 Methodology article
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nQuire: a statistical framework for ploidy estimation using next generation sequencing
BMC Bioinformatics volume 19, Article number: 122 (2018)
Abstract
Background
Intraspecific variation in ploidy occurs in a wide range of species including pathogenic and nonpathogenic eukaryotes such as yeasts and oomycetes. Ploidy can be inferred indirectly  without measuring DNA content  from experiments using nextgeneration sequencing (NGS). We present nQuire, a statistical framework that distinguishes between diploids, triploids and tetraploids using NGS. The commandline tool models the distribution of base frequencies at variable sites using a Gaussian Mixture Model, and uses maximum likelihood to select the most plausible ploidy model. nQuire handles large genomes at high coverage efficiently and uses standard input file formats.
Results
We demonstrate the utility of nQuire analyzing individual samples of the pathogenic oomycete Phytophthora infestans and the Baker’s yeast Saccharomyces cerevisiae. Using these organisms we show the dependence between reliability of the ploidy assignment and sequencing depth. Additionally, we employ normalized maximized log likelihoods generated by nQuire to ascertain ploidy level in a population of samples with ploidy heterogeneity. Using these normalized values we cluster samples in three dimensions using multivariate Gaussian mixtures. The cluster assignments retrieved from a S. cerevisiae population recovered the true ploidy level in over 96% of samples. Finally, we show that nQuire can be used regionally to identify chromosomal aneuploidies.
Conclusions
nQuire provides a statistical framework to study organisms with intraspecific variation in ploidy. nQuire is likely to be useful in epidemiological studies of pathogens, artificial selection experiments, and for historical or ancient samples where intact nuclei are not preserved. It is implemented as a standalone Linux command line tool in the C programming language and is available at https://github.com/clwgg/nQuireunder the MIT license.
Background
Polyploidy, the presence of more than two complete sets of chromosomes, can under certain circumstances accelerate evolutionary adaptation by influencing the generation and maintenance of genetic diversity [1, 2]. In addition, polyploidy also poses short and longterm challenges to organismal fitness, which are associated with increased nuclear and cellular volume, propensity to aneuploidy, and disruption of gene expression regulation [3]. Interspecific comparisons between eukaryotic genomes can identify ancient polyploidization events. In contrast, more recent polyploidization events result in intraspecific variation of ploidy and, in some cases, aneuploidy. The presence of individuals of different ploidy in a population can hinder mating. Therefore, intraspecific ploidy variation tends to occur  although not exclusively  in organisms that have the capacity to reproduce asexually [4–6], are selfcompatible or perennial [7].
Although ploidy traditionally has been investigated by measuring DNA content using flow cytometry [8], it can also be inferred from next generation sequencing (NGS) data either by examining kmer distributions, or by assessing the distribution of allele frequencies at biallelic single nucleotide polymorphisms (SNPs) [4]. This methodology has been used to estimate ploidy in newly assembled genomes in order to identify the number of likely collapsed haplotypes on a percontig basis [9], as well as to detect intraspecific variation of ploidy in the oomycete Phytophthora infestans [4, 6] and in the Baker’s yeast Saccharomyces cerevisiae [5]. It also was successfully used for ploidy estimation in P. infestans historic herbaria samples that are not suitable for flow cytometry, since they do not contain intact nuclei [4]. The method assumes that alleles present at biallelic SNPs occur at different ratios for different ploidy levels, that is, 0.5/0.5 in diploids, 0.33/0.67 in triploids, and a mixture of 0.25/0.75 and 0.5/0.5 in tetraploids. These ratios hold true for recent autopolyploids, as well as recent allopolyploids from heterozygous source genomes. To determine the ploidy level, the distribution of biallelic SNPs can be inspected visually [10], and/or qualitatively compared with simulated data [4]. However, this methodology does not provide summary statistics that permit quantifying how well the data fit the expected distributions, which is especially critical when dealing with noisy distributions typical for highlyrepetitive genomes. An additional disadvantage of this approach is that it is preceded by the identification of variable sites (“SNP calling”), which is carried out using methodologies that benefit from a previously known ploidy level [11]. In a further development Gompert et al. [12] used biallelic SNPs in a Bayesian statistical approach to distinguish between ploidy levels from genotypingbysequencing data. This method was primarily developed for resequencing studies, where typically multiple individuals from populations with ploidy variation are genotyped, as it benefits from preexisting knowledge about the ploidy levels that may be observed. Based on the posterior probabilities emitted by the Bayesian model, this approach separates samples into ploidy clusters, using dimensionality reduction methods such as PCA. Since it allows the inclusion of training data of known ploidy, test samples can be assigned a ploidy level if they belong to a cluster that includes samples of known ploidy.
Here we present a statistical model that aims to distinguish between the distribution of base frequencies at variable sites for diploids, triploids and tetraploids, directly from read mappings to a reference genome. It models base frequencies as a Gaussian Mixture Model (GMM), and uses maximum likelihood to assess empirical data under the assumptions of diploidy, triploidy and tetraploidy. We evaluated the performance of our method for different sequencing coverages using published genomes of S. cerevisiae [5], and highcoverage genomes of P. infestans produced for this study.
Methods
Model and Implementation
