Efficient moment-based inference of admixture parameters and sources of gene flow

Introduction

• Likelihood approaches can be computationally costly and often require a prespecified topology; moment methods are simpler but may have lower statistical power and estimator-design challenges.Phylogenetic-network methods can model hybridization but also tend to be computationally expensive.Source: Introduction · pp. 3–4
• MixMapper fits admixture trees by solving moment equations for f2 distances, then builds an unadmixed scaffold and automatically optimizes admixed populations’ placement without prespecified topology.f2 is the average squared allele-frequency difference between two populations.Source: Introduction · p. 4
• MixMapper obtained best-fit admixture parameters with bootstrap error estimates for 30 HGDP populations, 20 of them admixed, and found a significant event in all sampled Europeans, including Sardinians and Basques.The paper describes the European signal as involving ancient northern Eurasian ancestry.Source: Introduction · p. 5

New Approaches

• The workflow computes pairwise f2 distances, filters populations flagged by the 3-population test, and selects an approximately additive unadmixed subtree as a scaffold.The scaffold is intended to approximate the no-admixture expectation along its branches.Source: New Approaches · p. 5
• Each non-scaffold population is tested as a mixture of scaffold branches; after two-way fits, remaining populations are also tested as three-way mixtures involving inferred two-way mixtures.Algebraic relations among f2 statistics are checked against the data to identify likely sources and estimate mixture parameters.Source: New Approaches · pp. 5–6
• Bootstrap resampling yields ensembles and confidence intervals for model variables, while a new formula converts f2 distances into absolute drift units.For HGDP-scale data, fitting a chosen population takes seconds after initial setup on the order of an hour.Source: New Approaches · p. 6

Simulations

• MixMapper inferred correct gene-flow topologies with 100% confidence for pop4 and pop10, 98% for pop9, and 61% for pop8 in the complex simulation.TreeMix did not detect pop9 admixture or the pop2-related admixture in pop8, and two of its five inferred admixtures did not correspond to true events.Source: Simulations · p. 7

Application of MixMapper to HGDP data

• MixMapper obtained comprehensive results for 20 admixed HGDP populations.The paper notes that most HGDP groups showed signs of admixture detectable by the 3-population test.Source: Application of MixMapper to HGDP data · p. 8

Selection of a 10-population unadmixed scaffold tree

• The 10-population scaffold balanced additivity and geographic coverage, and analyses using nine-population subsets or an alternative 11-population scaffold gave robust results.The selected scaffold was the second-most additive of 21 similar trees, with maximum deviation 1.12 × 10^-3.Source: Selection of a 10-population unadmixed… · pp. 8–9

Ancient admixture in the history of present-day European populations

• 20–40% of each of eight HGDP European groups was fit as ancestry related to the Karitiana–Suruí common ancestor, with Sardinian and Basque among the lowest.An f4-ratio analysis estimated approximately 20–25% ancient northern Eurasian ancestry in Sardinian and Basque, in very good agreement with MixMapper.Source: Ancient admixture in the history of… · pp. 9–10

Two-way admixtures outside of Europe

• Daur, Hezhen, Oroqen, and Yakut were fit as North Asian–East Asian mixtures; Melanesians and Han Chinese also had two-way mixture fits.The first three North Asian populations had roughly 10–30% North Asian ancestry, Yakut 50–75%, and Melanesians roughly 80% Papuan-related ancestry; the Han fit was approximately equal between two East Asian sources.Source: Two-way admixtures outside of Europe · pp. 10–11

Recent three-way admixtures involving western Eurasians

• Mozabite, Bedouin, Palestinian, and Druze fit as three-way mixtures involving an African population and a Sardinian-related admixed western Eurasian population; Uygur and Hazara fit with East Asian-related sources.For the former groups, African ancestry decreases in the order listed.Source: Recent three-way admixtures involving… · p. 11

Estimation of ancestral heterozygosity

• Outgroup-ascertained SNPs yielded consistent ancestral heterozygosity estimates for population pairs sharing an MRCA, unlike original HGDP-array estimates affected by ascertainment bias.The heterozygosity estimates were used to express branch lengths in drift units.Source: Estimation of ancestral heterozygosity · pp. 11–12

