# General Framework of Pressure Effects on Structures Formed by Entropically Driven Self-Assembly

## Abstract

**:**

## 1. Introduction

_{g}, of staphylococcal nuclease has been studied using synchrotron X-ray small-angle scattering [2] and small-angle neutron scattering [5]. The application of high pressure leads to an approximate twofold increase of R

_{g}of the native structure. However, it is still much smaller than that of the random-coil structure [2]. In addition, some degree of β-like secondary structure is retained, even above 300 MPa [2], indicating that the pressure-denatured structure is different from the random-coil structure. Some authors have suggested that water has to penetrate into the protein interior to explain the experimental results [4,5,16]. The penetration of water into the protein interior has been observed using molecular dynamics (MD) simulations [13,14,15]–the number of the water molecules in the protein interior increases as the pressure becomes higher. According to these results, we can conclude that the pressure-denatured structure is characterized by swelling, water penetration into the protein interior, and only a moderate reduction of the compactness [6].

**Figure 1.**Schematic representation 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, leading to a gain of the water entropy.

## 2. Driving Force of Pressure Denaturation of Proteins: Translational Entropy of Water

_{I}+ Δμ − TΔS

_{C}

_{I}is the protein intramolecular energy, μ is the solvation free energy which is the free-energy change by the insertion process of a solute into solvent, S

_{C}is the conformational entropy of protein, and T is the absolute temperature. ΔB≡B

_{D}−B

_{N}denotes the change in a thermodynamic quantity B upon denaturation. The subscripts “N” and “D” represent the values for the native state and for the denatured state, respectively.

_{C}|, is expected to be independent of the pressure or smaller at high pressures due to the constraint caused by the denser solvent [42]. Within the framework of classical mechanics, the intramolecular energy for any structure remains unchanged against a pressure change. Thus, Δμ must decrease to a significant extent as the pressure P increases and eventually become negative for the denaturation to occur.

_{T}upon denaturation. According to the discussion in the previous paragraph, both of (∂ΔE

_{I}/∂P)

_{T}and (∂ΔS

_{C}/∂P)

_{T}are non-negative, therefore, the sign of ΔV is determined by (∂Δμ/∂P)

_{T}. The partial molar volume (PMV), which is denoted by V

_{PMV}, is the change of the system volume occurring upon the solute insertion in the isobaric process. Thermodynamically, the PMV is the pressure derivative of the solvation free energy expressed as:

_{PMV}= (∂μ/∂P)

_{T}

_{PMV}< 0). The solvation free energy μ is the excess chemical potential of the solute in the fluid of interest and is the same, irrespective of the solute insertion process (isochoric or isobaric) [43]. We consider the isochoric process hereafter. Under the isochoric condition, the solvation free energy is given by μ = U − TS where U is the solvation energy and S is the solvation entropy (SE). It has been observed that nonpolar side chains are more separated in a denatured structure with water molecules penetrating its hydrophobic core [5,14,15,16]. When the penetration occurs, the breakage of hydrogen bonds of water is unavoidable, leading to a loss in terms of the solvation energy. There must be an even larger gain in terms of the solvation entropy, a dominant increase in the solvent entropy.

EV (Å ^{3}) | ASA (Å ^{2}) | |
---|---|---|

Native | 11,600.3 | 3,670.90 |

Swelling | 11,926.9 | 4,210.81 |

Random coil | 14,002.9 | 5,947.13 |

_{B}ρ

_{S}ΔV

_{ex}where k

_{B}is the Boltzmann constant, ρ

_{S}the solvent density, ΔV

_{ex}(< 0) the decrease in the EV. Since ρ

_{S}becomes higher as the pressure increases, the structures with small EV should further be stabilized by applying high pressures. This statement clearly conflicts with the theoretical results [39,40] as well as the experimental observations [6]. It is also unclear whether the other phenomena such as the dissolution of amyloid fibril occurring at high pressures can also be explained in terms of the translational entropy of water. To solve these problems, we employ the morphometric approach described in the next section.

## 3. Models and Theoretical Approach for the Calculation of Solvation Thermodynamic Quantities

#### 3.1. Models

_{S}= 2.8 Å that is the molecular diameter of water, and the solutes (proteins and polypeptides) are modeled as a set of fused hard spheres. In this model system, all the configurations share the same energy and the system behavior is purely entropic in origin. The polyatomic structure, which is crucially important, is accounted for on the atomic level. The (x, y, z) coordinates of all the protein or polypeptide atoms (hydrogen, carbon, nitrogen, oxygen, etc.) in the backbone and side chains are taken into consideration.

