Prediction of geometrically feasible three-dimensional structures of pseudoknotted RNA through free energy estimation

Accurate free energy estimation is essential for RNA structure prediction. The widely used Turner''s energy model works well for nested structures. For pseudoknotted RNAs, however, there is no effective rule for estimation of loop entropy and free energy. In this work we present a new free energy estimation method, termed the pseudoknot predictor in three-dimensional space (pk3D), which goes beyond Turner''s model. Our approach treats nested and pseudoknotted structures alike in one unifying physical framework, regardless of how complex the RNA structures are. We first test the ability of pk3D in selecting native structures from a large number of decoys for a set of 43 pseudoknotted RNA molecules, with lengths ranging from 23 to 113. We find that pk3D performs slightly better than the Dirks and Pierce extension of Turner''s rule. We then test pk3D for blind secondary structure prediction, and find that pk3D gives the best sensitivity and comparable positive predictive value (related to specificity) in predicting pseudoknotted RNA secondary structures, when compared with other methods. A unique strength of pk3D is that it also generates spatial arrangement of structural elements of the RNA molecule. Comparison of three-dimensional structures predicted by pk3D with the native structure measured by nuclear magnetic resonance or X-ray experiments shows that the predicted spatial arrangement of stems and loops is often similar to that found in the native structure. These close-to-native structures can be used as starting points for further refinement to derive accurate three-dimensional structures of RNA molecules, including those with pseudoknots.     

Parsing nucleic acid pseudoknotted secondary structure: algorithm and applications.

Accurate prediction of pseudoknotted nucleic acid secondary structure is an important computational challenge. Prediction algorithms based on dynamic programming aim to find a structure with minimum free energy according to some thermodynamic ("sum of loop energies") model that is implicit in the recurrences of the algorithm. However, a clear definition of what exactly are the loops in pseudoknotted structures, and their associated energies, has been lacking. In this work, we present a complete classification of loops in pseudoknotted nucleic secondary structures, and describe the Rivas and Eddy and other energy models as sum-of-loops energy models. We give a linear time algorithm for parsing a pseudoknotted secondary structure into its component loops. We give two applications of our parsing algorithm. The first is a linear time algorithm to calculate the free energy of a pseudoknotted secondary structure. This is useful for heuristic prediction algorithms, which are widely used since (pseudoknotted) RNA secondary structure prediction is NP-hard. The second application is a linear time algorithm to test the generality of the dynamic programming algorithm of Akutsu for secondary structure prediction.Together with previous work, we use this algorithm to compare the generality of state-of-the-art algorithms on real biological structures.     

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.     

Novel and efficient RNA secondary structure prediction using hierarchical folding.

Algorithms for prediction of RNA secondary structure-the set of base pairs that form when an RNA molecule folds-are valuable to biologists who aim to understand RNA structure and function. Improving the accuracy and efficiency of prediction methods is an ongoing challenge, particularly for pseudoknotted secondary structures, in which base pairs overlap. This challenge is biologically important, since pseudoknotted structures play essential roles in functions of many RNA molecules, such as splicing and ribosomal frameshifting. State-of-the-art methods, which are based on free energy minimization, have high run-time complexity (typically Theta(n(5)) or worse), and can handle (minimize over) only limited types of pseudoknotted structures. We propose a new approach for prediction of pseudoknotted structures, motivated by the hypothesis that RNA structures fold hierarchically, with pseudoknot-free (non-overlapping) base pairs forming first, and pseudoknots forming later so as to minimize energy relative to the folded pseudoknot-free structure. Our HFold algorithm uses two-phase energy minimization to predict hierarchically formed secondary structures in O(n(3)) time, matching the complexity of the best algorithms for pseudoknot-free secondary structure prediction via energy minimization. Our algorithm can handle a wide range of biological structures, including kissing hairpins and nested kissing hairpins, which have previously required Theta(n(6)) time.     

Computational approaches for RNA energy parameter estimation

Methods for efficient and accurate prediction of RNA structure are increasingly valuable, given the current rapid advances in understanding the diverse functions of RNA molecules in the cell. To enhance the accuracy of secondary structure predictions, we developed and refined optimization techniques for the estimation of energy parameters. We build on two previous approaches to RNA free-energy parameter estimation: (1) the Constraint Generation (CG) method, which iteratively generates constraints that enforce known structures to have energies lower than other structures for the same molecule; and (2) the Boltzmann Likelihood (BL) method, which infers a set of RNA free-energy parameters that maximize the conditional likelihood of a set of reference RNA structures. Here, we extend these approaches in two main ways: We propose (1) a max-margin extension of CG, and (2) a novel linear Gaussian Bayesian network that models feature relationships, which effectively makes use of sparse data by sharing statistical strength between parameters. We obtain significant improvements in the accuracy of RNA minimum free-energy pseudoknot-free secondary structure prediction when measured on a comprehensive set of 2518 RNA molecules with reference structures. Our parameters can be used in conjunction with software that predicts RNA secondary structures, RNA hybridization, or ensembles of structures. Our data, software, results, and parameter sets in various formats are freely available at http://www.cs.ubc.ca/labs/beta/Projects/RNA-Params.     

