 Methodology Article
 Open Access
Joint haplotype assembly and genotype calling via sequential Monte Carlo algorithm
 Soyeon Ahn^{1}Email author and
 Haris Vikalo^{1}
https://doi.org/10.1186/s1285901506518
© Ahn and Vikalo; licensee BioMed Central. 2015
 Received: 25 December 2014
 Accepted: 26 June 2015
 Published: 16 July 2015
Abstract
Background
Genetic variations predispose individuals to hereditary diseases, play important role in the development of complex diseases, and impact drug metabolism. The full information about the DNA variations in the genome of an individual is given by haplotypes, the ordered lists of single nucleotide polymorphisms (SNPs) located on chromosomes. Affordable highthroughput DNA sequencing technologies enable routine acquisition of data needed for the assembly of single individual haplotypes. However, stateoftheart highthroughput sequencing platforms generate data that is erroneous, which induces uncertainty in the SNP and genotype calling procedures and, ultimately, adversely affect the accuracy of haplotyping. When inferring haplotype phase information, the vast majority of the existing techniques for haplotype assembly assume that the genotype information is correct. This motivates the development of methods capable of joint genotype calling and haplotype assembly.
Results
We present a haplotype assembly algorithm, ParticleHap, that relies on a probabilistic description of the sequencing data to jointly infer genotypes and assemble the most likely haplotypes. Our method employs a deterministic sequential Monte Carlo algorithm that associates single nucleotide polymorphisms with haplotypes by exhaustively exploring all possible extensions of the partial haplotypes. The algorithm relies on genotype likelihoods rather than on often erroneously called genotypes, thus ensuring a more accurate assembly of the haplotypes. Results on both the 1000 Genomes Project experimental data as well as simulation studies demonstrate that the proposed approach enables highly accurate solutions to the haplotype assembly problem while being computationally efficient and scalable, generally outperforming existing methods in terms of both accuracy and speed.
Conclusions
The developed probabilistic framework and sequential Monte Carlo algorithm enable joint haplotype assembly and genotyping in a computationally efficient manner. Our results demonstrate fast and highly accurate haplotype assembly aided by the reexamination of erroneously called genotypes.
A C code implementation of ParticleHap will be available for download from https://sites.google.com/site/asynoeun/particlehap.
Keywords
 Haplotype assembly
 Deterministic sequential Monte Carlo
 Genotype calling
Background
Increased affordability of highthroughput DNA sequencing has enabled studies of genetic variations and of the effects they have on health and medical treatments. In diploid organisms, such as humans, chromosomes come in pairs. The chromosomes in a pair of autosomes are homologous, i.e., they have similar composition and carry the same type of information but are not identical. The most common type of DNA sequence variation is a single nucleotide polymorphism (SNP), where a single base in the genome differs between individuals or paired chromosomes. Each of those variants is referred to as an allele; a SNP has at least two different alleles. If the two alleles at a SNP site are same, the SNP site is homozygous; if they are different, it is heterozygous. SNP calling is concerned with identification of the locations and types of such alleles, and is followed by genotype calling to decide the genotypes associated with the locations of the detected SNPs. Accurate SNP and genotype calling are challenging due to uncertainties caused by base calling and read alignment errors. The lowtomedium coverages typical of largescale sequencing projects are often associated with erroneous SNP and genotype calling [1]. As an illustration, in the lowcoverage (2−6×) 1000 Genomes Project pilot, the genotype accuracy at heterozygous sites was 90 % for the lowest allele frequencies (minor allele frequency (MAF) <3 %), 95 % for the intermediate frequencies (MAF 50 %), and 7080 % for the highest frequency variants (MAF >97 %) [2].
SNP and genotype calling do not assign alleles to specific chromosomes in the pairs. Such detailed information is provided by haplotypes, ordered collections of SNPs on the chromosomes. Haplotypes have been of fundamental importance for the studies of human diseases and effectiveness of drugs [3]. The International Haplotype Map Project’s pursuit of developing a haplotype map of the human genome reflects the significance of acquiring and understanding haplotype information [4]. Haplotype inference typically refers to the task of reconstructing haplotypes from the genotype samples of a population. Haplotype assembly, or single individual haplotyping, aims to reconstruct single individual haplotypes from highthroughput sequencing data. Since the SNP sites are assumed to be biallelic (i.e., each SNP site contains one of only two possible nucleotides), the alleles are labelled as 0 and 1 and the haplotypes are represented by binary sequences. Therefore, haplotype assembly is often cast as the problem of phasing two binary sequences from their short samples (i.e., reads) that are represented by ternary strings (where the third symbol denotes missing information). Majority of the existing haplotype assembly algorithms rely on this formulation of the problem [5].
