Model overview

To investigate the impacts of selection on many linked sites, we developed Ancestry-HMM Multi-Locus-Selection (AHMM-MLS). AHMM-MLS is an extension of Ancestry_HMM [44], the latter of which infers both local ancestry and time since admixture for admixed populations by modeling local ancestry in a set of samples from the admixed population as a hidden Markov model (HMM) using a neutral single or multi-pulse admixture model [45]. Our framework considers only single pulse admixture demographic models, where admixture occurs t generations prior to sampling. The hidden states of the HMM are the local ancestries of the samples at ancestry informative positions—i.e. the positions sampled from along the genome in genotype or pileup data whose allele frequencies differ between the ancestral populations—and the observed states are the alleles in the aligned reads at these positions. The emission probabilities of the HMM are computed based on the read alignment data from admixed samples and the allele frequencies in reference unadmixed populations (see [44] for details). AHMM-MLS uses the same emission probabilities as in prior work [44].

Our method introduces a new technique to infer the expected transition probabilities between ancestry types at adjacent ancestry informative sites along the chromosome under generalized models of multi-locus selection during admixture in a single pulse model (see below). Because alleles near selected sites tend to hitchhike to the selected site, the tract lengths for contiguous regions of local ancestry tend to be longer around selected sites [46], and this effect can be captured in the transition probabilities between ancestry types. We compute the expected transition probabilities using a numerical method (see below) and then use the forward algorithm to compute model likelihoods given the samples [47], and a direct search algorithm to optimize the multi-locus selection model. We do not consider epistatic interactions among sites in this framework, but we consider their effects on selection inference (see Verification and Comparison). Our approach also does not include the effects of genetic drift and instead assumes the admixed population is infinite. Below we evaluated the impacts of population size on resulting inferences (see Verification and Comparison), but as an approximation we expect that drift will have small effects in large or moderate admixed populations (S1 Fig).

Generating transition probabilities

To calculate the effects of multiple selected sites on the expected ancestry transition probabilities between two adjacent ancestry informative sites, we use a numerical method to track the expected distribution of haplotypes after t generations of admixture, including the local ancestries of the selected sites and the two ancestry informative sites. If there are n selected sites, then we track n + 2 total positions (i.e., selected positions and ancestry informative sites), leading to 2ⁿ⁺² haplotypes. We do not assume the selected sites are sampled as ancestry informative markers in the admixed population and their positions may lie anywhere along the chromosome. The haplotype distribution in the admixed population at generation 0 is modeled by a row vector H⁰, which undergoes a transformation in each generation, producing a sequence of vectors H⁰…H^t, one for each generation. The vector H^g is transformed into the vector H^g+1, representing the expected change in the haplotype distribution from generation g to g + 1. As a convention for this work, H^g₀ is the frequency of the haplotype consisting of positions all originating from ancestral population 0, (i.e. the ancestral origin of the sites along the chromosome is 00…00), and H^g₁ is the frequency of the haplotype 00…01, and H^g₂ of haplotype 00…10, and so on, counting in a binary fashion. At generation 0, only two haplotypes, H⁰₀ and , have non-zero frequencies. That is because we assume a single admixture pulse which begins from unadmixed population founders t generations before the time of sampling. These are the two haplotypes where all sites are of the same ancestry, and their initial values are dictated by the admixture fraction m.

In each generation, we take the tensor product of H^g with itself to produce a row vector, D. This vector corresponds to the expected distribution of diploid individuals that result from the haplotypes assuming an infinite population, random mating, and no segregation distortion.

Now, to go from this diploid genotype distribution to the haplotype distribution of the next generation, we apply the matrix operation M to D^g, the result of which is H′^g+1. H′^g+1 is then normalized to produce H^g+1 (Fig 1).

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Fig 1. Single generation process for updating haplotype distribution vector.

At generation 0, the haplotype vector H⁰ is initialized to only have non admixed haplotypes, and the matrix M is computed. In each generation, the diploid genotype distribution D is computed from the haplotype distribution by assuming random mating. When the M operation is applied to D, it results in the unnormalized haplotype distribution of the next generation, which is then normalized.

