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.     

Population model-based inter-diplotype similarity measure for accurate diplotype clustering

Classification of the individuals' genotype data is important in various kinds of biomedical research. There are many sophisticated clustering algorithms, but most of them require some appropriate similarity measure between objects to be clustered. Hence, accurate inter-diplotype similarity measures are always required for classification of diplotypes. In this article, we propose a new accurate inter-diplotype similarity measure that we call the population model-based distance (PMD), so that we can cluster individuals with diplotype SNPs data (i.e., unphased-diplotypes) with higher accuracies. For unphased-diplotypes, the allele sharing distance (ASD) has been the standard to measure the genetic distance between the diplotypes of individuals. To achieve higher clustering accuracies, our new measure PMD makes good use of a given appropriate population model which has never been utilized in the ASD. As the population model, we propose to use an hidden Markov model (HMM)-based model. We call the PMD based on the model the HHD (HIT HMM-based Distance). We demonstrate the impact of the HHD on the diplotype classification through comprehensive large-scale experiments over the genome-wide 8930 data sets derived from the HapMap SNPs database. The experiments revealed that the HHD enables significantly more accurate clustering than the ASD.     

Characterizing molecular flexibility by combining least root mean square deviation measures

The root mean square deviation (RMSD) and the least RMSD are two widely used similarity measures in structural bioinformatics. Yet, they stem from global comparisons, possibly obliterating locally conserved motifs. We correct these limitations with the so-called combined RMSD, which mixes independent lRMSD measures, each computed with its own rigid motion. The combined RMSD is relevant in two main scenarios, namely to compare (quaternary) structures based on motifs defined from the sequence (domains and SSE) and to compare structures based on structural motifs yielded by local structural alignment methods. We illustrate the benefits of combined RMSD over the usual RMSD on three problems, namely (a) the assignment of quaternary structures for hemoglobin (scenario #1), (b) the calculation of structural phylogenies (case study: class II fusion proteins; scenario #1), and (c) the analysis of conformational changes based on combined RMSD of rigid structural motifs (case study: one class II fusion protein; scenario #2). Based on these illustrations, we argue that the combined RMSD is a tool of choice to perform positive and negative discrimination of degree of freedom, with applications to the design of move sets and collective coordinates. Executables to compute combined RMSD are available within the Structural Bioinformatics Library ( http://sbl.inria.fr ).     

Conservation of substructures in proteins: interfaces of secondary structural elements in proteasomal subunits

It is observed that during divergent evolution of two proteins with a common phylogenetic origin, the structural similarity of their backbones is often preserved even when the sequence similarity between them decreases to a virtually undetectable level. Here we analyzed, whether the conservation of structure along evolution involves also the local atomic structures in the interfaces between secondary structural elements. We have used as study case one protein family, the proteasomal subunits, for which 17 crystal structures are known. These include 14 different subunits of Saccharomyces cerevisiae, 2 subunits of Thermoplasma acidophilum and one subunit of Escherichia coli. The structural core of the 17 proteasomal subunits has 23 secondary structural elements. Any two adjacent secondary structural elements form a molecular interface consisting of two molecular patches. We found 61 interfaces that occurred in all 17 subunits. The 3D shape of equivalent molecular patches from different proteasomal subunits were compared by superposition. Our results demonstrate that pairs of equivalent molecular patches show an RMSD which is lower than that of randomly chosen patches from unrelated proteins. This is true even when patch comparisons with identical residues were excluded from the analysis. Furthermore it is known that the sequential dissimilarity is correlated to the RMSD between the backbones of the members of protein families. The question arises whether this is also true for local atomic structures. The results show that the correlation of individual patch RMSD values and local sequence dissimilarities is low and has a wide range from 0 to 0.41, however, it is surprising that there is a good correlation between the average RMSD of all corresponding patches and the global sequence dissimilarity. This average patch RMSD correlates slightly stronger than the C(alpha)-trace RMSD to the global sequence dissimilarity.     

