Multivariate Mutation-Selection Balance With Constrained ...anterior-posterior polarity or segment...

Copyright 0 1989 by the Genetics Society of America

Multivariate Mutation-Selection Balance With Constrained Pleiotropic Effects

G. P. Wagner

Department of Ecology and Evolutionary Biology, Northwestern University, Evanston, Illinois 60201 Manuscript received August 24, 1988

Accepted for publication January 27, 1989

ABSTRACT A multivariate quantitative genetic model is analyzed that is based on the assumption that the

genetic variation at a locus j primarily influences an underlying physiological variable y,, while influence on the genotypic values is determined by a kind of “developmental function” which is not changed by mutations at this locus. Assuming additivity among loci the developmental function becomes a linear transformation of the underlying variables y onto the genotypic values x, x = By. In this way the pleiotropic effects become constrained by the structure of the B-matrix. The equilibrium variance under mutation-stabilizing selection balance in infinite and finite populations is derived by using the house of cards approximation. The results are compared to the predictions given by M. Turelli in 1985 for pleiotropic two-character models. It is shown that the B-matrix model gives the same results as Turelli’s five-allele model, suggesting that the crucial factor determining the equilibrium variance in multivariate models with pleiotropy is the assumption about constraints on the pleiotropic effects, and not the number of alleles as proposed by Turelli. Finally it is shown that under Gaussian stabilizing selection the structure of the B-matrix has effectively no influence on the mean equilibrium fitness of an infinite population. Hence the B-matrix and consequently to some extent also the structure of the genetic correlation matrix is an almost neutral character. The consequences for the evolution of genetic covariance matrices are discussed.

P LEIOTROPIC effects cause an association of heritable variation among different phenotypic char-

acters. Pleiotropic effects are among the major causes of genetic covariance among quantitative characters (for references see FALCONER 1981). Genetic covariances cause correlated responses to directional selection (FALCONER 1981; LANDE 1979) and indirect effects of stabilizing selection (LANDE and ARNOLD 1983).

However, pleiotropic effects lead to detectable genetic covariances only if different genes have a common bias towards positive or negative pleiotropic effects. If the pleiotropic effects have no common bias, positive and negative effects tend to cancel, leading to no net genetic covariance (LANDE 1980; CHEVERUD 1984; WAGNER 1984). Even in the absence of covariances, pleiotropic effects cause an association between the amount of genetic variance for different phenotypic characters. Even without detectable genetic covariance, pleiotropic effects can influence the consequences of directional and stabilizing selection. Sta- bilizing selection has confounding effects on the genetic variance of all those characters that are pleio- tropically connected to one another, if the allelic effects are not Gaussian (TURELLI 1985, 1988). In the case of a Gaussian distribution of allelic effects no confounding influences are possible in the absence of phenotypic correlations (LANDE and ARNOLD 1983).

Genetics 142: 223-234 (May, 1989)

The response to directional selection can also be affected by pleiotropic effects without genetic correlations if the characters are functionally interdepend- ent (WAGNER 1988a). If the functional significance of a character ZI depends on another character, say ZZ, then the correlation with fitness of z1 depends on the value of zZ. Therefore pleiotropic variation of z2 may deflate the intensity of directional selection on ZI. Hence, pleiotropic effects can have important consequences for the genetic properties of a population whether they lead to genetic covariances or not. Therefore it is important to understand how pleiotropic effects contribute to genetic variation.

One way pleiotropic effects contribute to the genetic composition of a population is a balance between pleiotropic mutations and stabilizing selection (LANDE 1980). Unfortunately the comparison of several models of mutation-selection balance with pleiotropy yields no simple picture (TURELLI 1985). TURELLI has shown that qualitatively different predictions are obtained based on different model assumptions. This contrasts with the theory of single character models, where fairly robust conclusions can be drawn from a variety of special models (LATTER 1960; BURGER 1986, 1988; SLATKIN 1987a; TURELLI 1984). As long as the per-locus mutation rate is low and selection not too weak, the predicted equilibrium variance is independent of assumptions about the number and effects

224 G . P. Wagner

of alleles or the distribution of mutational effects (TURELLI 1984; BARTON and TURELLI 1987, BURGER 1986, 1988). A similar robustness of predictions is not apparent if pleiotropic effects are taken into account.

In this paper a general model of “constrained pleiotropic effects” is presented which predicts the equilibrium genetic variance for any number of characters, and for large as well as small populations. The model is based on the assumption that the possible pleiotropic effects of mutations at a polygenic locus, i .e. , a locus that contributes to quantitative genetic variation, are constrained by the developmental genes which control the expression of these genes. Genes are assumed to be “held on the leash” by the genetic interaction system in which they occur (WAGNER 1988b). The biological motivation of this distinction is discussed in the next section.

The comparison of these predictions with those of TURELLI (1985) helps one to understand the reason for the qualitatively different results obtained from various models. It can be shown that the critical parameters determining the amount of genetic variance maintained by mutation-selection balance are the number of loci and the number of characters influenced by each gene. Finally it can be shown that the structure of the developmental system which con- strains the pleiotropic effects has effectively no influence on mean fitness at mutation-selection equilibrium.

