Derivation of the relationship between neutral mutation and fixation solely from the definition of selective neutrality.

A mutation whose fixation is independent of natural selection is termed a neutral mutation. Therefore selective neutrality of a mutation can be defined by independence of its fixation from natural selection. By the population genetic approach, Kimura [Kimura, M. (1962) Genetics 47, 713-719] predicted that the probability of fixation of a neutral mutation (u) is equal to the frequency of the new allele at the start (p). The approach traced the temporal sequence of the fixation process, and the prediction was obtained by assuming the selective equality of neutral mutant and wild-type alleles during the fixation process. The prediction, however, has not been verified by observation. In the present study, I search for the mathematical equation that represents the definition of selective neutrality. Because the definition concerns only mutation and fixation, an ideal approach should deal only with these and not with the intervening process of fixation. The approach begins by analysis of the state of fixation of a neutral mutation, and its relation with the initial state is deduced logically from the definition. The approach shows that the equality of the alleles during the fixation process is equivalent to the equality of probability of their ultimate fixation in a steady state. Both are manifestations of the definition of selective neutrality. Then, solely from this dual nature of the definition, the equality between u and p is derived directly. Therefore, the definition of selective neutrality can be represented by the equation u = p.