Models of general frequency-dependent selection and mating-interaction effects and the analysis of selection patterns in Drosophila inversion polymorphisms

Abstract

We investigate mechanisms of balancing selection by extending two deterministic models of selection in a one-locus two-allele genetic system to allow for frequency-dependent fitnesses. Specifically we extend models of constant selection to allow for general frequency-dependent fitness functions for sex-dependent viabilities and multiplicative fertilities, while non-multiplicative mating-dependent components remain constant. We compute protected polymorphism conditions that take the form of harmonic means involving both the frequency- and the mating-dependent parameters. This allows for a direct comparison of the equilibrium properties of the frequency-dependent models with those of the constant models and for an analysis of equilibrium of the general model of constant fertility. We then apply the theory to analyze the maintenance of inversion polymorphisms in Drosophila subobscura and D. pseudoobscura, for which data on empirical fitness component estimates are available in the literature. Regression on fitness estimates obtained at different starting frequencies enables us to implement explicit fitness functions in the models and therefore to perform complete studies of equilibrium and stability for particular sets of data. The results point to frequency dependence of fitness components as the main mechanism responsible for the maintenance of the inversion polymorphisms considered, particularly in relation to heterosis, although we also discuss the contribution of other selection mechanisms.

Figure 1.—

Sexual selection estimates of the Drosophila subobscura homokaryotypes relative to the heterokaryotype (Santos et al. 1986) and regressions on gene frequencies that best fit the data in Table 3. The fitness functions are m₁(X, Z) = 1.67 − 1.31p and m₃(X, Z) = 2.06 − 4.33q + 3.86q², where p and q are the adult gene frequencies of O_ST and O₃₊₄₊₇. The functions m₁ and m₃ depend indeed on X and Z, by substituting q = 1 − p, and p = X + (1 − X − Z).

Figure 2.—

Sexual selection estimates of the Drosophila subobscura homokaryotypes relative to the heterokaryotype (Santos et al. 1986) and regressions on genotype frequencies that best fit the data in Table 3. The fitness functions are m₁(X, Z) = 1.37 − 1.30X and m₃(X, Z) = 0.82 + (0.03/Z).

Figure 3.—

Viability estimates of the Drosophila pseudoobscura homokaryotypes relative to the heterokaryotype (Anderson et al. 1986) and regressions that best fit the data in Table 3. The fitness functions are and , where p_zy and q_zy are the zygotic gene frequencies of ST and CH. The functions v₁ and v₃ may be expressed directly in terms of the frequencies of the genotypes in an analogous way to m₁ and m₃ in Figure 1.

Figure 4.—

Sexual selection estimates of the Drosophila pseudoobscura homokaryotypes relative to the heterokaryotype (Anderson and Brown 1984) and regressions that best fit the data in Table 3. The fitness functions are m₁(X, Z) = 10.84 − 30.38p + 22.76p² and m₃(X, Z) = 2.65 − 7.74q + 6.03q², where p and q are as in Figure 1.