Table Of ContentInterdisciplinary Applied Mathematics
Volumes published are listed at the end of the book.
Springer-Science+Business Media, LLC
Interdisciplinary Applied Mathematics
Volume 27
Editors
S.S.Antman J.E. Marsden
L. Sirovich S. Wiggins
Geophysics and Planetary Sciences
Imaging, Vision, and Graphics
Mathematical Biology
L. Glass, J.D. Murray
Mechanics and Materials
R.V.Kohn
Systems and Control
S.S. Sastry, P.S. Krishnaprasad
Problems in engineering, computational science, and the physical and biological
sciences are using increasingly sophisticated mathematical techniques. Thus, the
bridge between the mathematical sciences and other disciplines is heavily
traveled. The correspondingly increased dialog between the disciplines has led to
the establishment of the series: Interdisciplinary Applied Mathematics.
The purpose of this series is to meet the current and future needs for the
interaction between various science and technology areas on the one hand and
mathematics on the other. This is done, firstly, by encouraging the ways that that
mathematics may be applied in traditional areas, as well as point towards new
and innovative areas of applications; and, secondly, by encouraging other
scientific disciplines to engage in a dialog with mathematicians outlining their
problems to both access new methods and suggest innovative developments
within mathematics itself.
The series will consist of monographs and high-level texts from researchers
working on the interplay between mathematics and other fields of science and
technology.
Warren J. Ewens
Mathematical
Population Genetics
I. Theoretical Introduction
Second Edition
, Springer
Warren J. Ewens
Department of Biology
University of Pennsylvania
Philadelphia, PA 19104
USA
[email protected]
Editors
S.S. Antman J.E. Marsden
Department of Mathematics Control and Dynamical Systems
and Mail Code 107-81
Institute for Physical Science and Technology California Institute of Technology
University of Maryland Pasadena, CA 91125
College Park, MD 20742 USA
USA [email protected]
[email protected]
L. Sirovich S. Wiggins
Division of Applied Mathematics School of Mathematics
Brown University University of Bristol
Providence, RI 02912 Bristol BS8 1T W
USA UK
chico@carne\ot.mssm.edu s.w [email protected]
Mathematics Subject Classification (2000): 92-02,92010, 6OJ70
ISBN 978-1-4419-1898-7 ISBN 978-0-387-21822-9 (eBook)
DOI 10.1007/978-0-387-21822-9
© 2004 Springer Science+8usiness Media New York
Originally published by Springer-Verlag New York. Inc in 2004
Softcover reprint of the hardcover 2nd edition 2004
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For Bronnie and Dave
Preface
This is the first of a planned two-volume sequence discussing mathematical
aspects of population genetics theory, with an emphasis on the evolution
ary theory. This first volume is intended to discuss the more introductory
aspects of the theory, with the second volume taking up more advanced
and more recent aspects. Because of this, this first volume draws heavily
on the first (1979) edition of this book, since the material in that edition
may now be taken, to a large extent, as introductory to the contemporary
theory. A second reason for drawing heavily on the 1979 edition is that
many present-day students have asked for access to earlier material not
now easily available. It is indeed remarkable how many results well-known
in the 1970's, and appearing in the literature of the time, are rediscovered
in the modern literature.
On the other hand, the subject has greatly expanded in scope and depth
over the last twenty-five years. Many topics have been introduced during
that time, or developed well beyond the level reached in the 1970's. No
doubt the most important of these is the development of the theory of
molecular population genetics. Introductory aspects of this theory molec
ular population genetics are taken up in the later chapters of this volume,
but a far more extensive description of the molecular theory will be given
in Volume II. As one example of this, the theory behind currently active
haplotype mapping projects will be discussed. To this extent, Volume II
will be largely data-based. It will thus also form connections between evo
lutionary genetics and currently active areas of problems of human genetics
and bioinformatics. On the other hand, developments of the evolutionary
theory itself will be considered also, taking up evolutionary questions relat-
viii Preface
ing to many species rather than evolutionary behavior within one species.
Other evolutionary topics such as the game theoretical approach to evo
lution, the analysis of gene-environment interactions, gene conversion and
the extended development of the concept of inclusive fitness, will also be
discussed in Volume II.
Despite the emphasis on evolutionary population genetics in this volume,
some material concerning human genetics, in particular those parts of the
theory that are best discussed in evolutionary terms, has been included.
One of the more pleasing developments over the last two decades has been
a convergence of work in mathematical human genetics and mathemati
cal evolutionary genetics, areas which in 1979 had very little overlap. A
manifestation of this convergence is the recent volume on mathematical
population genetics and human evolution by Donnelly and Tavare (1997).
The aim of the 1979 edition, namely to focus on the purely mathemat
ical aspects of population genetics theory, is retained in this book, even
though it is recognized that this provides a narrow and distorted view on
the subject of population genetics, and indeed of theoretical population ge
netics, as a whole. Thus, as in 1979, the book is intended as a complement
to broader and more balanced accounts of population genetics generally.
There are now many excellent books available devoted to this broader field,
but these often do not attempt any depth of mathematical treatment, so
that there is still a place for a narrowly focussed mathematical treatment.
Apart from this, there are now several excellent books on specific aspects
of population genetics theory. Of these it is appropriate to mention that by
Lynch and Walsh (1998) on quantitative traits, a topic not covered in this
volume, Epperson (2003) on geographical genetics and books by Chris
tiansen (2000) and Burger (2000) on multilocus theory. All these books
carry the theory beyond the introductory level aimed at in this volume.
One aim of the 1979 volume, not explicitly stated, was to induce
mathematically-trained workers to enter the population genetics field. This
aim is continued in this volume, and the mathematical beauty of many of
the formulas in the molecular genetics chapters of this book should help in
this endeavor.
