U(1) × U(1) × U(1) symmetry of the Kimura 3ST model and phylogenetic branching processes

An analysis of the Kimura 3ST model of DNA sequence evolution is given on the basis of its continuous Lie symmetries. The rate matrix commutes with a U(1) × U(1) × U(1) phase subgroup of the group G L (4) of 4 ×4 invertible complex matrices acting on a linear space spanned by the four nucleic acid base letters. The diagonal 'branching operator' representing speciation is defined, and shown to intertwine the U(1) × U(1) × U(1) action. Using the intertwining property, a general formula for the probability density on the leaves of a binary tree under the Kimura model is derived, which is shown to be equivalent to established phylogenetic spectral transform methods.