Base Pairing Probabilities Research Articles

Statistical and Bayesian approaches to RNA secondary structure prediction

Prediction of RNA secondary structure is a fundamental problem in computational structural biology. For several decades, free energy minimization has been the most popular method for prediction from a single sequence. In recent years, the McCaskill algorithm for computation of partition function and base-pair probabilities has become increasingly appreciated. This paradigm-shifting work has inspired the developments of extended partition function algorithms, statistical sampling and clustering, and application of Bayesian statistical inference. The performance of thermodynamics-based methods is limited by thermodynamic rules and parameters. However, further improvements may come from statistical estimates derived from structural databases for thermodynamics parameters with weak or little experimental data. The Bayesian inference approach appears to be promising in this context.

Complete probabilistic analysis of RNA shapes

BackgroundSoon after the first algorithms for RNA folding became available, it was recognised that the prediction of only one energetically optimal structure is insufficient to achieve reliable results. An in-depth analysis of the folding space as a whole appeared necessary to deduce the structural properties of a given RNA molecule reliably. Folding space analysis comprises various methods such as suboptimal folding, computation of base pair probabilities, sampling procedures and abstract shape analysis. Common to many approaches is the idea of partitioning the folding space into classes of structures, for which certain properties can be derived.ResultsIn this paper we extend the approach of abstract shape analysis. We show how to compute the accumulated probabilities of all structures that share the same shape. While this implies a complete (non-heuristic) analysis of the folding space, the computational effort depends only on the size of the shape space, which is much smaller. This approach has been integrated into the tool RNAshapes, and we apply it to various RNAs.ConclusionAnalyses of conformational switches show the existence of two shapes with probabilities approximately vs. , whereas the analysis of a microRNA precursor reveals one shape with a probability near to 1.0. Furthermore, it is shown that a shape can outperform an energetically more favourable one by achieving a higher probability. From these results, and the fact that we use a complete and exact analysis of the folding space, we conclude that this approach opens up new and promising routes for investigating and understanding RNA secondary structure.

Partition function and base pairing probabilities of RNA heterodimers.

BackgroundRNA has been recognized as a key player in cellular regulation in recent years. In many cases, non-coding RNAs exert their function by binding to other nucleic acids, as in the case of microRNAs and snoRNAs. The specificity of these interactions derives from the stability of inter-molecular base pairing. The accurate computational treatment of RNA-RNA binding therefore lies at the heart of target prediction algorithms.MethodsThe standard dynamic programming algorithms for computing secondary structures of linear single-stranded RNA molecules are extended to the co-folding of two interacting RNAs.ResultsWe present a program, RNAcofold, that computes the hybridization energy and base pairing pattern of a pair of interacting RNA molecules. In contrast to earlier approaches, complex internal structures in both RNAs are fully taken into account. RNAcofold supports the calculation of the minimum energy structure and of a complete set of suboptimal structures in an energy band above the ground state. Furthermore, it provides an extension of McCaskill's partition function algorithm to compute base pairing probabilities, realistic interaction energies, and equilibrium concentrations of duplex structures.AvailabilityRNAcofold is distributed as part of the Vienna RNA Package, .ContactStephan H. Bernhart – berni@tbi.univie.ac.at

Local RNA base pairing probabilities in large sequences

The genome-wide search for non-coding RNAs requires efficient methods to compute and compare local secondary structures. Since the exact boundaries of such putative transcripts are typically unknown, arbitrary sequence windows have to be used in practice. Here we present a method for robustly computing the probabilities of local base pairs from long RNA sequences independent of the exact positions of the sequence window. The program RNAplfold is part of the Vienna RNA Package and can be downloaded from http://www.tbi.univie.ac.at/RNA/.

Protein-DNA binding specificity predictions with structural models

Protein–DNA interactions play a central role in transcriptional regulation and other biological processes. Investigating the mechanism of binding affinity and specificity in protein–DNA complexes is thus an important goal. Here we develop a simple physical energy function, which uses electrostatics, solvation, hydrogen bonds and atom-packing terms to model direct readout and sequence-specific DNA conformational energy to model indirect readout of DNA sequence by the bound protein. The predictive capability of the model is tested against another model based only on the knowledge of the consensus sequence and the number of contacts between amino acids and DNA bases. Both models are used to carry out predictions of protein–DNA binding affinities which are then compared with experimental measurements. The nearly additive nature of protein–DNA interaction energies in our model allows us to construct position-specific weight matrices by computing base pair probabilities independently for each position in the binding site. Our approach is less data intensive than knowledge-based models of protein–DNA interactions, and is not limited to any specific family of transcription factors. However, native structures of protein–DNA complexes or their close homologs are required as input to the model. Use of homology modeling can significantly increase the extent of our approach, making it a useful tool for studying regulatory pathways in many organisms and cell types.