Several haplotype assembly criteria and algorithms to optimize them were considered in [6, 7]. The minimum error correction (MEC) criterion, in particular, has received a considerable amount of attention and has been broadly used in practice. Most of the haplotype assembly problem formulations have been shown to be NPhard [6–8], which has motivated numerous computationally efficient heuristic solutions [5]. FastHare, proposed in [9], was an early heuristic method that was followed by several approximate techniques in [10, 11]. In [12, 13], the use of clustering approaches for splitting reads into two sets, each associated with one chromosome in a pair, was proposed. In addition to the approximate methods, several algorithms that search for the exact solution to the MEC formulation of the problem were developed, including the branchandbound technique in [14]. However, as argued in [15], the exact algorithms are often infeasible in practice; the approach in [16] based on the Markov Chain Monte Carlo (MCMC) method, HASH, also incurs high computational cost while being more accurate than heuristics. As a followup to HASH, [17] presented a significantly faster heuristic algorithm, HapCUT, suffering only a minor loss of accuracy. To minimize the MEC score, HapCUT iteratively computes the maxcut in a graph that represents the assembly problem. In [18], another maxcut based heuristic, ReFHap, was proposed; ReFHap relies on a different graph structure to achieve higher speed while maintaining accuracy similar to that of HapCUT. Other methods include a dynamic programming solution in [19]; a method that solves an appropriate integer linear program [20]; and several other heuristics including [21], HBOP [22], HMEC [23], and HapCompass [24, 25].
A probabilistic framework for haplotype assembly was first introduced in [26]. There, in order to deal with inherently random errors in sequencing data, the probability that a site in the fragments is incorrectly sequenced is defined for each of the four nucleotide bases. The most likely haplotype phases between SNP sites are determined using joint posteriori probabilities whose calculation is limited to two or three adjacent SNPs due to the intensive computational cost that grows exponentially in the number of SNP sites. The locally estimated haplotype segments are linked if the corresponding confidence levels exceed a certain threshold. In the followup work [27], reconstruction of longer haplotype segments using the Gibbs sampling procedure was enabled. However, this iterative approach still first assembles short haplotype segments that are then connected, and requires runtimes infeasible for block lengths typically encountered in practice. More recently, [28] proposed a new probabilistic mixture model, MixSIH, which leads to a more efficient computation of the haplotype likelihoods than those in [26, 27]. However, MixSIH is still about 10fold slower than either HapCUT [17] or ReFHap [18] while having comparable accuracy, and the model there is restricted to the biallelic representation as in [625].
It is worth pointing out that most of the existing algorithms for haplotype assembly allow no more than two alleles at a SNP site and only deal with the errors caused by substitutions between those two alleles [625,28]. In practice, when sequencing errors lead to reads that report more than two alleles at a SNP site, either all of the tri or tetraallelic sites are discarded [10, 17] or the alleles that do not match the reference (or its alternative) are thrown away [24, 25]. The former drastically reduces not only the number of SNP sites to be reconstructed but also the chance of reliable full haplotype reconstruction (due to reducing the length of already short reads). The main drawback to the latter is that by fully trusting genotype information provided by SNP/genotype calling, the true haplotypes may be incorrectly reconstructed when the genotype calling is erroneous (i.e., when alleles corresponding to an incorrect genotype are preserved while the alleles corresponding to the true genotype are discarded).
