Abstract
Background
Recent methods have been developed to perform highthroughput sequencing of DNA by Single Molecule Sequencing (SMS). While NextGeneration sequencing methods may produce reads up to several hundred bases long, SMS sequencing produces reads up to tens of kilobases long. Existing alignment methods are either too inefficient for highthroughput datasets, or not sensitive enough to align SMS reads, which have a higher error rate than NextGeneration sequencing.
Results
We describe the method BLASR (Basic Local Alignment with Successive Refinement) for mapping Single Molecule Sequencing (SMS) reads that are thousands of bases long, with divergence between the read and genome dominated by insertion and deletion error. The method is benchmarked using both simulated reads and reads from a bacterial sequencing project. We also present a combinatorial model of sequencing error that motivates why our approach is effective.
Conclusions
The results indicate that it is possible to map SMS reads with high accuracy and speed. Furthermore, the inferences made on the mapability of SMS reads using our combinatorial model of sequencing error are in agreement with the mapping accuracy demonstrated on simulated reads.
Background
The first step in a resequencing study is to map reads from a sample genome onto a reference, accounting for sample variance and sequencing error. An accurate and sensitive approach is to use SmithWaterman [1] alignment; however, this is computationally infeasible for mapping to nearly any genome. Instead, methods have been created using heuristics and data structures that are appropriate for rapid mapping of the type of read considered. For example, reads produced by Sanger sequencing that are highly accurate and nearly 1000 bases long are successfully mapped using hashbased methods such as MEGABLAST [2], cross_match (Green P., http://www.phrap.org webcite, unpublished), and BLAT [3]. These methods are too inefficient to map read sets from next generation sequencing (NGS) instruments by Illumina (San Diego, CA, USA) and Life Technologies (Carlsbad, CA, USA), since they contain hundreds of millions of short reads. Instead, methods such as Bowtie, Bwa, and Soap2 are used [46]. These are based on querying the BurrowsWheeler Transform Fulltext Minutespace index (BWTFM) [7] of a genome. They are able to rapidly align reads when there is little variation between the read and the genome.
Sequencing methods based on single molecule sequencing (SMS) also produce large datasets that have high computational demands for mapping. SMS datasets do not have the length limitations of NGS or Sanger sequencing, but have a higher number of errors, and the errors are primarily insertions and deletions rather than substitutions. Thus, mapping methods created for NGS sequencing do not extend well to SMS reads. A recent study using the PacBioRS platform [8] included a large number of reads over 10 kilobases long. As reads become longer, the computational problem begins to resemble the whole genome alignment (WGA) problems that were examined when multiple mammalian genomes were sequenced [911]. The problem arises of how to align long (many kilobase) reads with moderate divergence from the genome (up to 20% divergence, concentrated in insertions and deletions) at the speed and sensitivity that NGS alignment methods operate.
Many alignment methods in similar application areas share related algorithmic approaches or data structures that are tailored to optimize the particular targeted application. The relationship between many existing alignment methods [1,35,1023] is qualitatively illustrated in Figure 1. We present an approach, Basic Local Alignment via Successive Refinement (BLASR), which maps reads using coarse alignment methods developed during WGA studies, while speeding up these methods by using the advanced data structures employed in many NGS mapping studies.
Figure 1. An illustration of relationships between alignment methods. The applications / corresponding computational restrictions shown are (green) short pairwise alignment / detailed edit model; (yellow) database search / divergent homology detection; (red) whole genome alignment / alignment of long sequences with structural rearrangements; and (blue) short read mapping / rapid alignment of massive numbers of short sequences. Although solely illustrative, methods with more similar data structures or algorithmic approaches are on closer branches. The BLASR method combines data structures from short read alignment with optimization methods from whole genome alignment.
Advances in isolation and detection of single molecules and reactions have enabled SMS methods [2426]. These SMS methods monitor processes in real time. The PacBioRS instrument produces reads by detecting which fluorescently labeled nucleotides are incorporated into a DNA chain as a template sequence is replicated by DNA polymerase. Other SMS methods have been proposed using detection of cleaved bases that pass through a protein nanopore [25], and identifying bases that have translocated through a nanopore fabricated in a graphene membrane [27]. In the case of the PacBioRS sequencing, a missing or weak signal of nucleotide incorporation results in a deleted base, and nucleotides that give fluorescence signal without being incorporated lead to insertions.
We propose aligning SMS reads with high indel rates to genomes as follows. First, find clusters of short exact matches between the read and the genome using either a suffix array or BWTFM index [7]. Then, perform a more detailed alignment of the regions where reads are matched to assign the alignment. To investigate the feasibility of doing this in the human genome, we need to determine two metrics: (1) the number of matches of minimal length expected to exist between a read and the genome at a given sequencing accuracy and read length, and (2) the number of false positive clusters the read is expected to have elsewhere in the genome. If the chances of finding a match between the read and the genome are low, or if there are many regions a read may map to incorrectly with high identity, our proposed approach would not be feasible. For a particular read length and accuracy, we present a method to determine the probability that the read contains a sufficient number of anchors to map; this method is based on counting integer compositions. We next examine the repeat structure of the human genome to determine how difficult it is to map to due to the repetitive nature of the genome. Rather than defining repeat content as the amount of sequence sharing high percent identity, we measure a different similarity metric on the human genome, the anchor similarity, where sequence similarity is measured as the number of shared anchors between the two sequences from the genome. We find that there are both a high number of expected matches between the read and the genome, and few false positive clusters of matches of the same size elsewhere in the genome, indicating that the proposed approach is feasible for mapping reads to the human genome.
We implemented our method in a program called BLASR (Basic Local Alignment with Successive Refinement), which combines the data structures used in short read mapping with alignment methods used in whole genome alignment. A BWTFM index or suffix array of a genome is queried to generate short exact matches that are clustered and give approximate coordinates in the genome for where a read should align. A rough alignment is generated using sparse dynamic programming on a set of short exact matches in the read to the region it maps to, and a final detailed alignment is generated using dynamic programming within an area guided by the sparse dynamic programming alignment.
