EQUILIBRIA UNDER INBREEDING AND SELECTION

B. S. WEIR2

University of California at Davis

Received October 23, 1969

HIS investigation arose from a need to determine whether or not a population barley, which practices both selfing and random mating, was going to remain polymorphic at several isozyme loci or was going to fixation for one of the many alleles at each locus. This of course is part of the general problem of accounting for the genetic polymorphism recently found in many species by the study of enzyme systems using electrophoretic techniques.

The general question of the effects of selection on equilibrium has received considerable attention recently and LI (1967) has given a comprehensive review of the subject. Random mating populations can be handled with any number of alleles at a locus (MANDEL 1959; TALLIS 1966; KIMURA 1956). Not only can equilibrium populations be located, but also the stability of the equilibria can be determined. For inbreeding populations, LEWONTIN (1958) introduced a set of general weights and

JAIN

and

WORKMAN

(1967) defined “partial fixation indices.” Neither of these methods lead to a straightforward method for locating equilibria for a mixture of selfing and random mating with a n arbitrary number

of alleles.

This note also extends the concept of fixation index (WRIGHT 195 1 ) and shows how equilibrium gene and genotype frequencies may be found. I t is restricted though to selfing and random mating, where any generation can be completely characterized from knowledge of the previous one.

RANDOM M A T I N G

To establish notation, the method of handling random mating is briefly re- viewed. The relative fitEess for the genotype with alleles a{ and ai is written as wii. The frequency of a homozygote is fii, while that of a heterozygote is 2fij. Allelic frequencies are written as pi. Assuming wij constant, and independent of allelic frequencies, the genotype transition equations are thus:

A

where summations are over the integers 1 to

k

for a k-allele system and ’ii is the

mean fitness:

(2)

0

-

33 Wijfij

.

- -

2 1

1 This work was supported in part by grants from the National Institutes of Health (GM-104.76) and from the National

2 Present address: Mathematics Department, Massey University, Palmerston North, New Zealand. Science Foundation (GB-6866).

3

For alleles then, the transition equations are:

As the system is random mating, f i j = pipj SO that

and ij =

.$?

wijpipj

+ 3

= p ” p ( 5 )

where fi is a

k

X

k

matrix with (i,j)th element equal to wij. The only require- ment for equilibrium is that pi’ = pi for all

i.

I n other words, we have the

k

simultaneous equations

( 6 ) Previous authors have pointed out that applying Cramer’s rule gives as solution to this set:

-

_{o i - 0 .}

--

where

Di

is the determinant of fi with the ith column replaced by 1’s. This assumes that

101

=

ziDi

= ij

g

D i

# 0. It is necessary that all of the

Di

have the same sign.

2 a

I N B R E E D I N G P O P U L A T I O N S

Inbreeding systems may be treated formally in the same way, provided the fitness matrix fi is modified appropriately. A series of fixation indices Fii is

defined soon after mating, before selection occurs as:

so that

(9)

f . . = p . . (1 -Fii)

2 3 zP1

With

k

alleles then the

k

(k+l ) /2-1 independent genotypic frequencies have been expressed in terms of k-1 independent allelic frequencies and

k

(k-I ) /2 independent fixation indices.

Note that Fij

I

1. The mean fitness is now given by

--

_o

_-

_T:

_{wii (piz}

₊

_{, X . pipjF”)}

₊

_{. E , wij pipj (1-F”)}

2 3 # % % I f 2

-

-p”*p where fi*ii = w.. 0 2 and

Although wii = wji and Fii = Fii, wij* # wji* so that fi* is not a symmetric matrix. It can also be shown that z does not increase monotonically over time until equilibrium as it does for random mating.

In any generation, of course, the p i j are functions of allelic frequencies and are thus not constant. Equilibrium now demands unchanging allelic frequencies and unchanging fixation indices. Nevertheless, once equilibrium has been reached the wij* are constant so that, for inbreeding systems which lead to

equations of the form (4)

,

the allelic frequencies given by (7) still hold. The

Di

are now determinants of

a*

with the ith column replaced by 1’s.

The solutions, for allelic frequencies, are thus functions of the fixation indices. Clearly some condition on these indices must be found before we can express allelic frequencies as functions of selection coefficients only. Such additional information must come from the particular matching scheme being used.

