Mutation and migration in models of microsatellite evolution∗

Dynamic and statistical properties of generalized stepwise mutation models are described and used to compare data on human di-, tri-, and tetranucleotide polymorphisms. The time-dependent behavior of an island model with stepwise mutation is analyzed and its equilibrium properties used to estimate the product Nm of the population size and the migration rate. Population statistics are derived from the complete equilibrium analysis of the model, and these are combined to give an estimate of Nm that is related to Slatkin’s 1995 estimate which used RST . When this new statistic is applied to a set of 85 human microsatellite polymorphisms, the resulting population clusters match the tree of Bowcock et al. 1994 quite well. Introduction Because of their high level of polymorphism and ubiquity in eukaryotic genomes, microsatellites are widely preferred markers in evolutionary and ecological studies. Databases of microsatellites isolated for population-level analyses are under development. Statistical evaluation and evolutionary interpretation of microsatellite polymorphisms demand models for the origin and maintenance of this variability, and the models used so far are variants of the stepwise mutation model (SMM), which was originally introduced by Ohta and Kimura (1973) to model electrophoretically detectable enzyme variation in finite populations. The first part of this paper describes these models and illustrates how stepwise mutation, in combination with genetic drift, affects important measures of withinand betweenpopulation variation. Using empirical estimates of overall mutation rates, these measures may then provide estimates of average population size or divergence times between populations. In the second part of the paper, the dynamics of a set of populations subject to migration according to Wright’s (1943) island model are developed. From the equilibrium structure of this model, estimators of the extent of migration based on population statistics are suggested. We shall assume throughout that the microsatellites are perfect; that is, each allele at a locus is completely specified by the number of times a motif is repeated. Generations are non-overlapping and each is produced from the previous by multinomial sampling from the parental generation’s array of alleles. Our analysis follows that of Moran (1975), who developed recursions for the population central moments under the simplest stepwise mutation model permitting alleles of arbitrary (positive and negative) repeat number. The modifications to this model that we discuss below include asymmetric mutations of arbitrary size, a simple model of linear bias in which the rate of mutation depends on the number of repeats in the allele, and range constraints on the permitted repeat score. Moran’s Formulation: A Generalized SMM. Consider a population of N diploid individuals and a locus at which each allele is characterized by its repeat score, which may take any positive or negative integer values. Let μc be the probability that mutation changes an allele by c repeat units irrespective of its original count. The total mutation rate is then μ = ∑ c 6=0 μc and the expected change in repeat score for any allele is c̄ = ∑ cμc. The variance in repeat score change is σ m = ∑ c(μc/μ)− c̄. We write w = μσ m. The standard one-step symmetric model has μ+1 = μ−1 = μ/2, say, and μ2 = 0 otherwise, in which case c̄ = 0 and w = μ. We shall refer to this special case as the one-step symmetric SMM. Let pi be the frequency of the allele carrying i repeats in the parental generation. Then, following mutation the frequency of this allele is