_{PMV}. The validity of the hard-body model in calculating S and V

_{PMV}has been shown in the previous papers [40,46]. For example, Imai et al. have considered the native structures of a total of eight peptides and proteins and calculated S using the three-dimensional reference interaction site model (3D-RISM) theory combined with the all-atom potentials and SPC/E water model [46]. Even when the protein-water electrostatic potentials, which are quite strong, are shut off and only the Lennard-Jones potentials are retained, S only decreases by less than 5%. Further, an approximate value of S can be obtained even by using the 3D integral equation theory [29,31,47] combined with the hard-body model. For example, −TS of human erythrocyte ubiquitin (PDB code: 3EBX) is 1,802 kcal/mol when the 3D-RISM theory combined with the all-atom potentials is applied, and it is 1,882 kcal/mol when the 3D integral equation theory combined with the hard-body model is applied [46]. This is because the contribution to the SE from the water molecules near the protein is much smaller than that from those in the system [26].

_{VH}, and energy U

_{VH}under the isochoric condition are calculated for a spherical solute with diameter 2.8 Å using the angle-dependent integral equation theory [48,49] combined with the multipolar water model [50,51] (when the solvent is water, the solvation free energy, entropy, and energy are referred to as the hydration free energy, entropy, and energy, respectively). This theory combined with the multipolar water model can reproduce well the experimental observations such as the dielectric constant of bulk water [48] and the hydrophobic [48] and hydrophilic [49] hydrations. For the hard-sphere solute with zero charge, the calculated values are μ = 5.95k

_{B}T, S

_{VH}= −9.22k

_{B}, and U

_{VH}= −3.27k

_{B}T [28]. When the point charge −0.5e (e is the electronic charge) is embedded at its center, the calculated values are μ = −32.32k

_{B}T, S

_{VH}= −10.11k

_{B}, and U

_{VH}= −42.43k

_{B}T [28]. Thus, S

_{VH}is fairly insensitive to the solute-water interaction potential while μ and U

_{VH}are largely influenced by it. This gives another justification of the protein model, a set of fused hard spheres.

_{PMV}of five different proteins in accordance with the Kirkwood-Buff formulation [52]:

_{PMV}= ∫∫∫{1−g

_{US}(x, y, z)}dxdydz

_{US}(x, y, z) represents the microstructure of the solvent near the protein surface and is referred to as the reduced density profile (hereafter, the subscripts, “S” and “U”, respectively, represent “solvent” and “solute”). It has the physical meaning that the number of solvent molecules within the volume element dxdydz is given by ρ

_{S}g

_{US}(x, y, z)dxdydz. g

_{US}(x, y, z) is calculated using the 3D integral equation theory combined with the hard-body model. The values of the PMV obtained are in accord with the experimentally measured values. For example, the PMV of lysozyme (PDB code: 1HEL) calculated is 11,600 cm

^{3}/mol that is in good agreement with the average of the experimentally measured value, 10,100 cm

^{3}/mol [40].

**Figure 2.**Reduced density profiles of hard-sphere solvent near a hard-sphere solute g

_{US}(r) at ρ

_{S}d

_{S}

^{3}= 0.2 (dotted line), ρ

_{S}d

_{S}

^{3}= 0.5 (2-dot dashed line), and ρ

_{S}d

_{S}

^{3}= 0.7 (solid line).

_{US}= 0. It follows that the integration inside the core region equals the EV, 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 the protein surface due to the packing force arising from the translational displacement of solvent molecules [40] (g

_{US}of hard-sphere solvent near a hard-sphere solute is shown in Figure 2 as an example). 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 outside the core region is roughly in proportion to the solvent-accessible surface area (ASA) denoted by A. Thus, we can write:

_{PMV}~ V

_{ex}− ξA

_{PMV}is affected not only by these two terms but also by the curvature terms as described in Section 3.2) The parameter ξ is related to the average solvent density within the dense layer. A hydrophilic group in the protein makes a large, positive contribution to ξ because g

_{US}>>1 on an average near it [49]. By contrast, near a group which is hydrophobic enough to overcome the packing force, g

_{US}~1 or g

_{US}<1 [49], with the result that the group makes a small, negative contribution to ξ. Since hydrophilic and hydrophobic groups are almost irregularly distributed on the protein surface, the overall value of ξ becomes positive, and V

_{PMV}is smaller than V

_{ex}. When the protein is modeled as a fused hard spheres and water is taken to be hard spheres, the water density near a hydrophilic group is underestimated while that near a hydrophobic group is overestimated, leading to a fortuitous cancellation of errors and a better result.

