Search

Markov state models (MSMs) are a powerful approach for analyzing the long-time behaviors of protein motion using molecular dynamics simulation data. However, their quantitative performance with respect to the physical quantities is poor. We believe that this poor performance is caused by the failure to appropriately classify protein conformations into states when constructing MSMs. Herein, we show that the quantitative performance of an order parameter is improved when a manifold-learning technique is employed for the classification in the MSM. The MSM construction using the K-center method, which has been previously used for classification, has a poor quantitative performance.

We compare the asymmetric packing of F1-ATPase and that of F1-ATPase containing the inhibitor protein IF1 (F1-ATPase(IF1)), which inhibits ATP hydrolysis. To analyze this asymmetric packing, we calculate the solvation entropy, S, of three subcomplexes. Subcomplexes I, II, and III contain βE, βTP, and βDP subunits, respectively. The order of |S| is subcomplex III

Cuprous oxide (Cu2O) particles obtained by surfactant-assisted liquid-phase synthesis have cuboid shapes but the internal structures are difficult to be visualized by electron microscopy. Herein, we investigated the internal structures of numerous individual Cu2O particles with submicrometer dimensions by X-ray diffraction imaging (XDI) using X-ray free-electron laser (XFEL) pulses. The reconstructed two-dimensional electron density maps, which displayed inhomogeneous internal structures, were divided into five classes characterized by the positions and shapes of high and low electron density areas. Further analysis of the maps in each class by a manifold learning algorithm revealed that the internal structures of Cu2O particles varied in correlation with total electron density while retaining the characteristics within each class. On the basis of the analyses, we proposed a growth mechanism to yield the inhomogeneity in the internal structures of Cu2O particles in surfactant-mediated liquid-phase synthesis.

Analyzing the structure of proteins in terms of their structural polymorphism has been recently performed using their two-dimensional electron-density maps obtained through cryo-electron microscopy...

On the basis of the free energy landscape theory, we develop a framework to calculate the structural relaxation time in supercooled liquids and glasses. By the framework, the relaxation time is obtained by an escaping time from a basin in a given free energy surface. In order to demonstrate its usefulness, we apply the framework to monodisperse hard-sphere glass systems. Then we show that the relaxation time increases drastically with the density. Additionally, we discuss an explicit picture of the cooperatively rearranging region by analyzing the spatial distribution of an activation free energy of one particle.

We have recently proposed the "packing exchange mechanism" for F1-ATPase, wherein the perturbation by a substrate binding/release or an ATP hydrolysis is followed by the reorganization of the asymmetric packing structure of the α3β3 complex, accompanying the γ subunit rotation. As part of a further investigation of this rotational mechanism, we performed all-atom molecular dynamics simulations for yeast F1-ATPase both before and after a 16° rotation of the γ subunit triggered by a Pi release. We analyzed the structural fluctuations, the subunit interface interactions, and the dynamics of the relative subunit arrangements before and after the rotation. We found that with the Pi release the αEβE subunit interface becomes looser, which also allosterically makes the αDPβDP subunit interface looser. This structural communication between these interfaces takes place through a tightening of the αTPβTP subunit interface. The γ subunit interacts less strongly with αDP and βDP and more strongly with αTP and βTP. After the Pi release, the tightly packed interfaces are reorganized from the interfaces around βDP to those around βTP, inducing the 16° rotation. These results, which are consistent with the packing exchange mechanism, allow us to deduce a view of the structural change during the 40° rotation.

The configurational entropy of solute molecules is a crucially important quantity to study various biophysical processes. Consequently, it is necessary to establish an efficient quantitative computational method to calculate configurational entropy as accurately as possible. In the present paper, we investigate the quantitative performance of the quasi-harmonic and related computational methods, including widely used methods implemented in popular molecular dynamics (MD) software packages, compared with the Clausius method, which is capable of accurately computing the change of the configurational entropy upon temperature change. Notably, we focused on the choice of the coordinate systems (i.e., internal or Cartesian coordinates). The Boltzmann-quasi-harmonic (BQH) method using internal coordinates outperformed all the six methods examined here. The introduction of improper torsions in the BQH method improves its performance, and anharmonicity of proper torsions in proteins is identified to be the origin of the superior performance of the BQH method. In contrast, widely used methods implemented in MD packages show rather poor performance. In addition, the enhanced sampling of replica-exchange MD simulations was found to be efficient for the convergent behavior of entropy calculations. Also in folding/unfolding transitions of a small protein, Chignolin, the BQH method was reasonably accurate. However, the independent term without the correlation term in the BQH method was most accurate for the folding entropy among the methods considered in this study, because the QH approximation of the correlation term in the BQH method was no longer valid for the divergent unfolded structures.

