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 saved on the brink of extinction. In any evolutionary system, there is significant evolutionary pressure to evolve sequences that are both fit and robust; at high mutation rates, individuals with greater mutational robustness can outcompete those with higher fitness, a concept that has been referred to as survival-of-the-flattest. Previous studies have suggested that population size does not influence the size of 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 landscape, we show that the size of mutation rate at which the high, narrow peak and the lower, broader peak are lost for increasing population sizes can be approximated by an exponential function (where a peak was considered to be lost when there were no individuals present anywhere in its range). In addition, there is evidence for a continuum of mutation rates representing a transition from survival-of-the-fittest to survival-of-the- flattest. This identifies a critical mutation rate representing the start of the transition, which is defined as the highest mutation rate where survival-of-the-fittest is still the predominant outcome, but where the population is no longer able to maintain the fittest peak indefinitely. The effect of population size on the critical mutation rate is shown to be particularly noticeable in small populations with 100 individuals or less. This provides new insight into the factors that can affect survival-of-the-flattest in small populations, and has implications for populations under threat of local extinction.