https://doi.org/10.1371/journal.pgen.1011062.g001

We call the matrix M the diploid to haploid transformation, as it converts the diploid genotype distributions of one generation to the expected haplotype distribution of the next generation, accounting for the effects of recombination and natural selection within diploid individuals. A specific entry, such as M_i,j, is the contribution of the diploid genotype i on the haplotype j, taking into account the fitness of the diploid genotype i, and the probability that a recombination event produces the haplotype j. The matrix depends on the location of the two adjacent ancestry informative sites, the location of every selected site, and the fitness coefficients of those selected sites. Therefore, M needs to be recomputed for each pair of adjacent ancestry informative sites, and each multi-locus model, but remains constant for each generation of the forward computation. Each selected position is determined by its position on the chromosome in Morgans, and two fitness coefficients. The two fitness coefficients are the dominance coefficient, h, and the selection coefficient, s. This would make the relative selection coefficients 1, 1—hs, 1—s. To simplify calculations, we assume all ancestry informative sites are neutral sites where s = 0. Using this procedure we can compute transition rates for around 30,000 sampled sites in a matter of seconds, making this method ideal for genome-wide data.

To generate M, we iterate through all possible diploid genotypes. We assume that fitness values combine across selected sites multiplicatively. For a particular diploid genotype i, it has an associated fitness S_i, which we compute by taking the product of the relevant selection coefficients for each site. For this genotype i, we iterate through each region where a recombination event may occur. If there are n sites, then there are n + 1 regions. Each of these regions has a corresponding recombination rate r. Recombination in a specific region would produce two haplotypes, k and l. So for each recombination event of this genotype, there would be a contribution proportional to D_i*S_i*r to haplotypes k and l. This contribution is reflected in M by adding S_i*r to the M_i,k and M_i,l entries. M is computed after we have iterated through all possible diploid genotypes and added their contributions to each haploid that they may produce through meiosis. By reducing the most computationally expensive parts of the numerical procedure to matrix multiplications, we are able to use the optimized linear algebra library armadillo [48,49] to quickly compute the transition probabilities for each pair of adjacent ancestry informative sites.

After iterating for t generations, we are left with the haplotype distribution that is expected under our model of selection assuming an infinite admixed population size. We directly calculate the transition probabilities between the local ancestries of the ancestry informative sites by iterating through the haplotype distribution and recording the rates of the four possible ancestry state combinations for the two ancestry informative sites.

Model optimization

Once we have the expected transition probabilities for a particular model, we use the forward algorithm to compute the likelihood of this model given the read pile-up or genotype data [47]. We optimize the parameters of the model to maximize its likelihood using the Nelder-Mead search simplex optimization algorithm [50]. For our hyperparameter values we used a reflection constant of 1, a contraction constant of 0.5, an expansion constant of 2, and a shrinkage constant of 0.5. For each site optimized with unrestricted selection, we optimize the location of the site and two selection coefficients, h and s. If h is known and fixed (e.g., if additive h = 0.5), then only one selection parameter needs to be optimized.

For each optimization, an initial starting point for the location of the selected sites must be supplied by the user. Each optimization is done in two stages, where in the first stage the simplex is centered around the supplied starting point, while in the second stage the simplex is centered around the optimum of the first stage. Each stage consists of multiple starts, with different simplex sizes and orientations. If the range of log likelihoods for every point in the simplex falls below a certain threshold, or if four shrinkage transformations occur in a row, then that search is stopped and the optimum of the simplex is taken to be the optimum of that search. In the first stage, the search is stopped if the range falls below 5 and it falls below a quarter of the initial simplex range. In the second stage, the search is stopped if the range falls below 0.1 and it falls below one 20th of the initial simplex range.

An iterative multi-locus model selection procedure

We developed, tested, and implemented an iterative procedure for fitting a multi-locus selection model to genotype data from admixed populations (Fig 2). Our method begins by identifying a set of potential selected sites using Ancestry_HMM-S [43], and applying basic local-optimum selection approaches to remove trivially close positions that may correspond to a single selected allele. We expect that even basic local-ancestry outlier analysis [51,52] might be sufficient for generating a set of candidate positions, but we do not evaluate alternatives here. We then sort the list of candidate selected sites by decreasing likelihood ratio. In the first iteration, we test the site with the highest likelihood ratio as a single site selection model against a null model consisting of a neutral admixture with similar demographic parameters. In each subsequent iteration, we add one additional selected position, constructing an alternative model that we test against the model obtained from the previous iteration.

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Fig 2. An example of our model selection procedure with four candidate selected positions.

