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One of the simplest ways to predict ligand binding sites is to identify pocket-shaped regions on the protein surface. Many programs have already been proposed to identify these pocket regions. Examination of their algorithms revealed that a pocket intrinsically has two arbitrary properties, "size" and "depth". We proposed a new definition for pockets using two explicit adjustable parameters that correspond to these two arbitrary properties. A pocket region is defined as a space into which a small probe can enter, but a large probe cannot. The radii of small and large probe spheres are the two parameters that correspond to the "size" and "depth" of the pockets, respectively. These values can be adjusted individual putative ligand molecule. To determine the optimal value of the large probe spheres radius, we generated pockets for thousands of protein structures in the database, using several size of large probe spheres, examined the correspondence of these pockets with known binding site positions. A new measure of shallowness, a minimum inaccessible radius, R(inaccess), indicated that binding sites of coenzymes are very deep, while those for adenine/guanine mononucleotide have only medium shallowness and those for short peptides and oligosaccharides are shallow. The optimal radius of large probe spheres was 3-4 A for the coenzymes, 4 A for adenine/guanine mononucleotides, and 5 A or more for peptides/oligosaccharides. Comparison of our program with two other popular pocket-finding programs showed that our program had a higher performance of detecting binding pockets, although it required more computational time., ((c) 2007 Wiley-Liss, Inc.)

We have investigated energy landscape of human lysozyme in its native state by using principal component analysis and a model, jumping-among-minima (JAM) model. These analyses are applied to 1 nsec molecular dynamics trajectory of the protein in water. An assumption embodied in the JAM model allows us to divide protein motions into intra-substate and inter-substate motions. By examining intra-substate motions, it is shown that energy surfaces of individual conformational substates are nearly harmonic and mutually similar. As a result of principal component analysis and JAM model analysis, protein motions are shown to consist of three types of collective modes, multiply hierarchical modes, singly hierarchical modes, and harmonic modes. Multiply hierarchical modes, the number of which accounts only for 0.5% of all modes, dominate contributions to total mean-square atomic fluctuation. Inter-substate motions are observed only in a small-dimensional subspace spanned by the axes of multiplyhierarchical and singly hierarchical modes. Inter-substate motions have two notable time components: faster component seen within 200 psec and slower component. The former involves transitions among the conformational substates of the low-level hierarchy, whereas the latter involves transitions of the higher level substates observed along the first four multiply hierarchical modes. We also discuss dependence of the subspace, which contains conformational substates, on time duration of simulation.

Molecular dynamics simulations were carried out to study what happens in a photoreceptor protein, photoactive yellow protein (PYP), immediately after the vertical transition of the chromophore from the ground to the excited state. A photon absorption simulation was performed to investigate the movement of amino acid residues upon photoexcitation. To calculate the excited state of the chromophore, SCF-CI calculation was carried out with INDO/S Hamiltonian. We observed that some amino acid residues have strong interactions with the chromophore. Most of these amino acid residues are conserved in PYPs from three different species of bacteria. This observation indicates the biological importance of these residues.

The mechanical response of a TIM-barrel protein to an applied pressure has been studied. We generated structures under an applied pressure by assuming the volume change to be a linear function of normal mode variables. By Delaunay tessellation, the space occupied by protein atoms is divided uniquely into tetrahedra, whose four vertices correspond to atomic positions. Based on the atoms that define them, the resulting Delaunay tetrahedra are classified as belonging to various secondary structures in the protein. The compressibility of various regions identified with respect to secondary structural elements in this protein is obtained from volume changes of respective regions in two structures with and without an applied pressure. We found that the beta barrel region located at the core of the protein is quite soft. The interior of the beta barrel, occupied by side chains of beta strands, is the softest. The helix, strand, and loop segments themselves are extremely rigid, while the regions existing between these secondary structural elements are soft. These results suggest that the regions between secondary structural elements play an important role in protein dynamics. Another aspect of tetrahedra, referred to as bond distance, is introduced to account for rigidities of the tetrahedra. Bond distance is a measure of separation of the atoms of a tetrahedron in terms of number of bonds along the polypeptide chain or side chains. Tetrahedra with longer bond distances are found to be softer on average. From this behavior, we derive a simple empirical equation, which well describes the compressibilities of various regions.

