D⁺ statistic

Patterson’s D statistic uses species and gene tree discordance within a 4-population tree with two populations as potential targets of introgression, population 1 (P₁) and population 2 (P₂). Population 3 (P₃) is a source of gene flow to either P₁ or P₂, and population 4 (P₄) serves as an outgroup (Fig 1). The patterns of biallelic single nucleotide polymorphisms (SNP) generated by the gene trees provide information on the shared ancestry between lineages in each population. Both the D and D⁺ statistic look at site patterns yielded when the gene tree does not match the species tree. A mutation will convert an ancestral allele (A), determined by the allele present in the outgroup, into a derived allele (B). An ABBA site (Fig 1A) describes a derived allele shared between P₃ and P₂, while a BABA site occurs when a derived allele is shared between P₃ and P₁. An ABBA or BABA site could arise due to incomplete lineage sorting (ILS) or gene flow. Under coalescent expectations, incomplete lineage sorting will generate equal numbers of gene trees with ABBA or BABA sites. An ABBA site can only be generated in a gene tree where P₃ and P₂ coalesce first before they find a common ancestor with P₁. On the other hand, a BABA site only occurs on gene trees where P₁ and P₃ coalesce first before they find a common ancestor with P₂. We expect an excess of ABBA sites when there is gene flow from P₃ to P₂.

The D statistic measures an excess of ABBA or BABA sites [4,5]. D is the normalized difference between ABBA and BABA sites, . The D statistic assumes that the frequency of ABBA and BABA sites due to ILS is approximately equal. Therefore, an excess of shared derived sites between P₃ and P₂, or ABBA sites, indicates gene flow from P₃ to P₂ as shown in Fig 1A. Conversely, an excess of BABA sites indicates gene flow from P₃ to P₁.

We extend this idea by making use of the fact that introgressed regions are inherited in chunks that contain both shared derived alleles and ancestral alleles that are introduced into the recipient population. D⁺ leverages the shared ancestral alleles between P₃ to P₂ to increase the amount of data about shared genetic variation in low nucleotide diversity regions. Sites where the ancestral allele is shared between P₃ and P₂ and the derived allele is only found in P₁ are BAAA sites (Fig 1B). In ABAA sites the ancestral allele is shared between P₃ and P₁ while P₂ has a derived allele. D⁺ incorporates both shared derived alleles and ancestral alleles to strengthen our inferences of introgression.

(1)

When the sample size is bigger than one, we can write down the equation for D⁺ as a function of the observed derived allele frequencies in populations P₁, P₂, P₃ or P₄. If the frequency of the allele at site i for population j is , and we have L sites, (2)

While in this paper, we mostly focus on comparisons between D⁺ and D, note that we could also define a statistic D_ancestral that measures the excess of shared ancestral alleles between P₃ and P₂ in a similar manner that the D statistic measures an excess of shared derived alleles between P₃ and P₂: (3)

D_ancestral is normalized and ranges from -1 to 1, with D_ancestral = 1 indicating gene flow from P₃ to P₂ and D_ancestral = −1 indicating gene flow from P₃ to P₁. D_ancestral approximates zero under the null hypothesis of no gene flow.

Durand et al. (2011) used a coalescent framework to derive the expectation of the D statistic under an instantaneous admixture model (IUA). The probability of getting an ABBA or BABA site is dependent on the mutation rate and the expected branch length of the branch where a mutation yields an ABBA site (T_ABBA) or the branch where a mutation yields a BABA site (T_BABA). The mutation rate μ is assumed to be constant. Therefore, the expected number of ABBA or BABA sites can be estimated by calculating the expectation of branch lengths of T_ABBA and T_BABA and multiplying by the mutation rate [5]. Similarly, we can compute the probability of getting an ABAA or BAAA site (see S1 Appendix), and we derived the expected lengths of T_BAAA and T_ABAA following the same framework. The full derivation of the expectation of T_BAAA and T_ABAA is in S1 Appendix. We find that the analytical expectation of D⁺ is (4)

As is true of ABBA and BABA sites, the expected number of BAAA and ABAA sites are equal when there is no gene flow. This is because, under no gene flow, we expect a similar amount of ancestral allele sharing between P₁ and P₃ and between P₂ and P₃. In the case of the BAAA and ABAA sites, we expect a similar amount of BAAA and ABAA sites under no gene flow assuming the same mutation rate in P₁ and P₂. As the admixture proportion from P₃ to P₂ increases, the number of BAAA sites exceeds the number of ABAA sites. The expected difference is a function of the admixture proportion f and the branch lengths of T_P3 and T_GF: (5)

