Vector set.

A k-ploidy phase of SNPs with genotypes is a tuple of vectors (not necessarily distinct) satisfying the genotype allele counts property, that is: for all . We will refer to this collection as a vector set and we think of each vector as a row vector.

We can build a phase by selecting a permutation of the alleles present for each SNP locus . Note that the number of distinct permutations, , is strictly dependent on the genotype of the SNP and in the diploid bi-allelic case is equivalent to selecting the chromosomes containing the alternative alleles, hence

For example, let , then . We enumerate the possible permutations below and include an example tetraploid genome.

The sample tetraploid genome featured above on the right has a genotype vector: ; recall this counts the number of alternative alleles present at each SNP site. For any SNP , let denote the set of distinct allele permutations at SNP locus . Throughout we are indifferent to the order of each chromosome, with this in mind we can see that the total number of phases is bounded below by .

Likelihood of a phase.

We formulate the haplotype reconstruction problem as identifying the most likely phase(s) given the read data , all SNP loci , as well as their genotypes, and sequencing error rates . We assume the sequencing errors are independent of each other, that is for all and all , that are independently correct with probabilities () and incorrect with probabilities . Let be a matrix containing all of these probabilities: . Given a vector set, , corresponding to a phase, , and ; the likelihood of the phase is determined by:(1)

As depends only on and the read set , it is therefore the same across all vector sets. Hence, we define a relative likelihood measure (RL) as

As for , there are several ways this can be modeled depending on the situation. For polyploid simulated data, we can assume that is equal for almost all vector sets, excluding ones containing duplicate vectors. Let be the set of the multiplicities in ; for example, if then . The probabilities will differ multiplicatively by multinomial coefficients . Specifically:

For real diploid data, there will never be duplicate vectors. To model , we might assume that since mutations tend to occur together, adjacent SNP sites are more likely to be phased in parallel or than switched or . Let and let denote the number of adjacent SNPs that are parallel in and the number of adjacent SNPs that are switched in (we must only consider as it determines ). For example, if , then and . For some (denoted as parallel bias) and , we model this vector set probability as

Finally, we consider . For a given and , let denote the positions of SNP loci where and agree and disagree respectively. For example, if and , then and We may now compute the desired probability, that is:

The goal of our haplotype reconstruction problem is to find the vector set(s) maximizing the product , equivalently . However, as the number of possible phases is on the order , checking all of these is intractable. Our solution is based on finding high likelihood phases for the first SNPs, conditioned on a collection of high likelihood phases for the first SNPs.

Semi-reads and sub-reads.

To properly describe our method we must first define the semi-reads of a SNP locus and the sub-reads of a subset .

Semi-reads. To form the set of semi-reads of , denoted , include each read that contains both and some ( is upstream of ) and ignore all information from on SNPs ( is downstream of ). Suppose the set of reads is:

{1∶1, 2∶1, 3∶1, 4∶1} {3∶1, 4∶1, 5∶0, 6∶0} {4∶0, 5∶1, 6∶1} {4∶0, 5∶1, 6∶1, 7∶0} {5∶0, 6∶0, 7∶1} {5∶1, 6∶1, 7∶0}

The corresponding semi-reads for each SNP locus would be:

None {1∶1, 2∶1} {1∶1, 2∶1, 3∶1} {1∶1, 2∶1, 3∶1, 4∶1} {3∶1, 4∶1}

{3∶1, 4∶1, 5∶0} {4∶0, 5∶1} {4∶0, 5∶1}

{3∶1, 4∶1, 5∶0, 6∶0} {4∶0, 5∶1, 6∶1} {4∶0, 5∶1, 6∶1} {5∶0, 6∶0} {5∶1, 6∶1}

{4∶0, 5∶1, 6∶1, 7∶0} {5∶0, 6∶0, 7∶1} {5∶1, 6∶1, 7∶0}

Sub-reads. The sub-reads of , denoted , are obtained by, for each , removing all keys to form , and then adding to if the length of is at least 2. Alternatively, corresponds to the set of reads relevant to the problem of only phasing . Continuing with the example above, if , then

Extension.

We first describe how to extend an existing a haplotype assembly on SNPs onto the SNP . Recall the set of permutations of is denoted and one particular permutation as . An extension of onto SNP locus can be defined by appending some permutation of alleles to ; . Note that it is possible for two distinct permutations to result in the same : . In these cases we do not include duplicates, as they are equivalent. Observe that if is empty, all allele permutations are the same as vector sets; we therefore include only one. For any , we can compute the probability of it being the correct haplotype (for the first SNPs) conditioning on being correct (for the first SNPs), as well as the semi-read data and error rate . We express this below:(2)

This computation is similar to those done above in equation (1). The EXTEND algorithm (Algorithm 1) is given below, which returns a list of all extensions of that occur with probability above a certain threshold, , given haplotype .

