[1967] sewall wright - surfaces of selective value

8/18/2019 [1967] Sewall Wright - Surfaces of Selective Value

1/8

SURFACES

OF SELECTIVE VALUE*

BY S E W A L L WRIGHT

LABORATORY

OF

GENETICS,

UNIVERSITY OF

WISCONSIN,

MADISON

C o m m u n i c a t e d

A p r i l 1 4 ,

1 9 6 7

T h e r e

h a v e

b e e n

many r e c e n t

p a p e r s o n e v o l u t i o n a r y c h a n g e i n

s y s t e m s o f

i n t e r a c t i n g

l o c i . B e c a u s e o f

n o n r a n d o m

c o m b i n a t i o n , i t h a s b e e n n e c e s s a r y

t o

u s e

e l e c t r o n i c

c o m p u t e r s

i n a l l

b u t

t h e s i m p l e s t c a s e s . I f ,

h o w e v e r , t h e s e l e c t i v e

d i f -

f e r e n c e s

a m o n g g e n o t y p e s

a r e

a s

s m a l l a s

s e e m s u s u a l l y t o

b e

t h e c a s e i n

n a t u r a l

v a r i a b i l i t y a n d t h e

l o c i

a r e

i n d i f f e r e n t c h r o m o s o m e s , a s i s u s u a l i n

o r g a n i s m s

w i t h

t y p i c a l n u m b e r s , o r e v e n i f

t h e y a r e l o o s e l y l i n k e d , u s e f u l

a p p r o x i m a t i o n s

may

b e

o b t a i n e d

b y

i g n o r i n g

t h e s m a l l

d e p a r t u r e s f r o m

r a n d o m c o m b i n a t i o n .

F o r m u l a e

u n d e r

Random

C o m b i n a t i o n . - U n d e r

r a n d o m

c o m b i n a t i o n , t h e g e n o -

t y p i c

f r e q u e n c y i s b y

d e f i n i t i o n t h e

p r o d u c t o f t h e

f r e q u e n c i e s o f t h e c o m p o n e n t

g e n e s ,

w i t h

d o u b l i n g f o r

e a c h h e t e r o z y g o u s

l o c u s .

U n d e r t h i s

a s s u m p t i o n , t h e r a t e

o f

c h a n g e o f

a g e n e f r e q u e n c y

P z p e r g e n e r a t i o n i s g i v e n

i n t e r m s o f

f r e q u e n c i e s

f ,

a n d

r e l a t i v e

s e l e c t i v e v a l u e s

w ,

o f

g e n o t y p e s

b y :

A P .

=

P . ( 1

-

P D )

Z

W

a

1 2 v .

1 )

I n

e v a l u a t i n g w h e r e

t h e r e a r e m u l t i p l e a l l e l e s 2

a p i -

p j / ( 1

-

P x ) .

2 )

W e

w i l l d e a l

h e r e

o n l y

w i t h

c a s e s

i n

w h i c h t h e s e l e c t i v e v a l u e s

o f

g e n o t y p e s a r e

i n d e p e n d e n t o f

t h e i r

f r e q u e n c i e s ,

u n d e r

w h i c h 3

A P X

=

P X

1

P X )

/ 2 w .

3 )

E f f e c t o f

D e p a r t u r e f r o m

Random

C o m b i n a t i o n , a S i m p l e C a s e . - T h e

d e p a r t u r e

a t

m e t a s t a b l e

e q u i l i b r i u m o f

t w o - f a c t o r g a m e t i c

f r e q u e n c i e s f r o m r a n d o m

c o m b i n a -

t i o n

w a s

g i v e n

many

y e a r s a g o 4

I i n

a s i m p l e b u t

i m p o r t a n t

t y p e

o f e x t r e m e

i n t e r a c -

t i o n

i n

w h i c h

t h e

o p t i m u m i s

a t

t h e

m i d p o i n t

o f

t h e

s c a l e .

G e n o t y p e

G r a d e

w

AABB

M

2 a

1-4s

AABb, A aBB

M

a

1-s

A A b b , A a B b ,

aaBB A I

1

A a b b ,

a a B b

AI-a

1-s

a a b b

M-2a

1-4s

H o m a l l e l i c

AAbb a n d aaBB a r e

b o t h a t t h e i n t e r m e d i a t e

o p t i m u m

a n d t h u s

a r e

a t

s e p a r a t e

s e l e c t i v e p e a k s

( 1 , 0 )

a n d

( 0 , 1 )

w i t h

r e s p e c t

t o

t h e

g e n e


p

a n d

q . T h e

e x t r e m e s , a a b b

a n d

AABB,

a r e

i n

s e l e c t i v e

p i t s

( 0 , 0 )

a n d

( 1 , 1 ) ,

r e s p e c t i v e l y .

T h e r e

i s

m e t a s t a b l e

e q u i l i b r i u m

a t t h e

s a d d l e

( 0 . 5 , 0 . 5 ) .

L e t t i n g

c

b e

t h e a m o u n t o f r e c o m b i n a t i o n

i n

d o u b l e

h e t e r o z y g o t e s ,

t h e

e q u a t i o n e x p r e s s i n g

t h e l a c k

o f

c h a n g e

i n


f A b

a n d

f a B

o f t h e

b a l a n c e d

g a m e t e s

a n d i n

t h o s e

1 6 5


2/8

GENETICS:

S .

WRI GHT

o f t h e u n b a l a n c e d g a m e t e s ,

f A B

a n d

f a b ,

i s

e a s i l y

f o u n d a t

m e t a s t a b l e e q u i l i b r i u m , a t

which

f a B

=

f A b and

f a b

=

f A B

=

( / 2 )

-

f A b .

