Pleiotropy and multilocus polymorphisms.

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Pleiotropy

and

Multilocus Polymorphisms

A. Gimelfarb

Department of Ecology and Evolution, The University of Chicago, Chicago, Illinois 60637 Manuscript received February 1, 199 1

Accepted for publication September 26, 199 1

ABSTRACT

It is demonstrated that systems of two pleiotropically related characters controlled by additive diallelic loci can maintain under Gaussian stabilizing selection a stable polymorphism in more than t w o loci. It is also shown that such systems may have multiple stable polymorphic equilibria. Stabilizing selection generates negative linkage disequilibrium, as a result of which the equilibrium phenotypic variances are quite low, even though the level of allelic polymorphisms can be very high. Consequently, large amounts of additive genetic variation can be hidden in populations at equilibrium under stabilizing selection on pleiotropically related characters.

A

model of two pleiotropically related quantitative characters was introduced by GIMELFARB (1 986). Each of the two characters, X1 and X2, in the model is controlled by the same two diallelic loci with additive allelic effects. While the action of one of the loci is similar on both characters, the action of the other locus is antagonistic, i.e., an allele in this locus increasing the value of X1 decreases at the same time the value of X2. Each character is assumed to be under its own stabilizing selection with the fitness functions wl(X1) and w4X2), and the total fitness of an individual, w(Xl,X2), is the product, wl(X1)wp (X2), i.e., selection on each character is independent of the other character. This system of two characters controlled by two loci with additive allelic effects has been shown to maintain a stable polymorphism in both loci.

Recently, HASTINCS and HOM (1 989) addressed the question of the number of loci contributing additively to several quantitative characters that can be maintained polymorphic under stabilizing selection. They have come out with the following result: “In a system in which m characters undergo weak Gaussian stabilizing selection, at most m loci will be polymorphic at a stable equilibrium, assuming linkage equilibrium.” In other words, the number of polymorphic loci cannot exceed the number of characters. An obvious question is whether this result is general, and, hence, the au- thors are correct in suggesting that “pleiotropy cou- pled with additive determination of characters will not lead to the levels of variability typically observed in natural populations, unless there are a large number of characters all controlled by the same loci” (HAS- TINGS and HOM 1989). That their result is not quite general follows from the examples presented by GALE and KEARSEY (1968) (also KEARSEY and GALE 1968) of a quantitative character controlled by two and three loci with additive but unequal effects on the character. Since the number of characters in this system is one,

Genetics 130: 223-227 (January, 1992)

the result by HASTINGS and HOM implies that only one locus can be maintained polymorphic under Gaus- sian selection, assuming linkage equilibrium. Yet, GALE and KEARSEY have demonstrated that a polymorphism can be maintained in two and three loci, if stabilizing selection is not in a Gaussian but in a “triangular” form. A substantial linkage disequilibrium is generated in their examples. However, NACYLAKI (1 989) has proven that a stable equilibrium with two loci polymorphic can exist under triangular selection, even if linkage disequilibrium is neglected.

T o determine how many loci can be maintained polymorphic in a system of two pleiotropically related characters, we shall expand the two locus model by GIMELFARB (1 986). But before doing so, let us briefly review the model and describe the computational methods used in the present work to investigate the dynamics of multilocus pleiotropic systems.

TWO CHARACTERS, TWO LOCI

Consider two quantitative characters, X1 and X2, controlled by two loci, L1 and L2, with two alleles, A and a, in each locus. T h e contribution by allele a in any locus to any character is zero, whereas the contribution of allele A is represented as follows:

L1 L2

x 1 1.00 _{1.00 (1)}

x2

1.00 -1.00

(2)

and X,,, are the minimum and the maximum of X among all possible genotypes, so that the values of any character are always between 0 and 1. Such a trans- formation (which amounts to a linear rescaling) does not change the addivity of allelic effects, but allows comparisons between models with different allelic contributions and even different numbers of loci.

Each of the two characters is assumed to be under its own Gaussian stabilizing selection (selection in a quadratic form was assumed by GIMELFARB, 1986). T h e total fitness function of an individual with characters X1 and X2 is

w(X1, X2) = exp -[SI(XI

-

01)’

+

-

&)‘I, (2)

ie., selection acts independently on each character. In

a symmetric case (0, = O2 = 0.5 and SI = S2), this model can be shown to yield for any recombination a stable equilibrium with the frequencies of alleles 0.5 in both loci.

