A kinetic approach to the prediction of RNA secondary structures

A new approach to the prediction of secondary RNA structures based on the analysis of the kinetics of molecular self-organisation is proposed herein. The Markov process is used to describe structural reconstructions during secondary structure formation. This process is modelled by a Monte-Carlo method. Examples of the calculation by this method of the secondary structures kinetic ensemble are given. Distribution of time-dependent probabilities within the ensembles is obtained. An effective method for search for the equilibrium ensemble is also suggested. This method is based on the construction of a tree of all possible secondary structures of RNA. By ascribing a probability for each structure (according to its free energy) the Boltzmann equilibrium ensemble can be obtained.     

A Kinetic Approach to the Prediction of RNA Secondary Structures

Abstract

A new approach to the prediction of secondary RNA structures based on the analysis of the kinetics of molecular self-organisation is proposed herein. The Markov process is used to describe structural reconstructions during secondary structure formation. This process is modelled by a Monte-Carlo method. Examples of the calculation by this method of the secondary structures kinetic ensemble are given. Distribution of time-dependent probabilities within the ensembles is obtained.

An effective method for search for the equilibrium ensemble is also suggested. This method is based on the construction of a tree of all possible secondary structures of RNA. By ascribing a probability for each structure (according to its free energy) the Boltzmann equilibrium ensemble can be obtained.     

Prediction of the ensembles of RNA secondary structures. Kinetic analysis of self-organization

A new approach to the problem of prediction of secondary structures of RNA, which is based on the kinetic analysis of self-organising molecules is proposed. Structural reconstructions that take place during formation of secondary structures are described in terms of Markov process. A set of states and probability transition were defined. Monte-Carlo methods were used to describe this process. Probability distributions of various secondary structures depending on time are given. Examples of calculations for ensembles of secondary structures of some tRNAs are described. An effective method of steady-state ensemble research, which is based on a quick RESETTING of all possible variance of the secondary structures of RNAs is given. By ascribing to each of these structures the value of probabilities as a function of free energy it was possible to obtain the Boltzmann ensemble of secondary structures.     

Combinatorics of saturated secondary structures of RNA.

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

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.     

Asymptotic Number of Hairpins of Saturated RNA Secondary Structures

In the absence of chaperone molecules, RNA folding is believed to depend on the distribution of kinetic traps in the energy landscape of all secondary structures. Kinetic traps in the Nussinov energy model are precisely those secondary structures that are saturated, meaning that no base pair can be added without introducing either a pseudoknot or base triple. In this paper, we compute the asymptotic expected number of hairpins in saturated structures. For instance, if every hairpin is required to contain at least θ=3 unpaired bases and the probability that any two positions can base-pair is p=3/8, then the asymptotic number of saturated structures is 1.34685?n ^?3/2?1.62178 ⁿ , and the asymptotic expected number of hairpins follows a normal distribution with mean $0.06695640 \cdot n + 0.01909350 \cdot\sqrt{n} \cdot\mathcal{N}$ . Similar results are given for values θ=1,3, and p=1,1/2,3/8; for instance, when θ=1 and p=1, the asymptotic expected number of hairpins in saturated secondary structures is 0.123194?n, a value greater than the asymptotic expected number 0.105573?n of hairpins over all secondary structures. Since RNA binding targets are often found in hairpin regions, it follows that saturated structures present potentially more binding targets than nonsaturated structures, on average. Next, we describe a novel algorithm to compute the hairpin profile of a given RNA sequence: given RNA sequence a ₁,…,a _n , for each integer k, we compute that secondary structure S _k having minimum energy in the Nussinov energy model, taken over all secondary structures having k hairpins. We expect that an extension of our algorithm to the Turner energy model may provide more accurate structure prediction for particular RNAs, such as tRNAs and purine riboswitches, known to have a particular number of hairpins. Mathematica^? computations, C and Python source code, and additional supplementary information are available at the website http://bioinformatics.bc.edu/clotelab/RNAhairpinProfile/.     

Computing the partition function and sampling for saturated secondary structures of RNA, with respect to the Turner energy model.

An RNA secondary structure is saturated if no base pairs can be added without violating the definition of secondary structure. Here we describe a new algorithm, RNAsat, which for a given RNA sequence a, an integral temperature 0 相似文献   

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.     

