Critical Mutation Rate Has an Exponential Dependence on Population Size

Populations of individuals exist in a wide range of sizes, from billions of microorganisms to fewer than ten individuals in some critically endangered species. In any evolutionary sys- tem, there is significant evolutionary pressure to evolve se- quences that are both fit and robust; at high mutation rates, individuals with greater mutational robustness can outcom- pete those with higher fitness, a concept that has been referred to as survival-of-the-flattest. Previous studies have not found a relationship between population size and the mutation rate that can be tolerated before fitter individuals are outcompeted by those that have a greater mutational robustness. However, using a genetic algorithm with a simple two-peak fitness land- scape, we show that the mutation rates at which the high, nar- row peak and the lower, broader peak are lost for increasing population sizes can be approximated by exponential func- tions. In addition, there is evidence for a continuum of muta- tion rates representing a transition from survival-of-the-fittest to survival-of-the-flattest and subsequently to the error catas- trophe. The effect of population size on the critical mutation rate is shown to be particularly noticeable in small popula- tions. This provides new insight into the factors that can af- fect survival-of-the-flattest in small populations, and has im- plications for populations under threat of local extinction.