 Methodology article
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HaploJuice : accurate haplotype assembly from a pool of sequences with known relative concentrations
BMC Bioinformatics volume 19, Article number: 389 (2018)
Abstract
Background
Pooling techniques, where multiple subsamples are mixed in a single sample, are widely used to take full advantage of highthroughput DNA sequencing. Recently, Ranjard et al. (PLoS ONE 13:0195090, 2018) proposed a pooling strategy without the use of barcodes. Three subsamples were mixed in different known proportions (i.e. 62.5%, 25% and 12.5%), and a method was developed to use these proportions to reconstruct the three haplotypes effectively.
Results
HaploJuice provides an alternative haplotype reconstruction algorithm for Ranjard et al.’s pooling strategy. HaploJuice significantly increases the accuracy by first identifying the empirical proportions of the three mixed subsamples and then assembling the haplotypes using a dynamic programming approach. HaploJuice was evaluated against five different assembly algorithms, Hmmfreq (Ranjard et al., PLoS ONE 13:0195090, 2018), ShoRAH (Zagordi et al., BMC Bioinformatics 12:119, 2011), SAVAGE (Baaijens et al., Genome Res 27:835848, 2017), PredictHaplo (Prabhakaran et al., IEEE/ACM Trans Comput Biol Bioinform 11:18291, 2014) and QuRe (Prosperi and Salemi, Bioinformatics 28:1323, 2012). Using simulated and real data sets, HaploJuice reconstructed the true sequences with the highest coverage and the lowest error rate.
Conclusion
HaploJuice provides high accuracy in haplotype reconstruction, making Ranjard et al.’s pooling strategy more efficient, feasible, and applicable, with the benefit of reducing the sequencing cost.
Background
With the rapid advancement of nextgeneration sequencing technologies, it is possible to obtain several gigabases of sequences in a single day. Given the huge volume of throughput, it is often costeffective to mix multiple subsamples in a single sample for sequencing, a process called pooling. Several approaches have been developed to demultiplex the sequencing reads from the mixture, i.e. assign reads to their respective subsamples. For example, a short unique identifiable sequence tag (i.e. barcode) is often appended to each DNA molecule of the same subsample before pooling and sequencing. Barcodes allow the reads to be separated into different groups according to their unique barcode sequences [1]. Each group is expected to originate from the same individual as with unpooled samples. Individual haplotypes can then be reconstructed by either by de novo assembly or computing the consensus sequence after aligning reads against one or more reference sequences. This approach cannot be applied to a mixture of reads without barcodes because the reads cannot be demultiplexed.
Nonetheless, in some instances, it may be useful to recover the constituent haplotype sequences from a mixture of haplotypes without using barcodes because the cost of the library preparation increases linearly with the number of required barcodes. Therefore, if it is possible to efficiently reconstruct haplotypes from mixtures of samples without using barcodes, this may reduce sequencing costs significantly.
Several methods have been designed to reconstruct the haplotypes from a mixture of reads without barcodes. The simplest of these approaches, developed by [2], aligns a mixture of reads against several reference sequences, allowing them to separate the reads to the different references. However, their method is only applicable for samples which are phylogenetically distant enough, e.g., for different species.
More sophisticated methods have also been developed to recover the constituent sequences from mixtures, when these sequences are genetically quite similar, e.g., haplotypes within populations or species. ShoRAH [3] implements localwindow clustering to recover the constituent haplotypes in a mixture. SAVAGE [4] uses an overlap graph and clique enumeration to reconstruct multiple haplotypes. PredictHaplo [5] uses Dirichlet prior mixture model, starts local reconstruction at the region of maximum coverage and progressively increases the region size until it covers the entire length of haplotypes. QuRe [6] uses sliding windows and reconstructs the haplotypes based on multinomial distribution matching heuristic algorithm [7]. However, ShoRAH, SAVAGE, PredictHaplo and QuRe assume that both the number and the proportion of the constituent haplotypes in the mixture are unknown and do not make use of these information in their algorithms.
