Enhancing the mathematical properties of new haplotype homozygosity statistics for the detection of selective sweeps

Abstract

Soft selective sweeps represent an important form of adaptation in which multiple haplotypes bearing adaptive alleles rise to high frequency. Most statistical methods for detecting selective sweeps from genetic polymorphism data, however, have focused on identifying hard selective sweeps in which a favored allele appears on a single haplotypic background; these methods might be underpowered to detect soft sweeps. Among exceptions is the set of haplotype homozygosity statistics introduced for the detection of soft sweeps by Garud et al. (2015). These statistics, examining frequencies of multiple haplotypes in relation to each other, include $H_{12}$, a statistic designed to identify both hard and soft selective sweeps, and $H_2/H_1$, a statistic that conditional on high $H_{12}$ values seeks to distinguish between hard and soft sweeps. A challenge in the use of $H_2/H_1$ is that its range depends on the associated value of $H_{12}$, so that equal $H_2/H_1$ values might provide different levels of support for a soft sweep model at different values of $H_{12}$. Here, we enhance the $H_{12}$ and $H_2/H_1$ haplotype homozygosity statistics for selective sweep detection by deriving the upper bound on $H_2/H_1$ as a function of $H_{12}$, thereby generating a statistic that normalizes $H_2/H_1$ to lie between 0 and 1. Through a reanalysis of resequencing data from inbred lines of Drosophila, we show that the enhanced statistic both strengthens interpretations obtained with the unnormalized statistic and leads to empirical insights that are less readily apparent without the normalization.

Introduction

A selective sweep, the process whereby beneficial mutations at a locus that contribute to the fitness of an organism rise in frequency to become prevalent in a population, can occur through two main mechanisms that leave distinct genomic signatures (Pritchard et al., 2010, Cutter and Payseur, 2013, Messer and Petrov, 2013). A relatively new adaptive allele can proliferate so that the single haplotype on which it has occurred reaches a high frequency, resulting in a signature of a “hard” selective sweep (Maynard Smith and Haigh, 1974, Kaplan et al., 1989, Kim and Stephan, 2002). Alternatively, a mutation that arises de novo multiple times or exists as standing genetic variation on several haplotype backgrounds before the onset of positive selection can increase in frequency; in these cases, multiple favored haplotypes have relatively high frequencies, generating a signature of a “soft” selective sweep (Hermisson and Pennings, 2005, Przeworski et al., 2005, Pennings and Hermisson, 2006a). Soft sweeps can provide an effective mechanism for natural selection and might explain a sizeable fraction of selective events in many systems (Orr and Betancourt, 2001, Innan and Kim, 2004, Pritchard et al., 2010, Messer and Petrov, 2013).

Most statistical methods that have been designed to detect selective sweeps from patterns of genetic polymorphism search for patterns expected under a hard-sweep model, such as the presence of a single common haplotype (Hudson et al., 1994), high haplotype homozygosity (Depaulis and Veuille, 1998, Sabeti et al., 2002, Voight et al., 2006), high-frequency derived variants and related features of site-frequency spectra (Tajima, 1989, Braverman et al., 1995, Fay and Wu, 2000, Nielsen et al., 2005), or local loss of variation near a putative selected site (Maynard Smith and Haigh, 1974, Begun and Aquadro, 1992, Kim and Stephan, 2002). Many methods that search for patterns expected with hard sweeps, however, can be less well suited to the problem of identifying soft sweeps (Pennings and Hermisson, 2006b, Teshima et al., 2006, Cutter and Payseur, 2013). Therefore, current genomic scans for selective sweeps might be limited in their ability to uncover an important class of adaptive events.

