Improvements of the hierarchical approach for predicting RNA tertiary structure

Computational prediction of RNA tertiary structures is a significant challenge, especially for longer RNA and pseudoknots. At present it is still difficult to do this by pure all-atom molecular dynamics simulation. One of possible approaches is through hierarchical steps: from sequence to secondary structure and then to tertiary structure. Here we present improvements of two key steps of this approach, the manual adjustment of atom clashes and bond stretches and molecular dynamics refinement. We provide an energy function to find the locations of atom clashes and bond stretches and to guide their manual adjustment and a new scheme of molecular dynamics refinement using a tested combination of solvent model and the ff98 Amber force field suitable for RNA. We predicted with higher accuracy the tertiary structures of nine typical RNA molecules of lengths from 12 to 52, including hairpins, duplex helices and pseudoknots.     

RNA loop structure prediction via bond scaling and relaxation

We have developed a method for predicting the structure of small RNA loops that can be used to augment already existing RNA modeling techniques. The method requires no input constraints on loop configuration other than end-to-end distance. Initial loop structures are generated by randomizing the torsion angles, beginning at one end of the polynucleotide chain and correlating each successive angle with the previous. The bond lengths of these structures are then scaled to fit within the known end constraints and the equilibrium bond lengths of the potential energy function are scaled accordingly. Through a series of rescaling and minimization steps the structures are allowed to relax to lower energy configurations with standard bond lengths and reduced van der Waals clashes. This algorithm has been tested on the variable loops of yeast tRNA-Asp and yeast tRNA-Phe, as well as the sarcin-ricin tetraloop and the anticodon loop of yeast tRNA-Phe. The results indicate good correlation between potential energy and the loop structure predictions that are closest to the variable loop crystal structures, but poorer correlation for the more isolated stem loops. The number of stacking interactions has proven to be a good objective measure of the best loop predictions. Selecting on the basis of energy and stacking, we obtain two structures with 0.65 and 0.75 Å all-atom rms deviations (RMSD) from the crystal structure for the tRNA-Asp variable loop. The best structure prediction for the tRNA-Phe variable loop has an all-atom RMSD of 2.2 Å and a backbone RMSD of 1.6 Å, with a single base responsible for most of the deviation. For the sarcin-ricin loop from 28S ribosomal RNA, the predicted structure's all-atom RMSD from the nmr structure is 1.0 Å. We obtain a 1.8 Å RMSD structure for the tRNA-Phe anticodon loop. © 1996 John Wiley & Sons, Inc.     

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.     

Fully differentiable coarse-grained and all-atom knowledge-based potentials for RNA structure evaluation

RNA molecules play integral roles in gene regulation, and understanding their structures gives us important insights into their biological functions. Despite recent developments in template-based and parameterized energy functions, the structure of RNA--in particular the nonhelical regions--is still difficult to predict. Knowledge-based potentials have proven efficient in protein structure prediction. In this work, we describe two differentiable knowledge-based potentials derived from a curated data set of RNA structures, with all-atom or coarse-grained representation, respectively. We focus on one aspect of the prediction problem: the identification of native-like RNA conformations from a set of near-native models. Using a variety of near-native RNA models generated from three independent methods, we show that our potential is able to distinguish the native structure and identify native-like conformations, even at the coarse-grained level. The all-atom version of our knowledge-based potential performs better and appears to be more effective at discriminating near-native RNA conformations than one of the most highly regarded parameterized potential. The fully differentiable form of our potentials will additionally likely be useful for structure refinement and/or molecular dynamics simulations.     

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.     

MINT: software to identify motifs and short-range interactions in trajectories of nucleic acids

Structural biology experiments and structure prediction tools have provided many high-resolution three-dimensional structures of nucleic acids. Also, molecular dynamics force field parameters have been adapted to simulating charged and flexible nucleic acid structures on microsecond time scales. Therefore, we can generate the dynamics of DNA or RNA molecules, but we still lack adequate tools for the analysis of the resulting huge amounts of data. We present MINT (Motif Identifier for Nucleic acids Trajectory) — an automatic tool for analyzing three-dimensional structures of RNA and DNA, and their full-atom molecular dynamics trajectories or other conformation sets (e.g. X-ray or nuclear magnetic resonance-derived structures). For each RNA or DNA conformation MINT determines the hydrogen bonding network resolving the base pairing patterns, identifies secondary structure motifs (helices, junctions, loops, etc.) and pseudoknots. MINT also estimates the energy of stacking and phosphate anion-base interactions. For many conformations, as in a molecular dynamics trajectory, MINT provides averages of the above structural and energetic features and their evolution. We show MINT functionality based on all-atom explicit solvent molecular dynamics trajectory of the 30S ribosomal subunit.     

