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.     

Statistics of RNA secondary structures

A statistical reference for RNA secondary structures with minimum free energies is computed by folding large ensembles of random RNA sequences. Four nucleotide alphabets are used: two binary alphabets, AU and GC, the biophysical AUGC and the synthetic GCXK alphabet. RNA secondary structures are made of structural elements, such as stacks, loops, joints, and free ends. Statistical properties of these elements are computed for small RNA molecules of chain lengths up to 100. The results of RNA structure statistics depend strongly on the particular alphabet chosen. The statistical reference is compared with the data derived from natural RNA molecules with similar base frequencies. Secondary structures are represented as trees. Tree editing provides a quantitative measure for the distance d_t, between two structures. We compute a structure density surface as the conditional probability of two structures having distance t given that their sequences have distance h. This surface indicates that the vast majority of possible minimum free energy secondary structures occur within a fairly small neighborhood of any typical (random) sequence. Correlation lengths for secondary structures in their tree representations are computed from probability densities. They are appropriate measures for the complexity of the sequence-structure relation. The correlation length also provides a quantitative estimate for the mean sensitivity of structures to point mutations. © 1993 John Wiley & Sons, Inc.     

Function annotation for pseudoknot using structure similarity

Many raw biological sequence data have been generated by the human genome project and related efforts. The understanding of structural information encoded by biological sequences is important to acquire knowledge of their biochemical functions but remains a fundamental challenge. Recent interest in RNA regulation has resulted in a rapid growth of deposited RNA secondary structures in varied databases. However, a functional classification and characterization of the RNA structure have only been partially addressed. This article aims to introduce a novel interval-based distance metric for structure-based RNA function assignment. The characterization of RNA structures relies on distance vectors learned from a collection of predicted structures. The distance measure considers the intersected, disjoint, and inclusion between intervals. A set of RNA pseudoknotted structures with known function are applied and the function of the query structure is determined by measuring structure similarity. This not only offers sequence distance criteria to measure the similarity of secondary structures but also aids the functional classification of RNA structures with pesudoknots.     

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.     

Energy landscape of k-point mutants of an RNA molecule

MOTIVATION: A k-point mutant of a given RNA sequence s = s(1), ..., s(n) is an RNA sequence s' = s'(1),..., s'(n) obtained by mutating exactly k-positions in s; i.e. Hamming distance between s and s' equals k. To understand the effect of pointwise mutation in RNA, we consider the distribution of energies of all secondary structures of k-point mutants of a given RNA sequence. RESULTS: Here we describe a novel algorithm to compute the mean and standard deviation of energies of all secondary structures of k-point mutants of a given RNA sequence. We then focus on the tail of the energy distribution and compute, using the algorithm AMSAG, the k-superoptimal structure; i.e. the secondary structure of a < or =k-point mutant having least free energy over all secondary structures of all k'-point mutants of a given RNA sequence, for k' < or = k. Evidence is presented that the k-superoptimal secondary structure is often closer, as measured by base pair distance and two additional distance measures, to the secondary structure derived by comparative sequence analysis than that derived by the Zuker minimum free energy structure of the original (wild type or unmutated) RNA.     

Full-length RNA structure prediction of the HIV-1 genome reveals a conserved core domain

A distance constrained secondary structural model of the ≈10 kb RNA genome of the HIV-1 has been predicted but higher-order structures, involving long distance interactions, are currently unknown. We present the first global RNA secondary structure model for the HIV-1 genome, which integrates both comparative structure analysis and information from experimental data in a full-length prediction without distance constraints. Besides recovering known structural elements, we predict several novel structural elements that are conserved in HIV-1 evolution. Our results also indicate that the structure of the HIV-1 genome is highly variable in most regions, with a limited number of stable and conserved RNA secondary structures. Most interesting, a set of long distance interactions form a core organizing structure (COS) that organize the genome into three major structural domains. Despite overlapping protein-coding regions the COS is supported by a particular high frequency of compensatory base changes, suggesting functional importance for this element. This new structural element potentially organizes the whole genome into three major domains protruding from a conserved core structure with potential roles in replication and evolution for the virus.     

