Abstract
Background
RNA viruses infecting a host usually exist as a set of closely related sequences, referred to as quasispecies. The genomic diversity of viral quasispecies is a subject of great interest, particularly for chronic infections, since it can lead to resistance to existing therapies. Highthroughput sequencing is a promising approach to characterizing viral diversity, but unfortunately standard assembly software was originally designed for single genome assembly and cannot be used to simultaneously assemble and estimate the abundance of multiple closely related quasispecies sequences.
Results
In this paper, we introduce a new Vi ral Sp ectrum A ssembler (ViSpA) method for quasispecies spectrum reconstruction and compare it with the stateoftheart ShoRAH tool on both simulated and real 454 pyrosequencing shotgun reads from HCV and HIV quasispecies. Experimental results show that ViSpA outperforms ShoRAH on simulated errorfree reads, correctly assembling 10 out of 10 quasispecies and 29 sequences out of 40 quasispecies. While ShoRAH has a significant advantage over ViSpA on reads simulated with sequencing errors due to its advanced error correction algorithm, ViSpA is better at assembling the simulated reads after they have been corrected by ShoRAH. ViSpA also outperforms ShoRAH on real 454 reads. Indeed, 7 most frequent sequences reconstructed by ViSpA from a real HCV dataset are viable (do not contain internal stop codons), and the most frequent sequence was within 1% of the actual open reading frame obtained by cloning and Sanger sequencing. In contrast, only one of the sequences reconstructed by ShoRAH is viable. On a real HIV dataset, ShoRAH correctly inferred only 2 quasispecies sequences with at most 4 mismatches whereas ViSpA correctly reconstructed 5 quasispecies with at most 2 mismatches, and 2 out of 5 sequences were inferred without any mismatches. ViSpA source code is available at http://alla.cs.gsu.edu/~software/VISPA/vispa.html.
Conclusions
ViSpA enables accurate viral quasispecies spectrum reconstruction from 454 pyrosequencing reads. We are currently exploring extensions applicable to the analysis of highthroughput sequencing data from bacterial metagenomic samples and ecological samples of eukaryote populations.
Background
Viral quasispecies
Many viruses (including SARS, influenza, HBV, HCV, and HIV) encode their genome in RNA rather than DNA. Unlike DNA viruses, RNA viruses lack the ability to detect and repair mistakes during replication [1] and, as a result, their mutation rate can be as high as 1 mutation per each 1,000100,000 bases copied per replication cycle [2]. Many of the mutations are well tolerated and passed down to descendants, producing a family of coexisting related variants of the original viral genome referred to as quasispecies, a concept that originally described a mutationselection balance [3–7].
The diversity of viral sequences in an infected individual can cause the failure of vaccines and virus resistance to existing drug therapies [8]. Therefore, there is a great interest in reconstructing genomic diversity of viral quasispecies. Knowing sequences of the most virulent variants can help to design effective drugs [9, 10] and vaccines [11, 12] targeting particular viral variants in vivo.
454 pyrosequencing technology
Briefly, the 454 pyrosequencing system shears the source genetic material into fragments of approximately 300800 bases. Millions of singlestranded fragments are sequenced by synthesizing their complementary strands. Repeatedly, nucleotide reagents are flown over the fragments, one nucleotide (A, C, T, or G) at a time. Light is emitted at a fragment location when the flown nucleotide base complements the first unpaired base of the fragment [13, 14]. Multiple identical nucleotides may be incorporated in a single cycle, in which case the light intensity corresponds to the number of incorporated bases. However, since the number of incorporated bases (referred to as a homopolymer length) cannot be estimated accurately for long homopolymers, it results in a relatively high percentage of insertion and deletion sequencing errors (which respectively represent 65%75% and 20%30% of all sequencing errors [15, 16]).
The software provided by instrument manufacturers were originally designed to assemble all reads into a single genome sequence, and cannot be used for reconstructing quasispecies sequences. Thus, in this paper we address the following problem:
Quasispecies Spectrum Reconstruction (QSR) problem
Given a collection of 454 pyrosequencing reads generated from a viral sample, reconstruct the quasispecies spectrum, i.e., the set of sequences and the relative frequency of each sequence in the sample population.
