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Biomathematics
Volume5
Edited by
K. Krickeberg . R. C. Lewontin· J. Neyman
M. Schreiber
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Albert Jacquard
The Genetic
Structure
of Populations
With 92 Figures
Springer-Verlag Berlin . Heidelberg . New York 1974
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Professor Albert Jacquard
Institut National d'Etudes D6mographiques Paris, France
Translated
by
D and
B Charlesworth
Department of Genetics, The University of Liverpool, England
Title of the French Edition:
Structures G6n6tiques des Populations
Masson Cie, Editeurs, Paris, 1970
AMS Subject Classifications (1970)
92-A-I0
ISBN 978-3-642-88417-7 ISBN 978-3-642-88415-3 (eBook)
DOI 10.1007/978-3-642-88415-3
The use of general descriptive names, trade marks, etc. in this publication, even if the former
are not especially identified, is not to be taken as a sign that such names, as understood by
the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.
This work is subject to copyright. All rights are reserved, whether the whole or part of the
material is concemed, specifically those of translation, reprinting, re-use of illustrations,
broadcasting, reproduction by photocopying, machine
or
sirnilar means, and storage in data
banks. Under § S4 of the German Copyright Law where copies are made for other than
private use, a fee
is
payable to the publisher, the arnount to the fee be deterrnined by agree
ment with the publisher. © by Springer-Verlag Berlin . Heidelberg 1974 Library of Congress
Catalog Card Nurnber 73-80868. Typesetting and printing:
Universitätsdmckerei
H
Stürtz AG, Würzburg
Softcover reprint ofthe hardcover 1st edition 1974
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Preface to the
English Edition
I t
is part of the ideology of science that
it
is an international enterprise,
carried out by a community that knows no barriers of nation
or
culture.
But the reality is somewhat different. Despite the best intentions of
scientists to form a single community, unseparated by differences of
national and political viewpoint, they are, in fact, separated by language.
Scientific literature in German is not generally assimilated by French
workers, nor that appearing in French by those whose native language
is English. The problem appears to have become more severe since the
last war, because the ascendance of the United States as the preeminent
economic power led, in a time of big and expensive science, to a pre
dominance of American scientific production and a growing tendency
(at least among English-speakers) to regard English as the international
language of science. International congresses and journals of world
circulation have come more and more to take English as their standard
or official language. As a result, students and scientific workers in the
English speaking world have become more linguistically parochial than
ever before and have been cut off from a considerable scientific literature.
Population genetics has been no exception to the rule. The elegant
and extremely innovative theoreticaI work of Malecot, for example, is
only now being properly assimilated by population biologists outside
France.
I t
was therefore with some sense of frustration that I read Prof.
Jacquard's "Structures Genetiques des Populations", for I realized that
this superb treatment of the theory of evolutionary genetics would be
unavailable
to
me
as
a teacher because it was inaccessible
to
my students
as readers. What I found so attractive in Jacquard's book was the Iucidity
and elegance of its presentation, but most especially the fusion of demo
graphie and genetical concepts with abundant examples from human
populations. The fusion of genetics and demography has been a slow
process since Fisher first briefly considered the problems of gene frequency
change and population growth in the "Genetical Theory of Natural
Selection" in
1929.
I t is still far from complete,
but
the point of view
represented by Prof. Jacquard, a geneticist and a demographer, is slowly
gaining. I was thus delighted
at
the prospect of
an
English translation
of "Structures Genetiques des Populations" so that this point of view
could be exposed
to
the widest possible audience.
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VI
Preface
Prof. Jacquard has been extremely fortunate in his choice of trans
lators. The Charlesworths have combined linguistic skill with scientific
contributions in demography and genetics to make
"The
Genetic
Structure of Populations" not merely a translation, but a new work of
even greater virtue than the original.
R. C. Lewontin
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Author s Preface
to the English Edition
This book was originally written for French students; its style of
reasoning and method of presentation were chosen to conform with
current usage in French universities. There is therefore a risk in publishing
an English edition. Professor Lewontin, however, feels
that
we should
take this risk; I should like to thank him for his favourable opinion of
mybook.
My hesitation in seeing my book exposed to a wider public has been
lessened by the help I have received from my translators, who have
criticised the book as weIl as performing the heavy work of the translation
itself; these criticisms have helped me to fill in some of the gaps
that
were left in the first edition, and to correct some errors. I am very grateful
to them.
A.
Jacquard
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1rranslators J»reface
This book is a translation of "Structures Genetiques des Popula
tions", which was published in 1970. There are a number of changes in
the present edition.
In
particular, there are three new chapters at the end
of the book; also, the original treatment of populations with overlapping
generations has been replaced by a new version written by
B.
Charles
worth (Chapter 7, and Section 3 of Chapter
10).
There is a new Appendix
on difference equations, and an additional Section (2.2.3) in Chapter 12,
by B. Charlesworth, and a number of smaller alterations throughout the
book, some by Professor Jacquard and some by the translators.
We would like to thank Professor Jacquard for his cooperation with us
throughout the work of translating his book, and especially for his
tolerance and patience when we have made criticisms.
D. Charlesworth
B.
Charlesworth
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Introduetion. .
The Individual .
The Population
General Bibliography .
Table of
Contents
Part 1
Basie Faets and Concepts
1
1
2
3
Cbapter
1.
The Foundations of Genetics . . . . . . . 6
1. The Mendelian Theory of Inheritance. . . . . . . 6
1.1. Mendel's First
Law.
The Law of Segregation. . 6
1.2. Mendel's Second
Law.
Independent Assortment 8
1.3. Restrietion of Mendel's Second
Law.
Linkage . 9
1.4. Some Definitions. . . . . . . . . . . . . . 11
2.
The Physical Basis of Mendelian Inheritance. The Chromosomes.
13
2.1. The Behaviour of the Chromosomes. Mitosis and Meiosis . .
13
2.2. Consequences of Chromosome Behaviour for Hereditary Transmission of
Characters. . . . . . . . 15
2.3. Linkage and Crossing Over . 16
2.4. Human Chromosomes . . .
17
2.5. The Sex Chromosomes . . . 19
2.6. Chromosome Strueture.
DNA
20
2.7. Mutation . . . . . 23
2.8. Individual Diversity . . . .
24
Cbapter
2.
Basic Concepts and Notation. Genetie Structure
of
Populations and of
IndividuaJs. . . . . . . . . . 25
1. Probability . . . . . . . . . . . . . . . .
25
1.1. Definition of Probability . . . . . . . .
26
1.2. Principle of Addition of Probabilities . . . 27
1.3. Principle of Multiplication of Probabilities. 27
1.4. Bayes' Theorem . . . . . . . . . . . . 28
1.5. Random Variables . . . . . . . . . . . 30
1.6. The Expectation and Variance of a Random Variable .
30
1.7. Examples of Random Variables . . . . . . . . . . 31
2. Genetie Struetures . . . . . . . . . . . . . . . . . .
33
2.1. The Definition of Genie and Genotypie Struetures . .
33
2.2. The Relation between Genie and Genotypie Structures
34
2.3. The Probability Structures of Populations 35
2.4. Probability Structures of Individuals . . . . . . . .
36
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XII
Table of Contents
3.
Sexual Reproduction . . . . . . . . . . . . .
3.1. Genie Structures of Parents and Offspring . .
3.2. Genotypie Structure of Parents and Offspring
Part
2
A Reference Model: Absence of Evolutionary Factors
37
37
39
Chapter 3. Tbe Hardy-Weinberg Equihörium for one Locus . 42
1. Populations . . . . . . . . . . . 42
2. The Hardy-Weinberg Principle. . . 43
2.1. Stability of the Genie Strueture 43
2.2. Genotypie Structure . . . . . 44
2.3. Panmixia and Perfeet Panmixia 47
2.4. The Hardy-Weinberg Principle . 48
3. The Classical Treatment of the Hardy-Weinberg Equilibrium 49
3.1. Establishment of the Equilibrium. . . . . . . 50
3.2. Random Union of Gametes . . . . . . . . . 52
3.3. Properties of the Hardy-Weinberg Equilibrium . 53
4. The Equilibrium for Sex-Linked Genes . . . . . .
55
4.1. Passage from One Generation to the Next. . . 55
4.2. The Equilibrium State . . .
. ' .
. . . . . .
56
5. The Hardy-Weinberg Principle in Human Populations 58
5.1. Autosomal Loci with Two Alleles .
58
5.2. Autosomal Loci with Three Alleles . 61
5.3. Sex-Linked Genes
63
5.4. Y-Linked Genes . . . . . . . .
65
Chapter 4. The Equilibrium for
Two
Loci 66
1. The Role of Individuals . . . . . . . 66
2.
