Beyond 2/3 and 1/3: The Complex Signatures of Sex-Biased Admixture on the X Chromosome

Abstract

Sex-biased demography, in which parameters governing migration and population size differ between females and males, has been studied through comparisons of X chromosomes, which are inherited sex-specifically, and autosomes, which are not. A common form of sex bias in humans is sex-biased admixture, in which at least one of the source populations differs in its proportions of females and males contributing to an admixed population. Studies of sex-biased admixture often examine the mean ancestry for markers on the X chromosome in relation to the autosomes. A simple framework noting that in a population with equally many females and males, two-thirds of X chromosomes appear in females, suggests that the mean X-chromosomal admixture fraction is a linear combination of female and male admixture parameters, with coefficients 2/3 and 1/3, respectively. Extending a mechanistic admixture model to accommodate the X chromosome, we demonstrate that this prediction is not generally true in admixture models, although it holds in the limit for an admixture process occurring as a single event. For a model with constant ongoing admixture, we determine the mean X-chromosomal admixture, comparing admixture on female and male X chromosomes to corresponding autosomal values. Surprisingly, in reanalyzing African-American genetic data to estimate sex-specific contributions from African and European sources, we find that the range of contributions compatible with the excess African ancestry on the X chromosome compared to autosomes has a wide spread, permitting scenarios either without male-biased contributions from Europe or without female-biased contributions from Africa.

Keywords: African-American genetics; X chromosome; admixture; mechanistic model; sex bias.

Figure 1

Sex-specific mechanistic model of admixture. At each generation g, females and males from each of two source populations, $S_1$ and $S_2,$ contribute to the admixed population, H. Contributions can vary in time. The fraction of admixture from source population $\alpha \in \left\{1,2\right\}$ for chromosomal type $\gamma \in \left\{A,X\right\}$ in an admixed individual of sex $\delta \in \left\{f,m\right\}$ is $H_{\alpha ,g,\delta }^{\gamma }.$ This model, by considering different chromosomal types, generalizes the model of Goldberg et al. (2014).

Figure 2

The mean of the X-chromosomal admixture fraction over time in females (red) and males (blue) from the admixed population, for a single admixture event (Equations 12 and 13). The mean X-chromosomal admixture (from source population 1) oscillates in approaching its limit (Equation 14). The limiting mean, shown in black, is the same for X chromosomes in both females and males. (A) $s_{1,0}^{\mathrm{f}}=0.25,$ $s_{1,0}^{\mathrm{m}}=\mathrm{0.5.}$ (B) $s_{1,0}^{\mathrm{f}}=0.5,$ $s_{1,0}^{\mathrm{m}}=\mathrm{0.25.}$

Figure 3

The difference between the expectation of the female X-chromosomal admixture fraction and its limit for $g\in \left[1,6\right]$ as a function of the difference between female and male contributions from $S_1$ for a single admixture event (Equation 15). As $g\to \mathrm{\infty },$ this quantity approaches zero. For small g, however, the limit $\left(2/3\right)s_{1,0}^{\mathrm{f}}+\left(1/3\right)s_{1,0}^{\mathrm{m}}$ (Equation 14) is a poor approximation. The difference oscillates so that the slope of the line is negative when g is odd and positive when g is even.

Figure 4

The expectation of the mean X-chromosomal and autosomal admixture fractions over time, with their associated limits, for constant ongoing admixture. (A) Male-biased admixture from population $S_1.$ (B) Female-biased admixture from population $S_1.$ The initial condition is $\left(s_{1,0}^{\mathrm{f}},s_{1,0}^{\mathrm{m}},s_{2,0}^{\mathrm{f}},s_{2,0}^{\mathrm{m}}\right)=\left(0.5,0.5,0.5,0.5\right).$ The autosomal admixture is constant over time because $s_{1,0}=s_{2,0}=1/2$ and $s_1=s_2;$ it is the same in both A and B because it does not depend on the sex-specific contributions. The X-chromosomal admixture is different in females and males; it is smaller than the autosomal mean for male-biased admixture from population 1 and larger for female-biased admixture. The X-chromosomal mean is plotted using Equations 17–20. The autosomal mean uses equation 37 from Goldberg et al. (2014).