#### 3.2. Morphometric Approach

_{B}and V

_{PMV}) are expressed using only four geometric measures of a complex (polyatomic) solute with a fixed structure and corresponding coefficients. The resultant expression is:

_{1}V

_{ex}+C

_{2}A+C

_{3}X+C

_{4}Y

_{ex}), the ASA (A), and the integrated mean and Gaussian curvatures of the accessible surface (X and Y) respectively. The water-accessible surface is the surface that is accessible to the centers of water molecules [22]. A, X, and Y are the surface area and the integrated curvatures of the water-accessible surface. The EV is the volume that is enclosed by the water-accessible surface area.

_{1}- C

_{4}) can be determined in a simple geometry. They are determined from the solvation thermodynamic quantities calculated for hard-sphere solutes with various diameters immersed in the hard-sphere solvent. The morphometric form applied to hard-sphere solutes reduces to:

_{1}{(4π/3)d

_{US}

^{3}}+C

_{2}(4πd

_{US}

^{2})+ 4πC

_{3}d

_{US}+4πC

_{4}

_{US}= (d

_{U}+d

_{S})/2 and d

_{U}is the solute diameter. The four coefficients are determined using the least squares fitting to Equation (6). Once the four coefficients are determined, solvation thermodynamic quantities for a solute with a fixed structure are obtained by calculating only the four geometric measures.

_{S}∫{h

_{US}(r)

^{2}/2− h

_{US}(r)c

_{US}(r)/2− c

_{US}(r)}r

^{2}dr

_{US}(r) and c

_{US}(r) are the total and direct correlation functions between the hard-sphere solute and hard-sphere solvent, respectively (g

_{US}(r) is h

_{US}(r)+1). The integration range is from 0 to ∞. In the hard-body model, μ is equal to −TS. The PMV is obtained using Equation (3). The correlation functions are calculated using the integral equation theory, elaborated statistical thermodynamics theory [38,43].

_{PMV}calculated by the 3D integral equation theory [29,31,47] applied to the hard-body model can be reproduced with sufficiently high accuracy by the morphometric approach applied to the same hard-body model [41,54]. For example, the deviation of the SE by the morphometric approach from that obtained by the 3D integral equation theory is less than ±2% [54]. V

_{PMV}of the complete α-helix structure composed of 20 alanines at ρ

_{S}d

_{S}

^{3}= 0.7 is 94d

_{S}

^{3}when the morphometric approach is applied, and it is 90d

_{S}

^{3}when the 3D integral equation theory is employed [41]. The high accuracy also indicates that the SE obtained using the all-atom potential and the PMV experimentally observed can well be reproduced by the morphometric approach applied to the hard-body model (in Section 3.3 we also show that the experimentally measured change in thermodynamic quantity upon apoplastocyanin folding is quantitatively reproduced using the morphometric approach).

#### 3.3. Quantitative Comparison between Experimental and Theoretical Results for Apoplastocyanin Folding

_{C})” where ΔH is the enthalpy loss and ΔS

_{C}is the conformational-entropy (CE) loss. We estimate ΔS

_{C}via following two different routes.

^{2}= 9 and the contribution to the CE is k

_{B}ln9. Based on the study by Doig and Sternberg [58], we regard the contribution from the side chain to the CE as 1.7k

_{B}per residue. It is assumed that the CE of the native structure is essentially zero. The CE loss ΔS

_{C}upon folding with N

_{r}residues is expressed as:

_{C}/k

_{B}= −N

_{r}(ln9+1.7)

_{C}/k

_{B}= −3N

_{r}ln(r

_{u}/ r

_{f})

_{u}and r

_{f}are the radius parameters for the unfolded state and for the folded state, respectively. The estimation of the radius parameters was made at three temperatures, 303K, 323K, and 343K. In order to estimate the CE loss at 298K, we first perform the linear fitting of the temperature dependence of r

_{u}and r

_{f}, and then obtain their values at 298K. The neutron scattering experiments cover mainly the picosecond time regime though the fluctuations in other time scales also affect the CE. For this reason, the use of Equation (9) results in a considerable underestimation of the CE loss. The actual CE loss should lie between the two values calculated from Equation (8) and Equation (9), respectively.

