Improved RNA secondary structure prediction by maximizing expected pair accuracy

Free energy minimization has been the most popular method for RNA secondary structure prediction for decades. It is based on a set of empirical free energy change parameters derived from experiments using a nearest-neighbor model. In this study, a program, MaxExpect, that predicts RNA secondary structure by maximizing the expected base-pair accuracy, is reported. This approach was first pioneered in the program CONTRAfold, using pair probabilities predicted with a statistical learning method. Here, a partition function calculation that utilizes the free energy change nearest-neighbor parameters is used to predict base-pair probabilities as well as probabilities of nucleotides being single-stranded. MaxExpect predicts both the optimal structure (having highest expected pair accuracy) and suboptimal structures to serve as alternative hypotheses for the structure. Tested on a large database of different types of RNA, the maximum expected accuracy structures are, on average, of higher accuracy than minimum free energy structures. Accuracy is measured by sensitivity, the percentage of known base pairs correctly predicted, and positive predictive value (PPV), the percentage of predicted pairs that are in the known structure. By favoring double-strandedness or single-strandedness, a higher sensitivity or PPV of prediction can be favored, respectively. Using MaxExpect, the average PPV of optimal structure is improved from 66% to 68% at the same sensitivity level (73%) compared with free energy minimization.     

A dynamic programming algorithm for finding alternative RNA secondary structures.

Dynamic programming algorithms that predict RNA secondary structure by minimizing the free energy have had one important limitation. They were able to predict only one optimal structure. Given the uncertainties of the thermodynamic data and the effects of proteins and other environmental factors on structure, the optimal structure predicted by these methods may not have biological significance. We present a dynamic programming algorithm that can determine optimal and suboptimal secondary structures for an RNA. The power and utility of the method is demonstrated in the folding of the intervening sequence of the rRNA of Tetrahymena. By first identifying the major secondary structures corresponding to the lowest free energy minima, a secondary structure of possible biological significance is derived.     

Abstract shapes of RNA

The function of a non-protein-coding RNA is often determined by its structure. Since experimental determination of RNA structure is time-consuming and expensive, its computational prediction is of great interest, and efficient solutions based on thermodynamic parameters are known. Frequently, however, the predicted minimum free energy structures are not the native ones, leading to the necessity of generating suboptimal solutions. While this can be accomplished by a number of programs, the user is often confronted with large outputs of similar structures, although he or she is interested in structures with more fundamental differences, or, in other words, with different abstract shapes. Here, we formalize the concept of abstract shapes and introduce their efficient computation. Each shape of an RNA molecule comprises a class of similar structures and has a representative structure of minimal free energy within the class. Shape analysis is implemented in the program RNAshapes. We applied RNAshapes to the prediction of optimal and suboptimal abstract shapes of several RNAs. For a given energy range, the number of shapes is considerably smaller than the number of structures, and in all cases, the native structures were among the top shape representatives. This demonstrates that the researcher can quickly focus on the structures of interest, without processing up to thousands of near-optimal solutions. We complement this study with a large-scale analysis of the growth behaviour of structure and shape spaces. RNAshapes is available for download and as an online version on the Bielefeld Bioinformatics Server.     

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.     

Complete suboptimal folding of RNA and the stability of secondary structures

An algorithm is presented for generating rigorously all suboptimal secondary structures between the minimum free energy and an arbitrary upper limit. The algorithm is particularly fast in the vicinity of the minimum free energy. This enables the efficient approximation of statistical quantities, such as the partition function or measures for structural diversity. The density of states at low energies and its associated structures are crucial in assessing from a thermodynamic point of view how well-defined the ground state is. We demonstrate this by exploring the role of base modification in tRNA secondary structures, both at the level of individual sequences from Escherichia coli and by comparing artificially generated ensembles of modified and unmodified sequences with the same tRNA structure. The two major conclusions are that (1) base modification considerably sharpens the definition of the ground state structure by constraining energetically adjacent structures to be similar to the ground state, and (2) sequences whose ground state structure is thermodynamically well defined show a significant tendency to buffer single point mutations. This can have evolutionary implications, since selection pressure to improve the definition of ground states with biological function may result in increased neutrality.     