Simulations to verify theoretical results for n = 1

To verify the theoretical results under the demographic model depicted in Fig 2, we calculate the expected branch lengths of T_ABBA, T_BABA, T_BAAA and T_ABAA and expectation of D and D⁺ using mspms simulations (see Figs 3 and S4) for a range of admixture proportions (f). We ran 1,000,000 simulations of independent loci and averaged the branch lengths of T_ABBA, T_BABA, T_BAAA and T_ABAA from the Newick tree file output of each locus. The branches T_ABBA, T_BABA, T_BAAA and T_ABAA are the branches where a mutation would yield an ABBA, BABA, BAAA or ABAA site, illustrated in S1 Fig. We simulated small, independent loci with 250 sites per loci and a mutation rate of 10⁻⁸ per bp per generation and no recombination. We also calculated D and D⁺ from the number of ABBA, BABA, BAAA and ABAA sites per locus and averaged D and D⁺ across all 1,000,000 loci. An example mspms simulation command for 1,000,000 independent loci with 1 sample taken from each of the 4 populations for an admixture proportion of 3% is:

mspms 4 1000000 -t 0.1 -I 4 1 1 1 1 -es 0.1 2 0.97 -ej 0.1 5 3 -ej 0.25 1 2 -ej 0.5 2 3 -ej 20 3 4 –T.

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Fig 2. Demographic model for msprime simulations.

(P₁) and (P₂) are sister populations that are closely related to (P₃). P₁ and P₂ diverged at time T_P2 (4,000 generations ago) and the ancestral population of P₁ and P₂ (P₁₂) diverged from P₃ at time T_P3 (16,000 generations ago). There is gene flow from (P₃) to (P₂) at time T_GF (1,600 generations ago) with an admixture proportion f = 3%. Divergence time of populations shown follow the demography of modern humans.

https://doi.org/10.1371/journal.pgen.1010155.g002

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Fig 3. Analytical and simulated expected branch lengths of T_ABBA, T_BABA, T_BAAA and T_ABAA.

The analytical (lines) and simulated (dots) expected branch lengths of T_ABBA, T_BABA, T_BAAA and T_ABAA for different proportions of admixture f between P₃ and P₂. The solutions to the analytical expectations match the simulated expectations. The branch length of T_ABBA is the branch that would produce an ABBA site pattern. The expectation of T_ABBA (E[T_ABBA]) can be used to calculate the expected number of ABBA sites. The same is true for T_BABA, T_BAAA, and T_ABAA for their respective site patterns. With no admixture (f = 0) the expected branch lengths for ABBA and BABA sites are equal (E[T_ABBA] = E[T_BABA]), as are the expected branch lengths for BAAA and ABAA sites (E[T_BAAA] = E[T_ABAA]) because the number of ABBA sites equals BABA sites and the number of BAAA sites equals the number ABAA sites due to ILS. As the admixture proportion increases, the expectation of T_ABBA and T_ABBA increases due to excess ABBA and BAAA sites. The difference in T_BAAA and T_ABAA (T_BAAA—T_ABAA) is equal to the difference in T_ABBA and T_BABA (T_ABBA—T_BABA).

https://doi.org/10.1371/journal.pgen.1010155.g003

Simulations to benchmark D⁺ using n = 1 for all populations

To evaluate the precision and recall of D and D⁺ we ran coalescent simulations using the software msprime [26]. The simulations followed the model depicting the evolutionary history of modern humans (Fig 2). The African and Eurasian populations are P₁ and P₂, respectively, and P₃ is the Neanderthal population. The African-Eurasian and Neanderthal divergence time T_P3 was set 16,000 generations ago and the Eurasian and African divergence time T_P2 was set 4,000 generations ago [16]. The time of gene flow (T_GF) between Neanderthals and Eurasians was set 1,600 generations ago [16]. We use an admixture proportion (f) of 3%. All simulations had a constant N_e of 10,000, a mutation rate of 1.5*10⁻⁸ per bp per generation and a recombination rate of 10⁻⁸ per bp per generation following [16]. We ran 100 simulations of 20 MB genomes with n = 1 for P₁, P₂ and P₃, and, in each run, we sampled a single haplotype to compute D⁺ using Eq 1. D⁺ can also be applied to populations with a sample size greater than 1. We ran 100 msprime simulations with n = 200 genomes for P₁ and P₂ and n = 2 for P₃ and computed D⁺ using derived allele frequencies (Eq 2). The full code for simulations can be found in a GitHub repository (https://github.com/LeslyLopezFang/Dplus).

To evaluate the performance of D⁺ under different values for the admixture proportion, recombination rate and mutation rate under this demography, we ran 100 msprime simulations with the new parameter value for a 20 MB genome for n = 1 for P₁, P₂ and P₃ under a model with no admixture and a model with admixture. We considered the following cases: f = 2%, 5% and 10%, a mutation rate of half the default mutation rate of 1.5*10⁻⁸ per bp per generation, a mutation rate that was double the default mutation rate of 1.5*10⁻⁸ per bp per generation, a recombination rate that was half the default recombination rate of 10⁻⁸ per bp per generation, and a recombination rate that was double the default recombination rate of 10⁻⁸ per bp per generation.