We used base frequencies at variable sites with only two bases segregating to distinguish between diploids, triploids and tetraploids (Fig. 1a). For that, we implemented a GMM that models the base frequency profiles as a mixture of three Gaussian distributions (Fig. 1b), which are scaled relative to each other. A loglikelihood can be calculated following:
Here, n describes the numbers of data points and x_{ i } describes the value of each data point (i.e. the base frequencies). At sites with two bases segregating, we use the frequencies of both bases to achieve symmetry in the frequency distribution for triploids and tetraploids. μ_{ j } and σ_{ j } are the parameters of the j^{th} of three Gaussian distributions N_{ j } that are scaled relative to each other through the parameter α_{ j }, with \(\sum _{j=1}^{3}\alpha _{j} = 1\).
This model allows for estimating the parameters of the Gaussian mixture components, as well as their mixture proportions, by maximizing the loglikelihood, either with or without constraints on the possible parameter space.
The likelihood maximization of the GMM is implemented through an ExpectationMaximization (EM) algorithm (Fig. 1b).
During the EM, we make use of the latent variables Z_{ i }. They represent the assignment of a data point to one of the mixture components. In the Estep, we use the current estimates for μ_{ j }, σ_{ j } and α_{ j } to calculate:
This probability is calculated for all x_{ i } and j∈{1,2,3} to form a n×3 matrix, the columns of which represent the probability of each data point belonging to either of the three mixture components. From this matrix we can calculate the column sum \(S_{j} = \sum _{i=1}^{n}\gamma _{Z_{i}}(\,j)\), which represents the size of each mixture component. In the following Mstep, we update our estimates of μ_{ j }, σ_{ j } and α_{ j }:
The loglikelihood is calculated after the Mstep, and the next Estep is initiated unless the loglikelihood has changed by less then ε=0.01 from the previous Mstep.
As shown above, the algorithm allows the estimation of μ_{ j }, σ_{ j } and α_{ j } simultaneously. Henceforth, we call this setup the “free model”. The loglikelihood of the free model upon convergence represents the optimum under the assumptions of the model.
We can also set certain parameters to fixed values and forgo their update in the Mstep. We use this to maximize the loglikelihood under the expectation of diploidy (one Gaussian with mean 0.5), triploidy (two Gaussians with means 0.33 and 0.67) and tetraploidy (three Gaussians with means 0.25, 0.5 and 0.75), and call these the three “fixed models”:
In these three models, we only estimate σ_{ j }, while μ_{ j } and α_{ j } are fixed as shown above. Since all fixed models are nested within the free model, it is possible to directly compute the loglikelihood ratios, following:
The ΔlogLs describe the distance between each fixed model and the best fit under the assumptions of the GMM. A substantially lower ΔlogL of one fixed model over the others supports the ploidy level described by this fixed model (Fig. 1c). Therefore, we used ΔlogL as summary statistics where the minimum value supports a given ploidy level.
Additionally, the GMM can be extended to a Gaussian Mixture Model with Uniform noise component (GMMU), by adding a uniform mixture component:
The constraint on the mixture proportions then becomes \(\sum _{j=1}^{4}\alpha _{j} = 1\).
The uniform noise component is used in our implementation to allow baseline noise removal (Additional file 1: Figure S1). This can be useful when the Gaussian peaks are observable but embedded in a basal noise, which could be caused by highly repetitive genomes or low coverage.
Multivariate Gaussian clustering
To cluster samples in three dimensions based on the normalized maximized loglikelihoods of the three fixed models, we used the mclust5 package [13] of the R programming language [14]. This package utilizes mixtures of multivariate Gaussian distributions to detect clusters in an arbitrary number of dimensions. mclust5 allows to set constraints on the volume, shape and orientation of each mixture component, by varying features of their covariance matrix either within each sample, or for all samples at once. For the analysis displayed in Fig. 4, we used clusters of equal volume, but varying shape and orientation. This configuration represented the data the best, as assessed by the recovery of ploidy levels from cluster assignments.
Phytophthora infestans libraries
The two benchmarking libraries from P. infestans were generated according to the protocol by Meyer and Kircher [15] from DNA extracted from lab cultures [16]. These libraries were sequenced to high coverage on an Illumina HiSeq 3000 machine in paired end 150 bp mode. This sequencing data is available at the European Nucleotide Archive (ENA) under study number PRJEB20998.
Results
Performance
nQuire directly processes BAM files [17] and is designed to be efficient in memory usage and runtime. To process a 1GB S. cerevisiae BAM file (100x coverage), nQuire needs 70 s to build appropriate data structures, 6 s to run the models and calculate the maximum likelihood estimates, and uses a maximum of 8 Mb of RAM, whereas for processing a 10GB P. infestans BAM file (100x coverage) it needs 760 s, 100 s and 60 Mb of RAM, respectively. These benchmarks were performed on a single core of a Intel®; Core™ i54670 CPU on a system with 16Gb of DDR31600 RAM and an SSD.
Analysis of individual samples
We evaluate nQuire’s performance using three S. cerevisiae samples at 100x coverage, which represent each of the three ploidy levels evaluated by the model, as well as two P. infestans samples, one diploid and one triploid, at 210x and 368x coverage, respectively. The ΔlogL of each of the fixed models to the free model at full coverage is shown in Table 1. At those coverages, the ΔlogL of the best model is more than two times closer to the free model than the second best. Additionally, it coincides in all samples with the ploidy level inferred by visually inspecting the empirical distributions of base frequencies at full coverage (Figs. 2ac and 3ab).
Coverage dependence