Evaluation of a sophisticated SCFG design for RNA secondary structure prediction

Predicting secondary structures of RNA molecules is one of the fundamental problems of and thus a challenging task in computational structural biology. Over the past decades, mainly two different approaches have been considered to compute predictions of RNA secondary structures from a single sequence: the first one relies on physics-based and the other on probabilistic RNA models. Particularly, the free energy minimization (MFE) approach is usually considered the most popular and successful method. Moreover, based on the paradigm-shifting work by McCaskill which proposes the computation of partition functions (PFs) and base pair probabilities based on thermodynamics, several extended partition function algorithms, statistical sampling methods and clustering techniques have been invented over the last years. However, the accuracy of the corresponding algorithms is limited by the quality of underlying physics-based models, which include a vast number of thermodynamic parameters and are still incomplete. The competing probabilistic approach is based on stochastic context-free grammars (SCFGs) or corresponding generalizations, like conditional log-linear models (CLLMs). These methods abstract from free energies and instead try to learn about the structural behavior of the molecules by learning (a manageable number of) probabilistic parameters from trusted RNA structure databases. In this work, we introduce and evaluate a sophisticated SCFG design that mirrors state-of-the-art physics-based RNA structure prediction procedures by distinguishing between all features of RNA that imply different energy rules. This SCFG actually serves as the foundation for a statistical sampling algorithm for RNA secondary structures of a single sequence that represents a probabilistic counterpart to the sampling extension of the PF approach. Furthermore, some new ways to derive meaningful structure predictions from generated sample sets are presented. They are used to compare the predictive accuracy of our model to that of other probabilistic and energy-based prediction methods. Particularly, comparisons to lightweight SCFGs and corresponding CLLMs for RNA structure prediction indicate that more complex SCFG designs might yield higher accuracy but eventually require more comprehensive and pure training sets. Investigations on both the accuracies of predicted foldings and the overall quality of generated sample sets (especially on an abstraction level, called abstract shapes of generated structures, that is relevant for biologists) yield the conclusion that the Boltzmann distribution of the PF sampling approach is more centered than the ensemble distribution induced by the sophisticated SCFG model, which implies a greater structural diversity within generated samples. In general, neither of the two distinct ensemble distributions is more adequate than the other and the corresponding results obtained by statistical sampling can be expected to bare fundamental differences, such that the method to be preferred for a particular input sequence strongly depends on the considered RNA type.     

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).     

Improved free energy parameters for RNA pseudoknotted secondary structure prediction

Accurate prediction of RNA pseudoknotted secondary structures from the base sequence is a challenging computational problem. Since prediction algorithms rely on thermodynamic energy models to identify low-energy structures, prediction accuracy relies in large part on the quality of free energy change parameters. In this work, we use our earlier constraint generation and Boltzmann likelihood parameter estimation methods to obtain new energy parameters for two energy models for secondary structures with pseudoknots, namely, the Dirks–Pierce (DP) and the Cao–Chen (CC) models. To train our parameters, and also to test their accuracy, we create a large data set of both pseudoknotted and pseudoknot-free secondary structures. In addition to structural data our training data set also includes thermodynamic data, for which experimentally determined free energy changes are available for sequences and their reference structures. When incorporated into the HotKnots prediction algorithm, our new parameters result in significantly improved secondary structure prediction on our test data set. Specifically, the prediction accuracy when using our new parameters improves from 68% to 79% for the DP model, and from 70% to 77% for the CC model.     

Thermodynamic characterization of RNA 2 × 3 nucleotide internal loops

To better elucidate RNA structure-function relationships and to improve the design of pharmaceutical agents that target specific RNA motifs, an understanding of RNA primary, secondary, and tertiary structure is necessary. The prediction of RNA secondary structure from sequence is an intermediate step in predicting RNA three-dimensional structure. RNA secondary structure is typically predicted using a nearest neighbor model based on free energy parameters. The current free energy parameters for 2 × 3 nucleotide loops are based on a 23-member data set of 2 × 3 loops and internal loops of other sizes. A database of representative RNA secondary structures was searched to identify 2 × 3 nucleotide loops that occur in nature. Seventeen of the most frequent 2 × 3 nucleotide loops in this database were studied by optical melting experiments. Fifteen of these loops melted in a two-state manner, and the associated experimental ΔG°(37,2×3) values are, on average, 0.6 and 0.7 kcal/mol different from the values predicted for these internal loops using the predictive models proposed by Lu, Turner, and Mathews [Lu, Z. J., Turner, D. H., and Mathews, D. H. (2006) Nucleic Acids Res. 34, 4912-4924] and Chen and Turner [Chen, G., and Turner, D. H. (2006) Biochemistry 45, 4025-4043], respectively. These new ΔG°(37,2×3) values can be used to update the current algorithms that predict secondary structure from sequence. To improve free energy calculations for duplexes containing 2 × 3 nucleotide loops that still do not have experimentally determined free energy contributions, an updated predictive model was derived. This new model resulted from a linear regression analysis of the data reported here combined with 31 previously studied 2 × 3 nucleotide internal loops. Most of the values for the parameters in this new predictive model are within experimental error of those of the previous models, suggesting that approximations and assumptions associated with the derivation of the previous nearest neighbor parameters were valid. The updated predictive model predicts free energies of 2 × 3 nucleotide internal loops within 0.4 kcal/mol, on average, of the experimental free energy values. Both the experimental values and the updated predictive model can be used to improve secondary structure prediction from sequence.     

