A DISCRETE FOURIER-ANALYSIS FOR EVOLUTIONARY TREES

Discrete Fourier transformations have recently been developed to model the evolution of two-state characters (the Cavender/Farris model). We report here the extension of these transformations to provide inverti ble relationships between a phylogenetic tree T (with three probabilit y parameters of nucleotide substitution on each edge corresponding to Kimura's 3ST model) and the expected frequencies of the nucleotide pat terns in the sequences. We refer to these relationships as spectral an alysis. In either model with independent and identically distributed s ite substitutions, spectral analysis allows a global correction for al l multiple substitutions (second- and higher-order interactions), inde pendent of any particular tree. From these corrected data we use a lea st-squares selection procedure, the closest tree algorithm, to infer a n evolutionary tree. Other selection criteria such as parsimony or com patibility analysis could also be used; each of these criteria will be statistically consistent for these models. The closest tree algorithm selects a unique best-fit phylogenetic tree together with independent edge length parameters for each edge. The method is illustrated with an analysis of some primate hemoglobin sequences.