A comparison of a normal mode analysis and principal component analysis of a 200-ps molecular dynamics trajectory of bovine pancreatic trypsin inhibitor in vacuum has been made in order to further elucidate the harmonic and anharmonic aspects in the dynamics of proteins. An anharmonicity factor is defined which measures the degree of anharmonicity in the modes, be they principal modes or normal modes, and it is shown that the principal mode system naturally divides into anharmonic modes with peak frequencies below 80 cm-1, and harmonic modes with frequencies above this value. In general the larger the mean-square fluctuation of a principal mode, the greater the degree of anharmonicity in its motion. The anharmonic modes represent only 12% of the total number of variables, but account for 98% of the total mean-square fluctuation. The transitional nature of the anharmonic motion is demonstrated. The results strongly suggest that in a large subspace, the free energy surface, as probed by the simulation, is approximated by a multi-dimensional parabola which is just a rescaled version of the parabola corresponding to the harmonic approximation to the conformational energy surface at a single minimum. After 200 ps, the rescaling factor, termed the "normal mode rescaling factor," has apparently converged to a value whereby the mean-square fluctuation within the subspace is about twice that predicted by the normal mode analysis.

We have developed theoretical models for analysis of X-ray diffuse scattering from protein crystals. A series of models are proposed to be used for experimental data with different degrees of precision. First, we propose the normal mode model, where conformational dynamics of a protein is assumed to occur mostly in a limited conformational subspace spanned by a small number of low-frequency normal modes in the protein. When high precision data are available, variances and covariances of the normal mode variables can be determined from experimental data using this model. For experimental data with lower degrees of precision, we introduce a series of simpler models. These models express the covariance matrix using relatively simple empirical correlation functions by assuming the correlation between a pair of atoms to be isotropic. As an application of these simpler models, we calculate diffuse-scattering patterns from a human lysozyme crystal to examine how each adjustable parameter in the models affects general features of the resulting patterns. The results of the calculation are summarized as follows. (1) The higher order scattering makes a significant contribution at high resolutions. (2) The resulting simulated patterns are sensitive to changes in correlation lengths of about 1 A, as well as to changes of the functional form of the correlation function. (3) But only the "average" value of the intra- and intermolecular correlation lengths seems to determine the gross features of the pattern. (4) The effect of the atom-dependent amplitude of fluctuations is difficult to observe.

Pressure effect on the equilibrium conformation in sperm whale deoxymyoglobin and its volume fluctuation are studied by the normal mode analysis and strain tensor analysis. The pressure-induced deformation of interhelix regions are found to be remarkably more compressed than the other parts of the molecule. The intrahelix compressibility is shown to be relatively small. We also calculate the compressibility of the three hydrophobic clusters, located at the bottom, distal, and proximal side of the heme. Its value is found to decrease in the indicated order. The average compressibility of these hydrophobic clusters is less than the average interhelix compressibility, even though there are large cavities in these clusters. In order to study overall deformation, we define a linear compressibility and calculate it for all pairs of C alpha atoms. The pressure-induced deformation is observed to be heterogeneous also in this analysis. The calculated root-mean-square displacement of the constituent atoms in the equilibrium conformation at 1,000 atm from those at 0 atm is 0.12 A, which is much smaller in magnitude than the average value of the atomic fluctuations at room temperature. In the water solvent, the volume excluded by the protein molecule in the equilibrium conformation is reduced by 0.9%, when the pressure is raised from 0 to 1,000 atm. The calculated magnitude of the root-mean-square volume fluctuation is 0.3% of the excluded volume at room temperature. The square of the volume fluctuation is given as a sum of contributions from individual normal modes. Contributions from low frequency normal modes are found to dominate over those from higher frequency normal modes. The estimated value of the isothermal compressibility of deoxymyoglobin is 9.37 x 10(-12) cm2/dyn.

Proteins consist of structural units such as globular domains, secondary structures, and modules. Modules were originally defined by partitioning a globular domain into compact regions, each of which is a contiguous polypeptide segment having a compact conformation. Since modules show close correlations with the intron positions of genes, they are regarded as primordial polypeptide pieces encoded by exons and shuffled, leading to yield new combination of them in early biological evolution. Do modules maintain their native conformations in solution when they are excised at their boundaries? In order to find answers to this question, we have synthesized modules of barnase, one of the bacterial RNases, and studied the solution structures of modules M2 (amino acid residues 24-52) and M3 (52-73) by 2D NMR studies. Some local secondary structures, alpha-helix, and beta-turns in M2 and beta-turns in M3, were observed in the modules at the similar positions to those in the intact barnase but the overall state seems to be in a mixture of random and native conformations. The present result shows that the excised modules have propensity to form similar secondary structures to those of the intact barnase.

