The maximality of circular codes in genes statistically verified

The maximality of circular codes in genes has 20 preferential trinucleotides in each frame. This combinatorial property is statistically verified in the genes of both bacteria and eukaryotes, and by two approaches computing the trinucleotide occurrence frequencies in the 3 frames at the gene population level (classical method) and at the gene level (recent method). Several remarks explain why the codon usage parameter is unable to identify the circular codes. Some historical and theoretical considerations on comma-free and circular codes are presented. An evolutionary process by trinucleotide permutation is proposed to describe the transformation of a circular code (and its motifs) into another circular code.