Efficient RMSD measures for the comparison of two molecular ensembles. Root-mean-square deviation

Quantitative measures are presented for comparing the conformations of two molecular ensembles. The measures are based on Kabsch's formula for the root-mean-square deviation (RMSD) and the covariance matrix of atomic positions of isotropically distributed ensembles (IDE). By using a Taylor series expansion, it is shown that the RMSD can be expressed solely in terms of the IDE matrices. A fast approximate method is introduced for the pairwise RMSD determination whose computational cost scales linearly with the number of structures. A similarity measure for two structural ensembles that is based on the trace metric of the differences of powers of the IDE matrices is presented. The measures are illustrated for conformational ensembles generated by a molecular dynamics computer simulation of a partially folded A-state analog of ubiquitin.     

Size-independent comparison of protein three-dimensional structures

Protein structures are routinely compared by their root-mean-square deviation (RMSD) in atomic coordinates after optimal rigid body superposition. What is not so clear is the significance of different RMSD values, particularly above the customary arbitrary cutoff for obvious similarity of 2–3 Å. Our earlier work argued for an intrinsic cutoff for protein similarity that varied with the number of residues in the polypeptide chains being compared. Here we introduce a new measure, ρ, of structural similarity based on RMSD that is independent of the sizes of the molecules involved, or of any other special properties of molecules. When ρ is less than 0.4–0.5, protein structures are visually recognized to be obviously similar, but the mathematically pleasing intrinsic cutoff of ρ>1.0 corresponds to overall similarity in folding motif at a level not usually recognized until smoothing of the polypeptide chain path makes it striking. When the structures are scaled to unit radius of gyration and equal principle moments of inertia, the comparisons are even more universal, since they are no longer obscured by differences in overall size and ellipticity. With increasing chain length, the distribution of ρ for pairs of random structures is skewed to higher values, but the value for the best 1% of the comparisons rises only slowly with the number of residues. This level is close to an intrinsic cutoff between similar and dissimilar comparisons, namely the maximal scaled ρ possible for the two structures to be more similar to each other than one is to the other's mirror image. The intrinsic cutoff is independent of the number of residues or points being compared. For proteins having fewer than 100 residues, the 1% ρ falls below the intrinsic cutoff, so that for very small proteins, geometrically significant similarity can often occur by chance. We believe these ideas will be helpful in judging success in NMR structure determination and protein folding modeling. © 1995 Wiley-Liss, Inc.     

SMS 2.0: an updated database to study the structural plasticity of short peptide fragments in non-redundant proteins

The function of a protein molecule is greatly influenced by its three-dimensional (3D) structure and therefore structure prediction will help identify its biological function. We have updated Sequence, Motif and Structure (SMS), the database of structurally rigid peptide fragments, by combining amino acid sequences and the corre-sponding 3D atomic coordinates of non-redundant (25%) and redundant (90%) protein chains available in the Protein Data Bank (PDB). SMS 2.0 provides information pertaining to the peptide fragments of length 5-14 resi-dues. The entire dataset is divided into three categories, namely, same sequence motifs having similar, intermedi-ate or dissimilar 3D structures. Further, options are provided to facilitate structural superposition using the pro-gram structural alignment of multiple proteins (STAMP) and the popular JAVA plug-in (Jmol) is deployed for visualization. In addition, functionalities are provided to search for the occurrences of the sequence motifs in other structural and sequence databases like PDB, Genome Database (GDB), Protein Information Resource (PIR) and Swiss-Prot. The updated database along with the search engine is available over the World Wide Web through the following URL http://cluster.physics.iisc.ernet.in/sms/.     

Determining and visualizing flexibility in protein structures

How to compare the structures of an ensemble of protein conformations is a fundamental problem in structural biology. As has been previously observed, the widely used RMSD measure due to Kabsch, in which a rigid‐body superposition minimizing the least‐squares positional deviations is performed, has its drawbacks when comparing and visualizing a set of flexible protein structures. Here, we develop a method, fleximatch, of protein structure comparison that takes flexibility into account. Based on a distance matrix measure of flexibility, a weighted superposition of distance matrices rather than of atomic coordinates is performed. Subsequently, this allows a consistent determination of (a) a superposition of structures for visualization, (b) a partitioning of the protein structure into rigid molecular components (core atoms), and (c) an atomic mobility measure. The method is suitable for highlighting both particularly flexible and rigid parts of a protein from structures derived from NMR, X‐ray diffraction or molecular simulation. Proteins 2015; 83:820–826. © 2015 Wiley Periodicals, Inc.     

