Research ArticleInfectious Disease

Classic reaction kinetics can explain complex patterns of antibiotic action

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Science Translational Medicine  13 May 2015:
Vol. 7, Issue 287, pp. 287ra73
DOI: 10.1126/scitranslmed.aaa8760
  • Fig. 1. Post-antibiotic effect observed with single-cell microscopy and model fit.

    (A and B) Time-lapse microscopy of single E. coli cells observed (imaging rate: one in 5 min) during and after a 16-hour exposure to two concentrations: 22 cells at a tetracycline dose of 6.25 mg/liter (0.4 MIC) (A) and 27 cells at a tetracycline dose of 12.5 mg/liter (0.8 MIC) (B), corresponding to 0.4 and 0.8 MIC, respectively. The solid black line indicates the mean growth rate of the observed bacterial population, and the dotted blue lines indicate the minimal and maximal observed growth rate. The model used to fit the data is described in Eq. 2 in Supplementary Materials and Methods. Kinetic parameters are derived from literature cited in table S2.

  • Fig. 2. Influence of bacterial density on antibiotic efficacy.

    These graphs show the relative bacterial population size over 12 hours with different tetracycline concentrations as compared to growth in the absence of antibiotics. (A) E. coli were grown in microtiter plates with different initial densities and exposed to different tetracycline concentrations (the MIC is 2 mg/liter); the graph shows relative population size in the presence versus in the absence of antibiotic. The average of three independent experiments is shown. CFU, colony-forming units. (B) Model prediction using parameters from single cells. (C) Predictive power of model. The y axis shows the average bacterial density over 12 hours as observed by turbidity measurements in microtiter plates. Bacterial cultures with different initial densities were exposed to varying concentrations of tetracycline (A). The x axis shows the theoretical prediction derived from our model parameterized with known binding data and the parameter estimates from fitting the model to single-cell data (Fig. 1, A and B). (D) Correlation between strength of inoculum effect (efficacy loss/log10 bacteria) and drug-target affinity (half-maximal target binding) in E. coli. For details, see Supplementary Materials and Methods, “Quantification of inoculum effect.” The experimental setup was the same as in (A). (E) Same as in (D) for Vibrio cholerae. Amp, ampicillin; Gen, gentamicin; Nal, nalidixic acid; Str, streptomycin; Tet, tetracycline; Cip, ciprofloxacin; Pen, penicillin; Rif, rifampicin.

  • Fig. 3. Model of persistence as tolerant subpopulations with phenotypic switch compared to alternative model.

    (A) This graph shows numerical simulations of the mathematical model proposed by Balaban et al. (23). In brief, this model assumes that there is a majority population of susceptible (“normal”) bacteria, n, and a subpopulation of persistent bacteria, p. According to the data in (23) (represented by red line), the majority population declines quickly with a rate μn of ~0.4 orders of magnitude per hour (μn = −1.84 hour−1) at an ampicillin dose of 100 mg/liter, the persister population is not affected by the antibiotic (μp = 0), and bacteria switch from a persistent to a normal state with a rate b = 0.07 hour−1. This can be described with the following mathematical model as presented in (23):Embedded Image10Embedded ImageThe black line represents an antibiotic activity that is 50% lower than 100 mg/liter; the green line indicates an antibiotic activity that is twice higher. (B) “Classical model” of persistence with a majority nonpersister population and a minority persister population. On the x axis, the antibiotic susceptibility of individual cells is given relative to the mean of the nonpersister population, and the y axis shows the number of bacteria with that specific susceptibility (total population size, 107). The relative proportion of persisters under this assumption is usually assumed to be several orders of magnitude lower than the majority population. For illustration purposes only, we adopt a persister frequency of 10−2 (the real frequency is much lower and would not be visible in this figure).

  • Fig. 4. A normal distribution of target molecules can lead to multiphasic bacterial killing.