Recently, Ranjard, et al. [8] proposed another pooling strategy without barcodes that can be applied for individuals of the same species. Their strategy consists of pooling in a single sample, individually amplified sequences in different known proportions. The proportions of these ‘subsamples’ induce different expected frequencies of the variants in the mixture, and hence, different expected sequencing read coverages. These frequencies, in turn, allow the subsampled sequences to be reconstructed accurately. Ranjard et al. applied their method to mitochondrial sequences from three kangaroo subsamples (each subsample consisting of an amplified fragment from a single kangaroo) mixed in proportions 62.5%, 25%, and 12.5%, and showed that the three haplotypes could be assembled effectively, thus reducing the cost of sequencing significantly. Hmmfreq [8], which was developed by Ranjard et al. to reconstruct the haplotypes under this scenario, is based on a Dirichletmultinomial model [9] and a Hidden Markov Model (HMM).
In this paper, we focus on the pooling strategy [8] proposed by Ranjard et al. but our method, however, does not assume any prior knowledge on the sample proportions; only the number of subsamples in the mixture is known a priori. We compute the subsample proportions directly from the mixture of reads using a maximum likelihood method. Based on the estimated sample proportions, we use a multinomial model and dynamic programming to reconstruct the multiple haplotypes simultaneously.
HaploJuice, which is an extension of Hmmfreq [8], considers all possible combinations for assigning local subsequences to haplotypes, and selects the combination with the highest overall likelihood. We evaluate HaploJuice against five different assembly algorithms, Hmmfreq [8], ShoRAH [3], SAVAGE [4], PredictHaplo [5] and QuRe [6], using simulated and real data sets in which three sequences are mixed in known frequencies. Based on our results, HaploJuice reconstructs sequences with the highest coverage of the true sequences and has the lowest error rate.
Results
HaploJuice first identifies the underlying subsample proportions from a mixture of reads and, second, reconstructs the haplotypes using these estimated proportions. As with Hmmfreq it requires an alignment of shortread sequences against a reference sequence. In our analysis, all reads are aligned to the reference sequence using Bowtie 2 [10].
Simulated datasets were used to evaluate our methods. Four hundred data sets were simulated and each data set was a mixture of three subsamples. The three subsamples were mixed under various proportions: 5:4:1, 5:3:2, 6:3:1, and 7:2:1 (100 data sets each). 150long pairended reads with total coverage 1500x were simulated by ART [11] with the default Illumina error model from three 10klong haplotypes, which were generated by INDELible [12] using JC [13] model from a 3tipped tree with 0.05 roottotip distance randomly created by Evolver [14] from PAML [15] package.
After using Bowtie 2 [10] to align the reads against the root sequence (also reported from INDELible [12]), we ran HaploJuice to estimate the subsample proportions in the mixture. As shown in Table 1, on average, the estimated subsample proportions were the same as the actual proportions with standard deviation 0.001. The method of estimation on the subsample proportions is, therefore, found to be effective on these simulated data sets.
HaploJuice was then used to reconstruct the haplotype sequences for each data set based on the estimated sample proportions. HaploJuice was compared to five different assembly algorithms, including Hmmfreq [8], ShoRAH [3], SAVAGE [4], PredictHaplo [5] and QuRe [6]. Note that SAVAGE, PredictHaplo and QuRe do not have prior assumptions on the number of haplotypes, whereas HaploJuice and Hmmfreq do. MetaQUAST [16] was then used with default parameters to evaluate the contigs, which were resulted by all the software, against the true sequences. By default, MetaQUAST discards all the contigs with length smaller than 500. Table 2 shows the summary of the performance of different methods on the simulated data sets. On average, HaploJuice reconstructed contigs over 99.7% haplotype coverage, which was the highest among all the methods. When checking the error rates (i.e. the percentage of bases in the contig sequences having mutations or indels when compared against with the real haplotypes), HaploJuice was less than 0.005% on average. It was the lowest among the software which reconstructed contigs over 90% haplotype coverage. In conclusion, HaploJuice is shown effective from the simulated data sets.