Branching.

Here we define branching a collection of haplotypes with threshold to SNP : BRANCH() (Algorithm 2). We assume all phase the first SNPs and that SNP is the SNP. The act of branching returns : a list of all extensions generated by EXTEND with threshold for all in . To initialize BRANCH we EXTEND the empty vector set to an arbitrary permutation of the alleles of the first SNP, as all permutations are equivalent as vector sets.

Pruning.

For a collection of haplotypes of SNPs , we can compute the relative likelihood of each haplotype conditioned on the sub-reads and error rate ; we write this as . The same computation as performed in equation 1 yields:

Since does not depend on :(3)

The goal of PRUNE() (Algorithm 3) is to return a subset containing only sufficiently probable haplotypes. It does so by computing the relative likelihood of the most probable , that is , and adding to if , where is between and . We note that that one can compute from by only looking at the semi-reads : we store the relative likelihood values for all and update them when branching to ; PRUNE is therefore no more costly than BRANCH.

Main algorithm.

Here we give a high-level description of our overall haplotype assembly method HapTree() (Algorithm 4) using the EXTEND, BRANCH, and PRUNE algorithms. We generate high likelihood phases for the first SNPs, BRANCH those phases to include (the SNP), then PRUNE the resulting phases, and repeat for . We begin with an arbitrary permutation of the first SNP, since all orderings result in the same vector set. For the final step, we PRUNE with , and therefore return only the maximally probable phases that we have found; if this set is of size greater than one, we choose a phasing from within it randomly. More generally, below we take and to be vectors, as and may depend on , the size of or other user-specified variables.

Relative Likelihood (RL) objective function vs. MEC score for polyploid genomes.

We assessed the effectiveness of our RL score by comparison to MEC score on simulated data. To do so, we simulated reads with error rate from a pair of phased -ploid SNP loci for different coverages (5×, 10×, 20×, 100×) and for . All possible phases were exhaustively enumerated, and phases of the maximal relative likelihood (RL) and phases of the minimal MEC score chosen. We computed the proportion of perfectly phased SNP pairs in both cases (perfect solution rate). Even with two SNP loci, RL significantly outperforms MEC for all (Figure 3A). It is also worth noting that MEC (in comparison to RL) deteriorates more seriously in accuracy as ploidy increases (Figure 3A). In addition, we also compared the vector error rate in both cases; for a pair of SNPs, this rate is the number of vectors from the proposed solution that cannot be matched with vectors from the true solution (Figure 3B).

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Figure 3. Proportion of perfectly phased SNP pairs and vector error rate for RL (solid line) and MEC (dashed line) optimization in 10000 trials over 5×, 10×, 20× and 100× coverage.

https://doi.org/10.1371/journal.pcbi.1003502.g003

The results demonstrate that the higher the ploidy, the better the relative likelihood (RL) score performs in comparison to MEC score for phasing a pair of SNPs (Figure 3). In fact, in simulations where , RL with 5× the coverage already outperforms MEC with 100× coverage. For the same coverage, RL always outperforms MEC for , and they are equivalent in the diploid case .

Comparisons of HapTree and HapCompass.

To evaluate the phasing capabilities of HapTree, we compare it with HapCompass [8] (latest version available at: www.brown.edu/Research/Istrail_Lab/hapcompass.php), to our knowledge the only other existing program that directly addresses polyploid haplotype assembly, over multiple depth coverage values and component sizes for triploid and tetraploid simulated genomes. We simulated triploid and tetraploid genomes with different block lengths (10, 20 or 40 SNP loci), different coverages (5×, 10×, 20× and 40×), SNP positions, and SNP densities. Throughout the simulations for both the triploid and tetraploid cases, our EXTEND module is run with threshold and PRUNE primarily with threshold . When the current number of haplotype options generated is above , we prune more aggressively with and when above , with . These parameters are chosen to ensure the efficiency of HapTree by only keep a tractable collection of promising solutions in each step. We also simulate a read set with uniform error rate and size dependent on coverage.

