RAG: an update to the RNA-As-Graphs resource

Computational tools to study RNA-protein complexes

RNA is the key player in many cellular processes such as signal transduction, replication, transport, cell division, transcription, and translation. These diverse functions are accomplished through interactions of RNA with proteins. However, protein–RNA interactions are still poorly derstood in contrast to protein–protein and protein–DNA interactions. This knowledge gap can be attributed to the limited availability of protein-RNA structures along with the experimental difficulties in studying these complexes. Recent progress in computational resources has expanded the number of tools available for studying protein-RNA interactions at various molecular levels. These include tools for predicting interacting residues from primary sequences, modelling of protein-RNA complexes, predicting hotspots in these complexes and insights into derstanding in the dynamics of their interactions. Each of these tools has its strengths and limitations, which makes it significant to select an optimal approach for the question of interest. Here we present a mini review of computational tools to study different aspects of protein-RNA interactions, with focus on overall application, development of the field and the future perspectives.

RNA-As-Graphs Motif Atlas—Dual Graph Library of RNA Modules and Viral Frameshifting-Element Applications

RNA motif classification is important for understanding structure/function connections and building phylogenetic relationships. Using our coarse-grained RNA-As-Graphs (RAG) representations, we identify recurrent dual graph motifs in experimentally solved RNA structures based on an improved search algorithm that finds and ranks independent RNA substructures. Our expanded list of 183 existing dual graph motifs reveals five common motifs found in transfer RNA, riboswitch, and ribosomal 5S RNA components. Moreover, we identify three motifs for available viral frameshifting RNA elements, suggesting a correlation between viral structural complexity and frameshifting efficiency. We further partition the RNA substructures into 1844 distinct submotifs, with pseudoknots and junctions retained intact. Common modules are internal loops and three-way junctions, and three submotifs are associated with riboswitches that bind nucleotides, ions, and signaling molecules. Together, our library of existing RNA motifs and submotifs adds to the growing universe of RNA modules, and provides a resource of structures and substructures for novel RNA design.

Fuzzy RNA recognition by the Trypanosoma brucei editosome

The assembly of high molecular mass ribonucleoprotein complexes typically relies on the binary interaction of defined RNA sequences or precisely folded RNA motifs with dedicated RNA-binding domains on the protein side. Here we describe a new molecular recognition principle of RNA molecules by a high molecular mass protein complex. By chemically probing the solvent accessibility of mitochondrial pre-mRNAs when bound to the Trypanosoma brucei editosome, we identified multiple similar but non-identical RNA motifs as editosome contact sites. However, by treating the different motifs as mathematical graph objects we demonstrate that they fit a consensus 2D-graph consisting of 4 vertices (V) and 3 edges (E) with a Laplacian eigenvalue of 0.5477 (λ₂). We establish that synthetic 4V(3E)-RNAs are sufficient to compete for the editosomal pre-mRNA binding site and that they inhibit RNA editing in vitro. Furthermore, we demonstrate that only two topological indices are necessary to predict the binding of any RNA motif to the editosome with a high level of confidence. Our analysis corroborates that the editosome has adapted to the structural multiplicity of the mitochondrial mRNA folding space by recognizing a fuzzy continuum of RNA folds that fit a consensus graph descriptor.

Thermostability, Tunability, and Tenacity of RNA as Rubbery Anionic Polymeric Materials in Nanotechnology and Nanomedicine-Specific Cancer Targeting with Undetectable Toxicity

RNA nanotechnology is the bottom-up self-assembly of nanometer-scale architectures, resembling LEGOs, composed mainly of RNA. The ideal building material should be (1) versatile and controllable in shape and stoichiometry, (2) spontaneously self-assemble, and (3) thermodynamically, chemically, and enzymatically stable with a long shelf life. RNA building blocks exhibit each of the above. RNA is a polynucleic acid, making it a polymer, and its negative-charge prevents nonspecific binding to negatively charged cell membranes. The thermostability makes it suitable for logic gates, resistive memory, sensor set-ups, and NEM devices. RNA can be designed and manipulated with a level of simplicity of DNA while displaying versatile structure and enzyme activity of proteins. RNA can fold into single-stranded loops or bulges to serve as mounting dovetails for intermolecular or domain interactions without external linking dowels. RNA nanoparticles display rubber- and amoeba-like properties and are stretchable and shrinkable through multiple repeats, leading to enhanced tumor targeting and fast renal excretion to reduce toxicities. It was predicted in 2014 that RNA would be the third milestone in pharmaceutical drug development. The recent approval of several RNA drugs and COVID-19 mRNA vaccines by FDA suggests that this milestone is being realized. Here, we review the unique properties of RNA nanotechnology, summarize its recent advancements, describe its distinct attributes inside or outside the body and discuss potential applications in nanotechnology, medicine, and material science.

