Construction of Dinucleotide Circular Codes Based on Nucleotide Probabilities

The construction of a circular code through a biological process, particularly a primitive one in the absence of the protein world, has remained an open problem since the discovery of a maximal [Formula: see text] self-complementary trinucleotide circular code in genes in 1996 (Arquès and Michel, 1996). Circular codes are defined by their ability to recover the correct reading frame of genes at any position. While a class of 216 such trinucleotide codes has been identified, the KL method (Koch and Lehman, 1997), based on nucleotide probability products, generates only a restricted subclass of 88 [Formula: see text]-codes (Lacan and Michel, 2001). Revisiting this probabilistic framework 25 years later, we demonstrate that various classes of dinucleotide circular codes can be generated using a nucleotide probability product model (called Construction 2). We introduce the concept of transitive dinucleotide codes and prove new theorems characterizing their circularity and comma-free properties. Using codon usage from bacteria, archaea, and eukaryotes, 2 "universal" maximal dinucleotide circular codes are observed: [Formula: see text] in the codon site [Formula: see text] and [Formula: see text] in the codon site [Formula: see text] which can be deduced from [Formula: see text] by 1-letter cyclical permutation [Formula: see text] or identically by reversing permutation [Formula: see text]. Unexpectedly, we then show that, under the independence assumption, the dinucleotide code [Formula: see text] through Construction 2 from nucleotide frequencies in the codon sites 1 and 2, is a maximal dinucleotide circular code and is equal to the observed dinucleotide code: [Formula: see text]. These findings support a theoretical model in which dinucleotide circular codes may have originated from statistical properties of primitive nucleotide distributions, providing insights into the possible emergence of the genetic code.