The mathematics of heredity

THE

MATHEMATICS

OF

HERIIDJTY

THE MATHEMATICS OF HEREDITY

Gustave

Malccor

D e m t u i o i M . Ycrnianos if CiiJift/r/j'j,

Ktitril.lt

W . H . Fterman an J Company JJ«

Fr.iftilife

Contents

Copyright ©

W- H. Freeman nnd t^WpflQjf,

No pitrt of this book may be reproduced by aciy mechanicul, ptatnBraphtc, nr. electronic prnce*.*, or in Pfc f i r m i>i"fl pJionogre-priic recording, nor may it be stored in n m r i t v a l ayi'&'ii, transmitted, or ttbflfWfH copied for public or privuee USE without Ihc Wiillcn pvnilisilan of the publisher, i'riiKcil in the Uniicd SLBirs oi' America-

Author's

Preface

Translator's Aufhttr'x

Preface

vil

Foreword

Preface

I*

to the French

edition

tit probability ct Hert'ditL

xi

liii

B The Mendeiiaa

rrBn^lnLtd tttxn French edition, enptright i£ by Masion ct Clc. I-Jiieurs.

Lottery

1

1,1 Heredity and the Law*, of Mendel I 1-2. Thu Chrnmosoine:, 4 I S . Resemblance Rclweiin Related Individuals

library of Conjircsi CaLik>a Caid Number W-lWO. Slmidaid Hook Number; 7j67-t)fiT3-l

I

B

2

Correlation Between Relative* in an Isoganiuus S t a t i o n a r y P o p u l a t i o n

]3

2 1 . ProbabilitiKofCicnca arid Genci>pci 13 2- 2. The Distribution o f Factors in an Isoganious Population 1 3 . Random Mendelian Variable? in i n isoBiinious Stationary Population IH

In

GMMUU

2.4. Correlations Bel wee u Relative* Without Dominance 2.5 Correlation* ileiween Unrelated Individual* with Dominance 2i r

Correlations Between Any fndividuah w j i h Dominance Author's

v o l u t i o n of ii M c n d c l i a n 3.1. 3.2. }.3, 3.4.

lull.,- ••• • o f PopuUnion Size on Neutral Gents I H I In: • • of Selection 41 influence of Migration M Appendi*: Discontinuous MiKruNOiVi 77

HMftgraphv Index

Population

Preface

31 31

85 %f

M a n y papers since my 1 H H h o o k have presented numerous appli¬ cation-, o f the ideas sketched in i t , p a r t i c u l a r l y about coancestry and m i g r a t i o n ; therefore, in this revised, English e d i l i o n . I have added a few explanatory fooinoles. and some formulas about the decrease o f coanceslry w i t h distance. For f u r t h e r i n f o r m a t i o n the reader may use the new references added to the o r i g i n a l bibliography, or my recent book [ 1 6 ] , [ am grateful to I"

•

•: [J M

Y c r m a n o s for his many sug-

gestions and corrections in revising this lent m d for the care w i t h which he has edited a m i translated i t . G,

MALk-tir

Translator'.r

Foreword

The need Tor an English translation o f Professor Gustavc Maleeol's classic w o r k . The Mtiihemtiths

of Heredity,

has been k n u w j i for some

time by students o f p o p u l a t i o n genetics interested i n !ii& approach to dealing w i t h problems o f p o p u l a t i o n structure. The lack, o f such a translation lias curtailed the dissemination English-speaking

o f his ideas

biologists. We ate now increasingly

among

concerned

with population science, yet there are few books i n this field. I hope that this revised, English e d i l i o n o f Professor Malecot's book

will

not only enrieh the literature now available, but also lielji b r i n g his work the recognition it deserves. The Preface by Professor N e w t o n M o r t o n to I'rubuhitites diifo,

el

fltre-

published i n 1966 hy the Presses l l n i v e r s i t a i r e s de France,

Summarizes well some o f ihe significant aspects o f Professor Mulecot's w o r k , and 1 have included it here w i t h the k i n d permission o f both Professor M o r t o n and Ihe Presses Llniversitaires de France. Stpttmberim

D

J

M

.

YERMANOS

Authors

Preface to

the French

Edition

The abjective o f this w o r k is the a p p l i c a t i o n o f p r o b a b i l i t y theory to prove a number o f classical formulas as w e l l as a few unpublished ones pertaining to genetics and the mathematical i h e o r y o f e v o l u t i o n . Instead o f suggesting a unique a p p r o a c h , w h i c h w o u l d have seemed too abstract to the biologist. I have preferred to present

various

methods, each adapted to a concrete p r o b l e m : once ihe fundamental concepts o f mathematical genetics arc ihus simplified, the f o u n d a ­ tions w i l l have been l a i d for experimentation, w h i c h is indispensable, and the way w i l l be clear for eventual synthesis. I apologize f o r the imperfections o f this first text, and I w i l l accept w i t h interest all remarks and criticism that anyone w o u l d care to make. I n p a r t i c u l a r , f w o u l d welcome comments

on whatever relates I n the theory o f

migration, published here for the first l i m e , and which must

be

matched w i t h experimental data. i express my gratitude to Professor G . Darmois and the Institute o r Statistics in Paris for m a k i n g this w o r k possible. A l s o , I express tny appreciation i o Professor L . Ularinghem f o r lus valuable en­ couragement and to Masson el Cie Tor the care w i t h w h i c h they have published this b o o k . iyon,

wh

G.

MALİÎCÜT

Preface U Probabílitcs

e t

H é r é d i t é

The probabilistic (henry o f genetic relationship and covar lance devel­ oped hy Malecot has been propaga ted by disciples in other countries, notably C r o w in the U n i t e d Status. Yasuda and K i m u r p in Japan, and Falconer in Great B r i t a i n , and is now universally ¡iceeplcd. The application o f his results for isolation by distance, begun by La mertte w i t h Ccpcii

and continued by Yasuda in man, promises to reveal

population structure and the forces that have acted on m a j o r genes. Maleeol's insight is the more remarkable because Fisher, Haldane, and W r i g h t , the great figure* o f p o p u l a t i o n genetics in the older generation, used correlation analysis and did not m i n d

thai the

derivation o f correlations f r o m probabilities is far easier than the reverse passage, liy mid-century a reaction was inevitable.

Major

genes For blood groups, serum p r o t e i n * , and other p o l y m o r p h i s m s , as w e l l as lethals and detrimentals, have become the heart o f p o p u ­ lation genetics, and for them correlation partitions are i n a p p r o p r i a t e . A t the same lime, the invalidity o f models o f p o p u l a t i o n structure based

on

genetic

"islands"

and

"neighborhoods"

has

become

apparent.

From Probab¡¡ltt<¡ a Ht
GIHUIVL:

MsJéCoE. Frases Un™*ilaİr« de

Xİv

Prıfaii la ProhibililM cl HircJilc A probabilistic approach

Pitfj't

begun by C o l t c r m a n , w h o in I 9 4 [

set f o r t h the c o n d i t i o n a l probabilities for many kinds o f relationship. H i s w o r k had l i t t l e i m p a c t , however, largely hecause the m a t e r i a l o f

f / P r o r u b i l i t c i ci Hctidhc

clcoiue, as an orderly presentation o f principles a n d as v i n d i c a t i o n I" the priority o f a great French savant w h o has t r a n s f o r m e d population genetics,

his thesis was published in s u m m a r y , but also because his f o r m u l a t i o n was designed f o r nearly pauniictic h u m a n populations and d i d not Malecot's thesis o f 1109 f o l l o w e d the classical approach o f Fisher and W r i g h t . His book o f I94-S, however, contains in a brief (S3 pages a p r o f o u n d l y o r i g i n a l treatment o f relationship, eovariance, a n d p o p u l a t i o n structure i n terms o f p r o b a b i l i t y t h e o r y . Every derivation began w i t h the genotypic probabilities f o r a single locus, and w i t h astonishing cluiity t h e most complicated properties o f M c n d e l i a n populations were revealed. M a l e c o t identified W r i g h t ' s coefficient o r inbreeding, one o f Ihc great u n i f y i n g concepts o f mathematical b i o l o g y , as the p r o b a b i l i t y that u n i t i n g gametes are identical by descent, and introduced the more general coefficient o f k i n s h i p {parosiv)

t o measure relationship o f individuals possibly separated

in t i m e , space, o r by o t h e r barriers, f r o m w h i c h m a t i n g pairs are not r a n d o m l y d r a w n . H e replaced W r i g h t ' s b e w i l d e r i n g diversity o f inbreeding coefficients relative t o different subpopulalinns. by one absolute measure o f isolation by distance, the relation between t h e mean coefficient o f k i n s h i p o r inbreeding a n d the m a r i t a l distance between b i r t h places o f potential mates. T h i s led W r i g h t t o reexamine his results and conclude that " n e i g h b o r h o o d size," on which his analysis o f p o p u l a t i o n structure is based, is almost independent o f Maleeai's basic relation between consanguinity a n d distance. There seems little d o u b t that research on p o p u l a t i o n structure i n the foreseeable f u t u r e w i l l f o l l o w t h e d i r e c t i o n set by MaEecot, (lis later w o r k o n p o p u l a t i o n structure was mathematically diffi­ c u l t , and publications i n the A wink's

pWftunrofG'™"?* tM**riity

reveal the f u l l power o f p r o b a b i l i t y methods.

tie i'Litiiwiilt-

d? Lyon

d i d not

receive immediate r e c o g n i t i o n . A s recenlly as I464 K i m u r a and Weiss rediscovered the f o r m u l a f o r two-dimensional isolation by distance w i d t h had been published by Malecot f l 5 " and earlier), Q

and believed that their result was new. This book is therefore d o u b l y

iv

i'f Hawaii

Ni:WTÛN E. M o K I U N

THE

MATHEMATICS

O F

HliltliDlTY

Chapter

The Mendelian Lottery

I.l

HEREDITY

A N D T H E LAWS O F

MENDtL

Let us retail Ihe laws o f M e n d e l , i n k i n g [lie f o u r o'clock jalapu)

(Mirabills

as an example. I f we cross a white-flowered plant w i t h a

red-flowered one. we obtain only pink-flowered plants. But i f we cross pink-flowered plants among themselves, we obtain progeny 1/4 of w h i c h , o n the average, have w h i l e (lowers. 1/2 p i n k flowers and 1/4 red flowers. T h e traits o f the grandparents reappear. T h i s is the phenomenon o f M e n d e l i a n d i s j u n c t i o n , o r segregation. I t can be explained by postulating that flower c o l o r i n the f o u r o'clock is determined by a pair o f hereditary units o r factors* each o f which can appear i n one or the other o f t w o states o r genes, w h i c h we w i l l designate by A o r n. T i m s , an i n d i v i d u a l could carry Ihe pairs AA, and so have red f l o w e r s ; An, and have pink Rowers; o r uu, and have white flowers. T h e three stales in which the pair can appear ore called genotypes o r zygotes. A A and tia are the homozygotes; is the heteroi.ygote. The outcome o f the cross can be interpreted by the f o l l o w i n g mechanism. T h e pair o f factors o f each plant resulting f r o m the cross. i.c, o f each " o f f s p r i n g , ' is obtained by d r a w i n g , at r a n d o m . 1

f

2

Üomoiygoîc- D o m i n a n c e can be incomplete, so that the helcrojygole i * e l

U 5 c r

1 0

""^

°^

l n e

homozygotes.

but is nevertheless

distinct. (B) Characteristics determined by several pairs o f factors arc called m u l t i f a c t o r i a l . For example, the shape o f a rooster's c o m b depends o n three pairs o f factors; Ihe first pair w i l h genes C (presence 'AAA

'/*Aa

'hau

FIGLTBH 1.

o f comb) d o m i n a n t over c ( r u d i m e n t a r y c o m b j ; the second pair R (rose) dominant over r (single); the t h i r d D (double) d o m i n a n t over d

nne Of Ihe two factors o f the father and one o f (he I w o factors o f ihe

(single). As a result o f dominance, the genotypes CCRRdd,

mother (see F i g . I ) , T h e cross o f an .-i.-l w i t h nn uu gives only Aa

CcRrdd*

(first generation), hut Ihe cross o f Aa genotypes a m o n g themselves

but CCffdd

gives: A A w i t h a p r o b a b i l i t y o f 1.4 ( o f d r a w i n g an d f r o m both

ccrrDd

parents); Aa. 1/2; and M,

and CCRrdd

CiRRdd,

produce lite same phenolype, Ihe rose c o m b ,

and Cordd

produce a single c o m b , and r c r r D D

and

produce the pea comb (double r u d i m e n t a r y c o m b ) .

1/4. T h i s interpretation agrees well with

The siudv o f crosses shows that segregation o f dill'crcnl pairs

the many observations o f frequencies for individuals in Ihe second

takes place independently. F o r example, crossing a breed o f chickens

generation.

Wilh a rose c o m b produced by the double

The laws o f M e n d e l explain remarkably well all Ihe phenomena

hctcrozyyule

with a breed having a pea c o m b o f genotype carDD

CRrdd produces

o f heredily and it can be said t h a i , w i l h a few rare exceptions, all

progeny which all have the pair Dd, but which have either C c or cc,

heredity follows the Mendelian process. One has l o a d m i t , however,

and either Rr or ri; each pair having a p r o b a b i l i t y o f 1/2. Therefore,

t h a i genes eati also act in a way different f r o m the one j u s t described.

since these two pairs undergo segregation independently, I , 4 o f the

T

( A ) Let us consider the example given by Mendel o f Ihe cross between peas w i t h round seeds and peas w i l h w r i n k l e d seeds. I n the

genotypes o f the progenv w i l l be, on the a\erage, CfRrDd, he tcRrDd.

first generation, we obtain only peas w i t h r o u n d seeds; when we

(C)

1/4 CtviDd,

and

\

1 '4 w i l l

4<
In the Ihree examples so far, the characteristics observed c o n -

cross these peas a m o n g themselves, 3/4 o f their progeny have round

stituted a discontinuous series. K a r l Pearson, w h o was, w i t h Frances

seeds and

Gallon, the founder o f biometry, distinguished f r o m ihis " a l t e r n a -

1/4 have w r i n k l e d seeds. T h i s result fits the previous

scheme perfectly i f we postulate that both A A and Aa have round

t i v e " heredity tile

seeds, and that only nn has w r i n k l e d seeds.

esample, lhat o f stature or skin c o l o r in the h u m a n species. I f one

continuous"

or " b l e n d i n g "

heredity, as, for

I n this case, ihe beletozygolc An has the same external appearance

Observes enough children f r o m a given couple, one finds that the

as-an A A homozygote, f r o m w h i c h it cannot be distinguished esoept

statures o f Ihe children are grouped a r o u n d a mean \alue w h i c h

by the characteristics o f its progeny; t h a i is, we musl distinguish

depends o n the statures o f the I w n parents, and c o n f o r m to a bell-

the genotype, or hereditary c o n s t i t u t i o n , f r o m the phenolype. or

shaped curve, w i l h extreme deviations being rare but possible. I :

external appearance. Here the three genotypes give only two pheno-

seems l h a i there is a blending o f the parental characters, complicated

types. The gene .1 is d o m i n a n t over the gene a, i.e.. a is recessive,

by

and the heterozygote exhibits the same phenolype as the d o m i n a n t

children can vary greally in skin c o l o r ; although most o f ihe c h i l d r e n

fluctuations.

In the sume way, when t w o muhttloes m a r r y , their

4

TI" ¡fitiuUlittn

Lellfj

w i l l b a t e skin c o l o r more or less like that o f the parents, f r o m time l o time . completely black or completely white genotype also occurs.* A l l these results are perfectly explained by Mendel's laws, on the basis that stature or skin color results f r o m the accumulated effects o f a large n u m b e r , FT. o f M e n d e l i a n factors which disjoin independ-

'ft

# \

i i. '

r r

I

ently. T o illustrate the p o i n t , if, in each pair o f factors, one o f the t w o possible genes adds 1 m m t o , and the o t h e r subtracts 1 nun f r o m , the mean stature, mid i f one crosses t w o individuals i n w h o m all pairs are heterozygous. A\u-\, A^a-, . . . A*a„

i n the offspring each

j ) .

k

I I

pair o f factors w i l l have the probabilities 1/4, 1, 2, and 1/4 o f being in the states 4uH«, A,Oi. and O , D , , and o f c o n t r i b u t i n g to the stature 2 m m , 0 m m , and - 2 m m , respectively, as m two independent rand o m choices, I f the : different pairs are stochastically independent, the stature o f a c h i l d w i l l be n o t h i n g but the sum o f gains and losses i n 2n inde-

i'iaoni! 2.

pendent series o r r a n d o m choices. T h i s Bum. as we k n o w , for an indefinitely large it, follows Gauss's law o f p r o b a b i l i t y , which fits

times exhibit differences a m o n g themselves which make it possible

the experimental bell-shaped curve. We shall see that the same is. true

t o recognize i n t w o different gametes the homnhgom

whether one suppose dominance to be generally complete or gener-

(see F i g , 1 f o r drosophila). W h e n a paternal gamete unites w i t h a

chromosomes

ally incomplete i n each pair or suppose different c o n t r i b u t i o n s for

maternal gamete, the fertilized egg has n pairs o f

different pairs. T h i s general scheme o f m u l t i f a c t o r i a l Mendelian

chromosomes. T h i s c h r o m o s o m a l c o n s t i t u t i o n persists i n all the

heredity w i l l be developed in detail in the second chapter, and w i l l

cells that the egg produces by d i v i s i o n and, finally, in a l l the cells

be shown to explain the results o f biometry as discovered hy G a l l o n

or the adult individual except the reproductive cells or gametes; the latter am produced hy a d i v i s i o n or disjunction

and Pearson |5, 17. l S j . i

homologous

w h i c h allows,

only one chromosome o f each pair to be included i n each gamete, this chromosome being taken al r a n d o m f r o m the t w o . U n i o n o f

1.2 T H E

CHROMOSOMES

these gametes, a l r a n d o m , w i t h the gametes o f the other parent

The physiological basis o f Mendelian heredity was discovered in the small

RHÍS

or clinnin'.mm'.'s

that arc constituents o f the nucleus o f the

reproductive cells o r ginne/cr. These chromosomes

have a

fixed

number, n, i n each species (23 in m a n . 4, i n d r o s o p h i l a ) and snme• To be precise, une Mart. j „ white child wilt U C C J I m every ]'i>urtcen children. On the uverayc D. M. V. I flukl J i m l l l i e n in hffK*C4l refer lu ihe J i [cm | u re cited ill the ertJ at t h e ]joOk. u r | l

n i :

produces the individuals o f the f o l l o w i n g generationThe laws o f Mendel are explained, therefore, by postulating t h a t the t w o factors o f a pair are carried by two homologous c h r o m o somes. T w o heterozygous parents, Aa, w i l l each f o r m gametes one half of which are A and one half a\ this is disjunction.

R a n d o m union

o f these gametes w i l l produce offspring i n the r a t i o 1/4 4 / 1 . 1/2 Aa, «nd l / 4 , z a .

6

TIM MtnJtitM

J .2 I i i Clmuiatwfı

Lftietr

A difficulty appear*, however; tlie different pairs o f factors can be independent o n l y i f each pair is carried by • different pair o f

7

o f genes can alfccl several characteristics. F o r example, the recessive e i :

n - for albinism produces, in Ihe homozygous recessive on. b o t h

chromosomes, each o f the l a l l c r being expected l o disjoin independ­

, ^ i i t e hair and red eyes (due l o lack o f pigment). <1n Ihe other hand.

ent

I single characleriilie. such as Mature, can be influenced hy many

Faelnrs carried by (he same pair 1 / chromosomes should be

eomplclely l i n k e d . However, given a genotype i n which A It m scm* t w o pair* o f

ICIOTS

M

i

linked o n t o t w o chromosome*, i f this

p p j n o f genes. A l t h o u g h , slriclly speaking, each characteristic de­ pends o n the entire gene complement

o n the l o u l "genetic con-

genotype is crossed w i t h a homozygote ahUh, [he genotype* actually

• d t u t i o n " - each characteristic is in fac( influenced appreciably by

found in the progeny are Atkib

and nbtib

only one pair o f genes, w h i c h made it possible f o r Mendel l o deduce

( I — r)/2, and Abab

each w i t h a frequency o f r/2, where

and tiBab,

r

each w i t h a frequency o f

r is, in general, LL small positive number, but run *ero u i it w o u l d be

his laws. We have so far been tacitly assuming that ihe genes occupying a

if linkage Were complete. These ratios can he explained only by

specific

postulating that the chromosomes break and exchange

by A and «, which wc call altclic

homologous

sections before disjunction. The f o u r types o f gamete* formed have the given frequencies because o f this exchange, whose p r o b a b i l i t y o f o c c u r r i n g is r. T h i s is tile phenomenon

d f "crossing i n c r . " A

IOCUH

can appear in only two different states, represented çenes. I n reality, they can also have

multiple stales A ' , A " , A ' " , . . . A ' K which we c;dJ a

There arc n h u m o i y g o i e s and .C? -

mitlllii/k-llsm.