Protein structure similarity from principle component correlation analysis

Background  

Owing to rapid expansion of protein structure databases in recent years, methods of structure comparison are becoming increasingly effective and important in revealing novel information on functional properties of proteins and their roles in the grand scheme of evolutionary biology. Currently, the structural similarity between two proteins is measured by the root-mean-square-deviation (RMSD) in their best-superimposed atomic coordinates. RMSD is the golden rule of measuring structural similarity when the structures are nearly identical; it, however, fails to detect the higher order topological similarities in proteins evolved into different shapes. We propose new algorithms for extracting geometrical invariants of proteins that can be effectively used to identify homologous protein structures or topologies in order to quantify both close and remote structural similarities.     

Information Content-Based Gene Ontology Functional Similarity Measures: Which One to Use for a Given Biological Data Type?

The current increase in Gene Ontology (GO) annotations of proteins in the existing genome databases and their use in different analyses have fostered the improvement of several biomedical and biological applications. To integrate this functional data into different analyses, several protein functional similarity measures based on GO term information content (IC) have been proposed and evaluated, especially in the context of annotation-based measures. In the case of topology-based measures, each approach was set with a specific functional similarity measure depending on its conception and applications for which it was designed. However, it is not clear whether a specific functional similarity measure associated with a given approach is the most appropriate, given a biological data set or an application, i.e., achieving the best performance compared to other functional similarity measures for the biological application under consideration. We show that, in general, a specific functional similarity measure often used with a given term IC or term semantic similarity approach is not always the best for different biological data and applications. We have conducted a performance evaluation of a number of different functional similarity measures using different types of biological data in order to infer the best functional similarity measure for each different term IC and semantic similarity approach. The comparisons of different protein functional similarity measures should help researchers choose the most appropriate measure for the biological application under consideration.     

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.     

The relationship between sequence and interaction divergence in proteins

There is currently a gap in knowledge between complexes of known three-dimensional structure and those known from other experimental methods such as affinity purifications or the two-hybrid system. This gap can sometimes be bridged by methods that extrapolate interaction information from one complex structure to homologues of the interacting proteins. To do this, it is important to know if and when proteins of the same type (e.g. family, superfamily or fold) interact in the same way. Here, we study interactions of known structure to address this question. We found all instances within the structural classification of proteins database of the same domain pairs interacting in different complexes, and then compared them with a simple measure (interaction RMSD). When plotted against sequence similarity we find that close homologues (30-40% or higher sequence identity) almost invariably interact the same way. Conversely, similarity only in fold (i.e. without additional evidence for a common ancestor) is only rarely associated with a similarity in interaction. The results suggest that there is a twilight zone of sequence similarity where it is not possible to say whether or not domains will interact similarly. We also discuss the rare instances of fold similarities interacting the same way, and those where obviously homologous proteins interact differently.     

Clustering algorithms for identifying core atom sets and for assessing the precision of protein structure ensembles

An important open question in the field of NMR-based biomolecular structure determination is how best to characterize the precision of the resulting ensemble of structures. Typically, the RMSD, as minimized in superimposing the ensemble of structures, is the preferred measure of precision. However, the presence of poorly determined atomic coordinates and multiple "RMSD-stable domains"--locally well-defined regions that are not aligned in global superimpositions--complicate RMSD calculations. In this paper, we present a method, based on a novel, structurally defined order parameter, for identifying a set of core atoms to use in determining superimpositions for RMSD calculations. In addition we present a method for deciding whether to partition that core atom set into "RMSD-stable domains" and, if so, how to determine partitioning of the core atom set. We demonstrate our algorithm and its application in calculating statistically sound RMSD values by applying it to a set of NMR-derived structural ensembles, superimposing each RMSD-stable domain (or the entire core atom set, where appropriate) found in each protein structure under consideration. A parameter calculated by our algorithm using a novel, kurtosis-based criterion, the epsilon-value, is a measure of precision of the superimposition that complements the RMSD. In addition, we compare our algorithm with previously described algorithms for determining core atom sets. The methods presented in this paper for biomolecular structure superimposition are quite general, and have application in many areas of structural bioinformatics and structural biology.     