On the left, each panel shows the estimated local ancestry proportion within the admixed population (black). On the right, each panel shows the distribution of log likelihood ratios between the alternative and the null model on simulations of the null model (black), and on the underlying data (blue). In the first iteration (top), we test a single site selection model (blue) against a neutral null model (orange). We then test a two-site selection model (blue, second row) against the previous single site selection model which is now the null model (orange, second row). In the third iteration, we test a three-site selection model (blue, third row) against the two-site selection model from the second iteration (orange, third row). Finally, in testing a four-site selection model (blue, bottom), the likelihood ratio (blue, bottom right) does not exceed the 95th percentile of null simulations (black line, bottom right) of the previously selected three-site model (orange, bottom). The method terminates and accepts the three-site selection model obtained in the third iteration. In each panel, vertical gray lines indicate the positions of selected sites in the alternative model considered.

https://doi.org/10.1371/journal.pgen.1011062.g002

Our method makes simplifying assumptions about the admixed and ancestral populations that may make it unsuitable for common model selection methods such as the Bayesian information criterion or the Akaike information criterion (S2 Fig) [53,54]. Instead, to obtain the expected distribution of likelihood ratios under the null model, we perform simulations of populations with selected positions that match this model, and we simulate the sampling of reads from these populations. Simulations under the null model show that population size is a contributor to additional variation not captured by the theoretically expected distribution (S2 Fig), and we expect that other unmodelled components may also affect inferences. On these simulated reads expected from the null model, we fit both the null model, and an alternative model which includes the same sites as the null but additionally includes the next site in our sorted list of candidate selected positions, and we note the likelihood ratio between these two estimated models. Essentially, evidence for selection at each site is evaluated in the context of all previously estimated selected sites. When an alternative model exceeds the 95th percentile of the simulated null model likelihood ratio distribution, we accept that position and this new multi-locus selection model becomes the null model in the next iteration. If a given position does not exceed the significance threshold, we discard it and attempt the same procedure with the next candidate position against the same null model. The procedure terminates once we have either accepted or rejected every candidate selected site (Fig 2). We use this same technique of simulating populations to obtain a null distribution for all of our verifications and applications below.

Model validation with forward simulations

To evaluate our method over a large variety of plausible introgression scenarios, we performed forward simulations. In each of our simulations, a diploid population received a single admixture pulse from another population carrying at least one selected allele, producing a large admixed population (n = 10000, unless stated otherwise). As in prior work [43–45], we first used the coalescent simulation program MaCS [55] to create the genotype data for unadmixed individuals. We simulated the local ancestry along the genome of admixed samples using SELAM [56]. This procedure is described in detail in prior work [44,45]. For each population and selection scenario, we simulated 20 admixture events that aligned with the null model and 20 that aligned with the alternative model. We used GNU parallel to run many batches of simulations at once [57]. We sampled 75 diploid individuals from each of these simulations, except in the simulations of the D. melanogaster and P. italiae populations below, where we sample the same number of individuals from our simulation as we have sampled from the population, which were 44 and 31 respectively.

In evaluating performance with simulated data, we found that our method could accurately distinguish between a single selected site and two nearby selected sites over varying population parameters (Fig 3). We varied the time since admixture t from 100 to 1000 generations, the admixture proportion from 0.05 to 0.5, and the distance in Morgans between the selected sites from 0.005 to 0.05. We found that one and two site selection models are easier to distinguish as t increases, and as the distance between the sites increases. This is because the selected positions have more time to rise in allele frequency and be broken up into separate haplotypes. Our method could also accurately predict the location of the two nearby sites (S3 Fig), when the number of generations is high enough (t ⪆ 500). The accuracy of the predicted locations was not affected by the distance between the sites (S3 Fig). In most cases, our method performed better as the time since admixture increased, and as the strength of selection increased.

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Fig 3. Performance of our method in detecting two nearby selected sites.

We evaluated AHMM-MLS in its ability to distinguish between the presence of two nearby sites under selection and the presence of a single site under selection. We ran simulations with varying minor ancestry fractions (0.05, 0.1, 0.2, and 0.5 from left to right), times since admixture (100–1000 generations), and distances between selected positions (0.005, 0.01, 0.02 and 0.05 Morgans, from top to bottom). In each simulation, the selection coefficient of both sites was 0.01. There were a total of 64 different combinations of demographic and selection model parameters. We also ran null model simulations, where there was only a single site under selection introduced in the admixture event, to establish a null model distribution. The points on the lines indicate the proportion of two site simulations in which the single site null model was correctly rejected.

https://doi.org/10.1371/journal.pgen.1011062.g003

We found that optimizing in the correct model space (e.g., with the correct number of selected alleles) gave more accurate predictions of the selection coefficients and of the positions of the sites under selection. We tested the prediction of selection coefficients in simulations where we varied the selection coefficients of the selected sites s from 0.005 to 0.05, and the distance in Morgans between the selected sites from 0.005 and 0.05. In most simulated scenarios, the selection coefficient was considerably overestimated if only a single selected site was fit, as the effects of linkage between selected sites were not considered. This effect was most prominent when the two simulated sites were closer to each other and when their selection was strong (Fig 4).