COMPUTING THE DYNAMICS OF PLEIOTROPIC SYSTEMS

The methods for investigating the dynamics of multilocus systems with pleiotropy used in this work are similar to those used by GIMELFARB (1 989) for multilocus systems with epistasis. Random mating is assumed, and the dynamics were obtained by iterating numerically on a computer the following recurrence equations for gametic frequencies:

1

Pk+l(g3) =- pk(gl)P~(gZ)w(gl,g2)H(g3lgl,g2),

Wk

gl g2

wk

= pk(gl)pk(g2)w(gl,g2)* (3) gl g2

Here, gl, g2, g3 are gametes, and

p k ,

p k + l are the

frequencies of gametes in generations

k

and

k

+

1. T h e fitness, w(gl,g*), of an individual whose genotype is made by gametes g, and g2 is obtained by computing the values of the characters X1 and X2 based on the contributions of alleles in the gametes, as described earlier, and substituting these values into (2). Function H(g3Jgl,g2) in (3) represents the probability that a gamete produced by an individual is g3, given that the individual’s own genotype is made by gametes gl and

g2. This function can be computed easily for any

number of loci and recombination pattern (it should be pointed out that crossingover without interference was assumed in all of the calculations).

At the end of each iteration, the distance,

d =

Jc

[pk+l(g)

-

Pk(g)l2, (4) g

between the gametic distributions in two consecutive generations was calculated, and it was decided that an equilibrium has been reached when d

<

10“’. Only

equilibria with all loci in the system being polymorphic were considered. T h e stability of such an equilibrium was tested in the following way. T h e equilibrium gametic frequencies were disturbed by adding to each of them its own small random number and then normalizing them to sum to unity. The iterations were repeated starting with the disturbed gametic frequencies. An equilibrium was classified as stable if the disturbed system returned to it, and it was classified as unstable otherwise. A justification for such method of testing the local stability has been discussed else- where (GIMELFARB 1989, Appendix C).

Initial conditions of two types: “central” and “corner” were used to start iterations. Given the number of diallelic loci under consideration, n , there are

N

=

2” different gametes. All of these gametes have ap- proximately equal frequencies at a central initial condition,

Po@,)

= (1

+

cjj>/C

(1

+

E j ) , (5)

j

where j = 1, 2,

. . .

N and c, is a random number between 0 and O.l/N. Whereas at a corner initial condition,

where, again, j = 1, 2,

.

. .

N and E, is a random

number between 0 and O.l/N, the frequency of a particular gamete, g j , is close to 1 whereas the frequencies of all remaining gametes are close to zero. One central and

2”

corner initial conditions corresponding to all possible gametes were investigated. A locally stable equilibrium that is approached from any of the initial conditions will be referred to as “globally” stable. It should be kept in mind, however, that such method of testing the global stability is not precise. T h e dynamics of multilocus systems can be very com- plex, and it is possible, in principle, for a locally stable equilibrium to exist in a system in which there is another locally stable equilibrium approachable from the center and from all of the corners. Because of that, we shall always use the term “global” in quotation marks.

Computations were conducted on a COMPA(V33 micro-computer having a mathematical coprocessor.

TWO CHARACTERS, THREE LOCI

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a in any locus is zero. T h e contributions of allele A , on the other hand, are as follows:

L1 L2 L3

x1

1.00 1.00 1.00 (7)

x 2

*

1.00 -1.00

T h e star indicates that the locus does not contribute to the corresponding character (i.e., the contribution by either allele A or a to the character is zero). T h e values of both characters are normalized, as described above, so that they are between 0 and 1 . Let the recombination pattern be r = (0.05, 0.51, i.e., the recombination coefficient is 0.05 between loci L 1 and L2 and 0.5 between L2 and L3. This can be interpreted as L2 and L3 being located on different chro- mosomes and L1 linked to L2. Under selection in the form (2) with = O2 = 0.5 (i.e., the optimum for each character is in the middle of its distributional range) and S1 = S2 = 30, the model has a “globally” stable equilibrium with the frequencies 0.5 of alleles in each locus. T h e mean fitness at the equilibrium, W = 0.37. Such selection is, of course, quite strong. It is possible, however, for the polymorphism to be maintained under much weaker selection, but with a tighter linkage between loci L2 and L3. If, for example, the recombination pattern is r = (0.05, 0.05) and S1 and S 2 = 5,

there is a “globally” stable equilibrium with allelic frequencies 0.5 and with the mean fitness, W = 0.85.

