Nature of protein family signatures: insights from singular value analysis of position-specific scoring matrices

Position-specific scoring matrices (PSSMs) are useful for detecting weak homology in protein sequence analysis, and they are thought to contain some essential signatures of the protein families. In order to elucidate what kind of ingredients constitute such family-specific signatures, we apply singular value decomposition to a set of PSSMs and examine the properties of dominant right and left singular vectors. The first right singular vectors were correlated with various amino acid indices including relative mutability, amino acid composition in protein interior, hydropathy, or turn propensity, depending on proteins. A significant correlation between the first left singular vector and a measure of site conservation was observed. It is shown that the contribution of the first singular component to the PSSMs act to disfavor potentially but falsely functionally important residues at conserved sites. The second right singular vectors were highly correlated with hydrophobicity scales, and the corresponding left singular vectors with contact numbers of protein structures. It is suggested that sequence alignment with a PSSM is essentially equivalent to threading supplemented with functional information. In addition, singular vectors may be useful for analyzing and annotating the characteristics of conserved sites in protein families.     

Optimized torsion-angle normal modes reproduce conformational changes more accurately than cartesian modes

We present what to our knowledge is a new method of optimized torsion-angle normal-mode analysis, in which the normal modes move along curved paths in Cartesian space. We show that optimized torsion-angle normal modes reproduce protein conformational changes more accurately than Cartesian normal modes. We also show that orthogonalizing the displacement vectors from torsion-angle normal-mode analysis and projecting them as straight lines in Cartesian space does not lead to better performance than Cartesian normal modes. Clearly, protein motion is more naturally described by curved paths in Cartesian space.     

Exploring the common dynamics of homologous proteins. Application to the globin family

We present a procedure to explore the global dynamics shared between members of the same protein family. The method allows the comparison of patterns of vibrational motion obtained by Gaussian network model analysis. After the identification of collective coordinates that were conserved during evolution, we quantify the common dynamics within a family. Representative vectors that describe these dynamics are defined using a singular value decomposition approach. As a test case, the globin heme-binding family is considered. The two lowest normal modes are shown to be conserved within this family. Our results encourage the development of models for protein evolution that take into account the conservation of dynamical features.     

A comprehensive vertebrate phylogeny using vector representations of protein sequences from whole genomes

We recently developed a method for producing comprehensive gene and species phylogenies from unaligned whole genome data using singular value decomposition (SVD) to analyze character string frequencies. This work provides an integrated gene and species phylogeny for 64 vertebrate mitochondrial genomes composed of 832 total proteins. In addition, to provide a theoretical basis for the method, we present a graphical interpretation of both the original frequency matrix and the SVD-derived matrix. These large matrices describe high-dimensional Euclidean spaces within which biomolecular sequences can be uniquely represented as vectors. In particular, the SVD-derived vector space describes each protein relative to a restricted set of newly defined, independent axes, each of which represents a novel form of conserved motif, termed a correlated peptide motif. A quantitative comparison of the relative orientations of protein vectors in this space provides accurate and straightforward estimates of sequence similarity, which can in turn be used to produce comprehensive gene trees. Alternatively, the vector representations of genes from individual species can be summed, allowing species trees to be produced.     

Biclustering via Sparse Singular Value Decomposition

Summary Sparse singular value decomposition (SSVD) is proposed as a new exploratory analysis tool for biclustering or identifying interpretable row–column associations within high‐dimensional data matrices. SSVD seeks a low‐rank, checkerboard structured matrix approximation to data matrices. The desired checkerboard structure is achieved by forcing both the left‐ and right‐singular vectors to be sparse, that is, having many zero entries. By interpreting singular vectors as regression coefficient vectors for certain linear regressions, sparsity‐inducing regularization penalties are imposed to the least squares regression to produce sparse singular vectors. An efficient iterative algorithm is proposed for computing the sparse singular vectors, along with some discussion of penalty parameter selection. A lung cancer microarray dataset and a food nutrition dataset are used to illustrate SSVD as a biclustering method. SSVD is also compared with some existing biclustering methods using simulated datasets.     

Simulating nanoscale functional motions of biomolecules

We are describing efficient dynamics simulation methods for the characterization of functional motion of biomolecules on the nanometer scale. Multivariate statistical methods are widely used to extract and enhance functional collective motions from molecular dynamics (MD) simulations. A dimension reduction in MD is often realized through a principal component analysis (PCA) or a singular value decomposition (SVD) of the trajectory. Normal mode analysis (NMA) is a related collective coordinate space approach, which involves the decomposition of the motion into vibration modes based on an elastic model. Using the myosin motor protein as an example we describe a hybrid technique termed amplified collective motions (ACM) that enhances sampling of conformational space through a combination of normal modes with atomic level MD. Unfortunately, the forced orthogonalization of modes in collective coordinate space leads to complex dependencies that are not necessarily consistent with the symmetry of biological macromolecules and assemblies. In many biological molecules, such as HIV-1 protease, reflective or rotational symmetries are present that are broken using standard orthogonal basis functions. We present a method to compute the plane of reflective symmetry or the axis of rotational symmetry from the trajectory frames. Moreover, we develop an SVD that best approximates the given trajectory while respecting the symmetry. Finally, we describe a local feature analysis (LFA) to construct a topographic representation of functional dynamics in terms of local features. The LFA representations are low-dimensional, and provide a reduced basis set for collective motions, but unlike global collective modes they are sparsely distributed and spatially localized. This yields a more reliable assignment of essential dynamics modes across different MD time windows.     