Normal mode analysis of mouse epidermal growth factor (mEGF) has been carried out at room temperature. The value of the lowest frequency is 4.1 cm-1. This mode corresponds to hinge-bending motion between the N-terminal and C-terminal domains of mEGF. This hinge-bending motion corresponds to the "mitten mode." In this motion, the N-terminal domain is almost rigid. However, the C-terminal domain is found to consist of three rigid segments. Two segments, C33-D46 and G51-R53, are observed moving in the same direction, but L47-W50 moves in the opposite direction. For this mode, the effective Young's modulus turned out to be 1.1 x 10(9) dyn.cm-2. This value is a little larger than that of the mode with the lowest frequency 4.4 cm-1 of BPTI. The difference may be related to the fraction of residues involved in beta-sheets in the molecule. Similar analyses are carried out for other low frequency modes.

A method is presented to describe the internal motions of proteins obtained from molecular dynamics or Monte Carlo simulations as motions of normal mode variables. This method calculates normal mode variables by projecting trajectories of these simulations onto the axes of normal modes and expresses the trajectories as a linear combination of normal mode variables. This method is applied to the result of the molecular dynamics and the Monte Carlo simulations of human lysozyme. The motion of the lowest frequency mode extracted from the simulations represents the hinge bending motion very faithfully. Analysis of the obtained motions of the normal mode variables provides an explanation of the anharmonic aspects of protein dynamics as due first to the anharmonicity of the actual potential energy surface near a minimum and second to trans-minimum conformational changes.

In our previous paper we reported the conformation of melittin bound to perdeuterated dodecylphosphocholine micelles as studied by 1H NMR experiment and distance geometry calculation. No hydrogen bonds were taken into consideration explicitly in the calculation. However, mostly alpha-helical conformations were obtained as results of the calculation even with no explicitly assumed hydrogen bonds. In the present paper we refined the distance geometry calculation by incorporating hydrogen bonds suggested by the previous calculation. As a result, we obtained the conformation of melittin, which was consistent with both NMR data and the additional hydrogen bonding data. The alpha-helical rod in the refined conformation also has a kink at Thr-11 and Gly-12, but its bent angle is now a bit narrowly distributed in 135 degrees +/- 15 degrees. In the present study another distortion at Trp-19 and IIe-20 becomes conspicuous. The average root-mean-square displacement of atoms is now much smaller and is 1.5 A for all backbone atoms. In the present paper side chain conformations are also analyzed.

A normal mode analysis of human lysozyme has been carried out at room temperature. Human lysozyme is an enzyme constituted of two domains separated by an active site cleft, the motion of which is thought to be relevant for biological function. This motion has been described as a hinge bending motion. McCammon et al. have determined the characteristics of the hinge bending motion but they assumed a prior knowledge of the hinge axis. In this work we propose a method which is free from this assumption and determines the hinge axis and root mean square (rms) rotation angle which give the best agreement with the pattern of changes in all the distances between nonhydrogen atoms in the two domains, obtained by the normal mode analysis. The hinge axis we found is notably different from the one previously determined and goes, roughly, through the C alpha 55 and C alpha 76, i.e., it is located at the base of the beta-sheet of the second domain. The rms value for the rotation angle is also twice as large as the previous one: 3.37 degrees. It is shown that this hinge bending motion provides a fairly good approximation of the dynamics of human lysozyme and that the normal mode with the lowest frequency has a dominating contribution to this hinge bending motion. A study of the accessible surface area of the residues within the cleft reveals that the motion does not result in a better exposure to the solvent of these residues. A characterization of the thermally excited state (under the hypothesis of the harmonicity of the potential energy surface) has been done using the concept of topology of atom packing. Under this hypothesis the thermal fluctuations result only in a small change of the topology of atom packing, leading therefore to nearly elastic deformations of the protein.

The normal mode analysis of conformational fluctuation is carried out for a small globular protein, bovine pancreatic trypsin inhibitor. Results are analyzed mainly to reveal the mechanical construction of the protein molecule. We take dihedral angles, including peptide omega angles, as independent variables for the normal mode analysis. There are 306 such angles in this molecule. Motions in modes with frequencies lower than 120 cm-1 are shown to involve atoms in the whole protein molecule, and spatial change of displacement vectors is continuous, i.e., those of atoms near in space are similar. To quantitate the observation of the continuity, a correlation function of direction vectors of atomic displacements is calculated. From this function we define a quantity that is interpreted as the wave length of an equivalent elastic plane wave. From this quantity we deduce effective Young's modulus for each mode. For the mode with the lowest frequency 4.4 cm-1, it turned out to be 0.8 x 10(9) dyn cm-2, the value two orders of magnitude softer than, for instance, alpha-helices. Prompted by this observation, the four lowest frequency modes and also the harmonic motions in the thermal equilibrium are analyzed further mainly to detect relatively rigid structural elements in the molecule. From this analysis emerges a mechanical picture of the protein molecule that is made up of relatively rigid elements held together by very soft parts.

