Approximate sampling formulae for general finite-alleles models of mutation

TL;DR

This paper develops approximate sampling formulas for finite-alleles mutation models in genetics, providing highly accurate results at low mutation rates, filling a gap where exact formulas are unknown for general models.

Contribution

It introduces approximate closed-form sampling formulas for complex mutation models using an urn construction related to the coalescent, applicable to models with finitely many alleles.

Findings

Formulas are highly accurate at low mutation rates

Applicable to models with up to three or four observed allele types

Provides a practical tool where exact formulas are unavailable

Abstract

Many applications in genetic analyses utilize sampling distributions, which describe the probability of observing a sample of DNA sequences randomly drawn from a population. In the one-locus case with special models of mutation such as the infinite-alleles model or the finite-alleles parent-independent mutation model, closed-form sampling distributions under the coalescent have been known for many decades. However, no exact formula is currently known for more general models of mutation that are of biological interest. In this paper, models with finitely-many alleles are considered, and an urn construction related to the coalescent is used to derive approximate closed-form sampling formulas for an arbitrary irreducible recurrent mutation model or for a reversible recurrent mutation model, depending on whether the number of distinct observed allele types is at most three or four, respectively. It is demonstrated empirically that the formulas derived here are highly accurate when the per-base mutation rate is low, which holds for many biological organisms.