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Single and simultaneous binary mergers in Wright-Fisher genealogies.

Andrew Melfi1, Divakar Viswanath1

  • 1Department of Mathematics, University of Michigan, United States.

Theoretical Population Biology
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Summary

This study extends the Wright-Fisher (WF) model to large, increasing sample sizes, proving convergence to the Kingman coalescent. It shows that large sample sizes in population genetics limit mergers in WF genealogies.

Keywords:
Convergence theoryKingman coalescentLarge samplesWright–Fisher model

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Area of Science:

  • Population genetics
  • Mathematical biology
  • Evolutionary genetics

Background:

  • The Kingman coalescent model is fundamental in population genetics, often derived from the Wright-Fisher (WF) model.
  • Existing proofs of WF convergence to the Kingman coalescent assume constant sample sizes, which is unrealistic for large human genetics datasets.

Purpose of the Study:

  • To develop a convergence theory for the Wright-Fisher model that accommodates increasing sample sizes relative to population size.
  • To investigate the impact of large sample sizes on the genealogical patterns predicted by the WF model and their convergence to the Kingman coalescent.

Main Methods:

  • Developed a new convergence theory for the Wright-Fisher model with sample sizes scaling as N^(1/3-ϵ) and N^(1/2-ϵ) with population size N.
  • Proved that for sample sizes N^(1/3-ϵ), WF genealogies converge to a process with at most one binary merger per generation, yielding the exact Kingman partition distribution.
  • Showed that for sample sizes N^(1/2-ϵ), WF genealogies avoid triple mergers in the large N limit.
  • Utilized numerical calculations to verify the asymptotic theory.
  • Implemented algorithmic approaches to handle variable population sizes, including bottlenecks.

Main Results:

  • With sample size N^(1/3-ϵ), WF genealogies exhibit at most a single binary merger per generation with high probability, exactly matching the Kingman partition distribution.
  • With sample size N^(1/2-ϵ), WF genealogies may feature simultaneous binary mergers but not triple mergers as population size N increases.
  • Numerical simulations confirm the asymptotic predictions.
  • Analysis reveals that population bottlenecks can increase the likelihood of simultaneous binary mergers and triple mergers in WF genealogies.

Conclusions:

  • The study provides a theoretical framework for understanding WF genealogies with large, increasing sample sizes, bridging a gap in current population genetics models.
  • The findings demonstrate that under specific scaling regimes of sample size with population size, the WF model converges to the Kingman coalescent, with implications for inferring demographic history.
  • The research highlights the influence of demographic events like bottlenecks on genealogical merger patterns.