Integrated gene and species phylogenies from unaligned whole genome protein sequences

MOTIVATION: Most molecular phylogenies are based on sequence alignments. Consequently, they fail to account for modes of sequence evolution that involve frequent insertions or deletions. Here we present a method for generating accurate gene and species phylogenies from whole genome sequence that makes use of short character string matches not placed within explicit alignments. In this work, the singular value decomposition of a sparse tetrapeptide frequency matrix is used to represent the proteins of organisms uniquely and precisely as vectors in a high-dimensional space. Vectors of this kind can be used to calculate pairwise distance values based on the angle separating the vectors, and the resulting distance values can be used to generate phylogenetic trees. Protein trees so derived can be examined directly for homologous sequences. Alternatively, vectors defining each of the proteins within an organism can be summed to provide a vector representation of the organism, which is then used to generate species trees. RESULTS: Using a large mitochondrial genome dataset, we have produced species trees that are largely in agreement with previously published trees based on the analysis of identical datasets using different methods. These trees also agree well with currently accepted phylogenetic theory. In principle, our method could be used to compare much larger bacterial or nuclear genomes in full molecular detail, ultimately allowing accurate gene and species relationships to be derived from a comprehensive comparison of complete genomes. In contrast to phylogenetic methods based on alignments, sequences that evolve by relative insertion or deletion would tend to remain recognizably similar.     

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.     

Formation of a direction map by projection learning using Kohonen's self-organization map

 In this paper, we propose a modification of Kohonen's self-organization map (SOM) algorithm. When the input signal space is not convex, some reference vectors of SOM can protrude from it. The input signal space must be convex to keep all the reference vectors fixed on it for any updates. Thus, we introduce a projection learning method that fixes the reference vectors onto the input signal space. This version of SOM can be applied to a non-convex input signal space. We applied SOM with projection learning to a direction map observed in the primary visual cortex of area 17 of ferrets, and area 18 of cats. Neurons in those areas responded selectively to the orientation of edges or line segments, and their directions of motion. Some iso-orientation domains were subdivided into selective regions for the opposite direction of motion. The abstract input signal space of the direction map described in the manner proposed by Obermayer and Blasdel [(1993) J Neurosci 13: 4114–4129] is not convex. We successfully used SOM with projection learning to reproduce a direction-orientation joint map. Received: 29 September 2000 / Accepted: 7 March 2001     

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.     

Fuzzy kernel clustering of RNA secondary structure ensemble using a novel similarity metric

Measuring the (dis)similarity between RNA secondary structures is critical for the study of RNA secondary structures and has implications to RNA functional characterization. Although a number of methods have been developed for comparing RNA structural similarities, their applications have been limited by the complexity of the required computation. In this paper, we present a novel method for comparing the similarity of RNA secondary structures generated from the same RNA sequence, i.e., a secondary structure ensemble, using a matrix representation of the RNA structures. Relevant features of the RNA secondary structures can be easily extracted through singular value decomposition (SVD) of the representing matrices. We have mapped the feature vectors of the singular values to a kernel space, where (dis)similarities among the mapped feature vectors become more evident, making clustering of RNA secondary structures easier to handle. The pair-wise comparison of RNA structures is achieved through computing the distance between the singular value vectors in the kernel space. We have applied a fuzzy kernel clustering method, using this similarity metric, to cluster the RNA secondary structure ensembles. Our application results suggest that our fuzzy kernel clustering method is highly promising for classifications of RNA structure ensembles, because of its low computational complexity and high clustering accuracy.     

Molecular dynamics simulations of peptides and proteins with amplified collective motions

We present a novel method that uses the collective modes obtained with a coarse-grained model/anisotropic network model to guide the atomic-level simulations. Based on this model, local collective modes can be calculated according to a single configuration in the conformational space of the protein. In the molecular dynamics simulations, the motions along the slowest few modes are coupled to a higher temperature by the weak coupling method to amplify the collective motions. This amplified-collective-motion (ACM) method is applied to two test systems. One is an S-peptide analog. We realized the refolding of the denatured peptide in eight simulations out of 10 using the method. The other system is bacteriophage T4 lysozyme. Much more extensive domain motions between the N-terminal and C-terminal domain of T4 lysozyme are observed in the ACM simulation compared to a conventional simulation. The ACM method allows for extensive sampling in conformational space while still restricting the sampled configurations within low energy areas. The method can be applied in both explicit and implicit solvent simulations, and may be further applied to important biological problems, such as long timescale functional motions, protein folding/unfolding, and structure prediction.     

