Difficulties

Direct sequencing is rather insensitive for minor DNA fractions that carry single-nucleotide substitutions (20–25% [3], [12], [14]); the sensitivity depends on the total DNA concentration in the sample [15]. Mixtures of significantly different DNA variants produce very complex chromatograms that are difficult to interpret with standard methods. Even when DNA variants are nearly identical, differing, for example, only by short insertions or deletions (indels), the chromatogram is still very complex.

All of these obstacles limit application of direct Sanger sequencing. It is not applicable for tasks like 16S rRNA gene sequencing for human specimens, determining HCV or HBV genotype in a case of mixed infection, and detection of indels.

Existing Approaches

Original methods for reading cloned DNA from a chromatogram (e.g. Phred [16], [17]) allow for identifying only one type of DNA. Thus, these methods do not, in general, work for direct sequencing of a DNA mixture. Due to the active growth of resequencing projects, some tools for processing chromatograms of heterozygous genomes have been developed: e. g., TraceTuner [18] re-analyzes chromatograms (i. e., performs base-recalling) after the primary sequence has been defined. To avoid misinterpreting sequencing artifacts as single nucleotide polymorphisms (SNPs), a number of algorithms, like Polyphred [19], [20], Polybayes [21], and AutoEditor [22], require pre-assembled DNA sequences of various individual genomes. These algorithms are designed for resequencing eukaryotic genomes with a low degree of variability, and they require relatively high coverage of each genome site. More complicated is the identification of indels using direct sequencing data. Currently, some algorithms allow indel detection in heterozygous genomes[23]–[27].

Many of the problems concerning direct sequencing of heterogenous DNA sequences are summarized in [28]. A clinical sample may contain any number of DNA variants, with any mutation types (single nucleotide variations (SNVs), indels, and substring substitutions). In order to decipher a structure of the mixture, the wild-type sequence and the vocabulary of all allowed mutations must be provided. This formulation is a special case of the set-cover problem [29]. Even after this generalization, the algorithm is not universal: it is not feasible to choose a single wild-type sequence in rapidly evolving organisms such as RNA viruses. Also, it is impossible to choose one for the sequencing of 16S rRNA PCR products of a polyspecies sample. In many cases, it is also impossible to compile an exact vocabulary of allowed mutations, although the task of compiling a vocabulary of sequences homologous to those in the sample seems realistic. The RipSeq WEB server is an application designed for analysis of complex direct sequencing chromatograms, which are obtained for 16S rRNA PCR products from clinical samples and which accurately identify the bacterial species [30]. However, it is not a universal application for sequence analysis because it can process only chromatograms that are read from specific sequencing primers. The software maps all possible short words on the {A,C,G,T} alphabet that can be extracted from a IUPAC [31] sequence of a chromatogram onto known 16S rRNA sequences from a vocabulary. The RipSeq software can detect the presence in the mixture of up to three species from a vocabulary, and it does not generate sequences at the output. Practical methods of indel detection in direct sequencing chromatograms without a known wild-type also exist [25], [27], [32]. The Indelligent tool [25] identifies the two most similar sequences containing deletions assuming heterozygota. The CHILD [27] aligns the primary and secondary sequences that were extracted from a chromatogram by Phred; the alignment process uses SSEARCH [33]. CHILD can detect indels in low fractions (5–10%) of DNA mixture. Other tools quantify the components of complex mixtures by analyzing direct-sequencing chromatograms [34], [35].

The BCV Algorithm

In this study, we propose a new method for analyzing population sequencing chromatograms, called base-calling with vocabulary (BCV). This method provides a comprehensive toolbox for Sanger chromatogram analysis. As input, this software uses the sequence of detected peaks in the chromatogram and the multiple alignment of nucleotide sequences similar to the expected DNA variants in the sample. We refer to this set of nucleotide sequences as the vocabulary. We did not assume any restriction on the number of possible mixed DNA variants. The BCV package contains tools for 3 main functions:

• base-calling,
• indel detection relative to the main consensus sequence and vocabulary sequences, and
• DNA mixture deconvolution.

Base-calling is sufficient for samples containing SNVs as the major mutation type. BCV does not require a vocabulary in this case.

The indel detection function is appropriate for analyzing chromatograms with a high proportion of degenerate positions when some of the sample variants carry indels (Fig. 1). A high homology of DNA variants is necessary for using the indel detection functionality. Otherwise, the most general functionality should be preferred.

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Figure 1. The shifting patterns in direct sequencing chromatograms.