Calculating precision, recall and false positive rate in simulated human data

We ran msprime simulations described in the Methods section titled “Simulations to benchmark using n = 1 for all populations” using the parameters shown in Fig 2 without an instance of admixture at T_GF to construct a null distribution for D and D⁺ by sampling a genome from each population and computing D and D⁺ in 50 kb non-overlapping windows. We take the significance threshold values for D and D⁺ from their respective null distributions. For a p-value of 0.05, we get a signal of gene flow from P₃ to P₂ from the significance thresholds defined at the top 2.5% values from the null distribution of D and D⁺ (see Fig 4). Undefined values (denominator divided by 0) of D or D⁺ where no informative sites were present in the window were dropped.

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Fig 4. Null distribution and false positive rate for D and D⁺ in simulations with no gene flow.

D and D⁺ were calculated in 50 kb windows of 100 runs of a 20 MB simulated genome under a model with no admixture. (A) The average of the null distribution of D and D⁺ is zero with a standard deviation of 0.74 for D and 0.26 for D⁺. The null distribution for D (red) is multi-modal at the tails with the tails (-1 and 1) accounting for 43.2% of the values of D. The null distribution of D⁺ (blue) is centered around its average of zero. (B) False positive rates for D (red) and D⁺ (blue) of null distribution. The p-value in the x-axis is used to set a significance threshold to get a false positive rate in the y-axis. D has a false positive rate of 43.2% with p-values less than 0.43. The false positive rate of D⁺ is similar to the corresponding p-values up until p-value> = 0.94.

https://doi.org/10.1371/journal.pgen.1010155.g004

When we sample a single lineage (n = 1) from each population, an introgressed window is defined as a window that has at least 10% of bases in the window overlapping with introgressed tracts from the chromosome sampled from P₂. A 50 kb window would then have at least 5 kb bases that are introgressed from P₃ for the sampled chromosome from P₂. Windows that have an overlap with introgressed tracts but that are less than 10% of the bases in the window are dropped. Most of the windows have an overlap of at least 50% of the window with introgressed tracts (S2 Fig). When the sample size is more than 1, we have to redefine what is an introgressed window, and compute D and D⁺ using our frequency-based definitions.

To compute recall, true positives are introgressed windows that are statistically significant, while the false negatives are introgressed windows that are not statistically significant. The false positives for the simulated data are windows that have no introgressed bases but are statistically significant. Precision measures the probability of a window truly being introgressed given that its D⁺ value is statistically significant. Precision is the percentage of true positives out of the sum of true positives and false positives, or 1 –false discovery rate. Recall measures how many of the introgressed windows are statistically significant and is the percentage of true positives out of the sum of true positives and false negatives. Here, false positive rate is the percentage of false positives from windows without introgression (false positives and true negatives; see Figs 4 and 5).

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Fig 5. Precision and recall of D and D⁺ in simulations.

The Precision-Recall of D and D⁺ for simulations with an admixture proportion of 3. D (red) and D⁺ (blue) were computed in non-overlapping 50 kb windows of 100 simulations of a 20 MB genome from each population with an admixture proportion of 3% (f = 0.03). (A) Precision and (B) recall are shown as a function of the p-value (0.01–1) used to get a significant threshold value of D and D⁺.

https://doi.org/10.1371/journal.pgen.1010155.g005

We also compute recall when the sample size is bigger than one. However, in this case, it will be harder to detect windows with introgressed tracts at low frequency in the recipient population. Therefore, in simulations where we sample n = 200 chromosomes from P₂, we redefine what is an introgressed window so that two conditions need to be true. First, a window needs to contain at least one introgressed tract that survives in at least 10% of the 200 chromosomes from P₂. Second, the length of the tract (or the sum of the tract lengths if more than one tract in a window pass the first condition) is at least 10% of the window (which is 10% of 50 kb or 5kb). For an example see S3 Fig. In S10A Fig shows recall for D, D⁺ and d_f as a function of the proportion of introgression. In S10B Fig, the proportion of introgression is set to f = 10% and recall is computed as a function of the required tract-frequency in P₂ within a window (first condition necessary to define a window as introgressed).

Testing violations of a strict molecular clock

D⁺ assumes a strict molecular clock such that all populations have the same mutation rate. When there is no gene flow, we can assume an equal number of ABBA and BABA sites, as well as an equal number of BAAA and ABAA sites. To test violations to this assumption we ran msprime simulations with n = 1 chromosome for P₁, P₂ and P₃ where we increase the mutation rate of either P₁ or P₂ by increasing all of the divergence times T_P2, T_P3 and T_GF by T_P2 in the model depicted in Fig 2. For example, to double the mutation rate of P₁, we increase all divergence times by T_P2 and sample P₁ at time t = 0 and sample P₂ at T_P2. P₃ is sampled right after the modified T_GF since P₃ is an archaic population.

