Data on the number of Open Reading Frames (ORFs) coded by genomes from the 3 domains of Life show some notable general features including essential differences between the Prokaryotes and Eukaryotes, with the number of ORFs growing linearly with total genome size for the former, but only logarithmically for the latter. Assuming that the (protein) coding and non-coding fractions of the genome must have different dynamics and that the non-coding fraction must be controlled by a variety of (unspecified) probability distribution functions, we are able to predict that the number of ORFs for Eukaryotes follows a Benford distribution and has a specific logarithmic form. Using the data for 1000+ genomes available to us in early 2010, we find excellent fits to the data over several orders of magnitude, in the linear regime for the Prokaryote data, and the full non-linear form for the Eukaryote data. In their region of overlap the salient features are statistically congruent, which allows us to: interpret the difference between Prokaryotes and Eukaryotes as the manifestation of the increased demand in the biological functions required for the larger Eukaryotes, estimate some minimal genome sizes, and predict a maximal Prokaryote genome size on the order of 8-12 megabasepairs. These results naturally allow a mathematical interpretation in terms of maximal entropy and, therefore, most efficient information transmission.
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