TY - JOUR

T1 - Phase-type distributions in population genetics

AU - Hobolth, Asger

AU - Siri-Jégousse, Arno

AU - Bladt, Mogens

PY - 2019

Y1 - 2019

N2 - 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 matrix manipulations. The application of phase-type theory in population genetics consists of describing the biological system 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 procedure by a number of examples: (a)Calculating the mean, (co)variance and even higher order moments of the site frequency spectrum in multiple merger coalescent models, (b)Analysing a sample of DNA sequences from the Atlantic Cod using the Beta-coalescent, and (c)Determining the correlation of 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, and in particular their formal manipulation and calculations, but also for further development or extensions.

AB - 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 matrix manipulations. The application of phase-type theory in population genetics consists of describing the biological system 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 procedure by a number of examples: (a)Calculating the mean, (co)variance and even higher order moments of the site frequency spectrum in multiple merger coalescent models, (b)Analysing a sample of DNA sequences from the Atlantic Cod using the Beta-coalescent, and (c)Determining the correlation of 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, and in particular their formal manipulation and calculations, but also for further development or extensions.

KW - Coalescent theory

KW - Multiple merger

KW - Phase-type theory

KW - Recombination

KW - Segregating sites

KW - Site frequency spectrum

UR - http://www.scopus.com/inward/record.url?scp=85063079391&partnerID=8YFLogxK

U2 - 10.1016/j.tpb.2019.02.001

DO - 10.1016/j.tpb.2019.02.001

M3 - Journal article

C2 - 30822431

AN - SCOPUS:85063079391

VL - 127

SP - 16

EP - 32

JO - Theoretical Population Biology

JF - Theoretical Population Biology

SN - 0040-5809

ER -