Diagrammatic approaches to RNA structures with trinucleotide repeats

Trinucleotide repeat expansion disorders are associated with the overexpansion of (CNG) repeats on the genome. Messenger RNA transcripts of sequences with greater than 60–100 (CNG) tandem units have been implicated in trinucleotide repeat expansion disorder pathogenesis. In this work, we develop a diagrammatic theory to study the structural diversity of these (CNG)_n RNA sequences. Representing structural elements on the chain’s conformation by a set of graphs and employing elementary diagrammatic methods, we have formulated a renormalization procedure to re-sum these graphs and arrive at a closed-form expression for the ensemble partition function. With a simple approximation for the renormalization and applied to extended (CNG)_n sequences, this theory can comprehensively capture an infinite set of conformations with any number and any combination of duplexes, hairpins, multiway junctions, and quadruplexes. To quantify the diversity of different (CNG)_n ensembles, the analytical equations derived from the diagrammatic theory were solved numerically to derive equilibrium estimates for the secondary structural contents of the chains. The results suggest that the structural ensembles of (CNG)_n repeat sequence with n ∼60 are surprisingly diverse, and the distribution is sensitive to the ability of the N nucleotide to make noncanonical pairs and whether the (CNG)_n sequence can sustain stable quadruplexes. The results show how perturbations in the form of biases on the stabilities of the various structural motifs, duplexes, junctions, helices, and quadruplexes could affect the secondary structures of the chains and how these structures may switch when they are perturbed.

Relative Information Gain: Shannon entropy-based measure of the relative structural conservation in RNA alignments

Structural characterization of RNAs is a dynamic field, offering many modelling possibilities. RNA secondary structure models are usually characterized by an encoding that depicts structural information of the molecule through string representations or graphs. In this work, we provide a generalization of the BEAR encoding (a context-aware structural encoding we previously developed) by expanding the set of alignments used for the construction of substitution matrices and then applying it to secondary structure encodings ranging from fine-grained to more coarse-grained representations. We also introduce a re-interpretation of the Shannon Information applied on RNA alignments, proposing a new scoring metric, the Relative Information Gain (RIG). The RIG score is available for any position in an alignment, showing how different levels of detail encoded in the RNA representation can contribute differently to convey structural information. The approaches presented in this study can be used alongside state-of-the-art tools to synergistically gain insights into the structural elements that RNAs and RNA families are composed of. This additional information could potentially contribute to their improvement or increase the degree of confidence in the secondary structure of families and any set of aligned RNAs.

Topological constraints of RNA pseudoknotted and loop-kissing motifs: applications to three-dimensional structure prediction

An RNA global fold can be described at the level of helix orientations and relatively flexible loop conformations that connect the helices. The linkage between the helices plays an essential role in determining the structural topology, which restricts RNA local and global folds, especially for RNA tertiary structures involving cross-linked base pairs. We quantitatively analyze the topological constraints on RNA 3D conformational space, in particular, on the distribution of helix orientations, for pseudoknots and loop-loop kissing structures. The result shows that a viable conformational space is predominantly determined by the motif type, helix size, and loop size, indicating a strong topological coupling between helices and loops in RNA tertiary motifs. Moreover, the analysis indicates that (cross-linked) tertiary contacts can cause much stronger topological constraints on RNA global fold than non-cross-linked base pairs. Furthermore, based on the topological constraints encoded in the 2D structure and the 3D templates, we develop a 3D structure prediction approach. This approach can be further combined with structure probing methods to expand the capability of computational prediction for large RNA folds.

Identification of novel RNA design candidates by clustering the extended RNA-As-Graphs library

BACKGROUND

We re-evaluate our RNA-As-Graphs clustering approach, using our expanded graph library and new RNA structures, to identify potential RNA-like topologies for design. Our coarse-grained approach represents RNA secondary structures as tree and dual graphs, with vertices and edges corresponding to RNA helices and loops. The graph theoretical framework facilitates graph enumeration, partitioning, and clustering approaches to study RNA structure and its applications.

METHODS

Clustering graph topologies based on features derived from graph Laplacian matrices and known RNA structures allows us to classify topologies into 'existing' or hypothetical, and the latter into, 'RNA-like' or 'non RNA-like' topologies. Here we update our list of existing tree graph topologies and RAG-3D database of atomic fragments to include newly determined RNA structures. We then use linear and quadratic regression, optionally with dimensionality reduction, to derive graph features and apply several clustering algorithms on our tree-graph library and recently expanded dual-graph library to classify them into the three groups.

RESULTS

The unsupervised PAM and K-means clustering approaches correctly classify 72-77% of all existing graph topologies and 75-82% of newly added ones as RNA-like. For supervised k-NN clustering, the cross-validation accuracy ranges from 57 to 81%.