A set of nearest neighbor parameters for predicting the enthalpy change of RNA secondary structure formation

A complete set of nearest neighbor parameters to predict the enthalpy change of RNA secondary structure formation was derived. These parameters can be used with available free energy nearest neighbor parameters to extend the secondary structure prediction of RNA sequences to temperatures other than 37°C. The parameters were tested by predicting the secondary structures of sequences with known secondary structure that are from organisms with known optimal growth temperatures. Compared with the previous set of enthalpy nearest neighbor parameters, the sensitivity of base pair prediction improved from 65.2 to 68.9% at optimal growth temperatures ranging from 10 to 60°C. Base pair probabilities were predicted with a partition function and the positive predictive value of structure prediction is 90.4% when considering the base pairs in the lowest free energy structure with pairing probability of 0.99 or above. Moreover, a strong correlation is found between the predicted melting temperatures of RNA sequences and the optimal growth temperatures of the host organism. This indicates that organisms that live at higher temperatures have evolved RNA sequences with higher melting temperatures.     

Rapid ab initio prediction of RNA pseudoknots via graph tree decomposition

The prediction of RNA secondary structure including pseudoknots remains a challenge due to the intractable computation of the sequence conformation from nucleotide interactions under free energy models. Optimal algorithms often assume a restricted class for the predicted RNA structures and yet still require a high-degree polynomial time complexity, which is too expensive to use. Heuristic methods may yield time-efficient algorithms but they do not guarantee optimality of the predicted structure. This paper introduces a new and efficient algorithm for the prediction of RNA structure with pseudoknots for which the structure is not restricted. Novel prediction techniques are developed based on graph tree decomposition. In particular, based on a simplified energy model, stem overlapping relationships are defined with a graph, in which a specialized maximum independent set corresponds to the desired optimal structure. Such a graph is tree decomposable; dynamic programming over a tree decomposition of the graph leads to an efficient optimal algorithm. The final structure predictions are then based on re-ranking a list of suboptimal structures under a more comprehensive free energy model. The new algorithm is evaluated on a large number of RNA sequence sets taken from diverse resources. It demonstrates overall sensitivity and specificity that outperforms or is comparable with those of previous optimal and heuristic algorithms yet it requires significantly less time than the compared optimal algorithms. The preliminary version of this paper appeared in the proceedings of the 6th Workshop on Algorithms for Bioinformatics (WABI 2006).     

The suboptimal structures find the optimal RNAs: homology search for bacterial non-coding RNAs using suboptimal RNA structures

Non-coding RNAs (ncRNAs) are regulatory molecules encoded in the intergenic or intragenic regions of the genome. In prokaryotes, biocomputational identification of homologs of known ncRNAs in other species often fails due to weakly evolutionarily conserved sequences, structures, synteny and genome localization, except in the case of evolutionarily closely related species. To eliminate results from weak conservation, we focused on RNA structure, which is the most conserved ncRNA property. Analysis of the structure of one of the few well-studied bacterial ncRNAs, 6S RNA, demonstrated that unlike optimal and consensus structures, suboptimal structures are capable of capturing RNA homology even in divergent bacterial species. A computational procedure for the identification of homologous ncRNAs using suboptimal structures was created. The suggested procedure was applied to strongly divergent bacterial species and was capable of identifying homologous ncRNAs.     

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.     

The evolution of 5S RNA secondary structures

We have applied the Pipas-McMahon algorithm based on free energy calculations to the search for a 5S RNA base-pair structure common to all known sequences. We find that a 'Y' shaped model is consistently among the structures having the lowest free energy using 5S RNA sequences from either eukaryotic or prokaryotic sources. Compaison of this 'Y' structure with models which have recently been proposed show these models to be remarkably similar, and the minor differences are explicable based on the technique used to obtain the model. That prokaryotic and eukaryotic 5S RNA can adopt a similar secondary structure is strong support for its resistance to change during evolution.     