    (A) Histogram of intracellular ribosomal concentration as experimentally determined in E. coli. Raw data were obtained from (42). Normal distribution was not rejected by Shapiro test, and the red line shows a normal distribution with mean and SD obtained from the data. (B) Predicted killing over time for bacterial population measured in (A). Kinetic rates of streptomycin binding to ribosomes were taken from the literature (table S3). Time of killing was calculated using Eq. 4. The colors indicate antibiotic concentrations relative to the MIC of E. coli MG1655. (C) Theoretical explanation of multiphasic kill curves. The dotted lines show the calculated equilibrium number of bound ribosomes for three different streptomycin concentrations. The solid lines show the time until a specific fraction of the target molecules is bound (Eq. 4, Materials and Methods). Black: Assumed killing threshold with a mean of 60% and variance of 10%. (D and E) These graphs show time-kill curves (E) resulting from a bimodal distribution of susceptibility (D, compare to fig. S6). The MIC was calculated as the drug concentration that achieves binding of 99% of the cells at equilibrium (Eq. 5). The vertical lines in (D) indicate the equilibria at given multiples of the MIC (green, 1× MIC; yellow, 4× MIC; orange, 8× MIC; red, 32× MIC). The same colors are used in the resulting calculated time-kill curves of a simulated population of 105 bacteria in (E). Kinetic rates are the same as in (A) to (C). The main “peak” of the bimodal distribution in (D) corresponds to the normal distribution in fig. S6A.

  • Fig. 5. Modeling reproduces concentration-dependent persistence.

    (A) Experimental time-kill curves of E. coli with different ciprofloxacin concentrations. The MIC is 8 mg/liter (table S1). The experiment was performed in biologically independent triplicates. (B) Numerical simulations of a model combining bacteriostatic and bacteriocidal effects (Eqs. 7 to 10) parameterized with in vitro kinetic parameters for ciprofloxacin binding. In the absence of known distributions of functional gyrase tetramers, we assume that the mean of this distribution is the average of the numbers given in the literature as cited in table S3. We assume a normal distribution in the absence of more detailed information (60, 61) and that the SD is the same as published for ribosomes.

  • Fig. 6. Concentration-dependent persistence in tuberculosis patients receiving increasing rifampicin dosages.

    (A to C) CFU counts of Mycobacterium tuberculosis in sputum during treatment as reported in (51). Patients with previously untreated, smear-positive pulmonary tuberculosis due to drug-sensitive strains received 14 days of initial monotherapy and then a standard multidrug regimen. Sputum was collected overnight on two consecutive nights before treatment was started and then after 2, 4, 6, 8, 10, 12, and 14 days of the allocated regimen. Patients received rifampicin at doses of (A) 5 mg/kg, (B) 10 mg/kg (rifampicin dosage in standard regimen), and (C) 20 mg/kg. (D) Calculated time-kill curves within patients (simulated population of 104 bacteria, Eqs. 4 and 5). The effective drug concentration in extracellular lining fluid was obtained from (52). The MIC for rifampicin in drug-susceptible strains was assumed to be 0.1 mg/liter (62). The available drug concentration was assumed to follow a normal distribution with a 22.5% SD (53).

  • Fig. 7. Model of isoniazid (INH) action and resulting kill curves.

    (A) INH forms a NAD (nicotinamide adenine dinucleotide) adduct (red spheres) with a rate, which depends on the enzyme KatG and O2 saturation. INH-NAD then reacts with its target InhA (blue crescents). (B) Time-kill curves resulting from calculating the expected time until 60% (with 10% SD) of the target molecules are bound (simulated population of 106 bacteria; Eq. 6; Materials and Methods). Black indicates the same intracellular oxygen content as in the cell-free experimental setup in (63), red 50% O2 saturation, and green 25% O2 saturation. (C) Time-kill curves for variance in intracellularly available KatG by 30% simulated population of 104 bacteria. All other parameters from (63). Black indicates the MIC for M. tuberculosis (28), red 2× MIC, and green 3× MIC.

  • Fig. 8. Schematic of the model.

    (A and B) Model overview. A bacterial cell with target molecules, T (in this case, ribosomes, blue crescents), and antibiotic molecules, A (red circles), is shown. Antibiotics must diffuse through the bacterial cell envelope with a diffusion constant, p, to bind their targets with an association rate, kf, to form a complex, AT, which may dissociate with a rate, kr (0 for irreversible reactions). (C) Different mechanisms can lead to bacterial killing. Bacterial cells are shown as black oval, target molecules are shown as blue crescents, and antibiotic molecules are shown as red circles. (D) Chemical reactions of bacterial cells. For an explanation of the parameters, see main text. The color of the reaction arrow is used to highlight the corresponding equations in Eq. 7 in the Supplementary Materials and Methods.