_{r}of apoPC is 99, ΔS

_{C}can be estimated to be in the range, 305 kJ/mol < −TΔS

_{C}< 956 kJ/mol [25]. It is assumed that the free-energy gain takes the most probable value shared by a number of proteins, −50 kJ/mol [60]. Using ΔH = 870 kJ/mol, the water-entropy gain is estimated to be in the range, −1876 kJ/mol<−TΔS<−1225 kJ/mol.

## 4. General Framework of Pressure Effects on Structures Formed by Self-assembly

_{3}and C

_{4}are not discussed here because in Equation (5) C

_{3}X+C

_{4}Y is much smaller than C

_{1}V

_{ex}+C

_{2}A.]. Figure 3(a) shows the density dependence (corresponding to the pressure dependence) of the first and second coefficients, C

_{1}and C

_{2}, in the morphometric form applied to the SE, −S/k

_{B}. It is found that C

_{1}and C

_{2}take positive and negative values, respectively, at any density. C

_{1}and |C

_{2}| increase remarkably with rising density, but the increase in the latter is larger: |C

_{2}| is much smaller than C

_{1}at low pressures, but they are comparable in magnitude at high pressures.

_{1}can easily be interpreted as the solvent-entropy loss caused by the solute insertion. The basic physics to give an interpretation of the negative value of C

_{2}is in the phenomenon that when a large hard-sphere solute is immersed in small hard spheres forming the solvent, the small hard spheres are enriched near the solute despite that there are no direct attractive interactions between the solute and solvent particles and this enrichment becomes greater as the pressure increases (see Figure 2). We note that the presence of a solvent molecule generates an EV for the other solvent molecules in the system [39,40]. Due to this solvent crowding, part of the solvent particles is driven to contact the solute surface. The contact brings the overlap of the EVs generated by the solute and the solvent particles in contact with the solute. As a consequence, the total volume available to the translational displacement of the other solvent particles (i.e., the solvent particles well outside the enriched layer in the vicinity of the solute) increases, leading to an entropic gain. Such an entropic gain becomes large when the solute takes the structure with large ASA. Therefore, C

_{2}takes negative value.

**Figure 3.**(a) C

_{1}(Å

^{−3}), C

_{2}(Å

^{−2}), and C

_{2}/C

_{1}(Å) of solvation entropy, −S/k

_{B}, plotted against solvent density corresponding to the pressure. (b) C

_{1}and C

_{2}(Å) of partial molar volume plotted against solvent density.

_{PMV}[Figure 3(b)], C

_{1}is constant at the value corresponding to 1. The density dependence of |C

_{2}| is similar to that of −S/k

_{B}–it increases as the density is raised. The positive value of C

_{1}represents the increment of the volume of the system by the EV of the solute. The negative value of C

_{2}arises from the reduction of the EVs by the contact of the solute and the solvent particles. V

_{PMV}is decreased as the ASA is increased.

_{3}X+C

_{4}Y is much smaller than C

_{1}V

_{ex}+C

_{2}A, the difference in the solvation thermodynamic quantity Z between the structures stabilized at high and low pressures can approximately be described as follows:

^{High}− Z

^{Low}~ C

_{1}(V

_{ex}

^{High}− V

_{ex}

^{Low})+C

_{2}(A

^{High}− A

^{Low})

_{B}or V

_{PMV}. At high pressures, the structure formed by the entropically driven self-assembly is usually dissolved: the protein is unfolded and the amyloid fibrils are dissolved. V

_{ex}

^{High}−V

_{ex}

^{Low}and A

^{High}−A

^{Low}are, respectively, changes in the EV and the ASA upon the destruction, and they are both positive. We consider −S/k

_{B}for Z. For the pressure-induced destruction to occur, C

_{2}(A

^{High}−A

^{Low}) (negative) must surpass C

_{1}(V

_{ex}

^{High}−V

_{ex}

^{Low}) (positive) at sufficiently high pressures. In other words, only the structure making the former larger can be stabilized at elevated pressures. Such structure should have a largest possible ASA together with sufficiently small EV (hereafter, we refer to these as “pressure-induced structures”). The pressure dependence of the structural stability is determined by a subtle balance between these two terms. In the case of the PMV, V

_{ex}

^{High}−V

_{ex}

^{Low}and C

_{2}(A

^{High}−A

^{Low}are positive and negative, respectively, and these signs are same as those of −S/k

_{B}. V

_{PMV}

^{High}−V

_{PMV}

^{Low}becomes negative when solute takes pressure-induced structure.

