dr. sewall wright - coefficients of inbreeding and relationship

7/27/2019 Dr. Sewall Wright - Coefficients of Inbreeding and Relationship

1/10

Coefficients of Inbreeding and RelationshipAuthor(s): Sewall Wright

Source: The American Naturalist, Vol. 56, No. 645 (Jul. - Aug., 1922), pp. 330-338Published by: The University of Chicago Pressfor The American Society of NaturalistsStable URL: http://www.jstor.org/stable/2456273.

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2/10

COEFFICIENTS OF INBREEDING

AND

RELATIONSHIP

DR.

SEWALL WRIGHT

BUREAU OF

ANIM2AL

INDUSTRY,

UNITED STATES DEPARTMENT

OF

AGRICULTURE

IN

the

breeding

of domestic animals

consanguineous

matings are frequently

made.

Occasionally matings are

made between very close relatives-sire and daughter,

brother

and

sister,

etc.-but

as

a.

rule

such close

inbreed-

ing

is

avoided and there s instead an attemptto concen-

trate

the

blood

of some

noteworthy

ndividual

by

what

is known as line breeding. No regular system of mating

such as might be

followed with laboratory animals is

practicable

as a

rule.

The importance

of having a coefficient y means of

which

the

degree

of

inbreeding may

be

expressed

has

been

brought

out

by

Pearl'

in

a number

of

papers pub-

lished between

1913

and

1917.

His

coefficient

s

based

on

the smallernumberof ancestors n each generationback

of an inbred

individual,

as

compared

with

the

maximum

possible

number.

A

separate

coefficient

s

obtained

for

each

generation by

the

formula

Z"

=

100

(I

-

)

100

(I

-2ttl

Pn--i 2fl1+1

where

q,,+,/2u+1

s the ratio of actual to maximum pos-

sible

ancestors in

the n + 1st generation. By finding he

ratio of a

summation of these coefficients o a similar

summation for the maximum possible inbreeding in

higher animals,

viz., brother-sistermating, he obtains a

single

coefficient

or

the.whole

pedigree.

This coefficient

as

the

defect, s Pearl himselfpointed

1 AMERICAN NATURALIST,

1917, 51: 545-559; 51: 636-639.

330

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3/10

No. 645] COEFFI CIENTS

OF INBREEDING

331

out, hat t may

comeout

the

same forsystems f breed-

ing whichwe

know re

radically

differents far as the

effects f inbreedingre concerned. For example, n

the continuous

mating

of double

first ousins, n

indi-

vidual

has

two parents,

four

grandparents,

our

great

grandparents

nd four n every

generation,

ack

to the

beginning

f

the

system.

Exactly

the same is

true

of

an individualproduced

by crossing

different

ines, in

each

of which

brother-sister

atinghas

been followed.

Yet in the first he ndividualwillbe homozygousn all

factors

f

the

system

has been

in progress

ufficiently

long;

in

the secondhe

will be

heterozygous

n

a

maxi-

num number

f respects.

In

order o

overcome his objection

earl has

devised

a

partial

nbreedingndex

which

s

intended

o express

the

percentage

f the nbreeding

hich

s

due to

relation-

shipbetween hesire and dam, nbreeding eingmeas-

ured

as above

described.

A

coefficient

f relationship

is used

in this

connection. These

coefficients

ave

been

discussed

by Ellinger2who suggestscertain

lterations

and extensions

y means of which

he total

inbreeding

coefficient,

total relationship oefficient

nd a total

re-

lationship-inbreeding

ndex

for

a

given

pedigree

an be

compared n thesamescale.

An inbreeding oefficient.

o be

of most

value should

measure

s directly

s possible he

effects

o

be.

xpected

o01

the average

from

he.

ystem

f mating

n the

given

pedigree.