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Fig 4. Comparing the inferred selection coefficients between single site and two site models.

We evaluated the estimated selection coefficients when using two different models to approach the same data. We ran simulations with varying selection coefficients (0.005, 0.01, and 0.05 from top to bottom) and distances between selected positions (0.005, 0.01, 0.02 and 0.05 Morgans, from left to right). On each simulation we fit a two-site model and a single site model. The dark blue and the light blue dots indicate the inferred selection coefficients of the two sites in the two-site model, while the yellow dots are the inferred selection coefficients of the single site model. The horizontal black lines indicate the simulated selection coefficients.

https://doi.org/10.1371/journal.pgen.1011062.g004

Our method could also distinguish between dominant selective pressure and additive selective pressure in certain admixture and selection scenarios (S4 Fig), although this was much more sensitive to the demographic parameters. We also obtained more accurate estimated selection coefficients in simulations including dominance when we model that site as having dominant, rather than additive, fitness effects (S5 Fig). For relatively recent admixture events (t = 100) and lower selection coefficients (s ≤ 0.02), this effect was most prominent, with the selection coefficients overestimated by nearly 100% when optimizing an additive model on simulations of a site under dominant selection. However, our method generally performs poorly in inferring dominance (S4 Fig), and we caution against strong interpretation of results obtained from this approach.

AHMM-MLS could accurately determine the number of linked sites under selection (Fig 5). In simulations, we varied the number of sites to be tested (3 to 4), the distance between the sites (1 to 2 centimorgans), the selection coefficients of the introduced sites (s = 0.005 to s = 0.01) and the number of generations since the admixture pulse (t = 100 to t = 1000). The simulated populations were the result of an introgression event where the minor population (m = 0.2) introduced multiple alleles spaced 1 to 2 centimorgans apart undergoing positive additive selection. For simulations where the time since the admixture pulse was around 500 generations or more, we could reliably estimate the correct number of sites. This implies that our method will be appropriate for many admixed populations including the population of sub-Saharan D. melanogaster, and the population of P. italiae that we consider below, but that caution is warranted for application to recently admixed populations with moderately weak selection.

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Fig 5. Performance of our method in detecting multiple nearby sites under selection.

We evaluated the ability of AHMM-MLS to distinguish three-site and four-site models from models with one fewer site. We ran simulations of admixed populations with an admixture proportion of 0.2, and with varying selection coefficients (0.005, and 0.01 from top to bottom), distances between selected positions (0.01 Morgans for the two columns on the left, and 0.02 Morgans for the two panels on the right), number of introgressed selected sites (three for the first and third columns, four for the second and fourth columns), and time since admixture (100–1000 generations). For the 32 different scenarios, we simulated both null and alternative models, where the null model had one fewer selected site. The points on the lines indicate the proportion of alternative model simulations in which the null model was correctly rejected.

https://doi.org/10.1371/journal.pgen.1011062.g005

Effect of small population size

For relatively moderate or long times since admixture (such as our default 500 generations) and with moderate selection coefficients (s = 0.005–0.05), our method starts to perform poorly when the effective population size of the admixed population drops below 2000 individuals (S1 Fig). We ran forward simulations of admixed populations with various numbers of individuals, and found that the ability to distinguish between two-site and single site models was severely limited once the number of individuals dropped below 2000 (S1A Fig). The accuracy of the inferred locations and selection coefficients of the two simulated sites were also hampered when the number of individuals was below 2000 (S1B and S1C Fig). This is because our method assumes an infinite population size, with no genetic drift, and when populations are too small and admixture is relatively ancient, this assumption will produce suboptimal results. This may be an issue when studying ancient human populations, as these populations tend to have a low effective population size, although some estimates for the effective population size of the Eurasian population that admixed with Neanderthals place it right at the edge of what our program can handle (Ne = 2,100) [58]. For larger admixed populations, such as e.g., some species of Heliconius butterflies (Ne = 792,000) [59], we expect the population size to be large enough for the effects to be negligible.

