Formal grammars and R-loops

A formal grammar consists of a finite set of symbols partitioned into variables V and terminals Σ, and a finite set of production rules . When applying the rule on a word xuy, the subword u is substituted by the subword v yielding a word xvy. A word derived by the grammar is obtained by a consecutive application of rules starting from S, a non-terminal symbol designated as a starting symbol. The language generated by the grammar consists of all words comprised of terminal symbols that can be derived by the rules starting from S [20–22].

We define the R-loop grammar as a formal grammar whose terminal symbols correspond to the basic structures of an R-loop. The three-strand sections α and ω correspond to those regions of branch migration that mark the initiation (RNA invasion) and termination (RNA dissociation) of the R-loop, respectively. The symbols σ and represent short DNA:DNA hybrids with a free RNA strand (the RNA transcript). The symbols τ and represent RNA:DNA hybrids with a free DNA strand (the non-template strand). The ‘ ^{^} ’ indicates a more stable configuration, i.e. a configuration that is not prone to changing state. Therefore, denotes a structure unlikely to transition from a DNA duplex to an RNA:DNA hybrid, and denotes a structure unlikely to transition from an RNA:DNA hybrid back into the DNA duplex. Fig 5 illustrates the main terminal symbols in the R-loop grammar. Fig 6 shows a word generated by the R-loop grammar and its corresponding R-loop structure. Note that if the sequence stability weakens within an R-loop, a less stable RNA:DNA duplex (indicated by τ) may follow after an initial string of one or more ’s. Intuitively, one or more consecutive τ’s may lead to an R-loop termination region.

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Fig 5. R-loop symbols.

Basic 3-strand structures found in an R-loop and their associated symbols in the R-loop grammar. We indicate less stable configurations σ and τ by breakage in the hydrogen bonds. This representation should not be taken as literal breakage of all bonds in that vicinity, but rather as an indication that this region is unstable and prone to opening of the helix. The color coding is as in Fig 1.

https://doi.org/10.1371/journal.pcbi.1013376.g005

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Fig 6. An example of an R-loop grammar word.

The figure illustrates an R-loop associated with the word . The colors are as in Fig 1. For simplicity, we omit the broken hydrogen bonds for σ and τ. We indicate stability with the symbols under the diagram.

https://doi.org/10.1371/journal.pcbi.1013376.g006

To determine probability assignments of the terminal symbols in the R-loop grammar and the application of each production rule, we first extract a training set from two thirds of the experimental data. We use the remaining third of the data for testing the method. The application of each production rule depends on a probability distribution generated from the data.

We apply this approach to the SMRF-seq data for plasmids pFC53 and pFC8 with three different plasmid topologies. The reader can find details of the method in section General approach and in section Materials and methods.

Symbol assignment and R-loop production rules

We use SMRF-seq data to determine the probabilities for the R-loop grammar production rules (all the experimental data are available on GitHub [23] and Zenodo [24]). To each block of k consecutive nucleotides (k-mer) in the gene sequence, we assign a terminal symbol according to the probability that it is contained in an R-loop. The words generated by the grammar correspond to R-loops (Fig 6).

Occasionally the basic terminal symbols cannot be unequivocally assigned to a k-mer based on the available experimental data. In those instances, we expand the set of terminal symbols to accommodate the corresponding k-mers. The symbol δ (respectively, β) represents ambiguous k-mers that, according to the statistical analysis of the training data, could be associated with both σ and (respectively, τ and ). The k-mers outside (respectively, within) an R-loop for which the statistical analysis does not provide enough information are indicated with γ (respectively, ρ). Moreover, to account for the fact that the experimental assignment of initiation and termination of each R-loop is not precise [8], the terminals α and ω corresponds to a segment of length . Hence, the R-loop grammar alphabet consists of 10 letters ( and is independent of the k-mer size k.