DINAMelt web server for nucleic acid melting prediction

The DINAMelt web server simulates the melting of one or two single-stranded nucleic acids in solution. The goal is to predict not just a melting temperature for a hybridized pair of nucleic acids, but entire equilibrium melting profiles as a function of temperature. The two molecules are not required to be complementary, nor must the two strand concentrations be equal. Competition among different molecular species is automatically taken into account. Calculations consider not only the heterodimer, but also the two possible homodimers, as well as the folding of each single-stranded molecule. For each of these five molecular species, free energies are computed by summing Boltzmann factors over every possible hybridized or folded state. For temperatures within a user-specified range, calculations predict species mole fractions together with the free energy, enthalpy, entropy and heat capacity of the ensemble. Ultraviolet (UV) absorbance at 260 nm is simulated using published extinction coefficients and computed base pair probabilities. All results are available as text files and plots are provided for species concentrations, heat capacity and UV absorbance versus temperature. This server is connected to an active research program and should evolve as new theory and software are developed. The server URL is .

A statistical analysis of RNA folding algorithms through thermodynamic parameter perturbation

Computational RNA secondary structure prediction is rather well established. However, such prediction algorithms always depend on a large number of experimentally measured parameters. Here, we study how sensitive structure prediction algorithms are to changes in these parameters. We found already that for changes corresponding to the actual experimental error to which these parameters have been determined, 30% of the structure are falsely predicted whereas the ground state structure is preserved under parameter perturbation in only 5% of all the cases. We establish that base-pairing probabilities calculated in a thermal ensemble are viable although not a perfect measure for the reliability of the prediction of individual structure elements. Here, a new measure of stability using parameter perturbation is proposed, and its limitations are discussed.

Using an RNA secondary structure partition function to determine confidence in base pairs predicted by free energy minimization.

A partition function calculation for RNA secondary structure is presented that uses a current set of nearest neighbor parameters for conformational free energy at 37 degrees C, including coaxial stacking. For a diverse database of RNA sequences, base pairs in the predicted minimum free energy structure that are predicted by the partition function to have high base pairing probability have a significantly higher positive predictive value for known base pairs. For example, the average positive predictive value, 65.8%, is increased to 91.0% when only base pairs with probability of 0.99 or above are considered. The quality of base pair predictions can also be increased by the addition of experimentally determined constraints, including enzymatic cleavage, flavin mono-nucleotide cleavage, and chemical modification. Predicted secondary structures can be color annotated to demonstrate pairs with high probability that are therefore well determined as compared to base pairs with lower probability of pairing.

An algorithm for computing nucleic acid base-pairing probabilities including pseudoknots.

Given a nucleic acid sequence, a recent algorithm allows the calculation of the partition function over secondary structure space including a class of physically relevant pseudoknots. Here, we present a method for computing base-pairing probabilities starting from the output of this partition function algorithm. The approach relies on the calculation of recursion probabilities that are computed by backtracking through the partition function algorithm, applying a particular transformation at each step. This transformation is applicable to any partition function algorithm that follows the same basic dynamic programming paradigm. Base-pairing probabilities are useful for analyzing the equilibrium ensemble properties of natural and engineered nucleic acids, as demonstrated for a human telomerase RNA and a synthetic DNA nanostructure.

Analyzing and enhancing mRNA translational efficiency in an Escherichia coli in vitro expression system

The dependence of efficiency of translation initiation on mRNA sequence parameters was investigated in an Escherichia coli in vitro expression system. We designed a large-scale expression experiment focussing on the influence of sequence variations in the translated region (TR) of the mRNA without changing the 5 ′-untranslated region (5 ′-UTR). The level of translated protein from 756 expression constructs was measured and the influence of a large number of possible effector attributes was statistically analyzed. Base exchanges immediately adjacent to the start codon up to nucleotide +25 had a profound effect on translational efficiency. Correlation analysis revealed a significant dependence on base pair probability and G + C content on the expression level, indicating that mRNA secondary structure in this region hampers translation. Using our training data, we developed a methodology to predict and improve the translation efficiency of open reading frames (ORFs).