In this paper, we propose a novel method that relies on a probabilistic model of the data to incorporate genotype calling in the haplotype assembly procedure. Unlike [26, 27], the proposed method infers both the most likely genotypes and haplotype phases by examining the complete set of SNP loci in a computationally efficient manner. To this end, we employ the sequential Monte Carlo (SMC) algorithm (i.e., a particle filter). Particle filters are capable of sequentially estimating the posterior density of unknown variables by representing them with a set of particles and associated weights [29]. When the solution space is discrete and finite, a deterministic form of SMC can be derived [30, 31]; this has been exploited for solving various problems in genomics [32, 33]. Noting that the set of possible haplotype pairs is discrete and finite, we develop a modified deterministic sequential Monte Carlo (DSMC) method for solving the haplotype assembly problem. Our algorithm, ParticleHap, relies on the 2 ^{ n d }order Markov model of the haploype sequence to search for the most likely association of the SNPs to haplotypes. Phasing of the SNPs is done sequentially: the posteriori probability of a partial haplotype comprising n SNPs is calculated using the read information about the SNP in the n ^{ t h } position of the haplotype sequence and the posteriori probability of the previously inferred (n−1)bases long partial haplotype. By working with SNP calls rather than their binary representations (the latter is typically used by stateoftheart haplotype assembly algorithms), ParticleHap can reliably infer the most likely genotypes. Our extensive computational studies demonstrate that the proposed scheme is more accurate and computationally more efficient than stateoftheart methods in [17, 18].
Methods
Problem formulation
In our model and the subsequently proposed haplotype assembly algorithm, we focus on the SNP sites where two or more alleles are observed. The sites with only one observed allele are declared homozygous and not used for the assembly. Assume there are m pairedend short reads covering n remaining SNP sites. Such data can be represented by an m×n matrix where the rows contain information provided by the reads while the columns correspond to the SNP sites.
We assume that each read is generated from one of the two haplotypes with probability \(\frac {1}{2}\), i.e., \(\text {Pr}(f_{i}=1)=\text {Pr}(f_{i}=2)=\frac {1}{2}\).
The ParticleHap algorithm
Following the adopted notation, the goal of haplotype assembly is to determine matrix S from the observation matrix X. A Bayesian approach to solving this problem involves maximization of the posteriori distribution p(SX). Let S _{·j } and X _{·j } denote the j ^{ t h } column vectors of S and X, respectively, and let us define S_{1:t }={S _{·1},S _{·2},⋯,S _{·t }} and X_{1:t }={X _{·1},X _{·2},⋯,X _{·t }}. Recursive Bayesian estimation (i.e., Bayesian filtering) is concerned with recursively finding the conditional probability density function p(S_{1:t }X_{1:t }). Having obtained the estimate \(\hat {p}(\mathrm {S}_{1:t}  \mathrm {X}_{1:t})\), we can determine the most likely S_{1:t }. However, finding an analytical form of this probability density function is often infeasible, as is the case for the haplotype assembly problem.
where \(\phantom {\dot {i}\!}s_{l}=(s_{l_{1}},s_{l_{2}})\), l=1,…,L. In particular, at step t, ParticleHap examines potential extensions of the partially reconstructed haplotype by adding a single SNP site, which requires no more than 12 likelihood calculations – one for each possible heterozygous pair \((s_{l_{1}},s_{l_{2}})\) at site t, with 12 such pairs when there are 4 different bases in the t ^{ t h } column of X.
When computing \(p\left (S_{\cdot t}=s_{l}  S_{\cdot \mathbf {Pos}^{t1}}^{(k)}, X_{\cdot \mathbf {Pos}^{t1}}\right)\), we assume that there is no correlation between the consecutive SNPs. Therefore, the state transition distribution is formed using the composition probabilities, e.g., \(p\left (S_{\cdot t}=s_{l}  S_{\cdot \mathbf {Pos}^{t1}}^{(k)},X_{\cdot \mathbf {Pos}^{t1}}\right) =\text {Pr}(S_{1t}=s_{l_{1}})\text {Pr}(S_{2t}=s_{l_{2}})\). Note that, in principle, side information such as genotype frequencies or the patterns of linkage disequilibrium (LD) can be incorporated in state transition probabilities.
Going back to the example illustrated in Fig. 2, the modification of the weights shown in (8) now allows ParticleHap to retrieve phase information at position t+2 from the reads (highlighted in orange) that have a gap in position t+1 but cover some SNPs in the positions left of the (t+1)^{ s t } one. This often has a beneficial effect on the switch error rate, defined as the ratio of the number of SNP positions where the two chromosomes of a resulting haplotype phase must be switched in order to reconstruct the true phase. As an illustration, consider column t in the observation matrix shown in Fig. 2. Since none of the reads that cover SNPs at column t provide any phasing information, the partially reconstructed haplotype pair would equally likely be extended by either (C,G) or (G,C) which might lead to the switch error at the position t. This ambiguity is resolved at position t+2 from the reads i _{1}=4 and i _{2}=13 (highlighted in orange), which do not cover sites t+1, t nor t−1, but do cover sites 4 and 1, respectively: ParticleHap relying on those reads in (8)(9) will assign larger weight to the particle propagated along the correct state path.
