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A shift of the frame in a polynucleotide sequence typically alters the codon content of the sequence. This provokes a question as to what sequence might be unaltered after shifting the frame. In fact, a linear sequence cannot exactly be so conserved – but there might be a possibility if it is a cyclic code subjected to a circular permutation, as we consider here. The solution is strikingly simple: A cyclic sequence of different nucleotides conserves a circular order of its codons under any shift of its frame if it has a length λ not divisible by 3 and is consecutively read κ times, or it is composed of κ repeated copies of a factor h of length λ, where κ is divisible by 3, while λ is not. For example, the sequence atcgatcgatcg has a factor atcg of length λ = 4 is repeated κ = 3 times. Translating this code without any shift gives isoleucine, aspartic acid, arginine, and serine, consecutively, or IDRS for short. The circular shift by 1 position results in SIDR, by 2 positions if produces RSID, and (here) at last, the circular shift by 3 positions gives DRSI. Apparently, all four translated codes of amino acids are the same relative to cyclic permutation. We conclude here discussing the cyclically invariant codes by noting that these can easily be enumerated using the famous Pólya’s theorem.

Keywords: nucleotide sequence, codonic, frame shift, cyclically invariant, permutable