Genie Strueture . . . . . . . . . . 67
2.1. The Recurrence Relation for the Transition from One Generation to the
Next . . . . . . . . . . . . . . 68
2.2. The Constancy of Gene Frequeneies
69
2.3. The Approach
to
Equilibrium
69
3.
Genotypie Strueture . . . . . .
70
4. Two Loci, Bach with Two Alleles 71
4.1. Gamete Frequencies . . . . 71
4.2. Fusion of Two Populations .
73
4.3. Instantaneous Attainment of Equilibrium
75
5. The Detection and Measurement of Linkage
77
5.1. Detection of Linkage-Penrose's Method . 77
5.2. Estimation of Reeombination Fraetions-Morton's Method . 80
5.3. Smith's "Bayesian" Method of Estimating Reeombination Fractions 82
5.4. The Linkage Map of Man. . . . . . . . . . . . . . . . . . . 85
Chapter S. Tbe Inheritance of Quantitative Characters . . . . . 86
1. The Mean. . . . . . . . . . . . . . . . . . . . . . . 87
1.1. Definiton of Additive Effects and Dominance Deviations 87
1.2. Determination of the Additive Effects and Dominance Deviations
89
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Table of Contents XIII
1.3. The Effect of a Small Change in Gene Frequency . 91
1.4. The
Case
of a Single Locus with Two Alleles . .
91
1.5. Characters Controlled by Several Loci . . . . . 95
1.6.
An
Example of a Character Controlled by Several Genes: Skin Colour 96
2. The Variance . . . . . . . 98
2.1. EnvironmentaI Variance. . . . . . . 98
2.2. Genotypie Variance . . . . . . . .
100
2.3. The
Case
of a Loeus with Two Alleles. 101
Chapter
6.
Genetie Re1ationships between ReJatives 102
1.
The Measure of Relatedness. . . . . . . .
103
1.1. Identity by Descent . . . . . . . . . . .
103
1.2. The Definition of Coefficients of Identity .
104
1.3. The CaIeulation of Coefficients of Identity .
109
1.4. Sex-Linked Genes . . . . . . . . . . .
114
2. The Genetie Structures of Related Individuals .
116
2.1. The Relation between the Genie Structures of Related Individuals
117
2.2. The Relation between the Genotypie Struetures of Related Individuals 120
2.3. The Relations between the Genie Structures of Inbred Individuals 128
2.4. Other
Points.
. . . . . . . . . . . . . . . . . . .
129
3. Resemblance between Relatives . . . . . . . . . . . . . 131
3.1.
The Determination of the Covariance between Relatives. 131
3.2. Some Partieular Relationships . . . . . . . . . . . .
135
3.3.
The Case of a Loeus with Two Alleles . . . . . . . .
136
3.4. The Interpretation of Observed Correlations between Relatives .
138
Chapter
7.
OverJapping Generations. . . . . . . . . . .
141
1. The Demographie Description of a Population. . . . .
141
1.1. Demographie Parameters . . . . . . . . . . . .
142
1.2. The Future Demographie Structure of a Population
146
1.3. The Intrinsie Rate of Natural Increase . . . . . .
150
1.4. The Male
Population.
. . . . . . . . . . . . .
152
2. The Equilibrium Genetie Strueture of a Population with Overlapping Genera-
tions . . • •............................ 153
2.1. Genie and Genotypie Struetures of Populations with Overlapping Genera-
tions . . . . . . . . . . . . . . . . . . . . . . . .
153
2.2. The Evolution
of
the Genetie Structure of a Population . .
155
2.3. The Evolution of the Genotypie Strueture of a Populat ion.
157
2.4. Conelusions . . . . . . . . . . . . . . . . . . . . .
158
Part 3
The Causes of Evolutionary Changes in Populations
Cbapter 8. Finite Populations. . . . . . . . . . . . . . . . . . 160
1. Identity by Descent of Genes in Finite Populations. . . . . . .
160
1.1. The Inbreeding Coefficient and Coeffieient of Kinship
of
a Population.
160
1.2. Increase of the Inbreeding Coefficient in a Finite Populat ion. 161
1.3. Constant Effective Population Size. 163
1.4. Changing Effective Population Size. . . . . . . . . . . .
166
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XIV
Table
of
Contents
1.5. Relations between Relatives in a Finite Population . . . . . . . . . . 167
1.6. The Effect
of
Variance
in
Number
of
Offspring
on
the Effective Population
Size . . . . . . . . . . . . . . . . . . . . . . . . • . . . .
171
1.7. The Effect
of
the Prohibition
of
Incest
on
Effective Population
Size. .
175
1.8. Effective Population Size
in
Populations with Overlapping Generations
178
2. Changes
in
the Genotypic Probability Structure . . . • . . . . 178
2.1.
The
Difference Equation for Genotypic Probability Structure 179
2.2.
The
Genotypic Probability Structure
at
Intermediate Stages 182
2.3. The Stages
of
Change
in
Genotypic Structure
183
2.4. Genetic Drift . . . . . . . . . . 184
2.5. The Disappearance
of
Heterozygotes . . . . 186
2.6. Sib-Mating . . . . . . . . . . . . . . .
188
2.7. Summary . . . . . . . . . . . . . . . . 190
3.
The
Transmission
of
Genes from One Generation to the Next
191
3.1.
The
Probability Distribution
of
the Number
of
Genes Transmitted
191
3.2. Changes
in
Gene Frequencies . . . . . . . 192
3.3. Genetic
Drift
. . . . . . . . . . . . . . 193
3.4.
The Rate of
Attainment of
Homozygosity.
. 195
4. Matings between Relatives
in
a Finite Population 197
4.1. Matings between
Sibs.
. . . . . . . . . . 197
4.2. Matings between First
Cousins.
. . . . . .
198
4.3. The Role
of
the Variance
in
Number
of
Offspring 200
5. Observations
on Human
Populations . . . . . . . . 202
5.1. The Frequency
of
Consanguineous Marriages . . 202
5.2. Consanguineous Marriages in France . • . . . . 204
5.3. Consanguineous Marriages
in
Several Catholic Countries
208
5.4. Consanguineous Marriages
in
some Non-Catholic Countries . 210
5.5. Mating between Relatives
in
Populations with Overlapping Generations .
211
6. Subdivision
of
a Population . . . . . . . . . . . . . . . 212
6.1. Changes
in
Gene Frequencies and Coefficients
of
Kinship 212
6.2. Effect
of
Limited SampIe
Sizes.
. . . . . 215
6.3. Sampling Variance of . . . . . . . . . 216
6.4. The Effect
of
Relationship between Groups • 218
Chapter 9. Deviations from Random Mating
. . . . . • . . . . . . . . . •
220
1. Genotype Frequencies Among the Offspring of Consanguineous and Assorta-
tive
Matings.
. . . . . . . . . . . . . • . . . . . . .
221
1.1.
An
Example
of
Non-Independence between Mates
.......... 221
1.2. The Offspring
of
a Consanguineous Mating . . . . . . . . . . . . .
223
1.3. The Biological Consequences of Consanguineous Mating . . . . . . .
225
1.4. The Frequency
of
Consanguineous Marriages among the Parents
of
Children Affected with Genetic Disorders 227
2. Choice
of
Mates Based
on
Relatedness 228
2.1. Sib-Mating . . . . . . . . 229
2.2. Parent-Offspring Mating. . .
231
2.3. Half-Sib Mating . . . . • . 232
2.4. Double First-Cousin Mating . 234
2.5. First-Cousin Mating . . . . 236
2.6. Second-Cousin Mating . . . 240
2.7. Number of
Ancestors
and
the Approach Towards Homozygosity 242
2.8. Avoidance of
or
Preference for Certain Types of
Marriage.
. . . 243
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3. Assortative Mating . . . . . . . . • . • 248
3.1. Total Positive Assortative Mating Based
on
Genotype . • 249
3.2. Partial Positive Assortative Mating Based
on
Genotype • 249
3.3. Total Positive Assortative Mating Based
on
Phenotype •
251
3.4. Partial Positive Assortative Mating Based on Phenotype . • 253
3.5. Total Negative Assortative Mating Based
on
Genotype . • 254
3.6. Partial Negative Assortative Mating Based on Genotype . . 257
3.7. Total Negative Assortative Mating Based
on
Phenotype . • 259
3.8. Partial Negative Assortative Mating Based on Phenotype • 260
4.
The
Offspring
of
Consanguineous Marr iages. . . . . . . . . 261
4.1. The American Medieal Association Study of 1856 . . . . 262
4.2. The Study in Morbihan and Loir-et-Cher
of
1952. Definition
of "Peri-
natal Mortality Rate" . . . . • 263
4.3. The Study in the Vosges
in
1968 • 265
4.4.
