Estimating scale-specific and localized spatial patterns in allele frequency.

Characterizing spatial patterns in allele frequencies is fundamental to evolutionary biology because these patterns contain evidence of underlying processes. However, the spatial scales at which gene flow, changing selection, and drift act are often unknown. Many of these processes can operate inconsistently across space, causing non-stationary patterns. We present a wavelet approach to characterize spatial pattern in allele frequency that helps solve these problems. We show how our approach can characterize spatial patterns in relatedness at multiple spatial scales, i.e. a multi-locus wavelet genetic dissimilarity. We also develop wavelet tests of spatial differentiation in allele frequency and quantitative trait loci (QTL). With simulation we illustrate these methods under different scenarios. We also apply our approach to natural populations of Arabidopsis thaliana to characterize population structure and identify locally-adapted loci across scales. We find, for example, that Arabidopsis flowering time QTL show significantly elevated genetic differentiation at 300 to 1300 km scales. Wavelet transforms of allele frequencies offer a flexible way to reveal geographic patterns and underlying evolutionary processes.