16 research outputs found
Phase-type distributions in population genetics
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
The Nested Kingman Coalescent:Speed of Coming Down from Infinity
The nested Kingman coalescent describes the ancestral tree of a population
undergoing neutral evolution at the level of individuals and at the level of
species, simultaneously. We study the speed at which the number of lineages
descends from infinity in this hierarchical coalescent process and prove the
existence of an early-time phase during which the number of lineages at time
decays as , where is the ratio of the coalescence rates
at the individual and species levels, and the constant is
derived from a recursive distributional equation for the number of lineages
contained within a species at a typical time.Comment: 24 page