Results and discussion
Our results are broken down into two sections; in the first, we examine characteristics of PacBioRS reads, and present theory on how these sequences contain matches that may be used to anchor alignments to the genome. In the next, we present a practical comparison of alignment methods on PacBioRS sequences.
Mapping feasibility
Our strategy to map SMS reads is to locate a relatively small number of candidate intervals where the read may map and then use detailed pairwise alignments to determine the best candidate. The candidate intervals may be found by locating all exact matches between the read and the genome, and then finding dense clusters of exact matches (anchors) in spans of similar length and the same (or reverse complement) order and orientation in both the genome and read, as described in detail in Methods. The feasibility of the method depends on the balance of having enough anchors to detect the correct interval to align a read to, vs. having so many anchors that clustering takes a prohibitive amount of time.
One approach to limiting the number of anchors is to limit to a set of anchors of
low multiplicity in the genome; this is commonly done by using longer anchors. When
the sequencing error rate is ρper position, without positional bias, the average length of an exact match is
Figure 2. The distribution of lengths of errorfree segments of reads. The line fitted to the points weighted by frequency has slope −0.071, corresponding to a geometric distribution with parameter 0.848, in close agreement with the 84.5% accuracy of the dataset used. Over 95% of segments are of length 20 less.
We may model SMS sequencing as a process that generates a series of errorfree words
with a geometric length distribution, each separated by a single erroneous base. With
this model, it is possible to determine how many words must be sequenced until there
is a high probability that a word of length K or greater (suitable for use in anchoring an alignment) has been sequenced. Denoting
the length of a word as W , Pr{W = K} = (1 − ρ)^{K}ρ, and Pr{W ≥ K} = (1−ρ)^{K}, where K≥0. In order to have a probability of 1 − εthat a word of length K or greater is sequenced without error, t words must be sequenced, where
The waiting lengths for words of size 15, 20, and 25 are shown for ε = 0.05 and varying ρin Figure 3. We refer to errorfree sequences of length K or greater as anchors.
Figure 3. Waiting length to sequence a word of length≥katε = 0.05. The waiting lengths to sequence a word of length ≥ k at ε = 0.05 at varrying accuracy. This gives an estimate of the number of bases required to sequence before having an error free stretch that may serve as an alignment anchor.
Other alignment methods such as Gapped BLAST [28] and BLAT [3] have shown that it is useful to initiate alignments at pairs of anchors. The waiting lengths may be used to compute the length of read required to be certain of having at least N anchors. Instead of using waiting lengths, it is possible to directly compute the probability of sequencing a certain number of anchors when the error rate is known. We do this with a model that approximates all errors as point mutations on a scan across a template. Given a fixed template length L, a minimal anchor length K, a number of errors M, and a number of anchors N, define NumConfigurations(M,N,K,L) as the number ways to distribute the positions of M errors when reading from the template such that there are at least N maximal substrings of length ≥ K not interrupted by error. In Appendix 1, we compute this using generating functions, allowing us to apply the result across the read lengths and error profiles found in SMS sequencing. Weese et al. [29] considered a similar problem for short reads and low error rates, and set bounds for filtering alignment hits in a qgram based mapping method by using a dynamic programming approach.
Assuming all permutations of errors are equally likely,
Figure 4. Values for
When a read is sampled from a repeat in the genome, there are likely to be many dense
clusters of anchors mapping the read across the genome. Assuming the repeat is divergent,
it is necessary to perform a detailed alignment (SmithWaterman) to all intervals
containing dense clusters of anchors in order to distinguish the correct mapping location
from other repeats. For copies of a repeat such as Alu or LINE in the human genome,
the computational demands are too prohibitive to align the read against all instances
of the repeat. On the other hand, if only a limited number of mapped locations are
aligned in detail, the chance of finding the correct location is small. The similarity
of repeats in a genome is typically defined by percent identity from a pairwise alignment
of the two sequences
[30]. However, sequences that have a high percent similarity may not share many long stretches
of exact matches, which is how they are compared when using anchorbased mapping.
To characterize repeats with respect to anchorbased mapping, we introduce an alternative
metric: the anchor similarity of two sequences is the maximum number of fixedlength, nonoverlapping, ordered
anchors, shared between two sequences, with certain constraints on anchor spacing.
If the anchor similarity is S, we also say the two sequences are Ssimilar, and ≥Ssimilar when two sequences have anchor similarity that is at least S. Using fixedlength
anchors simplifies the presentation, although the BLASR method uses variable length
anchors. Anchor similarity requires two parameters: K, the minimum anchor size; and
δ, the indel rate, which may change the spacing between anchors. The constraints reflect
the spacing one would expect between anchors of a read with indel errors and a genome.
For example, consider a sequence that contains anchors at coordinates a and b, matching
anchors at coordinates a^{′} and b^{′}in another sequence. If the ratio of the gaps between anchors is bounded by
Additional file 1. Supplementary Text S1. The supplementary text contains additional implementation details for the anchor similarity method, and description of the empirical model based read simulator.
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To characterize the repetitiveness by anchor similarity of sequences in the human genome, we took a sample of 1 million random intervals of length L=1 kb in the genome, and computed anchor similarity of each interval with all other intervals up to length (1 + δ)L = 1150 (assuming an indel rate δ = 0.15) in the rest of genome. We used anchors of lengths 15, 20, and 25. For each interval and anchor length, a histogram is generated for the number of times ≥Ssimilar intervals are found in the genome. A hypothetical sample sequence with K = 15 may have 50 thousand ≥1similar intervals in the genome; one thousand ≥2similar intervals; one hundred ≥3similar sequences; ten ≥4similar sequences; and one ≥5similar sequence. This results in one million histograms (for each anchor length). To summarize these, we examined the cumulative distribution of values of all histograms for ≥1, ≥5, ≥10, and ≥20similar sequences, as shown in Figure 5.