_{PMV}upon the transition to the pressure-induced structures is negative.

_{3}X+C

_{4}Y is fully incorporated in −S/k

_{B}and V

_{PMV}in Section 4.1 and Section 4.2.

#### 4.1. Microscopic Mechanism of Pressure Denaturation of Proteins

_{B}= (−S/k

_{B})

^{Unfold}−{(−S/k

_{B})

^{Native}} where the superscripts “Native” and “Unfold” represent the values for the native structure and for an unfolded structure of protein G, respectively (C

_{3}X+C

_{4}Y is fully incorporated here). All structures are the same as those shown in Table 1. In the present hard-sphere model, −ΔS/k

_{B}corresponds to the change in μ upon the transition from the native structure to an unfolded one. Figure 4(a) shows −ΔS/k

_{B}upon the structural transition to the swelling structure. As the density corresponding to the pressure increases, the swelling structure becomes more destabilized than the native structure in the low-pressure region. This is because C

_{1}is much larger than |C

_{2}|, with the result that any structure with larger EV is more destabilized. However, as the density increases further, it begins to decrease rather rapidly and they eventually turn more stable than the native structure. The change in the PMV upon the structural transition is 117 Å

^{3}at ρ

_{S}d

_{S}

^{3}= 0.2 and −195 Å

^{3}at ρ

_{S}d

_{S}

^{3}= 0.8, respectively. According to the experimental results [61], the volume change upon protein unfolding is positive at low pressures and negative at high pressures. Therefore, the present results are consistent with the experimental ones. The swelling structure can be regarded as the pressure-induced structure.

_{B}upon the structural transition to the random-coil structures in Figure 4(b). The SE of the random-coil structures is taken to be the average value calculated for the 32 random coils [62]. It follows that −ΔS/k

_{B}continues to increase upon raising density and thus the random-coil structures are unstable even for the high densities. The change in the PMV upon the structural transition to the random-coil structures is 464 Å

^{3}at ρ

_{S}d

_{S}

^{3}= 0.8 and is positive value. Therefore, the random-coil structure is not a pressure-induced one.

_{B}into C

_{1}ΔV

_{ex}, C

_{2}ΔA, and C

_{3}ΔX + C

_{4}ΔY in the case of Figure 4(a) is shown in Figure 4(c). The term C

_{1}ΔV

_{ex}is positive and increases as the solvent density becomes higher. The term C

_{3}ΔX+C

_{4}ΔY is also positive and is much smaller than the other two terms. Therefore, these terms prevent the transition. The structural transition comes primarily from the ASA term (C

_{2}ΔA) which is negative and decreases further as the density becomes higher. In the case of the random-coil structures where both of the EV and the ASA are much larger than those of the native structure (see Table 1), although their ASA term takes a very large negative value, −ΔS/k

_{B}is still positive due to the even larger positive value of the EV term (C

_{1}ΔV

_{ex}). Therefore, inversion of the relative stability occurs for the swelling structure because its ASA is considerably larger and its EV is only moderately larger than the native structure (see Table 1).

**Figure 4.**Negative of the entropy change of solvent scaled by k

_{B}upon the transition from the native structure to (a) the swelling structure and (b) the random-coil structures of protein G plotted against the bulk solvent density corresponding to the pressure P. −ΔS/k

_{B}= (−S/k

_{B})

^{Unfold}−{(−S/k

_{B})

^{Native}} where the superscripts “Native” and “Unfold” represent the values for the native structure and for the unfolded structure, respectively. (c) Decomposition of −ΔS/k

_{B}for swelling structure of protein G [Figure 4(a)] into C

_{1}ΔV

_{ex}, C

_{2}ΔA, and C

_{3}ΔX+C

_{4}ΔY at each bulk solvent density.

#### 4.2. Pressure-induced Helix-coil Transition of a Polypeptide

^{Helix}− S

^{Coil}and ΔS

_{C}= S

_{C}

^{Helix}− S

_{C}

^{Coil}denote the changes in the solvent entropy and in the conformational entropy upon the transition from the coil state to the helix state, respectively. The superscripts “Helix” and “Coil” represent the values for the helix state and for the coil state, respectively.