To initialize the algorithm at t=1 from k possible SNPs, all possible assignments are considered as \(S_{\cdot 1}^{(k)}, k=1,\dots,K\), with the corresponding weights \(w_{1}^{(l)}, l=1,\dots,L\), computed as \(w_{1}^{(l)}\propto p(X_{\cdot 1}  S_{\cdot 1}^{(l)}=s_{l})\). From t=2, all possible extensions of the (t−1)long haplotype are considered, and the extensions having nonzero weights are used to generate particles until K such particles are created. Once the set of K particles is formed, the subsequently generated particles are included in the set if their weight is greater than the weight of at least one particle that is already in the set; the latter then needs to be excluded from the set so that its cardinality remains K.

Step 1 (Initialization): For the first SNP position, compute \(w_{1}^{(l)}\propto p\left (X_{\cdot 1}  S_{\cdot 1}^{(l)}=s_{l}\right)\), l=1,…,L. Normalize \(w_{1}^{(l)}\) and store the corresponding possible haplotype pairs in \(S_{\cdot 1}^{(k)}\), k=1,2,⋯,K.

Step 2 (Run iterations for 2≤t≤n): For each step t, t=2,…,n, enumerate all possible extensions of the existing particles \(S_{\cdot t1}^{(k)}\), thus generating \(S_{\cdot t}^{(k,l)}=\left [S_{\cdot t1}^{(k)},s_{l}\right ]\), l=1,⋯,L. For all l, compute the weights \(w_{t}^{(k,l)}\) using (8).

Step 3 (Particle selection): Select and store K particles \(\left \{S_{\cdot t}^{(k)},k=1,\cdots,K\right \}\) with the highest importance weights \(\left \{w_{t}^{(k)},k=1,\dots,K\right \}\) from the set \(\left \{S_{\cdot t}^{(k,l)},w_{t}^{(k,l)},k=1,\dots,K,l=1,\dots,L\right \}\). Normalize the weights of the selected particles. Go back to Step 2 and repeat until t=n.

Step 4 (Haplotype reconstruction): At t=n, assemble the entire haplotype sequence by selecting the particle with the largest weight.
Complexity analysis of ParticleHap
Since ParticleHap searches for the most likely haplotypes by sequentially extending the partially reconstructed candidate haplotypes one position at a time, it is computationally very efficient and has complexity that scales linearly with the haplotype length, O(n). On average, the amount of calculations needed for haplotype assembly with ParticleHap is n K L _{ a } C _{ a }, where C _{ a } denotes the average number of bases covering heterozygous sites and L _{ a } denotes the average number of possible extensions of the partially reconstructed haplotype at a heterozygous SNP site. It is worth pointing out that while there are in principle 12 possible SNP pairs of \((s_{l_{1}},s_{l_{2}})\) at one site, the 3^{ r d } or 4^{ t h } most frequently reported nucleotides are never selected as potential SNPs’ genotypes based on the likelihood calculations. Therefore, ParticleHap can be implemented even more efficiently by retaining the two (or three in the case of ties) most frequently observed nucleotides at each site while the others, which are considered errors, are replaced by −; this step can significantly reduce the amount of likelihood calculations without compromising the accuracy of computing the most likely genotype.
Postprocessing
We can further improve the accuracy of assembled haplotypes by pooling the information obtained from multiple runs of ParticleHap whose performance may be affected by the choice of the starting position. In particular, we also run the algorithm in the opposite direction, e.g., perform sequential Monte Carlo reconstruction of the most likely haplotype starting from the site t=n and terminating the algorithm at the site t=1. For the sites where the haplotype pairs reconstructed by the two runs of ParticleHap differ from each other, we compare the likelihoods of the solutions (i.e., we compare the weights in (8)) and choose the one with larger likelihood. In case the sites are consecutive, we compare MEC scores for the two cases and choose the one with smaller MEC.
