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

On the optimal contact potential of proteins

We analytically derive the lower bound of the total conformational energy of a protein structure by assuming that the total conformational energy is well approximated by the sum of sequence-dependent pairwise contact energies. The condition for the native structure achieving the lower bound leads to the contact energy matrix that is a scalar multiple of the native contact matrix, i.e., the so-called Go potential. We also derive spectral relations between contact matrix and energy matrix, and approximations related to one-dimensional protein structures. Implications for protein structure prediction are discussed.Comment: 5 pages, text onl

Cooperative "folding transition" in the sequence space facilitates function-driven evolution of protein families

In the protein sequence space, natural proteins form clusters of families which are characterized by their unique native folds whereas the great majority of random polypeptides are neither clustered nor foldable to unique structures. Since a given polypeptide can be either foldable or unfoldable, a kind of "folding transition" is expected at the boundary of a protein family in the sequence space. By Monte Carlo simulations of a statistical mechanical model of protein sequence alignment that coherently incorporates both short-range and long-range interactions as well as variable-length insertions to reproduce the statistics of the multiple sequence alignment of a given protein family, we demonstrate the existence of such transition between natural-like sequences and random sequences in the sequence subspaces for 15 domain families of various folds. The transition was found to be highly cooperative and two-state-like. Furthermore, enforcing or suppressing consensus residues on a few of the well-conserved sites enhanced or diminished, respectively, the natural-like pattern formation over the entire sequence. In most families, the key sites included ligand binding sites. These results suggest some selective pressure on the key residues, such as ligand binding activity, may cooperatively facilitate the emergence of a protein family during evolution. From a more practical aspect, the present results highlight an essential role of long-range effects in precisely defining protein families, which are absent in conventional sequence models.Comment: 13 pages, 7 figures, 2 tables (a new subsection added

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

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

CRNPRED: highly accurate prediction of one-dimensional protein structures by large-scale critical random networks

BACKGROUND: One-dimensional protein structures such as secondary structures or contact numbers are useful for three-dimensional structure prediction and helpful for intuitive understanding of the sequence-structure relationship. Accurate prediction methods will serve as a basis for these and other purposes. RESULTS: We implemented a program CRNPRED which predicts secondary structures, contact numbers and residue-wise contact orders. This program is based on a novel machine learning scheme called critical random networks. Unlike most conventional one-dimensional structure prediction methods which are based on local windows of an amino acid sequence, CRNPRED takes into account the whole sequence. CRNPRED achieves, on average per chain, Q(3 )= 81% for secondary structure prediction, and correlation coefficients of 0.75 and 0.61 for contact number and residue-wise contact order predictions, respectively. CONCLUSION: CRNPRED will be a useful tool for computational as well as experimental biologists who need accurate one-dimensional protein structure predictions

Predicting residue-wise contact orders in proteins by support vector regression

BACKGROUND: The residue-wise contact order (RWCO) describes the sequence separations between the residues of interest and its contacting residues in a protein sequence. It is a new kind of one-dimensional protein structure that represents the extent of long-range contacts and is considered as a generalization of contact order. Together with secondary structure, accessible surface area, the B factor, and contact number, RWCO provides comprehensive and indispensable important information to reconstructing the protein three-dimensional structure from a set of one-dimensional structural properties. Accurately predicting RWCO values could have many important applications in protein three-dimensional structure prediction and protein folding rate prediction, and give deep insights into protein sequence-structure relationships. RESULTS: We developed a novel approach to predict residue-wise contact order values in proteins based on support vector regression (SVR), starting from primary amino acid sequences. We explored seven different sequence encoding schemes to examine their effects on the prediction performance, including local sequence in the form of PSI-BLAST profiles, local sequence plus amino acid composition, local sequence plus molecular weight, local sequence plus secondary structure predicted by PSIPRED, local sequence plus molecular weight and amino acid composition, local sequence plus molecular weight and predicted secondary structure, and local sequence plus molecular weight, amino acid composition and predicted secondary structure. When using local sequences with multiple sequence alignments in the form of PSI-BLAST profiles, we could predict the RWCO distribution with a Pearson correlation coefficient (CC) between the predicted and observed RWCO values of 0.55, and root mean square error (RMSE) of 0.82, based on a well-defined dataset with 680 protein sequences. Moreover, by incorporating global features such as molecular weight and amino acid composition we could further improve the prediction performance with the CC to 0.57 and an RMSE of 0.79. In addition, combining the predicted secondary structure by PSIPRED was found to significantly improve the prediction performance and could yield the best prediction accuracy with a CC of 0.60 and RMSE of 0.78, which provided at least comparable performance compared with the other existing methods. CONCLUSION: The SVR method shows a prediction performance competitive with or at least comparable to the previously developed linear regression-based methods for predicting RWCO values. In contrast to support vector classification (SVC), SVR is very good at estimating the raw value profiles of the samples. The successful application of the SVR approach in this study reinforces the fact that support vector regression is a powerful tool in extracting the protein sequence-structure relationship and in estimating the protein structural profiles from amino acid sequences

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

Better prediction of protein contact number using a support vector regression analysis of amino acid sequence

BACKGROUND: Protein tertiary structure can be partly characterized via each amino acid's contact number measuring how residues are spatially arranged. The contact number of a residue in a folded protein is a measure of its exposure to the local environment, and is defined as the number of C(β )atoms in other residues within a sphere around the C(β )atom of the residue of interest. Contact number is partly conserved between protein folds and thus is useful for protein fold and structure prediction. In turn, each residue's contact number can be partially predicted from primary amino acid sequence, assisting tertiary fold analysis from sequence data. In this study, we provide a more accurate contact number prediction method from protein primary sequence. RESULTS: We predict contact number from protein sequence using a novel support vector regression algorithm. Using protein local sequences with multiple sequence alignments (PSI-BLAST profiles), we demonstrate a correlation coefficient between predicted and observed contact numbers of 0.70, which outperforms previously achieved accuracies. Including additional information about sequence weight and amino acid composition further improves prediction accuracies significantly with the correlation coefficient reaching 0.73. If residues are classified as being either "contacted" or "non-contacted", the prediction accuracies are all greater than 77%, regardless of the choice of classification thresholds. CONCLUSION: The successful application of support vector regression to the prediction of protein contact number reported here, together with previous applications of this approach to the prediction of protein accessible surface area and B-factor profile, suggests that a support vector regression approach may be very useful for determining the structure-function relation between primary protein sequence and higher order consecutive protein structural and functional properties

Protein contact order prediction from primary sequences

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