As is common in formal grammars, the non-terminal symbols are written with capital symbols, and rules and with the same left side are written as . We define the following rules:

The sequence analysis in the Materials and methods section describes a way to map each k-mer to a terminal symbol. Such assignments allow each R-loop to be represented as a word over the symbols in this grammar. The word can be obtained by starting with the symbol S and applying a unique sequence of rules. For example, by applying the rules , , and in succession one obtains the word . We explain the probability assignments for each production rule in the Obtaining a model section and S1 Text. The probability assigned to each R-loop is computed as the product of the probabilities of the corresponding production rules. For example, the probability of the R-loop described by the word is defined as

(1)

General approach

One goal when analyzing experimental R-loop data on a given gene sequence is to identify genomic patterns specific to the initiation, elongation, and termination of R-loops. For each R-loop in the training set, we identify four regions of interest, r₁ and r₂ immediately upstream and downstream of the R-loop initiation, as well as r₃ and r₄ immediately upstream and downstream of the R-loop termination (Fig 6). For each i, we consider all of the k-mers that appear in region r_i and assign weights according to the relative frequency of the respective k-mer (see section Selecting the most relevant k-mers). We compile a list of k-mers (across all R-loops) in each region, and use the weights to associate a terminal grammar symbol to each k-mer, thus forming a dictionary that is then used to generate a grammar model.

To generate a grammar model–a probability assignment for each grammar rule–we translate the R-loops from a training set into words over the grammar symbols. Then we reverse-engineer the sequences of production rules that generate the words and assign probabilities to the rules according to the frequencies of each rule application (see section S1 Text). To make predictions, we first use the grammar to generate all possible R-loop words for a given dataset. The probability of a word w is proportional to the product of the probabilities of the sequence of rules that generate w (e.g. Eq 1). We compute the probability of each nucleotide being in an R-loop by summing the probabilities of the words where this event occurs. For example, let q_i(w) be 1 if the i-th nucleotide is in the R-loop represented by w, and 0 otherwise. The probability that the i-th nucleotide is in an R-loop is

(2)

where the sum is taken over all possible R-loop words in the the dataset.

For each plasmid and each starting topology, we take a portion of the experimental data as a holdout set. We take a random of the remaining data, use it for training and to generate a grammar model. We repeat this process 30 times to generate an ensemble of 30 grammar models (see sections Results, Materials and methods and S2 Fig). In the Results section, we show the average probability that each nucleotide is in an R-loop obtained from the ensemble of models.

Plasmid topology drives the probability of the production rules

Fig 7 shows the probability assignments for the production rules associated with symbols averaged over an ensemble of 30 grammar models. The probabilities of the other productions rules are included in S5 Fig, panel (a). A higher probability for a given rule implies that the training set contains a larger number of k-mers associated with that symbol. When a k-mer repeats, its multiplicity is taken into account. Note that the production rule probabilities change significantly with plasmid topology, which is consistent with the premise in [13].

As the supercoiling level increases towards hyper-negatively supercoiled, the probability of a stable DNA duplex () outside the R-loop also increases (rules and ). This suggests that the k-mers outside the R-loop are well determined. Once an R-loop starts, the pattern is reversed and the probability of elongating a stable R-loop decreases as supercoiling levels increase (rule ). We validated these observations using Kendall’s Tau correlation coefficient (S2 Table). Recall that when compared with linear and supercoiled plasmids, a significantly higher number of R-loops in hyper-negatively supercoiled plasmids appear closer to the transcription start (Fig 4). Because our dictionary assignment is focused on the regions at the start and the end of the R-loops, in hyper-negatively supercoiled plasmids a larger number of k-mers are spread throughout the R-loop containing regions (e.g., region between nt and nt for pFC53, see Fig 4). This spread of the R-loop starting points in hyper-negatively supercoiled cases implies that k-mers within those R-loops have a somewhat weaker association with R-loops. Hence, rule occurs with a much higher probability than . If we instead focus on linear and supercoiled plasmids, we observe that k-mers are mostly concentrated around the peak of the R-loop clusters (nt to nt). Accordingly, the difference between probabilities and is much smaller.

After an R-loop terminates, the probability of transitioning to a stable DNA duplex is high for all topologies. We observe the same trends in probabilities upon training the grammar on the data from each plasmid separately (S5 Fig(b) and S5 Fig(c)), as well as for the deterministic symbol assignment (S5 Fig(d)).