THE MODEL OF CONSTRAINED PLEIOTROPY

The biological motivation for the model described below is obtained from findings in developmental genetics [for references see RAFF and KAUFMAN (1983) or WALBOT and HOLDER (1987)l. The phenotypic effects of mutations depend on the stage in development during which the genes are expressed. Embryonic development can be roughly divided into two periods: on one hand morphogenesis and on the other hand growth and differentiation. During morphogenesis the basic attributes of the body plan are laid down, such as the dorsoventral axis and the anterior-posterior polarity or segment number and segment identity in arthropods, as well as the principal characters, such as legs, wings and body parts (tag- mata). During growth and differentiation the final size and the proportions among the characters are defined. Although there is no strict distinction between the phase of morphogenesis and the phase of growth and differentiation, there is at least a strong tendency that morphogenetic events are common in early development while growth and differentiation prevail in later stages of development.

Genetic variations which disturb the basic body plan, such as homeotic mutations or polarity mutations such as bicaudal in Drosophila, are caused by

genes which are expressed during morphogenesis or even earlier, as maternal effect mutations (NWSSLEIN- VOLHARD and WIESCHAUS 1980). The phenotypic effects of these mutations are not simply large quantitative effects but determine whether major parts of the body are present, or in which part of the body certain characters such as wings or legs are located. These genes can be called developmental genes as they determine which genes are activated in different parts of the body. For instance, the genes of the bithorax complex determine in which segment of the insect larvae the genes for making wings are activated and in which segment genes responsible for the growth and differentiation of abdominal segments are expressed.

The expression of quantitative genetic variation on the other hand is largely confined to the framework set up during morphogenesis by “developmental genes.” Genes that contribute to the genetic variation of wing size are expressed as heritable variation in wing size only if the homeotic genes provided the anlage of wings during morphogenesis. Although little is known about the physiology of quantitative genetic variation it seems plausible that the majority of quantitative variation is caused by variation of growth rate, size of the anlage and the “stopping rule” for the growth of these characters (RISKA 1986; SLATKIN 1987b; THOMPSON 1975).

The basic idea derived from these considerations is that quantitative genetic variation is expressed in a framework set early during development. In terms of quantitative genetic theory this means that the possible pleiotropic effects of quantitative genetic variation are most often not changed by mutations at the “polygenic loci,” i . e . , those which are responsible for the heritable variation of quantitative characters, but are determined by the genotype of other (e.g., homeotic) loci which control the tissue-specific expression of these polygenic loci.

This strict distinction between “developmental” and “polygenic” loci is to some extent artificial. What is known of homeotic genes does not exclude that subtle mutations at homeotic loci can have only quantitative effects and may thus contribute to quantitative genetic variation. Hence, it is not claimed that such a strict distinction actually exists. Nevertheless the distinction is useful to factor out the consequences of constrained pleiotropic effects and the results of changing pleiotropic effects at individual loci.

To put this model into mathematical form, it is assumed that genes responsible for quantitative variation do not influence the size of the phenotypic character directly, but produce a gene product that has a physiologically important activity. This product might be an enzyme with a specific catalytic activity or it may be an extracellular matrix protein that has

Constrained Pleiotropic Effects 225

a specific affinity to certain cell membrane components. Genetic variation at these loci then leads primarily to variation in the physiological properties of this gene product, while the influence of the underlying physiological variable on the phenotypic characters is determined by the developmental system in which these genes are expressed. This implies that the pleiotropic effects are not caused by the gene product directly but by the set of developmental genes that guide morphogenesis.

Let us consider a set of N quantitative phenotypic characters z = (zl, . , zN) which are influenced by n genes. These n genes are assumed to contribute to the quantitative genetic variation of the characters, but not to morphogenesis. Through most of this paper it is assumed that developmental genes, which determine the pleiotropic effect of these n “polygenic loci,” are not segregating. The case of segregating developmental genes will be considered in a separate paper (G. P. Wagner and D. Rutledge in preparation). Fi- nally the usual decomposition of the phenotypic value z, into its genotypic and environmental components is applied,

z = x + e ,

where e is a random vector with expected value zero and variance V , for each of its components.

Each locus j = 1, . . . , n is assumed to produce one gene product which has one physiological property relevant for the genotypic value of the characters x . Let us denote the value of this physiologic property by yj, where the index j refers to the locus that pro- duces this gene product. This variable y, cannot be observed directly, and is also not identical to allelic effects on phenotypic traits in classical quantitative genetic models. Instead they can either be considered as underlying variables, comparable to those used in quantitative genetic theory of threshold characters (see FALCONER 198 1, chapter 8), or as allelic effects on a physiological trait expressed during development.

Assuming additivity among alleles at the same locus the genetic value of this underlying variable then is

y . = y! + yj’ I f

where y,! and yj’ are the properties of the gene prod- ucts from paternal and maternal alleles.

Note that y is a vector in n-dimensional Euclidian space which is not identical with the N-dimensional space in which the phenotypic and genotypic values of the characters are scaled. The way the underlying variables y influence the genotypic values x of the phenotypic characters z is determined by a “developmental function”f(y) that maps the space of underlying variables y, R”, onto the phenotype space, RN,

f(Y) X =f(y), R“ - RN.

