Normal modes of vibration in bovine pancreatic trypsin inhibitor and its mechanical property

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 model for the calculation of the dielectric dispersion and the dipole moment of globular proteins in solution

We describe a new procedure whereby the magnitude of the dielectric dispersion of a solution of globular protein molecules can be calculated. The protein molecule is considered to have spherical symmetry and the charged residues are thought to be situated in a medium whose dielectric constant increases continuously as a function of the distance from the centre of mass. The dipole moment of the protein in the solution is made up of two parts: the intrinsic dipole moment due to the charge distribution of the protein and the dipole moment due to polarization of the medium and the ionic cloud. When the model is applied to solutions of cytochrome c it is found that polarization of the medium results in a decrease in the dielectric dispersion amplitude. The mean square dipole moment calculated with the help of this method indicates that the fluctuation of the configurations cannot be responsible for the large dispersion in the megahertz region.     

Conformational deformation in deoxymyoglobin by hydrostatic pressure

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^αatoms. 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 Å, 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 × 10^?12 cm²/dyn. © 1993 Wiley-Liss, Inc.     

Ligand-induced conformational change of a protein reproduced by a linear combination of displacement vectors obtained from normal mode analysis

The conformational change of a protein upon ligand binding was examined by normal mode analysis (NMA) based on an elastic-network model (ENM) for a full-atom system using dihedral angles as independent variables. Specifically, we investigated the extent to which conformational change vectors of atoms from an apo form to a holo form of a protein can be represented by a linear combination of the displacement vectors of atoms in the apo form calculated for the lowest-frequency m normal modes (m=1, 2,…, 20). In this analysis, the latter vectors were best fitted to the former ones by the least-squares method. Twenty-two paired proteins in the holo and apo forms, including three dimer pairs, were examined. The results showed that, in most cases, the conformational change vectors were reproduced well by a linear combination of the displacement vectors of a small number of low-frequency normal modes. The conformational change around an active site was reproduced as well as the entire conformational change, except for some proteins that only undergo significant conformational changes around active sites. The weighting factors for 20 normal modes optimized by the least-squares fitting characterize the conformational changes upon ligand binding for these proteins. The conformational changes sampled around the apo form of a protein by the linear combination of the displacement vectors obtained by ENM-based NMA may help solve the flexible-docking problem of a protein with another molecule because the results presented herein suggest that they have a relatively high probability of being involved in an actual conformational change.     

Electrostatics of proteins: Description in terms of two dielectric constants simultaneously

In the semi-continuum treatment of the energetics of charge formation (or transfer) inside a protein, two components of the energy are inevitably present: the energy of interaction of the ion with the pre-existing intraprotein electric field, and the energy due to polarization of the medium by the newly formed charge. The pre-existing field is set up by charges (partial or full) of the protein atoms fixed in a definite structure. The calculation of this field involves only the electronic polarization (the optical dielectric constant ϵ_o) of the protein because the polarization due to shifts of heavy atoms has already been accounted for by their equilibrium coordinates. At the same time, the aqueous surroundings should be described by the static constant ϵ_sw, as the positions of water molecules are not fixed. The formation of a new charge, absent in the equilibrium X-ray structure, results in shifts of electrons and polar atoms, i.e., it involves all kinds of medium polarization described by the static dielectric constant of protein ϵ_s. Thus, in calculations of the total energy, two different dielectric constants of the protein are operative simultaneously. This differs from a widely used algorithm employing one effective dielectric constant for both components of the ion's energy. Proteins: 28:174–182, 1997. © 1997 Wiley-Liss Inc.     