To investigate the impact of coverage on the performance of the GMM, we downsampled mapped reads from the S. cerevisiae (Fig. 2gi) and P. infestans (Fig. 3ef) strains shown in Table 1 to different coverage levels. In all cases the ΔlogLs of the two improper models increases with increasing coverage. In contrast, the ΔlogL of the best model stabilizes at low coverage and doesn’t increase further as coverage increases. The coverage at which the ΔlogL of the best model is stable will be different for genomes of different size and complexity, as shown in the difference between the two organisms used for benchmarking.
Analysis of population samples
In cases where multiple samples are sequenced simultaneously, it might be impractical to assess ploidy in each sample individually. In these cases, we propose to use maximized loglikelihoods of the three fixed models, normalized by that of the free model, to cluster samples into ploidy groups. The rationale is that within one species, the relative likelihoods of the fixed models will be similar within each ploidy level. As a proof of concept, we applied this to all di, tri and tetraploids from the S. cerevisiae test set [5], and clustered the samples into three groups in three dimensions using multivariate Gaussian clustering (see “Methods” section). The sample set was manually scored for ploidy, and the overlap between clusters and manually assessed ploidy level was calculated (Fig. 4). Running nQuire on raw data showed high recovery of ploidy level (93%, Fig. 4a), which was further improved through our denoising implementation utilizing the GMMU (96%, Fig. 4b).
Detection of Aneuploidies
Recent polyploidization is often associated with aneuploidies. To be able to detect those, nQuire allows to split the analysis of a sample by regions defined in BED format. We used this to reanalyze the sample YJM466 from the S. cerevisiae test set [5]. This sample had been shown to be triploid on whole genome level, but tetraploid for chromosome 6 and diploid for chromosome 9. The ΔlogLs for the three fixed models individually calculated for each of the 16 chromosomes of S. cerevisiae confirmed this observation (Fig. 5).
A natural extension of analyzing the genome by chromosomes is to use sliding windows to detect possible transitions between ploidies in aneuploid individuals. We used the three S. cerevisiae datasets shown in Table 1 at their full coverage to benchmark the number of randomly sampled positions needed to accurately assign ploidy (Fig. 6). For these test samples, 100200 random sites at 100x coverage are enough to correctly assign ploidy based on the ΔlogL. However, this will vary for regions of the genome with different complexity and repetitiveness.
Discussion
In addition to nucleotide and structural variation, certain organisms can also vary intraspecifically in their ploidy level, which constitutes another source of variation that selection might act upon. Using NGS data our method permits assessment of ploidy variation from data that is usually generated for variant detection. In contrast to previous methods that visually analyze the distributions of SNPs at biallelic heterozygous sites [4, 6, 10], we quantitatively distinguish between different ploidy levels based on the distribution of base frequencies at variable sites, using relative differences in likelihoods. In comparison to the approach proposed by Gompert and Mock [12], nQuire avoids the requirement of high quality SNP calls. The higher level of noise in the data resulting from this is accounted for by using Gaussian distributions. They approximate a binomial process, but are impacted less by the effects of high coverage outliers, which arise for example from misalignments of paralogous sequences (Additional file 1: Figure S2). Additionally, nQuire is a Linux command line tool that uses standard file formats as input and handles large genomes at high coverage efficiently.
In all test cases, triploids were the easiest to distinguish, most likely caused by the lack of probability density around 0.5 compared to the other two models. While diploids and tetraploids are more difficult to tease apart, our results on coverage dependence show that at sufficient coverage, the data fits the true model much better than either of the two alternatives (Fig. 2gi). For cases where the Gaussian peaks were largely overlapped by uniform noise, we extended our free model to include a uniform component, whose mixture proportion can be used  after likelihood maximization  for baseline removal. We show that this procedure improves the recovery of the true ploidy level when samples are clustered based on maximized likelihoods under the assumptions of the fixed models (Fig. 4b). We also show that few high quality positions are enough to estimate the correct ploidy level (Fig. 6). Such high quality positions can be selected by using stringent filtering criteria. Several filters are implemented in nQuire directly, thus no preprocessing of the BAM file (after duplicate removal) is necessary. They include minimum and maximum coverage, as well as mapping quality and base frequency filters. The default values of these filters have been configured to fit most applications. The exact coverage and number of positions needed for a reliable estimation of ploidy will however depend on the complexity and repetitiveness of the genome. Additionally, it is possible to obtain high quality positions by using BED files to define regions of low repetitiveness, where base frequencies can be more confidently assessed.
Conclusion
We present nQuire, a statistical approach to distinguish diploids, triploids and tetraploids of recent evolutionary origin based on the distribution of base frequencies at variable sites. The method facilitates analysis of ploidy in single samples, and we demonstrate how to apply it to population scale data, when available. nQuire can also interact with BED files, to limit the analysis to certain sequence features, or divide it by regions of the genome, for example to detect aneuploidies. Our approach will be useful to assess intraspecific variation in ploidy from both historic and modern samples, as well as in experimental evolution experiments.
Abbreviations
 BAM:

