Context-dependent substitution models for circular DNA

The most general context-dependent Markov substitution process, where each substitution event involves only one site and substitution rates depend on the whole sequence, is presented for the first time. The focus is on circular DNA sequences, where the problem of specifying the behaviour of the first and last sites in a linear sequence does not arise. Important special cases include (1) the established models where each site behaves independently, (2) models which are increasingly applied to non-coding DNA, where each site depends on only the immediate neighbouring sites, and (3) models where each site depends on two closest neighbours on both sides, such as the codon models. These special cases are classified and illustrated by published models. It is shown that the existing codon substitution models mix up the mutation and selection processes, rendering the substitution rates challenging to interpret. The classification suggests the study of a more interpretable codon model, where the mutation and selection processes are clearly delineated. Furthermore, this model allows a natural accommodation of possibly different selection pressures in overlapping reading frames, which may contribute to furthering the understanding of viral diseases. Also included are brief discussions on the stationary distribution of a context-dependent substitution process and a simple recipe for simulating it on a computer.