Results and discussion
We implemented ParticleHap in C and compared its performance with the publicly available implementations of HapCUT [17] and ReFHap [18]. All three methods are run on a Linux OS desktop with 3.06GHz CPU and 8Gb RAM (Intel Core i7 880 processor). Both real and simulated data are used for the experiments, as described in the remainder of this section.
genome project data
We first study the performance of ParticleHap on 454 sequencing data of CEU NA12878 genome (1000 Genomes Pilot Project [2]). The shortread data aligned with respect to a reference genome as well as variant and genotype calls for the individual are provided. The data contains a total of 2.58 million reads covering 1.65 million variants on 22 chromosomes. Due to the short lengths of reads and limited insert sizes, the data is split into a number of disconnected blocks. We use ParticleHap to reconstruct each such block.
To evaluate the performance of haplotype assembly, we adopt three measures: the number of phased SNPs (nPhased), the minimum error correction (MEC) score, and running time (Time). In particular, nPhased is the number of SNPs phased by a haplotype assembler; MEC score is the smallest number of entries in the data matrix which need to be changed so that the sequencing information is consistent with an errorfree haplotype pair (we report the total MEC score evaluated as the sum of the MEC scores obtained for each haplotype block); and, Time is the runtime of an algorithm in seconds, measured for each algorithm on the same processor.
We choose a different number of particles K for each block. Specifically, the longer the block length n, the larger the number of particles (12 for n≤12, \(\frac {n}{2}\) for 12<n<100, and 50 for n≥100). We assume that the bases in the sequence are equally likely, i.e., the composition probabilities of the 4 nucleotides are equal, 0.25. Note that we assume the error rate e=0.01, which is consistent with the typical sequencing accuracy of the 454 platform. Then, the sequencing error probabilities \(\phantom {\dot {i}\!}\text {Pr}(X_{\textit {ij}}=x_{\textit {ij}}  S_{\textit {kj}}=s_{l_{k}}) = e/3\) if \(\phantom {\dot {i}\!}x_{\textit {ij}}\neq s_{l_{k}}\) and are equal to 1−e otherwise.
The performance comparison on a CEU NA12878 data set sequenced using the 454 platform in the 1000 Genomes Project
ParticleHap  HapCUT  ReFHap  

chr  nPhased  MEC  Time(s)  nPhased  MEC  Time(s)  nPhased  MEC  Time(s) 
1  66661  2045  1.07  66616  2293  28.03  66490  2111  5.79 
2  78002  2742  1.11  77970  2857  35.71  77853  2698  7.78 
3  66217  2111  3.96  66178  2349  29.50  66071  2203  6.20 
4  69939  2386  3.94  69901  2591  37.16  69786  2410  9.01 
5  63723  1971  4.75  63693  2156  28.06  63605  2044  6.00 
6  69750  3312  5.25  69706  3544  58.60  69580  3318  13.39 
7  54330  1867  3.31  54302  2059  27.23  54202  1908  7.19 
8  56406  1700  3.89  56382  1828  25.87  56281  1690  5.60 
9  42244  1335  2.15  42230  1472  20.02  42157  1365  4.52 
10  50022  1618  2.73  49998  1814  23.44  49900  1662  5.22 
11  46141  1411  2.72  46124  1569  21.66  46051  1467  4.89 
12  43333  1467  2.32  43315  1581  20.00  43251  1495  3.92 
13  36952  1286  0.68  36937  1398  18.80  36872  1311  7.10 
14  30349  887  0.38  30334  982  13.25  30293  916  2.90 
15  26626  975  0.88  26614  1055  11.46  26567  968  2.60 
16  31675  1156  0.87  31662  1257  14.84  31612  1185  3.98 
17  21054  1206  0.59  21048  1223  11.19  21010  1172  5.35 
18  28784  851  0.37  28769  936  11.94  28717  855  2.67 
19  17018  653  0.25  17006  761  8.35  16961  687  4.25 
20  21679  737  0.43  21673  790  9.50  21635  735  2.84 
21  14737  485  0.41  14736  525  6.82  14714  500  2.09 
22  12929  388  0.28  12925  433  5.38  12891  395  1.74 
Simulated data
