An Analytical Model of Gene Evolution with 9 Mutation Parameters: An Application to the Amino Acids Coded by the Common Circular Code

Abstract

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 $\mathcal{C}$ of eukaryotes and prokaryotes and its 12 coded amino acids. The main property of this code $\mathcal{C}$ 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 $\mathcal{C}$ has never been studied. Very unexpectedly, this evolutionary model demonstrates that random substitutions in this code $\mathcal{C}$ 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.