8 s f A b

-

4 ( s

-

c ) f A b

-

c

=

0 ,

( 4 )

f A b

=

f a B

=

[ s

-

c

s2

c 2 ] / 4 s

t

/ 4 ) ( 1

s / 2 c )

i f

s < < c ,

( 5 )

f A B

=

f a b

=

[ S

C

-

V / S 2

C 2

] / 4 s

t

( 1 / 4 ) ( 1

-

s / 2 c )

i f

s < < c ,

W

=

1

-

4 8 f A R

B ( 6 )

E x t e n s i o n

h a s b e e n m a d e

t o

t h r e e e q u a l l y s p a c e d

l o c i . 6

A

s o m e w h a t

m o r e

g e n e r a l c a s e ,

t

f o r

t h e

s e l e c t i v e

d i s a d v a n t a g e o f AABB a n d o f

a a b b

a n d

s u p p l e m e n t a t i o n

b y

a d d i t i v e

h e t e r o s i s

h i

a n d

h 2 ,

a t

t h e

l o c i ,

h a s b e e n

p r e s e n t e d 5

6

a n d c o n f i r m e d f r o m a

d i f f e r e n t v i e w p o i n t . 7

T h e

e q u a t i o n

f o r g a m e t i c

f r e q u e n c y f A b a t

e q u i l i b r i u m ( m e t a s t a b l e o r s t a b l e ) b e c o m e s

8

( 4 s

-

t f A 3 b -

8 ( 3 s

-

t

f ~ b

2 ( 2 s

-

t

+

2 c )

f A b

-

=

O,

7 )

w h e r e

c

=

c (

h

h 2 ) .

G e n e r a l

E q u a t i o n s f o r

Two P a i r s

o f

A l l e l e s . - T h e

g e n e r a l e q u a t i o n s

f o r r a t e s

o f

c h a n g e

o f

g a m e t i c


i n

t h e

c a s e

o f

t w o

p a i r s

o f

a l l e l e s w e r e

g i v e n

b y

K i m u r a 8 f o r c o n t i n u o u s l y r e p r o d u c i n g

p o p u l a t i o n s .

T h e

c l o s e l y

s i m i l a r

d i s c r e t e

r a t e s p e r g e n e r a t i o n w e r e g i v e n

b y

L e w o n t i n a n d

K o j i m a . 9

A f A B

=

[ f A B

WAB

- )

-

C

D W A a B b I / D )

A f A b

=

[ f A b

( W A b I D 3

+

C

D W A a B b ] / W

8 )

A f a B

[ f a B

( W a B

-

W )

C

DWAabI/,I5

A f a b

=

[ f a b

( W a b

-

i)

-

C

D W A a b ] / b i

)

w h e r e

D

=

f A B f a b

-

f A b f a B

m e a s u r e s

d e p a r t u r e

f r o m

r a n d o m

c o m b i n a t i o n .

W AB

=

f A B W A B / A B

f A b W A B / A b

+

f a B W A B / a B

+

f a b W A B / a b ,

e t c .

W

=

fABWAB

f A b W A b

f a B W a B

f a b W a b .

Q u a s i - E q u i l i b r i u m . - K i m u r a 0

h a s shown

t h a t

t h e

r a t i o R =

f A B f a b / f A J f a B

a p -

p r o a c h e s c o n s t a n c y , q u a s i - e q u i l i b r i u m , d u r i n g

t h e

e v o l u t i o n

o f

s y s t e m s

u n d e r

w i d e l y o c c u r r i n g

c o n d i t i o n s .

I t

i s

c o n v e n i e n t

t o

u s e

s y m b o l s

t h a t

m a k e

R

g r e a t e r

t h a n

1

a s

f a r a s

p r a c t i c a b l e .

A f A

B

A f A b

A f a R

A f a b

A

l o g

R

-A

+ , 9

f A B

f A b f a B

f a b

w

A

l o g

R

t

[ W A B

-

W A b

-

W a B

W a b ]