Moments of the Boltzmann distribution for RNA secondary structures

We here present a dynamic programming algorithm which is capable of calculating arbitrary moments of the Boltzmann distribution for RNA secondary structures. We have implemented the algorithm in a program called RNA-VARIANCE and investigate the difference between the Boltzmann distribution of biological and random RNA sequences. We find that the minimum free energy structure of biological sequences has a higher probability in the Boltzmann distribution than random sequences. Moreover, we show that the free energies of biological sequences have a smaller variance than random sequences and that the minimum free energy of biological sequences is closer to the expected free energy of the rest of the structures than that of random sequences. These results suggest that biologically functional RNA sequences not only require a thermodynamically stable minimum free energy structure, but also an ensemble of structures whose free energies are close to the minimum free energy.     

An efficient algorithm to compute the landscape of locally optimal RNA secondary structures with respect to the Nussinov-Jacobson energy model.

We make a novel contribution to the theory of biopolymer folding, by developing an efficient algorithm to compute the number of locally optimal secondary structures of an RNA molecule, with respect to the Nussinov-Jacobson energy model. Additionally, we apply our algorithm to analyze the folding landscape of selenocysteine insertion sequence (SECIS) elements from A. Bock (personal communication), hammerhead ribozymes from Rfam (Griffiths-Jones et al., 2003), and tRNAs from Sprinzl's database (Sprinzl et al., 1998). It had previously been reported that tRNA has lower minimum free energy than random RNA of the same compositional frequency (Clote et al., 2003; Rivas and Eddy, 2000), although the situation is less clear for mRNA (Seffens and Digby, 1999; Workman and Krogh, 1999; Cohen and Skienna, 2002),(1) which plays no structural role. Applications of our algorithm extend knowledge of the energy landscape differences between naturally occurring and random RNA. Given an RNA molecule a(1), ... , a(n) and an integer k > or = 0, a k-locally optimal secondary structure S is a secondary structure on a(1), ... , a(n) which has k fewer base pairs than the maximum possible number, yet for which no basepairs can be added without violation of the definition of secondary structure (e.g., introducing a pseudoknot). Despite the fact that the number numStr(k) of k-locally optimal structures for a given RNA molecule in general is exponential in n, we present an algorithm running in time O(n (4)) and space O(n (3)), which computes numStr(k) for each k. Structurally important RNA, such as SECIS elements, hammerhead ribozymes, and tRNA, all have a markedly smaller number of k-locally optimal structures than that of random RNA of the same dinucleotide frequency, for small and moderate values of k. This suggests a potential future role of our algorithm as a tool to detect noncoding RNA genes.     

Predicting common foldings of homologous RNAs.

A new approach is proposed for determining common RNA secondary structures within a set of homologous RNAs. The approach is a combination of phylogenetic and thermodynamic methods which is based on the prediction of optimal and suboptimal secondary structures, topological similarity searches and phylogenetic comparative analysis. The optimal and suboptimal RNA secondary structures are predicted by energy minimization. Structural comparison of the predicted RNA secondary structures is used to find conserved structures that are topologically similar in all these homologous RNAs. The validity of the conserved structural elements found is then checked by phylogenetic comparison of the sequences. This procedure is used to predict common structures of ribonuclease P (RNAase P) RNAs.     

RNA secondary structure prediction based on free energy and phylogenetic analysis.

We describe a computational method for the prediction of RNA secondary structure that uses a combination of free energy and comparative sequence analysis strategies. Using a homology-based sequence alignment as a starting point, all favorable pairings with respect to the Turner energy function are identified. Each potentially paired region within a multiple sequence alignment is scored using a function that combines both predicted free energy and sequence covariation with optimized weightings. High scoring regions are ranked and sequentially incorporated to define a growing secondary structure. Using a single set of optimized parameters, it is possible to accurately predict the foldings of several test RNAs defined previously by extensive phylogenetic and experimental data (including tRNA, 5 S rRNA, SRP RNA, tmRNA, and 16 S rRNA). The algorithm correctly predicts approximately 80% of the secondary structure. A range of parameters have been tested to define the minimal sequence information content required to accurately predict secondary structure and to assess the importance of individual terms in the prediction scheme. This analysis indicates that prediction accuracy most strongly depends upon covariational information and only weakly on the energetic terms. However, relatively few sequences prove sufficient to provide the covariational information required for an accurate prediction. Secondary structures can be accurately defined by alignments with as few as five sequences and predictions improve only moderately with the inclusion of additional sequences.     

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).     

RNA二级结构预测系统构建

运用下列RNA二级结构预测算法:碱基最大配对方法、Zuker极小化自由能方法、螺旋区最优堆积、螺旋区随机堆积和所有可能组合方法与基于一级螺旋区的RNA二级结构绘图技术, 构建了RNA二级结构预测系统Rnafold. 另外, 通过随机选取20个tRNA序列, 从自由能和三叶草结构两个方面比较了前4种二级结构预测算法, 并运用t检验方法分析了自由能的统计学差别. 从三叶草结构来看, 以随机堆积方法最好, 其次是螺旋区最优堆积方法和Zuker算法, 以碱基最大配对方法最差. 最后, 分析了两种极小化自由能方法之间的差别.     

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.