Figure 5. ≥Ssimilar sequences measured in the human genome. 1 million query intervals, each 1000 bases long, were randomly sampled from the genome. Each query interval was searched against the human genome to determine the number of nonoverlapping 1000 base intervals in the genome that are ≥Ssimilar to the query. The cumulative distribution for the number of target intervals that are (A) ≥1similar, (B) ≥5similar, (C) ≥10similar, and (D) ≥20similar to these 1 million query intervals, is shown. Each panel uses minimum anchor lengths k = 15, 20, and 25 and indel rate δ = 0.15. From this, one may interpret the number of intervals that must be searched when mapping a read using anchors. For example, when mapping with a minimum of a single 25 base match, 80% of the queries match to 100 other intervals in the genome with at least one one 25 base match (point X). On the other extreme, the top 3% of queries map to over 1 million other with at least one matchpoint Y), due to the high repeat content of the genome. This indicates that 80% of sequences may be correctly mapped to the human genome using a single 25 base match by only searching 100 100 candidates, however for full sensitivity many more candidates must be searched. Points P and Q show a contrast of the fraction of intervals that have 100 or fewer matches in the genome when matching using 1 or more anchors versus 20 or more anchors, for an anchor length of 15. Only 20% of the samples are limited to 100 or fewer additional matching intervals with at least 1 anchor (point P), and 97.5% of the samples have 100 or fewer matches when requiring at least 20 anchors in a match (point Q).
We compared the distribution of values of anchor similarity from the human genome
with values of
To gauge the mapability of sequences to various genomes, we simulated reads from Escherichia coli, Arabidopsis thaliana, and human, for read lengths that vary from 100 to 10000 bases, and error rates from 20% down to 0%. We mapped them back to their reference genomes with BLASR (see The results are shown in Figure 6. We note for mapping to the human genome, while it is difficult to have precise predictions on the mapability of sequences, the results are in agreement with the inferences drawn from the distributions of number of anchors and anchorsimilarity measures. For example, 95% of 1000base reads from the human genome simulated with a 15% error rate map to the correct location in the genome.
Figure 6. The mapability of simulated sequences from theE. coli,A. thaliana, and human genomes. Mapping accuracy is shown on a Phred scale (
As shown in Figure 4B, a read with a 15% error rate has a 97% chance of having 10 anchors of length 15 or more. The anchor similarity corresponding to these reads uses parameters δ = 0.15,L = 1000, and k = 15, and is shown by the red curve in Figure 5A. Over 90% of the sampled intervals only have one location with at least 10 anchors of length 15, indicating they map uniquely under this repeat under this repeat metric. The other two genomes, E. coli, and A. thaliana, are shown for
Mapping benchmarks
We generated three datasets for evaluating mapping speed and accuracy of different aligners on SMS reads (see Table 1). For all E. coli datasets, reads were aligned to the genome of an isolate of the O104:H4 strain (doi: 10.5524/100001). The source reads are available at http://bix.ucsd.edu/projects/blasr. webcite Performance was measured additionally with both BLAT and the BWASW aligners [18]. BWASW was the first mapping method written that used both the BWTFM index used in short read mapping and methods that allow mapping long reads with indel error. This method is very compact (under 5 GB of memory for human genome alignments), and very sensitive to mapping reads with indel error, as compared to other existing methods. Other methods that were tested either did not run or produced insufficient results. This may be expected, as these methods are highly optimized for other types of data that is either short read or whole genome sequences. Of the programs that did not run, Soap2 and Lagan crashed, while Bowtie did not accept the read input due to read length, and the mapping sensitivity was low on reads truncated to the maximum allowed length. The Mosaik (Strömberg M., http://bioinformatics.bc.edu/marthlab/Mosaik webcite, unpublished), Mummer, and RazerS methods did execute, however the first two could only align to one chromosome of the human genome at a time due to space limitations, and were orders of magnitude slower than either BLASR or BWASW while finding very few hits. Finally, the RazerS method was only tested on E. coli reads, and found few hits across all tested parameters. Because of the low mapping sensitivity, these methods were excluded from benchmarking results. The BLAT method is included as a reference for comparison to methods optimized for mapping Sanger sequences, though it is slower and less sensitive than both BLASR and BWASW.
Table 1. Datasets used in benchmarking
The E. coliPacBioRS dataset contains 123,246 reads comprising 261.7 M bases after filtering, with lengths and error rate shown in Figure 7 (Short Read Archive accession numbers SRR305922, SRR305923, SRR305924, and SRR305925). The reads contain 10.7% insertion, 4.3% deletion, and 0.9% substitution error, though the details are sensitive to alignment penalty summary of the mapping statistics from each of the three programs is shown in Table 2. All programs were executed on a single core of a 2.9 GHz Xeon processor. The parameters used for each program are given in Additional file 2: Table S1.
Additional file 2. Supplementary Table S1. Supplementary Table S1 gives the command line parameters used to run the benchmarks.
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Figure 7. Statistics of reads fromE. coliO104:H4 produced by the PacBioRSsequencing platform. (Black) The fraction of reads with length at least x. This is roughly the survival curve of an exponential distribution. (Blue) The fraction of reads (of length at least x) that are correct at position x. Accuracy is nearly position independent, so the blue curve is roughly the constant 1−ρ, where ρis the error rate per position.
Table 2. A comparison of the BLASR, BWASW, and BLAT methods onE. colireads
To test the sensitivity and specificity of mapping, reads were simulated using an empirical model (described in Additional file 1: Text S1, Section 1.2) based on the measurement of error rates from reads aligned to E. coli. The results are shown in Table 3. The methods are largely in agreement on the reads that are correctly mapped, as well as in the number of bases from every read, and BLASR is marginally faster. The slight differences in mapping statistics between BLASRSA and BLASRBWT are due to implementation differences in the order anchors are generated: using a suffix array, sequences are searched left to right, but for a BWTFM index, sequences are searched right to left. One difference between BLASR and BWASW is that BWASW often produces several short alignments of possibly overlapping substrings of a read rather than one contiguous alignment. We consider the number of bases mapped by BWASW as the sum of uniquely mapped bases from each read. Usually this does not affect mapping and consensus quality, but occasionally there are subsequences from reads that are incorrectly mapped while the rest of the read is mapped correctly.
Table 3. A comparison of the BLASR, and BWASW methods on simulated reads