Is it possible for polymorphisms in all three loci to be maintained by each character separately, without pleiotropy? All loci affecting a character in system (7) contribute to it equally. Therefore, according to LEWONTIN (1964), character X1 can maintain only one locus polymorphic, whereas character X2 cannot maintain a polymorphism at all. Hence, the maintenance by system

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of polymorphisms in all three loci is due to pleiotropy.

TWO CHARACTERS, FOUR LOCI

Consider now a system of two characters affected by four loci with the contribution of allele a being again zero, and that of allele A as

L1 L2 L3 L4

x1

1.00 1.00 1.00

*

(8) x 2

*

1.00 -1.00 -1.00

This represents an expansion of the three-locus system

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by an addition of locus L4 contributing to character X 2 but not to character X1. T h e original two-locus model ( 1 ) also serves here as a base (loci L2 and L3). Let the recombination pattern be r = (0.05,0.5,0.05) which can be interpreted as two linked loci, L1 and L2, located on one chromosome, and two other linked loci, L3 and L4, located on a different chromosoqw. Under selection with O1 = 8 2 = 0.5 and S1 = S2 = 15, there is a stable equilibrium with the frequencies of

alleles 0.5 in all loci and with W = 0.63. However, this equilibrium is stable only locally, it is not reached from any of the corners. A corner initial condition results in a polymorphism in one of the. loci and fixation in the other three of them.

Even though system (8) involves four loci, only two of them are pleiotropic, i.e., contribute to both characters. Is it possible to maintain an equilibrium with polymorphisms in more than two pleiotropic loci? I was unable to obtain such an equilibrium in the case of equal contributions by the loci. Consider, however, the following system:

L1 L2 L3 L4

X1 0.50 1.00 1.00 0.50 (9) X2 0.50 1 .OO -1.00 -0.50

All loci are pleiotropic, but the contribution by L1 and L4 constitutes one half of that by L2 and L3. Let again r = {0.05, 0.5, 0.05). Under selection with 81 =

O2 = 0.5 and S1 = S2 = 6, there is a “globally” stable equilibrium with the frequencies of alleles 0.5 in all loci. T h e equilibrium mean fitness, W = 0.84. T h e equilibrium variance of each character, V1 = V2 = 0.0 16.

For the pleiotropic system (9) with the same pattern of recombination but under weaker selection (S1 = S2

= 4.5), four equilibria were found with the following frequencies of allele A in the loci:

L1 L2 L3 L4

Equilibrium 1 0.09 0.85 0.15 0.91

Equilibrium 2 0.09 0.85 0.85 0.09 (IO) Equilibrium 3 0.91 0.15 0.15 0.91 Equilibrium 4 0.91 0.15 0.85 0.09

All of these equilibria are locally stable. T h e mean fitness, W = 0.88, and the variances of the characters,

VI = V2 = 0.01, in each of the equilibria. It is

interesting that these variances are lower than those in the previous example (0.01 us. 0.016), in spite of weaker selection (SI = S2 = 4.5 vs. S1 = S2 = 6). Four locally stable equilibria exist also under selection that is still weaker, but the equilibrium allelic frequencies come very close to either 0 or 1 ( i e . , alleles become practically fixed).

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hand, the system (9) can maintain all four loci polymorphic under quite weak selection but with a tighter linkage between loci L2 and L3. If, for example, the recombination pattern is r = (0.01, 0.5, 0.011, there is a “globally” stable equilibrium with allelic frequencies 0.5 in all loci under selection with

S1

= S2 = 2. T h e equilibrium mean fitness, W = 0.96.

Computations show that neither of the two characters in system (9) can maintain even one locus polymorphic, if considered separately, i.e. in the absence of pleiotropy. Therefore, the maintenance of polymorphisms in all four loci is only because of pleiotropy.