The R-loop grammar accurately predicts R-loop formation for different topologies

The R-loop grammar model has two adjustable parameters, the tuple size k that is used for the dictionary and for the terminal symbol assignments, and the padding length p that is used to determine the size of regions (see section k-mer extraction). Due to experimental sensitivity, the initiation of the R-loop may vary up to 15 nucleotides from the location observed through SMRF-seq [8]. To account for this, we focused on padding parameters p=7,13 (see S3 Table). The k-mer plus the padding correspond approximately to one (p = 7), or one and a half (p = 13) turns of an A-DNA double-helix (bp). RNA:DNA hybrids are believed to have the same helical pitch as A-DNA. Furthermore, we assume that the k-mers in the vicinity of the experimental R-loop start/end locations are critical for accurate prediction. We add the padding (nucleotide segments) before/after the R-loop start/end, thus defining the k-mers in regions r₁ to r₄ (see Fig 9).

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Fig 9. k-mer parsing.

Sliding-window approach for extracting k-mers around the initiation and termination sites of an R-loop [i,j] in a training set .

https://doi.org/10.1371/journal.pcbi.1013376.g009

In order to rigorously assess the best choice of parameters (k, p), for each topology we use the R-loop grammar to predict the probability of R-loop formation on each plasmid as detailed below (Fig 8). We then use 3-fold cross-validation to evaluate the model predictions for all (k, p) with and (find details in section Dictionary: Grammar symbol assignment for the set ). For each topology, we select the pair (k, p) that produces the lowest average root mean square deviation (RMSD) and the highest average Pearson correlation coefficient computed from the three validation sets. We found k = 4 and p = 13 to be the optimal parameters (see S3 Table).

It is worth noting that when k = 3, the symbol assignment in the regions r₁ to r₄ (see Fig 6 and section Materials and methods) exhausts all 64 possible 3-mers and the probabilities for production rules going to γ or ρ are 0, resulting in an oversimplified grammar model. When k = 5, the sequences of the two plasmids provide insufficient information, leaving between and of the 5-mers () assigned to an indeterminate symbol γ or ρ (S4 Table and S5 Table). The choice k = 4 and p = 13 is optimal as it provides information for 86.6% to of all possible k-mers.

We generate the model predictions as follows. First, for each plasmid and topology, we randomly select one-third of the experimental data to serve as a holdout set for testing. The remaining two-thirds constitute the full training dataset. We produce an ensemble of 30 grammar models by training each model on a distinct 10% subsample (without replacement) drawn from the full training dataset (see section S1 Text). Hereafter we refer to this of the data as the training set for the corresponding grammar model (see section Training and holdout set). Next we use the union of training sets for the two plasmids to create a k-mer dictionary. We reverse engineer the rules used from the R-loop grammar to define a probability of a given R-loop (as in Eq 1). Finally, we compute the probability that each nucleotide in the gene region is inside an R-loop using (Eq 2).

The final predictions assigning the probability that each nucleotide is within an R-loop are the average probabilities taken over the ensemble of 30 models (Fig 8) [25]. We tested the stability of the ensemble of predictions using 3-fold cross-validation [26] (see section Materials and methods and S3 Fig).

The R-loop grammar shows overall better prediction capabilities than the existing thermodynamics-based model R-looper [13] for both plasmids and all topologies. When compared to R-looper, our approach reduces the Root Mean Square Deviation (RMSD) by up to 55% and, in most cases it improves the Pearson correlation coefficient by at least 2-fold. Note that the thermodynamics model R-looper [13] is not trained on data. Besides the probability prediction, R-looper also provides an energy landscape for the sequence, which may need to be taken into consideration together with the probability landscape. We refer the reader to the SI Appendix for details. See S6 Table for a comparison against the holdout set and S7 Table for comparison against the full dataset. S8 Table compares the holdout set with the full dataset, and S9 Table gives the same information as S6 Table but for the deterministic symbol assignments.

The grammar rule probabilities vary depending on the plasmid topologies, and thus produce different predictions (Fig 8). Overall the fit to the data is outstanding, with Pearson correlation values from 0.68952 to 0.95165 when compared to the holdout set (S6 Table). The R-loop grammar accurately identifies R-loop clusters along the gene regions in both plasmids and predicts the shift to the left as the supercoiling density increases. As noted in [13], experimental data for hyper-negatively supercoiled pFC8 plasmids present with a much larger number of R-loops near the promoter region as compared with the supercoiled or linear plasmids, where they are largely absent.

