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

  • Shashir B. Reddy
  • Noah A. RosenbergEmail author
Open Access


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.


Allele frequency Homozygosity Population genetics 

Mathematics Subject Classification (2000)




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.


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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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