In addition to the information encoding the alignment, BLASR produces a mapping quality value for every alignment. This value represents the PHRED scale probability that the coordinates the read is aligned to in the genome are incorrect, similar to the mapping quality values produced by Maq [20]. To test mapping quality values, we created three datasets of 10M simulated reads sampled from the genome with fixed read lengths of one, two, and three kilobases each. Errors were added to the reads using the empirical read simulator (Additional file 1: Text S1, Section 1.2). For each mapped read, we classified it as correctly and incorrectly mapped, allowing a measurement of accuracy of mapping quality value. The frequency of computed mapping quality values are shown in Figure 8A. The mapping quality values are largely binary, owing to the fact most reads contain sequences that align uniquely to the genome. The empirical mapping quality values are shown in Figure 8B.
Figure 8. Mapping quality values of reads simulated from the human genome. (A) The frequency of quality values for alignments of 10^{6}simulated 1000, 2000, and 3000 base sequences from the human genome. (B) The empirical mapping quality values of the alignments.
Conclusion
Methods to produce reads through single molecule sequencing were mostly theoretical a decade ago and are now produced in high throughput on an industrial platform. The different characteristics of the sequences produced by SMS relative to Next Generation sequencing (sequences several orders of magnitude longer than previous technologies, at the expense of a higher error rate concentrated in insertions and deletions), require new computational techniques to be efficiently analyzed. Here, we addressed the problem of mapping SMS reads to a reference genome by first examining the feasibility of mapping SMS reads, and then by benchmarking our new alignment method on reads produced by the PacBioRS instrument. The source code is available under the BSD license at https://github.com/PacificBiosciences/blasr webcite and is the default alignment method available to all running the PacBioRS.
There are many emerging problems for processing SMS sequences. As the length of the reads produced by SMS increases, the computational problem resembles whole genome alignment more than the read mapping problem. This increases the need to have methods that accurately detect structural rearrangements covered by single reads. Furthermore, with the inevitable exponential increase in sequencing throughput, the current methods will not be sufficient to align SMS reads without a large amount of time or computational resources, and further algorithmic improvements will be necessary. We did not address the issue of using multiple sequence alignment to produce a consensus sequence or variant calls. It has been shown that the additional information one may gain by observing the signal from singlemolecule events in real time may indicate DNA modifications such as methylation [25,31]. Thus, methods that produce consensus calls from SMS sequences may reveal more information about the sample sequence if this extra information is used. We aim to address many of these problem in subsequent iterations of the BLASR method.
Methods
We use a successive refinement approach to map SMS reads. This approach operates in three phases: (1) detecting candidate intervals by clustering short, exact matches; (2) approximate alignment of reads to candidate intervals using sparse dynamic programming; and (3) detailed banded alignment using the sparse dynamic programming alignment as a guide, as shown in Figure 9. It is not until the third step that read base positions are assigned to reference positions.
Figure 9. Overview of the BLASR method. (A) Candidate intervals are found by mapping short, exact matches as shown by colored arrows. Either a suffix array or BWTFM index of the genome are used to find the exact matches. Intervals are defined over clusters of matches and are ranked; intervals with score 3, 6, and 4 are shown. (B) Matches scoring above a threshold are aligned using sparse dynamic programming on shorter exact matches. (C) Alignments that have a highscoring sparsedynamic programming score are realigned by dynamic programming over a subset of cells defined using the sparse dynamic programming alignment as a guide.
Detecting candidate intervals
The input to the BLASR method is a read r with nucleotides r_{1},…,r_{R}; a genome g with nucleotides g_{1},…,g_{G}; and a minimum match length, K. Other parameters that modify small details of mapping are introduced in their context
later. We find all exact matches of substrings (of length at least K) from the read and the genome. An exact match of anchor a to the genome may be described by a triplet (Read(a), Genome(a), l(a)), where Read(a) is the start of the match in the read; Genome(a) is the start of the match in the genome; and l(a) is the length of the match. The set of all matches is
We use either a suffix array (SA) or BWTFM index on the genome to query for exact
matches, depending on time and space requirements. While some NGS alignment methods
such as mrFAST and RazerS match using hash tables on fixed width words (qgrams)
[29,32], the SA and BWTFM index allow matching long exact matches if they exist, and also
encode positions of shorter matches if a more sensitive search is required. The two
data structures support the same queries: c = Count(qt), the number of times a query sequence r occurs exactly in a text g; and
Descriptions of the implementation and methods for the Count and Locate queries using suffix arrays are given in [33]. Similar descriptions for the BWTFM index are in [7] and [4]. The COUNTLCP operation is about 1.5× faster using a suffix array than a BWTFM index, in our tests searching the human genome and limiting the number of times an LCP occurs to 10,000; however, the space usage for the index on a human genome is 12.8 GB with a suffix array, vs. 4.8 GB in our implementation of a BWTFM index. Our implementation of the Locate operation is faster for larger genomes using the BWTFM index than the suffix array when using SIMD hardware optimization. Because either index is shared across many threads, the amortized space usage is modest for both data structures.
Once the set of anchors
The clusters are assigned a frequency weighted score that is the sum
While limiting the number of clusters retained may miss alignments to repetitive regions, filtering clusters on this frequencyweighted score was shown to be highly discriminative in our tests.
Refining alignments
Each cluster
The read must be quickly aligned to a candidate interval, even if it is many tens
of kilobases long. Similar to the method of anchoring the interval to the genome but
on a smaller scale, a set of matches are found between the read and the candidate
interval. The matches used in SDP are of a short fixed length, K^{SDP} (typically 8–11 bases). Let
The SDP alignment does not align all bases in a read, and so it is necessary to realign
a final time using banded dynamic programming. For long reads with indels, the size
of the band used to contain the entire alignment becomes prohibitively large. The
set of anchors
In addition to the base sequences produced by the PacBioRS, each base in the read is also given three quality values (insertion, deletion, and
substitution) and two alternative base calls (substituted base and deleted base).
Let
The MISMATCHPRIOR and DELETIONPRIOR are PHRED scaled penalties that reflect the global mismatch and deletion rates. In practice, MISMATCHPRIOR is 20 and DELETIONPRIOR is 15.
Mapping quality values