Speed-up of DNA melting algorithm with complete nearest neighbor properties.

We describe an optimized algorithm, which is faster and more accurate compared to previously described algorithms, for computing the statistical mechanics of denaturation of nucleic acid sequences according to the classical Poland-Scheraga type of model. Nearest neighbor thermodynamics has been included in a complete and general way, by rigorously treating nearest neighbor interactions, helix end interactions, and isolated base-pairs. This avoids the simplifications of previous approaches and achieves full generality and controllability with respect to thermodynamic modeling. The algorithm computes subchain partition functions by recursion, from which various quantitative aspects of the melting process are easily derived, for example the base-pairing probability profiles. The algorithm represents an optimization with respect to algorithmic complexity of the partition function algorithm of Yeramian et al. (Biopolymers 1990, 30, 481-497): we reduce the computation time for a base-pairing probability profile from O(N2) to O(N), where N is the sequence length. This speed-up comes in addition to the speed-up due to a multiexponential approximation of the loop entropy factor as introduced by Fixman and Freire22 and applied by Yeramian et al. The speed-up, however, is independent of the multiexponential approximation and reduces time from O(N3) to O(N2) in the exact case. A method for representing very large numbers is described, which avoids numerical overflow in the partition functions for genomic length sequences. In addition to calculating the standard base-pairing probability profiles, we propose to use the algorithm to calculate various other probabilities (loops, helices, tails) for a more direct view of the melting regions and their positions and sizes. This can provide a better understanding of the physics of denaturation and the biology of genomes.

Prediction of RNA base pairing probabilities on massively parallel computers.

We present an implementation of McCaskill's algorithm for computing the base pair probabilities of an RNA molecule for massively parallel message passing architectures. The program can be used to routinely fold RNA sequences of more than 10,000 nucleotides. Applications to complete viral genomes are discussed.

ConStruct: a tool for thermodynamic controlled prediction of conserved secondary structure.

A tool for prediction of conserved secondary structure of a set of homologous single-stranded RNAs is presented. For each RNA of the set the structure distribution is calculated and stored in a base pair probability matrix. Gaps, resulting from a multiple sequence alignment of the RNA set, are introduced into the individual probability matrices. These 'aligned' probability matrices are summed up to give a consensus probability matrix emphasizing the conserved structural elements of the RNA set. Because the multiple sequence alignment is independent of any structural constraints, such an alignment may result in introduction of gaps into the homologous probability matrices that disrupt a common consensus structure. By use of its graphical user interface the presented tool allows the removal of such misalignments, which are easily recognized, from the individual probability matrices by optimizing the sequence alignment with respect to a structural alignment. From the consensus probability matrix a consensus structure is extracted, which is viewable in three different graphical representations. The functionality of the tool is demonstrated using a small set of U7 RNAs, which are involved in 3'-end processing of histone mRNA precursors. Supplementary Material lists further results obtained. Advantages and drawbacks of the tool are discussed in comparison to several other algorithms.

Discretized torsional dynamics and the folding of an RNA chain.

