EQUILIBRIA UNDER INBREEDING AND SELECTION1. B. S. WEIR2. University of California at Davis. Received October 23, 1969. HIS investigation arose from ...
EQUILIBRIA UNDER INBREEDING AND SELECTION1 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 (MANDEL1959; TALLIS 1966; KIMURA1956). 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 an arbitrary number of alleles. This note also extends the concept of fixation index (WRIGHT1951) and shows how equilibrium gene and genotype frequencies may be found. It 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 reviewed. 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: 0 - 33 Wijfij . (2)
--
21
1 This work was supported in part by grants from the National Institutes of Health (GM-104.76) and from the National Science Foundation (GB-6866). 2 Present address: Mathematics Department, Massey University, Palmerston North, New Zealand.
Genetics 65: 371-378 June 1970
372
B. S. WEIR
For alleles then, the transition equations are:
As the system is random mating,
and
fij
= pipj
ij =
SO
that
.$?wijpipj +3
=p”p (5) where fi is a k X k matrix with (i,j)th element equal to wij. The only requirement for equilibrium is that pi’ = pi for all i. In other words, we have the k simultaneous equations -o i - 0 . (6) Previous authors have pointed out that applying Cramer’s rule gives as solution to this set:
--
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 2 a the same sign. INBREEDING POPULATIONS
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) = p . . (1 -Fii) 23 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.. and fi*.1.3. = W 2. 3.* = wii + (wii - wii) p i . (10) 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. f..
--
+
02
+
%
EQUILIBRIA IN P O P U L A T I O N S
3 73
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. MIXED SELFING A N D RANDOM MATING
Suppose that in any generation there is a constant amount, s, of selfing and t = I--s of outcrossing. The genotypic transition equations are:
e
so that pi‘ = as in (4). Substituting in the genotypic frequencies from (9) w gives for equilibrium, when fij’ = fij:
For a nontrivial equilibrium then (1> p i > O ) :
Fij = s(2tj - W j j ) 25 - swij
S Zii-Wii
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, 0 pi Q*li (15) z
--
where the pi’s are functions of the single unknown, 3. In general this equation for i3 will have to be solved numerically.
3 74
B. S. WEIR 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 . In other words Q i i * = 1 - xi and a..* 23 = 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 - YIP Pi = - Y ( l - F)
+
1 where yi = -and xi
s(2c - 1) 2z - s
Y = z yi .From (13), F =--
,
F Yi
1-F Pi Yi so that the required polynomial for E is (2Y)zZ- [2(Y-1) + s Y + 2 s ( l - k ) ] n - s ( k - Y ) Yi Note that when s = 0 (random rating) ,F = 0 and pi = - , Y Y-1 as given by TALLIS (1966). 0-Y
andfrom (15):
n=(1
=o.
--
(ii) Equal homozygote fitnesses We now set wii = 1 and x i j = (1 - wii) (1 - Pi)and see that the elements of Q* are given by 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: %%
pi =
(wiz-wjj) - (W1 2-U)..) F12 22 (2w12-wll-w22) (1-F12)
i#j.
The quadratic for the mean fitness is (w11
+ w22 - 2W12)n2 +
{S(W12 WllfW22
- wllwz2)
4-s WlZ (WllWZ2
+ (w12 - WllW22) 1=
- WlZ
Wl1+W22)
= ()
2
(iv) Three alleles Even with three alleles the algebraic expressions for allelic frequencies and
3 75
EQUILIBRIA IN P O P U L A T I O N S
TABLE 1 Characterisiics of some tri-allelic equilibrium populations where s = 0.95, w I 1= 1.00, w Z Z = 0.75, wZ3= 1.00, w12= 2.00 Fitness values wla
U23
0.00 0.00 0.00 0.60 1.00 1.40 1.60 2.00 2.00 2.00
0.00 1.60 2.00 0.40 1.00 1.20 0.00 0.00 1.00 2.00
Allelic frequencies PI
P?