C

D W A a B

+

~~~~~~~~B

A b 1

f a B f a b

( 1 0 )

I n a

h a p l o i d p o p u l a t i o n ,

[ W A B

-

W A b

-

W a R W a b ]

i s

a

m e a s u r e

o f t h e

i n t e r a c t i v e

1 6 6

P R O C . N .

A . S .


3/8

GENETICS:

S .

W R I G H T

s e l e c t i o n ,

s I .

I n t h e d i p l o i d

c a s e , i t r e q u i r e s

f o u r

c o e f f i c i e n t s ,

e a c h

r e l a t e d t o o n e

o f t h e

f o u r k i n d s

o f g a m e t e s .

S I ( A B )

=

WAB/AB

-

W A B / A b

-

WAB/aB

W V A B / a b ,

e t c . ,

( 1 1 )

3 I

=

f A B 3 I ( A B )

A b 3 I ( A b )

a B 3 I ( a B )

a b 3 I ( a b ) ,

12)

w

A

lo g

R

3

C W A a B b

f A b f a B

R

1

±

)

( 1 3 )

f A

B f A b JaB f a l b

K i m u r a

s h o w e d

t h a t w i t h

s m a l l

s e l e c t i v e

d i f f e r e n c e s

a n d

l o o s e , i f a n y ,

l i n k a g e ,

R

c h a n g e s s o l i t t l e

i n a s i n g l e g e n e r a t i o n

t h a t

i \A l o g

R may b e t r e a t e d a s

z e r o t o

o b t a i n

a

u s e f u l

a p p r o x i m a t e

r e s u l t .

He

g a v e

a

n u m b e r

o f e x a m p l e s , c a l c u l a t e d

t h r o u g h h u n d r e d s

o f

g e n e r a t i o n s

b y e l e c t r o n i c

c o m p u t e r ,

i n

w h i c h R

a p p r o a c h e d

c o n s t a n c y . He

a l s o

i l l u s t r a t e d t h e

f a i l u r e o f t h e p r i n c i p l e

w h e r e

m o d e r a t e l y g r e a t


s e l e c t i o n i s

a s s o c i a t e d

w i t h

v e r y

t i g h t

l i n k a g e

a n d

R

i n c r e a s e d

w i t h o u t

l i m i t

a s

f i x a t i o n

o f o n e o f

t h e

g e n o t y p e s

w a s

a p p r o a c h e d .

I t i s c o n v e n i e n t

t o

l e t x

=

R

-

1 ,

a n d u s e K

=

S I / C W A B b

a s a n i n d e x t h a t e x -

p r e s s e s

t h e

r e l a t i o n

b e t w e e n


s e l e c t i o n a n d

r e c o m b i n a t i o n

w h e r e

t h e

t e r m

w A

l o g R

i s n e g l i g i b l e .

X 2

( A

b

f a B )

X

( 1 -

K)

-

K

=

0 .

I f

i v A l o g

R

i s

n o t

n e g l i g i b l e i n c o m p a r i s o n

w i t h

3 , ,

i t

i s

p o s s i b l e t o make

p r o -

v i s i o n a l e s t i m a t e s a n d r e d e f i n e

K a s

3 ,

-

i t

A

l o g

R ) / C W A a B b

i n

t h e

i t e r a t i o n p r o c e s s

o f

s o l u t i o n :

x

=

[ V / ( 1

-

K ) 2

4 K ( f A b

f a B )

-

( 1

-

K)

] /

[ 2 ( f A b

f a B )

]

( 1 4 )

I f K

i s

a l w a y s s m a l l ,

x

i s

a p p r o x i m a t e l y

K/ 1

-

K ) , R

i s

n e v e r much l a r g e r

t h a n

1 a n d

s u f f i c i e n t l y

u n i f o r m

t h a t

D v

A

l o g

R

i s u n i m p o r t a n t ,

a n d t h e r e i s c o n s e q u e n t l y

q u a s i - e q u i l i b r i u m

i n

K i m u r a s

s e n s e .

B u t e v e n

i f K

a p p r o a c h e s 1 ,

a n d

R b e c o m e s

i n d e f i n i t e l y l a r g e

a s p

a p p r o a c h e s

0 o r

1 ,

t h e r e

may b e q u a s i - e q u i l i b r i u m

i n

a b r o a d e r

s e n s e .

T h e

Q u a s i - E q u i l i b r i u m

S u r f a c e o f

Mean

S e l e c t i v e

V a l u e s . - T h e

s y s t e m

o f

g a m e t i c


i n

t h e

c a s e

o f

p a i r s

o f

a l l e l e s c a n

b e

r e p r e s e n t e d b y

p o i n t s

i n

a n

e q u i l a t -

e r a l

t e t r a h e d r o n

o f

u n i t

h e i g h t ,

t o

e a c h

o f

w h i c h

a

mean

s e l e c t i v e

v a l u e

c a n

b e

a s s i g n e d .

A s s u m i n g

t h a t

g a m e t i c


c h a n g e

o n l y

s l o w l y

a n d

K

i s

n o t

t o o

l a r g e ,

t h e r e

i s

a s u r f a c e

w i t h i n

t h i s

s p a c e ,

b o u n d e d b y

t h e

e d g e s

a b

-

A b ,

Ab

-

A B ,

AB

-

a B ,

a n d aB

-

a b ,

o n

w h i c h

D

A

l o g

R

i s

l e s s

t h a n

f o r

p o i n t s

o n

