Struct-f4: a Rcpp package for ancestry profile and population structure inference from f₄-statistics

Abstract

Summary

Visualization and inference of population structure is increasingly important for fundamental and applied research. Here, we present Struct-f4, providing automated solutions to characterize and summarize the genetic ancestry profile of individuals, assess their genetic affinities, identify admixture sources and quantify admixture levels.

Availability and implementation

Struct-f4 is written in Rcpp and relies on f₄-statistics and Markov Chain Monte Carlo (MCMC) optimization. It is freely available under GNU General Public License in Bitbucket (https://bitbucket.org/plibradosanz/structf4/).

Supplementary information

Supplementary data are available at Bioinformatics online.

1 Introduction

Next-generation sequencing has opened for the routine characterization of genome variation at the population scale, including in non-model organisms, which provides invaluable insights into evolution (Nielsen et al., 2017). Enhanced characterization of population structure has also found a range of applications in medicine, forensics, conservation biology and more. Many statistical methods are available to visualize (e.g. Principle Component Analysis; Patterson et al., 2006) and model population structure, e.g. as combinations of K ancestry clusters (e.g. ADMIXTURE; Alexander et al., 2009). However, these methods can be biased by the amount of genetic drift exclusive to single populations (Lawson et al., 2018). Other methods based on shared drift aimed to overcome such limitations and increasingly gained popularity in the last decade (Patterson et al., 2012). For example, qpGraph and qpAdm leverage patterns of allele sharing between population quartets (the so-called f₄-statistics) to model evolutionary histories, including admixture coefficients. This methodology is, however, highly supervised through the specification of homogeneous groups, potentially acting as admixture sources, and requires to assess alternative models individually. This becomes practically challenging as the number of populations and/or admixture events increase, rapidly exceeding the current capacity of automated solutions (Leppälä et al., 2017).

To remediate this situation, we developed Struct-f4, a package leveraging the power of f₄-statistics and automating statistical inference within an MCMC framework. Struct-f4 first estimates the shared drift across pairs of individuals, allowing visualization of population structure through Multi-Dimensional Scaling (MDS). It also models individual genetic profiles as mixtures from K ancestral populations, not assumed to follow Hardy–Weinberg equilibrium, and accommodates both supervised and unsupervised analyses.

2 Materials and methods

Struct-f4 was originally proposed by Fages et al. (2019) to visualize the genetic structure within ancient and modern horse populations. The methodology involved maximum likelihood optimization to place individuals within the 3D-Euclidean space that best fits the observed combination of f₄-statistics. Here, we redesigned the underlying statistical model to retrieve direct estimates of the shift in allele frequency that occurred between pairs of individuals, as follows:

$$f_4\left(\mathrm{H}1,\mathrm{ }\mathrm{H}2;\mathrm{ }\mathrm{H}3,\mathrm{ }\mathrm{H}4\right)=\left(p_{H1}-p_{H2}\right)\left(p_{H3}-p_{H4}\right)=d_{H1H2}{d}_{H3H4},$$

where d_H1H2 the difference in allele frequency between individuals H₁ and H₂. Assuming f₄-statistics follow normal distributions, the likelihood of the d_ij parameters can be calculated, allowing for their optimization within an adaptive Metropolis-Hastings MCMC framework (Supplementary Information). This approach can be generalized to model individual profiles as mixtures of K ancestral populations, where K is user-defined

$$f_4\left(\mathrm{H}1,\mathrm{ }\mathrm{H}2;\mathrm{ }\mathrm{H}3,\mathrm{ }\mathrm{H}4\right)=\left(\sum _{i=1}^K{Q}_{\mathrm{i}¯H1}\sum _{j=1}^K{Q}_{j¯H2}{d}_{ij}\right)\left(\sum _{i=1}^K{Q}_{i¯H3}\sum _{j=1}^K{Q}_{j¯H4}{d}_{ij}\right)$$

and d_ij now represents the allele frequency shift that occurred between the ancestral components i and j, while Q_{i_H1} the proportion of the i ancestry inherited by individual H₁ (i.e. its mixture coefficient).

Struct-f4 is implemented in Rcpp for computational efficiency and requires a matrix of f₄-statistics as input. We also provide the Calc-f4 C program, which was parallelized and optimized for fast computation of f₄-statistics. This reduces the running time to calculate 82 215 f₄-statistics on a single 2700 MHz core to 48ʹ49″, versus 1011ʹ11″ for qpDstat. Struct-f4 outputs posterior mean values and credible intervals for each estimated parameter, together with the full MCMC sample and the corresponding probability used to assess convergence. It also provides (i) an MDS plot of genetic affinities between individuals (Supplementary Fig. S1), (ii) an unsupervised clustering based on the allele frequency shifts that occurred across pairs of individuals and/or K ancestral populations (Supplementary Fig. S2) and (iii) a barplot representation of ancestry profiles (Fig. 1).

Fig. 1.

(a) Simulated tree including 18 populations, and three sampled individuals per population. One admixture was simulated (m) into population 14 [0,0.25]. (b) Admixture proportions estimated by Struct-f4. For consistency with Harney et al. (2021), we simulated the full population history but restricted inference to 10 populations only. The name of each sample is composed of its corresponding population of origin, concatenated to a unique sample identifier. Samples from population 14 show an increasing ancestry from populations 0, 5 and 7 (colored in red) for greater simulated introgression proportions, as expected, and despite the true donor population remained unsampled

3 Results

We evaluated Struct-f4 using the same simulation framework as that implemented by (Harney et al., 2021) for assessing qpAdm performance. Individuals simulated as belonging to populations either closely related or increasingly connected by gene-flow appeared next to each other in the MDS space. Each individual was found to cluster according to the phylogenetically closest cladal group in the simulated model and showed genetic profiles consistent with the intensity of admixture (Fig. 1). Slight underestimates of the admixture proportions were returned for unsupervised analyses, if sampling only three haploid individuals per population. Supervised inference, nevertheless, completely fixed this bias (Fig. 1). Therefore, Struct-f4 can be used even when sampling efforts are limited, advantageously expanding the analytical toolkit in statistical genomics by providing a robust, flexible and user-friendly platform to automatically characterize population genetic structure. We successfully applied Struct-f4 to characterize the population structure underlying 284 ancient and modern horse genomes from over 11 million permutations of f₄-statistics (Librado et al., 2021).

Funding

This work was received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme [Grant 681605].

Conflict of Interest: none declared.