In this paperf(y) is assumed to be constant. Little if anything can be said about the develop-

mental function f(y). It can assume any degree of nonlinearity, depending on the epistatic interactions among the loci, and thus also on the scaling system used to measure the phenotypic characters. However, if we consider metric characters and only small variations in the underlying variables y it might be suffi- cient to approximate the developmental function by a linear transformation. A linear transformation from R” onto R N is sufficiently characterized by a N X n matrix, say B. In this case the coefficients of the matrix can be considered as the partial derivates off with respect to the underlying variables, ie., the Jacobian matrix of the functionf(y).

x = By. (1)

The coefficients bq of the B-matrix determine the phenotypic effect of mutations at the locusj onto the character i

n

x; = b . YYI (2) j= 1

Equation 2 reveals the biological meaning of the B- matrix more clearly than the matrix notation (1). If one takes for instance a row vector of the B-matrix

bi.= (bil , - * , bin),

its components determine how much the variation in the underlying variables y ~ , . . , yn influences the genotypic values xi of character i. On the other hand the column vectors of the B-matrix

b.j = (b l j , - * * , b y )

determine the relative magnitudes of the pleiotropic effects at locusj (see Figure 1).

Replacing the developmental functionf(y) by a matrix B is identical to the assumption of additivity of allelic effects among loci widely used in quantitative genetic theory. Given the scaling of the phenotypic values, we are free to choose the scaling of the underlying values and the coefficients of the B-matrix, such that they match the scaling of the phenotypic values. In this paper the y values are scaled to make the variance of mutational effects equal to 1. In addition it is assumed that the distribution of mutational effects is Gaussian with zero mean and a variance that is independent of y. In this case the covariance matrix M of mutational effects on the genotypic values assumes the form (WAGNER 1984)

M = BBT. (3)

The genetic variance and covariance of the char-

226 G . P. Wagner

t

4 b) FIGURE 1 .-These diagrams show how variation in an underlying

parameter,y, oryi, influence the genotypic values of two quantitative characters xI and x2 according to the B-matrix model. a) In this case the effect ofy, the genotypic values is about the same, ie., blJ = b,. Note that all pairs o f genotypic values lie on a straight line with the slope b,/b,,. The ratio of the genotypic values is always equal to bpJ/ 61,. b) In this case the underlying variable has a stronger influence on xp than on XI, ie., bpJ > b,,; nevertheless, the ratio ofxs/xl remains the same.

acters is easily obtained as n

V, = Var(x;) = b$V, (4)

c,, = Cov(x;, XI) = bljbljV, (5)

j= 1

n

j= 1

with Vj = Var(yj).

The inclusion of linkage disequilibrium is straightforward, but irrelevant in the present context where only the equilibrium variance is considered. The expected equilibrium variance is not much influenced by linkage disequilibrium (BULMER 1972; BURGER 1989; LANDE 1976; LYNCH and HILL 1986; TURELLI 1984).

The main advantage of this model lies in the fact that quantitative genetic variation contributed by one locus can be treated as a single character problem, i . e . , a problem for which a great deal of results are already available (BARTON and TURELLI 1987; LAT- TER 1960; BURGER 1986, 1989; SLATKIN 1987a; TUR- ELLI 1984). The complete solution for the multichar- acter case is then simply obtained by a linear combi- nation of the single character solutions.

RESULTS

So far the model described above puts restrictions only on the possible phenotypic effects of mutations affecting quantitative traits. The phenotypic effects of mutations at some locus j are essentially restricted to a one-dimensional subspace of the multidimensional phenotype space, defined by the appropriate column vector b.j of the B-matrix. This implies that the ratios of the pleiotropic effects on any pair of characters are

the same for all alleles at a particular locus. The model is still open to any assumptions about the number of alleles, the allelic effects on the underlying variable y,, the selection regime, population size and so on. One can assume the continuum-of-alleles model of KIMURA (1 965), the ladder model Of OHTA and KIMURA (1 973) or the two-allele model of LATTER (1960). T o for- mulate the model, a continuum of alleles is assumed for the underlying parameter of each locus. However, this assumption does not influence the predicted equilibrium variance as long as the per locus mutation rate is low (<0.0001), the average mutational effect is not too small and selection not too weak. Under these conditions the house of cards approximation of the continuum of alleles model holds (TURELLI 1984), and the equilibrium variance does not depend on the effects of the alleles and are identical with the predictions from other models.

Following the analyses of LANDE (1 980) and TUR- ELLI (1985) the fitness function is assumed to be a Gaussian function

w(z) = exp(-(Yz)(z - O)O"(Z - o ) ~ ) (6)

where 0 is a positive definite symmetrical matrix determining the intensity of stabilizing selection and o is the optimum of w(z). If the environmental effects are Gaussian, then the fitness function of the genotypic vector x is again a Gaussian function with the matrix W = 0 + E, where E is the environmental covariance matrix. Without loss of generality the cal- culations can be very much simplified by choosing a coordinate system of the phenotype space which has its origin right at the optimum of the fitness function and has base vectors that coincide with the eigenvec- tors of the matrix W (the so-called canonical represen- tation). The fitness function then reads

N w(x) = exp - (V2) (.x;/w,)' (7)

where w' are the eigenvalues of W. The results based on fitness function (7) are still entirely general but mathematically more tractable. The change in the coordinate system does change the coefficients of the B-matrix but leaves the scaling of the underlying parameters y unaffected. The predictions for the more general form of the fitness function (6) can be gained from those from (7) by a linear transformation.