We ran 100 msprime simulations using n = 1 chromosome for P₁, P₂ and P₃ under the new divergence times with no introgression and with introgression (f = 3%). The performance of D⁺ can be calculated using the null distribution with the new divergence times to assess statistical significance. Introgressed windows are windows where at least 10% of the bases in the 50 kb window are introgressed tracts for the haplotype from P₂. We considered four cases: 1) P₁ with a mutation rate double the mutation rate of P₂, 2) P₁ with a mutation rate ten times the mutation rate of P₂, 3) P₂ with a mutation rate double the mutation rate of P₁ and 4) P₂ with a mutation rate ten times the mutation rate of P₁.

Application of D⁺ in modern-day humans

To evaluate the performance of D⁺ at identifying introgressed regions in empirical data we apply D⁺ to previously detected regions of Neanderthal introgression in modern-day humans. We assume that introgressed segments inferred in [7] using the Altai Neanderthal genome [27] are the true introgressed segments. From the 1000 Genomes Project [28] we used an individual from the YRI (Yoruba in Ibadan, Nigeria) population for P₁ and an individual from the GBR (British from England and Scotland) population for P₂. P₃ is the Altai Neanderthal genome [27]. The ancestral allele of each position was taken from the ancestral allele listed in the 1000 Genome Project. For the GBR individual we used a Neanderthal introgression map including all the haplotypes inferred to be Neanderthal with a probability > 90% in [7]. We calculated D and D⁺ in non-overlapping 50 kb windows using one autosomal chromosome of each individual from all three populations, discarding the first and last window of each chromosome. Note that here we assume that the phasing of the GBR individual is perfect, so we are able to compute two D⁺ and two D values for each window in the genome. However, we use the same chromosome in the YRI individual to compute the two D⁺ values for the GBR individual. Since phase is unavailable for the Neanderthal genome, we randomly sampled one of the Neanderthal alleles at each site. Each window had two D and D⁺ values, one for each autosomal chromosome of the sampled GBR individual.

To find significance thresholds for the empirical data, we use all of the D and D⁺ values from all of our windows (two per window) to build the empirical distributions for D and D⁺. If the maximum of the two D (or D⁺) values in a window is in the top 2.5% of D and D⁺ values for the empirical distribution, then it will be called statistically significant. To compute recall, we need to define a true introgressed window that will be called a true positive or false negative based on the empirical distribution of D and D⁺ values. We defined a true introgressed window as a window with a set minimum percentage of bases that overlap with a Neanderthal introgressed segment (inferred in [7]). We used different values for the minimum percentage of bases needed to overlap with the Neanderthal segment for a window to be called introgressed. Using the empirical distribution, we call a window “introgressed” (or a true positive) when the maximum of its two D⁺ values are statistically significant (i.e. values are in the top 2.5% of the distribution) and it has at least a pre-defined percentage of overlap with a Neanderthal introgressed segment (e.g. 5% to 100% in intervals of 5%, x-axis of Fig 6). Recall was then calculated by dividing the number of true positives by the total number of windows defined as true introgressed windows (based on having at least a pre-defined percentage of overlap with a Neanderthal introgressed segment). We assume that the introgression maps capture true positives or a subset of them; however, we cannot assume that regions not included in the introgression maps are true negatives. Therefore, we do not assess false positives or precision.

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Fig 6. Recall of D and D⁺ in human data.

The recall of D and D⁺ in non-overlapping 50 kb windows. Windows overlap with Neandertal introgression maps [7] from 5% to 100%. The populations are as follows: P₁: YRI, P₂: GBR, P₃: Altai Neandertal, P₄: Ancestral Alleles. Data for humans from 1000 Genomes Project [28] and data for Altai Neandertal from [27].

https://doi.org/10.1371/journal.pgen.1010155.g006

We also took computed recall mimicking the empirical approach in simulated data where we know what the ground truth is. Specifically, in our simulations we sampled one individual (n = 2 chromosomes) from P₁, P₂ and P₃ and computed D and D⁺ for each chromosome from the individual in P₂, and also obtained the maximum D and D⁺ value per window. In the simulated data, we only considered one value (10%) for the minimum percentage of overlap between a window and an introgressed tract. We computed precision and recall using all the D and D⁺ values from all the windows as the null distribution.

Finally, we assess how recall is affected when phasing is unavailable. In this case, we use the same YRI, GBR and Neanderthal individuals and randomly sample a haplotype (n = 1 for all 3 populations) at every position to create a single haploid genome for each population. We partitioned the haploid genomes in non-overlapping 50 kb windows and compute D⁺ values for each window. We do this experiment 100 times and compute recall for each experiment.

