Error Introduced by the Infinite - Sites Model 1

In a recent paper, Henry Harpending and I (Rogers and Harpending 1992) interpret variation in human mitochondrial DNA by using a model by Li ( 1977 ) . The model is unrealistic in several respects, one of which is its use of the “infinite sites” model (Kimura 1969 ) of molecular evolution. This model assumes that no nucleotide (or restriction) site mutates more than once, an assumption that is clearly violated in human data (Kocher and Wilson 199 1). The infinite-sites model may nonetheless be useful as an approximation, provided that the error it introduces is small. This note evaluates the relative error of this approximation. Harpending and I make use of the infinite-sites assumption at only one point in our analysis: we assume that differences between a pair of individuals are introduced at a rate, 2u per generation, which is constant in time. With a finite number of sites, this assumption cannot hold exactly, because, after a nucleotide site has been struck once by mutation, later mutations need not add to the count of differences between our pair of individuals. The more sites there are with prior mutations, the lower will be the rate at which new differences accumulate. Thus, differences accumulate at a decreasing, rather than a constant, rate. Let D(t) denote the expected number of differences over a finite number, K, of nucleotide sites between a pair of lineages that have been separated for t generations. Under the infinite-sites model, D(t) = 2Kyt, where l.t is the mutation rate per nucleotide site, and where K --* cc while p + 0, such that Kl.t remains finite. Here, I assume instead that K is finite and that mutation at the ith site follows a Poisson process with rate l.ti per generation. Thus, the number of mutations at site i since time zero is Poisson with mean 2pLit. In human data, transversions are rare. Thus, I assume that all mutations are transitions. Consider the comparison, between two lineages, at a single site, say site i. Initially, at time 0, the two lineages will be identical, since they share a common ancestor then. The first mutation along the path connecting the two lineages will make them different at site i, but the second will restore their identity, because of the assumption that all mutations are transitions. Each successive mutation at site i toggles the two lineages back and forth between the states of identity and nonidentity. The probability that the two lineages differ at site i is thus equal to the probability that the number of prior mutations there is odd. This probability is ( 1-e-4”1t)/2 (Haldane 1919). The expected sum of differences over all K sites is