Pleiotropic Models of Quantitative Variation

Pleiotropic Modelsof Quantitative Variation

N. H. Barton Department of Genetics and Biometry, University College London, London hW12HE, England Manuscript received May 23, 1989 Accepted for publication November 2 1, 1989

ABSTRACT It is widely held that each gene typically affects many characters, and thateach character is affected by many genes. Moreover, strong stabilizing selection cannot act on an indefinitely large number of independent traits. This makes it likely that heritable variation in any one trait is maintained as a side effect of polymorphisms which have nothing to do with selection on that trait. This paper examines the idea that variation is maintained as the pleiotropic side effect of either deleterious mutation, or balancing selection. If mutation is responsible, it must produce alleles which are only mildly deleterious (s = lo-'), but nevertheless have significant effects on the trait. Balancing selection can readily maintain high heritabilities; however, selection must be spread over many weakly selected polymorphisms if large responses to artificial selection are to be possible. In both classes of pleiotropic model, extreme phenotypes are less fit, giving the appearance of stabilizing selection on the trait.However, it is shown that this effect isweak (of the same order as the selection on each gene): the strong stabilizing selection which is often observed is likely to be caused by correlations with a limited number of directly selected traits. Possible experiments for distinguishing the alternatives are discussed.

HE main application of quantitative genetics to Following ROBERTSON(1967), we can contrast two T artificial and natural populations has been to kinds of explanation for these apparently contradic- use the pattern of genetic variances and covariances tory observations. The simplest possibility invokes di- to predictthe response of the meanphenotype to rect selection on the characterof interest: there might selection. This prediction only requires the assump- be a balance between mutation and stabilizing selection thatthe joint distribution of phenotypes and tion (LANDE 1975);selection on the character might breeding values is Gaussian, and so is fairly robust to induce overdominance on the underlying loci (GIL- the underlying genetics. However, tounderstand LESPIE and TURELLI1989); or frequency-dependence morphological evolution in the longer term, we must might maintain diversitya of phenotypes (ROUGHGAR- find out what maintains genetic variation, and how DEN 1972; SLATKIN1979). Such direct explanations variation will change under selection. This has proved have received most attention, because they are math- difficult and contentious. The central difficulty is that, ematically tractable, and because they offer the pos- although the methods of Mendelian genetics cannot sibility that variation might be understoodin terms of be used directly to study alleles with small effects, the measurable parameters. However, even without going evolution of the phenotypic variance does depend on into genetic details, the sheer number of quantitative the numbers and distribution of effects of individual characters and gene loci makes directexplanations genes (BARTONand TURELLI1987; TURELLIand BAR- seem implausible. Suppose we accept the view that TON 1989). quantitative variation can be understood characterby There is a further problem, in that the basic obser- character. Can Gaussian stabilizing selection of vations conflict: we see both abundant polygenic var- strength V, 20Vg [the typical value suggested by iation, andstrong stabilizing selection thatshould TURELLI(1 984)] act independently on a largenumber rapidly eliminate that variation. Genetic variation is (m) of characters? (Here, the fitness of an individual reflected directly in correlationsbetween relatives, with phenotype z is w(z) = exp(-z2/2Vs); the optimum and indirectly in sustained responses to directional is arbitrarily set at zero). Genetic variation around the selection thattake the phenotype well beyond its optimum reduces fitness by a factor dVS/(Vs + V,) z original range. Evidence for stabilizing selection exp(-Vg/2V,) for each character, and so a naive load comes from the reduced fitness of extreme pheno- argument suggests a net reductionby exp(-mVg/2V,). types [reviewed by CHARLESWORTH, LANDEand SLAT- This sets a limit of at most 100 independent charac- KIN (1982) and ENDLER (1986)], and perhaps more ters. convincingly, from theconstancy of form overgeolog- Such arguments have been criticized because they ical time (CHARLESWORTH, LANDEand SLATKIN 1982; assume that fitnesses are constant and combine mul- MAYNARDSMITH 1983). tiplicatively (SVED,REED and BODMER1967; EWENS

Genetics 124 773-782 (March, 1990) 774 N. H. Barton

