The spiderweb of error-detecting codes in the genetic information

Nature possesses inherent mechanisms for error detection and correction during the translation of genetic information, as demonstrated by the discovery of a self-complementary circular C³-code called X₀ in various organisms such as bacteria, eukaryotes, plasmids, and viruses (Arquès and Michel, 1996; Michel, 2015, 2017). Since then, extensive research has focused on circular codes, which are believed to be remnants of ancient comma-free codes. These codes can be regarded as an additional genetic code specifically optimized for detecting and preserving the proper reading frame in protein-coding sequences. A study by Fimmel et al. in 2014 identified that a total of 216 maximal self-complementary C³-codes can be grouped into 27 equivalence classes with eight codes in each class. In this work, we study how the 27 equivalence classes are related to each other. While the codes in each equivalence class obtained by Fimmel et al. in 2014 are permutations of each other, i.e. one code can be obtained from the other by applying a permutation of the bases, it has not been clear how the equvalence classes are connected. We show that there is an ordering of the equivalence classes such that one gets from one class to the next one by substituting only one pair of codon/anticodon in the corresponding codes, i.e. the corresponding codes have a maximal intersection of 18 codons. To perform this analysis, we define two graphs, G₂₁₆ and G₂₇, whose vertices are, respectively, all 216 maximal self-complementary C³-codes and 27 equivalence classes. Several properties of the graphs are obtained. Most surprisingly, it turns out that G₂₇ contains Hamiltonian paths of length 27. This fact ultimately leads to a representation of the set of all 216 maximal self-complementary C³-codes as a kind of spider web. Finally, we define dinucleotide cuts of such codes by projecting each codon to its first two bases and show that the paths of lengths 27 in G₂₁₆ can even be chosen so that all the codes contain a special subset of dinucleotides defined by Rumer's roots. These observations raise a lot of new questions about the biological function of such structures.