Application of D⁺ in Heliconius butterflies

D was applied to Heliconius butterflies and found to have high variance in areas of low nucleotide diversity [25]. To assess whether D⁺ reduces variance in these areas of low nucleotide diversity we recreated Fig 3 from [25] using the same Heliconius genome data from [29]. They show values of D as a function of nucleotide diversity π for P₂ (the recipient population) in non-overlapping regions of 5 kb. Only biallelic alleles were used. D was computed using derived allele frequencies and we also use the frequencies from the four populations to compute D⁺ (Eq 2).

[25] and d_f [24] were also computed for the 5 kb non-overlapping windows. was only applied to windows where D is positive. The equation for written in terms of derived allele frequencies with as the maximum of and is d_f incorporates BBAA sites where only P₁ and P₂ share a derived allele. The equation for d_f in terms of allele frequencies is

Four samples were used, one each from H. melpomene aglaope (P₁), the recipient population H.m. amaryllis (P₂), the donor population H. timareta thelxinoe (P₃). The outgroup (P₄) consisted of a sample from species in the silvaniform clade including H. hecale, H. ethilla, H. paradalinus sergestus and H. pardalinus ssp. nov. The ancestral state of an allele was determined by the outgroup if the allele was fixed within the outgroup. Otherwise, it was the major allele of all four populations. The wing pattern loci HmB and HmYb are defined in [25]. Code was adapted from [25] with details in GitHub repository (https://github.com/LeslyLopezFang/Dplus).

Similar to the method to find recall described in “Calculating precision, recall and false positive rate in simulated human data.”, we calculate recall for D, D⁺, , d_f and D_ancestral. The windows that overlap the HmB and HmYb loci are considered introgressed windows. Recall here is the number of these introgressed windows that the introgression statistic identifies as introgressed. Each window has one value for each statistic, and we find the statistical thresholds for each statistic by finding the top 2.5% value of the distribution.

Theoretical results

The expectation for the values of D and D⁺ is dependent on the lengths of the branches that produce each site pattern. T_ABBA is the length of the branch starting from the time of the most recent common ancestor of P₂ and P₃ until that lineage coalesces with P₁ (which happens in the ancestral population P₁₂₃ under the instantaneous admixture model). The average length of the T_ABBA branch increases with the migration rate (Fig 3). A mutation on this branch produces an ABBA site pattern. T_BABA is then the length of the branch from the time of the most recent common ancestor of P₁ and P₃ until that lineage coalesces with P₂. T_BAAA and T_ABAA are the external branches of P₁ and P₂, respectively. When there is no gene flow, the average length of the external branches of P₁ or P₂ are equal. With gene flow between P₂ and P₃, the external branch of P₁ will be longer than the external branch of P₂; therefore, the expectation of T_BAAA increases with the admixture proportion f.

The analytical and theoretical expectation of T_ABBA, T_BABA, T_BAAA and T_ABAA are shown in Fig 3. The theoretical expectation of each branch takes into account all scenarios that could produce each site pattern, including gene flow and no gene flow (S1 Appendix). The simulated expected branch lengths approximate the theoretical expected branch lengths at all the admixture proportions (f) calculated. When there is no admixture, the number of ABBA sites is equal to the number of BABA sites as any sharing of derived alleles between P₃ and P₂ (or P₃ and P₁) is due to incomplete lineage sorting. In the case of ancestral sharing and under a model of no admixture, the number of BAAA sites and ABAA sites will be equal because we assume equal mutation rates in P₁ and P₂.

For all values of migration between P₂ and P₃, the expected branch lengths that can lead to a BAAA (T_BAAA) or a ABAA (T_ABAA) site are always greater than the expected branch lengths that can lead to an ABBA (T_ABBA) or BABA site (T_BABA). Therefore, if we assume a constant mutation rate, we expect to see more ABAA sites than BABA sites and more BAAA sites than ABBA sites. In Fig 3, assuming a constant mutation rate multiplied with the analytical and simulated expected branch lengths, there are 5–6 times more BAAA and ABAA sites than ABBA and BABA sites.

Interestingly, our theoretical results also show that even though the number of BAAA and ABAA is higher (than ABBA or BABA), the difference between T_BAAA and T_ABAA (T_BAAA—T_ABAA) is equal to the difference (T_ABBA—T_BABA). Therefore, for all admixture proportions between P₂ and P₃, the expected difference of BAAA and ABAA sites (BAAA—ABAA) is equal to the expected difference of ABBA and BABA sites (ABBA—BABA). These observations suggest that leveraging ancestral shared variation can be informative about introgression and provides justification for defining D⁺ which leverages both ancestral and derived allele sharing to maximize the number of informative sites used in a genomic window. This increase in informative sites can provide greater predictive accuracy for detecting local gene flow.