Comparative sequence analysis and patterns of covariation in RNA secondary structures

A novel method of RNA secondary structure prediction based on a comparison of nucleotide sequences is described. This method correctly predicts nearly all evolutionarily conserved secondary structures of five different RNAs: tRNA, 5S rRNA, bacterial ribonuclease P (RNase P) RNA, eukaryotic small subunit rRNA, and the 3' untranslated region (UTR) of the Drosophila bicoid (bcd) mRNA. Furthermore, covariations occurring in the helices of these conserved RNA structures are analyzed. Two physical parameters are found to be important determinants of the evolution of compensatory mutations: the length of a helix and the distance between base-pairing nucleotides. For the helices of bcd 3' UTR mRNA and RNase P RNA, a positive correlation between the rate of compensatory evolution and helix length is found. The analysis of Drosophila bcd 3' UTR mRNA further revealed that the rate of compensatory evolution decreases with the physical distance between base-pairing residues. This result is in qualitative agreement with Kimura's model of compensatory fitness interactions, which assumes that mutations occurring in RNA helices are individually deleterious but become neutral in appropriate combinations.     

A general edit distance between RNA structures.

Arc-annotated sequences are useful in representing the structural information of RNA sequences. In general, RNA secondary and tertiary structures can be represented as a set of nested arcs and a set of crossing arcs, respectively. Since RNA functions are largely determined by molecular confirmation and therefore secondary and tertiary structures, the comparison between RNA secondary and tertiary structures has received much attention recently. In this paper, we propose the notion of edit distance to measure the similarity between two RNA secondary and tertiary structures, by incorporating various edit operations performed on both bases and arcs (i.e., base-pairs). Several algorithms are presented to compute the edit distance between two RNA sequences with various arc structures and under various score schemes, either exactly or approximately, with provably good performance. Preliminary experimental tests confirm that our definition of edit distance and the computation model are among the most reasonable ones ever studied in the literature.     

Topological structure of the space of phenotypes: the case of RNA neutral networks

The evolution and adaptation of molecular populations is constrained by the diversity accessible through mutational processes. RNA is a paradigmatic example of biopolymer where genotype (sequence) and phenotype (approximated by the secondary structure fold) are identified in a single molecule. The extreme redundancy of the genotype-phenotype map leads to large ensembles of RNA sequences that fold into the same secondary structure and can be connected through single-point mutations. These ensembles define neutral networks of phenotypes in sequence space. Here we analyze the topological properties of neutral networks formed by 12-nucleotides RNA sequences, obtained through the exhaustive folding of sequence space. A total of 4(12) sequences fragments into 645 subnetworks that correspond to 57 different secondary structures. The topological analysis reveals that each subnetwork is far from being random: it has a degree distribution with a well-defined average and a small dispersion, a high clustering coefficient, and an average shortest path between nodes close to its minimum possible value, i.e. the Hamming distance between sequences. RNA neutral networks are assortative due to the correlation in the composition of neighboring sequences, a feature that together with the symmetries inherent to the folding process explains the existence of communities. Several topological relationships can be analytically derived attending to structural restrictions and generic properties of the folding process. The average degree of these phenotypic networks grows logarithmically with their size, such that abundant phenotypes have the additional advantage of being more robust to mutations. This property prevents fragmentation of neutral networks and thus enhances the navigability of sequence space. In summary, RNA neutral networks show unique topological properties, unknown to other networks previously described.     

Nucleotide sequence, secondary structure and evolution of the 5S ribosomal RNA from five bacterial species

The nucleotide sequences of the 5S ribosomal RNAs of the bacteria Agrobacterium tumefaciens, Alcaligenes faecalis, Pseudomonas cepacia, Aquaspirillum serpens and Acinetobacter calcoaceticus have been determined. The sequences fit in a generally accepted model for 5S RNA secondary structure. However, a closer comparative examination of these and other bacterial 5S RNA primary structures reveals the potential of additional base pairing and of multiple equilibria between a set of slightly different alternative secondary structures in one area of the molecule. The phylogenetic position of the examined bacteria is derived from a 5S RNA sequence alignment by a clustering method and compared with the position derived on the basis of 16S ribosomal RNA oligonucleotide catalogs.     