A. Sequences of 2 mixed DNA types (1 and 2) and their alignment. B and C. Chromatogram sequences (results of base-calling) for both reading directions are named bc-fw and bc-rev. The final Indels are assigned relatively to the main subgroup that is comprised of the DNA type, which has the higher fraction in the mixture. Italic underlined font shows shifting patterns. Bold shows the sequence portions that precede indel positions in each reading direction. Italics show the sequence portions of 2 DNA types that are aligned with the given coordinate shift.

https://doi.org/10.1371/journal.pone.0054835.g001

The most general functionality of the BCV package is the mixture deconvolution. The software can predict sequences of DNA variants that altogether explain the amplitude profile of the chromatogram, provided that a vocabulary of sequences that are similar to actual mixture components is available. This functionality is effective only if the vocabulary contains sequences that are more similar to mixture components than the mixture components are similar to each other. When BCV is used for the mixture deconvolution, the predicted DNA variants can be further explored by Blast [36] homology searches in biological sequence databases, or by a kind of phylogenetic analysis. This functionality can be applied to complex mixtures of DNA variants, like direct sequences of 16S rRNA genes from clinical samples, and for detection of viral genotypes (e.g. HCV, HBV, HIV) by sequencing assays in the cases of mixed infections. Table 1 summarizes empirical rules for selecting the appropriate BCV functionality for a given sample.

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Table 1. The guide for selection of the BCV usecase.

https://doi.org/10.1371/journal.pone.0054835.t001

We consider sequencing chromatogram processing to consist of the following steps: peak detection, base-calling, mixture deconvolution, and indel detection. The peak detection step is required to extract information from the 4 raw fluorescent traces obtained by a sequencer. We use the TraceTuner [18] program to determine the primary nucleotide sequence, and the Polyscan [24] program for re-base-calling. Since the source code for PolyScan is available, we add an option to output all the detected peaks, their physical properties, and the corresponding peaks’ probabilities into text files (refer to [24] for further details); thus, we don’t use the PolyScan’s grouping of peaks into sequence positions.

For sequencing of mixed DNAs, a chromatogram and its peak interpretation (partitioning) is represented as a hidden Markov model (HMM; generalized from [37]). Each partitioning represents an assignment of sequence positions to peaks, filtering out artifact peaks. A DNA base occurs only once in a sequence position; thus, each position can be described by an IUPAC DNA code. The base-calling step is a search for an optimal partition by the Viterbi algorithm [38], [39] for HMM on the chromatogram’s peak sequence. The mixture deconvolution step is intended to determine the DNA variants of the mixture that produced optimal partitioning. The variants are constructed by a greedy method that is based on preliminary DNA variants production by pairwise alignments of the partition with vocabulary sequences. Alignment scores are set up according to the standard HMM pairwise alignment schema, weighted by the peaks’ amplitudes. At each step, the sequence is determined from the best alignment; then the peaks that correspond to the found sequences are lowered (by subtracting the sequence from the mixture). Then the greedy search is repeated. Finally, the greedy-predicted DNA variants are combined by expectation maximization (EM) clustering procedure [40]. The distance measure for a pair of sequences, which is used for clustering, is a monotonically increasing function (see Methods) of the probability that the pair has been obtained from different chromatogram profiles. Initial clustering is made by a procedure similar to DOTUR [41]. The term ‘DNA variant’ refers to the consensus sequence found by the EM algorithm, unless explicitly stated otherwise. Expectations for DNA variant frequencies are evaluated from the clustering.

In the indel detection step, we use another clustering paradigm. All chromatograms for a sample are analyzed together. We suppose that all the greedy-generated (preliminary) DNA variants are classified into dense subgroups: those containing indels (one group per indel), and the group without indels (main subgroup). In each group, the preliminary variants are aligned with each other and with the vocabulary. After the main subgroup is identified for each chromatogram, we search for shifting patterns. A shifting pattern is a fragment on the chromatogram sequence that corresponds to a segment on preliminary DNA variants and that is aligned without indels against the main subgroup (Figure 1). Indels we are looking for occur before the start of shifting patterns (towards a read direction). Indel positions are calculated relative to the vocabulary sequences. Then we combine all the chromatogram’s main groups into the main consensus sequence that corresponds to the dominating pool of DNA variants in the sample.

We demonstrate the applicability of BCV in several situations. First, we use base-calling for analyzing direct sequencing traces of the hepatitis A and D viruses. Second, we use BCV to detect indels in the pncA gene of Mycobacterium tuberculosis. Finally, we apply BCV to deciphering a sample containing 2 subtypes of the hepatitis B virus (HBV) and complex mixtures of bacterial 16S RNA in human gastric mucosa by using direct sequencing.

The M. tuberculosis pcnA gene encodes the enzyme pyrazinamidase, which converts the pro-drug pyrazinamide to its active form, pyrazinoic acid [42]. In a considerable portion of cases, pyrazinamide resistance in tuberculosis occurs due to mutations disrupting the open reading frame of the the pncA gene [43]–[45]. Mutations found in resistant strains include amino acid substitutions, nonsense mutations, frameshifts, mutations in the promoter region, and even complete deletions of the pncA gene [43], [44]. Currently, over 350 mutations associated with resistance to pyrazinamide have been identified [46]. Traditional methods of sequence analysis work for detecting pyrazinamide resistant strains of M. tuberculosis in samples, but often fail to identify the presence of resistant strains in samples also containing wild-type bacteria.

Samples and Cultures

All patients signed an informed consent form in accordance with the institutional review board of the Federal State Institution of Science Central Research Institute of Epidemiology, Moscow. The study was approved by the institutional ethics committee.