The
Study
in
Japan
in
1958-60 . • 266
4.5. Sex-Linked Genes 267
4.6. Conclusions • 267
Further
Reading .
268
Chapter 10. SeIection . . . . . . . . 269
1. Some Simple Models of Selection . 270
1.1. Definition
of
Selective Values • 270
1.2. Change in Gene Frequencies . • 272
1.3. Loci with Two Alleles . • 273
1.4. Constant Selective Values . . . 277
1.5. Some Partieular Cases . . . • 278
1.6. Variable Selective Values . . 291
1.7. Constant Selection for a Sex-Linked Gene. 297
1.8. Selection
in
the Multi-Loeus Case
. • • •
301
2. The Consequences of Selection for the Mean Fitness of Populations 301
2.1. Constant Selective Values . . . . . . . . . . . . 302
2.2. Variable Selective Values . . . . . . . . . . . . 306
3. Selection
in
Populations with OverIapping Generations . 307
3.1. Demographie Parameters and Selective Differences . 308
3.2. Some Examples
of
Selection
in Human Populations.
311
4.
The
Study
of
Selection
in
Human
Populations.
. . . . 316
4.1. Difficulties in Detecting Selective Effects . . . • . 316
4.2. Direet Evidence for Selective Differences Associated with
Human
Poly-
morphisms. . . . . . . . . . . . . . . 318
4.3. Indirect Evidence for
Selection.
. . . . . 319
4.4. The Index of the Opportunity for Selection 321
Further Reading . . .
330
Chapter 11. Mutation. . . . . . . • . . . . . . . . . . . . 331
1. The Probability
of
Survival of a
Mutant
Gene . . . . . . . . 331
1.1. Elimination of a Neutral Allele. . . . . . . . . . . . . 332
1.2. Survival of a Neutral Mutant Gene
in
a Finite Population . • 334
1.3. The Probability
that
an
Advantageous New Mutant Gene will be Main-
tained
in
the Population. . . . . . . . . . . . . . . . . . . . . . 334
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XVI
Table of Contents
2. Recurrent Mutations • . . . . . . . . . . . . 335
2.1. Change
in
Genie Structure
due to
Recurrent
Mutation
336
2.2. The Case of a Locus with two Alleles . . . . . . . . 337
3.
The
Resultant Effect
of
Selection
and
Mutation
at
a Locus with Two Alleles 339
3.1.
The
Equilibrium between
Mutation and
Selection. . . . 339
3.2. Constant Selective Values . . . . . . . . . . . . . . 340
4.
The Human Mutation Rate
. . . . . . . . . . . . . . . 343
5.
The
Spread of a
Mutation:
Congenital Dislocation
of the Hip
349
Further Reading . . . . . . . . . . . . . . . . . . . . . 350
Cbapter 12.
Migration.
. . . . . . . .
351
1. Deterministie Models with Migration. 351
1.1. Changes
in
Genie Strueture . . . 352
1.2. Changes in Genotypie Strueture . 355
1.3. Applications
to
Actual Populations . 357
1.4. Deterministie Models of Migration when Other Forces
for
Change are
Acting
....................•........
358
2. Stoehastie Models with Migration . . . . . . . . . . . . . . . . . . . 362
2.1.
Migration.
. . . . . . . . . . . . . . . . . . . . . . . . . . . 363
2.2. Stoehastie Models of Migration with Other Evolutionary Forces also
Acting . . . . . . . . . . . . . . . . . . . . . . . . . 371
2.3. Migration and
Mutation
in a Spatially Continuous
Population.
376
3.
Data on
Migration
in Human
Populations . . . . . . . . . . . .
381
3.1. Models of the Migration Process . . . . . . . . . . . . . . . 381
3.2. Comparison
of the
Genetie Models with
the
Models of Migration 384
4. Conc1usions . 385
Further
Reading 386
Cbapter 13. The Combined Effects 01 Different Evolutionary Forces . 388
1. Wright's Model . . . . . . . . . . . . . . . . . . . . . . 389
1.1. Change in Gene Frequeney
from One Generation to
the
Next
389
1.2. The Fundamental Equation . . . . . . . . . . . . . . . 391
1.3. The Asymptotie Probability Dist ribution . . . . . . . . . 393
1.4. Some
Further
Results
on
Selection
and Mutation in
Finite Populations 398
2. Simulation . . . . . . . . . . . . . . . 400
2.1. The Prineiples of
Monte
Carlo
Methods.
. . . . . 401
2.2. The Use of
Monte
Carlo Methods . . . . . . . . 403
2.3. Simulation of the Genetie Strueture of a Population 405
3. Maintenance of Polymorphisms. Genetie Load. . . 406
3.1.
The
Equilibrium
under
Mutation
and
Selection 406
3.2. Maintenance of Variability by Neutral Mutat ion 408
3.3. Heterotie
Equilibrium.
. . . . . . . . . 409
3.4. The Genetie Load of a Locus . . . . . . 410
3.5.
The
Total Genetie
Load.
. . . . . . . . 412
3.6.
The
Effect
of
Inbreeding
on
Selective Value 415
3.7. Conc1usion: "Neo-Darwinian" Versus
"Non-Darwinian" Evolution.
417
Further Reading . . . . . . . . . . . . . . . . . . . . 418
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Table of Contents
XVII
Part 4
Tbe Study of
Human
Population Strueture
Cbapter
14.
Genetie Distance. I. Basic Concepts and
Methods.
. . 420
1. Tbe Idea
of Distance . . . . . . . . . . . . • . . . . . .
420
1.1.
Tbe Definition of Distance . . . . . . . . . . . . . .
422
1.2.
Distance between Objects Characterised
by
Measurements .
422
1.3.
Distance between Objects Characterised by Qualitative
Attributes. 428
2.
Distance between Individuals of
Known
Ancestry .
433
2.1.
Inadequacy of the Coefficient of Kinship . . .
433
2.2.
Genotypie Distance between Relatives
. .
. .
434
2.3. Other
Measures of Distance between Relatives . 444
3. Distances between Populations. . . . . . . . . .
449
3.1.
Distance between
the
Genetie Structures of Populations .
449
3.2.
Distance between Populations of Known Ancestry . . .
453
3.3.
Biometrical Estimation of
the
Relatedness of Two Populations .
456
3.4.
ConcIusion 461
Further Reading .
462
Cbapter
15.
Genetie Distance. n. The Representation of Sets of Objects 463
1. Prineipal Components Analysis
465
1.1.
Tbe First Principal
Axis.
. 465
1.2.
Tbe First
Principal Surface
470
1.3.
Generalisat ion . . . . . • .
473
1.4.
Normalisation of Measures .
475
1.5.
Interpretation of the Projections Obtained. Representation of Charaeters
476
2.
Prineipal Components Analysis of Contingeney Tables . . . • . . . . . .
479
2.1.
Tbe x2Metrie ....................•.....
480
2.2.
Tbe Projection of the Object-Points
Onto the
Prineipal Plane . . . . .
482
2.3. Tbe
Principal Plane of the Character-Points . . . . . . . . . . . . .
484
2.4.
Interpretation
of the
Simultaneous Representation
of
Objects
and
Charac-
ters.
. . . . . . . . . .
485
3. Cluster Analysis . . . . . . . . . . . . . . . . . . . . . . .
. . 487
3.1.
Information and Variance. . . . . . . . . . . . . . . . .
. . 488
3.2.
Aggregation
of
Two
Objeets.
. . . . . . . . . . . . . . .
. .
489
3.3.
Interpretation of the Decrease in Variance: Tbe Diameter of a Class 491
3:4.
Phylogenetie Trees . . . . . . . . . . . . . . . . . . . .
. .
492
Cbapter
16.
Some Studies of Human Populations . . . . . .
1. Tbe Jieaque Indians of the Montaiia de la Flor, Honduras
1.1. History of the Group. . . . . . . . . . . . .
1.2.
Inbreeding
among
the Jicaque Indians . . . . .
1.3.
Changes in the Genetie Composition of the Group
2.
Tbe Bedik of Eastem Senegal . . . . . . . . . . .
2.1.
History
and
Ecology . . . . . . . . . . . . .
2.2.
Marriages
among
the Bedik . . . . . . . . . .
2.3.
Haematologieal
Charaeters-Distances
between Villages.
2.4.
Representation of the Strueture of the Populat ion. . . .
494
495
495
496
499
501
501
502
504
506
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XVIII Table of Contents
3.
The Kel Kummer Tuareg of Mali . . . . . .
