Conditions for positive and negative correlations between fitness and heterozygosity in equilibrium populations

The past decades have witnessed extensive efforts to correlate fitness traits with genomic heterozygosity. While positive correlations are revealed in most of the organisms studied, results of no/negative correlations are not uncommon. There has been little effort to reveal the genetic causes of these negative correlations. The positive correlations are regarded either as evidence for functional overdominance in large, randomly mating populations at equilibrium, or the results of populations at disequilibrium under dominance. More often, the positive correlations are viewed as a phenomenon of heterosis, so that it cannot possibly occur under within-locus additive allelic effects. Here we give exact genetic conditions that give rise to positive and negative correlations in populations at Hardy-Weinberg and linkage equilibria, thus offering a genetic explanation for the observed negative correlations. Our results demonstrate that the above interpretations concerning the positive correlations are not complete or even necessary. Such a positive correlation can result under dominance and potentially under additivity, even in populations where associated overdominance due to linked alleles at different loci is not significant. Additionally, negative correlations and heterosis can co-occur in a single population. Although our emphasis is on equilibrium populations and for biallelic genetic systems, the basic conclusions are generalized to non-equilibrium populations and for multi-allelic situations.