A method of estimating from two aligned present-day DNA sequences their ancestral composition and subsequent rates of substitution, possibly different in the two lineages, corrected for multiple and parallel substitutions at the same site

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Abstract
The course of evolutionary change in DNA sequences has been modeled as a Markov process. The Markov process was represented by discrete time matrix methods. The parameters of the Markov transition matrices were estimated by least-squares direct-search optimization of the fit of the calculated divergence matrix to that observed for two aligned sequences. The Markov process corrected for multiple and parallel substitutions of bases at the same site. The method avoided the incorrect assumption of all previously described methods that the divergence between two present-day sequences is twice the divergence of either from the common and unknown ancestral sequence. The three previous methods were shown to be equivalent. The present method also avoided the undesirable assumptions that sequence composition has not changed with time and that the substitution rates in the two descendant lineages were the same. It permitted simultaneous estimation of ancestral sequence composition and, if applicable, of different substitution rates for the two descendant lineages, provided the total number of estimated parameters was less than 16. Properties of the Markov chain were discussed. It was proved for symmetric substitution matrices that all elements of the equilibrium divergence matrix equal 1/16, and that the total difference in the divergence matrix at epoch k equals the total change in the common substitution matrix at epoch 2k for all values of k. It was shown how to resolve an ambiguity in the assignment of two different substitution rates to the two descendant lineages when four or more similar sequences are available. The method was applied to the divergence matrix for codon site 3 for the mouse and rabbit beta-globins. This observed divergence matrix was significantly asymmetric and required at least two different substitution rates. This result could be achieved only by using different asymmetric substitution matrices for the two lineages.