An analytical model of gene evolution with 9 mutation parameters: an application to the amino acids coded by the common circular code

We develop here an analytical evolutionary model based on a trinucleotide mutation matrix 64 x 64 with nine substitution parameters associated with the three types of substitutions 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 and six parameters. It determines at some time t the exact occurrence probabilities of trinucleotides mutating randomly according to these nine substitution parameters. An application of this model allows an evolutionary study of the common circular code [Formula: see text] of eukaryotes and prokaryotes and its 12 coded amino acids. The main property of this code [Formula: see text] 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. However, since its identification in 1996, amino acid information coded by [Formula: see text] has never been studied. Very unexpectedly, this evolutionary model demonstrates that random substitutions in this code [Formula: see text] and with particular values for the nine substitutions parameters retrieve after a certain time of evolution a frequency distribution of these 12 amino acids very close to the one coded by the actual genes.