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.     

Aligning two RNA secondary structures with l-block

RNA sequences can form structures which are conserved throughout evolution and the question of aligning two RNA secondary structures has been extensively studied. Most of the previous alignment algorithms require the input of gap opening and gap extension penalty parameters. The choice of appropriate parameter values is controversial as there is little biological information to guide their assignment. In this paper, we present an algorithm which circumvents this problem. Instead of finding an optimal alignment with predefined gap opening penalty, the algorithm finds the optimal alignment with exact number of aligned blocks.     

HotKnots: heuristic prediction of RNA secondary structures including pseudoknots

We present HotKnots, a new heuristic algorithm for the prediction of RNA secondary structures including pseudoknots. Based on the simple idea of iteratively forming stable stems, our algorithm explores many alternative secondary structures, using a free energy minimization algorithm for pseudoknot free secondary structures to identify promising candidate stems. In an empirical evaluation of the algorithm with 43 sequences taken from the Pseudobase database and from the literature on pseudoknotted structures, we found that overall, in terms of the sensitivity and specificity of predictions, HotKnots outperforms the well-known Pseudoknots algorithm of Rivas and Eddy and the NUPACK algorithm of Dirks and Pierce, both based on dynamic programming approaches for limited classes of pseudoknotted structures. It also outperforms the heuristic Iterated Loop Matching algorithm of Ruan and colleagues, and in many cases gives better results than the genetic algorithm from the STAR package of van Batenburg and colleagues and the recent pknotsRG-mfe algorithm of Reeder and Giegerich. The HotKnots algorithm has been implemented in C/C++ and is available from http://www.cs.ubc.ca/labs/beta/Software/HotKnots.     

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.     

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.     

Conserved RNA secondary structures in Picornaviridae genomes

The family Picornaviridae contains important pathogens including, for example, hepatitis A virus and foot-and-mouth disease virus. The genome of these viruses is a single messenger-active (+)-RNA of 7200–8500 nt. Besides coding for the viral proteins, it also contains functionally important RNA secondary structures, among them an internal ribosomal entry site (IRES) region towards the 5′-end. This contribution provides a comprehensive computational survey of the complete genomic RNAs and a detailed comparative analysis of the conserved structural elements in seven of the currently nine genera in the family Picornaviridae. Compared with previous studies we find: (i) that only smaller sections of the IRES region than previously reported are conserved at single base-pair resolution and (ii) that there is a number of significant structural elements in the coding region. Furthermore, we identify potential cis-acting replication elements in four genera where this feature has not been reported so far.     

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.     

Pseudoknots in RNA secondary structures: representation, enumeration, and prevalence.

A number of non-coding RNA are known to contain functionally important or conserved pseudoknots. However, pseudoknotted structures are more complex than orthodox, and most methods for analyzing secondary structures do not handle them. I present here a way to decompose and represent general secondary structures which extends the tree representation of the stem-loop structure, and use this to analyze the frequency of pseudoknots in known and in random secondary structures. This comparison shows that, though a number of pseudoknots exist, they are still relatively rare and mostly of the simpler kinds. In contrast, random secondary structures tend to be heavily knotted, and the number of available structures increases dramatically when allowing pseudoknots. Therefore, methods for structure prediction and non-coding RNA identification that allow pseudoknots are likely to be much less powerful than those that do not, unless they penalize pseudoknots appropriately.     

Prediction of RNA secondary structure, including pseudoknotting, by computer simulation.

A computer program is presented which determines the secondary structure of linear RNA molecules by simulating a hypothetical process of folding. This process implies the concept of 'nucleation centres', regions in RNA which locally trigger the folding. During the simulation, the RNA is allowed to fold into pseudoknotted structures, unlike all other programs predicting RNA secondary structure. The simulation uses published, experimentally determined free energy values for nearest neighbour base pair stackings and loop regions, except for new extrapolated values for loops larger than seven nucleotides. The free energy value for a loop arising from pseudoknot formation is set to a single, estimated value of 4.2 kcal/mole. Especially in the case of long RNA sequences, our program appears superior to other secondary structure predicting programs described so far, as tests on tRNAs, the LSU intron of Tetrahymena thermophila and a number of plant viral RNAs show. In addition, pseudoknotted structures are often predicted successfully. The program is written in mainframe APL and is adapted to run on IBM compatible PCs, Atari ST and Macintosh personal computers. On an 8 MHz 8088 standard PC without coprocessor, using STSC APL, it folds a sequence of 700 nucleotides in one and a half hour.     