Likelihood approximation

To reduce the time taken to optimize each proposed model, we only calculate the effects of a selected site on the transition probabilities between two ancestry informative sites if those sites are less than a specified distance from the selected site (with a default of 2 centimorgans). When this distance is smaller than the distance between two sites, and those sites have a strong selection (e.g., selection coefficient of 0.05), then the selection coefficients for these sites can be overestimated (S6 Fig). We recommend increasing this distance when estimating models with strong selection. We also only calculate the transition probabilities between every k-th pair of adjacent ancestry informative sites, as an adjustable parameter (default of 4). We found that calculating the transition probabilities for only every k-th pair of adjacent ancestry informative sites does not affect the inference of selection coefficients when k is relatively small. If the local ancestry must be decoded for a given MLS model, then we calculate the effects of sites throughout the entire chromosome. This can be costly, as the size of the matrix M is exponential with respect to the number of sites on each tracked haplotype, so we don’t recommend doing this for models with more than 6 sites. We instead recommend that users split the model into pieces with sites that are decently separated, as we have done with chromosome 3R of D. melanogaster below.

Robustness to demographic model misspecification

We ran simulations of two nearby selected sites and, in different simulations, we misspecified both the time since admixture and the admixture fraction by factors of 0.5, 0.8, 1.2, and 2. As the true time since admixture increased, our method was generally more robust to time misspecification. When the true time since admixture was 500 generations or higher, misspecifying by a factor of 0.5 or 2 produced only small effects for the ability of AHMM_MLS to correctly support the alternative model (S7 Fig). The estimated strength of selection was affected when the true time since admixture was small (t = 100), with most estimated selection coefficients being off by more than 50% (S8 Fig). This effect decreased as the true time since admixture increases to 500 or 1000, where most estimated selection coefficients had an error less than 30%. When the true admixture fraction was relatively small (m ≤ 0.2), our method was not strongly impacted by misspecifications of the admixture fraction by a factor of 0.5 or 2 (S9 Fig).

Robustness to recombination map misspecification

Recombination occurring between linked selected sites allows us to discern their individual effects, and recombination around a selected site allows us to infer its location and strength. As such, a misspecified recombination map may confound our method’s ability to infer the number of selected positions and their selection parameters. To address these potential confounding effects, we ran simulations of two cases of recombination map misspecification.

In both of these cases we ran simulations of populations which admixed 500 generations ago, with an admixture proportion of 0.2, with a two-site or a single site selection model, and used our method to discriminate between these two scenarios. In the first case, we scaled the provided recombination rate in 100kb regions by varying amounts, to test the effect of a recombination map with high error rates. We found that our method was robust to these errors in simulations of two-site versus single site tests. Even when the error rates of the recombination maps were up to 75%, the two scenarios were accurately discriminated, and the location and selection coefficients were accurately inferred (S10 Fig). In the second case, we simulated cases where the recombination map had systematic correlated errors. We tested this by scaling the entire recombination map by a varying scalar, in simulations like those just described. We found that we could still accurately discriminate between the two cases, even with the recombination map scaled by up to 2 times the simulated rate (S11 Fig). The locations were accurately confirmed but the selection coefficients had some bias. When the map was scaled by 2x, the selection coefficients were inferred to be about 150% of their simulated values, and when the map was scaled by 0.5x, they were about 75% of their simulated values.

Recurrent migration and selection on a cline

Our method assumes that the admixed population is the result of a single admixture pulse happening at some point in the past, and that the resulting admixed population receives no subsequent migration after that pulse. These assumptions may be difficult to meet, so we explored the effects of applying our tools to two simulated cases that break these assumptions. In both cases, we tested our method’s ability to distinguish between the presence of two selected positions and a single selected position. In the first case, we simulated ancestry on a cline, where we sample the center of the cline at various times after initial hybridization (t = 100, 200, 500, 1000) (S12 Fig). The clines were simulated as 20 sub-populations arranged on a line, with half of the subpopulations originating from one ancestral population and the other half from another. These sub-populations had a migration rate of 5% per generation between adjacent sub-populations. We found that our method could detect the two selected positions as well as their locations once 500 generations had passed since initial hybridization. However, even after 500 generations, the selection coefficients were still generally underestimated. In the second case, we explored the effects of recurrent migration. We simulated populations with our standard demographic parameters, 500 generations since admixture and an admixture fraction of 0.2, with varying amounts of migration from the ancestral populations into the admixed population (S13 Fig). We found that migration rates of greater than 0.0005 per generation tended to confound the signal of two-locus selection, and biased the estimation of selection coefficients.