e a c h

s i d e .

P o p u l a t i o n s

o n t h i s s u r f a c e move a l o n g

i t

i n

q u a s i - e q u i l i b r i u m ,

i n a

b r o a d

s e n s e ,

e v e n

t h o u g h

R

may

c h a n g e g r e a t l y

w i t h t h e

c h a n g e s

i n

t h e s e t

o f

g e n e f r e q u e n c i e s .

I f t h e r e

i s

s y m m e t r y

o f t h e s e l e c t i v e

v a l u e s

a b o u t t h e

l i n e

q

=

p ,

t h e

g a m e t i c


a n d t h e mean

s e l e c t i v e

v a l u e s

a l o n g

t h i s

l i n e

c a n

r e a d i l y

b e c a l c u l a t e d .

W e

w i l l

c o n s i d e r

c a s e s

i n w h i c h

t h e

s e l e c t i v e

p e a k

o r

p e a k s

a r e

on t h i s l i n e .

T h e s e

v a l u e s ,

i n

c o n j u n c t i o n

w i t h

t h o s e

o f h o m a l l e l i c

AAbb

a n d

a a B B ,

i n d i c a t e

f a i r l y

w e l l

t h e

n a t u r e

o f

t h e

s u r f a c e .

A l o n g

t h e

l i n e

q

=

P f a B

=

f A b i

f A B

= p

-

f A b ,

a n d f a

b

=

1

-

p

-

f A b ,

( 1 5 )

= 1

x

=

( p

-

f A b ) ( 1

-

f A b ) / f A b ,

V O L .

5 8 , 1 9 6 7

1 6 7


4/8

G E N E T I C S : S .

WRIGHT

X f A b

f A b

-

p ( 1

-

p )

=

0 ,

( 1 6 )

f A b

=

[ I 1

4 p ( l -

p ) x -

1 ] / 2 x ,

( 1 7 )

R

-

1

=

x

=

[ V / ( 1

-

K ) 2

8 K f A b

-

( 1

-

K )

] / 4 f A b

f r o m

( 1 4 ) .

( 1 8 )

T h e g a m e t i c f r e q u e n c i e s

a n d t h e v al u e s

o f

K

i f

n o t

c o n s t a n t ) ,

o f

R ,

a n d o f

W

c a n

b e

f o u n d b y

i t e r a t i o n

o f e q u a t i o n s

( 1 7 ) , ( 1 2 ) i f n e c e s s a r y ,

a n d

( 1 8 )

f o r e a c h

d e s i r e d

v a l u e

o f

q

=

p f o r s y s t e m s

w i t h g i v e n

s e l e c t i v e v a l u e s

w a n d g i v e n

a m o u n t s

o f

r e c o m b i n a t i o n

f o r

c o m p a r i s o n w i t h

t h o s e g i v e n u n d e r t h e

a s s u m p t i o n o f r a n d o m

c o m b i n a t i o n ( R

=

1 ) .

T h e

f o l l o w i n g

s y s t e m s

o f c o n s t a n t

g e n o t y p i c

v a l u e s

( a a b b a t l o w e r l e f t , AABB

a t

u p p e r r i g h t )

w i l l

b e c o n s i d e r e d .

I

( ,

- =

0 . 2 0 )

I I

3 ,

=

0 . 2 0 )

I I I

S I

=

0 . 1 5 )

IV

0 . 6 0 . 9

1

0 . 6

1 . 1 1 0 . 7 5

1

1 . 1 0

1

1

1

0 . 9

1

0 . 9

1 . 1 1 .

4 1 . 1 0 . 9 5 1 . 0 5 1

1

1 . 2 5

1

1

0 . 9 0 . 6

1 1 . 1 0 . 6

1

0 . 9 5

0 . 7 5

1

1 1

V

VI

V I I

( S I

=

0 . 2 5 )

0 . 8 1

1

0 . 9 7

0 . 6 4

1

0 . 9 6

1

1 . 5 0

2

1 1 . 0 3

1

1 1 . 0 5 1

1 1 . 2 5 1 . 5 0

0 . 9 7

1

0 . 8 1 0 . 9 4

1

0 . 6 4

1

1

1

C a s e

I ,

s y m m e t r i c a l

a b o u t

b o t h

d i a g o n a l s ,

i s

o f

t h e

s o r t

i n

w h i c h t h e r e i s

a n

o p t i m u m

a t

t h e

m i d p o i n t

o f

t h e

p h e n o t y p i c

s c a l e ,

d i s c u s s e d

a b o v e ,

b u t

h e r e

w i t h

e x c h a n g e

o f

B

a n d

b .

C a s e I I

i s t h e

s a m e

e x c e p t

f o r

a

b o n u s

( 0 . 2 )

f o r

h e t e r o s i s

a t

e a c h

l o c u s .

C a s e

I I I

i s s o m e w h a t

l i k e

c a s e

I

e x c e p t

t h a t t h e s e l e c t i v e

p e a k s

a r e

u n e q u a l .

I n

e a c h

o f t h e s e

t h r e e

c a s e s ,

t h e i n t e r a c t i v e

s e l e c t i o n

i s

u n i f o r m . C a s e

I V

i s a

s i m p l e

e x a m p l e

o f

n o n a d d i t i v e

h e t e r o s i s .

T h e i n t e r a c t i v e

s e l e c t i o n

i s

v a r i a b l e

( 4

0 . 2 5

i n

t h e

f o u r

c o r n e r s )

s o

t h a t

s 1

m u s t

b e c a l c u l a t e d

f o r

e a c h

p

i n

t h e

i t e r a t i o n p r o c e s s .

I t

i s a

s p e c i a l

c a s e

o f a

t y p e

d i s c u s s e d

b y

L e w o n t i n

a n d

K o j i m a . 9

C a s e s

V a n d

V I