Indel Detection in Clinical Samples

If DNA variants that contain indels were directly sequenced, the chromatogram was highly degenerate downstream from the indel sites (Figure 1). The results of the BCV indel detection analysis for the HIV, HBV, and M. tuberculosis clinical samples are shown in Table 2. Most samples in table 2 were obtained from our study (see below) of prevalence of pncA mutations in M. tuberculosis. Indel lengths varied from 1 to 12 nucleotides. For five samples, predictions made by the BCV were also confirmed by clone sequencing; for one sample the cloning experiment was not available (see Methods S1 and Table 2). The data produced by BCV indel detection analysis for the HIV sample GEN014DR.01A (Table 2) are shown in Figure 2. Sequencing the 3′ end of the HIV gag gene and the complete protease was done in both forward and reverse directions, using primers hiv-pf2 and hiv-pr2 correspondingly (see table S1). For the HIV sample GEN014DR.01A, the indel location was detected with minor error (−6 b.p) from the reverse sequencing primer, and the same indel location was detected with a larger error (+36) from the forward primer. The sample contained a mixture of HIV DNA variants; some of them carried tandem repeats relative to other strains that were in excess in the mixture. This error occurred due to some unrecognized minor peaks at the beginnings of the high polymorphism density regions of the chromatograms.

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Figure 2. BCV-predicted shifting patterns and tandem repeat positions for the human immunodeficiency virus (HIV) sample GEN014DR.01A show the multiple alignment of sequenced clones and consensus sequences for shifting patterns including the main consensus sequence (A) and chromatogram trace images (B).

Tandem repeats were highlighted by a frame on the sequence of clone 3. The beginning of shifting patterns, as for example for the hiv-pf2|shift +12 pattern, is marked by arrows. Sequencing primers are hiv-pf2 (forward) and hiv-pr2 (reverse). HXB2– is a reference sequence.

https://doi.org/10.1371/journal.pone.0054835.g002

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Table 2. Detected insertions and deletions (indels).

https://doi.org/10.1371/journal.pone.0054835.t002

We assessed how well the BCV indel detection script could determine the consensus sequence corresponding to a mixture of DNA variants containing indels. The BCV built a main consensus sequence that represented a pool of DNA variants that did not carry indels relative to each other. Preferably, the pool constituted the major portion of the mixture (main subgroup); in the case of an inability to select a subgroup of variants that clearly dominated in the mixture, the main subgroup was defined as the subgroup of variants with higher similarity to vocabulary sequences (see Methods). The phylogenetic tree that was shown at the Figure 3 had the following leaves:

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Figure 3. The comparison of BCV main sequence assembling results with sequences of cloned PCR products.

Phylogenetic tree shows relationships between consensus sequences (black squares) assembled from direct reads of the HIV protease gene fragment with sequences of clones (black circles) for sample GEN014DR.01A. The consensus assembled from two opposite direct reads with trimmed degenerate parts is denoted as D.vqa01; the one that is assembled by the BCV indel detection script is FR.main. F.main is the dominating DNA type extracted from a direct read in the forward direction by the BCV indel detection script; R.main is the same read in the opposite direction. H61 is the blastn best hit to sequence D.vqa01 used for scaling quasispecies variation (black circles).Reads in forward and reverse directions have different fractions of non-degenerate positions: F: 56/503 = 11%; R –430/492 = 87%. B: a node in the tree corresponding to HIV subtype B branch. The phylogenetic tree is constructed by the Minimum Evolution method [66] for the Maximum Composite Likelihood [67] distance matrix by the MEGA 5 software [68].

https://doi.org/10.1371/journal.pone.0054835.g003

1. clone sequences (black circles)
2. the consensus sequence obtained in the traditional way by assembling two sequences (reads); high density polymorphism areas corresponding to shifting patterns were trimmed (named D.vqa01)
3. the main consensus sequences predicted for the hiv-pf2 and hiv-pr2 chromatograms by separate (F.main, R.main) and simultaneous analysis (FR. main)
4. the closest homologue found by Blastn [36] search in GenBank [49] of clone sequences (H61-white circle)
5. the sequence from the BCV vocabulary, which was used for predicting mixture content (HXB2)
6. other reference sequences, used to define the distance variation within HIV subtype B

A high degree of heterogeneity was found in HIV quasi-species in the given sample (black circles). Consensus sequences (black squares) were located in the tree within quasi-species heterogeneity range around the best blastn hit H61. We wanted to emphasize that approximately 90% of the chromatogram derived with the forward primer (hiv-pf2 ) was degenerate. Thus, the algorithm was able to restore with reasonable accuracy the dominant DNA variant on a chromatogram with a highly degenerate sequence.

Base-calling

The results of comparing base-calling accuracies of four applications (BCV, ABI base-caller [52], TraceTuner v. 3.01 [18], and PolyScan [24]) are shown in Table 3. BCV displayed an advantage over the other applications in specificity on the test datasets. The advantage reached 1% on the HAV dataset and 4% on the HDV dataset as compared with ABI base-caller, and 1% compared with PolyScan. The data models on which BCV and PolyScan are based allowed degeneracy of up to 4 peaks in each chromatogram position, unlike the other two applications. BCV and PolyScan were more specific on the HDV testing set of moderately degenerate sequences than ABI base-caller and TraceTuner (Table 3). All the programs showed a similar sensitivity exceeding 97% on both datasets. Table 3 shows the identities of predicted and annotated sequences. Identity statistics for BCV, ABI base-caller, and PolyScan were 89% on the HAV testing set. This value was significantly better than the identity obtained by TraceTuner. BCV and PolyScan had an advantage over the other two programs on the HDV testing set.

