Assortative mating by population of origin in a mechanistic model of admixture

Abstract

Populations whose mating pairs have levels of similarity in phenotypes or genotypes that differ systematically from the level expected under random mating are described as experiencing assortative mating. Excess similarity in mating pairs is termed positive assortative mating, and excess dissimilarity is negative assortative mating. In humans, empirical studies suggest that mating pairs from various admixed populations - whose ancestry derives from two or more source populations - possess correlated ancestry components that indicate the occurrence of positive assortative mating on the basis of ancestry. Generalizing a two-sex mechanistic admixture model, we devise a model of one form of ancestry-assortative mating that occurs through preferential mating based on source population. Under the model, we study the moments of the admixture fraction distribution for different assumptions about mating preferences, including both positive and negative assortative mating by population. We demonstrate that whereas the mean admixture under assortative mating is equivalent to that of a corresponding randomly mating population, the variance of admixture depends on the level and direction of assortative mating. We consider two special cases of assortative mating by population: first, a single admixture event, and second, constant contributions to the admixed population over time. In contrast to standard settings in which positive assortment increases variation within a population, certain assortative mating scenarios allow the variance of admixture to decrease relative to a corresponding randomly mating population: with the three populations we consider, the variance-increasing effect of positive assortative mating within a population might be overwhelmed by a variance-decreasing effect emerging from mating preferences involving other pairs of populations. The effect of assortative mating is smaller on the X chromosome than on the autosomes because inheritance of the X in males depends only on the mother's ancestry, not on the mating pair. Because the variance of admixture is informative about the timing of admixture and possibly about sex-biased admixture contributions, the effects of assortative mating are important to consider in inferring features of population history from distributions of admixture values. Our model provides a framework to quantitatively study assortative mating under flexible scenarios of admixture over time.

Keywords: Admixture; Assortative mating; Mechanistic models; X chromosome.

Figure A1:

The variance of ancestry as a function of the sex-specific contributions for a single admixture event in a randomly mating population. (A) g = 1. (B) g = 2. (C) g = 3. (D) g = 8. In each panel, $\mathbb{V}\left[H_{1,g,f}^X\right]$ is plotted over the range of permissible values of $s_{1,0}^f$ and $S_{1,0}^m$, using eq. (44).

Figure 1:

Schematic of the mechanistic model of assortative mating by population. Two source populations, S₁ and S₂, contribute females and males to the next generation of the hybrid population H, potentially with time-varying proportions. The fractional contributions of the source populations and the hybrid population to the next generation g are s_1,g, s_2,g, and h_g, respectively. Sex-specific contributions from the populations are $s_{1,g}^f$, $s_{2,q}^f$ and $h_g^f$, and $S_{1,g}^m$, $S_{2,g}^m$ and $h_g^m$, for females and males, respectively. $H_{\alpha ,g,\delta }^{\gamma }$ represents the fraction of admixture from source population α ∈ {1, 2} in generation g for a random individual of sex δ ∈ {f, m} in population H for chromosomal type γ ∈ {A, X}. Within the admixed population, at every generation, parents from generation g – 1 pair according to one of three mating models. Individuals from S₁ are represented by triangles, S₂ by pentagons, and H by squares. (A) Random mating. The probability of a pairing is given by the product of the proportional contributions of the two populations. (B) Positive assortative mating. Individuals are more likely to mate with individuals from their own population. (C) Negative assortative mating. Individuals are more likely to mate with individuals from a different population. In each panel, a mating pair is indicated by a pair of adjacent symbols. Each panel considers the same values for the contributions from the three populations to generation g + 1.

Figure 2:

Variance of ancestry under a single-admixture scenario with assortative mating in the founding generation. (A) Autosomes, $\mathbb{V}\left[H_{1,g,\delta }^A\right]$. (B) X chromosomes in a female, $\mathbb{V}\left[H_{1,g,f}^X\right]$. (C) X chromosomes in a male, $\mathbb{V}\left[H_{1,g,m}^X\right]$. Each panel considers positive assortative mating (green), c_11,0 = 0.1, negative assortative mating (purple), c_11,0 = −0.1, and random mating (black), c_11,0 = 0. The plots use eqs. (41), (43), and (44) with $s_{1,0}^f=s_{1,0}^m=0.5$. For positive assortment, the variance is higher than for random mating. For negative assortative mating, the variance is lower.