w e r e p r e s e n t e d

i n a

p r e v i o u s p a p e r

t o

i l l u s t r a t e

t h e

p o s s i b i l i t y

o f

t w o

s e l e c t i v e

p e a k s

t h a t

may

b e h e t e r a l l e l i c

a t b o t h l o c i .

I n

b o t h , S I

v a r i e s .

I n

V I I ,

a s

i n c a s e s

c o n s i d e r e d

b y F e l s e n s t e i n , 1 2

mean

s e l e c t i v e

v a l u e

r i s e s

t o w a r d o n e

e x t r e m e ,

b u t

d o e s

s o

n o n l i n e a r l y .

I n t e r a c t i v e

s e l e c t i o n

i s

u n i f o r m .

C a l c u l a t i o n s

f o r

c

=

0 . 5

a n d

0 . 2

w e r e

m a d e ,

f i r s t

t r e a t i n g

w

A

l o g

R a s

n e g l i g i b l e .

H a v i n g e s t i m a t e d

R

f o r

e a c h

v a l u e

o f

p ,

a t

i n t e r v a l s o f

0 . 1 ,

t h i s

t e r m

w a s

e s t i m a t e d

i n e a c h

c a s e

f r o m t h e

f o r m u l a

w - A

p ( d l o g R / d p )

i n

w h i c h

W

A

P

=

[ f A B W A B

f A b W A b

P@]

T h e

s u m s

o f

t h e

n i n e

a b s o l u t e e s t i m a t e s

o f

W

A

l o g

R

a n d

o f

S I

a n d t h e

r a t i o s

a r e

g i v e n

i n

T a b l e

1 .

T h e

l a r g e s t

r a t i o s

a r e

i n

c a s e

I V

b u t

i n v o l v e

o n l y

t r i v i a l d e v i a -

t i o n o f

M

f r o m

i t s

v a l u e u n d e r

r a n d o m

c o m b i n a t i o n .

O f more

i n t e r e s t

i s

c a s e

I

w i t h

r e l a t i v e l y

l a r g e

d e v i a t i o n s

o f

u v .

R e c a l c u l a t i o n s

w e r e

made

i n t h e s e c a s e s

u s i n g

K

=

I

-

W

A

l o g

R ) / C W A a B b .

T h e

c h a n g e s

w e r e

t r i v i a l

e x c e p t

f o r R i n

a

f e w

c a s e s .

T a b l e s

2

a n d

3

g i v e

t h e

q u a n t i t i e s

i n d i c a t e d

i n

t h e

h e a d i n g s ,

u s i n g

t h e r e c a l c u l a -

t i o n s w h e r e

m a d e .

F i g u r e

1 s h o w s

t h e v a l u e s

o f

a v

f o r c =

0 . 5 , 0 . 2 ,

a n d 0

i n

c a s e s

I t o VI i n s o l i d

l i n e s ,

1 6 8

- P R O . - o . N . A . S .


5/8

GENETICS: S .

W R I G H T

C a s e

I

I I

I I I

IT

V I

V

I I

TABLE 1

R A T I O OF

TH E

TERM

WA l o g R TO

3 I

C=0.5 c = 0 . 2

2 I l i v

A

l o g

R I

2 : 1 1 1

R a t i o

; l Z s v

l o g

R I

x t i I l

R a t i o

0 . 0 1 2

1 . 8 0 0 . 0 0 7

0 . 0 5 8

1 . 8 0

0 . 0 3 2

0 . 0 0 6 1 . 8 0 0 . 0 0 4 0 . 0 5 4 1 . 8 0 0 . 0 3 0

0 . 0 0 4

1 . 3 5 0 . 0 0 3

0 . 0 2 7 1 . 3 5

0 . 0 2 0

0 . 0 4 3 0 . 6 2 0 . 0 6 8

0 . 1 2 9

0 . 6 6 0 . 1 9 3

0 . 0 0 2

0 . 5 1

0 . 0 3 1

0 . 0 0 4 0 . 4 9 0 . 0 0 9

0 . 0 0 5 0 . 9 7

0 . 0 0 5 0 . 0 1 4

0 . 9 1

0 . 0 1 5

0 . 0 6 0

2 . 2 5

0 . 0 2 7

-

f o r

c o m p a r i s o n

w i t h t h o s e

u n d e r

t h e

a s s u m p t i o n o f r a n d o m c o m b i n a t i o n ( b r o k e n

l i n e s ) . T h e l a t t e r a r e a l l o u t s i d e t h e r a n g e o f

t r u e v a l u e s c

=

0 . 5 t o

c

=

0 )

e x c e p t

a t

t h e e x t r e m e

v a l u e s

o f

p , a n d a t p

=

0 . 5 ,

i n

c a s e I V .

I n

c a s e s

I

t o

I I I ,

t h e c u r v e s

r e p r e s e n t L v a l o n g a

r i d g e o n

t h e q u a s i - e q u i l i b r i u m

s u r f a c e ( a a b b

t o

AABB)

b e t w e e n

d e e p d e p r e s s i o n s

a t

AAbb a n d

a a B B . T h e

c u r v e s

f o r

c =

0 . 5

d i f f e r

l i t t l e

f r o m

t h e

a p p r o x i m a t i o n

s h o w n b y

t h e

b r o k e n

l i n e .

T h a t f o r

t h e

c u r v e

o n

t h e s u r f a c e a t r i g h t

a n g l e s t o t h i s , p a s s i n g

t h r o u g h t h e s a m e

v a l u e

a t

( 0 . 5 , 0 . 5 ) ,

c a n

d i f f e r l i t t l e

f r o m t h e

a p p r o x i m a t i o n .

E v e n w i t h

c

=

0 . 2

t h e r e

i s

r o u g h

a p p r o x i m a t i o n .

T h e v a l u e s f o r w a t q

