Structure of the G-matrix in relation to phenotypic contributions to fitness

Classical theory in population genetics includes derivation of the stationary distribution of allele frequencies under balance between selection, genetic drift, and mutation. Here we investigate the simplest generalization of these single locus models to quantitative genetics with many loci, assuming simple additive effects on a set of phenotypes and a linear approximation to the fitness function. Genetic effects and pleiotropy are simulated by a prescribed stochastic model. Our goal is to analyze the structure of the G-matrix at stasis when the model is not very close to being neutral. The smallest eigenvalue of the G-matrix is practically zero by Fisher's fundamental theorem for natural selection and the fitness function is approximately a linear function of the corresponding eigenvector. Evolution of genetic trade-offs is closely linked to the fitness function. If a single locus never codes for more than two traits, then additive genetic covariance between two phenotype components always has the opposite sign of the product of their coefficients in the fitness function under no mutation, a pattern that is likely to occur frequently also in more complex models. In our major examples only 1-2 percent of the loci are over-dominant for fitness, but they still account for practically all dominance variance in fitness as well as all contributions to the G-matrix. These analyses show that the structure of the G-matrix reveals important information about the contribution of different traits to fitness.