Figure 3:

Special case of the variance of ancestry under a single-admixture scenario with negative assortative mating in the founding generation. (A) Autosomes $\mathbb{V}\left[H_{1,g,\delta }^A\right]$. (B) X chromosomes in a female, $\mathbb{V}\left[H_{1,g,f}^X\right]$. (C) X chromosomes in a male, $\mathbb{V}\left[H_{1,g,m}^X\right]$. In each panel, the variance is plotted for negative assortative mating, c_11,0 = −0.25, using eqs. (41), (43), and (44) with $s_{1,0}^f=s_{1,0}^m=0.5$. The plot highlights a special case in which the variance of autosomal admixture is zero because the numerator of eq. (41) is zero, even though $s_{1,0}^f=s_{1,0}^m=0.5$ is a maximum of the variance with respect to the sex-specific contributions in a randomly mating population. The c_11,0 = 0 case is copied from Figure 2.

Figure 4:

Variance of ancestry under a single-admixture scenario as a function of the assortative mating parameter c_11,0. (A) Autosomes, $\mathbb{V}\left[H_{1,g,\delta }^A\right]$. (B) X chromosomes in a female, $\mathbb{V}\left[H_{1,g,f}^X\right]$. (C) X chromosomes in a male, $\mathbb{V}\left[H_{1,g,m}^X\right]$. In each panel, the variance is plotted over the range of c_11,0, [−0.25, 0.25], for values of g between 1 and 6 (from darkest to lightest gray), with $s_{1,0}^f=s_{1,0}^m=0.5$, using eqs. (41), (43), and (44).

Figure 5:

Variance of ancestry under a constant-admixture scenario with assortative mating in producing g ≥ 2. (A) Autosomes, $\mathbb{V}\left[H_{1,g,\delta }^A\right]$. (B) X chromosomes in a female, $\mathbb{V}\left[H_{1,g,f}^X\right]$. (C) X chromosomes in a male, $\mathbb{V}\left[H_{1,g,m}^X\right]$. Each panel considers positive assortative mating (greens), negative assortative mating (purples), and random mating (black dashed). The plots use eqs. (20), (21), and (29)–(32). We fix c₁₁ = c₂₂ and c_1h = 0, with $s_1^f=s_1^m=s_2^f=s_2^m=0.2$. For initial conditions, $s_{1,0}^f=s_{1,0}^m=s_{2,0}^f=s_{2,0}^m=0.5$ and c_11,0 = 0. Positive assortative mating increases the variance relative to a randomly mating population, whereas negative assortative mating decreases it.

Figure 6:

Example of the variance of ancestry under a constant-admixture scenario with assortative mating in producing g ≥ 2 when populations are allowed to differ in the direction of their mating preference. (A) Autosomes, $\mathbb{V}\left[H_{1,g,\delta }^A\right]$. (B) X chromosomes in a female, $\mathbb{V}\left[H_{1,g,f}^X\right]$. (C) X chromosomes in a male, $\mathbb{V}\left[H_{1,g,m}^X\right]$. In all scenarios, populations S₁ and H experience positive assortative mating, c₁₁,c_hh = 0.02. Green curves show a scenario in which S₂ also experiences positive assortative mating, c₂₂ = 0.06; purple curves show a scenario in which S₂ experiences negative assortative mating, c₂₂ = −0.04. The scenario of random mating with the same contribution parameters is the black dashed line. The plots use eqs. (20), (21), and (29)–(32). We set $s_1^f=s_1^m=s_2^f=s_2^m=0.2$, and use initial conditions $s_{1,0}^f=s_{1,0}^m=s_{2,0}^f=s_{2,0}^m=0.5$ and c_11,0 = 0. In this example, negative assortative mating in S₂ offsets the positive assortative mating in S₁ and H, producing lower variances of admixture in S₁, $\mathbb{V}\left[H_{1,g,\delta }^{\gamma }\right]$, in the assortatively mating population than in the randomly mating population.

Figure 7:

Timing of admixture under assortative mating. The number of generations since admixture, g, for a single admixture event is plotted as a function of c_11,0 using the variance in autosomal admixture, eq. (51), with V ^A = 0.0625 and $s_{1,0}^s=s_{1,0}^m=0.5$. The plot traverses the range of possible c_11,0 values for g ≥ 0. Recalling that the values of the c_ij,0 are bounded such that the probability of each given parental pairing takes its values in the interval [0, 1], and such that each probability is no greater than the probability of one of its constituent components, for c_11,0 > 0, we have $c_{11,0}\le \text{min}\left(s_{1,0}^f,s_{1,0}^m\right)-s_{1,0}^f{s}_{1,0}^m$. In this case, c_11,0 ≤ 0.25. For c_11,0 < 0, we have $c_{11,0}\ge s_{1,0}^f{s}_{1,0}^m$. In this case, we have c_11,0 ≥ −0.25. However, g cannot be negative; therefore, in this case, we truncate the domain at the value of c_11,0 that produces g = 0. The value of g for a randomly mating population with the same contributions and variance is shown in the dashed line.