Modeling the covarion hypothesis of nucleotide substitution

A "covarion" model for nucleotide substitution that allows sites to turn "on" and "off" with time was proposed in 1970 by Fitch and Markowitz. It has been argued recently that evidence supports such models over later, alternative models that postulate a static distribution of rates across sites. However, in contrast with these latter well-studied models, little is known about the analytic properties of the former model. Here we analyze a covarion-style model and show (i) how to obtain the evolutionary distance between two species from the expected proportion of sites where two species differ, (ii) that the covarion model gives identical results to a suitably chosen rates-across-sites model if several sequences are compared in pairs by using only the expected proportion of sites at which they differ, (iii) conditions under which the two models will give identical results if the full joint probability matrix is examined, and (iv) that the two models can, in principle, be distinguished when there are at least four monophyletic groups of species. This last result is based on a distance measure that is tree additive under certain versions of the covarion model but, in general, will not be additive under a rates-across-sites model. The measure constructed does not require knowledge of the parameters of the model and so shows that sequences generated by the covarion model do in fact contain information about the underlying tree.