Experimental data

We use experimental R-loop data detected in [13] by SMRF-seq, a method that profiles individual R-loops at ultra-deep coverage [8,18]. The nucleotide sequences of the plasmids pFC53 and pFC8 (previously reported in [13]) share the same backbone and incorporate specific regions known to be prone to R-loop formation [4]. More specifically, pFC53 contains a 1.3-kb portion of the murine Airn CpG island, and pFC8 contains a 942-bp portion of the human SNRPN CpG island. We make the complete SMRF-seq experimental data and software available on GitHub [23] and Zenodo [24]. The template strand of each plasmid is in FASTA format. The corresponding R-loop locations for each of the three starting plasmid topologies are included in BED files. The data consist of the following: for pFC53 there are 79 co-transcriptional R-loops within the linear, 612 within the supercoiled and 408 within the hyper-negatively supercoiled datasets; for pFC8 there are 116 R-loops within the linear, 104 within the supercoiled and 1044 within the hyper-negatively supercoiled datasets. In total, the data contain 2363 R-loops. Since the gene region in pFC53 is 1749nt long, and that of pFC8 is 1432nt long, the experimental per nucleotide probability of R-loop formation is reported for a total of 3,784,504 nucleotides.

Training set, holdout set and parameter choice

k-mer extraction

We consider each plasmid P in the direction of the non-template strand. R-loop locations are specified by their initiation i and termination j indices, with j>i. We denote the R-loop segment as the interval of nucleotides [i,j], i.e. the sequence of nucleotides . In order to have each R-loop length as a multiple of k, we modify the termination indices of each R-loop as needed (see section S3 Text). The k-mer extraction for the dictionary takes place around the initiation and termination sites of an R-loop (Fig 9). This is done separately for each plasmid’s training subset.

Given a P-training set for a given plasmid P, we employ a sliding-window approach to extract the k-mers specific to the initiation, elongation, and termination of R-loops. Let be a given padding parameter and let [i,j] be an R-loop in . The regions of interest are given by r₁ = [i−k−p,i−1], r₂ = [i,i + k + p−1], r₃ = [j−k−p + 1,j] and r₄ = [j + 1,j + k + p].

We take the k-mer [i–k,i–1] containing the k nucleotides before the beginning of the R-loop and shift this window to the left, one nucleotide at a time, for a total of p shifts. We perform the shifting as long as the k-mer remains in the gene sequence. We discard any extracted k-mers that are not fully contained within the gene sequence. The collection of k-mers obtained in this way is the set of k-mers within region r₁ and is denoted . Similarly, we construct the remaining three collections , , and ℛ₄ that correspond to k-mers within regions r₂, r₃ and r₄, respectively. The set ℛ₁ consists of the k-mers preceding the beginning of an R-loop in , while (respectively, and ) consists of the k-mers at the beginning (respectively, before and after the end) of an R-loop in . Fig 9 illustrates the sliding-window approach.

Selecting the most relevant k-mers

We select the k-mers used for the dictionary from the collections , . To each k-mer s in we associate a weight w_i(s) with respect to the region r_i as follows:

where N is the number of R-loops in the training set , n_i(s) is the number of occurrences of s in the region r_i across all the R-loops in (counted with multiplicities), and m_s is the number of occurrences (counted with multiplicity) of s within the gene region of the given plasmid. The weight w_i(s) quantifies the prevalence of s in region r_i, across all the R-loops in .

The k-mers in are ordered in decreasing order by weight so that s_i has the highest weight. Table 1 shows a portion of all 115 4-mers in from the hyper-negatively supercoiled pFC53 R-loop data set.

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Table 1. Selection of the most relevant 4-mers. Sample in from the hyper-negatively supercoiled pFC53 R-loop data set. The first column is the ranking n of the 4-mers in after ordering them by weight. The second column lists all the 4-mers where 4-mers with the same ranking are listed in the same field. In this sample the total number of 4-mers in is 115. The third column indicates the weight of s_n. The last three columns illustrate the steps of the selection procedure for determining the cutoff point: weight rescaling, entropy calculation, and average entropy computation. The cutoff point–highlighted–is the maximum of the average entropy values.

https://doi.org/10.1371/journal.pcbi.1013376.t001

In order to identify the most relevant k-mers, for each we determine a cutoff point for thresholding the ordered list using a procedure that relies on entropy reduction [31]. We rescale all k-mer weights in by normalizing with respect to the highest weight (i.e., s₁) with . We define the entropy of the k-mer as and compute the average entropy of as . The (global) maximum of is set to be the threshold for [31]. The threshold is achieved when , i.e. when adding a new value h_n is ‘not significant’ with respect to the sum of the already added values.

