Statistics capturing the population genetic signature of introgression

We set out to assemble a set of statistics that could be used in concert to reliably determine whether a given genomic window had experienced recent gene flow. Several statistics that have been designed to this end ask whether there is a pair of samples exhibiting a lower than expected degree of sequence divergence within the window of interest. The most basic of these is d_min, the minimum pairwise divergence across all cross-population comparisons (S1 Fig; [50]). The reasoning behind d_min is that even if only a single sampled individual contains an introgressed haplotype, d_min should be lower than expected and the introgression event may be detectable. A related statistic is G_min, which is equal to d_min/d_xy [51]; the presence of this term in the denominator is meant to control for variation in the neutral mutation rate across the genome. RND_min accomplishes this by dividing d_min by the average divergence of all sequences from either species to an outgroup sequence [52]. The name of this statistic is derived from its constituent parts, d_min, and RND [75].

As described in the following section, we incorporated a number of previously devised statistics into our classification approach, including some of those based on d_min. We also included some novel statistics that we designed to have improved sensitivity to particularly recent introgression. The first of these is defined as: where π₁ is nucleotide diversity [49] in population 1. Similarly, d_d2 = d_min/π₂. d_d1 and d_d2 statistics are so named because they compare d_min to diversity within populations 1 and 2, respectively. The rationale behind these statistics is that, if there has been recent introgression from population 1 into population 2, and at least one sampled chromosome from population 2 contains the introgressed haplotype, then the cross-population pair of individuals yielding the value of d_min should both trace their ancestry to population 1. Thus, the sequence divergence between these two individuals should on average be equal to π₁. Similarly, if there was introgression in the reverse direction d_min would be on the order of π₂. Following similar rationale, we devised two related statistics: d_d-Rank1 and d_d-Rank2. d_d-Rank1 is the percentile ranking of d_min among all pairwise divergences within population 1; the value of this statistic should be lower when there has been introgression from population 1 into population 2. d_d-Rank2 is the analogous statistic for introgression from population 2 into population 1. We also included a statistic comparing average linkage disequilibrium within populations to average LD within the global population (i.e. lumping all individuals from both species together), as follows: where Z_nS1, and Z_nS2 measure average LD [76] between all pairs of variants within the window in population 1 and population 2, respectively, and Z_nSG which measures LD within the global population. The reasoning behind this statistic is based on the assumption that, in the presence of gene flow, LD may be elevated within the recipient population(s) but not in the global population. S2 Fig shows that the distributions of these statistics do indeed differ substantially between genealogies with and without introgression (simulation scenarios described below), especially when this introgression occurred recently. In addition to these and other statistics summarizing diversity across the two population samples, we also incorporated several single-population statistics into our classifier (see below), as these may also contain information about recent introgression. For example, separate measures of nucleotide diversity in our two population samples would contain useful information because introgression is expected to increase diversity in the recipient population, especially if the source population was large or if the two populations split long ago.

Description of FILET classifier

We used a supervised machine learning approach to assign a genomic window to one of three distinct classes on the basis of a “feature vector” consisting of a number of statistics summarizing patterns of variation within the window from two closely related populations. These three classes are: introgression from population 1 into population 2, introgression from population 2 into population 1, and the absence of introgression. Specifically, we used an Extra-Trees classifier [53], which is an extension of random forests [77], an ensemble learning technique that creates a large ensemble of semi-randomly generated binary decision trees [78], before taking a vote among these decision trees in order to decide which class label should be assigned to a given data instance (i.e. genomic window in our case). In an Extra-Trees classifier, the tree building process is even more randomized than in typical random forests: in addition to selecting a random subset of features when generating a tree, the separating threshold for each feature is randomly chosen, rather than selected the threshold that optimally separates the data classes. We require example regions for each class in order to train the Extra-Trees classifier, so we used coalescent simulations to generate these training examples (described below). Our ultimate goal was to detect introgression within 10 kb windows in Drosophila, so to train our classifier properly we simulated chromosomal regions approximating this length (simulation details are given below). The target window size could easily be altered by changing the length of the regions simulated for training (i.e. by adjusting the recombination and mutation rates, θ and ρ).

FILET's feature vector contains a number of single-population summaries of per-base pair genetic variation: π, the variance in pairwise distances among individuals, the density of segregating sites, the density of polymorphisms private to the population, Fay and Wu's H and θ_H statistics [79], and Tajima's D [80]. The feature vector also includes two single-population summary statistics that are not normalized per base pair: Z_nS (which is averaged across all pairs of SNPs), and the number of distinct haplotypes observed in the window. Each feature vector included values of these 9 statistics for each population, yielding 18 single-population statistics in total. In addition, the two-population statistics included in FILET's feature vector were as follows: F_ST (following ref. [81]), Hudson's S_nn [82], per-bp d_xy, per-bp d_min, G_min, d_d1, d_d2, d_d-Rank1, d_d-Rank2, Z_X, IBS_MaxB (the length of the maximum stretch of identity by state, or IBS, among all pairwise between-population comparisons), and IBS_Mean1 and IBS_Mean2 which capture the average IBS tract length when comparing all pairs of sequences within populations 1 and 2, respectively. These IBS statistics are calculated by examining all pairs of individual sequences within a population (or across populations in the case of IBS_MaxB), noting the positions of differences, and examining the distribution of lengths between these positions (as well as between the first position and the beginning of the window and between the last position and the end of the window). Note that we did not include RND_min or other measures such as Patterson’s D and F₄ statistics [83] that require alignment to one or more additional species along with the focal pair, because we designed FILET so that it would not require outgroup information. We note however that through its use of supervised machine learning, FILET could easily be extended to incorporate such data. Instead, in order to improve robustness to mutational variation, we adopted the approach of drawing the mutation rate from a wide range of values when generating training examples to train FILET to classify data from our Drosophila samples (see below). All code necessary to run the FILET classifier (including calculating summary statistics on both simulated and real data sets, training, and classification) along with the full results of our application to D. simulans and D. sechellia (described below) are available at https://github.com/kern-lab/FILET/.

