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Methods for Analyzing RNA Pseudoknots via Chord Diagrams and Intersection Graphs

Rayan Ibrahim, Allison H. Moore

TL;DR

The paper develops a rigorous graph-theoretic framework for analyzing RNA pseudoknots using chord diagrams and intersection graphs, introducing a distance-based $\tau$ to capture 3D topology. It defines pseudoknots via weighted vertex covers on segment-intersection graphs and presents the $\tau$-segment graph method to interpolate between segment-based and augmented representations. The approach is validated on bpRNA-1m data, showing genus and vertex-cover metrics robustly quantify pseudoknot complexity, and revealing insights into partition persistence and secondary-structure classification. Overall, the method provides a topologically informed, computable toolkit for enumerating, classifying, and predicting pseudoknotted RNA structures with practical MATLAB implementations.

Abstract

RNA molecules are known to form complex secondary structures including pseudoknots. A systematic framework for the enumeration, classification and prediction of secondary structures is critical to determine the biological significance of the molecular configurations of RNA. Chord diagrams are mathematical objects widely used to represent RNA secondary structures and to analyze structural motifs, however a mathematically rigorous enumeration of pseudoknots remains a challenge. We introduce a method that incorporates a distance-based metric $τ$ to analyze the intersection graph of a chord diagram associated with a pseudoknotted structure. In particular, our method formally defines a pseudoknot in terms of a weighted vertex cover of a certain intersection graph constructed from a partition of the chord diagram representing the nucleotide sequence of the RNA molecule. In this graph-theoretic context, we introduce a rigorous algorithm that enumerates pseudoknots, classifies secondary structures, and is sensitive to three-dimensional topological features. We implement our methods in MATLAB and test the algorithm on pseudoknotted structures from the bpRNA-1m database. Our findings confirm that genus is a robust quantifier of pseudoknot complexity.

Methods for Analyzing RNA Pseudoknots via Chord Diagrams and Intersection Graphs

TL;DR

The paper develops a rigorous graph-theoretic framework for analyzing RNA pseudoknots using chord diagrams and intersection graphs, introducing a distance-based to capture 3D topology. It defines pseudoknots via weighted vertex covers on segment-intersection graphs and presents the -segment graph method to interpolate between segment-based and augmented representations. The approach is validated on bpRNA-1m data, showing genus and vertex-cover metrics robustly quantify pseudoknot complexity, and revealing insights into partition persistence and secondary-structure classification. Overall, the method provides a topologically informed, computable toolkit for enumerating, classifying, and predicting pseudoknotted RNA structures with practical MATLAB implementations.

Abstract

RNA molecules are known to form complex secondary structures including pseudoknots. A systematic framework for the enumeration, classification and prediction of secondary structures is critical to determine the biological significance of the molecular configurations of RNA. Chord diagrams are mathematical objects widely used to represent RNA secondary structures and to analyze structural motifs, however a mathematically rigorous enumeration of pseudoknots remains a challenge. We introduce a method that incorporates a distance-based metric to analyze the intersection graph of a chord diagram associated with a pseudoknotted structure. In particular, our method formally defines a pseudoknot in terms of a weighted vertex cover of a certain intersection graph constructed from a partition of the chord diagram representing the nucleotide sequence of the RNA molecule. In this graph-theoretic context, we introduce a rigorous algorithm that enumerates pseudoknots, classifies secondary structures, and is sensitive to three-dimensional topological features. We implement our methods in MATLAB and test the algorithm on pseudoknotted structures from the bpRNA-1m database. Our findings confirm that genus is a robust quantifier of pseudoknot complexity.
Paper Structure (13 sections, 3 theorems, 4 equations, 14 figures, 4 tables, 1 algorithm)

This paper contains 13 sections, 3 theorems, 4 equations, 14 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Combinatorial Theory
  3. Chord Diagrams
  4. Pseudoknotted Structures
  5. Crossingless Secondary Structures
  6. Qualitatively Similar Pseudoknotted Structures
  7. Methods
  8. The tau-Segment Graph Method
  9. Pseudoknot Quantification Process
  10. Discussion
  11. Maximum Values of tau and Persistence of Partitions
  12. Classifying Bases
  13. Calculation of Genus and Clique Numbers

Key Result

Theorem 2.6

Let $D$ be a chord diagram and let $G$ be the intersection graph of $D$. If $G$ is acyclic, then $\gamma(D) = \beta(G)$.

Figures (14)

  • Figure 1: Top left: A $3$-nesting. Top right: A $3$-crossing. Bottom left: Two $2$-crossings. Bottom right: A $2$-nesting and two $2$-crossings.
  • Figure 2: Two methods of calculating the genus of a given chord diagram.
  • Figure 3: (Top) Linear chord diagram of transfer RNA molecule of type 76-MER from Escherichia coli (PDB_652) with segments labeled. (Bottom) Segment graph $G_{\mathcal{S}}$ with weights.
  • Figure 4: An illustrated example of secondary structures in a crossingless chord diagram. There are eight stems in this structure
  • Figure 5: Two closely related instances of bonding between two hairpins
  • ...and 9 more figures

Theorems & Definitions (18)

  • Definition 2.1
  • Definition 2.2: Chord Obstructed
  • Definition 2.3: Segment
  • Definition 2.5: Genus
  • Theorem 2.6
  • proof
  • Remark 2.7
  • Theorem 2.8: bpRNA
  • proof
  • Definition 2.9
  • ...and 8 more