Figure 5

The expectation of the mean X-chromosomal and autosomal admixture fractions over time for constant admixture, with female-biased contributions from both source populations. The initial condition is $\left(s_{1,0}^{\mathrm{f}},s_{1,0}^{\mathrm{m}},s_{2,0}^{\mathrm{f}},s_{2,0}^{\mathrm{m}}\right)=\left(0.5,0.5,0.5,0.5\right).$ The mean X-chromosomal admixture from $S_1,$ $\mathbb{E}\left[H_{1,g,\delta }^{\mathrm{X}}\right],$ is smaller than the mean autosomal admixture, $\mathbb{E}\left[H_{1,g,\delta }^{\mathrm{A}}\right],$ even though $S_1$ has an excess of females, $s_1^{\mathrm{f}}>s_1^{\mathrm{m}}.$ The expectation of X-chromosomal admixture is plotted using Equations 17 and 18. The autosomal mean uses equation 37 from Goldberg et al. (2014).

Figure 6

The expectation of the mean X-chromosomal and autosomal admixture fractions over time for constant admixture, in a case with small continuing contributions. The initial condition is $\left(s_{1,0}^{\mathrm{f}},s_{1,0}^{\mathrm{m}},s_{2,0}^{\mathrm{f}},s_{2,0}^{\mathrm{m}}\right)=\left(0.25,0.5,0.75,0.5\right).$ With sex bias in the founding generation and small continuing contributions, the mean X-chromosomal admixture has a pattern resembling the oscillating behavior seen for a single admixture event. The expectation of X-chromosomal admixture is plotted using Equations 17 and 18. The autosomal mean uses equation 37 from Goldberg et al. (2014).

Figure 7

Sex-specific contributions estimated from the data of Cheng et al. (2009). (A) The range, median, and 25th and 75th percentiles of the sets of sex-specific contributions for which the Euclidean distance D (Equation 25) between model-predicted admixture and observed admixture from Cheng et al. (2009) was at most 0.01. The range of values for $s_2^{\mathrm{f}}$ and $s_2^{\mathrm{m}},$ the contributions representing Europeans $\left(S_2\right),$ is smaller than that representing Africans $\left(S_1\right).$ (B) Female contributions as a function of D. (C) Male contributions as a function of D. For B, the male contributions are fixed at their median values producing $D\le 0.01,$ $\left(s_1^{\mathrm{m}},h^{\mathrm{m}},s_2^{\mathrm{m}}\right)=\left(0.47,0.34,0.19\right).$ For C, the female contributions are fixed in a corresponding manner, at $\left(s_1^{\mathrm{f}},h^{\mathrm{f}},s_2^{\mathrm{f}}\right)=\left(0.64,0.29,0.07\right).$ Each side of the triangles in B and C represents one of three parameters that sum to 1, $\left(s_1^{\mathrm{m}},h^{\mathrm{m}},s_2^{\mathrm{m}}\right)$ and $\left(s_1^{\mathrm{f}},h^{\mathrm{f}},s_2^{\mathrm{f}}\right)$ in B and C, respectively. Parameter values were tested at increments of 0.01 for each quantity.

Figure 8

The natural logarithm of the ratio of male to female contributions in African-Americans, as inferred from the data of Cheng et al. (2009). (A) The range (excluding infinity, produced when a parameter value is zero), median, and 25th and 75th percentiles of the natural logarithm of the ratio of male to female contributions from $S_1$ (Africans) and $S_2$ (Europeans) separately for the sex-specific contributions that produced $D\le 0.01$ (Equation 25). Values from Africans, $S_1,$ are largely negative, or female biased, whereas contributions from Europeans, $S_2,$ are mostly positive and male biased. Approximately 26.15% of the contributions from Africans are male biased and 9.25% from Europeans are female biased. This pattern is typically observed when a still larger sex bias occurs in the other population. (B) The logarithm of the ratio of male and female contributions from $S_2$ on the y-axis and the corresponding ratio for $S_1$ on the x-axis, plotted by the density of points in 0.05 square bins. For the cases with male bias in Africa $\left(\text{ln}\left(s_1^{\mathrm{m}}/s_1^{\mathrm{f}}\right)>0\right),$ the level of male bias in Europe is also positive; for the cases with female bias in Europe $\left(\text{ln}\left(s_2^{\mathrm{m}}/s_2^{\mathrm{f}}\right)<0\right),$ the level of female bias from Africa is also negative. Parameter sets in which at least one parameter is 0, and therefore we have values of +∞ or −∞ for $\text{ln}\left(s_1^{\mathrm{m}}/s_1^{\mathrm{f}}\right)$ or $\text{ln}\left(s_2^{\mathrm{m}}/s_2^{\mathrm{f}}\right),$ appear in bins on the edge of the plot for convenience. These bins contain a substantial number of parameter sets.