Supplementary material for: nQMaker: estimating time non-reversible amino acid substitution models

Amino acid substitution models are a key component in phylogenetic
analyses of protein sequences. All commonly-used amino acid models
available to date are time-reversible, an assumption designed for
computational convenience but not for biological reality. Another
significant downside to time-reversible models is that they do not allow
inference of rooted trees without outgroups. In this paper, we introduce a
maximum likelihood approach nQMaker, an extension of the recently
published QMaker method, that allows the estimation of time non-reversible
amino acid substitution models and rooted phylogenetic trees from a set of
protein sequence alignments. We show that the non-reversible models
estimated with nQMaker are a much better fit to empirical alignments than
pre-existing reversible models, across a wide range of datasets including
mammals, birds, plants, fungi, and other taxa, and that the improvements
in model fit scale with the size of the dataset. Notably, for the recently
published plant and bird trees, these non-reversible models correctly
recovered the commonly estimated root placements with very high
statistical support without the need to use an outgroup. We provide
nQMaker as an easy-to-use feature in the IQ-TREE software
(http://www.iqtree.org), allowing users to estimate non-reversible models
and rooted phylogenies from their own protein datasets.