3.1. History, Ecology and Soeial Organisation .
3.2. The Genealogy of the Kel Kummer People
3.3. Changes in the Genetie Make-up of the Population .
3.4. Haematologieal Studies of the Kel Kummer Population .
4. Classifieation of Populations Using the HL-A Systems
4.1.
Data
and Methods of Caleulation
4.2. Results . . . . . . . . . . . . . . . . . . .
509
509
511
513
518
524
524
525
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530
Appendix A. Linear Difference Equations 533
1. Definitions . . . . . . . . . . . .
533
2. The Solution of Linear Difference Equations 534
2.1. The General Solution of a Homogeneous Equation of Order
m . 534
2.2. The General Solution of an Inhomogeneous Equation of Order m 535
2.3. The Solution of the Homogeneous Equation of Order m, with Constant
Coefficients . . . . . . . . . . 535
2.4. The Renewal Equation of Order m . . . . . . . . . . . . . . . . . 536
Appendix B. Some Definitions and Results in Matrix Algebra 538
1. Definitions . . . . . . . . . . 538
1.1. Types of Matrix . . . . . . . . .
538
1.2. The Determinant of a Matrix . . .
539
1.3. Matrix Addition
and
Multiplication 540
2. Diagonalisation of a Square Matrix 542
2.1. The Powers of a
Matrix.
. . 542
2.2. The Eigenvalues of a Matrix .
543
3. The Spectral Analysis of a Matrix 546
4. Real Symmetrie Matrices . . . . 547
4.1. The Eigenvalues of a Real Symmetrie Matrix are aIl Real 547
4.2. The Eigenvectors of a Real Symmetrie Matrix Corresponding to Distinet
Eigenvalues are Orthogonal . . . . .
548
5. Stoehastie Matrices. . . . . . . . . . . . . . 549
5.1. The Eigenvalues of Stoehastie Matrices . . .
549
5.2. The Speetral Analysis of a Stoehastie
Matrix.
551
References. .
553
Subject Index 561
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Introduction
The
Individual
From the very ftrst moment of the creation of a new human being
at fertilisation, the single cell which
is
the new individual
is
already
endowed with its full complement of hereditary information. This single
cell will divide and form millions of new cells, which will each be adapted
for speciftc functions; millions of chemical compounds will be synthesised,
which will be used in the cells themselves,
or
for communications
between cells; mechanisms for the precise regulation
of
all kinds of
processes will develop. All these events take place in a determined
sequence: embryonic development, growth, senescence, then death.
All the information which will ensure that these events take place
is
contained in the original single cell; half
of
the information comes
from the egg, and half from the spermatozoon which fertilises it. The
chromosomes of the cell carry all the instructions for the development
of an individual, for example a human being, or more exactly,
this
particular
human being.
The genetic information which the new individual receives
is
the
basis ofhis individuality. It distinguishes him from all other individuals;
no man now or in the past has had precisely the same genes as another.
Of course, the development of the individual, his ftnal size and whether
he grows up to be strong
or
weak are greatly affected by his environment.
This includes the state
of
health of his mother before he
was
born,
the nutrition he receives, the temperatures and irradiations to which
he
is
subjected, and many other factors. But the effects which environ
mental factors will have are themselves determined by the genetic
information, since this determines the internal processes which can, in
fact, take place. The end result of the individual's developmental process
depends on random events in his environment, but the effects
of
these
events obey probability distributions that depend on the genetic informa
tion brought together at his conception.
At each cell division, the new cell has a structure which corresponds
to the function it will carry out in the organism (e.g. nerve cell, bone
cell,
blood cell,
etc.).
Whatever the function of the cell, however, it will
receive a full copy of the genetic information of the individual of which
it
is a member. Each cell carries all the information which determines the
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2
Introduction
whole individual, inc1uding all the special characteristics that are unique
to this individual. Bach single cell of a particular individual
is,
above
all, a cell of
this
individual, and not just a cell with a certain function,
such as a nerve cell,
or
a bone cell.
There
is
one exception to this. The reproductive cells, eggs or
spermatozoa, carry only half
of
the genetic information of the individual
they come from. Nearly all these cells are destined to die. A tiny propor
tion
of
them will find another reproductive
cell
of the opposite sort,
and fuse with it. In the new cell which results, there is once again a
complete set of the genetic instructions which are necessary for a new
human being to develop.
The Population
We have seen that each individual passes only half of his genes to
any one of his offspring.
I t
is a matter of chance which
of
the two genes
for a character the offspring receives.
In a sufficiently large population, the genetic heritage
is
maintained
through the generations, despite the fact that, in each generation, the
genes are segregated into the gametes, and then come together in new
combinations in the new individuals. A common heritage
is
therefore
concealed behind the diversity
of
the individuals of a population.
I f
the chromosomes of an individual carry allthe information
(" genotype") necessary for the organism to synthesise a certain metabolic
product, then he may, in fact, synthesise it, and exhibit the characteristic
"phenotype", which could be advantageous or disadvantageous. Next
generation, it may happen that none
of
his offspring receives all the
information for this synthesis, and so the corresponding phenotype has
disappeared. However, the information for the synthesis
of
this product
is
not lost; in a later generation, it could once again come together
into a single individual, and this individual would have the same pheno
type as his ancestor.
At
the population level, the fundamental reality
is
the information
that
is
carried by the chromosomes coiled up in the reproductive cells,
not the characters which the individuals manifest. The units
of
informa
tion carried in the chromosomes are handed down through the genera
tions, often unexpressed and hidden, alternately separated and paired
up together according to the chances offertilisation,
but
alwayspreserved
unchanged.
Above and beyond the particular group of individuals which exist
at any one time, the fundamental property of a population
is
its common
genetic heritage.
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Introduetion
3
Each individual can have a small effect on this common heritage;
the genetic information which
he
bears
will
form a larger or smaller
proportion of the total in the next generation, depending on whether
he
has many or few offspring. In this way, the mean frequency of any
particular unit of information can change from one generation to the
next.
Population genetics is chiefly concerned with the study of these
changes. Its aim is to answer the question: "What factors affect the
genetic heritage of populations, and what are their effects?"
General Bibliography
Burdette, W.: Methodology
in
human genetics. San Franeiseo: Holden-Day 1961.
Cavalli-Sforza, L. L., Bodmer, W. F.: The genetics of human populations. San Franciseo:
W. H. Freeman
1971.
Crow, J. F., Kimura, M.: An introduetion to population genetics theory. New York:
Harper and Row
1970.
Elandt-Johnson, R.C.: Probability models and statistical methods in genetics. New York:
Wiley 1971.
Ewens, W.J.: Population genetics. London: Methuen
1969.
Faleoner, D.S.: Introduetion to quantitative genetics. Edinburgh: Oliver
&
Boyd 1960.
Fisher, R.A.: The genetieal theory of natural selection. Oxford: Clarendon Press 1930,
(Reprinted. New York: Dover Publications
1958).
Fisher, R.A.: The theory ofinbreeding. Edinburgh: Oliver
&
Boyd
1949.
Kempthorne,
0.: An
introduetion to genetie statistics. New York: Wiley
1957.
Kimura, M.: Diffusion models in population geneties. London: Methuen
1964.
Kimura, M., Ohta, T.: Theoretieal aspects of population geneties. Princeton, New Jersey:
Princeton University Press
1971.
Li, C. C.: Population genetics. Chieago, Illinois: Univ.
of
Chicago Press
1955.
Maleeot, G.: Les mathematiques de l'heredite. Paris: Masson 1948.
Maleeot, G.: Probabilites et heredite. Paris: Presses Universitaires de France
1966.
Maleeot, G.: The mathematics
of
heredity, (Translation and revised version of "Les
mathematiques de l'heredite"). San Franciseo: W. H. Freeman 1969.
MeKusiek, V.A.: Human genetics. Englewood Cliffs, New Jersey: Prentiee Ha111964.
Moran, P. A. P.: The statistieal processes of evolutionary theory. Oxford: Clarendon Press
1962.
Stern, C.: Prineiples of human geneties 3rd ed. San Franeiseo: W.H. Freeman 1973.
Wright, S.: Evolution and the genetics
of
populations. Vol. I (1968): Genetie and biometrie
foundations. Vol. II (1969): The theory of gene frequeneies. Chieago, Illinois: Univ.
of Chieago Press.
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PARTl
Basic Facts and Concepts
The aim of population genetics is to study changes in the genetic
heritage,
at
the population level. I t may be useful to the reader to be
reminded of certain biological facts
at
the level of the individual, or even
of the single cell, which relate to the transmission of this heritage from
one generation to the next.
