In his three-substitution types model (K3ST) of nucleotide evolution Kimura proposed substitutions at a site could be classified into transitions, and transversion of two types, with substitutions within the classes occurring at independent rates α;β and γ. Over time period t, the expected numbers of the substitutions are qα=αt; qβ=βt and qγ=γt respectively. We have shown that time-dependency can be removed by specifying the expected numbers of substitutions of each type qα(e),qβ(e) and qγ(e) across each edge e of a phylogenetic tree T, as the parameters for the model. The probabilities of each possible pattern of nucleotides observed at the tips of T at that site can then be derived by 4-state Hadamard conjugation from these parameters. The invertibility of Hadamard conjugation means that T and all of its edge-length parameters can be easily derived from these probabilities which can be estimated from the frequencies of the nucleotide patterns in an alignment of homologous sequences.
Using the nucleotide pairings of R/Y (puRine, pYrimidine), W/S (Weak, Strong) or M/K (aMino, Ketone), a 4-state sequence can be projected to a 2-state sequences in three ways. In this presentation I will show that a further application of a Hadamard matrix allows us to parameterise the K3ST model by the expected numbers of substitutions of these 2-state sequences. Although the parameters are independent, the 2-state sequences evolved on the same tree T, giving the opportunity for 3 independent tree reconstructions (using your preferred tree-builder) from your sequence data, as a test of accuracy of the method.