Priors and Implementation

We used the same priors as in previous works (Lartillot et al. 2013; Rodrigue and Lartillot 2014):

Branch lengths are i.i.d. exponential of rate λ, itself exponential of rate 0.1.

We use a Dirichlet process over amino-acid fitness profiles, with base distribution a Dirichlet(α_i), where the α_i are i.i.d. exponential of rate 1.

The granularity parameter of the Dirichlet process is exponential of rate 0.1 (mean 10).

Nucleotide exchangeability parameters and nucleotide frequency parameters are each flat Dirichlets.

Non-synonymous rate factors ω and ω_* are ratios of two exponential random variables (Huelsenbeck et al. 2006).

The use of PhyloBayes-MPI with the mutation–selection model is explained within the online manual, and activating the ω_* parameter is done by adding the option -freeomega to the command. To obtain the plain MG model, the options -freeomega and -catfix uniform are applied, whereas to obtain the MutSelYN model, the options -freeomega, -rigidbaseprior, and -ncat 1 are applied. For simulated data, inferences based on MCMC calculations were conducted under fixed tree topology, as originally used for the simulations, and were run for 1,100 cycles, discarding the first 100 as burn-in. Note that each cycle itself includes hundreds of Gibbs and Metropolis-Hastings updates within PhyloBayes-MPI. Real data analyses were run with 5,500 cycles (500 as burn-in), treating the topology (with uniform priors) as a nuisance variable of the inference. Source code is freely available within the PhyloBayes-MPI package, distributed at www.phylobayes.org.