Evaluation of the influence of the internal aqueous solvent structure on electrostatic interactions at the protein-solvent interface by nonlocal continuum electrostatic approach

The dielectric properties of the polar solvent on the protein-solvent interface at small intercharge distances are still poorly explored. To deconvolute this problem and to evaluate the pair-wise electrostatic interaction (PEI) energies of the point charges located at the protein-solvent interface we used a nonlocal (NL) electrostatic approach along with a static NL dielectric response function of water. The influence of the aqueous solvent microstructure (determined by a strong nonelectrostatic correlation effect between water dipoles within the orientational Debye polarization mode) on electrostatic interactions at the interface was studied in our work. It was shown that the PEI energies can be significantly higher than the energies evaluated by the classical (local) consideration, treating water molecules as belonging to the bulk solvent with a high dielectric constant. Our analysis points to the existence of a rather extended, effective low-dielectric interfacial water shell on the protein surface. The main dielectric properties of this shell (effective thickness together with distance- and orientation-dependent dielectric permittivity function) were evaluated. The dramatic role of this shell was demonstrated when estimating the protein association rate constants.     

Analyzing the normal mode dynamics of macromolecules by the component synthesis method.

This paper presents a general method for studying the harmonic dynamics of large biomolecules and molecular complexes. The performance and accuracy of the method applied to a number of molecules are also reported. The basic approach of the method is to divide a macromolecule into a number of smaller components. The local normal modes of the components are first calculated by treating individual components and the interactions between nearest neighboring components. The physical displacements of all atoms are then represented in the local normal mode space, in which a selected range of high-frequency local modes is neglected. The equation of motion of the molecule in the local normal mode space will then have a smaller dimension, and consequently the normal modes of the whole structure, particularly for large molecules, can be solved much more easily. The normal modes of two polypeptides--(Ala)6 and (Ala)12--and a double-helical DNA--d(ATATA).d(TATAT)--are analyzed with this method. Reductions on the dimensions of harmonic dynamic equations for these molecules have been made, with the fraction of the deleted high-frequency modes ranging from 1/2 to 5/6. The calculated low-frequency normal modes are found to be very accurate as compared to the exact solutions by standard procedure. The major advantage of the present approach on macromolecule harmonic dynamics is that the reduction on the dimensionality of the eigenvalue problems can be varied according to the size of molecules, so the method can be easily applied to large macromolecules with controlled accuracy.     

Theory of excitable membranes. I. A simple model for a three-state artificial membrane.

The effect of the thermal fluctuation on the orientation distribution pattern of globular protein molecules in a two-dimensional lattice was investigated by the method of computer simulation. A set of interaction parameters was assigned to interaction sites on each molecule and the interaction energy between two molecules was given by the product of the parameters of facing sites. Orientation fluctuation was assumed to take place with the probability proportional to the Boltzmann factor. Patterns having different degrees of order appeared with the change of temperature. The entropy and other thermodynamic quantities of these patterns were calculated, and gradual and transitional changes of the pattern were discussed in comparison with the case of simple atoms or molecules.     

Protein conformational dynamics of crambin in crystal, solution and in the trajectories of molecular dynamics simulations

Atomic displacement parameters — B factors of the eight crambin crystal structures obtained at 0.54–1.5 Å resolution and temperatures of 100–293 K have been analyzed. The comparable contributions to the B factor values are the intramolecular motions which are modeled by the harmonic vibration calculations and derived from the molecular dynamics simulation (MD) as well as rigid body changes in the position of a protein molecule as a whole. In solution for the average NMR structure of crambin the amplitudes of the backbone atomic fluctuations of the most residues of the segments with the regular backbone conformations are close to the amplitudes of the small scale harmonic vibrations. For the same residues the probability of the medium scale fluctuations fixed by the hydrogen exchange method is very low. The restricted conformational mobility of those segments is coupled with the depressed amplitudes of the fluctuation changes of the tertiary structure registered by the residue accessibility changes in an ensemble of NMR structures that forms the average NMR structure of crambin. The amplitudes of temperature fluctuations of backbone atoms and the tertiary structure raise in the segment with the irregular conformations, turn and loops. In the same segments the amplitudes of the calculated harmonic vibrations also increase, but to a lesser extent and especially in the interhelical loop with the most strong and complicated fluctuation changes of the backbone conformation. In solution for the NMR structure in this loop the conformational transitions occur between the conformational substates separated by the energy barriers, but they are not observed even in the long 100 ns trajectories from the MD simulation of crambin. These strong local fluctuation changes of the structure may play a key role in the protein functioning and modern performance improvements in the MD simulation techniques are oriented to increase the probability of protein appearance in the trajectories from the MD simulations.     

