Selection and Coalescence in a Finite State Model

TL;DR

This paper introduces a finite state model incorporating selection into coalescence processes, revealing how selection influences genealogical tree structures and coalescence times through a novel partition-based approach.

Contribution

It presents a new model using modified integer partitions and a transition matrix to analyze selection effects in small populations, capturing multiple collisions and tree asymmetry.

Findings

Selection causes multiple collisions per generation.

Selection influences coalescence times and tree shape.

Model aligns with higher order coalescences.

Abstract

To introduce selection into a model of coalescence, I explore the use of modified integer partitions that allow the identification of a preferred lineage. I show that a partition-partition transition matrix, along with Monte Carlo discrete time kinetics, treats both the neutral case and a wide range of positive and negative selection pressures for small population sizes. Selection pressure causes multiple collisions per generation, short coalescence times, increased lengths of terminal branches, increased tree asymmetry, and dependence of coalescence times on the logarithm of population size. These features are consistent with higher order coalescences that permit multiple collisions per generation. While the treatment is exact in terms of the simplified Wright-Fisher model used, it is not easily extended to large population size. Keywords: Selection, Coalescence, Integer Partitions, Multiple Collisions, Tree Asymmetry.