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The multivariate Wright-Fisher process with mutation: Moment-based analysis and inference using a hierarchical Beta

Asger Hobolth1, Jukka Siren2

  • 1Bioinformatics Research Center, Aarhus University, Denmark.

Theoretical Population Biology
|November 28, 2015
PubMed
Summary

This study analyzes the Wright-Fisher process with mutations, finding the Dirichlet distribution insufficient for complex models. A new hierarchical Beta distribution is proposed for nucleotide-level allele frequencies.

Keywords:
Allele frequencyDiffusionDirichlet modelHierarchical BetaMomentsMultivariate Wright–Fisher

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Area of Science:

  • Population Genetics
  • Mathematical Biology
  • Evolutionary Dynamics

Background:

  • The Wright-Fisher model is a cornerstone of population genetics, describing random genetic drift.
  • Understanding allele frequency dynamics is crucial for inferring evolutionary parameters.
  • Existing models often simplify mutation processes, limiting their applicability.

Purpose of the Study:

  • To derive analytically tractable formulas for moments of allele frequency distribution in the multivariate Wright-Fisher process with mutation.
  • To assess the suitability of the Dirichlet distribution for modeling allele frequencies under different evolutionary scenarios.
  • To propose an improved distribution for allele frequencies that accounts for nucleotide-level mutation specifics.

Main Methods:

  • Utilizing diffusion approximation for the multivariate Wright-Fisher process.
  • Deriving analytical formulas for first- and second-order moments of allele frequencies.
  • Investigating expected homozygosity and evaluating distribution fitting.

Main Results:

  • Derived formulas for key moments of allele frequency distribution.
  • Demonstrated that the Dirichlet distribution is adequate for pure drift but not for general mutation models.
  • Identified limitations of the Dirichlet distribution in capturing complex evolutionary dynamics.

Conclusions:

  • The Dirichlet distribution is insufficient for modeling allele frequencies in the Wright-Fisher process with general mutation models.
  • A novel hierarchical Beta distribution is proposed for nucleotide-level allele frequencies, distinguishing transitions and transversions.
  • This new distribution offers greater flexibility for more realistic evolutionary modeling.