Demand theory of gene regulation. I. Quantitative development of the theory.

The study of gene regulation has shown that a variety of molecular mechanisms are capable of performing this essential function. The physiological implications of these various designs and the conditions that might favor their natural selection are far from clear in most instances. Perhaps the most fundamental alternative is that involving negative or positive modes of control. Induction of gene expression can be accomplished either by removing a restraining element, which permits expression from a high-level promoter, or by providing a stimulatory element, which facilitates expression from a low-level promoter. This particular design feature is one of the few that is well understood. According to the demand theory of gene regulation, the negative mode will be selected for the control of a gene whose function is in low demand in the organism's natural environment, whereas the positive mode will be selected for the control of a gene whose function is in high demand. These qualitative predictions are well supported by experimental evidence. Here we develop the quantitative implications of this demand theory. We define two key parameters: the cycle time C, which is the average time for a gene to complete an ON/OFF cycle, and demand D, which is the fraction of the cycle time that the gene is ON. Mathematical analysis involving mutation rates and growth rates in different environments yields equations that characterize the extent and rate of selection. Further analysis of these equations reveals two thresholds in the C vs. D plot that create a well-defined region within which selection of wild-type regulatory mechanisms is realizable. The theory also predicts minimum and maximum values for the demand D, a maximum value for the cycle time C, as well as an inherent asymmetry between the regions for selection of the positive and negative modes of control.