CONCLUSIONS

Using linear regression with unsupervised clustering, or quadratic regression with supervised clustering, provides better accuracies than supervised/linear clustering. All accuracies are better than random, especially for newly added existing topologies, thus lending credibility to our approach.

GENERAL SIGNIFICANCE

Our updated RAG-3D database and motif classification by clustering present new RNA substructures and RNA-like motifs as novel design candidates.

Inverse folding with RNA-As-Graphs produces a large pool of candidate sequences with target topologies

We present an RNA-As-Graphs (RAG) based inverse folding algorithm, RAG-IF, to design novel RNA sequences that fold onto target tree graph topologies. The algorithm can be used to enhance our recently reported computational design pipeline (Jain et al., NAR 2018). The RAG approach represents RNA secondary structures as tree and dual graphs, where RNA loops and helices are coarse-grained as vertices and edges, opening the usage of graph theory methods to study, predict, and design RNA structures. Our recently developed computational pipeline for design utilizes graph partitioning (RAG-3D) and atomic fragment assembly (F-RAG) to design sequences to fold onto RNA-like tree graph topologies; the atomic fragments are taken from existing RNA structures that correspond to tree subgraphs. Because F-RAG may not produce the target folds for all designs, automated mutations by RAG-IF algorithm enhance the candidate pool markedly. The crucial residues for mutation are identified by differences between the predicted and the target topology. A genetic algorithm then mutates the selected residues, and the successful sequences are optimized to retain only the minimal or essential mutations. Here we evaluate RAG-IF for 6 RNA-like topologies and generate a large pool of successful candidate sequences with a variety of minimal mutations. We find that RAG-IF adds robustness and efficiency to our RNA design pipeline, making inverse folding motivated by graph topology rather than secondary structure more productive.

VfoldLA: A web server for loop assembly-based prediction of putative 3D RNA structures

RNA three-dimensional (3D) structures are critical for RNA cellular functions. However, structure prediction for large and complex RNAs remains a challenge, which hampers our understanding of RNA structure-function relationship. We here report a new web server, the VfoldLA server (http://rna.physics.missouri.edu/vfoldLA), for the prediction of RNA 3D structures from nucleotide sequences and base-pair information (2D structure). This server is based on the recently developed VfoldLA, a model that classifies the single-stranded loops (junctions) into four different types and according to the loop-helix connections, assembles RNA 3D structures from the loop/junction templates. The VfoldLA web server provides a user-friendly online interface for a fully automated prediction of putative 3D RNA structures using VfoldLA. With a single-RNA or RNA-RNA complex sequence and 2D structure as input, the server generates structure(s) with the JSmol visualization along with a downloadable PDB file. The output result may serve as useful scaffolds for future structure refinement studies.

bpRNA: large-scale automated annotation and analysis of RNA secondary structure

While RNA secondary structure prediction from sequence data has made remarkable progress, there is a need for improved strategies for annotating the features of RNA secondary structures. Here, we present bpRNA, a novel annotation tool capable of parsing RNA structures, including complex pseudoknot-containing RNAs, to yield an objective, precise, compact, unambiguous, easily-interpretable description of all loops, stems, and pseudoknots, along with the positions, sequence, and flanking base pairs of each such structural feature. We also introduce several new informative representations of RNA structure types to improve structure visualization and interpretation. We have further used bpRNA to generate a web-accessible meta-database, ‘bpRNA-1m’, of over 100 000 single-molecule, known secondary structures; this is both more fully and accurately annotated and over 20-times larger than existing databases. We use a subset of the database with highly similar (≥90% identical) sequences filtered out to report on statistical trends in sequence, flanking base pairs, and length. Both the bpRNA method and the bpRNA-1m database will be valuable resources both for specific analysis of individual RNA molecules and large-scale analyses such as are useful for updating RNA energy parameters for computational thermodynamic predictions, improving machine learning models for structure prediction, and for benchmarking structure-prediction algorithms.

An extended dual graph library and partitioning algorithm applicable to pseudoknotted RNA structures

Exploring novel RNA topologies is imperative for understanding RNA structure and pursuing its design. Our RNA-As-Graphs (RAG) approach exploits graph theory tools and uses coarse-grained tree and dual graphs to represent RNA helices and loops by vertices and edges. Only dual graphs represent pseudoknotted RNAs fully. Here we develop a dual graph enumeration algorithm to generate an expanded library of dual graph topologies for 2-9 vertices, and extend our dual graph partitioning algorithm to identify all possible RNA subgraphs. Our enumeration algorithm connects smaller-vertex graphs, using all possible edge combinations, to build larger-vertex graphs and retain all non-isomorphic graph topologies, thereby more than doubling the size of our prior library to a total of 110,667 dual graph topologies. We apply our dual graph partitioning algorithm, which keeps pseudoknots and junctions intact, to all existing RNA structures to identify all possible substructures up to 9 vertices. In addition, our expanded dual graph library assigns graph topologies to all RNA graphs and subgraphs, rectifying prior inconsistencies. We update our RAG-3Dual database of RNA atomic fragments with all newly identified substructures and their graph IDs, increasing its size by more than 50 times. The enlarged dual graph library and RAG-3Dual database provide a comprehensive repertoire of graph topologies and atomic fragments to study yet undiscovered RNA molecules and design RNA sequences with novel topologies, including a variety of pseudoknotted RNAs.

