




Probability and its Applications / Théorie des probabilités
et leurs applications
(Sponsored by the National Programme Committee (CRM, Fields, PIMS) /
Parrainée par le Comité du Programme Nationale (CRM, Fields, PIMS)) (Martin Barlow, Rick Durrett, Claudia Neuhauser and Edwin Perkins,
Organizers)
 DAVID BRILLINGER, CaliforniaBerkeley

 PETER CALABRESE, Cornell University, Ithaca, New York 14853, USA
Microsatellite models

Microsatellites are a type of genetic marker: specifically they are
simple sequence tandem repeats in DNA, for example AT repeated 25 times
in a row. Numerous models for microsatellite mutation have been
proposed. We use entire genome sequence data and the likelihood ratio
test to sort out which model is best.
 DON DAWSON, Carleton University, Ottawa, Ontario K1S 5B6
Some aspects of the infinitely many Alleles model

The infinitely many alleles model plays a significant role in
population genetics. In this lecture we describe the analysis of this
and some related models. In particular, we describe recent joint
results with Shui Feng on large deviations and joint work with Andreas
Greven on meanfield migration models with selection based on the
infinitely many alleles model.
 JOE FELSENSTEIN, University of Washington, Seattle, Washington, USA
Constructing a robotic evolutionary geneticist

Much of the data collected by evolutionary geneticists consists of
population samples of molecular sequences or microsatellite genotypes.
Many of the past papers in the field have involved choosing a somewhat
arbitrary statistic, obtaining its expectation as a function of some
parameters by great heroics, and then vaguely hoping that someone would
use this to estimate the parameters. Using coalescent approaches, we
are using Markov Chain Monte Carlo methods to approximate the
likelihood functions for data involving a variety of evolutionary
forces, including genetic drift, mutation, migration, population
growth, and natural selection. Work on LAMARC, a unified computer
program to allow analysis of complicated combinations of these forces,
is well advanced in our laboratory. The program will use
objectoriented methods and will be available for free. We hope that
this will free evolutionary geneticists to spend their time on data
collection and on the interpretation of the resulting likelihood
surfaces, so that they will not be too offended at having their former
activities replaced by a black box.
 STEVE KRONE, Department of Mathematics, University of Idaho, Moscow,
Idaho 83844, USA
When can one detect overdominant selection in the infinite
alleles model?

Gillespie (1999), using simulations of the infinite alleles model with
heterozygote advantage, noticed that when the scaled mutation rate and
selection intensity get large together the behavior appears to be
indistinguishable from that of the corresponding neutral model. He
explains how these results for the infinite alleles diffusion imply
something about the difficulty of detecting selection in large
populations with fixed mutation and selection parameters.
We attempt to explain and refine these somewhat surprising observations
by analyzing the limiting behavior of the likelihood ratio of the
stationary distributions for the model under selection and neutrality,
as the mutation rate and selection intensity go to infinity in a
specified manner. In particular, we show that the likelihood ratio
tends to one (i.e., the selective model cannot be distinguished
from the neutral model) as the mutation rate goes to infinity, provided
the selection intensity grows like the mutation rate raised to any
power less than 3/2. When the power is greater than 3/2 selection
can be detected. These results rely on Gaussian limit theorems for the
PoissonDirichlet distribution, which appear to be of interest in their
own right. (Joint work with Paul Joyce and Thomas G. Kurtz.)
 NEAL MADRAS, Department of Mathematics and Statistics, York University,
Toronto, Ontario M3J 1P3
A model of segregation

In 1998, the New York Times ran a story about mathematical models that
try to explain why racial segregation is such a persistent problem. In
one version of the model, particles of two different colours live at
the sites of a large grid. A fixed fraction of sites are empty. Each
unit of time, some particle chooses an empty site at random and moves
there if the eight nearest sites are occupied by at least two particles
having the same colour as the particle that wants to move. Simulations
show that large clusters occur (i.e., segregation). A weakness
in this model led me to modify the rule so that there is a nonzero
probability of moving to a square with too few like neighbours.
Simulations and heuristics indicate a certain critical value of this
nonzero probability above which large scale segregation does not
occur. This has implications about the adequacy of the original
model.
 PETER MARCH, Ohio State
To be announced

 JEREMY QUASTEL, University of Toronto, Toronto, Ontario
Internal DLA

Internal DLA is a stochastic growth model in which particles are
produced at the origin of a multidimensional lattice and then perform
independent random walks until they reach the first site which has not
been previously visited. Then they stop at that forever. We show how
methods of hydrodynamic limits can be applied to study the asymptotic
shape, growth rate and profile of live particles in appropriate
regimes.
 BYRON SCHMULAND, Alberta
To be announced


