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A dual process for the coupled Wright-Fisher diffusion.

Martina Favero1, Henrik Hult2, Timo Koski2

  • 1Department of Mathematics, KTH, 11428, Stockholm, Sweden. mfavero@kth.se.

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Summary
This summary is machine-generated.

This study introduces a dual ancestral process for the coupled Wright-Fisher diffusion, modeling multi-locus genetic frequencies. The derived process accounts for complex genetic interactions, including pairwise selection, providing new tools for population genetics.

Keywords:
Ancestral graphsDualityMarkov processesPopulation geneticsWright–Fisher diffusion

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Area of Science:

  • Population Genetics
  • Mathematical Biology
  • Evolutionary Dynamics

Background:

  • The coupled Wright-Fisher diffusion models multi-locus, multi-allelic genetic frequencies using stochastic differential equations.
  • Inter-locus selection captures pairwise interactions among genetic loci.

Purpose of the Study:

  • To derive an ancestral process dual to the coupled Wright-Fisher diffusion.
  • To analyze the structure and rates of this dual process, incorporating single- and double-locus selection.
  • To obtain explicit results for simplified cases.

Main Methods:

  • Derivation of a dual ancestral process.
  • Characterization of the dual process as a block counting process of coupled ancestral selection graphs.
  • Analysis of jump processes including coalescence, mutation, single-branching, and double-branching.

Main Results:

  • The dual process features coalescence and mutation rates similar to the Kingman coalescent.
  • Single-branching rates generalize ancestral selection graphs by including one- and two-locus selection parameters.
  • Double-branching rates reflect pairwise selection structures.
  • Explicit stationary density and transition rates were obtained for a two-locus, two-allele model.

Conclusions:

  • The derived dual process provides a powerful framework for studying the coupled Wright-Fisher diffusion.
  • This approach effectively incorporates complex genetic interactions, including pairwise selection.
  • The findings offer new analytical tools for understanding evolutionary dynamics at multiple loci.