  Phylogenetic data arising on two possibly different tree topologies might be
mixed through several biological mechanisms, including incomplete lineage
sorting or horizontal gene transfer in the case of different topologies, or
simply different substitution processes on characters in the case of the same
topology. Recent work on a 2-state symmetric model of character change showed
such a mixture model has non-identifiable parameters, and thus it is
theoretically impossible to determine the two tree topologies from any amount
of data under such circumstances. Here the question of identifiability is
investigated for 2-tree mixtures of the 4-state group-based models, which are
more relevant to DNA sequence data. Using algebraic techniques, we show that
the tree parameters are identifiable for the JC and K2P models. We also prove
that generic substitution parameters for the JC mixture models are
identifiable, and for the K2P and K3P models obtain generic identifiability
results for mixtures on the same tree. This indicates that the full
phylogenetic signal remains in such mixtures, and that the 2-state symmetric
result is thus a misleading guide to the behavior of other models.
