An analytical model of gene evolution with six mutation parameters: an application to archaeal circular codes

We develop here an analytical evolutionary model based on a trinucleotide mutation matrix 64 x 64 with six substitution parameters associated with the transitions and transversions in the three trinucleotide sites. It generalizes the previous models based on the nucleotide mutation matrices 4 x 4 and the trinucleotide mutation matrix 64 x 64 with three parameters. It determines at some time t the exact occurrence probabilities of trinucleotides mutating randomly according to six substitution parameters. An application of this model allows an evolutionary study of the common circular code COM and the 15 archaeal circular codes X which have been recently identified in several archaeal genomes. The main property of a circular code is the retrieval of the reading frames in genes, both locally, i.e. anywhere in genes and in particular without a start codon, and automatically with a window of a few nucleotides. In genes, the circular code is superimposed on the traditional genetic one. Very unexpectedly, the evolutionary model demonstrates that the archaeal circular codes can derive from the common circular code subjected to random substitutions with particular values for six substitutions parameters. It has a strong correlation with the statistical observations of three archaeal codes in actual genes. Furthermore, the properties of these substitution rates allow proposal of an evolutionary classification of the 15 archaeal codes into three main classes according to this model. In almost all the cases, they agree with the actual degeneracy of the genetic code with substitutions more frequent in the third trinucleotide site and with transitions more frequent that transversions in any trinucleotide site.