' heleroiygotcs." The

sludy o r (his phenomenon leads to the assumption thai each factor

origin o f allelic genes is to he traced back l o Ihe phenomenon

of

is locali/ed al a specific p i a n l o n (he c h r o m o s o m e : ibis p o i n t is its

m u t a t i o n , which appears 1 0 he (nr. least according l o pre*enl ohsena-

litciix ( p l u r a l , t o r i ) . I -t i w o factors f o u n d on lite same chromosome,

lions. since one cannot guess w h a t l o o k place in palconlological

i l Is eiidkml (hal Ihe farther apart Ihcy are located o n Ihe c h r o m o ­

times) the only inheritable way i n w h i c h l o i n g organisms can be

some, the greater w i l l be (he probability r o f their crossing over.

modified and. therefore, the only one thai affects (he c v u l u l i o n o f

T h i s phenomenon

»pecie». M u t a t i o n is an n b n i p ( change -ı

made i l possible t o map the loci o f the different

ı • • one o f t w o h o m o l ­

factors for the f o u r pairs o f chromosomes o f drosophila. We shall

ogous loci in an i n d i v i d u a l ; (his change is, therefore. transmitted

sec in the f o l l o w i n g chapter t h a i , because o f crossing over, linkage

t o one half its gamete*. T h u s , in a p o p u l a t i o n

o f factors on ihe name chromosome does not p ^ e r i l them

in ihe

A A individuals, which are indistinguishable in terms o f ı ı. pair o f

independent.

genes, there may appear an An heLcro/ygote. which, even though

long r u n . f r o m being as reshulflcd as i f they were

H

Sex is determined hy a pair o f chromosomes, t w o .V chromosomes 1"i

femali , all V and a )' L h r o m o s o n i i l " i thi Utah (CKKpt in

lepidopiera ami birds.), called helcrosomes; ihe other chromosome* arc called autosomes; The iaclors carried o n the helcrosomes arc called "aexdinkcd'*L we k n o w p r i m a r i l y tho^e carried o n (he X c h r o ­ mosome. These other factors arc never masked in males, h i l l can be in females (us w i t h d a l t o n i s m and hemophilia). Whalc*cf

(be physiological process hy w h i c h penes alfecl (he

development o f individuals, it is to he expected (ha( a single pair

of

homozygous

A is dominant, can be identified by its progeny. M u t a t i o n produced the new gene a, which is allelic l o the old gene A . Repealed m u t a ­ tions alfeciing the same locus can continue l o create the same gene n {recurrent inula I i un I, or cause gene a l o revert to A (reverse mutnt i o n ) or crcale olher alleles ( m u l l i a l l e l i s m i . (

F t " cMmpk. Hur I..,- hkHHİ in iron l i t JctnmıruJ Iw ihree :ilh-k-s. * i B. and O, A ami U r^mp tlnnmunl aver (t, which KHX Ilır umr phcnoljrvt A (Bin.'U,-, I I toy, B\Blt or Bin, -ut l u n n a ^ ı l ft.ii.kni>. and O yOOt (imiv trial dunun

El

Thı Miiidt¡iiy¡

U

IMitn

RESTAM BLANCH R E L A T E D

BETWEEN

INDIVIDUALS

T w o individuals i n R p o p u l a t i o n Lire related İJ ihey have one or more L

common

ancestors. I f they do, their genetic ditforencc must

he

smaller, o n the average, Ehan t h a i between t w o individuals t a k e n at r a n d o m , because Home o f the genes o f the first t w o are derived f r o m the corresponding

genes o f the c o m m o n

ancestor.

Disregarding

Fjouan I .

mutations, these genes cannot he different, whereas they often could be i n unrelated individuals. F o r precision i n t e r m i n o l o g y , let us distinguish between genes,

and tod.

L e t us call gates

furtvrs,

ability thai these two loci come f r o m unrelated ancestors, i.e., t h a t

the different slates i n which each

they are stochastically independent, since k n o w i n g the gene which

factor can appear w i t h o u t regard to the i n d i v i d u a l i n which they arc

occupies one locus does not provide any i n f o r m a t i o n about the gene

observed. T w o genes corresponding to the same factor and observed

which occupies the o t h e r ; these t w o genes can be identical or differ­

ciLher in the same i n d i v i d u a l or i n t w o different individuals w i l l be

ent, but their probabilities are independent.

called identical or different, depending o n whether they appear i n

We shall designate as the coefficient

nf inbreeding,

fu,

o f an i n d i ­

the same stale, for example, .-I, or i n t w o allelic states, for example,

vidual M the probability that its two homologous loci are identical.

A and u, I lowever, t w o ÍOCÍ w i l l be called identical only i f Lhey were

Since one locus is derived f r o m its father and the other f r o m its

derived by M e n d c l i a u descent

mother, fu

common

ancestor;

f r o m the same locus oT the same

otherwise Ihey w i l l

be called different. T w q

identical loci are by necessity occupied

by t w o identical genes, i f

is nothing but the coefficient o f eoancestry o f its t w o

parents. Let lis evaluate the coefficient o f eoancestry, f , tL

there is no m u t a t i o n , but t w o different loci can be occupied

by

cither t w o identical or two differenL genes,

or more c o m m o n ancestors,

A n i n d i v i d u a l , A has t w o parents, f o u r grandparents, . . . 2" an­

of two indi­

viduals, I and L . It differs f r o m zero only where / and L have one J,, J,,

etc., w h i c h we w i l l assume

they have. Let us suppose at first that there is only one

common

cestors o f order n. A locus o f / has a p r o b a b i l i t y o f 1/2 o f being

ancestor, / , o f order ;r Tor / and o f order p for L along two d i s t i n c t

derived f r o m the father, 1,2 f r o m the mother,

chains or descent, which together constitute a chain o f eoancestry

1/4 f r o m each o f

the grandparents, 1/2" f r o m each ancestor oT order ft, along a given chain o f descent. A n ancestor o f / can be l i n k e d to i t by several

l i n k i n g I and L . The probability that one locus o f / and one homologous locus o f

chains o f descent; for example, J in Figure 3 is a m u l t i p l e ancestor,

i are both derived from J is ( I 2VIf

and could even be an ancestor o f different order i n ditTerent chains.

they liave

W e w i l l designate as the encfjicietri

o f eoancestry,

fju

individuals / and L the p r o b a b i l i t y that t w o homologous

c

?

t

w

o

loci, one

f r o m / and the other f r o m L , are identical, i.e., are descended f r o m the same locus. T h e complementary p r o b a b i l i t y , 1 — fa„

is the prob-

a

they are b o t h derived f r o m J,

probability o f 1 2 o f being derived f r o m the same locus

o f J and a p r o b a b i l i t y o f 1/2 o f being derived f r o m different l o c i ; "f they are f r o m different loci, the probability that Ihey w i l l be Identical i s /

F r o m this, / „ . = ( 1 , ' 2 ) " ' H - I - / , ) • • ! I n particular, +

J r

tbfi coefficient of eoancestry o f an i n d i v i d u a l and an ancestor

of

10

Tht M.iitJtUjn

1.3

luxury

order n is given by letting p = 0 ; t h t coefficient o f eoancestry o f

F

an i n d i v i d u a l w i t h itaeif

ytA-FCBDEG

given by I d l i n g n = p = 0 .

Let us n o w consider the general ease, in w h i c h / and L are con-

r

A

E

G

=

\,P-:GC

Let as s u p p ^

ADF.I

K.tsiinbtiinci Bflu'ttn lit}j-nl

-

I 64; FCA

DEC

= 1 u4. Therefore. /,,.• l , C J W

l

=

Imhi'toJwIj'

1J

\/6A,GCBDEF

=

I . ' 22.

' ' a l Ihe loci t o n s i d c r e d may have an average

nected by i n j number of chains o f coaneestry. each chain being the

te of m u t a t i o n , it. per generation. The p r o b a b i l i t y that a Locus o f

c o m b i n a t i o n o f t w o chains o f descent leading f r o m / and f r o m L to

individual reproduces w i t h o u t m o d i f i c a t i o n the parental locus

a c o m m o n ancestor 7, and having no other c o m m o n p o i n t except J j j

f r o m which it v,as derived is 1 - u; therefore, the p r o b a b i l i t y that

two chains o f coaneestry are considered distinct, even i f they have

Ihis locus may be transmitted w i t h o u t m o d i f i c a t i o n along a specified

links in c o m m o n , provided they differ by at least one l i n k . Since the

chain o f descent having 'i links is [ { I - u ) / 2 ] \ The coefficient o f

transmission o f identical loci along a spceilicd chain o f relationship

coaneestry of two related individuals then heconics

excludes their transmission along any other, the principle o f toljd p r o b a b i l i t y gives'

_„y,+p,i+/, 2

/

I t

=^l/2j"'+pCl+W/2-

/ and L ; the it It chain has n, -f p, links ascending f r o m / and f r o m h ancestor J,, whose coefficient o f inbreeding is

f

Jr

For example, i f we assume that all chains o f relationship are shown in F i g u r e 4 and t h a t the ancestors A arid B are n o t related, there are the f o l l o w i n g distinct chains and respective contributions to the coefficient f ;

G C F = l/S; G E F = 5/32: G C A E F

r/F

,1

n

G

i

HÜUK& 4.

)

2

J

"

The correction thus introduced is insignificant for close rcladves,

The sum 2 extends over all distinct chains o f relationship connecting to the c o m m o n

•]

=

1/32;

since u is extremely small; it becomes i m p o r t a n t , as we shall see. only when very distant ancestors are i n v o l v e d .

CbapW

Correlations Between Relatives in an Isogamous Stationary Population 2 1 PROllAhiLITri-S O f

GENES

A N D GENOTYPES Let us classify i h e indi^iduuls o f ¡1 p o p u l a t i o n F according 1 0 Lhc slates o f a specified pair o f factors. Let us : • 1 suppose t h a i there arc only t w o alleles, A and ti. a n d . therefore, the three genotypes AA,

Aa. nu, w i t h

the respective frequencies /\ ZQ.

ft. where

P + 2Q + H - 1 Let us define the frequencies o f genes A a n d a as the quantities p and u. where p » t* + Q, q = Q + R, a n d These quantities are the probabilities that a gene taken a i r a n d o m f r o m a n y i n d i v i d u a l i n p o p u l a t i o n / is i n slate A o r a, respectively. I n each i n d i v i d u a l / o f the p o p u l a t i o n F. each o f the t w o homologous loci w i l l be occupied by gene A o r a w i t h the p r o b a b i l i t y p o r q. However, there will generallv he a relationship between the p r o b a bilities o f these t w o loci, that is. a c o r r e l a t i o n between the slates o f these t w o loci, because k n o w i n g which gene occupies one o f these loci affects the probabilities f o r the oiher locus. I n fact, the I w o parents o f t h e preceding generation f r o m

which these loci are

descended c o u l d have been selected according 10 their relationship

14

Ca"tL'\tuil\ BitKltN

Rilaiftrs

in JII tsmg^mtm .\usi\jrwry Popn!j,'i<.rt

l.i

Profufxlhtrl

vf GtUci -in,! dfiie'jfi)

35

(consanguinity) or according to t h e i r resemblance f a s s o r t J l i v e mat-

types w i l l occur w i t h the probabilities, f p 4- f I — / ) / > " = p" 4~ fp^

ing), or they could have left o n l y selecLcd descendants because o f

2(1

differential f e c u n d i t y ; i f so any i n f o r m a i i n n on the genotype o f one

the first genotype, the t w o loci should he identical and one o f them

parent modifies the probabilities for the o t h e r In this chapter we

should be A , or fhey should be independent and both o f [hem s h o u l d

shall deal w i t h the f o l l o w i n g t w o eases.

be A).

T

{ A ) T h e parents mate at r a n d o m ; the p r o b a b i l i t y o f finding a

f)P*ii

'^ dn

f q + O — f)q"

= q" 4- fpq- ( F o r example, to ha\e

Consanguinity, therefore, causes an appreciable increase in the

mate is the same for a l l i n d i v i d u a l s ; and fecundity is the same for

probability o f h o m o z y g o u s

a l l couples. T h i s is " r a n d o m m a t i n g . " p a n m i ' i a . I n this Case, k n o w -

heierozygoles. T h i s fact explains the danger o f marriages between

ing the gene w h i c h occupies one o f t w o loci o f / gives us no i n f o r m a -

related persons; latent defects in the h u m a n species arc generally

t i o n about the o t h e r ; the states o r these two loci are stochastically

determined by rare recessive genes, anil appear o n l y i n homozygous

independent. T h e r e f o r e , f may

have one o f the three genotypes

reeessives an. \K q is the frequency, presumably l o w , o f a defective

A A , An. tia, w i t h probabilities p". 2pq, q", I f the p o p u l a t i o n is large,

gene a, Ihe p r o b a b i l i t y that an i n d i v i d u a l / carries the defect, i.e..

the observed frequencies P 1Q,

and a decrease in the p r o b a b i l i t y o f

R. must be close to these quantities.

that if is o f the genotype tin, w i l l be equal to q- ( w h i c h is extremely

T o prove this, it is suHLeicni to show that Q- - PR is close to zero

low) i f the parents of / are not r e l a t e d ; but this p r o b a b i l i t y inereases

[ H a r d y ' s l a w ) , because we can set P — p* + X

to q' 4- fl»{

T

T

2Q = 2pq — 1?,

fq i f U is rather h i g h . F o r c\arnplc, a defect brought

R = q -|- v, and h'ince w e have set P 4- Q = p, Q + ft = y, and

about by a gene w i l h frequency r/ = 1 0 " w i l l appear w i t h the p r o b -

p + q = 1. we have \ = p = v therefore,

ability I0~* i n an offspring w i t h o u t i n b r e e d i n g , but w i l h the p r o b a -

:

r

Q* -

PR

^

(pq

- J * - (p* + MO? + M = 1

-X,

1

bility

JCT-yift in an

danger is doubled

offspring o f first cousins ( / =

1/16).*

for d o u b l e first cousins (J = |/S)

r

The

I t is lhas

which equals 0 only when X = • . N a t u r a l populations actually exist

Unreasonable to tolerate marriage between d o u b l e first cousins and

in w h i c h H a r d y ' s law is c o n f i r m e d , e.g., the p o p u l a t i o n o f eoleoptera

between uncle and niece, and to f o r b i d marriage between half-sibs

Dermestes

which presents exactly the same danger ( / =

\'uipsnu.\ observed by Philip

19J (the p a i r o f factors

studied determines w i n g c o l o r ) . W e shall see that there are such populations i n the h u m a n species. l o o for b l o o d groups. T

Let us consider n o w Lhe more general case o f m u l t i a l l e l i s m . Suppose that the allelic genes A . have the frequencies p

l


T

1,'S)

{-p, = 1),

(1) W i t h r a n d o m mating, the probabilities o f different genotypes

considering t h e i r genotypes or resemblance; the p r o b a b i l i t y of finding

are p'f Tor an A , A , homozygotc. 2p,p,

a mate is the same for all i n d i v i d u a l s ; and a l l couples have the same

these probabilities being coefficients i n the expansion o f ( 2 p , r , ) .

fecundity. T h i s is pure consanguinity or isogumy. T h e r e f o r e , a locus

These f o r m u l a s a p p r o x i m a t e well the frequencies o f b l o o d groups

in any i n d i v i d u a l , whether derived f r o m a consanguineous cross or

in a homogeneous p o p u l a t i o n (p- 4- 2pr, tj- 4- 2qr, 2pq, r-).

n o t . has always the same probabilities, p and q, o f c a r r y i n g the genes A

or a;

f u r t h e r m o r e , for any

i n d i v i d u a l / whose coefficient o f

for an A . A , heterozygole, r

(2) i n the more general case o f isogamy, these probabilities, f o r an iDdividua] w i t h coefficient o f i n b r e e d i n g / are, respectively.

inbreeding f, = f. is k n o w n , the two homologous loci have, as we have seen, the p r o b a b i l i t y / o f being identical and the p r o b a b i l i t y 1 — / o f being stochastic ally independent; therefore, the three geno-

* I n 4!lTi.'ct. a has L U L U teuad Mint 1.2 of ihe caws nf h'rlcdrckb** ;Uniiii, ui W L I I •* 1/1 uf the raw* ef albinism, aw derived from mama^cs hviwixn rtluLivLs.

]fi

Ce"f/pIi"Hi

ü'tı"ita

Kilittim

i" •>'• Iwjpmws

İP- + O ~ f ) p f = P< + f P ^ - P>)

JfJriWLnJ

and

these being coefficients i n the expansion oT/lpA'f

1 2 Tin Di\irrhaTİün oj FiU'cn

fopui^lia'i

TH£ IN

Let

DISTRIBUTION

AN

ISOGAMOUS

U S call " i s o g a m o u s "

Of

+ [1 - / K ^ / v , ) . 7

FACTORS

plav as i m p o r t a n t a role as the close ones. A f t e r considering the segregation o f one pair o f factors. let us

L

J p o p u l a t i o n , F, derived f r o m parents

ing all h o m o g a m y ) , and i n which all pairs have (he same fecundity. having a coefficient o f

coancestry f, is ie, (the p r o p o r l i o n \v, corresponds to r a n d o m mating, w i t h f« = 0 } ; u', is, therefore, the frequency o f individuals i n the p o p u l a t i o n w i t h inbreeding uoelfieicnt

sludy the simultaneous segregation in the population F o f t w o

probabilities p, and

POPULATION

Lhat the p r o p o r t i o n o f couples

17

pairs o f factors occupied by genes having the states A and Ü. w i t h

chosen either at r a n d o m or because o f their coancestry ( h u l exclud­ Assume

Populjtiaa

coefficient, because the distant relationships, w h i c h are o v e r l o o k e d ,

2[1 -/)/>,;>„

H t 2.1

in au İmainour

and 2Z,w, = 1, We

have

seen thai the probabilities o f the alleles A and a (assuming only i w o o r them Tor s i m p l i c i t y ) are the same among these individuals as i n the t o t a l p o p u l a t i o n , e,g., p and q. The probabilities o f the three genotypes in the entire p o p u l a t i o n , and. accordingly, their frequencies, P, 2Q. fi, i f the p o p u l a t i o n is large, are

An two

respectively.

individual / mken at r a n d o m in / results prom the union o f

gametes f r o m the preceding generation. F\ L e t us call Pi, the

probability Lhat any gamete I" c o m i n g f r o m generation F' has i n its chromosomes the genes A , and U

/m

and /',, the p r o b a b i l i t y that the

same w i l l be true for any gamete V f r o m /-', i.e., for a gamete p r o ­ duced bv I: and let us find the relation between P,, and P',;. W h e n the gamete 1' produced by / has the genes .-1, and ft,, either both arc derived from the same gamete I " or each came f r o m one o f the t w o gameles 1" which made up /. These two possibilities each have the p r o b a b i l i t y I .'2. i f the two genes are f o u n d on two different c h r o m o ­ somes, because o f independent segregation; but they have the prob­ abilities 1 - r and r i f the t w o genes are located o n the same chromosome, because o f ' crossing o\er/" The first possibility w i l l be included P

u

with

=, (1 - riPlj

the second

when

r = 1 1

Then

we

have:

-b n r . j . where r,, is the p r o b a b i l i t y t h a t , i n genera­

t i o n F, > gamete carrying A , may unite w i t h a gamete carrying B , . which can also be w r i t t e n as pip

-\-

setting a = Sw.f,;

2/irrfl — a } ,

q{q -f- op),

o is the mean inbreeding coefficient of the popula­

t i o n , the mean o f the coefficients ° f ' introduced a pi-hri

Different pairs, therefore, arc n o t generally stochastically inde­

E s

individuals. İl had been

by Bernstein [ 2 ] to measure deviations f r o m

panmixia. His approximate evaluation has been tested on some h u m a n populations w i t h the help o f state census data o n con­

pendent, since their d i s t r i b u t i o n depends

on the d i s t r i b u t i o n i n

preceding generations, i.e., on an initial d i s t r i b u t i o n w h i c h m i g h t be a r b i t r a r y . It w i l l be shown, however, that there is an " a s y m p t o t i c independence" under the f o l l o w i n g hypotheses. (1) The p o p u l a t i o n considered is very large, so lhat frequencies and probabilities in each generalion are essentially equal, (2) T h e p o p u l a t i o n is isogamous. so Lhat. as we have seen, no gene

marriages. I n general, this coefficient is s m a l l : i n a

is f a v o r e d ; therefore, i n cuds generation, the gene probabilities w i l l

r u r a l A u s t r i a n p o p u l a t i o n , Rcudingcr found n to be O.h per c e n t ;

remain equal to their frequencies in the preceding generation. A s a

in a Jewish population Orel f o u n d a l o be a little over I per cent.

result, the frequencies p, will remain constant over generations.

These estimates, however, are probably much

These will be the character is tie constants o f the p o p u l a t i o n and o f

sanguineous

below the actual

IS

GjtntjrioiT

Biiwttti

Rıijiim

in J S 'jdgjmcnr Suimury

Papui^iim

J.3

R.JfJ^". MinJi/iuw 1

V.ni.thiti

in JIJ ha^Aiiiaw

SljUvllary

Pv pit f.i rial

19

the System o f alleles considered. F r o m these, cine can derive the

measurable, or qualitative and arbitrarily assigned to values o n a

probabilities o f the three genotypes f o r i n d i v i d u a l * w i t h coefficient

numerical scale. C a l l y the numerical value thus attributed to the

o f i n b r e e d i n g / or their frequencies i f they are sufficiently numerous.

trait i n each i n d i v i d u a l . F o r an i n d i v i d u a l / taken at r a n d o m f r o m

(3j

The mating system adopted, although it implies a relationship

the p o p u l a t i o n , y is a r a n d o m variable, We shall regard j - as being

between the two gametes that unite, leaves their probabilities o f

the sum o f a r a n d o m variable, i " , which represents the influence o f

carrying dilfcrenL genes independent. T h i s consequence, evidenL for

the genetic c o n s t i t u t i o n o f / o n the trait considered, and o f another

panmixia, is not always valid i n crosses between relatives, e g.. when

r a n d o m variable, -. which represents the influence o f chance and

the p o p u l a t i o n is divided inro groups between which crosses are

environment on the development o f this t r a i t , z being stochastically

impossible. I t can be shown, for example. thaL it applies to brother-

independent o f X" Consider A" the sum o f c o n t r i b u t i o n s made Lo (he

sister minings, i f all individuals in each generation are brothers and

trait by a certain number

sisters o f one f a m i l y ; i f n o t , the p o p u l a t i o n w o u l d be d i s t r i b u t e d

c o n t r i b u t i o n Jt! o f one o f its pairs w i l l he equal l o i.j. or A, depending

into several groups, and differences between genes existing i n these

on whether this pair has the state A A , An, or tin, whose probabilities

groups w o u l d continue to exist indefinitely. L e t us assume, therefore,

are p- 4- fpq, 2(1 - f}pif,

r

R

R

of pairs o f factors. F o r example, the

and tj(q + f p ) , respectively, where p and q

that the mating system chosen is such that it leaves independent the

are the frequencies o f .-1 and a, and / is the inbreeding coefficient

probabilities o f one u n i t i n g gamete c a r r y i n g gene A , , the other

o f /. T V will he called the gcnolypic r a n d o m variable associated w i t h

gene B

T h e n the p r o b a b i l i t y T „ o f the union o f a gamete c a r r y ­

the t r a i t and w i t h the pair o f factors considered.* I f one o f the

ing A , w i t h a gamete c a r r y i n g B, w i l l be constant and equal to p,x,.

alleles has complete dominance, j = i ar j = k. I f there is no d o m ­

The above recurrence equation may be w r i t t e n as

inance, i.e., when the helcrozygote is exactly intermediate between

r

the t w o homozygotes,./ = (/ + d)-2

ı

-

P<X, = ( 1 - r){Pu

-

p^,y

» one can l e l i = 2 t J = s d- i , r

T

and £ - I Î ; one can readily verify that the three-valued r a n d o m

If, therefore, the P,, o f one generation is equal to p x„ t

i t w i l l always

variable X is the sum o f the two-valued r a n d o m variables 1

and //',

remain equal i n the f o l l o w i n g generation; we say then thai the p o p ­

each o f which has the value I or s w i t h probabilities p or q, and

ulation is stationary, and we note that the genes o f the different

which have Ihe p r o b a b i l i t y f o f being identical and the p r o b a b i l i t y

pairs are stochastically independent. I n a nonstationary p o p u l a t i o n .

