On the inference of large phylogenies with long branches: How long is too long?

The accurate reconstruction of phylogenies from short molecular sequences is an important problem in computational biology. Recent work has highlighted deep connections between sequence-length requirements for high-probability phylogeny reconstruction and the related problem of the estimation of ancestral sequences. In Daskalakis et al. (in Probab. Theory Relat. Fields 2010), building on the work of Mossel (Trans. Am. Math. Soc. 356(6):2379-2404, 2004), a tight sequence-length requirement was obtained for the simple CFN model of substitution, that is, the case of a two-state symmetric rate matrix Q. In particular the required sequence length for high-probability reconstruction was shown to undergo a sharp transition (from O(log n) to poly(n), where n is the number of leaves) at the "critical" branch length g (ML)(Q) (if it exists) of the ancestral reconstruction problem defined roughly as follows: below g (ML)(Q) the sequence at the root can be accurately estimated from sequences at the leaves on deep trees, whereas above g (ML)(Q) information decays exponentially quickly down the tree.Here, we consider a more general evolutionary model, the GTR model, where the q×q rate matrix Q is reversible with q≥2. For this model, recent results of Roch (Preprint, 2009) show that the tree can be accurately reconstructed with sequences of length O(log (n)) when the branch lengths are below g (Lin)(Q), known as the Kesten-Stigum (KS) bound, up to which ancestral sequences can be accurately estimated using simple linear estimators. Although for the CFN model g (ML)(Q)=g (Lin)(Q) (in other words, linear ancestral estimators are in some sense best possible), it is known that for the more general GTR models one has g (ML)(Q)≥g (Lin)(Q) with a strict inequality in many cases. Here, we show that this phenomenon also holds for phylogenetic reconstruction by exhibiting a family of symmetric models Q and a phylogenetic reconstruction algorithm which recovers the tree from O(log n)-length sequences for some branch lengths in the range (g (Lin)(Q),g (ML)(Q)). Second, we prove that phylogenetic reconstruction under GTR models requires a polynomial sequence-length for branch lengths above g (ML)(Q).