Molecular mimicry: quantitative methods to study structural similarity between protein and RNA

With rapidly increasing availability of three-dimensional structures, one major challenge for the post-genome era is to infer the functions of biological molecules based on their structural similarity. While quantitative studies of structural similarity between the same type of biological molecules (e.g., protein vs. protein) have been carried out intensively, the comparable study of structural similarity between different types of biological molecules (e.g., protein vs. RNA) remains unexplored. Here we have developed a new bioinformatics approach to quantitatively study the structural similarity between two different types of biopolymers--proteins and RNA--based on the spatial distribution of conserved elements. We applied it to two previously proposed tRNA-protein mimicry pairs whose functional relatedness between two molecules has been recently determined experimentally. Our method detected the biologically meaningful signals, which are consistent with experimental evidence.     

The Poisson Index: a new probabilistic model for protein ligand binding site similarity

MOTIVATION: The large-scale comparison of protein-ligand binding sites is problematic, in that measures of structural similarity are difficult to quantify and are not easily understood in terms of statistical similarity that can ultimately be related to structure and function. We present a binding site matching score the Poisson Index (PI) based upon a well-defined statistical model. PI requires only the number of matching atoms between two sites and the size of the two sites-the same information used by the Tanimoto Index (TI), a comparable and widely used measure for molecular similarity. We apply PI and TI to a previously automatically extracted set of binding sites to determine the robustness and usefulness of both scores. RESULTS: We found that PI outperforms TI; moreover, site similarity is poorly defined for TI at values around the 99.5% confidence level for which PI is well defined. A difference map at this confidence level shows that PI gives much more meaningful information than TI. We show individual examples where TI fails to distinguish either a false or a true site paring in contrast to PI, which performs much better. TI cannot handle large or small sites very well, or the comparison of large and small sites, in contrast to PI that is shown to be much more robust. Despite the difficulty of determining a biological 'ground truth' for binding site similarity we conclude that PI is a suitable measure of binding site similarity and could form the basis for a binding site classification scheme comparable to existing protein domain classification schema.     

SIDEseq: A Cell Similarity Measure Defined by Shared Identified Differentially Expressed Genes for Single-Cell RNA sequencing Data

One goal of single-cell RNA sequencing (scRNA seq) is to expose possible heterogeneity within cell populations due to meaningful, biological variation. Examining cell-to-cell heterogeneity, and further, identifying subpopulations of cells based on scRNA seq data has been of common interest in life science research. A key component to successfully identifying cell subpopulations (or clustering cells) is the (dis)similarity measure used to group the cells. In this paper, we introduce a novel measure, named SIDEseq, to assess cell-to-cell similarity using scRNA seq data. SIDEseq first identifies a list of putative differentially expressed (DE) genes for each pair of cells. SIDEseq then integrates the information from all the DE gene lists (corresponding to all pairs of cells) to build a similarity measure between two cells. SIDEseq can be implemented in any clustering algorithm that requires a (dis)similarity matrix. This new measure incorporates information from all cells when evaluating the similarity between any two cells, a characteristic not commonly found in existing (dis)similarity measures. This property is advantageous for two reasons: (a) borrowing information from cells of different subpopulations allows for the investigation of pairwise cell relationships from a global perspective and (b) information from other cells of the same subpopulation could help to ensure a robust relationship assessment. We applied SIDEseq to a newly generated human ovarian cancer scRNA seq dataset, a public human embryo scRNA seq dataset, and several simulated datasets. The clustering results suggest that the SIDEseq measure is capable of uncovering important relationships between cells, and outperforms or at least does as well as several popular (dis)similarity measures when used on these datasets.     

Combining functional and topological properties to identify core modules in protein interaction networks

Advances in large-scale technologies in proteomics, such as yeast two-hybrid screening and mass spectrometry, have made it possible to generate large Protein Interaction Networks (PINs). Recent methods for identifying dense sub-graphs in such networks have been based solely on graph theoretic properties. Therefore, there is a need for an approach that will allow us to combine domain-specific knowledge with topological properties to generate functionally relevant sub-graphs from large networks. This article describes two alternative network measures for analysis of PINs, which combine functional information with topological properties of the networks. These measures, called weighted clustering coefficient and weighted average nearest-neighbors degree, use weights representing the strengths of interactions between the proteins, calculated according to their semantic similarity, which is based on the Gene Ontology terms of the proteins. We perform a global analysis of the yeast PIN by systematically comparing the weighted measures with their topological counterparts. To show the usefulness of the weighted measures, we develop an algorithm for identification of functional modules, called SWEMODE (Semantic WEights for MODule Elucidation), that identifies dense sub-graphs containing functionally similar proteins. The proposed method is based on the ranking of nodes, i.e., proteins, according to their weighted neighborhood cohesiveness. The highest ranked nodes are considered as seeds for candidate modules. The algorithm then iterates through the neighborhood of each seed protein, to identify densely connected proteins with high functional similarity, according to the chosen parameters. Using a yeast two-hybrid data set of experimentally determined protein-protein interactions, we demonstrate that SWEMODE is able to identify dense clusters containing proteins that are functionally similar. Many of the identified modules correspond to known complexes or subunits of these complexes.     

