Bistable Bacterial Growth Rate in Response to Antibiotics with Low Membrane Permeability

We demonstrate that growth rate bistability for bacterial cells growing exponentially at a fixed external antibiotic concentration can emerge when the cell wall permeability for the drug is low and the growth rate sensitivity to the intracellular drug concentration is high. Under such conditions, an initially high growth rate can remain high, due to dilution of the intracellular drug concentration by rapid cell volume increase, while an initially low growth rate can remain low, due to slow cell volume increase and insignificant drug dilution. Our findings have implications for the testing of novel antibiotics on growing bacterial strains.

Figure 1

Graphical illustration of the constraints on $a_{\mathrm{fr}}(a)$ and $\mu (a)$. (A). Examples of how the free intracellular antibiotic concentration, $a_{\mathrm{fr}}$, may depend on the intracellular antibiotic concentration, $a$. (B). Examples of how the flow over the membrane, $J_{\mathrm{mem}}$, may change with $a$. (C). Examples of how the growth rate may depend on $a$. (D). Examples of dilution flows, $J_{\mathrm{gr}}$, compatible with bistability. (E). Examples of how intersections between $J_{\mathrm{gr}}$ and $J_{\mathrm{mem}}$ give rise to two stable (filled discs) and one unstable (unfilled disc) steady state(s). (F). Graphical demonstration of why multiple steady states always emerge when the convex $J_{\mathrm{mem}}$-curve intersects the inflexion point of the $J_{\mathrm{gr}}$-curve, which is concave to the right of its maximum.

Figure 2

(A). The figure shows the flow in trough the membrane, $J_{\mathrm{mem}}$ (dashed line) and the dilution flow, $J_{\mathrm{gr}}$ (solid line) for different intracellular concentration of antibiotic, $a$. The intersections indicate the stationary states. (B). The stationary states are plotted for three different permeability values, $c$. Solid lines correspond to stable and dashed lines to unstable states. The stationary states for $a_{\mathrm{ex}}=2\mu \mathrm{M}$ on the $c=0.0001{\mathrm{s}}^{-1}$ curve (vertical dashed line) correspond to the situation in Fig. 2a. Other parameters are $r=10\mu \mathrm{M}$, $K_d=10\mathrm{nM}$, $v_m=20{\mathrm{s}}^{-1}$.

Figure 3

(A). The steady state growth rate is plotted as a function of the external antibiotic concentration. The solid lines indicate stable and the dashed line indicates unstable steady states. The arrows indicate the direction of growth rate change in different regions for a fixed external concentration of antibiotic. (B). This panel illustrates the times to establish steady state growth for different external drug concentrations and starting out with small (solid lines) or a large (dashed lines) internal drug concentrations. Parameters: $r=10\mu \mathrm{M}$, $K_d=10\mathrm{nM}$, $c=0.0001{\mathrm{s}}^{-1}$, $v_m=20{\mathrm{s}}^{-1}$.