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.     

Fast calculation of the infrared spectra of large biomolecules

Vibrational spectra of proteins potentially give insight into biologically significant molecular motion and the proportions of different types of secondary structure. Vibrational spectra can be calculated either from normal modes obtained by diagonalizing the mass-weighted Hessian or from the time autocorrelation function derived from molecular dynamics trajectories. The Hessian matrix is calculated from force fields because it is not practical to calculate the Hessian from quantum mechanics for large molecules. As an alternative to molecular dynamics the spectral response can be calculated from a time autocorrelation derived from numerical solution of the harmonic equations of motion, resulting in calculations at least 4 times faster. Because the calculation also scales linearly with number of atoms, N, it is faster than normal-mode calculations that scale as N ³ for proteins with more then 4,700 atoms. Using this method it is practical to perform all-atom calculations for large biological systems, for example viral capsids, with the order of 10⁵ atoms.     

A critical review of the nucleosidic vibration modes appearing in the 800-500-cm-1 spectral region, based on new harmonic dynamics calculations

On the basis of a harmonic dynamics calculation, it is shown that in the 800–500-cm^?1 spectral region of DNA vibrational spectra, the characteristic Raman peaks and ir bands do not arise from the same nucleosidic motions. The Raman spectra involve mainly the ring-breathing modes of nucleic bases while the ir spectra reveal essentially their out-of-plane vibrations. Moreover, the calculated results show the splitting of the guanine- and adenine-residue breathing modes upon their coupling with the sugar-pucker motions. This fact is in agreement with the poly[d(G-C)] and poly[d(A-T)] Raman spectra.     

Filtering molecular dynamics trajectories to reveal low-frequency collective motions: phospholipase A2

A novel method for analysing molecular dynamics trajectories has been developed, which filters out high frequencies using digital signal processing techniques and facilitates focusing on the low-frequency collective motions of proteins. These motions involve low energy slow motions, which lead to important biological phenomena such as domain closure and allosteric effects in enzymes. The filtering method treats each of the atomic trajectories obtained from the molecular dynamics simulation as a "signal". The trajectories of each of the atoms in the system (or any subset of interest) are Fourier transformed to the frequency domain, a filtering function is applied and then an inverse transformation back to the time domain yields the filtered trajectory. The filtering method has been used to study the dynamics of the enzyme phospholipase A2. In the filtered trajectory, all the high frequency bond and valence angle vibrations were eliminated, leaving only low-frequency motion, mainly fluctuations in torsions and conformational transitions. Analysis of this trajectory revealed interesting motions of the protein, including concerted movements of helices, and changes in shape of the active site cavity. Unlike normal mode analysis, which has been used to study the motion of proteins, this method does not require converged minimizations or diagonalization of a matrix of second derivatives. In addition, anharmonicity, multiple minima and conformational transitions are treated explicitly. Thus, the filtering method avoids most of the approximations implicit in other investigations of the dynamic behaviour of large systems.     

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.     

A coarse-grained normal mode approach for macromolecules: an efficient implementation and application to Ca(2+)-ATPase

A block normal mode (BNM) algorithm, originally proposed by Tama et al., (Proteins Struct. Func. Genet. 41:1-7, 2000) was implemented into the simulation program CHARMM. The BNM approach projects the hessian matrix into local translation/rotation basis vectors and, therefore, dramatically reduces the size of the matrix involved in diagonalization. In the current work, by constructing the atomic hessian elements required in the projection operation on the fly, the memory requirement for the BNM approach has been significantly reduced from that of standard normal mode analysis and previous implementation of BNM. As a result, low frequency modes, which are of interest in large-scale conformational changes of large proteins or protein-nucleic acid complexes, can be readily obtained. Comparison of the BNM results with standard normal mode analysis for a number of small proteins and nucleic acids indicates that many properties dominated by low frequency motions are well reproduced by BNM; these include atomic fluctuations, the displacement covariance matrix, vibrational entropies, and involvement coefficients for conformational transitions. Preliminary application to a fairly large system, Ca(2+)-ATPase (994 residues), is described as an example. The structural flexibility of the cytoplasmic domains (especially domain N), correlated motions among residues on domain interfaces and displacement patterns for the transmembrane helices observed in the BNM results are discussed in relation to the function of Ca(2+)-ATPase. The current implementation of the BNM approach has paved the way for developing efficient sampling algorithms with molecular dynamics or Monte Carlo for studying long-time scale dynamics of macromolecules.     