Comparison with previous approaches

• MixMapper generalizes the 3-population test and f4 ratio estimation, automatically testing tree topologies and gene-flow sources rather than requiring an admixture location in advance.With five populations in the specified topology, its f2 system for a single mixture becomes equivalent to f4 ratio estimation; more populations add constraints.Source: Comparison with previous approaches · pp. 12–13
• In the second simulated tree, MixMapper accurately fit complicated histories, whereas TreeMix recovered only portions of the true tree and added two inaccurate mixtures.For HGDP, both methods fit several Middle Eastern and Central Asian populations as admixed, but MixMapper suggested three-way admixture; each method also confidently fit populations the other did not.Source: Comparison with previous approaches · p. 14
• MixMapper offers a middle ground between qpgraph’s precise fitting of small population sets and TreeMix’s large-scale automation, which may be less precise in complex admixture scenarios.Compared with qpgraph, MixMapper automatically considers topologies, supports larger trees, and provides drift-unit conversion and bootstrap uncertainty estimates.Source: Comparison with previous approaches · pp. 13–15

Ancient European admixture

• All HGDP European populations were best modeled as mixtures of ancestry related to the common ancestors of Native Americans and non-Africans in the scaffold tree.The authors interpret the result as evidence that most, if not all, modern Europeans descend from at least one large-scale ancient admixture event involving some combination of hunter-gatherers, Neolithic farmers, and/or other migrants.Source: Ancient European admixture · p. 15
• MixMapper detected the same admixture signal in Sardinian and Basque populations as in the rest of Europe, despite their not being confirmed as admixed by the 3-population test.The authors attribute that test’s difficulty in part to lower ancient northern Eurasian ancestry and greater post-admixture drift; similar inferred ancestral branch points across Europeans support a shared-history interpretation.Source: Ancient European admixture · p. 16

Future directions

• Twenty-two of 52 HGDP populations, excluding San, could not be fit reasonably on the unadmixed tree or as admixtures.The authors cite the limits of an instantaneous-admixture model, ongoing low-level migration, insufficiently informative sampled populations, and complex ancestry sources as factors; archaic-human admixture was also excluded.Source: Future directions · p. 17
• The authors describe MixMapper as valuable alongside full-genome sequencing because SNP arrays remain easier and less expensive to genotype, while sequencing is currently more error-prone.They report that the study’s array contained over 100,000 loci and provided substantial statistical power.Source: Future directions · p. 17

Model assumptions and f-statistics

• The model assumes neutral, biallelic, autosomal SNPs, no double mutations, and one-time admixture in which parental allele frequencies combine as a weighted average.The authors characterize this as a first-order approximation to more complicated mixture histories.Source: Model assumptions and f-statistics · p. 18
• MixMapper uses f2 as the mean squared allele-frequency difference between populations, with f3 and f4 formed as linear combinations of f2 statistics.The paper uses unbiased estimators for its f-statistic computations.Source: Model assumptions and f-statistics · p. 18

Constructing an unadmixed scaffold tree

• To build a scaffold, MixMapper removes populations with any negative f3 test result, then builds neighbor-joining trees from f2 distances among subsets of the remaining populations.Without admixture, f2 distances are additive along tree paths.Source: Constructing an unadmixed scaffold tree · p. 18
• MixMapper ranks candidate scaffolds by maximum deviation between fitted tree distances and observed f2 distances, retaining a user-specified number of best subsets at each size in a beam search.The beam search avoids exponential complexity; real-data deviations from additivity correspond to non-zero 4-population-test results.Source: Constructing an unadmixed scaffold tree · p. 19
• Users choose a scaffold that balances additivity with coverage, since smaller trees are more additive but less informative and broader continental representation offers more potential admixture branch points.Users can inspect ranked trees, constraints, and the benefit of removing a population; MixMapper then re-optimizes branch lengths while preserving the neighbor-joining topology.Source: Constructing an unadmixed scaffold tree · p. 19
• The scaffold is a modeling device rather than a claim that its human populations were entirely unmixed, and interactive scaffold changes can test robustness.The authors report qualitatively unchanged conclusions across different scaffolds.Source: Constructing an unadmixed scaffold tree · p. 20

Two-way admixture fitting

• For two-way admixture, MixMapper expresses f2 distances from an admixed population to unadmixed references using four unknowns: two parental split locations, a combined terminal branch length, and the mixture fraction.At least four independent constraints are needed; with more references, noisy and slightly inconsistent equations are solved by least squares.Source: Two-way admixture fitting · pp. 20–21
• MixMapper tests every pair of scaffold branches as candidate sources and selects the branch pair and parameters with the smallest residual norm.With the mixture fraction fixed, the remaining three variables are solved by constrained linear least squares; the fraction is optimized over 0 to 1.Source: Two-way admixture fitting · p. 21