There

are twoclasses

of effects

hich

re ascribed o

inbreeding: irst, a decline

n

all elements f vigor,

s

weight, ertility,itality, tc., nd second, n increasen

uniformity ithin the

inbred

stock, correlated

with

which

s

an

increase

n

prepotency

n outside

crosses.

Both

of these

kinds of

effects ave

ample experimental

support

s average not

necessarily

navoidable)

conse-

quences

of

inbreeding.

The

best explanation

f the

de-

crease n

vigor

s

dependent

n the view

that Mendelian

2

AMERICAN NATURALIST,

1920, 54: 540-545.



4/10

332

THE AMERICAN NATURALIST

[VOL.

LVI

factors

unfavorable

to

vigor in any respect

are more

frequently

ecessive than

dominant, .

situation which

is

the logical consequence of the two propositions that

mutations

are more

likely to

injure

than

improve

the

complex

adjustments

within

n organism

and

that

injuri-

ous dominant mutations

will

be relatively

promptly

weeded

out,

leaving the recessive

ones to accumulate,

especially

if they

happen to be

linked

with

favorable

dominant

factors.

On

this view

it

may

readily

be

shown

that the decreasee n vigor on starting inbreeding in a

previously

random-bred

stock

should be directly

pro-

portional

to

the increase

in the

percentageof

homozygo-

sis. Numerous

experiments

with plants and lower

animals are

in

harmony

with this

view.

Extensive

ex-

periments

with guinea-pigs

conducted

by the Bureau

of

Animal

Industry

are in

close

quantitative

agreement.

As for the othereffects f inbreeding,fixationof cha-r-

acters

and

increased prepotency,

hese

are of course

in

direct proportion

to the percentage

of homozygosis.

Thus,

if we can calculate

the percentage

of homozygosis

which would follow on

the

average

from

a given system

of mating,

we can

at once form

the

most.natural

coeffi-

cient

of

inbreeding.

The

writer3

as recentlypointed

out

a methodof calculating this percentage of honmozygosis

which

is

applicable

to the

irregular

systems

of mating

found in actual pedigrees

as well

as to regular

systems.

This method, t mav

be

said.

gives

results widely

different

from Pearl's

coefficient,

n many

cases

even as regards

the

relative

degree

of inbreeding

of two animals.

Taking the

typical

case in which

there

a-rean equal

number of dominant. nd recessive genes (A and a) in

the

population,

the random-ibred

tockwill be

composed

of

25

per cent.

AA,

50

per cent.

Act and

25 per

cent.

aca.

Close inbreeding

will tend

to convert the

proportions

to

50

per cent.

AA,

50

per

cent. aa,

a change

from 50

per

cent.

homozygosis

to

100

per

cent.homozygosis.

For

a

natural

coefficient

f inbreeding,

we want

a scale which

3

Genetics,1921, 6: 111-178.



5/10

No.

645] COEFFICIENTS OF

INBREEDING 333

runs from 0 to 1, while the percentage of homozygosis

is

runningfrom,

0

per cent. to

100

per cent.

The for-

mula.2h-1, whereh is the proportionof complete homo-

zygosis, gives the required value.

This

can also be

written 1-2p where p

is

the proportion

of

heterozygo-

sis.

In

the

above-mentionedpaper

it was shown

that

the

coefficient

f

correlation between

uniting egg and

sperm is expressed by this same formula,

f

1-2p.

We

can thus obtain the coefficient

f inbreeding

fb

for

a

given individualB, by the use of themethods thereout-

lined.

The symbol rbc, for the coefficient

f the correlation

between

B

and C, may be used as a coefficientf relation-

ship. It has the value 0 in the case

of two

random

indi-

viduals, .50 for brothers in a random

stock and ap-

proaches

1.00

for individuals belonging

to a closely in-

bred subline of the general population.

In

the general case in

whiicli

ominants

and recessives

are not

equally numerous, ihe omipositioll f the random-

bred stock

s

of the form

x2

_4l 2xy Ala,

y2

aa. The per-

centage

of

homozygosis

s

here

greater

than

50

per cent.

