Predicting large-scale conformational changes in proteins using energy-weighted normal modes

We report the development of a method to improve the sampling of protein conformational space in molecular simulations. It is shown that a principal component analysis of energy-weighted normal modes in Cartesian coordinates can be used to extract vectors suitable for describing the dynamics of protein substructures. The method can operate with either atomistic or user-defined coarse-grained models of protein structure. An implicit reverse coarse-graining allows the dynamics of all-atoms to be recovered when a coarse-grained model is used. For an external test set of four proteins, it is shown that the new method is more successful than normal mode analysis in describing the large-scale conformational changes observed on ligand binding. The method has potential applications in protein-ligand and protein-protein docking and in biasing molecular dynamics simulations.     

Quasi-harmonic method for studying very low frequency modes in proteins

A quasi-harmonic approximation is described for studying very low frequency vibrations and flexible paths in proteins. The force constants of the empirical potential function are quadratic approximations to the potentials of mean force; they are evaluated from a molecular dynamics simulation of a protein based on a detailed anharmonic potential. The method is used to identify very low frequency (～1 cm^?1) normal modes for the protein pancreatic trypsin inhibitor. A simplified model for the protein is used, for which each residue is represented by a single interaction center. The quasi-harmonic force constants of the virtual internal coordinates are evaluated and the normal-mode frequencies and eigenvectors are obtained. Conformations corresponding to distortions along selected low-frequency modes are analyzed.     

Comparison of normal mode analyses on a small globular protein in dihedral angle space and Cartesian coordinate space

Normal mode analyses on the protein, bovine pancreatic trypsin inhibitor, in dihedral angle space and Cartesian coordinate space are compared. In Cartesian coordinate space it is found that modes of frequencies lower than 30 cm(-1) contribute 80% of the total mean-square fluctuation and are represented almost completely by motions in the dihedral angles. Bond angle and length fluctuations dominate in modes above 200 cm(-1), but contribute less than 2% to the total mean-square fluctuation. In the low-frequency modes a good correspondence between patterns of atomic displacements was found, but on average the root-mean-square fluctuations of the Cartesian coordinate modes are 13% greater than their dihedral angle counterparts. The main effect of fluctuations in the bond angles and lengths, therefore, is to allow the dihedral angles to become more flexible. As the important subspaces determined from the two methods overlap considerably, dihedral angle space analysis can be applied to proteins too large for Cartesian coordinate space analysis.     

Terahertz vibration modes in Na/K-ATPase

Mechanical vibration in the Terahertz range is believed to be connected with protein functions. In this paper, we present the results of a normal-mode analysis (modal analysis) of a Na/K-ATPase all-atom model, focusing the attention on low-frequency vibration modes. The numerical model helps in the interpretation of experimental results previously obtained by the authors via Raman spectroscopy of Na/K-ATPase samples, where several unassigned peaks were found in the sub-500 cm^?1 range. In particular, vibration modes corresponding to peaks at 27, 190 and 300 cm^?1, found experimentally, are confirmed here numerically, together with some other modes at lower frequencies (wavenumbers) that were not possible to observe in the experimental test. All the aforementioned modes correspond to vibrations involving the protein ends, i.e. portions directly related to the operating mechanism of the sodium-potassium pump.     

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     

Can conformational change be described by only a few normal modes?

We suggest a simple method to assess how many normal modes are needed to map a conformational change. By projecting the conformational change onto a subspace of the normal-mode vectors and using root mean square deviation as a test of accuracy, we find that the first 20 modes only contribute 50% or less of the total conformational change in four test cases (myosin, calmodulin, NtrC, and hemoglobin). In some allosteric systems, like the molecular switch NtrC, the conformational change is localized to a limited number of residues. We find that many more modes are necessary to accurately map this collective displacement. In addition, the normal-mode "spectra" can provide useful information about the details of the conformational change, especially when comparing structures with different bound ligands, in this case, calmodulin. Indeed, this approach presents normal-mode analysis as a useful basis in which to capture the mechanism of conformational change, and shows that the number of normal modes needed to capture the essential collective motions of atoms should be chosen according to the required accuracy.     