Simulated test scenarios

Following Rosenzweig et al. [52], we used the coalescent simulator msmove (https://github.com/geneva/msmove) to simulate data for testing FILET’s power to detect introgression in populations with four different values of T_D (the time since divergence): 0.25×4N, 1×4N, 4×4N, and 16×4N generations ago, where N is the population size. For each of these simulations the population size was held constant (i.e. the ancestral population size equals that of both daughter populations). We developed a classifier for each of these scenarios of population divergence. Supervised machine learning techniques such as the Extra-Trees classifier require training data consisting of examples from each of the three classes, but in practice a large number of example loci known to have experienced introgression may not be available. We therefore simulated training data sets for each of the four values of T_D. Again following Rosenzweig et al. [52], the relevant parameters for each of these simulations include: T_M, the time since the introgression event, which we drew from {0.01×T_D, 0.05×T_D, 0.1×T_D, 0.15×T_D, …, 0.9×T_D} (i.e. multiples of 0.05×T_D up to 0.9, and also including 0.01×T_D); and P_M, the probability that a given lineage would migrate from the source population to the sink population during the introgression event, which we drew from {0.05, 0.1, 0.15, …, 0.95}. We simulated an equal number of training examples for each combination of these two parameter values for both directions of gene flow, yielding 10⁴ simulations in total for both of these classes, conditioning that each of these instances must have contained at least one migrant lineage. Finally, we simulated an equivalent number of samples without introgression, yielding a balanced training set (10⁴ examples for each class).

Next, we computed feature vectors as described above for each of these training examples, and proceeded with training our Extra-Trees classifiers by conducting a grid search of all training parameters precisely as described in Schrider and Kern [54], and setting the number of trees in the resulting ensemble to 100. All training and classification with the Extra-Trees classifier was performed using the scikit-learn Python library (http://scikit-learn.org; [84]). We also calculated feature importance and rankings thereof by training an Extra-Trees classifier of 500 decision trees on the same training data (using scikit-learn’s defaults for all other learning parameters), and then using this classifier’s “feature_importances_” attribute. Briefly, this feature importance score is the average reduction in Gini impurity contributed by a feature across all trees in the forest, always weighted by the probability of any given data instance reaching the feature’s node as estimated on the training data [85]; this measure thus captures both how well a feature separates data into different classes and how often the feature is given the opportunity to split (i.e. how often it is visited in the forest). The values of these scores are then normalized across all features such that they sum to one.

For each T_D, we evaluated the appropriate classifier against a larger set of 10⁴ simulations generated for each parameter combination along a grid of values of T_M and P_M. The values of P_M were drawn from the same set as those in training as described above, while one additional possible value of T_M was included: 0.001×T_D. Also note that for these simulations we did not require at least one migrant lineage as we had done for training (information that is recorded by msmove). For test simulations with bidirectional migration, we did not require each replicate sample to contain at least one migrant lineage, though we modified msmove to record which samples did in fact experience migration. In addition to test examples for each direction of gene flow, we simulated 10⁴ examples where no migration occurred in order to assess false positive rates. In order to examine the performance of FILET when an unsampled ghost lineage was the source of introgression, we generated test simulations with the same values of T_D, T_M, and P_M as above, but where the source of the introgression was a third, unsampled population that separated from the two sampled populations at time T_D. In all of our simulations, both for training and testing, we set locus-wide population mutation and recombination rates θ and ρ to 50 and 250, respectively, similar to autosomal values in 10 kb windows in D. melanogaster [86] and sampled 15 individuals from each population. We also experimented with different window sizes, simulating training and test data (1,000 replicates for each class for each set) with window sizes corresponding to 1 kb, 10 kb, 5 kb, and 50 kb by multiplying θ and ρ by the appropriate scalar. When testing the sensitivity of our method on these data, we considered a window to be introgressed if FILET’s posterior probability of the no-introgression class was <0.05, except for the scenario with T_D equal to 16×4N generations ago in which case we used a posterior probability cutoff of 0.01, as we found that this step mitigated the elevated false positive rate under this scenario (reducing the rate from >10% to the estimate of 6% shown in S3 Fig). In windows labeled as introgressed, the direction of gene flow was determined by asking which of the two introgression classes had a higher posterior probability. Note that we used the same demographic scenario for both the training and test data for each T_D, and discuss the implications of demographic model misspecification in the Results and Discussion.

In order to compute receiver operator characteristic (ROC) curves we constructed balanced binary training sets composed of 10⁴ examples with no introgression, and 10⁴ examples allowing for introgression (with equal representation to each combination of T_M, P_M, and direction of introgression. The score that we obtained for each test example in order to compute the ROC curve was one minus the posterior probability of no introgression as generated by the Extra-Trees classifier (i.e. the classifier’s estimated probability of introgression, regardless of directionality).

Comparison to ChromoPainter

We compared FILET’s accuracy to that of ChromoPainter [46], a software program that utilizes a variant of the copying model first proposed by Li and Stephens [87]. In this model each sample haplotype is a mosaic composed of chromosomal segments chosen from a set of possible ancestral haplotypes, allowing for differences caused by mutation and the potential for changes in ancestry at recombination breakpoints. Such an approach can thus be used to predict the ancestry of each individual at each polymorphism—these predictions are referred to as “paintings” by ChromoPainter. To this end we repeated our simulations above but increased the size of the chromosomal segments to 1 Mb by increasing θ and ρ to 5000 and 25000. In these simulations only gene flow from population 2 to population 1 was allowed, and we modified msmove to record the coordinates of introgressed segments, and to restrict introgression events to those involving segments falling entirely within the central 100 kb of the chromosome. For each combination of T_M and P_M we generated 10 replicate simulations, including 10 replicates without introgression.