Selection on the underlying parameters: Given the fitness function (7) and the relationship between the underlying parameters y and the genotypic values X

(2) the fitness function of the underlying parameters is

w(y) = exp(-('/2)(By)W"(By)T). (8)

In order to calculate the equilibrium variance of the underlying parameters we need to know the marginal

I= 1


fitness for the y-values at each locusj r r

T o obtain a solution of (9) it is convenient to intro- duce a new vector

gj = ( b Ij /Wl, * . 9 b N j / W N )

which is an N-dimensional vector associated with locus j. Then the fitness function can be written as

W(Y) = e~p(-(G)b,%&, + 2yj C (gk, &)yk k#j

+ x Y& goYlf) (10) kJ#j

where (.,.) is the scalar product of two vectors. As long as n is large but nu << 1, where u is the

average per locus mutation rate, the marginal fitness is approximately given by (see APPENDIX)

W ( y j ) = exp(- (~z)~ .hI~) (1 1)

where [gj] is the Euclidian norm of vector gj

C N I

The fitness function is again approximately Gaus- sian with the fitness weight

Mutation-selection balance in large populations: Since the genetic variation at each locus essentially influences a one-dimensional trait y,, the single-character theory of mutation-selection balance can directly be applied to the B-matrix model. The mutation- selection equation is then

where

P'(yj9 t ) = W(yjlP(yp t ) /w( t )

and G ( t ) is the mean fitness at generation t , and m(x) is the distribution of mutational effects with variance 1 and mean 0. Approximate stationary solutions of this equation have been obtained for the case of high per-locus mutation rate and weak selection [the Gaus- sian approximation (KIMURA 1965; LANDE 1976)] and for low per-locus mutation rate ul and not too weak selection, ie., the house of cards approximation of TURELLI (1 984). In this paper the more conservative assumption of low per-locus mutation rate is adopted.

The consistency criterion for the application of the house of cards approximation is

uj << a$/Vsj (14)

where af is the variance of mutational effects at locus j on character i. In terms of the complete multivariate model this criterion reads

N

uj << (bij /Wi)2 (15) i= 1

Criterion (15) is far less restrictive than the single- character criterion (written in terms of the B-matrix model)

U, << (bv/Wi)' (1 6)

or the two-character criterion given by TURELLI (1 985)

uj << a14 ~ ~ J / ( w I w ~ )

for the continuum of alleles model and

uj << max(al,/wl, a2&~2)~

for the five allele model. Hence with pleiotropic effects and stabilizing selec-

tion on many characters the house of cards approximation seems to be fairly robust. This fact is not surprising since under the house of cards conditions all pleiotropic effects are selected against even if they might not lead to phenotypic correlations (TURELLI 1985). Thus any pleiotropic effect on a character under stabilizing selection increases the selection intensity at the locus and makes the house of cards approximation even more precise, because it assumes selection intensity at the individual loci to be high compared to mutation rate.

Using the single character house of cards approximation, the equilibrium variance for the underlying parameter is

Vir(yj) = Vj = 4 uj[gr]-'. (1 7)

Using the equations for the genetic variance and covariance of the characters (4) and (5) the genetic equilibrium variance and covariance are

n

Qggl = 4 ~j(bv/[&])' (18) j= 1

n

6, = 4 2 ~ , b q b l j l [ g r ] ~ . ( 1 9) j = 1

As expected for a house of cards solution, the expected equilibrium variance does not depend on the average effect of mutations, ie., formulas (1 8) and (1 9) do not change if each coefficient of the B-matrix bq is multiplied by the same factor.

House of cards approximations of the equilibrium variance in two-character models have been derived by TURELLI (1985). He analyzed two models with

228 G. P. Wagner

different assumptions about the number of alleles. On one hand he used the continuum of alleles model of LANDE ( 1 980) but used the house of cards equilibrium approximation. On the other hand he analyzed a five- allele model, whose equilibrium was also approximated by the house of cards method. Assuming no correlations in the mutational effects at each locus, and choosing a coordinate system where the fitness matrix W is a diagonal matrix, as done in this paper, the house of cards solution for the continuum of alleles model is (TURELLI 1985, formula 24),

n

fgI(HC) = 4w: uj/(l + d) , (20) J= I

where

and

where uq is the standard deviation of mutational effects on character i of mutations at locus j. For the five-allele model the solution is

n

Ggg1(5) = 4w: uj/(l + d 2 ) (21)

(TURELLI’S formula 38). TURELLI concluded that in the two-character case the equilibrium variance strongly depends on assumptions about the number of alleles. However, a comparison with the two-character solution of the B-matrix model

j = 1

Qgl(B) = 4 u j b ? j / ( ( b ~ j / w ~ ) ~ + (hj/w2)2) (22) j = 1

reveals that the number of alleles is not the factor that accounts for the difference between formulas (20) and (21). In the two-character case, the B-matrix model gives the same result as TURELLI’S five-allele model even though prediction (22) was derived under the assumption of a continuum of alleles at each locus.