=

p

=

0 . 5 ,

c

=

0 . 1 0 , 0 . 0 5 , 0 . 0 2 ,

a n d 0 . 0 1

a r e

s h o w n

b y d o t s

i n

c a s e s

I

a n d

I I

( f o r m u l a

5 ) .

I n

c a s e I V, t h e t r u e c u r v e s , e v e n f o r c

=

0 . 2 , d i f f e r v e r y

l i t t l e

f r o m t h e

a p p r o x i m a -

t i o n ,

t h e r e

b e i n g n o d i f f e r e n c e

a t

( 0 . 5 , 0 . 5 ) .

C a s e

V

h a s

t h e

s m a l l e s t

s e l e c t i v e d i f f e r e n c e s a n d t h e l e a s t i n t e r a c t i v e s e l e c t i o n

o n

t h e a v e r a g e . T h e s h a l l o w s a d d l e b e t w e e n h e t e r a l l e l i c

p e a k s i s ,

h o w e v e r ,

c l o s e

t o t h e

R - 1

R u t

c - o

l--

I

I

0 2

A

A

a 8

1 . 0

0 . 2

A

8

1 . 0

p

F i G .

1.-Mean s e l e c t i v e v a l u e s

? b

on t h e s u r f a c e

b e t w e e n

a a b b

a n d

AABB

i n

s i x t w o - f a c t o r

s y s t e m s

as

d e s c r i b e d

i n

t h e

t e x t .

1 2

1 1

W

I D

0 9 1

I

1 1

W

I D

1 . 0 0

0.8

0 . 9 6

0 . 9 4

C - o

x

CPO

V O L .

5 8 ,

1 9 6 7

1 6 9

I

C-0


6/8

GENETICS:

S . W R I G HT

TABLE

DATA ON TH E Q U A S I - E Q U I L I B R I U M

S U R F A C E S

OF ME AN S E L E C T I V E

VALUE

q

=

X

r w

Ap

1 . 0 0 0 0

0 . 9 6 4

- 0 . 0 0 7 5

0 . 9 3 6

- 0 . 0 1 0 3

0 . 9 1 6 - 0 . 0 0 9 2

0 . 9 0 4

- 0 . 0 0 5 3

0 . 9 0 0 0

1 . 0 0 0 0

1 . 0 3 6

+ 0 . 0 0 6 9

1 . 0 6 4

+ 0 . 0 0 9 0

1 . 0 8 4 + 0 . 0 0 7 7

1 . 0 9 6 + 0 . 0 0 4 4

1 . 1 0 0 0

1 . 0 0 0 0

0 . 9 8 3

- 0 . 0 0 3 2

0 . 9 7 2

- 0 . 0 0 3 3

0 . 9 6 7 - 0 . 0 0 1 1

0 . 9 6 8

+ 0 . 0 0 2 5

0 . 9 7 5

+ 0 . 0 0 6 3

0 . 9 8 8 + 0 . 0 0 9 7

1 . 0 0 7

+ 0 . 0 1 1 5

1 . 0 3 2

+ 0 . 0 1 0 9

1 . 0 6 3

+ 0 . 0 0 7 2

1 . 1 0 0

0

1 . 0 0 0 0

1 . 0 0 8

+ 0 . 0 0 3 2

1 . 0 2 6 + 0 . 0 0 7 5

1 . 0 4 4

+ 0 . 0 0 8 4

1 . 0 5 8 + 0 . 0 0 5 4

1 . 0 6 3

0

0 . 9 7 0 0

0 . 9 7 8

+ 0 . 0 0 1 1

0 . 9 8 1

+ 0 . 0 0 0 4

0 . 9 8 1 - 0 . 0 0 0 3

0 . 9 8 0

- 0 . 0 0 0 4

0 . 9 8 0

0

0 . 9 4 0

0

0 . 9 5 6

+ 0 . 0 0 2 4

0 . 9 6 2

+ 0 . 0 0 0 9

0 . 9 6 2

- 0 . 0 0 0 5

0 . 9 6 1

- 0 . 0 0 0 5

0 . 9 6 1

0

0 . 9 6 3

+ 0 . 0 0 1 9

0 . 9 6 7

+ 0 . 0 0 2 1

0 . 9 7 0

+0.0008

0 . 9 7 0

- 0 . 0 0 1 0

0 . 9 6 0

0

q =

1-p

1

0 . 6 0 0

0 . 7 0 8

0 . 7 9 2

0 . 8 5 2

0 . 8 8 8

0 . 9 0 0

0 . 6 0 0

0 . 7 8 0

0 . 9 2 0

1 . 0 2 0

1 . 0 8 0

1 . 1 0 0

0 . 7 5 0

0 . 8 3 1

0 . 8 9 4

0 . 9 3 9

0 . 9 6 6

0 . 9 7 5

0 . 9 6 6

0 . 9 3 9

0 . 8 9 4

0 . 8 3 1

0 . 7 5 0

1 . 0 0 0

1 . 0 0 8

1 . 0 2 6

1 . 0 4 4

1 . 0 5 8

1 . 0 6 3

0 . 8 1 0

0 . 8 7 6

0 . 9 2 3

0 . 9 5 5

0 . 9 7 4

0 . 9 8 0

0 . 6 4 0

0 . 7 6 5

0 . 8 5 5

0 . 9 1 5

0 . 9 5 0

0 . 9 6 1

0 . 9 5 0

0 . 9 1 5

0 . 8 5 3

0 . 7 6 5

0 . 6 4 0

_--C

=

0 . 5

f A b

0 1 . 0 0 0

0 . 0 8 6 0 . 9 6 6

0 . 1 4 8 0 . 9 4 1

0 . 1 9 2

0 . 9 2 3

0 . 2 1 7 0 . 9 1 3

0 . 2 2 5

0 . 9 1 0

0 1 . 0 0 0

0 . 0 8 7 1 . 0 3 7

0 . 1 5 2 1 . 0 6 7

0 . 1 9 7

1 . 0 8 9

0 . 2 2 3 1 . 1 0 3

0 . 2 3 2

1 . 1 0 7

0 1 . 0 0 0

0 . 0 8 7

0 . 9 8 4

0 . 1 5 2

0 . 9 7 4

0 . 1 9 7 0 . 9 7 1

0 . 2 2 3

0 . 9 7 3

0 . 2 3 2

0 . 9 8 0

0 . 2 2 3

0 . 9 9 3

0 . 1 9 7 1 . 0 1 1

0 . 1 5 2

1 . 0 3 4

0 . 0 8 7

1 . 0 6 4

0 1 . 1 0 0

0 1 . 0 0 0

0 . 0 8 7

1 . 0 0 9

0 . 1 5 6

1 . 0 2 6

0 . 2 0 6

1 . 0 4 4

0 . 2 3 9

1 . 0 5 8

0 . 2 5 0

1 . 0 6 3

0

0 . 9 7 0

0 . 0 9 0

0 . 9 7 8

0 . 1 5 7 0 . 9 8 1

0 . 2 0 4

0 . 9 8 2

0 . 2 3 2 0 . 9 8 2

0 . 2 4 1 0 . 9 8 1

0

0 . 9 4 0

0 . 0 8 9 0 . 9 5 6

0 . 1 5 5

0 . 9 6 3

0 . 2 0 0 0 . 9 6 5

0 . 2 2 5

0 . 9 6 6

0 . 2 3 3

0 . 9 6 6

0 . 2 2 4 0 . 9 6 8

0 . 1 9 9

0 . 9 7 0

0 . 1 5 5

0 . 9 7 1

0 . 0 8 9

0 . 9 7 0

0

0 . 9 6 0

t h r e s h o l d ,

a n d

d i s a p p e a r s

b y

c o a l e s c e n c e

o f