M I X E D S E L F I N G A N D R A N D O M M A T I N G

Suppose that in any generation there is a constant amount, s, of selfing and

t = I--s of outcrossing. The genotypic transition equations are:

so that pi‘ =

e

as in ( 4 ) . Substituting in the genotypic frequencies from (9) gives for equilibrium, when fij’ = f i j :

w

For a nontrivial equilibrium then (1 > p i > O ) : s(2tj

-

W j j )

25 - swij

Fij =

S

Z i i - W i i

.

2

From equation (13) we see that each element of Q* at equilibrium can now be written in terms of the amount of selfing, the selection coefficients (all assumed known) and the mean fitness. The equilibrium allelic frequencies thus involve only one unknown: i3. We solve for ti using one of the equilibrium conditions

tji = 3. For example,

( 1 5 )

--

0

-

pi Q*li

z

where the pi’s are functions of the single unknown, 3.

SPECIAL CASES (i) Equal heterozygote fitnesses

A nice feature of the fixation indices given by (13) is that any two of them are equal when the corresponding heterozygotes have equal fitness. An often used example is wij = 1, wii = 1 - xi

.

I n other words Q i i * = 1 - xi and

a..*

2 3 = 1 - xiF where F is the common value of the Fii’s. With these values and

our mixture of selfing and random mating, at equilibrium from equation (7), for

k

alleles:

yi

+

[(k--l)yi - Y I P Y ( l - F)

Pi

= -

1 s(2c

-

1 )

where yi = -and Y =

z

yi .From (13), F =--

,

2z - s xi

F 1 - F

Yi

Y i

andfrom (15): n = ( 1

Pi

so that the required polynomial for E is

Note that when s = 0 (random rating)

,

F = 0 and pi =

-

,

(2Y)zZ- [ 2 ( Y - 1 ) + s Y + 2 s ( l - k ) ] n - s ( k - Y )

= o .

Yi Y

Y-1

Y as given by TALLIS (1966).

--

0 - -

(ii) Equal homozygote fitnesses of Q* are given by

We now set wii = 1 and x i j = (1 - wii) (1

-

Pi)

and see that the elements

Q * . . % % 1

a*..-

2 3 - 1 - x . . r 3 i # j , (16)

The matrix is now symmetric but no great simplification occurs unless the wij are also equal, in which case the equilibrium gene frequencies are given by ( l / k )

.

(iii) Two alleles

The case of two alleles has been discussed by WORKMAN and JAIN (1966). The two equilibrium allelic frequencies are given by:

(wiz-wjj) - (W -U)..) F12

i # j .

1 2 2 2

pi = (2w12-wll-w22) (1-F12)

The quadratic for the mean fitness is (w11

+

w22 - 2W12) n2

+

{S(W12 WllfW22 - wllwz2)

+

(w12 - WllW22)

1

=

Wl1+W22) = ()

4-

s WlZ (WllWZ2 - WlZ 2

(iv) Three alleles

TABLE 1

Characterisiics of some tri-allelic equilibrium populations where s = 0.95, w I 1 = 1.00, w Z Z = 0.75, w Z 3 = 1.00, w12 = 2.00

Fixation indices Mean fitness

F3i Fl2

-

Fitness values Allelic frequencies

U 2 3 wla P I P? Pa F?3

0.00 0.00 0.00 1.60 0.00 2.00 0.60 0.40 1.00 1.00

1.40 1.20 1.60 0.00 2.00 0.00

2.00 1.00 2.00 2.00

0.18 0.02 0.80 0.95 0.95 -0.24 0.99

0.51 0.01 0.48 0.95 0.82 0.50 1.05

0.17 0.02 0.81 0.93 0.94 -0.15 0.99

no nontrivial equilibrium

no nontrivial equilibrium no nontrivial equilibrium

0.08 0.04. 0.88 0.79 0.95 0.03 1 .00

0.36 0.29 0.35 0.62 0.95 0.62 1.09

0.11 0.26 0.63 0.63 0.91 0.63 1.10

no nontrivial equilibrium

mean fitness are too cumbersome to display. Some numerical example will be given instead. For an amount of selfing of s = 0.95 four fitness values were fixed a t wlI 1.0, wZ2 = 0.75, wS3 = 1.0 and w12 = 2.0. The remaining two, w23 and w13 were allowed to vary between 0.0 and 2.0. Table 1 displays the characteristics of the equilibrium population for a few such combinations. The table illustrates the equality of fixation indices when corresponding heterozygotes have equal fitness, and also shows that when a heterozygote has zero fitness the fixation index equals the amount of selfing.

