Evolution, 40(5), 1986, pp. 946-957 THE FITNESS CONSEQUENCES OF MULTIPLE-LOCUS HETEROZYGOSITY UNDER THE MULTIPLICATIVE OVERDOMINANCE AND INBREEDING DEPRESSION MODELS PETER E. SMOUSE Department ofHuman Genetics, University ofMichigan, Ann Arbor, M148109 Abstract,- There is a growing body ofliterature suggesting that the fitness ofan individual increases with the observed number of heterozygous loci. Broad theoretical considerations indicate that under various sorts ofbalancing selection, this is what one should generally expect in a population ofmultiple-locus genotypes. To date, however, it has not been possible to distinguish between two potential explanations of the phenomenon, The first explanation is that the loci examined are themselves responsible for the fitness differences observed (or, equivalently, are very closely linked to those that do). The genetic variation in question is thought to be maintained in polymorphic equilibrium by some form ofbalancing selection. The second explanation assumes that the observed loci are themselves selectively irrelevant but that their heterozygosity reflects that of the total genome. Genomic heterozygosity is thought to be predictive offitness, being an obverse measure ofgeneralized inbreeding depression. We provide a formal derivation of an explicit relationship between fitness and multiple-locus genotype for a simple form of the first explanation, the mul­ tiplicative overdominance model. The inbreeding depression model is a degenerate special case of this more general formulation. A formal estimation and testing framework is constructed that should facilitate evaluation of the two models with empiric data on heterozygosity and fitness. Received June 7, 1985. Accepted June 3, 1986 The accumulating allozyme data from attractive nor a feasible route to wide gen­ natural populations have established con­ eralization about the maintenance of ob­ vincingly that genetic variation is wide­ served genetic variability; some other ap­ spread in a great many organisms, but we proach is needed. still have no compelling evidence to suggest Single-locus analysis of the relation be­ how much of that variation is maintained tween genetic variation and fitness has not by natural selection. Quite a bit ofeffort has been very rewarding, and a number ofwork­ been expended on the search for meaningful ers have been led to consider the impact of selective differences among alleles (or ge­ multiple-locus genotypes on fitness. In par­ notypes) at particular loci, with at least ticular, several authors have attempted to modest success (e.g., Tsakas and Krimbas, correlate fitness or some surrogate measure 1970; Vigue and Johnson, 1973; Cavener with the number ofheterozygous loci in an and Clegg, 1981a, 1981b; DiMichele and individual. Some workers find an increase Powers, 1982; Hilbish et al., 1982; Oake­ in fitness with increasing heterozygosity shott et al., 1982; Burton and Feldman, (Schall and Levin, 1976; Singh and Zouros, 1983; Watt, 1983; Watt et al., 1983), but 1978; Bottini et al., 1979; Zouros et al., 1980; the vast majority ofsegregating loci in most Mitton et al., 1981; Koehn and Shumway, of the species surveyed so far have never 1982; Pierce and Mitton, 1982; Ledig et al., been examined in enough detail to charac­ 1983; Leary et al., 1984), but others do not terize selective factors in satisfactory detail. (e.g., Mukai et al., 1974; Gaines et al., 1978; Indeed, the effort required to "prove" the Knowles and Grant, 1981; Mitton et al., selective relevance ofeven a single locus in 1981). even a single species is so prohibitive that Turelli and Ginzburg (1983) have theo­ we can never hope to do so for more than retically demonstrated that if allozyme a small fraction of the locus/species com­ polymorphisms are maintained by any of binations of interest. The careful effort to several forms of balancing selection, then evaluate particularcases has been rewarding we should observe a general increase in fit­ in a variety ofways, and is to be encouraged, ness with an increase in the number ofhet­ but exhaustive enumeration is neither an erozygous loci. Whether the variation in 946 HETEROZYGOSITY AND FITNESS 947 question is actually maintained by balanc­ present an assessment procedure for eval­ ing selection remains to be seen, ofcourse, uating the hypothesis. I argue that the in­ but the observations are generally in agree­ breeding depression hypothesis is a degen­ ment with predictions based on a broad erate special case of the more general concept of balancing selection. formulation. In a later paper (Bush et al., Alternatively, it has been suggested that unpubl.), the predictions ofthe two models measured heterozygosity may merely reflect are compared for a series ofpopulation sam­ overall genomic heterozygosity and that ples from pitch pine (Pinus rigida Mill.). highly homozygous genotypes suffer from inbreeding depression (Ledig et al., 1983). A Set ofTwo-Allele Loci Mitton and Pierce (1980) and Chakraborty Considera diploid population with K seg­ (1981) have shown, however, that hetero­ regating loci, unlinked and with two alleles zygosity from a small sample of loci is a each. I assume that the two alleles at each poor indicator of total genomic heterozy­ locus are codominant, as is usual with al­ gosity for an individual, suggesting that there lozyme electromorphs; denote them as A I should be little or no association ifinbreed­ and A 2, B 1 and B 2, and their allele frequen­ ing depression is the root cause ofthe phe­ cies as PAand QA, P B and QB,etc. I presume nomenon. On the other hand, inbreeding that the population has already attained depression can be a powerful force (cf. multiple-locus Hardy-Weinberg equilibri­ Franklin, 1970), so even a small correlation um. between observed and genomic heterozy­ I envisage a broad sort of balancing se­ gosity might be sufficient to account for the lection that maintains the genetic variation loose association observed between hetero­ in equilibrium. The marginal fitness values zygosity and fitness. for anyone locus are functions ofgenotypic It is difficult to choose between these al­ frequencies and of the fitness values of the ternative explanations, because our current multiple-locus genotypes. I postulate that at assessment procedures are almost hopeless­ equilibrium the net balance offorces is such ly crude; they are notdesigned to distinguish that the marginal fitness array at each locus between the two selective hypotheses. In shows overdominance. Karlin and Liber­ current practice, no attention is paid to man (1979a, 1979b) have shown that for which loci are heterozygous, and most of either additive or multiplicative fitness the potentially useful information is sup­ across a set of "loosely linked" loci, stable pressed in the interest of convenient sum­ polymorphic equilibria will exhibit margin­ marization. The objective of this paper is al overdominance. More general fitness to develop a predictive model for the rela­ schemes leading to multiple-locus stable tionship between fitness and the particular polymorphism are not guaranteed to yield multiple-locus genetic state of a single in­ marginal overdominance at equilibrium dividual. The model employed is the sim­ (Lewontin and Kojima, 1960; Ewens and plest I could use, the multiplicative over­ Thomson, 1977; Hastings, 1981, 1982), but dominance model for unlinked loci. I present Turelli and Ginzburg (1983) have shown an evaluation procedure: (1) that is very eas­ that marginal overdominance is the most ily implemented, (2) that provides a test of frequent consequence of multiple-locus the null hypothesis of no selective differ­ polymorphism. For purposes ofexposition, ences among observed genotypes, (3) that then, let the A -locus be represented as shown distinguishes between genomic inbreeding in the panel below. and specific-locus effects, (4) that permits a determination ofthe marginal effect ofeach Genotype locus, and (5) that facilitates an assessment Frequency of the multiplicative overdominance as­ Fitness (WJ sumption. I begin with a consideration ofthe pattern The values of SA and T A are assumed to be of fitness to be expected from a set of two­ small and positive, compatible with the no­ allele loci, then describe the pattern to be tion that marginal selective differentials are expected of multiple-allele loci, and then small. It will be convenient to use the fact 948 PETER E. SMOUSE that (1 - SA) :::::: exp{-SA} and (1 - T A ) :::::: The fitness values (WA ) above are marginal exp{- T A} for small SA and T A, and I shall genotypic averages, ofcourse, and we really switch back and forth between these nota­ desire values for individuals. For the jth tional forms whenever convenient. individual, the natural logarithm of fitness I have assumed Hardy-Weinberg fre­ (denoted YA ) and adaptive distance (XA ) are quencies for expository purposes, and this simply related: warrants some comment. Not all types of Y (11) = 10g[WA(Il)] heterotic selection lead to Hardy-Weinberg A proportions at equilibrium, but where se­ = -SA + ~j lective differentials are small, departures = -a·XA(II) + ~j from Hardy-Weinberg proportions will be =[-a/PA]+~j small as well. Moreover, there is little evi­ (2a) dence in natural populations for large de­ YA(12) = log[WA(12)] partures from Hardy-Weinberg proportions = -a' X + ~j that could be attributed to large selective A(12) differentials, although the difficulties of de­ = ~j (2b) tecting departures from Hardy-Weinberg Y A (22) = log[ WA(22)] proportions