Effect of epistatic interactions between loci

Our method does not model epistatic interactions, which occur frequently in admixed populations. To investigate how our method reacts to epistatic interactions, we simulated simple cases of epistasis between two distant loci on a single chromosome. These populations were the result of an admixture event occurring 500 generations ago, with an admixture proportion of 0.5, and we simulated recessive epistasis with a strength of s = 0.05 and s = 0.1, and dominant epistasis with a strength of s = 0.01 and s = 0.05. We used our method to fit a model including both of these sites, and inferred their selection coefficients as well as their locations. Unsurprisingly, the selection coefficients were underestimated, as our method is treating these sites as independent selected positions (S14 Fig), but the locations were still accurate. Our method infers that one of the sites is positively selected, while the other one is negatively selected, and cannot discern the interaction between them.

Multi-locus selection in Drosophila melanogaster

In order to investigate the potential impacts of interference between linked sites on inferences of natural selection in real data, we applied our method to chromosome 3R of an admixed population of Drosophila melanogaster from South Africa. This population shows signals of admixture which have been noted in previous studies [44,45,60]. The admixture history is consistent with a one-pulse model, with admixture parameters that suggest that this population is suitable for our program [45]. In a previous study, we found evidence that chromosome 3R may contain multiple nearby selected alleles [43]. This study had found 13 putative sites under selection on 3R, most of which were fewer than 5 centimorgans away from another selected site. Here we fit models with multiple nearby selected positions, to determine whether and to what extent interference may have impacted our prior estimates of the number of selected sites and their selection coefficients. We used publicly available datasets of D. melanogaster collected from South Africa [61]. In a previous study, the data was prepared so that it could be analyzed by the AHMM programs [43]. This included removing the known large chromosomal inversions found on some of the chromosome arms [60,62,63]. We used a publicly available fine-scale recombination map of chromosome 3R [64]. However, we note that differences in the assumed map and the true recombination rates in this population may impact the accuracy of inferences we obtained here. We used the FlyBase sequence coordinates converter to convert assembly 5 base pair coordinates to assembly 6 [65].

Obtaining a demographic model

We first used AHMM [44] on chromosome arm 3L of this population of D. melanogaster, to estimate a demographic model for this population. This is the other arm on chromosome 3, opposing the arm 3R, and it showed no evidence for the presence of alleles that experienced strong positive selection during admixture [43]. We inferred the admixture fraction and admixture time, obtaining values (m = 0.138 and t = 466) similar to what we obtained in prior work, which inferred these parameters on 3R directly [45], but we note that here the admixture fraction is slightly lower and the number of generations since admixture is slightly higher. Both differences are consistent with the notion that the presence of selected positions along other chromosome arms may have slightly impacted our prior demographic modeling efforts [44]. We do not account for other forms of selection that may be present, such as background selection, but the demographic inference method is generally robust to a large range of weak selection effects [44]. This supplied an estimated demographic history that we used as a baseline in our models of selection.

Identifying candidate selected positions

We ran AHMM-S [43] to evaluate evidence for positive selection for cosmopolitan ancestry along chromosome arm 3R (Fig 6). This program evaluates a neutral model and optimizes an additive selection model and outputs the likelihoods for each. At each position, we recorded the log likelihood ratio between each model. To identify local optima, we performed a simple peak finding algorithm, where we recorded each site that had the highest maximum likelihood of the nearest 1,400 sampled sites and a log likelihood ratio above 15. This gave us 17 candidate selected positions to examine, which we ordered by decreasing likelihood ratio. We then applied the iterative procedure described above to produce a multi-locus selection model, fixing the position of each selected position and optimizing their selection coefficients under an additive selection model. When performing simulations under the null model to determine the expected likelihood ratio distributions, we used a population size of 10,000. This is probably smaller than the effective population size in natural D. melanogaster populations. Presumably, because smaller populations are more impacted by drift, we expect that this will be a conservative choice when evaluating evidence for positive selection in this natural population (S2 Fig). If the true population size is larger in reality, additional candidate positions might exceed the significance threshold, but the selection coefficients are not expected to change substantially.

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Fig 6. Candidate selected positions on chromosome 3R of D. melanogaster.

