Center for Systems Biology, Department of Molecular and Cellular Biology, and School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, 02138, USA

Abstract

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

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.

Background

The resistance of bacteria to antibiotics has prompted intense scientific research in the last several decades because it directly underlies the clinical treatment of infections

To address these questions, here we study inducible resistance mediated by the MAR (multiple antibiotic resistance) system. The MAR system, present in many bacterial species, consists of an operon that confers efflux-mediated

Results

Experimental characterization of salicylate-induced antibiotic resistance

To study the trade-off between drug toxicity and induced resistance, we first measured the effects of salicylate and two protein synthesis inhibitors, chloramphenicol and tetracycline, on cell growth. As expected, the effect of each drug alone is to slow cell growth as concentration increases. To quantitatively characterize this effect, we define growth cost as the reduction in growth rate of cells treated with one drug relative to the growth rate of untreated cells. In general, we find that cost functions are well-described by Hill functions with K_{i}, the concentration of drug i at which cost is half maximal, and n, the Hill coefficient (Figure

**Supplemental Figures, Supplemental Methods, and Supplemental Notes [**
**]. **

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Single and Multiple Drug Growth Costs

** Single and Multiple Drug Growth Costs.****a**. Growth cost is defined as the reduction in growth rate of cells treated with a drug relative to the growth rate of untreated cells. Solid lines, best fits to Hill functions h(x) = x^{n}/(K_{i}^{n} + x^{n}), with K_{i} equal to the concentration of drug i (i = S for salicylate, i = A for both antibiotics) at which growth is inhibited by 50%. K_{A} = 1.80 (1.66, 1.93) μg/mL, n = 1.97 (1.67, 2.26) for chloramphenicol; K_{A} = 0.41 (0.37, 0.45) μg/mL, n = 1.90 (1.61, 2.19) for tetracycline; and K_{S} = 6.06 (5.50, 6.62) mM, n = 1 for salicylate. 95% confidence intervals from nonlinear least squares fitting in parentheses. Error bars are standard deviation of four replicates. **b**. In the presence of high concentrations of chloramphenicol (Cm), growth is maximal for a nonzero concentration of salicylate (upper insets: OD time series). Contours of constant growth in concentration space indicate suppression. Dashed line, contours of 45% and 55% maximum growth.

Phenomenological cost-benefit model for MAR-induced drug resistance

To quantitatively model the interplay between inducer cost and benefit in a two-drug environment that includes an inducer, we assume that the effects of the two drugs, in the absence of MAR induction, are independent in the sense that the relative growth rate of cells in the presence of both drugs (g_{SA}) equals the product of individual relative growth rates of cells in the presence of each drug alone (g_{SA} = g_{S} g_{A}). This assumption, known as Bliss independence

Here, we extend the concept of Bliss independence to include drug interactions mediated by the induction of the MAR system. Specifically, we assume that the presence of one drug (S) provides a fitness benefit by reducing the effective concentration of the second drug (A), thereby coupling the effects of the drugs. We incorporate this rescaling and re-write the model in terms of the combined growth cost, C_{SA} (by definition 1- g_{SA}) to arrive at

where f(A) and g(S) are the _{eff}. Therefore, the approximate additivity of drug costs implied by Bliss independence is modified to an approximate additivity of _{eff}. However, because the antibiotics used are poor inducers of the MAR system (Additional file

Cost-Benefit Model for Interactions between MAR inducers and Antibiotics

** Cost-Benefit Model for Interactions between MAR inducers and Antibiotics.****a**. The combined growth cost of two drugs (S and A) is assumed to be Bliss independent, as long as the effective concentration of one drug is reduced by the presence of the second drug. Thus, the combined cost of both drugs is approximately equal to the sum of effective growth costs, f and g, for each individual drug. The reduction of each drug's growth cost depends on the concentration of the other drug, which in turn dictates the growth benefit provided. **b**.