0.18 0.51
0.02 0.01
0.17
0.02
0.08 0.36 0.11
0.04. 0.29 0.26
Pa
F?3
Fixation indices F3i
Mean fitness Fl2
0.95 -0.24 0.80 0.95 0.50 0.95 0.82 0.48 no nontrivial equilibrium 0.94 -0.15 0.81 0.93 no nontrivial equilibrium no nontrivial equilibrium 0.03 0.79 0.95 0.88 0.62 0.95 0.35 0.62 0.63 0.63 0.91 0.63 no nontrivial equilibrium
-
0.99 1.05 0.99
1.00 1.09 1.10
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 w13were 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 determinants have the same sign. This condition, from the transition equation (11) for mixed selfing and random mating leads to
> swii
?ii
(19)
These conditions are of use in defining the range of ti when numerical methods must be used to solve (15 ) . T H E N A T U R E O F EQUILIBRIA
Determining the nature of equilibria is more complicated for inbreeding populations than for random mating populations. In the latter case, only changes in allelic frequencies in the neighborhood of equilibria need be considered and 1959). appeal can be made to the monotonicity of the mean fitness (e.g., MANDEL This procedure does not carry over to inbreeding systems. In these, changes in both allelic frequencies and fixation indices in the neighborhood of equilibria must be considered. This of course is equivalent to looking at changes in genotype
376
B. S. WEIR
TABLE 2 Stability of some tri-allelic equilibrium populations where 0.95, wI1= 1.00, wZ2= 0.75, w~~= 1.00, wlZ=2.00
s
w13
2.0 1.8 1.6 1.4 1.2 1.o 0.8 0.6 0.4 0.2 0.0
N N S N N N U U U U U 0.0
N N S N N N U U U U U 0.2
N N N N N N U U U U U 0.4
N N N N N N U U U U U 0.6
N N N N N N U U U U U 0.8
N N N N N N U U U U U 1.0
N N N N N N U U U U U 1.2
N N N N N N U U U U U 1.4
N N N N N N U U U U U 1.6
N S S N N N N N N N N 1.8
N S S S S S U U U U U 2.0
wz3
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 in specific cases. Each genotypic frequency can be given a small deviation from its equilibrium value (the mean of the deviations being zero) and the population observed 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 3and wgBincrease) 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 (1953). The method of locating equilibria Model I, also considered by HAYMAN is as described above, and will just be outlined briefly here. For Model I selection and mixed selfing and random mating, the .genotypic transition equations are:
EQUILIBRIA IN POPULATIONS
377
where the divisor h ensures that ?? fij' = 1. It 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 h
> swii > ( ~ / 2 )wij
The allelic transition equations are:
pip = --pi hi h
where
x=qpixi. 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 FIIijreplaced by s( 1 27Iij)/2. This relation, for two alleles, was noted by WORKMAN and JAIN (1966).
+
SUMMARY
A matrix method for locating equilibria in random mating has been adapted to apply to mixtures of selfing and random mating for a locus with an arbitrary number of alleles. Genotypic frequencies are written in terms of allelic frequencies and fixation indices-each distinct heterozygote requiring one distinct fixation index. The fixation indices appear in the off-diagonal elements of the fitness matrix. The equilibrium values of these indices are given by a simple expression involving the amount of selfing and the mean fitness, while the equilibrium allelic frequencies are functions just of the selection coefficients and fixation indices. The problem is thus reduced to that of determining the mean
378
B. S. WEIR
fitness at equilibrium. An equation for this quantity is found and requires numerical 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 Mixed selfing and random mating when homozygotes are a t a disadvantage. Heredity 7: 185-192.
JAIN,S. K. and P. L. WORKMAN, 1967 Generalized F-statistics and the theory of inbreeding and selection. Nature 214: 674-678. KIMURA,M., 1956 Rules for testing stability of a selective polymorphism. Proc. Natl. Acad. Sci. U.S. 42: 336-340. LEWONTIN, R. C., 1958 A general method for investigating the equilibrium of gene frequency in a population. Genetics 43 :419-434.
LI, C. C., 1967 Genetic equilibrium under selection. Biometrics 23: 397-484. S. P. H., 1959 The stability of a multiple allelic system. Heredity 13: 289-302. MANDEL, TALLIS,G. M., 1966 Equilibria under selection for k alleles. Biometrics 22: 121-127. TURNER, J. R. G., 1969 The basic theorems of natural selection: A naive approach. Heredity 24: 75-84.
P. L. and S. K. JAIN,1966 Zygotic selection under mixed random mating and selfWORKMAN, fertilization: theory and problems of estimation. Genetics 54: 159-171. WRIGHT,S., 1951 The genetical structure of populations. Ann. Eugenics 15: 323-354.
