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¡¡ltturtcen 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). " = 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

)

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 Hpul\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 + 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 loTcnun '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..--..' . ' 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


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 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 " / 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) 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.:•'

.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 «sJf 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


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

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=*+-

(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

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.|pusitc V A I Ü L I , .- EinJ - t . j

by "Thai i i . uli valUL-s ,., fiucli rhni 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