(1 — / ) of being independent. H and H\ w h i c h represent the respec­

PI

—piXj

— y - 0 as (1 - r ) — 0

when the number, n, o f genera­

tive stales o f the two loci o f the pair, will be referred to as genie

tions tends to intiniry, mid the p o p u l a t i o n tends t o become station­

r a n d o m variables.. I f there İs dominance (complete or incomplete),

ary; l e t us assume i n the remainder o f the chapter that ibis state o r

we can still keep Ihe r a n d o m variables H and H' by t a k i n g appro­

e q u i l i b r i u m has been attained, and, i n particular, that there is

priate values for a and r. and letting X = H -+- H' 4- d, the c o m ­

stochastic independence o f the different factors.

ponent o f dominance,

n

being equal to i — 2 j j — .v — i , or k — 2J, r

according to whether l i -\- i V is equal l o 2r, s + I, or 2s (the most 2.J R . A . N D G M AN Let

MHNDJEL1AN

VARIABLES

ISOGAMOUS STATIONARY

INJ

POPULATION

us consider a specific t r a i t , e.g., stature, o f the individuals that

make up the p o p u l a t i o n ; this t r a i t can he cither quantitative and

* Unless niticrwiie ineoiied. rhc samnllrit unit on which this randum variüblı: depend-, is JIIL i.. • •' i I laken al tandem front among tbdfO inliret:dmg eoelllcİL-nl /

2d

C""'il^iiifl

Df'U!"i

Rttiiititi

i'l an liagu"ten.\ iterivnary

F\?paLi/it/ii

convenient values Tor .v and t w i l l be discussed later). We

have;

therefore:

J j SjfiJuw MmJi/iiin Variablu

IH ja J.rpjjfrjenj Statmury

Pnfuiialian

2l

acleristics. The theories o f " b l e n d i n g i n h e r i t a n c e " that certain bionictricians tended to accept w o u l d i m p l y t h a i , i f the hereditary

y

=

x + z

=

J

=

S C / f +

fl'-E-

¿ 0 +

z,

p o r d o n A o f a trait i n the two parents was equal to

and Xj, it

w o u l d be equal to {x% + * ) / 2 i n the offspring, the remainder o f a

w i t h ^ designating ¿1 sum over fill pairs o f factors influencing the

variance being a t t r i b u t e d to chance and to the environment. G i v e n

trait under cousideration (for a mo nu facto rial t r a i l , £ covers onlv

panmixia, and assuming that rrtl(.v) = 0 the variance o f X i n the

one t e r m ) . Since the p o p u l a t i o n

entire progeny o f a p o p u l a t i o n , t a k i n g M f o f t ) - 0, w o u l d

under study is assumed, to- be

H

be

stationary, Ihe different terms o f ^ are, as we have seen, independent

3Ti[(.V] -f x*)/2]-

r a n d o m variables; - is also assumed to be independent.

thus, the genetic variance o f .v w o u l d tend rapidly t o w a r d ?ero after

= J l i t . t ) / ! . i,e., h a l f the v a r i a b i l i t y In the parents; 1

T o s i m p l i f y matters, we shall assume, henceforth, that each o f

several generations. F i n a l l y , the o n l y variance left w o u l d be caused

these r a n d o m variables is given its mean value i n the p o p u l a t i o n

either by chance and environment (but the experiments o f Johannsen

( o r i n a specified subgroup o f this p o p u l a t i o n ) as an o r i g i n . T h i s

[9] on pure lines have shown that such variance is small f o r most

assumption is n o t restrictive as long as we agree l o stipulate that,

traits) or by m u t a t i o n s , which w o u l d then have to be very frequent

in measuring ihe characteristic, we take its expectation as equal to U,

(but this conclusion contradicEs our experience). " B l e n d i n g inher-

which is approximately the same as subtracting the general mean in

itance" is thus inadmissible, and the M e n d e l i a n scheme, w i t h indef-

the p o p u l a t i o n ( o r i n the subgroup} i f it contains a large number o f

inite disjunction of parental traits, is one o f the simplest o f those that

i n d h Lduals. I f we symbolize the expectation o f a r a n d o m variable

have the conservation o f hereditary variance as a consequence

by ¡«7. the stipulation w h i c h we made will be expressed by -nz(.iC) = U, ;iM{.\) - 0, aHUJ = 0, and $n(y) values o f j and !, w(&)

w 0, and. by selecting appropriate

- 0, :m[<J) = 0

[4].

Let us now show how the Mendelian scheme leads to the same results as b i o m e t r y . Assume, henceforth, that the t r a i t sladied is multifactorial and depends on a large n u m b e r o f genes, each m a k i n g

The variance o f t r a i t y (i.e.. the square o f its standard deviation)

a c o n t r i b u t i o n o f the same order o f magnitude. Therefore, A is the

in the p o p u l a t i o n ( o r i n the subgroup) because o f independence w i l l

sum o f a great many independent r a n d o m variables, each o f which

be:

is small in relation l o the standard d e v i a t i o n . d- o f J ; according to JT

Liapounov's

theorem, the p r o b a b i l i t y o f x follows Gauss's law,

( l / \ / 2 i f l ) exp ( — A " , 2ffxd ttx. I f Ihe effects o f chance and e n v i r o n r

which we w r i t e as

ment on development come f r o m m u l t i p l e and independent sources, J

Oy

—

J

,

Us T

J

°ii

being the standard d e v i a t i o n .

z and v w i l l also be almost Gaussian, which result agrees w i t h the observations o f G a l l o n and Pearson on stature. L e t us measure the t r a i t .i f o r t w o related individuals, h and 1

A l l these formulas are also valid for n i u l l i a l l e l i s m .

and let y, and ft be the t w o respective values, ft can be shown that

The fact t h a i a p o p u l a t i o n js s i a i i o n a r y imposes the c o n d i t i o n

the probability o f ihe sum o f the t w o r a n d o m variables Vi and y?

that variance r e m a i n constant over generations. Thus, the fact that

follows closely Gauss's law and, therefore, can be expressed by its

variance is conserved, as we k n o w by experience, may be considered

coeflicient o f c o r r e l a t i o n . The experimental d e t e r m i n a t i o n o f Ibis

c o n t i n u a t i o n o f the Mendelian hypothesis o f ihe inheritance o f char-

coefficient f o r a large number o f pairs o f individuals w i t h Ihe same

11

CotrrtatltM.r Bt'ufia

Rd.ıfirtt

ırt «

IttgflWtftl

ST^t'taary

PapnİMıan

ancestry i r a large p o p u l a t i o n was made f o r different populations

T h a s . / i s the c o r r e l a t i o n coefficient o f Hi and / / . T h e r e f o r e , :

by G a l t o n " a ] , Pearson [17, IH , and Snow |2f}]. We shall now c o n ­ sider its theoretical value. On the o t h e r hand, i f / , and /

:

are the inbreeding eoellicienls o f

/, and J;, w e have 1A

COR R H L ATİ ONS WITHOUT

BETWEEN

RELATIVES my')

DOMINANCE

W i t h o u t dominance, we have the relationships y =

nc + i =

= E;>ni//, -f- my

since WiH.ffH

4-

= ftJULlfi)-

= *%i 4- fnm$$

+

a^OT,

Therefore, the c o r r e l a t i o n eoeHieiettl

I o u g h t is

+ I I ' ) A- z

and •MM

4- / / ' ) - W t t f f l = Upt

4- flf).

catling £ the ratio ; i l i ( z - l . ' ? ^ i n f t i t ) . :

T h e r e f o r e . Ihe convention :iTTf."FsJ] - 0 is equivalent to :mf//J = 0, that is, pi -\-qs = 0. Let

For a t r a i t determined by heredity only and for unrelated i n d i v i d ­ uals, r reduces to t% = If,

which w i l l be called the f u n d a m e n t a l

Vi and T J he the r a n d o m variables representing ihe i r a i l s o f

correlation. T h i s gives the familiar coefficients: 1,2 for parent and

t w o individuals, /, and I:, w i l h cocllieieiit o f coaneestry f, W i t h o u t

olTspring and for f u l l sibs: I . 4 for half-sihs. o r f o r grandfather and

Jon un a nee,

grandson, or for uncle and nephew, or for d o u b l e first cousins;

y

t

=

+

4- z, = Hy + H[ -b ffij + Ki +

- 4- H ,

and

1/8 for first cousins; and so o n . But any coaneestry between the individuals c o m p a r e d , and any effects o f environment upon t h e m , will m a k e / , , / . , and

>': = ^ ( / / i 4 - / f i ) 4 - ^ =

unequal to zero* and w i l l reduce the f u n d a ­

mental c o r r e l a t i o n .

---¬

T o find t h e i r coefficient o f c o r r e l a t i o n , r. let us calculate the mean 2,5 C O R . R f f L A T l . O N S

value o f t h e i r p r o d u c t which is reduced to

BETWEEN

INDIVIDUALS WITH

UNRELATED

DOMINANCE

Given that the probabilities o f genes 4 and a are p and y, the p r o b ­ since, because o f independence, i m f o / / , ) = :Ut£z,J:m(//-) = 0, and so oni

âRCMvJ •

0 : and. i f K and A" represent the genie

variables for :my other pair, an (KM

= m^ûm^i

random

= 0. and so o n .

F u r t h e r m o r e , each t e r m , such as : i | i ( t f [ / / ) , is calculated on the

abilities o f the three genotypes in the p o p u l a t i o n w i l l he p'

T

2pq,

and q". Let us still consider that the r a n d o m variables jV have origins such l h a l :iTi(,TC J = pV 4- 2pqj :

i — 2l, $

4- tf-k = 0 ; d takes Ihe values

f, and k — 2s, w i t h probabilities p", 2pq, q"-\ t and .i

:

basis that the r a n d o m variables Hi and H. reflect the slate o f t w o

are Ihe values thai each o f the r a n d o m variables H and H' mav lake

homologous loci taken at r a n d o m on A and /•; i.e.. they have a

(vaJues w h i c h , so Tar, are a r b i t r a r y } . A l o n g w i l h Fisher [,V. let us

p r o b a b i l i t y f o f being identical and

choose values which m i n i m i z e :VP.[d-); we obtain

F r o m this.

I —f

o f being

independent.

rf;-2,)4-fl(j-i-0

= 0,

p i i - 5 - 0 4 - ^ - 2 i ) - 0 ,

H

Cornİjl'^H

Rılaan

III J'I [•^.ı-ı./.aı i'titlrau.ıri

Rtt,ı"rt>

Pıpııhrion

2 i Ca'rtUsioni

by selling the p a r t i a l derivatives w i t h respect to t and 5 equal t u /.em. fffl

myiyJ

indin1

,,'ifi- Dominant,

= 3[*Rf/f,//.) 4- Mttim)

Ihns obtain the fixed values. / = pi 4- qj. s = pj -4 qk. which

satisfy the equation

Mfft—g l-nfti^ttd

= ft 4-

= 0 (because pr + q> = d j . Therefore,

¿5

4- arctorf,)]

4- B M 0 f t £ h

and everything goes hack t o the c o m p u l a t i o n o f Mföffök • o n

=

>

i

w -

f ^ ^ i -

=

°-

F u r t h e r m o r e , equations (2,1) indicate Lhal t h e meflu value o f i f

2.^.1 T h e T w o I n d i v i d u a l s A r c Related

is zero when t h e value o f H ( o r o f H ' ) is fixed. I f we set // equal

by O n l y O n e o f T h e i r L o c i

to 1, H' [ w h i c h is independent o f H because the individuals are n o t related} w i l l lake the values r or v w i l h probabilities /> or q. and d w i l l take ihe values o f i - 2i o r J - I — t, whose tnean value is equnl t o zero i n accordance w i t h (2.5.1). I t follows that M { d H ) = sm££f//') = 0. T h u s 1

u

then ¡1- depends only on Hi and becomes independent o f H<, M'<, İî

3

t

is i i x e d ; therefore, f o r any t u l a e o f H

Ml(ıltd:)

u

Nfrjfe) " ft«tifiA) eienl o f and pt is

4" ^ ( 2 * ) .

a H ( y ) = EZnTLf// ) + 1

Let

:

cient, which depends o n their degree o f relationship. I f we fix J i and 4s, a n d its mean value is equal to zero. T h e n 'Jti(r/j^) = 0 when

altCaC ) - im
*

I n this s i t u a t i o n , only t w o o f Ihe f o u r genie r a n d o m variables are not independent, e.g., Ji, and H \ let $ be their c o r r e l a t i o n coeffi­

m

jXMitify

= U, T h u s .

F r o m this, the correlation coeFh-

us take Tor t w o related individuals, h a n d /., the values

= $ 4 , 4- M

+ ¿1) +

and

=

ŞfjtÇgyi)

W.: 4- ft) 4 - - By

f

:

hypothesis. İl, and J/İ arc independent, as are H a n d fi- H :

u

anCyij

there­

= =

rf. 2(1 + V -h i ) ' J

fore, cannot be positively correlated at t h e same t i m e w i t h both

where T,- = S T C i r f a ^ î î ( t f ) , and t ' = s m f z ^ n K r y j . T h i s can also

H- a n d H'<. L e i us suppose, f o r example, that 11, is correlated w i t h

be w r i t t e n as r = [ [ > / 2 ) r M , V , where r- is the "genie additive

H- o n l y ; i n lhat case H\ can only be correlated w i t h H'-\ let us designate the respective c o r r e l a t i o n eoellieieiits by p ami <*>'. We have - B ( « X # ) f â ) 4-

TC[/M)

+

+ ani*^*)

:

variance." ZZNllT*),

and o- the " t o t a l

v a r i a n c e , " 3SBW(İr*Î +

1\WLIÜ~ )

+ ! H ( f Q , T o avoid having t o evaluate 0, we note that,

since

does not depend on dominance, this f o r m u l a can be w r i t t e n

;

as r = r„T .V. F being t h e " f u n d a m e n t a l c o r r e l a t i o n " previously T

4- iiTTtatfi) 4- M d - t f t i 4-

titytöttl.

(2.5.2)

B

defined. T h e r e f o r e , Tor individuals related by one locus o n l y ,

because terms such i K t i T t f o t f : ) , illiUif/-}. o r M f ö f t ) are equal t o zero,

dominance plays exactly t h e same role as chance a n d the e n v i r o n ­

since the r a n d o m variables they include are independent a n d have

ment i n reducing a l l the " f u n d a m e n t a l c o r r e l a t i o n s " by a fixed ratio

mean values equal l o zero,

which is less than u n i t y ; This f o r m u l a , in particular, gives, f o r simple

F u r t h e r m o r e . Ihe last four lerms o f (2.5.2) are also equal t o zero.

correlations i n direct o r collateral line o f descenl, r = ( l / 2 ) M / f f \

I f . f o r example, t h e value o f ti± is fixed. J . depends only on H' \

w i t h n = 1 f o r pa rent-offspring c o r r e l a t i o n , n = 2 f o r grandparent-

therefore, it İs independent o f H„ a n d W^ftA)

bat

grandson, half-sibs. o r ancle-nephew, ı> = i f o r first cousins, and

being equal

so o n . I t does not apply t o f u l l sibs o r l o d o u b l e lirst cousins, w h o

to zero when tf-± is fixed, is also equal to zero l o r any value o f H .

are related by two loci at the same time. We shall show {see S2.5.2»

The same is'true f o r the other three terms. Thus

that i n these cases, the reduction o f the f u n d a m e n t a l c o r r e l a t i o n is

:

- MH^M^),

we k n o w - M d ) = ü Therefore, the mean value o f ll,d-, :

r

:

20

C<"*r!.ı'i3t; tirlıı et" UfLılını

iti .JPJ lıv^t'isıt

P S'ıilim.ı-y V->pul\Hİw>

less i m p o r t a n t because there is a positive c o r r e l a t i o n . Ml{d d ) s

> 0,

:

2.5.2 I n d i v i d u a l s A r c R e l a t e d

helwecn Ihe residual dominance, d\ and J-., of Ihe Lwo i n d i v i d u a l s .

by T w o o f T h e i r L o c i

F i n a l l y , i f we w i s h to find the partial c o r r e l a t i o n between [ r a i l y in an i n d i v i d u a l , h. and i n one o f his ancestors. /•-, assuming ihe value o f ıha t İra İl as lived İn an inler mediate ancestor, h- w h o is separated f r o m i h e m by rj and p links, respectively, we

CÎLII

apply ihe

f o l l o w i n g Classical f o r m u l a i f the regressions are linear (as Ihey are when the r a n d o m v a r i a b l e s y are Gaussian and i n Gaussian r e l a t i o n ) :

Let rti = H, + H\ + di and

= H

late P u > t i W ) , k n o w i n g l h a l H

and H

t

:

+ ffj + d

lt

Let us calcu­

have a c o r r e l a t i o n coeffi­

:

cient tf. that H\ and H' ha\c a coefficient e> and that these two seis :

h

o f r a n d o m variables are independent o f each other. The generating f u n c t i o n V { x , y , u , r j o f a l l ihtSe f o u r functions V,{x,y)

and V4\u, r ) o f the l w o sets //, and

Hi and fc&

Let us recall that the generating f u n c t i o n o f r a n d o m (l/2Y^r"-/^

^(1

- ||g -

variables,

taking the respective values a, j i . and so o n , is, by d e f i n i t i o n , the

(.i/2)"( yo(|/2)'-(^) T

random

variables taken together is, therefore, Ihe product o f the generating

5)

expectation ofJf»>* . . . (instead or the characteristic f u n c t i o n w h i c h is the expectation o f i P f l f f i . ,

T h i s coefficient, in general positive, is n o l zero except i f r'fa-

Therefore,

•= \,

t h a t is, i f there is neither dominance nor influence o f the e n v i r o n ­

y&.y)

= pip

ment; it is only i n this case that, i f we k n o w the t r a i l in an ancestor

=

4- ftpty

-r pqi\ -

(P*' 4- qx^ipy

# X * 5 r + x y ) + q{q 4-

+ q y ) + pq[x' - x H y -

1

J

tp)xy

f%

o f It, similar i n f o r m a t i o n f r o m previous ancestors i n the same line o f descent w o u l d n o t give us any more i n f o r m a t i o n about h

(no

and Kjf«. r) may be expressed in terms o f ;

1

by replacing x w i t h u,

"ancestral i n h e r i t a n c e " ) . But there is almost always dominance or

y w i t h r, and 0 w i t h * ' We k n o w that the generating f u n c t i o n o f Ihe

influence o f the environment, and because o f this, knowledge about

t w o variables taken together, H\ + Hi and H- 4- Hi,

a t r a i l i n an ancestor allows a positive c o r r e l a t i o n a m o n g earlier

tained by setting x = u and y = r i n the p r o d u c t V\V-\ it is then

ancestors anil ihe descendants. T h i s " l a w o f ancestral i n h e r i l a n c e , "

W{x,y)

shown experimentally by G a l t o u ami I'earsou, is then n o l at all i n

W(x,

c o n t r a d i c t i o n fas Bateson

and

Weld o n believed] to the laws o f

M e n d e l . F r o m Mendel's laws i t f o l l o w s , indeed, t h a t i n

making

=

Hx.y)^x,

y i m tt^jpf^i

may be ob­

the coefficient P^

o f x'y*

in

y) representing, by d e f i n i t i o n , the p r o b a b i l i t y o f also having

Ht 4- HI - c and H 4- Hi = 3, a n d , therefore, o f x , and 3C having ;

;

deternaued values f{r,) and f { S ) . K n o w i n g J f enables us t o calculate

predictions about olYspring, knowledge o f the genetic c o n s t i t u t i o n

iiHijCj.^-] = I * W f c i / < # b y replacing ( i n W) x- by f { a ) and y* b y

o f one ancestor makes all knowledge about earlier ancestors

un­

f[H),

i m p o r t a n t . Our study, however, simply shows that knowledge

of

i.e., x-' and y" by t\ and so o n . Let us calculate, then, y ) = (px

1

4- qx-npy'

+

qyf

t r a i l v in a given ancestor, when there is dominance or environmental 4- pqfo

effects, provides insufficient i n f o r m a t i o n about its genetic c o n s t i t u ­

4-

4- qx-ftx-

-

x^py-

-f- qy'){y'

- y )

t i o n , and more precise i n f o r m a t i o n can be derived f r o m knowledge a h o u l earlier ancestors. by replacing x-' and obtain.

by f,

and f**

1

by j , x-' and y - by k, we :

i 1

ZD

i- • BtHıttH r ' . p p > J M / ı tfAmtu ı i r , i ; P* pjıf pj P'pHİH'—t

1

t i CmrUtfrv*

Bet*""

.lit

l*.iııı.ttjlı

*rtb

DtwM»,,

29

correlation i\. therefore, higher t h a n t h a t belween parent and off¬ spring when ihere is d o m i n a n c e ; another reason for (his higher

- 2J + i ) ' .

4

T h i s is a symmetric bilinear f o r m o f 0 a n d i n w h i c h Ihı: «ıcflicicnu arc well-del cr mined i n • given p o p u l a t i o n and are independent f r o m # and

I n lite M

way,

correlation is (hat (he effects o f e m i r o n n i e i i L o n t w o brothers cannot be regarded as independent double cousins.

i f they

are b r o u g h t up together. F o r

1 4 . and t = f f l . ' 4 ) r - 4 - ( < V 4 ) ) , V ; thtis

c u r r e l a l h m U higher than that belween uncle and nephew. T h e phenomenon L I | dominance is, thus, statistical!) expressed by

-

4- *')[pi

4-íw - í t í -

+

- 2J + U ' J r V -

correlation coef+icients w h i c h are higher for the double relationships than for the corresponding simple relationships. This higher correla­

Let i n calculate the coefficients by giving 0 and rp' spec i lie values.

tion decreases rapidly, however, as the relationship becomes

We have seen that, for 4,* = U, r is reduced 10 < # / 2 } r ' , r . We can

distant, because ihe product >b
1

more

write, therefore, 2.5.1 Various

r = [(*4-#V/l' i + *#V.V. ,

where A * = M y ' ) = s L L ^ f t f " ! 4- i
Ihe t o t a l v a r i ­

r

ance; r

1

- ZXrtty/FA. the genk>:*ddiLİvc variance; and 1- -

(he dominance variance. I f we set

-Md*l

I . .\\ and .V? become

•»

7

= =

smjjc-> áé£r(

s f i m C f P ) + tmíífil "

?

