Population Density Modulates Drug Inhibition and Gives Rise to Potential Bistability of Treatment Outcomes for Bacterial Infections

Abstract

The inoculum effect (IE) is an increase in the minimum inhibitory concentration (MIC) of an antibiotic as a function of the initial size of a microbial population. The IE has been observed in a wide range of bacteria, implying that antibiotic efficacy may depend on population density. Such density dependence could have dramatic effects on bacterial population dynamics and potential treatment strategies, but explicit measures of per capita growth as a function of density are generally not available. Instead, the IE measures MIC as a function of initial population size, and population density changes by many orders of magnitude on the timescale of the experiment. Therefore, the functional relationship between population density and antibiotic inhibition is generally not known, leaving many questions about the impact of the IE on different treatment strategies unanswered. To address these questions, here we directly measured real-time per capita growth of Enterococcus faecalis populations exposed to antibiotic at fixed population densities using multiplexed computer-automated culture devices. We show that density-dependent growth inhibition is pervasive for commonly used antibiotics, with some drugs showing increased inhibition and others decreased inhibition at high densities. For several drugs, the density dependence is mediated by changes in extracellular pH, a community-level phenomenon not previously linked with the IE. Using a simple mathematical model, we demonstrate how this density dependence can modulate population dynamics in constant drug environments. Then, we illustrate how time-dependent dosing strategies can mitigate the negative effects of density-dependence. Finally, we show that these density effects lead to bistable treatment outcomes for a wide range of antibiotic concentrations in a pharmacological model of antibiotic treatment. As a result, infections exceeding a critical density often survive otherwise effective treatments.

Fig 1. Computer-controlled continuous culture devices can measure population growth at constant cell density.

A. Bacterial cultures (15 mL) are grown in glass vials with customized Teflon tops that allow inflow and outflow of fluid via silicone tubing. Cell density is monitored by light scattering using infrared LED/Detector pairs on the side of each vial holder. At the onset of the experiment, stationary phase bacterial cultures are diluted 500X and allowed to grow in the culture vials until a specific density is reached. At that point, drug is manually added at the desired concentration to both the culture vial and a connected chamber with fresh media. Flow between the media chamber, the culture vial, and a waste vial is managed by a series of computer-controlled peristaltic pumps that maintain constant cell density according to the pictured schematic. The entire system is controlled by custom Matlab software, and up to 18 cultures can be grown simultaneously using a multi-position magnetic stirrer. See also Figure A in S1 Text. B. Examples of a bacterial growth curve (red) and a constant cell-density experiment in which feedback from light scattering is used to maintain a constant cell density (blue). Lower inset: time series showing the status of the inflow/outflow pump, which provides fresh media and removed waste, during constant density (blue) experiment. Because total culture volume remains constant, the pump status time series can be used to calculate per capita growth rate, g, as a function of time when cell density is held constant: g = F/V, where F is the (time dependent) pump flow rate (mL/min) and V is the (constant) culture volume (mL). Upper inset: Calibration plot showing that voltage output from IR detectors is linearly related to optical density. C. Time dependent population growth rate is estimated from the relative pump flow rate F(t), which is the flow rate of the pumps required to maintain cell density (flow rates measured relative to maximum possible flow rate of approximately 1 ml/min). Drug is added at time 0, and following transient growth rate dynamics of approximately 200 minutes, growth rate reaches a steady state that is dependent on cell density. To reduce high-frequency noise, F(t) is estimated with a moving-average filter with window size of 15 minutes. D. To estimate growth rate relative to untreated cells, the steady state growth rate F(t) is averaged in the steady state and normalized by the same measurement in the absence of drug. Upper left inset, full growth curve for E. faecalis in the absence of drug. The densities measured here (0.2≤OD≤0.8) correspond to exponential phase growth, represented by a straight line (red) on a semi-log plot. Upper right inset, growth rate (not normalized) with and without drug. Without drug, growth varies by approximately 6% around the mean over these density ranges.

Fig 2. Cell density modulates the inhibitory effects of multiple antibiotics.

A-I: Steady state population growth rate (relative to untreated cells) as a function of cell density for multiple drug concentrations. Drug concentrations are A. Tigecycline concentration = 15 (green), 25 (blue), 50 (red), 100 (black) ng/mL; B. Spectinomycin concentration = 50 (green), 100 (blue), 150 (red), 400 (black) μg/mL; C. Daptomycin concentration = 1.0 (green), 1.25 (blue), 1.50 (red), 3.0 (black) μg/mL; D. Nitrofurantoin concentration = 50 (green), 100 (blue), 125 (red), 250 (black) μg/mL; E. Ciprofloxacin concentration = 100 (green), 150 (blue), 200 (red), 300 (black), 400 (cyan) ng/mL; F. Linezolid concentration = 0.1 (green), 0.5 (blue), 4 (red), 5 (black) μg/mL; G. Ampicillin concentration = 200 (green), 300 (blue), 400 (red), 500 (black) ng/mL; Note that ampicillin growth does not reach steady state on the time-scale of our experiment, so these measurements are effective growth rates averaged over a non-steady state (Figure A in S1 Text). H. Ceftriaxone concentration = 5 (green), 50 (blue), 200 (red), 300 (black) μg/mL; I. Doxycycline concentration = 33 (green), 100 (blue), 333 (red), 500 (black) ng/mL. Statistically significant differences between growth at lowest and highest densities (0.2 and 0.8), intermediate densities (0.4 and 0.6), or both are indicated by *, **, and ***, respectively. Error bars are +/- 1.96 standard error (95% confidence intervals). See also Figures B, C in S1 Text.