A large data set comparison of protein structures determined by crystallography and NMR: statistical test for structural differences and the effect of crystal packing

The existence of a large number of proteins for which both nuclear magnetic resonance (NMR) and X-ray crystallographic coordinates have been deposited into the Protein Data Bank (PDB) makes the statistical comparison of the corresponding crystal and NMR structural models over a large data set possible, and facilitates the study of the effect of the crystal environment and other factors on structure. We present an approach for detecting statistically significant structural differences between crystal and NMR structural models which is based on structural superposition and the analysis of the distributions of atomic positions relative to a mean structure. We apply this to a set of 148 protein structure pairs (crystal vs NMR), and analyze the results in terms of methodological and physical sources of structural difference. For every one of the 148 structure pairs, the backbone root-mean-square distance (RMSD) over core atoms of the crystal structure to the mean NMR structure is larger than the average RMSD of the members of the NMR ensemble to the mean, with 76% of the structure pairs having an RMSD of the crystal structure to the mean more than a factor of two larger than the average RMSD of the NMR ensemble. On average, the backbone RMSD over core atoms of crystal structure to the mean NMR is approximately 1 A. If non-core atoms are included, this increases to 1.4 A due to the presence of variability in loops and similar regions of the protein. The observed structural differences are only weakly correlated with the age and quality of the structural model and differences in conditions under which the models were determined. We examine steric clashes when a putative crystalline lattice is constructed using a representative NMR structure, and find that repulsive crystal packing plays a minor role in the observed differences between crystal and NMR structures. The observed structural differences likely have a combination of physical and methodological causes. Stabilizing attractive interactions arising from intermolecular crystal contacts which shift the equilibrium of the crystal structure relative to the NMR structure is a likely physical source which can account for some of the observed differences. Methodological sources of apparent structural difference include insufficient sampling or other issues which could give rise to errors in the estimates of the precision and/or accuracy.     

Evaluating Protein Similarity from Coarse Structures

To unscramble the relationship between protein function and protein structure, it is essential to assess the protein similarity from different aspects. Although many methods have been proposed for protein structure alignment or comparison, alternative similarity measures are still strongly demanded due to the requirement of fast screening and query in large-scale structure databases. In this paper, we first formulate a novel representation of a protein structure, i.e., Feature Sequence of Surface (FSS). Then, a new score scheme is developed to measure the similarity between two representations. To verify the proposed method, numerical experiments are conducted in four different protein data sets. We also classify SARS coronavirus to verify the effectiveness of the new method. Furthermore, preliminary results of fast classification of the whole CATH v2.5.1 database based on the new macrostructure similarity are given as a pilot study. We demonstrate that the proposed approach to measure the similarities between protein structures is simple to implement, computationally efficient, and surprisingly fast. In addition, the method itself provides a new and quantitative tool to view a protein structure.     

Protein homologous cores and loops: important clues to evolutionary relationships between structurally similar proteins

Background  

To discover remote evolutionary relationships and functional similarities between proteins, biologists rely on comparative sequence analysis, and when structures are available, on structural alignments and various measures of structural similarity. The measures/scores that have most commonly been used for this purpose include: alignment length, percent sequence identity, superposition RMSD and their different combinations. More recently, we have introduced the "Homologous core structure overlap score" (HCS) and the "Loop Hausdorff Measure" (LHM). Along with these we also consider the "gapped structural alignment score" (GSAS), which was introduced earlier by other researchers.