Applying ecological resistance and resilience to dissect bacterial antibiotic responses

Applying an ecological framework to bacterial antibiotic responses improves understanding of population recovery.


Model development Section S
We model the dynamics of a system of 4 variables: the cell density (N ), the nutrient concentration (S), the antibiotic concentration (A) and the external Bla concentration (B).
1. Left: A cell, plus an equal mass of nutrients, produces another cell with a Monod dynamics. Right: Cells lyse at a rate L (which depends on A and S), releasing a quantity B in of Bla and a proportion of nutrients ξ.
2. The antibiotic and Bla decay respectively at rates d A and d B .

Enzymatic degradation of the antibiotic by Bla (outside the cells).
Units In order to derive differential equations straightforwardly from this reaction network, adequate units need to be chosen for the state variables and parameters. We will count N and S in the same unit, number of (equivalent) bacteria per litre, and A and B in moles per litre. In the following, as a visual aid, we will note |X| all quantities in the unit of N , and [X] all quantities in the unit of A A quasi-steady-state approximation on the transitory complex BA yields . Replacing this expression into the system gives Dimensionless model In order to make the model more suitable for simulations, and factor out some unidentifiable parameters, we made it dimensionless by the following change of variables Resulting in this dimensionless model Heterogeneous model Heterogeneity was introduced into the model by dividing the population into subpopulations (i) with unique growth rates selected from a normal distribution of values (π) and equal starting densities Parameter values and typical ranges Initial values In all the experiments, a 3-hour delay is observed before the crash becomes visible (see S. Fig. 3 for example). We didn't model this delay, but assumed that the growth was nominal during this initial phase, and started all the simulations from the state reached after 3 hours of normal exponential growth. Because of collective antibiotic tolerance, to conserve a potent antibiotic on a higher cell density, we had to underestimate the efficiency of the antibiotic hydrolysis by Bla (κ b ). This is indeed the parameter that deviates the most from the literature values. Analytical expression of resistance It is possible to derive from this model an analytical expression for the resistance. Indeed, the resistance is defined as , where ρ A is the maximum net lysis rate observed, and ρ 0 is the maximum growth rate observed without antibiotic. The maximum lysis rate is always observed at the introduction of the antibiotic (after the 3-hour delay), when the antibiotic concentration in the medium is still unaltered. The net lysis rate is A simple analytical expression does not seem to exist for the resilience.

Sensitivity analysis
Sobol sensitivity analysis was used to determine which parameters influenced resistance and resilience under a range of antibiotic concentrations (see the tutorial by Zhang [22] for a good explanation). Briefly, the total-order sensitivity index, ST, reflects how much a parameter contributes to the variation of resistance or resilience, alone or in coordination with any number of others. The sensitivity analysis was performed on the dimensionless model, on a parameter space spanning a factor of 4 centered around each parameter's default value (x ∈ [ x0 2 , 2x 0 ]). The sampling of the parameter space and analysis of the results was done with the open-source Python library SALib[43].

Resistance
The main effector of resistance is the maximum lysis rate γ, which is given a total-effect index of 1 at 1, 10 and 100 µg/mL.
From the expression derived earlier 1, resistance will be 1, regardless of parameter variation. When a 1, resistance becomes dependent on γ as resistance approaches 1−γ. When a ≈ 1, h should have a minor influence, which is shadowed by γ's. These parameters are characteristic of single cell level behavior, suggesting that resistance is a single cell level trait.

Section S 2.2 Resilience
Results from the sensitivity analysis for resilience revealed that all parameters contributed to varying degrees, depending on the antibiotic concentration. At low concentrations of antibiotic, the most influential parameters are the maximum lysis rate (ST γ = 0.81 ± 0.02) and the catalytic activity of Bla (ST κ b = 0.145 ± 0.005). At higher concentrations, γ and κ b 's total-effect indexes persist as the two mainly influential parameters, with the addition of the decay rate of Bla which appears at ST d b = 0.040 ± 0.002 at A = 10 µg/mL and at ST d b = 0.053 ± 0.003 at A = 100 µg/mL. Although γ is the main effector, κ b and d b are related to population-level behaviors, suggesting that resilience could be a population-level trait. . Collective antibiotic tolerance. The population survival to antibiotic exposure depends on the initial cell density. The starting density was diluted from 500 to 10000 times and then exposed to a range of cefotaxime doses. The time courses show that increasing the antibiotic concentration resulted in a longer recovery time because more cells were lysed. If the cell density is too low, then there will be too few cells to produce the amount of Bla needed to degrade the antibiotic and allow the population to recover. e. (a-d) Four ESBLproducing isolates were exposed to 50 µg/mL cefotaxime in combination with exogenous Bla ranging from 1 to 100 µg/mL. Increasing Bla concentrations reduced the recovery time and increased resilience (e) for all isolates in a dosedependent fashion. (f) Sensitive (MG1655) cells were exposed to 100 µg/mL of carbenicillin in combination with exogenous Bla ranging from 1 to 20 µg/mL. Increasing Bla concentrations reduced the recovery time resilience (g).

