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

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.

A Fiedler Vector Scoring Approach for Novel RNA Motif Selection

Novel RNA motif design is of great practical importance for technology and medicine. Increasingly, computational design plays an important role in such efforts. Our coarse-grained RAG (RNA-As-Graphs) framework offers strategies for enumerating the universe of RNA 2D folds, selecting "RNA-like" candidates for design, and determining sequences that fold onto these candidates. In RAG, RNA secondary structures are represented as tree or dual graphs. Graphs with known RNA structures are called "existing", and the others are labeled "hypothetical". By using simplified features for RNA graphs, we have clustered the hypothetical graphs into "RNA-like" and "non-RNA-like" groups and proposed RNA-like graphs as candidates for design. Here, we propose a new way of designing graph features by using Fiedler vectors. The new features reflect graph shapes better, and they lead to a more clustered organization of existing graphs. We show significant increases in K-means clustering accuracy by using the new features (e.g., up to 95% and 98% accuracy for tree and dual graphs, respectively). In addition, we propose a scoring model for top graph candidate selection. This scoring model allows users to set a threshold for candidates, and it incorporates weighing of existing graphs based on their corresponding number of known RNAs. We include a list of top scored RNA-like candidates, which we hope will stimulate future novel RNA design.

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.

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.

Going beyond base-pairs: topology-based characterization of base-multiplets in RNA

Identification and characterization of base-multiplets, which are essentially mediated by base-pairing interactions, can provide insights into the diversity in the structure and dynamics of complex functional RNAs, and thus facilitate hypothesis driven biological research. The necessary nomenclature scheme, an extension of the geometric classification scheme for base-pairs by Leontis and Westhof, is however available only for base-triplets. In the absence of information on topology, this scheme is not applicable to quartets and higher order multiplets. Here we propose a topology-based classification scheme which, in conjunction with a graph-based algorithm, can be used for the automated identification and characterization of higher order base-multiplets in RNA structures. Here, the RNA structure is represented as a graph, where nodes represent nucleotides and edges represent base-pairing connectivity. Sets of connected components (of n nodes) within these graphs constitute subgraphs representing multiplets of "n" nucleotides. The different topological variants of the RNA multiplets thus correspond to different nonisomorphic forms of these subgraphs. To annotate RNA base-multiplets unambiguously, we propose a set of topology-based nomenclature rules for quartets, which are extendable to higher multiplets. We also demonstrate the utility of our approach toward the identification and annotation of higher order RNA multiplets, by investigating the occurrence contexts of selected examples in order to gain insights regarding their probable functional roles.

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.

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.

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.

Opportunities and Challenges in RNA Structural Modeling and Design

We describe opportunities and challenges in RNA structural modeling and design, as recently discussed during the second Telluride Science Research Center workshop organized in June 2016. Topics include fundamental processes of RNA, such as structural assemblies (hierarchical folding, multiple conformational states and their clustering), RNA motifs, and chemical reactivity of RNA, as used for structural prediction and functional inference. We also highlight the software and database issues associated with RNA structures, such as the multiple approaches for motif annotation, the need for frequent database updating, and the importance of quality control of RNA structures. We discuss various modeling approaches for structure prediction, mechanistic analysis of RNA reactions, and RNA design, and the complementary roles that both atomistic and coarse-grained approaches play in such simulations. Collectively, as scientists from varied disciplines become familiar and drawn into these unique challenges, new approaches and collaborative efforts will undoubtedly be catalyzed.

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.

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.