Journal of Mathematical Biology

, Volume 64, Issue 1–2, pp 87–108 | Cite as

Refining the relationship between homozygosity and the frequency of the most frequent allele

Open Access
Article

Abstract

Recent work has established that for an arbitrary genetic locus with its number of alleles unspecified, the homozygosity of the locus confines the frequency of the most frequent allele within a narrow range, and vice versa. Here we extend beyond this limiting case by investigating the relationship between homozygosity and the frequency of the most frequent allele when the number of alleles at the locus is treated as known. Given the homozygosity of a locus with at most K alleles, we find that by taking into account the value of K, the width of the allowed range for the frequency of the most frequent allele decreases from \({2/3-\pi^2/18 \approx 0.1184}\) to \({1/3-1/(3K)-\{K/[3(K-1)]\}\sum_{k=2}^K 1/k^2}\). We further show that properties of the relationship between homozygosity and the frequency of the most frequent allele in the unspecified-K case can be obtained from the specified-K case by taking limits as K → ∞. The results contribute to a greater understanding of the mathematical properties of fundamental statistics employed in population-genetic analysis.

Keywords

Allele frequency Homozygosity Population genetics 

Mathematics Subject Classification (2000)

92D10 

Notes

Acknowledgments

The authors gratefully acknowledge two reviewers for detailed comments, T. Pemberton for technical assistance, and the Burroughs Wellcome Fund and NIH grant GM081441 for financial support.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. Hedrick PW (2005) A standardized genetic differentiation measure. Evolution 59: 1633–1638Google Scholar
  2. Jost L (2008) G ST and its relatives do not measure differentiation. Mol Ecol 17: 4015–4026CrossRefGoogle Scholar
  3. Long JC, Kittles RA (2003) Human genetic diversity and the nonexistence of biological races. Hum Biol 75: 449–471CrossRefGoogle Scholar
  4. Rosenberg NA, Calabrese PP (2004) Polyploid and multilocus extensions of the Wahlund inequality. Theor Pop Biol 66: 381–391CrossRefMATHGoogle Scholar
  5. Rosenberg NA, Jakobsson M (2008) The relationship between homozygosity and the frequency of the most frequent allele. Genetics 179: 2027–2036CrossRefGoogle Scholar
  6. Rosenberg NA, Mahajan S, Ramachandran S, Zhao C, Pritchard JK, Feldman MW (2005) Clines, clusters, and the effect of study design on the inference of human population structure. PLoS Genet 1: 660–671CrossRefGoogle Scholar
  7. Van Liere JM, Rosenberg NA (2008) Mathematical properties of the r 2 measure of linkage disequilibrium. Theor Pop Biol 74: 130–137CrossRefGoogle Scholar
  8. Weir BS (1996) Genetic data analysis II. Sinauer, SunderlandGoogle Scholar
  9. Wray NR (2005) Allele frequencies and the r 2 measure of linkage disequilibrium: impact on design and interpretation of association studies. Twin Res Hum Genet 8: 87–94CrossRefGoogle Scholar

Copyright information

© The Author(s) 2011

Authors and Affiliations

  1. 1.Center for Computational Medicine and BioinformaticsUniversity of MichiganAnn ArborUSA
  2. 2.Department of Human Genetics, Center for Computational Medicine and Bioinformatics, and the Life Sciences InstituteUniversity of MichiganAnn ArborUSA

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