The mathematics of the genetic code reveal that frequency degeneracy leads to exponential scaling in the DNA codon distribution of Homo sapiens

The nature of the quantitative distribution of the 64 DNA codons in the human genome has been an issue of debate for over a decade. Some groups have proposed that the quantitative distribution of the DNA codons ordered as a rank-frequency plot follows a well-known power law called Zipf's law. Others have shown that the DNA codon distribution is best fitted to an exponential function. However, the reason for such scaling behavior has not yet been addressed. In the present study, we demonstrate that the nonlinearity of the DNA codon distribution is a direct consequence of the frequency recurrence of the codon usage (i.e., the repetitiveness of codon usage frequencies at the whole genome level). We discover that if frequency recurrence is absent from the human genome, the frequency of occurrence of codons scales linearly with the codon rank. We also show that DNA codons of both low and high frequency of occurrence in the genome are best fitted by an exponential function and provide strong evidence to suggest that the coding region of the human genome does not follow Zipf's law. Information-theoretic methods and entropy calculations are applied to the DNA codon distribution and a new approach, called the lariat method, is proposed to quantitatively analyze the DNA codon distribution in Homo sapiens.