We ran ChromoPainter with the following parameters: the “-a 0 0” switch to model each individual haplotype as a mosaic of each other individual rather than using a set of predefined reference haplotypes, “-i 10” and “-ip” options to estimate copying proportions over 10 Expectation-Maximization (EM) iterations, and the default “-s 10” switch to stochastically draw 10 chromosome paintings for each individual from the HMM following EM. We then used the output from ChromoPainter to predict introgressed chromosomal segments as follows:

For each polymorphism, we examine each haplotype i among our n haplotypes, and record which of the other n-1 haplotypes serves as the best ancestor for i in each of our 10 paintings. We then examine each individual in population 2 (the recipient population), and count the number of paintings for which the ancestral haplotype is from population 1. If this number is > 5 (i.e. a majority) for any of our individuals in population 2, then we consider this focal polymorphism to be introgressed. If two adjacent polymorphisms are predicted to be introgressed, all sites between them are also considered to be introgressed. If only 1 polymorphism is predicted, then just that one site is considered introgressed. We also produced a more stringent version of these predictions by only retaining introgressed segments consisting of at least 25 consecutive introgressed polymorphisms. Note that ChromoPainter requires base pair positions, and msmove uses an infinite sites model where polymorphisms are located in a continuous space between zero and one. Thus in order to perform this analysis we had to map msmove’s continuous locations to physical locations, which we accomplished by multiplying by 10⁶ and rounding to the nearest available position.

We compared ChromoPainter to a sliding-window application of FILET’s classifier for 10 kb windows (with 1 kb step sizes). We also produced finer-scale FILET predictions using a 1 kb classifier (with 100 bp step sizes) to refine predictions made by the 10 kb classifier: only sliding windows predicted as introgressed by the 1 kb classifier and lying within introgressed segments predicted by the 10 kb classifier were retained as candidates by this version. For the refinement step, FILET’s posterior probability cutoff for introgression was relaxed to 0.5 (i.e. introgression more probable than not); a more lenient cutoff is appropriate here because this classifier was only applied within regions already predicted to be introgressed by the 10 kb classifier.

Drosophila sechellia collection

Drosophila sechellia flies were collected in the islands of Praslin, La Digue, Marianne and Mahé with nets over fresh Morinda fruit on the ground. All flies were collected in January of 2012. Flies were aspirated from the nets by mouth (1135A Aspirator–BioQuip; Rancho Domingo, CA) and transferred to empty glass vials with wet paper balls (to provide humidity) where they remained for a period of up to three hours. Flies were then lightly anesthetized using FlyNap (Carolina Biological Supply Company, Burlington, NC) and sorted by sex. Females from the melanogaster species subgroup were individualized in plastic vials with instant potato food (Carolina Biologicals, Burlington, NC) supplemented with banana. Propionic acid and a pupation substrate (Kimwipes Delicate Tasks, Irving TX) were added to each vial. Females were allowed to produce progeny and imported using USDA permit P526P-15-02964. The identity of the species was established by looking at the taxonomical traits of the males produced from isofemale lines (genital arches, number of sex combs) and the female mating choice (i.e., whether they chose D. simulans or D. sechellia in two-male mating trials).

Sequence data and variant calling and phasing

We obtained sequence data from 20 D. simulans inbred lines [74] from NCBI’s Short Read Archive (BioProject number PRJNA215932). We also sequenced wild-caught outbred D. sechellia females (see above) from Praslin (n = 7 diploid genomes), La Digue (n = 7), Marianne (n = 2), and Mahé (n = 7). These new D. sechellia genomes are available on the Short Read Archive (BioProject number PRJNA395473). For each line we then mapped all reads with bwa 0.7.15 using the BWA-MEM algorithm [88] to the March 2012 release of the D. simulans assembly produced by Hu et al. [89] and also used the accompanying annotation based on mapped FlyBase release 5.33 gene models [90]. Next, we removed duplicate fragments using Picard (https://github.com/broadinstitute/picard), before using GATK’s HaplotypeCaller (version 3.7; [91–93]) in discovery mode with a minimum Phred-scaled variant call quality threshold (-stand_call_conf) of 30. We then used this set of high-quality variants to perform base quality recalibration (with GATK’s BaseRecalibrator program), before again using the HaplotypeCaller in discovery mode on the recalibrated alignments. For this second iteration of variant calling we used the—emitRefConfidence GVCF option in order to obtain confidence scores for each site in the genome, whether polymorphic or invariant. Finally, we used GATK’s GenotypeGVCFs program to synthesize variant calls and confidences across all individuals and produce genotype calls for each site by setting the—includeNonVariantSites flag, before inferring the most probable haplotypic phase using SHAPEIT v2.r837 [94]. The genotyping and phasing steps were performed separately for our D. simulans and D. sechellia data, and for each of step in the pipeline outlined above we used default parameters unless otherwise noted. In order to remove potentially erroneous genotypes (at either polymorphic or invariant sites), we considered genotypes as missing data if they had a quality score lower than 20, or were heterozygous in D. simulans. After throwing out low-confidence genotypes, we masked all sites in the genome missing genotypes for more than 10% of individuals in either species’ population sample, as well as those lying within repetitive elements as predicted by RepeatMasker (http://www.repeatmasker.org). Only SNP calls were included in our downstream analyses (i.e. indels of any size were ignored). These phased and masked data are available at https://zenodo.org/record/1166021.

Demographic inference

Having obtained genotype data for our two population samples, we used ∂a∂i [95] to model their shared demographic history on the basis of the folded joint site frequency spectrum (downsampled to n = 18 and n = 12 in D. simulans and D. sechellia, respectively); using the folded spectrum allowed us to circumvent the step of producing whole genome alignments to outgroup species in D. simulans coordinate space in order to attempt to infer ancestral states. We used an isolation-with-migration (IM) model that allowed for continual exponential population size change in each daughter population following the split. This model includes parameters for the ancestral population size (N_anc), the initial and final population sizes for D. simulans (N_{sim_0} and N_sim, respectively), the initial and final sizes for D. sechellia (N_{sech_0} and N_sech, respectively), the time of the population split (T_D), the rate of migration from D. simulans to D. sechellia (m_sim→sech), and the rate of migration from D. sechellia to D. simulans (m_sech→sim). We also fit our data to a pure isolation model that was identical to our IM model but with m_sim→sech and m_sech→sim fixed at zero. Finally, we fit our data to an admixture model identical to the isolation model but with the addition of two parameters: the time of a pulse admixture event from D. simulans into D. sechellia (T_AD) and the proportion of individuals in D. sechellia migrating from D. simulans during this event (P_AD). Our optimization procedure consisted of an initial optimization step using the Augmented Lagrangian Particle Swarm Optimizer [96], followed by a second step of optimization refining the initial model using the Sequential Least Squares Programming algorithm [97], both of which are included in the pyOpt package for optimization in Python (version 1.2.0; [98]) as in Schrider et al. [99]. We performed ten optimization runs fitting both of these models to our data, each starting from a random initial parameterization, and assessed the fit of each optimization run using the AIC score. Code for performing these optimizations can be obtained from https://github.com/kern-lab/miscDadiScripts, wherein 2popIM.py, 2popIsolation.py, 2popIsolation_admixture.py fit the IM, isolation, and admixture models described above, respectively. For scaling times by years rather than numbers of generations, we assumed a generation time of 15 gen/year as has been estimated in D. melanogaster [100].