  • Table 1. Overview of phenomena with previous explanations and explanations suggested here.
    PhenomenonCurrent explanationsAlternative explanation
    Post-antibiotic effectBacterial stressExtended target occupancy due to transmembrane diffusion
    and unspecifically bound reservoir
    Inoculum effectDensity-dependent growth stages, quorum-sensingReduced ratio of antibiotics/target at higher bacterial densities
    Concentration-
    dependent
    persistence
    Exposure to antibiotics at sub-MIC concentrations
    stimulates generation of persisters
    Small variance in drug-target binding
    Concentration-
    independent
    persistence
    Phenotypic switch between persistent and
    “normal” populations
    Small variance in processes up- or downstream of drug-target
    binding

Supplementary Materials

  • www.sciencetranslationalmedicine.org/cgi/content/full/7/287/287ra73/DC1

    Materials and Methods

    Fig. S1. Analysis of single-cell time-lapse microscopy data E. coli MG1655 cells exposed to tetracycline.

    Fig. S2. Baseline growth rate affects length of post-antibiotic growth suppression.

    Fig. S3. Time-kill curves of E. coli CAB1 exposed to five different classes of antibiotics in various concentrations.

    Fig. S4. Analysis of single-cell time-lapse microscopy data for E. coli MG1655 cells exposed to streptomycin.

    Fig. S5. Simulated time-kill curves under different assumptions.

    Fig. S6. Distribution of drug susceptibility and expected time-kill curve.

    Fig. S7. Effect of variance in target molecule content.

    Fig. S8. Overview of kill mechanisms, chemical reactions, and compartmental models used in this study.

    Table S1. Ciprofloxacin persisting E. coli have not acquired higher resistance to the antibiotic.

    Table S2. Antibiotic concentrations used for experiments.

    Table S3. Kinetic parameters for different antibiotics.

    Table S4. Explanation of variables and parameters.

    Movie S1. Single-cell microscopy data for E. coli MG1655 cells exposed to tetracycline (12.5 mg/liter) (0.8× MIC).

    Movie S2. Single-cell microscopy data for E. coli MG1655 cells exposed to streptomycin (25 mg/liter) (2× MIC).

    References (6482)

  • Supplementary Material for:

    Classic reaction kinetics can explain complex patterns of antibiotic action

    Pia Abel zur Wiesch,* Sören Abel, Spyridon Gkotzis, Paolo Ocampo, Jan Engelstädter, Trevor Hinkley, Carsten Magnus, Matthew K. Waldor, Klas Udekwu, Ted Cohen

    *Corresponding author. E-mail: pzw{at}daad-alumni.de

    Published 13 May 2015, Sci. Transl. Med. 7, 287ra73 (2015)
    DOI: 10.1126/scitranslmed.aaa8760

    This PDF file includes:

    • Materials and Methods
    • Fig. S1. Analysis of single-cell time-lapse microscopy data E. coli MG1655 cells exposed to tetracycline.
    • Fig. S2. Baseline growth rate affects length of post-antibiotic growth suppression.
    • Fig. S3. Time-kill curves of E. coli CAB1 exposed to five different classes of antibiotics in various concentrations.
    • Fig. S4. Analysis of single-cell time-lapse microscopy data for E. coli MG1655 cells exposed to streptomycin.
    • Fig. S5. Simulated time-kill curves under different assumptions.
    • Fig. S6. Distribution of drug susceptibility and expected time-kill curve.
    • Fig. S7. Effect of variance in target molecule content.
    • Fig. S8. Overview of kill mechanisms, chemical reactions, and compartmental models used in this study.
    • Table S1. Ciprofloxacin persisting E. coli have not acquired higher resistance to the antibiotic.
    • Table S2. Antibiotic concentrations used for experiments.
    • Table S3. Kinetic parameters for different antibiotics.
    • Table S4. Explanation of variables and parameters.
    • Legends for movies S1 and S2
    • References (6482)

    [Download PDF]

    Other Supplementary Material for this manuscript includes the following:

    • Movie S1 (.avi format). Single-cell microscopy data for E. coli MG1655 cells exposed to tetracycline (12.5 mg/liter) (0.8× MIC).
    • Movie S2 (.avi format). Single-cell microscopy data for E. coli MG1655 cells exposed to streptomycin (25 mg/liter) (2× MIC).

    [Download Movies S1 and S2]

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