Apart from the simulated data sets, mixtures of reads from three kangaroo subsamples [8] were also used to evaluate the performance of the methods. These reads [8] were obtained by short read sequencing of three mitochondrial amplicons on an Illumina platform. The subsamples were mixed in the proportions: 0.625, 0.25, and 0.125 during the library preparation, and the total coverage of reads is 1600x. There is a total of 30 data sets; 10 data sets for each amplicon (three amplicons in total).
All the reads were aligned against the corresponding amplicon regions on the reference mitochondrial sequence [17] (Genbank accession number NC_027424) by Bowtie 2 [10]. The alignment file is the input of HaploJuice and the estimated subsample proportions are listed in Table 3. Although the subsamples were intentionally mixed in the proportions 0.625, 0.25 and 0.125, variations on the estimated proportions were noticed. For example, for the data sets of amplicon 3, the estimated proportions were 0.646, 0.251, and 0.103 on average. The variation between the estimated proportions and the expected proportions was 6.2% on average, ranging from 0.3% to 17.9%. This revealed the fact that the actual subsample proportions in the mixture may be differ from expectation, when the subsamples are mixed manually during the library preparation.
HaploJuice as well as the other five methods, including Hmmfreq [8], ShoRAH [3], SAVAGE [4], PredictHaplo [5] and QuRe [6], were used to reconstruct the three haplotypes for each amplicon region from the mixture of kangaroo reads. MetaQUAST [16] with default parameters was used to evaluate the resulting contigs against the true haplotypes inferred by deep sequencing [8]. Table 4 shows the summary on the performance of different methods. On average, HaploJuice resulted in contigs with the highest haplotype coverage for all amplicons (97% for amplicon 2 and over 99% for amplicon 1 and 3) among all the methods, and with the lowest (or one of the lowest) error rate among the methods with contigs over 90% haplotype coverage (on average, 0.05% for amplicon 1, 0.02% for amplicon 2, and 0.01% for amplicon 3). Thus, HaploJuice is shown to be effective at recovering the constituent haplotypes from the real data sets, even though the read coverage in the data sets fluctuates considerably along the mitochondrial genome (as shown in [8]).
To understand how the performance of HaploJuice varies with different genetic distances between the subsamples, another one hundred data sets were simulated. Each data set was a mixture of three subsamples under the proportions 1:2:5. For each triplet, the roottotip genetic distance of the tree was fixed at 0.05, and the genetic distance of the ancestor of the two most closely related sequences was a uniform random variable between 0.001 and 0.05. Similar to the previous simulated data sets, 150long pairended reads with total coverage 1500x were simulated and they were aligned to the root sequence. The haplotype sequences were reconstructed using HaploJuice from the read alignments. Figure 1 shows that the resulting haplotype coverage of the contigs is higher than 99.55% in all data sets, and the resulting error rates of the contigs are less than 0.001% with the exception of in one data set, where the error rate was 0.1% (data not shown). The results indicates that HaploJuice performs consistently with different distances between the haplotypes.
The performance of HaploJuice was also evaluated under different subsample proportions. A total of 833 datasets were simulated to cover all possible unique combinations of three subsample proportions with range between 1% and 98%, with a step size of 1%. As before, the 150long pairended reads with total coverage 1500x were simulated and they were aligned to the root sequence. HaploJuice was used to reconstruct the haplotype sequences from the read alignments. Figure 2 shows the performance of HaploJuice with different combinations of subsample proportions (i.e. x%, y%, z%). Figure 2a indicates that the haplotype coverage is close to 100%, but decreases when either x, y, or z are too small (i.e. less than 5%). The haplotype coverage also decreases when x≈y≈z (e.g., when subsample proportions are 33%, 33%, 34%). Similarly, Fig. 2b shows that the error rates are generally very low, except when two of the subsample proportions are close (e.g., x≈y, y≈z, x≈z or x≈y≈z). This result is in line with our expectations, because the algorithm uses proportions to reconstruct haplotypes, and haplotypes having similar proportions will naturally confound the process. From Fig. 2a and b, we found that the haplotype proportions have to be at least 5% different for HaploJuice to perform effectively.
