The expected neutral frequency spectrum of linked sites

TL;DR

This paper derives exact formulas for the expected neutral site frequency spectrum of linked sites without recombination, extending classical models and applicable to biological scenarios like structural variants and inversions.

Contribution

It provides the first exact, closed-form expressions for the joint frequency spectrum of two linked neutral sites without recombination, extending classical single-site models.

Findings

Derived exact formulas for 2-site frequency spectrum without recombination.

Provided formulas for linked sites with known focal mutation frequency.

Applied results to biological cases like chromosomal inversions.

Abstract

We present an exact, closed expression for the expected neutral Site Frequency Spectrum for two neutral sites, 2-SFS, without recombination. This spectrum is the immediate extension of the well known single site $\theta/f$ neutral SFS. Similar formulae are also provided for the case of the expected SFS of sites that are linked to a focal neutral mutation of known frequency. Formulae for finite samples are obtained by coalescent methods and remarkably simple expressions are derived for the SFS of a large population, which are also solutions of the multi-allelic Kolmogorov equations. Besides the general interest of these new spectra, they relate to interesting biological cases such as structural variants and introgressions. As an example, we present the expected neutral frequency spectrum of regions with a chromosomal inversion.