Phase-type distributions in population genetics

TL;DR

This paper demonstrates how phase-type distributions can be applied to model DNA sequence evolution in population genetics, simplifying complex calculations through matrix-based Markov models.

Contribution

It introduces the use of phase-type theory for coalescent models, enabling more efficient analysis of genetic variation and recombination in population genetics.

Findings

Calculated moments of the site frequency spectrum in multiple merger coalescents.

Analyzed mean and variance of segregating sites in two-locus ancestral recombination graphs.

Showed phase-type theory's potential for simplifying population genetics models.

Abstract

Probability modelling for DNA sequence evolution is well established and provides a rich framework for understanding genetic variation between samples of individuals from one or more populations. We show that both classical and more recent models for coalescence (with or without recombination) can be described in terms of the so-called phase-type theory, where complicated and tedious calculations are circumvented by the use of matrices. The application of phase-type theory consists of describing the stochastic model as a Markov model by appropriately setting up a state space and calculating the corresponding intensity and reward matrices. Formulae of interest are then expressed in terms of these aforementioned matrices. We illustrate this by a few examples calculating the mean, variance and even higher order moments of the site frequency spectrum in the multiple merger coalescent models, and by analysing the mean and variance for the number of segregating sites for multiple samples in the two-locus ancestral recombination graph. We believe that phase-type theory has great potential as a tool for analysing probability models in population genetics. The compact matrix notation is useful for clarification of current models, in particular their formal manipulation (calculation), but also for further development or extensions.