Heritable Variation by Mutation-Selection Balance

The role of linkage in influencing heritable variation maintained through a balance between mutation and stabilizing selection is investigated for two different models. In both cases one trait is considered and the interactions within and between loci are assumed to be additive. Contrary to most earlier investigations of this problem no a priori assumptions on the distribution of genotypic values are imposed. For a deterministic two-locus two-allele model with recombination and mutation, related to the symmetric viability model, a complete nonlinear analysis is performed. It is shown that, depending on the recombination rate, multiple stable equilibria may coexist. The equilibrium genetic and genic variances are calculated. For a polygenic trait in a finite population with a possible continuum of allelic effects a simulation study is performed. In both models the equilibrium genetic and genic variances are roughly equal to the house-of-cards prediction or its finite population counterpart as long as the recombination rate is not extremely low. However, negative linkage disequilibrium builds up. If the loci are very closely linked the equilibrium additive genetic variance is slightly lower than the house-of-cards prediction, but the genic variance is much higher. Depending on whether the parameters are in favor of the house-of-cards or the Gaussian approximation, different behavior of the genetic system occurs with respect to linkage. THE question of how much heritable variation in quantitative traits can be maintained through a balance between mutation and stabilizing selection has received much attention during recent years. Basically, two kinds of models have been treated analytically, namely diallelic multilocus models (LATTER 1960; BULMER 1972, 1980; BARTON 1986) and continuum-of-alleles models (KIMURA 1965; LATTER 1970; LANDE 19’75; FLEMING 1979; TURELLI 1984, 1986; NAGYLAKI 1984; BURGER 1986, 1988a,b; FOLEY 1987). TURELLI (1984) and SLATKIN (1987) also considered models with three and five alleles per locus. These analyses, except BULMER’S, were primarily devoted to deterministic models not taking into account random drift in finite populations. For the continuum-of-alleles model two different approximations have been derived, the Gaussian approximation (KIMURA 1965; LANDE 1975) and the house-of-cards approximation (TURELLI 1984). The Gaussian approximation yields G%(G) = 2ndpa2Vs as the equilibrium variance, where n denotes the number of loci affecting the trait, p the per locus mutation rate, a2 the variance of mutational effects and V, the inverse measure of the strength of stabilizing selection (see Equation 9 below). The house-of-cards approximation leads to an equilibrium variance of GE(HC) = 4npVs. The latter agrees with the diallelic and triallelic results of LATTER (1960), BULMER (1972), TURELLI (1984) and SLATKIN (1987). The Gaussian and the Genetics 121: 175-184 (January, 1989) house-of-cards approximations are, however, extrapolations from the haploid one-locus model under the assumption of global linkage equilibrium. The haploid model has been analyzed rigorously by BURGER (1 986, 1988a,b), where existence, uniqueness and global stability of a stationary frequency distribution of types have been proved under very general assumptions. In particular, an upper bound for the true equilibrium variance has been derived which is almost identical to the house-of-cards prediction. It follows from these results that the Gaussian approximation is applicable only if a* 5 4pV,, but in this case it is lower than the house-of-cards prediction. In fact, LANDE (1975) had noted that the validity of the Gaussian approximation rests on the assumption that a2 << pVs and the numerical results of TURELLI ( 1 984, 1986) suggested that the Gaussian approximation can be correct only if the variance of mutational effects per locus is much smaller than the existing variance at this locus. It is the aim of the present paper to go beyond extrapolations of haploid models and to investigate the influence of linkage in models of mutation-stabilizing selection balance. LANDE (1975, 1977) investigated the role of linkage and found that the equilibrium expressed genetic variance is independent of the linkage relation of the loci unless linkage is extremely tight. His analysis, however, is based on the assumption of a normal distribution of allelic effects. On the basis of a second order approximation, but based on