When comparing the running time between different methods on the Kangaroo data sets, HaploJuice was the fastest, averaging 0.14 min for each data set, while other software took from 4 to 139 min. The summary is shown in Table 5.
Discussion
In order to decrease the cost of sequencing, Ranjard et al. [8] proposed a pooling strategy to mix subsamples in specific known proportions thus simplifying library preparation by removing the need for barcode sequences. According to their experiments on mitochondrial amplicons from three kangaroo subsamples mixed in proportions 0.625, 0.25, and 0.125, they found that the three haplotypes could be reconstructed effectively using these known frequencies. However, they found that variation of the ratios of subsamples when mixing due to stochastic experimental effects can decrease the accuracy of haplotype reconstruction. Our research provides an alternative haplotype reconstruction algorithm for Ranjard et al.’s pooling strategy. We show that estimating the empirical proportions of the mixed subsamples, prior to the reconstruction the haplotype sequences, significantly increases the accuracy of the approach. As shown from the simulated data sets and the real data sets, our method can, first, accurately identify the underlying subsample proportions from a mixture of reads and, second, reconstruct the haplotypes according to these estimated proportions.
The pooling strategy can be applied on a greater number of sequences. Consider a total of n subsamples. A group of three subsamples of the same species can be mixed in the specific known proportions and applied the same barcode. Thus only \(\frac {n}{3}\) barcodes are required and the cost of the library preparation can be greatly reduced. After sequencing, HaploJuice can be used to assemble the reads associated with the same barcode and reconstruct the three haplotypes for each group of the subsamples. As shown from the simulated data sets and the real data sets, the high accuracy of assembled haplotypes makes the suggested pooling strategy [8] become more realistic, feasible, and applicable.
Our method relies on aligning reads against a reference sequence. The accuracy of the read alignments affects the effectiveness of our method. In our evaluations, we only used alignments reported by Bowtie 2 [10] with mapping quality of at least 20. Whereas we understand that coverage varies along the haplotype, but we assume that ratios of the read coverage for each haplotype at each location follows the same multinomial distribution. If a region on some haplotypes is very different from the reference sequence, reads from this region may not align to the reference, and the induced read coverage for those haplotypes may decrease substantially. The bias in the induced read coverage ratio can cause misleading results, because of its deviation from the common multinomial distribution. Therefore, this method is designed for the pooling strategy applied on the subsamples that align well with the reference sequence.
HaploJuice assumes that the number of haplotypes is known in advance. There is no equivalent assumption with ShoRAH [3], SAVAGE [4], PredictHaplo [5] and QuRe [6]. Nonetheless, these are the only available software for haplotype reconstruction from a pool of reads originating from a mixture of different subsamples. We expect that the effectiveness of haplotype reconstruction using these methods are also likely to be improved if the number of haplotypes is known in advance. One reasonable approach to assemble the reads from a sample with unknown number of haplotypes is therefore to develop a statistical method to estimate the number of haplotypes from a mixture of reads, and then reconstruct the haplotypes using our method according to this estimated number of haplotypes.
Conclusions
HaploJuice is designed for the reconstruction of three pooled haplotypes from a mixture of short sequencing reads obtained under the strategy proposed by Ranjard et al. [8]. As shown from the simulated data sets and the real data sets, HaploJuice provides high accuracy in haplotype reconstruction, thus increasing the estimation efficiency of Ranjard et al.’s pooling strategy.
Methods
HaploJuice is designed for the pooling strategy [8] proposed by Ranjard et al., assuming the number of subsamples is known and the subsamples have different proportions. Figure 3 shows the work flow in HaploJuice. HaploJuice first estimates the subsample proportions from a mixture of reads using maximum likelihood method. The algorithm then reconstructs the haplotype sequences using a dynamic programming method. The following subsections describes the details of the algorithm.
Estimation of sample proportions
HaploJuice requires an alignment of shortread sequences against a reference sequence. All reads are aligned to the reference sequence using Bowtie 2 [10]. Only the reads which are aligned at unique positions on the reference are considered. The alignment of each read has a starting and an ending position on the reference. A sliding window approach is used.
