Trade-offs between drug toxicity and benefit in the multi-antibiotic resistance system underlie optimal growth of E. coli

Background Efflux is a widespread mechanism of reversible drug resistance in bacteria that can be triggered by environmental stressors, including many classes of drugs. While such chemicals when used alone are typically toxic to the cell, they can also induce the efflux of a broad range of agents and may therefore prove beneficial to cells in the presence of multiple stressors. The cellular response to a combination of such chemical stressors may be governed by a trade-off between the fitness costs due to drug toxicity and benefits mediated by inducible systems. Unfortunately, disentangling the cost-benefit interplay using measurements of bacterial growth in response to the competing effects of the drugs is not possible without the support of a theoretical framework. Results Here, we use the well-studied multiple antibiotic resistance (MAR) system in E. coli to experimentally characterize the trade-off between drug toxicity (“cost”) and drug-induced resistance (“benefit”) mediated by efflux pumps. Specifically, we show that the combined effects of a MAR-inducing drug and an antibiotic are governed by a superposition of cost and benefit functions that govern these trade-offs. We find that this superposition holds for all drug concentrations, and it therefore allows us to describe the full dose–response diagram for a drug pair using simpler cost and benefit functions. Moreover, this framework predicts the existence of optimal growth at a non-trivial concentration of inducer. We demonstrate that optimal growth does not coincide with maximum induction of the mar promoter, but instead results from the interplay between drug toxicity and mar induction. Finally, we derived and experimentally validated a general phase diagram highlighting the role of these opposing effects in shaping the interaction between two drugs. Conclusions Our analysis provides a quantitative description of the MAR system and highlights the trade-off between inducible resistance and the toxicity of the inducing agent in a multi-component environment. The results provide a predictive framework for the combined effects of drug toxicity and induction of the MAR system that are usually masked by bulk measurements of bacterial growth. The framework may also be useful for identifying optimal growth conditions in more general systems where combinations of environmental cues contribute to both transient resistance and toxicity.

2 b. Sodium benzoate interacts suppressively with chloramphenicol. To estimate growth rate, stationary phase cell cultures were diluted 1000x at time t 1 and grown in LB media supplemented with various drugs concentrations until time t 2 = 9-12 hours. Based on the dilution factor (1000) and the final optical density, an average growth rate was estimated as where OD t is the optical density at time t. Growth rate contours are determined by cubic spline interpolation of 48 approximately equally separated data points in chloramphenicol-sodium benzoate space. Each data point is a mean of two replicates.    a. Contour plot of growth rate of ΔTolC mutant exposed to salicylate and chloramphenicol (see also phase diagram in Figure 4). 8 b. Contour plot of growth rate of tetracycline mutant exposed to salicylate and chloramphenicol (see also phase diagram in Figure 4).  light blue (0 µg/mL). It is clear that Tet and Cm are much weaker inducers of the mar system than salicylate, which strongly induces the mar system after approximately 100 minutes, even at concentrations (< 2 mM) that only slightly decrease growth (compare to inset, Figure 2b). Error bars represent sample standard deviations from 6 independent trials.