Training FILET to detect introgression between D. simulans and D. sechellia

Having obtained a demographic model that provided an adequate fit to our data, we set out to simulate training examples under this demographic history for each of our three classes (i.e. no migration, migration from D. simulans to D. sechellia, and from D. sechellia to D. simulans). For training examples including introgression, T_M was drawn uniformly from the range between zero generations ago and T_D/4, while P_M raged uniformly from (0, 1.0]. In addition, in order to make our classifier robust to uncertainty in other parameters in our model, for each training example we drew values of each of the remaining parameters from [x−(x/2), x+(x/2)], where x is our point estimate of the parameter from ∂a∂i. In addition to the parameters from our demographic model (T_D, ρ, N_anc, N_sim, and N_sech), these include the population mutation rate θ = 4Nμ (where μ was set to 3.5×10⁻⁹), and the ratio of θ to the population recombination rate ρ (which we set to 0.2). Continuous migration rates were set to zero (i.e. the only migration events that occurred were those governed by the T_M and P_M parameters, and the no-migration examples were truly free of migrants). In total, this training set comprised of 10⁴ examples from each of our three classes.

As described above, we masked genomic positions having too many low confidence genotypes or lying within repetitive elements (described above) before proceeding with our classification pipeline. While doing so, we recorded which sites were masked within each 10 kb window in the genome that we would later attempt to classify. In order to ensure that our masking procedure affected our simulated training data in the same manner as our real data, for each simulated 10 kb window we randomly selected a corresponding window from our real dataset and masked the same sites in the simulated window that had been masked in the real one. We then trained our classifier in the same manner as described above.

In order to ensure that this classifier would indeed be able to reliably uncover loci experiencing recent gene flow between our two populations, we assessed its performance on simulated test data. First, we applied the classifier to test examples simulated under this same model (again, 10⁴ for each class). Next, to address the effect of demographic model misspecification, we applied our classifier to an isolation model with a different parameterization and no continuous size change in the daughter populations. For this model we simply set N_sim and N_sech to π_sim/4μ and π_sech/4μ, respectively, where π for a species is the average nucleotide diversity among all windows included in our analysis after filtering, and μ was again set to 3.5×10⁻⁹. We then set N_anc to be equal to N_sim, and set T to d_xy/(2μ) − 2N_anc generations where d_xy is the average divergence between D. simulans and D. sechellia sequences across all windows. This latter value is simply the expected TMRCA for cross-species pairs of genomes, minus the expected waiting time until coalescence during the one-population (i.e. ancestral) phase of the model. This simple model thus produces samples with similar levels of nucleotide diversity for the two daughter populations as those produced under our IM model, but that would differ in other respects (e.g. the joint site frequency spectrum and linkage disequilibrium, which would be affected by continuous population size change after the split). For both test sets we masked sites in the same manner as for our training data before running FILET.

Classifying genomic windows with FILET

We examined 10 kb windows in the D. simulans and D. sechellia genomes, summarizing diversity in the joint sample with the same feature vector as used for classification (see above), ignoring sites that were masked as described above. We omitted from this analysis any window for which >25% of sites were masked, and then applied our classifier to each remaining window, calculating posterior class membership probabilities for each class. We then used a simple clustering algorithm to combine adjacent windows showing evidence of introgression into contiguous introgressed elements: we obtained all stretches of consecutive windows with >90% probability of introgression as predicted by FILET (i.e. the probability of no-introgression class <10%), and retained as putatively introgressed regions those that contained at least one window with >95% probability of introgression. In order to test for enrichment of these introgressed regions for genic/intergenic sequence or particular Gene Ontology (GO) terms from the FlyBase 5.33 annotation release [90], we performed a permutation test in which we randomly assigned a new location for each cluster or introgressed windows (ensuring the entire permuted cluster landed within accessible windows of the genome according to our data filtering criteria). We generated 10,000 of these permutations.

FILET detects introgressed loci with high sensitivity and specificity

We sought to devise a bioinformatic approach capable of leveraging population genomic data from two related population samples to uncover introgressed loci with high sensitivity and specificity. In the Materials and Methods, we describe several previous and novel statistics designed to this end. However, rather than preoccupying ourselves with the search for the ideal statistic for this task, we took the alternative approach of assembling a classifier leveraging many statistics that would in concert have greater power to discriminate between introgressed and non-introgressed loci. Supervised machine learning methods have proved highly effective at making inferences in high-dimensional datasets and are beginning to make inroads in population genetics [101]. In this vein, we designed FILET, which uses an extension of random forests called an Extra-Trees classifier [53]. We previously succeeded in applying Extra-Trees classifiers for a separate population genetic task—finding recent positive selection and discriminating between hard and soft sweeps [54, 55]—though other methods such as support vector machines [102] or deep learning [103] could also be applied to this task.

Briefly, FILET assigns a given genomic window to one of three distinct classes—recent introgression from population 1 into population 2, introgression from population 2 into 1, or the absence of introgression—on the basis of a vector of summary statistics that contain information about the two-population sample’s history. This feature vector contains a variety of statistics summarizing patterns of diversity within each population sample, as well as a number of statistics examining cross-population variation (see Materials and Methods for a full description). FILET must be trained to distinguish among these three classes, which we accomplish by supplying 10,000 simulated example genomic windows of each class, with each example represented by its feature vector. Because we expect that the majority of introgression events to be non-adaptive, these simulations did not include natural selection. Once this training is complete, FILET can then be used to infer the class membership of additional genomic windows, whether from simulated or real data.