TWO CHARACTERS, SIX LOCI

Can a system of two pleiotropically related characters maintain a polymorphism in more than four loci? Consider a system involving six loci with the following contributions by allele A :

L1 L2 L3 L4 L5 L6

X1 0.25 0.50 1.00 1.00 0.50 0.25 (1 1) X2 0.25 0.50 1.00 -1.00 -0.50 -0.25

T h e contributions by allele a are zero, as before. T h e base of this system is again the original two-locus model (loci L3 and L4). Assume the recombination pattern, r = (0.05, 0.05, 0.5, 0.05, 0.051, which can be interpreted as three linked loci, L 1, L2, L3, on one chromosome and three linked loci, L4, L5, L6, on a different chromosome. Under selection with O1 = 02

= 0.5 and SI = S2 = 50, this system has a “globally” stable equilibrium with allelic frequencies 0.5 in all loci. T h e mean fitness at the equilibrium, W = 0.72, and the variances of the characters, V1 = V2 = 0.005. Computations show that neither of the two characters can maintain a polymorphism in even one locus, if considered separately, i.e., without pleiotrophy.

Computations involving six loci are extremely time consuming. Because of that, no other sets of parame- ters were investigated in the six-locus case.

DISCUSSION

All of the systems of two pleiotropically related characters considered above have under Gaussian selection a stable equilibrium with a polymorphism maintained in more than two loci. T h e level of the polymorphism in a locus can be quite high (frequencies of alleles either equal or close to 0.5). Yet, the equilibrium variances of characters are low. This is due to the negative linkage disequilibrium generated in the system by stabilizing selection. For example, the four-locus system (9) maintains under selection with S I and S2 = 6 allelic frequencies 0.5 in all four loci. T h e equilibrium variance of a character, however, is only 0.016, i e . , 4.3 times lower than the variance 0.069 that is maintained in a population with the same

allelic frequencies but in linkage equilibrium. For the six-locus system (1 1) which also maintains allelic frequencies in all loci at 0.5, the equilibrium variance of a character is only 0.005, i.e., 10.8 times below the linkage equilibrium variance of 0.054. Hence, a population at equilibrium under stabilizing selection on pleiotropically related characters can be phenotypically quite uniform, in spite of the high level of allelic polymorphism. This also means that a substantial amount of additive genetic variation can be “hidden” in such a population. T h e hidden variation will be “released” by recombination if selection is relaxed.

T h e uniformity of phenotypes in a population is not solely due to the linkage disequilibrium generated by stabilizing selection, but also due to the constraints imposed on phenotypes by a particular pattern of pleiotropy. Consider, for example, phenotypes such that the values of both characters, X1 and X2, are at the extreme (either 0 or 1). Individuals with these phenotypes cannot exist for any of the pleiotropic systems discussed above. If one of the characters of an individual has an extreme value, its other character must have the intermediate value of 0.5.

As a consequence of the phenotypic uniformity, the pressure of selection on a population at equilibrium is not very high, even when selection on individuals is strong. For example, consider the four-locus system (9) under selection (2) with S1 = S2 = 6. T h e “worst” (having the lowest fitness) phenotype possible for this system has one character at an extreme value (0 or 1) and the other character at the intermediate value of 0.5. T h e fitness of an individual with such a phenotype is 0.22. This is only 22% of the fitness of an individual having the “best” phenotype (X1 = X2 = 0.5) which is 1. On the other hand, the mean fitness of a population at equilibrium is 0.84, which, given that the maximum fitness of an individual is 1, means that the “genetic load” (CROW and KIMURA, 1970) in the population is 16%. Regarding the load as a measure of the selection pressure on a population, we can say that, even though not very low, it is also not terribly high. Such a load cannot be precluded in a natural population, especially for species in which only a very small proportion of individuals survive and reproduce. T h e contrast between the strength of selection on individuals and its pressure on an equilibrium population is even more dramatic in the case of the six- locus system (1 1). T h e fitness of a “worst” phenotype under selection with S 1 = S2 = 50 is only one millionth of the fitness of the “best” one. Yet, the mean fitness of the equilibrium population, W = 0.72, implying the load of 28%, which, again, although not very low, is not extremely high.