A clue to the physical origin of the hydrophobicity is in the experimental observations, which show that it is weakened at low temperatures. By considering a solvophobic model protein immersed in water and three species of simple solvents, we analyze the temperature dependence of the changes in free energy, energy, and entropy of the solvent upon protein unfolding. The angle-dependent and radial-symmetric integral equation theories and the morphometric approach are employed in the analysis. Each of the changes is decomposed into two terms, which depend on the excluded volume and on the area and curvature of the solvent-accessible surface, respectively. The excluded-volume term of the entropy change is further decomposed into two components representing the protein–solvent pair correlation and the protein–solvent–solvent triplet and higher-order correlation, respectively. We show that water crowding in the system becomes more serious upon protein unfolding but this effect becomes weaker as the temperature is lowered. If the hydrophobicity originated from the water structuring near a nonpolar solute, it would be strengthened upon lowering of the temperature. Among the three species of simple solvents, considerable weakening of the solvophobicity at low temperatures is observed only for the solvent where the particles interact through a strong attractive potential and the particle size is as small as that of water. Even in the case of this solvent, however, cold denaturation of a protein cannot be reproduced. It would be reproducible if the attractive potential was substantially enhanced, but such enhancement causes the appearance of the metastability limit for a single liquid phase.

In this study, free-energy function (FEF) for discriminating the native fold of a protein from misfolded decoys was investigated. It is a physics-based function using an all-atom model, which comprises the hydration entropy (HE) and the total dehydration penalty (TDP). The HE is calculated using a hybrid of a statistical–mechanical theory applied to a molecular model for water and the morphometric approach. The energetic component is suitably taken into account in a simple manner as the TDP. On the basis of the results from a careful test of the FEF, which have been performed for 118 proteins in representative decoy sets, we show that its performance is distinctly superior to that of any other function. The FEF varies largely from model to model for the candidate models for the native structure (NS) obtained from nuclear magnetic resonance experiments, but we can find models or a model for which the FEF becomes lower than for any of the decoy structures. A decoy set is not suited to the test of a free-energy or potential function in cases where a protein isolated from a protein complex is considered and the structure in the complex is used as the model NS of the isolated protein without any change or where portions of the terminus sides of a protein are removed and the percentage of the secondary structures lost due to the removal is significantly high. As these findings are made possible, we can assume that our FEF precisely captures the features of the true NS. Proteins 2011; © 2011 Wiley-Liss, Inc.

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 βE, βTP, and βDP 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.

Markov state models (MSMs) have enabled us to analyze the long-time behaviors of protein motions using molecular dynamics (MD) simulation data. To construct the model, protein conformations must be...

We have developed a free-energy function based on an all-atom model for proteins. It comprises two components, the hydration entropy (HE) and the total dehydration penalty (TDP). Upon a transition to a more compact structure, the number of accessible configurations arising from the translational displacement of water molecules in the system increases, leading to a water-entropy gain. To fully account for this effect, the HE is calculated using a statistical-mechanical theory applied to a molecular model for water. The TDP corresponds to the sum of the hydration energy and the protein intramolecular energy when a fully extended structure, which possesses the maximum number of hydrogen bonds with water molecules and no intramolecular hydrogen bonds, is chosen as the standard one. When a donor and an acceptor (e.g., N and O, respectively) are buried in the interior after the break of hydrogen bonds with water molecules, if they form an intramolecular hydrogen bond, no penalty is imposed. When a donor or an acceptor is buried with no intramolecular hydrogen bond formed, an energetic penalty is imposed. We examine all the donors and acceptors for backbone-backbone, backbone-side chain, and side chain-side chain intramolecular hydrogen bonds and calculate the TDP. Our free-energy function has been tested for three different decoy sets. It is better than any other physics-based or knowledge-based potential function in terms of the accuracy in discriminating the native fold from misfolded decoys and the achievement of high Z-scores.