We began by assessing FILET’s power on a number of simulated datasets, examining windows roughly equivalent to 10 kb in length in Drosophila (Materials and Methods). In particular, because the power to detect introgression depends on the time since their divergence, T_D, we measured FILET’s performance under four different values of T_D, training a separate classifier for each. In Fig 1 (T_D = 0.25×4N) and S3 Fig (T_D values of 1, 4, and 16×4N), we compare FILET’s power to that of two related statistics that have been devised to detect introgressed windows, d_min and G_min (Materials and Methods). These figures show that FILET has high sensitivity to introgression across a much wider range of introgression timings (T_M) and intensities (P_M) than either of these statistics under each value of T_D, and that this disparity is amplified dramatically for smaller values of T_D. Furthermore, these figures demonstrate that FILET infers the correct directionality of recent introgression with high accuracy, whereas d_min and G_min contain no information about the direction of gene flow. Finally, FILET does not appear especially sensitive when the source of gene flow is an unsampled ghost population rather than one of the two sequenced populations (S4 Fig), though it could potentially be trained to detect such cases if desired.

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Fig 1. Heatmaps showing several methods’ sensitivity to detect introgression.

We show the fraction of simulated genomic regions with introgression occurring under various combinations of migration times (T_M, shown as a fraction of the population divergence time T_D) and intensities (P_M, the probability that a given lineage will be included in the introgression event) that are detected successfully by each method. (A) Accuracy of d_min and G_min statistics, where a simulated region is classified as introgressed if the values of these statistics are found in the lower 5% tail of the distribution under complete isolation (from simulations). Thus, the false positive rate is fixed at 5%. (B) The accuracy of FILET on these same simulations. On the left we show the fraction of regions correctly classified as introgressed (compare to panel A). On the right, we show the fraction of all simulated regions that are not only classified as introgressed, but also for which the direction of gene flow was correctly inferred (i.e. if the direction is inferred with 100% accuracy for a given cell in the heatmap, the color shade of that cell will be identical to that in the heatmap on the left). The false positive rate, as determined from applying FILET to a simulated test set with no migration, is also shown.

https://doi.org/10.1371/journal.pgen.1007341.g001

We also note that for d_min and G_min we established 95% significance thresholds from our simulated training data without introgression, thereby achieving a false positive rate of 5%. In order to assess FILET’s false positive rate, we classified a set of test simulations without introgression and found that FILET’s false positive rate was considerably lower (Fig 1 and S3 Fig) except for our largest value of T_D, where it was initially higher (0.4% for T_D = 0.25×4N but >10% for T_D = 16×4N), despite our selection of a posterior probability cutoff of 95% (Methods). This illustrates an important issue with posterior probability estimates produced by supervised machine learning methods: they may occasionally be miscalibrated. We therefore adjusted the cutoff for the T_D = 16×4N scenario (to 99% probability of introgression) which lowered our false positive rate to 6% as shown in S3 Fig. Thus, when an appropriate posterior probability cutoff is chosen—a task that can be performed in a straightforward manner by testing on simulated data—FILET achieves much greater sensitivity to introgression than d_min and G_min often at a much lower false positive rate. We also demonstrate the FILET’s greater power than these statistics via ROC curves (S5 Fig), where it outperforms each statistic under each scenario. Specifically, the difference in power between FILET and d_min is dramatic for smaller values of T_D (area under curve, or AUC, of 0.85 versus 0.73 when T_D = 0.25×4N for FILET and d_min, respectively) but comparatively miniscule for our largest T_D (AUC of 0.94 versus 0.93 when T_D = 16×4N). It therefore appears that FILET’s performance gain relative to single statistics is highest for the more difficult task of finding introgression between very recently diverged populations, while for the easier case of detecting introgression between highly diverged populations some single statistics may perform nearly as well.

We also experimented with smaller training sets, finding similar classification power (measured by AUC) as above when we trained FILET using only 1000 or even 100 simulated examples per class (S6 Fig), though in the latter case estimated class posterior probabilities were far less accurate. In addition, we examined the effect of altering the target window size used when training and testing FILET (S7 Fig).

A comparison of the power and resolution of FILET and ChromoPainter

Methods designed to uncover changes in ancestry along a recombining chromosome within admixed populations can also be used to recover introgressed regions. To this end we used ChromoPainter [46] which has the advantage of not requiring a set of “reference haplotypes” known to be free of introgression from each population, and can instead predict for each haplotype, which of all the other haplotypes in the sample (from either population) is most closely related. We simulated two-population samples for 1 Mb chromosomes where introgression from population 2 to population 1 was allowed in the central 100 kb window, and used ChromoPainter to identify introgressed loci (see Methods). We then ran FILET on these simulations, this time using a sliding-window approach to detect introgressed segments (Methods).

Fig 2 shows that FILET has substantially higher sensitivity than ChromoPainter—summing across the entire parameter space (including many scenarios where introgression is quite difficult to detect) FILET recovered 27.7% of introgressed base pairs compared to 19.4% for ChromoPainter—while having a roughly 20-fold lower false positive rate (0.42% for FILET versus 9.31% for ChromoPainter). For scenarios with more ancient and less intense introgression, we did observe somewhat higher sensitivity in ChromoPainter’s predictions. However, this seems to be driven largely by ChromoPainter’s propensity to identify a larger fraction of base pairs as introgressed regardless of their true ancestry, as evidenced by its higher false positive rate. To demonstrate this further we show for the positive predictive value (the number of base pairs correctly predicted to be introgressed divided by the total number of base pairs predicted to be introgressed) for each method in S8 Fig. This figure shows that FILET’s positive predictive is consistently far higher than ChromoPainter’s. We sought to improve this by adopting a more stringent threshold for ChromoPainter’s predictions, requiring at least 25 adjacent polymorphisms to be called introgressed in order to retain the candidate region. This approach did succeed at reducing the false positive rate to 1.15%, though this is still substantially higher than FILET’s, but this improvement came at the cost of ChromoPainter’s sensitivity, which was reduced to 8.6%, roughly one-third that of FILET (Fig 2 and S8 Fig). We also tried an intermediate threshold (5 polymorphisms), but did not observe a substantial increase in specificity compared to our initial more lenient approach (8.49% false positive rate). Thus, while we cannot rule out that it may be possible to devise a method to leverage ChromoPainter’s model to predict introgression that exceeds the performance of our application of ChromoPainter here, our results suggest that it is unlikely that such an approach would eclipse the performance of FILET. We note that ChromoPainter does have the advantage of not requiring simulated training data. ChromoPainter also has the potential to identify donor and recipient haplotypes, which FILET currently does not, but the far greater power of FILET demonstrated above will make it preferable to many researchers who are interested in identifying introgressed regions. Moreover our above results imply that predictions of the span and origin of introgressed haplotypes made directly from ChromoPainter’s output may not always be particularly reliable.