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rium population being zero, in spite of the fact that both X1 and X 2 are controlled by the same loci. This means that the coefficient of genetic correlation between characters may not be a good indicator of the degree of pleiotropy between them.

Even though the genetic systems that we dealt with so far were symmetric, the symmetry is not necessary for the maintenance of polymorphisms. Indeed, consider the following four-locus system:

L1 L 3 L 2 L4

X1 0.50 0.90 1.00 0.45 (1 2 )

X 2 0.40 1.00 -0.95 -0.55

with the recombination pattern r = (0.05, 0.5, 0.05). Under selection with dl = 0.51,dz = 0.49 and SI = 9,

Sz = 15, there is a “globally” stable equilibrium with the following frequencies of allele A in the loci:

L 1 L 2 L 3 L4

0.52 0.49 0 . 3 0 0.71 ( 1 3)

Thus, in spite of the obvious asymmetry of the system, a stable polymorphism is maintained in all four loci.

An important property of pleiotropic systems (first indicated by HASTINGS and HOM 1990) is the possibil- ity of multiple polymorphic equilibria, even if the effects of genes are strictly additive and the system is symmetric. This is illustrated by the four-locus system (9) having under selection with S1 = S 2 = 4.5 four stable equilibria ( 1 0). T h e equilibria differ quite dra- matically with regard to the frequency of an allele in a particular locus. An allele most frequent in one equilibrium can be the least frequent in another one. Judging by the differences in the allelic frequencies between populations at different equilibria, one may be easily led to believe that they are under different selection forces, and, hence, experience different eco- logical conditions. Yet, in reality all of them are under the same selection forces, and their differences are attributable to their respective histories.

CONCLUSIONS

1 . A system of pleiotropically related characters can

maintain under Gaussian selection a stable polymorphism in a number of loci exceeding the number of characters. T h e level of the polymorphism can be quite high.

2 . This requires that the selection pressure on the population (as measured by genetic load) be not very weak. On the other hand, it need not be exceedingly strong either. T h e actual strength of selection de- pends on the particularities of the patterns of pleiotropy and recombination between the loci.

3 . A negative linkage disequilibrium is generated between the loci. As a result, the equilibrium population can be quite uniform phenotypically, in spite of the high level of allelic polymorphism. Large amounts of genetic variability can be hidden in the equilibrium population.

4. Multiple stable equilibria are possible under stabilizing selection on pleiotropically related characters, even if the action of genes is strictly additive. Popu- lations at different equilibria may differ substantially with regard to the maintained allelic frequencies.

5. Pleiotropy cannot be ruled out as a mechanism responsible for the maintenance of polymorphisms by many loci in natural population.

I am grateful to THOMAS NAGYLAKI, RUSSELL LANDE and

MICHAEL WADE for their interest and useful discussions. Support for this work was provided by U.S. Public Health Service grant GM27 120.

LITERATURE CITED

CROW, J. F., and M. KIMURA, 1970 An Introduction to Population Genetics Theory. Harper & Row, New York.

GALE, J. S., and M. J. KEARSEY, 1968 Stable equilibria in the absence of dominance. Heredity 23: 553-561.

GIMELFARB, A,, 1986 Additive variation maintained under stabi- lizing selection: a two-locus model of pleiotropy for two quan- titative characters. Genetics 112: 71 7-725.

GIMELFARB, A., 1989 Genotypic variation for a quantitative char- acter maintained under stabilizing selection without mutations: epistasis. Genetics 123: 21 7-227.

HASTINGS, A., and C. L. HOM, 1989 Pleiotropic stabilizing selec- tion limits the number of polymorphic loci to at most the number of characters. Genetics 122: 459-463.

HASTINGS, A., and C. L. HOM, 1990 Multiple equilibria and maintenance of additive genetic variance in a model of pleio- tropy. Evolution 4 4 1153-1 163.

KEARSEY, M. J., and J. S. GALE, 1968 Stabilizing selection in the absence of dominance: an additional note. Heredity 23: 6 17- 620.

LEWONTIN, R. C., 1964 The interaction of selection and linkage.

11. Optimal model. Genetics 5 0 757-782.

NAGYLAKI, T., 1989 The maintenance of genetic variability in two-locus models of stabilizing selection. Genetics 122: 235- 248.