A hint for resolving this seeming paradox is obtained from results based on the moment equations derived by BARTON and TURELLI (1987). TURELLI (1988) gives a two-character extension of the model in BARTON and TURELLI (1987). In this general approach it can be shown that the five-allele result (21) is obtained whenever

1 1

Vlj lV2j = (alJ/a2j)2,

i .e . , if the ratio of the equilibrium variances at each locus is equal to the ratio of the variances of mutational effects at each locus. This is true for the five- allele model because of the restrictions introduced by the small number of alleles and in the B-matrix model because of constrained pleiotropy.

This suggests that the five-allele B-matrix result has

to be also derivable from the TURELLI-LANDE continuum of alleles model by introducing the assumption that the correlation between the pleiotropic effects is equal to one at each locus. In fact it is easy to show that this is the case by a straightforward evaluation of TURELLI’S integral (2.5 on page 193 of TURELLI 1985). Hence the two results obtained by TURELLI from the five-allele model and the continuum of alleles model are not arbitrarily different results depending on assumptions about the number of alleles. Rather they appear to be two extremes on a continuum. On one hand there are models with constrained pleiotropy, i . e . , the five-allele model and the B-matrix model, and on the other hand, LANDE’S continuum of alleles model with the additional assumption that there are no constraints on the possible pleiotropic effects, i.e., the assumption that there is no correlation between the pleiotropic effects.

Given our ignorance about the genetic basis of heritable quantitative variation and the statistics of the pleiotropic effects of mutations at individual genes, there is no way to discriminate between TUR- ELLI’S house of cards approximation of LANDE’S continuum of alleles model and the B-matrix model. However, the results presented here seem to delineate the range of possible outcomes for mutation-stabilizing selection equilibrium. One may confidently infer that in the case of intermediate levels of correlation between pleiotropic effects the mutation-selection equilibrium has to be somewhere between the continuum of alleles prediction (20) and the B-matrix prediction (22).

Based on the analogy between the five-allele model and the B-matrix model for the case of two characters one can infer that the major conclusions TURELLI obtained from his five-allele model also hold true for the B-matrix model. Single-character models tend to overestimate the true equilibrium variance maintained by mutation-selection balance with pleiotropic effects. If the number of characters connected by pleiotropic effects increases the situation even gets worse.

If we consider the completely symmetrical case, where bij = a, uj = u, and wi = w for all characters i and all loci j the genetic equilibrium variance becomes

= 4nuw2/N (23) which is 1/N of the equilibrium variance predicted from the single character model (TURELLI 1984). Hid- den pleiotropic effects on characters under stabilizing selection deflate the amount of genetic variance that can be maintained by mutation-selection balance. The critical parameter seems to be the number of characters that are influenced by an individual gene.

The conclusions that can be based on these and TURELLI’S results are threefold: (i) One may either assume that the pleiotropic effects are universal, as do


WRIGHT and TURELLI, and conclude that mutation- selection balance is unlikely to account for the observed amount of heritable variation in natural populations; (ii) another way to reconcile the present results with observed heritabilities is to assume that pleiotropic effects are not as universal as thought and that there are many genes that can contribute to the variation of a quantitative character; and (iii) finally one may argue that the result (23) is an artefact of the symmetry assumption, that all genes have the same pleiotropic effects. The last possibility is explored in a separate paper using a statistical approach to de- scribe pleiotropic effects.

Finite populations: The fact that the complete solution of the multivariate B-matrix model can be obtained from the single character mutation selection theory invites generalizing the results to finite populations.

The asymptotic variance for a single character under mutation and drift has been derived by CLAYTON and ROBERTSON (1955), CHAKRABOTRY and NEI ( 1 982), and LYNCH and HILL (1 986). Applying these results to the B-matrix model without selection and employing the assumption Neu << 1 , the variance of the underlying variables ~j becomes

e, = 4u,Ne (24)

and the genetic covariance matrix

G = 4NeuBBT. (25)

Several approximate solutions of the single character mutation, selection and drift problem have been derived. One solution by BULMER (1972) performs very well in a comparison with stochastic simulations (BURGER, WAGNER and STETTINGER 1989) but is algebraically not very revealing since it is essentially a rational function of confluent hypergeometric func- tions. An equally precise but algebraically much more simple formula was obtained recently by BURGER (1989b) and KEICHTLEY and HILL (1988). Applied to the underlying parameters of the B-matrix model, the formula reads

c;. = 4ujNe/(1 + Ne[%]‘). (26)

Consequently the equilibrium variance and covariance is

n

t&+ = 4Ne c bijUj/(l + Ne[%]’) (27) j= 1

n

del = 4N, b,b,uj/(l + N.[g,]’). (28)

These formulas have been tested by stochastic simulations. The simulation model used is essentially the same as that described in BURGER, WAGNER and STET- TINGER ( 1 989), except that two characters with constrained pleiotropy were simulated.

J = 1

In the model the life cycle consists of three stages: 1 . Random sampling of breeding pairs from the base

population. The base population is the surviving offspring of the preceeding generation. To keep the population size constant it was assumed that there is a fixed number of “nesting places” that limit the number of breeding pairs and the maximal population size Np. Two individuals, chosen randomly from the base population, constitute the breeding pair of each nesting place.

