Estimating Effective Number of Breeders Under the Heterozygote‐Excess Equilibrium

Estimating the effective number of breeders per reproductive cycle or cohort ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics> <mml:mrow><mml:msub><mml:mi>N</mml:mi> <mml:mi>b</mml:mi></mml:msub> </mml:mrow> <mml:annotation>$$ {N}_b $$</mml:annotation></mml:semantics> </mml:math> ) can contribute to understanding the viability of wild populations. However, previous <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics> <mml:mrow><mml:msub><mml:mi>N</mml:mi> <mml:mi>b</mml:mi></mml:msub> </mml:mrow> <mml:annotation>$$ {N}_b $$</mml:annotation></mml:semantics> </mml:math> estimators based on heterozygote-excess assume that parental genotypes are in accordance with Hardy-Weinberg equilibrium (HWE), and can only be applied to diploids. When these criteria are not fulfilled, estimates of expected heterozygosity may thus be biased. We extended a previous <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics> <mml:mrow><mml:msub><mml:mi>N</mml:mi> <mml:mi>b</mml:mi></mml:msub> </mml:mrow> <mml:annotation>$$ {N}_b $$</mml:annotation></mml:semantics> </mml:math> estimator to autopolyploids and account for parental genotypes either in HWE or heterozygote-excess equilibrium (HEE). Results include the following: the distributions of genotypes converge to HEE within nine generations (with a relative error of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics> <mml:mrow><mml:msup><mml:mn>10</mml:mn> <mml:mrow><mml:mo>-</mml:mo> <mml:mn>3</mml:mn></mml:mrow> </mml:msup> </mml:mrow> <mml:annotation>$$ {10}^{-3} $$</mml:annotation></mml:semantics> </mml:math> ), the corrected estimators can asymptotically estimate <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics> <mml:mrow><mml:msub><mml:mi>N</mml:mi> <mml:mi>b</mml:mi></mml:msub> </mml:mrow> <mml:annotation>$$ {N}_b $$</mml:annotation></mml:semantics> </mml:math> without bias, and estimations for autopolyploids are slightly better than those for diploids. The relationships between <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics> <mml:mrow> <mml:msub><mml:mover><mml:mi>N</mml:mi> <mml:mo>̂</mml:mo></mml:mover> <mml:mi>b</mml:mi></mml:msub> </mml:mrow> <mml:annotation>$$ {\hat{N}}_b $$</mml:annotation></mml:semantics> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics> <mml:mrow><mml:msub><mml:mi>N</mml:mi> <mml:mi>b</mml:mi></mml:msub> </mml:mrow> <mml:annotation>$$ {N}_b $$</mml:annotation></mml:semantics> </mml:math> and the influence of variable effective breeding numbers across generations on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics> <mml:mrow> <mml:msub><mml:mover><mml:mi>N</mml:mi> <mml:mo>̂</mml:mo></mml:mover> <mml:mi>b</mml:mi></mml:msub> </mml:mrow> <mml:annotation>$$ {\hat{N}}_b $$</mml:annotation></mml:semantics> </mml:math> were clarified. We find <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics> <mml:mrow> <mml:msub><mml:mover><mml:mi>N</mml:mi> <mml:mo>̂</mml:mo></mml:mover> <mml:mi>b</mml:mi></mml:msub> </mml:mrow> <mml:annotation>$$ {\hat{N}}_b $$</mml:annotation></mml:semantics> </mml:math> mainly reflects <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics> <mml:mrow><mml:msub><mml:mi>N</mml:mi> <mml:mi>b</mml:mi></mml:msub> </mml:mrow> <mml:annotation>$$ {N}_b $$</mml:annotation></mml:semantics> </mml:math> in the previous generation in diploids but is a weighted harmonic mean of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics> <mml:mrow><mml:msub><mml:mi>N</mml:mi> <mml:mi>b</mml:mi></mml:msub> </mml:mrow> <mml:annotation>$$ {N}_b $$</mml:annotation></mml:semantics> </mml:math> in previous generations in polyploids. Our methods are implemented in a software package, polygene v1.7, which is freely available at https://github.com/huangkang1987/polygene.