Zlpq[pi+{ti

and f are

bilinear symmetrical f o r m s o f * and o' whose coefficients are dcler'.' • T h e

I.

•' • c o u l d be extended t o the case where ihe elfecls o f

the different pairs o f genes o n the trails considered are n o l additive I genera I i A H ion of d o m i n a n c e ) [ 3 , 1 1

These calculations, inciden la My. lead l o , =

111 The results are valid i f there is multia|lch»m. because V,{x, y) is still a linear f o r m o r * , therefore, W(.\, y ) . Ml(X,X-A mined by selling 0' - (I, and then £ =

identical, and so We h a v e T ^ + E

Intensions

[ } ) T h e calculations could be m o d i f i e d to lake i n t o consideration

-

the resemblance between parents I h o m o g a m y ) ; the effecl o f d o i n g

p\}-qk\*

so is to increase all ihe correlations [ 3 , I I . 22. 1İ, 2 4 . . atid

{A) -

Comparing

this w i t h

the f o r m u l a •tn()' y .) l

2:ill((fa/J, we note that : i R ( 4 4 ) = » V

I f we separate the sexes in the statistical measurements o f Ihe

correlation, we Ihid, in general, dilferent results for each sex* because

2 / +

-r «'1-HI(//-) -4

:

- « ' . V í t ^ J ; the c o r r e l a t i o n

of the c o n t r i b u t i o n of sex-linked genes to the t r a i l considered, and the same calculation as i n fr2.4 could be applied |7, K;.

coefheient between i h e dominance eorupoijcnts if, and d- o f I, and is llfccreforc

w h i c h is the p r o d u c t o f the correlation coefficients

helwecn (he genie r a n d o m variables, l ( is / e r o i f i and /? arc related

2,6 C O R R E L A T I O N S BfiTWJiEiN INDIVIDUALS

WITH

ANY

DOMINANC1.

by only one locus, b u l positive i f h and / are related by t w o o f their

F o r t w o i n d i v i d u a l * , t, and t-, w i t h a eucnieieni o f inbreeding m i l

loci, and this results i n an increase o f (heir c o r r e l a t i o n . F o r example,

equal to * e r o , the calculation o f correlations is m u c h less simple

f o r brothers, * - 1 , 2 , * ' =

when there is dnminuncc. because the f o u r r a n d o m variables, ft,.

:

I 2 , a n d r = [ ( I . 2 ) r + («'/2)^/V. T h i s r

Hf,

"IH

Jl'u

Cjırrıijtianı

Bclrrnri

Kr/xttW

İB -"I Irtgtwaiti

SlıfirUrj

Populjlroi

'C't w i l l be related a m o n g tbcmscbes. It then b W W i

indis­

pensable t o determine the generating function o f all f o u r r a n d o m Chapter

variables, which, for a given type o f Relationship, eun only be done

3

step by step hy the f o l l o w i n g m e t h o d : given a g r o u p o f individuals. A, &

- • - , I», let us designate by / ^ - w . . .

the j o i n t p r o b a b i l i t y

that their In homologous loci are in the stales represented by o. a',

Evolution of a Mendelian Population

d, d', - - (each one o f these quantities having one o f the values i or t% The generating f u n c t i o n f o r the ~h\ loci w i l l then be: tffli.

" . h,. b, :

) = l$feift* - - - Ö r f İ f i f

w i t h the f o l l o w i n g properties. I f we bring together t w o groups o f individuals w i t h no correla­ t i o n , the functions <£ are m u l t i p l i e d . I f we disregard one o f the individuals, f o r example, l

ÎT

the gen­

erating f u n c t i o n f o r the remaining individuals can be deduced f r o m £ by setting o, = Hi = 1,

W e have discussed, thus far, only a stationary

Mendeliun

popu­

I f we add to the g r o u p an offspring, f . f r o m a couple o f the g r o u p ,

lation. i,e,„ a p o p u l a t i o n in which the frequency o f any given genes

for example, an offspring o f /( and A., the generating f u n c t i o n o f the

does not change f r o m one generation to the neM. a circumstance

group thus increased will include t w o more variables, related to E,

that can occur only if.the p o p u l a t i o n is very large and i f the different

say, A and h; according to Mendel's laws the generating f u n c t i o n

alleles do not give their carriers either an advantage or a disadvantage

w i l l then be

in the struggle for existence (i.e., all neutral genes). We shall now consider, first, a p o p u l a t i o n o f l i m i t e d size, and later, a p o p u l a t i o n I

2

i/^E^fl]/,, o-, W., M + i(«ı/ı,0* h,, bA)

in which there is selection o f genes. We shall see (hut in such p o p u ­ lations the frequency o f genes (JpeK m i l remain constant hut changes m the course o f time. We shall then have to answer two questions: Where does this e v o l u t i o n lead'? A t what rate does it take place?

W e can then proceed gradually f r o m the probabilities o f a given i n i t i a l g r o u p to the probabilities o f any g r o u p which was derived f r o m it by given malings. The calculations, however, are rarely simple.

3 1 INFLUENCE O F O N

N E U T R A L

P O P U L A T I O N

SIZE

G E N E S

3-1.1 C o n s t a n t P o p u l a t i o n Size Let us examine a p o p u l a t i o n made up o f a constant number o f individuals, A , reproducing by r a n d o m m a t i n g , and consider, first, genes whose m u t a t i o n

rate is negligible. Starting w i t h an

initial

generation, F^ we designate the successive generations, which we

32

Eflkiim

af -i Mi"<İı/tafi

i.!

Papulation

Inffitittu

\if Population

5 % sn Nculr
Jl

shall ussu mı: l o be no n overlapping, by F,. F», und so o n ; i f genera­

females. Since there is panmi\ia, the two homologous

tions overlap, and mating between different generations is possible,

i n d i v i d u a l , /„, o f /"„ are taken at r a n d o m , one f r o m i F , , . , . the other

computations

from

become more complicated, b u l the resulls are not

essentially modified. I n spiLe of r a n d o m mating, the individuals of the nib generation, F„, w i l l certainly present some consanguinity

T h e p r o b a b i l i t y that they come f r o m the same i n d i v i d u a l

o f O ^ - T or o f jF^-n is

tf'it 1

is sulTieienlly large, because each one will have at the most K distinct ancestors o f order ff, rather than the theoretical 2" ancestors.

loci o f a n

1

1

2 ^ 2

_L

I + 2^,2 1

1

1 N

We

could calculate the coefficient o f eoa uccstry o f an i n d i v i d u a l only i f

(by designating ,V as the h a r m o n i c

mean o f 1N

S

we knew all the chains o f relationship connecting his two parents,

1 [ 4 J V I ] 4- I.'i4,Vv).

i.e., i f we had complete records o f mating since the beginning.

the p r o b a b i l i t y t h a i they come f r o m different individuals o f F^.*. I n

w i l l be the p r o b a b i l i t y , evaluated a prion,

probability,

1 - {\/N).

is

the average

these two cases, the probabilities that the two loci are identical are

lly definition,/,,

( I + /,--)/2 and / „ _ ! (because /„-i represents the p r o b a b i l i t y that

shall see, however, that one can characterize ti priori coaneestry o f the " l h generation by a n u m b e r / , ,

We

T h e complementary

and 2N-. \/N =

t h a t the two

homologous

two homologous

loci taken f r o m t w o deferent individuals o f F__

T

loci o f an i n d i v i d u a l taken at random i n F, arc identical, i.e., they

are identical*. Therefore. / , . the p r o b a b i l i t y that the t w o loci o f / .

come f r o m die same locus o f a c o m m o n ancestor. In each experiment

are identical, is given by

conducted, the a posteriori

coefficient o f coaneestry w i l l depend on

the i n d i v i d u a l considered, but given a large number o f individuals, f„ w i l l approximate the mean value. Since the genes being considered are neutral, the Q priori

proba­

bilities o f the different alleles will be the same for all generations, that is. p and M.

Ipqtl

q. i f we assume two alleles o n l y . T h e

formalas

- A ) , and q\q 4 / - i ) w i l l represent the a

priori

F r o m this linear recurrence we can easily deduce./"„. First o f a l l . we return to a homogeneous recurrence by n o t i n g that the equation is verified f o r / , - constant = I and

probabilities o f the three genotypes for the m h generation, and also their frequencies, g o c i l enough experiments, in which we w o u l d always start with the same frequencies, p and q. for genes .-I and a. We shall c a l c u l a t e / , i n d i l l e r c n l cases, disregarding m u t a t i o n s . "

Ik = [ i - U W K - i

Dk'Udous

!miividihii}i.

Consider

first

an a n i m a l

population

w i t h separate sexes, made up oT constant numbers |V| o f males and N- o f females, f o r m i n g the snbpopulations

i f o f males and F o f r

• The atunv fntnmln* arc Nm Lİ "<ı Ilıt nsHuiniulon tlial tliere Is only riiaJnm ınkreiNİıns (connj.lent »alı rwnriiii'iJi IT fere n ut*o fyitcrnatk in^retLhns [see n. i.l), the funnului ITIH.V \V WHMM. L

= 1 —/ „

from

+

El/2*)"w

Here t i will be a linear c o m b i n a t i o n o l t w o solutions o f the f o r m -V", n

k being given bv the characteristic equallon k* -

A.

by letting

which

[I -

{}/X)]k

-

0,

Thus, =

\[(X -

l/N

-J- v ' l 4¬ 4- a [ f l -

1/iV - V I 4- l / A ) / 2 V ' ri

\ , and v being determined by the t w o initial values o f og and «].

r

31

I\\yİUttan of J M/rıJtJiıifi

a* = \ + u,

PefaÎıJltaH

oı = au[£l -

Let us note that ıh e

u

nn

1W2]

+ (>• -

rfyl

+

in which the three genotypes coexist indefinitely w i t h frequenciesp",

I/WZ

2ptt. i/ . B u i i t must be noted that the asymptotic homogeneity is 1

brother-sister mating, studied i n

reached extremely slowly i f ,V is large; f o r a . =

I — f„ to be reduced

detail by Haldanc and by Fisher, becomes a special case i n this

to one-tenth o f i l s value, a number, i\, o f generations is required such

f o r m u l a i f we let W = 2;

that e*p ( — ' I / 2 - V } = 0.1, and therefore tt = I N In I d ; to appreciably

a

„ =

d- V % P Î h 4- « 1

-

reduce the difference I —

VSj/4]-.

I f , V i s large, we have fcVj

the deviation f r o m

viduals i n the p o p u l a t i o n . These results have i m p o r t a n t biological

+ 1/np = [_(V] 4- l/JP X •

which measures

homogeneity, requires about as many generations as there are i n d i ­

- 1 -f-

consequences; several biologists have insisted o n Ihe role o f chance

4- ijiş,

in the e l i m i n a t i o n o f neutral genes. We observe, i n fact, i n many

a. 4- $ f 2 $

animal and plant species. Ihe divergence o f "geographical races." A 0, t h a t is, i f the

w h i c h , after having been separated by a barrier, such as u hodv o f

initial p o p u l a t i o n is n o t f o r m e d by identical homozygoles. The t e r m

water or a range o f mountains, evolve toward different homozygous

in a becomes rapidly negligible w i t h respect to the t e r m i n K, because

states, one having finally o n l y genotypes A A . the other having only

their ratio is equivalent to ( — 1/2A0'>.'X

titt. T h a t divergence could certainly be explained by disruptive selec­

Therefore X has f o r Us p r i n c i p a l part a,, i f n

W | f

t

have, then, as n starts

tion depending on the geographic situation, the gene A being advan­

to increase,

tageous i n one [oculion. the gene 0 in another. A s frequently happens m

= I - A ^ < * , ( l - WIN)*

-^aitr*" , 1

for neutral genes, however, it must be admitted that ihis evolution

and we are led to the f o l l o w i n g i m p o r t a n t c o n c l u s i o n : / , , tends t o w a r d I when rr tends toward infinity. T h u s , we tend asymptotically t o w a r d a p o p u l a t i o n i n w h i c h the t w o homologous loci o f each i n d i v i d u a l w o u l d have ihe p r o b a b i l i t y 1 o f being identical, and, therefore, a p o p u l a t i o n in which all die loci w o u l d be identical, made up o f identical hoiiiozvgotcs. Kor neulrul genes and w i t h no m illations, indefinite panmixia i n a l i m i l e d p o p u l a t i o n always leads to complete homogeneity. T h i s result, surprising at Irrst, stems f r o m Ihe fact that a gene can be eliminated when the r a n d o m drawing o f the 2N loci 1

o f the f o l l o w i n g generation happens to always favor Ihe same one o f the t w o alleles; on the other hand, a gene, once eliminated, never

results f r o m a small p o p u l a t i o n becoming homogeneous; this homogcnizaiion arises f r o m r a n d o m e l i m i n a t i o n , which i n its course eliminates sometimes one o f the t w o genes, sometimes Ihe other. T h i s explanation has at times been used i m p r o p e r l y ; i t musl be emphasized that r a n d o m e l i m i n a t i o n cannot lake place in such a short period o f time unless Ihe p o p u l a t i o n is scry small. Consider the blood group o f the American Indians, A l l these Indians seem to come f r o m ihe saute ancestors, in spite o f their m o r p h o l o g i c a l varia­ bility and l i m i l e d intermarriage w i t h immigrants f r o m Oceania and Melanesia. They are the only race i n the World to have exclusively only one b l o o d g r o u p , Ihe group O { 0 0 ) .

F

The blood groups A and 8, however, result f r o m extremely old

generation are certainly always constant and equal to p and r/, but

mutations, since they probably existed before the separation o f the

this n o w means that ihe final p o p u l a t i o n has the p r o b a b i l i t y p o f

lines o f chimpanzees and men, and must have always existed i n far

containing only A A ' s and the probability y o f containing only

i/u's.

eastern Asia, where the great human migrations probably o r i g i n a t e d .

Here is a large difference f r o m the case o f an unlimited p o p u l a t i o n ,

The different b l o o d groups seem to be w i t h o u t selective value, be-

reappears. The a priori

probabilities o f genes A and 0 i n the

a

36

EiolwlmU

af -> Mt"Jı/İ-"l

VaftUtün

cause ihc-y coexist i n Asia and i n Europe under all climates nid at all

B. Unnoerbm

fm#r0UOts.

Consider now a plant p o p u l a t i o n o f

latitudes- It seems, i h e n , that Indians derive f r o m a group o f Asiatic

A monoecious individuals, in w h i c h both sexes occur on the same

i m m i g r a n t s in w h i c h the genes A and

plant. Self-fertilization is now possible, but suppose 1 hat it is no more

R disappeared

by

random

1

e l i m i n a t i o n . T h e g r o u p must have developed rapidly, however, after

or less probable than cross-fertilization. The two homologous loci

its a r r i v a l i n the new w o r l d ; it could not have remained small t o r

o f an i n d i v i d u a l l„ o f /'„ then have the p r o b a b i l i t y \/N o f c o m i n g

more than a lew generations, after w h i c h the change i n gene f r e ­

f r o m the same i n d i v i d u a l o f F„.

quencies had to he very slow. T o reach homozygosity w i t h i n a few

p r o b a b i l i t y o f being identical is ( I - \ - a n d

generations, the g r o u p w o u l d have consisted of" only a very few

bility I - 1 N o f c o m i n g f r o m different individuals, in w h i c h ease

i n d i v i d u a l s T h e hypothesis o f r a n d o m e l i m i n a t i o n o f genes A and B

their c o n d i t i o n a l p r o b a b i l i t y o f being idenlicu] we denote by * , ; then.

i n w h i c h case their c o n d i t i o n a l

İT

have the p r o b a ­

in A m e r i c a thus leads us to consider that most o f the A m e r i c a n Indians

derive

genetically

from

a

small

number

of

Asiatics

& = [
itm*.

( M o n g o l o i d s ) , w h o came l i t A m e r i c a perhaps by crossing o^cr the Bering straits, and to c o n f i r m the thesis o f A m e r i c a n ethnographers, but n o t the thesis that the A m e r i c a n

Indian race resulted f r o m

h y b r i d i z a t i o n a m o n g M o n g o l o i d s . Australians, and Melanesians w h o

W e huve dap clad <j> as the p r o b a b i l i t y t h a t two homologous loci r

taken f r o m two dilTercnt individuals o f /•„ , are identical. Because o T p a n m i t i a . *„ = A-t', therefore

came at different times by sea. I m m i g r a t i o n f r o m Melanesia has had A

obvious influence only on very isolated regions, such as the S i r i o n o area { t h e v i r g i n Forest o f A m a z o n i a ) . Our hypothesis is f u r t h e r cor­ r o b o r a t e d by the observation that, a l t h o u g h the A"A and ftfN

=

l / 2 j V 4- ( I -

\?lN)f*-u

from which

blood

groups o f the MA/-:Y/.V-;YA series occur i[Uite frequently a m o n g all

= ! - / .

= £!-

l / 2 / V l a , . , = <^i-

l/2N)\

races, they hardly ever occur a m o n g Indians, Certainly, there are many other genes for w h i c h the A m e r i c a n Indian p o p u l a t i o n includes

Thus

hctcrozyjioles. but these genes may have originated, f o r the most p a r i ,

o f indefinite self-fertilization is obtained

f r o m mutations w h i c h occurred after the occupation o f A m e r i c a .

decreases by h a l f i n each generation, and almost complete homozy­

T h e slowness o f r a n d o m e l i m i n a t i o n o f genes i n a p o p u l a t i o n w h i c h

gosity is reached quite rapidly. Repeated self-fertilization o f a plant

numbers even a few hundred individuals is confirmed by the example

species is a rapid procedure f o r obtaining a line homozygous

o f the Gypsies, nomads w h o came f r o m India i n t o Europe more than

almost all factors. But i f rV is large, homogeneity is established very

a thousand years ago. and w h o have conserved r e m a r k a b l y

the

slowly;

still tends t o w a r d zero, and j„

t o w a r d I . T h e classical case for A'

is then o f the order o f e*p {- n/2tf).

I ; 1 — /„

then

for

as w i t h dioecious

I Inula type, because they m a r r y almost exclusively a m o n g t h e m ­

individuals. T h e occurrence o f both sexes o n the same plant modifies

selves. T h e few thousand individuals that these isolated p o p u l a t i o n s

the e v o l u t i o n o f the p o p u l a t i o n to an insignificant extent, provided

number i n Germany and in France have conserved the same f r e ­

sell-fertilization is not favored more than cross-fertilization, since

quency o f b l o o d groups as the H i n d u s , in spite o f the thousand-year

we have already shown that exclusive ielf-fertdization rapidly leads

separation, being 40 per cent 2?, the highest p r o p o r t i o n in the w o r l d .

to homogeneity.

VJ

3 . 1 . 2 P o p u l a r ı n n Size N u t

"n:

Constant

Suppose n o * thai p o p u l a t i o n size is not constant hut varies over the course o f l i m e . Consider [he ease in which the sexes are separate. The numbers ,V und A' will be functions o f the order, i . or the t

generation f..

a

Let us sel 1/4AT, + 1/47^ =

I/W).

= [ l . ^ . V W ] [ F - 2 u . , ) / t l - «„+«)] <

(1) I f

i/2mt

M

Tbereiore M

A + J

N{n)

13

Influcurc of PcpMİMıcu Size m A W F V T / Giun

— > - 0 if ,V0JJ —>- ^ .

remains

It follows t h a t : It—i-

finite when

Mm log k« < U ; therefore log

q

c.

Ilm k

tl

= — c t . f, = [). and /

<

I,

mid

tends to¬

ward I .

The f o r m u l a

(2)

I f N(n) tends t o w a r d infinity along w i t h n, fr„ = 1 —

—*

I,

and log * , = log (1 — u ) — ^ 0. T o study the series w i t h general n

j({ = [(1 + A _ . V 2 A ' J - r - ( l -

terms log k„, lei us note t h a i

holds, p r o v i d i n g we substitute for ft the value N(ti - 2). F r o m this

h

tin

by increasing the indices by t w o to s i m p l i f y the f o r m u l a , we deduce

-

(i

-

* )/mo, n

where - [1 -

tfflt*8*»*

4-

aJ2N(n). ΄ =

— 0 .

J΄ L/Î1 + T

W e have this time a linear homogeneous recurrence w i t h variable coefficients. W e shall solve it by setting ft = W i - - - k„. the

M

Therefore

being constants to be determined, w h i c h are related as f o l l o w s : ItaJta

= 0 - WHSSSk^

+

and the series logA\,_- = lug ( t — u ^-) converges i f the series

(CRN*

n

u„

i:

converges.

that is.

(3) I f A'fa) increases al the most as a linear f u n c t i o n o f n, the t ^ = [ l - l / ^ J ] + l / î 2 ^ ^ , j ,

scries diverges, and f—t-

I.

C.4> I f N(it) increases al least as n'" "* (k > 0), ihe series converges, 1

w h i c h enables us to calculate gradually the *,s. starting w i t h A, > 0. Then

k,+,

>

0:

for

Jc . < 1, because k ,. n

3

K

n ?

0,

ft,+,

>

I -

I / , V ( M ) ; and,

< 1 is equivalent to A

h l l

>

for

w J

1,

1/2. T h e r e f o r e

lim/<

I. and ihere is no complete disappearance o f hcicro/ygotcs.

The same results w o u l d obtain f o r monoecious plants.

the values taken by « . are positive and decreasing, and, when u tends toward infinity, tend t o w a t d a l i m i t , tr İ 0 ; and / therefore tends t o w a r d the l i m i t {1 - a) ^ 1.