Three-way admixture fitting

• For three-way fitting, MixMapper models a second admixed population as a mixture of the first admixed population’s terminal branch and another scaffold branch.The system relates distances from both admixed populations to the unadmixed references, plus the distance between the admixed populations, to two mixture fractions and six branch lengths.Source: Three-way admixture fitting · pp. 21–22
• Although the framework could extend to more ancestral admixture events, this MixMapper version limits fitting to three-way admixtures to reduce overfitting risk.Source: Three-way admixture fitting · p. 22

Expressing branch lengths in drift units

• Although f2 distances are additive along unadmixed lineages and convenient for fitting, they are not directly interpretable like genetic drift D.Source: Expressing branch lengths in drift units · p. 22
• MixMapper converts each branch by dividing twice its f2 length by the inferred heterozygosity at the ancestral population at the branch’s top.The conversion corrects qualitatively for heterozygosity-related stretching of f2 branches; accurate ancestral heterozygosity inference requires SNPs ascertained in an outgroup.Source: Expressing branch lengths in drift units · p. 22
• MixMapper tests possible root branches and checks consistency among heterozygosity estimates from pairs of descendants on opposite sides of each internal node.It infers the root and ancestral heterozygosities before converting the fitted f2 branch lengths as a post-processing step.Source: Expressing branch lengths in drift units · pp. 22–23

Bootstrapping

• Each of 500 replicates resamples 50 genomic blocks and individuals within each population, then yields percentile 95% confidence intervals for branch lengths and mixture fractions.The intervals are the 2.5th and 97.5th percentiles across replicates.Source: Bootstrapping · p. 23

Evaluating fit quality

• Negative f3 values and comparisons across reference pairs provide an independent, simpler check on MixMapper’s inferred admixture and likely sources.The 3-population test uses similar data, but its simpler form can help confirm whether MixMapper results are reasonable.Source: Evaluating fit quality · p. 24
• Large bootstrap error bars, especially when inferred source branches change across replicates, signal less reliable fits.Uncertainty depends on sample sizes, SNP density, within-population homogeneity, and other data characteristics.Source: Evaluating fit quality · p. 24
• Fits with one source contributing over 90% are often unreliable, particularly when the minor source shifts branches, sits at a leaf tip, or seems historically implausible.This pattern can arise when a non-admixed population is fitted as an admixture.Source: Evaluating fit quality · pp. 24–25
• An inferred event must join contemporaneous mixing populations, but confounded terminal drift and branch lengths tied to both population size and time prevent a definitive check in every case.An event can be rejected when no realization of the variables could satisfy contemporaneity, such as a very recent and a very old branch paired with small total mixed drift.Source: Evaluating fit quality · p. 25
• Historical or other external knowledge can flag best-fitting models as implausible, and novel, surprising findings receive extra scrutiny.Such cases may also trigger one or more of the other fit-evaluation criteria.Source: Evaluating fit quality · p. 25

Data set and ascertainment

• The data comprise 934 HGDP individuals in 53 populations genotyped on the Affymetrix Axiom Human Origins Array, which was designed for population-genetic inference.The authors note that ascertainment bias can distort estimated population divergences.Source: Data set and ascertainment · p. 25
• The authors favor a priori bias-free data because ascertainment based on modern-population properties can distort divergence estimates and typical array ascertainment schemes may be opaque.Methods to correct ascertainment bias exist, but the paper regards using suitable data as preferable.Source: Data set and ascertainment · p. 26
• Panel 4 contains 163,313 SNPs ascertained as heterozygous in a San individual; treating the San as an outgroup makes ascertainment nearly neutral for the other populations.They exclude 61,369 gene-region SNPs because their analyses assume neutrality and these loci show a frequency-spectrum shift indicative of selection.Source: Data set and ascertainment · p. 26

Simulations

• For the second simulated data set, TreeMix was fit with five admixtures, chosen using pairwise-distance residuals of about three standard errors and the known number of admixtures in the true tree.The analyzed data set contained 96,258 SNPs after ascertainment.Source: Simulations · p. 27

Software

• MixMapper source code is available from the project website.Source: Software · p. 27