A major challenge in solving the QSR problem is that the quasispecies sequences are only slightly different from each other. The amount and distribution along the genome of differences between quasispecies varies significantly between virus species, as different species have different mutation rates and genomic architectures. In particular, due to the lower mutation rate and longer conserved regions, HCV quasispecies are harder to reconstruct than quasispecies of HBV and HIV. Additionally, the QSR problem is made difficult by the limited read length and relatively high error rate of high throughput sequencing data generated by current technologies.
Related work
The QSR problem is related to several wellstudied problems: de novo genome assembly [17–19], haplotype assembly [20, 21], population phasing [22] and metagenomics [23]. As noted above, de novo assembly methods are designed to reconstruct a single genome sequence, and are not wellsuited for reconstructing a large number of closely related quasispecies sequences. Haplotype assembly does seek to reconstruct two closely related haplotype sequences, but existing methods do not easily extend to the reconstruction of a large (and a priori unknown) number of sequences. Computational methods developed for population phasing deal with large numbers of haplotypes, but rely on the availability of genotype data that conflates information about pairs of haplotypes. Metagenomic samples do consist of sequencing reads generated from the genomes of a large number of species. However, differences between the genomes of these species are considerably larger than those between viral quasispecies. Furthermore, existing tools for metagenomic data analysis focus on species identification, as reconstruction of complete genomic sequences would require much higher sequencing depth than that typically provided by current metagenomic datasets.
In contrast, achieving high sequencing depth for viral samples is very inexpensive, owing to the short length of viral genomes. Mapping based approaches to QSR are naturally preferred to de novo assembly since reference genomes are available (or easy to obtain) for viruses of interest, and viral genomes do not contain repeats. Thus, it is not surprising that such approaches were adopted in the two pioneering works on the QSR problem [24, 25]. Eriksson et al. [24] proposed a multistep approach consisting of sequencing error correction via clustering, haplotype reconstruction via chain decomposition, and haplotype frequency estimation via expectationmaximization, with validation on HIV data. In Westbrooks et al. [25], the focus is on haplotype reconstruction via transitive reduction, overlap probability estimation and network flows, with application to simulated errorfree HCV data. Recently, the QSR software tool ShoRAH was developed [26] and applied to HIV data [27]. Another combinatorial method for QSR was also developed and applied to HIV and HBV data in [28], with results similar to those of ShoRAH. Our contributions in this paper are as follows:

A novel QSR tool called V iral S pectrum A ssembler (ViSpA) taking into account sequencing errors at multiple steps,

Comparison of ViSpA with ShoRAH on HCV synthetic data both with and without sequencing errors, and

Statistical and experimental validation of the two methods on real 454 pyrosequencing reads from HCV and HIV samples.
Methods
Our method for inferring the quasispecies spectrum of a virus sample from 454 pyrosequencing reads consists of the following steps (see Fig. 1):

Constructing the consensus virus genome sequence for the given sample and aligning the reads onto this consensus,

Preprocessing aligned reads to correct sequencing errors,

Constructing a transitively reduced read graph with vertices representing reads and edges representing overlaps between them,

Selecting paths in the read graph that correspond to the most probable quasispecies sequences, and assembling candidate sequences for selected paths by weighted consensus of reads, and

Estimating candidate sequence frequencies by EM
Below we describe each step separately.
Read alignment and consensus genome sequence construction
We assume that a reference genome sequence of the particular virus strain is available (e.g., from NCBI [29]). Since viral genomes do not have sizable repeats and the quasispecies sequences are usually close enough to the reference sequence, the majority of reads can typically be uniquely aligned onto the reference genome. However, a significant number of reads may remain unaligned due to differences between the reference genome and sequences in the viral sample. In order to recover as many of these reads as possible, we iteratively construct a consensus genome sequence from aligned reads.
In particular, we first align 454 pyrosequencing reads to the reference sequence using the SEGEMEHL software [30]. Then we extend the reference sequence with a placeholder I for each nucleotide inserted by at least one uniquely aligned read. Similarly, we add a placeholder D to the read sequence for each reference nucleotide missing from the aligned read. Then we perform sequential multiple alignment of the previously aligned reads against this extended reference sequence. Finally, the consensus genome sequence is obtained by (1) replacing each nucleotide in the extended reference with the nucleotide or placeholder in the majority of the aligned reads and (2) removing all I and D placeholders, respectively corresponding to rare insertions and to deletions found in a majority of reads. Reads may contain a small portion of unidentified nucleotides denoted by N’s – we treat N as a special allele value matching any of nucleotides A, C, T, G, as well as placeholders I, and D.