D has a high false positive rate in small genomic windows

We calculated D and D⁺ for 50 kb windows on simulated genomes following the demography in Fig 2 with no admixture event at T_GF to get the null distribution of D and D⁺ (Fig 4A). The null distribution of D is a multimodal distribution with large peaks at the tails as well as zero. The average of D is 0 with a standard deviation of 0.74. The tails (D = 1 and D = −1) account for 43.2% of the distribution. These peaks at the tails cause a high false positive rate of 43.2% for D at p-values less than 0.43 (Fig 4B) because the significance threshold for D is 1 or -1. Therefore, we have low power to assess statistically significant values of D. In contrast D⁺ has a null distribution centered on zero. The average of D⁺ is 0 and the standard deviation is 0.26. The null distribution is much narrower than the null distribution of D and does not have peaks at the tails. As expected, the false positive rate of D⁺ approximates the p-value set to find significant values of D⁺ up until a significance threshold approaches 0 for high p-values (p-values > = 0.94) (Fig 4A).

D⁺ has better precision than D in simulated data

We calculated precision and recall for 50 kb windows of 100 simulations with a 20 MB simulated genome shown in Fig 5 following the demography in Fig 2. Undefined values were dropped (see Methods) so more windows were analyzed for D⁺ than D because D had more undefined values. While precision measures the accuracy of windows giving a signal of gene flow from P₃ to P₂ through statistical significance, recall measures how many introgressed windows the statistic can detect without considering false positives. We obtained precision and recall for p-values from 0.01–1 (Fig 5). Each p-value has a corresponding significant threshold value from the null distribution in Fig 4A in which values of D or D⁺ greater than the threshold are statistically significant. For realistic p-values (i.e. p-values 0.01, 0.02, 0.03, 0.04 and 0.05), D⁺ has better precision than D; At these realistic p-values, precision for D⁺ ranges from 29.48% to 53.30% and the precision of D is 7.65% (Fig 5A). For these p-values, D has better recall than D⁺ (Fig 5B) with recall for D⁺ ranging from 12.46% to 23.33% and recall for D equaling 34.33%. For D, precision and recall are the same (7.65% and 34.33% p-values < 0.43, because the D value is 1 since the null distribution is multimodal with peaks at the tails (Fig 4A). It should be noted that these results are robust to different window thresholds—i.e., introgressed tracts covering at least 5%, 10%, and 25% of a 50kb window (S24 Fig). Notably, when we consider a more complex human demography (see S5 Fig), the recall and precision is 58.71% and 37.34% at a p-value of 0.05 (S6 Fig). One of the reasons the recall is higher under the more complex model is because the effective population size of Neanderthals is smaller. This means that the Neanderthal sequenced used to compute the D statistics is more closely related to the actual Neanderthals that introgressed into modern humans.

To see how changes in the admixture proportion, mutation and recombination rate affect the performance of D⁺, we also simulated under different admixture proportions, recombination rates and mutation rates. We find that precision is sensitive to the admixture proportion (S7 Fig). For a p-value of 0.05, a higher admixture proportion of f = 5% increases precision by 16% and an admixture proportion of f = 10% increases precision by 36% in comparison to an admixture proportion of f = 3% shown in Fig 5. In contrast, decreasing the admixture proportion from f = 3% to 2% decreases the precision by 6%. Recall is less sensitive to the admixture proportion than precision with the biggest change happening for f = 10% with an increase of 4%. The recall and precision for other p-values is shown in S7 Fig.

Changing the mutation rate affects the number of informative sites per window and changing the recombination rate affects the length of the introgressed segments and the number of windows that count as introgressed. We show precision and recall for different p-values in S8 and S9 Figs for a larger and smaller mutation rate and recombination rate. For a p-value of 0.05, increasing the mutation rate by a factor of two increases precision by approximately 4% and increases recall by approximately 2% (S8 Fig compared to Fig 5). In contrast, when the mutation rate is decreased by a factor of two, the precision and recall drop by 4% and 7% respectively (S8 Fig). Decreasing the recombination rate decreases the number of windows that are introgressed. These windows contain more overlap with an introgressed segment since the introgressed segments are longer. By contrast, increasing the recombination rate by a factor of 2 almost doubled the number of windows that are introgressed; however, the introgressed segments within a window are shorter in length. When the recombination rate decreases by a factor of two, recall increases by 2% (S9B Fig) and precision increases by 7% (S9A Fig). When the recombination rate is doubled, recall decreased by 7% (S9E Fig) and precision increases by 2% (S9D Fig). For p-values of 0.01, 0.02, 0.03, 0.04 and 0.05, the false positive rate for a model with no gene flow remains relatively constant as the admixture proportion, mutation rate or recombination rate change (S7C, S7F, S7I, S8C, S8F, S9C and S9F Figs).