Dynalign: an algorithm for finding the secondary structure common to two RNA sequences

With the rapid increase in the size of the genome sequence database, computational analysis of RNA will become increasingly important in revealing structure-function relationships and potential drug targets. RNA secondary structure prediction for a single sequence is 73 % accurate on average for a large database of known secondary structures. This level of accuracy provides a good starting point for determining a secondary structure either by comparative sequence analysis or by the interpretation of experimental studies. Dynalign is a new computer algorithm that improves the accuracy of structure prediction by combining free energy minimization and comparative sequence analysis to find a low free energy structure common to two sequences without requiring any sequence identity. It uses a dynamic programming construct suggested by Sankoff. Dynalign, however, restricts the maximum distance, M, allowed between aligned nucleotides in the two sequences. This makes the calculation tractable because the complexity is simplified to O(M(3)N(3)), where N is the length of the shorter sequence.The accuracy of Dynalign was tested with sets of 13 tRNAs, seven 5 S rRNAs, and two R2 3' UTR sequences. On average, Dynalign predicted 86.1 % of known base-pairs in the tRNAs, as compared to 59.7 % for free energy minimization alone. For the 5 S rRNAs, the average accuracy improves from 47.8 % to 86.4 %. The secondary structure of the R2 3' UTR from Drosophila takahashii is poorly predicted by standard free energy minimization. With Dynalign, however, the structure predicted in tandem with the sequence from Drosophila melanogaster nearly matches the structure determined by comparative sequence analysis.     

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。     

Secondary structure prediction of interacting RNA molecules

Computational tools for prediction of the secondary structure of two or more interacting nucleic acid molecules are useful for understanding mechanisms for ribozyme function, determining the affinity of an oligonucleotide primer to its target, and designing good antisense oligonucleotides, novel ribozymes, DNA code words, or nanostructures. Here, we introduce new algorithms for prediction of the minimum free energy pseudoknot-free secondary structure of two or more nucleic acid molecules, and for prediction of alternative low-energy (sub-optimal) secondary structures for two nucleic acid molecules. We provide a comprehensive analysis of our predictions against secondary structures of interacting RNA molecules drawn from the literature. Analysis of our tools on 17 sequences of up to 200 nucleotides that do not form pseudoknots shows that they have 79% accuracy, on average, for the minimum free energy predictions. When the best of 100 sub-optimal foldings is taken, the average accuracy increases to 91%. The accuracy decreases as the sequences increase in length and as the number of pseudoknots and tertiary interactions increases. Our algorithms extend the free energy minimization algorithm of Zuker and Stiegler for secondary structure prediction, and the sub-optimal folding algorithm by Wuchty et al. Implementations of our algorithms are freely available in the package MultiRNAFold.     

Revolutions in RNA secondary structure prediction

RNA structure formation is hierarchical and, therefore, secondary structure, the sum of canonical base-pairs, can generally be predicted without knowledge of the three-dimensional structure. Secondary structure prediction algorithms evolved from predicting a single, lowest free energy structure to their current state where statistics can be determined from the thermodynamic ensemble. This article reviews the free energy minimization technique and the salient revolutions in the dynamic programming algorithm methods for secondary structure prediction. Emphasis is placed on highlighting the recently developed method, which statistically samples structures from the complete Boltzmann ensemble.     

Efficient parameter estimation for RNA secondary structure prediction

MOTIVATION: Accurate prediction of RNA secondary structure from the base sequence is an unsolved computational challenge. The accuracy of predictions made by free energy minimization is limited by the quality of the energy parameters in the underlying free energy model. The most widely used model, the Turner99 model, has hundreds of parameters, and so a robust parameter estimation scheme should efficiently handle large data sets with thousands of structures. Moreover, the estimation scheme should also be trained using available experimental free energy data in addition to structural data. RESULTS: In this work, we present constraint generation (CG), the first computational approach to RNA free energy parameter estimation that can be efficiently trained on large sets of structural as well as thermodynamic data. Our CG approach employs a novel iterative scheme, whereby the energy values are first computed as the solution to a constrained optimization problem. Then the newly computed energy parameters are used to update the constraints on the optimization function, so as to better optimize the energy parameters in the next iteration. Using our method on biologically sound data, we obtain revised parameters for the Turner99 energy model. We show that by using our new parameters, we obtain significant improvements in prediction accuracy over current state of-the-art methods. AVAILABILITY: Our CG implementation is available at http://www.rnasoft.ca/CG/.     

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。     

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.     