1979). The genetic variance in fitness leads to a more bristle number in Drosophila melanogaster. The weak robust (albeit weaker) limit. Assuming Gaussian genetic correlation between these characters (SHERI- breeding values for the trait, the variance of squared DAN and BARKER1974; DAVIESand WORKMAN 1971) deviations, var(z‘), would be 2Vi. Since fitness declines does not in itself imply that there is no correlation in with z2/2V,, the total genetic variance in fitness would theeffects of individual genes. However, DAVIES be =m(Vg/V,)’/2. (I assume that selection on each (1 97 1)located the differences between selected lines character is weak enough that exp(-z2/2Vs) = 1 - z2/ to chromosome regions that primarily affected only 2V.s.) This formula differs from that given by CHAR- one or other character. The nature and extent of LESWORTH (1 987),who calculated the (much smaller) pleiotropy is an important open question. At present, additive component under a two-allele model; it is the strongest evidence of its general importance lies consistent with TACHIDAand COCKERHAM’S (1988) in the complex biochemical and developmental proc- results, in the limit of large numbers of loci, and with esses through which genotype affects thearbitrary truly independent characters. metriccharacters which we choose tomeasure Though we know very little about the genetic vari- (WRIGHT 1968). ance in relative fitness (as opposed to its components), Widespread pleiotropy does not necessarily imply it is hardto believe that it is much greater than that variation in one trait will be affected by selection z 0.25. It must lie between the additive genetic vari- on other traits: in LANDE’S (1 980)model of Gaussian ance,and the total variance. Fisher’s fundamental allelic effects, a locus can have an arbitrary combina- theorem implies that the former will be small (CHAR- tion of effects on many traits. However,only a limited LESWORTH 1987), while the total variance averages number of alleles can segregate at each locus, and so z 1 across the range of species surveyed by CLUTTON- it is unreasonable to suppose that selection on one BROCK(1988). Thisadmittedly rough figureof =O.25 locus could give independent responses in any of an would suggest a limit ofat most 200 or so independent arbitrarilylarge number of phenotypic directions characters. (TURELLI1985). Moreover, if variation at a locus Genetic data set other kinds of constraints. Com- primarily affects just the netactivity of some enzyme, parison of mutation rates for quantitative characters its effects will be constrained to a single degree of (Zp 0.01 or more) (TURELLI1984) with thenet freedom (cf: WAGNER 1989;KEIGHTLEY and KACSER mutationrate to deleterious alleles (Zp = 0.25 for 1987). Patternsof expression in different tissues or at egg-to-adult viabilityin Drosophila) (SIMMONSand different times might vary, but effects will still be CROW 1977; CROW andSIMMONS 1983) suggest that constrained to a few degrees of freedom at each locus, a significant fraction of genes (0.01/0.25 = 1/25) may however many alleles segregate. affect each trait, so that each gene must affect many One could reconcile these observations with “di- traits. The argument is somewhat weakened if the rect” explanations by supposing that there is in fact mutation rate to all genes reducing fitness is larger variation for only a few independently evolving prin- thanthe rate quoted here from measurements of cipal components, such as size: variation in measured viability (CHARLESWORTH CHARLESWORTH and 1987); characters might simply reflect selection on a much however, 2p cannotbe very muchlarger without smaller set of traits (cj KIRKPATRICKand LOFSVOLD producing an excessive mutation load. (In the spirit 1989). This possibility cannot be excluded. However, of this paper, I am assuming here that most mutations the fact that selection has produced a profusion of affectingquantitative traits are in fact deleterious: delicate adaptations, each involving many traits, and there could of course be any number of genes with the fact that artificial selection can sculpt almost all little effect on fitness, and withspecific effects on aspects of a plant or animal, has led to the view that metriccharacters.) Differences between species or evolution can proceed along any of a large numberof selected linesin any onetrait can involve alarge phenotypic degrees of freedom (DARWIN, 1859, Ch. number (“1 0’) of genes (WRIGHT 1968;BARTON and 6; FISHER1930, pp. 38-41; CHARLESWORTH, LANDE CHARLESWORTH 1984),again suggesting widespread and SLATKIN 1982). It therefore seems more likely pleiotropy. that quantitative variation is based on the overlapping Conversely, major mutants commonly have pleio- effects of many genes on many characters, and that tropic effects on a significant fraction of morphologi- any one character will be strongly influenced by selec- cal traits(DOBZHANSKY and HOLZ 1943; WRIGHT tion on others (cj CHARLESWORTH 1984). 1968). Evidence here is not as clear as one would wish: If pleiotropy is indeed widespread, then the varia- it is not obvious that minor mutantswill have similarly tion of any one character may be a mere side-effect wide pleiotropic effects, and therehas been no system- of polymorphisms maintained for quiteother reasons. atic survey of the effects of spontaneous mutants on This is likely to be the case even if one considers sets multivariate morphology. An interesting counterex- of characters, as in the multivariate analyses developed ample is provided by abdominal and sternopleural by LANDEand ARNOLD (1983). Onecan still measure Pleiotropic Models 775 only a tiny fraction of the traits which affect fitness, and KEIGHTLEY1988). KEIGHTLEYand HILL (1983, and statistical analysisof more than a dozen or SO 1988, 1989) have made extensive studies of this kind characters requires inordinate effort(MITCHELL-OLDS of model: they assume that allelic effects are drawn and SHAW 1987). Polygenic variation could be ana- froma reflected gamma distribution of theform lysed by postulating that fitness effects are mediated lail’”exp(-laii/w). In the present notation, this gives by large numbers of hypothetical Characters. How- CY’= w‘@(@ + l), K = (@+2)((?