It has been shown in a recent experimental study for an alanine-based peptide that the transition from the coil state to the helix state occurs at high pressures. Here we show that this result can be elucidated by our method wherein the solvent entropy, which is calculated using an elaborate statistical–mechanical theory, is treated as an essential factor. The structural stability is described by the competition between the solvent-entropy gain and the conformational-entropy loss of the polypeptide upon the transition. Though the former is smaller at low pressures, it prevails over the latter at high pressures.

We have recently shown that the water entropy is the key quantity in elucidating the folding/unfolding mechanisms for proteins. Here we consider thermal denaturation. The water-entropy gain upon the transition from the random-coil state to the native structure is calculated for some representative proteins by employing the angle-dependent integral equation theory combined with the multipolar water model and the morphometric approach. It is found that the water-entropy gain at 25 °C divided by the number of residues is a good measure of the thermal stability. A protein with a larger value of this measure tends to have a higher denaturation temperature.

The hydration free energy (HFE) is a crucially important physical quantity to discuss various chemical processes in aqueous solutions. Although an explicit-solvent computation with molecular dynamics (MD) simulations is a preferable treatment of the HFE, huge computational load has been inevitable for large, complex solutes like proteins. In the present paper, we propose an efficient computation method for the HFE. In our method, the HFE is computed as a sum of 〈UUV〉/2 (〈UUV〉 is the ensemble average of the sum of pair interaction energy between solute and water molecule) and the water reorganization term mainly reflecting the excluded volume effect. Since 〈UUV〉 can readily be computed through a MD of the system composed of solute and water, an efficient computation of the latter term leads to a reduction of computational load. We demonstrate that the water reorganization term can quantitatively be calculated using the morphometric approach (MA) which expresses the term as the linear combinations of the four geometric measures of a solute and the corresponding coefficients determined with the energy representation (ER) method. Since the MA enables us to finish the computation of the solvent reorganization term in less than 0.1 s once the coefficients are determined, the use of the MA enables us to provide an efficient computation of the HFE even for large, complex solutes. Through the applications, we find that our method has almost the same quantitative performance as the ER method with substantial reduction of the computational load.

Coherent x-ray diffraction imaging (CXDI) enables us to visualize noncrystalline sample particles with micrometer to submicrometer dimensions. Using x-ray free-electron laser (XFEL) sources, two-dimensional diffraction patterns are collected from fresh samples supplied to the irradiation area in the "diffraction-before-destruction" scheme. A recent significant increase in the intensity of the XFEL pulse is promising and will allow us to visualize the three-dimensional structures of proteins using XFEL-CXDI in the future. For the protocol proposed for molecular structure determination using future XFEL-CXDI [T. Oroguchi and M. Nakasako, Phys. Rev. E 87, 022712 (2013)10.1103/PhysRevE.87.022712], we require an algorithm that can classify the data in accordance with the structural polymorphism of proteins arising from their conformational dynamics. However, most of the algorithms proposed primarily require the numbers of conformational classes, and then the results are biased by the numbers. To improve this point, here we examine whether a method based on the manifold concept can classify simulated XFEL-CXDI data with respect to the structural polymorphism of a protein that predominantly adopts two states. After random sampling of the conformations of the two states and in-between states from the trajectories of molecular dynamics simulations, a diffraction pattern is calculated from each conformation. Classification was performed by using our custom-made program suite named enma, in which the diffusion map (DM) method developed based on the manifold concept was implemented. We successfully classify most of the projection electron density maps phase retrieved from diffraction patterns into each of the two states and in-between conformations without the knowledge of the number of conformational classes. We also examined the classification of the projection electron density maps of each of the three states with respect to the Euler angle. The present results suggest that the DM method is imperative for future applications of XFEL-CXDI experiments for proteins, and clarify issues to be taken care of in the future application.

