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.     

A new approach for determining low-frequency normal modes in macromolecules

A new method for calculating a set of low-frequency normal modes in macromolecules is proposed and applied to the case of proteins. In a first step, the protein chain is partitioned into blocks of one or more residues and the low-frequency modes are evaluated at a low-resolution level by combining the local translations and rotations of each block. In a second step, these low-resolution modes are perturbed by high-frequency modes explicitly calculated in each block, thus leading to the exact low-frequency modes. The procedure is tested for three cases–decaalanine, icosaleucin, and crambin–using a perturbation-iteration scheme in the second step. Convergence properties and numerical accuracy are assessed and tested for various partitions. The low-resolution modes obtained in the first step are always found to be good starting approximations. Potential advantages of the method include a central processing unit time roughly N² dependent on the size of the problem (N being the number of degrees of freedom), the possibility of using parallel processing, the nonrequirement for loading the complete mass-weighted second-derivative input matrix into central memory, and the possibility of introducing in the procedure further structural hierarchy, such as secondary structures or motifs. In addition, any improvement or refinement of the algorithm benefits from the efficient formalism of the effective Hamiltonian theory. © 1994 John Wiley & Sons, Inc.     

Building-block approach for determining low-frequency normal modes of macromolecules

Normal mode analysis of proteins of various sizes, ranging from 46 (crambin) up to 858 residues (dimeric citrate synthase) were performed, by using standard approaches, as well as a recently proposed method that rests on the hypothesis that low-frequency normal modes of proteins can be described as pure rigid-body motions of blocks of consecutive amino-acid residues. Such a hypothesis is strongly supported by our results, because we show that the latter method, named RTB, yields very accurate approximations for the low-frequency normal modes of all proteins considered. Moreover, the quality of the normal modes thus obtained depends very little on the way the polypeptidic chain is split into blocks. Noteworthy, with six amino-acids per block, the normal modes are almost as accurate as with a single amino-acid per block. In this case, for a protein of n residues and N atoms, the RTB method requires the diagonalization of an n x n matrix, whereas standard procedures require the diagonalization of a 3N x 3N matrix. Being a fast method, our approach can be useful for normal mode analyses of large systems, paving the way for further developments and applications in contexts for which the normal modes are needed frequently, as for example during molecular dynamics calculations.     

Diagonalization in a mixed basis: A method to compute low-frequency normal modes for large macromolecules

Low-frequency collective motions in proteins are generally very important for their biological functions. To study such motions, harmonic dynamics proved most useful since it is a straightforward method; it consists of the diagonalization of the Hessian matrix of the potential energy, yielding the vibrational spectrum and the directions of internal motions. Unfortunately, the diagonalization of this matrix requires a large computer memory, which is a limiting factor when the protein contains several thousand atoms. To circumvent this limitation we have developed three methods that enable us to diagonalize large matrices using much less computer memory than the usual harmonic dynamics. The first method is approximate; it consists of diagonalizing small blocks of the Hessian matrix, followed by the coupling of the low-frequency modes obtained for each block. It yields the low-frequency vibrational spectrum with a maximum error of 20%. The second method consists, after diagonalizing small blocks, of coupling the high- and low-frequency modes using an iterative procedure. It yields the exact low-frequency normal modes, but requires a long computational time with convergence problems. The third method, DIMB (Diagonalization in a Mixed Basis), which has the best performance, consists of coupling the approximate low-frequency modes with the mass-weighted cartesian coordinates, also using an iterative procedure. It reduces significantly the required computer memory and converges rapidly. The eigenvalues and eigenvectors obtained by this method are without significant error in the chosen frequency range. Moreover, it is a general method applicable to any problem of diagonalization of a large matrix. We report the application of these methods to a deca-alanine helix, trypsin inhibitor, a neurotoxin, and lysozyme. © 1993 John Wiley & Sons, Inc.     

A theorem on amplitudes of thermal atomic fluctuations in large molecules assuming specific conformations calculated by normal mode analysis

An exact theorem is proved and its implication is discussed. The theorem states that, if a large molecule, typically biological macromolecules such as proteins, undergoes small-amplitude conformational fluctuations around its native conformation in such a way that within the range of conformational fluctuations at thermal equilibrium the conformational energy surface can be approximated by a multidimensional parabola, then the mass-weighted mean-square displacement of constituent atoms is given by the sum of the contributions from each normal mode of conformational vibration, which in turn is proportional to the inverse of the square of its frequency. This theorem provides a firm theoretical basis for the fact hitherto empirically recognized in the conformational dynamics of, for instance, native proteins that very-low-frequency normal modes make dominant contributions to the conformational fluctuations at thermal equilibrium. Discussion is given on the implication of this theorem, especially on the importance of the concept of the low-frequency normal modes, even in the case where the basic assumption of the harmonicity of the energy surface does not hold.     