In this section, we shall also give some definitions of the terms which
will be used in this book, and introduce some fundamental ideas and
notation that
will
be used frequently.
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Chapter 1
The Foundations of Genetics
1. The Mendelian Theory
of
Inheritance
All our ideas about the inheritance
of
particular characters, and
also about changes in the characteristics
of
populations, make use
of
the concepts introduced a century ago by Gregor Mendel
(1822-1884).
W orking in a small
1
experimental garden in his monastery in Brno,
in Moravia, and using peas as his experimental material, Mendel suc
ceeded in formulating a hypothesis which explained the results of his
crosses in a simple fashion.
His hypothesis was contrary to the accepted ideas
of
his time, and
his paper published in 1865
2
remained unknown. In 1900, sixteen years
after Mendel's death, his work was rediscovered, frrst by the Dutchman
Hugo de Vries, then by the German Correns and the Austrian von
Tschermak.
Prom
then onwards, the science of genetics could develop.
The school of Morgan, in the U.S.A., was especially important in this.
But thirty-five precious years had been lost.
1.1. Mendel's First Law. The Law
of
Segregation
When he crossed peas with yellow cotyledons with peas with green
cotyledons, Mendel found that all the hybrids had yellow cotyledons.
When he crossed the hybrids among themselves (or, rather, selfed them),
he obtained two types of peas again; the character green cotyledons,
which was
not
expressed in the first generation hybrids, reappeared in
the second generation: "segregation" of the characters
had
occurred.
Mendel also found that, in the second generation, he always obtained
proportions elose to 3 yellow to 1 green. These experiments were done
with six other characters (round or wrinkled seeds, axial or terminal
inflorescences, etc.), and similar results were obtained.
Mendel's hypothesis to explain these results can be expressed as
follows (using a different terminology from Mendel's):
1
250
m
2
in area.
2
"Versuche über Pflanzenhybriden" was published in the Verhandlungen des natur
forschenden Vereines in Brünn 4
(1865).
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The Mendelian Theory of Inheritance
7
1. Each character of an individual is controlled
by
two "factors", the
"genes", one of which the individual receives from its male parent, and one
from its female parent.
2.
"When an individual carries two different genes for a particular
character, one of them will be expressed
("dominant"),
while the effect of
the other may not
be
apparent ("recessive").
3. A reproductive cell produced by an individual bears, for each
character, one and only one
of the two genes which the individual carries.
yy
Initial
generation
1
st
hybrid
generation
2
nd
hybrid
generation
Fig. 1.1. Mendel's fundamental experiment
In the example given above, plants from the line with yellow cotyledons
carry only "yellow" genes,
and
the genetic constitution
of
these plants
can be written as (Y /Y). Plants from the line with green cotyledons have
the genetic constitution (y/y). (In what follows, genetic constitution will
be designated by pairs ofletters, and will be enclosed in brackets. Lower
case letters will stand for recessive genes.)
In the cross between the yellow and green lines, individuals
of
the
first hybrid generation receive one gene from a parent whose genetic
constitution is (Y/Y), and the other from a (y/y) parent. The genetic
constitution ofthese individuals is therefore (Y/y). The Y gene
is
dominant
to the y gene, so
that
these individuals are all yellow.
The second generation
is
the result of crossing the hybrids,
and
this
cross can
be
written as:
(Y/y) x
(Y
y).
Each parent will produce gametes
of
which half carry the gene Y,
and
half the gene
y. The
off pring can therefore be of three sorts:
one quarter will be
two quarters will be
one quarter will be
(Y/y)
(Y y)
or
(y /y)
(y/y).
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8
The Foundations of Genetics
Because Y is dominant over y, the (Y/y) or (yjY) peas are yellow, so we
obtain the ratio 3 yellow peas to 1 green. which was the observed ratio.
Mendel's fundamental insight was the realisation that the character
which was not manifest in the first hybrid generation (the character
"green cotyledons ", in this example) must somehow remain present in
this generation, since it reappears in the next generation.
In order to explain this paradox,
we
have to accept the duplication
of the information controlling characters. This duplication exists in all
organism which reproduce sexually, although it may be only a transitory
phase of the life-cycle as in many lower plants.
Mendel's fundamental hypothesis is that the hereditary material has
the properties of discontinuity and stability. The genes controlling a
character separate from one another, and come together in pairs in new
individuals. They co-exist within individuals, in whom the action of one
gene can mask the action of another,
but
they are themselves unaltered
by this co-existence. They remain indivisible, and do not exchange parts
with one another.
In
1865,
such a "quantum" concept of a biological phenomenon was
unacceptable. Even in 1900, Kar Pearson, in his studies of hereditary
phenomena, was still using Galton's hypo thesis of the fusion (and not
the co-existence) of the maternal and paternal hereditary contributions.
I t
is
therefore not surprising that Mendel's hypothesis was ignored when
it was first proposed.
1.2. Mendel s Second Law. Independent Assortment
Mende1 also studied crosses between lines which differed for two
characters: one line had yellow cotyledons and round seeds, while the
other line had green cotyledons and wrinkled seeds. The first hybrid
generation seeds were all yellow and round, but the second generation
consisted of four types, showing the four possible combinations of the
characters. The proportions were: 9/16 yellow, round, 1/16 green, wrin
kled, 3/16 yellow, wrinkled, and 3/16 green, round. Mendel obtained
similar results with seven different characters altogether.
Assuming that the first law
is
valid, these results can be explained by
the single further hypothesis :
Genes controlling different characters segregate independently.
We
can
write the genetic constitutions of the original lines as (Y /Y R/R) and
(y/y r/r), and the first hybrid generation as (Y/y R/r). The gene Y is
dominant to
y,
and R is dominant to r, so the hybrid peas are yellow
and round. The hybrid plants produce four types of gametes: (YR), (yR),
(Yr) and (yr). Since the genes controlling cotyledon colour are assumed
to segregate independently of the genes controlling the form of the seed,
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The Mendelian Theory of Inheritance 9
Table 1.1. Independent segregation oftwo characters
Gametes
Y R
Y r
yR yr
YR
(YfY R/R)
(YjY
Rjr) (Yjy RjR)
(Yjy Rjr)
Yellow
Yellow
Yellow Yellow
Round Round Round Round
Yr (YfY Rjr) (YfY rjr) (Yjy Rjr)
(Y jy rjr)
Yellow
Yellow
Yellow Yellow
Round Wrinkled Round Wrinkled
yR
(Yjy
R/R) (Yjy Rjr) (yjy RjR)
(yjy Rjr)
Yellow Yellow Green Green
Round Round Round Round
yr
(Yjy
Rjr)
(Yjy
rjr) (yjy Rjr) (yjy rjr)
Yellow
Yellow Green
Green
Round Wrinkled Round Wrinkled
we
expect the frequencies
of
all four types
of
gametes to be the same
t
of the total. Table
1.1
shows the genotypes obtained by fusion
of
the
four types of male (d') gametes with each of the four types of female (2)
gametes. The genotypes are shown in brackets, and the phenotypes, which
are obtained using the dominance relations of the genes, are given below
them. ,
;
Re 1lmbering that the gene Y is dominant to
y,
and R to r,
we
have:
1. Four genotypes correspond to yellow, round seeds. They are:
(Y/Y R/R),
(YjY
R/r), (Y/y R/R) and (Y/y R/r). These are formed by
nine out of the 16 possible combinations of gamete types.
2. Two genotypes-(YjY r/r) and (Y/y
r/r)-give
yellow, wrinkled
seeds. These correspond to three of the combinations of gamete types.
3.
Two genotypes -
(y
/y R/R) and
(y
/y R/r) - give green, round seeds.
These correspond to three
of
the combinations
of
gamete types.
4. Finally, the green, wrinkled seeds must be ofthe genotype (y/y r/r),
which can be produced by only one combination of gamete types.
Since fertilisation occurs at random, the frequencies
of
all the
combinations
of
gametes in Table
1.1
are e q u a ~ so the frequencies of the
four types of pea will be 9/16, 3/16, 3/16, and 1/16.
1.3. Restriction of Mendel's
Second
Law. Linkage
Mendel's second law is often found not to hold, unlike the first law,
which
is
very generally valid. Morgan, using the fruit
fly,
Drosophila
melanogaster, discovered that independent segregation
of
characters is
by no means a general rule. He found that it was possible to group the
characters
of
an organism into a number
of
"linkage groups"; two
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10 The Foundations
of
Genetics
genes are considered to belong to the same linkage group if there
is
a
tendency for the parental associations of the characters to carry over
into the offspring; a gene
is
considered not to belong to a linkage group
if it segregates independently of all the genes of that group. The degree
oflinkage between genes can be measured by the strength of the tendency
for the parental associations of the characters to carry over into the
offspring; the proportion of the offspring that show the new character
combinations
is
used as a measure of linkage.