Analyzing the normal mode dynamics of macromolecules by the component synthesis method: Residue clustering and multiple-component approach

A study of the component synthesis method (CSM) for analyzing the normal mode dynamics of macromolecules is reported. The procedure involves a reduction of the dimensions of the normal mode problems for large molecular systems and the accurate extraction of the low-frequency modes. A macromolecule is divided into small components based on a hierarchical clustering of the residues in the structure. Interactions between coupled components are treated by the method of static correlation. The normal modes of the components are obtained first, and a fraction of the low-frequency normal modes of the components under mutual correlations are then used as a reduced basis for solving for the normal modes of the whole molecule. Multiple components are introduced for large macromolecules so that the dimensions of the eigenvalue problems at the component level are small. The method is applied to the protein crambin. In test calculations in which the dimensions of the eigenvalue equations are reduced to 1/6 of their natural size, the errors in the normal mode frequencies calculated by the CSM procedure are only about 1–2% when compared with the exact values. The rms fluctuations of all atoms in crambin calculated by the CSM procedure are basically identical to the exact results. The CSM procedure is shown to be accurate for calculating the normal modes of large macromolecules with a significant reduction of the size of the problem. © 1994 John Wiley & Sons, Inc.     

A microscopic electrostatic model for the amphotericin B channel.

A microscopic model of an amphotericin B channel is proposed. The structure of the pores is generated using the atomic coordinates of the molecule in the structure determined experimentally by X-ray diffraction. The net charges of the atoms are determined by Mulliken analysis. With these charges the electrostatic energy profiles are calculated for a monovalent ion passing through the channels formed by different number of antibiotic molecules having different radii. The water inside the channel was considered through a continuum medium using the dielectric constant of the bulk, and the membrane contribution was included using the virtual images of the pore in a dielectric slab of epsilon = 3. The model satisfactorily explains the permeability and selectivity characteristics as well as other observations yet unexplained. The electrostatic profiles obtained reinforce the hypothesis of the existence of channels formed by a variable number of units.     

Explicit solvent models in protein pKa calculations.

Continuum methods for calculation of protein electrostatics treat buried and ordered water molecules by one of two approximations; either the dielectric constant of regions containing ordered water molecules is equal to the bulk solvent dielectric constant, or it is equal to the protein dielectric constant though no fixed atoms are used to represent water molecules. A method for calculating the titration behavior of individual residues in proteins has been tested on models of hen egg white lysozyme containing various numbers of explicit water molecules. Water molecules were included based on hydrogen bonding, solvent accessibility, and/or proximity to titrating groups in the protein. Inclusion of water molecules significantly alters the calculated titration behavior of individual titrating sites, shifting calculated pKa values by up to 0.5 pH unit. Our results suggest that approximately one water molecule within hydrogen-bonding distance of each charged group should be included in protein electrostatics calculations.     

Dielectric properties of hemoglobin and myoglobin. II. Dipole moment of sperm whale myoglobin

This paper is concerned with the molecular origin of the dipole moment of sperm whale myoglobin as it can be calculated from the dielectric dispersion at 1 Mcps on the basis of a mechanism of orientational polarization. It was possible to compare the dielectric increment of native myoglobin and its change during the reaction with bromo acetate with dipole moments calculated according to the known coordinates of the charged groups of the molecule. The agreement between the two shows that in myoglobin only the permanent dipole moment due to these charged groups is important, and that contributions from other possible sources remain within the limits of experimental error.     