Structural alignment of proteins by a novel TOPOFIT method, as a superimposition of common volumes at a topomax point

Similarity of protein structures has been analyzed using three-dimensional Delaunay triangulation patterns derived from the backbone representation. It has been found that structurally related proteins have a common spatial invariant part, a set of tetrahedrons, mathematically described as a common spatial subgraph volume of the three-dimensional contact graph derived from Delaunay tessellation (DT). Based on this property of protein structures, we present a novel common volume superimposition (TOPOFIT) method to produce structural alignments. Structural alignments usually evaluated by a number of equivalent (aligned) positions (N(e)) with corresponding root mean square deviation (RMSD). The superimposition of the DT patterns allows one to uniquely identify a maximal common number of equivalent residues in the structural alignment. In other words, TOPOFIT identifies a feature point on the RMSD N(e) curve, a topomax point, until which the topologies of two structures correspond to each other, including backbone and interresidue contacts, whereas the growing number of mismatches between the DT patterns occurs at larger RMSD (N(e)) after the topomax point. It has been found that the topomax point is present in all alignments from different protein structural classes; therefore, the TOPOFIT method identifies common, invariant structural parts between proteins. The alignments produced by the TOPOFIT method have a good correlation with alignments produced by other current methods. This novel method opens new opportunities for the comparative analysis of protein structures and for more detailed studies on understanding the molecular principles of tertiary structure organization and functionality. The TOPOFIT method also helps to detect conformational changes, topological differences in variable parts, which are particularly important for studies of variations in active/ binding sites and protein classification.     

Integration of small-angle X-ray scattering data into structural modeling of proteins and their assemblies

A major challenge in structural biology is to determine the configuration of domains and proteins in multidomain proteins and assemblies, respectively. All available data should be considered to maximize the accuracy and precision of these models. Small-angle X-ray scattering (SAXS) efficiently provides low-resolution experimental data about the shapes of proteins and their assemblies. Thus, we integrated SAXS profiles into our software for modeling proteins and their assemblies by satisfaction of spatial restraints. Specifically, we modeled the quaternary structures of multidomain proteins with structurally defined rigid domains as well as quaternary structures of binary complexes of structurally defined rigid proteins. In addition to SAXS profiles and the component structures, we used stereochemical restraints and an atomic distance-dependent statistical potential. The scoring function is optimized by a biased Monte Carlo protocol, including quasi-Newton and simulated annealing schemes. The final prediction corresponds to the best scoring solution in the largest cluster of many independently calculated solutions. To quantify how well the quaternary structures are determined based on their SAXS profiles, we used a benchmark of 12 simulated examples as well as an experimental SAXS profile of the homotetramer d-xylose isomerase. Optimization of the SAXS-dependent scoring function generally results in accurate models if sufficiently precise approximations for the constituent rigid bodies are available; otherwise, the best scoring models can have significant errors. Thus, SAXS profiles can play a useful role in the structural characterization of proteins and assemblies if they are combined with additional data and used judiciously. Our integration of a SAXS profile into modeling by satisfaction of spatial restraints will facilitate further integration of different kinds of data for structure determination of proteins and their assemblies.     

Integrating Multimeric Threading With High-throughput Experiments for Structural Interactome of Escherichia coli

