March 22, 2018
The sample frequency spectrum (SFS), or histogram of allele counts, is an important summary statistic in evolutionary biology, and is often used to infer the history of population size changes, migrations, and other demographic events affecting a set of populations. The expected multipopulation SFS under a given demographic model can be efficiently computed when the populations in the model are related by a tree, scaling to hundreds of populations. Admixture, back-migration, and introgression are common natural processes that violate the assumption of a tree-like population history, however, and until now the expected SFS could be computed for only a handful of populations when the demographic history is not a tree. In this article, we present a new method for efficiently computing the expected SFS and linear functionals of it, for demographies described by general directed acyclic graphs. This method can scale to more populations than previously possible for complex demographic histories including admixture. We apply our method to an 8-population SFS to estimate the timing and strength of a proposed “basal Eurasian” admixture event in human history. We implement and release our method in a new open-source software package momi2.
[Introduction, Method and Background sections omitted. Please see the paper for these sections.]
We tested our method on a demographic inference problem in human genetics that is currently of interest. Lazaridis et al. (2014) showed that genetic variation in present-day Europeans suggests an admixture model involving three ancestral meta-populations: Ancient North Eurasian (ANE), Western Hunter Gatherers (WHG), and Early European Farmers (EEF). They also showed that EEF contains ancestry from a source that is an outgroup to all non-African populations, and yet shares much of the genetic drift common to non-African populations; they dubbed this ancestry component as “Basal Eurasian” ancestry. Later work (Lazaridis et al., 2016) showed that Basal Eurasian ancestry is shared by ancient and contemporary Middle Eastern populations, and is correlated with a decrease in Neanderthal ancestry, implying that Basal Eurasian ancestry contains lower levels of Neanderthal admixture when compared with non-Basal ancestry. The results from Lazaridis et al. (2014, 2016) were based on several related methods for modeling covariances in population allele frequencies, most notably qpGraph and qpAdm (Patterson et al., 2012; Haak et al., 2015). These methods are computationally efficient and robust, but are unable to infer the timing of demographic events.
We applied momi2 to estimate the strength and timing of basal Eurasian admixture into early European farmers, and the split time of the basal Eurasian lineage. To do this, we built a demographic model relating 12 samples from 8 populations, shown in Figure 2. These samples consisted of the Altai Neanderthal (Prüfer et al., 2014); the 45,000 year old Ust’Ishim man from Siberia (Fu et al., 2014); 3 present-day populations (Mbuti, Sardinian, Han) with 3, 2, and 2 samples respectively; and 3 ancient samples representing the European ancestry components identified by Lazaridis et al. (2014): a 7,500 year old sample from the Linearbandkeramik (LBK) culture (representing EEF), an 8,000 year old sample from the Loschbour rock shelter in Luxembourg (representing WHG), and the 24,000 year old Mal’ta boy (“MA1”) from Siberia (representing ANE). After data cleaning, our dataset consisted of 2:4 106 autosomal transversion SNPs. See Appendix A.3 for more details about the data.
To construct the topology of the model in Figure 2, we first obtained a tree by neighbor joining (Saitou and Nei, 1987), then added 3 extra admixture events reflecting prior knowledge, as well as a recent Neanderthal population decline starting at the Mbuti-Eurasian split. We inferred split times, population sizes (including the Neanderthal decline), and admixture times and proportions by maximizing a composite likelihood, given by the product of the likelihoods at every SNP:
and was computed by momi2. The low-coverage of the MA1 sample and the deep divergence of the Neanderthal sample may cause bias in SFS entries where only these samples contain derived alleles; we thus excluded these entries and corrected the normalizing constant appropriately (see Appendix A.3).
We used automatic differentiation to compute the gradient (, which we used to search for the optimum of log likelihood. We constructed nonparametric bootstrap confidence intervals by splitting the genome into 100 equally sized blocks, resampling these blocks to create 300 bootstrap datasets, and re-inferring the demography for each bootstrap dataset. We also used 300 parametric bootstraps to assess how well we could infer the demography under simulated data; for each parametric bootstrap dataset, we used msprime (Kelleher et al., 2016) to simulate ten 300 Mb chromosomes from our initial point estimate, and re-inferred the demography. Note the nonparametric bootstrap is better able to account for model misspecification, and we use it for all confidence intervals reported below.
Our inferred demography, along with nonparametric bootstrap re-estimates, are shown in Figure 2 and Table 2. Our parametric bootstrap estimates are shown in Figure 3. We inferred a pulse of 0.094 (95% CI of 0.049-0.174) from the ghost Basal Eurasian population to EEF ancestry (LBK), substantially less than the 0.44 inferred by (Lazaridis et al., 2014). This admixture was inferred to occur 33.7 kya (95% CI of 10.8-41.1 kya), shortly after the Loschbour-LBK split at 37.7 kya (95% CI of 32.2-42.3 kya). The split time of the ghost Basal Eurasian lineage from other Eurasians was inferred at 79.8 kya (95% CI of 67.4-101 kya). Other parameters were broadly in line with previous estimates, such as a Mbuti-Eurasian split of 96 kya, a Han-European split of 50 kya, a Neanderthal split of 696 kya, and Eurasians deriving 0.03 of their ancestry from Neanderthal (Terhorst et al., 2017; Green et al., 2010; Meyer et al., 2016).
[Please see the paper for the remainder of the Application discussion.]