ncRNA consensus secondary structure derivation using grammar strings

Many noncoding RNAs (ncRNAs) function through both their sequences and secondary structures. Thus, secondary structure derivation is an important issue in today's RNA research. The state-of-the-art structure annotation tools are based on comparative analysis, which derives consensus structure of homologous ncRNAs. Despite promising results from existing ncRNA aligning and consensus structure derivation tools, there is a need for more efficient and accurate ncRNA secondary structure modeling and alignment methods. In this work, we introduce a consensus structure derivation approach based on grammar string, a novel ncRNA secondary structure representation that encodes an ncRNA's sequence and secondary structure in the parameter space of a context-free grammar (CFG) and a full RNA grammar including pseudoknots. Being a string defined on a special alphabet constructed from a grammar, grammar string converts ncRNA alignment into sequence alignment. We derive consensus secondary structures from hundreds of ncRNA families from BraliBase 2.1 and 25 families containing pseudoknots using grammar string alignment. Our experiments have shown that grammar string-based structure derivation competes favorably in consensus structure quality with Murlet and RNASampler. Source code and experimental data are available at http://www.cse.msu.edu/~yannisun/grammar-string.     

Predicting RNA secondary structures with pseudoknots by MCMC sampling

The most probable secondary structure of an RNA molecule, given the nucleotide sequence, can be computed efficiently if a stochastic context-free grammar (SCFG) is used as the prior distribution of the secondary structure. The structures of some RNA molecules contain so-called pseudoknots. Allowing all possible configurations of pseudoknots is not compatible with context-free grammar models and makes the search for an optimal secondary structure NP-complete. We suggest a probabilistic model for RNA secondary structures with pseudoknots and present a Markov-chain Monte-Carlo Method for sampling RNA structures according to their posterior distribution for a given sequence. We favor Bayesian sampling over optimization methods in this context, because it makes the uncertainty of RNA structure predictions assessable. We demonstrate the benefit of our method in examples with tmRNA and also with simulated data. McQFold, an implementation of our method, is freely available from http://www.cs.uni-frankfurt.de/~metzler/McQFold.     

Stochastic modeling of RNA pseudoknotted structures: a grammatical approach

MOTIVATION: Modeling RNA pseudoknotted structures remains challenging. Methods have previously been developed to model RNA stem-loops successfully using stochastic context-free grammars (SCFG) adapted from computational linguistics; however, the additional complexity of pseudoknots has made modeling them more difficult. Formally a context-sensitive grammar is required, which would impose a large increase in complexity. RESULTS: We introduce a new grammar modeling approach for RNA pseudoknotted structures based on parallel communicating grammar systems (PCGS). Our new approach can specify pseudoknotted structures, while avoiding context-sensitive rules, using a single CFG synchronized with a number of regular grammars. Technically, the stochastic version of the grammar model can be as simple as an SCFG. As with SCFG, the new approach permits automatic generation of a single-RNA structure prediction algorithm for each specified pseudoknotted structure model. This approach also makes it possible to develop full probabilistic models of pseudoknotted structures to allow the prediction of consensus structures by comparative analysis and structural homology recognition in database searches.     

PseudoViewer: automatic visualization of RNA pseudoknots

MOTIVATION: Several algorithms have been developed for drawing RNA secondary structures, however none of these can be used to draw RNA pseudoknot structures. In the sense of graph theory, a drawing of RNA secondary structures is a tree, whereas a drawing of RNA pseudoknots is a graph with inner cycles within a pseudoknot as well as possible outer cycles formed between a pseudoknot and other structural elements. Thus, RNA pseudoknots are more difficult to visualize than RNA secondary structures. Since no automatic method for drawing RNA pseudoknots exists, visualizing RNA pseudoknots relies on significant amount of manual work and does not yield satisfactory results. The task of visualizing RNA pseudoknots by hand becomes more challenging as the size and complexity of the RNA pseudoknots increase. RESULTS: We have developed a new representation and an algorithm for drawing H-type pseudoknots with RNA secondary structures. Compared to existing representations of H-type pseudoknots, the new representation ensures uniform and clear drawings with no edge crossing for any H-type pseudoknots. To the best of our knowledge, this is the first algorithm for automatically drawing RNA pseudoknots with RNA secondary structures. The algorithm has been implemented in a Java program, which can be executed on any computing system. Experimental results demonstrate that the algorithm generates an aesthetically pleasing drawing of all H-type pseudoknots. The results have also shown that the drawing has high readability, enabling the user to quickly and easily recognize the whole RNA structure as well as the pseudoknots themselves.     

RNA pseudoknot prediction in energy-based models.