Extraction of the energetics of selected types of motion from molecular dynamics trajectories by filtering

A novel method for analyzing molecular dynamics trajectories has been developed which enables the study of selected motions and the corresponding energetics. In particular, it is possible to filter out the high-frequency motions and focus on the structural and energetic features of low-frequency collective motions. The trajectories of the properties of interest are Fourier transformed to the frequency domain, a filtering function is applied, and then an inverse transformation back to the time domain yields the filtered trajectory. The method is demonstrated for harmonic fluctuations and conformational transitions of acetamide and N-acetylalanine N-methylamide, as models for peptides and proteins.     

Dynamic properties of some β-chain mutant hemoglobins

The thermal behavior of the Soret band relative to the carbonmonoxy derivatives of some β-chain mutant hemoglobins is studied in the temperature range 300–10 K and compared to that of wild-type carbonmonoxy hemoglobin. The band profile at various temperatures is modeled as a Voigt function that accounts for homogeneous broadening and for the coupling with high- and low-frequency vibrational modes, while inhomogeneous broadening is taken into account with a gaussian distribution of purely electronic transition frequencies. The various contributions to the overall bandwidth are singled out With this analysis and their temperature dependence, in turn, gives information on structural and dynamic properties of the system studied. In the wildtype and mutant hemoglobins, the values of homogeneous bandwidth and of the coupling constants to high-frequency vibrational modes are not modified with respect to natural human hemoglobin, thus indicating that the local electronic and vibrational properties of the heme–CO complex are not altered by the recombinant procedures. On the contrary, differences in the protein dynamic behavior are observed. The most relevant are those relative to the “polar isosteric” βVal-67(Ell) →Thr substitution, localized in the heme pocket, which results in decreased coupling with low-frequency modes and increased anharmonic motions. Mutations involving residue βLys-144(HC1) at the C-terminal and residue βCys-112(G14) at the α₁β₁ interface have a smaller effect consisting in an increased coupling with low-frequency modes. Mutations at the β-N-terminal and at the α₁β₂ interface have no effect on the dynamic properties of the heme pocket. © 1995 Wiley-Liss, Inc.     

Local dynamics of DNA probed with optical absorption spectroscopy of bound ethidium bromide.

We have studied the local dynamics of calf thymus double-helical DNA by means of an "optical labeling" technique. The study has been performed by measuring the visible absorption band of the cationic dye ethidium bromide, both free in solution and bound to DNA, in the temperature interval 360-30 K and in two different solvent conditions. The temperature dependence of the absorption line shape has been analyzed within the framework of the vibronic coupling theory, to extract information on the dynamic properties of the system; comparison of the thermal behavior of the absorption band of free and DNA-bound ethidium bromide gave information on the local dynamics of the double helix in the proximity of the chromophore. For the dye free in solution, large spectral heterogeneity and coupling to a "bath" of low-frequency (soft) modes is observed; moreover, anharmonic motions become evident at suitably high temperatures. The average frequency of the soft modes and the amplitude of anharmonic motions depend upon solvent composition. For the DNA-bound dye, at low temperatures, heterogeneity is decreased, the average frequency of the soft modes is increased, and anharmonic motions are hindered. However, a new dynamic regime characterized by a large increase in anharmonic motions is observed at temperatures higher than approximately 280 K. The DNA double helix therefore appears to provide, at low temperatures, a rather rigid environment for the bound chromophore, in which conformational heterogeneity is reduced and low-frequency motions (both harmonic vibrations and anharmonic contributions) are hindered. The system becomes anharmonic at approximately 180 K; however, above approximately 280 K, anharmonicity starts to increase much more rapidly than for the dye free in solution; this can be attributed to the onset of wobbling of the dye in its intercalation site, which is likely connected with the onset of (functionally relevant) DNA motions, involving local opening/unwinding of the double helix. As shown by parallel measurements of the melting curves, these motions precede the melting of the double helix and depend upon solvent composition much more than does the melting itself.     