Genome-wide protein–protein interaction (PPI) determination remains a significant unsolved problem in structural biology. The difficulty is twofold since high-throughput experiments (HTEs) have often a relatively high false-positive rate in assigning PPIs, and PPI quaternary structures are more difficult to solve than tertiary structures using traditional structural biology techniques. We proposed a uniform pipeline, Threpp, to address both problems. Starting from a pair of monomer sequences, Threpp first threads both sequences through a complex structure library, where the alignment score is combined with HTE data using a naïve Bayesian classifier model to predict the likelihood of two chains to interact with each other. Next, quaternary complex structures of the identified PPIs are constructed by reassembling monomeric alignments with dimeric threading frameworks through interface-specific structural alignments. The pipeline was applied to the Escherichia coli genome and created 35,125 confident PPIs which is 4.5-fold higher than HTE alone. Graphic analyses of the PPI networks show a scale-free cluster size distribution, consistent with previous studies, which was found critical to the robustness of genome evolution and the centrality of functionally important proteins that are essential to E. coli survival. Furthermore, complex structure models were constructed for all predicted E. coli PPIs based on the quaternary threading alignments, where 6771 of them were found to have a high confidence score that corresponds to the correct fold of the complexes with a TM-score >0.5, and 39 showed a close consistency with the later released experimental structures with an average TM-score = 0.73. These results demonstrated the significant usefulness of threading-based homologous modeling in both genome-wide PPI network detection and complex structural construction.     

Robust probabilistic superposition and comparison of protein structures

Background  

Protein structure comparison is a central issue in structural bioinformatics. The standard dissimilarity measure for protein structures is the root mean square deviation (RMSD) of representative atom positions such as α-carbons. To evaluate the RMSD the structures under comparison must be superimposed optimally so as to minimize the RMSD. How to evaluate optimal fits becomes a matter of debate, if the structures contain regions which differ largely - a situation encountered in NMR ensembles and proteins undergoing large-scale conformational transitions.     

Clustering biomolecular complexes by residue contacts similarity

Inaccuracies in computational molecular modeling methods are often counterweighed by brute-force generation of a plethora of putative solutions. These are then typically sieved via structural clustering based on similarity measures such as the root mean square deviation (RMSD) of atomic positions. Albeit widely used, these measures suffer from several theoretical and technical limitations (e.g., choice of regions for fitting) that impair their application in multicomponent systems (N > 2), large-scale studies (e.g., interactomes), and other time-critical scenarios. We present here a simple similarity measure for structural clustering based on atomic contacts--the fraction of common contacts--and compare it with the most used similarity measure of the protein docking community--interface backbone RMSD. We show that this method produces very compact clusters in remarkably short time when applied to a collection of binary and multicomponent protein-protein and protein-DNA complexes. Furthermore, it allows easy clustering of similar conformations of multicomponent symmetrical assemblies in which chain permutations can occur. Simple contact-based metrics should be applicable to other structural biology clustering problems, in particular for time-critical or large-scale endeavors.     

Prediction of the conformation and geometry of loops in globular proteins: testing ArchDB, a structural classification of loops

In protein structure prediction, a central problem is defining the structure of a loop connecting 2 secondary structures. This problem frequently occurs in homology modeling, fold recognition, and in several strategies in ab initio structure prediction. In our previous work, we developed a classification database of structural motifs, ArchDB. The database contains 12,665 clustered loops in 451 structural classes with information about phi-psi angles in the loops and 1492 structural subclasses with the relative locations of the bracing secondary structures. Here we evaluate the extent to which sequence information in the loop database can be used to predict loop structure. Two sequence profiles were used, a HMM profile and a PSSM derived from PSI-BLAST. A jack-knife test was made removing homologous loops using SCOP superfamily definition and predicting afterwards against recalculated profiles that only take into account the sequence information. Two scenarios were considered: (1) prediction of structural class with application in comparative modeling and (2) prediction of structural subclass with application in fold recognition and ab initio. For the first scenario, structural class prediction was made directly over loops with X-ray secondary structure assignment, and if we consider the top 20 classes out of 451 possible classes, the best accuracy of prediction is 78.5%. In the second scenario, structural subclass prediction was made over loops using PSI-PRED (Jones, J Mol Biol 1999;292:195-202) secondary structure prediction to define loop boundaries, and if we take into account the top 20 subclasses out of 1492, the best accuracy is 46.7%. Accuracy of loop prediction was also evaluated by means of RMSD calculations.     