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Table 3. Comparison of the base-calling accuracy statistics of Base-Caller with Vocabulary program (BCV) and other programs.

https://doi.org/10.1371/journal.pone.0054835.t003

Mixture Deconvolution

Calculating the quality of correspondence between predicted and actual sample components.

A very relevant advantage of BCV over other methods of direct sequencing chromatogram analysis, which are based on subtraction of the reference sequence [23], [24], [28], [53], [54], is that BCV does not require an exact match between vocabulary sequences and actual sample components, and that, moreover, the number of variant DNAs does not need to be known. We assessed the effects of maximal distance (the number of differences) between vocabulary sequences and the actual DNA variants on the Quality of Correspondence (QC), which was defined as a measure of the relation of predicted DNA variants to the real components of the mixture. The QC measure was calculated on a phylogenetic tree that also contained both predicted, actual DNA types and vocabulary sequences. If the QC = 0, none of the predicted variants is situated within a subtree that grew from the most recent common ancestor (MRCA) of actual mixture components if QC = 1, all predicted sequences are placed in the subtrees of sister branches of actual DNA variants (see Methods S1 and Figure S1). For this, a model experiment was prepared. We directly sequenced a revertase gene fragment from a 1∶2 mixture of two complete HBV genome plasmids (F2 and D1 subtypes) using two primers “hbv-rt-F” and “hbv-rt-S”, and then used phylogenetic analysis to compare the predicted and actual DNA sequences as described in (Methods S1). Figure 4 shows two phylogenetic trees in which the predicted BCV DNA variants are located at different distances from the corresponding sample components. For clarity, only a small subset of the vocabulary is shown. The black squares in Figure 4A mark the sequences predicted by BCV using the complete vocabulary HBVRT (see Methods). Each of two predicted sequences belonged to the same subtype of HBV as the corresponding actual DNA (black circles). The sequences (black squares) in Figure 4B were predicted using a dictionary containing only two sequences; both of the sequences were an approximate distance of 0.028 substitutions per site from real sample components. One predicted sequence belongs to the same genotype as the corresponding mixture component, and the other belongs to the same subtype. The genotypes also included the actual corresponding sequences (black circles). Figure 5 is a plot of QC versus the maximal distance between vocabulary sequences and the actual DNA variants for two reads. The QC exceeded 0.85 in the [0, 0.028] distance range that roughly corresponds to intra-subtype variations of the HBV revertase gene. The distance between sample components F2 and D1 was 0.1. For both reads, the QC value decreased slowly from 1.0 to the 0.5 and its variation increased with distance.

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Figure 4. DNA types predicted by BCV for the sample composed from 2 components of D and F hepatitis B virus (HBV) genotypes.

Black squares show predicted DNA types; black circles show actual sample components (identical to the GenBank sequences X02496, and X69798). Suffixes of sequence names correspond to HBV subtypes. Branches containing a mixture component are shown in bold. Right square brackets mark branches that contain predicted DNA types. The tics below the panels show the time scale. A and B correspond to two different vocabularies. A. Tree with DNA types predicted by BCV using the HBVRT vocabulary composed from 639 sequences of HBV genotypes A–H. B. Tree with DNA types predicted by BCV with vocabulary composed from 2 sequences approximately 0.028 substitution per site distant from components of the df7 sample. Phylogenetic trees are constructed by the Minimum Evolution method [66] for the Maximum Composite Likelihood [67] distance matrix by the MEGA 5 software [68].

https://doi.org/10.1371/journal.pone.0054835.g004

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Figure 5. Dependence of mixture reconstruction accuracy on the level of similarity between vocabulary sequences and real components of the sample.

The sample df7 that comprised a mixture of two HBV genome fragments of different genotypes (the same as on the figure 4) was sequenced from two primers “hbv-rt-F” and “hbv-rt-S” (see Table S1); each read was processed by the BCV using vocabularies of sequences that were on the different distances to the real mixture components. The Quality of Correspondence (QC) value of predicted and real components of the mixture is shown (see Methods S1).

https://doi.org/10.1371/journal.pone.0054835.g005

16S rRNA analysis of clinical sample microbial communities.