Spectral broadening of the Soret band in myoglobin: an interpretation by the full spectrum of low-frequency modes from a normal modes analysis

In this work the temperature dependence of the Soret band line shape in carbon-monoxy myoglobin is re-analyzed by using both the full correlator approach in the time domain and the frequency domain approach. The new analyses exploit the full density of vibrational states of carbon-monoxy myoglobin available from normal modes analysis, and avoid the artificial division of the entire set of vibrational modes coupled to the Soret transition into "high-frequency" and "low-frequency" subsets; the frequency domain analysis, however, makes use of the so-called short-times approximation, while the time domain one avoids it. Time domain and frequency domain analyses give very similar results, thus supporting the applicability of the short-times approximation to the analysis of hemeprotein spectra; in particular, they clearly indicate the presence of spectral heterogeneity in the Soret band of carbon-monoxy myoglobin. The analyses also show that a temperature dependence of the Gaussian width parameter steeper than the hyperbolic cotangent law predicted by the Einstein harmonic oscillator and/or a temperature dependence of inhomogeneous broadening are not sufficient to obtain quantitative information on the magnitude of an-harmonic contributions to the iron-heme plane motion. However, the dependence of the previous two quantities may be used to obtain semiquantitative information on the overall coupling of the Soret transition to the low-frequency modes and therefore on the dynamic properties of the heme pocket in different states of the protein.     

Normal Mode Analysis Theoretical and Applications to Biological and Chemical Systems.

The realization that many of the important biochemical phenomenaoccur in the micro- to millisecond timescale has resulted inthe need for computational methods that would extend beyondthe nanosecond regime that is currently accessible with explicit-solventmolecular dynamics simulations. The normal mode analysis method,initially developed in solid-phase physics, met that need byenabling the exploration of large scale collective motions thatoccur in systems ranging from small proteins and nucleic acidsto the bacterial ribosome. This method requires the derivationof eigenvectors and eigenvalues through the diagonalizationof the Hessian matrix, which is composed of mass-weighed secondderivatives of the potential energy function. The     

Hydrodynamic modes of arbitrarily shaped flexible macromolecules: Influence on dynamic light scattering

The diffusional motions of flexible macromolecules are analyzed with an increasingly realistic Rouse–Zimm model, i.e., by modeling the molecule as an arbitrary set of spheres connected by nearly harmonic springs. New features include (1) nearly arbitrary arrangements of spheres, (2) arbitrary arrangements of translational and torsional springs, (3) significant anharmonic corrections to the elastic potential surface, and (4) inclusion of torsional damping and various hydrodynamic cross-coupling effects (including two types of translational-rotational coupling) with no additional fitted parameters. The hydrodynamic interactions [R. F. Goldstein (1985) Journal of Chemical Physics, Vol. 83, pp. 2390–2397] contain no adjustable parameters other than temperature, viscosity, and the radii and positions of the spheres. These hydrodynamic interactions allow accurate calculations of rigid body diffusion as well as flexible motions. Given the positions, radii, and spring constant matrix, one can calculate a full set of three-dimensional diffusional modes. Because one uses an off-diagonal hydrodynamic resistance matrix instead of a diagonal mass matrix, the diffusional modes are different in structure from vacuum normal modes, and give rise to different rms motions in the laboratory frame. These hydrodynamic modes include the effects of vibrational-translational cross-coupling (i.e., motion along a vibrational coordinate may give rise to a translational force, and vice versa). The diffusional modes are used to simulate dynamic light scattering (DLS). I examine various molecules with different shapes, flexibilities, and with different scattering vectors. Radial and angular motions influence DLS decays differently. These effects are dependent upon the molecular shape (straight, bent, or curved) and type of flexibility (stretching or bending). Furthermore, small cubic corrections to the potential surface can be significant for DLS of certain geometries such as straight rods and semicircles. © 1993 John Wiley & Sons, Inc.     

