Zipf-like behavior in procaryotic protein expression

The relative rates of synthesis $p_r$ of proteins present in various procaryotic organisms have been found to follow the simple canonical law $p_r\sim (r+\rho {)}^{-1/\theta },$ where r is the rank. The parameter $\rho$ is interpreted as the bias characterizing the mode of control (i.e., the overall preference for positive or negative control) of gene expression. By analogy with thermodynamics, and drawing parallels with the abstract theory of messages, $\theta$ is the informational temperature, which characterizes the extent to which the organism’s genome is used to produce proteins. The quantity of selective information H (analogous to thermodynamic entropy) was calculated for the distribution of synthesis rates using Shannon’s formula. For all the organisms investigated, H was approximately 8 bits/protein.