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.     

The equilibrium partition function and base pair binding probabilities for RNA secondary structure.

A novel application of dynamic programming to the folding problem for RNA enables one to calculate the full equilibrium partition function for secondary structure and the probabilities of various substructures. In particular, both the partition function and the probabilities of all base pairs are computed by a recursive scheme of polynomial order N3 in the sequence length N. The temperature dependence of the partition function gives information about melting behavior for the secondary structure. The pair binding probabilities, the computation of which depends on the partition function, are visually summarized in a "box matrix" display and this provides a useful tool for examining the full ensemble of probable alternative equilibrium structures. The calculation of this ensemble representation allows a proper application and assessment of the predictive power of the secondary structure method, and yields important information on alternatives and intermediates in addition to local information about base pair opening and slippage. The results are illustrated for representative tRNA, 5S RNA, and self-replicating and self-splicing RNA molecules, and allow a direct comparison with enzymatic structure probes. The effect of changes in the thermodynamic parameters on the equilibrium ensemble provides a further sensitivity check to the predictions.     

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.     

A statistical sampling algorithm for RNA secondary structure prediction

An RNA molecule, particularly a long-chain mRNA, may exist as a population of structures. Further more, multiple structures have been demonstrated to play important functional roles. Thus, a representation of the ensemble of probable structures is of interest. We present a statistical algorithm to sample rigorously and exactly from the Boltzmann ensemble of secondary structures. The forward step of the algorithm computes the equilibrium partition functions of RNA secondary structures with recent thermodynamic parameters. Using conditional probabilities computed with the partition functions in a recursive sampling process, the backward step of the algorithm quickly generates a statistically representative sample of structures. With cubic run time for the forward step, quadratic run time in the worst case for the sampling step, and quadratic storage, the algorithm is efficient for broad applicability. We demonstrate that, by classifying sampled structures, the algorithm enables a statistical delineation and representation of the Boltzmann ensemble. Applications of the algorithm show that alternative biological structures are revealed through sampling. Statistical sampling provides a means to estimate the probability of any structural motif, with or without constraints. For example, the algorithm enables probability profiling of single-stranded regions in RNA secondary structure. Probability profiling for specific loop types is also illustrated. By overlaying probability profiles, a mutual accessibility plot can be displayed for predicting RNA:RNA interactions. Boltzmann probability-weighted density of states and free energy distributions of sampled structures can be readily computed. We show that a sample of moderate size from the ensemble of an enormous number of possible structures is sufficient to guarantee statistical reproducibility in the estimates of typical sampling statistics. Our applications suggest that the sampling algorithm may be well suited to prediction of mRNA structure and target accessibility. The algorithm is applicable to the rational design of small interfering RNAs (siRNAs), antisense oligonucleotides, and trans-cleaving ribozymes in gene knock-down studies.     

A kinetic model of RNA folding

The RNA folding process is represented as a Markov process with states corresponding to RNA secondary structures and transition probabilities corresponding to transformations of a secondary structure caused by formation or disintegration of a helix. Transition probabilities (kinetic constants) are determined. A notion of a group of structures is introduced, and it allows to reduce the state space. Energetic and kinetic parameters of pseudoknots are estimated. Algorithms for computation of a kinetic ensemble for structures and groups of structures are presented, as well as their modifications that take into account pseudoknots. The described algorithms are implemented as a procedure for prediction of RNA secondary structure that is included in the package DNA-SUN.     

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.     

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.     