Binary alignment map
 BED:

Browser extensible data
 CPU:

Central processing unit
 EM:

Expectation maximization
 ENA:

European nucleotide archive
 GMM:

Gaussian mixture model
 GMMU:

Gaussian mixture model with uniform noise component
 NGS:

Next generation sequencing
 RAM:

Random access memory
 SNP:

Single nucleotide polymorphism
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Acknowledgments
We thank Michael Dannemann, Richard Neher, Thomas Mailund, Oliver Kohlbacher, Moises ExpositoAlonso, Kay Pruefer, and members of the Research Group for Ancient Genomics and Evolution (AGE) for useful discussions and input on model implementation; Patricia Lang, the AGE group and Michael Dannemann for comments on the manuscript.
Funding
This work was supported by the Presidential Innovation Fund of the Max Planck Society.
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CLW, SK and HAB conceived the study. MP, LMC and CLW generated sequencing data. CLW developed the software application and analyzed the data. CLW and HAB wrote the the manuscript with input from all authors. All authors read and approved the final manuscript.;
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Availability of data and materials
The Phytophthora infestans sequencing data generated for this study is available at the European Nucleotide Archive (ENA) under study number PRJEB20998. The nQuire source code is available at https://github.com/clwgg/nQuireunder the MIT license.
Additional file
Additional file 1
Supplementary figures addressing the effect of denoising, as well as the effect of high coverage outliers on the likelihood of Gaussian and binomial mixtures. (PDF 144 kb)
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Weiß, C.L., Pais, M., Cano, L.M. et al. nQuire: a statistical framework for ploidy estimation using next generation sequencing. BMC Bioinformatics 19, 122 (2018). https://doi.org/10.1186/s128590182128z
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Keywords
 Ploidy
 Probabilistic modeling
 Next generation sequencing