The aim of this work is to implement a discrete coarse codification of local torsional states of the RNA chain backbone in order to explore the long-time limit dynamics and ultimately obtain a coarse solution to the RNA folding problem. A discrete representation of the soft-mode dynamics is turned into an algorithm for a rough structure prediction. The algorithm itself is inherently parallel, as it evaluates concurrent folding possibilities by pattern recognition, but it may be implemented in a personal computer as a chain of perturbation-translation-renormalization cycles performed on a binary matrix of local topological constraints. This requires suitable representational tools and a periodic quenching of the dynamics for system renormalization. A binary coding of local topological constraints associated with each structural motif is introduced, with each local topological constraint corresponding to a local torsional state. This treatment enables us to adopt a computation time step far larger than hydrodynamic drag time scales. Accordingly, the solvent is no longer treated as a hydrodynamic drag medium. Instead we incorporate its capacity for forming local conformation-dependent dielectric domains. Each translation of the matrix of local topological constraints (LTM's) depends on the conformation-dependent local dielectric created by a confined solvent. Folding pathways are resolved as transitions between patterns of locally encoded structural signals which change within the 1 ns-100 ms time scale range. These coarse folding pathways are generated by a search at regular intervals for structural patterns in the LTM. Each pattern is recorded as a base-pairing pattern (BPP) matrix, a consensus-evaluation operation subject to a renormalization feedback loop. Since several mutually conflicting consensus evaluations might occur at a given time, the need arises for a probabilistic approach appropriate for an ensemble of RNA molecules. Thus, a statistical dynamics of consensus formation is determined by the time evolution of the base pairing probability matrix. These dynamics are generated for a functional RNA molecule, a representative of the so-called group I ribozymes, in order to test the model. The resulting ensemble of conformations is sharply peaked and the most probable structure features the predominance of all phylogenetically conserved intrachain helices tantamount to ribozyme function. Furthermore, the magnesium-aided cooperativity that leads to the shaping of the catalytic core is elucidated. Once the predictive folding algorithm has been implemented, the validity of the so-called "adiabatic approximation" is tested. This approximation requires that conformational microstates be lumped up into BPP's which are treated as quasiequilibrium states, while folding pathways are coarsely represented as sequences of BPP transitions. To test the validity of this adiabatic ansatz, a computation of the coarse Shannon information entropy sigma associated to the specific partition of conformation space into BPP's is performed taking into account the LTM evolution and contrasted with the adiabatic computation. The results reveal a subordination of torsional microstate dynamics to BPP transitions within time scales relevant to folding. This adiabatic entrainment in the long-time limit is thus identified as responsible for the expediency of the folding process.

Automatic detection of conserved base pairing patterns in RNA virus genomes

Almost all RNA molecules—and consequently also almost all subsequences of a large RNA molecule—form secondary structures. The presence of secondary structure in itself therefore does not indicate any functional significance. In fact, we cannot expect a conserved secondary structure for all parts of a viral genome or a mRNA, even if there is a significant level of sequence conservation. We present a novel method for detecting conserved RNA secondary structures in a family of related RNA sequences. The method is based on combining the prediction of base pair probability matrices and comparative sequence analysis. It can be applied to small sets of long sequences and does not require a prior knowledge of conserved sequence or structure motifs. As such it can be used to scan large amounts of sequence data for regions that warrant further experimental investigation. Applications to complete genomic RNAs of some viruses show that in all cases the known secondary structure features are identified. In addition, we predict a substantial number of conserved structural elements which have not been described so far.

Using reliability information to annotate RNA secondary structures.

A number of heuristic descriptors have been developed previously in conjunction with the mfold package that describe the propensity of individual bases to participate in base pairs and whether or not a predicted helix is "well-determined." They were developed for the "energy dot plot" output of mfold. Two descriptors, P-num and H-num, are used to measure the level of promiscuity in the association of any given nucleotide or helix with alternative complementary pairs. The third descriptor, S-num, measures the propensity of bases to be single-stranded. In the current work, we describe a series of programs that were developed in order to annotate individual structures with "well-definedness" information. We use color annotation to present the information. The programs can annotate PostScript files that are created by the mfold package or the PostScript secondary structure plots produced by the Weiser and Noller program XRNA (Weiser B, Noller HF, 1995, XRNA: Auto-interactive program for modeling RNA, The Center for Molecular Biology of RNA, Santa Cruz, California: University of California; Internet: ftp://fangio.ucsc.edu/pub/XRNA). In addition, these programs can annotate ss files that serve as input to XRNA. The annotation package can also handle structure comparison with a reference structure. This feature can be used to compare predicted structure with a phylogenetically deduced model, to compare two different predicted foldings, and to identify conformational changes that are predicted between wild-type and mutant RNAs. We provide several examples of application. Predicted structures of two RNase P RNAs were colored with P-num information and further annotated with comparative information. The comparative model of a 16S rRNA was annotated with P-num information from mfold and with base pair probabilities obtained from the Vienna RNA folding package. Further annotation adds comparisons with the optimal foldings obtained from mfold and the Vienna package, respectively. The results of all of these analyses are discussed in the context of the reliability of structure prediction.