Chromosome 3R of D. melanogaster shows signs of many nearby selected sites that are in close linkage. The likelihood ratio outputs of AHMM-S, which test each site for additive selection using a single selected site model, indicate high variation in models of natural selection (blue). Using a simple peak finding algorithm, we identified 17 sites that may be experiencing selection in this admixed population (yellow dots).

https://doi.org/10.1371/journal.pgen.1011062.g006

After applying the procedure to iteratively construct a model of multiple selected sites for chromosome arm 3R, we identified 9 selected sites (Fig 7). We then fine-tuned the position of these sites, and then re optimized the selection coefficients to arrive at our final model (Table 1). Among the 9 sites included in the multi-locus model is the Ace locus on 3R, which has common alleles that confer resistance to insecticides [66]. Our model does not include the CHKov1 gene on 3R, which was also not recovered from our lab’s previous study [43], perhaps because the variants responsible for infection resistance are thought to have been present in both ancestral African populations [67,68]. Comparing our model to our previous approach that treated each site as a separate hypothesis test, we found that selection coefficients on 3R may have been overestimated. Indeed, the selection coefficients estimated by AHMM-S were up to 49% higher than those found by our method (Table 1). Presumably this occurs because each positively selected site is hitchhiking to an extent on linked positively selected sites, thereby increasing their frequencies in aggregate to a larger degree than would be expected given the same selection coefficient at an isolated selected position. This effect would also confound other methods to estimate the selection coefficient, such as those based on excess of local ancestry. We therefore conclude that linkage among selected positions has likely impacted analyses of selection during admixture. More generally, the tool we present in this work is a useful approach for disentangling these potentially complex effects.

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Fig 7. The local ancestry (LA) expected from the model roughly follows the estimated local ancestry of the samples.

Due to our method having a time complexity that is exponential with respect to the number of sites, we only calculate these expected local ancestries for half of the chromosome arm at a time (subdivided on the left and right). (A) The identified sites and their selection coefficients from our method (yellow) along with 14 out the 17 candidate selected positions from AHMM-S (blue). (B) The mean local ancestries from the samples (blue) and the local ancestries expected from the model (yellow). (C) Error between the model’s predicted local ancestry and what we infer from the empirical data.

https://doi.org/10.1371/journal.pgen.1011062.g007

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Table 1. Comparison of selection coefficients inferred for selected sites by AHMM-MLS and AHMM-S.

https://doi.org/10.1371/journal.pgen.1011062.t001

Additionally, we found that many positions that when analyzed individually are consistent with positive selection on admixed ancestry, are no longer supported when we consider the effects of linkage between selected sites. Of the original 17 candidate sites we identified using AHMM-S, 8 were not included in the final model selected by our iterative procedure. As might be expected, these 8 positions are primarily those with lower selection coefficients as estimated by AHMM-S (Table 1). However, when analyzed individually using AHMM-S, we found that each had a large likelihood ratio when comparing a single site selection model to a neutral model (16.56–73.74, Table 1). Support for sites that are relatively distant from other selected positions, such as the candidate site at 31371682 base pairs, were still impacted. This result emphasizes the importance of evaluating multi-locus selection models to capture the evolutionary dynamics of natural selection in admixed populations.

Application to Passer italiae

We applied our method to another admixed system to confirm its wide applicability to other recombining admixed populations. This system is a population of Italian Sparrows (Passer italiae), which is an admixed population with P. domesticus bactrianus and P. hispaniolensis ancestry [69]. Importantly for our work, this population has a previously estimated demographic history [70], and the ancestry tract length distribution in this population shows a single peak, possibly consistent with a single pulse admixture (S15 Fig). We collected whole genome resequencing data for 122 specimens of European sparrow (Passer domesticus bactrianus, P. italiae and P. hispaniolensis) from the SRA repository [69,70]. We then used the SNParcher workflow [71] to generate a multi-sample VCF. In short, SNParcher trims the fastq files using fastp and aligns the trimmed reads to the P. domesticus genome (GCA_001700915.1) using BWA mem [72], duplicate reads are marked and removed using sambamba [73] and a multisample VCF is produced using GATK [74]. Individuals used to construct the parental panels were identified using ancestry estimates from Admixture analysis by selecting the subset of individuals displaying nearly complete, 0.95 or greater, ancestry from one of the two populations [75]. We obtained the admixed population by selecting the subset of individuals with a very similar admixture fraction (0.25–0.31) (S1 Table). This is necessary because samples in the whole dataset were taken across a geographic range and display different ancestry fractions in some cases. For admixed panels we used a flat recombination rate of 2 centimorgan per Mbp. For null model simulations, we used MaCS [55] to simulate genotypes for unadmixed individuals based on the demographic history of the parental species [70] using the following command line:

macs 400 112673505 -t 0.00409 -r 0.00772 -I 2 200 200 0 -n 2 0.951 -en 0.0182 1 0.0144 -en 0.0327 1 0.327 -en 0.0293 2 0.541 -ej 2.368 1 2 -en 2.368 2 0.0562

Similarly to the D. melanogaster population, we simulated admixed populations with an effective population size of 10,000.