Figure 6 and Table S2 presented the results of taxonomical assignment of microbial populations from human gastric mucosa clinical samples, based on 16S rRNA analysis by two different methods: cloning and subsequent sequencing and classification using RDP classifier [55], [56], and direct sequencing followed by analysis by BCV, filtering out of rare (<5%) or short (<50% of chromatogram sequence length) predictions and classification using STAP (see Methods S1). For both types of analysis, sequences were searched by blastn [36] against the rRNA Greengenes database [57]. Figure 6 shows that the diversity of bacteria obtained by sequencing a small number of clones (10–15) coincided well with the results of direct sequencing followed by BCV analysis of three chromatograms per each sample. Table S2 shows the results of the search of the predicted sequences in the 16S rRNA Greengenes database. blastn hits with different taxonomy assignments were shown for a sequence if the assignment scores differed by not more than 2 bits. The similarity between predicted sequences and database sequences was within the 84–99% interval (the median is 95%); 9 of 12 BCV predicted sequences had identities with the best blast hit that were higher than 90%. All sequences except one were assigned by the STAP method into families or into more special categories - genera or species (table S2). Blastn was a suitable method to refine the STAP classification of predicted sequences, as hits had sufficiently high identities and had no contradictions with STAP (see Methods S1 and Figure S2 about the tolerance to errors of these classification methods).

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Figure 6. Comparing classification of DNA sequences of sequenced clones and BCV predictions of the 16S rRNA PCR product from a gastric mucosa biopsy.

Each line corresponds to a single taxonomic category. Parentheses contain the number of sequences of clones classified using the RDP Classifier (first value) and the number of best alignments using blastn on the 16S rRNA database Greengenes (second value); brackets contain the number of BCV predictions classified by the method based on STAP (first value) and the number of best alignments using blastn on the 16S rRNA database Greengenes unambiguously assigned to that category (second value, see Table S2). Taxonomic tree represents the RDP classification. The species names of the best blastn hits are marked with circles. Inconsistencies in categorization between BCV and cloning are shown in bold. A. Sample 95. B. Sample97.

https://doi.org/10.1371/journal.pone.0054835.g006

There was a good correspondence between bacterial community characterizations by two methods: sequencing after cloning and direct sequencing followed by the BCV analysis. There was a (7/10) correspondence at the family level and a (7/10) at the genus level (figure 6). Two of three discordant taxonomy categories comprised only clone sequences: g. Streptococcus and Unclassified_Lachnospiraceae group, and one category (family Enterococcaceae) comprised a predicted sequence only. Contradictions between two methods could be explained at least in part by a small number of selected clones: e.g. five genera were observed only by one clone sequence per genus, and it was just by chance that these genera had not been missed.

Four 16S rRNA sequences obtained by cloning and two predicted by BCV in sample “97”, and no sequences in sample “95”, were classified into the Helicobacter genus (Figure 6). H. pylori was also detected in sample “97” by PCR and the rapid urease test; neither test detected H. pylori in sample “95”.

BCV Software Data Flow

Figure 7 represents the data flow of the BCV software. The main BCV application BCV::proc receives tree input files: FPOLY file containing the peaks’ physical properties, BQS file containing the probability scores of the corresponding peaks in the positions and a vocabulary in FASTA format. The FPOLY and BQS files are generated by PolyScan [24] that received the primary sequence from TraceTuner [18] as input. The scores of the peak probability from the BQS file are used for trimming the 3′ artifacts at the end of the chromatogram; peaks in the 3′ tail with probability scores lower than 5% are excluded. Indel detection is done using the Perl script bcv_indels.pl. The script’s output (*.indel.txt) consists of shift patterns, the main consensus sequence, and the indel coordinates, which can be determined with respect to the main consensus sequence and to a specified vocabulary sequence. The script receives a multiple alignment of DNA variants in the GFAS format, obtained by the greedy deconvolution procedure. The GFAS format is similar to the FASTA format; it carries additional information ascribing sequences to chromatograms. Preparation of such an alignment is done by the Perl script bcv_run.pl. The script receives BCV::proc output files (*.strains.fasta and *.decomplog.gfas) for the same sample as the input. The dataflow is enveloped in the bcv_run.pl script that takes a path to a folder with ABI chromatogram files and a project XML file with specification of the correspondence of the chromatograms to samples and required functionality (usecase).

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Figure 7. BCV dataflow.

Rectangles depict software applications; rolls depict files; black arrows are the pipeline input and output streams with the corresponding input and output file extensions shown in italic bold. The file extensions are as follows: The input ABIF (*.ab1) file contains the chromatogram itself and the ABI base-calling. TraceTuner files (PHRED compatible): *.scf contains the chromatogram; *.phd.1 is the chromatogram sequence, and *.poly is the secondary peak calling results. PolyScan files: *.fpoly contains minor peak calls around the primary sequence, and *.bqs contains the peak likelihoods. BCV pipeline output files: *.viterbi.fasta contains the chromatogram sequence; *.cluster.fasta is the DNA type reconstruction and *.indels.txt is the indel report. The configuring and calling of TraceTuner, BCV::PolyScan and BCV::proc applications is enveloped in the bcv_run.pl script. For indel detection functionality the call of the bcv_indels.pl script is followed of the bcv_run.pl. The bcv_run.pl prepares an alignment of raw predicted DNA variants (from the *.strains.fasta file) with similar sequences from the vocabulary that are listed in the *.decomplog.gfas file. Both files are generated by the BCV::proc application. The input file for the indel detection script bcv_indels.pl has the grouped FASTA format and corresponding.gfas file name extension.

https://doi.org/10.1371/journal.pone.0054835.g007

BCV HMM Training

Parameters for the HMM of chromatogram base-calling were trained by a Baum-Welch algorithm [61] on a training set of 80 sequencing traces. For each trace in the set, an annotating sequence was available. The annotation was obtained by the manually curated assembling of two traces of HAV genome fragment 2C in both directions (NC_001489: positions 3796–4443). The training annotation sequences did not contain degenerate positions.