Homology modeling by the ICM method

Five models have been built by the ICM method for the Comparative Modeling section of the Meeting on the Critical Assessment of Techniques for Protein Structure Prediction. The targets have homologous proteins with known three-dimensional structure with sequence identity ranging from 25 to 77%. After alignment of the target sequence with the related three-dimensional structure, the modeling procedure consists of two subproblems: side-chain prediction and loop prediction. The ICM method approaches these problems with the following steps: (1) a starting model is created based on the homologous structure with the conserved portion fixed and the noncon-served portion having standard covalent geometry and free torsion angles; (2) the Biased Probability Monte Carlo (BPMC) procedure is applied to search the subspaces of either all the nonconservative side-chain torsion angles or torsion angles in a loop backbone and surrounding side chains. A special algorithm was designed to generate low-energy loop deformations. The BPMC procedure globally optimizes the energy function consisting of ECEPP/3 and solvation energy terms. Comparison of the predictions with the NMR or crystallographic solutions reveals a high proportion of correctly predicted side chains. The loops were not correctly predicted because imprinted distortions of the backbone increased the energy of the near-native conformation and thus made the solution unrecognizable. Interestingly, the energy terms were found to be reliable and the sampling of conformational space sufficient. The implications of this finding for the strategies of future comparative modeling are discussed. © 1995 Wiley-Liss, Inc.     

Local versus global optimal sports techniques in a group of athletes

Various optimization algorithms have been used to achieve optimal control of sports movements. Nevertheless, no local or global optimization algorithm could be the most effective for solving all optimal control problems. This study aims at comparing local and global optimal solutions in a multistart gradient-based optimization by considering actual repetitive performances of a group of athletes performing a transition move on the uneven bars. Twenty-four trials by eight national-level female gymnasts were recorded using a motion capture system, and then multistart sequential quadratic programming optimizations were performed to obtain global optimal, local optimal and suboptimal solutions. The multistart approach combined with a gradient-based algorithm did not often find the local solution to be the best and proposed several other solutions including global optimal and suboptimal techniques. The qualitative change between actual and optimal techniques provided three directions for training: to increase hip flexion–abduction, to transfer leg and arm angular momentum to the trunk and to straighten hand path to the bar.     

Computing the partition function for kinetically trapped RNA secondary structures

An RNA secondary structure is locally optimal if there is no lower energy structure that can be obtained by the addition or removal of a single base pair, where energy is defined according to the widely accepted Turner nearest neighbor model. Locally optimal structures form kinetic traps, since any evolution away from a locally optimal structure must involve energetically unfavorable folding steps. Here, we present a novel, efficient algorithm to compute the partition function over all locally optimal secondary structures of a given RNA sequence. Our software, RNAlocopt runs in O(n3) time and O(n2) space. Additionally, RNAlocopt samples a user-specified number of structures from the Boltzmann subensemble of all locally optimal structures. We apply RNAlocopt to show that (1) the number of locally optimal structures is far fewer than the total number of structures--indeed, the number of locally optimal structures approximately equal to the square root of the number of all structures, (2) the structural diversity of this subensemble may be either similar to or quite different from the structural diversity of the entire Boltzmann ensemble, a situation that depends on the type of input RNA, (3) the (modified) maximum expected accuracy structure, computed by taking into account base pairing frequencies of locally optimal structures, is a more accurate prediction of the native structure than other current thermodynamics-based methods. The software RNAlocopt constitutes a technical breakthrough in our study of the folding landscape for RNA secondary structures. For the first time, locally optimal structures (kinetic traps in the Turner energy model) can be rapidly generated for long RNA sequences, previously impossible with methods that involved exhaustive enumeration. Use of locally optimal structure leads to state-of-the-art secondary structure prediction, as benchmarked against methods involving the computation of minimum free energy and of maximum expected accuracy. Web server and source code available at http://bioinformatics.bc.edu/clotelab/RNAlocopt/.     

