Using molecular markers for detecting domestication, improvement, and adaptation genes

Abstract

Development of statistical tests to detect selection (strictly speaking, departures from the neutral equilibrium model) has been an active area of research in population genetics over the last 15 years. With the advent of dense genome sequencing of many domesticated crops, some of this machinery (which heretofore has been largely restricted to human genetics and evolutionary biology) is starting to be applied in the search for genes under recent selection in crop species. We review the population genetics of signatures of selection and formal tests of selection, with discussions as to how these apply in the search for domestication and improvement genes in crops and for adaptation genes in their wild relatives. Plant domestication has specific features, such as complex demography, selfing, and selection of alleles starting at intermediate frequencies, that compromise many of the standard tests, and hence the full power of tests for selection has yet to be realized.

Appendix: Details of frequency spectrum-based tests

Fu and Li’s D ^* and F ^* tests

Table 1 provides three different estimators of θ under the infinite-sites model. Tajima’s D is based on the contrast between two of these, but this leaves two other contrasts, which Fu and Li (1993) used as the basis for two new tests. Their D ^* test compares the segregating sites (S) versus singletons (η) estimator of θ,

$$ D^{\ast}=\frac{\widehat{\theta}_S-\widehat{\theta}_{\eta}} {\sqrt{\alpha_{\ast}S +\beta_{\ast} S^2}} $$

(A1a)

$$ \alpha_{\ast}=\frac{1}{a_n}\left({\frac{n+1}{n}-\frac{1}{a_n}}\right)-\beta_{\ast} $$

(A1b)

$$ \beta_{\ast} =\frac{1}{a_n^2+b_n}\left({\frac{b_n}{a_n^2}-\frac{2}{n}\left({1 + \frac{1}{a_n}-a_n+\frac{a_n}{n}}\right)- \frac{1}{n^2}}\right). $$

(A1c)

In contrast, their F ^* test compares the average pair-wise divergence (k) versus singletons (η) estimator of θ,

$$ F^{\ast}= \frac{\widehat{\theta}_k-\widehat{\theta}_{\eta}}{\sqrt{\alpha_FS +\beta_F S^2}} $$

(A2a)

$$ \alpha_F= \frac{1}{a_n}\left({\frac{4n^2+19n+3-12(n+1)a_{n+1}}{3n(n-1)}}\right) -\beta_F $$

(A2b)

$$ \beta_F =\frac{1}{a_n^2+b_n}\left({\frac{2n^4+110n^2-255n+153}{9n^2(n-1)} + \frac{2(n-1)a_n}{n^2} - \frac{8b_n}{n}} \right). $$

(A2c)

These expression are from Simonsen et al. (1995), with Eq. A2c correcting a typo in the original Fu and Li paper. Critical values for both tests are tabulated by Fu and Li (1993). While these tests are fairly widely used, Simonsen et al. (1995) found that they are not as powerful as Tajima’s test for detecting a selective sweep or population structure departures (bottlenecks or population subdivision). However, Fu (1997) found that both tests have more power than Tajima’s D for detecting signals of background selection.

Fu’s W and F _S tests

Fu (1996, 1997) proposed several more refined tests for specific settings, such as too few alleles or too many alleles. These tests use the infinite alleles (as opposed to infinite sites) framework for sequence analysis (see Fig. 4). To develop these, we first need to introduce Ewen’s Sampling Formula (Evens 1972), which gives the probability (under the infinite alleles model) that we see K alleles (haplotypes) in a sample of size n as

$$ \Pr( K=k)=\frac{|\,S^k_n|\theta^k}{S_n(\theta)} $$

(A3a)

where

$$ S_n(\theta)=\theta(\theta+1)(\theta+2) \cdots (\theta+n-1) $$

(A3b)

and S ^k _n is the coefficient on the θ^k term in the polynomial given by S _n (θ). (S ^k _n is called a Stirling number of the first kind). For example, the probability that only a single allele is seen in our sample is

$$ \Pr(K=1)=\frac{(n-1)!}{(\theta+1)(\theta+2) \cdots (\theta+n-1)} $$

(A4)