Figures

• Figure 1 shows the workflow: select an unadmixed scaffold from trees ranked by f2-additivity, then fit remaining populations as two- or three-way mixtures with bootstrap ensembles.The scaffold selection is user-driven, while bootstrap resampling supports confidence estimation.Source: Figure 1 · p. 32
• Figure 2 diagrams fitted branch locations, ancestry proportions, and mixed drift for two-way admixture, then adds parameters for a three-way mixture.In the two-way case, α is ancestry from Branch1 and β = 1 − α from Branch2; MixedDrift combines the drift lengths.Source: Figure 2 · p. 33
• Figure 3 juxtaposes true histories and MixMapper and TreeMix fits for one- and four-admixed-population simulations; continuous 40-generation gene flow is depicted as point admixture by both methods.The terminal drift edges for admixed populations in the MixMapper panels represent half the total mixed drift.Source: Figure 3 · p. 34
• Figure 4 compares a neighbor-joining tree for 30 high-confidence HGDP populations with MixMapper’s tree, where dashed lines mark admixture and drift lengths are not additive.The majority of populations in the MixMapper tree are inferred to be admixed.Source: Figure 4 · p. 35
• Figure 5 depicts Sardinian admixture from branches related to the common ancestors of Native Americans and all non-Africans, with a map sketching possible ancestral movements.The branch lengths a, b, and c are confounded, so the figure shows one plausible combination.Source: Figure 5 · p. 36
• Figure 6 marks direct modern-population heterozygosities in black and inferred ancestral values in green, and compares ancestral estimates from Illumina and San-ascertained SNPs.For unbiased data, pairs sharing a common ancestor should have equal ancestral heterozygosity values.Source: Figure 6 · p. 37

• Table 1 reports European mixture parameters, including ancient northern Eurasian ancestry proportions, branch split locations, and summed drift lengths.Source: Table 1 · p. 38, Tables · p. 38
• Table 2 reports non-European two-way admixture parameters, including optimal split points, bootstrap support, ancestry proportions, branch locations, and summed drift lengths.Source: Table 2 · p. 39, Tables · p. 39
• Table 3 reports three-way admixture parameters, including the third component’s split point, topology support, European-related ancestry proportion, and terminal drift.The notes also identify the common ancestral branch of five East Asian populations.Source: Table 3 · p. 40, Tables · p. 40

of admixture parameters and sources of gene flow

• European estimates remain similar under the alternative scaffold, with slightly higher α and reduced support for the ancient Eurasian pairing, while overall conclusions remain robust.Most remaining bootstrap replicates place the first ancestral population along the Pima branch, a pattern the authors relate to evidence of European-related admixture in Pima.Source: Table S2 · p. 49, of admixture parameters and sources of… · p. 49
• For non-European two-way fits, α is roughly 20% higher for the first four populations, Melanesian is fit essentially identically, and Han appears nearly unadmixed.The authors suspect Han’s fit reflects removal of Japanese, leaving no appropriate northern East Asian population related to one ancestor.Source: Table S3 · p. 50, of admixture parameters and sources of… · p. 50
• Three-way admixture parameters under the alternative scaffold are quite similar to the original estimates, with only some relative changes in bootstrap support among alternative topologies.In all cases, one parent splits from the admixed Sardinian branch and the other from Branch3.Source: Table S4 · p. 51, of admixture parameters and sources of… · p. 51
• An f4-ratio calculation was used to validate Sardinian and Basque mixture proportions, with standard errors from 500 bootstrap replicates and a topology assumption inherited from MixMapper.The estimated fraction is the ancient northern Eurasian ancestry proportion; the calculation uses Asian, European, American, and outgroup populations.Source: Table S5 · p. 52, of admixture parameters and sources of… · p. 52
• Because f2 distances are additive on unadmixed drift trees but not for admixed populations, MixMapper fits admixture parameters using distance equations, with two distinct outgroups sufficient to identify key quantities.The system solves for the ancestry fraction, split locations, and combined drift term; for fixed α, it is linear in the other variables.Source: of admixture parameters and sources of… · pp. 55, 57
• Because f2 values depend on SNP frequency distributions, non-adjacent tree edges are not directly comparable; the supplement derives a heterozygosity-based conversion to absolute drift lengths.The authors note that the underlying equation is equivalent to one in Nei (1987), but say it had not previously been applied to estimate ancestral heterozygosity.Source: of admixture parameters and sources of… · pp. 58–59
• ˆd is a nearly unbiased moment estimator of drift: infer ancestral heterozygosity from f2 tree lengths, then divide to convert branch lengths into absolute drift.Ancestral heterozygosity can be estimated from a present-day population and its inferred f2 distance to an ancestor.Source: of admixture parameters and sources of… · p. 60
• For a terminal admixed branch, twice the inferred mixed drift is divided by estimated heterozygosity plus twice that drift; this approximation should have very small error.The conversion uses the present-day population’s heterozygosity for three disjoint branches.Source: of admixture parameters and sources of… · p. 60
• For the specified biallelic-locus formula, FST is exactly half of ˆd; as a general rule, drift estimates are approximately twice reported FST values.The comparison concerns FST calculated as (ΠD − ΠS)/ΠD.Source: of admixture parameters and sources of… · p. 61