Optimal control of gene expression for fast proteome adaptation to environmental change

Bacterial populations growing in a changing world must adjust their proteome composition in response to alterations in the environment. Rapid proteome responses to growth medium changes are expected to increase the average growth rate and fitness value of these populations. Little is known about the dynamics of proteome change, e. g., whether bacteria use optimal strategies of gene expression for rapid proteome adjustments and if there are lower bounds to the time of proteome adaptation in response to growth medium changes. To begin answering these types of questions, we modeled growing bacteria as stoichiometrically coupled networks of metabolic pathways. These are balanced during steady-state growth in a constant environment but are initially unbalanced after rapid medium shifts due to a shortage of enzymes required at higher concentrations in the new environment. We identified an optimal strategy for rapid proteome adjustment in the absence of protein degradation and found a lower bound to the time of proteome adaptation after medium shifts. This minimal time is determined by the ratio between the Kullback-Leibler distance from the pre- to the postshift proteome and the postshift steady-state growth rate. The dynamics of optimally controlled proteome adaptation has a simple analytical solution. We used detailed numerical modeling to demonstrate that realistic bacterial control systems can emulate this optimal strategy for rapid proteome adaptation. Our results may provide a conceptual link between the physiology and population genetics of growing bacteria.