+3)/[3@(@+1)].Note that ever, it seems more promising to consider just the net one has some choice over thevariance a‘, the kurtosis, effects of alleles on fitness and on the character of K, and the numberof loci, n. Onecould either consider interest, as suggested by HILL andKEIGHTLEY (1 988): all loci, in which case n and K would be high, and CY’ the effects of alleles on hidden metric characters,and low, or only those loci with significant effects, in which the consequences of those characters for fitness, are case n and K would belower, and CY’higher. The summarized in the net effect on fitness. This distinc- overall constraint is that the increase in variation due tion between direct and pleiotropic selection is to to mutation is V, = 2npa2 per zygote per generation. some extent semantic, since it depends on whether it Maintenance of variation: The equilibrium genetic is more helpful to suppose that the effects of genes on variance is Ea’ = (2npa2/s). The gametic mutation fitness act through a set of intermediate “traits.” For rate to alleles which reduce viability when homozy- example, GILLESPIEand TURELLI(1989) show that if gous has been estimated for the second chromosome allelic effects fluctuate at random,stabilizing selection of D.melanogaster (SIMMONSand CROW1977; CROW on a character can induce overdominance, and can and SIMMONS1983; CHARLESWORTH198’7). Extrapo- maintain variation. However, if one extends this ar- lating to the whole genome, np = 0.01 3 for lethals, gument to stabilizing selection on many characters, and ~0.13- 0.43 fordetrimentals. The effecton influenced by many genes, then the model converges fitness in theheterozygous state has been inferred toone of pleiotropic overdominance (GILLESPIE from allele frequencies in nature, on the assumption 1984). Though the overdominance at each locus is of negligible local inbreeding. Estimates for recessive caused here by stabilizing selection onquantitative lethals and detrimentals are similar, with s = 2% characters, the outcome would be the same with any (CROWand SIMMONS1983; p. 27). Thus, substantial kind of overdominance of the same strength. heritabilities (V, = Ve)are expected if the variance of We now need consider only two broad classes of effects, CY’,across all deleterious alleles, is =0.03Ve. pleiotropic models: variation may be maintained by At first sight, such a value is not implausible: random deleteriousmutations, or by balancing selection. It mutations are expected to have a wide range of pleio- might seem that because we cannot measure selection tropic effects (WRIGHT 1968),and responses to arti- on individual polygenes (or even count these poly- ficial selection may involve recessive lethals or detri- genes), investigation of pleiotropic variation would be mentals (e.g. Yo0 1980).Support comes from Rus- hopeless. However, observationsof high heritabilities, SELL, SPRAGUE and PENNY’S (1963)observations of strong stabilizing selection, and large responses to spontaneous mutations in maize. RUSSELLet al. meas- selection do impose surprisingly strong constraints on ured a mutation rate to alleles affecting the trait of both classes of pleiotropic models. np = 6 X lo-’; these alleles had a variance of effects ofat least a’ = 0.03Ve,and perhaps considerably more MUTATION/SELECTION BALANCE (LANDE1975; TURELLI1984). These values would Suppose that mutations at any of the 2n genes in a account for substantial genetic variability. However, diploid individual can affect a neutral character, z. evidenceon both the variance of effects, andthe Mutations occur at a rate p per locus, and each mul- mutation rate, of alleles affecting quantitative traits is tiplies the fitness of heterozygotes by a factor (1 - s). flimsy (TURELLI1984). A much stronger argument We assume that p << s << 1, so that any particular can be made from the rate at which mutation intro- allele is rare, and the number of deleterious alleles duces genetic variance: since V, = 2npa2, Vg = V,/s. per individual, k, follows a Poisson distribution with A range of experiments has shown that V, a 10-3Ve mean K = 2np/s. The mean fitness of the population (LANDE 1975; HILL 1982; LYNCH 1988).With selec- is exp(-2np). tion against deleterious alleles of a few percent, this Allele i has effect ai on the character of interest; CY, mutational input could not maintain significant vari- is drawn from a symmetric distribution with variance ability. a2 and fourth moment 3~a~.K is a measure of the In reality, the selection coefficient will vary between kurtosis of the distribution of mutant effects: if all alleles: the genetic variance is determined by the av- effects are equal in absolute value, K = 1/3, while if erage of a’/s, weighted by mutation rate. One would they are drawn from a normal distribution,K = 1. For expect mutants with a large effect on the character to simplicity, I consider only variation in mutant effects, have large effects on fitness: if the relation is strong, a. Variation in s would also affect the results (HILL so that s increases faster than a*, then most variation 776 N. H. Barton