Using Eq. A3a, the mean and variance for the number of alleles can be found to be

$$ E(K) =1+ \theta\cdot \sum\limits_{j=2}^{n} \frac{1}{\theta+j-1} $$

(A5a)

and

$$ \sigma^2(K)=\theta\cdot \sum\limits_{j=1}^{n-1} \frac{j}{(\theta+j)^2} $$

(A5b)

Fu’s W test (1996) is based on Ewen’s sampling formula, and is as follows. Suppose we have an estimate $\widehat{\theta}$ of θ and we observe k alleles in our sample. The probability of seeing k (or fewer) alleles in our sample under the null hypothesis is just

$$ W=\Pr( K \le k)=\sum\limits_{i=1}^k \Pr( K= k|\widehat{\theta})= \sum\limits_{i=1}^k \frac{|S^k_n|\widehat{\theta}^{k}} {S_n(\widehat{\theta})} $$

(A6)

The W test uses the Watterson estimator $\widehat{\theta}=S/a_n$ so that

$$ S_n(\widehat{\theta})=S/a_n(S/a_n+1)(S/a_n+2) \cdots (S/a_n+n-1) $$

This is a test for a deficiency of rare alleles, and hence W is a one-sided test statistic. Fu (1996) showed that the W test is more powerful that Tajima’s D and Fu and Li’s D ^* and F ^* tests for detecting samples from a structured population (as also occurs with overdominant selection).

Fu’s F _S test (1997) is the complement of his W statistic, being a test for excess rare alleles. It starts by computing the probability of seeing m or more alleles in our sample,

$$ S^{\prime}=\Pr( K \ge m)=\sum\limits_{i=m}^n \frac{|S^m_n|\widehat{\theta}^{m}} {S_n(\widehat{\theta})} $$

(A7a)

but now using $\widehat{\theta}=k,$ the estimator of θ based on average number of pair-wise differences. Fu notes that S′ is not an optimal test statistic because its critical points are often too close to zero. Because of this, the test statistic S is the logit of S′,

$$ F_S=\ln \left({ \frac{S^{\prime}}{1-S^{\prime}}}\right) $$

(A7b)

F _S is negative when there is an excess of rare alleles (as occurs with an excess of recent mutations as would occur with a selective sweep or population expansion), with a sufficiently large negative value being evidence for selection. Hence, F _S is also a one-sided test statistic. Fu (1997) showed that F _S is more powerful that Tajima’s and Fu and Li’s tests for detecting population growth/selective sweeps. Conversely, Fu and Li’s tests are more powerful for detecting background selection.

Fay and Wu’s H test

Fay and Wu (2000) and Kim and Stephan (2000) note that a distinct signal is left by a selective sweep that is not left by background selection. Specifically, it is common to see alleles that have newly arisen by mutation at high frequency following a sweep (as they hitched along for the ride). With background selection, this feature is not expected. This is the basis for Fay and Wu’s H test, which disproportionately weights derived alleles at high frequency. Their test requires an outgroup so that one can access whether an allele occurs in the outgroup or has recently been derived by mutation. Such derived alleles are expected to be at lower frequency (as under neutrality, the frequency of an allele is a rough indicator of its age, with older alleles being more frequent). The test proceeds as follows. Let S _i denote the number of derived mutants foundi times in our sample of size n. For example, if there are five unique (derived) alleles, four alleles each appearing twice, and one allele appearing five times in our sample of size 18, then S ₁ = 5, S ₂ = 4, S ₅ = 1. The estimate of θ from the average pair-wise difference expressed in terms of the S _i is

$$ \widehat{\theta}_k=2\sum\limits_{i=1}^{n-1} \frac{S_i i(n-i)}{n(n-1)} $$

(A8a)

while an estimate of θ weighted by homozygosity is

$$ \widehat{\theta}_H=2\sum\limits_{i=1}^{n-1} \frac{S_i i^2}{n(n-1)} $$

(A8b)

Fay and Wu’s H test is given by the scaled difference of $\widehat{\theta}_H- \widehat{\theta}_k.$

Given that Fay and Wu’s test weights derived allele at high frequency, a significant H and D test is consistent with a selective sweep, while a significant D test, but not a significant H test suggests background selection or demographic features more likely account for the departure from neutrality. While widely used, the H test is not without problems. While it is largely insensitive to population bottlenecks, it is highly sensitive to population structure. Further, the power of H rapidly decreases over time following a sweep, while the D test retains substantial power over a much longer time after a sweep (Przeworski 2002).