Due to the repetitive nature of genomes, a read often maps with a high alignment score
to many locations. It is informative to calculate the probability that the interval
a read is mapped to by an alignment is the correct location in the genome. This probability
may be interpreted as a mapping quality value
A Bayesian probability technique was presented in [20] to compute the mapping quality for short reads with base calling quality values. We present the formulation in [20] using the notation in this paper: we are given read r and a position m that it is mapped to in a sequence g. The posterior mapping probability that a read r is sampled from m is computed as
where i runs over all positions in the genome. The probability that position i is sampled by the sequencer is denoted Pr(i), and is considered to be uniform both here and in
[20]. The quantity Pr(rg_{i,…,i + R−1}) is the probability of observing the read r if the sequence at positions i,…,i + R − 1 in the genome is read by the sequencer. For reads that include base quality values
q, let q_{i} denote the probability that a base in a read is incorrect. Then Pr(rg) may be replaced by Pr(rgq). In
[20], Pr(rgq) is rapidly approximated by summing the quality values of bases that mismatch in
the ungapped alignment between r and g_{i,…,i + R − 1}. When there are insertions and deletions in the sequence, the value Pr(rg_{i,…,i + R − 1}) may be computed as
The denominator of Equation 1 gives the marginal probability that the read is observed from anywhere in the genome. Evaluating this full sum is computationally infeasible even for short reads and ungapped alignments. Since the probability of observing a read given a template sequence drops geometrically with divergence, most positions in the genome do not contribute significantly to the sum. For short reads, the sum is approximated in [20] as the sum of the probability of the top scoring alignment and all second best alignments.
In BLASR, the mapping quality value is calculated in a similar manner. The sum in Equation 1 is limited to the top MAXCANDIDATES alignments, and is then scaled by a factor that reflects the limited sample size by aligning only at most MAXCANDIDATES clusters. When the read is sampled from a unique region of the genome, there will be few clusters of high score, and the highest scoring cluster will likely contain the true match to the genome. However, when the read is sampled entirely from a repetitive sequence, there will be many high scoripng clusters. In this case, it is possible the cluster from the correct interval on the genome will not have high enough score to be retained in MAXCANDIDATES clusters. To account for this, we assume that the correct interval in the genome may correspond to any significantly highly scoring cluster, and multiply the sum in Equation 1 by the ratio of the number of significant clusters found in the genome to MAXCANDIDATES, as long as the number of significantly highly scoring clusters is greater than MAXCANDIDATES. The significance of a cluster may be measured by comparing the number of anchors in a cluster to the number of anchors expected at the correctly mapped location. The distributions of numbers of anchors expected to correctly map were found using simulations of error processes for different error rates and read lengths; however, it is possible to model this theoretically (see the next section). The expected number of anchors a read has when mapped to the correct location is genomeindependent: it depends only on the error rate, length of the read, and minimum anchor length. We use a slightly different metric, the number of anchorbases (the total number of bases in all anchors) to measure cluster significance, and this is similarly genomeindependent. For efficiency, in BLASR we precompute the expectation and variance for the number of anchorbases for a range of feasible accuracies, read lengths, and minimum match lengths, and minimum match size. The accuracy of the highest scoring alignment is used as a proxy for the true accuracy of the read. Given the accuracy, the length of the aligned sequence, and the minimum match length, we look up the mean μ and variance σ^{2} for number of anchor bases, and count all clusters with more than μ−2σ anchor bases as significant.
Appendix 1
Enumeration of configurations with specified numbers of errors and anchors
In this section, we will show how to explicitly compute NumConfigurations(M,N,K,L).
Consider a read of length L with exactly M errors, at positions 1 ≤ x_{1} < x_{2} < … < x_{M} ≤ L. Also set x_{0} = 0 and x_{M + 1} = L + 1.
For the sake of simplicity, we assume all sequencing errors are of length 1, but this can be generalized to insertions and deletions that change the length of the read.
The error positions split the read into parts of sizes λ = x_{i} − x_{i−1} ≥ 1 for i = 1,…,M + 1. Each part λ_{i} (i = 1,…,M) consists of λ_{i} − 1 matches followed by one mismatch. The last part consists of λ_{M + 1} − 1 matches. Note that if there are two consecutive mismatches, there will be a part λ_{i} = 1 corresponding to 0 matches followed by one mismatch.
Part sizes λ_{i} are related to the notation W of the Results section by λ_{i} = W + 1. Note that W counted only the correct positions, and we did not have a subscript (W_{i}) to specify the word number. In this section, λ_{i} counts the correct bases and also counts one incorrect base at the end, based on our simplification that all sequencing errors are of length one.
Set λ = (λ_{1},λ_{2},…,λ_{M + 1}). These are positive integers that add up to L + 1. In Combinatorics, this is called a strict composition of L + 1 into M + 1 parts. Let K be the minimum anchor length (a parameter).
Consecutive errors greater than K apart (λ_{i} > K) give segments that are anchors while consecutive errors shorter than this (λ ≤ K) give segments called short matches.
In Figure 10, we illustrate a read of length L = 7 with M = 2 error positions. For a minimum anchor length K = 3, there are 6 compositions where the first part is an anchor:
Figure 10. Toy example for counting components. A read of length L = 7 with M = 2 errors is shown, with errors in red. In general, M errors splits the read into M + 1 parts, some of which may be null; in this case, the third part is null. For anchor length threshold K = 3 (meaning parts of size >3 are anchors, parts of size ≤3 are not), we have N = 1 anchor (the first part).
For reads of length 7 with 2 errors, and minimum anchor length 3, the number of compositions with exactly one anchor (allowing it to be any of the parts, via permutations of these compositions) is 6·3 = 18.
For arbitrary values of the parameters, we first compute the number of configurations
where all N anchors come first and all M + 1−N short parts come last. Then we multiply the count of these by
Let N ≥ 0 be an integer. For given parameters M,N,K,L, we will enumerate the number of arrangements of error positions that result in exactly
N anchors. This is equivalent to the combinatorial problem of counting integer compositions of L + 1 with certain restrictions on the sizes of the parts. We will use generating function
techniques from combinatorics to count arrangements of M error positions that give exactly N anchors (so the other N + 1−M parts are short fragments). Let c_{M,N,K}(L) denote the number of arrangements of M error positions that result in exactly N anchors, where the read length is L and anchors are defined as parts λ_{i} > K. Let
Note that