A simple model, which assumes that the concentration of antibiotic A is reduced by the induction of efflux pumps (Additional file 1), suggests a form for A_{eff}:

We call the function β(S) the _{max}, so that _{ind} = 0.8 is the concentration of inducer that yields half maximal promoter activity (Figure _{max} links

The cost-benefit theory assumes that the effects of the inducing drug and the antibiotic are independent, up to a reduction in the concentration of the antibiotic. The aim of our model is not to achieve a microscopic theoretical description of the system, but rather to provide a minimal phenomenological model that quantitatively captures the measured behavior. The success or failure of the model must therefore be determined experimentally. Specifically, our model does not attempt to elucidate the microscopic variables governing the multi-drug effects—which would require dozens, if not hundreds, of microscopic parameters—but rather posits that a simple relationship should exist between cell growth in the presence of one drug (which is a function of the cell’s internal state) and cell growth in the presence of two drugs (which is a function of an entirely different intracellular state). Specifically, the model requires equality between cell growth in the presence of A and cell growth in the presence of the combination S and A' (with A'> A), once we account for the costs of drug S. To directly verify this hypothesis, we can rearrange equation 1 to express this concept as

The left hand side represents the "adjusted" multi-drug cost, once the toxic effects of S (cost) have been removed. The model implies that this adjusted cost of the drug combination, as a function of A_{eff}, is functionally equivalent to the cost function f(x) for a single drug. Hence, if one removes from the multi-drug costs (C_{SA}) the effects of g(S) according to the left-hand side of Equation 3 and then properly chooses the concentration reduction A→A_{eff} (determined by the single parameter β_{max}), all two-drug data from a given drug combination should collapse to the single curve determined by f(x). Note that because a Hill function describes costs for many individual drugs as well as slices through many drug combination effect surfaces_{max}. Once this parameter is determined, the beneficial effects of different concentrations of salicylate are linked by the _{max}, using two-drug cost curves for salicylate and chloramphenicol (β_{max} = 1.15 +/− 0.15) and salicylate and tetracycline (β_{max} = 6.04 +/− 0.16). In all cases, the model provides an excellent fit to the data (R^{2} = 0.98; see Additional file

Cost-Benefit Analysis Quantitatively Describes Multi-Drug Growth Cost Functions

** Cost-Benefit Analysis Quantitatively Describes Multi-Drug Growth Cost Functions.****a**. Multi-drug growth cost curves for salicylate and tetracycline (top) and salicylate and chloramphenicol (bottom) are increasing functions of antibiotic concentration, but also depend on the concentration of salicylate. Each growth cost curve represents growth in increasing concentrations of antibiotic at a single salicylate concentration, ranging from 0 to 7 mM. Black lines, ([salicylate] = 0 mM), which correspond to the single drug growth cost functions. Data points, means of four replicates. Error bars, +/− one sample standard deviation. **b**. The cost-benefit model requires rescaled versions of the multi-drug cost functions to have the same mathematical form as single drug cost functions (Equation 3). To directly test this assumption, we rescaled the multi-drug cost functions in panel **a** by removing the contribution from salicylate toxicity (cost) to each multi-drug growth cost function. Then, we rescaled the concentration of each antibiotic (chloramphenicol or tetracycline) using the single parameter β_{max}, which accounts for the salicylate-induced reduction of intracellular antibiotic concentration (benefit). We trivially exploit the common Hill form of the single drug costs functions to show data from both drug pairs on a single plot, achieved by replacing drug concentration with effective toxicity, defined as ([A]/K_{A})^{n}, where K_{A} and n characterize the single drug growth cost function for the antibiotics (see Figure _{max}) specific to the two antibiotics. See also Additional file

Transition from salicylate as toxic to salicylate as beneficial

Our results uncover an interesting range of cellular behavior that emerges from the trade-offs of salicylate toxicity and the simultaneous induction of multi-drug resistance. First, for each concentration of antibiotic, we find that growth is maximal at a single salicylate concentration S*. In addition, we see an apparent transition between two regimes—one where the presence of the inducing drug is harmful, and another where it is beneficial—as A eclipses a threshold A_{crit} (Figure _{crit}. In other words, the cell benefits from the presence of salicylate, but only when the antibiotic environment is sufficiently toxic to offset the inherent toxicity of the inducer. Interestingly, we find that