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Fig 2. A comparison of the power and resolution of FILET and ChromoPainter using simulations of a 1 Mb chromosome where introgression was allowed within the central 100 kb region.

As in Fig 1, the population split time was set to N generations ago, and the darkness of the heatmap shows sensitivity to introgression. Unlike Fig 1, here we are measuring sensitivity at the level of the individual base pair rather than evaluating the question of whether a window at large was recovered as containing introgressed alleles. The “coarse” version of FILET refers to a FILET classifier trained to detect introgression in 10 kb windows, which was applied to sliding windows (1 kb step size) across the chromosome. The “fine” version of FILET applied a classifier trained on 1 kb windows to sliding windows (100 bp step size) within those regions classified as introgressed by the FILET classifier. The lenient version of ChromoPainter required evidence of introgression at a single SNP to identify introgression, while the stringent version required candidate regions to contain at least 25 consecutive SNPs supporting introgression.

https://doi.org/10.1371/journal.pgen.1007341.g002

It is important to note that in the above simulations many introgressed regions will be considerably smaller than our 10 kb window size. This fact, combined with our use of accuracy measurements counting the number of individual base pairs correctly classified, makes the results presented above useful for evaluating FILET’s resolution and the impact of window size on our predictions. By these measures FILET outperforms ChromoPainter, which does not use windows and is only limited in scale by the density of polymorphisms. This suggests that when using sliding windows FILET is able to achieve adequate resolution regardless of its use of a predefined window size. Nonetheless we sought to improve our resolution further by using a finer-scale FILET classifier trained on 1 kb windows as described above to refine the location of putatively introgressed regions identified by the 10 kb classifier (see Methods). While this did marginally reduce our false positive rates and increase our positive predictive values (see Fig 2 and S8 Fig), sensitivity was also somewhat reduced (to 25.7%; Fig 2). The relatively minor effect of adding this refinement step reinforces the notion that a predefined window size is not a major hindrance to our methods’ effectiveness. Thus for most applications a window size that yields enough polymorphisms to reliably calculate the statistics included in our feature vectors may suffice.

Overall, FILET detects introgressed regions with greater power and resolution than ChromoPainter, a method designed to detect ancestry tracks along recombining chromosomes. However we note that many methods of this class exist, and it is possible that some may achieve greater accuracy in some circumstances (e.g. if reference haplotype panels are used).

Sensitivity to continuous gene flow

While FILET is designed for identifying particular genomic windows that experienced introgression as a result of a pulse migration event, genomic windows with genealogies that include introgression may of course also be produced by continuous migration, with the timing of geneflow varying from window to window. We therefore simulated genomic windows experiencing a variety of bidirectional migration rates under each of our values of T_D and recorded the fraction of windows for which our sampled individuals contained at least one migrant lineage. Next, for each simulated window we applied the FILET classifier trained under the appropriate divergence time as described above, recording the fraction of windows with at least one migrant lineage that were classified as introgressed by FILET. The results of these tests (S1 Table) show that for each value of T_D, the lowest bidirectional migration rates that we tested do not produce migrant lineages, while higher rates will produce a small to modest fraction migrants, most of which are undetected (e.g. when m = 0.01, 23% and 59% of windows contain at least one migrant when T_D = 0.25 and T_D = 1, respectively, but <5% are detected by FILET). Thus, FILET, as currently trained, may not be sensitive to gene flow produced by low levels of continuous migration. However, as the migration rate increases further, more and more of these migrant windows will be detected (e.g. when m = 1, 70% and 100% of windows are detected as migrants by FILET when T_D = 0.25 and T_D = 1, respectively).

Ranking the importance of statistics for detecting introgression

Although our goal was to use a set of statistics to perform more accurate inference than possible using individual ones, another benefit of our Extra-Trees approach is that it also allows for a natural way to evaluate the extent to which different statistics are informative under different scenarios of introgression. To this end, we used the Extra-Trees classifier to calculate feature importance, which captures the ability of each statistic to separate the data into its respective classes (Materials and Methods). We find that for our lowest T_D (a split N generations ago) the top four features, all with similar importance, are the density of private alleles in population 1, the density of private alleles in population 2, d_d-Rank1, and d_d-Rank2. For our next-lowest T_D (4N generations ago), the top four, again with similar importance score estimates, are F_ST, Z_X, d_d1, and d_d2. Thus our newly devised d_d and Z_X statistics seem to be particularly informative in the case of recent introgression between closely related populations. For the larger values of T_D, d_xy and d_min rise to prominence. The complete lists of feature importance for each T_D are shown in S2 Table.

The exceptional accuracy with which FILET uncovers introgressed loci underscores the potential for machine learning methods to yield more powerful population genetic inferences than can be achieved via more conventional tools which are often based on a single statistic. Indeed, machine learning tools have been successfully leveraged in efforts to detect recent positive selection [54, 104–107], to infer demographic histories [108], or even to perform both of these tasks concurrently [109].

Joint demographic history of D. simulans and D. sechellia

As described in the Materials and Methods, we used publically available D. simulans sequence data [74], and collected and sequenced a set of D. sechellia genomes. We mapped reads from these genomes to the D. simulans assembly [89], obtaining high coverage >28× for each sequence (see sampling locations, mapping statistics, and Short Read Archive identifier information listed in S3 Table). We do not expect that our reliance on the D. simulans assembly resulted in any appreciable bias, as reads from our D. sechellia genomes were successfully mapped to the reference genome at nearly the same rate as reads from D. simulans (S3 Table).