A pipeline for computational design of novel RNA-like topologies

Designing novel RNA topologies is a challenge, with important therapeutic and industrial applications. We describe a computational pipeline for design of novel RNA topologies based on our coarse-grained RNA-As-Graphs (RAG) framework. RAG represents RNA structures as tree graphs and describes RNA secondary (2D) structure topologies (currently up to 13 vertices, ≈260 nucleotides). We have previously identified novel graph topologies that are RNA-like among these. Here we describe a systematic design pipeline and illustrate design for six broad design problems using recently developed tools for graph-partitioning and fragment assembly (F-RAG). Following partitioning of the target graph, corresponding atomic fragments from our RAG-3D database are combined using F-RAG, and the candidate atomic models are scored using a knowledge-based potential developed for 3D structure prediction. The sequences of the top scoring models are screened further using available tools for 2D structure prediction. The results indicate that our modular approach based on RNA-like topologies rather than specific 2D structures allows for greater flexibility in the design process, and generates a large number of candidate sequences quickly. Experimental structure probing using SHAPE-MaP for two sequences agree with our predictions and suggest that our combined tools yield excellent candidates for further sequence and experimental screening.

Dual Graph Partitioning Highlights a Small Group of Pseudoknot-Containing RNA Submotifs

RNA molecules are composed of modular architectural units that define their unique structural and functional properties. Characterization of these building blocks can help interpret RNA structure/function relationships. We present an RNA secondary structure motif and submotif library using dual graph representation and partitioning. Dual graphs represent RNA helices as vertices and loops as edges. Unlike tree graphs, dual graphs can represent RNA pseudoknots (intertwined base pairs). For a representative set of RNA structures, we construct dual graphs from their secondary structures, and apply our partitioning algorithm to identify non-separable subgraphs (or blocks) without breaking pseudoknots. We report 56 subgraph blocks up to nine vertices; among them, 22 are frequently occurring, 15 of which contain pseudoknots. We then catalog atomic fragments corresponding to the subgraph blocks to define a library of building blocks that can be used for RNA design, which we call RAG-3Dual, as we have done for tree graphs. As an application, we analyze the distribution of these subgraph blocks within ribosomal RNAs of various prokaryotic and eukaryotic species to identify common subgraphs and possible ancestry relationships. Other applications of dual graph partitioning and motif library can be envisioned for RNA structure analysis and design.

Adventures with RNA graphs

The structure of RNA has been a natural subject for mathematical modeling, inviting many innovative computational frameworks. This single-stranded polynucleotide chain can fold upon itself in numerous ways to form hydrogen-bonded segments, imperfect with single-stranded loops. Illustrating these paired and non-paired interaction networks, known as RNA's secondary (2D) structure, using mathematical graph objects has been illuminating for RNA structure analysis. Building upon such seminal work from the 1970s and 1980s, graph models are now used to study not only RNA structure but also describe RNA's recurring modular units, sample the conformational space accessible to RNAs, predict RNA's three-dimensional folds, and apply the combined aspects to novel RNA design. In this article, we outline the development of the RNA-As-Graphs (or RAG) approach and highlight current applications to RNA structure prediction and design.

Hierarchical Assembly of RNA Three-Dimensional Structures Based on Loop Templates

The current RNA structure prediction methods cannot keep up the pace of the rapidly increasing number of sequences and the emerging new functions of RNAs. Template-based RNA three-dimensional structure prediction methods are restricted by the limited number of known RNA structures, and traditional motif-based search for the templates does not always lead to successful results. Here we report a new template search and assembly algorithm, the hierarchical loop template-assembly method (VfoldLA). The method searches for templates for single strand loop/junctions instead of the whole motifs, which often renders no available templates, or short fragments (several nucleotides), which requires a long computational time to assemble and refine. The VfoldLA method has the advantage of accounting for local and nonlocal interloop interactions. Benchmark tests indicate that this new method can provide low-resolution predictions for RNA conformations at different levels of structural complexities. Furthermore, the VfoldLA-predicted conformations may also serve as reliable putative models for further structure prediction and refinements. VfoldLA is accessible at http://rna.physics.missouri.edu/vfoldLA .