will be contributed by alleles of small effect, whereas A further prediction of the pleiotropic model is that if it is weak, major mutants will be more important. individuals extreme for one trait will also tend to be Thus, it is not clear how variation between alleles extreme for other, uncorrelated, traits. Consider two would affect the genetic variance. It is also unclear traits, z1 and z2, with mean zero, which are affected whether it is safe to extrapolate from the Drosophila by the same set of loci; each allele has random effects data on alleles with noticeable homozygous effects: we all, az, onthe two traits. The genetic covariance have no direct way of investigating alleles with unde- between them will be Pnpcov(a1, a2)/s, and so if the tectable effects on homozygotes. However, the rela- allelic effects are uncorrelated, the traits will be un- tively good estimates of V, = 10-3V, show that if any correlated.However, one can easilyshow thatthe kind of mutation/selection balance (direct or pleio- genetic covariance between the squares of the traits, tropic) is to maintain quantitative variation, then al- standardized relative to their geneticvariances, is just leleswith effects onthe trait must be only mildly cov(z:, z~)/[var(r,)var(z2)]= 1/Z. deleterious (s VJV, = This is quite possible, z Notethat TURELLI(1985, Eq. 1 lb) argued that but would implythat the mutationsrevealed by studies selection on one trait could not give the appearance on Drosophila viability are largely distinct from those of selection on another unless the traits were corre- responsible for variation in other metric characters. lated. However, this argument assumed a normal dis- Moreover, the extra influx of very mildly deleterious tribution of phenotypes. The selection is here induced alleles cannot be great: the measured mutation rate to alleles which reduce Drosophila viability is already because the Poisson distribution of deleterious alleles large enough to cause a substantial load (CROWand is not quite normal. More generally, covariances be- SIMMONS 1983). tween deviations (cov(z:, z;)) involve fourth mo- Apparentstabilizing selection: Since individuals ments, which may be significant when the distribution with extreme values of the trait will tend to carry of allelic effects is leptokurtic, as will be the case if the more deleterious alleles, we expect a negative corre- relevant loci are close to fixation (BARTONand TUR- lation between fitness and deviation from the mean. ELLI 1987). This will give the false appearance of stabilizing selec- For a given value of V,, this model predicts a lower tion on the characteritself. The regression of relative genetic variance thandoes TURELLI’S(1984) “rare fitness on squared deviation, 7,is a practical measure alleles” approximation, in which selection acts directly of the strength of stabilizing selection (LANDE and on the character (V, = 4npV,/(3~+ 44s)< 4nwVs, ARNOLD 1983).An alternative, used more often in since K > 1/3). The different predictions of the two theoretical work, is to write fitness as W = exp(-(z - models illustrate the important point that one cannot zoPJz/2Vs);now, y corresponds to 1/2Vs, if one makes calculate the equilibrium variance from a purely phe- the approximation that the load is not large (m 1). notypic model which balances the erosion of variation With pleiotropic mutations, the regression gives: by the observed selection against the introduction of variation by mutation: the same values of V, and V, = v, = cU2(3K + 4np/s)/2s (la) 2npa2 give different predictions, depending on the This can be rewritten in a dimensionless form, in underlying genetics. With pleiotropy, selection differ- terms of the apparent genetic load associated with the entials on the character alonedo not suffice to predict character-that is, the difference between the fitness the equilibrium variance. of individuals with the optimal breeding value, and How much stabilizing selection will in fact be in- the mean fitness: duced by the pleiotropic effects of deleterious mutations? The Drosophila data discussed above suggest Vg - 2np (Ib) that thetotal number of deleterious alleles per diploid 2V, (3~+ 4np/s) genome is large (E = 2np/s 30). Observations of stabilizing selection in natural and artificial popula- If an individual typically carries rather few delete- tions give typicalloads of Vg/2V, 2.5% (LANDE1975; rious alleles that can affect the character (K = 2np/s z << l), the apparent load is =2np/3~,whereas if this TURELLI1984), which is comparable with estimates number is large, it is ~/2.There is a simple expla- of s/2 zs 1% from Drosophila. Vg/2V, might be ex- nation for these alternative limits. If individuals carry pected to be greater than estimates of s/2, for two at most one deleterious allele, then individuals ex- reasons. The apparent load should be given by an treme for one trait will be extreme for all traits, and average of s which is weighted towards alleles with so the whole genetic load (~2nw)will appear to be large effects on the character, and will therefore be associated with each trait. On the other hand, ifis larger than the average of s for spontaneous muta- large, deviations in any one trait will be a poor pre- tions. Also, the Drosophila estimates are derived from dictor of the number of deleterious alleles: the appar- the persistence of recessive lethals and detrimentalsin ent load will be smaller by a factor zl/E. nature (CROWand SIMMONS1983, p. 27) Since these Pleiotropic Models Pleiotropic 777 are a selected sample, they will have milder average in which variation is based on rare alleles. The vari- effects on heterozygotes than spontaneous mutations. ance will only change slowly if the number of loci is This argument assumes that variation (Vg)is main- large (BULMER1980), and furthermore, the distribu- tained by a mutation/selection balance. However, we tion of allele frequencies in the base and the selected have seen that the low value of V,,, makes this implau- populations will be distorted by drift. KEICHTLEYand sible: it also makes it unlikely that deleterious muta- HILL (1989) show that these effects can easily mask tions contribute significantly to measures of stabilizing the expected rise and fall in variance. selection. This can be seen by assuming a Gaussian HILL and KEIGHTLEY (1988) have examined the distribution of breeding values (so that K is small). The response