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.     

Evaluating elastic network models of crystalline biological molecules with temperature factors, correlated motions, and diffuse x-ray scattering

In this study, the variance-covariance matrix of protein motions is used to compare several elastic network models within the theoretical framework of x-ray scattering from crystals. A set of 33 ultra-high resolution structures is used to characterize the average scaling behavior of the vibrational density of states and make comparisons between experimental and theoretical temperature factors. Detailed investigations of the vibrational density of states, correlations, and predicted diffuse x-ray scatter are carried out for crystalline Staphylococcal nuclease; correlations and diffuse x-ray scatter are also compared to predictions from the translation, libration, screw model and a liquid-like dynamics model. We show that elastic network models developed to best predict temperature factors without regard for the crystal environment have relatively strong long-range interactions that yield very short-ranged atom-atom correlations. Further, we find that the low-frequency modes dominate the variance-covariance matrix only for those models with a physically reasonable vibrational density of states, and the fraction of modes required to converge the correlations is higher than that typically used for elastic network model studies. The practical implications are explored using computed diffuse x-ray scatter, which can be measured experimentally.     

Low-frequency vibrational modes and infrared absorbance of red, blue and green opsin

Vibrational excitations of low-frequency collective modes are essential for functionally important conformational transitions in proteins. We carried out an analysis of the low-frequency modes in the G protein coupled receptors (GPCR) family of cone opsins based on both normal-mode analysis and molecular dynamics (MD) simulations. Power spectra obtained by MD can be compared directly with normal modes. In agreement with existing experimental evidence related to transmembrane proteins, cone opsins have functionally important transitions that correspond to approximately 950 modes and are found below 80 cm⁻¹. This is in contrast to bacteriorhodopsin and rhodopsin, where the important low-frequency transition modes are below 50 cm⁻¹. We find that the density of states (DOS) profile of blue opsin in a solvent (e.g. water) has increased populations in the very lowest frequency modes (<15 cm⁻¹); this is indicative of the increased thermostability of blue opsin. From our work we found that, although light absorption behaves differently in blue, green and red opsins, their low-frequency vibrational motions are similar. The similarities and differences in the domain motions of blue, red and green opsins are discussed for several representative modes. In addition, the influence of the presence of a solvent is reported and compared with vacuum spectra. We thus demonstrate that terahertz spectroscopy of low-frequency modes might be relevant for identifying those vibrational degrees of freedom that correlate to known conformational changes in opsins. An erratum to this article can be found at     

Quasi-harmonic analysis of mode coupling in fluctuating native proteins

Mode coupling and anharmonicity in a native fluctuating protein are investigated in modal space by projecting the motion along the eigenvectors of the fluctuation correlation matrix. The probability distribution of mode fluctuations is expressed in terms of tensorial Hermite polynomials. Molecular dynamics trajectories of Crambin are generated and used to evaluate the terms of the polynomials and to obtain the modal energies. The energies of a few modes exhibit large deviations from the harmonic energy of kT/2 per mode, resulting from coupling to the surroundings, or to another specific mode or to several other modes. Slowest modes have energies that are below that of the harmonic, and a few fast modes have energies significantly larger than the harmonic. Detailed analysis of the coupling of these modes to others is presented in terms of the lowest order two-mode coupling terms. Finally, the effects of mode coupling on conformational properties of the protein are investigated.     

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.     

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.