Fuzzy kernel clustering of RNA secondary structure ensemble using a novel similarity metric

Measuring the (dis)similarity between RNA secondary structures is critical for the study of RNA secondary structures and has implications to RNA functional characterization. Although a number of methods have been developed for comparing RNA structural similarities, their applications have been limited by the complexity of the required computation. In this paper, we present a novel method for comparing the similarity of RNA secondary structures generated from the same RNA sequence, i.e., a secondary structure ensemble, using a matrix representation of the RNA structures. Relevant features of the RNA secondary structures can be easily extracted through singular value decomposition (SVD) of the representing matrices. We have mapped the feature vectors of the singular values to a kernel space, where (dis)similarities among the mapped feature vectors become more evident, making clustering of RNA secondary structures easier to handle. The pair-wise comparison of RNA structures is achieved through computing the distance between the singular value vectors in the kernel space. We have applied a fuzzy kernel clustering method, using this similarity metric, to cluster the RNA secondary structure ensembles. Our application results suggest that our fuzzy kernel clustering method is highly promising for classifications of RNA structure ensembles, because of its low computational complexity and high clustering accuracy.     

Fuzzy Kernel Clustering of RNA Secondary Structure Ensemble Using a Novel Similarity Metric

Abstract

Measuring the (dis)similarity between RNA secondary structures is critical for the study of RNA secondary structures and has implications to RNA functional characterization. Although a number of methods have been developed for comparing RNA structural similarities, their applications have been limited by the complexity of the required computation. In this paper, we present a novel method for comparing the similarity of RNA secondary structures generated from the same RNA sequence, i.e., a secondary structure ensemble, using a matrix representation of the RNA structures. Relevant features of the RNA secondary structures can be easily extracted through singular value decomposition (SVD) of the representing matrices. We have mapped the feature vectors of the singular values to a kernel space, where (dis)similarities among the mapped feature vectors become more evident, making clustering of RNA secondary structures easier to handle. The pair-wise comparison of RNA structures is achieved through computing the distance between the singular value vectors in the kernel space. We have applied a fuzzy kernel clustering method, using this similarity metric, to cluster the RNA secondary structure ensembles. Our application results suggest that our fuzzy kernel clustering method is highly promising for classifications of RNA structure ensembles, because of its low computational complexity and high clustering accuracy.     

Pfold: RNA secondary structure prediction using stochastic context-free grammars

RNA secondary structures are important in many biological processes and efficient structure prediction can give vital directions for experimental investigations. Many available programs for RNA secondary structure prediction only use a single sequence at a time. This may be sufficient in some applications, but often it is possible to obtain related RNA sequences with conserved secondary structure. These should be included in structural analyses to give improved results. This work presents a practical way of predicting RNA secondary structure that is especially useful when related sequences can be obtained. The method improves a previous algorithm based on an explicit evolutionary model and a probabilistic model of structures. Predictions can be done on a web server at http://www.daimi.au.dk/~compbio/pfold.     

The theoretical analysis of the process of RNA molecule self-assembly

The Kinetic approach to the problem of the RNA structure prediction based on the analysis of the molecule self-formation is proposed. Re-structurization that occurs during processing is described in terms of Markov processes. A new formalism designating nucleotides by complex numbers is proposed, leading to the complex unitary space of nucleic vectors. Properties of structure and transition matrices are discussed in relation to the analysis of RNA structural formation processes. The non-linear dynamic behavior of secondary structure transitions is analyzed. Soliton-like oscillations of RNA and DNA tertiary structures are predicted. The Monte-Carlo simulation of the RNA structure self-formation is used to calculate the ensemble of the secondary structures of the tRNA^Ala precursor from Bombix mori formed during processing.     

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.     

Visualizing the global secondary structure of a viral RNA genome with cryo-electron microscopy

The lifecycle, and therefore the virulence, of single-stranded (ss)-RNA viruses is regulated not only by their particular protein gene products, but also by the secondary and tertiary structure of their genomes. The secondary structure of the entire genomic RNA of satellite tobacco mosaic virus (STMV) was recently determined by selective 2′-hydroxyl acylation analyzed by primer extension (SHAPE). The SHAPE analysis suggested a single highly extended secondary structure with much less branching than occurs in the ensemble of structures predicted by purely thermodynamic algorithms. Here we examine the solution-equilibrated STMV genome by direct visualization with cryo-electron microscopy (cryo-EM), using an RNA of similar length transcribed from the yeast genome as a control. The cryo-EM data reveal an ensemble of branching patterns that are collectively consistent with the SHAPE-derived secondary structure model. Thus, our results both elucidate the statistical nature of the secondary structure of large ss-RNAs and give visual support for modern RNA structure determination methods. Additionally, this work introduces cryo-EM as a means to distinguish between competing secondary structure models if the models differ significantly in terms of the number and/or length of branches. Furthermore, with the latest advances in cryo-EM technology, we suggest the possibility of developing methods that incorporate restraints from cryo-EM into the next generation of algorithms for the determination of RNA secondary and tertiary structures.     