_{C}|, is expected to be independent of the pressure or decreasing function of the pressure. We assume that ΔS

_{C}is independent of the pressure hereafter. The pressure-induced helix-coil transition of a polypeptide [17,18] can be understood by the structural stability described by the competition between the solvent-entropy gain and the conformational-entropy loss of the polypeptide upon the transition [41]. At low pressures, ΔS is smaller than |ΔS

_{C}| and the coil state is stabilized. However, since ΔS is a monotonically increasing function of the pressure [see Figure 4(b)], the inversion occurs at a sufficiently high pressure, leading to the transition from the coil state to the helix state.

_{B}and ΔV

_{PMV}= V

_{PMV}

^{Helix}− V

_{PMV}

^{Coil}using the morphometric approach to show the validity of this physical picture [41] (C

_{3}X+C

_{4}Y is fully incorporated here). The polypeptide we consider is composed of 20 alanine residues. The hard-body model described in Section 3.1 is employed. We assume that the helix state is represented by a complete α-helix structure and that the coil state is an ensemble of random coils. Random coil structures are generated by assigning a random number to the dihedral angles for the main chain [62]. The SE and the PMV of the coil state are taken to be the average value of those for the nine random coils generated. We estimate the range of ΔS

_{C}/k

_{B}using the method described in Section 3.3. Since N

_{r}is 20, the actual CE loss should lie in the range, 24.9 < |ΔS

_{C}/k

_{B}| < 78.0.

_{B}, and the conformational-entropy loss, |ΔS

_{C}/k

_{B}|, upon the transition from the coil state to the helix state. At low densities ΔS/k

_{B}is smaller than the lower limit of |ΔS

_{C}/k

_{B}| and thus the polypeptide is in the coil state. On the other hand, ΔS/k

_{B}prevails over the upper limit of |ΔS

_{C}/k

_{B}| at sufficiently high densities. Therefore, the transition from the coil state to the complete α-helix structure occurs. The change in the PMV upon the transition, ΔV

_{PMV}, is −134 Å

^{3}at ρ

_{S}d

_{S}

^{3}= 0.7 and is negative value. The experimental results by Kato et al. [17,18] have thus been reproduced qualitatively. Even when the conformational-entropy loss is assumed to be a decreasing function of the pressure, our conclusions are not altered: The transition occurs simply at a slightly lower solvent density.

**Figure 5.**ΔS and |ΔS

_{C}| plotted against the bulk solvent density corresponding to the pressure. |ΔS

_{C}| lies between the two dashed lines. The SE gain and the CE loss upon the transition from the coil state (an ensemble of random coils) to the complete α-helix structure are compared.

_{r}. For the polypeptide whose N

_{r}is very small, as shown by Harano and Kinoshita [24], at low pressures the solvent entropy gain cannot prevail the conformational-entropy loss and the peptide takes the coil state. On the other hand, a protein with large N

_{r}folds into the native structure even at low pressures because the solvent-entropy gain dominates [24]. As shown in Figure 4(b), the native structure becomes more stable than the random coil structure at high pressures.

#### 4.3. Comment on Formation/Dissociation Process of Amyloid Fibrils

_{ex}

^{Low}is significantly large while A

^{Low}is fairly small. Therefore, upon the dissolution of the amyloid fibrils, V

_{ex}

^{High}−V

_{ex}

^{Low}can be kept sufficiently small, even though A

^{High}−A

^{Low}becomes quite large. The dissolution to monomers whose structures feature like pressure-induced ones can be the best solution. The monomers cannot be random coils because of the unacceptably large EV-increase despite the largest ASA-increase. The dissolution of the other protein complexes can be understood in a similar manner. Thus, the folding/unfolding transition of a protein and the formation/dissociation process of amyloid fibrils can be discussed within the same framework, pending theoretical verification in future studies for the latter.

## 5. Conclusions

_{1}-ATPase [72]. Therefore, it should be emphasized that the water-entropy effect is imperative for a variety of self-assembling and aggregation processes in biological systems sustaining life.

## Acknowledgements

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Yoshidome, T.
General Framework of Pressure Effects on Structures Formed by Entropically Driven Self-Assembly. *Entropy* **2010**, *12*, 1632-1652.
https://doi.org/10.3390/e12061632

**AMA Style**

Yoshidome T.
General Framework of Pressure Effects on Structures Formed by Entropically Driven Self-Assembly. *Entropy*. 2010; 12(6):1632-1652.
https://doi.org/10.3390/e12061632

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2010. "General Framework of Pressure Effects on Structures Formed by Entropically Driven Self-Assembly" *Entropy* 12, no. 6: 1632-1652.
https://doi.org/10.3390/e12061632