Hudson–Kreitman–Aguade (HKA) test

Consider two species (or distant populations) A and B that are at mutation-drift equilibrium with population sizes N _A  = N and N _B  = α N, respectively. Further assume they separated T = τ/(2N) generations ago from a common population of size $N^{\ast}=(N_A+N_B)/2=N(1+\alpha)/2,$ the average of the two current population sizes. Suppose $i=1, \cdots, L$ unlinked loci are examined in both species. The amount of polymorphism for locus i in A is a function of $\theta_i=4N_e\mu_i,$ while the amount of polymorphism for the same locus in B is a function of $4N_B\mu_i=4(\alpha N_e)\mu_i=\alpha\theta_i.$ The resulting summary statistics used are LS ^A _i values, for the number of segregating sites at locus i in A, another LS ^B _i for the same loci in B, and L D _i values, for the amounts of divergence (measured by the average number of differences between a random gamete from A and a random gamete from B). Given these 3L summary statistics, the HKA test statistic X ² is given by

$$ X^2=\sum\limits_{i=1}^L \frac{\left({S_i^A - \widehat{E}(S_i^A)}\right)^2}{\widehat{Var}(S_i^A)} + \sum\limits_{i=1}^L \frac{\left({S_i^B - \widehat{E}(S_i^B)}\right)^2} {\widehat{Var}(S_i^B)} + \sum\limits_{i=1}^L \frac{\left({D_i - \widehat{E}(D_ii)}\right)^2}{\widehat{Var}(D_i)} $$

(A9)

where for n _A samples from A and n _B samples from B,

$$ \widehat{E}(S_i^A)= \widehat{\theta_i}a_{n_A},\quad \widehat{E}(S_i^A)=\widehat{\alpha}\widehat{\theta_i}a_{n_B} $$

(A10a)

$$ \widehat{Var}(S_i^A) =\widehat{\theta_i}a_{n_A} + \widehat{\theta_i}^2b_{n_A}, \widehat{Var}(S_i^B )= \widehat{\alpha} \widehat{\theta_i}a_{n_A} + \widehat{\alpha}^2\widehat{\theta_i}^2b_{n_B} $$

(A10b)

$$ \widehat{D_i}= \widehat{\theta_i}\left({\widehat{T} + \frac{1+\widehat{\alpha}}{2}}\right) $$

(A10c)

$$ \widehat{Var}(D_i)=\widehat{\theta_i}\left({\widehat{T} + \frac{1+\widehat{\alpha}}{2}}\right) +\left({\frac{ \widehat{\theta_i}(1+\widehat{\alpha})}{2}}\right)^2 $$

(A10d)

and a _n and b _n are given by Eq. 2. Equations A10a and (A10b) follow from Eq. 3, while Eq. A10c follows by re-writing

$$ \theta_i\left({T + \frac{1+\alpha}{2}}\right)=4N\mu_i\left( {\frac{\tau}{2N} + \frac{1+\alpha}{2}}\right)=2\mu_i\tau + 4\mu_i \frac{N (1+\alpha)}{2} $$

where the first term is the between-population divergence due to new mutations and the second term the divergence from partitioning of the polymorphism $4N^{\ast}\mu_i$ in the ancestral population. Thus, the HKA test has L + 2 parameters to estimate, the $L\theta_i^x$ values and two demographic parameters, T and α. The HKA test estimates these parameters and then (using Eq. A10) computes the goodness of fit X ² statistic (Eq. A9), which is approximately χ² distributed with 3L − (L + 2) = 2L − 2 degrees of freedom.