RNA molecules are sequences of nucleotides that serve as more than mere intermediaries between DNA and proteins, e.g., as catalytic molecules. Computational prediction of RNA secondary structure is among the few structure prediction problems that can be solved satisfactorily in polynomial time. Most work has been done to predict structures that do not contain pseudoknots. Allowing pseudoknots introduces modeling and computational problems. In this paper we consider the problem of predicting RNA secondary structures with pseudoknots based on free energy minimization. We first give a brief comparison of energy-based methods for predicting RNA secondary structures with pseudoknots. We then prove that the general problem of predicting RNA secondary structures containing pseudoknots is NP complete for a large class of reasonable models of pseudoknots.     

PSEUDOVIEWER2: Visualization of RNA pseudoknots of any type

Visualizing RNA pseudoknot structures is computationally more difficult than depicting RNA secondary structures, because a drawing of a pseudoknot structure is a graph (and possibly a nonplanar graph) with inner cycles within the pseudoknot, and possibly outer cycles formed between the pseudoknot and other structural elements. We previously developed PSEUDOVIEWER for visualizing H-type pseudoknots. PSEUDOVIEWER2 improves on the first version in many ways: (i) PSEUDOVIEWER2 is more general because it can visualize a pseudoknot of any type, including H-type pseudoknots, as a planar graph; (ii) PSEUDOVIEWER2 computes a drawing of RNA structures much more efficiently and is an order of magnitude faster in actual running time; and (iii) PSEUDOVIEWER2 is a web-based application program. Experimental results demonstrate that PSEUDOVIEWER2 generates an aesthetically pleasing drawing of pseudoknots of any type and that the new representation offered by PSEUDOVIEWER2 ensures uniform and clear drawings, with no edge crossing, for all types of pseudoknots. The PSEUDOVIEWER2 algorithm is the first developed for the automatic drawing of RNA secondary structures, including pseudoknots of any type. PSEUDOVIEWER2 is accessible at http://wilab.inha.ac.kr/pseudoviewer2/.     

A new method to predict the consensus secondary structure of a set of unaligned RNA sequences

MOTIVATION: To predict the consensus secondary structure, possibly including pseudoknots, of a set of RNA unaligned sequences. RESULTS: We have designed a method based on a new representation of any RNA secondary structure as a set of structural relationships between the helices of the structure. We refer to this representation as a structural pattern. In a first step, we use thermodynamic parameters to select, for each sequence, the best secondary structures according to energy minimization and we represent each of them using its corresponding structural pattern. In a second step, we search for the repeated structural patterns, i.e. the largest structural patterns that occur in at least one sequence, i.e. included in at least one of the structural patterns associated to each sequence. Thanks to an efficient encoding of structural patterns, this search comes down to identifying the largest repeated word suffixes in a dictionary. In a third step, we compute the plausibility of each repeated structural pattern by checking if it occurs more frequently in the studied sequences than in random RNA sequences. We then suppose that the consensus secondary structure corresponds to the repeated structural pattern that displays the highest plausibility. We present several experiments concerning tRNA, fragments of 16S rRNA and 10Sa RNA (including pseudoknots); in each of them, we found the putative consensus secondary structure.     

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).     

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.     

Predicting RNA secondary structures with arbitrary pseudoknots by maximizing the number of stacking pairs.

The paper investigates the computational problem of predicting RNA secondary structures. The general belief is that allowing pseudoknots makes the problem hard. Existing polynomial-time algorithms are heuristic algorithms with no performance guarantee and can handle only limited types of pseudoknots. In this paper, we initiate the study of predicting RNA secondary structures with a maximum number of stacking pairs while allowing arbitrary pseudoknots. We obtain two approximation algorithms with worst-case approximation ratios of 1/2 and 1/3 for planar and general secondary structures, respectively. For an RNA sequence of n bases, the approximation algorithm for planar secondary structures runs in O(n(3)) time while that for the general case runs in linear time. Furthermore, we prove that allowing pseudoknots makes it NP-hard to maximize the number of stacking pairs in a planar secondary structure. This result is in contrast with the recent NP-hard results on psuedoknots which are based on optimizing some general and complicated energy functions.     