BCV Vocabularies

The BCV application uses a multiple alignment of target genome fragments in FASTA format as the vocabulary for DNA mixture deconvolution. To analyze direct sequencing chromatograms of the HIV genome fragment containing the 3′ fragment of the gag and complete protease genes (K03455∶2052–2623), we created the vocabulary HIVPRT (HIV protease). The genome fragments were cut from whole genome sequences used in the NCBI Viral Genotyping Tool [62] for HIV-1 subtyping. Another vocabulary, HBVRT (HBV revertase), was used to analyze the revertase domain of the polymerase gene (X04615∶336–1161). It consisted of 669 sequences extracted from the complete HBV genome sequence alignment obtained from the HVDB [63] database, and assigned to 1of the 8 HBV genotypes (A–H). The vocabulary for analyzing the M. tuberculosis pncA gene contained one sequence of the corresponding genome fragment (NC_002755∶2291656–2290984). BCV vocabulary of human bacterial community 16S rRNA sequences was obtained from the Greengenes [57] server by merging two multiple alignments: HMP_strains_16S_aligned.fasta (Human Microbiom Project [64]) and human_assoc_gold_strains_gg16S_aligned.fasta.

Algorithms

This chapter thoroughly describes the algorithms implemented in the BCV software.

Main Definitions

Hidden Markov Model (HMM)

The expression for the probability of a chromatogram partition B is.

(I),

where. The task of chromatogram decomposition can be represented as the optimization of the functional (I):

.

If we assume that the prior distribution P(B) is uniform, then the problem reduces to finding maximum likelihood. Likelihood is computed by an algorithm that generalizes the work of Andrade-Cetto (2005) [37] to the case allowing ambiguity in chromatogram sequence positions (the position’s degeneration). We assume that the correct peaks in positional frames are in first-order Markov dependence. We introduce an additional notation for true peak sets:. Letting be the right-most true peak in the positional frame that is mapped to sequence position , the operator […] assigns a sub-sequence, like a chromatogram sub-sequenceor a false peak sub-sequence (sequencing artifacts in the [i,j] range of peaks). Then the likelihood of the chromatogram can be written as:(III)

The first term on the right side of the expression is the initial probability of the Markov model as a product of two terms: one is concerning the TP peaks subset for the chromatogram start and the second is concerning the FP subset. The product on the second line accounts similar terms in a first order Markov chain manner.

The functional (III) can be considered as an HMM, so it can be optimized (II) by the Viterbi algorithm [38], [39].

DNA Mixture Deconvolution Algorithm

Setting partition B as the optimal solution to task (II), we find a set of DNA variants (D6) that is the most likely to correspond to this partition. We align sequences w from vocabulary W against the partition. The algorithm of such an alignment is similar to an ordinary pairwise global alignment algorithm in the space. The third dimension corresponds to the nucleotide composition of the positions in the partition. We then choose a scheme of alignment weights proportional to relative nucleotide frequencies in positional frames, so for each sequence in the vocabulary, the best alignment will mostly pass through the highest peaks in frames that contain no peaks matched to these bases in the sequence. For each positional frame α, we determine the relative amplitude of each symbol . If a peak in this partition is treated as false, . We represent relative amplitudes as array F with dimension equal to the number of peaks in the chromatogram. The mixture deconvolution algorithm determines a set of DNA variants . DNA variants will be represented by a sequence of elements, where 0 means that a peak does not emit a base for a DNA variant sequence. Below, we write the pseudo-code for the mixture deconvolution algorithm:

Input. (Z, B, W).

Parameters:

K_max - the maximum number of components in the mixture, so 1/K_max is the minimum frequency with which the DNA variant may be detected. This frequency is determined by the minor variant’s detection limit of the sequencing method ().

- rate of the amplitude descent. A reasonable value is .

Output. the DNA mixture U = U(B) and the vector of the mixture’s component fractions γ.

Calculations:

1. Calculate the peak’s relative amplitudes in positions. Set up the initial DNA variant fractions .
2. Iterate for each sequence w_k in vocabulary W
   2.1. Align sequence w_k with (B, S, F). Alignment weight for a true peak corresponding to the symbol s in positional frame α is set to - , where t is an alignment position, and corresponds to emission and transition probabilities for a pairwise alignment HMM in state π_t = (match, opening an X insertion, opening an Y ins., X ins. elongation, Y ins. elongation, etc).
3. Take the alignment with the highest score. Set up elements for a new DNA variant u_i: u_i,j =1 for each peak j used in the alignment, and u_i,j =0 otherwise.
4. Find peak j’ with the minimal amplitude from all peaks used in the alignment.
5. For each peak where , change the relative amplitudes to.
6. Add new variant u into the hash table U. Update the variant’s fraction vector
7. Terminate if there is a positional frame, where all peak frequencies are below the threshold - , else go to 2.
8. Output (U, γ).