3.1.3 T h e R u l e o f M u t a r i o n s It is obvious that Ihe genelie heterogeneity o f a p o p u l a t i o n , i.e.,

T o m a J t e l i m / = (1 - n) < I , that is. for the hetero/ygotes never

the presence o f numerous

heterozygotes. does not usually result

to be completely eliminated, it is necessary and sufficient that the

f r o m Ihe fact t h a i the p o p u l a t i o n is extremely large, but f r o m new

scries log <, = log fc, + l o g k, + - . , + l o g k„ -f . . - converge. F o r

genes appearing f r o m time to t i m e , either by m u t a t i o n or by migra­

this A-„ must lend toward I . which necessitates, by the recurrence

t i o n o f individuals f r o m a different p o p u l a t i o n . Let a, be the mean

f o r m u l a , ihut both ,V(fj> and n be i n f i n i t e ; this last c o n d i t i o n sullices

frequency o f m u l a l i o u per generation

for jfc — h 1, because, by letting k„,- = 1 -

w, the mean frequency o f migrants per generation, these migrants

tt

I fN(>i),

L

the r e e u r r e i i L L - may

be w r i t t e n

where Ü < n „ - < t

Tor a specified locus,

and

c o m i n g f r o m a p o p u l a t i o n large enough t h a i we can assume Ihere

•I

! i :'\.:ı. •! of ,i Mt*J*ti*M

1

J; 2 I'tilticKt

"apkiillltl

is no relationship a m o n g t h t m . Lei U S sel u = u\ + bility that a locus, - 4

M

T h e proba­

comes f r o m I n o n m u l a l e d locos o f an " i n ­

digenous." i n d i v i d u a l (i.e., a n n n m i g r a n l ) o f ihc preceding generation it

I — Ut — u- "

I — u.

A s y result. In Ihc ease o f

oj Stltithn

41

The e q u i l i b r i u m value is reached more rapidly when there is mulat i o n or migration and much more rapidly i f 4 A ' n is large. I n monoe­ cious plants, we find again the same results. I n s u m m a r y , we note thai I he l o c l l i c i c n l o f eoanccslry /., (ends

dioecious

always a w a r d a ünite l i m i t / . I T it is equal l o I . the p op u lati on w i l l

individuals, / , = U -

")'[<• + /- >/2iV + ( I

almost certainly become genetically homozygous,

-

s

C. i representing the p r o b a b i l i t y that tw<« h v i taken f r o m t w o differ­

in the linal p o p u l a t i o n , the u priori

ent indigenous individuals o f F„ j arc i d e n t i c a l But Ihc cocllicictfl

taken

is e v i d e n l l y ( I - ¡1)*$,.

o f e o a n c e ^ r y . / , . . ] , o f an i n d i v i d u a l o f ,

given snliicicnl

time, f r i t is different f r o m I . usually sonic heierozygoles w i l l persist at

random

2pt/\\ - / J .

p r o b a b i l i t y that an i n d i v i d u a l

f r o m this p op u lati on

is heterozygous

I f ihc population size cannot

being

be taken as increasing

i n d e f i n i t e l y . / - 1/(1 4/ 4\aJ ik considerably less than I . p r o v i d e d

F r o m this we deduce

that 4,\'u is not small, or that the frequency a o f m u t a t i o n

and

migration per generation is o n the order of 1. A , at least, or that the 1

Since we can assume n

to he negligible, the e q u i l i b r i u m value

:

tolal number. 2;Vu. o f new genes introduced in each generation by

of / . i i

m u t a t i o n or m i g r a t i o n is one or more. Whatever ihc

(1 - 4 * X 1 + / J / 2 J V + CI - 2 - X I - l / W -

might be, i f m u t a t i o n or m i g r a t i o n affect some individuals in each

- ,

p j g f

generation, a considerable number

p op u lati on

o f heterozygnies may persist.

T o see h o w / l e n d s t o w a r d Ihis l i m i t , let us sel a . = f — f-- W e t

have

yl n * = ( I — 4 ı ı h F , _ / 2 j V + ( l - 2i*XL r

rtfh^i.

H

(I -

2.^1

-

MW)*

-

(I -

4rO/2A/ =

u.

factors having only t w o alleles. A and ti. Let us designate by p and i f , p probabiliiıes p -\-fipand q \ bq (flr/ tion,

The

- (I -

SELECTION

hffl

-

\/N)±

v (1 -

2ıı)V

-

I W

+ 21.1

-

are J i f f i«f the t w o r o n l s is given by:

2k - I - 2it - l / V + f l - m - 2 / t f + b f . A + 2 / A T - Bıı/.V)''* f

1

q. the probabilities o f A

and ,i in the adult breeding individuals o f ihe generation F„. The

Therefore 2*

Oh

IA-I U S study the d i s t r i b u t i o n , o t c r the course o f time, o f a pair o f

This equation has [ w o sohılions. <». - fc» * being, a r o o l o f k* -

INFLUENCE

&p) in the f o l l o w i n g genera­

w i l l be. in general, different f r o m p a u d y . The change Ay

results f r o m several causes, each producing a small change (such thai their squares and prodnciv w o u l d be negligible). A.

'•' • .

Because o f recurrent and reversible mutations,

there is. in each generalimi, in the reproductive cells o f F> a mean t = I - 2a-

l / 2 , V 4 - 0 ( ü > + Ü(l. Af J. ,

,

p r o p o r t i o n , u,, o f a genes transformed to . 4 . and another p r o p o r ­ t i o n , r i . or" .4 genes t r a n s f o r m e d lo
TİKrefore

a* Ihe average frequency o f a in Ihe repr*Kİı*eti*e cells; the change o f q produced by m u t a t i o n is a linear f u n c t i o n o f q.

42

uf .1 MttíJttiM

Euoialien

B . Migration.

S.2 influence af Seia'tien

Pupfhl'ot

We must lake m i g r a t i o n i n t o account as soon as

we consider a local p o p u l a t i o n instead o f all the individuals o f a

43

the F„ generation, because o f m i g r a t i o n , c a r r y the gene u w i t h the frequency 5 , = q + S ij = q — uq + t [ l - q), t

species, since a local p o p u l a t i o n is almost never completely isolated: it always exchanges individuals w i t h [he neighboring p o p u l a t i o n s .

T.

Gametic

Sett-cllon.

Assume that the gametes

produced

by

I t f o l l o w s t h a t , i f w h a t we .called F„ designates all the indigenous

these reproductive cells do not c a r r y the gene i? w i t h a p r o b a b i l i t y q

individuals h o r n at a certain place, the breeding p o p u l a t i o n

any longer, but w i t h a different p r o b a b i l i t y . q ,

will

t

because this gene

:

differ f r o m F,.\ it w i l l be f o r m e d , on the average, o f only a f r a c t i o n ,

presents an advantage

or a disadvantage

for the gametes

which

I — k, o f F* individuals, the r e m a i n i n g Traction, k, consisting o f

carry it (gametic selection); assume, also, that the probabilities o í

m i g r a n t individuals. I f we assume (hat these individuals come f r o m

the two genes, instead o f being q¡ and

a group o f populations whose c o m p o s i t i o n can be considered c o n ­

I * - f l U

stant over the course o f time and characterized by a frequency, q,„,

designated by I - 1. T h i s ratio characterizes the degree o f v i a b i l i t y

o f the gene u, the mean frequency o f ti i n the breeding p o p u l a t i o n

o f the gametes, i.e., the intensity o f ''gametic selection" fwe can, i f

w i l l be

necessary, consider g positive, by calling a the unfavorable genej.

I - q

are $ = q¡

u

and

a

~ ? I ) H <* n d d having a constant r a t i o close to 1, a

Since we must have (1 - k)q -3- kq„ - q f

k{q

-

m

q).

™vi -h

T h e change i n q caused by m i g r a t i o n is, therefore, a linear func­ t i o n o f y, as w i t h m u t a t i o n s . It can be w r i t t e n i n the same f o r m , —u q

-f- i-L-t 1 - q),

ti- =

— q ).

r

by setting kq

n

= >- and

k = ¡1, + r ,

T h e change produced j o i n d y

m

:

—

by m u t a t i o n

-

ß{\

and

l/d

+

ui + ii; =

-

- q ) - M * 4- J ' J U - q ) =

-i>q

-h i f I - ' / ) .

a

/a

= 1 - s,

we have

= 1-

= 1, iq , t

and

by setting u =

with

that is.

m i g r a t i o n is then ttiq =

I - qit = 1,

* < r V l - r + sift + Qfi'J. lit + k [ l - q ) m

and

r = KM -h

Î'J

= f'ı

+

Let us pulq-

= q a

t

= q¡ |- ^ . T h e n f i ^ = {a -

l)q,

= -.\q (] t

-

q,).

This change is therefore a linear f u n c t i o n o f q. The inclusion o f both m u t a t i o n and m i g r a t i o n ctTeci* i n the same

D.

Ctmswi<
Let us assume pure consanguinity, because

f o r m u l a is based on a simplified, gross model w h i c h assumes that

6f W'hieh each gamete c o n t r i b u t i n g to r e p r o d u c t i o n , whatever its

the migrants come f r o m outside populations whose

c o n s t i t u t i o n may be, has the same p r o b a b i l i t y o f u n i t i n g w i t h another

remains constant

in

time.

composition

In reality, these populations

undergo

gamete; the eventual consanguinity, however, w i l l increase the p r o b ­ ability that the other gamete carries the same gene as the first one

r

be studied, then, is the e v o l u t i o n o f a g r o u p o f populations inter­

I f w e c a l l / the average coefficient o f inbreeding o f generation F

,

acting w i t h each other bv m i g r a t i o n (sec |3.3).

the gametes that unite to f o r m the individuals b o r n , o r " z y g o t e s "

e v o l u t i o n , and are themselves affected by m i g r a t i o n . Whar

should

For the time being we shall assume that the reproductive cells o f

o f generation F^.¡.

B + I

have a m o n g themselves o n the average a condi-

44

Eıelidı**

wf a Mt*JcItJt

J.Z Iwfimtx, wf idnftem

Ptfuhum

4J

t i o m d ' c o r r e l a t i o n coefficient L ami each o f Lbem carries A or a w i t h i h c probabilities p and q-\ the i h r c c a y g o t e i AA, An. aa. have i h e n

= *p-q [(2k-

:

i

I KI - r t y + W - r - A j + i X V h

ill- probabilities we can replace q< by q and jfc by P - HP-^^I 5. Zygotic

1Q = 2 p * ( l " Selection.

'J-

* =

/p-h

Ti - flı " I " DU); * " L

Assume thaL Ihe ihrcc zygotes do not have

T h e three quantities r.

T

+• * r f = ?i( I - f l X l +

and - R. r

and -, have constant ratios close to ] :

(disregarding the second-order terms in * and c ) . w i t h

- m | — f: = I - ha T h e t w o conslunls o and / i e h a r a t l e n z e the p r M

degree o f v i a b i l i t y o f the z y g o t e s or Ihe i n t e n t l y o i " z y g o t i c selec­ t i o n . " T h e helcrozvgoies wilt he i n l c r m c d i u l c i n viability between ihe t w o honnuvgvHes i f 0 < h < I : ihey w i l l be superior l o b o t h homn/ygiHes i f A < O f a n d inferior to b o t h i f A > I ) and i f * > 0 , and vice versa \( a < 0. We must have t

W ~

4A4rM

and

<Ö*- W

-

r>

I f i . o and h i have the same sign. I w i l l also have Ihe same s i g n ; t w i l l be called d i e coefficient o f l o t j l ^election. These I w o selections together produce, therefore, a change w h i c h is a third-degree func­ tion in q, becoming equal \<> / e m fur q

0 and q = I . I n fact, the

İq = I ) . A n essential difference exists between Ihe above change and

I.

l

-

Î - - I - / . - M I - I ) ,

selection ceases to operate when J is eliminated (q = 0 ) or is fixed

p+2 iQ+- R= t

then that the total change. S.q and &,q due

to gametic and zygotic selection, is o f the f o r m

the same p r o b a b i l i t ) of" developing and reaching the ad aiL rcprodue* live Mage and that the probabilities o f the adult* are rP. lv.Q

w h i c h d i f f e r by just Off*) [since

lu\Q + P\ -

1,

f r o m whieh

the change produced by mutations

or m i g r a t i o n . T h e latter is a

first-degree f u n c t i o n , and d o c t not become equal to zero at the limits because it is always affected by a gene even i f it is lived or eliminated.

> - I ( I - oH - 2 A o p ) - I +

ffff-

T h e f u n c t i o n is reduced to the second degree i f w = 0, i.e., i f

lAoG + Of^). v -

0 {selection cuthisivcly gametic), or i f h = 1/2 ( h c t c r o i y g o l c

"Hie p r o b a b i l i t y q- + o.q o f « i n Ihe adult breeding i n d i v i d u a l s o f

exactly intermediate f r o m Ihe point o f view o f v i a b i l i t y ) , or i f X =

F p * i is thexefore

( p o p u l a t i o n consisting o f homozygoles, cschısivcly).

I

W e can replace q, by q i n ihe f o r m u l a i f Ihe second-order terms = - ifc -

- M Q

4- (1 -

and by grouping the terms in Q and

riftv

in i i , r, i , and a are disregarded. We then o b t a i n f o r the t o t a l change.

-Iff] - irfl -

= Arf + fitf + i j t f . i n one generation, a # i a

and reducing the denominator

10 1,

Iq

-

-iiq

^

+ r ( l - q\ + tfl - qXl -h » V ) . v ' »————* mutation selection J

and m i g r a t i o n * Thb i i W i ' j - K J j / LOncLnkm n n - l l k m . *hh.h n ı u ı h zero in ease of random niaiing, n noı ılıt HMV as i l v u prıtm LDritkaml " i J.VI. ~:M.-. uns larger Hun zero.

T h i s llurd-degrce pvilynimii.il we w i l l call ifql

l i s eocilîeients arc 10

'

*ma|| ihat their p r o d u c t s m i ^ r c * can be disregarded." I i goes

1,2

fl trnrt */ StlmSıan

T b e c u n e rises f r o m ıhe p*ıint ( 0 , — ^ ) l o ıhe p o i n l f l ,

j ;

because

i b e ehanges in t h e prubukfir)) <-<\u'. 3* well as i n ils frequency i n a very large p o p u l a t i o n . caused by mutation, m i g r a t i o n , and selection.

3 . 2 . 1 T h e Case o f a Very larr* Papular i o n T h e difference- between tic j i i * i b i l i l y and i h e frequency o f a gene i n a very l a r g e popuLn^:- Cifligihlc T h e frequency q varies frinu

one g e n e r a t i o n to t h c t U b j a quantity Hq). supposed to be

s m a l l ; q is a f u n c t i o n o f M R figured in generations. w h i * e finite difference is the f u n c t i o n a l Iktategration o f if*/) is approximately reduced [n the f o l l o w i n g quadratic form!

% l heing Ihe inılı.ıl valuc in ıfe rokialion J -

a

We shall p r o e e c d l o t-bun ıhe ümit of 9 when i — ^

-f«

by a

graphic dîscuMİon (ıtsymptutif düttibution o f the genes). Let us assume [hat the rıılcs of equa| to z e m . W e

muutuMi

anıl m i g r a t i o n .

and r, are n o t

n o l e ihtn t u l ılıe dıird-degrce p o l y n o m i a l Mq)

equal l o r > 0 l o r q = 0 Aİ10 - M < 0 for q •

I , İt doev not

Teıhıee l o zero Tor q - 0 o r i f i = I- and i l allı-w - c i l h e r

ur

ihree mlersections bclwcerti>ifci I . T o h e nHire speeifıe. ( A ) Let us assume that curve C meets the straight line D i n only une p o i n t , Q, o f the abscissa, q. Since Jty) Let us represem İn a plmriy- rUhe slraiuhl line />, generuted by Yt •= r + n v , and t h e cuoc l,?Liı^aied by

0. an initial frequency,

| h that was equal l o q w o u l d remain constant t h r o u g h the genera­ tions S t a t i o n a r y f r e q u e n c y ! I n Ihe general case y, - Jfc Iherefuce. M.tf \ > ti it'q < q and < 0 i(q > r j : fl.y) is, therefore, always opposite in n g n Lo q — q. T h e difference q - q =^ r decreases eiKistanily i n absolute value f r o m \\t initial value, r. - y„

' We cnıau. o f rour«=, formdün Wi *ıUigj[ rıukuın ıhan aunruvın>uLii'iı%. tıui t hi; c> prensi nıı nhiulnrd " '• • • •• • 1 m ııtl *pıxuı| cauı* as iMHlf '• lellıııl !•• ••• •fr>Tcnun 'it 1. M u, ı.. ı.. n.: ., • |n" »<mU n«ri İv iwuliuiKe, HL! L K I L Î T - I I »ınıkl octertfrcLcs* t v vmpWml. hnauxr ıhne ikiıuld İn; «ıl> umfa-i.ı^preKnt1

:

to see i f it tends

t o w a i d zero, and at what rate, let us study the u u o t i e u l

which q-q

is a p o l y n o m i a l o f at most (he second degree. positive, and never equal l o zero. Lei us call m > 0 its m i n i m u m i n Ihe range o f values

4^

Hipltiîiiul ./ 'à MflWWfjUi

Population

•

small thaï Lheir products and squares C M be disregarded.* U give*

i.i

JV'J..--..' . ' <<:t

..• •

-47

The curve rises f m m ihe point ( 0 , — x ) Lu the p o i n l ( l , + * J, because

the change* in the p r o b a b i l i t y y o f a. as well as i n ils frequency i n a verv large p o p u l a t i o n , caused by n i u t a i i n n . m i g r a t i o n , and selection. 5,2,1

T h e Case a l a V e r y L a r g e P o p u l a t i o n

I t crosses t l w ,v-axis ai the point

=

'

(Sec Figure 5.)

The différente between Lbe probahility and Ihe frequency o f a gene i n a very large p o p u l a t i o n is negligible. The frequency q varies f r o m one generation t o the next by a quantity Htf), supposed l o be small; q is a f u n c t i o n o f t i m e , measured in generations, whose finite difference is the function o(q). The integration of % ) is approximately reduced l i t the f o l l o w i n g quadratic f o r m ;

?t

m

-

1

- L

m

q,. being the initial value in the generation 1 = 0. We shall proceed to o b t a i n Ihe limit o r q when r — *

+T.

by a

graphic discussion (asymptotic d i s t r i b u t i o n o f the genes). Lei us assume ( l u i Ihe rates o f m u i a r i o n and m i g r a t i o n , u ami r, are m>t equal 10 / c m

We note then that the Ihird-dcprcc p o l y n o m i a l Mq)

is equal to r > fl f o r q - fl and l o — u < 0 for q reduce l o zero for q

fl

I . I l does not

or for q = 1 and ¡1 allows either one nr T

I'm

three intersections between Ik and 1. T o be more specific,

/ V ' -

- t r M H

"

Lei us represent m a plane tq. r ) the straight hue il, generated by

[A» L e i us assume that curve C" meets the straight line Î) in only q. Mq)

. L

\ -q

q

Q < q < *

I.

' v\ toukl, nf course, tormulau: a ^ i v\nh,>ui muLing th™- JiviuviniAlK'iiï. l u i (IL . >i h . otHaiiml «oukl he unn^tn»yL'uHL- i u i | H m w h vinaul H W P a* KVK VEUVI» ni lethal f i r m * h> I E A M C T E l l ' , mini.: uu iv luimphli; 4iml « - I ; m> wuukl tm \\- ttug\if-i\An, K J I iht duTimb Miukl nricnhdc» ta MmrlaiiO. befauu' U L L T V W i w k J bt txilr t * o gcnmvpti pHHHt t

V

one p o i n t , Q, o f the abscissa, q. Since &{q) - 0, an initial frequency, that was equal l o q w o u l d remain constant t h r o u g h the genera-

tion', {stationary frequency!. I n Ihe general cane J'I

i'i = 1 •• »q, initl ihe curve C. generated by „m

O U IL

t'j, therefore,

> fl i f q - q and < fl i f q > q: Hq) is, therefore, always opposite

in sign i n q - q. The difference q - q - r decrease* constantly i n oh\olule value f r o m ils i n i i i a l value, r, = q,

q: to *c< i( it tends

t o w a r d zero, and at w h a t rule, |eL us study the q u o t i e n t

'''

i s a p u l y i i o m i a l n f al mosl the second degree,

and never

•

,u-.

which

equal t o zero, l.el lis call rn > 0 its m i n i m u m i n Ihe range o f lulues

lEfàltttioti tıj ¡1 Sicultliaii

fl&

• 1 . 7 infint'ist

L'afitihrt ttm

uf StUttiw

4?

Uiken by q, i.e., between q„ and q; i f q , and consequently y, is sulTi-

zontaL w i t h ordinate t (coefficient o f t o t a l selection); / < 0 i f the

cıendy close to q, one could write essentially 5(q] = a'tq){q

gene o is selected against; and q tends t o w a r d the asymptotic \alue q,

u

and. Lhu.s, IBke approximately ,"r = b'(jj).

Thus, w i t h Ar = /

— q). — ;•

designating the change in r f r o m one generation to the nest, we have —y-

> m

"~â

T

^

A

!'1

<

-

"

r r

w h i c h is lower Hum q< =

b{q)

!'

'

• Let us calculate q. We have +- (t - " - '<)q + >\

m -lq~

the roots of w h i c h are |/| - \r\ + A f t e r fj generations. |r| < ( 1 -

< ( 1 - w)|r|. fli)" !^!; 1

-1

therefore, r = q — q tends

+

M +

•? ±

vfy

-

u — ?)* +

4f /

-If

l o w a r d zero at least as fast as ( I — m j " does. T i l e stationary f r e ­ quency q = q. considered earlier, is stable, and any other frequency

Since öfl) < U, t h e r e f o r e ^ , w h i c h lies between 0 and I . is the smallest

tends asymptotically toward i t . the deviation I' = q. — q being m u l t i ­

r o o t ; the other r o u t , q is obtained by t a k i n g the positive value o f

plied after a generations by a q u a n t i t y certainly less than ( 1 — WJJ".

the radical, and we have

There are two important specific cases. < 1 > In the first one there is no selection; m u t a t i o n s and m i g r a t i o n

S(q)

=

- K l -q)0f-

%)-

act a l o n e ; iv = f = 0 ; and D coincides w i t h the .v-axis. The asymp­ Therefore we w i l l lake Tor m. the m i n i m u m o f

totic value, q, is equal to

^ \ the m i n i m u m q-q

o f ~ K q — q.}> Which is the smallest o f the two quantities and - r ( f l - qô} Hq)

- -\iq + r ( l - q) = - { j + g g -

Therefore. B) •

g

— flts reduced i n n generations to a quan­

tity less than ( I — u — lift. T h i s reduction is not significant unless u is o n the order o f

- q)

r

I n the specific and usual ease where a and r (reduced to Ihe m u t a t i o n rate w i t h o u t any m i g r a t i o n } are small compared w i t h the coefficient o f t o t a l selection, I, the roots are given by

' ; i f a and I'- are reduced to the rate o f if 4- (•

m u t a t i o n , w h i c h is extremely l o w ( o n the order o f 10 ) , ij does not l

noticeably approach the asymptotic value unless the number n o f generations is on the order o f IÜ\ l i w i l l be almost impossible to

which is équivalent to ( 3 / 2 ) ^ 1 ± ^ 1 + y ^ ] i thererore q

observe a p o p u l a t i o n lhat became stationary under the action o f

^ ^

mutations alone. M o r e o v e r , the irregularity i n the rate o f m u t a t i o n s ,

éliminâtes almost eompletely the unfavorahle gene a; ils complète

as well as i n the rate o f m i g r a t i o n , restricts the validity o f the f o r m u l a ,

disappearanec

hut in practice selection usually plays the p m i c i p a l role.

q

(2)

I n the second case there is gametic selection only, w i t h helero-

zygotes being exactly intermediate i n v i a b i l i t y ; Mr = 0 ; D is hori-

a

1 + r / f , and the asymptolic value q = —r/l

-r/f,

is s m a l l . Sélection

is prevenled by the m u t a t i o n rate, r. ulone. Unies*

is n o t close to q, i.e., close to 1, m is on the order o f - f, and

w o u l d not equal u +- >' unlcss there was sélection; the asymptotic value is, tbereforc, reaehed mi.eh more l a p i d l y .

