Solving the selection-recombination equation: Ancestral lines under selection and recombination

TL;DR

This paper develops a novel analytical and genealogical approach to explicitly solve the deterministic selection-recombination equation for an arbitrary number of neutral sites linked to a selected site, providing a stochastic representation of the solution.

Contribution

It introduces a recursive integral representation and an efficient genealogical structure for solving the selection-recombination equation with multiple linked sites.

Findings

Derived a recursive integral solution for arbitrary neutral sites

Developed a genealogical structure based on ancestral selection-recombination graph

Established duality and stochastic representation of the deterministic solution

Abstract

The deterministic selection-recombination equation describes the evolution of the genetic type composition of a population under selection and recombination in a law of large numbers regime. So far, an explicit solution has seemed out of reach; only in the special case of three sites with selection acting on one of them has an approximate solution been found, but without an obvious path to generalisation. We use both an analytical and a probabilistic, genealogical approach for the case of an \emph{arbitrary} number of neutral sites linked to one selected site. This leads to a recursive integral representation of the solution. Starting from a variant of the ancestral selection-recombination graph, we develop an efficient genealogical structure, which may, equivalently, be represented as a weighted partitioning process, a family of Yule processes with initiation and resetting, and a family of initiation processes. We prove them to be dual to the solution of the differential equation forward in time and thus obtain a stochastic representation of the deterministic solution, along with the Markov semigroup in closed form.