Full reconstruction of non-stationary strand-symmetric models on rooted phylogenies

Understanding the evolutionary relationship among species is of fundamental importance to the biological sciences. The location of the root in any phylogenetic tree is critical as it gives an order to evolutionary events. None of the popular models of nucleotide evolution currently used in likelihood or Bayesian methods are able to infer the location of the root without exogenous information. It is known that the most general Markov models of nucleotide substitution also cannot identify the location of the root or be fitted to multiple sequence alignments with fewer than three sequences. We prove that the location of the root and the full model can be identified and statistically consistently estimated for a non-stationary, strand-symmetric substitution model given a multiple sequence alignment with two or more sequences. We also generalise earlier work to provide a practical means of overcoming the computationally intractable problem of labelling hidden states in a phylogenetic model.