F-RAG: Generating Atomic Coordinates from RNA Graphs by Fragment Assembly

Coarse-grained models represent attractive approaches to analyze and simulate ribonucleic acid (RNA) molecules, for example, for structure prediction and design, as they simplify the RNA structure to reduce the conformational search space. Our structure prediction protocol RAGTOP (RNA-As-Graphs Topology Prediction) represents RNA structures as tree graphs and samples graph topologies to produce candidate graphs. However, for a more detailed study and analysis, construction of atomic from coarse-grained models is required. Here we present our graph-based fragment assembly algorithm (F-RAG) to convert candidate three-dimensional (3D) tree graph models, produced by RAGTOP into atomic structures. We use our related RAG-3D utilities to partition graphs into subgraphs and search for structurally similar atomic fragments in a data set of RNA 3D structures. The fragments are edited and superimposed using common residues, full atomic models are scored using RAGTOP's knowledge-based potential, and geometries of top scoring models is optimized. To evaluate our models, we assess all-atom RMSDs and Interaction Network Fidelity (a measure of residue interactions) with respect to experimentally solved structures and compare our results to other fragment assembly programs. For a set of 50 RNA structures, we obtain atomic models with reasonable geometries and interactions, particularly good for RNAs containing junctions. Additional improvements to our protocol and databases are outlined. These results provide a good foundation for further work on RNA structure prediction and design applications.

Using sequence signatures and kink-turn motifs in knowledge-based statistical potentials for RNA structure prediction

Kink turns are widely occurring motifs in RNA, located in internal loops and associated with many biological functions including translation, regulation and splicing. The associated sequence pattern, a 3-nt bulge and G-A, A-G base-pairs, generates an angle of ∼50° along the helical axis due to A-minor interactions. The conserved sequence and distinct secondary structures of kink-turns (k-turn) suggest computational folding rules to predict k-turn-like topologies from sequence. Here, we annotate observed k-turn motifs within a non-redundant RNA dataset based on sequence signatures and geometrical features, analyze bending and torsion angles, and determine distinct knowledge-based potentials with and without k-turn motifs. We apply these scoring potentials to our RAGTOP (RNA-As-Graph-Topologies) graph sampling protocol to construct and sample coarse-grained graph representations of RNAs from a given secondary structure. We present graph-sampling results for 35 RNAs, including 12 k-turn and 23 non k-turn internal loops, and compare the results to solved structures and to RAGTOP results without special k-turn potentials. Significant improvements are observed with the updated scoring potentials compared to the k-turn-free potentials. Because k-turns represent a classic example of sequence/structure motif, our study suggests that other such motifs with sequence signatures and unique geometrical features can similarly be utilized for RNA structure prediction and design.

Partitioning and Classification of RNA Secondary Structures into Pseudonotted and Pseudoknot-free Regions Using a Graph-Theoretical Approach

Dual graphs have been applied to model RNA secondary structures with pseudoknots, or intertwined base pairs. In this paper we present a linear-time algorithm to partition dual graphs into maximal topological components called blocks and determine whether each block contains a pseudoknot or not. We show that a block contains a pseudoknot if and only if the block has a vertex of degree 3 or more; this characterization allows us to efficiently isolate smaller RNA fragments and classify them as pseudoknotted or pseudoknot-free regions, while keeping these sub-structures intact. Applications to RNA design can be envisioned since modular building blocks with intact pseudoknots can be combined to form new constructs.

Capturing RNA Folding Free Energy with Coarse-Grained Molecular Dynamics Simulations

We introduce a coarse-grained RNA model for molecular dynamics simulations, RACER (RnA CoarsE-gRained). RACER achieves accurate native structure prediction for a number of RNAs (average RMSD of 2.93 Å) and the sequence-specific variation of free energy is in excellent agreement with experimentally measured stabilities (R2 = 0.93). Using RACER, we identified hydrogen-bonding (or base pairing), base stacking, and electrostatic interactions as essential driving forces for RNA folding. Also, we found that separating pairing vs. stacking interactions allowed RACER to distinguish folded vs. unfolded states. In RACER, base pairing and stacking interactions each provide an approximate stability of 3-4 kcal/mol for an A-form helix. RACER was developed based on PDB structural statistics and experimental thermodynamic data. In contrast with previous work, RACER implements a novel effective vdW potential energy function, which led us to re-parameterize hydrogen bond and electrostatic potential energy functions. Further, RACER is validated and optimized using a simulated annealing protocol to generate potential energy vs. RMSD landscapes. Finally, RACER is tested using extensive equilibrium pulling simulations (0.86 ms total) on eleven RNA sequences (hairpins and duplexes).

