Divergence and Shannon Information in Genomes

Shannon information (SI) and its special case, divergence, are defined for a DNA sequence in terms of probabilities of chemical words in the sequence and are computed for a set of complete genomes highly diverse in length and composition. We find the following: SI (but not divergence) is inversely proportional to sequence length for a random sequence but is length independent for genomes; the genomic SI is always greater and, for shorter words and longer sequences, hundreds to thousands times greater than the SI in a random sequence whose length and composition match those of the genome; genomic SIs appear to have word-length dependent universal values. The universality is inferred to be an evolution footprint of a universal mode for genome growth.

Figure 1

The 5-spectra of the genome of C. acetobutylicum (black lines) ($p\approx 0.7$) and that of its random match (green or gray lines) normalized to that of a 2 Mb sequence. The random-match spectrum is composed of sharply peaking subspectra ${\mathcal{F}}_m^{\{k\}}$, $m=0$ to 4. The center of the $m=5$ subspectrum is off scale at 10 500. Ordinates $n_f$ are not integers because (for better viewing only) large fluctuations in the spectra are smoothed out by averaging over a range of 21 frequencies.

Figure 2

SI in the $k$ spectra for $k=2$ (◇), 3 (⊲), 4 (*), 5 (▽), 6 (+), 7 (△), 8 (×), 9 (⊳), and 10 (□) of the genomes of 12 organisms: E cun, E. cuniculi (Chromosome I; length 0.198 Mb, $p=0.533$); CM, C. muridarum; MG, M. genitalium (0.580 Mb, 0.683); SC, yeast (S cerevisiae, IV; 1.53 Mb, 0.621); MJ, M. jannaschii (1.66 Mb, 0.686); PA, P. aerophilum; E col, E. coli K12 (4.64 Mb, 0.492); SA, S. avermitilis; CE, worm (C. elegans, I; 15.1 Mb, 0.643); DM, fly (D. melanogaster, X; 21.8 Mb, 0.574); MM, mouse (M. musculus, IX; 98.9 Mb, 0.562); HS, human (H. sapiens, I; 228 Mb, 0.582). The lines give SI expected in random sequences for $k=10$ (open circles connected by dotted line) and 100 times the expected value for $k=2$ (solid circles connected by solid line). The $k$-mer SI of a sequence is not given when sequence length is less than twice $4^k$.