Iteratively, we replace the reference with the consensus and try to align the reads, for which we could not find any acceptable alignment previously. Our experiments on a dataset consisting of approximately 31,000 454 pyrosequencing reads generated from a 5.2kblong HCV fragment (see data description in Results and Discussions) show that 85% of reads are uniquely aligned onto the reference sequence and an additional 9% of the reads are aligned onto the final consensus sequence. Reads that cannot be aligned onto the final consensus are removed from the further consideration.
Preprocessing of aligned reads
Since aligned reads contain insertions and deletions, we use placeholders I and D to simplify position referencing among the reads. All placeholders are treated as additional allele values but they are removed from the final assembled sequences. First, we substitute each deletion in the aligned reads with placeholder D. Deletion supported by a single read is replaced either with the allele value, which is present in all other reads overlapping this position, or with N, signifying an unknown value, otherwise. Next, we fill with placeholder I each gap in a read corresponding to the insertions in the other reads. All insertions supported by a single read are removed from consideration.
Read graph construction
We begin with the definition of the read graph, introduced in [25] and independently in [24], and then describe the adjustments that need to be made to read graph construction and edge weights to account for sequencing errors as well as the high mutation rate between quasispecies.
The read graph G = (V, E) is a directed graph with vertices corresponding to reads aligned with the consensus sequence. For a read u, we denote by b(u), respectively e(u), the genomic coordinate at which the first, respectively the last, base of u gets aligned. A directed edge (u, v) connects read u to read v if a suffix of u overlaps with a prefix of v and they coincide across the overlap. Two auxiliary vertices  a source s and a sink t – are added such that s has edges into all reads with zero indegree and t has edges from all reads with zero outdegree. Then each s – tpath corresponds to a possible candidate quasispecies sequence. The read graph is transitively reduced, i.e., each edge e = (u, v) is removed if there is a u – vpath not including edge e. Note that certain reads can be completely contained inside other reads. Let a superread refer to a read that is not contained in any other read and let the rest of the reads be called subreads. Subreads are not used in the construction of the read graph, but are taken into account in the final assembly of candidate sequences and frequency estimation.
Since the number of different s – tpaths is exponential, we wish to generate a set of paths that have high probability to correspond to real quasispecies sequences. In order to estimate path probability, we independently estimate for each edge e the probability p(e) that it connects two reads from the same quasispecies, and then multiply estimated probabilities for all edges on the path. Under the assumption of independence between edges, if we assign to each edge e a cost equal to – log(p(e)) = log(1/p(e)), then the minimumcost s – tpath will have the maximum probability to represent a quasispecies sequence.
For reads without errors, [25] estimated the probability that two reads u and v connected by edge (u, v) belong to the same quasispecies as
where Δ = b(v) – b(u) is the overhang between reads u and v[25], N = #reads, q = #quasispecies, and L = # starting positions. Thus, in this case the cost of an edge with overhang Δ can be approximated by Δ ∝ log(1/p_{Δ}).
To account for sequencing errors, we adjust the construction of the read graph to allow for mismatches. We use three parameters: (1) n = # mismatches allowed between a read and a superread, (2) m = #mismatches allowed in the overlap between two adjacent reads, and (3) t = # mismatches expected between a read and a random quasispecies. The probability that two reads u and v with j mismatches within an overlap of length o = e(u) – b(v) belong to the same quasispecies can be estimated as:
where ε is the estimated 454 sequencing error rate. As in the case of errorfree reads, defining the edge costs as ensures that s – tpaths with low cost correspond to most likely quasispecies sequences.
Candidate path selection
To generate a set of highprobability (lowcost) paths that are rich enough to explain observed reads, we compute for each vertex in the read graph the minimum cost s – tpath passing through it. Finding these paths is computationally fast. Indeed, we only need to compute two shortestpaths trees in G, one outgoing from s and one incoming into t; the shortest s – tpath passing through a vertex v is the concatenation of the shortest s – v and v – tpaths.
Preliminary simulation experiments (see Additional File 1) show that better candidate sets are generated when edge costs c defined by (1) and (2) are replaced by e^{c}. In fact, if we use even faster dependency on c then we obtain better candidate sets. The fastest growing cost effectively changes the shortest path into so called maxbandwidth path, i.e., paths that minimizes maximum edge cost for the entire path and for each subpath. So, ViSpA generates candidate paths using this strategy.
