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 ...
Copyright 0 1989 by the Genetics Societyof 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 variationat a locusj primarily influencesan underlyingphysiological 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. T h e equilibrium variance under mutation-stabilizing selection balance in infinite and finite populations is derived by using the house of cards approximation. T h e results are compared to the predictions given by M. Turelli in 1985 forpleiotropic 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 theequilibrium variance in multivariate models with pleiotropy is the assumption about constraints on the pleiotropic effects, and not the number of alleles as proposedby Turelli. Finally it is shown that underGaussian 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. T h e consequences for the evolution of genetic covariance matrices are discussed.
The response to directional selection can also be LEIOTROPIC effects cause an association of heritable variation among different phenotypic char- affected by pleiotropic effects without genetic correlations if the characters are functionally interdependacters. Pleiotropic effects are among the majorcauses ent (WAGNER 1988a). If the functional significance of of genetic covariance among quantitative characters a character ZI depends on another character, say ZZ, (for references see FALCONER 1981). Genetic covarithen the correlation with fitness of z1 depends on the ances cause correlated responses to directional selecvalue of zZ. Therefore pleiotropic variation of z2 may tion (FALCONER 1981; LANDE 1979) and indirect effects of stabilizing selection (LANDEand ARNOLD deflate the intensity of directional selection on ZI. Hence, pleiotropic effects can have important conse1983). quences for the genetic properties of a population However, pleiotropic effects lead to detectable gewhether they lead to genetic covariances or not. netic covariances only if different genes have a comTherefore it is important to understand how pleiomon bias towards positive or negative pleiotropic eftropic effects contribute to geneticvariation. fects. If the pleiotropic effects have no common bias, One way pleiotropic effects contribute to the gepositive and negative effects tend to cancel, leading to no netgenetic covariance (LANDE 1980; CHEVERUDnetic composition of a populationis a balance between 1984; WAGNER 1984). Even in the absence of covaripleiotropic mutations and stabilizing selection (LANDE ances, pleiotropiceffects cause an association between 1980). Unfortunately the comparison of several the amount of genetic variance for different phenomodels of mutation-selection balance with pleiotropy yields no simple picture (TURELLI 1985). TURELLI has typic characters. Even without detectable genetic covariance, pleiotropic effects can influence the conseshown that qualitatively different predictions are obquences of directional and stabilizing selection. Statained based on different model assumptions. This bilizing selection has confounding effects on the contrasts with the theory of single character models, genetic variance of all those characters thatare pleiowhere fairly robust conclusions can be drawn from a tropically connected tooneanother, if the allelic variety ofspecial models (LATTER 1960; BURGER effects are not Gaussian (TURELLI 1986, 1988; SLATKIN1987a; TURELLI 1985, 1988). In the 1984). As long case of a Gaussian distribution of allelic effects no as the per-locus mutation rate is low and selection not confounding influences are possible in the absence of too weak, the predicted equilibrium variance is indephenotypic correlations (LANDE and ARNOLD 1983). pendent of assumptions about the number and effects
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ofalleles orthe 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 generalmodel 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 theassumption that thepossible 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 onthe leash” by the genetic interaction systeminwhich they occur(WAGNER1988b). 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 amountof 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 systemwhich constrains the pleiotropic effects has effectively no influence on mean fitness at mutation-selection equilibrium. THE MODEL OF CONSTRAINEDPLEIOTROPY
The biological motivation for the model described below is obtainedfrom findings in developmental genetics [forreferences see RAFF and KAUFMAN (1983) or WALBOT and HOLDER(1987)l. The phenotypic effects of mutationsdependonthestage 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 thedorsoventral axis andthe anterior-posterior polarity or segment number and segment identity in arthropods, as well as theprincipal characters, such as legs, wings and body parts (tagmata). Duringgrowth and differentiationthe 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 disturbthe 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 (NWSSLEINVOLHARD 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. Forinstance, 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 forthe 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 geneticvariation of wing size are expressed as heritable variation in wingsizeonly if the homeotic genes provided the anlage of wingsduring 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, sizeof theanlage and the “stoppingrule”for the growth of thesecharacters (RISKA 1986;SLATKIN 1987b; THOMPSON 1975). The basic idea derived from these considerationsis 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 geneticvariation are most oftennotchanged by mutations atthe “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. T o put this model into mathematical form, it is assumed that genes responsible for quantitative variation do notinfluence the sizeof 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 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 genesare 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 ofthis paper it is assumed that developmental genes, which determine the pleiotropic effect of these n “polygenic loci,” are not segregating. T h e case of segregating developmental genes will be considered in a separate paper (G. P. Wagner and D. Rutledge in preparation). Finally the usual decomposition of the phenotypic value z, into its genotypic and environmental components is applied,
225
In this paperf(y) is assumed to be constant. Little if anything can be said about the developmentalfunction 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 sufficient 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.