Consider another cross between lines of peas: if plants from a line
with purpie flowers and long pollen grains are crossed with plants from
a line with red flowers and short pollen, the first hybrid generation
plants have purpie flowers and long pollen. The genetic formulae for
the parents are (P/P L/L) and (p/p
1/1), and that for the hybrids is (P/p L/l).
P is dominant to
p,
and L to
1,
so the hybrids have purpie flowers and
long pollen. In the second generation, four types appear, as expected:
the two types showing the parental combinations
of
the characters,
purpie, long and red, short, and the two types with new combinations of
the characters purpie, short and red, long. But the two types with the
parental associations of the characters are much more frequent than
Mendel's second law would predict. Instead of the expected frequencies
9/16=55%, 1/16=7%,
3/16=19%
and
3/16=19%,
the observed
frequencies are 70
%,20 %, 5 % and 5 %.
Table
1.2
shows how we can explain these results by s u p p o · ~ i n g that
the gametes formed by the first generation hybrid plants (P /p t,/l) are
more frequently types (P
L)
and
(p
1), like the parental gametes, than
recombinant types (P 1) and
(p L).
We can obtain quantitative agreement
Table 1.2. Segregation
oftwo
Iinked genes
Gametes
PL
PI
pL pI
Frequency
10/22
1/22 1/22 10/22
PL
10/22
0.207
0.021 0.021
0.207
PurpIe PurpIe PurpIe PurpIe
Long Long Long Long
PI
1/22
0.021 0.002
0.002
0.021
PurpIe
PurpIe PurpIe PurpIe
Long
Short Long Short
pL
1/22
0.021 0.002 0.002 0.021
PurpIe PurpIe Red Red
Long Long Long Long
pI
10/22
0.207
0.021 0.021 0.207
PurpIe PurpIe Red
Red
Long Short Long Short
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The Mendelian Theory
of
Inheritance
11
with the experimental data if, of every 11 gametes which carry the gene P,
10
carry the gene
L,
and only one carries the gene 1; and if, of every 11
gametes carrying
p,
one carries
Land
10
1.
The two characters, purpie or red flower colour, and long or short
length of the pollen grains, therefore belong to the same linkage group.
The ratio
10:1
is very different from the 1:1 ratio which would be given
by independent segregation of the genes. We therefore say that these
genes are tightly linked.
By a large number of studies of the simultaneous segregation of genes,
it has been possible to establish the existence of four linkage groups in
Drosophila melanogaster, seven in peas, and ten in maize.
Mendel studied seven characters in peas. By a remarkable piece of
luck, each
of
these seven genes belonged to
aseparate
linkage group
(which has an
apriori
prob ability of only one in
163,
approximately);
these genes therefore segregated independently. This unlucky chance
meant that Mendel did not disco ver linkage, though it did, of course,
make his results easier to interpret.
1.4. Some Definitions
Genetics, like every independent field of study, has developed a
terminology, which is sometimes imprecise and sometimes over-precise
and which can vary from author to author. This can confuse the novice,
who may not realise that he is being confused
by
ambiguous use ofwords,
rather than by the difficulty of the arguments themselves.
It
therefore
seems important to define the "key-words" which will be used in this
book.
A "gene" is a unit of information concerning a unit character, which
is
transmitted by a parent to his offspring. In sexually-reproducing
organisms, individuals carry two genes for each unit character, one from
each parent.
The set of genes carried by an individual
is
called his
"genome".
Genes which act on the same unit character are said to be
"at
the
same
loeus".
One could equally
weIl
say that genes at the same locus are
homologous (Gillois, 1964).
At each locus, an individual has two genes, one from his mother and
one from his father. He transmits a copy of one of the two genes to each
of his offspring.
For
the purposes of population genetics, the locus can
be considered as the basic, indivisible unit of hereditary transmission.
The set of genes of one 10cus, i. e. that act on the same unit character,
is called a set of "alleles ".
For a given locus, the number of alleles
is
the same
as
the number
of modes of action on the character. So
far,
we have only considered 10ci
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12
The Foundations of Genetics
with two alleles (green or yellow cotyledons, round or wrinkled seeds,
etc.), but large numbers of alleles are possible; Iod with more than ten
known alleles are not uncommon. In what folio
ws,
alleles will be des
ignated
by
the symbols
Al' ... , Ai' ... ,
An'
The two genes which an individual carries at a given locus can be
the same allele, or different; these two possible states are called the
"homozygous"
and
"heterozygous"
states of the locus. The genotypic
formula of a homozygote is written
(Ai Ai)'
and a heterozygote
is
written
(Ai A
j
).
If there are n alleles at a locus, there are n possible homozygotes:
n(n-l)
.
(Al
Al)
...
(Ai Ai) ...
(An
An)'
Also, 2 dIfferent heterozygotes are pos-
sible:
(A
l
A
2
)
.• .
(A
l
A
n
)(A
2
A
3
)
.••
(An_lA
n
.
The set of all these homo
zygous and heterozygous types constitutes the
"genotypes"
that are
.
n(n-l) n(n+l)
possIble at the locus. There are n+ 2 2 of them.
The characteristic which an individual manifests, with respect to a
unit character, is called his" phenotype". It is observable, and sometimes
measurable, unlike the genotype.
The phenotypic effects of different genotypes may be the same,
because the alleles at a locus can show dominance-recessivity relations.
Allele
Ai
is said to be dominant to
A
j
(or, equivalently,
A
j
is said to be
recessive to Ai)' when the action of Ai' but not that of A
j
, is manifest in
the
(AiA
j
) heterozygote.
When the heterozygous genotype
(AiA)
has the same phenotype
as
the
homozygous genotype
(Ai Ai)'
for
the character concerned, dominance is
said to be total.
Dominance is said to be incomplete when the (AiA) genotype is closer
in phenotype to
(AiAi)
than to (AjA).
Clearly, the presence
ofthe
recessive
A
j
gene
is
never entirely without
effect on the organism; dominance is, in reality, merely a function ofthe
aspect of the phenotype that is observed, and the precision of the
observations.
For a unit character controlled
by
a locus with n alleles, the number
of phenotypes is determined
by
the dominance relations of the alleles;
there
will
be n phenotypes ifthe alleles can be ordered in sequence, with
each one fully dominant to the lower ones in the sequence; if there is no
.
n(n+l)
dommance, there
wIll
be as many phenotypes as genotypes - 2 ;
in other cases, there will be an intermediate number of phenotypes.
A
unit character
is
one which
is
inherited according to Mendel's
first law. In other words, it is a character controlled by the alleles of a
single locus.
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The
Physical
Basis
of Mendelian Inheritance. The Chromosomes 13
This definition exposes the underlying tautology in the theory as so far
described.
If
it were not
for
the discovery that the words "gene", "locus"
(and also "linkage group") correspond to concrete biological entities,
and not only to theoretical concepts that
we
use to explain the facts of
hereditary transmission, the whole of the above theory would remain a
purely abstract, logical model.
As often happens, a hypothetical model which would explain the
observed phenomena of heredity was developed before the physical
entities concemed were known. Subsequent cytological and biochemical
studies have shown that the Mendelian model is not merely a construct
which agrees with the facts it was designed to explain, but also corre
sponds to the behaviour of the actual hereditary mechanism.
2.
The Physical Basis
of
Mendelian
Inheritance.
The Chromosomes
In the decades following the publication ofMendel's paper, cytologists
established the fact that the nuclei of cells contain filamentous structures
which can
be
stained
at
the time of cell division - the chromosomes. Living
cells contain a set of 2n chromosomes, where n is a number characteristic
ofthe species: four in Drosophila
melanogaster,
seven in peas, ten in maize.
The regular behaviour of the chromosomes during cell division
suggested to cytologists that they might be involved in hereditary
transmission. At the end of the 19th century, cytologists had formulated
the rules of inheritance that would appIy if the chromosomes were the
bearers of the hereditary determinants. When Mendel's work was
rediscovered, the correspondence with the cytologists' expectations was
apparent.
Without going into details in this rapidIy changing field,
we
shall now
give a summary of the physical basis of inheritance, on which population
genetics is founded.
2.1. The
Behaviour
of
the
Chromosomes.
Mitosis and Meiosis
Each
ofthe
millions of cell divisions which occur during the deveIop
ment
of
an individual consists of a compiex series of events which chiefly
involve the chromosomes. The chromosomes are invisible in the non
dividing nucleus, and they become visible as the cell prepares to divide,
in the form of a set of pairs of rod-shaped objects (Fig.1.2, stage
1).