Identifying complete RNA structural ensembles including pseudoknots

The close relationship between RNA structure and function underlines the significance of accurately predicting RNA structures from sequence information. Structural topologies such as pseudoknots are of particular interest due to their ubiquity and direct involvement in RNA function, but identifying pseudoknots is a computationally challenging problem and existing heuristic approaches usually perform poorly for RNA sequences of even a few hundred bases. We survey the performance of pseudoknot prediction methods on a data set of full-length RNA sequences representing varied sequence lengths, and biological RNA classes such as RNase P RNA, Group I Intron, tmRNA and tRNA. Pseudoknot prediction methods are compared with minimum free energy and suboptimal secondary structure prediction methods in terms of correct base-pairs, stems and pseudoknots and we find that the ensemble of suboptimal structure predictions succeeds in identifying correct structural elements in RNA that are usually missed in MFE and pseudoknot predictions. We propose a strategy to identify a comprehensive set of non-redundant stems in the suboptimal structure space of a RNA molecule by applying heuristics that reduce the structural redundancy of the predicted suboptimal structures by merging slightly varying stems that are predicted to form in local sequence regions. This reduced-redundancy set of structural elements consistently outperforms more specialized approaches.in data sets. Thus, the suboptimal folding space can be used to represent the structural diversity of an RNA molecule more comprehensively than optimal structure prediction approaches alone.     

Topological Organization of Picornaviral Genomes: Statistical Prediction of RNA Structural Signals

We have analyzed 11 picornaviral RNA genomic sequences by optimal and suboptimal minimum free energy folding algorithms. The systematic summation of all pairing partners for each base in the suboptimal structures (P-num value) shows a distinct pattern of alternating low and high values when plotted against the sequence length and indicate regions within each genome where secondary structure(s) are likely to play a significant role in virus biology. The individual folds augmented by data from phylogenetic folds, collectively suggest some revisions of existing models for 5′-untranslated regions of cardioviruses and enteroviruses that might better explain the functions of these regions.     

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.     

The application of statistical decision theory to animal behaviour

Statistical decision theory is discussed as a general framework for analysing how animals should learn. Attention is focused on optimal foraging behaviour in stochastic environments. We emphasise the distinction between the mathematical procedure that can be used to find optimal solutions and the mechanism an animal might use to implement such solutions. The mechanisms might be specific to a restricted class of problems and produce suboptimal behaviour when faced with problems outside this class. We illustrate this point by an example based on what is known in the literature on animal learning as the partial reinforcement effect.     

RnaPredict—An Evolutionary Algorithm for RNA Secondary Structure Prediction

This paper presents two in-depth studies on RnaPredict, an evolutionary algorithm for RNA secondary structure prediction. The first study is an analysis of the performance of two thermodynamic models, Individual Nearest Neighbor (INN) and Individual Nearest Neighbor Hydrogen Bond (INN-HB). The correlation between the free energy of predicted structures and the sensitivity is analyzed for 19 RNA sequences. Although some variance is shown, there is a clear trend between a lower free energy and an increase in true positive base pairs. With increasing sequence length, this correlation generally decreases. In the second experiment, the accuracy of the predicted structures for these 19 sequences are compared against the accuracy of the structures generated by the mfold dynamic programming algorithm (DPA) and also to known structures. RnaPredict is shown to outperform the minimum free energy structures produced by mfold and has comparable performance when compared to suboptimal structures produced by mfold.     

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.     

Protein fold recognition by total alignment probability

We present a protein fold-recognition method that uses a comprehensive statistical interpretation of structural Hidden Markov Models (HMMs). The structure/fold recognition is done by summing the probabilities of all sequence-to-structure alignments. The optimal alignment can be defined as the most probable, but suboptimal alignments may have comparable probabilities. These suboptimal alignments can be interpreted as optimal alignments to the "other" structures from the ensemble or optimal alignments under minor fluctuations in the scoring function. Summing probabilities for all alignments gives a complete estimate of sequence-model compatibility. In the case of HMMs that produce a sequence, this reflects the fact that due to our indifference to exactly how the HMM produced the sequence, we should sum over all possibilities. We have built a set of structural HMMs for 188 protein structures and have compared two methods for identifying the structure compatible with a sequence: by the optimal alignment probability and by the total probability. Fold recognition by total probability was 40% more accurate than fold recognition by the optimal alignment probability. Proteins 2000;40:451-462.