We further test the performance of our proposed method on the simulated data set. In particular, we examine how the genotype calling errors affect the performance of haplotype assembly. The data are generated using a similar strategy to the one in [15] except that a genotype calling error is included in our simulation data. We generate a pair of phased heterozygous SNP sequences of length n, which have genotype calling errors with probability g _{ e }. The parameter g _{ e } is judiciously chosen as 0.04 and 0.08 in order to emulate practical scenarios reported in [1] (there, the genotype call accuracy for high call rates was up to 96 % with the use of LD information, from 78 % and 87 % with the use of single sample and multiple individuals, respectively, for the 62 CEU individuals). The true haplotype sequences are generated by emulating genotyping errors; each base in the erroneous heterozygous SNP sequences is flipped to one of the other three nucleotides with equal probability, g _{ e }/3. To generate the observation data matrix, instead of sampling (with reads) from true haplotype sequences c times as in [15], we sample true haplotype sequences \(\frac {c}{2}\) times and sequences containing genotype calling errors \(\frac {c}{2}\) times. Each replicate is randomly partitioned into nonoverlapping fragments of length between 3 and 7 (the lengths typical of benchmarking data sets in [15]). In order to simulate pairedend (or matepair) sequences, we randomly merge some of the generated fragments (fragments whose SNPs are in the first half of haplotype sequence are merged with those whose SNPs in the last half of sequence; as a result, half of the fragments in the dataset are pairedend sequences). Once the fragments are arranged in a SNP matrix, we emulate sequencing errors by randomly flipping a base to one of the other three nucleotides with equal probability. The probability that each base is flipped is 0.03 and 0.01 for g _{ e }=0.04 and g _{ e }=0.08, respectively, and thus the total error rate for the entries in the SNP matrix is e=0.05. To explore the performance of the algorithm over a broad range of experimental parameters, we generate datasets with different SNP lengths (n=100, 200 and 300) and vary the coverage rate (c=4, 6, 8 and 10) for each genotype calling error rate (g _{ e }=0.04 and g _{ e }=0.08). For each of the 24 combinations of the parameters, the experiment is repeated 100 times and the results averaged over the 100 instances are reported for each case.
The performance comparison on a simulated data set for g e=0.04
ParticleHap  HapCUT  ReFHap  

n  c  ImpGeAc  ReconRate  Time(s)  ReconRate  Time(s)  ReconRate  Time(s) 
100  4  0.6254  0.9785  0.02  0.9598  0.66  0.9497  0.11 
6  0.6252  0.9794  0.02  0.9570  0.84  0.9481  0.25  
8  0.5977  0.9792  0.03  0.9590  1.01  0.9524  0.56  
10  0.5737  0.9780  0.03  0.9582  1.17  0.9517  1.23  
200  4  0.5935  0.9779  0.07  0.9594  1.78  0.9499  0.26 
6  0.5977  0.9783  0.08  0.9597  2.26  0.9518  0.88  
8  0.5840  0.9757  0.09  0.9596  2.71  0.9524  2.56  
10  0.5758  0.9777  0.11  0.9593  3.13  0.9528  5.80  
300  4  0.6013  0.9715  0.17  0.9591  3.39  0.9493  0.53 
6  0.5848  0.9720  0.20  0.9596  4.34  0.9511  2.12  
8  0.5842  0.9695  0.22  0.9598  5.14  0.9525  6.35  
10  0.5671  0.9703  0.24  0.9599  5.90  0.9537  14.68 
The performance comparison on a simulated data set for g e=0.08
ParticleHap  HapCUT  ReFHap  

n  c  ImpGeAc  ReconRate  Time(s)  ReconRate  Time(s)  ReconRate  Time(s) 
100  4  0.6211  0.9618  0.02  0.9193  0.66  0.9009  0.11 
6  0.5941  0.9610  0.02  0.9184  0.86  0.8997  0.25  
8  0.5970  0.9585  0.03  0.9184  1.04  0.9017  0.56  
10  0.5845  0.9572  0.03  0.9193  1.19  0.9041  1.26  
200  4  0.6262  0.9615  0.08  0.9186  1.80  0.8979  0.27 
6  0.5938  0.9418  0.09  0.9198  2.30  0.9021  0.89  
8  0.6050  0.9389  0.10  0.9193  2.75  0.9019  2.53  
10  0.5997  0.9438  0.12  0.9193  3.16  0.9039  5.84  
300  4  0.6245  0.9432  0.18  0.9197  3.43  0.8995  0.53 
6  0.6069  0.9397  0.21  0.9198  4.49  0.9009  2.11  
8  0.6058  0.9315  0.23  0.9186  5.54  0.9009  6.29  
10  0.5792  0.9192  0.27  0.9187  6.37  0.9029  15.00 
Conclusions
In this paper, we presented a novel deterministic sequential Monte Carlo (i.e., particle filtering) algorithm for solving the haplotype assembly problem. ParticleHap sequentially infers the haplotype sequence, one SNP site at a time, by exhaustively searching for the most likely extension of the partially assembled haplotype in each step, examining both the possible genotypes and phase. We tested the performance of ParticleHap on 1000 Genomes Project data, showing that it achieves better minimum error correction scores and phases more heterozygous sites than two of the most accurate existing methods while being significantly more computationally efficient. The results of testing ParticleHap on the simulated dataset also demonstrate that the proposed method can reconstruct haplotypes with higher accuracy and efficiency than those of competing techniques over a wide range of the haplotype assembly problem parameters.
