The Genetic Structure Od Populations -Albert_Jacquard

8/19/2019 The Genetic Structure Od Populations -Albert_Jacquard

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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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Table of Contents :xv

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


14/583

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


15/583

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


16/583

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


17/583

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.


18/583

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.


19/583

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.


20/583

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).


21/583

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).


22/583

8


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,


23/583

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


24/583

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


25/583

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


26/583

12


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.


27/583

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).


28/583

14


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:


29/583

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


30/583

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


31/583

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.


32/583

18


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


33/583

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.


34/583

20


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.


35/583

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


36/583

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.


37/583

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-


38/583

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.


39/583

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

Documents

The Genetic Structure Od Populations -Albert_Jacquard