Candidate sequence assembly
When no mismatches are allowed in the construction of the read graph, finding the candidate sequence corresponding to a s – tpath is trivial, since by definition adjacent superreads coincide across their overlap. When mismatches are allowed, we first assemble a consensus sequence from superreads used by the s – tpath. It may be not the best choice, especially when the coverage with superreads is low. Hence, we replace each initial candidate sequence with a weighted consensus sequence obtained using both superreads and subreads of the path, as described below.
For each read r, we compute the probability that it belongs to a particular initial candidate sequence s as:
where l and L denote the lengths of the read and initial candidate sequence, respectively, k is the number of mismatches between the read and the initial candidate sequence s, and t/L is the estimated mutation rate. Then final candidate sequence is computed as the weighted consensus over all reads, where the weight of a read is the probability that it belongs to the sequence. Note that, unlike the case without mismatches, the same candidate sequence can be obtained from different candidate s – tpaths, so we remove duplicates at the end of this step.
Estimation of candidate quasispecies sequence frequencies
We assume that reads R with observed frequencies where generated from a quasispecies population Q as follows. First, a quasispecies sequence q ∈ Q is randomly chosen accordingly to its unknown frequency f_{ q }. A read starting position is generated from the uniform distribution and then a read r is produced from quasispecies q with j sequencing errors. The probability of this event is calculated as , where l is the read length and ε is the sequencing error rate. Thus, the probability of observing the read r under this model is .
Quasispecies frequencies are estimated by maximizing the loglikelihood function:
using an EM algorithm [31] (see Additional File 1 for details). Currently, convergence of the EM algorithm is determined at the tolerance level 0.005.
Results and discussions
In our simulation studies we use the following read data sets.
Reads simulated from known HCV quasispecies
In order to perform crossvalidation on the assembly method, we simulate reads data from 1739bp long fragment from the E1E2 region of 44 HCV sequences [32] when sequence frequencies are generated according to some specific distribution. In our simulation experiments, we use geometric distribution (ith sequence is constant factor more frequent than the (i + 1)th sequence) to create sample quasispecies populations with different number of randomly selected abovementioned quasispecies sequences.
We first simulate reads without sequencing errors: the length of a read follows normal distribution with a particular mean value and variance 400, and a starting position follows the uniform distribution. This simplified model of reads generation has two parameters: number of the reads that varies from 20K up to 100K and the average read length that varies from 200bp up to 600bp.
Additionally, we simulate 454 pyrosequencing reads from 10 quasispecies sequences (following geometric distribution of frequencies) out of 44 HCV sequences [32] using FlowSim [33]. We generated 30K reads with average length 350bp.
454 pyrosequencing reads from HCV samples
The data set Data1 has been received from HCV Research Group in Institute of Biomedical Research, at University of Birmingham. Data1 contains 30,927 reads obtained from the 5.2kblong fragment of HCV1a genome (which is more than a half of the entire HCV genome). The average (aligned) read length average is 292bp but it significantly varies as well as the depth of position coverage (see Additional File 1 for details). The depth of reads coverage variability is due to a strong bias in the sequence start points, reflecting the secondary structure of the template DNA or RNA used to generate the initial PCR products. As a result, shorter reads are produced by GCrich sequences. Data1 is available upon request from the authors.
454 pyrosequencing reads from HIV samples
The HIV dataset [27] contains 55,611 reads from mixture of 10 different 1.5kblong region of HIV1 quasispecies, including pol protease and part of the pol reverse transcriptase. The aligned reads length varies from 35bp to 584bp with average about 345bp (see Additional File 1 for details). In contrast to [27], we do not filter out reads with lowquality scores.
Experimental validation on simulated data
In all our experimental validations, we compare the proposed algorithm ViSpA with the stateoftheart tool ShoRAH as well as with ViSpA on ShoRAHcorrected reads (ShoRAHreads + ViSpA). We say the quasispecies sequence is captured if one of the candidate sequences exactly matches it. We measure the quality of assembling by portion of the real quasispecies sequences being captured by candidate sequences and its portion among candidate sequences (positive predictive value ) in crossvalidation tests. Both sensitivity and PPV are analyzed as functions of the number of quasispecies in underlying sample population (see Fig. 2 (left)). ViSpA can correctly assemble all sequences out of 10 quasispecies and 29 sequences out of 40 quasispecies if average read length is at least 300bp. If the average read length is smaller (for example, in range from 250bp till 299bp), the method can assemble at least 8 out of 10 sequences and 20 out of 40 sequences. Here, we see advantage of ViSpA over ShoRAH.