A computer experiment of protein dynamics is carried out, which consists of two steps: (1) A Monte Carlo simulation of thermal fluctuations in the native state of a globular protein, bovine pancreatic trypsin inhibitor; and (2) a simulation of the quick freezing of fluctuating conformations into energy minima by minimization of the energy of a number of conformations sampled in the Monte Carlo simulation. From the analysis of results of the computer experiment is obtained the following picture of protein dynamics: multiple energy minima exist in the native state, and they are distributed in clusters in the conformational space. The dynamics has a hierarchical structure which has at least two levels. In the first level, dynamics is restricted within one of the clusters of minima. In the second, transitions occur among the clusters. Local parts of a protein molecule, side chains and local main chain segments, can take multiple locally stable conformations in the native state. Many minima result from combinations of these multiple local conformations. The hierarchical structure in the dynamics comes from interactions among the local parts. Protein molecules have two types of flexibility, each associated with elastic and plastic deformations, respectively.

An analysis is carried out of differences in the minimum energy conformations obtained in the previous paper by energy minimization starting from conformations sampled by a Monte Carlo simulation of conformational fluctuations in the native state of a globular protein, bovine pancreatic trypsin inhibitor. Main conformational differences in each pair of energy minima are found usually localized in several side chains and in a few local main chain segments. Such side chains and local main chain segments are found to take a few distinct local conformations in the minimum energy conformations. Energy minimum conformations can thus be described in terms of combinations of these multiple local conformations.

Conformational fluctuations in a globular protein, bovine pancreatic trypsin inhibitor, in the time range between picoseconds and nanoseconds are studied by a Monte Carlo simulation method. Multiple energy minima are derived from sampled conformations by minimizing their energy. They are distributed in clusters in the conformational space. A hierarchical structure is observed in the simulated dynamics. In the time range between 10(-14) and 10(-10) seconds dynamics is well represented by a superposition of vibrational motions within an energy well with transitions among minima within each cluster. Transitions among clusters take place in the time range of nanoseconds or longer.

Differences in atom packing are studied in the minimum energy conformations derived from the record of the Monte Carlo simulation of conformational fluctuation in the native state of a globular protein, bovine pancreatic trypsin inhibitor. It is found that local deformations observed among the minima which are found in the previous paper are accompanied by rearrangement of atom packing. Spatial locations of the local deformations in the three-dimensional folded structure are also studied. It is found that the local deformations are distributed in space in several clusters in the folded structure. The size and location of the clusters characterize the respective fluctuations of the first and the second levels observed in the simulation. In the fluctuations of the first level local deformations, each of which usually involves a few side chains and one main chain local segment, are thermally exited independently of each other near the surface of the molecule. The observed fluctuation of the second level involves a cooperative deformation involving many side chains and local main chain segments all in one cluster, which goes though the core of the molecule. The collective local deformations observed both in the first and second levels are plastic in the sense that they are accompanied with rearrangement of atom packing.

Distances between centers of gravity of individual residues are compared among the minimum energy conformations derived from the record of the Monte Carlo simulation of conformational fluctuations in the native state of a globular protein, bovine pancreatic trypsin inhibitor. It is found that local deformations originating from the multiplicity of local conformations cause deformations of the whole structure of the molecule in various ways, which can be classified into two types. Type 1: When a local deformation occurs in a region consisting of a few residues near the surface of the molecule, the whole shape of the molecule responds by deforming elastically. The magnitude of this deformation is in the range of thermal fluctuations calculated by the harmonic approximation around a single minimum. Type 2: We have observed one case belonging to the second type in which local deformations occur cooperatively in an extended region. This region goes across the whole molecule and divide the remaining parts into two. Atom packing changes in and around the extended region of local deformations. For this reason deformation in this region is plastic. Relative location and orientation between the divided two parts change very much. Deformation of the whole shape in this case, associated with the plastic deformation in an extended region, demonstrates that protein molecules have a flexibility beyond the harmonic limit.