As a Landau-type theory for the glass transition, we present a free energy landscape (FEL) picture which provides a unified understanding of glass transition singularities and show that the FEL can really be calculated by a theoretical and a computational approach. We first give a clear definition of the FEL and argue that there are two kinds of cooperative rearranging region. One is the region defined by the difference between two adjacent basins which could be called simultaneously rearranged region and the other is the atoms involved in the excited state between two adjacent basins. Exploiting the density functional theory, we obtain the FEL for a relaxation process which is characterized by a string motion and determine the size of the cooperatively rearranging region. We also show that the FEL can be determined by the principal component analysis for time dependent configuration obtained by the MD simulation.

A general frame work is devised to obtain the specific heat of nonequilibrium systems described by the energy-landscape picture, where a representative point in the phase space is assumed to obey a stochastic motion which is governed by a master equation. The specific heat depends on the observation time and becomes quenched one for short observation time and annealed one for long observation time. In order to test its validity, the frame work is applied to a two-level system where the state goes back and forth between two levels stochastically. The specific heat is shown to increase from zero to the Schottky form as the observation time is increased from zero to infinity. The anomaly of specific heat at the glass transition is reproduced by a system with a model energy-landscape, where basins of the landscape form a one-dimensional array and jump rate between adjacent basins obeys a power-law distribution. It is shown that the glass transition can be understood as a transition from an annealed to a quenched system and that the glass transition temperature becomes lower when the observation time is increased.

The hydration free energy (HFE) is an essential factor to characterize various chemical processes in solvent. It is then needed to construct computational methods of the HFE. In the computational point of view, not only accuracy but efficiency is necessary when large and complex molecules such as proteins are analyzed. For example, the explicit treatment of solvent is effective for accuracy, however, the treatment for protein solutions sacrifices computational efficiency. To construct a computational method satisfying both accuracy and efficiency, we decompose the HFE into two parts called the direct and indirect parts: one depends on the solute-solvent interaction, and the other is the remaining determined by the geometry of solute. The former is computed by the all-atom treatment. The latter is computed by a hybrid method of the energy representation (ER) method and the morphometric approach (MA). The ER method provides the same accuracy as standard methods such as the thermodynamic integration and free energy perturbation methods and achieves efficient computation by the end-point calculation. The ER method has computational advantageous, however, its protocols can be reduced by the hybrid method. The MA provides thermodynamic quantities primarily governed by the geometry of solute and achieves rapid computation by the fitting function constructed by geometric measures and its corresponding coefficients. Since the indirect part of the HFE depends on the geometry of solute, the MA can be applied to the indirect part and can provide same accuracy as the ER method when the coefficients of the MA are determined by fitting to values calculated by the ER method. We show the accuracy and efficiency of the hybrid method, and demonstrate performance in the discrimination of the native structure from decoys.

Coherent x-ray diffraction imaging enables us to collect huge amounts of diffraction images of a single particle for a short time. Since images represent projections of snapshots of a particle, they reflect its dynamics. Thus, dynamics can be analyzed using the images. One of the requirements for the analysis is a classification of images. Usual classification methods, however, would generate a great deal of classes because there are images arising from snapshots with a subtle difference as well as those with a large difference. This makes the analysis difficult. Here we propose a classification protocol of projection images using the concept of manifold, by which the issue above can be solved. We demonstrate the usefulness of the protocol using the images constructed in a computer.

Coherent X-ray diffraction imaging (CXDI) experiments enable us to visualize the structures of non-crystalline particles with micrometer to sub-micrometer dimensions. Using the X-ray free-electron laser (XFEL), two-dimensional diffraction patterns are collected in the diffraction-before-destruction scheme. To reconstruct three-dimensional electron density maps of a particle, it is necessary to classify the diffraction patterns in terms of structural polymorphism before the reconstruction. Here, we performed a simulation for XFEL-CXDI experiment of a protein predominantly adopting two states. After classifying the data using the custom-made software “ENMA” [1] in which the manifold theory is implemented, we successfully reconstructed the three-dimensional electron density maps of the states using the EMC algorithm [2].[1] T. Yoshidome, T. Oroguchi, M. Nakasako, and M. Ikeguchi, Phys. Rev. E 92, 032710 (2015).[2] N.-T. Duane Loh and V. Elser, Phys. Rev. E 80, 026705 (2009).