Assessment of a model for intron RNA secondary structure relevant to RNA self-splicing--a review

A widespread class of introns is characterized by a particular RNA secondary structure, based upon four conserved nucleotide sequences. Among such "class I" introns are found the majority of introns in fungal mitochondrial genes and the self-splicing intron of the large ribosomal RNA of several species of Tetrahymena. A model of the RNA secondary structure, which must underlie the self-splicing activity, is here evaluated in the light of data on 16 further introns. The main body or "core structure" of the intron always consists of the base-paired regions P3 to P9 with the associated single-stranded loops, with P2 present also in most cases. Two minority sub-classes of core structure occur, one of which is typical of introns in fungal ribosomal RNA. Introns in which the core structure is close to the 5' splice site all have an internal guide sequence (IGS) which can pair with exon sequences adjacent to the 5' and 3' splice sites to align them precisely, as proposed by Davies et al. [Nature 300 (1982) 719-724]. In these cases, the internal guide model allows us to predict correctly the exact location of splice sites. All other introns probably use other mechanisms of alignment. This analysis provides strong support for the RNA splicing model which we have developed.     

Base-pair probability profiles of RNA secondary structures

Dynamic programming algorithms are able to predict optimal andsuboptimal secondary structures of RNA. These suboptimal oralternative secondary structures are important for the biologicalfunction of RNA. The distribution of secondary structures presentin solution is governed by the thermodynamic equilibrium betweenthe different structures. An algorithm is presented which approximatesthe total partition function by a Boltzmann–weighted summationof optimal and suboptimal secondary structures at several temperatures.A clear representation of the equilibrium distribution of secondarystructures is derived from a two-dimensional bonding matrixwith base–pairing probability as the third dimension.The temperature dependence of the equilibrium distribution givesthe denaturation behavior of the nucleic acid, which may becompared to experimental optical denaturation curves after correctionfor the hypochromicities of the different base-pairs. Similarly,temperature-induced mobility changes detected in temperature-gradientgel electrophoresis of nucleic acids may be interpreted on thebasis of the temperature dependence of the equilibrium distribution.Results are illustrated for natural circular and synthetic linearpotato spindle tuber viroid RNA respectively, and are comparedto experimental data.     

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 locally optimal RNA secondary structures

It is a classical result of Stein and Waterman that the asymptotic number of RNA secondary structures is $1.104366 \cdot n^{-3/2} \cdot 2.618034^n$ . Motivated by the kinetics of RNA secondary structure formation, we are interested in determining the asymptotic number of secondary structures that are locally optimal, with respect to a particular energy model. In the Nussinov energy model, where each base pair contributes $-1$ towards the energy of the structure, locally optimal structures are exactly the saturated structures, for which we have previously shown that asymptotically, there are $1.07427\cdot n^{-3/2} \cdot 2.35467^n$ many saturated structures for a sequence of length $n$ . In this paper, we consider the base stacking energy model, a mild variant of the Nussinov model, where each stacked base pair contributes $-1$ toward the energy of the structure. Locally optimal structures with respect to the base stacking energy model are exactly those secondary structures, whose stems cannot be extended. Such structures were first considered by Evers and Giegerich, who described a dynamic programming algorithm to enumerate all locally optimal structures. In this paper, we apply methods from enumerative combinatorics to compute the asymptotic number of such structures. Additionally, we consider analogous combinatorial problems for secondary structures with annotated single-stranded, stacking nucleotides (dangles).     

Search for conservative secondary structures of RNA

We suggest a new algorithm to search a given set of the RNA sequences for conserved secondary structures. The algorithm is based on alignment of the sequences for potential helical strands. This procedure can be used to search for new structured RNAs and new regulatory elements. It is efficient for the genome-scale analysis. The results of various tests run with this algorithm are shown.     

Predicting a set of minimal free energy RNA secondary structures common to two sequences

MOTIVATION: Function derives from structure, therefore, there is need for methods to predict functional RNA structures. RESULTS: The Dynalign algorithm, which predicts the lowest free energy secondary structure common to two unaligned RNA sequences, is extended to the prediction of a set of low-energy structures. Dot plots can be drawn to show all base pairs in structures within an energy increment. Dynalign predicts more well-defined structures than structure prediction using a single sequence; in 5S rRNA sequences, the average number of base pairs in structures with energy within 20% of the lowest energy structure is 317 using Dynalign, but 569 using a single sequence. Structure prediction with Dynalign can also be constrained according to experiment or comparative analysis. The accuracy, measured as sensitivity and positive predictive value, of Dynalign is greater than predictions with a single sequence. AVAILABILITY: Dynalign can be downloaded at http://rna.urmc.rochester.edu     

Effects of secondary structures in RNA on interlocking probabilities

In 1967 Wang and Schwartz reported on the formation of interlocked rings between linear or circular DNA molecules by the enzyme topoisomerase. We propose viewing the secondary structured loop in RNA (or single stranded DNA) as analogous to a circular DNA molecule. Formation of a catenane between such an RNA loop with a DNA molecule may constitute a probe of the secondary and general three dimensioanl structure of the RNA molecule. The experimental results may be compared with the theoretical calculation. We suggest here a method for estimating linkage probabilities and calculate them for several cases for which secondary structures of the RNA have been proposed.     

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.