Comparison of mode analyses at different resolutions applied to nucleic acid systems

More than two decades of different types of mode analyses has shown that these techniques can be useful in describing large-scale motions in protein systems. A number of mode analyses are available and include quasiharmonics, classical normal mode, block normal mode, and the elastic network model. Each of these methods has been validated for protein systems and this variety allows researchers to choose the technique that gives the best compromise between computational cost and the level of detail in the calculation. These same techniques have not been systematically tested for nucleic acid systems, however. Given the differences in interactions and structural features between nucleic acid and protein systems, the validity of these techniques in the protein regime cannot be directly translated into validity in the nucleic acid realm. In this work, we investigate the usefulness of the above mode analyses as applied to two RNA systems, i.e., the hammerhead ribozyme and a guanine riboswitch. We show that classical normal-mode analysis can match the magnitude and direction of residue fluctuations from the more detailed, anharmonic technique, quasiharmonic analysis of a molecular dynamics trajectory. The block normal-mode approximation is shown to hold in the nucleic acid systems studied. Only the mode analysis at the lowest level of detail, the elastic network model, produced mixed results in our calculations. We present data that suggest that the elastic network model, with the popular parameterization, is not best suited for systems that do not have a close packed structure; this observation also hints at why the elastic network model has been found to be valid for many globular protein systems. The different behaviors of block normal-mode analysis and the elastic network model, which invoke similar degrees of coarse-graining to the dynamics but use different potentials, suggest the importance of applying a heterogeneous potential function in a robust analysis of the dynamics of biomolecules, especially those that are not closely packed. In addition to these comparisons, we briefly discuss insights into the conformational space available to the hammerhead ribozyme.     

Vibrational normal modes and dynamical stability of DNA triplex poly(dA). 2poly(dT): S-type structure is more stable and in better agreement with observations in solution.

A normal-mode and statistical mechanical calculation was carried out to determine the vibrational normal modes, contribution of internal fluctuations to the free energy, and hydrogen bond disruption of DNA triplex poly(dA).2poly(dT). The calculation was performed on both the x-ray fiber diffraction model with a N-type sugar conformation, and a newly proposed model with a S-type sugar conformation. Our calculated normal modes for the S-type structure are in better agreement with observed IR spectra for samples in D2O solution. We also find that the contribution of internal fluctuations to free energy, premelting hydrogen bond disruption probability, and hydrogen bond melting temperatures for the Hoogsteen and Watson-Crick hydrogen bonds all show that the S-type structure is dynamically more stable than the N-type structure in a nominal solution environment. Therefore our calculation supports experimental findings that the triplex d(T)n.d(A)nd(T)n most likely adopts a S-type sugar conformation in solution or at high humidity. Our calculations, however, do not preclude the possibility of an N-type conformation at lower humidities.     

Enhancing backbone sampling in Monte Carlo simulations using internal coordinates normal mode analysis

Normal mode methods are becoming a popular alternative to sample the conformational landscape of proteins. In this study, we describe the implementation of an internal coordinate normal mode analysis method and its application in exploring protein flexibility by using the Monte Carlo method PELE. This new method alternates two different stages, a perturbation of the backbone through the application of torsional normal modes, and a resampling of the side chains. We have evaluated the new approach using two test systems, ubiquitin and c-Src kinase, and the differences to the original ANM method are assessed by comparing both results to reference molecular dynamics simulations. The results suggest that the sampled phase space in the internal coordinate approach is closer to the molecular dynamics phase space than the one coming from a Cartesian coordinate anisotropic network model. In addition, the new method shows a great speedup (∼5–7×), making it a good candidate for future normal mode implementations in Monte Carlo methods.     