Fig 3. Density dependence of antibiotic inhibition partially due to local pH changes.

A. Top row: Steady state population growth was measured as a function of cell density (here schematically represented by low, medium, and high density) by holding each vial at a constant density while exposing cells to constant drug concentration in highly buffered media. Bottom row: Different culture vials were all held at low-density (OD = 0.2) but grown in BHI supplemented with HCl to achieve pH = 7.5, 6.8, and 6.0, which correspond to pH of steady state cultures held at OD = 0.2, 0.5, and 0.8, respectively. B. Red curves, regular media. Black dashed curves, buffered media. Blue dotted curves, external pH modulation. Tigecycline concentration 50 ug/mL; Ampicillin concentration 200 ng/mL;C. Red curves, regular media. Black dashed curves, buffered media. Ciprofloxacin concentration 200 ng/mL, Spectinomycin concentration 150 ug/mL. Statistically significant differences between growth at lowest and highest densities (0.2 and 0.8), intermediate densities (0.4 and 0.6), or both are indicated by *, **, and ***, respectively. See also Figure D in S1 Text. Error bars are +/- 1.96 standard error (95% confidence intervals).

Fig 4. Density dependent growth modulates time-to-threshold and optimal antibiotic treatment for in-vitro growth model.

A. Time for population to grow from OD = 0.2 to OD = 0.8 (“threshold”) in the presence of tigecycline, relative to time with no drug. Curves represent theoretical predictions with (ε = 0.9, black, solid) and without (ε = 0, red, dashed) density dependence. Points represent experimental measurements in regular media (black) and highly buffered media (red). Inset: comparison of theory (smooth lines) and experimental time series of optical density in regular media (blue) and buffer (red) for [tigecycline] = 1.2 (units of IC₅₀). B. Decrease in final population size when naïve dosing (upper right inset, dashed red line) at initial concentration D₀ is replaced by optimal step-like dosing (upper right inset, dashed blue line). Dashed white line: ε = 0.9, as for tigecycline. Small inset: Fraction decrease in the population as a function of <D> for tigecycline (ε = 0.9). The step-like therapy introduces drug at initial concentration D₀/τ and then sets drug concentration to zero at time t = τT. The parameter τ is chosen to minimize the cell density n(T) at time T, the end of the treatment (0≤τ≤1). In the absence of density dependence, both therapies result in a time-averaged drug concentration $(\langle D\rangle =\frac{1}{T}{\displaystyle {\int }_0^{T}D(t)dt=D_0})$. Upper right inset: drug concentration over time with (“actual”; solid lines) and without (“expected”; dashed lines) density dependence for naïve (red) and step-like (blue) dosing. Lower right inset: final population size (relative to the case with no density dependence) for the naïve treatment (red) and the optimal step-like treatment (blue) as a function of ε. At a given value of ε, density-dependence can significantly increase n(T) (magenta arrow), but the optimal step dosing can often reduce the effects by 50% or more (green arrow).

Fig 5. Density dependence of antibiotic leads to bistable treatment outcomes and potential treatment failure in a pharmacokinetic / pharmakodynamic (PK/PD) model of infection.

A. Main panel: Theoretical (solid and dashed lines) and numerical (shaded region) phase diagrams indicate treatment outcomes in PK/PD model as a function of initial cell density (ranging from 0 to the carrying capacity, C) and initial antibiotic concentration D₀. Solid red lines, stable fixed points of population density (theory). Dashed red lines, unstable fixed points (theory). The curved dashed red line is the phase boundary (separatrix) indicating the critical density above which a population will survive. A region of growth bistability, where treatment can lead to success or failure depending on initial cell density, exists for antibiotic concentrations K₀γ(0) ≤ D₀ ≤ K₀γ(C), where K₀ is the MIC and γ(n) is a nonlinear function that depends on the maximum drug kill rate (g_min), the Hill coefficient (h), the drug decay rate (k_d), and the dosing period T (S1 Text). Shaded regions indicate treatment failure in numerical solutions of the PK/PD model. Upper right inset: numerical solution of PK/PD equations for five different initial densities (indicated by red and black squares on the phase diagram). Lower inset: temporal dynamics of antibiotic concentration. For numerical phase diagram and simulations, g_min = -0.05 and ε = 0.9. Simulations in insets correspond to D₀ = 200 (in units of MIC, K₀). B. Phase diagrams from both theory (solid and dashed lines) and numerical simulations (shaded region) for increasing maximum kill rates (g_min = -0.25, -1, -2 from top to bottom) and populations densities on the order of 10⁸ cells/mL (corresponding to the OD ranges measured here). g_min is measured in units of g_max; biologically, g_max≈1 hr^-1 for bacteria, so one can also view these units as inverse hours. C. Initial dose of antibiotic (units of MIC, K₀) required to clear infections of density OD = 0 (dashed red), OD = 0.4 (blue line), and OD = 0.8 (red line) for different maximum kill rates for the case with no density dependence (ε = 0, left), modest density dependence (ε = 0.5, middle), and strong density dependence (ε = 0.9, right). In all panels, the Hill coefficient h = 2, k_d = ½, and T = 8, corresponding to a treatment period of 8 hours and a natural drug decay rate of ½ hr^-1. Qualitatively similar results are found for other parameters (Figure G of S1 Text).