Genome Galaxy Identified by the Circular Code Theory

The genome galaxy identified in bacteria is studied by expressing the reading frame retrieval (RFR) function according to the YZ-content (GC-, AG- and GT-content) of bacterial codons. We have developed a simple probabilistic model for ambiguous sequences in order to show that the RFR function is a measure of the gene reading frame retrieval. Indeed, the RFR function increases with the ratio of ambiguous sequences and the ratio of ambiguous sequences decreases when the codon usage dispersion increases. The classical GC-content is the best parameter for characterizing the upper arm, which is related to bacterial genes with a low GC-content, and the lower arm, which is related to bacterial genes with a high GC-content. The galaxy center has a GC-content around 0.5. Then, these results are confirmed by expressing the GC-content of bacterial codons as a function of the codon usage dispersion. Finally, the bacterial genome galaxy is better described with the GC3-content in the 3rd codon site compared to the GC1-content and GC2-content in the 1st and 2nd codons sites, respectively. Whereas the codon usage is used extensively by biologists, its dispersion, which is an important parameter to reveal this genome galaxy, is surprisingly little known and unused. Therefore, we have developed a mathematical theory of codon usage dispersion by deriving several formulæ. It shows three important parameters in codon usage: the minimum and maximum codon probabilities and the number of codons with high frequency, i.e. with a probability at least 1/64. By applying this theory to the evolution of the genetic code, we see that bacteria have optimised the number of codons with high frequency to maximise the codon dispersion, thus maximising the capacity to retrieve the reading frame in genes. The derived formulæ of dispersion can be easily extended to any weighted code over a finite alphabet.