D⁺ can also be applied to sample sizes greater than one using derived allele frequencies with Eq 2. For simulations with n>1, we use admixture proportions greater than 3% (see Methods). We compute D⁺ for simulations with admixture proportions of 10%, 20%, 30%, 40% and 50% and computed recall (S10A Fig). Note that we have a different definition of what an introgressed window is, which we explained in the Methods section titled “Calculating precision, recall and false positive rate in simulated human data” and an example is provided in S3 Fig. For a p-value of 0.05 D⁺ has higher recall than D for all admixture proportions. The recall of D⁺ increases as the admixture proportion increases. As two conditions need to be met to call a window introgressed (see Methods), we considered relaxing the first assumption involving the frequency of the introgressed tract in the recipient population (P₂). When we change the frequency of the introgressed tract(s) in P₂, recall increases as the frequency of the tract increases (see S10B and S10C Fig). Furthermore, we ran additional simulations with a realistic admixture proportion of 3% and computed precision and recall for a different set of chromosome and window thresholds (see S1 Text section “Comparing performance of D and D+ for different chromosome and window thresholds”). We find that for realistic p-values D+ will always have a higher precision and recall than D, which demonstrates that for a realistic admixture proportion, increased number of sampled chromosomes, and for all pairwise possibilities of chromosome and window thresholds D+ consistently outperforms D for detecting signals of introgression at a local scale (S25 and S26 Figs).

D⁺ performs well under moderate violations of the molecular clock

Under a strict molecular clock, we expect the number of ABBA and BABA sites and the number of BAAA and ABAA sites to be equal under the null model with no gene flow. To assess the performance of D⁺ when the mutation rate of P₁ and P₂ are not equal, we increased the mutation rate for P₁ and P₂ in comparison to each other by a factor of 2 and 10.

When P₁ has a higher mutation rate the amount of BAAA sites is greater than the amount of ABAA sites under no admixture (S11A Fig). This skews the distribution and average of D⁺ towards 1, a signal of introgression from P₃ into P₂. Similarly, when P₂ has a larger mutation rate, the amount of ABAA sites is greater than the amount of BAAA sites under the model of no admixture and makes the average of D⁺ negative instead of 0, indicating gene flow from P₃ to P₁ (S11B Fig).

For a p-value of 0.05, precision of D⁺ is 31.60% when the mutation rate of P₁ is double the mutation rate of P₂ and precision is 37.22% when the mutation rate of P₂ is double the mutation rate of P₁ (S12 Fig). In a more extreme scenario, increasing the mutation rate of either P₁ or P₂ by a factor of ten decreases the precision in comparison to either P₁ or P₂ having double the mutation rate. At a p-value of 0.05, precision decreases to 23.95% when P₁ has a higher mutation rate than P₂ and precision decreases to 29.36% when P₂ has a higher mutation rate than P₁ (S12 Fig). In general, these results suggest that precision is only mildly affected by the differences in mutation rate between P₁ and P₂.

When P₂ is the population with double the mutation rate, recall of D⁺ is 18.35%, and 23.77% when the mutation rate of P₁ is doubled (S12 Fig). When P₁ is ten times the mutation rate of P₂, recall is 16.32%, and 10.00% when P₂ is ten times the mutation rate of P₁ (S12 Fig). It makes sense that recall is worse when the mutation rate is higher in P₂ since this increases the number of ABAA sites, so the difference (BAAA-ABAA) is smaller which means the signal of introgression is smaller.

D⁺ identifies Neanderthal introgressed regions in modern-day humans

To investigate the behavior of D⁺ in real data, we applied D⁺ to modern-day humans [28] and an Altai Neanderthal [27] to find if signals of gene flow corresponded to previously identified Neanderthal introgressed regions. Unlike simulated data, in real human genomes we do not know the ground truth, and to compare the performance of D and D⁺, we assumed that the Neanderthal introgressed regions from [7] were the truth. We calculated D and D⁺ windows for the two phased chromosomes of a single GBR individual from [28] to compute the recall of D and D⁺ (Fig 6). Since the 50 kb windows will sometimes only partially contain an introgressed segment, we defined a window as introgressed if the window had a minimum percentage of bases overlapping with an introgressed segment (see Methods). Statistical significance was computed using the genome-wide distribution of D⁺ values (or D values) as the null distribution. Recall is the number of these “true” introgressed windows that were called statistically significant over the total number of introgressed windows (see Methods).

Fig 6 shows that recall for D⁺ was consistently better than D as a function of the minimum percentage of introgressed bases in a window. The recall decreases as the minimum overlap between an inferred introgressed segment and a window increases. This happens because the number of introgressed windows used to calculate recall decreases when we increase the amount of overlap to call a window introgressed. We note that for this analysis the D⁺ value assigned for each window was the maximum of the two D⁺ values for each of the two phased chromosomes in the GBR individual. We tested this method of choosing the maximum of D⁺ values per window on simulated data (under the demographic history in Fig 2) and computed the precision and recall for this scenario (see S13 Fig). We found that for a p-value of 0.05 the recall is 23.56% and precision is 44.69%. Taking the maximum of D⁺ values per window did not affect the recall in comparison to the recall of D⁺ when only one chromosome from P₂ is used to compute D⁺ (see Fig 5). By comparison, the corresponding recall in the empirical data is around 59% (recall when the point on the x-axis is 10% in Fig 6) which is similar to the recall under a more complex human demographic model (S6 Fig). The complex human demographic model (shown in S5 Fig) includes a smaller effective population size for the Neanderthal population.

