Complete families of linear invariants for some stochastic models of sequence evolution, with and without the molecular clock assumption.

For various models of sequence evolution, the set of linear functions of the frequencies of the nucleotide patterns forms a vector space, the invariant space. Here we distinguish between the model of nucleotide substitution, and the phylogenetic tree T describing the paths on which these changes occur. We describe a procedure to construct a basis of the invariant space for those models that are extensions of models incorporating Kimura's three substitution model of nucleotide change, including both the Jukes-Cantor and Cavender-Farris models. The dimension of the invariant space is determined, for those models where it is independent of the tree topology, as a function of the number of sequences. These are calculated where the nucleotide distribution at the root is unspecified, and both with, and without, the assumption of the molecular clock hypothesis. The invariants have a number of potential applications, including tree identification, and testing the fit of models (which could include the molecular clock) to sequence data.