Sequence determinants of quaternary structure in lumazine synthase

Riboflavin, an essential cofactor for all organisms, is biosynthesized in plants, fungi and microorganisms. The penultimate step in the pathway is catalyzed by the enzyme lumazine synthase. One of the most distinctive characteristics of this enzyme is that it is found in different species in two different quaternary structures, pentameric and icosahedral, built from practically the same structural monomeric unit. In fact, the icosahedral structure is best described as a capsid of twelve pentamers. Despite this noticeable difference, the active sites are virtually identical in all structurally studied members. Furthermore, the main regions involved in the catalysis are located at the interface between adjacent subunits in the pentamer. Thus, the two quaternary forms of the enzyme must meet similar structural requirements to achieve their function, but, at the same time, they should differ in the sequence traits responsible for the different quaternary structures observed. Here, we present a combined analysis that includes sequence-structure and evolutionary studies to find the sequence determinants of the different quaternary assemblies of this enzyme. A data set containing 86 sequences of the lumazine synthase family was recovered by sequence similarity searches. Seven of them had resolved three-dimensional structures. A subsequent phylogenetic reconstruction by maximum parsimony (MP) allowed division of the total set into two clusters in accord with their quaternary structure. The comparison between the patterns of three-dimensional contacts derived from the known three-dimensional structures and variation in sequence conservation revealed a significant shift in structural constraints of certain positions. Also, to explore the changes in functional constraints between the two groups, site-specific evolutionary rate shifts were analyzed. We found that the positions involved in icosahedral contacts suffer a larger increase in constraints than the rest. We found eight sequence sites that would be the most important icosahedral sequence determinants. We discuss our results and compare them with previous work. These findings should contribute to refinement of the current structural data, to the design of assays that explore the role of these positions, to the structural characterization of new sequences, and to initiation of a study of the underlying evolutionary mechanisms.     

TASSER-based refinement of NMR structures

The TASSER structure prediction algorithm is employed to investigate whether NMR structures can be moved closer to their corresponding X-ray counterparts by automatic refinement procedures. The benchmark protein dataset includes 61 nonhomologous proteins whose structures have been determined by both NMR and X-ray experiments. Interestingly, by starting from NMR structures, the majority (79%) of TASSER refined models show a structural shift toward their X-ray structures. On average, the TASSER refined models have a root-mean-square-deviation (RMSD) from the X-ray structure of 1.785 A (1.556 A) over the entire chain (aligned region), while the average RMSD between NMR and X-ray structures (RMSD(NMR_X-ray)) is 2.080 A (1.731 A). For all proteins having a RMSD(NMR_X-ray) >2 A, the TASSER refined structures show consistent improvement. However, for the 34 proteins with a RMSD(NMR_X-ray) <2 A, there are only 21 cases (60%) where the TASSER model is closer to the X-ray structure than NMR, which may be due to the inherent resolution of TASSER. We also compare the TASSER models with 12 NMR models in the RECOORD database that have been recalculated recently by Nederveen et al. from original NMR restraints using the newest molecular dynamics tools. In 8 of 12 cases, TASSER models show a smaller RMSD to X-ray structures; in 3 of 12 cases, where RMSD(NMR_X-ray) <1 A, RECOORD does better than TASSER. These results suggest that TASSER can be a useful tool to improve the quality of NMR structures.     

Tertiary structure predictions on a comprehensive benchmark of medium to large size proteins

We evaluate tertiary structure predictions on medium to large size proteins by TASSER, a new algorithm that assembles protein structures through rearranging the rigid fragments from threading templates guided by a reduced Calpha and side-chain based potential consistent with threading based tertiary restraints. Predictions were generated for 745 proteins 201-300 residues in length that cover the Protein Data Bank (PDB) at the level of 35% sequence identity. With homologous proteins excluded, in 365 cases, the templates identified by our threading program, PROSPECTOR_3, have a root-mean-square deviation (RMSD) to native < 6.5 angstroms, with >70% alignment coverage. After TASSER assembly, in 408 cases the best of the top five full-length models has a RMSD < 6.5 angstroms. Among the 745 targets are 18 membrane proteins, with one-third having a predicted RMSD < 5.5 A. For all representative proteins less than or equal to 300 residues that have corresponding multiple NMR structures in the Protein Data Bank, approximately 20% of the models generated by TASSER are closer to the NMR structure centroid than the farthest individual NMR model. These results suggest that reasonable structure predictions for nonhomologous large size proteins can be automatically generated on a proteomic scale, and the application of this approach to structural as well as functional genomics represent promising applications of TASSER.     