Analysis of conformational motions and related key residue interactions responsible for a specific function of proteins with elastic network model

Protein collective motions play a critical role in many biochemical processes. How to predict the functional motions and the related key residue interactions in proteins is important for our understanding in the mechanism of the biochemical processes. Normal mode analysis (NMA) of the elastic network model (ENM) is one of the effective approaches to investigate the structure-encoded motions in proteins. However, the motion modes revealed by the conventional NMA approach do not necessarily correspond to a specific function of protein. In the present work, a new analysis method was proposed to identify the motion modes responsible for a specific function of proteins and then predict the key residue interactions involved in the functional motions by using a perturbation approach. In our method, an internal coordinate that accounts for the specific function was introduced, and the Cartesian coordinate space was transformed into the internal/Cartesian space by using linear approximation, where the introduced internal coordinate serves as one of the axes of the coordinate space. NMA of ENM in this internal/Cartesian space was performed and the function-relevant motion modes were identified according to their contributions to the specific function of proteins. Then the key residue interactions important for the functional motions of the protein were predicted as the interactions whose perturbation largely influences the fluctuation along the internal coordinate. Using our proposed methods, the maltose transporter (MalFGK₂) from E. Coli was studied. The functional motions and the key residue interactions that are related to the channel-gating function of this protein were successfully identified.     

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.     

Normal modes for predicting protein motions: a comprehensive database assessment and associated Web tool

We carry out an extensive statistical study of the applicability of normal modes to the prediction of mobile regions in proteins. In particular, we assess the degree to which the observed motions found in a comprehensive data set of 377 nonredundant motions can be modeled by a single normal-mode vibration. We describe each motion in our data set by vectors connecting corresponding atoms in two crystallographically known conformations. We then measure the geometric overlap of these motion vectors with the displacement vectors of the lowest-frequency mode, for one of the conformations. Our study suggests that the lowest mode contains useful information about the parts of a protein that move most (i.e., have the largest amplitudes) and about the direction of this movement. Based on our findings, we developed a Web tool for motion prediction (available from http://molmovdb.org/nma) and apply it here to four representative motions--from bacteriorhodopsin, calmodulin, insulin, and T7 RNA polymerase.     

Protein–protein alternative binding modes do not overlap

Proteins often bind other proteins in more than one way. Thus alternative binding modes is an essential feature of protein interactions. Such binding modes may be detected by X‐ray crystallography and thus reflected in Protein Data Bank. The alternative binding is often observed not for the protein itself but for its structural homolog. The results of this study based on the analysis of a comprehensive set of co‐crystallized protein–protein complexes show that the alternative binding modes generally do not overlap, but are spatially separated. This effect is based on molecular recognition characteristics of the protein structures. The results are also in excellent agreement with the intermolecular energy funnel size estimates obtained previously by an independent methodology. The results provide an important insight into the principles of protein association, as well as potential guidelines for modeling of protein complexes and the design of protein interfaces.     

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.     

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.     

High-integrity human intervention in ecosystems: Tracking self-organization modes

Humans play major roles in shaping and transforming the ecology of Earth. Unlike natural drivers of ecosystem change, which are erratic and unpredictable, human intervention in ecosystems generally involves planning and management, but often results in detrimental outcomes. Using model studies and aerial-image analysis, we argue that the design of a successful human intervention form calls for the identification of the self-organization modes that drive ecosystem change, and for studying their dynamics. We demonstrate this approach with two examples: grazing management in drought-prone ecosystems, and rehabilitation of degraded vegetation by water harvesting. We show that grazing can increase the resilience to droughts, rather than imposing an additional stress, if managed in a spatially non-uniform manner, and that fragmental restoration along contour bunds is more resilient than the common practice of continuous restoration in vegetation stripes. We conclude by discussing the need for additional studies of self-organization modes and their dynamics.     

Stability of Alfvén modes in a collisionless anisotropic plasma confined by a highly curved magnetic field

The stability of Alfvén modes in a collisionless plasma with an anisotropic pressure in a highly curved magnetic field is studied. A linearized equation for describing longitudinally nonuniform MHD perturbations with frequencies below the bounce frequency is derived. In this equation, the perturbations of longitudinal and transverse pressures are calculated using a collisionless kinetic equation. It is shown that longitudinal fluxes of the transverse and longitudinal plasma energies give rise to pressure perturbations different from those in the Chew-Goldberger-Low collisionless hydrodynamics. The corresponding energy principle is constructed. A stability criterion for Alfvén modes is obtained and is found to be more stringent than that in the Chew-Gold-berger-Low model.     

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.