to selection due to new mutations with del- part of the regression of squared deviations on the eterious effects, takinginto account the effects of character which is attributable to deleterious muta- drift. They find that,as here, strong directionalselec- tions is then Vg/2V,= V,,,/2Vg,which is negligibly small. tion will overwhelm the effects of pleiotropy: rapid Pleiotropic mutation is made even less likely as an responses are possible, despite the associated loss of explanation for variation and apparent stabilizing se- fitness. Because they include recurrentmutation, lection by ENDLER’S(I 986) survey of natural popula- there is no selection limit: in the model set out here, tions: he finds that selection may beremarkably the limit would be reached when all favorable alleles strong, so that V, is often much less than the figureof in the base population have been fixed. In practice, 20V, used here. one would expect a continued response as new muta- The response to selection: Ignoring for the mo- tions occurred;one would also expectthat if the ment the difficulties in accepting mutation-selection original variation had not been entirely exhausted, balance as the main cause of polygenic variation, we the trait would return toward its original value. can ask whethera model of pleiotropic mutation/ selection balance would be compatible with sustained BALANCINGSELECTION responses to directional selection. If is large, the Suppose now thatquantitative variation is main- distribution of breeding values will be approximately tained as a side effect of balanced polymorphisms at Gaussian, and so the initial response will be the prod- n loci; for simplicity, suppose that the effects of all the uct of the heritability and the selection differential. alleles on the trait are additive. The polymorphisms The selection coefficient at a locus produced by direc- might be maintained either by overdominance or by tional selection of intensity i is approximately ia/fi frequency-dependent selection. However, since the = ia/mwhere V is the phenotypic variance, and relation between fitness and genotype varies consid- h2 = V,/V KIMURAand CROW1978). Substituting for erably between models with frequency-dependence, V, gives i +h2s/2np;if artificial selection is intense, and only overdominance will be analysed in detail. Fur- heritability high (i = 1, h2 = 0.5), this will be much thermore, since overdominance cannot easily main- larger than natural selection against the deleterious tain more than afew alleles at each locus (LEWONTIN, allele (im7 G).Thus, directional selection is GINZBURGand TULJAPURKAR1978), two alleles per likely to overwhelm intrinsic effects, and fix deleteri- locus are assumed. Only results on the limits to selec- ous alleles. The maximum response will be (nrcu), tion are sensitive to the form of the polymorphism. where rcu is half the mean effect of “+” alleles; for a Overdominance has been considered as a mechanism normal distribution of a, F is I/& = 0.20. Relative formaintaining quantitative variability by WRIGHT to theinitial genetic standard deviation, themaximum (1935a), ROBERTSON(1956), BULMER(1 973), andGIL- response is rn/C. This could belarge, and so is LESPIE (1984). All these authorsshowed that overdom- consistent with observations. At the selection limit, inance couldaccount for high heritabilities despite half the loci will be fixed for deleterious alleles. Del- stabilizing selection directly on the character. More- eterious alleles are of course likely to be partially over, GILLESPIE(1984, Eq. 4) showed (using a mul- recessive: ratherthan beingfixed, they may be tiallelic model) that overdominance introduced phe- broughtto high frequency, in a balance between notypic variation in a manner analogous to mutation: artificial andnatural selection. There would bea the ratio between the selective advantage of homozy- substantial fitness loss, since entirely homozygous flies gotes and the numberof alleles plays the same role as have very low fitness (SIMMONSand CROW 1977; V,. ROBERTSON’S(1956) treatment produced all the CROWand SIMMONS1983). results needed for thebiallelic case: in this section, the A further prediction is that as rare advantageous aim is to present these in the same framework as for alleles increase in frequency, the genetic variance mutation/selection balance. should increase; it should eventually fall as advanta- With random mating, the three genotypes at locus geous alleles are fixed. Such changes in variance are i have frequencies q?: 2piqi: p?, their relative fitnesses rarely seen under directional selection. However, this are 1 - si: 1: 1 - ti (si, t, << l), so that selection maintains is not decisive evidence against this model, or others a polymorphism with stability Si = siti/(si + t,) and 778 N. H. Barton equilibrium Po, = S,/ti. (Si is the rate of return toward be somewhat reduced by variation across loci, but the equilibrium: dp,/dt z - &(pi- poi)when p = po. It since overdominance cannot easily maintain extreme is also the segregation load associated with the poly- polymorphisms, this reduction will be small, Thus, if morphism. The contributions of the genotypes to the pleiotropic overdominance is to account for observed trait are -ai: 0: +a,. As before, the allelic effect ai stabilizing selection, 5 must be large: ~5%for equal varies randomly across loci, and is drawnfrom a allele frequencies, andmore if frequenciesdiffer. symmetric distribution with variance a' and fourth Because the overall variance in fitness is limited, this moment 3~a~.a: is assumed to be independent of the is only plausible if thenumber of polymorphisms