AN ASYMPTOTIC SAMPLING FORMULA FOR THE COALESCENT WITH RECOMBINATION

Area of Science:

• Population Genetics
• Mathematical Biology
• Statistical Genetics

Background:

• The Ewens sampling formula (ESF) is a key probability distribution in population genetics, describing genetic diversity under the infinite-alleles model.
• Extending the ESF to incorporate recombination, particularly at two loci, has remained a significant challenge in the field.
• Existing models lack general closed-form solutions for two-locus sampling distributions with recombination.

Purpose of the Study:

• To derive a general closed-form sampling formula for a two-locus model with recombination.
• To investigate the behavior of sampling distributions when the population-scaled recombination rate (ρ) is large.
• To provide a mathematical tool for analyzing genetic variation in populations with recombination.

Main Methods:

• Developed an asymptotic expansion of the two-locus sampling formula in inverse powers of the population-scaled recombination rate (ρ).
• Derived closed-form expressions for the leading terms of this asymptotic expansion.
• The methodology is applicable to arbitrary sample sizes and genetic configurations.

Main Results:

• Obtained a novel asymptotic sampling formula for two-locus models applicable when the recombination rate is large.
• Provided explicit closed-form expressions for the initial terms of the expansion in terms of ρ.
• The derived formula offers a practical approach to analyzing genetic data in the presence of recombination.

Conclusions:

• The asymptotic sampling formula offers a tractable solution for analyzing two-locus genetic data under high recombination.
• This work bridges a gap in theoretical population genetics by providing a closed-form approximation for complex models.
• The findings facilitate a deeper understanding of genetic diversity patterns influenced by both mutation and recombination.