Accurate Classification of RNA Structures Using Topological Fingerprints

While RNAs are well known to possess complex structures, functionally similar RNAs often have little sequence similarity. While the exact size and spacing of base-paired regions vary, functionally similar RNAs have pronounced similarity in the arrangement, or topology, of base-paired stems. Furthermore, predicted RNA structures often lack pseudoknots (a crucial aspect of biological activity), and are only partially correct, or incomplete. A topological approach addresses all of these difficulties. In this work we describe each RNA structure as a graph that can be converted to a topological spectrum (RNA fingerprint). The set of subgraphs in an RNA structure, its RNA fingerprint, can be compared with the fingerprints of other RNA structures to identify and correctly classify functionally related RNAs. Topologically similar RNAs can be identified even when a large fraction, up to 30%, of the stems are omitted, indicating that highly accurate structures are not necessary. We investigate the performance of the RNA fingerprint approach on a set of eight highly curated RNA families, with diverse sizes and functions, containing pseudoknots, and with little sequence similarity-an especially difficult test set. In spite of the difficult test set, the RNA fingerprint approach is very successful (ROC AUC > 0.95). Due to the inclusion of pseudoknots, the RNA fingerprint approach both covers a wider range of possible structures than methods based only on secondary structure, and its tolerance for incomplete structures suggests that it can be applied even to predicted structures. Source code is freely available at https://github.rcac.purdue.edu/mgribsko/XIOS_RNA_fingerprint.

Predicting Large RNA-Like Topologies by a Knowledge-Based Clustering Approach

An analysis and expansion of our resource for classifying, predicting, and designing RNA structures, RAG (RNA-As-Graphs), is presented, with the goal of understanding features of RNA-like and non-RNA-like motifs and exploiting this information for RNA design. RAG was first reported in 2004 for cataloging RNA secondary structure motifs using graph representations. In 2011, the RAG resource was updated with the increased availability of RNA structures and was improved by utilities for analyzing RNA structures, including substructuring and search tools. We also classified RNA structures as graphs up to 10 vertices (~200 nucleotides) into three classes: existing, RNA-like, and non-RNA-like using clustering approaches. Here, we focus on the tree graphs and evaluate the newly founded RNAs since 2011, which also support our refined predictions of RNA-like motifs. We expand the RAG resource for large tree graphs up to 13 vertices (~260 nucleotides), thereby cataloging more than 10 times as many secondary structures. We apply clustering algorithms based on features of RNA secondary structures translated from known tertiary structures to suggest which hypothetical large RNA motifs can be considered "RNA-like". The results by the PAM (Partitioning Around Medoids) approach, in particular, reveal good accuracy, with small error for the largest cases. The RAG update here up to 13 vertices offers a useful graph-based tool for exploring RNA motifs and suggesting large RNA motifs for design.

RAG-3D: a search tool for RNA 3D substructures

To address many challenges in RNA structure/function prediction, the characterization of RNA's modular architectural units is required. Using the RNA-As-Graphs (RAG) database, we have previously explored the existence of secondary structure (2D) submotifs within larger RNA structures. Here we present RAG-3D—a dataset of RNA tertiary (3D) structures and substructures plus a web-based search tool—designed to exploit graph representations of RNAs for the goal of searching for similar 3D structural fragments. The objects in RAG-3D consist of 3D structures translated into 3D graphs, cataloged based on the connectivity between their secondary structure elements. Each graph is additionally described in terms of its subgraph building blocks. The RAG-3D search tool then compares a query RNA 3D structure to those in the database to obtain structurally similar structures and substructures. This comparison reveals conserved 3D RNA features and thus may suggest functional connections. Though RNA search programs based on similarity in sequence, 2D, and/or 3D structural elements are available, our graph-based search tool may be advantageous for illuminating similarities that are not obvious; using motifs rather than sequence space also reduces search times considerably. Ultimately, such substructuring could be useful for RNA 3D structure prediction, structure/function inference and inverse folding.

Physics-based RNA structure prediction

Despite the success of RNA secondary structure prediction for simple, short RNAs, the problem of predicting RNAs with long-range tertiary folds remains. Furthermore, RNA 3D structure prediction is hampered by the lack of the knowledge about the tertiary contacts and their thermodynamic parameters. Low-resolution structural modeling enables us to estimate the conformational entropies for a number of tertiary folds through rigorous statistical mechanical calculations. The models lead to 3D tertiary folds at coarse-grained level. The coarse-grained structures serve as the initial structures for all-atom molecular dynamics refinement to build the final all-atom 3D structures. In this paper, we present an overview of RNA computational models for secondary and tertiary structures’ predictions and then focus on a recently developed RNA statistical mechanical model—the Vfold model. The main emphasis is placed on the physics behind the models, including the treatment of the non-canonical interactions in secondary and tertiary structure modelings, and the correlations to RNA functions.