A Comparison Study on Similarity and Dissimilarity Measures in Clustering Continuous Data

Similarity or distance measures are core components used by distance-based clustering algorithms to cluster similar data points into the same clusters, while dissimilar or distant data points are placed into different clusters. The performance of similarity measures is mostly addressed in two or three-dimensional spaces, beyond which, to the best of our knowledge, there is no empirical study that has revealed the behavior of similarity measures when dealing with high-dimensional datasets. To fill this gap, a technical framework is proposed in this study to analyze, compare and benchmark the influence of different similarity measures on the results of distance-based clustering algorithms. For reproducibility purposes, fifteen publicly available datasets were used for this study, and consequently, future distance measures can be evaluated and compared with the results of the measures discussed in this work. These datasets were classified as low and high-dimensional categories to study the performance of each measure against each category. This research should help the research community to identify suitable distance measures for datasets and also to facilitate a comparison and evaluation of the newly proposed similarity or distance measures with traditional ones.     

Boltzmann ensemble features of RNA secondary structures: a comparative analysis of biological RNA sequences and random shuffles

Ensemble-based approaches to RNA secondary structure prediction have become increasingly appreciated in recent years. Here, we utilize sampling and clustering of the Boltzmann ensemble of RNA secondary structures to investigate whether biological sequences exhibit ensemble features that are distinct from their random shuffles. Representative messenger RNAs (mRNAs), structural RNAs, and precursor microRNAs (miRNAs) are analyzed for nine ensemble features. These include structure clustering features, the energy gap between the minimum free energy (MFE) and the ensemble, the numbers of high-frequency base pairs in the ensemble and in clusters, the average base-pair distance between the MFE structure and the ensemble, and between-cluster and within-cluster sums of squares. For each of the features, we observe a lack of significant distinction between mRNAs and their random shuffles. For five features, significant differences are found between structural RNAs and random counterparts. For seven features including the five for structural RNAs, much greater differences are observed between precursor miRNAs and random shuffles. These findings reveal differences in the Boltzmann structure ensemble among different types of functional RNAs. In addition, for two ensemble features, we observe distinctive, non-overlapping distributions for precursor miRNAs and random shuffles. A distributional separation can be particularly useful for the prediction of miRNA genes.     

Expected distance between terminal nucleotides of RNA secondary structures

In "The ends of a large RNA molecule are necessarily close", Yoffe et al. (Nucleic Acids Res 39(1):292-299, 2011) used the programs RNAfold [resp. RNAsubopt] from Vienna RNA Package to calculate the distance between 5' and 3' ends of the minimum free energy secondary structure [resp. thermal equilibrium structures] of viral and random RNA sequences. Here, the 5'-3' distance is defined to be the length of the shortest path from 5' node to 3' node in the undirected graph, whose edge set consists of edges {i, i + 1} corresponding to covalent backbone bonds and of edges {i, j} corresponding to canonical base pairs. From repeated simulations and using a heuristic theoretical argument, Yoffe et al. conclude that the 5'-3' distance is less than a fixed constant, independent of RNA sequence length. In this paper, we provide a rigorous, mathematical framework to study the expected distance from 5' to 3' ends of an RNA sequence. We present recurrence relations that precisely define the expected distance from 5' to 3' ends of an RNA sequence, both for the Turner nearest neighbor energy model, as well as for a simple homopolymer model first defined by Stein and Waterman. We implement dynamic programming algorithms to compute (rather than approximate by repeated application of Vienna RNA Package) the expected distance between 5' and 3' ends of a given RNA sequence, with respect to the Turner energy model. Using methods of analytical combinatorics, that depend on complex analysis, we prove that the asymptotic expected 5'-3' distance of length n homopolymers is approximately equal to the constant 5.47211, while the asymptotic distance is 6.771096 if hairpins have a minimum of 3 unpaired bases and the probability that any two positions can form a base pair is 1/4. Finally, we analyze the 5'-3' distance for secondary structures from the STRAND database, and conclude that the 5'-3' distance is correlated with RNA sequence length.     