The molecular nature of current data implies that statistical methods
are used far more frequently than was the case in 1979, with the molecular
data being used to test various hypotheses about the evolutionary process.
For the statistical analyses discussed I have adopted the standard conven
tion of employing upper-case letters to denote random variables and the
corresponding lower-case letters to denote their observed values, except in
cases where this seemed pedantic. This has also sometimes implied replac
ing Greek letters sometimes used in the literature for random variables by
Roman letters. Probability distributions and density functions are written
in lower case.
Despite the fact that the earlier chapters of this book are based heavily
on the 1979 edition, the discussion does sometimes differ substantially from
Preface IX
that in the 1979 edition, especially where the 1979 viewpoint now seems
to be misguided or out of date. As one example, the discussion of the Fun
damental Theorem of Natural Selection is now quite different from that of
the 1979 edition. The 1979 interpretation of the theorem, standard at the
time, is now seen as incorrect and has been discarded. However I have no
illusions about its ability to continue to exist as the textbook interpreta
tion, offered to students, especially since the correct interpretation requires
greater mathematical depth than does the textbook version.
Current theory in mathematical population genetics emphasizes retro
spective analyses rather then the prospective analyses making up much
of the classical theory. In particular, theory surrounding the Kingman co
alescent process forms, quite appropriately, a significant part of current
research. An introduction to this theory is given in Chapter 10, and a more
extensive discussion will be given in Volume II. One of the aims of this
book is to make connections between the prospective theory that much
of the book considers with this retrospective theory. Apart from this, the
classical prospective theory, considering properties of forward-going evolu
tionary processes, is still relevant to retrospective analyses. As one example
of this, the theory surrounding the coalescent is often best developed by
considering a process moving forward in time from a common ancestor to
a sample of genes in the present generation, rather than by starting with
the contemporary sample and moving backward in time to the common
ancestor.
Despite the natural current emphasis on the retrospective theory, there
are several reasons for discussing the prospective theory in some detail in
this book. The Darwinian theory of evolution continues to be attacked by
various interest groups, and these attacks are sometimes helped by incorrect
statements about the prospective evolutionary theory made sometimes even
by biologists. The many extraordinary statements made by L0vtruP (1987),
for example, illustrate this.
Even professionals in population genetics contribute to this problem.
Arguments against evolution as a Darwinian process have been based the
concept of the substitutional genetic load, which I believe has been delete
rious concept that should be dropped from into the theory. Substitutional
load "theory", as well as segregational load "theory", is discussed, and I
hope debunked, in Section 2.11. The "blind watchmaker" paradigm, peri
odically raised by outsiders to population genetics theory as refuting the
Darwinian process and indeed evolution generally, is discussed, and I hope
also debunked, in Section 1.6. On two more narrow points where those
active in areas close to population genetics theory frequently abuse the
theory, the correct as opposed to the text book version of the Fundamental
Theorem of Natural Selection, mentioned above, is described in Sections
2.9 and 7.4.5. A discussion of the much-misunderstood expression "effective
population size", often incorrectly used in with reference to the history of
the human population, is given in Section 3.7.
x Preface
The recent and welcome infusion of population genetics theory into a
variety of disciplines associated with the evolutionary process has not been
without some problems. Perhaps the most important of these is that it
has led to an uncritical use of some formulas from the theory without
due assessment of whether the formulas are appropriate to the situation
at hand. All formulas in population genetics theory derive some model of
the evolutionary process, and in some cases this model can be no more
than a very rough approximation to reality. For this reason a new section
has been added, in this volume, discussing the modeling process and what
may reasonably be concluded from the models discussed in the population
genetics literature.
On more technical matters, it has not always been possible to use the no
tation of various published papers whose results are described here, since in
some cases the notation used in different papers for the same quantity dif
fer, and in other cases different authors use the same symbol for different
quantities. As in the 1979 version of this book, the notation is not con
sistent, so that the symbol "x;" might variously mean the frequency of an
allele in generation i, in subpopulation i, the frequency of the allele Ai, and
so on. On a similar point, I have adopted American spelling but English
punctuation conventions: The latter are more suited to a mathematical
text.
It is a pleasure to acknowledge the inspiration I have received from my
long-time colleagues Bob Griffiths and Geoff Watterson. It is also a pleasure
to thank Peter Donnelly, John Kingman and Simon Tavare for an equally
close, albeit long-range, collegial association. I thank various colleagues for
pointing out typographical errors in the 1979 edition of this book, and
Alan Rogers for pointing out an error concerning exchangeable model cal
culations. Any errors observed in this volume will be gratefully received
at [email protected] and an archive of these will be maintained at
www.textbook-errata.org
Philadelphia, Pennsylvania, USA Warren J. Ewens
October 2003
Contents
Preface vii
Introduction xvii
1 Historical Background 1
1.1 Biometricians, Saltationists and Mendelians 1
1.2 The Hardy-Weinberg Law 3
l.3 The Correlation Between Relatives 6
l.4 Evolution. 11
1.4.1 The Deterministic Theory 11
1.4.2 Non-Random-Mating Populations 18
1.4.3 The Stochastic Theory 20
1.5 Evolved Genetic Phenomena . 31
1.6 Modelling 35
1.7 Overall Evolutionary Theories 38
2 Technicalities and Generalizations 43
2.1 Introduction 43
2.2 Random Union of Gametes. 44
2.3 Dioecious Populations . 44
2.4 Multiple Alleles 49
2.5 Frequency-Dependent Selection 54
2.6 Fertility Selection 54
2.7 Continuous-Time Models . 57
