Table Of ContentBiomathematics
Volume 1
Edited by
K. Krickeberg· R. C. Lewontin . J. Neyman
M. Schreiber
Mathematical Topics
in Population Genetics
Edited by
Ken-ichi Kojima
With 55 Figures
Springer-Verlag Berlin· Heidelberg· New York 1970
ISBN -13 :978-3-642-46246-7 e-ISBN-13 :978-3-642-46244-3
DOl: 10.1007/978-3-642-46244-3
This work is subject to copyright. All rights are reserved, whether the whole or part of the
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with the publisher. © by Springer-Verlag Berlin· Heidelberg 1970. Library of Congress
Catalog Card Number 78-103329.
Softcover reprint of the hardcover 1st edition 1970
Title No. 7880.
Preface
A basic method of analyzing particulate gene systems is the proba
bilistic and statistical analyses. Mendel himself could not escape from an
application of elementary probability analysis although he might have
been unaware of this fact. Even Galtonian geneticists in the late 1800's
and the early 1900's pursued problems of heredity by means of mathe
matics and mathematical statistics. They failed to find the principles of
heredity, but succeeded to establish an interdisciplinary area between
mathematics and biology, which we call now Biometrics, Biometry, or
Applied Statistics.
A monumental work in the field of popUlation genetics was published
by the late R. A. Fisher, who analyzed "the correlation among relatives"
based on Mendelian gene theory (1918). This theoretical analysis over
came "so-called blending inheritance" theory, and the orientation of
Galtonian explanations for correlations among relatives for quantitative
traits rapidly changed. We must not forget the experimental works of
Johanson (1909) and Nilsson-Ehle (1909) which supported Mendelian
gene theory. However, a large scale experiment for a test of segregation
and linkage of Mendelian genes affecting quantitative traits was, prob
ably for the first time, conducted by K. Mather and his associates and
Panse in the 1940's.
By 1930 or thereabout, R. A. Fisher and J. B. S. Haldane in England,
and Sewall Wright in the United States of America were identified as the
founders of population genetics. A series of papers was published by
Haldane on mathematical analyses of selection, mutation, etc., in popula
tions of Mendelian genes, and the most of these analyses were assembled
in his The Causes of Evolution (1932). Fisher's book called The Genetical
Theory of Natural Selection was published in 1930. The inspiration one
gets from reading these books is unmeasurable.
Sewall Wright has been contributing extensively to the development
of popUlation genetics ever since the publication dealing with the in
heritance of coat color characteristics in guinea pigs (1916). In 1921, he
published five papers in a row concerning "Systems of Mating". Then
came a historical paper by Wright Evolution in Mendelian Populations
in 1931. In this paper he introduced the concept of "Effective Population
Size", and "Genetic Random Drift" in finite size populations. Undoubt-
VI Preface
edly, a three-volume series by Sewall Wright, of which the first volume
appeared in 1968, will make another cornerstone in population genetics.
I am extremely grateful to have Professor Sewall Wright as one of
the contributors in this volume.
There are dozens of very capable investigators whose backgrounds
vary. Just mentioning several, Samuel Karlin and his group at Stanford
University with bona fide mathematics backgrounds. There is Walter
Bodmer, also at Stanford University, who is a brilliant investigator in
both theoretical and experimental genetics. There are Oscar Kempthorne,
Howard Levene, R. E. Comstock, Everett Dempster and many others
with statistical backgrounds. However, there is a limit for this kind
of undertaking, assembling authors, topics and matching them. Conse
quently, I had to reluctantly drop several investigators from a list of
potential contributors, letting a higher priority go to those with biological
backgrounds.
My original aim of this book was to document major issues and
ideas that appeared during the development of population genetics in
the past half century. However, some papers in this book are the results
of original researches, and others are reviews of some aspects of popula
tion genetics.
While I have been editing this book, I have begun to feel a new trend
and "revival" (according to I. M. Lerner) in population genetics. The
new generation of population geneticists recognizes various factors such
as population density, non-constant fitness values of a given genotype
including frequency-dependent selection, biochemical aspects of popula
tion genetics, and ecological conditions for a population, and so forth.
Thus, I am convinced that the studies of genetic populations have entered
into a new phase, and this is a good reason to document various mathe
matical studies made up-to-date by geneticists in one volume.
Several authors in this book dedicated their papers to Professor
Th. Dobzhansky for his seventieth birthday.
