Table Of ContentInstitute of Mathematical Statistics
LECTURE NOTES-MONOGRAPH SERIES
Statistics in Molecular Biology
and Genetics
Selected Proceedings of a 1997 Joint AMS-IMS-SIAM Summer Conference on Statistics
in Molecular Biology
Francoise Seillier-Moiseiwitsch, Editor
Volume 33
Published by the Institute of Mathematical Statistics
and the American Mathematical Society
Institute of Mathematical Statistics
LECTURE NOTES-MONOGRAPH SERIES
Volume 33
Statistics in Molecular Biology
and Genetics
Selected Proceedings of a 1997 Joint AMS-IMS-SIAM
Summer Conference on Statistics in Molecular Biology
Francoise Seillier-Moiseiwitsch, Editor
American Mathematical Society
Providence, Rhode Island
Institute of Mathematical Statistics
Hayward, California
Institute of Mathematical Statistics
Lecture Notes-Monograph Series
Editorial Board
Andrew A. Barbour, Joseph Newton, and David Ruppert (Editor)
The production of the IMS Lecture Notes-Monograph Series is
managed by the IMS Business Office: Julia A. Norton, IMS
Treasurer, and Elyse Gustafson, IMS Business Manager.
This volume was co-published with the American Mathematical Society
Library of Congress Catalog Card Number: 99-076060
International Standard Book Number 0-940600-47-1
Copyright © 1999 Institute of Mathematical Statistics
All rights reserved
Printed in the United States of America
Ill
TABLE OF CONTENTS
Preface v
F. Seillier-Moiseiwitsch
Genetic Mechanisms
On a Markov Model for Chromatid Interference 1
H. Zhao and T. Speed
Population Genetics
Some Statistical Aspects of Cytonuclear Disequilibria 21
5. Datta
Diffusion Process Calculations for Mutant Genes in Nonstationary
Populations 38
R. Fan and K. Lange
The Coalescent with Partial Selfing and Balancing Selection:
An Application of Structured Coalescent Processes 56
M. Nordborg
Human Genetics
Statistical Aspects of the Transmission/Disequilibrium Test (TDT) 77
W. Ewens
Estimation of Conditional Multilocus Gene Identity among Relatives 95
E. Thompson and S. Heath
Quantitative Genetics
A Review of Methods for Identifying QTL's in Experimental Crosses 114
K. Broman and T. Speed
Evolutionary Genetics
Markov Chain Monte Carlo for the Bayesian Analysis of Evolutionary
Trees from Aligned Molecular Sequences 143
M. Newton, B. Man and B. Larget
Likelihoods on Coalescents: A Monte Carlo Sampling Approach to
Inferring Parameters from Population Samples of Molecular Data 163
J. Felsenstein, M. Kuhner, J. Yamato and P. Beerli
IV
Uses of Statistical Parsimony in HIV Analyses 186
K. Crandall
Linear Estimators for the Evolution of Transposable Elements 207
P. Joyce, L. Fox, N. Casavant and H. Wichman
A Conditional Approach to the Detection of Correlated Mutations 221
M. Karnoub, F. Seillier-Moiseiwitsch and P.K. Sen
Correlated Mutations in Protein Sequences: Phylogenetic and Structural
Effects 236
A. Lapedes, B. Giraud, L. Liu and G. Stormo
Sequence Motifs
Compound Poisson Approximations for Occurences of Multiple Words ... 257
G. Reinert and 5. Schbath
Protein Structure
Deriving Interatomic Distance Bounds from Chemical Structure 276
M. Trosset and G. Phillips
Protein Fold Class Prediction is a New Field for Statistical Classification
and Regression 288
L. Edler and J. Grassmann
PREFACE
The papers in this volume were presented at one of the 1997 Summer Re-
search Conferences in the Mathematical Sciences jointly sponsored by the In-
stitute of Mathematical Statistics, the American Mathematical Society and the
Society for Industrial and Applied Mathematics. The theme of the meeting was
"Statistics in Molecular Biology and Genetics". That this volume is published
jointly by the Institute of Mathematical Statistics and the American Mathe-
matical Society reflects the emerging importance of Statistics in these fields.
These papers fall into broad categories: population genetics, evolutionary
genetics, protein structure, genetic mechanisms, quantitative genetics, human
genetics and sequence motifs. While some of these areas have a long history
of statistical input (and have motivated some mainstream statistical devel-
opments), others are new statistical applications. The talks by Professors D.
Botstein, M.-C. King and M. Olson underlined the great need for statistical
expertise in cutting-edge biological technology. Their stimulating presentations
treated us with wonderfully clear overviews of current directions in important
areas of genetic research (namely, physical mapping, genetic mapping and func-
tional genetics).
The manuscripts underwent a rigorous review process: each was scrutinized
by two anonymous referees. For their critical reviews, my gratitude goes to:
D. Balding, L. Edler, W. Ewens, J. Felsenstein, J. Hein, P. Joyce, A. Kong,
M. Kuhner, K. Lange, A. Lapedes, M. Man, P. Marjoram, K. Mengersen, M.-
S. McPeek, M. Moehle, D. Nelson, M. Nordborg, I. Painter, A. Pluzhnikov, A.
Ramaswami, G. Reinert, S. Sawyer, M. Stephens, E. Thompson and M. Trosset.