At a later stage, each rod is seen to be divided into two identical
"chromatids", which remain attached at one point, the "centromere".
MeanwhiIe, the nuclear membrane disappears (stage
2).
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14
The Foundations of Genetics
1
2
3
1
2
4
3
5
4
6
5
)
/---0
,
,
" \
I Y.
\
..
_____
I
\ ~ / ( /
"
/
_--,'
7
6
Fig.
1.2. Mitosis
/ , - . r - -
I \
I§s
8\
~
T ;
\ I I
'"
\
/
\
...
_- \;.-,
~ ~ r:
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The Physical Basis of Mendelian Inheritance. The Chromosomes IS
The double chromosomes next come to lie on the equator of the cell
spindie, which forms at this time. The centromeres appear to be responsi
ble for the movement
of
the chromosomes to the equatorial plane. Then
the centromeres double and move apart towards the two poles of the
spindie, pulling the sister chromatids with them (stage 4). A membrane
forms around each of the groups of chromosomes, which both contain
one chromosome of each of the original types (stages 5 and 6); finally,
the chromosomes gradually lose their property of staining. This beauti
fully
precise mechanism gives each nucleus ofthe two daughter cells a full
and perfect copy of each of the chromosomes of the original cello
A variant of this set of manoeuvres, meiosis, occurs during gamete
formation. The sequence begins just like mitosis: the chromosomes
become visible (stage 1 of Fig. 1.3), double to form a pair of chromatids
joined
by
a centromere (stage 2), and move to the equatorial plane. But
during this stage the two homologous chromosomes of each pair come
together and form groups of four chromatids (stage
3).
A first division now takes place; the centromeres, each with a pair of
chromatids attached, move to the poles, in such a way that each group
receives only one (doubled) member of each pair of chromosomes
(stages 4 and
5).
The two temporary cells at the poles now divide again,
and in this division the sister chromatids separate (stages 6 and 7). The
final result of meiosis
is
four reproductive cells, each of which contains
only one member of each of the pairs of chromosomes that were present
in the original cello
2.2.
Consequences
of
Chromosome
Behaviour
for
Hereditary Transmission of Characters
The regularity of chromosome behaviour in meiosis supports the
following model: the transmission of characters from parent to offspring
is mediated by elements borne on the chromosomes, and for each
character there correspond two such elements, one on each of the pair
of homologous chromosomes.
Each individual has two corresponding series of n chromosomes, one
set
of
wh ich came from his father, and one from his mother.
I f
he has
received two different elements controlling a given character, the action
of one may mask that of the other. When the individual reproduces, the
homologous chromosomes, maternal and paternal, separate and the
descendants may again manifest the character that was masked in their
ancestor: this is segregation.
Furthermore, the migration of the chromosomes to the two poles of
the spindie, in stages 3, 4 and 5 of meiosis, occurs independentIy of their
origin (paternal or maternal). The simple example shown in Fig. 1.3, where
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16 The
Foundations
of Genetics
n=2, shows how the two temporary cells (stage 6) can receive either
copies of the two matemal chromosomes, or of the two patemal chromo
somes, or one patemal and one matemal chromosome. This
is
independent
segregation of
chromosomes.
The two ideas of segregation and independence, which we have just
demonstrated for the chromosomes, are exacdy analogous to Mendel's
two laws. Mendel's purely formal explanation of the facts of heredity is
thus in perfect agreement with what we know about the physical
mechanism of hereditary transmission.
Mendel's "genes
",
which he proposed as the "factors" mediating
heredity, can therefore be equated with real "elements" carried on the
chromosomes.
The only other phenomenon which remains for us to explain is
linkage.
2.3.
Linkage and
Crossing Over
In the above description of the behaviour of chromosomes in meiosis,
and its implications
for
genetics, we have been supposing that two
characters will segregate independendy if they are controlled by genes
carried on different chromosomes, and will be completely linked if the
genes are on the same chromosome.
In reality, exchanges can take place between the chromatids of a pair
of homologous chromosomes during the 3rd stage of meiosis, when the
chromosomes are aligned with one another.
1
2
3
A
8
( )
~
:8
( )
a:
:b
a
b
a
b
o
A
Ä
a:
a
Fig. 1.4. Crossing-over
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The Physical Basis of Mendelian Inheritance. The Chromosomes 17
Fig.
1.4
shows an exchange between a paternal chromosome carrying
genes A and B,
and a maternal chromosome carrying a and
b.
After the
exchange of segments between chromatids
("
crossing-over
"),
some
of
the chromosomes which migrate to the poles will still have the parental
associations of genes (AB and ab) but others will be "recombinants"
(Ab and
aB).
The greater the distance between the Ioci
of
the two genes on the
chromosome, the greater the chance that crossing-over will occur between
them 3, and thus the greater the proportion of recombinant gametes.
The degrees
of
linkage measured by segregation of characters can
therefore serve to estimate distance between points on the chromosomes.
Chromosome maps have been made in several organisms, in which
distances are measured,
not
in any of the ordinary units
of
distance, but
by the percentages of recombinants given in crossing experiments.
14
I
\
A 8
I,
I
"
C
,I
6
8
8
' ' - - - - - - - - . . 6 : - - - - - J '
Fig.
1.5.
Part of a chromosome map
In order to use recombination fractions as measures
of
distance,
these fractions must clearly be additive.
Ir,
for example, the distance
between A and B is 6
%,
and that between
Band
C
is
8
%,
the distance
between A and C must be either 14 %or 2 %, depending on the order of
these three genes on the chromosome (Fig. 1.5). This requirement is met
in practice, to a good approximation.
2.4.
Human Cbromosomes
The number
n
of chromosomes is constant for each species, and is
therefore a fundamental fact about the species. The human chromosome
number was not, however, correct1y established until1956.
Until this time it was thought that n= 24. The lack
of
a good enough
technique, and probably also excessive respect for established opinions,
had the result that the number n=24 was accepted without dispute until
3 We are ignoring the possibility of multiple cross-overs here.
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18
The Foundations of Genetics
A
B
-
----------
111111
Al
Al
2
3
4
5
C
-----
--
J1Illllli11A
Al
6
7
8
9
10
I1
12
D
E
ÄÄ
ÄÄ Al
xx
I1 1
13
14 15 16
17
18
F
G
A
-----
X XX
19 20
21
22 X
Fig.
1.6.
The karyotype of a man
•
Tjio and Levan (1956), using an improved method, showed that n was
really only 23.
Man
therefore has 46 chromosomes.
The 23 pairs have sufficiently distinct sizes and shapes to be classified
according
to
an international convention. In this classification, the sex
chromosome pair
XX
or XV,
is
distinguished from the 22 autosomal
pairs. The autosomes are numbered according to size and the position of
the centromere, starting with 1 for the largest chromosomes, which have
a median centromere, and going
up
to 22 for the smallest chromosomes,
whose centromere
is
near the end
of
the chromosome.
The
whole set
of
an individual's chromosomes
is
called his" karyotype".
Variations of the normal
human
karyotype are known, and some well
known abnormalities are associated with such variations. In 1959,
Lejeune, Turpin and Gautier showed
that
mongolism is due to the
presence
of
an
extra
chromosome 21 ("trisomy 21", i.e. 3 examples
of
chromosome 21, instead of the normal 2). Many other chromosome
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The Physical Basis
of
Mendelian Inheritance. The Chromosomes 19
abnormalities are now known. In particular, certain
types
of cancer and
leukaemia are associated with abnormal karyotypes.
2.5. The Sex Chromosomes
The sex chromosomes distinguish the male karyotype from the female
one. In women, the 23rd chromosome pair is represented by
two
identical
chromosomes, called the X chromosomes, which are about the same
length as chromosome
6.
In men, the
23
rd pair consists of
two
chromo
somes
of
very
different
size:
one of them is just the same as the X chromo
some of females, but the other, called
Y,
is small, like a chromosome
21
or 22.
Mother
Father
Daughter
Son
Fig.
1.7.
Sex-linked inheritance
The X and Y chromosomes carry genes affecting all sorts of characters,
not merely
genes
concerning sexual differences. Because of the difference
in the sex chromosomes in men and women, characters controlled by
genes on these chromosomes
will
be inherited in a different way from
those controlled
by
autosomal
genes:
a son must inherit
his
Y chromo
some from
his
father, and his X chromosome must come from
his
mother.
Therefore a
gene
carried on
his
father's X chromosome cannot be
transmitted to hirn. A daughter, on the other hand, receives an X from
her father and one
from
her mother, and cannot inherit a gene carried
on the Y chromosome of her father
4
•
This asymmetrical type of inheritance means that the
genes
of the X
and Y chromosomes, or "sex-linked ", genes, have to be treated separately
from the others.