The vibrational normal modes of beta-barrels in an IgG antibody molecule.

Based on the quasi-continuity model, and using the method of group theory, we studied the normal vibrations of the VL- and the CHL-beta-barrels in an IgG molecule. We put emphasis on the Raman- and the infrared-active normal modes. The Raman modes we obtained include both the breathing motion mode (or the dominant low-frequency mode) which corresponds to the maximum peak in the Raman spectrum, and the normal modes that correspond to the lower peaks. Our calculated vibration frequencies are found to be in good agreement with the experimental results observed by Painter et al. (Biopolymers 20 (1981) 243). The method and work presented in this paper may improve Chou's quasi-continuity theory in calculating the vibrational modes of a beta-barrel protein.     

Dynamic light-scattering studies of DNA. I. The coupling of internal modes with anisotropic translational diffusion in congested solutions

A model for the coupling between internal modes, or molecular rotation, and anisotropic translational diffusion in congested solutions is proposed to account for the anomalously slow component that has appeared ubiquitously in reported autocorrelation functions of Rayleigh scattered light from solutions of DNA's with molecular weights greater than about 10⁷. The predicted existence of an anomalously slow mode in addition to a faster “normal” mode, as well as the predicted relative amplitudes of both fast and slow components, are qualitatively in agreement with the observations. For sufficiently long-wavelength fluctuations all of the amplitude appears in the slower mode, which then exhibits an appropriately averaged translational diffusion coefficient. In support of the model it is shown in the Appendix that nonideal central interactions between macromolecules are by themselves insufficient to generate isolated internal mode relaxation terms in the autocorrelation function, unless translational ordering of the macromolecules extends over the illuminated observation region.     

Langevin modes of macromolecules: applications to crambin and DNA hexamers

Langevin modes describe the behavior of atoms moving on a harmonic potential surface subject to viscous damping described by a classical Langevin equation. We present applications to the protein crambin and to the DNA duplex d(CpGpCpGpCpG)2 and its complex with ethidium. Our friction matrix is weighted according to surface area exposed to solvent, and results are reported for various values of the solvent viscosity and models for hydrodynamic interactions. Even for relatively small solvent friction (eta = 0.3 cp) a substantial number of modes are overdamped, and time correlation functions decay smoothly without the oscillations characteristic of gas-phase calculations. Perturbation theory starting from the gas-phase modes is accurate for many low-frequency modes (which are overdamped in the presence of solvent), but fails badly for higher modes. For correlation functions of interest to fluorescence depolarization or nmr relaxation, the plateau values are insensitive to solvent viscosity, but the relaxation times are not. The advantages and limitations of this analysis of macromolecular motions are discussed.     

Normal mode refinement: crystallographic refinement of protein dynamic structure. I. Theory and test by simulated diffraction data.

A dynamic structure refinement method for X-ray crystallography, referred to as the normal mode refinement, is proposed. The Debye-Waller factor is expanded in terms of the low-frequency normal modes whose amplitudes and eigenvectors are experimentally optimized in the process of the crystallographic refinement. In this model, the atomic fluctuations are treated as anisotropic and concerted. The normal modes of the external motion (TLS model) are also introduced to cover the factors other than the internal fluctuations, such as the lattice disorder and diffusion. A program for the normal mode refinement (NM-REF) has been developed. The method has first been tested against simulated diffraction data for human lysozyme calculated by a Monte Carlo simulation. Applications of the method have demonstrated that the normal mode refinement has: (1) improved the fitting to the diffraction data, even with fewer adjustable parameters; (2) distinguished internal fluctuations from external ones; (3) determined anisotropic thermal factors; and (4) identified concerted fluctuations in the protein molecule.     

The role of shape in determining molecular motions

We examined the role of molecular shape in determining the patterns of low-frequency deformational motions of biological macromolecules. The low-frequency subspace of eigenvectors in normal mode analysis was found to be robustly similar upon randomization of the Hessian matrix elements as long as the structure of the matrix is maintained, which indicates that the global shape of molecules plays a more dominant role in determining the highly anisotropic low-frequency motions than the absolute values of stiffness and directionality of local interactions. The results provided a quantitative foundation for the validity of elastic normal mode analysis.     

An analysis of core deformations in protein superfamilies

An analysis is presented on how structural cores modify their shape across homologous proteins, and whether or not a relationship exists between these structural changes and the vibrational normal modes that proteins experience as a result of the topological constraints imposed by the fold. A set of 35 representative, well-populated protein families is studied. The evolutionary directions of deformation are obtained by using multiple structural alignments to superimpose the structures and extract a conserved core, together with principal components analysis to extract the main deformation modes from the three-dimensional superimposition. In parallel, a low-resolution normal mode analysis technique is employed to study the properties of the mechanical core plasticity of these same families. We show that the evolutionary deformations span a low dimensional space of 4-5 dimensions on average. A statistically significant correspondence exists between these principal deformations and the approximately 20 slowest vibrational modes accessible to a particular topology. We conclude that, to a significant extent, the structural response of a protein topology to sequence changes takes place by means of collective deformations along combinations of a small number of low-frequency modes. The findings have implications in structure prediction by homology modeling.     

