Recoverable One-dimensional Encoding of Three-dimensional Protein Structures

Protein one-dimensional (1D) structures such as secondary structure and contact number provide intuitive pictures to understand how the native three-dimensional (3D) structure of a protein is encoded in the amino acid sequence. However, it has not been clear whether a given set of 1D structures contains sufficient information for recovering the underlying 3D structure. Here we show that the 3D structure of a protein can be recovered from a set of three types of 1D structures, namely, secondary structure, contact number and residue-wise contact order which is introduced here for the first time. Using simulated annealing molecular dynamics simulations, the structures satisfying the given native 1D structural restraints were sought for 16 proteins of various structural classes and of sizes ranging from 56 to 146 residues. By selecting the structures best satisfying the restraints, all the proteins showed a coordinate RMS deviation of less than 4\AA{} from the native structure, and for most of them, the deviation was even less than 2\AA{}. The present result opens a new possibility to protein structure prediction and our understanding of the sequence-structure relationship.Comment: Corrected title. No Change In Content

Properties of contact matrices induced by pairwise interactions in proteins

The total conformational energy is assumed to consist of pairwise interaction energies between atoms or residues, each of which is expressed as a product of a conformation-dependent function (an element of a contact matrix, C-matrix) and a sequence-dependent energy parameter (an element of a contact energy matrix, E-matrix). Such pairwise interactions in proteins force native C-matrices to be in a relationship as if the interactions are a Go-like potential [N. Go, Annu. Rev. Biophys. Bioeng. 12. 183 (1983)] for the native C-matrix, because the lowest bound of the total energy function is equal to the total energy of the native conformation interacting in a Go-like pairwise potential. This relationship between C- and E-matrices corresponds to (a) a parallel relationship between the eigenvectors of the C- and E-matrices and a linear relationship between their eigenvalues, and (b) a parallel relationship between a contact number vector and the principal eigenvectors of the C- and E-matrices; the E-matrix is expanded in a series of eigenspaces with an additional constant term, which corresponds to a threshold of contact energy that approximately separates native contacts from non-native ones. These relationships are confirmed in 182 representatives from each family of the SCOP database by examining inner products between the principal eigenvector of the C-matrix, that of the E-matrix evaluated with a statistical contact potential, and a contact number vector. In addition, the spectral representation of C- and E-matrices reveals that pairwise residue-residue interactions, which depends only on the types of interacting amino acids but not on other residues in a protein, are insufficient and other interactions including residue connectivities and steric hindrance are needed to make native structures the unique lowest energy conformations.Comment: Errata in DOI:10.1103/PhysRevE.77.051910 has been corrected in the present versio

Predicting Secondary Structures, Contact Numbers, and Residue-wise Contact Orders of Native Protein Structure from Amino Acid Sequence by Critical Random Networks

Prediction of one-dimensional protein structures such as secondary structures and contact numbers is useful for the three-dimensional structure prediction and important for the understanding of sequence-structure relationship. Here we present a new machine-learning method, critical random networks (CRNs), for predicting one-dimensional structures, and apply it, with position-specific scoring matrices, to the prediction of secondary structures (SS), contact numbers (CN), and residue-wise contact orders (RWCO). The present method achieves, on average, $Q_3$ accuracy of 77.8% for SS, correlation coefficients of 0.726 and 0.601 for CN and RWCO, respectively. The accuracy of the SS prediction is comparable to other state-of-the-art methods, and that of the CN prediction is a significant improvement over previous methods. We give a detailed formulation of critical random networks-based prediction scheme, and examine the context-dependence of prediction accuracies. In order to study the nonlinear and multi-body effects, we compare the CRNs-based method with a purely linear method based on position-specific scoring matrices. Although not superior to the CRNs-based method, the surprisingly good accuracy achieved by the linear method highlights the difficulty in extracting structural features of higher order from amino acid sequence beyond that provided by the position-specific scoring matrices.Comment: 20 pages, 1 figure, 5 tables; minor revision; accepted for publication in BIOPHYSIC

Wang-Landau molecular dynamics technique to search for low-energy conformational space of proteins

Multicanonical molecular dynamics (MD) is a powerful technique for sampling conformations on rugged potential surfaces such as protein. However, it is notoriously difficult to estimate the multicanonical temperature effectively. Wang and Landau developed a convenient method for estimating the density of states based on a multicanonical Monte Carlo method. In their method, the density of states is calculated autonomously during a simulation. In this paper we develop a set of techniques to effectively apply the Wang-Landau method to MD simulations. In the multicanonical MD, the estimation of the derivative of the density of states is critical. In order to estimate it accurately, we devise two original improvements. First, the correction for the density of states is made smooth by using the Gaussian distribution obtained by a short canonical simulation. Second, an approximation is applied to the derivative, which is based on the Gaussian distribution and the multiple weighted histogram technique. A test of this method was performed with small polypeptides, Met-enkephalin and Trp-cage, and it is demonstrated that Wang-Landau MD is consistent with replica exchange MD but can sample much larger conformational space.Comment: 8 pages, 7 figures, accepted for publication in Physical Review

Composite structural motifs of binding sites for delineating biological functions of proteins