In the previous analysis, we assumed that the chromosomes of the GBR individual are perfectly phased. However, as the phasing is inferred, there could be phasing errors and/or often phased chromosomes may not be available. When phasing is unavailable, studies randomly sample a single allele to create a haploid genome (e.g. as in the Neanderthal genome). We tested how this works in both real and simulated data. We implemented this approach with the individuals we used for Fig 6 and compute recall 100 times and find that recall for a p-value of 0.05 is on average 24.07% (see S14 Fig), similar to the recall in the simulated data, with recall ranging from 22.39–25.67% for all 100 runs. Therefore, we recommend that when the phase is not available or is uncertain, the user can randomly sample a chromosome from each individual as is often done in ancient DNA studies.

D⁺ can detect introgression events in regions of low nucleotide diversity

One of the main reasons the D statistic is not useful for detecting introgression in small regions of the genome is that the variance of D is high in areas of low nucleotide diversity [25]. To address this [25] proposed as an alternative approach to quantify and detect introgression in small genomic regions. The numerator of is in the same form as that of D; however, the denominator of replaces the derived allele frequency of P₂ and P₃ with the maximum derived allele frequency of P₂ and P₃. This leads to having a lower variance in areas of low nucleotide diversity, thus reducing spurious results in comparison to D. Like , d_f is also designed to quantify the admixture proportion of small genomic regions [24]. The approach in d_f is to incorporate BBAA sites as fewer sites with this pattern are expected when introgression occurs between P₂ and P₃ or between P₁ and P₃.

Both , d_f are estimates of the admixture proportion while D and D⁺ are used to detect and not quantify introgression. To compare D⁺ to and d_f we used the same Heliconius genome data from [29]. Heliconius butterflies have strong evidence for both genome-wide and adaptive introgression between species, including mimicry loci for wing patterns [14,29,30]. We use these data to compute these statistics in windows as a function of nucleotide diversity, since the relationship between D and nucleotide diversity observed in [29] inspired the developments of new statistics to detect and quantify introgression in small windows of the genome. For the four populations, we use H. melpomene aglaope as P₁, H. melpomene amaryllis as P₂, H. timareta thelxinoe as P₃ and the H. hecale, H. ethilla, H. paradalinus sergestus and H. pardalinus ssp. nov. species in the silvaniform clade as the outgroup (P₄). We compute nucleotide diversity π, , d_f, D and D⁺ in non-overlapping 5 kb windows. Windows from the candidate introgressed loci responsible for the red wing pattern (HmB) and the yellow and white wing pattern (HmYb) are shown in red and yellow, respectively, in Fig 7. We find similar results as [25]; D has a high variance and a wide distribution in regions of low nucleotide diversity (Fig 7A). As nucleotide diversity increases the distribution of D narrows. reduces the high variance of values in areas of low nucleotide diversity (Fig 7B). d_f also reduces variance with most of the d_f values centered around zero, including windows with the HmB and HmYb loci (Fig 7C). D⁺ has smaller variance with fewer outliers than D and similar variance to d_f (Fig 7D). D⁺ detects candidate regions of introgression, including windows not detected by D (see S1 Table). Many of the windows that D⁺ detects as introgressed correspond to the candidate regions for introgression that have been previously suggested in Heliconius butterflies and are associated with wing patterning (red and yellow points in Fig 7). In fact, D⁺ detects approximately 68.4% of these candidate introgressed windows. In comparison, D detects 52.6%, d_f detects 27.6% and detects 63.1%. We also computed D_ancestral which only uses the ancestral shared patterns (ABAA and BAAA), and it detects 63.2% of the windows (S15 Fig), suggesting that using the ancestral site patterns alone is better behaved than the D statistic, which shows the utility of using ancestral shared variation.

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Fig 7. Application of D, , d_f, and D⁺ in Heliconius butterfly.

(A) D, (B) , (C) d_f and (D) D⁺ as a function of nucleotide diversity in P₂ in non-overlapping 5 kb windows. P₁: H. melpomene aglaope, P₂: H. melpomene amaryllis, P₃: H. timareta thelxinoe, P₄: H. hecale, H. ethilla, H. paradalinus sergestus and H. pardalinus ssp. nov. from the silvaniform clade. Red and yellow circles correspond to windows with introgressed loci HmB and HmYb, respectively. Methods follow Fig 3 from [25] with Helicionius genome data from [29].

https://doi.org/10.1371/journal.pgen.1010155.g007