A clique-based method for the edit distance between unordered trees and its application to analysis of glycan structures

Background

Measuring similarities between tree structured data is important for analysis of RNA secondary structures, phylogenetic trees, glycan structures, and vascular trees. The edit distance is one of the most widely used measures for comparison of tree structured data. However, it is known that computation of the edit distance for rooted unordered trees is NP-hard. Furthermore, there is almost no available software tool that can compute the exact edit distance for unordered trees.

Results

In this paper, we present a practical method for computing the edit distance between rooted unordered trees. In this method, the edit distance problem for unordered trees is transformed into the maximum clique problem and then efficient solvers for the maximum clique problem are applied. We applied the proposed method to similar structure search for glycan structures. The result suggests that our proposed method can efficiently compute the edit distance for moderate size unordered trees. It also suggests that the proposed method has the accuracy comparative to those by the edit distance for ordered trees and by an existing method for glycan search.

Conclusions

The proposed method is simple but useful for computation of the edit distance between unordered trees. The object code is available upon request.

     

Use of linear regression model to compare RNA secondary structures

With more and more ribonucleic acid (RNA) secondary structures accumulated, the need for comparing different RNA secondary structures often arises in function prediction and evolutionary analysis. Numerous efficient algorithms were developed for comparing different RNA secondary structures, but challenges remain. In this paper, six new models based on the linear regression model were proposed for the comparison of RNA secondary structures. The proposed models were tested on a mixed data, containing six secondary structures from RNase P RNAs, three secondary structures from SSU rRNA and five secondary structures from 16S ribosomal RNAs. The results have shown the effectiveness of the proposed models. Moreover, the time complexity of our models is favorable by comparing with that of the existing methods which solve the similar problem.     

RNAdigest: A Web-Based Tool for the Analysis and Prediction of Structure - Specific RNAse Digestion Results

Despite recent developments in analyzing RNA secondary structures, relatively few RNA structures have been determined. To date, many investigators have relied on the traditional method of using structure-specific RNAse enzymes to probe RNA secondary structures. However, if these data were combined with novel computational approaches, investigators would have an informative and valuable tool for RNA structural analysis. To this end, we created the web server “RNAdigest.” RNAdigest uses mfold RNA structural models in order to predict the results of RNAse digestion experiments. Furthermore, RNAdigest also utilizes both RNA sequence and the experimental digestion patterns to formulate the constraints for predicting secondary structures of the RNA. Thus, RNAdigest allows for the structural interpretation of RNAse digestion experiments. Overall, RNAdigest simplifies RNAse digestion result analyses while allowing for the identification of unique fragments. These unique fragments can then be used for testing predicted mfold structures and for designing structural-specific DNA/RNA probes.     

Mutational patterns in RNA secondary structure evolution examined in three RNA families

The goal of this work was to study mutational patterns in the evolution of RNA secondary structure. We analyzed bacterial tmRNA, RNaseP and eukaryotic telomerase RNA secondary structures, mapping structural variability onto phylogenetic trees constructed primarily from rRNA sequences. We found that secondary structures evolve both by whole stem insertion/deletion, and by mutations that create or disrupt stem base pairing. We analyzed the evolution of stem lengths and constructed substitution matrices describing the changes responsible for the variation in the RNA stem length. In addition, we used principal component analysis of the stem length data to determine the most variable stems in different families of RNA. This data provides new insights into the evolution of RNA secondary structures and patterns of variation in the lengths of double helical regions of RNA molecules. Our findings will facilitate design of improved mutational models for RNA structure evolution.     

Transient RNA structure features are evolutionarily conserved and can be computationally predicted

Functional RNA structures tend to be conserved during evolution. This finding is, for example, exploited by comparative methods for RNA secondary structure prediction that currently provide the state-of-art in terms of prediction accuracy. We here provide strong evidence that homologous RNA genes not only fold into similar final RNA structures, but that their folding pathways also share common transient structural features that have been evolutionarily conserved. For this, we compile and investigate a non-redundant data set of 32 sequences with known transient and final RNA secondary structures and devise a dedicated computational analysis pipeline.