Optimal transcriptional regulation of dynamic bacterial responses to sudden drug exposures

Cellular responses to the presence of toxic compounds in their environment require prompt expression of the correct levels of the appropriate enzymes, which are typically regulated by transcription factors that control gene expression for the duration of the response. The characteristics of each response dictate the choice of regulatory parameters such as the affinity of the transcription factor to its binding sites and the strength of the promoters it regulates. Although much is known about the dynamics of cellular responses, we still lack a framework to understand how different regulatory strategies evolved in natural systems relate to the selective pressures acting in each particular case. Here, we analyze a dynamical model of a typical antibiotic response in bacteria, where a transcriptionally repressed enzyme is induced by a sudden exposure to the drug that it processes. We identify strategies of gene regulation that optimize this response for different types of selective pressures, which we define as a set of costs associated with the drug, enzyme, and repressor concentrations during the response. We find that regulation happens in a limited region of the regulatory parameter space. While responses to more costly (toxic) drugs favor the usage of strongly self-regulated repressors, responses where expression of enzyme is more costly favor the usage of constitutively expressed repressors. Only a very narrow range of selective pressures favor weakly self-regulated repressors. We use this framework to determine which costs and benefits are most critical for the evolution of a variety of natural cellular responses that satisfy the approximations in our model and to analyze how regulation is optimized in new environments with different demands.