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.     

Ab initio construction of protein tertiary structures using a hierarchical approach

We present a hierarchical method to predict protein tertiary structure models from sequence. We start with complete enumeration of conformations using a simple tetrahedral lattice model. We then build conformations with increasing detail, and at each step select a subset of conformations using empirical energy functions with increasing complexity. After enumeration on lattice, we select a subset of low energy conformations using a statistical residue-residue contact energy function, and generate all-atom models using predicted secondary structure. A combined knowledge-based atomic level energy function is then used to select subsets of the all-atom models. The final predictions are generated using a consensus distance geometry procedure. We test the feasibility of the procedure on a set of 12 small proteins covering a wide range of protein topologies. A rigorous double-blind test of our method was made under the auspices of the CASP3 experiment, where we did ab initio structure predictions for 12 proteins using this approach. The performance of our methodology at CASP3 is reasonably good and completely consistent with our initial tests.     

A Parallel Implementation of the Wuchty Algorithm with Additional Experimental Filters to More Thoroughly Explore RNA Conformational Space

We present new modifications to the Wuchty algorithm in order to better define and explore possible conformations for an RNA sequence. The new features, including parallelization, energy-independent lonely pair constraints, context-dependent chemical probing constraints, helix filters, and optional multibranch loops, provide useful tools for exploring the landscape of RNA folding. Chemical probing alone may not necessarily define a single unique structure. The helix filters and optional multibranch loops are global constraints on RNA structure that are an especially useful tool for generating models of encapsidated viral RNA for which cryoelectron microscopy or crystallography data may be available. The computations generate a combinatorially complete set of structures near a free energy minimum and thus provide data on the density and diversity of structures near the bottom of a folding funnel for an RNA sequence. The conformational landscapes for some RNA sequences may resemble a low, wide basin rather than a steep funnel that converges to a single structure.     

A coarse-grained protein force field for folding and structure prediction

We have revisited the protein coarse-grained optimized potential for efficient structure prediction (OPEP). The training and validation sets consist of 13 and 16 protein targets. Because optimization depends on details of how the ensemble of decoys is sampled, trial conformations are generated by molecular dynamics, threading, greedy, and Monte Carlo simulations, or taken from publicly available databases. The OPEP parameters are varied by a genetic algorithm using a scoring function which requires that the native structure has the lowest energy, and the native-like structures have energy higher than the native structure but lower than the remote conformations. Overall, we find that OPEP correctly identifies 24 native or native-like states for 29 targets and has very similar capability to the all-atom discrete optimized protein energy model (DOPE), found recently to outperform five currently used energy models.     

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.     

Visualizing RNA base-pairing probabilities with RNAbow diagrams

There are many effective ways to represent a minimum free energy RNA secondary structure that make it easy to locate its helices and loops. It is a greater challenge to visualize the thermal average probabilities of all folds in a partition function sum; dot plot representations are often puzzling. Therefore, we introduce the RNAbows visualization tool for RNA base pair probabilities. RNAbows represent base pair probabilities with line thickness and shading, yielding intuitive diagrams. RNAbows aid in disentangling incompatible structures, allow comparisons between clusters of folds, highlight differences between wild-type and mutant folds, and are also rather beautiful.     

Equilibria in 5-S ribosomal RNA secondary structure. Bulges and interior loops in 5-S RNA secondary structure may serve as articulations for a flexible molecule

The basic assumption in this paper is that the secondary structure of a 5-S ribosomal RNA cannot be represented by a single model. We propose that the molecule can adopt, at least within the ribosome, a series of slightly different structures of nearly equal stability. The different structures arise from the existence of ambiguous base-pairing opportunities in bulged helices and the adjacent interior loops. In eubacterial 5-S RNAs there is one such an area, in eukaryotic 5-S RNAs two such areas that can give rise to structural switches. We explain how a change in secondary structure in these areas may influence the relative orientation of the surrounding helices, in other words how bulges and interior loops may serve as articulations and give rise to a flexible tertiary structure.     

Predicting Ion Binding Properties for RNA Tertiary Structures

Recent experiments pointed to the potential importance of ion correlation for multivalent ions such as Mg²⁺ ions in RNA folding. In this study, we develop an all-atom model to predict the ion electrostatics in RNA folding. The model can treat ion correlation effects explicitly by considering an ensemble of discrete ion distributions. In contrast to the previous coarse-grained models that can treat ion correlation, this new model is based on all-atom nucleic acid structures. Thus, unlike the previous coarse-grained models, this new model allows us to treat complex tertiary structures such as HIV-1 DIS type RNA kissing complexes. Theory-experiment comparisons for a variety of tertiary structures indicate that the model gives improved predictions over the Poisson-Boltzmann theory, which underestimates the Mg²⁺ binding in the competition with Na⁺. Further systematic theory-experiment comparisons for a series of tertiary structures lead to a set of analytical formulas for Mg²⁺/Na⁺ ion-binding to various RNA and DNA structures over a wide range of Mg²⁺ and Na⁺ concentrations.     

