Prediction of consensus RNA secondary structures including pseudoknots

Most functional RNA molecules have characteristic structures that are highly conserved in evolution. Many of them contain pseudoknots. Here, we present a method for computing the consensus structures including pseudoknots based on alignments of a few sequences. The algorithm combines thermodynamic and covariation information to assign scores to all possible base pairs, the base pairs are chosen with the help of the maximum weighted matching algorithm. We applied our algorithm to a number of different types of RNA known to contain pseudoknots. All pseudoknots were predicted correctly and more than 85 percent of the base pairs were identified.     

Prediction of alternative RNA secondary structures based on fluctuating thermodynamic parameters.

In this paper we present a new method for predicting a set of RNA secondary structures that are thermodynamically favored in RNA folding simulations. This method uses a large number of 'simulated energy rules' (SER) generated by perturbing the free energy parameters derived experimentally within the range of the experimental errors. The structure with the lowest free energy is computed for each SER. Structural comparisons are used to avoid multiple generation of similar structures. Computed structures are evaluated using the energy distribution of the lowest free energy structures derived in the simulation. Predicted be graphically displayed with their occurring frequencies in the simulation by dot-plot representations. On average, about 90% of phylogenetic helixes in the known models of tRNA, Group I self-splicing intron, and Escherichia coli 16 S rRNA, were predicted using the method.     

Modeling RNA secondary structures. I. Mathematical structural model for predicting RNA secondary structures.

A mathematical model for analyzing the secondary structures of RNA is developed that is based on the connection matrix associated with the planar p-h graph. The classification of the elementary structures allows the introduction of the basis of structural space from which to build the global secondary structure. All admissible solutions belong to the configuration space and can be obtained directly from its basis.     

Prediction of locally stable RNA secondary structures for genome-wide surveys

MOTIVATION: Recently novel classes of functional RNAs, most prominently the miRNAs have been discovered, strongly suggesting that further types of functional RNAs are still hidden in the recently completed genomic DNA sequences. Only few techniques are known, however, to survey genomes for such RNA genes. When sufficiently similar sequences are not available for comparative approaches the only known remedy is to search directly for structural features. RESULTS: We present here efficient algorithms for computing locally stable RNA structures at genome-wide scales. Both the minimum energy structure and the complete matrix of base pairing probabilities can be computed in theta(N x L2) time and theta(N + L2) memory in terms of the length N of the genome and the size L of the largest secondary structure motifs of interest. In practice, the 100 Mb of the complete genome of Caenorhabditis elegans can be folded within about half a day on a modern PC with a search depth of L = 100. This is sufficient example for a survey for miRNAs. AVAILABILITY: The software described in this contribution will be available for download at http://www.tbi.univie.ac.at/~ivo/RNA/ as part of the Vienna RNA Package.     

Metrics on RNA secondary structures.

Many different programs have been developed for the prediction of the secondary structure of an RNA sequence. Some of these programs generate an ensemble of structures, all of which have free energy close to that of the optimal structure, making it important to be able to quantify how similar these different structures are. To deal with this problem, we define a new class of metrics, the mountain metrics, on the set of RNA secondary structures of a fixed length. We compare properties of these metrics with other well known metrics on RNA secondary structures. We also study some global and local properties of these metrics.     

Combinatorial properties of RNA secondary structures.

The secondary structure of an RNA molecule is of great importance and possesses influence, e.g., on the interaction of tRNA molecules with proteins or on the stabilization of mRNA molecules. The classification of secondary structures by means of their order proved useful with respect to numerous applications. In 1978, Waterman, who gave the first precise formal framework for the topic, suggested to determine the number a(n,p) of secondary structures of size n and given order p. Since then, no satisfactory result has been found. Based on an observation due to Viennot et al., we will derive generating functions for the secondary structures of order p from generating functions for binary tree structures with Horton-Strahler number p. These generating functions enable us to compute a precise asymptotic equivalent for a(n,p). Furthermore, we will determine the related number of structures when the number of unpaired bases shows up as an additional parameter. Our approach proves to be general enough to compute the average order of a secondary structure together with all the r-th moments and to enumerate substructures such as hairpins or bulges in dependence on the order of the secondary structures considered.     

Computer-aided prediction of RNA secondary structures.

A brief survey of computer algorithms that have been developed to generate predictions of the secondary structures of RNA molecules is presented. Two particular methods are described in some detail. The first utilizes a thermodynamic energy minimization algorithm that takes into account the likelihood that short-range folding tends to be favored over long-range interactions. The second utilizes an interactive computer graphic modelling algorithm that enables the user to consider thermodynamic criteria as well as structural data obtained by nuclease susceptibility, chemical reactivity and phylogenetic studies. Examples of structures for prokaryotic 16S and 23S ribosomal RNAs, several eukaryotic 5S ribosomal RNAs and rabbit beta-globin messenger RNA are presented as case studies in order to describe the two techniques. Anm argument is made for integrating the two approaches presented in this paper, enabling the user to generate proposed structures using thermodynamic criteria, allowing interactive refinement of these structures through the application of experimentally derived data.     

Combinatorics of saturated secondary structures of RNA.

