Fig 1.
RNA secondary structure, dot-bracket notation and rooted tree representation.
Panel A shows the secondary structure of a 41 nucleotides (nt) long RNA molecule. Each node represents a nucleotide (A, U, G, or C). The colors indicate different regions of the RNA secondary structure. Integers indicate the nucleotide position along the RNA sequence. Solid black edges represent covalent or hydrogen bonds, and gray edges represent bonds between distant nucleotides forming a pseudoknot. The bottom insert shows the sequence of the RNA molecule and the dot-bracket notation. Panel B shows the loop-stem tree representation of the RNA secondary structure in panel A without the pseudoknot. Here, every vertex represents a loop, and the colors of the vertices correspond to different regions. The blue vertex representing the opening region is the root of the tree. Every edge in the tree represents a stem of the RNA secondary structure.
Fig 2.
The computational pipeline for investigating and predicting R-loop formation.
This figure shows a brief pictorial description of the computational pipeline for predicting R-loop formation using DrTransformer and tree-polynomial. This pipeline takes a DNA sequence (panel A) as an input and covert the DNA sequence into the corresponding RNA sequence (panel B). Then, we use DrTransformer to predict co-transcriptional RNA secondary structures (panel C) of the RNA sequence. We use tree-polynomials (panel D) to represent the secondary structures obtained from DrTransformer, and compute the scaled sums of the tree-polynomials (panel E). To test the performance of the pipeline in predicting R-loop formation, we compare the scaled sums to the experimental probabilities of R-loop formation (panel F) obtained using SMRF-seq. See S8 Fig for more detail.
Fig 3.
Tree-polynomial representations of an RNA secondary structure.
The figure shows the first four types of tree-polynomial representations of the RNA secondary structure in Fig 1. Each tree-polynomial representation of an RNA secondary structure consists of a rooted tree representation and its corresponding polynomial. Vertices in the trees have the same colors as the loops and stem regions that they represent. The black round vertices represent unpaired nucleotides. The black square vertices in the type 2 and the type 4 tree representations indicate artificial internal vertices introduced to group unpaired nucleotides. In each rooted tree representation, we illustrate the recursive process of computing the corresponding polynomial from leaf vertices to the root vertex. The polynomial at the root vertex (blue) of a tree is the polynomial that represents the tree. Polynomial P is associated with type 1 and type 3 trees, and polynomial Q is associated with type 2 and type 4 trees.
Table 1.
Definition and performance of the eight tree-polynomial representations of RNA secondary structures.
Fig 4.
The correlations between the scaled sums and the R-loop formation probabilities of supercoiled pFC8 and pFC53 plasmids (see Data and Experiments section).
The figure shows the experimental probability of R-loop formation for the supercoiled pFC8 plasmid with the type 1 (panel A) and the type 4 (panel B) scaled sums, and the experimental probability of R-loop formation for the supercoiled pFC53 plasmid with the type 1 (panel C) and the type 4 (panel D) scaled sums. The experimental probabilities of R-loop formation are from [6]. The displayed scaled sums have the highest PCC in the last ten transcription steps.
Fig 5.
Secondary structures of RNA segments with the largest type 1 coefficient sums.
The figure shows the secondary structures of RNA segments of pFC8 (panel A and B) and of pFC53 (panel C and D) with the two largest type 1 coefficient sums at the transcription step with the highest PCC in the last 10 transcription steps. Type 1 tree representations follow the corresponding secondary structures.