Incorporating global features of RNA motifs in predictions for an ensemble of secondary structures for encapsidated MS2 bacteriophage RNA

The secondary structure of encapsidated MS2 genomic RNA poses an interesting RNA folding challenge. Cryoelectron microscopy has demonstrated that encapsidated MS2 RNA is well-ordered. Models of MS2 assembly suggest that the RNA hairpin-protein interactions and the appropriate placement of hairpins in the MS2 RNA secondary structure can guide the formation of the correct icosahedral particle. The RNA hairpin motif that is recognized by the MS2 capsid protein dimers, however, is energetically unfavorable, and thus free energy predictions are biased against this motif. Computer programs called Crumple, Sliding Windows, and Assembly provide useful tools for prediction of viral RNA secondary structures when the traditional assumptions of RNA structure prediction by free energy minimization may not apply. These methods allow incorporation of global features of the RNA fold and motifs that are difficult to include directly in minimum free energy predictions. For example, with MS2 RNA the experimental data from SELEX experiments, crystallography, and theoretical calculations of the path for the series of hairpins can be incorporated in the RNA structure prediction, and thus the influence of free energy considerations can be modulated. This approach thoroughly explores conformational space and generates an ensemble of secondary structures. The predictions from this new approach can test hypotheses and models of viral assembly and guide construction of complete three-dimensional models of virus particles.     

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.     

RNA二级结构的最小自由能算法

RNA(即tRNA,rRNA,mRNA和SnRNA)有两大主要功能：一是某些病毒的遗传物质;二是参与蛋白质的合成,这些与细胞分化、代谢、记忆的储存等有重要关系,这些功能与RNA二级结构的稳定性。自由能密切相关．常用的计算自由能的方法有热力学微扰法及热力学微积分法等．本文以寻找最小自由能二级结构为目的,给出了RNA二级结构的最小自由能算法,该算法的时间复杂性不超过O(n^4)。     

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.     

The computer simulation of RNA folding involving pseudoknot formation.

The algorithm and the program for the prediction of RNA secondary structure with pseudoknot formation have been proposed. The algorithm simulates stepwise folding by generating random structures using Monte Carlo method, followed by the selection of helices to final structure on the basis of both their probabilities of occurrence in a random structure and free energy parameters. The program versions have been tested on ribosomal RNA structures and on RNAs with pseudoknots evidenced by experimental data. It is shown that the simulation of folding during RNA synthesis improves the results. The introduction of pseudoknot formation permits to predict the pseudoknotted structures and to improve the prediction of long-range interactions. The computer program is rather fast and allows to predict the structures for long RNAs without using large memory volumes in usual personal computer.     

A Test and Refinement of Folding Free Energy Nearest Neighbor Parameters for RNA Including N6-Methyladenosine

RNA folding free energy change parameters are widely used to predict RNA secondary structure and to design RNA sequences. These parameters include terms for the folding free energies of helices and loops. Although the full set of parameters has only been traditionally available for the four common bases and backbone, it is well known that covalent modifications of nucleotides are widespread in natural RNAs. Covalent modifications are also widely used in engineered sequences. We recently derived a full set of nearest neighbor terms for RNA that includes N⁶-methyladenosine (m⁶A). In this work, we test the model using 98 optical melting experiments, matching duplexes with or without N⁶-methylation of A. Most experiments place RRACH, the consensus site of N⁶-methylation, in a variety of contexts, including helices, bulge loops, internal loops, dangling ends, and terminal mismatches. For matched sets of experiments that include either A or m⁶A in the same context, we find that the parameters for m⁶A are as accurate as those for A. Across all experiments, the root mean squared deviation between estimated and experimental free energy changes is 0.67 kcal/mol. We used the new experimental data to refine the set of nearest neighbor parameter terms for m⁶A. These parameters enable prediction of RNA secondary structures including m⁶A, which can be used to model how N⁶-methylation of A affects RNA structure.     

Expanded sequence dependence of thermodynamic parameters improves prediction of RNA secondary structure.

An improved dynamic programming algorithm is reported for RNA secondary structure prediction by free energy minimization. Thermodynamic parameters for the stabilities of secondary structure motifs are revised to include expanded sequence dependence as revealed by recent experiments. Additional algorithmic improvements include reduced search time and storage for multibranch loop free energies and improved imposition of folding constraints. An extended database of 151,503 nt in 955 structures? determined by comparative sequence analysis was assembled to allow optimization of parameters not based on experiments and to test the accuracy of the algorithm. On average, the predicted lowest free energy structure contains 73 % of known base-pairs when domains of fewer than 700 nt are folded; this compares with 64 % accuracy for previous versions of the algorithm and parameters. For a given sequence, a set of 750 generated structures contains one structure that, on average, has 86 % of known base-pairs. Experimental constraints, derived from enzymatic and flavin mononucleotide cleavage, improve the accuracy of structure predictions.