The compositions of L + 1 into M + 1 parts, where the first N parts are anchors and the remaining M + 1 − N parts are short, have the following constraints:
• λ_{1},…,λ_{N} ∈ 1,2,…,K(short parts).
• λ_{N + 1},λ_{N + 2},…,λ_{M} ∈ K + 1,K + 2,… (anchors).
• λ_{1} + ⋯ + λ_{M} = L + 1.
The generating functions for short parts, S(t), and anchors, A(t), are
Standard methods for enumerating compositions with generating functions give that
where we expand the left side in a MacLaurin series (Taylor series centered at t = 0) to obtain the right side. The counts
To compute c_{M,N,K}(L), we use Taylor series methods to compute the coefficient of t^{L + 1}in (A5). We present two methods to do this.
First Taylor series method: The coefficient of tin (A5) may be determined by polynomial multiplication. We truncate the middle expression in (A3) to terms of degree ≤ L + 1, which turns it into a polynomial; the middle expression of (A2) is already a polynomial. We take powers and products of the polynomials, truncating terms of degree > L + 1 at intermediate steps. The coefficient of t^{L + 1} in the result is c_{M,N,K}(L). All intermediate products and sums involve only nonnegative integers.
Second Taylor series method: We present an exact closedform solution. Mathematically, closedform solutions are usually preferred. However, the first method above may be preferable for computation because intermediate steps of this second method require much higher precision, as discussed in Appendix 2.
Theorem A1
ForK = 0: if N = M + 1 then
ForK ≥ 1, setD = L − NK − Mandi_{max} = min(⌈D/K),M + 1−N⌉.
IfD < 0 ori_{max} < 0 thenc_{M,N,K}(L) = 0. Otherwise,
Proof
For K=0, there are no short parts; all parts are anchors. This is equivalent to counting
the number of strict compositions of L + 1 into M + 1 parts, which is wellknown to be
For K ≥ 1, note that we may write
The binomial theorem and the negative binomial series give
Plugging these into (A7), we obtain
In (A5), the coefficient of t^{L + 1} is c_{M,N,K}(L). Collecting together the terms in (A8) where the exponent of t is L + 1 gives (A6). We omit the detailed but straightforward derivation. □
Appendix 2
Numerical precision of the closed form solution for the number of anchors
Theorem A1 (also called the “Second Taylor series method”) gives a closed form expression
(A6) to compute c_{M,N,K}(L). This closed form solution has only a small number of terms. However, for practical
parameter values, it may require more bits of precision than are available in a finite
precision computation, even if the final answer does not overflow the variable size.
This is because the expression has an alternating sum with terms of much higher absolute
value than the final answer. Consider this part of the summation in Theorem A1, omitting
the coefficient
For M = 75, N = 1, K = 15, L = 1000, this has 61 alternating terms of magnitude between 2^{93} and 2^{401}, while the value of the sum is much smaller, with magnitude 2^{294}. Using high precision floating point, we need at least 110 bits for the mantissa to get the first decimal digit correct. This is significantly more bits than is currently standard: the current standard for floating point, IEEE 754, provides for a 53 bit mantissa in double precision. Alternatively, using high precision integers, we would need 294 bits of integer precision, plus a sign bit. However, software for arbitrary precision integers, such as Maple or Mathematica, will handle this example correctly.
By contrast, the “First Taylor series method” only involves sums and products of positive integers, each bounded above by the value of c_{M,N,K}(L). Thus, if the integer precision is adequate to store the value of c_{M,N,K}(L), it is also adequate to perform all intermediate calculations.
Appendix 3
Statistics of number of anchors
We may estimate the number of anchors using the following theorem.
Theorem A2
Fix M,K,L. Under the uniform distribution on compositions of L + 1 into M + 1 parts, the mean number of anchors and its variance are given by
For fixed M and K, consider the twovariable generating function
For fixed M,K,L, the probability of exactly N anchors is c_{M,N,K}(L)/T, where
Note that T counts the total number of compositions of L + 1 into M + 1 parts, with 0 or more anchors. Thus, it actually counts the total number of compositions of L + 1 into M + 1 parts, without regard to sizes of parts. So we have:
and thus the probability of exactly N anchors is
Next, for fixed M,K,L, we evaluate E[N], the mean number of anchors under the uniform distribution of compositions of L + 1 into M + 1 parts.
Using standard generating function properties, the numerator
First we evaluate the derivative; second, we plug in u = 1; third, we extract the coefficient of t^{L + 1}; and fourth, we use this to compute E[N]:
1. The derivative in Eq. (A14) is
2. Plug in u = 1:
3. Expand the Taylor series and extract the coefficient of t^{L + 1}:
The term t^{L + 1}occurs when j = L − M − K.If j < 0, this coefficient is 0. If j ≥ 0, this coefficient is
4.Evaluate E[N] to obtain Equation (A9):
Note that if L − K < M, then E[N] = 0.
Next we compute the variance of N, using a similar generating function technique. The generating function will enable us to compute E[N(N − 1)], so we will compute the variance in the form
which is equivalent to the more common formula σ^{2} = E[N^{2}] − E[N]^{2}. We have:
The numerator
We evaluate this in a fashion similar to E[N]:
1. The derivative in Eq. (A15) is
2. Plug in u = 1:
3. Expand the Taylor series and extract the coefficient of t^{L + 1}:
The term t^{L + 1} occurs when j = L − (M + 2K). This coefficient is
4. Evaluate E[N(N − 1)]:
5. Evaluate σ^{2} = Var[N] to prove Equation (A10):
Appendix 4
Asymptotic number of anchors
Theorem A3
Let μ,σ^{2} be given by Theorem A2. For sufficiently large M,
where
Proof
For fixed M,K, Eq. (A12) gives the generating function H_{M,K}(t,u) as a rational function in t,u, raised to the power M + 1. When M is sufficiently large, the Central Limit Theorem gives that the coefficients c_{M,N,K}(L) in its Taylor series, Eq. (A11), are wellapproximated by a bivariate normal distribution.
Restricting to the coefficients of t^{L + 1} for fixed L gives that the coefficients of
Thus, we obtain Eqs. (A16) and (A17) as approximations for the coefficients c_{M,N,K}(L) and the survival function NumConfigurations(M,N,K,L). In Eq. (A17), note that
In Figure 11, we plot c_{M,N,K}(L) and NumConfigurations(M,N,K,L). The solid markers are the true values computed from the generating function. The curve is the estimate computed by the preceding theorem, and does indeed approximate the true values well.
Figure 11. The fraction of configurations with exactly and at leastNanchors. (A) Plot of the fraction of configurations with exactly N anchors,
Competing interests
MJC is a fulltime employee at Pacific Biosciences, a company commercializing singlemolecule, realtime nucleic acid sequencing technologies. GT was partially supported by a grant from the National Institutes of Health, USA (NIH grant 3P41RR02485102S1). NIH had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Author’s contributions
MJC proposed and implemented the mapping method, performed the analysis, and wrote the manuscript. GT solved and implemented the combinatorial analysis, and wrote the manuscript. Both authors read and approved the final manuscript.
Acknowledgements
We thank Jon Sorenson, James Bullard, Eric Schadt, and Jonas Korlach for useful comments in writing this manuscript.
References