Ab initio RNA folding

RNA molecules are essential cellular machines performing a wide variety of functions for which a specific three-dimensional structure is required. Over the last several years, the experimental determination of RNA structures through x-ray crystallography and NMR seems to have reached a plateau in the number of structures resolved each year, but as more and more RNA sequences are being discovered, the need for structure prediction tools to complement experimental data is strong. Theoretical approaches to RNA folding have been developed since the late nineties, when the first algorithms for secondary structure prediction appeared. Over the last 10 years a number of prediction methods for 3D structures have been developed, first based on bioinformatics and data-mining, and more recently based on a coarse-grained physical representation of the systems. In this review we are going to present the challenges of RNA structure prediction and the main ideas behind bioinformatic approaches and physics-based approaches. We will focus on the description of the more recent physics-based phenomenological models and on how they are built to include the specificity of the interactions of RNA bases, whose role is critical in folding. Through examples from different models, we will point out the strengths of physics-based approaches, which are able not only to predict equilibrium structures, but also to investigate dynamical and thermodynamical behavior, and the open challenges to include more key interactions ruling RNA folding.

RiboFSM: frequent subgraph mining for the discovery of RNA structures and interactions

Frequent subgraph mining is a useful method for extracting meaningful patterns from a set of graphs or a single large graph. Here, the graph represents all possible RNA structures and interactions. Patterns that are significantly more frequent in this graph over a random graph are extracted. We hypothesize that these patterns are most likely to represent biological mechanisms. The graph representation used is a directed dual graph, extended to handle intermolecular interactions. The graph is sampled for subgraphs, which are labeled using a canonical labeling method and counted. The resulting patterns are compared to those created from a randomized dataset and scored. The algorithm was applied to the mitochondrial genome of the kinetoplastid species Trypanosoma brucei, which has a unique RNA editing mechanism. The most significant patterns contain two stem-loops, indicative of gRNA, and represent interactions of these structures with target mRNA.

RNA graph partitioning for the discovery of RNA modularity: a novel application of graph partition algorithm to biology

Graph representations have been widely used to analyze and design various economic, social, military, political, and biological networks. In systems biology, networks of cells and organs are useful for understanding disease and medical treatments and, in structural biology, structures of molecules can be described, including RNA structures. In our RNA-As-Graphs (RAG) framework, we represent RNA structures as tree graphs by translating unpaired regions into vertices and helices into edges. Here we explore the modularity of RNA structures by applying graph partitioning known in graph theory to divide an RNA graph into subgraphs. To our knowledge, this is the first application of graph partitioning to biology, and the results suggest a systematic approach for modular design in general. The graph partitioning algorithms utilize mathematical properties of the Laplacian eigenvector (µ2) corresponding to the second eigenvalues (λ2) associated with the topology matrix defining the graph: λ2 describes the overall topology, and the sum of µ2's components is zero. The three types of algorithms, termed median, sign, and gap cuts, divide a graph by determining nodes of cut by median, zero, and largest gap of µ2's components, respectively. We apply these algorithms to 45 graphs corresponding to all solved RNA structures up through 11 vertices (∼ 220 nucleotides). While we observe that the median cut divides a graph into two similar-sized subgraphs, the sign and gap cuts partition a graph into two topologically-distinct subgraphs. We find that the gap cut produces the best biologically-relevant partitioning for RNA because it divides RNAs at less stable connections while maintaining junctions intact. The iterative gap cuts suggest basic modules and assembly protocols to design large RNA structures. Our graph substructuring thus suggests a systematic approach to explore the modularity of biological networks. In our applications to RNA structures, subgraphs also suggest design strategies for novel RNA motifs.

Viral RNAs are unusually compact

A majority of viruses are composed of long single-stranded genomic RNA molecules encapsulated by protein shells with diameters of just a few tens of nanometers. We examine the extent to which these viral RNAs have evolved to be physically compact molecules to facilitate encapsulation. Measurements of equal-length viral, non-viral, coding and non-coding RNAs show viral RNAs to have among the smallest sizes in solution, i.e., the highest gel-electrophoretic mobilities and the smallest hydrodynamic radii. Using graph-theoretical analyses we demonstrate that their sizes correlate with the compactness of branching patterns in predicted secondary structure ensembles. The density of branching is determined by the number and relative positions of 3-helix junctions, and is highly sensitive to the presence of rare higher-order junctions with 4 or more helices. Compact branching arises from a preponderance of base pairing between nucleotides close to each other in the primary sequence. The density of branching represents a degree of freedom optimized by viral RNA genomes in response to the evolutionary pressure to be packaged reliably. Several families of viruses are analyzed to delineate the effects of capsid geometry, size and charge stabilization on the selective pressure for RNA compactness. Compact branching has important implications for RNA folding and viral assembly.