For the triploid case, we observed that HapTree finds a perfect solution at a rate independent of the number of SNPs used in the simulation; in contrast, HapCompass declines in performance the larger the block size (Figure 4). While both HapTree and HapCompass improve steadily the higher the coverage, in every case HapTree significantly outperforms HapCompass; the least significant improvement of occurs in the case of 10 SNP loci and 10× coverage, whereas the most significant improvement occurs in the case of 40 SNP loci and 40× coverage. For both vector error rate and likelihood of perfect solution, we find that HapTree substantially outperforms HapCompass.

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Figure 4. HapTree (solid lines) and HapCompass (dashed lines) on simulated triploid genomes: Likelihood of Perfect Solution and Vector Error Rates, 1000 Trials, Block lengths: 10, 20, and 40.

https://doi.org/10.1371/journal.pcbi.1003502.g004

For tetraploid simulations, HapTree significantly outperforms HapCompass with block length of 10 SNP loci (Figure 5). For larger block lengths HapCompass arrives at the perfect solution at a rate of less than ; HapTree however does so at a rate between and depending on block size and coverage at least 20×.

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Figure 5. HapTree (solid line) and HapCompass (dashed line) on simulated tetraploid genomes: Likelihood of Perfect Solution and Vector Error Rates, 1000 Trials, Block length: 10.

https://doi.org/10.1371/journal.pcbi.1003502.g005

We varied the allele error rates ( and ) and observed decreases in accuracy that vary approximately linearly with the (uniform) allele error rates (Figure 6). The allele error rate is the likelihood of the sequencing technology to report the incorrect allele for a given position in one read. We ran 10000 trials for simulated triploid genomes of block size 10, with coverages 10×, 20×, and 40×.

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Figure 6. HapTree performance over varied error rates (.001, .02, .05, .1) and coverages (10×, 20×, 40×) on simulated triploid genomes: Likelihood of Perfect Solution and Vector Error Rates, 10000 Trials, Block length: 10.

https://doi.org/10.1371/journal.pcbi.1003502.g006

For the simulations above in Figure 6, we modeled our read data on Illumina sequencing technologies; for more details, please see section Simulated polyploid data generation below. We also ran simulations on longer read data, modeled after 454 sequencing technologies and found almost identical results.

The primary reasons for HapTree's superior performance are, first, that HapTree's relative likelihood is more effective than HapCompass's MEC score (see Relative likelihood vs MEC); and second, that HapTree's inference algorithm is more accurate than the approximation algorithm used by HapCompass.

Reads.

To generate a paired-end read, we uniformly choose a starting point on the genome (we make sure the genome starts sufficiently before the first SNP and ends at the last). We fix the read-end length (read_len) to be 150. The fragment length (frag_len) is normally distributed with a mean of 550 and standard deviation of 30, but with min and max lengths of 500 and 600 respectively. The insert length (insert_len) is determined by the fragment length and read-end length, that is, insert_len  =  frag_len - 2read_len. Once we know the start and fragment length, we must choose from which chromosome to read; we do so uniformly from the chromosomes. Finally, we add uniform error to the read; we choose a rate of , based on the reported error rate of Illumina sequencing technologies. For every SNP that the read covers, independently with probability we flip the allele to any other allele; two-thirds of the time when we have this error, we can see that the allele present is neither the reference nor the alternative, and therefore we delete it. Hence, conditional on seeing a SNP in a read, it is incorrect with probability and correct with probability .

Genomes.

To simulate a genome, we fix a ploidy and the number of SNPs . We determine the positions for the SNPs by randomly generating the distance between each pair of adjacent SNPs. We do so using a geometric random variable with parameter (SNP density); this choice is equivalent to assuming that any position is a SNP independently with probability . For phasing purposes, once one has generated the reads, the exact genomic positions are no longer relevant; they were only needed to simulate more accurate read data. We therefore refer to SNPs by their position amongst the SNPs, not their position in the genome. For each SNP, we randomly generate its haplotype, assuming for each chromosome, that the alternative and reference alleles are equally likely; if we generate a homozygous SNP, we try again. This procedure results in the likelihood of genotype equal to , and all orderings being equally likely. For the simulations discussed we use this model. Note, however, that HapTree is not dependent on this model. When running HapTree on real data, different assumptions ought to be made regarding the distributions of vector sets.

Coverage.

For any genome, to generate a read set with x coverage we need each base pair to be on average covered by reads. To determine the number of reads to generate, we must know the length of the genome and the read length (read_len). The expected length of the genome is for SNP density , and the read_len is 150 for each end (of which there are two); therefore we simulate reads for x coverage. Note that many of these reads will see only zero or one SNP(s), thus for x coverage the number of useful reads for any SNP will be less than .