Most biological processes are described as a series of interactions between proteins and other molecules, and interactions are in turn described in terms of atomic structures. To annotate protein functions as sets of interaction states at atomic resolution, and thereby to better understand the relation between protein interactions and biological functions, we conducted exhaustive all-against-all atomic structure comparisons of all known binding sites for ligands including small molecules, proteins and nucleic acids, and identified recurring elementary motifs. By integrating the elementary motifs associated with each subunit, we defined composite motifs which represent context-dependent combinations of elementary motifs. It is demonstrated that function similarity can be better inferred from composite motif similarity compared to the similarity of protein sequences or of individual binding sites. By integrating the composite motifs associated with each protein function, we define meta-composite motifs each of which is regarded as a time-independent diagrammatic representation of a biological process. It is shown that meta-composite motifs provide richer annotations of biological processes than sequence clusters. The present results serve as a basis for bridging atomic structures to higher-order biological phenomena by classification and integration of binding site structures.Comment: 34 pages, 7 figure

SeSAW: balancing sequence and structural information in protein functional mapping

Motivation: Functional similarity between proteins is evident at both the sequence and structure levels. SeSAW is a web-based program for identifying functionally or evolutionarily conserved motifs in protein structures by locating sequence and structural similarities, and quantifying these at the level of individual residues. Results can be visualized in 2D, as annotated alignments, or in 3D, as structural superpositions. An example is given for both an experimentally determined query structure and a homology model

Amino acid "little Big Bang": Representing amino acid substitution matrices as dot products of Euclidian vectors

<p>Abstract</p> <p>Background</p> <p>Sequence comparisons make use of a one-letter representation for amino acids, the necessary quantitative information being supplied by the substitution matrices. This paper deals with the problem of finding a representation that provides a comprehensive description of amino acid intrinsic properties consistent with the substitution matrices.</p> <p>Results</p> <p>We present a Euclidian vector representation of the amino acids, obtained by the singular value decomposition of the substitution matrices. The substitution matrix entries correspond to the dot product of amino acid vectors. We apply this vector encoding to the study of the relative importance of various amino acid physicochemical properties upon the substitution matrices. We also characterize and compare the PAM and BLOSUM series substitution matrices.</p> <p>Conclusions</p> <p>This vector encoding introduces a Euclidian metric in the amino acid space, consistent with substitution matrices. Such a numerical description of the amino acid is useful when intrinsic properties of amino acids are necessary, for instance, building sequence profiles or finding consensus sequences, using machine learning algorithms such as Support Vector Machine and Neural Networks algorithms.</p

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. The presented method may be used to separate functionally important sites from structurally important ones, and thus it may be a useful tool for predicting protein functions.Comment: 22 pages, 7 figures, 4 table

Selective Constraints on Amino Acids Estimated by a Mechanistic Codon Substitution Model with Multiple Nucleotide Changes

Empirical substitution matrices represent the average tendencies of substitutions over various protein families by sacrificing gene-level resolution. We develop a codon-based model, in which mutational tendencies of codon, a genetic code, and the strength of selective constraints against amino acid replacements can be tailored to a given gene. First, selective constraints averaged over proteins are estimated by maximizing the likelihood of each 1-PAM matrix of empirical amino acid (JTT, WAG, and LG) and codon (KHG) substitution matrices. Then, selective constraints specific to given proteins are approximated as a linear function of those estimated from the empirical substitution matrices. Akaike information criterion (AIC) values indicate that a model allowing multiple nucleotide changes fits the empirical substitution matrices significantly better. Also, the ML estimates of transition-transversion bias obtained from these empirical matrices are not so large as previously estimated. The selective constraints are characteristic of proteins rather than species. However, their relative strengths among amino acid pairs can be approximated not to depend very much on protein families but amino acid pairs, because the present model, in which selective constraints are approximated to be a linear function of those estimated from the JTT/WAG/LG/KHG matrices, can provide a good fit to other empirical substitution matrices including cpREV for chloroplast proteins and mtREV for vertebrate mitochondrial proteins. The present codon-based model with the ML estimates of selective constraints and with adjustable mutation rates of nucleotide would be useful as a simple substitution model in ML and Bayesian inferences of molecular phylogenetic trees, and enables us to obtain biologically meaningful information at both nucleotide and amino acid levels from codon and protein sequences.Comment: Table 9 in this article includes corrections for errata in the Table 9 published in 10.1371/journal.pone.0017244. Supporting information is attached at the end of the article, and a computer-readable dataset of the ML estimates of selective constraints is available from 10.1371/journal.pone.001724

Barriers to Diffusion in Dendrites and Estimation of Calcium Spread Following Synaptic Inputs

The motion of ions, molecules or proteins in dendrites is restricted by cytoplasmic obstacles such as organelles, microtubules and actin network. To account for molecular crowding, we study the effect of diffusion barriers on local calcium spread in a dendrite. We first present a model based on a dimension reduction approach to approximate a three dimensional diffusion in a cylindrical dendrite by a one-dimensional effective diffusion process. By comparing uncaging experiments of an inert dye in a spiny dendrite and in a thin glass tube, we quantify the change in diffusion constants due to molecular crowding as Dcyto/Dwater = 1/20. We validate our approach by reconstructing the uncaging experiments using Brownian simulations in a realistic 3D model dendrite. Finally, we construct a reduced reaction-diffusion equation to model calcium spread in a dendrite under the presence of additional buffers, pumps and synaptic input. We find that for moderate crowding, calcium dynamics is mainly regulated by the buffer concentration, but not by the cytoplasmic crowding, dendritic spines or synaptic inputs. Following high frequency stimulations, we predict that calcium spread in dendrites is limited to small microdomains of the order of a few microns (<5 μm)