DNA Variant Clustering

The sequences of DNA variants and their number depend on the parameters that are used for mixture deconvolution – r, K_max. Since the sequence of each of the virtual DNA variants may contain various errors, it is important to combine similar DNA variants into clusters and to build consensus sequences for each cluster that decreases the error level in predicted sequences of DNA variants. Thus, the algorithm would make it possible to identify significantly different DNA variants by their genetic distance. To solve the clustering problem, we adapted the algorithm proposed in [40] for 454 reads.

We assume that the mixture deconvolution algorithm at each iteration i generates a unique variant. Consider a DNA variant (here we used superscript indices to emphasize the parameters depended on an iteration number) and a nucleotide sequence w not necessarily belonging to vocabulary W. To use the algorithm proposed by Quince [40], we need to build a model of the distance between the DNA variant and the sequence . To consider a possible alternative choice of peaks in positional frames by the mixture deconvolution algorithm, we introduce a variant profile.

.

The variant profile assigns probability that each true peak emits a base for the variant u⁽ⁱ⁾. The profile represents an uncertainty model determining the chance that the peak is included into a variant sequence generated at iteration i. The distance between the variant and the sequence will be calculated under the condition that the profile is also known: . We will further assume that we also have a pairwise global alignment of a DNA variant and a sequence, and, strictly speaking, the distance will be a function of the alignment. Here and below, unless otherwise noted, variant u⁽ⁱ⁾ is regarded as a nucleotide sequence. Let us also set the base substitution model that determines the probability that base w_k depends on base s. We then define distance.(1)through the probability of sequence w to be obtained from the profile of DNA variant P⁽ⁱ⁾ and the substitution model. Expressiondenotes the length of the pairwise alignment ignoring indels. Distance is normalized to alignment length in order to accommodate the different lengths of DNA variants. We now write an expression for calculating the probability that a sequence was derived from the specified variant profile:

(2)Where

(3)

The sum in expression (3) is taken over all true peaks in a positional frame to which peak j belongs. The product in expression (2) is taken over all matches in the alignment of sequences w and u⁽ⁱ⁾ . Equation (1) can be rewritten as the sum of positional distances in the alignment of a DNA variant u⁽ⁱ⁾ and sequence w using expression (2): Z

(1.1),

where

(1.2).

Probabilitieswere calculated using expression (3). The final output of the expectation maximization algorithm described in [40] depends on the initial conditions. The initial variant grouping was done by the hierarchical clustering algorithm similar to the DOTUR [41]. We determine the distance between 2 DNA variants using the formula.

(4)where the expressiondefines the probability of true homology between DNA variants u′ and u′′ (i.e., the probability that both variants were derived from a single profile):

(5)Used in formula (4), the expression is evaluated in accordance with (2), assuming that w = u⁽ⁱ⁾.

Below we write a pseudo-code for the expectation maximization algorithm for clustering DNA variants:

Input. (U, {P⁽ⁱ⁾}).

Output. The set C of consensus sequences corresponding to set U of DNA variants.

Calculations:

1. Construct a multiple alignment of DNA variant nucleotide sequences. Denote representing the nucleotide sequence of the DNA variant at i-th iteration of the mixture deconvolution algorithm in the multiple alignment. The length of this sequence is equal to the alignment length. In each sequence position, the following characters are valid: .
2. Make initial associations of DNA variants into clusters. Pairwise distances between variants are calculated in accordance with expression (5). Pairwise alignments of variants are retrieved from the multiple alignment (column containing characters ‘-’ were removed). As a result, we obtain matrix Z, which is a correspondence table for variants and clusters. See [40] for details.
3. Iterate where matrix Z changes more than the specified accuracy threshold from one iteration to the next:
   3.1. The M-step: calculate the consensus nucleotide sequences for DNA variant clusters U_j and their corresponding weights τ_j. Expressions (6–8) use the following notations: C^j_l is a character at position l in the consensus sequence of cluster j; K is the number of DNA variants; i is a specified DNA variant; z^{i, j} are elements of correspondence matrix Z; and distances d′ are calculated according to expression (3). Consensus sequences are calculated taking into account the contributions of mixture fractions γ of DNA variants (see 8).
   (6)
   (7)
   (8)
   3.2. The E-step: Calculating the expected values of elements z^{i, j} of correspondence matrix Z. The expression for distances d was calculated in accordance with expression (1.1).

(9)

Indel Detection Algorithm

Our next task is to use BCV to analyze a sample containing homologous DNA variants, including some with indels. The challenge here is to determine indel positions with respect to a reference sequence from the vocabulary. We introduce the following definitions:

D8. Main group of DNA variantsis a set of variants that align with each other variant from the main group without internal deletions and that represent a sufficient proportion of the mixture (γ^main). The main group is formed as follows:

• All the groups of the DNA variants that align with each other without internal deletions are sorted in order of their weights decreasing.
• The first group in the list, whose weight is less than 60% of the weight of previous group, is found. All the preceding groups are referred as the “head of the list”.
• If there is a single group at the head of the list, it is defined as the main group; otherwise the main group is defined as a group from the head of the list that contains the DNA variant that was generated at the first iteration of the greedy deconvolution algorithm (“strain_1”). This variant is the most similar to the vocabulary sequences. If the “strain_1” is not a member of any group in the head of the list, then the first group (with highest weight) is defined as the main group.

