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× 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× 4 and the trinucleotide mutation matrix 64× 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 of eukaryotes and prokaryotes and its 12 coded amino acids. The main property of this 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. However, since its identification in 1996, amino acid information coded by has never been studied. Very unexpectedly, this evolutionary model demonstrates that random substitutions in this code 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.