Coarse-grained modeling of large RNA molecules with knowledge-based potentials and structural filters

Understanding the function of complex RNA molecules depends critically on understanding their structure. However, creating three-dimensional (3D) structural models of RNA remains a significant challenge. We present a protocol (the nucleic acid simulation tool [NAST]) for RNA modeling that uses an RNA-specific knowledge-based potential in a coarse-grained molecular dynamics engine to generate plausible 3D structures. We demonstrate NAST's capabilities by using only secondary structure and tertiary contact predictions to generate, cluster, and rank structures. Representative structures in the best ranking clusters averaged 8.0 ± 0.3 Å and 16.3 ± 1.0 Å RMSD for the yeast phenylalanine tRNA and the P4-P6 domain of the Tetrahymena thermophila group I intron, respectively. The coarse-grained resolution allows us to model large molecules such as the 158-residue P4-P6 or the 388-residue T. thermophila group I intron. One advantage of NAST is the ability to rank clusters of structurally similar decoys based on their compatibility with experimental data. We successfully used ideal small-angle X-ray scattering data and both ideal and experimental solvent accessibility data to select the best cluster of structures for both tRNA and P4-P6. Finally, we used NAST to build in missing loops in the crystal structures of the Azoarcus and Twort ribozymes, and to incorporate crystallographic data into the Michel–Westhof model of the T. thermophila group I intron, creating an integrated model of the entire molecule. Our software package is freely available at https://simtk.org/home/nast.     

Thermodynamics of three-way multibranch loops in RNA

RNA multibranch loops (junctions) are loops from which three or more helices exit. They are nearly ubiquitous in RNA secondary structures determined by comparative sequence analysis. In this study, systems in which two strands combine to form three-way junctions were used to measure the stabilities of RNA multibranch loops by UV optical melting and isothermal titration calorimetry (ITC). These data were used to calculate the free energy increment for initiation of a three-way junction on the basis of a nearest neighbor model for secondary structure stability. Imino proton NMR spectra were also measured for two systems and are consistent with the hypothesized helical structures. Incorporation of the experimental data into the mfold and RNA structure computer programs has contributed to an improvement in prediction of RNA secondary structure from sequence.     

Improved prediction of RNA tertiary structure with insights into native state dynamics

The importance of RNA tertiary structure is evident from the growing number of published high resolution NMR and X-ray crystallographic structures of RNA molecules. These structures provide insights into function and create a knowledge base that is leveraged by programs such as Assemble, ModeRNA, RNABuilder, NAST, FARNA, Mc-Sym, RNA2D3D, and iFoldRNA for tertiary structure prediction and design. While these methods sample native-like RNA structures during simulations, all struggle to capture the native RNA conformation after scoring. We propose RSIM, an improved RNA fragment assembly method that preserves RNA global secondary structure while sampling conformations. This approach enhances the quality of predicted RNA tertiary structure, provides insights into the native state dynamics, and generates a powerful visualization of the RNA conformational space. RSIM is available for download from http://www.github.com/jpbida/rsim.     

Higher plant 5S rRNAs share common secondary and tertiary structure. A new three domains model

A new model of secondary and tertiary structure of higher plant 5S RNA is proposed. It consists of three helical domains: domain alpha includes stem I; domain beta contains stems II and III and loops B and C; domain gamma consists of stems IV and V and loops D and E. Except for, presumably, a canonical RNA-A like domain alpha, the two remaining domains apparently adopt a perturbed RNA-A structure due to irregularities within internal loops B and E and three bulges occurring in the model. Bending of RNA could bring loops B and E and/or C and D closer making tertiary interactions likely. The model differs from that suggested for eukaryotic 5S rRNA, by organization of domain gamma. Our model is based on the results of partial digestion obtained with single- and double-strand RNA specific nucleases. The proposed secondary structure is strongly supported by the observation that crude plant 5S rRNA contains abundant RNA, identified as domain gamma of 5S rRNA. Presumably it is excised from the 5S rRNA molecule by a specific nuclease present in lupin seeds. Experimental results were confirmed by computer-aided secondary structure prediction analysis of all higher plant 5S rRNAs. Differences observed between earlier proposed models and our proposition are discussed.