We performed molecular dynamics simulations for crystal structures of yeast F1-ATPase whose crystallographic unit contains two different states. One complex binds Pi in the βE subunit, and the other liberated it. In the latter structure, the position of central stalk is rotated +16° in the hydrolysis direction. Because single molecule experiments have proven that Pi release occurs at the βE subunit, and the γ subunit is rotated after ATP hydrolysis and Pi release, these structures are supposed to represent snapshots before and after Pi releases for the 40° substep, respectively.We describe how the F1-ATPase complex changes the structure after Pi release, based on the results of the MD simulations for yF1II and yF1I. Then, we propose a possible mechanism of the 40° rotation of the γ subunit.

We show how to characterize the native-structure models of a protein using our free-energy function F which is based on hydration thermodynamics. Ubiquitin is adopted as an example protein. We consider models determined by the X-ray crystallography and two types of NMR model sets. A model set of type 1 comprises candidate models for a fixed native structure, and that of type 2 forms an ensemble of structures representing the structural variability of the native state. In general, the X-ray models give lower F than the NMR models. There is a trend that, as a model deviates more from the model with the lowest F among the X-ray models, its F becomes higher. Model sets of type 1 and those of type 2, respectively, exhibit two different characteristics with respect to the correlation between the deviation and F. It is argued that the total amount of constraints such as NOEs effectively taken into account in constructing the NMR models can be examined by analyzing the behavior of F. We investigate structural characteristics of the models in terms of the energetic and entropic components of F which are relevant to intramolecular hydrogen bonding and to backbone and side-chain packing, respectively.

We propose a novel picture of the rotation mechanism of F1-ATPase [1]. In the proposal, the key factor is the translational entropy of water, which has been shown to drive a variety of self-assembly processes in biological systems [2]. We calculate the hydration entropies of three different sub-complexes comprising the gamma subunit, one of the beta subunits, and two alpha subunits adjacent to them. The calculation is made using the angle-dependent integral equation theory combined with the multipolar water model [3] and morphometric approach [4]. The major finding is that the packing in F1-ATPase is highly asymmetrical and this asymmetry is ascribed to the water-entropy effect. We discuss how the rotation of the gamma subunit is induced by such chemical processes as ATP binding, ATP hydrolysis, and release of the products. In our picture, the asymmetrical packing plays crucially important roles and the rotation is driven by the water-entropy effect. As part of the demonstration of the validity of our rotation mechanism, we also analyze the water-entropy change in yeast F1-ATPase during 16-degree rotation of the gamma subunit [5]. The result demonstrates the validity of the water-entropy mechanism proposed in Ref 1.[1] T. Yoshidome, Y. Ito, M. Ikeguchi, and M. Kinoshita, J. Am. Chem. Soc. 133, 4030 (2011).[2] M. Kinoshita, Front. Biosci. 14, 3419 (2009).[3] M. Kinoshita, J. Chem. Phys. 128, 024507 (2008).[4] R. Roth, Y. Harano, and M. Kinoshita, Phys. Rev. Lett. 97, 078101 (2006).[5] T. Yoshidome, Y. Ito, N. Matubayasi, M. Ikeguchi, and M. Kinoshita, Soft Matter (to be submitted).

“Hot spots” are residues accounting for the majority of the protein–protein binding free energy (BFE) despite that they comprise only a small fraction of the protein–protein interface. A hot spot can be found experimentally by measuring the BFE change upon mutating it to alanine: the mutation gives rise to a significantly large increase in the BFE. Theoretical prediction of hot spots is an enthusiastic subject in biophysics, biochemistry, and bioinformatics. For the development of a reliable prediction method, it is essential to understand the physical origin of hot spots. To this end, we calculate the water-entropy gains upon binding both for a wild-type complex and for its mutant complex using a hybrid method of the angle-dependent integral equation theory applied to a molecular model for water and the morphometric approach. We note that this type of calculation has never been employed in the previously reported methods. The BFE change due to alanine mutation is evaluated only from the change in the water-entropy gain with no parameters fitted to the experimental data. It is shown that the overall performance of predicting hot spots in our method is higher than that in Robetta, a standard free-energy-based method using fitting parameters, when the most widely used criterion for defining an actual hot spot is adopted. This result strongly suggests that the water-entropy effect we calculate is the key factor governing basic physics of hot spots.

