An evolutionary analytical model of a complementary circular code simulating the protein coding genes, the 5′ and 3′ regions

Abstract

The self-complementary subset $\mathcal{T}_0 = \mathcal{X}_0$∪{AAA,TTT} with $\mathcal{X}_0$ = {AAC, AAT, ACC, ATC, ATT, CAG, CTC, CTG, GAA, GAC, GAG, GAT, GCC, GGC, GGT, GTA, GTC, GTT, TAC, TTC} of 22 trinucleotides has a preferential occurrence in the frame 0 (reading frame established by the ATG start trinucleotide) of protein (coding) genes of both prokaryotes and eukaryotes. The subsets $\mathcal{T}_1 = \mathcal{X}_1$∪{CCC} and $\mathcal{T}_2 = \mathcal{X}_2$∪{GGG} of 21 trinucleotides have a preferential occurrence in the shifted frames 1 and 2 respectively (frame 0 shifted by one and two nucleotides respectively in the 5′-3′ direction). $\mathcal{T}_1$ and $\mathcal{T}_2$ are complementary to each other. The subset $\mathcal{T}_0$ contains the subset $\mathcal{X}_0$ which has the rarity property (6 × 10⁻⁸) to be a complementary maximal circular code with two permutated maximal circular codes $\mathcal{X}_1$ and $\mathcal{X}_2$ in the frames 1 and 2 respectively. $\mathcal{X}_0$ is called a C³ code.

A quantitative study of these three subsets $\mathcal{T}_0 ,\mathcal{T}_1 ,\mathcal{T}_2$ in the three frames 0, 1, 2 of protein genes, and the 5′ and 3′ regions of eukaryotes, shows that their occurrence frequencies are constant functions of the trinucleotide positions in the sequences. The frequencies of $\mathcal{T}_0 ,\mathcal{T}_1 ,\mathcal{T}_2$ in the frame 0 of protein genes are 49, 28.5 and 22.5% respectively. In contrast, the frequencies of $\mathcal{T}_0 ,\mathcal{T}_1 ,\mathcal{T}_2$ in the 5′ and 3′ regions of eukaryotes, are independent of the frame. Indeed, the frequency of $\mathcal{T}_0$ in the three frames of 5′ (respectively 3′) regions is equal to 35.5% (respectively 38%) and is greater than the frequencies $\mathcal{T}_1$ and $\mathcal{T}_2$, both equal to 32.25% (respectively 31%) in the three frames.

Several frequency asymmetries unexpectedly observed (e.g. the frequency difference between $\mathcal{T}_1$ and $\mathcal{T}_2$ in the frame 0), are related to a new property of the subset $\mathcal{T}_0$ involving substitutions. An evolutionary analytical model at three parameters (p, q, t) based on an independent mixing of the 22 codons (trinucleotides in frame 0) of $\mathcal{T}_0$ with equiprobability (1/22) followed by t ≈ 4 substitutions per codon according to the proportions p ≈ 0.1; q ≈ 0.1 and r = 1 − p − q ≈ 0.8 in the three codon sites respectively, retrieves the frequencies of $\mathcal{T}_0 ,\mathcal{T}_1 ,\mathcal{T}_2$ observed in the three frames of protein genes and explains these asymmetries. Furthermore, the same model (0.1, 0.1, t) after t ≈ 22 substitutions per codon, retrieves the statistical properties observed in the three frames of the 5′ and 3′ regions. The complex behaviour of these analytical curves is totally unexpected and a priori difficult to imagine.