Redefining and interpreting genomic relationships of metafounders release_uepl3sjtorcvjbghh4tep4ja44

Abstract

<jats:title>Abstract</jats:title>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, <jats:italic>F</jats:italic><jats:sub><jats:italic>ST</jats:italic></jats:sub> 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 <jats:inline-formula><jats:alternatives><jats:tex-math>$${\Gamma }_{\left(b,{b}^{\prime}\right)}=\left(\frac{2}{n}\right)\left(2{\mathbf{p}}_{b}-\mathbf{1}\right)\left(2{\mathbf{p}}_{{b}^{\prime}}-\mathbf{1}\right)^{\prime}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>Γ</mml:mi> <mml:mfenced> <mml:mi>b</mml:mi> <mml:mo>,</mml:mo> <mml:msup> <mml:mrow> <mml:mi>b</mml:mi> </mml:mrow> <mml:mo>′</mml:mo> </mml:msup> </mml:mfenced> </mml:msub> <mml:mo>=</mml:mo> <mml:mfenced> <mml:mfrac> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:mfrac> </mml:mfenced> <mml:mfenced> <mml:mn>2</mml:mn> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>b</mml:mi> </mml:msub> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mfenced> <mml:msup> <mml:mfenced> <mml:mn>2</mml:mn> <mml:msub> <mml:mi>p</mml:mi> <mml:msup> <mml:mrow> <mml:mi>b</mml:mi> </mml:mrow> <mml:mo>′</mml:mo> </mml:msup> </mml:msub> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mfenced> <mml:mo>′</mml:mo> </mml:msup> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> with <jats:inline-formula><jats:alternatives><jats:tex-math>$$\mathbf{p}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> being vectors of allele frequencies at <jats:inline-formula><jats:alternatives><jats:tex-math>$$n$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> markers in populations <jats:inline-formula><jats:alternatives><jats:tex-math>$$b$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>b</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$$b^{\prime}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>b</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:math></jats:alternatives></jats:inline-formula>. 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; <jats:italic>F</jats:italic><jats:sub><jats:italic>ST</jats:italic></jats:sub> differentiation coefficients; segregation variance; and Nei's genetic distance) can be obtained from elements of matrix <jats:inline-formula><jats:alternatives><jats:tex-math>$${\varvec{\Gamma}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Γ</mml:mi> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>.
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