Following [24], we measure the prediction quality of frequency distribution with KullbackLeibler divergence, or relative entropy. Given two probability distributions, relative entropy measures the ”distance” between them, or, in the other words, the quality of approximation of one probability distribution by the other distribution. Formally, the relative entropy between true distribution P and approximation distribution Q is given by the formula:
where summation is over all reconstructed original sequences I = {i  P(i) > 0, Q(i) > 0} , i. e., over all original sequences that have a match (exact or with at most k mismatches) among assembled sequences. The relative entropy is decreasing with increasing of the average read length. It is expected since sensitivity is increasing with increasing of the average read length and EM predicts underlying distribution more accurately. ViSpA algorithm considerably outperforms ShoRAH (see Fig. 2 (right)).
However, ShoRAH has a significant advantage over ViSpA on a read data simulated by FlowSim both in prediction power and in robustness of results (see Table 1). Indeed, ShoRAH correctly infers 3 out of 10 real quasispecies sequences whereas ViSpA reconstructs only 1 sequence. Additionally, 10 most frequent assemblies inferred by ShoRAH are more robust with repeating up to 45% of times on 10%reduced data versus 1% of times for ViSpA’s assemblies. This advantage can be explained by superior read correction in ShoRAH. If ViSPA is used on ShoRAHcorrected reads, the results drastically improves: 5 quasispecies sequences are inferred and exactly 95% of times are repeated on reduced data, confirming that ViSpA is better in assembling sequences (see Table 1).
Experimental validation on 454 pyrosequencing reads from HCV samples
We first discuss the choice of parameters of the read graph and candidate sequence assembly from s – tpaths. Then we give statistical validation for obtained 10 most frequent quasispecies sequences.
We infer quasispecies spectrum based on the read graphs constructed with various numbers n and m (numbers of mismatches allowed for superreads and overlaps corresponding to edges). We sort the estimated frequencies in descending order and count the number of sequences which cumulative frequency is 85%, 90%, and 95%. Fig. 3 reports these numbers as a percent of the total number of candidate sequences. There is an obvious drop in percentage for all three categories if we allow up to n = 6 mismatches to cluster reads and up to m = 15 mismatches to create edges. In this case, the constructed read graph has no isolated vertices.
To refine assembled candidate sequences, we use all reads and parameter t varying from 80bp till 350bp, or, in the other words, mutation rate varying from 1.75% up to 8% per sequence (which is in the range observed in [34]). Out of 3207 maxbandwidth paths, we obtain as much as 938 distinct sequences (t = 80) and as low as 755 sequences (t = 350) for different values of t ∈ [80; 350].
The neighborjoining tree for the most frequent 10 candidate sequences obtained by ViSpA and ShoRAH (see Fig. 4) reminds a neighborjoining tree for HCV quasispecies evolution. Additionally, the most frequent candidate sequence found by ViSpA is 99% identical to one of the actual ORFs obtained by cloning the quasispecies.
Viral sequences containing internal stop codons are not viable since the entire HCV genome consists of a single coding region for a large polyprotein. So the number of reconstructed viable sequences can serve as an accuracy measure for quasispecies assembly. Out of 10 most frequent sequences reconstructed by ViSpA, only 3 are not viable while ShoRAH is able to reconstruct only one viable sequence. This sequence has 99.94% similarity with the ViSpA’s fourth most frequent assemblies. Both methods returned similar frequency estimations for this sequence: 0.017% (ShoRAH) and 0.019% (ViSpA).
Both ShoRAH and ViSpA (n = 6, m = 15) are run on eight 2.66GHzCPUs with 8M cache. They take around 40 minutes to assemble sequences and estimate their frequencies. Smaller value of n increases ViSpA’s runtime since its bottleneck (candidate sequences assembling) is proportional to the number of reads times number of paths. Indeed, smaller value of n results in larger number of superreads in built read graph, thus, in larger set of candidate paths. For example, ViSpA runs 90 minutes for n = 2, m = 2.
Statistical validation of the quasispecies spectrum
The plot on Fig. 5 shows validation results for 10 most frequent quasispecies sequences with respect to EM estimations assembled on Data1 by ShoRAH and ViSpA (n = 6, m = 15, and t = 120). Repeatedly, 100 times we have deleted randomly chosen 10% of reads and run both methods on each reduced read instance to reconstruct quasispecies spectrum.
