## Ancestral Processes with Selection

* Stephen M. Krone* and *Claudia Neuhauser*
**Abstract**

In this paper, we show how to construct the genealogy of a sample of genes
for a large class of models with selection and mutation. Each gene corresponds
to a single locus at which there is no recombination. The genealogy of the
sample is embedded in a graph which we call the * ancestral selection
graph*. This graph contains all the information about the ancestry;
it is the analogue of Kingman's coalescent process which arises in the case
with no selection. The ancestral selection graph can be easily simulated and
we outline an algorithm for simulating samples. The main goal is to
analyze the ancestral selection graph and to compare it to Kingman's
coalescent process. In the case of no mutation, we find that the distribution
of the time to the most recent common ancestor does not depend on the selection
coefficient, and hence is the same as in the neutral case. When the mutation
rate is positive, we give a procedure for computing the probability that two
individuals in the sample are identical by descent and the Laplace transform
of the time to the most recent common ancestor of a sample of two individuals;
we evaluate the first two terms of their respective power series in terms
of the selection coefficient. The probability of identity by descent depends
on both the selection coefficient and the mutation rate and is different
from the analogous expression in the neutral case. The Laplace transform
does not have a linear correction term in the selection coefficient.
We also provide a recursion formula that can be used to approximate the
probability of a given sample by simulating backwards along the sample paths
of the ancestral selection graph, a technique developed by Griffiths and
Tavare (1994).