Smith T, Waterman M: Identification of Common Molecular Subsequences.

Zhang Z, Schwartz S, Wagner L, Miller W: A greedy algorithm for aligning DNA sequences.
J Comput Biol 2000, 7:203214. PubMed Abstract  Publisher Full Text

Kent W: BLAT–the BLASTlike alignment tool.
Genome Res 2002, 12:656664. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Langmead B, Trapnell C, Pop M, Salzberg S: Ultrafast and memoryefficient alignment of short DNA sequences to the human genome.
Genome Biol 2009, 10:R25. PubMed Abstract  BioMed Central Full Text  PubMed Central Full Text

Li H, Durbin R: Fast and accurate short read alignment with BurrowsWheeler transform.
Bioinformatics 2009, 25:17541760. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Li R, Yu C, Li Y, Lam T, Yiu S, Kristiansen K, Wang J: SOAP2: an improved ultrafast tool for short read alignment.
Bioinformatics 2009, 25:19661967. PubMed Abstract  Publisher Full Text

Ferragina P, Manzini G: Opportunistic data structures with applications.
Proc. of the 41st IEEE Symp on Found of Comput Sci 2000, 390398.

Rasko D, Webster D, Sahl J, Bashir A, Boisen N, Scheutz F, Paxinos E, Sebra R, Chin C, Iliopoulos D, Klammer A, Peluso P, Lee L, Kislyuk A, Bullard J, Kasarskis A, Wang S, Eid J, Rank D, Redman J, Steyert S, FrimodtMller J, Struve C, Petersen A, Krogfelt K, Nataro J, Schadt E, Waldor M: Origins of the E. coli strain causing an outbreak of hemolyticuremic syndrome in Germany.
New England J Med 2011, 365:709717. Publisher Full Text