After completing variant calling and phasing (Materials and Methods), we performed principal components analysis on intergenic SNPs at least 5 kb away from the nearest gene in order to mitigate the bias introduced by linked selection [99, 110, 111]. While this is unlikely to completely eliminate the confounding effect of linked selection in Drosophila, the fraction of mutations that are deleterious is far greater in coding regions than in intergenic regions (~90% versus <50% according to [112]); thus it is reasonable to presume that the impact of linked selection is reduced several kilobases away from genes [113]. We observed evidence of population structure within D. sechellia. In particular, the samples obtained from Praslin clustered together, while all remaining samples formed a separate cluster (S9A Fig). Running fastStructure [114] on this same set of SNPs yielded similar results: when the number of subpopulations, K, was set to 2 (the optimal value for K selected by fastStructure’s chooseK.py script), our data were again subdivided into Praslin and non-Praslin clusters. Indeed, across all values of K between 2 and 8, fastStructure’s results were suggestive of marked subdivision between Praslin and non-Praslin samples, and comparatively little subdivision within the non-Praslin data (S9B Fig). This surprising result differs qualitatively from previous observations from smaller numbers of loci [71, 115], and underscores the importance of using data from many loci—preferably intergenic and genome-wide—in order to infer the presence or absence of population structure.

Next, we examined the site frequency spectra of the Praslin and non-Praslin clusters, noting that both had an excess of intermediate frequency alleles in comparison to that of the D. simulans dataset (S10 Fig), in line with our expectations of D. sechellia’s demographic history. We also note that the Praslin samples contained far more variation (50,243 sites were polymorphic within Praslin) than non-Praslin samples (4,108 SNPs within these samples). This difference in levels of variation may reflect a much lesser degree of population structure and/or inbreeding on the island of Praslin than across the other islands, or may result from other demographic processes. Additional samples from across the Seychelles would be required to address this question. In any case, in light of this observation we limited our downstream analyses of D. sechellia sequences to those from Praslin.

Because we required a model from which to simulate training data for FILET, we next inferred a joint demographic history of our population samples using ∂a∂i [95]. In particular, we fit three demographic models to our dataset: an isolation-with-migration (IM) model allowing for continuous population size change and migration following the population divergence, an isolation model with the same parameters but fixing migration rates at zero, and an isolation model with one burst of pulse migration from D. simulans into D. sechellia (Materials and Methods). In S4 Table we show our model optimization results, which show clear support for the IM model over the other models. The IM model that provided the best fit to our data (Fig 3A) includes a much larger population size in D simulans than D. sechellia (a final size of 9.3×10⁶ for D. simulans versus 2.6×10⁴ for D. sechellia), consistent with the much greater diversity levels in D. simulans [10, 71]. This model also exhibits a modest rate of migration, with a substantially higher rate of gene flow from D. simulans to D. sechellia (2×N_ancm = 0.086) than vice-versa (2×N_ancm = 0.013). Thus, the results of our demographic modeling are consistent with the observation of hybrid males in the Seychelles [73], and the possibility of recent introgression between these two species across a substantial fraction of the genome (see refs. [72, 116]).

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Fig 3. Inferred joint population history of D. simulans and D. sechellia, and power to detect introgression under this model.

(A) The parameterization of our best-fitting demographic model. Migration rates are shown by arrows, and are in units of 2×N_ancm, where m is the probability of migration per individual in the source population per generation. (B) Confusion matrix showing FILET’s classification accuracy under this model as assessed on an independent simulated test set. Perfect accuracy would be 100% along the entire diagonal from top-left to bottom-right, and the false positive rate is the sum of top-middle and top-right cells.

https://doi.org/10.1371/journal.pgen.1007341.g003

An interesting characteristic of the model shown in Fig 3A is that, assuming 15 generations per year, the estimated time of the D. simulans- D. sechellia population split is ~86 kya, or 1.3×10⁶ generations ago. This contrasts with a recent estimate of 2.5×10⁶ generations ago from Garrigan et al. [72] which was based on single genomes rather than population genomic data, but did account for ancestral polymorphism, as did estimates from Obbard et al. [117] which yielded even older split times. Supporting our inference, we note that our average intergenic cross-species divergence of 0.017 yields an average TMRCA of ~2.5×10⁶ generations ago, assuming a mutation rate of 3.5×10⁻⁹ mutations per generation as observed in D. melanogaster [112, 118], and this estimate would include the time before coalescence in the ancestral population. Unless the mutation rate the D. simulans species complex is substantially lower than in D. melanogaster, a population split time of 2.5×10⁶ generations ago therefore seems unlikely given that the ancestral population size (and therefore the period of time between the population divergence and average TMRCA) was probably large (>500,000 by our estimate). Thus, we conclude that the D. simulans and D. sechellia populations may have diverged more recently than previously appreciated, perhaps within the last 100,000 years.

Although the specific parameterization of our model should be regarded as a preliminary view of these species’ demographic history that is adequate for the purposes of training FILET, future efforts with larger sample sizes will be required to refine this model. That being said, the basic features of this model—a much larger D. simulans population size than sechellia, and a fairly large ancestral population size—are unlikely to change qualitatively.

Widespread introgression from D. simulans to D. sechellia

Application to population genomic data.

We applied FILET to 10,185 non-overlapping 10 kb windows that passed our data quality filters (101.85 Mb in total, or 86.7% of the five major chromosome arms; Materials and Methods). FILET classified 267 windows as introgressed with high-confidence, which we clustered into 94 contiguous regions accounting for 2.93% of the accessible portion of the genome (2.99 Mb in total; Materials and Methods). This finding is qualitatively similar to a previous estimate (4.6%) by Garrigan et al. [72] based on comparisons of single genomes from each species in the D. simulans complex. Unlike this previous effort, FILET is able to infer the directionality of introgression with high confidence (Fig 3B), and we find evidence that the majority of this introgression has been in the direction of D. simulans to D. sechellia: only 21 of the 267 (7.9%) putatively introgressed windows were classified as introgressed from sechellia to D. simulans. This finding is not a result of a detection bias, as we have greater power to detect gene flow from D. sechellia to D. simulans than in the reverse direction. Given that our D. simulans sequences are from the mainland, one interpretation of this result is that although there has been recent gene flow from D. simulans into the Seychelles, where D. simulans and D. sechellia occasionally hybridize, there does not appear to be an appreciable rate of back-migration to the mainland of D. simulans individuals harboring haplotypes donated from D. sechellia. On the other hand, D. sechellia alleles may often be purged from D. simulans by natural selection. This may be in part due to the reduced ecological niche size of D. sechellia, such that any alleles which may introgress into D. simulans and lead to preference for or resistance to Morinda fruit may prove deleterious in other environments. More generally, D. sechellia haplotypes introgressing into D. simulans may harbor more deleterious alleles due to their smaller population size, which will be more effectively purged in the larger D. simulans population if mutations are not fully recessive [28]. Tests of these hypotheses will have to wait for a population sample of genomes from D. simulans collected in the Seychelles.