The rate of increase, however, under

a given system of

mating, is always exactly proportional

to that in the

case of equality. The coefficients thus of general ap-

plication.

If an individual is

inbred,

his sire

and dam are con-

nected

in

the pedigree by lines of

descent from a com-

mon

ancestor or ancestors. The coefficientf inbreeding

is

obtained by a summation of coefficientsor every line

by which the parents are connected, each line tracing

back fromthe sire to a commonancestor and thencefor-

ward

to the

dam,

and

passing through

no individual

more than once. The

same ancestor

may of course be

involved

in

more than

one

line.

The path coefficient,

or

the

path,

sire (S) to offspring

(0), is given by the formulapo.s

-

I/(I + fs)

/(1

+

fo),

where

fs

and

fo

are

the

coefficients

f

inbreeding

for

sire



6/10

334

THE AMERICAN

NATURALIST

[VOL.LVI

and offspring,

espectively. The

coefficientor the

path,

dam

to

offspring,

s

similar.

In the case of sire's sire (G) and individual, we have

po.g Po.s

ps.g

4V

(1

+

fg)/

(

+

fo),

and

forany ances-

tor

(A) we

have for the

coefficient

ertaining to a given

line

of descent

os

(4)V(1 +

fa)l(l

+

fo),

where

1

is

the

number of

generations between

them

n this line.

The

correlationbetween two

individuals (r ) is ob-

tained

by

a summation

of

the

coefficients

or all connect-

ing paths.

Thus

Tbc

=

vPbaPca

=

,

+71

1 + ba

2

(1

+

bb)

I +

be)

where

u

and n' are

the

number of

generations

in

the

paths from

A

to

B

and

from

A

to

C,

respectively.

The formula for the correlation between uniting

gametes,which

s also

the required

coefficientf

inbreed-

ing, s

fo

=24'saV\/(t

+ fs)

(1

f

)

where

rad is

the correlation

between

sire

and dam and

fs

and

fd are coefficients

f

inbreeding of sire and

dam.

Substituting he value of

rsaz

we obtain

fo

(2)

?1+(1

+

fa)

If

the ancestor (A)

is not

inbred,

the component

for

the given path

is simply

(0)?1+1"'

where

n

and

n'

are

the

number

of

generations *from ire

and dam

respectively

to the ancestor in question. If the common ancestor is

inbred

himself, his coefficient f

inbreeding (fa)

must

be

worked out

fromhis pedigree.

This

formula gives the

departure from the

amount of

homozygosis

under

random

mating toward

complete

homozygosis.

The

percentage

of

homozygosis (assum-

ing

50

per

cent. under

random

mating)

is

I

(1+

fo)

X 100.



7/10

No.

6451

COEFFICIENTS OF

INBREEDING

335

4o -4-

4

CC

4-

r-~

c

C

4CC

-

-j~

--

c:j

co

In

r-j :

CA

41

rC

Fo

|

2~~~~~~~~c

-

h

temU$>

;

>;

S

;

>

;

H I

; > 0

V hV

e

~ ~

~ ~~~~~~~~~~~~~~~~~~~~~

.

r

rd

r4

d

bon

t,

NDV

C~~~~~~~~~~~~~~~~~~~~y

C)

biC

'-

Ct

o~~~~

~ ~

c

CC

N~~~~~~~~C

CC

O

& '

CC

-:

?O

0

H

001

CC~~~~~~~-

~ ~

C

CCO

CC

0

CC

*---



8/10

336

THE AMERICAN NATURALIST

[VOL.