I am personally grateful for the encouragement and leniency for the
time schedule for publication of this book given by Dr. Klaus Peters of
Springer-Verlag, and the National Institutes of Health of the United
States for their summer supports through the project program grant
GM 15769. My personal thanks go to Mrs. JoBeth Porterfield who typed
up almost all of the manuscripts in the final form and assisted with
editorial work. I also acknowledge Dr. Yoshiko Tobari and Mrs. Myrtle
Wing who supervised my laboratory work while I was heavily involved
in editorial work.
Austin, September 1970 Ken-ichi Kojima
Table of Contents
S. WRIGHT Random Drift and the Shifting Balance Theory
of Evolution . . . . . . . . . . . . . . .
1. R. G. TURNER Changes in Mean Fitness under Natural Selection 32
R. H. RICHARDSON Models and Analyses of Dispersal Patterns. . 79
C. C. COCKERHAM Avoidance and Rate of Inbreeding. . . . . . 104
1. F. CROW Genetic Loads and the Cost of Natural Selection 128
M. KIMURA Stochastic Processes in Population Genetics,
with Special Reference to Distribution of Gene
Frequencies and Probability of Gene Fixation 178
W. G. HILL Theory of Limits to Selection with Line Crossing 210
A. ROBERTSON A Theory of Limits in Artificial Selection with
Many Linked Loci . . . . . . . . 246
1. A. SVED and
O. MAYO The Evolution of Dominance 289
H. E. SCHAFFER Survival of Mutant Genes as a Branching Process 317
C. C. LI The Incomplete Binomial Distribution.. 337
K. KOJIMA and
R. C. LEWONTIN Evolutionary Significance of Linkage and
Epistasis. . . . . . . . 367
R. LEVINS Fitness and Optimization. . . 389
List of Contributors
C. CLARK COCKERHAM, Department of Experimental Statistics, North
Carolina State University, Raleigh, North Carolina/USA.
JAMES F. CROW, Laboratory of Genetics, Genetics Building, Madison,
Wisconsin/USA.
WILLIAM G. HILL, Institute of Animal Genetics, Edinburgh/Scotland.
MOTOO KIMURA, National Institute of Genetics, Mishima/Japan.
KEN-IcHi KOJIMA, Department of Zoology, University of Texas, Austin,
Texas/USA.
RICHARD LEVINS, Committee on Mathematical Biology, University of
Chicago, Chicago, Illinois/USA.
R. C. LEWONTIN, Committee on Mathematical Biology, University of
Chicago, Chicago, Illinois/USA.
C. C. LI, Graduate School of Public Health, University of Pittsburgh,
Pittsburgh, Pennsylvania/USA.
OLIVER MAYO, Department of Genetics, University of Adelaide,
Adelaide/South Australia.
R. H. RICHARDSON, Department of Zoology, University of Texas,
Austin. Texas/USA.
ALAN ROBERTSON, Institute of Animal Genetics, Edinburgh/Scotland.
HENRY E. SCHAFFER, Department of Genetics, North Carolina State
University, Raleigh, North Carolina/USA.
JOHN A. SVED, School of Biological Sciences, University of Sydney,
Sydney/Australia.
JOHN R. TURNER, Department of Biology, University of York, Hes
lington/England.
SEWALL WRIGHT, Department of Genetics, University of Wisconsin,
Madison, Wisconsin/USA.
Random Drift and the Shifting Balance Theory
of Evolution*'**
S. WRIGHT
Introduction
The word "drift" has been used rather frequently in population
genetics, but not always in the same sense. The author is probably respon
sible for its introduction by using it in papers in 1929, though with no
intention of giving it a technical meaning. It has, however, come to be so
used and thus is in need of careful definition.
My first use of "drift" in connection with population genetics was in
a paper on R. A. Fisher's (1928) theory of the evolution of dominance of
type alleles of deleterious mutations.
"In consequence of these facts there will be a gradual drift of the
heterozygote toward wild type" (Wright, 1929b).
"Drift" referred here to the results of a directed process, selection. It
was used in a second note on the same subject in a different sense.
"Being random, such variations largely neutralize each other but
there is a second order drift which cannot be ignored" (Wright, 1929 c).
In this case "drift" referred to the cumulative effects of a random pro
cess, accidents of sampling. I continued to use the term occasionally, in
such expressions as "drifting at random" and it was taken up by others
as a technical term in this second sense. I adopted it in a later paper
(Wright, 1956 b) but pointed out the importance of always distinguishing
"steady drift" and "random drift" and in the latter case of specifying
whether from accidents of sampling or otherwise (as from fluctuations in
selection pressure). Cain and Currey (1963) have suggested the convenient
term "sampling drift" for the component due to accidents of sampling.