Fuzzy Kernel Clustering of RNA Secondary Structure Ensemble Using a Novel Similarity Metric

Abstract

Measuring the (dis)similarity between RNA secondary structures is critical for the study of RNA secondary structures and has implications to RNA functional characterization. Although a number of methods have been developed for comparing RNA structural similarities, their applications have been limited by the complexity of the required computation. In this paper, we present a novel method for comparing the similarity of RNA secondary structures generated from the same RNA sequence, i.e., a secondary structure ensemble, using a matrix representation of the RNA structures. Relevant features of the RNA secondary structures can be easily extracted through singular value decomposition (SVD) of the representing matrices. We have mapped the feature vectors of the singular values to a kernel space, where (dis)similarities among the mapped feature vectors become more evident, making clustering of RNA secondary structures easier to handle. The pair-wise comparison of RNA structures is achieved through computing the distance between the singular value vectors in the kernel space. We have applied a fuzzy kernel clustering method, using this similarity metric, to cluster the RNA secondary structure ensembles. Our application results suggest that our fuzzy kernel clustering method is highly promising for classifications of RNA structure ensembles, because of its low computational complexity and high clustering accuracy.     

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.     

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.     

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.     

A comprehensive whole genome bacterial phylogeny using correlated peptide motifs defined in a high dimensional vector space

As whole genome sequences continue to expand in number and complexity, effective methods for comparing and categorizing both genes and species represented within extremely large datasets are required. Methods introduced to date have generally utilized incomplete and likely insufficient subsets of the available data. We have developed an accurate and efficient method for producing robust gene and species phylogenies using very large whole genome protein datasets. This method relies on multidimensional protein vector definitions supplied by the singular value decomposition (SVD) of a large sparse data matrix in which each protein is uniquely represented as a vector of overlapping tetrapeptide frequencies. Quantitative pairwise estimates of species similarity were obtained by summing the protein vectors to form species vectors, then determining the cosines of the angles between species vectors. Evolutionary trees produced using this method confirmed many accepted prokaryotic relationships. However, several unconventional relationships were also noted. In addition, we demonstrate that many of the SVD-derived right basis vectors represent particular conserved protein families, while many of the corresponding left basis vectors describe conserved motifs within these families as sets of correlated peptides (copeps). This analysis represents the most detailed simultaneous comparison of prokaryotic genes and species available to date.     

Coherent conformational degrees of freedom as a structural basis for allosteric communication

Conformational changes in allosteric regulation can to a large extent be described as motion along one or a few coherent degrees of freedom. The states involved are inherent to the protein, in the sense that they are visited by the protein also in the absence of effector ligands. Previously, we developed the measure binding leverage to find sites where ligand binding can shift the conformational equilibrium of a protein. Binding leverage is calculated for a set of motion vectors representing independent conformational degrees of freedom. In this paper, to analyze allosteric communication between binding sites, we introduce the concept of leverage coupling, based on the assumption that only pairs of sites that couple to the same conformational degrees of freedom can be allosterically connected. We demonstrate how leverage coupling can be used to analyze allosteric communication in a range of enzymes (regulated by both ligand binding and post-translational modifications) and huge molecular machines such as chaperones. Leverage coupling can be calculated for any protein structure to analyze both biological and latent catalytic and regulatory sites.     

Development of a computational model for astronaut reorientation

The ability to model astronaut reorientations computationally provides a simple way to develop and study human motion control strategies. Since the cost of experimenting in microgravity is high, and underwater training can lead to motions inappropriate for microgravity, these techniques allow for motions to be developed and well-understood prior to any microgravity exposure. By including a model of the current space suit, we have the ability to study both intravehicular and extravehicular activities. We present several techniques for rotating about the axes of the body and show that motions performed by the legs create a greater net rotation than those performed by the arms. Adding a space suit to the motions was seen to increase the resistance torque and limit the available range of motion. While rotations about the body axes can be performed in the current space suit, the resulting motions generated a reduced rotation when compared to the unsuited configuration.     

Approximation and visualization of large-scale motion of protein surfaces

We present a method for the approximation and real-time visualization of large-scale motion of protein surfaces. A molecular surface is represented by an expansion of spherical harmonic functions, and the motion of protein atoms around their equilibrium positions is computed by normal mode analysis. The motion of the surface is approximated by projecting the normal mode vectors of the solvent-accessible atoms to the spherical harmonic representation of the molecular surface. These surface motion vectors are represented by a separated spherical harmonic expansion. Representing the surface geometry and the surface motion vectors by spherical harmonic expansions allows variable-resolution analysis and real-time display of the large-scale surface motion. This technique has been applied to interactive visualization, interactive surface manipulation, and animation.