511

therefore, to an unstable whether q

n

stationary stale, w h i c h , depending

is smaller or larger than q . :

on

lends toward the stable

Stationary values iy, o r i f r . (C-) Lei us study directly any type o f selection, when m u t a t i o n s and m i g r a t i o n are negligible, i . e . ti = r = 0 ; this case docs not come directly under the preceding presentation, because under these c o n ditions, curve C degenerates. W e have °(q) = tfO — q)U + tvq) = n f l U — tfXtf -

w

f n could he inside or outside the internal 0 values are q = 0, q = 1 and

0

=

-*/"'

. . I ) . The stationary

I or o < 0 , tXq) has a constant sign; if, f o r example,

0 ) ir-r > in

l E l

q=aifQ
r

(BJ The curve c can be met by a straight line, such us

i

the sign is negative, q always decreases;

-a{q)iq

has a positive

three points (act Figure h), when C" bus t w o rent tangents w i t h the

m i n i m u m nr. We deduce f r o m this that q lends toward zero faster

same slope, a-, as D

than 1 1 — m)

h

and when D, fulls heiwccn these two tangents.

T i l e tangents parallel t o D

then, have their points o f contact given

u

by fy~_

"b ^

* i an equation basing t w o real solotions, q, be-

=

tween 0 and I i f W is greater than the m i n i m u m . ( u

l ,J

+ r' )\ o f the 2

first member between 0 and I . If, moreover. I is w i t h i n the interval !i . . . Kg)

m

frequency was [ i f there were a very low rate o f m u t a t i o n , u w o u l d persist w i t h a l o w frequency, as happens w i t h gametic selection). I f f f a ) is positive, q — • - 1 and gene a is fixed Whatever its i n i t i a l frequency was. [2)

f r o m the ordinates t o the o r i g i n o f the tangents, die equation

= f) w i l l have three solutions heiwccn U and

[. i n order o f

does, and so gene it is eliminated whatever its i n i t i a l

I f 0 < o < 1, t w o cases must be distinguished;

(a) I f u' > (1, f>\q) always has the same sign as q — a. The change in q, and therefore i n q — ^

has the same sign as q - a ; q - a i n -

magnitude £/•, if-.-, q>. Lach o f these sulucs results i n a stationary

creases i n absolute value f r o m ils initial value o f q„ - c. As pre-

d i s t r i b u t i o n t h a t is maintained indefinitely, but ¡1" we start wills a

viously, we note that q tends t o w a r d zero i f 41, < a, and q tends

different value o f y,, Figure b shows t h a t :

t o w a r d 1 i f ^ i i > I T . One o f the genes is still eliminated, but this time

( 1 1 I f t/,. <

a{q) = n —

is opposite i n sign t o q — ^ L ; the

which gene is eliminated depends u n the initial frequency,

difference r = q - 5 , decreases i n absolute value f r o m its initial one, fit = M — Wt', i f we take tn > 0 as the m i n i m u m o f — I n the inteim

q-q

val q . , .iyi the difference 1 is still reduced after n generations by' a a

L

r

q u a n t i t y less than f l — m ) . and q tends t o w a r d the asymptotic h

value •:• {1}

I f {(„ > q-^ the same reasoning shows that q tends toward the

asymptotic value g

Sl

The intermediate m o t , q.^, o f

corresponds,

l b ) I f »' < 0, Hqf is always opposite i n sign to q - a. W'e note uyain t h a i the dilîerence r = q — a decreases i n absolute value and tends to zero. In the asymptotic d i s t r i b u t i o n , the t w o genes 1 and A coexist w i t h the stationary frequencies 1 , there is exclusively zygotic selection, and the hétérozygote is superior in viability to either homozygote, provided consanguinity is

not

too

high.

In

fact, we

have

w < U, and

o = — t/w -=

^Z

Et"ıiattan "f J Mtu.'ıiıJFI

Pcpııl.ılıcn

}.!

infkf'lie

if Selfclian

51

f/i, is possible, then t?(ij, q \ is always greater than zero. T h i s assump­ ir 4- ^

I

* —

X

~ ^

^

^ul

P ' ^ -' 0 5

1

1

1 l L i l

'

L L

*

S

C

'

L

J

R

'

ı

^ L ^ P

1

;

ir

tion implies that l h c rates o f m u t a t i o n fi and r are n o t equal to zero, because otherwise we could not pass f r o m q = t) o r g = 1 to different

< (7ı — I ) . which makes necessary l h a l i ' f i — X) < tı, [hat

is, X < I -

values. M a r k o v ' s theorem indicates then thai the rj priori

law o f

p r o b a b i l i t y . *,.([/) i/q. o f the frequency o f t / in I he generation F„ tends

\/h.

toward a l i m i t law. &q\dq, HEM ARK.

w h i c h is independent o f the initial value

o f q, when ti tends t o w a r d infiivity.

Wc can easily verify [hat ıh e case rr = » = O o f f C ) ı.v, ı • us a s|iecial ease in Ihc graphic discussion o l (A) or o f [H). i f we consider lhc curve C l o have degenerated into lhc broken line detined by (q = I), y < Ûf 0 < < * 0 ; ç =• L > Iı lollops that i f u ıınd ı lire small with reaped to f and • bul not equal 1u BCfO fdoued linet, the discussion will be the H P V C as in f O , the only difference hein^ lhal elimination ami fluuion will he rcplııucd by an asymptotic equilibrium correspond]nj; lo a frequency of 3, close to fi or 1, f l

It is possihle to f o r m u l a i c ihcse laws i n terms o f certain hypotheses concerning lhc law o f t r a n s i t i o n , Gt.q.q^dq,.

w h i c h is the law o f

p r o b a b i l i t y o f q when q is lixed. Let us assume it to be a f o r m o f t

y

Gauss's law w i t h mean value q f- öf;;), o{q\ being small and such that ¿(0; £ 0 and a{\) ^ 0, and w i t h a s m a l l variance.

= w{q) £

0.

b c i n ^ equal l o zero only for q •- 0 and q - 1. L e i us assume, for instance, that the 2N gametes which produce lhc F „ n

Qfe taken al r a n d o m f r o m an infinitely large n u m b e r

of

gametes produced by F, and have essentially the frequencies, q and 3 . 2 . 1 T h e Case o f a F i n i t e Let

Population

f l — q) f o r ci and A . We k n o w that the law o f p r o b a b i l i t y o f the

,V be the number o f individuals i n each generation. Ef q is

frequency o f o i n F,,,,

w i l l be practically Ciaussian. and that the

the frequency o f a i n /-,„ we have seen that Lhc probability o f a i n

c o n d i t i o n a l variance o f this frequency w i t h respect to its mean value

F„+, will be q +-

w i l l be
fi<#X

being represented (as a first a p p r o x i m a ­

t i o n ) by a third-degree p o l y n o m i a l . Uut the frequency. q , t

of a in

In

lH

represents only the mean

general, i f because o f systematic consanguinity

A' zygotes o f F ^, n

v alue. W h e n the law o f p r o b a b i l i t y o f q is k n o w n as a f u n c t i o n o f q. t

e.g., Siq. q\) dqi, the frequencies o f

i n successive generations appear

w h i c h does not quite become zero

except i f q - fl or q = 1.

f . . [ w i l l differ f r o m the p r o b a b i l i t y , q -r Hq\ because ihis frequency is a r a n d o m variable for which q +

q(\ — q):'2N.

i n F„

the

are each taken at r a n d o m w i t h lhc c o n d i t i o n a l

probabilities /' -- p(p

f X4J. 2Q ^= 2pq(\ - \% and R = q{q +- \p)

for the three states A A An, and uu. X being the c o n d i t i o n a l inbreed­ t

as r a n d o m

variables i n the simple

M a r k o v chain whose law

of

i n g coefficient o f /\, L, the c o n d i t i o n a l variance oT the frequency qi t

transition is rliq, q^dq,,

w h i c h is assumed to be independent o f the

rank, n. o f the (feneration considered, as is possible i f A is constant.

of n in

w i t h rcspecl to its mathematical expectation q w i l l be

r

'

:

a

= q(l

-q){\

+

W

If we assume that the t r a n s i t i o n in one generation, or i n a certain A . Fiuidi/wila!

number o f generations, o f any frequency, q. to any other frequency. to * C will I v nK't at only erne |ionu by the ttraiRnt line U if i > 1 or n < o feme 11 or i f 0 < i, < I and »' < 0 4casc Jbj, bul in iJinrc ptnats If 0 < n < I and • > Qtcnsu 2a).

*„(£/) dq

Equation.

the n priori

I n the t r a n s i t i o n f r o m generation /-'„

law o r p r o b a b i l i t y o f the frequency changes f r o m

lo tf,,ıitfL)i/tfı

= dq

y

/'

{qMq.qi)dq. r

54

F.ıalutiûi Ü/ ÉI M.etııit!i.ıl

Ptfxiittfin

If we call W, and Ml the moments o f the ti priori

law o f p r o b a b i l i t y

verified] exactly by the specific f o r m s which we have indicated, wc shall w r i t e :

in F„ and in r .,-!. we have: 7

%)

=

X- Atq>\

Mq)

t>0

=

^

fjii

Ufa'.

By comparing the small variance, M', - A/,, to a derivative

tlMddt

( t i m e , J, being measured in generations}, equation [3.2.1) is trans­ formed to a differential system for the m o m e n t s :

=

¿

= ^ , . i i ,

(q)^(qhq

1

Ul

1

(

, +

i

'->

l l

rfl,A/,

: T

,

f j . 2 2)

(bv inverting the integrations, w h i c h İs legitimate for functions [hat

T h i s system cannot he sobed directly, because in the second

are bounded and ean he integrated w i t h i n finite intervals).

o f the equation there are moments o f higher order than in the first;

[ F j j , ( $ arc the moments o f Gauss's law. Q{q,q¡)dq¡, and variance are q + &{q) and uiqt,

whose mean

it enables us, however, to obtain a partial derivative equation for the

respectively, and İT fi and tv are

characteristic function f o r Laplace t r a n s f o r m a t i o n ) o f the proba­

small, these moments are calculated by developing the characteristic

bility law flg, f)ttq."

function according to the powers o f its variable ?:

t r a n s f o r m a t i o n is

cxp [(q 4- fl)r 4 wtfffl

part

= 1 -h {g 4- ÍJ-- + * r V 2 f 4- -• - -

F(s.t)

for which the moments are A f , ( i ) . I n fact, this

=

1'

PWq*

0 <*q =

35

MJfiP/gt

*• Kff + Ú J 4 - > " / 1 1 ' / Í I 4-- - -. Í

T

W e sec that, by disregarding the terms in

w i t h derivative

and fi-'.

Eg +

and

İte* S í *

+

J ( f

T

¡ i

%*%

these functions always exist since we integrate only between 0 and I. By m u l t i p l y i n g equation {J.2.2J by f-*/P,

+ 0£w>) +

and s u m m i n g over i f r o m

d to 4 - ^ , we obtain O Î ^ ' Î + 9cwfl¡

therefore, the variance o f the moments f r o m one generation E O the next is

F o l l o w i n g the Laplace t r a n s f o r m a t i o n , bv setting

=

i j\q)q->Uq)dq

+ ' ^ f ^

j

a

u{q)q^Uq)
•This i'anuuoa ilioaJU I * miqtrabk: when u • 1/ :. I lor nil valin. » 01 ¡. It he o ideal ironi equjilunv f J.i.to ami ( 1 . J . 1 U I t hut flu condition bus to be hunr""cil true only for r = t\ provided u > n arid p' > n. 1

«•¡11

If we assume that b{q) and n f g l can be represented by p o l y n o m i a l s .

T

56

Eral'tiaB

af J Mtfaİfİ'jtl

Î.2 ['ijlumst

PıpıtUlian

vj Sthctwn

%~)

i t is, therefore, the law f o r whieh ihe p r o b a b i l i t y density is

we have

*£g) - \ K M q ) ] e $

ft

[3.2.50

KN

In particular, when „• = tfl - f,)/2fl. I Hi-

¿14,

I

e'**"?

dq

=•

'

J

ÖV

p"i.-—

and

fjn

- iJ + ( W * +

i 0 7 ) / r t l - fl) = -

- [

e^Ydq

1

+ tifc

we have

-

hy selling w i t h A'ı determined in such a way Ibat the integral between (J and I fit

ö7

is equal t o J.

II

This f o r m u l a , given by W r i g h t [22, 23, 1 4 ] f o r specific cases but

und n o l i n g that V = U fcır g - 0 and Tnr ^ = 1.

w i t h o u t general demonstration, represents the probability t h a i , i n

Sinire t w o functions t o r w h i c h t h e Laplace transformations are

a l i m i t e d p o p u l a t i o n o f N individuals, a gene a, w i t h given coeffi­

Ih e same arc identical almost everywhere, we obtain [Vom equa­

cients of m u t a t i o n , m i g r a t i o n , and selection, after an infinitely large

tion { 3 1 3 ) :

number o f generations, has a frequency between q and q + dq. I t also represents, therefore, the law o r asymptotic d i s t r i b u t i o n o f gene a. after an i n l i n i l e l y l o n g time i n an infinitely large number; o f

Defy

populations o f the same size J V a n d i n w h i c h ail t h e coefficients t

chat is,

w o u l d be Ihe same. Let us indicate some specific cases. tl) 0-2.4)

i f a = 0, o r r = t). K is by necessity zero, since the integral t

between 0 and I o f \/q o r o f 1/(1 — q) is infinite. T h i s result i n d i ­ cates that, eventually, genes not affected by m u t a t i o n o r m i g r a t i o n w i l l certainly be either eliminated o r

Such is the fundamental equal ion.

filed.

(2) I M . W i n d 4Nt are less ıhan 1, i.e., i f t h e p o p u l a t i o n size is i

large enough, and t h e m u t a t i o n or m i g r a l i o n rates are n o t t o o l o w , E.

Asymptotic

Probability

Law.

I f we consider ^ ( y ) f/y ihe law

o f asymptotic probability f o r infinite i. [hen, according t o M a r k o v ' s

Piq)

= 0 f o r q - 0 and q = 1, a n d is represented by a bell- or

double-be U-shaped curve (Figure 7) w i t h one o r more d o m i n a n t q,

theor>, Ihe law o f stationary p r o b a b i l i t y , verifying (3.2.4), w i l l be given by the equation - = 0 t h a i is, &

T

f3.2ö:j

4/V*Î?L)

-t

2?t —

1 - 0. which,

dq

for a very large ,V, becomes a
5B &

-

BiAkp,

Akd)*

and its moments are given by f o r m u l a (3.2.2), w h i c h can be w r i u c n

0 - i(-k\f.

Jf we starl w i t h

F I G L'HE 7.

+ kqM.

J + '('-^(M,-

-

t

MX

= ] , Mi = q, and

equation or ihe asymptotic \ulucs£ m a very large p o p u l a t i o n t h a i were studied in $3.2,1.

f r o m which we can deduce ( r r being the variance) that :

Simple result* are obtained by assuming that there i * only one d o m i n a n t yı. and t h a i q remains close l o it w i t h a p r o b a b i l i t y n o t far ( r o m 1; as u h'rst a p p r n x h n a t i o n , let us replace St_q) by a linear function o f t / , i.e.,

i f 4.-VA- is large, i.e., i f the order o f magnitude o f I is greater than that o f I / J V ;

is then small, the d i s t r i b u t i o n is concentrated, and

it is legitimate to admit that q varies, practically. In an interval o f ğ being, bv necessity, ihe asymptotic value i n a very Luge popula­ tion,

Limited range and Ihat h{q\ is linear.

as i n P 2 . 1 C A ) , 3 , 2 . U C U ) , or 3.2.UC.2.hl. and k being equal

For a large number o f populations under the same conditions,

to - û ' l ğ j and therefore being o n the order o f magnitude o f the

all having the same size. ,V. sufficiently large f o r ANk to be large,

largest of the numbers ıı, ı . n , r, according l o Taylor's f o r m u l a .

the asymptotic frequencies observed will almost

F o r m u l a (3.2.7) w o u l d be exact i f % J was linear. i.c , i f there was

around the value that corresponds to an infinite p o p u l a t i o n . The

no selection, as i n $3.2. J(A. I ) , The d o m i n a n t q is given, i h e n , by

experimental estimate o f the variance o f these frequencies w i l l

1

r

t

-AHkiq,

- '!)

r 2g, -

1 =

y

f r o m which

+ r j or - O ' / i J o r T - T - H X / U - *]f(.2ti (3) I f 4.Vw and 4.Vt are less lhan one.

- Î

grouped

enable us l o determine k, j f we litiow N and the variance ott

0,

vf(u

tittf

all be

gives

- 1). k infinite f o r q = 0

and q = 1 and is represented by a U-shaped curve.(see Figure S).

_

The smaller u and v gel, ihe smaller A" becomes. It is the frequencies ;

and i f ANk is large, ihe d o m i n a n t coincides essentially w i t h

ihe

asymptotic value in an infinite p o p u l a t i o n . The asymptotic d i s t r i b u ­ t i o n (3.2.5'J is w r i t t e n

close l o q = 0 and q = 1 t h a i have by far the highest p r o b a b i l i t y . M o s t o f the genes are approaching fixation or e l i m i n a t i o n , and the only t h i n g t h a t stops the approach is recurring mutations or renewed m i g r a t i o n . There is a basic difference, then, between the case o f a p o p u l a t i o n (hat is very small or that has very l o w rates o f m u t a t i o n

with

Q *lunU* f»r the l-altrian uilcgruE.

1.1

40

1)

A^MW

*/Stitllian

il

and

w i t h A(G) = A T I ) * 0 , T h e last equation cannot be satisfied w u h these conditions ai the l i m i t i unless the ennslant A belongs [o [he scries o f " p r o p e r values." \._ w h i c h are real, positive, and. presumably, arranged i n increasing

i

order o f magnitude.

By calling K.iq)

the " p r o p e r s o l u t i o n " corre­

s p o n d i n g 1 0 \„ any scries and ">.:•'

.it.-'

and whose gene* Lend i n w a r d I: -.

-11

- •-• of

i . m n ilin Riq.l)

and lhe ease o f J large p o p u l a t i o n w i t h each gene almost stabilized

=

HA - 'K,{q\ l

lt

a r o u n d a determined frequency. Satisfies, simultaneously, both 13.2,7) and the conditions aLthe l i m i t s . C\ Evolution

of the Probability

Law over

Time.

e q u a t i o n (3.2. J ) , let us cull * t i / . /> and -tig, if = jj

I n verification

Qfq* ')dq

of

the law

o f elementary p r o b a b i l i t y and the integral al (ime t, respectively;

In

a d d i t i o n , it satisfies the initial c o n d i t i o n Hlq, Ü) = f i i q )

coefficients A , are chonen so that i A . K A q )

a R,[q),

i f the

i.e., i f they are

given by the cvpan-don o f the f u n c t i o n R¿qt m series o f f u n c t i o n s KAq\- We k n o w that >neh on cspansion is possible f o r t f u n c t i o n R¿q)

lei us call dq)

and Mq)

- jj

4iq)dq

the asymptotic l a w . deduced

w h i c h is c o n t i n u o u s ,ind e q u a l l o Í-.TO at the limits q - O a n d ? = |

f r o m ( 3 . 2 . 5 ) ; w e designate by A t y , t) = -\^q, '1 — Mq) the difference

T o evpress the cspansion, it suflices t o w r i t e equation

between the integral law ami the asymptotic Jaw al instant t. T h i s

reduced f o i in

difference is given for the initial instant us R{q ()) • m

conditions at the limits A l t ) , t] =

ff(l

T

j

n

r

the

R„{q)i it satisfies

t) m H and İt verifies, evi-

0"

*iqW{q)

'

denify, the equation "obtained while d e r i v i n g (3.2.5) f r o m U.2.4JJ I «> T . .âRl

AR

r

.

,tR

designating the new variable. / * o{q)dq,

by r. w h i c h is the f u n c t i o n

o f t o t a l p r o b a b i l i t y M.q\. W e k n o w , then, t h a i the proper solutions The difference w i l l be d e t e r m i n e d , therefore, by o b t a i n i n g the solu­ tions o f f 3-2.7) w h i c h become zero for q - 0 and q - I and are o f the f o r m R = K(q)-L{t\. L\i) L{t)

are o r t h o g o n a l l a n d can be taken i n be n o r m a l i z e d )

respect to the f u n c t i o n I .

• •. i . i.e.. that

These solutions must satisfy wK'Xq)

"

Kir)

2 k\q\

for w h i c h i l is necessary I hat

¡w' +

\ 2

_

\ K'{q) }

Kiq)'

or. by going back to [he variable q, t

1

J,,

K.tq)K,yq) *Uq)

with

T h e solutions that equal zero f o r | = 0 correspond [ o C • 0.

where

(

There w i l l be. therefore, " p r o p e r s o l u t i o n s " becoming equal t o i c r o either when q • I» w Jıı

wtftfl

w h i c h , according

T h e eocllieicni». A. oV ıhc expansion « f M f ) PfCi Iherefore, o f ıhe form

f^'!l t

when , J = I . provided that f W * # * 1 * ı •

t o Gauss's theory o f equalions,

_ i J'

i

i

T

w

,

s

is e q i u l

io

**• iWs e q u i r e s (hat . o r rf be equal t o r

whole n u m b e r , H > 1. i.e.. that equation ( 3 . 1 1 1 ) h- . a whole,

positive root n w h i c h gives f o r \ t h e " p r o p e r v a l u e i " \ - n'-4S +n

nffc — I.. 4iV), values Ihi.it increase f r o m k t o 4 ^ * . T h e corresponding proper standardized solutions are the hyper-

which given the noliition t o t h e problem as

geometric functions %,/)

»

£

AirV«fM

(3.2.10)

which is a u n i f o r m l y converging series We nole thai the magnitude

fC.iq)

- tuF(n

+ 4NkJf.

