Multiple equilibria and maintenance of additive genetic variance in a model of pleiotropy.

-We describe a multilocus model that incorporates pleiotropic stabilizing selection on a large number of characters. We find many different stable equilibria with different levels of polymorphism and additive genetic variability. The results lend support to Wright's concept of a complex adaptive surface with many peaks of different heights. The model assumes that alleles contribute additively to the characters. We analyze the multilocus model by first considering a two-locus model. The two-locus model depends critically on having loci of different effect and on having the optimum phenotype not be that of a completely heterozygous individual. The effects of different loci need to differ only by less than a factor of two. For the multilocus, multicharacter model, we assume that completely heterozygous individuals do not have the optimum phenotype. By restricting attention to a two-allele model, we also assume that there are no alleles that can affect all characters in all possible combinations of directions. Received July 25, 1989. Accepted December 10, 1989. One of the primary components of Sewall Wright's work on evolution was his notion of an adaptive landscape with many peaks of different heights. However, Wright was unable to explain the mechanism by which multiple adaptive peaks are maintained in a population: simple genetic models (Wright, 1977) that have multiple peaks or stable equilibria always lead to fixation. In contrast, in many natural systems, there is substantial additive genetic variability for many traits (e.g., Mousseau and Roff, 1987). These three notions combined suggest that the genetics underlying the variability observed in natural populations is more complex than that described by Wright's simple models. Over the past several decades, many more elaborate models have been developed to investigate potential mechanisms for the maintenance of additive genetic variability in natural populations. These mechanisms include mutation-selection balance (e.g., Bulmer, 1972; Lande, 1975; Turelli, 1984; Barton and Turelli, 1987; Burger, 1989), temporally varying selection regimes (Turelli, 1988), pleiotropic overdominance (Gillespie, 1984), environmental variability (e.g., Gillespie and Turelli, 1989), drift (e.g., Lynch and Hill, 1986), and antagonistic pleiotropy (e.g., Rose, 1985; Gimelfarb, 1986; Hastings and Hom, 1989; Wagner, 1989). However, regardless of the mechanism examined in each model, these studies generally share several common features. Most, for example, assume first, that the genetic variance ultimately obtained does not depend on history, i.e., populations with different initial gene frequencies will arrive at the same equilibrium; second, that there are equal effects of all loci; and third, that there are trait optima at the character value corresponding to an individual heterozygous at all loci. These models of evolutionary dynamics have generated important insights about maintenance of additive genetic variance, although they do not lead to an adaptive surface with many peaks of different heights. This, in and of itself, is not a weakness. However, the genetic systems of real organisms may not follow the simple assumptions incorporated in these models. In particular, the assumption that loci have equal effects has been examined closely in only a very few empirical systems and, for some characters, may be erroneous. For example, Shrimpton and Robertson (1988) suggested that bristle number, a quantitative character in Drosophila melanogaster, may be controlled by a few major genes with additional contributions from numerous other genes of smaller effect. Lande (1983) reviewed experimental evidence for loci affecting quantitative characters and found several examples that indicated loci of different effects. One striking example is Batesian mimicry (Ford, 1975). Keightley and Kacser (1987) developed a model to explain