2. Production of offspring: each breeding pair pro- duces the same number of offspring, namely 10. The genotype of each descendant is obtained from that of its parents by recombination according to the specified recombination rate rc of adjacent loci. Hence each locus is assumed to be located on a separate chromosome, if rc = 0.5, or on one chromosome, if rc < 0.5. This corresponds to the so-called “mouse” and “Dro- sophila’’ models of BULMER ( 1 976). The distribution of mutational effects is Gaussian with zero mean and variance 1 .

3. Viability selection was imposed by assigning fitness according to the Gaussian fitness function. Its values vary between 0 and 1 and were interpreted as proba- bilities of survival. After selection the surviving offspring served as the base population for the next parental generation, as described above.

In this model the effective population size is given by

Ne = lONp(1 + 3/(10(Np - 1)))/9 (see BURGER, WAGNER and STETTINGER 1989).

Tables 1 and 2 summarize the simulation results. Each datum in Tables 1 and 2 is a mean value over the time averages of 20 or 40 runs. The time averages were taken from generation 140 1 to generation 2000.

In Table 1 the average equilibrium variances are compared to the predictions of the B-matrix model @), formula (27), and the single character prediction q(l) , which ignores the pleiotropic effects. The simulations are in good agreement with the predictions of the B-matrix model. The results show that pleiotropic effects reduce the average equilibrium variance, although the reduction is not as severe as in large populations. Note that for instance in case 2 of Table 1 the prediction of the multivariate model would be ‘/2 that of the single character model (see formula 23), while in finite populations the reduction due to pleiotropic effects is only by a factor of 0.74.

In Table 2 simulations with free recombination of genes are compared with simulations where adjacent loci had a recombination rate of 0.01. None of the differences is significant according to the Mann-Whit- ney U test (P > 0.3).

Influence of the developmental function on mean equilibrium fitness: In single-character models of mutation-selection balance under Gaussian stabilizing

230 G . P. Wagner

TABLE 1

Influence of pleiotropic effects on the equilibrium variance of one character in finite populations

Caae Matrix W I WP C(0bs) (SEW W ) licl, NO.

1 M 1 100 100 0.0961 (0.0086) 0.1003 0.1056 20 2 M 1 10 10 0.05 18 (0.0042) 0.0527 0.07 16 ‘LO 3 MI 100 10 0.067 1 (0.0089) 0.07 13 0.1065 20 4 M3 100 100 0.09 17 (0.0 109) 0.1024 0.1056 20 5 M3 10 10 0.0558 (0.0053) 0.0603 0.07 16 20 6 M3 100 10 0.0851 (0.0068) 0.0837 0.1056 4 0

M1: all coefficients of the B-matrix are 0.2236 M3: blJ = 0.2236 for all j less equal 50

bzl = (-1)’ 0.2236 for j>20, bzJ = 0 else

The simulation results c(0bs) are compared with the prediction of the B-matrix model c ( B ) and the single character prediction c( l) , which ignores pleiotropic effects. The effective population size in all simulations is 11 1.5, the number of loci 50, the per locus mutation rate is 0.0001, the recombination rate 0.5, and the environmental variance is 1. No. is the number of simulation runs, and SEM is the standard error of the mean.

TABLE 2

The equilibrium variance ( ~ S E M ) of trait 1 with pleiotropic effects to trait 2 and low recombination rate rc

M4 o r M5 M 4 M5 W i WP rc = 0.5 rc = 0.01 rc = 0.01

~ ~~

10 10 0.0623 0.0646 0.0595 (0.0037) (0.0046) (0.0040)

100 10 0.0924 0.0982 0,1099 (0.0074) (0.0078) (0.0162)

M4: bl, = 0.2236 for all j

M5: blJ = 0.2236 for all j bpJ = rt0.2236 if j is even and bzJ = 0 if j is odd

bxJ = rt0.2236 for j>25 and b, = 0 else

Each mean value is obtained as an average over 40 simulation runs. None of the differences seen in the lines of this table is statistically significant according to the Mann-Whitney U test. As in Table 1 the number of loci is 50, the effective population size is 100, and the per locus mutation rate 0.0001.

selection, the house of cards approximation predicts that mean fitness is independent of the allelic effects and the intensity of stabilizing selection. This is most easily demonstrated by assuming that the number of genes is large enough to justify the assumption of a Gaussian distribution of genotypic values. Then mean fitness in equilibrium is

2 = (w2/(w2 + IQ”. (29)

Because the equilibrium variance predicted by the house of cards approximation is Fg = 42inw2, the mean fitness is simply

3 = (1 + 4nG)”’Z. (30)

This implies that 6 is a function only of the genomic mutation rate. This result is not uncommon for models of mutation-selection balance. Other examples where mean equilibrium fitness is a function only of genomic mutation rate come from a deterministic treatment of Felsenstein’s model of Muller’s ratchet

(HAIGH 1978) and the one-locus two-allele models with selection and mutation (for reference see CROW and KIMURA 1970, section 6.12).

A similar principle applies to the multivariate B- matrix model, and most probably for all models with constrained pleiotropic effects. This result is demonstrated below in two ways: (i) by the aid of a weak selection approximation of mean fitness, and (ii) for N = 2 under the assumption that the genotypic values are Gaussian.