I particularly want to thank Professors M. Waterman and P. Donnelly for
their help in organizing the conference. Financial support for the meeting was
provided by the National Science Foundation and the National Institutes of
Health. Professor P. Donnelly deserves a double thank-you for dealing with
the papers with which I had a conflict of interest. I extend my thanks to the
anonymous referees he selected.
Finally, I am enormously endebted to R. Budrevich for his tireless help in
the preparation of this volume.
F. Seillier-Moiseiwitsch
Chapel Hill
VI
Statistics in Molecular Biology
IMS Lecture Notes - Monograph Series (1999) Volume 33
ON A MARKOV MODEL FOR CHROMATID
INTERFERENCE
BY HONGYU ZHAO AND TERENCE P. SPEED
Yale University School of Medicine and University of California, Berkeley
Meiotic exchange between homologous chromosomes takes place after the
formation of a bundle of four chromatids. Crossovers are precise breakage-and-
reunion events. Random strand involvements (no chromatid interference) and
random distribution of crossovers (no chiasma interference) are usually assumed
in analyzing genetic data. In this paper, we discuss a Markov model for chro-
matid interference. Closed form expression for the probability of any multilocus
recombination/tetrad pattern is derived. Both chromatid interference and chi-
asma interference can be studied together using this model. In particular, we
discuss chi-square models for chiasma interference.
1. Introduction. In diploid cells, each chromosome is paired with its ho-
mologue during meiosis . Each member of a given homologous pair has two
identical sister chromatids, so that each synapsed paired structure consists of
four chromatids. Usually one or more crossovers occur among the four chro-
matids. A crossover is a precise breakage-and-reunion event occurring between
two nonsister chromatids.
The types of genetic data considered here are single spore data, in which
the products of a single meiosis are recovered separately, and tetrad data, in
which all four meiotic products are recovered together. A tetrad consists of four
spores, each of which is haploid, encased in a structure called an ascus. In some
organisms, such as Neurospora crassa (red bread mold), tetrads consist of four
spores in a linear order corresponding to the meiotic divisions; these are called
ordered tetrads. In other organisms, such as Saccharomyces cerevisiae (baker's
yeast), the four spores are produced as a group without order and are called
unordered tetrads. Griffiths et al. (1996) covers relevant genetic background.
In this paper, genes (markers, loci) are denoted by script letters. For exam-
ple, we use A and B to denote two genes. Alleles of A are denoted by A and
a, while alleles of B are denoted by B and b. Suppose that A and B are on the
same chromosome arm, and consider a diploid cell having AB and ab on ho-
mologous chromosomes. There are four possible products at these loci resulting
from meiosis of this cell, namely, AB, ab, Ab, and aB. The first two are called
Supported in part by NIH grant HG-01093.
AMS 1991 subject classifications. Primary 60J20; secondary 60P10.
Key words and phrases. Chromatid interference, chiasma interference, Markov model, chi-square
model, single spore data, tetrad data.
H. ZHAO AND T. P. SPEED
parental types or nonrecombinants, the other two types, Ab and aB, are called
recombinants. If two markers are recombined by crossovers in a meiotic product,
then during meiosis an odd number of crossovers must have occurred between
the two markers on the strand carrying them. The proportion of recombinants,
r^^, is called the recombination fraction. Because recombination fractions are
not additive, genetic (or map) distance is used as an additive measure of distance
between loci. Genetic distance between two markers is defined as the average
number of crossovers per strand per meiosis between these markers. The unit
of genetic distance is Morgan (M). Two markers are 1M apart if on average
there is one crossover occurring on a single strand per meiosis between these
two markers. In practice, centiMorgan (cM = 0.01M) is more commonly used
in genetic mapping.
The occurrence of crossovers cannot be observed directly and has to be in-
ferred from observed recombination events. In the case of single spore data, a
given meiotic product may be scored as recombinant or non-recombinant for
each pair of markers. The map distance between two markers can be estimated
from the observed recombination fraction. In the case of unordered tetrad data,
there are three possible observed outcomes for each pair of markers: parental di-
type when all four strands are non-recombinants, tetratype when exactly two of
the four strands show recombination between the two markers, and nonparental
ditype when all four strands are recombinants. The map distance between two
markers can be estimated from the observed proportions of these three tetrad
types. In the case of ordered tetrads bearing one marker Λ with alleles A and
a, there are six distinguishable configurations:
1 2 3 4 5 6
A a a A A a
A a A a a A
a A a A a A
a A A a A a
Configurations 1 and 2 are called first division segregation (FDS) patterns and
configurations 3 to 6 are called second division segregation (SDS) patterns.
Because of random spindle to centromere attachments during meiosis, configu-
rations 1 and 2 have the same probability, and the four configurations showing
SDS pattern also have the same probability (Griffiths et al. 1996). The map
distance between a marker and the centromere can then be estimated from the
observed SDS proportion.
To estimate map distance from the observed data, a model is needed which
connects the process of crossover to the observed outcomes. Any model has to