Tetraclycine Resistant Mutants
To select for tetracycline-resistance MAR mutants, we grew 96 individual 150 µl cultures of wild type cells in a high concentration (1 µg/mL) of tetracycline. After 48 h, we randomly chose a culture that grew to stationary phase and also exhibited increased mar promoter activity (as measured by YFP expression) relative to wild type cells. We subsequently isolated a single mutant, here called a Tet-mutant, by selecting one colony from the culture.
While this randomly selected mutant with constitutive mar promoter activity eliminated suppression between salicylate and chloramphenicol (Figure 4b), it was not clear how commonly similar mutations affecting the MAR system arise following antibiotic exposure. However, it is well-known that mutations related to the MAR system can be selected by tetracycline 1,2 and such mutations have also been isolated from clinical samples 3,4,5 . To verify the prevalence of mar promoter activity in our selection experiments, we measured the growth and mar promoter activity of the remaining 95 cultures grown in 1 µg/mL tetracycline. After 48 h, 40 of the cultures had reached stationary phase (Fig. S6a). These cultures presumably contained the resistant mutants which would dominate a large culture, and over half (22/35) showed substantially 12 increased fluorescence, corresponding to high mar promoter activity (Fig. S6b). To verify the significance of mutations affecting the MAR system in larger cultures and characterize the cross resistance of such mutants to chloramphenicol, we grew 3 mL cultures of wild-type cells for 48 h in various concentrations of tetracycline up to approximately 3 times the MIC. Adapted cells grown in tetracycline concentrations greater than 0.5 µg/mL grew at the same rate as wild type cells and developed cross resistance to chloramphenicol (Fig. S6c). In addition, these resistant cells showed high levels of mar promoter activity (Fig. S6d). Thus, the MAR system was a common target of resistance-conferring mutations in cells exposed to tetracycline. This result suggests that, similar to the Tet mutant, cells grown in high levels of tetracycline for several days adapt to exploit the resistance conferred by the MAR system without the associated cost and toxicity of an inducing drug.

Measuring mar Promoter Activity
To monitor mar promoter activity, we used the YFP reporter plasmid pZS*2 MAR-venus 6-8 , which contains the mar promoter as well as Kan r and a SC101* ori that maintains the copy number at 3-4 copies/cell.
Mar promoter activity was determined by first correcting raw YFP fluorescence by subtracting a background fluorescence curve (fluorescence vs. absorbance) obtained from untreated cells.
Temporal profiles of mar background-corrected fluorescence concentration (fluorescence/absorbance) were generated from means of two replicates (Fig. 2). Mar promoter activity was taken to be the background-corrected fluorescence concentration (fluorescence/absorbance), averaged over steady state, times the growth rate k. Fluorescence concentration alone is a sufficient measure of relative promoter activity in strains with similar growth rates.

Drug degradation model for inducible benefit
To determine a functional form for A eff , we assume that the internal antibiotic concentration a is governed by where A is the external concentration of antibiotic, k 1 is the rate constant governing passive influx of drug into the cell, k 2 is the rate constant governing drug degradation and/or efflux in the absence of inducer, and Δk 2 captures the change in drug degradation activity imparted by the presence of inducer. By definition, Δk 2 = 0 in the absence of inducer. Equation S1 assumes that dilution from cell growth is slow on the timescale of efflux pumping and can therefore be neglected. In the steady state, the internal concentration a is a function of Δk 2 , a ss !k 2 suggests that we define an effective antibiotic concentration A eff as With this definition, equation S2 simplifies to a ss !k 2 Since Δk 2 /(k 1 +k 2 ) is assumed to contain the entire dependence on inducer concentration S, we can generalize S3 by writing where β(S) is defined as the inducible benefit. While the functional form of β(S) can, in general, be arbitrarily complex, physical arguments suggest that inducible benefit will be a saturating function of S. In the case of the MAR system, we experimentally verify that the model can quantitatively describe several multi-drug combinations if β(S) is taken to be proportional to the normalized activity of the mar promoter, with β max a scaling constant equal to the asymptotic value of β as S à ∞ (Figure 3, Figure S2). Equivalently, we are assuming that the increase in efflux rate Δk 2 is proportional to the relative mar promoter activity. Such proportionality would be expected, for example, if the mar promoter activity was proportional to the number of efflux pumps synthesized in response to inducer. More generally, the form S5 captures the notion that increasing inducible benefit β(s) decreases the effective concentration A eff .
The model S1 is not critical to our overall hypothesis, but we nevertheless note some of its limitations. First, we do not account for dilution of intracellular antibiotic by cell growth.
Second, we assume that the internal antibiotic concentration can be used to approximate the internal "free" (unbound) antibiotic concentration. We make the preceding two assumptions to reduce the number of parameters and simplify the interpretation of our experiments, but we cannot rule out more complex behavior in other experimental regimes. Relaxing these two assumptions gives rise to a much more complex situation. For example, in cases where cell permeability (passive influx) is very low, there is the possibility of bistable growth. While we never observed any experimental evidence of such bistability, this possibility has been considered in more detailed theoretical models 9 . Our rescaling model is a specific instance of this more general model that includes bistability.

