Microsatellite genetic distances with range constraints: Analytic description and problems of estimation

Statistical properties of the symmetric stepwise-mutation model for microsatellite evolution are studied under the assumption that the number of repeats is strictly bounded above and below. An exact analytic expression is found for the expected products of the frequencies of alleles separated by k repeats. This permits characterization of the asymptotic behavior of our distances D₁ and (σμ)² under range constraints. Based on this characterization we develop transformations that partially restore linearity when allele size is restricted. We show that the appropriate transformation cannot be applied in the case of varying mutation rates (β) and range constraints (R) because of statistical difficulties. In the special case of no variation in β and R across loci, however, the transformation simplifies to a usable form and results in a distance much more linear with time than distances developed for an infinite range. Although analytically incorrect in the case of variation in β and R, the simpler transformation is surprisingly insensitive to variation in these parameters, suggesting that it may have considerable utility in phylogenetic studies.