Many chromosomes show evidence of multiple sites under selection for P. domesticus bactrianus ancestry according to AHMM-S, making this a suitable system for use with our method. In particular, chromosome 1 shows evidence of a few strong peaks in the likelihood ratio, potentially indicating the presence of selected sites throughout the whole chromosome, which may be in linkage with each other. Because we lacked a neutral chromosome arm on chromosome 1, we used AHMM [44] on the chromosomes 1 through 6 and 16 to infer the demographic parameters of the admixed population. We chose these chromosomes for their larger size, as smaller chromosomes may have recombination rates that differ from our assumed uniform rate and could affect our demographic inference. We obtained values for the admixture fraction and time since admixture that were suitable for our program (m = 0.322, and t = 438).

Selection model choice for P. italiae

We carried out the same procedure for choosing and evaluating selected positions as we did for D. melanogaster. Candidate selected positions on chromosome 1 were chosen using AHMM-S, where positions with the highest maximum likelihood of the nearest 300 sites and a log likelihood ratio above 40 were candidates (Fig 8). Even with a high log likelihood ratio cutoff, most candidate positions were rejected by our iterative method. Of the 12 candidate positions, only 4 were identified in our multi-locus model (Fig 9). As was the case with the D. melanogaster population, the selection coefficients of the candidate positions appear to have been overestimated when fitting single site models using AHMM-S, when compared to the multi-site model that account for linkage using AHMM-MLS (Table 2).

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Fig 8. Candidate selected positions for chromosome 1 of P. italiae.

The maximum likelihood of a single site model fit for every sampled position along the chromosome (blue), and the local maxima that we take to be candidate selected positions for our iterative method (yellow dots).

https://doi.org/10.1371/journal.pgen.1011062.g008

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Fig 9. Local ancestry expected from our model follows broad peaks in the local ancestry inferred from the P. italiae population.

(A) Selection coefficients of sites identified using AHMM-S (blue) and AHMM-MLS (yellow). (B) Expected local ancestry proportion from multi-locus model (yellow) and the mean local ancestry from the samples (blue). (C) Error between the model’s predicted local ancestry and what we infer from the empirical data.

https://doi.org/10.1371/journal.pgen.1011062.g009

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Table 2. Multi-locus model inferred for chromosome 1 of P. italiae.

https://doi.org/10.1371/journal.pgen.1011062.t002

Caveats

Although our method provides a promising means for quantifying and investigating the impacts of interference and natural selection in admixed populations, there are several important caveats. First, the state space of multi-locus selection models is extraordinarily complex, and there is no evaluation procedure that could exhaustively attempt all possibilities. For example, even in the scenario that we considered for D. melanogaster with 17 candidate selected positions, there are 2¹⁷ (~130,000) possible sets of sites in multi-locus selection models. This is an intractable number of models to test, as the time taken to optimize a single model with three sites can take 10 minutes on a single core with an M2 processor. The iterative procedure we present is an appealing way to prioritize model space and we expect that it will perform well in a variety of scenarios, but undoubtedly there are other plausible models that we could not evaluate. Second, our approach will accommodate scenarios where there are a modest set of loci of relatively large effect. However, some authors have proposed that the aggregate effect of hundreds of weakly selected linked mutations might shape the landscape of admixed ancestry in natural populations [9]. Our approach is not well suited to such scenarios because each site is unlikely to reach significance in itself and because the time taken to compute the expected transitions for each model is exponential with respect to the number of sites. Third, our approach is also not suitable for populations with a small number of individuals (Ne < 2000), or which were admixed recently (t ⪅ 200), except possibly if selection is very strong (e.g., s > 0.025). We expect that AHMM-MLS will typically perform best with populations that are only somewhat genetically divergent and where strong selection affects the dynamics of introduced alleles.

Conclusion

Admixture has the potential to simultaneously introduce multiple linked selected sites, but this phenomenon is rarely addressed in empirical investigations. To meet this need, we created AHMM-MLS. In validating our method over simulated data, we found that it could identify multiple nearby selected sites, and estimate the selection coefficients better when the linkage between these sites was accounted for. We found that our previous study of adaptive introgression on chromosome arm 3R of an admixed D. melanogaster population may have overestimated the number and strength of selected sites along the chromosome. We similarly found that failing to account for linkage may overestimate the number and strength of selected sites in an admixed P. italiae population. Because divergent populations may introduce many selected alleles at once, analyzing the effects of linkage between these sites is critical for understanding the evolutionary dynamics of admixed populations. We hope that our method can be applied to the many examples of adaptive introgression that have already been identified, and can better quantify cases where multiple advantageous sites have been introduced at once.