Phenotype switching and mutations in random environments

Cell populations can benefit from changing phenotype when the environment changes. One mechanism for generating these changes is stochastic phenotype switching, whereby cells switch stochastically from one phenotype to another according to genetically determined rates, irrespective of the current environment, with the matching of phenotype to environment then determined by selective pressure. This mechanism has been observed in numerous contexts, but identifying the precise connection between switching rates and environmental changes remains an open problem. Here, we introduce a simple model to study the evolution of phenotype switching in a finite population subject to random environmental shocks. We compare the successes of competing genotypes with different switching rates, and analyze how the optimal switching rates depend on the frequency of environmental changes. If environmental changes are as rare as mutations, then the optimal switching rates mimic the rates of environmental changes. If the environment changes more frequently, then the optimal genotype either maximally favors fitness in the more common environment or has the maximal switching rate to each phenotype. Our results also explain why the optimum is relatively insensitive to fitness in each environment.