The Shifting Balance Theory
Most of the use of "random drift" has been in connection with the
shifting balance theory of evolution proposed in 1929a (more fully in
1931 and 1932) and much of the confusion in its use seems to have been
* Paper No. 1203 from the Department of Genetics, University of Wisconsin.
** Aid is acknowledged from grant no. G-4920 from the National Science Foundation.
2 s. Wright
due to misunderstanding of the latter. In this theory, random processes
('lOt necessarily sampling drift) play an essential role but only as an ad
junct of selection. Many of those who have referred to the theory have
insisted that random drift (specifically sampling drift) had been proposed
as an alternative to selection, instead of merely as a source of raw material
for selection at a higher level than that provided by mutation. In order
to clarify this matter, it is desirable to begin with the major premises of
the theory since in fairly recent papers some of these have been asserted
to have been the opposite of what they actually were.
Premises
A. Multiple Factor Theory of Quantitative Variability. The aspect of
evolution that is considered primarily is that of adaptive transformation
based on the cumulative effects of changes in the frequencies of multiple
minor factors, apparently responsible at any given time for merely quan
titative variability. Species are assumed to carry numerous isoalleles at
most loci, at fairly high frequencies, instead of being almost hom allelic
for a single type allele at each. The occasional utilization of major muta
tions is by no means excluded, but it is assumed that the establishment of
one in the species depends on an adaptive process in the arra y of modifiers.
B. Universal Pleiotropy. The ramifications from the primary effect
of any gene replacement through the network of metabolic reactions and
of intercellular developmental processes to the characters which have
selective differences, insure that each such replacement have effects on
numerous characters (cf. Wright, 1963 a).
e.
Multiple Selective Peaks. The occurrence of joint reactions in the
above networks implies extensive interaction effects among the effects of
genes on the various characters. Moreover, all characters contribute to
a single one, "selective value". Because of pleiotropy and interaction,
there is always the potentiality for a vast number of different more or less
harmonious combinations. These correspond to different "selective peaks"
in the "surface" of selective values relative to the multidimensional field
of gene frequencies. This contrasts with the single best type which would
necessarily be present if each gene were favorable or unfavorable in itself.
Finally, even where there is apparently additivity with respect to a quanti
tatively varying character, the usual intermediacy of the optimum grade
insures the potentiality for a great many selective peaks, and pleiotropy
insures that these be at diverse values.
D. Multiple Partially Isolated Demes. Most species contain many
small, random breeding local populations (demes) that are sufficiently
isolated, if only by distance, to permit differentiation of their sets of gene
Random Drift and the Shifting Balance Theory of Evolution 3
frequencies, but that are not so isolated as to prevent the gradual spread
ing of favorable gene complexes throughout the species from their centers
of origin. This differentiation need not be associated with conspicuous
phenotypic differences.
Phases of the Evolutionary Process
Change of gene frequency is treated as the elementary evolutionary
process since it permits reduction of the effects of all factors to a common
basis.
According to the theory built on the above premises, each significant
step in the evolutionary process involves, in general, the conjunction of
three phases. In principle, these would permit a continuing process as
long as there is any ecological opportunity for this, with very little muta
tion and without necessarily involving any environmental differences
among localities or any systematic environmental changes in time. In the
first statement of these three phases, no such differences or changes are
assumed.
1. Phase of Random Drift. In each deme the set of gene frequencies
drifts at random in a multidimensional probability distribution about
the equilibrium characteristic of a particular selective peak. The set of
equilibrium values for each gene frequency is the resultant of three sorts
of "pressures", those due to rccurrent mutation, to recurrent immigration
from other demes, and to selection. The fluctuations responsible for the
random drift are in this special case the cumulative effects of accidents of
sampling.
2. Phase of Mass Selection. From time to time, the set of gene fre
quencies drifts across one of the many two-factor saddles in the proba
bility distribution in a deme. There ensues a period of relatively rapid
change, dominated by selection among individuals (or families), until
the set approaches the equilibrium associated with the new selective
peak about which it now drifts at random, and thus returns to the first
phase, but in general at a higher level.
3. Phase of Interdemic Selection. A deme that comes under control
of a selective peak, superior to those controlling the neighboring demes,
produces a greater surplus population and by excess dispersion system
atically shifts the positions of equilibrium of these toward its own posi
tion, until the same saddle is crossed in them and they move autono
mously to the same peak. This process spreads in concentric circles. Two
such circles spreading from different centers may overlap and give rise to