We asked whether our candidate introgressed loci were enriched for particular GO terms using a permutation test (Materials and Methods), finding no such enrichment. We did observe a deficit in the number of genes either partially overlapping or contained entirely within introgressed regions in our true set versus the permuted set; although a paucity of introgressed genes would be consistent with introgressed functional sequence often being deleterious, this difference was not significant (297 vs. 373.2, respectively; P = 0.083; one-sided permutation test).

One notable introgressed region on 3R that FILET identified had been previously found by Garrigan et al. as containing a 15 kb region of introgression. We show that gene flow in this region actually extends for over 200 kb (Fig 4). When Brand et al. [119] sequenced the 15 kb region originally flagged by Garrigan et al. in a number of D. simulans and D. sechellia individuals, they also uncovered evidence of a selective sweep in D. sechellia originating from an adaptive introgression from D. simulans. Our data set also supports the presence of an adaptive introgression event at this locus: a 10 kb window lying within the putative sweep region (highlighted in Fig 4) is in the lower 5% tail of both d_min (consistent with introgression) and π_sech (consistent with a sweep in sechellia); this is the only window in the genome that is in the lower 5% tail for both of these statistics. This region contains two ionotropic glutamate receptors, CG3822 and Ir93a, which may be involved in chemosensing among other functions [120], and the latter of which appears to play a role in resistance to entomopathogenic fungi [121]. Also near the trough of variation within D. sechellia are several members of the Turandot gene family, which are involved in humoral stress responses to various stressors including heat, UV light, and bacterial infection [122, 123], and perhaps parasitoid attack as well [124]. On the other hand, Brand et al. [119] hypothesize that the target of selection may be a transcription factor binding hotspot between RpS30 and CG15696, and the phenotypic target of this sweep remains unclear.

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Fig 4. A large genomic region on 3R classified by FILET as introgressed from D. simulans to D. sechellia.

Values of the d_d-sim and d_min (upper two panels) within each 10 kb window in the region are shown, along with the posterior probability of introgression from FILET (i.e. 1 –P(no introgression)). Clustered regions classified as introgressed are shown as gray rectangles superimposed over these probabilities. Also shown are windowed values of π in D. sechellia, with the sweep region highlighted in red, and the locations of annotated genes with associated FlyBase identifiers [90].

https://doi.org/10.1371/journal.pgen.1007341.g004

Interestingly, this particular window is the only one in this region that is classified by FILET as having recent gene flow from D. sechellia to D. simulans. However this classification may be erroneous as one might expect FILET, which was not trained on any examples of adaptive introgression, to make an error in such a scenario because rather than gene flow increasing polymorphism in the recipient population, diversity is greatly diminished if the introgressed alleles rapidly sweep toward fixation. We note that this window is immediately flanked by a large number of windows classified as introgressed from D. simulans to D. sechellia and which show a large increase in diversity in the recipient population as expected. Moreover, Brand et al.’s phylogenetic analysis of introgression in this region also supported gene flow in this direction. Brand et al. also found evidence suggesting that the introgressed haplotype began sweeping to higher frequency in D. simulans (though it has not reached fixation in this species) prior to the timing of the introgression and subsequent sweep in D. sechellia. Thus we conclude that the adaptive allele probably did indeed originate in D. simulans before migrating to D. sechellia, and FILET’s apparent error in this case underscores the genealogical differences between adaptive gene flow and introgression events involving only neutral alleles.

Concluding remarks

Here we present a novel machine learning approach, FILET, that leverages population genomic data from two related populations in order to determine whether a given genomic window has experienced gene flow between these populations, and if so in which direction. We applied FILET to a set of D. simulans genomes as well as a new set of whole genome sequences from the closely related island endemic D. sechellia, confirming widespread introgression and also inferring that this introgression was largely in the direction of D. simulans to D. sechellia. Future work leveraging D. simulans data sampled from the Seychelles will be required to determine whether this asymmetry is a consequence of low rate of migration of D. simulans back to mainland Africa (where our D. simulans data were obtained), or whether the directionality of gene flow is biased on the islands themselves. In addition to creating FILET, we devised several new statistics, including the d_d statistics and Z_X which our feature rankings show to be quite useful for uncovering gene flow.

Despite the success of FILET on both simulated data sets and real data from Drosophila, there are several improvements that could be made. First, by framing the problem as one of parameter estimation (i.e. regression) rather than classification, we may be able to precisely infer the values of relevant parameters of introgression events (i.e. the time of the event and the number of migrant lineages). Deep learning methods, which naturally allow for both classification and regression, may prove particularly useful for this task [103]. Indeed, Sheehan and Song [109] used deep learning to infer demographic parameters (regression) while simultaneously identifying selective sweeps (classification). Another step we have not taken is to explicitly handle adaptive introgression, which could potentially greatly improve our approach’s power to detect such events.

While population genetic inference has traditionally relied on the design of a summary statistic sensitive to the evolutionary force of interest, a number of highly successful supervised machine learning methods have been put forth within the last few years [54, 104–109]. These methods are often thought of as black boxes, a characterization that may not always be fair [125]. Indeed in the context of evolutionary genetics such machine learning approaches are easily interpreted as we have strong generative models that guide our intuition. Nonetheless, classical statistical estimation from parametric models may often be more interpretable. Hybrid approaches combining machine learning techniques with Bayesian approaches to estimate posterior distributions of evolutionary parameters (e.g. [126]) thus represent an attractive alternative to either approach in their “pure” form. As genomic data sets continue to grow, we argue that machine learning approaches—in whatever shape they eventually take—leveraging high dimensional feature spaces have the potential to revolutionize evolutionary genomic inference.