The plot reports the percentage of runs when each of 10 most frequent sequences assembled on Data1 are reproduced among the 10 most frequent quasispecies inferred on the reduced instances with no mismatches (k = 0), or with k = 1, 2, 5 mismatches. For example, for k = 0 ShoRAH repeatedly (35% of times) reconstructs only the third most frequent sequence while ViSpA reconstructs 7 sequences in at least 15% times, and the most frequent sequence is reconstructed 40% times. This plot shows that the found sequences are pretty much reproducible for ViSpa.
Experimental validation on 454 pyrosequencing reads from HIV samples
In order to compare ViSpA and ShoRAH, we run both of the methods on HIV dataset, used in the first experiment in [27]. As said above, we do not preprocess reads with respect to its 454 quality score, and it can explain poorer performance of ShoRAH. Indeed, ShoRAH correctly infers only 2 quasispecies sequences with at most 4 mismatches: one assembly has 3 mismatches with real quasispecies sequence, and the other has 4 mismatches.
ViSpA correctly reconstructs 5 quasispecies with at most 2 mismatches (3 of them among 10 most frequent assemblies): two sequences are inferred without any mismatches (one is among 10 most frequent assemblies), one assembly has 1 mismatch with real quasispecies sequence (and it is among 10 most frequent assemblies), and the rest sequences have 2 mismatches (one is among 10 most frequent assemblies). The assemblies correspond to a viable protein sequences.
If ViSpA is applied to ShoRAHcorrected reads, it can successfully infer three real quasispecies without any mismatches.
Conclusions
In this paper, we have proposed and implemented ViSpA, a novel software tool for quasispecies spectrum reconstruction from highthroughput sequencing reads. The ViSpA assembler takes into account sequencing errors at multiple steps, including mappingbased read preprocessing, path selection based on maximum bandwidth, and candidate sequence assembly using probabilityweighted consensus techniques. Sequencing errors are also taken into account in ViSpA’s EMbased estimation of quasispecies sequence frequencies.
We have validated our method on simulated errorfree reads, FlowSimsimulated reads with sequencing errors, and real 454 pyrosequencing reads from HCV and HIV samples. We are currently exploring extensions of ViSpA to pairedend reads; the main difficulty is selection of pairaware candidate paths. We also foresee application of ViSpA’s techniques to the analysis of highthroughput sequencing data from microbial communities [23] and ecological samples of eukaryote populations [35].
Availability
The ViSpA source code is available at http://alla.cs.gsu.edu/~software/VISPA/vispa.html.
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Acknowledgements
IA, BT, SM and KW were supported in part by Molecular Basis of Disease Fellowship, Georgia State University. IM was supported in part by NSF awards IIS0546457, IIS0916948, and DBI0543365. IA, BT, SM and AZ were supported in part by NSF award IIS0916401. IA, BT, SM, IM and AZ were supported by Agriculture and Food Research Initiative Competitive Grant no. 20116701630331 from the USDA National Institute of Food and Agriculture.
This article has been published as part of BMC Bioinformatics Volume 12 Supplement 6, 2011: Proceedings of the First Annual RECOMB Satellite Workshop on Massively Parallel Sequencing (RECOMBseq). The full contents of the supplement are available online at http://www.biomedcentral.com/14712105/12?issue=S6.
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IA designed algorithms, developed software, performed analysis and experiments, wrote the paper. BT performed analysis and experiments. SM contributed to developing software. KW designed algorithms and developed software. IM contributed to designing the algorithms and writing the paper. PB supplied the HCV data and contributed to performing the analysis. AZ designed the algorithms, wrote the paper and supervised the project. All authors have read and approved the final manuscript.
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The authors declare that they have no competing interests.
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Additional file 1: Supplementary Materials. The file contains derivation of edge cost formula (2) and EM algorithm, example of read graph construction and analysis of 454 pyrosequencing data. (PDF 609 KB)
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Astrovskaya, I., Tork, B., Mangul, S. et al. Inferring viral quasispecies spectra from 454 pyrosequencing reads. BMC Bioinformatics 12 (Suppl 6), S1 (2011). https://doi.org/10.1186/1471210512S6S1
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DOI: https://doi.org/10.1186/1471210512S6S1