Following Zuker (1986), a saturated secondary structure for a given RNA sequence is a secondary structure such that no base pair can be added without violating the definition of secondary structure, e.g., without introducing a pseudoknot. In the Nussinov-Jacobson energy model (Nussinov and Jacobson, 1980), where the energy of a secondary structure is -1 times the number of base pairs, saturated secondary structures are local minima in the energy landscape, hence form kinetic traps during the folding process. Here we present recurrence relations and closed form asymptotic limits for combinatorial problems related to the number of saturated secondary structures. In addition, Python source code to compute the number of saturated secondary structures having k base pairs can be found at the web servers link of bioinformatics.bc.edu/clotelab/.     

An algebraic representation of RNA secondary structures.

This paper develops mathematical methods for describing and analyzing RNA secondary structures. It was motivated by the need to develop rigorous yet efficient methods to treat transitions from one secondary structure to another, which we propose here may occur as motions of loops within RNAs having appropriate sequences. In this approach a molecular sequence is described as a vector of the appropriate length. The concept of symmetries between nucleic acid sequences is developed, and the 48 possible different types of symmetries are described. Each secondary structure possible for a particular nucleotide sequence determines a symmetric, signed permutation matrix. The collection of all possible secondary structures is comprised of all matrices of this type whose left multiplication with the sequence vector leaves that vector unchanged. A transition between two secondary structures is given by the product of the two corresponding structure matrices. This formalism provides an efficient method for describing nucleic acid sequences that allows questions relating to secondary structures and transitions to be addressed using the powerful methods of abstract algebra. In particular, it facilitates the determination of possible secondary structures, including those containing pseudoknots. Although this paper concentrates on RNA structure, this formalism also can be applied to DNA.     

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.     

A method for rapid similarity analysis of RNA secondary structures

Background  

Owing to the rapid expansion of RNA structure databases in recent years, efficient methods for structure comparison are in demand for function prediction and evolutionary analysis. Usually, the similarity of RNA secondary structures is evaluated based on tree models and dynamic programming algorithms. We present here a new method for the similarity analysis of RNA secondary structures.     

Detection of common motifs in RNA secondary structures.

We describe a novel computerized system for comparison of RNA secondary structures and demonstrate its use for experimental studies. The system is able to screen a very large number of structures, to cluster similar structures and to detect specific structural motifs. In particular, the system is useful for detecting mutations with specific structural effects among all possible point mutations, and for predicting compensatory mutations that will restore the wild type structure. The algorithms are independent of the folding rules that are used to generate the secondary structures.     

Elastic network models capture the motions apparent within ensembles of RNA structures

The role of structure and dynamics in mechanisms for RNA becomes increasingly important. Computational approaches using simple dynamics models have been successful at predicting the motions of proteins and are often applied to ribonucleo-protein complexes but have not been thoroughly tested for well-packed nucleic acid structures. In order to characterize a true set of motions, we investigate the apparent motions from 16 ensembles of experimentally determined RNA structures. These indicate a relatively limited set of motions that are captured by a small set of principal components (PCs). These limited motions closely resemble the motions computed from low frequency normal modes from elastic network models (ENMs), either at atomic or coarse-grained resolution. Various ENM model types, parameters, and structure representations are tested here against the experimental RNA structural ensembles, exposing differences between models for proteins and for folded RNAs. Differences in performance are seen, depending on the structure alignment algorithm used to generate PCs, modulating the apparent utility of ENMs but not significantly impacting their ability to generate functional motions. The loss of dynamical information upon coarse-graining is somewhat larger for RNAs than for globular proteins, indicating, perhaps, the lower cooperativity of the less densely packed RNA. However, the RNA structures show less sensitivity to the elastic network model parameters than do proteins. These findings further demonstrate the utility of ENMs and the appropriateness of their application to well-packed RNA-only structures, justifying their use for studying the dynamics of ribonucleo-proteins, such as the ribosome and regulatory RNAs.     

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.     

RNA secondary structures and their prediction

This is a review of past and present attempts to predict the secondary structure of ribonucleic acids (RNAs) through mathematical and computer methods. Related areas covering classification, enumeration and graphical representations of structures are also covered. Various general prediction techniques are discussed, especially the use of thermodynamic criteria to construct an optimal structure. The emphasis in this approach is on the use of dynamic programming algorithms to minimize free energy. One such algorithm is introduced which comprises existing ones as special cases. Issued as NRCC No. 23684.     

Theory of protein molecule self-organization. I. Thermodynamic parameters of local secondary structures in the unfolded protein chain

This paper presents the results of a stereochemical analysis of local interactions in unfolded protein chains (sterical repulsions, hydrogen, and hydrophobic bonds, etc.) by means of space-filling modeles. On the basis of this analysis, an evaluation is made of thermodynamic parameters controlling the building-in of all the 20 natural amino acid residues in all the physically possible position of local secondary structures (α-helices, including α-helices with short fragments of helices 3₁₀ at the C-terminus; β-bends of different types, helices 3₁₀, and their combinations) as well as thermodynamic parameters of separate hydrogen bonds of polar side groups with the neighbor peptide groups (“local contacts”). The accuracy of the obtained results is discussed.     

A theoretical analysis of structural restructuring during formation of secondary RNA structures

An improved method for predicting the RNA secondary structure is proposed. The process of self-organization of structure is considered as a Markov chain. The kinetic of secondary structure is analysed by the Monte-Carlo method. The topological compatibility of helices is discussed. On the base of analysis it follows that the dynamical process of secondary structure formation is so that it is impossible to define a static set of complementary pairs. The method was used for predicting the mRNA secondary structure of a series of recombinant plasmids, containing the cro gene. The observed variation in expression can be explained by secondary structure.