Autocorrelation of gene frequencies under isolation by distance

Spatial autocorrelation statistics are used for description of geographic variation of gene frequencies, but the relationship of these indices with the parameters describing the genetic structure of populations is not established. A simple relation is derived here between kinship coefficient and a measure of spatial autocorrelation, Moran's I. The autocorrelation coefficient of gene frequencies at a given distance is a direct function of the kinship at that distance, and an inverse function of the standardized gene frequency variance, Fst. Under isolation by distance, the expected values of Moran's I for any allele may be calculated by means of Malécot-Morton function, which predicts an exponential decline of genetic similarity in space. This allows comparison of observed gene frequency patterns with the patterns that should be caused by interaction of short range migration and random genetic drift.