Copyright 0 1990 by the Genetics Society of America 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 polymor- phisms 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 predict the response of the mean phenotype to rect selection on the characterof interest: there might selection. This prediction only requires the assump- be a balance between mutation and stabilizing selec- tion that the 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 direct explanations 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 mutation rate 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