A dependent-rates model and an MCMC-based methodology for the maximum-likelihood analysis of sequences with overlapping reading frames

We present a model and methodology for the maximum-likelihood analysis of pairwise alignments of DNA sequences in which two genes are encoded in overlapping reading frames. In the model for the substitution process, the instantaneous rates of substitution are allowed to depend on the nucleotides occupying the sites in a neighborhood of the site subject to substitution at the instant of the substitution. By defining the neighborhood of a site to extend over all sites in the codons in both reading frames to which a site belongs, constraints imposed by the genetic code in both reading frames can be taken into account. Due to the dependency of the instantaneous rates of substitution on the states at neighboring sites, the transition probability between sequences does not factorize and therefore cannot be obtained directly. We present a Markov chain Monte Carlo procedure for obtaining the ratio of two transition probabilities between two sequences under the model considered, and we describe how maximum-likelihood parameter estimation and likelihood ratio tests can be performed using the procedure. We describe how the expected numbers of different types of substitutions in the shared history of two sequences can be calculated, and we use the described model and methodology in an analysis of a pairwise alignment of two hepatitis B sequences in which two genes are encoded in overlapping frames. Finally, we present an extended model, together with a simpler approximate estimation procedure, and use this to test the adequacy of the former model.