Can the site-frequency spectrum distinguish exponential population growth from multiple-merger coalescents?

Abstract

The ability of the site-frequency spectrum (SFS) to reflect the particularities of gene genealogies exhibiting multiple mergers of ancestral lines as opposed to those obtained in the presence of population growth is our focus. An excess of singletons is a well-known characteristic of both population growth and multiple mergers. Other aspects of the SFS, in particular, the weight of the right tail, are, however, affected in specific ways by the two model classes. Using an approximate likelihood method and minimum-distance statistics, our estimates of statistical power indicate that exponential and algebraic growth can indeed be distinguished from multiple-merger coalescents, even for moderate sample sizes, if the number of segregating sites is high enough. A normalized version of the SFS (nSFS) is also used as a summary statistic in an approximate Bayesian computation (ABC) approach. The results give further positive evidence as to the general eligibility of the SFS to distinguish between the different histories.

Keywords: approximate Bayesian computation; approximate maximum likelihood test; coalescent; multiple mergers; population growth; site-frequency spectrum.

Figure 1

Matching ${\varphi }_{\text{1}}^{(n\text{,Π})}$ [see Equation 2] for the different coalescent processes Π ∈ {A, B, D, E} with number of leaves n as shown. Expected values were computed exactly. The processes and their associated parameters are algebraic growth (A, γ), beta(2 − α, α)-coalescent (B, α), Dirac coalescent (D, ψ), and exponential growth (E, β). The values with label 5+ represent the collapsed tail ${\displaystyle {\sum }_{i>\text{5}}{\varphi }_i^{\left(n\text{,Π}\right)}}$.

Figure 2

(A) Estimate of ${\tilde{G}}_{\left(\text{E,B;}s\right)}$ from Equation 10 based on the approximate likelihood from Equation 12 as a function of α (no lumping) with number of leaves n as shown and s = 50. (B) Estimate of ${\tilde{G}}_{\left(\text{B,E;}s\right)}$ from Equation 10 based on the approximate likelihood from Equation 12 as a function of β (no lumping) with number of leaves n as shown and s = 50. The symbols denote the size of the test, as shown in the legend. The interval hypotheses are discretized to ${\mathrm{\Theta }}_s^{\text{E}}=\left\{\beta :\beta \in \left\{\text{0,1,2,…,10,20,…,1000}\right\}\right\}$ and. ${\text{Θ}}_s^{\text{B}}=\left\{\alpha :\alpha \in \left\{\text{1,1.025,…,2}\right\}\right\}$. In A, the beta(2 − α, α)-coalescent is the alternative; in B, exponential growth is the alternative.

Figure 3

The ${\ell }_{\text{2}}$ distance $d_{\text{2}}^{\left(n\right)}\left(\text{E,X}\right)$ for $\text{X}\in \left\{\text{B,D}\right\}$ of the normalized expected spectra ${\varphi }_i^{\left(n\text{,E}\right)}$ [see Equation 2] and ${\varphi }_i^{\left(n\text{,Π}\right)}$ as a function of α (X = B) [resp. ψ (X = D)] and β (E) for number of leaves n = 100. Expected values were computed exactly. The grid points are α ∈ {1,1.025,…,2} and $\psi \in \left\{\text{0.01,0.02,}\dots \mathrm{,}\text{0.1,0.15,0.2,}\dots ,\text{0.95}\right\}$; for β as shown.

Figure 4

The ${\ell }_2$ distance $d_{\text{2}}^{\left(n\right)}\left(\text{A,X}\right)$ for X ∈ {B, D} of the normalized expected spectra ${\varphi }_i^{\left(n\text{,A}\right)}$ [see Equation 2] and ${\varphi }_i^{(n\text{,X})}$ as a function of α (X = B) [resp. ψ (X = D)] and γ (A) for number of leaves n = 100. Expected values were computed exactly. The grid points are $\psi \in \left\{\text{0.01,0.02,}\dots ,\text{0.1,0.15,0.2,}\dots ,\text{0.95}\right\}$ and α ∈ {1, 1.025, …, 2}; for γ as shown.