Let W be the set of overlapping windows. For each window w∈W, we collect the reads that are aligned across the whole window. We extract the corresponding subsequences according to the window’s bounds, and obtain the set of unique subsequences T_{w}={t_{w1},t_{w2},...} and the frequencies G_{w}={g_{w1},g_{w2},...} where g_{wi} is the number of reads with subsequence t_{wi}. The subsequences inside T_{w} are sorted in decreasing order of frequencies.
Say n subsamples are pooled with unknown proportions f_{1},f_{2},...,f_{n} where f_{1}>f_{2}>...>f_{n}. When there is no sequencing error and each subsample is from a unique haploid sequence, each subsample should produce only one subsequence in T_{w}. In those regions where two or more subsamples are identical, the subsequences originating from these subsamples will be the same. For each sliding window, the number of possible combinations of n samples producing subsequences, i.e. the number of possible partitions of a set with n different elements (where each element represents a subsample, and the elements in the same partition are regarded as the subsamples producing the same subsequences), is the Bell number B_{n} [18]. Each case will lead to different expected frequencies of the subsequences.
However, under real sequencing conditions, the number of subsequences in each window may be greater than n, because some erroneous subsequences are created by sequencing errors. We assume that the frequencies of erroneous subsequences are always lower than that of real subsequences. For each window, we only consider the topn most frequent subsequences. Table 6 lists the expected frequencies of the subsequences for all cases when n=3.
Let p_{ki} be the ith expected frequency for case k. Assume the observed frequencies of the subsequences in a window w∈W follow a multinomial distribution. The likelihood value for the window w, (L(w)), is computed as follows:
The probability of the case k (i.e. prob(case k)) is estimated by the following equation:
And the overall loglikelihood value (logL) for all the windows w∈W is:
The optimal values of \(f_{1},f_{2},...,f_{n},f_{e},f_{e^{\prime }}\) are computed such that the overall loglikelihood value (logL) is maximum. In practice, the following constraints are used: \(f_{1} \geq f_{2} \geq \cdots f_{n} \geq f_{e} \geq f_{e^{\prime }}\) and f_{e}≤b, where b is an upper limit for the frequency of an erroneous subsequence. The estimated sample proportions are the optimal values of f_{1},f_{2},...,f_{n}. The time complexity is: O(B_{n}∗n∗W), where B_{n} is the nth Bell number, n is the number of haplotypes, and W is the number of windows.
Reconstruction of haplotype sequences
The next step is to reconstruct the haplotype sequences according to the subsample proportions estimated in the previous step. We assume that each subsample is generated from a unique haploid sequence (i.e. haplotype). If we can identify the corresponding subsequence of each haplotype for every sliding window, then the haplotype sequences can be reconstructed by combining the subsequences from all the windows. However, in practice, it is not obvious, because the real subsequences are usually mixed with erroneous subsequences caused by sequencing errors. Moreover, multiple haplotypes may share the same subsequence and the observed frequencies of the subsequences may deviate from expectation at some positions.
A dynamic programming approach was used to reconstruct multiple haplotype sequences simultaneously, by considering all the cases for each window, and choosing the best arrangement with the maximum likelihood value.
Consider a sliding window w∈W and the topn most frequent subsequences (i.e. t_{w1},t_{w2},...,t_{wn}) in the window. Since each haplotype can generate one subsequence, there are n^{n} possible cases to generate n different subsequences by n haplotypes (considering that multiple haplotypes can generate the same subsequence and some subsequences can be erroneous), and each case will lead to a different set of expected frequencies of the subsequences. Table 7 lists all 27 possible cases and the expected frequencies of the subsequences when n=3.
Define A(w,k)=(t_{1},⋯,t_{n}) as an assignment of the haplotypes to the subsequences in sliding window w when case k is considered (i.e. ith haplotype generates subsequence t_{i},1≤i≤n). For example, as shown in Table 7, for n=3 and case 7, A(w,7)=(t_{w1},t_{w1},t_{w2}) (i.e. the observed subsequence with the highest frequency in window w is generated from both the first and the second haplotypes, while the observed alignment with the second highest frequency is generated from the third haplotype).