Dielectric behavior of DNA in water–organic co-solvent mixtures

The radiowave dielectric dispersions of DNA in different water–organic co-solvent mixtures have been measured in the frequency range from 100 kHz to 100 MHz, where the polarization mechanism is generally attributed to the confinement of counterions within some specific lengths, either along tangential or perpendicular to the polyion chain. The dielectric dispersions have been analyzed on the basis of two partially different dielectric models, a continuum counterion fluctuation model proposed by Mandel and a discrete charged site model, proposed by Minakata. The influence of the quality of the solvent on the dielectric parameters has been investigated in water–methanol and water–glycerol mixtures at different composition, by varying the permittivity ?_m and the viscosity η of the solvent phase. The analysis of the dielectric spectra in solvents where electrostatic and hydrodynamic interactions vary with the solvent composition suggests that both the two models are able, in principle, to account for the observed high-frequency dielectric behavior. However, while some certain assumptions are necessary about the polyion structure within the Mandel model, no structural prerequisite is needed within the Minakata model, where the polarization mechanism invoked considers a radial counterion exchange with the outer medium, which is largely independent of the local polyion conformation.     

Temperature dependence of the polarization and the dielectric constant near the paraelectric-ferroelectric transitions in BaTiO3

Temperature dependences of the spontaneous polarization and the dielectric constant are calculated near the paraelectric-ferroelectric (cubic-tetragonal) transition in BaTiO₃ using our mean field model. By expanding the free energy in terms of the spontaneous polarization (order parameter), expressions for the temperature dependence of the spontaneous polarization and the dielectric constant are derived. By considering the temperature dependence of the Raman frequencies for the lattice mode (～310 cm^?1) which is related to the spontaneous polarization, the experimental data from the literature is analyzed near the first order paraelectric-ferroelectric transition in BaTiO₃. The dielectric constant is then calculated as a function of temperature for the cubic-tetragonal transition in BaTiO₃. Our results show that the observed behavior of the spontaneous polarization in the ferroelectric phase (T<T_C) and that of the dielectric constant in both paraelectric (T>T_C) and ferroelectric phases, can be described adequately by the mean field model studied here for BaTiO₃.     

Influence of the solvent structure on the electrostatic interactions in proteins

The proper estimation of the influence of the many-body dynamic solvent microstructure on a pairwise electrostatic interaction (PEI) at the protein-solvent interface is very important for solving many biophysical problems. In this work, the PEI energy was calculated for a system that models the interface between a protein and an aqueous solvent. The concept of nonlocal electrostatics for interfacial electrochemical systems was used to evaluate the contribution of a solvent orientational polarization, correlated by the network of hydrogen bonds, into the PEI energy in proteins. The analytical expression for this energy was obtained in the form of Coulomb's law with an effective distance-dependent dielectric function. The asymptotic and numerical analysis carried out for this function revealed several features of dielectric heterogeneity at the protein-solvent interface. For charges located in close proximity to this interface, the values of the dielectric function for the short-distance electrostatic interactions were found to be remarkably smaller than those determined by the classical model, in which the solvent was considered as the uniform dielectric medium of high dielectric constant. Our results have shown that taking into consideration the dynamic solvent microstructure remarkably increases the value of the PEI energy at the protein-solvent interface.     

Frequency-Dependent Polarization Properties of Local Electric Field in Gold–Dielectric Multi-Nanoshells

The polarization properties of the local electric field in the gold–dielectric–gold multilayer nanoshells are investigated by theoretical calculation based on the quasi-static approximation. The calculation results show that the complete polarized incident light does not only stimulate the same directional polarized local electric field. The polarized angle of the local field may changes from 0° to 90° as the wavelength and location are changed. The distributions of local field polarization are different in dielectric layer or gold shell and display different features in different plasmonic hybridization mode. As the incident wavelength is increased, the hot spot of polarizing angle moves monotonously in the middle dielectric shell, whereas moves nonmonotonously in the gold shell and surrounding environment. In the gold shell, the gap between hot spots of polarizing angle may occur at the resonance frequency. However, the hot spots of polarizing angle always occur at the resonance frequencies in the surrounding environment. These interesting results show that the single-molecule detection based on metal nanostructure induced surface-enhanced Raman scattering and surface enhanced fluorescence could be optimized by adjusting the incident light polarization and frequency.     

