The genealogy, site frequency spectrum and ages of two nested mutant alleles

In this paper we consider the genealogy of two nested mutant alleles, assuming the constant-size neutral coalescent model with infinite sites mutation. We study the conditional genealogy and derive explicit formulas for the joint and marginal site frequency spectra for the double, single and zero mutant allele. In addition, we find the mean ages of the two mutations. We show that the age of the youngest mutation does not depend on the frequency of the single mutant allele and that the frequency spectra for the single mutant allele and the zero mutant allele are the same.