Fig. S3
(a) Time courses were measured for a range of antibiotic concentrations (added at time = 0). At low concentrations (A < 1.5 µg/mL), the growth curves looked very similar to the control (A = 0) and are considered to be "resistant" to these conditions. At A = 1.5 µg/mL, the growth curve starts to deviate from the control due to an increase in cell lysis, showing a decrease in resistance. Once the concentration was high enough (A ≥ 5 µg/mL), the population started displaying the crash and recovery dynamics. The time to half maximum density (T 50% , blue circle) was determined for each concentration and used to calculate resilience. (b) Resistance was calculated from the net growth rate. ρ A is calculated as the point of maximum deviation from the untreated population, as indicated by the black crosses. The initial 3 hours were not included because the ratio of noise to actual measurement was too high. (c) The time courses of two colonies (solid and dashed lines) are compared for a selection of cefotaxime concentration to demonstrate reproducibility. Each curve represents the average of four technical replicates from the same colony. . Time courses and rate of change curves generated by the f Fig. S5 Once the cefotaxime concentration was strong enough to induce significant lysis (1 µg/mL cefotaxime), the amount of Bla activity observed in the supernatant exceeds that from the whole cells. Even if the cells are sonicated, the Bla released from the periplasm is about an order of magnitude less active than the Bla present in the supernatant.
. Quantifying Bla activity in different components of culture.

Fig. S6
and resilience. (a) Resistance is most sensitive to the maximum lysis rate (γ) for all antibiotic concentrations tested. (b) Resilience is affected by numerous parameters. Regardless of antibiotic concentration, lysis rate has the greatest impact followed by Bla activity level (κ b ). As antibiotic concentration increases, the turnover rate of Bla (d b ) and nutrient recycling (ξ) become more influential and the Hill coefficient (h) becomes less influential. Of interest are the parameters affecting Bla activity, κ b and d b , because they can be targeted clinically by a Bla inhibitor.
. Sensitivity analysis reveals parameters ecting resistance aff Fig. S7. Time courses showing the e ects of Bla inhibition. Isolate I was exposed to the same range of cefotaxime (0 µg/mL to 300 µg/mL) in combination with clavulanic acid ranging from 0 µg/mL to 0.5 µg/mL. With increasing clavulanic acid concentrations, the isolate became increasingly sensitive to the cefotaxime as more Bla activity was inhibited. . .  Adjust the OD to 0.5, dilute the culture 1000x, and add the different concentrations of antibiotic. Set up a 96-well plate and insert into a microplate reader that collects OD measurements every 10 minutes. MATLAB was used to sort and plot the time courses. (b) Antibiotic degradation assay: A single colony of sensitive (MG1655) and ESBL-producing cells (Isolate I) were inoculated in 2 mL of M9 and incubated for 12 hours at 30 • C. Each culture was adjusted to 0.5 OD and diluted 1000x before cefotaxime was added for a final concentration of 10 or 100 µg/mL. The cultures were incubated for 6 hours at 30 • C before centrifuging to separate the supernatant. Sensitive cells were spread on fresh agar plates. Then 5 µL of the supernatant from each culture was dropped into the middle of separate plates spread with sensitive cells. The plates were then incubated for 16 hours at 37 • C. (c) Methods for re-exposing survivors to test for selection: The methods were the same as described in ig. S8a. After the first time course was completed, cells that had grown back were diluted 10x in fresh M9, grown for 3 hours at 37 • C, before repeating the methods for setting up a time course experiment. f