Likelihood analysis of phylogenetic networks using directed graphical models

A method for computing the likelihood of a set of sequences assuming a phyl ogenetic network as an evolutionary hypothesis is presented. The approach a pplies directed graphical models to sequence evolution on networks and is a natural generalization of earlier work by Felsenstein on evolutionary tree s, including it as a special case. The likelihood computation involves seve ral steps. First, the phylogenetic network is rooted to form a directed acy clic graph (DAG). Then, applying standard models for nucleotide/amino acid substitution, the DAG is converted into a Bayesian network from which the j oint probability distribution involving all nodes of the network can be dir ectly read. The joint probability is explicitly dependent on branch lengths and on recombination parameters (prior probability of a parent sequence). The likelihood of the data assuming no knowledge of hidden nodes is obtaine d by marginalization, i.e., by summing over all combinations of unknown sta tes. As the number of terms increases exponentially with the number of hidd en nodes, a Markov chain Monte Carlo procedure (Gibbs sampling) is used to accurately approximate the likelihood by summing over the most important st ates only. Investigating a human T-cell lymphotropic virus (HTLV) data set and optimizing both branch lengths and recombination parameters, we find th at the likelihood of a corresponding phylogenetic network outperforms a set of competing evolutionary trees. In general, except for the case of a tree , the likelihood of a network will be dependent on the choice of the root, even if a reversible model of substitution is applied. Thus, the method als o provides a way in which to root a phylogenetic network by choosing a node that produces a most likely network.