Brudno M, Poliakov A, Salamov A, Cooper G, Sidow A, Rubin E, Solovyev V, Batzoglou S, Dubchak I: Automated WholeGenome Multiple Alignment of Rat, Mouse, and Human.
Genome Res 2004, 14:685692. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Schwartz S, Kent W, Smit A, Zhang Z, Baertsch R, Hardison R, Haussler D Miller: HumanMouse Alignments with BLASTZ.
Genome Res 2003, 13:103107. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Kurtz S, Phillippy A, Delcher A, Smoot M, Shumway M, Antonescu C, Salzberg S: Versatile and open software for comparing large genomes.

Altschul S, Gish W, Miller W, Myers E, Lipman D: Basic local alignment search tool.
J Mol Biol 1990, 215:403410. PubMed Abstract

Lipman D, Pearson W: Rapid and sensitive protein similarity searches.

Bray N, Dubchak I, Pachter L: AVID: A global alignment program.
Genome Res 2003, 13:97102. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Darling A, Mau B, Blatter F, Perna N: Mauve: multiple alignment of conserved genomic sequence with rearrangements.
Genome Res 2004, 14:13941403. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Brudno M, Do C, Cooper G, Kim M, Davydov E, Green E, Sidow A, Batzoglou S: LAGAN and MultiLAGAN: efficient tools for largescale multiple alignment of genomic DNA.
Genome Res 2003, 13:721731. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Kent WJ, Baertsch R, Hinrichs A, Miller W, Haussler D: Evolution’s cauldron: Duplication, deletion, and rearrangement in the mouse and human genomes.
Proc Nat Acad Sci 2003, 100:1148411489. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Li H, Durbin R: Fast and accurate longread alignment with Burrows Wheeler transform.
Bioinformatics 2010, 26:589595. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Li R, Li Y, Kristiansen K, Wang J: SOAP: short oligonucleotide alignment program.
Bioinformatics 2008, 24:713714. PubMed Abstract  Publisher Full Text

Li H, Ruan J, R D: Mapping short DNA sequencing reads and calling variants using mapping quality scores.
Genome Res 2008, 18:18511858. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Rumble S, Lacroute P, Dalca A, Fiume M, Sidow A, et al.: SHRiMP: Accurate Mapping of Short Colorspace Reads.
PLoS Comput Biol 2009, 5:e1000386. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Durbin R, Eddy S, Krobh A, Mitchison G: Biol Sequence Anal: Probabilistic Models of Proteins and Nucleic Acids. The Edinburgh Building, Cambridge, CB2 2RU. United Kingdom: Cambridge University Press; 1998.

Needleman S, Wunsch C: A general method applicable to the search for similarities in the amino acid sequence of two proteins.
J Mol Biol 1970, 48:443453. PubMed Abstract  Publisher Full Text

Eid J, Fehr A, Korlach J, Turner S, et al.: Realtime DNA sequencing from single polymerase molecules.
Science 2009, 323:133138. PubMed Abstract  Publisher Full Text

Clarke J, Wu H, Jayasinghe L, Patel A, Reid S, Bayley H: Continuous base identification for singlemolecule nanopore DNA sequencing.
Nat Nanotechnol 2009, 4:265270. PubMed Abstract  Publisher Full Text

Cherf G, Lieberman K, Rashid H, Lam C, Karplus K, Akeson M: Automated forward and reverse ratcheting of DNA in a nanopore at 5Å precision.

Garaj S, Hubbard W, Reina A, Kong J, Branton D, Golovchenko J: Graphene as a subnanometre transelectrode membrane.
Nature 2010, 467:190193. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Altschul S, Madden T, Schäffer A, Zhang J, Zhang Z, Miller W, Lipman D: Gapped BLAST and PSIBLAST: a new generation of protein database search programs.

Weese D, Emde A, Rausch T, Döring A, Reinert K: RazerSfast read mapping with sensitivity control.
Genome Res 2009, 19:16461654. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Bao Z, Eddy S: Automated de novo identification of repeat sequence families in sequenced genomes.
Genome Res 2002, 12:12691276. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Flusberg B, Webster D, Lee J, Travers K, Olivares E, Clark T, Korlach J, SW T: Direct detection of DNA methylation during singlemolecule, realtime sequencing.
Nat Methods 2010, 7:461465. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Alkan C, Kidd J, MarquesBonet T, Aksay G, Antonacci F, Hormozdiari F, Kitzman J, Baker C, Malig M, Mutlu O, Sahinalp S, Gibbs R, Eichler E: Personalized CopyNumber and Segmental Duplication Maps using NextGeneration Sequencing.
Nat Genet 2009, 41:10611067. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Myers E, Manber U: Suffix arrays: A new method for online string searches.
SIAM J Comput 1993, 22:935948. Publisher Full Text

Abouelhoda M, Ohlebusch E: A Local Chaining Algorithm and its Applications in Comparitive Genomics.

Eppstein D, Galil Z, Giancarlo R, Italiano G: Sparse Dynamic Programming I: Linear cost functions.
J Assoc Comput Machinery 1992, 39:519545. Publisher Full Text