Evolutionary Triplet Models of Structured RNA

The reconstruction and synthesis of ancestral RNAs is a feasible goal for paleogenetics. This will require new bioinformatics methods, including a robust statistical framework for reconstructing histories of substitutions, indels and structural changes. We describe a “transducer composition” algorithm for extending pairwise probabilistic models of RNA structural evolution to models of multiple sequences related by a phylogenetic tree. This algorithm draws on formal models of computational linguistics as well as the 1985 protosequence algorithm of David Sankoff. The output of the composition algorithm is a multiple-sequence stochastic context-free grammar. We describe dynamic programming algorithms, which are robust to null cycles and empty bifurcations, for parsing this grammar. Example applications include structural alignment of non-coding RNAs, propagation of structural information from an experimentally-characterized sequence to its homologs, and inference of the ancestral structure of a set of diverged RNAs. We implemented the above algorithms for a simple model of pairwise RNA structural evolution; in particular, the algorithms for maximum likelihood (ML) alignment of three known RNA structures and a known phylogeny and inference of the common ancestral structure. We compared this ML algorithm to a variety of related, but simpler, techniques, including ML alignment algorithms for simpler models that omitted various aspects of the full model and also a posterior-decoding alignment algorithm for one of the simpler models. In our tests, incorporation of basepair structure was the most important factor for accurate alignment inference; appropriate use of posterior-decoding was next; and fine details of the model were least important. Posterior-decoding heuristics can be substantially faster than exact phylogenetic inference, so this motivates the use of sum-over-pairs heuristics where possible (and approximate sum-over-pairs). For more exact probabilistic inference, we discuss the use of transducer composition for ML (or MCMC) inference on phylogenies, including possible ways to make the core operations tractable.     

基于最小自由能和协变信息预测带伪结RNA二级结构的迭代化方法

多数RNA分子的结构在进化中是高度保守的, 其中很多包含伪结。而RNA伪结的预测一直是一个棘手问题, 很多RNA 二级结构预测算法都不能预测伪结。文章提出一种基于迭代法预测带伪结RNA 二级结构的新方法。该方法在给潜在碱基对打分时综合了热力学和协变信息, 通过基于最小自由能RNA折叠算法的多次迭代选出所有的碱基对。测试结果表明: 此方法几乎能预测到所有的伪结。与其他方法相比, 敏感度接近最优, 而特异性达到最优。     

Pair stochastic tree adjoining grammars for aligning and predicting pseudoknot RNA structures

MOTIVATION: Since the whole genome sequences of many species have been determined, computational prediction of RNA secondary structures and computational identification of those non-coding RNA regions by comparative genomics become important. Therefore, more advanced alignment methods are required. Recently, an approach of structural alignment for RNA sequences has been introduced to solve these problems. Pair hidden Markov models on tree structures (PHMMTSs) proposed by Sakakibara are efficient automata-theoretic models for structural alignment of RNA secondary structures, although PHMMTSs are incapable of handling pseudoknots. On the other hand, tree adjoining grammars (TAGs), a subclass of context-sensitive grammars, are suitable for modeling pseudoknots. Our goal is to extend PHMMTSs by incorporating TAGs to be able to handle pseudoknots. RESULTS: We propose pair stochastic TAGs (PSTAGs) for aligning and predicting RNA secondary structures including a simple type of pseudoknot which can represent most known pseudoknot structures. First, we extend PHMMTSs defined on alignment of 'trees' to PSTAGs defined on alignment of 'TAG trees' which represent derivation processes of TAGs and are functionally equivalent to derived trees of TAGs. Then, we develop an efficient dynamic programming algorithm of PSTAGs for obtaining an optimal structural alignment including pseudoknots. We implement the PSTAG algorithm and demonstrate the properties of the algorithm by using it to align and predict several small pseudoknot structures. We believe that our implemented program based on PSTAGs is the first grammar-based and practically executable software for comparative analyses of RNA pseudoknot structures, and, further, non-coding RNAs.     

An iterated loop matching approach to the prediction of RNA secondary structures with pseudoknots

MOTIVATION: Pseudoknots have generally been excluded from the prediction of RNA secondary structures due to its difficulty in modeling. Although, several dynamic programming algorithms exist for the prediction of pseudoknots using thermodynamic approaches, they are neither reliable nor efficient. On the other hand, comparative methods are more reliable, but are often done in an ad hoc manner and require expert intervention. Maximum weighted matching, an algorithm for pseudoknot prediction with comparative analysis, suffers from low-prediction accuracy in many cases. RESULTS: Here we present an algorithm, iterated loop matching, for reliably and efficiently predicting RNA secondary structures including pseudoknots. The method can utilize either thermodynamic or comparative information or both, thus is able to predict pseudoknots for both aligned and individual sequences. We have tested the algorithm on a number of RNA families. Using 8-12 homologous sequences, the algorithm correctly identifies more than 90% of base-pairs for short sequences and 80% overall. It correctly predicts nearly all pseudoknots and produces very few spurious base-pairs for sequences without pseudoknots. Comparisons show that our algorithm is both more sensitive and more specific than the maximum weighted matching method. In addition, our algorithm has high-prediction accuracy on individual sequences, comparable with the PKNOTS algorithm, while using much less computational resources. AVAILABILITY: The program has been implemented in ANSI C and is freely available for academic use at http://www.cse.wustl.edu/~zhang/projects/rna/ilm/ Supplementary information: http://www.cse.wustl.edu/~zhang/projects/rna/ilm/     

RAGA: RNA sequence alignment by genetic algorithm.

