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2, each DNA strand has a well-defined positive direction (the 5 → 3 sense), along which successive nucleotides can be joined to each other [29]. The two strands of a DNA double helix have opposite positive directions. DNA molecules have a very “patchy” structure with intertwined coding and noncoding segments oriented in both direct and inverse sense [36]. For most currently sequenced genomes, the information about the direct or inverse orientation of the coding regions—the ORF—has been identified and is available in the genomic databases [2].

7) by the projection on the xOz plane, and by taking Ox as the real axis and Oz as the imaginary one. 8) can also be seen as representing the nucleotides in two (bipolar) binary systems, with mutually orthogonal complex units. For methionine, to which corresponds the codon ATG that starts any gene, the vector representation is M = 4a + 2t + g = 5i + j + 3k, while the complex representation is M = 4a + 2t + g = 5 + 3 j. 14. 13. Genetic code vectorial representation. 15 is obtained [23]. 11. 12 on the xOz plane.

38 shows the cumulated and unwrapped phase of the genomic signal obtained by concatenating the 4393 reoriented coding regions of Salmonella typhi genome (accession AL 5113382 [2]). Each inverse coding region (inverse ORF) has been reversed and complemented, that is, the nucleotides inside the same W (adenine-thymine) or S (cytosine-guanine) class have been replaced with each other, to take into account the switching of the strands that accompanies the segment reversal. As expected from the model, the breaking points in the cumulated phase disappear and the absolute values of the slopes increase, as there is no longer interweaving of direct and inverse ORFs.

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