Derivation of general phase diagram
To derive a phase diagram, we begin with the definition of Lowe additivity of two drugs, which says 5 where S δ and A δ are the concentrations of drug 1 and 2, respectively, in a mixture that results in a fractional growth inhibition δ. Similarly, S 0,δ and A 0,δ are the concentrations of drugs 1 and 2 alone that result in a growth inhibition δ. For simplicity, in what follows we take δ = 1/2; that is, we define the drug interaction based on the contour line in drug concentration space defined by 50% growth inhibition. In this case, S 0,δ and A 0,δ reduce to K 1 and K 2 , respectively, which are the binding constants characterizing the single drug cost functions. Deviations from this additivity result in synergy (left hand side of Equation S6 < 1) or antagonism (left hand side of Equation S6 > 1).
The contour separating drug synergy from antagonism--that is, the contour of additivity--can be found in the (S, β max ) space by setting κ = 1/2 in Equation 3, using Equations 1,2, 4 and 5 to solve for A 1/2 , and then plugging into Equation S6. The contour separating synergy from antagonism is then given by While the shape of the phase boundary will, in general, depend on the specific parameters, it is straightforward to show that lim S!0 meaning that the phase boundary intersects the vertical axis at a value of β max proportional to K ind /K 1 . In general, for a given value of S, increasing β max beyond a threshold given by S7 will lead to a transition from synergistic to antagonistic interactions. Interestingly, there is a range of β max values for which the nature of the drug interaction depends on S (Fig. S3).
Drug suppression is an extreme form of antagonism where the effect of two drugs is less than that of one drug alone. In our simple model with drug 1 chosen to be an inducing drug, suppression will arise when a maximum exists at S=S* in the the growth contour in drug concentration space and, additionally, A(S*) > A(0). Using Equations 1-5, it is straightforward to show that the slope ∂ A ∂ S characterizing the contour of constant growth (50%) in drug concentration space is a monotonically decreasing function of S. Furthermore, it is clear that the contour A(S) approaches zero at S = K 1 , the MIC. In order for a maximum in the contour A(S) to exist, it is therefore necessary and sufficient that lim S!0 Any maximum will have A(S*) > A(0). We therefore have the following additional condition for drug suppression: Generally, the nature of a drug interaction is determined by a balance between the cost of the physiological response, which determines the phase boundaries, and the benefit conferred by this response, which is governed by β max . Specifically, the phase boundary separating antagonism from suppression depends on the ratio K ind /K 1 , where the constant K 1 characterizes the inducer cost and K ind the induction of physiological components which potentially provide benefit. The ratio K ind /K 1 therefore measures the cost of inducing beneficial elements in response to drug 1.
In addition, the phase boundary for suppression decreases with increasing n, the Hill coefficient governing the steepness of the antibiotic cost function. For large n>>1, the antibiotic cost function approaches a step function. Therefore, even a slight shift in drug concentration can result in a significant benefit, as the cost drops abruptly from 1 to 0 as concentration is decreased across the threshold value K 2 . As a result, the onset of suppression requires only a nonzero β max .
However, in practice, n is typically on the order of 1, so the phase boundary is not significantly dependent on n. Two examples of phase diagrams for different values of n are shown in Figure   S3. While the phase boundary separating synergy from antagonism depends on n, in all cases increasing β max at a given concentration S leads to increasingly antagonistic and eventually suppressive behavior.