We describe a new approach for accurately aligning two homologous RNA sequences when the secondary structure of one of them is known. To do so we developed two software packages, called RAGA and PRAGA, which use a genetic algorithm approach to optimize the alignments. RAGA is mainly an extension of SAGA, an earlier package for multiple protein sequence alignment. In PRAGA several genetic algorithms run in parallel and exchange individual solutions. This method allows us to optimize an objective function that describes the quality of a RNA pairwise alignment, taking into account both primary and secondary structure, including pseudoknots. We report results obtained using PRAGA on nine test cases of pairs of eukaryotic small subunit rRNA sequence (nuclear and mitochondrial).     

Design,implementation and evaluation of a practical pseudoknot folding algorithm based on thermodynamics

Background  

The general problem of RNA secondary structure prediction under the widely used thermodynamic model is known to be NP-complete when the structures considered include arbitrary pseudoknots. For restricted classes of pseudoknots, several polynomial time algorithms have been designed, where the O(n ⁶)time and O(n ⁴) space algorithm by Rivas and Eddy is currently the best available program.     

基于动态权重匹配的RNA折叠算法

本文在最大权重匹配（Maximum Weighted Matching,MWM）算法的基顾础上引入与茎区长度相关的动态权重,采用一种递归算法逐步寻找具有最大权重和的茎区。从而最终确定RNA的二级结构．该算法避开了繁杂的自由能计算,同样也能达到较高的预测精确度并且还能预测到大多数类型的潜在假结（pseudoknots）．     

A range of complex probabilistic models for RNA secondary structure prediction that includes the nearest-neighbor model and more

The standard approach for single-sequence RNA secondary structure prediction uses a nearest-neighbor thermodynamic model with several thousand experimentally determined energy parameters. An attractive alternative is to use statistical approaches with parameters estimated from growing databases of structural RNAs. Good results have been reported for discriminative statistical methods using complex nearest-neighbor models, including CONTRAfold, Simfold, and ContextFold. Little work has been reported on generative probabilistic models (stochastic context-free grammars [SCFGs]) of comparable complexity, although probabilistic models are generally easier to train and to use. To explore a range of probabilistic models of increasing complexity, and to directly compare probabilistic, thermodynamic, and discriminative approaches, we created TORNADO, a computational tool that can parse a wide spectrum of RNA grammar architectures (including the standard nearest-neighbor model and more) using a generalized super-grammar that can be parameterized with probabilities, energies, or arbitrary scores. By using TORNADO, we find that probabilistic nearest-neighbor models perform comparably to (but not significantly better than) discriminative methods. We find that complex statistical models are prone to overfitting RNA structure and that evaluations should use structurally nonhomologous training and test data sets. Overfitting has affected at least one published method (ContextFold). The most important barrier to improving statistical approaches for RNA secondary structure prediction is the lack of diversity of well-curated single-sequence RNA secondary structures in current RNA databases.     

Prediction of RNA pseudoknots using heuristic modeling with mapping and sequential folding

Predicting RNA secondary structure is often the first step to determining the structure of RNA. Prediction approaches have historically avoided searching for pseudoknots because of the extreme combinatorial and time complexity of the problem. Yet neglecting pseudoknots limits the utility of such approaches. Here, an algorithm utilizing structure mapping and thermodynamics is introduced for RNA pseudoknot prediction that finds the minimum free energy and identifies information about the flexibility of the RNA. The heuristic approach takes advantage of the 5' to 3' folding direction of many biological RNA molecules and is consistent with the hierarchical folding hypothesis and the contact order model. Mapping methods are used to build and analyze the folded structure for pseudoknots and to add important 3D structural considerations. The program can predict some well known pseudoknot structures correctly. The results of this study suggest that many functional RNA sequences are optimized for proper folding. They also suggest directions we can proceed in the future to achieve even better results.     

基于堆积协变信息与最小自由能预测含伪结的RNA二级结构

RNA伪结预测是RNA研究的一个难点问题。文中提出一种基于堆积协变信息与最小自由能的RNA伪结预测方法。该方法使用已知结构的RNA比对序列(ClustalW比对和结构比对)测试此方法, 侧重考虑相邻碱基对之间相互作用形成的堆积协变信息, 并结合最小自由能方法对碱基配对综合评分, 通过逐步迭代求得含伪结的RNA二级结构。结果表明, 此方法能正确预测伪结, 其平均敏感性和特异性优于参考算法, 并且结构比对的预测性能比ClustalW比对的预测性能更加稳定。文中同时讨论了不同协变信息权重因子对预测性能的影响, 发现权重因子比值在l1: l2=5:1时, 预测性能达到最优。