The threshold reduced list of k-mers , called highly weighted, comprises of all k-mers in such that is greater than or equal to the threshold value of . Note that there may be several k-mers corresponding to the threshold cutoff. In this case, all such k-mers are included in the highly weighted list .

Total number of 4-mers = 115

Grammar symbol assignments to the k-mers

For each plasmid and each set of experimental conditions, we use the list of k-mers and their associated weights to assign grammar symbols. This assignment enables us to represent each R-loop with an R-loop grammar word.

Given an R-loop [i,j] in within the gene region [b,e] of P, we focus on three segments [b,i–1], [i,j], and which comprise the sequences preceding, within, and following the R-loop. We subdivide each of the three segments into consecutive and non-overlapping parsing blocks. These blocks are k-mers, except possibly for the block that ends with i–1 and the one that starts with , which could be shorter (i.e., of length <k). By construction, the R-loop segment [i,j] is always a multiple of k (see Fig 10 and section S3 Text).

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Fig 10. Parsing blocks.

Subdivision of the gene region into parsing blocks. We indicate initiation and termination of the R-loop with i and j, respectively. Each parsing block is a k-mer, with the possible exception of the block immediately preceding the R-loop (labeled α, in purple) and the block immediately after the R-loop (labeled ω, in purple), which could be shorter.

https://doi.org/10.1371/journal.pcbi.1013376.g010

We represent each R-loop in the training set as a word in the R-loop grammar by establishing a correspondence between the parsing blocks within [b,i–1], [i,j], and and the grammar symbols. The correspondence between parsing blocks and grammar symbols is obtained through a symbol assignment function depending on the location of the first nucleotide of the k-mer s.

The grammar symbol assignment depends on the parsing block weights generated by the P-training set . A parsing block can be highly weighted in a region r_i (for some ; Fig 6), it can appear in two or more regions but not be highly weighted in any, or not appear in any of the regions. For example, highly weighted parsing blocks in region r₄ are treated as stable DNA:DNA duplexes, and those within region r₂ are treated as stable RNA:DNA duplexes. However, the weighted values can result in ambiguous assignments thus requiring more complex symbol assignment maps. We define the symbol assignment map for highly weighted k-mers below and that for not highly weighted blocks in section S2 Text.

Dictionary: Grammar symbol assignment for the set

Let (respectively, , ) denote the symbol assignments for the union training set (respectively, plasmid training sets , ). The symbol assignment for is based on the symbol assignments and . If the symbol assignment for both and is the same, then . If the symbol assignments by and differ, then we apply two approaches to resolve the conflict - a deterministic one and a stochastic one (see section S2 Text for details).

R-loop grammar words for to generate a model

To obtain a model for the R-loop grammar, i.e., to specify the probabilities of each of the grammar production rules, we write all R-loops in the union training set as words over the alphabet .

This is done by using the splitting of the gene region into parsing blocks according to the R-loop initiation and termination indices [i,j] as discussed above, see Fig 10. All but two of the blocks are k-mers. To a given k-mer, a grammar symbol is assigned according to function for the union training set , depending on whether the block precedes, is within, or follows the positions [i,j]. The lengths of the blocks α and ω for the transitions can vary between 0 and k–1 depending on the values i,j,k. Note that this means that each R-loop word contains exactly one α and ω even when their corresponding lengths are 0.

Obtaining a model

A grammar model for the union training set is obtained by assigning probabilities to each of the production rules. Detailed formulas are included in section S1 Text. Using Eq 1 for all words in union training set , and then Eq 2, we obtain the final probability for a given nucleotide of a plasmid to be within an R-loop. We produce an ensemble of (in our case 30) models and take the average over all models (S2 Fig).

Assessing model stability

We use 3-fold cross-validation to assess the stability of model predictions. To do so, we repeat a total of three times the process described under the Training and holdout set subsection, each time selecting a different non-overlapping holdout set and creating an ensemble of models from the respective full training dataset (see S3 Fig).