Trinucleotide [formula omitted]-circular codes II: Biology

Abstract

A code X is (⩾k)-circular if every concatenation of words from X that admits, when read on a circle, more than one partition into words from X, must contain at least k+1 words. In other words, the reading frame retrieval is guaranteed for any concatenation of up to k words from X. A code that is (⩾k)-circular for all integers k is said to be circular. Any code is (⩾0)-circular and it turns out that 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 et al., 2022).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 apply 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 trinucleotide k-circular codes are given for the cases maximal, minimal, self-complementary k-, (k,k,k)- and self-complementary (k,k,k)-circular.