Page 418 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 418
MATHEMATICS IN BIOLOGY AND MEDICINE (MS-35)
Solving the selection-recombination equation
Ellen Baake, ebaake@techfak.uni-bielefeld.de
Bielefeld University, Germany
Coauthor: Frederic Alberti
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, only the special case of three sites with selection acting on one of them has
been treated, but only approximately and without an obvious path to generalisation. We use
both an analytical and a probabilistic, genealogical approach for the case of an arbitrary num-
ber 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 de-
velop 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 ini-
tiation 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.
Phylogeny and population genetics: The mutation process on the ancestral
line
Enrico Di Gaspero, en.digaspero@gmail.com
Bielefeld University, Germany
Coauthor: Ellen Baake
We consider a well-known observation at the interface of phylogeny and population genetics:
mutation rates estimated via phylogenetic methods tend to be much smaller than direct esti-
mates from pedigree studies. To understand this, we consider the Moran model with two types,
mutation, and selection, and investigate the line of descent of a randomly-sampled individual
from a contemporary population. We trace this ancestral line back into the distant past, far be-
yond the most recent common ancestor of the population (thus connecting population genetics
to phylogeny) and analyze the mutation process along this line. We use a probabilistic tool,
namely the pruned lookdown ancestral selection graph, which consists of the set of potential
ancestors of the sampled individual at any given time. A crucial observation is that the mutation
process on the ancestral line is not a Markov process by itself, but it becomes Markov when
consindering a broader state space. Relative to the neutral case (that is, without selection), we
obtain a general bias towards beneficial mutations. These results shed new light on previous
analytical findings of Fearnhead (2002).
416
Solving the selection-recombination equation
Ellen Baake, ebaake@techfak.uni-bielefeld.de
Bielefeld University, Germany
Coauthor: Frederic Alberti
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, only the special case of three sites with selection acting on one of them has
been treated, but only approximately and without an obvious path to generalisation. We use
both an analytical and a probabilistic, genealogical approach for the case of an arbitrary num-
ber 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 de-
velop 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 ini-
tiation 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.
Phylogeny and population genetics: The mutation process on the ancestral
line
Enrico Di Gaspero, en.digaspero@gmail.com
Bielefeld University, Germany
Coauthor: Ellen Baake
We consider a well-known observation at the interface of phylogeny and population genetics:
mutation rates estimated via phylogenetic methods tend to be much smaller than direct esti-
mates from pedigree studies. To understand this, we consider the Moran model with two types,
mutation, and selection, and investigate the line of descent of a randomly-sampled individual
from a contemporary population. We trace this ancestral line back into the distant past, far be-
yond the most recent common ancestor of the population (thus connecting population genetics
to phylogeny) and analyze the mutation process along this line. We use a probabilistic tool,
namely the pruned lookdown ancestral selection graph, which consists of the set of potential
ancestors of the sampled individual at any given time. A crucial observation is that the mutation
process on the ancestral line is not a Markov process by itself, but it becomes Markov when
consindering a broader state space. Relative to the neutral case (that is, without selection), we
obtain a general bias towards beneficial mutations. These results shed new light on previous
analytical findings of Fearnhead (2002).
416