CONDITIONS FOR E X I S T E N C E O F EQUILIBRIA

As already noted, a necessary condition for a valid equilibrium population is that all of the allelic frequencies have positive sign, so that all the

Di

determi- nants have the same sign. This condition, from the transition equation (11) for mixed selfing and random mating leads to

?ii

>

swii (19)

These conditions are of use in defining the range of ti when numerical methods

must be used to solve (1 5 ) .

T H E N A T U R E O F EQUILIBRIA

TABLE 2

Stability of some tri-allelic equilibrium populations where s 0.95, w I 1 = 1.00, w Z 2 = 0.75, w~~ = 1.00, wlZ = 2.00

2.0 1.8 1.6 1.4 1.2

0.8 0.6 0.4 0.2 0.0

w 1 3 1

.o

N N N N N N N N N N N

N N N N N N N N N S S

S S N N N N N N N S S

N N N N N N N N N N S

N N N N N N N N N N S

N N N N N N N N N N S

U U U U U U U U U N U

U U U U U U U U U N U

U U U U U U U U U N U

U U U U U U U U U N U

U U U U U U U U U N U

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

wz 3

N: no nontrivial equilibrium.

S: stable equilibrium.

U: unstable equilibrium.

frequencies. Methods such as that of looking at “haploid variance” (TURNER

1969) are difficult to apply because of the dependence of gene and genotypic frequencies on unknown fixation indices away from equilibria.

It is always possible however to resort to numerical techniques i n specific cases. Each genotypic frequency can be given a small deviation from its equi- librium value (the mean of the deviations being zero) and the population ob- served in subsequent generations. Should the displaced frequencies tend to return to their equilibrium values, the equilibrium is said to be stable.

Table 2 shows the nature of equilibria for the three allele case mentioned earlier. It should be noted that many of the unstable equilibria (particularly as

w I 3 and wgB increase) could, for all practical purposes, be described as neutral since the subsequent deviations from equilibrium genotypic frequencies increase very slowly, and may even decrease for the first few generations.

OTHER SELECTION MODELS

All of the preceding work has been for the Model I1 of WORKMAN and

JAIN

(1966)

,

in which census for genotypic frequency data occurs soon after matings. When census is taken prior to mating and after all selection, we have their Model I, also considered by HAYMAN (1953). The method of locating equilibria

is as described above, and will just be outlined briefly here.

where the divisor h ensures that

??

fij' = 1. I t may be written as 1 . 3

where

These last relations are deduced from the expression of (21) in terms of allelic frequencies and fixation indices. Such expressions show that, at equilibrium:

and that it is necessary that

h

>

swii h > ( ~ / 2 ) wij The allelic transition equations are:

pi hi pip = --

h

where x = q p i x i .

1.

Here hi corresponds to i3i in equation

(4).

Equilibrium is thus given by

where Ei is the determinant of A with the ith column replaced by 1's. As before, the fixation indices

P i i

are replaced by functions of h to produce a polynomial in h via (26).

Note that if we suffix the present fixation indices by I, and the previous ones by 11, the present computations are just the previous ones with FIIij replaced by

s ( 1

+

27Iij)/2. This relation, for two alleles, was noted by WORKMAN and JAIN

(1966).

S U M M A R Y

fitness at equilibrium. An equation for this quantity is found and requires nu- merical methods of solution for all but the simplest cases. A numerical example is given for the tri-allelic case, and the question of stability is briefly considered.

LITERATURE CITED

HAYMAN, B. I., 1953

JAIN, S. K. and P. L. WORKMAN, 1967

KIMURA, M., 1956

U.S. 42: 336-340. LEWONTIN, R. C., 1958

LI, C. C., 1967

MANDEL, S. P. H., 1959 TALLIS, G. M., 1966

TURNER, J. R. G., 1969

24: 75-84.

WORKMAN, P. L. and S. K. JAIN, 1966

WRIGHT, S., 1951

Mixed selfing and random mating when homozygotes are a t a disadvantage.

Generalized F-statistics and the theory of inbreeding and

Rules for testing stability of a selective polymorphism. Proc. Natl. Acad. Sci.

A general method for investigating the equilibrium of gene frequency Heredity 7: 185-192.

selection. Nature 214: 674-678.

in a population. Genetics 43 : 419-434.

Genetic equilibrium under selection. Biometrics 23: 397-484.

The stability of a multiple allelic system. Heredity 13: 289-302. Equilibria under selection for k alleles. Biometrics 22: 121-127.

The basic theorems of natural selection: A naive approach. Heredity

Zygotic selection under mixed random mating and self- fertilization: theory and problems of estimation. Genetics 54: 159-171.