Assessing the reliability of RNA folding using statistical mechanics

We have analyzed the base-pairing probability distributions of 16 S and 16 S-like, and 23 S and 23 S-like ribosomal RNAs of Archaea, Bacteria, chloroplasts, mitochondria and Eukarya, as predicted by the partition function approach for RNA folding introduced by McCaskill. A quantitative analysis of the reliability of RNA folding is done by comparing the base-pairing probability distributions with the structures predicted by comparative sequence analysis (comparative structures). We distinguish two factors that show a relationship to the reliability of RNA minimum free energy structure. The first factor is the dominance of one particular base-pair or the absence of base-pairing for a given base within the base-pairing probability distribution (BPPD). We characterize the BPPD per base, including the probability of not base-pairing, by its Shannon entropy ( S). The S value indicates the uncertainty about the base-pairing of a base: low S values result from BPPDs that are strongly dominated by a single base-pair or by the absence of base-pairing. We show that bases with low S values have a relatively high probability that their minimum free energy (MFE) structure corresponds to the comparative structure. The BPPDs of prokaryotes that live at high temperatures (thermophilic Archaea and Bacteria) have, calculated at 37°C, lower S values than the BPPDs of prokaryotes that live at lower temperatures (mesophilic and psychrophilic Archaea and Bacteria). This reflects an adaptation of the ribosomal RNAs to the environmental temperature. A second factor that is important to consider with regard to the reliability of MFE structure folding is a variable degree of applicability of the thermodynamic model of RNA folding for different groups of RNAs. Here we show that among the bases that show low S values, the Archaea and Bacteria have similar, high probabilities (0.96 and 0.94 in 16 S and 0.93 and 0.91 in 23 S, respectively) that the MFE structure corresponds to the comparative structure. These probabilities are lower in the chloroplasts (16 S 0.91, 23 S 0.79), mitochondria (16 S-like 0.89, 23 S-like 0.69) and Eukarya (18 S 0.81, 28 S 0.86).

Thermodynamic Prediction of Conserved Secondary Structure: Application to the RRE Element of HIV, the tRNA-like Element of CMV and the mRNA of Prion Protein

An algorithm for prediction of conserved secondary structure of single-stranded RNA is presented. For each RNA of a set of homologous RNAs optimal and suboptimal secondary structures are calculated and stored in a base-pair probability matrix. A multiple sequence alignment is performed for the set of RNAs. The resulting gaps are introduced into the individual probability matrices. These homologous probability matrices are summed to give a consensus probability matrix emphasizing the conserved secondary structure elements of the RNA set. Thus the algorithm combines the advantages of thermodynamic structure prediction by energy minimization with the information obtained from phylogenetic alignment of sequences. The algorithm is applied to three examples. The REV-responsive element of HIV, the structure of which is well known from the literature, was chosen to test the algorithm. The second example is the 3′ terminal segment of genomic single-stranded RNAs of cucumber mosaic viruses; a structure similar to that of the related brome mosaic virus was expected and was confirmed. The third example is the prion-protein mRNA from different organisms; the structure of this mRNA is not known. By application of the algorithm highly conserved hairpins were found in the prion-protein mRNA.

Base pairing probabilities in a complete HIV-1 RNA.

We have calculated the base pair probability distribution for the secondary structure of a full length HIV-1 genome using the partition function approach introduced by McCaskill (1990). By analyzing the full distribution of base pair probabilities instead of a restricted number of secondary structures, we gain more complete and reliable information about the secondary structure of HIV-1. We introduce methods that condense the information in the probability distribution to one value per nucleotide in the sequence. Using these methods we represent the secondary structure as a weighted average of the base pair probabilities, and we can identify interesting secondary structures that have relatively well-defined base pairing. The results show high probabilities for the known secondary structures at the 5'-end of the molecule that have been predicted on the basis of biochemical data. The Rev response element (RRE) appears as a distinct element in the secondary structure. It has a meta-stable domain at the high affinity site for the binding of Rev. The overall structure decomposes into fairly small independent structures in the first 4,000 bases of the molecule. The remaining 5,000 bases (excluding the terminal repeat) form a single, large structure, on top of which the RRE is located.

RNAdraw: an integrated program for RNA secondary structure calculation and analysis under 32-bit Microsoft Windows.

The use of algorithms for calculation and analysis of RNA secondary structures has largely been limited to mainframe computers, mainly due to the 16-bit memory restrictions imposed by MS-DOS. The program presented here, RNAdraw, moves calculations to the 32-bit Microsoft Windows environments with an intuitive user interface with extensive viewing, editing and printing possibilities. The algorithms for secondary structure/basepair probability matrix/heat curve calculation have been ported directly from a 32-bit Unix environment. RNAdraw also offers novel features such as the options to edit energy parameters, extract structures of different probability levels, create de novo secondary structures interactively, and combine viewing of structures and basepair probabilities.