Recently, it has been shown that statistics based on haplotype homozygosity can identify both hard and soft sweeps from population-genomic data (Ferrer-Admetlla et al., 2014, Garud et al., 2015). Garud et al. (2015) developed a haplotype homozygosity statistic, $H_{12}$, relying on the principle that in a soft sweep, the most frequent haplotype might not predominate in frequency, and instead, multiple frequent haplotypes might be present. In terms of frequencies $p_i\ge 0$ for $i=1,2,3,\dots$ with ${\sum }_{i=1}^{\infty }{p}_i=1$ and $p_1\ge p_2\ge p_3\ge \cdots$, Garud et al. (2015) defined $H_{12}$ as

$$H_{12}={(p_1+p_2)}^2+\sum _{i=3}^{\infty }{p}_i^2\text{.}$$

This statistic calculates homozygosity by combining the two largest haplotype frequencies into a single value and then computing a haplotype homozygosity. Garud et al. (2015) determined that $H_{12}$ has reasonable power to detect both hard and soft sweeps, applying the statistic to Drosophila population-genomic data and identifying abundant signatures of natural selection.

To determine whether the genomic regions with the highest values of $H_{12}$ were compatible with either a hard-sweep or soft-sweep pattern, Garud et al. (2015) examined a second statistic, $H_2/H_1$, a ratio of a haplotype homozygosity $H_2$ that excludes the most frequent haplotype and a haplotype homozygosity $H_1$ that includes this haplotype:

$$H_1=p_1^2+p_2^2+\sum _{i=3}^{\infty }{p}_i^2$$$$H_2=p_2^2+\sum _{i=3}^{\infty }{p}_i^2\text{.}$$

For high values of $H_{12}$, hard sweeps are expected to produce relatively low values of $H_2/H_1$ because they produce a single high-frequency haplotype (very high $p_1$, low $p_2$). Soft sweeps, on the other hand, produce multiple high-frequency haplotypes (high $p_1$, $p_2$, and perhaps others), and are expected to produce higher values of $H_2/H_1$.

Garud et al. (2015) found that this two-step process–identification of regions with high $H_{12}$ followed by examination of $H_2/H_1$–could both detect selective sweeps in general and distinguish hard and soft sweeps. As we will show, however, a complication in the approach is that the permissible range of $H_2/H_1$ varies with the value of $H_{12}$. Thus, the magnitude of $H_2/H_1$ that might be regarded as indicative of a soft or hard sweep can depend on the associated values of $H_{12}$. This potential difference in interpretations for values of $H_2/H_1$ as a function of $H_{12}$ can present a particular challenge when comparing $H_2/H_1$ at multiple loci with a wide range of $H_{12}$ values.

In a line of work separate from the use by Garud et al. (2015) of homozygosity-based soft sweep statistics, Rosenberg and Jakobsson (2008) and Reddy and Rosenberg (2012) analyzed the properties of homozygosity statistics in relation to the frequency of the most frequent allele, identifying upper and lower bounds on homozygosity given the frequency of the most frequent allele. This work, along with related work on other statistics (Long and Kittles, 2003, Hedrick, 2005, Jost, 2008, VanLiere and Rosenberg, 2008, Maruki et al., 2012, Jakobsson et al., 2013), seeks to understand mathematical bounds on population-genetic statistics, so that their application and interpretation can be suitably informed by the mathematical constraints on their numerical values.

Here, to facilitate the interpretation of the statistics of Garud et al. (2015) and to enhance comparisons among values of these statistics at loci with different haplotype homozygosities, we use a result from Rosenberg and Jakobsson (2008) to determine the upper and lower bounds on $H_2/H_1$ as a function of $H_{12}$. The upper bound provides a basis for normalization of $H_2/H_1$ to produce a statistic with the same range, from 0 to 1, irrespective of the value of $H_{12}$. Using the upper bound and the new normalized statistic, we reexamine Drosophila data analyzed by Garud et al. (2015), demonstrating that the upper bound, ${(H_2/H_1)}_{\max }$, and the normalized statistic, ${(H_2/H_1)}^{\prime }$, enable improved insights regarding soft selective sweeps on the basis of genetic polymorphism data.