D9. The d-shifting pattern is a set of non-overlapping intervals on the chromatogram sequence corresponding to the optimal partition B:, where d is a pattern shift size. Intervals of the shifting pattern obey the rule:

, where A(u_i, u_j) is the optimal global pairwise alignment of DNA variant sequences.

Ideally, a shifting pattern consists of a single segment. If d >0, then the pattern corresponds to an insertion, and if d <0, it corresponds to a deletion relative to the main group. We define a shifting pattern d = 0 as the main group of DNA variants; a value d is called the shift size.

The beginning of a shifting pattern is the position on the chromatogram sequence that corresponds to the minimal left coordinate of the pattern’s segments (if reading in the forward direction), or to the maximal right coordinate of the pattern’s segments (if reading in the reverse direction).

The end of a shifting pattern is analogously defined. The end of a shifting pattern exists if there is another shifting pattern for the same chromatogram that begins at the 3′ end to the right of the pattern in the appropriate reading direction. Thus, a single shifting pattern for a chromatogram has no end, and the last pattern in the reading direction has no end. This restriction takes into account the fact that peak amplitudes decrease towards the end, so more positions lose minor peaks, making establishing the true positions where a shifting pattern ends unreliable.

D10. The consensus sequence of a d-shifting pattern is the sequence of IUPAC symbols , where F^cons is the consensus-building function that projects bases from aligned DNA variants and their fractions on the IUPAC alphabet.

D11. An indel event is the deletion or insertion relative to the main group of DNA variants. If the indel event corresponds to the beginning or end of a shifting pattern, we set that event type d equals to the pattern’s shift size. If d >0, then the indel event corresponds to an insertion, and if d <0, it corresponds to a deletion relative to the main group.

For events corresponding to consecutive patterns, the type is defined as follows. If 2 successive patterns in the chromatogram reading direction have type values of d1 and d2, then the corresponding event has type value.

Additional conditions may be superimposed on the generation of events, e.g, a limit on the maximum distance between consecutive patterns.

For each event, we associate a pair of coordinates (lpos, rpos) on the alignment, defining an interval in which an indel has occurred; a pair of shift size values for shifting patterns (d1, d2) that caused the event; and a likelihood (L) expressed in terms of alignment weight of profiles corresponding to the main group and the shifting pattern at each event boundary in the window of a given size: .

To introduce the indel detection algorithm, we assume that pairwise global alignments of DNA variants are taken from the global multiple alignment of DNA variants predicted for a sample for all covering chromatograms. We assume that this multiple alignment also contains reference sequences relative to which indel coordinates can be determined.

The pseudo-code for the indel detection algorithm is described below.

Input. A multiple alignment of DNA variants MSA({U}) and DNA variant frequency vectors {γ} predicted by the deconvolution algorithm. Here, {…} means a set of elements corresponding to chromatograms of a sample.

Output. Indel events with the maximum local likelihood.

Calculations:

1. DNA variants with pairwise alignments that have no internal deletions join separately into groups for each chromatogram.
2. For each chromatogram, determine the main group of DNA variants.
3. Build shifting patterns using definition 9.
4. Generate indel events using definition 11.
5. Estimate event likelihoods.
6. For each chromatogram, separately make groups of events overlapping on alignment coordinates. Only one situation is possible when for an event , corresponding to the beginning or end of a pattern, there is overlap with another event corresponding to consecutive patterns ; if holds, then event e is rejected. This rule is based on the Parsimony assumption to obtain the minimum number of events while explaining all shifting patterns for a sample. If it is possible to construct an event corresponding to consistent patterns for a sample’s chromatogram, then this event is always within a set with minimal cardinality.
7. Group all overlapping events of equal type values (d) that correspond to different chromatograms.
8. For each group, determine the maximum likelihood event e*, which is included in the final dataset.

Availability and Future Directions

The source code for the BCV software package is available at the BCV website (http://basecv.sourceforge.net/) for gcc compiler that is available for all UNIX-like (Linux, MAC, Cygwin) platforms. See the website for the examples of the three functionalities and for the documentation. The system requirements are quite moderate: the gcc compiler with installed Boost and GSL libraries are necessary to compile/link; Perl5 is necessary for the scripts to run. The program can work on any standard desktop computer. The executable binary files for Win32 and Linux are also available upon request.

The main limitation of the Sanger sequencing method is the high variation of peak amplitudes in chromatograms, which restricts the accuracy of predicted DNA variants for which the vocabulary has no proper homologue and limits the determination of a sample’s component fractions. Modifying BCV for pyrosequencing data (i.e., not Next Generation Sequencers) that possess a much better signal-to-noise ratio can significantly improve the predictive ability of the method. Another promising direction of the method development is incorporation of a priori statistical model of the target DNA if the model is known, e.g. from RNA structure or from a known transcript function, primary or secondary structure.