after the 120° rotation, especially the conformation corresponding to that shown in Figure 3(b), are required. However, the crystal structure corresponding to Figure 3(b) has not been determined yet. Instead, the conformation after 16° rotation from Figure 3(a) is found in the crystal structures of yeast F₁-ATPase13. As part of the demonstration of the validity of our rotation mechanism, we are now analyzing the water-entropy change in yeast F₁-ATPase during 16° rotation of the γ subunit. The compensation in terms of the water entropy, which is explained in the text, actually occurs during 16° rotation (details will be presented in a future article). As a future work, the conformations of the α₃β₃γ complex which appear between the conformations before and after the 120° rotation could be generated using an MD simulation.

We have focused on the water-entropy effect in the present review. The strong correlation between the results obtained by our theoretical method and by an MD simulation (Fig. 4) validates the predominance of the water-entropy effect. In order to construct the complete rotation mechanism, however, the effect of the energetic component on the rotation scheme should be clarified. At present, we have the following picture of the energetic component. Decreases in the intramolecular energy by tight packing occur in sub-complexes I and II upon rotation of the γ subunit. However, such decreases are somewhat compensated by increases in the protein-water interaction energy. This is also true for sub-complex III: an increase in the intramolecular energy by a looser packing is compensated by a decrease in the protein-water interaction energy. Thus, the differences among the sub-complexes in the energetic component should be substantially smaller than those in the water entropy: The differences in the free energy are still considerably large due to the large difference in the water entropy. In order to validate the picture, we are now analyzing the energetic terms for subunit pairs and sub-complexes by the MD simulation coupled with the method of energy representation38. Details will be presented in a future article.

Recently, using high-speed atomic force microscopy (AFM), Uchihashi et al. have shown that even in the α₃β₃ complex (i.e., without the γ subunit) the three β subunits undergo the same cyclic conformational changes in the same rotary direction as in the case of the α₃β₃γ complex39. In accordance with this result, we calculate the HEs of three sub-complexes without the γ subunit19 (in this case, sub-complex III, for example, comprises β_DP, α_DP, and α_E). We have found that the order of the HE is the same as that in the case with the γ subunit shown in Table 1. Our result and the experimental one by Uchihashi et al. suggest that the asymmetric nature of the α₃β₃ complex is crucially important for the rotation.

The water-entropy effect has been shown to be the key factor in elucidating the mechanisms of folding and unfolding (pressure27,40, cold37,41, and heat42,43 denaturations) of proteins. We have recently reported that the unidirectional movement of a linear-motor protein along a filament44 and the insertion of a big solute into and from a vessel constructed by biopolymers45,46 are also driven by the water-entropy effect. Thus, the water-entropy effect is imperative for a variety of biological processes sustaining life.

Supporting Information

Theoretical framework of the partial molar volume

The partial molar volume V can be obtained by the Kirkwood-Buff formula1

$$V=\int \int \int \{1-g_{\text{US}}(x,y,z)\}dxdydz,$$ (1)

where g_US(x, y, z) represents the microstructure of the solvent near the protein surface and is referred to as the reduced density profile. It has the physical meaning that the number of solvent molecules within the volume element dxdydz is given by ρ_Sg_US(x, y, z)dxdydz. At distances which are sufficiently far from the protein surface, g_US→1. No overlap of an atom in the protein and a water molecule is allowed, and at positions where such an overlap occurs g_US=0. The values of V obtained by Eq. (1) are in accord with the experimentally measured values2. For example, V of lysozyme calculated is 11,600 cm³/mol that is in good agreement with the average of the experimentally measured value, 10,100 cm³/mol.

It is convenient to divide the term in the right hand of Eq. (1) into the integrations inside and outside the core regions2. Inside the core region, due to the overlap of the protein and solvent, the protein-solvent potential is infinitely large and g_US=0. It follows that the integration inside the core region equals the excluded volume, V_ex, which the centers of solvent molecules cannot enter. The integration outside the core region takes a negative value because a layer within which the solvent density is higher than in the bulk is formed near both of hydrophobic3 and hydrophilic surfaces4. Since the higher density is almost limited to the first layer (i.e., the thickness of the denser layer reaches only about half of the solvent diameter), the integration over region 2 can be approximated by −ζA where ζ (>0) is a parameter representing the average number density of water within the layer and A is the water-accessible surface area (ASA). Thus, we can express Eq. (1) as

$$V\approx V_{\text{ex}}-\zeta A$$ (2)

Equation (2) indicates that V is smaller than V_ex due to the second term in the right-hand side of Eq. (2).