Under weak stabilizing Gaussian selection the mean fitness can be approximated by

W = exp(mean(1og w ) ) (31)

[see, for instance, BULMER (1972)l. Then N

mean(Iog w ) = (-Vi!) v&J?. (32)

Using the equilibrium solution of the B-matrix model

,= 1

(18) N n

mein(Iog w ) = (-Vz) I/W? ujb&312. (33)

A trivial rearrangement of the summation order reveals

i= I j = 1

n N

mein(1og w ) = (-VZ)G [gJ” b$/w?. (34)

Note that the inner sum in (34) is [&I2, and this gives

W E exp(-2nG) (35)

which is approximately equal to the result of the single-character model (30) as long as 4nii << 1.

According to this result, the B-matrix (= the “developmental function”) has to be considered as an almost neutral character, ie., the structure of B does not much influence the mean equilibrium fitness. This result holds only among those B-matrices which have the same number of column vectors with norms

j= I i= 1

Constrained Pleiotropic Effects 23 1

greater than zero. Biologically this means that the structure of the developmental system has no influence on mean fitness as long as the number of loci contributing to quantitative genetic variation remains the same and no aspect of development other than the expression of quantitative genetic variation is con- cerned.

However, it is important to keep in mind the conditions under which this result has been obtained: large population size (deterministic description), and low per-locus mutation rate (house of cards approximation). It is obvious from the results presented above that the average mean fitness of small populations depends at least slightly on the B-matrix. Also a solution of the B-matrix model with the Gaussian approximation ( i e . , for high per-locus mutation rates) depends on the B-matrix.

An estimate of how large the factors are that are neglected by approximation (3 1) can be obtained from an analysis of the two-character case, using the assumption that the genotypic values are Gaussian. Note, however, that the results do not require a Gaussian distribution of allelic effects at individual loci. Then the mean fitness in this case is

3 = ( q + h + l)+

h = egl /w: + eg2/w;

q = egl(l - r’)/w:w: (36)

where r is the genetic correlation. As shown above h is independent of B,

h = 4nG (37)

and q depends on B, but is always <(h/2)‘, because the geometric mean is always smaller than the arith- metic mean.

q < 4 G2n2. (38)

Relation (38) also places an upper limit on the selection coefficient s of a gene that may change the structure of the B-matrix

s < 2ii2n2. (39)

The maximal selective advantage associated with a change in the structure of the B-matrix is of the order of the squared genomic mutation rate, if only stabilizing selection is acting on a set of polygenic loci which are themselves close to equilibrium. But note that the upper limit (39) has been derived from an approximate solution of the equilibrium variance (1 8 and 19). It might well be that the influence on equilibrium fitness of the B-matrix, i e . , the mutational effects, is actually zero as in the case of single locus mutation- selection balance results [see CROW and KIMURA (1970) pp. 299-3031.

DISCUSSION

The model of constrained pleiotropy is a limiting case of LANDE’S multivariate continuum of alleles model (LANDE 1980). This model was studied by LANDE using the Gaussian approximation and for N = 2 by TURELLI (1985) using the so called house of cards approximation. Hence the results of LANDE (1980) and TURELLI (1985) are based on the same model. The different conclusions reached by them are due to the use of different approximation proce- dures. Therefore I will refer here to the general model as LANDE’S continuum of alleles model, and to the result of TURELLI (1985) as the house of cards approximation of LANDE’S model.

In LANDE’S continuum of alleles model the alleles can have any kind of pleiotropic effects. The distribution of phenotypic effects is determined by a locus- specific covariance matrix. In contrast, the mutational effects in the B-matrix model are constrained to a linear subspace of the phenotype space, i.e., for all alleles at a locus the phenotypic effects have the same ratio. The assumption of constrained pleiotropy is motivated by the fact that the possible phenotypic effects of a gene are determined by the developmental framework in which it is expressed (for instance by the genes which regulate the tissue specific expression of the polygenes).

The assumption of constrained pleiotropy is easily incorporated into LANDE’S general model by assuming that the locus-specific mutational effects are all correlated with correlations equal to one. Under this assumption TURELLI’S house of cards treatment of LANDE’S model (for N = 2) yields the same predictions for the equilibrium variance as the B-matrix model. Hence the B-matrix model is fully compatible with TURELLI’S treatment of the continuum of alleles model.

However, the main advantage of the B-matrix model is that it is easily generalized to any number of characters and to finite populations. The reason for this flexibility of the B-matrix model is the notation used, where the variation at a particular locus is primarily mapped on a one-dimensional underlying variable y,. Hence, it is possible to derive the complete multivariate solution by solving n single-character problems, which are already well understood.

Interestingly the assumption of constrained pleiotropy ( i e . , the B-matrix model and LANDE’S continuum of alleles model with correlated mutational effects) give the same predictions as TURELLI’S (1985) five-allele model. On the other hand, the house of cards solution of LANDE’S continuum of alleles model gives qualitatively different results if the pleiotropic effects are totally uncorrelated (TURELLI 1985). TUR- ELL1 attributed this discrepancy to the assumption about the number of alleles, which would be in con-

232 G. P. Wagner

trast to the properties of the single-character models, which are independent of this assumption. However, the results presented in here show that the crucial difference is not the number of alleles, as the five- allele result can be derived from the B-matrix model as well as from LANDE’S continuum of alleles model with correlated mutational effects. The crucial factor is the assumption of constraints on the possible pleiotropic effects. In the five-allele model the possible pleiotropic effects are constrained because of the low number of alleles, while in the B-matrix model and the continuum of alleles model correlations between the mutational effects have the same effect. This suggests that the two results obtained by TURELLI (1985) for N = 2 are two extremes on a continuum. At one side there is the result for stringently constrained pleiotropic effects and on the other side the result for totally uncorrelated mutational effects.

