THE
MATHEMATICS
OF
HERIIDJTY
THE MATHEMATICS OF HEREDITY
Gustave
Malccor
D e m t u i o i M . Ycrnianos if CiiJif...
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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