equilibrium frequency,but may becorrelated with involved is small, and creates serious difficulties for the strength of selection, as measured by the segre- the model. gation load Si. The response to selection:Is the observed response Maintenance of variation: The equilibrium genetic to selection consistent with pleiotropic overdomi- variance is 2npoqoa2, wherethe overbar denotes a nance? Notethat although the long-term response mean across loci. Clearly, a great deal of variability may be largely due to new mutations (HILL 1982),the can be maintained by this mechanism. This is true large initial responses which are often seen over the even if only a small fraction of molecular polymor- first ten or twenty generations do constrain variation phisms are maintained by balancing selection: 100 or in the base population. When the favoured allele is so loci with equal allele frequencies and variance of close to fixation, the selection coefficient due to nat- effects a2 = 0.02Ve would suffice. The result would ural selection is -S,/po, andthe coefficient due to not be greatly altered by allowing more- alleles per artificial selection is ia&. (Here Po is the frequency locus, since the average heterozygosity (2poqo)cannot of the favoured allele in the absence of artificial selec- be greater than 1. tion.) Directional selection will overwhelm balancing Apparent stabilizing selection:Individuals with ex- selection if its effect on a locus is greater than the treme phenotypes will tend to be more homozygous, average strength of balancing selection at extreme and so will tend to have lower fitness. The covariance frequencies: ia& > = F/po. With equal allele fre- between relative fitness and the squared deviation of quencies, this leads to the condition id? > s&. the trait is cov(W/w, r') = -2na2Spoqo. The variance Since there cannot be a very large number of poly- of 2' is n(2n - 3)mz + nm4, where m2 and m4 are the morphisms under strongbalancing selection (note that second and fourth central moments of effects of in- the last term is proportional tothe coefficient of dividual loci. Using the assumption that allelic effects variation in fitness caused by balancing selection), are independent of the equilibrium frequency, this directional selection is likely to dominate. The maxi- reduces to: mum response to selection, assuming equal allele frequencies, is na/2; relative to the initial genetic stand- var(r') = 4a4n(2n - 3)(Poq0)' ard deviation, this is so that hundreds of poly- (24 a morphisms are needed to allow responses of several + 3~na~poq0[2- 15p0qo(P: + q:]. genetic standard deviations. ROBERTSON(1 956) This unpleasant expression differs from that derived showed that the loss of fitness due to small deviations by ROBERTSON(1956), because he assumed a normal in the mean Afi is %A2/2h2.This may not be exces- distribution of breeding values, and because he cal- sive for small changes. However, at theselection - limit, culated the curvature of the relation between fitness fitness could be reduced by =n?/2, where F is the and the trait, rather than the regression of relative unweighted averageof Si. This would be largeif there fitness on squared deviation. The expression can be were enough strongly selected polymorphisms to ac- simplified by noting thatthe second term will be countfor polygenic variation and substantial re- negligible if the number of loci is large relative to the sponses to selection. kurtosis of allelic effects. The apparent genetic load Frequency-dependent selection:Variability in hap- is then: loids or predominantly self-fertilizing species cannot bemaintained by overdominance. The model dis- VK/2V, = -vKcov(W/m, z')/[2var(z')] cussed above is restricted in other ways. Overdomi- (2b) nance can maintain rare allelesonly if it is highly = F/[2( 1 + C')] symmetric (LEWONTIN, GINZBURCTULJAPURKAR and where s is the average segregation load per locus, 1978). In contrast,if frequency-dependence maintains weighted by poqo, and C is the coefficient of variation the polymorphisms, alleles can be keptat low frequen- of poqo across loci. This is the same result as ROBERT- cies, allowing a much larger response to selection. SON (1956). It is remarkably similar to that for muta- A difficulty with frequency-dependence is that, in tion/selection balance: the apparent load is at most the simplest models, all genotypes will have equal half the segregation load at a single locus, s/2. It may fitnesses at equilibrium: no stabilizing selection can be Pleiotropic Models 779 induced on the trait. Stabilizing selection will appear mutation at a rate as low as V,,,= 1O-V, cannot induce only if fitness effects are nonadditive. Suppose that a significant amount of apparent stabilizing selection. the lzth allele at a locus has frequency pk, and adds Balancing selection could maintain considerable effect ffk to the trait. At equilibrium, let the fitness of amounts of quantitative variation. However, in order genotype kl be wkl; the marginal fitnesses of all alleles to account for observed selection responses, it must must bethe same: = zi, for all k. As before,the be spread over many weakly selected loci. If this is so, effects of alleles on the trait are drawn from a sym- then it cannot create the appearance of stabilizing metric distribution with kurtosis K, and are independ- selection on neutral traits. Measurements of strong ent of effects on fitness. The regression of fitness on stabilizing selection are therefore likely to reflect se- squared deviations gives an apparent load of Vg/2V,= lection caused by those traits, or by a limited number 6W/3[(K - 1) + (K + l)/H]. Here, H