I - M - 4 M + 4 A % 1 f Iffltfif),

T h e constant*; h., j r e chosen lo give

o f Ihe d e t r c J H - o f the difference Rtq. li between the asymptotic low at instant • a n d Ihc integral law is o n ihe order e

v

. \i being I he

or .i proper value, unless i n t h e f u n c t i o n A'ı|ıj) ihe initial d e l a t i o n , fUiO

is not o r t h o g o n a l I O I --mi: T h e rate o f i h e process ıs thus

T h e coefficients .-f_ are g o en by

ctuiractcri/cd. It is easy t o resolve the p r o b l e m completely i n the case pre­

'

viously studied, where Mq\ C M be r e p r i c e d by the linear f u n c t i o n 4( 1 - - k ( q - q). T h e n equation (3*1-4}. where w - q\\ - q), IN V

Jo q' 0 u;

~

qr -' ,

ít,,

T h e difference is given by t h e f o r m u l a (3.2.10). Since Ai - k, the order o f magnitude

becomes t ilium's equation

o f the decrease o f i b i s

ditference will be. in general, that o f e~' ; t h e number t o f generar

qi I - tf)K"

4

[1 —

2g + ANk(q

- q)]K'

+ ANhK

- 0.

(3-2.9')

ti-QM needed l o approach the state o f asymptotic e q u i l i b r i u m appre­ ciably, therefore, will he on t h e order o f magnitude o f I k. We have

The UaıivıiLm parameters here are • and J , the roots 0"f

seen [f3.2.J(A)J t h a i when Hqi has the general f o r m derived at ihe i i " + (ASk

-

I ) o - 4JVX

Calling M M . J . i ~ * q ) t i o n o f i}.!})')

- 0

and

7

=

I - 4,V*#

(3.2.11)

Uie I i j p e i g e o m e t r i c series, Ihe geueral solu­

is

end o f $J.2(ti), but the d i s t r i b u t i o n remain*, over lime, siillieienlly concentrated a r o u n d the value q. we t a k e k

i ' ( f l j - « + p - ( I — lq)t - wq\2 — 3?>:

k is, then, on (he order o f magnitude o f Üıe Lu^gest ( i n absolute value) where

o f the quantities i l . r, I. i r . W h e n a l l these quantities are s m a l l , I / * IS o' - a + I - f ,

tf'

= a + I - y,

T' •

2 — 7,.

large, a n d the numher o f generations needed ( o approach e q u i l i b r i u m

İ-İ lafllirBcr

i * considerable. W e cannot ü n ü m e , therefore, iluıi a n a t u r a l popula­ tion

ha\ reached the state o f e q u i l i b r i u m unless c o n d i t i o n s have

remained Ihe same d u r i n g a very long period o| l i m e .

65

depending o n the p o i n t C and die r a n k ti o f the g e n e r a t i o n : ihe X*t

relative to t w o different points C w i l l have a stochastic rela-

l i o n , " T h e r a n d o m variables.V„, J D) relative l o the f o l l o w i n g gener­

T h e preceding m e t h o d does not apply any longer i n eases where

a t i o n will have c o n d i t i o n a l probabilities Weil-delcrmbed on the basis

there are neither m u t a t i o n i nor migrations, i.e.. when u = t = l\

ol the XJ£\

because ıhcıi K = 0 and

follows t h a i the a priori

1

tfqtittf.

of Mifraliim

the density of asymptotic p r o b a b i i i l y ,

values, A c c o r d i n g t o the theory o f M a r k o v chains, it probabilities o f lite A ; ( C ) s and their rela­

equals *cro at any point between II and I . A l l probability is

tionships w i l l tend eventually t o w a r d a stationary slate, independent

Concentrated at the two extremes, q • 0 and q - L T h e manner i n

o| the r a n k , n, o f the generation. Jl is this stationary stale we propose

w h i c h this, asymptotic slate is reached can be studied by a d i t t c r c n l

l o study.

method [ H I -

I f if and i- are Ihe probabililies o f m u l a t i o n o f a i n t o A and o f A

3.1

INFLUtiNCE OF

MIGRATION

i n t u a i n each generation, the c o n d i t i o n a l espectation o f the

r a n d o m variable X' relative l o a locus o i a n o f l s p r i n g o f a specified

T h e h y p o i l t s i s by w h i c h W r i g h t [21. 13. 24] explains the effects o f

parent w i l l be

m i g r a t i o n w o u l d a p p l y well only l o uu island p o p u l a t i o n receiving

WX ) 1

ınigıants fı o m a large continental p o p u l a t i o n w i t h constant composi­

- ( I - u)X+

i(l - A),

t i o n . A scheme closer to the actual s i t u a t i o n , w h i c h takes i n t o

X being Ihc specified value o f the r j i n d o m variable attached to tile

account ihc interaction o r one group w i t h another by m i g r a t i o n ,

corresponding locus in the pareni, T h i s can be written

w o u l d be the f o l l o w i n g , l i t a population be distributed over an Jtttf')

urea -I w i t h a density HP) at point P w i t h coordinates (.*,>")- Let us

= (I -

k)X+

kc,

assume thai each i n d i v i d u a l , f r o m the t i m e o f b i u h to the reproduc­

calling • Ihe quantity q = r/[u

tive stage, has a k n o w n probability . / ( f , Q) dS^

sponding t o Ihe m u l a t i o n pressure.! Since there is no stochastic

the point P t o an

relaiion a m o n g c h i l d r e n other I f u n the one resulting f r o m the even­

Q) dSu = I J . A c c o r d i n g to BayeVs f o r m u l a , each parent

tual relation a m o n g their parents, ihe jetini moments ;iR[JT'(C)A"'( D}\

o f a n i n d i v i d u a l b o r n at point Q w i l l have the k n o w n p r o b a b i l i t y .

' l i Ine coelfkKiu of eoanecslrj r\-l*t*u u v l o i d m k Invited in plates i and ft anıt \ J f i i have i n u nrkvl nioEutnUf- o l uf being h l t i i l u l ami a ptflkİMHty of ' - 0 . n l temt Nfc H. hiiilJLj 11> nulu]VmJi:nl, Afc juvrs, a s ıh* vıüuı I^KH „ ptum tonelation HVltkiLiU. o„JC, 0|. Tin- asjlnptptiv wcaie, t>), pf Ihis lUritictcrU WiU 1* enL'uUucJ iurther; a i i u ^ b i l to km** ıhuı a u ıh? .tunv * i . ilv i,*tfKie*ri of

tfiP,

of

being

Q) dS

born

r

- • t W C f t Q)

in

an

area

area. dS^

tâ'

dSi-

fff

centered at point

r) and A ihe q u a n t i t y ı- + u c o r r e ­

Q

fit'.

elementary

of migrating f r o m

, HflfiP.

centered

Q)

around

dS,.,

point

P

is vjllcd «sJ<~. O j . ihe minium saiuıhk» \ J O

mmmrr

Let

XAC)

Mendelian

be the r a n d o m

variable representing the stale o f a

locus i n an i n d i v i d u a l o f the n t h generation b o r n at

point C. A priori,

.Y (f_| w i l l lake the values I or 0. corresponding n

t o the allelic states u or A, w i l h apriori

probabilities £j and p = 1 - q.

hrm-tvi, c unJ ti.

I II i l i c n it n ccuuiuai svlinion itr-omre In i'mur (if ıht hcieı-o/inmc, w kncnv Hut it will he eıpreuud. Jinni^lmıılcl>-. Inr I lnr^L mımEıyr of IaJıvıdujıl^ Hi ii letriiHiLa ui ihc sumo ftm it. and r naiuria> luvina nther vaiunt The vuliulntkiMs * i shall perform MM, ıHcrctmc. K n l u i i spprpunvlkiii ant^heuhln i*y eıuıuant IBİMÎufc h«r ilır> c u l u k , macı.-.ıı n- *"jpi*jrBptoc • • , n . . . iltia Ocpends on lotatinn.

o f a certain number o f r a n d o m variable« A" o l the (ir + I Hb f u ü ı * -

distinct (ihey may be. i n case o f r a n d o m m a t i n g , ihe t w o h o m o l o g o u s

l i o n w i l l be linear combinations o f die* products o f ihe variables 1

loci o f Ihe v i m e i n d i v i d u a l ) .

o f Ibe parents, i f the latter arc k n o w n ; i f they arc u n k n o w n , the

Two

loci,

n.jiCl

and

Y.^D).

at

(wo

individuds

in

ihe

X' w i l l be linear e o m hi nal ions o f lb e mathematical especial i ons ol

fn + i H h generation b o m

these products, i.e., o f the j o i n t moments or the Nib generation. By

j i f f . C)KV\ D)d5,f/SV

equaling ihe j o i n t moments o f the two generations, we shall o b t a i n

and Ihe p r o b a b i l i t y xtE, C)g[f\ DidSi

Linear integral equations for d e t e r m i n i n g these moment*. W e shall

b o r n in the same n c i g h h o r h i u k l o f a single site £ : i n the laiter case

i n t i k a l e only the calculations for ihe moments o f orders I and 1.

Ihey w i l l have the c o n d i t i o n a l p r o b a b i l i t y I

The mathematical expectation, : i f i ( 0 . o f XiVt

w i l l he given by

from

the same

in C ami ft w i l l have the p r o b u b i l h y

o f c o m i n g f r o m parents b o r n in E and P o f c o m i n g f r o m parents b o l h 2ME)dS \

I - 1 İ2UE )dS ^

i>f c o m i n g

K

locus o f ihe same parent

and

the p r o b a b i l i t y

o r c o m i n g f r o m loci infinitely dose b u l d i s t i n c t . "

£

W e have, i l t c r c f o r c . when the places o r b i n h , £ a n d t

o f ihe p a r e n t

T

are k n o w n ( c o n d i t i o n a l expectation), t h a i is, j * rf .,(Or^,(/)) r

H i i 0 = ff 0 A

- AWHsv^ 0dSr

l -

and when I hey are u n k n o w n (n prhri

an equation whose only s o l u t i o n , i f k - f i f + r) > Ü. is

r

t

a

expectation).

•m[Y^C)Y^(D)] =

!li(r*) = constant - c -

=

-.m[-Ml .[Y.JC)Y. ^D}]\ r

[ i

The nuoheoiaoeal cspectotion İs therefore independent o f Ibe geo­ graphical p o s i t i o n . I n Ihe calculations

k\--YjEiY {Fl;

+ Ac,

lti.it f o l l o w X - c =

:ul( K) - IÎ. and f r o m one generation l o the nc*lüfi( Y') - ( 1 —

Y. k)Y.

T h e variance o r X, o r o f Y, w i l l be

~

k ) :

l

//.//. ^ W ) l W t t K W R

dS . r

M [ y j f j r . f f ) ; should b e t a k e n as equal t o t * * 4 £ , O i f the elements of

area

dS

and

L

M[YJi£)YjFi\

are

d i s t i n c t ; i f they

are

noi

distinct,

should he taken J I e q u a l l o

T i l e j o i n t lirsi m o m e n t o f the t w o r a n d o m variables W,C) and or the sanu' generation w i l l be designated V'«(f", />); giC

ft)

by l l t f Y(C\ i ( D ) J

=

that is, equal l o

is h o l h coetTicieht o f eoancestry and u J>riW ^

c o r r e l a t i o n coefficient of these t w o r a n d o m variable* and also o f X[C)

and X{D).'

one, when

f

£

£tq.

,

a

n

-

syfi.£)L

Let us call tfC C) its l i m i t , obviously less than

D gets infinitely close t o t\ the t w o loci

remaining

* A h n ft ılıt bCpl ini|ucniieivi aı*hj . m plants * and If. bMfeM inesc »rf Vxal draltiivia: means of such raıukım vnıahk'». r

d i v i d i n g by s we have the " F r e d h o l r u i t e r a t i o n " ; ;

• Korumla Inr nton-vcum* randiwi nmnnif. in ca>e o l vpurait M v s . * £ > a l * n r i l i i hbimonic mean n i inak and k i r a k dcn^ilin in £.

3 3 Injuria

$ (E, £) = n

híl

- *)

^

a

' ' ^ ^ s ^ n g f E ,

ft&j

D) + {CE Vr + W ' . R i i V e C

MiamijH

69

"f- • - -

D)dS** using the symbol £ E V, for the operator

In the ajationary state, iftf>( JT, £1 = l i m ^„(E, tion,

(3,3-1} w o u l d

be.

for the

was a k n o w n func-

unknown

function

cUC, D) =

lim 0 „ , , ( C D ) , a Fred h o l m equation w h h un integrable kernel o f

its powers k-ms dchned as u^ual

=(

— \ ' {

V I

fl

n o r m {1 — k f < I ( i f k > 0 ) : i l w o u l d Ihen have a unique solution given, whatever the initial values, by the same integration as f o r ¿ero i n i t i a l values:

It iv now easy lo express; die double area integral in the second term o f (3 3.1) as a function o f tbe partial derivatives of ^ i C , D), the coefficients heing-thc moments. w „ calculated from place C. and the similar moments, calculaied from place D\ tlte betrinnine of this M f f t f i l is (considering u symmetrical case, for the sake of simplicity, because the I»IJL| moments are then equal I D zernj: r

n

iT

by setting

•ji/m

=

K¿E,

-

O

\j g(E FMF.Od$r. A

jJ

g,-:(E.

A

1

FMF,

O

JSr.

By taking E = C, we obtain a second FYedholm equation for the determination o f

£}:

This equation in general (when its kernel is integrable and o f n o r m < 1) has a single s o l u t i o n , obtain

E ) ; by p u t t i n g it into (3-3.2J, we

£>). 1

ff the moment:; and I heir products are negligible from some order, and if we replace *.(£', D\and tfv.rfC D) by their equilihnuw expression, * [ C , ¿J), this last function is a solution (which (ends lo ¿ere when distance CD ten da to inJinity) of a linear partial dilferential equation, o f Which the nouhomogeneous term jj

-—-^^j—

g{E, €)tf{E.

D}tiS

E

itself lends to ¿tro when CD tends lo infinity.

I • • i . • L. D o n O V T U l f E Q U A T I O N .•Vw-flOKOIATTHO CJiJ-Tj, We may imrodace tbe moments of the migration taw, i,e.,

K E M A J L K

[The fomiulu far aiiidimensiomd or tridimensional casca is naturally o f die same form.)

A

3.3.1 S p e c i a l C a s t u l

"Homogeneous

and Isotropic" M i g r a t i o n »J^r =

ix

R

-

xcfty,

- 3&0S<

C)

by replacing, in Hie second term of [3.3.1). i J E . F f development.

ttSs.

by its Taylor

Let

us suppose that the area occupied by the p op u lati on can be

considered u n l i m i t e d , that the density 3(E) is constant ( i n space and time) and that j\F, (?) depends only on the distance F Q = r ; then

7LI

C M » f i i l ^/ - ^Uıiıiıiiuiı

gtfı

Q) is equal to f\P, Q).

"3.J

Püp'Aat'O's

Let U S sel tfP, (*) =

fljr).

so tTmt ¡1

$ =

Ki",

jf

Ittflurnit

»i Miffjtioa

71

-r y'-) d.x dy,

becomes a function o f a single variable, no longer o f f o u r : similarly 0

=

because we have

F r o m ÎU.3.1 we get

* Î C C )

fi

=

1

" İ l "

£

)

Î

d

-

n=U

- I

un integral equation whose solution by successive goes cKC CJ = constant =

=

approximations

-WF-'/lX

-

(I

-

It f o l l o w s l l u l a forniula which is also obtained by applving the Fourier t r a n s f o r m directly to f o r m u l a (3.3.1). T h u s K is expressed as a function o f F i t ' . ' ) ,

from wbieli

which is k n o w n .

Front this, by inversion o f the Fourier transform w i t h t w o variables, we have

where

(fJv selling .V = v = 0 . we hnd again Lhe linear c q u u i i o n for
^

Let 4 =
These calculations can be carried still f u r t h e r hy assuming that lhe displacement o f each i n d i v i d u a l is a r a n d o m movement f o l l o w i n g

i

^

f f

the scheme o f Polyn. i.e., that the \\iwf{ridS

-ky^^ECiUmdSs.

Gi^)dS=

(3.3.4) which

This is the correlation coefficient between two loci whose distance is C D ,

We can express, in algebraic terms, the ""products of c o m ­

1-

• R[r)dS

is an iso­

tropic n o r m a l law, (l/2-n ")i.--"' ' dS, , ,

n

|Jw

FUI. <•)•=

r " ^ ^

4

i

,

,

(

p o s i t i o n " f o r " c o n v o l u t i o n s " ) which appear in (3,3.4) by considering 2fl

the Fourier transforms,

fir-.

>) =

j j

e

r

'itvV

p-i

F r o m this we deduce scries (3.3.4). w h i c h is easier to calculate than +

y"ydx
(3-3*5), and f r o m which

and M • tir tin cndhcırnl nf cmncesıry hüiwceıı i ' muJ P \ p. 65up, For jandom rnüiinjs. <;<[£'. i'ı = io(Eoancesir> bcl*ecn eloncb Jocaled laed iiülso llif Nihrıredw& liOflHhiuiH djnnaceury bcrwcco (he I W D Imıııoteısouı Icei ot ene individuul). 1

= «

M l p-t

- ky«Gt2r°-).

T h i s series is u n i f o r m l y convergent when A > 0, since

P-14'J

|J

In/Jutnn of Miftjtmn

73

who*c sum i% f " - * » l a t i f i - * • ) JT]->

rf*

F o r m u l a £3.3.3) can he f o u n d here by n u k i n g r - 0. w h i c h Leads us l o calculate we also dcılıuc from I his thai the numerator or (3.3.-1"» is equal to H -

s (i -

m p / t a r f

dx

-

-log [I - (I -

fcfi^W

bcin* the llessel function. Uy Idling r = (I, we liııd a^uri the denominalor U

= - l o g f Z A - A ), 3

W .

f r o m Which ft W e can calculate

- I [1 - K " i . l o ( 7 j t - k * ) l !

E

{3.3.3")

easily, f r o m l h e pressure A ( o f ovcrdnminnncc

REM Aft K 111 I I t, lends toward ^ero. die mnnerator ami [tie denominator o f [ 1 A 4 " ) tend toward IriMnily, hul their rfitrercnec remains rinite (.mordinu to the [HQIrtfca of J , j ; therefore, H —v- • . and * —»- 1; and the population lends toward complete homu[rcneity. which is Inevitable in any population w i l b a linile l i a in lhe absence nf m u i m i o r u .

O J or m u t a t i o n ) and f r o m the number rv-'l oY individuals i n a eirele o r rudius o. i n w h i c h resides, o n Lhe average, -HI per cent o f lite individuals b o r n at its c e n t e r ) ; the smaller these t w o quantities i r e , the closer o. is 1 0 I (local quasi homogeneity!; next, we deduce

HTMAkJs I V We may, i n lhe partial differential equation sliown lo iipprroimaic ( 3 . 3 d ) , when a is u n t i l with t e r c e l to r\ k. keep only the tecond momenti M I * - mm ° ">L = IH'„, = m = m\, - 11 (the higher moments, bcin$ higher powers o f o. p i i v m ^ l t g m k i r h i r ^ l i t f u l k roots). lt

from (3.3.4'). Z

il - * p f •

-,)c-"'W

r heina luiyc wiih re*pcil to a.

ff

i-Trt^. r ) | H negligible and

gW' . /Ji erfj-j is a solution, null at infinity, n f ılıt Inninjyta^ou^ HeJmhuli/ equation which shows that the c o r r e l a t i o n to the distance r decreases f r o m *

tftr)

u

- (I - t ] ' | * J +

**4#ft

to () when F increases f r o m 0 to •/-. T h e numerical value o f this rutin depends only o n t w o quantities, k and r/a-, i t is, therefore, easy t o set up tables that w i l l enable us to i n t e r p r e t the c\pcrimcnl.d result*

- 1 * heiny ıl>e "'Laplutian" ^ and 0. tx.

tike

| p - which, in polar coordinates r

iiklcnrndent n f 0 and equal to Ef

I J^*;

so

w i i h the help o f this f o r m u l a . (when nrBfciiing A j we obtain the Bessel equation J

KLMAKk

II

T o ealeuLate O.i.A") numerically, we can develop the .. . -:, 11- .• l u tliu powers of r\ arriving at the series 1 (t p-l

ftjfeftfe

namcramr trr*

r Br

»*

O f the two distinel solutions. /. and A.. unJ> A . is IxxinUVd. ihus i i v i n n lhe correlation |or eodheienl u f coanersiry);

74

EM/MHOM

of a MCHUHJH

t'ofoijriaa

JJ

Itffaıuıt

af Mi^uiiaa

75

o f distance, f o r v e r i f i c a t i o n o f this theory, can be done in several JirTerent ways. where " is u constant tint! r is much greater than o_ "Ihe same equation, and the same result, is true foe every migration l a * all of whose reduced moments art hounded [43], and the I t d m h o l t ' equation is valid for an isolropie migration of any dimensionality. 1

(1) We can measure the frequency, i / . , o f a Mendelian gene (with¬ out geographic selection) at a large number o f points. J*,, p f a w i d e t e r r i t o r y ; we shall take the general mean o f these frequencies us an estimate o f c, and Lite mean o f aJI the quantities *

f^

—

.dl

So, in u nidi mensi anal cases. ^~ — ^ # = 1) gives an exponential

? J

- c)

calculated f r o m t w o points, P, and Pj, whose distance is r. as an decrease proportional lo c*p — \ Ikr• 17. This exponential decrease lias tfoç been llmnd in discontinuous eases [13. 151. Wciis and Kimura [25] extended the Formulas to Ihe tridimensional case; A * — BlVci

&*&

ef

+

^ CAP — .r '

2

2k

r Or

of


o-

. ,

: * = " . Mivjiiu ¡1 decrease proportional to

estimate o f *
rtwrittrults

[10, 16].

(2) W e can measure o n different individuals a biornetrieal i r a i t , neutral for selection, whose intensity can be considered the additive etVcct o f a certain number o f independent Mendelian r a n d o m var­ iables, A', w i t h cspectations M, (each .V, assuming values v, and I,

Vlkrio.

