Trinucleotide k-circular codes II: biology

A code X is (≥ k)-circular if any concatenation of at most k words from X, when read on a circle, admits exactly one partition into words from X. A code that is (≥ k)-circular for all integers k is said to be circular. Any code is (≥ 0)-circular and a code of trinucleotides is circular as soon as it is (≥ 4)-circular. A code is k-circular if it is (≥ k)-circular and not (≥ k+1)-circular. The theoretical aspects of trinucleotide k-circular codes have been developed in a companion article [Michel, C.J., Mouillon, B., Sereni, J.-S., Trinucleotide k-circular codes I: theory, submitted for publication]. Trinucleotide circular codes always retrieve the reading frame, leaving no ambiguous sequences. On the contrary, trinucleotide k-circular codes, for k ∈ {0, 1, 2, 3} all have ambiguous sequences, for which the reading frame cannot always be retrieved. However, such a trinucleotide k-circular code is still able to retrieve the reading frame for a number of sequences, thereby exhibiting a partial circularity property. We describe this combinatorial property for each class of trinucleotide k-circular codes with k ∈ {0, 1, 2, 3}. The circularity, i.e. the reading frame retrieval, is an ordinary property in genes. In order to consider the different cases of ambiguous sequences, we derive a new and general formula to measure the reading frame loss, whatever the trinucleotide k-circular code. This formula allows us to study the evolution of any trinucleotide k-circular code of (maximal) cardinality 20 to the genetic code, based on the reading frame retrieval property. We applied this approach to analyse the evolution of the trinucleotide circular code X observed in genes to the genetic code. The (≥ 1)-circular codes of maximal size 20 necessarily have the same number of each nucleotide, specifically 15 = 3 • 20/4. This balanceness property can also be achieved by trinucleotide codes of cardinality 4, 8, 12 and 16. We call such trinucleotide codes balanced. We develop a general mathematical method to compute the number of balanced trinucleotide codes of each size, which also applies to self-complementary trinucleotide codes. We establish and quantify a relation between this balanceness property and the self-complementarity property. The combinatorial hierarchy of trinucleotide k-circular codes is updated with the growth function results. The numbers of amino acids coded by the maximal, minimal, self-complementary trinucleotide k-or (k, k, k)-circular codes are given