RNA Sampler: a new sampling based algorithm for common RNA secondary structure prediction and structural alignment

MOTIVATION: Non-coding RNA genes and RNA structural regulatory motifs play important roles in gene regulation and other cellular functions. They are often characterized by specific secondary structures that are critical to their functions and are often conserved in phylogenetically or functionally related sequences. Predicting common RNA secondary structures in multiple unaligned sequences remains a challenge in bioinformatics research. Methods and RESULTS: We present a new sampling based algorithm to predict common RNA secondary structures in multiple unaligned sequences. Our algorithm finds the common structure between two sequences by probabilistically sampling aligned stems based on stem conservation calculated from intrasequence base pairing probabilities and intersequence base alignment probabilities. It iteratively updates these probabilities based on sampled structures and subsequently recalculates stem conservation using the updated probabilities. The iterative process terminates upon convergence of the sampled structures. We extend the algorithm to multiple sequences by a consistency-based method, which iteratively incorporates and reinforces consistent structure information from pairwise comparisons into consensus structures. The algorithm has no limitation on predicting pseudoknots. In extensive testing on real sequence data, our algorithm outperformed other leading RNA structure prediction methods in both sensitivity and specificity with a reasonably fast speed. It also generated better structural alignments than other programs in sequences of a wide range of identities, which more accurately represent the RNA secondary structure conservations. AVAILABILITY: The algorithm is implemented in a C program, RNA Sampler, which is available at http://ural.wustl.edu/software.html     

RNA structure prediction from evolutionary patterns of nucleotide composition

Structural elements in RNA molecules have a distinct nucleotide composition, which changes gradually over evolutionary time. We discovered certain features of these compositional patterns that are shared between all RNA families. Based on this information, we developed a structure prediction method that evaluates candidate structures for a set of homologous RNAs on their ability to reproduce the patterns exhibited by biological structures. The method is named SPuNC for ‘Structure Prediction using Nucleotide Composition’. In a performance test on a diverse set of RNA families we demonstrate that the SPuNC algorithm succeeds in selecting the most realistic structures in an ensemble. The average accuracy of top-scoring structures is significantly higher than the average accuracy of all ensemble members (improvements of more than 20% observed). In addition, a consensus structure that includes the most reliable base pairs gleaned from a set of top-scoring structures is generally more accurate than a consensus derived from the full structural ensemble. Our method achieves better accuracy than existing methods on several RNA families, including novel riboswitches and ribozymes. The results clearly show that nucleotide composition can be used to reveal the quality of RNA structures and thus the presented technique should be added to the set of prediction tools.     

Monte Carlo simulation for single RNA unfolding by force

Using polymer elastic theory and known RNA free energies, we construct a Monte Carlo algorithm to simulate the single RNA folding and unfolding by mechanical force on the secondary structure level. For the constant force ensemble, we simulate the force-extension curves of the P5ab, P5abc deltaA, and P5abc molecules in equilibrium. For the constant extension ensemble, we focus on the mechanical behaviors of the RNA P5ab molecule, which include the unfolding force dependence on the pulling speed, the force-hysteresis phenomenon, and the coincidence of stretching-relaxing force-curves in thermal equilibrium. We particularly simulate the time traces of the end-to-end distance of the P5ab under the constant force in equilibrium, which also have been recorded in the recent experiment. The reaction rate constants for the folding and unfolding are calculated. Our results show that the agreement between the simulation and the experimental measurements is satisfactory.