t h e

p e a k s

i f c

=

0 . 2 . C a s e

VI w i t h

u n e q u a l p e a k s

i s

even more

s e n s i t i v e ,

i n

s p i t e

o f

g r e a t e r

s e l e c t i v e

d i f f e r e n c e s .

T h e r e

i s l o s s o f t h e

s a d d l e

i f

c

=

0 . 5 0 ,

as

b r o u g h t o u t

i n

a c o m p u t e r s t u d y b y

J a i n

a n d

A l l a r d . 3

The

s a d d l e w o u l d

p e r s i s t , h o w e v e r ,

i f

s e l e c t i v e

d i f f e r e n c e s

from

W A a B b

were

l e s s

t h a n

o n e

t h i r d

as

g r e a t .

I n

c a s e V I I , t h e d e v i at i o n

o f

W I ,

c

=

0 . 5 ,

f r o m

t h a t

u n d e r

random c o m b i n a t i o n

w o u l d

b e b a r e l y

p e r c e p t i b l e

i n

a

f i g u r e

o f

t h e

s c a l e

o f

t h e

o t h e r s

i n

s p i t e

o f

t h e

l a r g e

u n i f o r m

v a l u e

o f

K

( 0 . 4 0 ) .

The

a p p r o x i m a t i o n

u n d e r

random

c o m b i n a t i o n

f a l l s

o u t s i d e

t h e t r u e

range, h o w e v e r ,

b y

a b o u t

1 1

per

c e n t

a t

p

=

0 . 5

( l e s s e l s e w h e r e ) ,

w h i c h

i s

s o m e w h a t

s i m i l a r

t o o t h e r

c a s e s .

I n

a l l

o f

t h e s e c a s e s ,

t h e

c a l c u l a t i o n

o f

T w

u n d e r t h e

a s s u m p t i o n

o f

random

com-

C a s e

p

I

0

0 . 1

0 . 2

0 . 3

0 . 4

0 . 5

I I

0

0 . 1

0 . 2

0 . 3

0 . 4

0 . 5

I I I

0

0 . 1

0 . 2

0 . 3

0 . 4

0 . 5

0 . 6

0 . 7

0 . 8

0 . 9

1 . 0

IV

0

0 . 1

0 . 2

0 . 3

0 . 4

0 . 5

V

0

0 . 1

0 . 2

0 . 3

0 . 4

0 . 5

V I

0

0 . 1

0 . 2

0 . 3

0 . 4

0 . 5

0 . 6

0 . 7

0 . 8

0 . 9

1 . 0

f A b

0

0 . 0 9

0 . 1 6

0 . 2 1

0 . 2 4

0 . 2 5

0

0 . 0 9

0 . 1 6

0 . 2 1

0 . 2 4

0 . 2 5

0

0 . 0 9

0 . 1 6

0 . 2 1

0 . 2 4

0 . 2 5

0 . 2 4

0 . 2 1

0 . 1 6

0 . 0 9

0

0

0 . 0 9

0 . 1 6

0 . 2 1

0 . 2 4

0 . 2 5

0

0 . 0 9

0 . 1 6

0 . 2 1

0 . 2 4

0 . 2 5

0

0 . 0 9

0 . 1 6

0 . 2 1

0 . 2 4

0 . 2 5

0 . 2 4

0 . 2 1

0 . 1 6

0 . 0 9

0

T P R ~ o c .

N . A . S .

7 0


7/8


8/8

GENETICS:

S .

W R I G H T

TABLE 3

DATA

O N

TH E Q U A S I - E Q U I L I B R I U M SURFACE

OF MEAN S E L E C T I V E

VALUE

1 1

I N

CASE

V I I F O R R

=

1,

C

=

0 . 5 ,

AND

C

=

0

R

=

1

-

c

q

=

p

q

=-p

0. 5

fA b

0

f A b 3

f A b

i S

R K 1 7 3

0

0

1 . 0 0 0 0

1 . 0 0 0

0 1.000 1 . 6 7

0.400

1.000

0 . 1

0 . 0 9

1 . 0 1 0 +0.0045

1 . 0 9 0

0 . 0 8 6 1 . 0 1 2 1 . 5 7

1.055

0 . 2 0 . 1 6 1 . 0 4 0

+0.0160

1 . 1 6 0 0.148

1 . 0 4 6 1.53

1.120

0 . 3 0 . 2 1

1 . 0 9 0 +0.0318 1 . 2 1 0 0 . 1 9 2 1 . 0 9 9

1 . 5 0

1 . 1 9 5

0 . 4 0 . 2 4 1 . 1 6 0 +0.0493 1 . 2 4 0 0 . 2 1 7 1 . 1 7 2 1 . 4 9

1.280

0 . 5 0 . 2 5 1 . 2 5 0 +0.0666

1 . 2 5 0 0 . 2 2 5

1 . 2 6 2

1 . 4 9

1 . 3 7 5

0 . 6 0 . 2 4 1 . 3 6 0

+0.0828 1 . 2 4 0 0 . 2 1 7

1 . 3 7 2

1 . 4 9

1.480

0 . 7 0 . 2 1 1 . 4 9 0

+0.0967

1 . 2 1 0

0 . 1 9 2

1 . 4 9 9

1 . 5 0

1.595

0 . 8 0 . 1 6 1 . 6 4 0

+0.1085

1 . 1 6 0

0 . 1 4 8

1 . 6 4 6 1 . 5 3

1 . 7 2 0

0 . 9 0 . 0 9 1 . 8 1 0 +0.1178 1 . 0 9 0 0 . 0 8 6 1 . 8 1 2 1 . 5 7

1 . 8 5 5

1 . 0 0

2 . 0 0 0

0 1 . 0 0 0

0

2.000 1 . 6 7

2 . 0 0 0

b r e e d i n g a c c o r d i n g t o t h e same s y s t e m l o n g e n o u g h t h a t t h e r e i s

d i v e r g e n c e f r o m

r a n d o m c o m b i n a t i o n

o n l y

a s

f o r c e d b y i n t e r a c t i v e

s e l e c t i o n .

A

c r o s s b e t w e e n t w o i n b r e d l i n e s i s l i k e l y t o h a v e a s e l e c t i v e v a l u e

h i g h e r

t h a n

t h e n e a r e s t

p e a k o n t h e e q u i l i b r i u m s u r f a c e . The

randomly

b r e d d e s c e n d a n t s

w i l l

t h u s m o v e d o u w t o w a r d

t h e l a t t e r .

B e c a u s e o f t h i s

s o r t o f

s i t u a t i o n ,