is the homozygos- of correlated traits. ity, Zp;, and 6w is the average fitness excess of heter- These pleiotropic models have been presented as a ozygotes (6w = Z(1 - Wkk/zi,)P:). Thus, extreme phe- general alternative to the popular models inwhich notypes will only be less fit if there is some element of variation in a trait is explained solely by the effects on overdominance: heterozygotes must be, on average, fitness of that trait. Both are caricatures ofreality; the fitterthan homozygotes. Whilst frequency-depend- main aim of this paper is to focus attention on an ence can maintain polymorphisms despite underdom- alternative class of explanations. In the introduction, inance, the polymorphism is more likely to be stable it was argued that direct models cannot explain mor- with some overdominance. Stabilizing selection is phological variation in all of an indefinite number of therefore expected as a side-effect of polymorphisms traits. However,some traits are known to directly cause maintained by frequency-dependence.However, it large fitness differences (e.g., bill shape in the Gala- will be weak, since the apparent load is roughly the pagos finches, male secondary sexual characters) (EN- product of the fitness excess of heterozygotes, and the DLER 1986). This is trivially true for life history traits homozygosity (6w % (1 - w~/zi,)H).By the same rea- which are components of fitness. It is quite reasonable soning as foroverdominance, 6w cannotbe large to suppose that stabilizing selection of strength V, z without producing an excessive genetic variance in 20V, acts on a few tens of independentcharacter fitness. Although frequency-dependence could main- combinations. This would still be compatible with tain substantial quantitative variability, and could al- observations of strong stabilizing selection on any low a greaterresponse to selection than with overdom- arbitrarily chosen character: one would generally ex- inance, it cannot easily account for observations of pect some correlation with some directly selected trait. stabilizing selection. An organism can bedescribed by an essentially infinite number of characters, and many evolutionists DISCUSSION believe that changes can occur in any of a very large number of degrees of freedom. The extreme view Neither mutation/selection balance nor balancing would be that each segregating allele has a unique selection alone can easily account for both the high vector of effects on the organism: a population could heritabilities and the strongstabilizing selection which then evolve along a number of dimensions given by are commonly observed. Mutation producespolygenic the number of alternative alleles (minus the number variation so slowly that the net selection against the of loci). Even if each locus was constrained to have alleles involved must be weak if they are to reach effects along a single direction (WAGNER 1989),sev- significant frequencies (s VJV, = lo-’): there must eral thousand degrees of freedom would still be avail- bea class of alleles with significant effects on the able. Strong stabilizing selection cannot act on all of character, butvery little effect on fitness. If (following these variables: for themajority, genetic variance must TURELLI1985) oneimagines that all fitness effects are be explained by pleiotropic effects of the kind de- mediated by a number of additive traits, each under scribed here. This is not to propose a neutral theory stabilizing selection such that V, = 20V,, this requires of morphological variation: the argumentis that while that thetotal variance of effects of each allele, summed every trait may affect fitness, its variability is deter- over all traits, be very small: Za2 = SV,V,/V, = 0.04. mined primarily by thenet selection on the genes This paper gives the obvious generalization of TUR- involved. The prediction is that selection on a large ELLI’S (1985) argument, by accounting for all pleio- number of traits could be factored into strong stabi- tropic effects on fitness, however mediated. It shows lizing selection on only a few principal components that mutation-selection balance is an unlikely cause of (their number being limited by the overall load); and quantitative variation, even if one considers rather that on the remaining components, one would see at few traits: one must suppose that thereis an abundant most weak stabilizing selection, of the same order as supply of mutations that affect metric characters, but the selection on the underlying polymorphisms. [The which hardlyreduce fitness (s z lo-’). Moreover, strength of stabilising selection on each component is 780 N. H. Barton given by the eigenvalues of the matrix of selection titative characters in Drosophila would show whether gradientson variance components, y (LANDEand the belief in widespread pleiotropy was justified, and ARNOLD 1983);these are the elements of A in Eq. 9 would establish the contribution of deleterious muta- of PHILLIPSand ARNOLD(1 989).] tions to quantitative variation. Experimental manipu- How might one distinguish the many possible expla- lations or mechanistic arguments may demonstrate nations for quantitative variation? There are two im- direct effects (e.g., ANDERSON1982; GRANT 1986), portant issues. First, what is the distribution of effects though it is not easy to show quantitatively that all of individual loci? This question cuts across the dis- apparent selection can beaccounted for by direct tinction between direct and pleiotropic models, and is selection. independent of the mechanism maintaining variation. In one of the few attempts to investigate selection Amongdirect models, the