LVI

- zD; n

?e

t $

17

s G

a;

E

, ;

~~~~~~ct

-1Z

cq

0

~ ~ ~

Q _

00 m0

C)

CA

--

-

-

0

Co

0

~CKl

o

8>

CA8

y~~~~

o0,

040

4W

0

0'-'

0~~~~~~~~~.

*H C:1

b0



9/10

No.

645]

COEFFICIENTS OF

INBREEDING

337

By

this means the

inbreedingn an actual

pedigree,

however rregular he system

of

mating,

an

be

com-

pared accuratelywith hatunder ny regular ystem f

mating.

As

an

illustration,ake

thepedigree f

Roan

Gauntlet,

a famousShorthorn

ire, bred by

Amos

Cruickshank.

This bulltracesback n

every ine to a mating

fChain-

pion

of

England with a

daughter

r

granddaughter

f

Lord

Raglan. For the

presentpurposewe will assume

that thesebullswerenot at all inbredthemselvesnd

not related

o each other. Since the sire'traces

wice

o

Champion

f

England and

twice o

Lord

Raglan and

the

dam

once

o each bull, here

re

in

all four

ines

by

which

the

sire

and

dam are

connected.

Common

Ancestors

Individual of Sire n/a

21

f' (t)fl?7l'?1

and Dam

X

(1

+

ba)

Roan

Gauntlet

Champion of England

45,276

(35,284)

(17,526)

..........0

2 1

.062500

2

.062500

Lord

Raglan

(13,244). .0 3 3

.007812

3 .007813

.140625

The coefficientfinbreeding omes out 14.1per

cent.,

a

rather ow figurewhen

compared o such

systems s

brother-sister

ating one

generation 5 per

cent., wo

generations 7.5 per cent., hree enerations 0

per cent.,

ten

generations 8.6 per

cent.) or parent-offspringa-

ting,

one generation 5 per cent., wo

generations 7.5

per cent., hreegenerations 3.8 per cent., pproaching

50

per

cent. s a limit).

As an

exampleof closer nbreeding,ake

the pedigree

of Charles

Collings'bull,

Comet.

The

sire

was the bull

Favorite

and the dam

was from matingof Favorite

with

his

own

dam.

As

Favorite was

himself

nbred

o

some

extent,t

is

necessaryto calculate

first

his

own

coefficientfinbreeding.



10/10

338

THE

AMERICAN NATURALIST [VOL.

LVI

Common

Ancestors

Individual of Sire fa

t n

(l

f

+n(+l

and Dam

X

(1

+

fa)

Favorite 252) Foljambe (263) 0

1

1 .1250

Favorite

cow) .......

0

2

1

.0625

.1875

Cornet 115)

Favorite 252)

.

1875 0

1

.2969

Phcenix

0

1

1 .1250

Fojaminbe

0

2

2

.0312

Favorite cow) 0 3 2 .0156

.4687

In

the case of Comet, Foljambe and Favorite (cow)

each appears

twice in the pedigree of the sire and three

times

in

the pedigree of the dam. However, only those

pedigree paths

which connect ire and dam and which

do

not pass through he same animal twice are counted. The

listing of Favorite (252) and Phcenixas common ances-

tors eliminates all

but

one

path

in

each case

as

regards

Foljambe and

Favorite cow.

'The

remaining paths are

those

due to the

common descent of Bolingbroke,

the

sire's sire and Phcenix as the dam's

dam

from the above

two animals.

By tracing the

pedigrees back to the beginningof the

herd book, the coefficientsf inbreeding are slightly n-

creased. This meant

going back to the seventh genera-

tion

for one common

ancestor of the sire and dam of

Favorite. The coefficientn the case of Favorite be-

comes

.192

instead

of

.188

and that of Comet 471 instead

of

.469.

Remote

common

ncestors

in

general have

little

effect

on

the

coefficient.

t

will

be

noticed that

Comet

has a degree of inbreedingalmost equal to threegenera-

tions

of

brother-sister

mating

or an

indefinite mount of

sire-daughter

mating

where the sire

is

not

himself nbred.

dr. sewall wright - coefficients of inbreeding and relationship

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