Taking variation of evolutionary rates between sites into account in inferring phylogenies

As methods of molecular phylogeny have become more explicit and more biolog ically realistic following the pioneering work of Thomas Jukes, they have h ad to relax their initial assumption that rates of evolution were equal at all sites. Distance matrix and likelihood methods of inferring phylogenies make this assumption; parsimony, when valid, is less limited by it. Nucleot ide sequences, including RNA sequences, can show substantial rate variation ; protein sequences show rates that vary much more widely. Assuming a prior distribution of rates such as a gamma distribution or lognormal distributi on has deservedly been popular, but for likelihood methods it leads to comp utational difficulties. These can be resolved using hidden Markov model (HM M) methods which approximate the distribution by one with a modest number o f discrete rates. Generalized Laguerre quadrature can be used to improve th e selection of rates and their probabilities so as to more nearly approach the desired gamma distribution. A model based on population genetics is pre sented predicting how the rates of evolution might vary from locus to locus . Challenges for the future include allowing rates at a given site to vary along the tree, as in the "covarion" model, and allowing them to have corre lations that reflect three-dimensional structure, rather than position in t he coding sequence. Markov chain Monte Carlo likelihood methods may be the only practical way to carry out computations for these models.