We propose a novel picture of the rotation mechanism of F(1)-ATPase, a rotary-motor protein complex. Entropy, which originates from the translational displacement of water molecules, is treated as the key factor in the proposal. We calculate the water entropy gains upon formation of the α-β, α-γ, and β-γ subunit pairs. The gain is given as the difference between the hydration entropy of a subunit pair and the sum of the hydration entropies of the separate subunits forming the pair. The calculation is made using a hybrid of a statistical-mechanical theory for molecular liquids and morphometric approach. The water entropy gain is considered as a measure of tightness of the packing at each subunit interface. The results are highly correlated with the numbers of stable contacts at the subunit interfaces estimated by a molecular dynamics simulation. We also calculate the hydration entropies of three different subcomplexes comprising the γ subunit, one of the β subunits, and two α subunits adjacent to them. The major finding is that the packing in F(1)-ATPase is highly asymmetrical, and this asymmetry is ascribed to the water entropy effect. We discuss how the rotation of the γ subunit is induced by such chemical processes as ATP binding, ATP hydrolysis, and release of the products. In our picture, the asymmetrical packing plays crucially important roles, and the rotation is driven by the water entropy effect.

We have recently proposed a measure of the thermal stability of a protein: the water-entropy gain at 25 ◦ C upon folding normalized by the number of residues, which is calculated using a hybrid of the angle-dependent integral equation theory combined with the multipolar water model and the morphometric approach. A protein with a larger value of the measure is thermally more stable. Here we extend the study to analyses on the effects of heme on the thermal stability of four cytochromes c (PA c551 ,P Hc552 ,H Tc552, and AA c555) whose denaturation temperatures are considerably different from one another despite that they share significantly high sequence homology and similar threedimensional folds. The major conclusions are as follows. For all the four cytochromes c, the thermal stability is largely enhanced by the heme binding in terms of the water entropy. For the holo states, the measure is the largest for AA c555. However, AA c555 has the lowest packing efficiency of heme and the apo polypeptide with hololike structure, which is unfavorable for the water entropy. The highest stability of AA c555 is ascribed primarily to the highest efficiency of side-chain packing of the apo polypeptide itself. We argue for all the four cytochromes c that due to covalent heme linkages, the number of accessible conformations of the denatured state is decreased by the steric hindrance of heme, and the conformational-entropy loss upon folding becomes smaller, leading to an enhancement of the thermal stability. As for the apo state modeled as the native structure whose heme is removed, AA c555 has a much larger value of the measure than the other three. Overall, the theoretical results are quite consistent with the experimental observations (e.g., at 25 ◦ Ct heα-helix content of the apo state of AA c555 is almost equal to that of the holo state while almost all helices are collapsed in the apo states of PA c551 ,P Hc552, and HT c552). © 2011 American Institute of Physics.

We report a new progress in elucidating the mechanism of the unidirectional movement of a linear-motor protein (e.g., myosin) along a filament (e.g., F-actin). The basic concept emphasized here is that a potential field is entropically formed for the protein on the filament immersed in solvent due to the effect of the translational displacement of solvent molecules. The entropic potential field is strongly dependent on geometric features of the protein and the filament, their overall shapes as well as details of the polyatomic structures. The features and the corresponding field are judiciously adjusted by the binding of adenosine triphosphate (ATP) to the protein, hydrolysis of ATP into adenosine diphosphate (ADP)+Pi, and release of Pi and ADP. As the first step, we propose the following physical picture: The potential field formed along the filament for the protein without the binding of ATP or ADP+Pi to it is largely different from that for the protein with the binding, and the directed movement is realized by repeated switches from one of the fields to the other. To illustrate the picture, we analyze the spatial distribution of the entropic potential between a large solute and a large body using the three-dimensional integral equation theory. The solute is modeled as a large hard sphere. Two model filaments are considered as the body: model 1 is a set of one-dimensionally connected large hard spheres and model 2 is a double helical structure formed by two sets of connected large hard spheres. The solute and the filament are immersed in small hard spheres forming the solvent. The major findings are as follows. The solute is strongly confined within a narrow space in contact with the filament. Within the space there are locations with sharply deep local potential minima along the filament, and the distance between two adjacent locations is equal to the diameter of the large spheres constituting the filament. The potential minima form a ringlike domain in model 1 while they form a pointlike one in model 2. We then examine the effects of geometric features of the solute on the amplitudes and asymmetry of the entropic potential field acting on the solute along the filament. A large aspherical solute with a cleft near the solute-filament interface, which mimics the myosin motor domain, is considered in the examination. Thus, the two fields in our physical picture described above are qualitatively reproduced. The factors to be taken into account in further studies are also discussed.