The coefficients bq of the B-matrix determine the phenotypic effect of mutations at the locusj onto the character i n
+ yj’
where y,! and yj’ are the properties of the gene products 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, R N , =f(y),
R“
(2)
j= 1
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 produces this gene product. This variable y, cannot be observed directly, and is also not identical to allelic effects on phenotypictraits inclassical quantitative genetic models. Instead they can either be considered as underlying variables, comparable to those used in quantitativegenetictheory of thresholdcharacters (see FALCONER198 1, chapter 8), or as allelic effects ona physiological traitexpressed during development. Assuming additivity among alleles at thesame locus the genetic value of this underlying variable then is
X
bY.YI
x; =
z=x+e,
y.I = y!f
(1)
f(Y)
RN.
Equation 2 reveals the biological meaning of the Bmatrix 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 ,
-
* *
,by)
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 tochoose 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 thedistribution 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 as1984) sumes the form (WAGNER M = BBT.
(3)
The genetic variance and covariance of the char-
G . P. Wagner
226
t
4
b)
FIGURE1 .-These diagrams show how variation in an underlying parameter,y, oryi, influence the genotypic values of two quantitative characters xI and x2 according to theB-matrix model. a) In this case the effect ofy, the genotypic valuesis about the same, ie., blJ = b,. Note that all pairs o f genotypic values lie on a straight line with the slope b,/b,,. T h e ratio of the genotypic valuesis 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)
the same for all alleles at a particularlocus. The model is still open to any assumptions about the number of alleles, the allelic effects on the underlyingvariable y,, the selection regime, population size and so on. One can assume the continuum-of-alleles model of KIMURA (1965), the ladder model Of OHTA and KIMURA(1 973) or the two-allele model of LATTER(1960). T o formulate the model, a continuum of alleles is assumed for the underlying parameterof each locus. However, this assumption does not influence the predicted equilibrium variance as long as the perlocus mutation rate is low (20, bzJ = 0 else
c(
l), which T h e simulation results c(0bs) are comparedwith the prediction of the B-matrix model c ( B ) and the single character prediction ignores pleiotropic effects. T h e 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 ofsimulation 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
Wi
WP
M 4 o r M5 rc = 0.5
M5
M4
rc
= 0.01
rc = 0.01 ~
10
10
100
0.0623 (0.0037)
0.0646 (0.0046)
10 0.0982 0.0924 (0.0074)
(0.0162) (0.0078)
~~
0.0595 (0.0040) 0,1099
= 0.2236 for all j bpJ = rt0.2236 if j is even and bzJ = 0 if j is odd M5: blJ = 0.2236 for all j bxJ = rt0.2236 for j>25 and b, = 0 else
M4:
bl,
W
Each mean value is obtained as an average over 40 simulation runs.None of thedifferences seen in the lines ofthis table is statistically significant according to theMann-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 theequilibrium variance predicted by the house of cards approximationis Fg= 42inw2, the mean fitness is simply
3 = (1 + 4nG)”’Z.
(HAIGH1978)andthe one-locus two-allele models with selection and mutation (for reference see CROW and KIMURA1970, section 6.12). A similar principle applies to the multivariate Bmatrix model, and most probably for all models with constrained pleiotropic effects. This result is demonstrated belowin 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
(30)
This implies that 6 is a function only of the genomic mutationrate. This result is notuncommonfor models of mutation-selection balance. Other examples where mean equilibrium fitness is a function only of genomic mutation rate comefromadeterministic treatment of Felsenstein’s model of Muller’s ratchet
= exp(mean(1og w ) )
(31)
[see, for instance, BULMER(1972)l. Then N
mean(Iog w ) = (-Vi!)
,=1
v&J?.
(32)
Using the equilibrium solution of the B-matrix model (18) N
mein(Iog w ) = (-Vz)
n
I/W? i= I
ujb&312.
(33)
j= 1
A trivial rearrangement of the summation order reveals N
n
mein(1og w ) = (-VZ)G
[gJ” j=I
Note that the inner sum in (34) is
W
E
exp(-2nG)
b$/w?. (34) i= 1
[&I2,
and this gives (35)
which is approximately equal tothe result of the single-character model (30) as long as 4nii 1 , mean
q1j
is approximately Gaussianwith
Since q 2 , is a quadratic form of the same random variablesas q ~ j , q l jand q 2 j are approximately independent. (If y would be Gaussian, q1j and q 2 , would be exactly independent.)Hencethe relative marginal fitness of y j is independent of q 2 , and the integral (9)
234
G. P. Wagner
which shows that the fitness function of the underlying variables is also approximately Gaussian with the parameter
can be approximated by w( yj) = ( ~ a ~ a r ( q l j ) - ” e x p +m
- (y2)(%,g)y:
S_,
exp{2yjqljjexp
- {q?j/2Var(qlj)Jdq1j which can explicitly be evaluated: w(yj)
1:
exp -
(Y2){(%,
gJ
- Var(qlJJy;,
(gp &) - Var(q1J as long as bq cc wir uj(n - 1 ) cc 1 , and u,