Predicting 3D structures of transient protein-protein complexes by homology

The paper reports a homology based approach for predicting the 3D structures of full length hetero protein complexes. We have created a database of templates that includes structures of hetero protein-protein complexes as well as domain-domain structures (), which allowed us to expand the template pool up to 418 two-chain entries (at 40% sequence identity). Two protocols were tested-a protocol based on position specific Blast search (Protocol-I) and a protocol based on structural similarity of monomers (Protocol-II). All possible combinations of two monomers (350,284 pairs) in the ProtCom database were subjected to both protocols to predict if they form complexes. The predictions were benchmarked against the ProtCom database resulting to false-true positives ratios of approximately 5:1 and approximately 7:1 and recovery of 19% and 86%, respectively for protocols I and II. From 350,284 trials Protocol-I made only approximately 500 wrong predictions resulting to 0.5% error. In addition, though it was shown that artificially created domain-domain structures can in principle be good templates for modeling full length protein complexes, more sensitive methods are needed to detect homology relations. The quality of the models was assessed using two different criteria such as interfacial residues and overall RMSD. It was found that there is no correlation between these two measures. In many cases the interface residues were predicted correctly, but the overall RMSD was over 6 A and vice versa.     

RNAMotifScanX: a graph alignment approach for RNA structural motif identification

RNA structural motifs are recurrent three-dimensional (3D) components found in the RNA architecture. These RNA structural motifs play important structural or functional roles and usually exhibit highly conserved 3D geometries and base-interaction patterns. Analysis of the RNA 3D structures and elucidation of their molecular functions heavily rely on efficient and accurate identification of these motifs. However, efficient RNA structural motif search tools are lacking due to the high complexity of these motifs. In this work, we present RNAMotifScanX, a motif search tool based on a base-interaction graph alignment algorithm. This novel algorithm enables automatic identification of both partially and fully matched motif instances. RNAMotifScanX considers noncanonical base-pairing interactions, base-stacking interactions, and sequence conservation of the motifs, which leads to significantly improved sensitivity and specificity as compared with other state-of-the-art search tools. RNAMotifScanX also adopts a carefully designed branch-and-bound technique, which enables ultra-fast search of large kink-turn motifs against a 23S rRNA. The software package RNAMotifScanX is implemented using GNU C++, and is freely available from http://genome.ucf.edu/RNAMotifScanX.     

Families and the structural relatedness among globular proteins.

Protein structures come in families. Are families “closely knit” or “loosely knit” entities? We describe a measure of relatedness among polymer conformations. Based on weighted distance maps, this measure differs from existing measures mainly in two respects: (1) it is computationally fast, and (2) it can compare any two proteins, regardless of their relative chain lengths or degree of similarity. It does not require finding relative alignments. The measure is used here to determine the dissimilarities between all 12, 403 possible pairs of 158 diverse protein structures from the Brookhaven Protein Data Bank (PDB). Combined with minimal spanning trees and hierarchical clustering methods, this measure is used to define structural families. It is also useful for rapidly searching a dataset of protein structures for specific substructural motifs. By using an analogy to distributions of Euclidean distances, we find that protein families are not tightly knit entities.     

Analysis of zinc binding sites in protein crystal structures

The geometrical properties of zinc binding sites in a dataset of high quality protein crystal structures deposited in the Protein Data Bank have been examined to identify important differences between zinc sites that are directly involved in catalysis and those that play a structural role. Coordination angles in the zinc primary coordination sphere are compared with ideal values for each coordination geometry, and zinc coordination distances are compared with those in small zinc complexes from the Cambridge Structural Database as a guide of expected trends. We find that distances and angles in the primary coordination sphere are in general close to the expected (or ideal) values. Deviations occur primarily for oxygen coordinating atoms and are found to be mainly due to H-bonding of the oxygen coordinating ligand to protein residues, bidentate binding arrangements, and multi-zinc sites. We find that H-bonding of oxygen containing residues (or water) to zinc bound histidines is almost universal in our dataset and defines the elec-His-Zn motif. Analysis of the stereochemistry shows that carboxyl elec-His-Zn motifs are geometrically rigid, while water elec-His-Zn motifs show the most geometrical variation. As catalytic motifs have a higher proportion of carboxyl elec atoms than structural motifs, they provide a more rigid framework for zinc binding. This is understood biologically, as a small distortion in the zinc position in an enzyme can have serious consequences on the enzymatic reaction. We also analyze the sequence pattern of the zinc ligands and residues that provide elecs, and identify conserved hydrophobic residues in the endopeptidases that also appear to contribute to stabilizing the catalytic zinc site. A zinc binding template in protein crystal structures is derived from these observations.     