We have recently shown that protein folding is driven by the water-entropy gain. When the alpha-helix or beta-sheet is formed, the excluded volumes generated by the backbone and side chains overlap, leading to an increase in the total volume available to the translational displacement of water molecules. Primarily by this effect, the water entropy becomes higher. At the same time, the dehydration penalty (i.e., the break of hydrogen bonds with water molecules) is compensated by the formation of intramolecular hydrogen bonds. Hence, these secondary structures are very advantageous units, which are to be formed as much as possible in protein folding. The packing of side chains, which leads to a large increase in the water entropy, is also crucially important. Here we investigate the roles of the side-chain packing in the second structural preference in protein folding. For some proteins we calculate the hydration entropies of a number of structures including the native structure with or without side chains. A hybrid of the angle-dependent integral equation theory combined with the multipolar water model and the morphometric approach is employed in the calculation. Our major findings are as follows. For the structures without side chains, there is an apparent tendency that the water entropy becomes higher as the alpha-helix or beta-sheet content increases. For the structures with side chains, however, a higher content of alpha-helices or beta-sheets does not necessarily lead to larger entropy of water due to the effect of the side-chain packing. The thorough, overall packing of side chains, which gives little space in the interior, is unique to the native structure. To accomplish such specific packing, the alpha-helix and beta-sheet contents are prudently adjusted in protein folding.

We have developed a free-energy function based on an all-atom model for proteins. It comprises two components, the hydration entropy (HE) and the total dehydration penalty (TDP). Upon a transition to a more compact structure, the number of accessible configurations arising from the translational displacement of water molecules in the system increases, leading to a water-entropy gain. To fully account for this effect, the HE is calculated using a statistical-mechanical theory applied to a molecular model for water. The TDP corresponds to the sum of the hydration energy and the protein intramolecular energy when a fully extended structure, which possesses the maximum number of hydrogen bonds with water molecules and no intramolecular hydrogen bonds, is chosen as the standard one. When a donor and an acceptor (e.g., N and O, respectively) are buried in the interior after the break of hydrogen bonds with water molecules, if they form an intramolecular hydrogen bond, no penalty is imposed. When a donor or an acceptor is buried with no intramolecular hydrogen bond formed, an energetic penalty is imposed. We examine all the donors and acceptors for backbone-backbone, backbone-side chain, and side chain-side chain intramolecular hydrogen bonds and calculate the TDP. Our free-energy function has been tested for three different decoy sets. It is better than any other physics-based or knowledge-based potential function in terms of the accuracy in discriminating the native fold from misfolded decoys and the achievement of high Z-scores.

We investigate the molecular mechanism of pressure denaturation of proteins using the angle-dependent integral equation theory combined with the multipole water model and the morphometric approach. We argue that the hydration entropy of a protein is the key quantity. It is verified that at an elevated pressure, a swelling structure--which has only moderately less compact than the native structure but has a much larger water-accessible surface area--turns more stable than the native structure in terms of the water entropy. The swelling structure is characterized by the penetration of water into the interior. The hydration entropy is decomposed into contributions from the translational and rotational restrictions for the molecular motions of water. Each contribution is further decomposed into the water-protein pair correlation component and the water-water-protein triplet and higher-order correlation components. The pair correlation component in the translational contribution is divided into two terms arising from the excluded volume and the water structure near the protein, respectively. It is found that pressure denaturation accompanies a loss of the translational and rotational entropies at the pair correlation level but a much larger gain of the translational entropy at the triplet and higher-order correlation levels. Although the translational and rotational motions of water molecules penetrating the protein interior and contacting the protein surface are constrained, the translational restriction for the water molecules well outside the protein is greatly reduced. The latter entropic gain dominates, leading to the denaturation.