The main goal of ParticleHap is accurate haplotype assembly rather than genotype calling. However, methods that can improve accuracy of genotype calling can be incorporated in the proposed algorithm. For example, the prior information about allele and genotype frequencies or linkage disequilibrium patterns can be incorporated in the proposed algorithm, which may further improve the accuracy of genotype calling and thus of haplotype assembly. In conclusion, the proposed method provides a framework for joint genotyping and haplotyping that leads to accurate haplotype assembly.
Declarations
Acknowledgements
This work was funded in part by the National Science Foundation CCF1320273.
Authors’ Affiliations
References
 Nielsen R, Paul JS, Albrechtsen A, Song YS. Genotype and snp calling from nextgeneration sequencing data. Nat Rev Genet. 2011; 12(6):443–51.View ArticlePubMedPubMed CentralGoogle Scholar
 1000 Genomes Project Consortium, Abecasis GR, Altshuler D, Auton A, Brooks LD, Durbin RM, et al.A map of human genome variation from populationscale sequencing. Nature. 2010; 467(7319):1061–73.View ArticleGoogle Scholar
 Hoehe MR, Köpke K, Wendel B, Rohde K, Flachmeier C, Kidd KK, et al.Sequence variability and candidate gene analysis in complex disease: association of μ opioid receptor gene variation with substance dependence. Hum Mol Genet. 2000; 9(19):2895–908.View ArticlePubMedGoogle Scholar
 Gibbs RA, Belmont JW, Hardenbol P, Willis TD, Yu F, Yang H, et al.The international hapmap project. Nature. 2003; 426(6968):789–96.View ArticleGoogle Scholar
 Schwartz R, et al.Theory and algorithms for the haplotype assembly problem. Commun Inf Syst. 2010; 10(1):23–38.Google Scholar
 Lancia G, Bafna V, Istrail S, Lippert R, Schwartz R. Snps problems, complexity, and algorithms. Lecture Notes Comput Sci. 2001; 2161:182–93.View ArticleGoogle Scholar
 Lippert R, Schwartz R, Lancia G, Istrail S. Algorithmic strategies for the single nucleotide polymorphism haplotype assembly problem. Brief Bioinform. 2002; 3(1):23–31.View ArticlePubMedGoogle Scholar
 Cilibrasi R, van Iersel L, Kelk S, Tromp J. On the complexity of several haplotyping problems. Algorithms Bioinformatics. 2005; 3692:128–39.Google Scholar
 Panconesi A, Sozio M. Fast hare: A fast heuristic for single individual snp haplotype reconstruction. Algorithms Bioinformatics. 2004; 3240:266–77.View ArticleGoogle Scholar
 Levy S, Sutton G, Ng PC, Feuk L, Halpern AL, Walenz BP, et al. The diploid genome sequence of an individual human. PLoS Biol. 2007; 5(10):254.View ArticleGoogle Scholar
 Chen Z, Fu B, Schweller R, Yang B, Zhao Z, Zhu B. Linear time probabilistic algorithms for the singular haplotype reconstruction problem from snp fragments. J Comput Biol. 2008; 15(5):535–46.View ArticlePubMedGoogle Scholar
 Zhao YY, Wu LY, Zhang JH, Wang RS, Zhang XS. Haplotype assembly from aligned weighted snp fragments. Comput Biol Chem. 2005; 29(4):281–7.View ArticlePubMedGoogle Scholar
 Wang Y, Feng E, Wang R. A clustering algorithm based on two distance functions for mec model. Comput Biol Chem. 2007; 31(2):148–50.View ArticlePubMedGoogle Scholar
 Wang RS, Wu LY, Li ZP, Zhang XS. Haplotype reconstruction from snp fragments by minimum error correction. Bioinformatics. 2005; 21(10):2456–462.View ArticlePubMedGoogle Scholar
