Bounding measures of genetic similarity and diversity using majorization

J Math Biol. 2018 Sep;77(3):711-737. doi: 10.1007/s00285-018-1226-x. Epub 2018 Mar 22.

Abstract

The homozygosity and the frequency of the most frequent allele at a polymorphic genetic locus have a close mathematical relationship, so that each quantity places a tight constraint on the other. We use the theory of majorization to provide a simplified derivation of the bounds on homozygosity J in terms of the frequency M of the most frequent allele. The method not only enables simpler derivations of known bounds on J in terms of M, it also produces analogous bounds on entropy statistics for genetic diversity and on homozygosity-like statistics that range in their emphasis on the most frequent allele in relation to other alleles. We illustrate the constraints on the statistics using data from human populations. The approach suggests the potential of the majorization method as a tool for deriving inequalities that characterize mathematical relationships between statistics in population genetics.

Keywords: Genetic diversity; Homozygosity; Majorization; Shannon and Rényi entropies.

Publication types

  • Research Support, N.I.H., Extramural
  • Research Support, U.S. Gov't, Non-P.H.S.

MeSH terms

  • Computer Simulation
  • Gene Frequency
  • Genetic Variation*
  • Genetics, Population / statistics & numerical data*
  • Homozygote
  • Humans
  • Mathematical Concepts
  • Microsatellite Repeats
  • Models, Genetic*