ESS gene expression of X-linked imprinted genes subject to sexual selection

We define ESS (Evolutionary Stable Strategy) conditions for the evolution of genomic imprinting at an X-linked locus. The system analysed is designed for mammalian imprinting in which X-linked genes typically undergo random X-inactivation and lack Y-linked homologues. We consider two models that map cellular gene expression to fitness in females subject to random X-inactivation. In the first model, female fitness is simply a function of the average gene expression across all cells. In the second model, each cell contributes independently to fitness, and female fitness is assessed as the average of these contributions across all cells. In both models, imprinting readily evolves when sexual selection favours different levels of gene expression in the two sexes. Imprinting is beneficial as it improves adaptation in both sexes. There are limits to the improvement in adaptation when sexual selection is strong and favours greater gene expression in males (the heterogametic sex). We also consider the consequences of an active Y-linked homologue on the evolution of imprinting. Our analysis suggests that restrictive conditions apply for the evolution of polymorphic ESSs at an X-linked imprinted loci.