 Geraci F. A comparison of several algorithms for the single individual snp haplotyping reconstruction problem. Bioinformatics. 2010; 26(18):2217–225.View ArticlePubMedPubMed CentralGoogle Scholar
 Bansal V, Halpern AL, Axelrod N, Bafna V. An mcmc algorithm for haplotype assembly from wholegenome sequence data. Genome Res. 2008; 18(8):1336–46.View ArticlePubMedPubMed CentralGoogle Scholar
 Bansal V, Bafna V. Hapcut: an efficient and accurate algorithm for the haplotype assembly problem. Bioinformatics. 2008; 24(16):153–9.View ArticleGoogle Scholar
 Duitama J, Huebsch T, McEwen G, Suk EK, Hoehe MR. Refhap: A reliable and fast algorithm for single individual haplotyping. In: Proceedings of the First ACM International Conference on Bioinformatics and Computational Biology, BCB ’10. New York, NY, USA: ACM: 2010. p. 160–9.Google Scholar
 He D, Choi A, Pipatsrisawat K, Darwiche A, Eskin E. Optimal algorithms for haplotype assembly from wholegenome sequence data. Bioinformatics. 2010; 26(12):183–90.View ArticleGoogle Scholar
 Chen ZZ, Deng F, Wang L. Exact algorithms for haplotype assembly from wholegenome sequence data. Bioinformatics. 2013; 29(16):349.View ArticleGoogle Scholar
 Deng F, Cui W, Wang L. A highly accurate heuristic algorithm for the haplotype assembly problem. BMC Genomics. 2013; 14(Suppl 2):2.View ArticleGoogle Scholar
 Xie M, Wang J, Jiang T. A fast and accurate algorithm for single individual haplotyping. BMC Syst Biol. 2012; 6(Suppl 2):8.View ArticleGoogle Scholar
 Bayzid MS, Alam MM, Mueen A, Rahman MS. Hmec: A heuristic algorithm for individual haplotyping with minimum error correction. ISRN Bioinformatics. 2013; 2013:10.View ArticleGoogle Scholar
 Aguiar D, Istrail S. Hapcompass: a fast cycle basis algorithm for accurate haplotype assembly of sequence data. J Comput Biol. 2012; 19(6):577–90.View ArticlePubMedPubMed CentralGoogle Scholar
 Aguiar D, Istrail S. Haplotype assembly in polyploid genomes and identical by descent shared tracts. Bioinformatics. 2013; 29(13):352–60.View ArticleGoogle Scholar
 Li LM, Kim JH, Waterman MS. Haplotype reconstruction from snp alignment. J Comput Biol. 2004; 11(23):505–16.View ArticlePubMedGoogle Scholar
 Kim JH, Waterman MS, Li LM. Diploid genome reconstruction of ciona intestinalis and comparative analysis with ciona savignyi. Genome Res. 2007; 17(7):1101–10.View ArticlePubMedPubMed CentralGoogle Scholar
 Matsumoto H, Kiryu H. Mixsih: a mixture model for single individual haplotyping. BMC Genomics. 2013; 14(Suppl 2):5.View ArticleGoogle Scholar
 Arulampalam MS, Maskell S, Gordon N, Clapp T. A tutorial on particle filters for online nonlinear/nongaussian bayesian tracking. Signal Process IEEE Trans. 2002; 50(2):174–88.View ArticleGoogle Scholar
 Fearnhead P. Sequential monte carlo methods in filter theory. Ph. D. thesis, University of Oxford. 1998.Google Scholar
 Punskaya E. Sequential monte carlo methods for digital communications. Ph. D. thesis, University of Cambridge. 2003.Google Scholar
 Liang KC, Wang X, Anastassiou D. A profilebased deterministic sequential monte carlo algorithm for motif discovery. Bioinformatics. 2008; 24(1):46–55.View ArticlePubMedGoogle Scholar
 Liang KC, Wang X. A deterministic sequential monte carlo method for haplotype inference. Selected Topics Signal Process IEEE J. 2008; 2(3):322–31.View ArticleGoogle Scholar
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