Graph-based sampling for approximating global helical topologies of RNA

A current challenge in RNA structure prediction is the description of global helical arrangements compatible with a given secondary structure. Here we address this problem by developing a hierarchical graph sampling/data mining approach to reduce conformational space and accelerate global sampling of candidate topologies. Starting from a 2D structure, we construct an initial graph from size measures deduced from solved RNAs and junction topologies predicted by our data-mining algorithm RNAJAG trained on known RNAs. We sample these graphs in 3D space guided by knowledge-based statistical potentials derived from bending and torsion measures of internal loops as well as radii of gyration for known RNAs. Graph sampling results for 30 representative RNAs are analyzed and compared with reference graphs from both solved structures and predicted structures by available programs. This comparison indicates promise for our graph-based sampling approach for characterizing global helical arrangements in large RNAs: graph rmsds range from 2.52 to 28.24 Å for RNAs of size 25-158 nucleotides, and more than half of our graph predictions improve upon other programs. The efficiency in graph sampling, however, implies an additional step of translating candidate graphs into atomic models. Such models can be built with the same idea of graph partitioning and build-up procedures we used for RNA design.

Network Theory Tools for RNA Modeling

An introduction into the usage of graph or network theory tools for the study of RNA molecules is presented. By using vertices and edges to define RNA secondary structures as tree and dual graphs, we can enumerate, predict, and design RNA topologies. Graph connectivity and associated Laplacian eigenvalues relate to biological properties of RNA and help understand RNA motifs as well as build, by computational design, various RNA target structures. Importantly, graph theoretical representations of RNAs reduce drastically the conformational space size and therefore simplify modeling and prediction tasks. Ongoing challenges remain regarding general RNA design, representation of RNA pseudoknots, and tertiary structure prediction. Thus, developments in network theory may help advance RNA biology.

Predicting helical topologies in RNA junctions as tree graphs

RNA molecules are important cellular components involved in many fundamental biological processes. Understanding the mechanisms behind their functions requires knowledge of their tertiary structures. Though computational RNA folding approaches exist, they often require manual manipulation and expert intuition; predicting global long-range tertiary contacts remains challenging. Here we develop a computational approach and associated program module (RNAJAG) to predict helical arrangements/topologies in RNA junctions. Our method has two components: junction topology prediction and graph modeling. First, junction topologies are determined by a data mining approach from a given secondary structure of the target RNAs; second, the predicted topology is used to construct a tree graph consistent with geometric preferences analyzed from solved RNAs. The predicted graphs, which model the helical arrangements of RNA junctions for a large set of 200 junctions using a cross validation procedure, yield fairly good representations compared to the helical configurations in native RNAs, and can be further used to develop all-atom models as we show for two examples. Because junctions are among the most complex structural elements in RNA, this work advances folding structure prediction methods of large RNAs. The RNAJAG module is available to academic users upon request.

Visualizing large RNA molecules in solution

Single-stranded RNAs (ssRNAs) longer than a few hundred nucleotides do not have a unique structure in solution. Their equilibrium properties therefore reflect the average of an ensemble of structures. We use cryo-electron microscopy to image projections of individual long ssRNA molecules and characterize the anisotropy of their ensembles in solution. A flattened prolate volume is found to best represent the shapes of these ensembles. The measured sizes and anisotropies are in good agreement with complementary determinations using small-angle X-ray scattering and coarse-grained molecular dynamics simulations. A long viral ssRNA is compared with shorter noncoding transcripts to demonstrate that prolate geometry and flatness are generic properties independent of sequence length and origin. The anisotropy persists under physiological as well as low-ionic-strength conditions, revealing a direct correlation between secondary structure asymmetry and 3D shape and size. We discuss the physical origin of the generic anisotropy and its biological implications.

Classifying multigraph models of secondary RNA structure using graph-theoretic descriptors

The prediction of secondary RNA folds from primary sequences continues to be an important area of research given the significance of RNA molecules in biological processes such as gene regulation. To facilitate this effort, graph models of secondary structure have been developed to quantify and thereby characterize the topological properties of the secondary folds. In this work we utilize a multigraph representation of a secondary RNA structure to examine the ability of the existing graph-theoretic descriptors to classify all possible topologies as either RNA-like or not RNA-like. We use more than one hundred descriptors and several different machine learning approaches, including nearest neighbor algorithms, one-class classifiers, and several clustering techniques. We predict that many more topologies will be identified as those representing RNA secondary structures than currently predicted in the RAG (RNA-As-Graphs) database. The results also suggest which descriptors and which algorithms are more informative in classifying and exploring secondary RNA structures.