Why are large conformational changes well described by harmonic normal modes?

Low-frequency normal modes generated by elastic network models tend to correlate strongly with large conformational changes of proteins, despite their reliance on the harmonic approximation, which is only valid in close proximity of the native structure. We consider 12 variants of the torsional network model (TNM), an elastic network model in torsion angle space, that adopt different sets of torsion angles as degrees of freedom and reproduce with similar quality the thermal fluctuations of proteins but present drastic differences in their agreement with conformational changes. We show that these differences are related to the extent of the deviations from the harmonic approximation, assessed through an anharmonic energy function whose harmonic approximation coincides with the TNM. Our results indicate that mode anharmonicity is more strongly related to its collectivity, i.e., the number of atoms displaced by the mode, than to its amplitude; low-frequency modes can remain harmonic even at large amplitudes, provided they are sufficiently collective. Finally, we assess the potential benefits of different strategies to minimize the impact of anharmonicity. The reduction of the number of degrees of freedom or their regularization by a torsional harmonic potential significantly improves the collectivity and harmonicity of normal modes and the agreement with conformational changes. In contrast, the correction of normal mode frequencies to partially account for anharmonicity does not yield substantial benefits. The TNM program is freely available at https://github.com/ugobas/tnm.     

Effects of surface water on protein dynamics studied by a novel coarse-grained normal mode approach

Normal mode analysis (NMA) has received much attention as a direct approach to extract the collective motions of macromolecules. However, the stringent requirement of computational resources by classical all-atom NMA limits the size of the macromolecules to which the method is normally applied. We implemented a novel coarse-grained normal mode approach based on partitioning the all-atom Hessian matrix into relevant and nonrelevant parts. It is interesting to note that, using classical all-atom NMA results as a reference, we found that this method generates more accurate results than do other coarse-grained approaches, including elastic network model and block normal mode approaches. Moreover, this new method is effective in incorporating the energetic contributions from the nonrelevant atoms, including surface water molecules, into the coarse-grained protein motions. The importance of such improvements is demonstrated by the effect of surface water to shift vibrational modes to higher frequencies and by an increase in overlap of the coarse-grained eigenvector space (the motion directions) with that obtained from molecular dynamics simulations of solvated protein in a water box. These results not only confirm the quality of our method but also point out the importance of incorporating surface structural water in studying protein dynamics.     

Thorough validation of protein normal mode analysis: a comparative study with essential dynamics

The deformation patterns of a large set of representative proteins determined by essential dynamics extracted from atomistic simulations and coarse-grained normal mode analysis are compared. Our analysis shows that the deformational space obtained with both approaches is quite similar when taking into account a representative number of modes. The results provide not only a comprehensive validation of the use of a low-frequency modal spectrum to describe protein flexibility, but also a complete picture of normal mode limitations.     

Lattice vibrations in crystalline L-alanine

Zwitterionic L -alanine forms crystals containing strong hydrogen-bonding and methyl-methyl interactions. Well-defined low-frequency lattice vibrations exist in the crystals involving correlated intermolecular motions on the picosecond timescale. A characterization of these vibrations is expected to provide useful information on the nature of nonbonded interactions in peptides and proteins. We examine some of the vibrations using coherent inelastic neutron scattering and computer simulation techniques. The neutron scattering measurements are used to determine phonon dispersion relations for the acoustic and some low-frequency optic modes in the crystal. There is evidence for interaction between the two lowest frequency optical phonons and the longitudinal acoustic mode. The velocity of sound is anisotropic and can be correlated with the hydrogen-bonding arrangement in the crystal. Corresponding phonon dispersion relations are derived from normal mode analyses of the crystal using the program CHARMM. Although some calculated vibrational frequencies are somewhat too high, the form of the calculated dispersion relations are in good agreement with experiment. © 1993 John Wiley & Sons, Inc.     

Normal mode refinement: crystallographic refinement of protein dynamic structure applied to human lysozyme.

A new method of dynamic structure refinement of protein x-ray crystallography, normal mode refinement, is developed. In this method the Debye-Waller factor is expanded in terms of the low-frequency normal modes and external normal modes, whose amplitudes and couplings are optimized in the process of crystallographic refinement. By this method, internal and external contributions to the atomic fluctuations can be separated. Also, anisotropic atomic fluctuations and their interatomic correlations can be determined experimentally even with a relatively small number of adjustable parameters. The method is applied to the analysis of experimental data of human lysozyme to reveal its dynamic structure.