4
Many genes on the X-chromosomes are known (e.g. haemophilia, colour-blindness),
but very
few
have been found to
be
carried on the Y chromosome.
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20
The Foundations of Genetics
2.6. Chromosome Structure.
DNA
During the last twenty years there has been great progress in
understanding how the structure of chromosomes is related to their two
basic functions:
1. To reproduce themselves at cell division, and
2.
To
control biochemical reactions.
Watson and Crick, in 1953, showed that the chromosome consists of
two long molecules, coiled together in a double helix (DNA).
Each
of
the two strands is aseries of nuc1eotides, of which there are
four types, depending on which of the four bases (adenine A, guanine G,
thymine T
or
cytosine
C)
the nuc1eotide contains.
Astrand
of
DNA
therefore has adefinite sequence, e.g. TCGAGCAAGCC .. ,
Fig. 1.8. The
DNA
double-helix
The nuc1eotides are like four "Ietters " of the alphabet, in which the
message of the DNA is "written".
Furthermore, the two strands
of
the
DNA
double helix are com
plementary: an A in one corresponds to aT in the other, and aG corre
sponds to a
C.
The complementary sequence to the one written above
would be AGCTCGTTCGG ...
Auto-replication
of
DNA. In the earliest stages of mitosis
or
meiosis,
the two strands of the DNA separate. Then a complementary strand to
each
of
the two is synthesised, using the cell's pool
of
nuc1eotides; this
synthesis
is
catalysed by
an
enzyme. The final result
is
a pair of DNA
double helices, each identical with the original one.
Control
of
biochemical reactions. Proteins, which are the basis of all
living matter, are extremely large molecules, with molecular weights
of
up to several million. Proteins consist of one or more long polypeptide
chains, which are compact1y folded
up
in a way which is determined by
the amino acid sequence
of
the chains. Twenty types of amino acid enter
into the composition of proteins.
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The Physical Basis of Mendelian Inheritance. The Chromosomes
21
1
2
3
---@
©
Fig.
1.9.
DNA replication
@
--
®
/ ®
©
Just as astrand of DNA is characterised by its base sequence, so a
polypeptide chain is characterised by its amino acid sequence. The
amino acids are therefore like the 20 "Ietters " of an alphabet in which
the message of the proteins is "written".
The two sequences, the
DNA
sequence and the protein sequence, are
related to one another by the "genetic code". An amino acid corresponds
to a sequence of three nucleotides.
The mechanism by which the chromosome governs protein synthesis
has become understood in recent years. The main stages are as folIows:
1. A molecule of RNA is synthesised alongside a
DNA
helix, by a
process which must be like DNA synthesis itself. The molecule
ofRNA
is
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22
Tbe Foundations of Genetics
... A G C Tee A A C G T... DNA
. . .
U C G A G G U U G C A. ..
MI Ssenger
RNA
2
_U
C G A G G U U G C A .. " 5 ~ ' " RNA
Trans ;
RNA
3
4
Fig. 1.10. Polypeptide syntbesis
like a single strand
ofDNA,
but contains uracil bases instead ofthymine.
The sequence AGCTCGAA ... in DNA therefore gives rise to the RNA
sequence UCGAGCUU ...
2. This complementary
RNA
molecule, which
is
called messenger
RNA, goes out of the nucleus into the cytoplasm of the cello Ribosomes
become attached to one end of the messenger RNA, and then they move
along the messenger.
3. A pool of the different amino acids, attached to molecules of
"transfer RNA",
is
present in the cello The transfer RNA molecules are
shorter than messen ger RNA, and each type is specific both for the amino
acid that can be attached to it, and also for a sequence of 3 bases
in
messenger RNA.
For
example, one transfer
RNA
type can have phenyl
alanine attached to it, and also recognises the sequence UUU in messenger
RNA; a different type is specific for serine and for the sequence UCA.
The transfer RNA molecules thus form the mechanism whereby the
genetic code is translated - the 20 amino acids are brought into corre
spondence with the 4
3
=
64 nucleotide triplets.
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The Physical Basis of Mendelian Inheritance. The Chromosomes 23
4. As
a ribosome moves along the messenger RNA, each triplet in
turn is
recognised by a transfer RNA molecule, which has its amino acid
attached to it, and, one by one, the amino acids are added to the growing
polypeptide chain. The sequence of amino acids in the chain
is
thus
determined by the base sequence of the messenger RNA.
The
genetic code.
It
s
now known that the genetic code
is
"degenerate":
several of the
20
amino acids are coded for by more than one base triplet,
e.g.
UUG
and CUC both code for leucine. However, three triplets do
not code for amino acids, but are "nonsense " triplets, which serve as
"punctuation marks"; these triplets are known to signal termination of
the polypeptide chain.
It is
possible that the degeneracy of the genetic code
is
such
as
to
minimise the effects of mutations on the amino acids which are most
important for protein structure and function.
2.7.
Mutation
Chromosomes are not perfectly stable entities; changes occur in
them with a small, but not negligible frequency. The nucleotide sequence
of
DNA
can be changed by irradiation with ultra-violet light or X-rays,
by certain chemicals or viruses, or simply by an error in the replication
process. The types of changes produced are the replacement of one
nucleotide by another, and the deletion or insertion of one or more
nucleotides.
The descendants of a mutated cell will all carry the same change in
their DNA. I f the mutation occurred in a somatic c e l ~ the individual in
which it happened will be a "mosaic", with some ofhis cells genetically
different from the others (this seems to be the case in some cancers);
if the mutation occurs in a reproductive
c e l ~
the offspring who receives
the mutant gene will differ genetically from both parents, and will
transmit the mutation to his descendants.
A change of a single nucleotide of the DNA can have important
consequences for the whole organism. A change in the
DNA
sequence
which codes for a pro tein can abolish the synthesis of a functional
protein. I f he changed pro tein is an enzyme, for example, the biochemical
process it
is
involved in will be abolished.
However, each cell contains two copies of the gene controlling each
function, one on a paternal and one on a maternal chromosome. This
greatly decreases the effects of mutations on the organism. I f one of the
two genes is altered and its action abolished, this
is
usually not apparent,
since the other gene contains the information needed for the normal
functioning of the
cello
Therefore this heterozygosity has slight conse-
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24
The
Foundations of
Genetics
quences, or none at all. This explains the phenomenon of recessivity and
dominance; a mutant gene which cannot give rise to the synthesis of a
protein, or codes for an inactive protein, will
be
recessive to a gene
which has these capacities. I fa single functional gene gives rise to enough
protein for normal cellular function, the recessivity
is
total. I f not, the
non-functional gene
is
partially recessive, and the phenotype of the
heterozygote differs from that of the dominant homozygote.
In a later generation, however,
agamete
carrying the mutant gene
may unite with another gamete which also carries this mutation, and so
produce a homozygote for the mutation, who will manifest its
effect.
2.8.
Individual Diversity
Mutations are not necessarily harmful to the organism. They are
random changes, which sometimes can be beneficial. Mutations are the
source of individual diversity.
To understand the extent of this diversity, consider a population
whose members can differ for N characters, each of which can have one
of two states. The total number of possible types will be 2
N
.
At present, the total human population of the earth
is
about 3 x
10
9
,
while the total number of human beings who have ever lived must
probably be less than 10
11
• So if humans differ for only 40 characters,
every individual who has ever lived could be different from every other.
The number of spermatozoa that have been formed by all the men
who have ever lived
is
of the order of
10
22
• A man who
is
heterozygous
for 75
genes, which
is
not a particularly large number, would produce
2
75
= 10
23
genetically different types of spermatozoa.
How
is
this diversity maintained? How do the proportions of the
different genes change in relation to environmental conditions and the
behaviour ofthe individual population members? These are the questions
that population genetics tries to answer.
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Chapter 2
Basic Concepts and
Notation.
Genetic
Structure
of
Populations
and of
Individuals
1.
Probability
When the members of the pairs of chromosomes separate from one
another during meiosis, the distribution of patemal and matemal
chromosomes to the reproductive cells that are formed takes place
"at random ". Out of tens of millions of spermatozoa emitted, which one
succeeds in fertilising the egg is the result of "chance".
The techniques which have been developed for handling processes
that involve "random", or "chance" events
(i.
e.
probability theory) are
therefore frequently used in studying hereditary processes.
Of course, there
is
always a deterministic reason for the particular
outcome of the
"random"
event that does, in fact, occur, but these
causes are inaccessible to us, and we can only study them by observing