Define δ(A(w,k),A(w^{′},k^{′})) as the compatibility between two assignments A(w,k)=(t_{1},⋯,t_{n}) and A(w^{′},k^{′})=(t^{′}_{1},⋯,t^{′}_{n}) and δ(A(w,k),A(w^{′},k^{′}))=1 if, for all 1≤i≤n, two subsequences t_{i} and t^{′}_{i} are exactly the same in their overlapped region. Mathematically, if the window size is d, the two windows overlap l bases, and window w is before window w^{′},
We begin from a starting window w_{s}∈W and consider all possible n^{n} assignments in w_{s}. Then we consider the left and the right windows besides w_{s}, and continue until all the windows have been considered. The optimal reconstruction of n haplotypes is the set of compatible assignments for all the windows with the maximum loglikelihood value. The following dynamic programming approach is used to compute the optimal compatible assignments for all the windows.
Given a starting window w_{s}∈W, define ζ(k_{s},k_{t},w_{t}), where w_{t}∈W,1≤k_{s},k_{t}≤n^{n}, as the maximum loglikelihood value of the optimal compatible assignments for the consecutive windows from w_{s} to w_{t} with assignment A(w_{s},k_{s}) in window w_{s} and assignment A(w_{t},k_{t}) in window w_{t}. If s<t, the assignment is proceeded from left to right, while if t<s, the assignment is proceeded from right to left.
Without loss of generality, considering the situation that the haplotype assignment is proceeded from left to right, the recursive formula of ζ(k_{s},k_{t},w_{t}) is defined as:
where like(w_{t},k_{t}) is the likelihood value of the observed frequencies of the subsequences in window w_{t} when assignment A(w_{t},k_{t}) is selected.
Let q_{ki} be the ith largest expected frequency for case k.
Therefore,
In order to increase the accuracy of the haplotype reconstruction, we reconstruct the haplotypes starting from a relatively reliable window \(w_{\hat {s}}\) with much dissimilarity between the haplotypes. When n=3, we locate the window \(w_{\hat {s}}\) which have the greatest value of likelihood value for the case when each haplotype is assigned to different subsequence. Let the first and the last window on the haplotype region be w_{1} and w_{last}. The haplotypes are reconstructed in both directions from the window \(w_{\hat {s}}\) to the beginning and to the ending of the haplotypes, respectively. Considering the different case \(k_{\hat {s}}\) for the starting window \(w_{\hat {s}}\), the loglikelihood value of the optimal set of compatible assignments for the whole haplotype region is:
Since k_{s} and k_{t} have n^{n} possible values (where n is the number of haplotypes), the overall time complexity of the method is: O(n^{2n}∗W). The method explores all the possible cases and is an exact algorithm. The time is growing exponentially with the number of haplotypes. For higher number of haplotypes, a heuristic approach should be developed accordingly.
Abbreviations
 B _{ n } :

nth of the Bell numbers
 HMM:

Hidden Markov Model
 JC:

Jukes and Cantor model
 N50:

A weighted median statistic such that 50% of the entire assembly is contained in contigs longer than or equal to this value
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Acknowledgements
We thank two anonymous reviewers for their constructive comments, which helped to improve the manuscript.
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This research was supported by the Australian Research Council Discovery Project Grant #DP160103474.
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TW, LR and AR proposed the initial idea and designed the methodology. TW implemented the concept and processed the results, under the help of LR, YL and AR. TW, LR and AR wrote the manuscript. All authors read and approved the final manuscript.
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The software HaploJuice and the simulated datasets are available in OSF repository: https://osf.io/b8nmf/ (https://doi.org/10.17605/OSF.IO/B8NMF).
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Wong, T., Ranjard, L., Lin, Y. et al. HaploJuice : accurate haplotype assembly from a pool of sequences with known relative concentrations. BMC Bioinformatics 19, 389 (2018). https://doi.org/10.1186/s1285901824247
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DOI: https://doi.org/10.1186/s1285901824247