The mutation process in colored coalescent theory.

The mutation process is introduced into the colored coalescent theory. The mutation process can be viewed as an independent Poisson process running on the colored genealogical random tree generated by the colored coalescent process, with the edge lengths of the random tree serving as the time scale for the mutation process. Moving backward along the colored genealogical tree, the color of vertices may change in two ways, when two vertices coalesce, or when a mutation happens. The rule that governs the coalescent change of color involves a parameter x; the rule that governs the mutation involves a parameter micro. Explicit computations of the expectation of the coalescent time (the first hitting time), and the coalescent probabilities (the first hitting probabilities) are carried out. For example, our calculation shows that when x = 1/2, for a sample of n colored individuals, the expected time for the colored coalescent process with the mutation process superimposed to first reach a black MRCA or a white MRCA, respectively, is 3 -- 2/n with probability 1/2 for any value of the parameter micro. On the other hand, the expected time for the colored coalescent process with mutation to first reach a MRCA, either black or white, is 2 -- 2/n for any values of the parameters micro and x, which is the same as that for the standard Kingman coalescent process.