We elucidate the microscopic mechanism of the weakening of the hydrophobicity at low temperatures by investigating cold denaturation of a protein. We employ an elaborate statistical-mechanical theory combined with a realistic water model. At low temperatures, the ordered structure with enhanced hydrogen bonds of water molecules is formed near nonpolar groups, leading to entropic loss and energy gain which are both quite large. However, they are canceled out and make no contribution to the free-energy change. We argue that a different factor, which is responsible for the weakening of the hydrophobicity at low temperatures, induces cold denaturation.

We propose a general framework of pressure effects on the structures formed by the self-assembly of solute molecules immersed in solvent. The integral equation theory combined with the morphometric approach is employed for a hard-body model system. Our picture is that protein folding and ordered association of proteins are driven by the solvent entropy: At low pressures, the structures almost minimizing the excluded volume (EV) generated for solvent particles are stabilized. Such structures appear to be even more stabilized at high pressures. However, it is experimentally known that the native structure of a protein is unfolded, and ordered aggregates such as amyloid fibrils and actin filaments are dissociated by applying high pressures. This initially puzzling result can also be elucidated in terms of the solvent entropy. A clue to the basic mechanism 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 and this enrichment becomes greater as the pressure increases. We argue that "attraction" is entropically provided between the solute surface and solvent particles, and the attraction becomes higher with rising pressure. Due to this effect, at high pressures, the structures possessing the largest possible solvent-accessible surface area together with sufficiently small EV become more stable in terms of the solvent entropy. To illustrate this concept, we perform an analysis of pressure denaturation of three different proteins. It is shown that only the structures that have the characteristics described above exhibit interesting behavior. They first become more destabilized relative to the native structure as the pressure increases, but beyond a threshold pressure the relative instability begins to decrease and they eventually become more stable than the native structure.

It has been experimentally shown that the folding of apoplastocyanin (apoPC) accompanies a very large enthalpic loss [N. Baden et al., J. Chem. Phys. 127, 175103 (2007)]. This implies that an even larger entropic gain occurs in stabilizing the folded structure to overcome the enthalpic loss. Here, we calculate the water-entropy gain upon the folding of apoPC using the angle-dependent integral equation theory combined with the multipolar water model and the recently developed morphometric approach. It is demonstrated that the calculated value is in quantitatively good accord with the value estimated from the experimental data by accounting for the conformational-entropy loss. According to a prevailing view, the water adjacent to a hydrophobic group is unstable especially in terms of the rotational entropy and the folding is driven primarily by the release of such unfavorable water to the bulk through the burial of nonpolar side chains. We show, however, that the resultant entropic gain is too small to elucidate the experimental result. The great entropic gain observed is ascribed to the reduction in the restriction for the translational motion of water molecules in the whole system.

Exploiting the thermodynamic potential functional provided by density functional theory, we determine analytically the free-energy landscape (FEL) in a hard-sphere fluid. The FEL is represented in the three-dimensional coordinate space of the tagged particle. We also analyze the distribution of the free-energy barrier between adjacent basins and show that the most provable value and the average of the free-energy barrier are increasing functions of the density. Since the size of the cooperatively rearranging region (CRR) is also increased as the density is raised [Yoshidome, Phys. Rev. E 76, 021506 (2007)], the present result is consistent with the Adam-Gibbs theory in which the increase of the activation energy is due to the increase of the size of the CRR.

Exploiting the density functional theory, we calculate the free energy landscape (FEL) of the hard sphere glass in three dimensions. From the FEL, we estimate the number of the particles in the cooperatively rearranging region (CRR). We find that the density dependence of the number of the particles in the CRR is expressed as a power law function of the density. Analyzing the relaxation process in the CRR, we also find that the string motion is the elementary process for the structural relaxation, which leads to the natural definition of the simultaneously rearranging region as the particles displaced in the string motion.