The fragment transformation method to detect the protein structural motifs

To identify functional structural motifs from protein structures of unknown function becomes increasingly important in recent years due to the progress of the structural genomics initiatives. Although certain structural patterns such as the Asp-His-Ser catalytic triad are easy to detect because of their conserved residues and stringently constrained geometry, it is usually more challenging to detect a general structural motifs like, for example, the betabetaalpha-metal binding motif, which has a much more variable conformation and sequence. At present, the identification of these motifs usually relies on manual procedures based on different structure and sequence analysis tools. In this study, we develop a structural alignment algorithm combining both structural and sequence information to identify the local structure motifs. We applied our method to the following examples: the betabetaalpha-metal binding motif and the treble clef motif. The betabetaalpha-metal binding motif plays an important role in nonspecific DNA interactions and cleavage in host defense and apoptosis. The treble clef motif is a zinc-binding motif adaptable to diverse functions such as the binding of nucleic acid and hydrolysis of phosphodiester bonds. Our results are encouraging, indicating that we can effectively identify these structural motifs in an automatic fashion. Our method may provide a useful means for automatic functional annotation through detecting structural motifs associated with particular functions.     

Visualization of conformational distribution of short to medium size segments in globular proteins and identification of local structural motifs

Analysis of the conformational distribution of polypeptide segments in a conformational space is the first step for understanding a principle of structural diversity of proteins. Here, we present a statistical analysis of protein local structures based on interatomic C(alpha) distances. Using principal component analysis (PCA) on the intrasegment C(alpha)-C(alpha) atomic distances, the conformational space of protein segments, which we call the protein segment universe, has been visualized, and three essential coordinate axes, suitable for describing the universe, have been identified. Three essential axes specified radius of gyration, structural symmetry, and separation of hairpin structures from other structures. Among the segments of arbitrary length, 6-22 residues long, the conservation of those axes was uncovered. Further application of PCA to the two largest clusters in the universe revealed local structural motifs. Although some of motifs have already been reported, we identified a possibly novel strand motif. We also showed that a capping box, which is one of the helix capping motifs, was separated into independent subclusters based on the C(alpha) geometry. Implications of the strand motif, which may play a role for protein-protein interaction, are discussed. The currently proposed method is useful for not only mapping the immense universe of protein structures but also identification of structural motifs.     

Flexible protein alignment and hinge detection

Here we present a novel technique for the alignment of flexible proteins. The method does not require an a priori knowledge of the flexible hinge regions. The FlexProt algorithm simultaneously detects the hinge regions and aligns the rigid subparts of the molecules. Our technique is not sensitive to insertions and deletions. Numerous methods have been developed to solve rigid structural comparisons. Unlike FlexProt, all previously developed methods designed to solve the protein flexible alignment require an a priori knowledge of the hinge regions. The FlexProt method is based on 3-D pattern-matching algorithms combined with graph theoretic techniques. The algorithm is highly efficient. For example, it performs a structural comparison of a pair of proteins with 300 amino acids in about 7 s on a 400-MHz desktop PC. We provide experimental results obtained with this algorithm. First, we flexibly align pairs of proteins taken from the database of motions. These are extended by taking additional proteins from the same SCOP family. Next, we present some of the results obtained from exhaustive all-against-all flexible structural comparisons of 1329 SCOP family representatives. Our results include relatively high-scoring flexible structural alignments between the C-terminal merozoite surface protein vs. tissue factor; class II aminoacyl-tRNA synthase, histocompatibility antigen vs. neonatal FC receptor; tyrosine-protein kinase C-SRC vs. haematopoetic cell kinase (HCK); tyrosine-protein kinase C-SRC vs. titine protein (autoinhibited serine kinase domain); and tissue factor vs. hormone-binding protein. These are illustrated and discussed, showing the capabilities of this structural alignment algorithm, which allows un-predefined hinge-based motions.