Weak selection of phenotypic memory in periodic environments

Offspring can inherit a physiological or molecular state from their parents, and the carry-over in phenotypes across generations that results is often called epigenetic or phenotypic "memory''. Most theory on the evolution of such memory asks how the rate of switching between pre-existing phenotypic variants evolves in a fluctuating environment. Here we ask instead how the variants themselves originate. We model an asexual population in a periodically alternating environment in which a genotype specifies two inherited states, the phenotypic value each produces and the probability with which each is retained from parent to offspring. Starting from an ancestor that produces a single phenotype, mutations creating small differences between inherited states have no first-order selective advantage. Their spread depends on second-order selection and requires strong selection against phenotype-environment mismatch. A maternal effect of the same small size, conditioned on the maternal environment, is under first-order selection in the same model. Once phenotypes have diverged, genotypes whose retention probabilities match the duration of the environmental states are strongly favored, and the evolved strategy persists under conditions in which it could not originate. Memory is a consequence of individual selection and its origin probably follows earlier selection for cue-dependent maternal effects.