The results on mutation-selection balance in the B- matrix model generally support the conclusions of TURELLI (1985) concerning the influence of pleiotropic effects on the equilibrium genetic variance. Pleiotropic effects on characters that are themselves under stabilizing selection reduce the equilibrium variance, whether there are phenotypic correlations or not.

The B-matrix model is an attempt to include developmental constraints on the phenotypic expression of genetic variation in quantitative genetic models, but the structure of the model is not based on a mecha- nistic model of development. Recently some attempts have been made to include explicit developmental mechanisms into quantitative genetic models (RISKA 1986; SLATKIN 198713). The derivations of SLATKIN (1987b) are based on the model of heterochrony by ALBERCH et al. (1979). The resulting model, which relates the underlying developmental parameters to phenotypic variation, is nonlinear and thus not directly comparable to the B-matrix model. However, if we consider the evolution of the genetic variance, as in this paper, SLATKIN’S model can be approximated by a linear function of the underlying parameters, where the coefficients are the population means of the underlying parameters (see for instance Equation 5 in SLATKIN 1987b). Hence, with respect to the evolution of variances, SLATKIN’S model can be rep- resented by a B-matrix model.

In addition, SLATKIN has shown that under stabilizing selection the mean values of the developmental parameters are not influenced by selection. Hence the mean values do not influence the mean fitness under stabilizing selection. This is analogous to the result that the coefficients of the B-matrix are almost neutral characters under Gaussian stabilizing selection. How- ever, under directional selection the situation is very different. Then the mean values of the developmental

parameters are changed and the analogy to the B- matrix model breaks down.

These properties of the B-matrix model resemble those of Fisher’s dominance modification model. Dominance modifiers have virtually no influence on mean equilibrium fitness as long as the primary alleles are close to equilibrium. However, if the primary alleles are in the process of substitution the modifier may experience considerable selection (WAGNER and BURGER 1985).

These results have interesting consequences for the evolution of genetic covariance matrices. The genetic equilibrium covariance matrix depends on the fitness function, the per-locus mutation rate and the B-matrix (or the mean values of the developmental parameters in SLATKIN’S model). Since the B-matrix as well as the mean values of the developmental parameters are almost neutral, many covariance matrices can be re- alized under the same regime of stabilizing selection, depending on the structure of the B-matrix. For any given B-matrix or any given set of mean values of the developmental parameters in SLATKIN’S model, the genetic covariance is determined by the selection function (as proposed by CHEVERUD 1984), but within the limits set by the given B-matrix.

Changes of the B-matrix or the mean values of the developmental parameters are possible only by directional selection or random drift. Hence, the genetic covariance matrices of different populations will to a large extent be determined by the history of directional selection pressures on these populations, as proposed by RISKA (1986). This fact may explain why the comparison of genetic or phenotypic correlation matrices among related populations does not reveal any discernable pattern [RISKA 1985; LOFSVOLD 1986; J. M. CHEVERUD (on monkey skulls) unpublished data; G. P. WAGNER (on honey bees), unpublished data]. Even though the genetic and phenotypic correlation matrices are significantly similar to one another, the differences between them are not related to any known aspect of the evolution of the populations. This fact is understandable if the genetic covariance matrix is in part influenced by a neutral character, such as the B-matrix or the mean values of the developmental parameters, as in SLATKIN’S model.

The author is indebted to R. BURGER, J . CHEVERUD, T. NAGY- LAKI, P. PHILLIPS, M . SLATKIN, M . TURELLI and C. VOGL for reading the manuscript and many helpful suggestions, and to F. STETTINGER for his help in performing the simulations. The finan- cial support by the Austrian Fonds zur Forderung der wissenschaft- lichen Forschung (project No. P5994) is gratefully acknowledged.

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APPENDIX

To obtain an approximate equation for the marginal fitness of the underlying variables one has to evaluate the integral (9). To simplify this task two new variables are introduced:

q l j = (&, g k ) y k W

which are independent of y,. q l j is a linear function of (n - 1) y k values and 9 2 , is a quadratic form of the same random variables. Formula ( 1 0) simplifies to

If (n - 1) >> 1 , q 1 j is approximately Gaussian with mean

Since q 2 , is a quadratic form of the same random variables as q ~ j , q l j and q 2 j are approximately independent. (If y would be Gaussian, q1j and q 2 , would be exactly independent.) Hence the relative marginal fitness of y j is independent of q 2 , and the integral (9)

234 G. P. Wagner

can be approximated by w( yj) = (~a~ar(qlj)-”exp

+m

- (y2)(%, g)y: S_, exp{2yjqljjexp

- {q?j/2Var(qlj)Jdq1j which can explicitly be evaluated:

which shows that the fitness function of the underlying variables is also approximately Gaussian with the parameter

V s j = (gp &) - Var(q1J

as long as bq cc wir uj(n - 1 ) cc 1 , and u, << l / w ? , Var(q1j) can be ignored.

w(yj) 1: exp - (Y2){(%, gJ - Var(qlJJy;,

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