Coarse Grained Normal Mode Analysis vs. Refined Gaussian Network Model for Protein Residue-Level Structural Fluctuations

We investigate several approaches to coarse grained normal mode analysis on protein residual-level structural fluctuations by choosing different ways of representing the residues and the forces among them. Single-atom representations using the backbone atoms C _α , C, N, and C _β are considered. Combinations of some of these atoms are also tested. The force constants between the representative atoms are extracted from the Hessian matrix of the energy function and served as the force constants between the corresponding residues. The residue mean-square-fluctuations and their correlations with the experimental B-factors are calculated for a large set of proteins. The results are compared with all-atom normal mode analysis and the residue-level Gaussian Network Model. The coarse-grained methods perform more efficiently than all-atom normal mode analysis, while their B-factor correlations are also higher. Their B-factor correlations are comparable with those estimated by the Gaussian Network Model and in many cases better. The extracted force constants are surveyed for different pairs of residues with different numbers of separation residues in sequence. The statistical averages are used to build a refined Gaussian Network Model, which is able to predict residue-level structural fluctuations significantly better than the conventional Gaussian Network Model in many test cases.     

An analysis of non-bonded energy of proteins

Non-bonded energy of 16 proteins was calculated using the atomic co-ordinates obtained by X-ray crystallography. The curve of total energy against the number of atoms in proteins is approximately linear with a slight concaved shape. According to a linear equation to fit the curve, the extrapolated length of a polypeptide chain of a globular shape is expected to be 18 residues, which corresponds conceivably to an approximate size of nucleus for a folding of the polypeptide chain. Contributions from short-range and medium-range energies are always much greater than those from long-range energy for all the proteins and there seems to exist a change of each contribution in a range from 1200 to 1700 atoms. The energies with a lag less than four residues are a major part of the total energy and the contribution of energy from main-chain atoms is considerably higher than that from side-chain atoms. Side-chain atoms of a residue have a tendency to interact more strongly with main-chain atoms of N-terminal, than with those of C-terminal side of the residue, indicating asymmetry of the interaction in a protein. Amino acid residues in proteins may be divided into three groups by the order of strength of average energy. The first group exhibiting strong interaction consists mainly of hydrophobic amino acids and the third group consists of hydrophilic ones corresponding to the location in a protein molecule. Cys, val, leu and met are important for medium-range and long-range energies; gly and ala for medium-range energy; ile, trp, phe, tyr and arg for long-range energy. One simple application of the average energy of amino acid residues is illustrated to estimate local energy of a segment of nine residues given by a protein sequence. There is a good correlation between the curve computed by the average energy and the experimental curve for myoglobin.     

Dielectric properties of the polar head group region of zwitterionic lipid bilayers.

A theoretical model describing the dielectric properties of the lipid membrane-water interface region was developed. The rotating polar head groups (e.g. phosphatidylcholine) were simulated as a collection of interacting dipoles imbedded in a nonhomogeneous dielectric. The interactions between the nearest neighborhood were explicitly taken into account, while the other interactions were evaluated by means of the continuum theories. The values of the dielectric constant, its anisotropy and the spontaneous polarization of the interface were evaluated. As an application, we calculated the energy of interaction between an ion and the membrane polar head group region. The results indicate a small spontaneous polarization of the interface (1-1.7 Debyes per lipid molecule) due to the tilting angle of the choline residue with respect to the membrane surface. This dipolar field partially compensates that of opposite orientation originating from the ester group region, giving calculated overall dipolar potentials in better agreement with the experimental data. Our model suggests also a very strong dielectric anisotropy of the interface region, the component of the dielectric constant perpendicular to the membrane plane being much smaller than the parallel component.