On an approximate formula for the distribution of 2-locus 2-allele model with mutual mutations

An asymptotic approximation of the density function of 2-locus 2-allele model with mutual neutral mutations was obtained invoking the small disturbance asymptotic theory. It was shown by comparing the approximate formula with simulations that this asymptotic method gives a good approximation over the whole time evolution when the mutation rates are high, though it does not give good approximations near the stationary state when the mutation rates are low. On the stationary state, the squared standard linkage deviation made up by using the approximate formula was compared with the exact one obtained by Ohta and Kimura (1969b). It gave a good approximation when the recombination rate is high, even under low mutation rates. Furthermore, as an application of the asymptotic method, The Ancestral Recombination Graph (ARG) was considered. [INTRODUCTION] In population genetics, it is very fundamental to derive the distribution of allele frequency or haplotype frequency. This is because one of the goals of population genetics is to understand the fate of mutants in Mendelian populations, and the frequency distribution is itself the sequel of the fate of mutants.