In nil these fur inula a, • is the standard deviation of llıc migration Bloua each axis of Loordiııaiçs fnugratinu may nol be normal); the eurrelation <&r\ with large distance r depends only on the ratio r.'o and on the rate i ,

w i t h probabilities tf, a n d p, as functions o f the l o c a t i o n ) ; the mean c o n elation between t w o individuals, / and

situated at a distance r

will be an estimate o f

These asymptotic formulas f . V M and its varieties) are Independent o f density. 6, so we can use Ihem i n u p o p u l a t i o n whose density

that is, will be an estimate o f 4>[r) i f we postulate that t h e rate o f

varies considerably over the years fas With Chetverikov's waves o f

n u t a t i o n , k, is the same f o r a l l the genes c o n c e r n e d . ' We must

vitality),

remember, however, that i h e c o r r e l a t i o n decreases i f a fraction o f

If Ihe individuals show a tendency to stay grouped i n " c o l o n i e s " or " s w a r m s , " we shall take that into account by postulating that each

Ihe variability iü n o t genetic (We then have l o multiply by the " heri tabii i t y " ) .

i n d i v i d u a l has respective probabilities n and (1 - u ) o f m a k i n g an inhnilcly small displacement w i t h variance *- o r o f m a k i n g a migra­ t i o n t o a distant p o i n t w i t h variance o-, i.e., by taking p[r| = L

ftG{t

)

-T

(I

—

[i)(j(ir-(.

*KT')

which gives

= ac- wn <<.•*+*•> _i_ ( i _

3-3.2 O t h e r A p p l i c a t i o n s (A]

Panmixia i n a finite isolated p o p u l a t i o n o f ,\' individuals can

be studied by assuming that the occupied area. 4, is equal t o 1 , ojp-r.yîîı*'*^

f r o m which we have a f o r m u l a f o r flt(rj, i.e . the same asymptotic r

• "Hu', b *ml> npnrnMnihiL- ' f * • cnruidtr the fact dial die bhti rnndom vuriulılcs .V, o[ each individual have, « a h random ınaiınp isee p. T ' i , n coeilicieni of inbreeding enual m the nunsLj-nLiiir and iltnominmerarc A^ir^'iXi — flf.-)' and 2\i 4- ıftıll^'t.t, Mi} , the nummuiinn ~' titiny nnw extended only to noahLimnlogous locc, (FORI nhük Utt LMfufclboB htlwetn die genetic com­ ponent of metrical trails {without diuninanLe rıur cni*fflsi>o i * — ^—71

expression, but w i t h variance at" + (1 — j.)n-. The experimental d e t e r m i n a t i o n o f the correlation as a f u n c t i o n

=l

1

T

L

i is equal to JV, and g(P, £ l

"f

u

j

eilhcJ to I . i f P a i x l y

l

w i t h i t H . o f to 0. i f Ihcy are outside. We have, i h o n . for C and M.C,

D) - » ( C , C ) = - " f f i - ^

2

U -

are

0in.i,

*W-Q>~

fj

2*

1

£,£j

[i

41

AJ^ In p a r t i c u l a r , i f we take for &P,

which ga^cs 4

cunsianl =

W

D

Q) Gauss's no n o o t r o p i c l a w ,

(J - W 2A [l - ( I r

- OT + fl - A )

r

By equaling this expression to em g h e n by (3.3,3) or ( J , 3 , J " ) , we

we find

£ | • constant = * t , and we have

obtain the si/.c N o f a patuuiclie group "'equivalent" to u g r o u p occupying a very small area and c o n s t i t u t i n g part o f u p o p u l a t i o n w i t h r a n d o m isotropic m i g r a t i o n . Let f r o m w h i c h we calculate
distances, w e obi i n a homogeneous p a r t i a l dilferenli.il equation, ( o r elliptic t j p e j . W e could introduce an analogous scheme w i i h thfee T h i s concept o f " e q u i v a l e n t effective n u m b e r ' i n t r o d u c e d by W r i g h t

dimensions l o represent ihe v a r i a b i l i t y o f an aquatic

population

2 1 , 2 3 . 2 4 , f o l l o w i n g entirely different reasoning, does not have the

according l o ihe i w o coordinates o f surface and d e p t h . T l i e a p p r o x -

weight that he attributes to i t . because it does not account for the

imate p a r t i a l dhFerenlial e q u a t i o n is easy to w r i t e , since it is a

c o r r e l a t i o n w i t h distance.

generalization o f ihe H c l m h o l l / . e q u a t i o n o n p, 73.

i H i We can t r y to f o r m u l a i c a scheme o f homogeneous but Honisotropic m i g r a t i o n I i n an u n l i m i t e d p o p u l a t i o n
3.4

APPENDIX:

DISCONTINUOUS

MIGRATIONS

two

independent displacements w i t h different laws o f p r o b a b i l i t y , in

T h e case o f discontinuous migrations refers l o the model in w h i c h

two

rectangular directions. i,c„ that

ihe individuals o f each generation do n o t i n h a b i t a continuous area but ralher a discrete set o f places (still called .4), each place being

gif,

Q\dS,,

= mix,

- mfffji

- y-J Nb| J v i .

looked al as u p o i n t (still culled C or D for t w o offsprings, / and J.

Dcshmulmg the coordinates o f P and Q by Jfe ft, jr.. y - {m and U m

being t w o functions each o i one variable. whu*c integral f r o m — * l o + * • is equal t o I . . and setting

lakeu i n F , , and £ or F for ihe possible places o f I heir parents. p l

P,

and P j ) . T h e i n l e g r a l i o n * i n f o r m u l a 1 3 J . 1 ) have to he replaced

by summations ••tf

o n the lattice A o f all possible places £ and F\

C ) is the p r o b a b i l i t y , i n place C, o f an i n d i v i d u a l c o m i n g f r o m

place E. We have

s i m i l a r l y for n), f o r m u l a (3-3,2) becomes

2

g « C i -

L;

Tfa

Ltı/ÎMle* tj - A J r V j U j J t i

i 4 Affrmii.x. Di\to*tinatas

F»paİMı*u

be supposed 1.1 be integers p and a.*

which components ouv tü Â î t —m. t J ı / J (

1 0

replaced by ıhc c o n d i t i o n a l p r o b a b i l i t y f o r L W O

parents, successively Liken i n place t , d» he identical, i.e., by l / ? A -

us put (<£". O

Mı/>r4ttt*t

Let

- i ^ p . ^ i . The m i g r a t i o n Jaw may then be defined by

the "generating f u n c t i o n "

t

d illere are A i equally 1

•• •

monoecious parent*;

When there arc An- and A'^e equally probable lather* J U J mothers, ıhc c o n d i t i o n a l |iı j bubi lily is sull 1, l . V the harmonic mean, I , A * - l ^ 4 ( ] / A u

A

n' we cull A ' the double o f

which

1- 1/A'aW-

r> - e

t

fcquatiou

13J.İJ

may now he w r i t t e n

converges ,J

absolutely

when

a! =

I and

|j

- I

(putting

and ¡1 •= c ^ , we have a F o u r i e r series).

In the same manner, we p u l Ifcfafti

t " _ J i'»_Ji w h e r e * ı j / f . q) neJE,

-

IUP-^JJ .

(3.4,3)

1

C ) expressed u s u f u n c t i o n o f the c o m p o n e n t s

p und 1/ o r the vector C £; i.e.. by i n t r o d u c i n g i n 0 , 4 . 2 ' ) the c o m ­ L

I f we call atC", D ) some solunun +*+AL\ M U

independent

- (HC, O ) - f ^ i t C , I>) is related to

o f 11, ibe difference fcfö

/ J by a recurrence n o w homogeneous, and suj»

- *£, F) = F} tends

p o n e n t s / / and*/' o f r i a n d / v a n d v < o r C £ , related hv p - p' + /%, Î

e

i ' + io,

t o zero when • goes [ J h i r h i i t y ; so « C , İJ) [when LL is supposed l o exist) is unique, and is ihe h m i t o f 0„ when u goes to iuHoily. C o n ­

S

versely, J JirniL existing tor some particular i n i t i a l eondıiion is u solution indepeudenl

o f n, j n d ıhıs is die same .is vhe l i m i t for any

i mu J I condition.

t'ta,

* . - , [ £ . Fte'pgtF,

CM**

i'ca

Since there is absolute convergence, we may make the summations in any order. By s u m m i n g first w i t h respect to E. since Ihe last three

So, Lo ublaıu ıhc U n t i l , l l is s u l l i u e n t t o lake Lite p a r i i e u k u i n i t i a l

f a c t o r * d o mH depend o n £ , wc have

c o n d i t i o n * * £ , F) = 0, w h i c h give* n

«C,D)-il-kr

- ^ f - ^ Z O - k ^ - ^ C i s ^ D ) ,

r

.1

G"_-i may be made a factor i n the s u m m a t i o n ,

which I hen gives

(JA2) pulling so we have C>V

£

* _ ! < £ , P W £ , O-

(SAî'j

L e i us m m supoose t h a i the m i g r a t i o n is homogeneous, i.e., that

• If t i * : notoiknh wmi Ut M A j f u n i eon.>n ı>f M ( M P « i " i n u n c r • > itrdinBiet"!. p ınU v arv ilu: t-n. vn» of lit.' E O D T U L I V I I C S o l £o*cr ıhc •• «•-:••• •• • uff. L

C ) depend* o n l y o n the c o m p o n e n t s id" Ihe vector CE, each o f

H

EnÍMliai

tf j

Mtu.íiitjB

Vapularían

and hv i t e r a t i o n ,

aeries is. as ï ( l • (•to* H

l i q u a t i o n Í.1.4.2J f o r tfC", / i ) m a )

•

ky**-,

-

absolutely convergent; so

may

we

put

'I

Ifl •

then he transformed i n t o an J.L.

equation g i v i n g a "generating f u n c t i o n " o r * ; it is sufficient to note W h e n s u m m i n g up the right-hand side o f f_V_V4j, we may begin

t h i t l i f we complete the definition o f homogeneous m i g r a t i o n by putting !Vg = N imli'pvtiiiiiil

r/pía,;

1

E, tf £ . £ J = constant goes- a

by noticing that gJE,

solution l o r tfC, D}. a solution w h i c h is k n o w n l o be u n i q u e ; so we may put tf £ . £ ) = constan! -

like the left-hand side, depend* only o n the c o m p o n e n t * o f /'< w i l l m i w t v called

which may he called M and v; : n I f we m u l t i p l y ( 3 A 2 ) by a'd* t e r m o r the s u m í

£ ÑJ

T h e right-hand side o f (3,4.2), C0

tf-W*tU.P

D) = nJ.p

- x q-y)

- x.tf

y), and calculating

- C . f l / a , 1/flJ - [ G ( l / < * , 1/0)].

t

A f t e r w a r d s , the s u m m a t i o n

over f

gives a factor G^{a, ,i) m

h

tfi.jr},*

this amounts to m u l t i p l y i n g each

hy n ' t f ' a ' - ' t f * - ' , p and ^ being the components

[ Q a . JJ,".

and the s u m m a t i o n

w h i c h gives the same

formula

over m gives a geometric series, as that obtained f u r the

Fourier

t r a n s f o r m in Ihe continuous case (but extended n o w to nonsymmetrical m i g r a t i o n J.

< i f C £ . x - / i a n d . i - q being the components o f CD

— C£-

- 0£,

^

-it

0

- *K
We then have

=

^

;

ci

—

Jn

A>=*+-

(3.4,4)

the

symmetrical

case,

where

giE.

C) - g{C, £ J ,

we

have

G ( l / r t , 1/jJ) H Gíít.tí). Hut Ihe " i n v e r s i o n , " i.e., Ihe p r o b l e m o f going hack f r o m Ihe F o u r i e r series " U n , J) to its c o e f f i c i e n t tf v.\) may be simpler l h a n

s e l l i n g ^ = Efr

using

!:•.(•.:!

f o r m u l a t i o n s o f these coefficients [ f o r m u l a ()_3.3)

w r i t t e n w i t h e " ' = a. tf' * = (J and integrated over A- f£ ( 0 . 2 s } and 1

T h e right-hand side may. i f * > ti, be summed up o u r all i aloes o f x and W

•

r {components

o f CO).

i,c.

flW

all points

D.

when

JdJ - I ; i l i s a m u l t i p l e series whose general l e r m (indexed by

" i , L . D ) has a modulus bounded

by (1 - A )

: m | J

((k 2w) |. F o r instance, let us sludy ihe Miiiinwiuiunal

ea&e when H

the coefficient o f eoarieesiry. for algebraic distance x. is called tf A ) and has a generating f u n c t i o n . I ^ n ) = I

..MI.

given by

c J £ , C)í,.,|£, 0 ) ;

but because o f homogeneity, Í J Í _ ( E . D) is the same as I gj£. 0), n /: and thus equal to I ; thus X m j f i DjlgjE. D) = 1. and then the £

y £

,. *

, f

l

J

a "

- m a

-fci/^/vjfJWvín,^)

i-[i-A)=ti( )C(iM n

li

I n mosi cases w here G f o H s a p o l y n o m i a l * kno* I hat tliu h ihe inhrcfitinp «-rilVitni m any place ' ^ . kmp " • u n i r mime foi the funclwn or Ihc I H W miniifrln: they i r e scalnrs. noi palms, and Í Í H H I L I m» K - euiuusal.

as i n the case o f m i g r a t i o n

between adjacent groups o n l y , where G i n ) •

1 - 2nt -r- ma + m a¬

w e shall sec t h a i the expansion i n t o p a r t i a l fractions g i v i s only t w o

B2

Izıılmiv"

terms w i t h

af ¡1 Mrt)dt!r:iM

İ'.ıj'itlu'ruıt

i J Apj'ttfiir.Y.'

large residues, i.e.. those corresponding I n the t w o

solu l i o n s near 1 o f the equation G"(a)G ( = These t w o W n - k\solutions, raj and u are obtained by developing G[n) = İ ^ / J ^ J ' : r

[ C £ = p and

/>> =

^

=

(1 -

k)-[{[ -

Miyratii/tii

fo) 2N}GMGn-' ) !

ai

-IL -

ky-£[G( )G(Y/ ,]* a

a

(where the Lİenomimıtoj equals a,)-,

O J i n t o the moments w,, o f the m i g r a t i o n

(1-W2A

law, using formulas

o

G(\)=Zrip)=

Diıcanrin/attı

_

1

- in -

I -ft.

iym

oa

I,

(where the d e n o m i n a t o r o f the lefthand fractions equals T h i s show* lhat the residues A\ and A cor res p on ding l o the t w o :

roots ( u j and « ) near I are much larger than the others: I f we suppose :

i l l <. 1 <. a . the only terms w i l h negative exponents in Ihe expansion

T J - G

I

- £p(P a

j — i

;

I K / ' ) - İftr — -«in

ü

A,_

f

O —

f r o m which

+

i»i

_A*_

w M l

h

c

» —

| h ( i i j ;

giving, for negative values

G f a ) = ] + mfa

- 1H- ( f f l - mdfa -

G ( l / a ) = I - unfa -

+ 0[( f t

L > / + tm- - Wi)(r* - \y-/2a

+

Q [ >

-

=

+.-

u

ii — Q|

- i

L

L

H

U

S

» _ 0

T

a

J

with

1) +

-

IJ= -f- (m, - m^(

a

- I f / l

A M I

- f a ) / [ 2 A a ( I - at)] ~ f l r

&)/4H V2İL

i

<

a

(and .v > I}). Similarly, we obtain positive exponents by expansion o f :

! ) " ] ,

-

I)- -f- 0 [ ( r t

!) J. l

tı —

which introduces only (even in the nonsymmetrical case) ihe variance so we obtain, when A is small and when we equate

G ( a ) G i l / d } i n 1/(1 —

. _£fj_

n

G f u j G U . "J = 1 + 0": -

o" = mi — nü:

(>|

l) J,

-I- 0 [ ( « - - l ) * ] = 1 - mifu -

,

flit

1 + 2k, two solutions near I . a, and u-

T

(dven hy

{-A=fai)

+

flj

3

E

_u

afa­

which give* f w h e n ' * ^ 0) tfx)

- ^ W

1

1

— .4,(1 -h

>

and p e c n l i a r l ^ f £., - 1 =

* { 0 ) - #a - ,4, - £1 - ^ ) / 4 A ' ^ 2 Â ,

±V2k/a . !

D

I

Let us recall that the espunsion of.|<ö) i n l o partial fractions uses 1 +

t/ii roots i i . o f iih d e n o m i n a t o r ' anil is S

ANa^lk

. each A . heinc niven ' Tburv niuv lv mher pooU ıhan <M of modulus < 1, Hut they HHi Nüi(toH ınuca M M D limn 'J, 1 f[>r i > 0, HfB linpusitc V A I Ü L I , .- EinJ - t . j

by "Thai i i . uli valUL-s ,., fiucli rhni
84

L< • •

•• •

pf

J .'• İr- i. •

I'-

I .•'

'If

So we obtain Use general fDi'umkı (ıhe sign — meaning

"only

when A- İs s m a l l " ) :

Bibliography •

—

I H- 4A nv'2fe r

The

numerical decrease w i t h distance is the same as in die u n i ­

dimensional continuons case; i[ may be seen that e* also is the same; but İn the two-dimensional case.

is very different, (depending, as

we have seen, on log 2k and not on

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Eirxwir'

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13.

14.

;

. IftU). "QLickjue* schémas probuhi Lilies Mir la M di) i t i des <<• ı : ı . ı I • " -lurnr/ı'.ı /'mr. uV Ivıki, Seleaces. strict A .

—

. I W i r " U n ütmemeni si udia etique des problèmes lincııircs çn

:

ténu [Uf ile." Annuls 15. —

, l

r

•

İdi i |i

Univ. ,fc Luvt. lı

íi-rTj/1'wnnf i*n Malhrnbtlmıı

16. —

ıı i

-Scirıuvs. A.. 14:7^-117. b t i o n d ı j p " Pur.

SíaltMin

. 1066. Pmtmiutitès eıhem/Hf.

Fifth

Rtrhvtry

ıimf Pn>tnıhrftty, I V . J J T.

Presses i •:.. • -ı.ı.ı-. •:• l-'rıınce.

IT ï*ciir*on. K. I *H)4. "< In u generalised t h e o r y nf allem il l i v e i n her i lame r

w i l h h|xvi:il retírente l u Mendrl's laws." Plut. Trans.

fio\ik¡ Sue.,

w r m A . 203:53 KA. 18.

, 101 W. " U n ıhe ancestral ^ n i k i r correlations t$S Mmdeliun pupuJjlion maline n rundum."

A m u / S i d /.«»iv/««. »eiies El.

iy. Philip, U- ıy.3W. " M u t i n a şi^lertMi in wild populations o f m/pinu-i jnıl l i n ruwuiiv" J. G r n r u r j * 36:107-21 I .

Dtrmatts

30. .Snow. J : , C. I U I . " f h i the delerini nu lion of lite chief correlations tvlween colliUerati in ıhı- ease o f a simple Mendelian populdlion muJiruj al random." Proc. Ruyttt S"t. Um fort, series U , 83:37 55, 21. Teissier, i i - 1 0 4 4 '-J jjurkhre ües ıierıeı lethaus les popuI¡1 liona sJjiionıiires punmiciiqne." fiVi. Sıirntif. 82:145 54. 22. U ' j i f h i . S. 1931. " L m l u i i o n m Stcndelian imputations." I*:"ï7-150. 23.

. 103U. Sruitiiiiru!

gerretta

in rrfarhm

íirnritei,

w ı
SdvMttkjUcs el Intiu-> iridies, BflZ l'.ıris: Hermunn. 24. —

. İVjfı.

I•.

LAoluliun. 31 M jnil 3S, 37 án
altınının. 7. I i AincfKin IcHİuriv H M

b y dkiaiwc under doersv sysicnt. oí malina."

25. Wdv», ti. I i . , and M . Kimura. 1065. "Sieppinusiıuıe model of jjeneU'. correlation." J. Applied Pra/mftflfiy, 2:129.

,ll| a ıı1UIJIJÜıL.

HuyiV formula. M Ucumun, P.. it> Hrsul function, 7 X Nood Kfoufh. 7n. 15. W i'd'pin'n utufful't-

'

ftétt, K A.. p, 3J F m r i B ifïnïlorio. 71*. Kl trirdWilin iKiilH>n< " ' I rK*kiel^s « I J I M I I , 15 :

7S

Oulieii. H. *f, 2lf, 36 h.. 36, l<2 63.77 tiw ni .1 i " • nf 4. 21, 5JI

e o e l l h ^ r u «f, H - L1 - I i i . tt 32, « 4 Q p H l t

66. Ttb. 75. RI eomjnnuinil>, 14 16. * ¿nriebiuin. 1J-3U

*'

eadhCH.-cn of. 21 25. 17 2*. 70

;L pnofi. İ H M» cinidi1»*nul, 43 JJ lırnkün-enıul, 2İ. 2Í

pAilinl. 2h ( r m i i r l t ÉMf« 6. l ï /íriíWH-oc> yutptifi. M disjuncuon, s,fi,21 dominant. 3 4, 1«, ^ 10) 57-3H jn<] rT>üiiUauu 7 ix|uiMHiunı. H , *W , .u,, . 52. 63-64

•

r

. . . I -

r

MjkUne. J. B S . 34 h ^ i d ^ i law. 14 Hi-lniKull/ n|Lii>linn, 73f, 77 hemophilia. 6

. • . i . - - • • m hfcndmn- 3. -1 disenniinunn. 1 niuUifKturuil. 3f. 31 ItticrnttOTKS. tnlTrHdii^

(ih'ltwwiit ^

l

"- - ' lù

ar

and radiKomi ni coonemrj. v. t i . 41,70

HU

rW*-

t

iHh-Éi.ii

mie of. I I . 4 0 . Í I . 33. 57. W. 76 and w l n f i n i i . 4| » / » I I H Î I » . 17. »

5)

anıl Lim^angıiınıi). I-I Ih mıpaım. 13 JO

21 n j i w r . 3| il pnwlm 44*67 in ı rinde baUlı-tl v U I I N H I , 73-76

I M I M..- • M

Julianen, W l . . . I l r

IM

K. mı ur A . M., î J

I V J Î M - I , k . Jt, 2lt.

LwrKUhf. M.. 73 I .-i [i I ikL t- ıiqinİPrni4iiKHi. JflJ Liıt|iuupuVı I Î H M I L T I , ? |

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