Metafounders are a useful concept to characterize relationships within and across populations, and to help genetic evaluations because they help modelling the means and variances of unknown base population animals. Current definitions of metafounder relationships are sensitive to the choice of reference alleles and have not been compared to their counterparts in population genetics—namely, heterozygosities, FST coefficients, and genetic distances. We redefine the relationships across populations with an arbitrary base of a maximum heterozygosity population in Hardy–Weinberg equilibrium. Then, the relationship between or within populations is a cross-product of the form $${\Gamma }_{\left(b,{b}^{\prime}\right)}=\left(\frac{2}{n}\right)\left(2{\mathbf{p}}_{b}-\mathbf{1}\right)\left(2{\mathbf{p}}_{{b}^{\prime}}-\mathbf{1}\right)^{\prime}$$ with $$\mathbf{p}$$ being vectors of allele frequencies at $$n$$ markers in populations $$b$$ and $$b^{\prime}$$ . This is simply the genomic relationship of two pseudo-individuals whose genotypes are equal to twice the allele frequencies. We also show that this coding is invariant to the choice of reference alleles. In addition, standard population genetics metrics (inbreeding coefficients of various forms; FST differentiation coefficients; segregation variance; and Nei’s genetic distance) can be obtained from elements of matrix $${\varvec{\Gamma}}$$ .

中文翻译：

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重新定义和解释元创始人的基因组关系

元创建者是一个有用的概念，可以描述群体内部和群体之间的关系，并有助于遗传评估，因为它们有助于对未知基础群体动物的均值和方差进行建模。目前元创建者关系的定义对参考等位基因的选择很敏感，并且尚未与群体遗传学中的对应等位基因（即杂合性、FST 系数和遗传距离）进行比较。我们以哈代-温伯格平衡中最大杂合性群体的任意基数重新定义了群体之间的关系。然后，群体之间或群体内部的关系是以下形式的叉积：$${\Gamma }_{\left(b,{b}^{\prime}\right)}=\left(\frac{2} {n}\right)\left(2{\mathbf{p}}_{b}-\mathbf{1}\right)\left(2{\mathbf{p}}_{{b}^{\prime }}-\mathbf{1}\right)^{\prime}$$ 其中 $$\mathbf{p}$$ 是群体 $$b$$ 和 $$ 中 $$n$$ 标记处等位基因频率的向量b^{\prime}$$ 。这只是两个伪个体的基因组关系，其基因型等于等位基因频率的两倍。我们还表明，这种编码对于参考等位基因的选择是不变的。此外，标准群体遗传学指标（各种形式的近交系数；FST 分化系数；分离方差；以及 Nei 遗传距离）可以从矩阵 $${\varvec{\Gamma}}$$ 的元素获得。