Modeling the Kinetics of the Permeation of Antibacterial Agents into Growing Bacteria and Its Interplay with Efflux

A mathematical model of the passive permeation of a novel solute into bacteria that explicitly accounts for intracellular dilution through growth was developed. A bacterial cell envelope permeability coefficient of approximately >10 -8 cm 2 · s -1 is predicted to ensure passive permeation into rapidly replicating bacterial cells. The relative importance of the permeability coefficients of the cytoplasmic and outer membranes of Gram-negative bacteria in determining the overall envelope permeability coefficient was analyzed quantitatively. A mathematical description of the balance between passive influx and active efflux was also developed and shows that bacterial expansion through growth can usually be neglected for compounds likely to be prepared in antibacterial drug discovery programs and the balance between passive inward permeation and active outwardly directed efflux predominates. A new parameter, efflux efficiency (η, where η is equal to k / P , in which k is the rate coefficient for the efflux pump and P is the permeability coefficient for the membrane across which the pump acts), is introduced, and the consequences for the efficiency of efflux pumping by a single pump, two pumps in parallel across either the cytoplasmic or the outer membrane, and two pumps in series, one across the cytoplasmic membrane and one across the outer membrane of Gram-negative bacteria, are explored. The results, showing additive efficiency for two pumps acting across a single membrane and multiplicative efficiency for two pumps acting in series across the cytoplasmic and outer membranes, can be quantitatively related to the ratios between MICs measured against pump-sufficient and pump deletion strains and agree with those of previous experimental and theoretical studies.