Single--crossover recombination and ancestral recombination trees

TL;DR

This paper analyzes the ancestral recombination trees in a Wright-Fisher model with single-crossover recombination, providing a probabilistic description of tree topologies and a semi-explicit solution to the associated deterministic equation.

Contribution

It introduces a novel approach to describe ancestral recombination trees and derives a semi-explicit solution to the long-standing deterministic single-crossover equation.

Findings

Probabilistic characterization of ancestral tree topologies

Explicit formulas for tree topology probabilities

Solution to the deterministic single-crossover equation

Abstract

We consider the Wright-Fisher model for a population of $N$ individuals, each identified with a sequence of a finite number of sites, and single-crossover recombination between them. We trace back the ancestry of single individuals from the present population. In the $N \to \infty$ limit without rescaling of parameters or time, this ancestral process is described by a random tree, whose branching events correspond to the splitting of the sequence due to recombination. With the help of a decomposition of the trees into subtrees, we calculate the probabilities of the topologies of the ancestral trees. At the same time, these probabilities lead to a semi-explicit solution of the deterministic single-crossover equation. The latter is a discrete-time dynamical system that emerges from the Wright-Fisher model via a law of large numbers and has been waiting for a solution for many decades.