Mutation-selection balance: ancestry,load, and maximum principle

We analyze the equilibrium behavior of deterministic haploid mutation-selection models. To this end, both the forward and the time-reversed evolution processes are considered. The stationary state of the latter is called the ancestral distribution, which turns out as a key for the study of mutation-selection balance. We find that the ancestral genotype frequencies determine the sensitivity of the equilibrium mean fitness to changes in the corresponding fitness values and discuss implications for the evolution of mutational robustness. We further show that the difference between the ancestral and the population mean fitness, termed mutational loss, provides a measure for the sensitivity of the equilibrium mean fitness to changes in the mutation rate. The interrelation of the loss and the mutation load is discussed. For a class of models in which the number of mutations in an individual is taken as the trait value, and fitness is a function of the trait, we use the ancestor formulation to derive a simple maximum principle, from which the mean and variance of fitness and the trait may be derived; the results are exact for a number of limiting cases, and otherwise yield approximations which are accurate for a wide range of parameters. These results are applied to threshold phenomena caused by the interplay of selection and mutation (known as error thresholds). They lead to a clarification of concepts, as well as criteria for the existence of error thresholds.