An evolutionary model of a complementary circular code.

The subset X0 = [sequence: see text] of 20 trinucleotides has a preferential occurrence in frame 0 (a reading frame established by the ATG start trinucleotide) of protein (coding) genes of both prokaryotes and eukaryotes. This subset X0++ has the rarity property (6 x 10(-8)) to be a complementary maximal circular code with two permutated maximal circular codes X1 and X2 in frames 1 and 2 respectively (frame 0 shifted by one and two nucleotides respectively in the 5'-3' direction). X0 is called a C3 code. A quantitative study of these three subsets X0, X1 and X2 in the three frames 0, 1 and 2 of eukaryotic protein genes shows that their occurrence frequencies are constant functions of the trinucleotide positions in the sequences. The frequencies of X0, X1 and X2 in frame 0 of the eukaryotic protein genes are 48.5%, 29% and 22.5% respectively. These properties are not observed in the 5' and 3' regions of eukaryotes where X0, X1 and X2 occur with variable frequencies around the random value (1/3). Several frequency asymmetries unexpectedly observed, e.g. the frequency difference between X1 and X2 in the frame 0, are related to a new property of the C3 code X0 involving substitutions. An evolutionary model at three parameters (p, q, k) based on an independent mixing of the 20 codons (trinucleotides in frame 0) of X0 with equiprobability (1/20) followed by k approximately 5 substitutions per codon in the three codon sites in proportions p approximately 0.1, q approximately 0.1 and r = 1-p-q approximately 0.8 respectively, retrieves the frequencies of X0, X1 and X2 observed in the three frames of protein genes and explains these asymmetries.