Gaussian approximation via pleiotropic effects, LINNEY, BARNES KEARSEYand (LANDE 1975)involves many alleles of small effect, (1 971) examined stabilizing selection on bristle num- whereas the “houseof cards” approximation(TURELLI ber. Selection clearly does not act on bristle number 1984) leads toa highly leptokurticdistribution of itself, since it occurs at the larval stage (KEARSEYand effects. Amongthe pleiotropic models, overdomi- BARNES1970). By showing that a population consist- nance would involve a moderate number of loci with ing of a variety of homozygous flies showed the same common alleles, while mutation/selection balance re- reduction in variance as the outbred base population, lies on a large number of loci, each near fixation. It they eliminated any role for overdominance. How- might seem that since we can hardly count or map the ever, the alternativesof frequency-dependence or del- polygenes, we cannot expect to measure the distribu- eterious mutation remain open. tion of their effects. However,indirect approaches ROBERTSON(1 967) suggested anotherapproach, offer some comfort. TURELLI(1 984) argued thatbe- using a kind of genetic manipulation. Suppose that cause the production of genetic variability, relative to when selection for (say) increased bristle number is the standing geneticvariance, is much larger than per- relaxed, the population returns towards its original locus mutation rates (Vm/Vg>> p), mutant alleles must value. This could be because too many bristles are have effects which are large relative to the standing inherently bad, or because alleles which increase bris- variance, and hence must be rare. Another argument tle number are deleterious for other reasons. If one is that if lociin the base population are close to places a chromosome from the selected line into a fixation, one would expecta transient increase in homozygous background that reducesbristle number, variance under directional selection, and a large var- the mean could be brought to its original value. If iance in response. While drift and linkage disequi- selection acts directly on bristles, one would see no librium can obscure these patterns (KEIGHTLEYand further change, whereas if it acts on pleiotropic ef- HILL 1989),they might be observed in large popula- fects, one would see a further decrease below the tions, if care was taken to eliminate disequilibrium initial value. SPIERS(1974) reported little change in beforemeasurements were taken. TACHIDAand lines manipulated in this way, supporting apleiotropic COCKERHAM (1 988)have suggested another possibil- interpretation. However, these lines were maintained ity: extending work by WRIGHT (1985b),they show at low density; since stabilising selection might only that under stabilizing selection, the variance in fitness be significant under crowded conditions, he repeated will be largely additive if alleles are rare, but largely theexperiment at high density. Two of thefour epistatic (additive by additive) if alleles are common. manipulated lines showed no return to the original It is extremely hard to measure genetic variation in mean, whilst two showed a ~50%return. However, fitness (CHARLESWORTH 1987),and rash to suppose SPIERS(1974) found no consistent relation between that much fitness variation can be attributed to stabi- bristle number and survival or competitive ability: he lizing selection on additive characters at equilibrium. concluded that the relaxations were best explained by However, TACHIDAand COCKERHAM’Ssuggestion linkage to deleterious alleles, rather thanby stabilising could be applied more rigorously to a biometricanaly- selection on bristle number or characters correlated sis of the squares of deviations of the character, rather with it. than to fitness. This is essentially an analysis of the If one accepts that variation in most quantitative higher moments of the distribution of breeding val- traits is to be explained by the pleiotropic effects on ues, originally suggested by FISHER,IMMER and TEDIN fitness of the underlying genes, then the problem is (1932) as a way of investigating dominance. to understand the nature and causes of fitness varia- The second issue is whether fitness differences can tion. This is an important but difficult problem, on be attributed to the character under study, to char- which we have little reliable information. The possi- acters correlated with it, or to pleiotropic effects of bility that most additive variance in fitness is simply the underlying genes. A systematic survey of the ef- due to mutation cannot be rejected (CHARLESWORTH fects of spontaneous mutationson fitness and on quan- 1987). If that is so, the slow generation of quantitative Pleiotropic Models 78 1 variation by mutationrequires that the selection ENDLER,J. A., 1986 Natural Selection in the Wild. Princeton Uni- against the underlying polygenes be correspondingly versity Press, Princeton, N.J. EWENS,W. J., 1979 MathematicalPopulation Genetics. Springer- weak. On the other hand, if substantial fitness varia- Verlag, Berlin. tion is due tobalanced polymorphisms (as is suggested FISHER,R. A,, 1930 TheGenetical Theory of Natural Selection. by MUKAI and NAGANO’S(1983) dataon viability Oxford University Press, Oxford. variation in southern populations of Drosophila mela- FISHER,R. A,, F. R. IMMERand 0. TEDIN,1932 The genetical nogaster), then these polymorphisms could maintain interpretation of statistics of the third degree in the study of quantitative characters. Genetics 17: 107-1 24. substantial heritabilities. The models discussed here GILLESPIE,J. 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