M o r a n 5

h a s d r a w n t h e c o n c l u s i o n i n d i c a t e d

b y

t h e

t i t l e o f h i s paper,

The

n o n e x i s t e n c e

o f

a d a p t i v e t o p o g r a p h i e s .

T h i s was

b a s e d on

a m i s u n d e r s t a n d i n g o f t h e

c o n c e p t .

S u m m a r y . - K i m u r a s

c o n c e p t

o f

q u a s i - e q u i l i b r i u m w i t h

r e s p e c t t o

a

r a t i o o f

g a m e t i c


i s

u s e d t o c l a r i f y t h e c o n c e p t

o f a s u r f a c e o f

mean

s e l e c t i v e

v a l u e s ,

t h e

g r a d i e n t o f w h i c h t e n d s t o

c o n t r o l

e v o l u t i o n a r y c h a n g e w i t h i n

a

pan-

m i c t i c

p o p u l a t i o n .

I t i s s h o w n t h a t t h e a c t u a l

s u r f a c e ,

w h e r e t h e r e

i s

d e v i a t i o n

f r o m

r a n d o m c o m b i n a t i o n b e c a u s e o f i n t e r a c t i v e

s e l e c t i o n , d i f f e r s l i t t l e

f r o m

t h a t

c a l c u -

l a t e d

u n d e r

t h e a s s u m p t i o n o f

r a n d o m c o m b i n a t i o n i f s e l e c t i v e d i f f e r e n c e s ,

i n c l u d i n g

t h e


c o e f f i c i e n t s ,

a r e a s

s m a l l

as

i s

p r o b a b l y

u s u a l l y t h e c a s e i n n a t u r e ,

a n d

t h e l o c i

a r e i n

d i f f e r e n t c h r o m o s o m e s ,

as

i s u s u a l i n o r g a n i s m s w i t h t y p i c a l

n u m b e r s ,

o r i f i n

t h e s a m e

c h r o m o s o m e , are

o n l y

l o o s e l y

l i n k e d .

* P a p e r n o .

1119

from

the Laboratory

of

Genetics, University of

Wisconsin,

Madison,

W i s -

c o n s i n .

T h i s

w o r k h a s b e e n s u p p o r t e d

b y

a

g r a n t

f r o m

t h e

N a t i o n a l

S c i e n c e

F o u n d a t i o n

( G B -

1 3 1 7 ) .

1

W r i g h t ,

S . ,

B u l l . Am. M a t h .

S o c . ,

4 8 , 2 2 3 - 2 4 6 ( 1 9 4 2 ) .

2

W r i g h t , S . ,

i n G e n e t i c s , P a l e o n t o l o g y a n d E v o l u t i o n , e d .

G .

L . J e p s e n , G .

G . S i m p s o n ,

a n d E .

Mayr

( P r i n c e t o n ,

N .

J . :

P r i n c e t o n

U n i v e r s i t y

P r e s s , 1 9 4 9 ) ,

pp.

3 6 5 - 3 8 9 .

W r i g h t ,

S . , J . G e n e t i c s ,

3 0 , 2 4 3 - 2 5 6 ,

2 5 7 - 2 6 6

( 1 9 3 5 ) .

W r i g h t , S . , E c o l o g y ,

2 6 , 4 1 5 - 4 1 9 ( 1 9 4 5 ) .

W r i g h t , S . , i n Q u a n t i t a t i v e I n h e r i t a n c e ,

A g r i c u l t u r a l

R e s e a r c h C o u n c i l ( L o n d o n :

H e r

M a j e s t y s

S t a t i o n a r y

O f f i c e , 1 9 5 2 ) ,

pp. 5 - 4 1 .

6

W r i g h t ,

S . , P r o c .

R o y . S o c .

L o n d o n , B 1 6 2 , 80-104 ( 1 9 6 5 ) .

7 B o d m e r ,

W. F . , a n d

P . A . P a r s o n s , A d v .

G e n e t . , 1 1 ,

1 - 1 0 0

( 1 9 6 2 ) .

8

K i m u r a , M . ,

E v o l u t i o n , 1 0 , 2 7 8 - 2 8 7 ( 1 9 5 6 ) .

L e w o n t i n , R . C . , a n d K .

K o j i m a ,

E v o l u t i o n ,

1 4 ,

4 5 8 - 4 7 2

( 1 9 6 0 ) .

1 0

K i m u r a ,

M., Genetics, 5 2 , 875-890

1 9 6 5 ) .

1 1

W r i g h t ,

S . , i n E v o l u t i o n

a f t e r

D a r w i n ,

e d . S o l Tax

( C h i c ag o : U n i v e r s i t y o f C h i c a g o

P r e s s ,

t 9 6 0 ) , V o l . 1 ,

pp.

4 2 9 - 4 7 5 .

1 2

F e l s e n s t e i n ,

J . ,

G e n e t i c s ,

5 2 ,

3 4 9 - 3 6 3

( 1 9 6 5 ) .

J a i n , S . K . , a n d R . W. A l l a r d ,

t h e s e P R O C E E D I N G S ,

5 4 , 1 4 3 6 - 1 4 4 3

( 1 9 6 5 ) .

W r i g h t ,

S . ,

P r o c . 6 t h I n t e r n a t .

C o n g r .

G e n e t i c s , 1 , 3 5 6 - 3 6 0

( 1 9 3 2 ) .

15

M o r a n , P . A . P . , A n n .

H u m a n G e n e t . , 2 7 ,

3 8 3 - 3 9 7

( 1 9 6 4 ) .

1 7 2

. P . - . o c .

N .

A .

S .

[1967] sewall wright - surfaces of selective value

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