Phase Diagram for Salicylate-Chloramphenicol Interactions

** Phase Diagram for Salicylate-Chloramphenicol Interactions.****a**. The concentration of salicylate, S* (in mM), at which growth is maximal depends on the external concentration of antibiotic. A transition occurs between the low antibiotic regime, where the presence of salicylate is harmful, to the high antibiotic where the presence of salicylate increases the growth of cells. Red, tetracylcline; Blue, chloramphenicol. Concentrations for both drugs measured in μg/mL. Curves, numerical calculation from model; circles, estimates of S* from experiments at single concentrations of A. Maxima were estimated using cubic spline interpolation between data points at different concentrations of salcilyate. Error bars, concentrations corresponding to +/− 1% of maximum. **b**. Apparent MIC, defined as the minimum concentration of antibiotic (red, tetracycline; blue, chloramphenicol; both in units of μg/mL) at which growth is reduced to δ = 0.5. While the quantitative results depend on the precise definition of MIC, the qualitative features do not depend on the choice of δ. **c**. Cost-benefit theory predicts general properties of drug interactions as a function of the maximum inducible benefit β_{max} and the concentration of inducer, S. Solid line, phase boundary between synergistic and antagonistic drug interactions. Dashed line, phase boundary between antagonistic and suppressive interactions. In general, a higher value of the β_{max} increases the antagonism between drug pairs at a given concentration of drug S. Parameters characterizing individual drugs include K_{S} and K_{ind}, which characterize, respectively, the cost of drug S and the corresponding induction of resistance systems (e.g. efflux pumps), and K_{A} and n, which characterize the cost of drug A. By contrast, β_{max} couples the individual effects of two drugs. Insets, heat maps of two-dimensional growth surfaces for 3 cell strains in the presence of Salicylate (Sal) and Chloramphenicol (Cm); top: β_{max} = 1.15 (WT cells; suppressive), β_{max} = 0.19 (mar mutant, antagonistic), and β_{max} = −0.15 (tolC mutant, synergistic). See also Additional file

Contributions of inducer cost and benefit to apparent MIC’s of antibiotics

Given our characterization of salicylate-induced multi-drug resistance, it is straightforward to calculate the apparent MIC of an antibiotic in the presence of any concentration of salicylate. Here, we define the apparent MIC to be the minimum concentration of drug A at which relative growth is reduced to a value of δ. For small concentrations of salicylate, the apparent MIC’s of both tetracycline and chloramphenicol increase dramatically (Figure _{max}, which provides a quantitative measure of inducible benefit that is not masked by the effects of inducer cost.

Phase diagram for MAR-mediated drug interactions

The interactions between salicylate and tetracycline, salicylate and chloramphenicol, and sodium benzoate and tetracycline are all suppressive, but this class of models describes a range of interactions ranging from synergistic to suppressive, based on the interplay of induction benefit and drug toxicity. Using Equations 1 and 2, we quantitatively determine a phase diagram (Figure _{ind} and β_{max}), the cost of inducer (K_{S}), and the steepness of the antibiotic dose-response curve (n). Specifically, the cellular response to the first drug (S) alone includes both the growth cost of the drug (characterized by K_{S}) and the induction of MAR system (characterized by K_{ind}). The dynamics of the efflux pumps and their specificity for drug A determine the concentration reduction (A to A_{eff}), which is governed by β_{max}. The phase diagram demonstrates that properties of the drugs alone (n, K_{S}, K_{ind}) determine the level of drug coupling, contained in β_{max}, required to achieve antagonism or suppression. Drugs that strongly induce growth benefit and have low associated cost (K_{ind} <<K_{S}) are always suppressive. By contrast, high cost inducers (K_{ind} >> K_{S}) can never be suppressive because the cost of induction is too high (Figure

The inducible benefit parameter (β_{max} = 1.15) characterizing the salicylate and chloramphenicol combination in wild type cells is far above the suppressive-antagonistic boundary (β_{max} > > 2K_{ind}/(K_{1}n) = 0.45), and the interaction is clearly suppressive for all concentrations of salicylate. To explore different regions of the phase diagram experimentally, we measured the effects of two different mutations on the suppressive drug interaction between salicylate and chloramphenicol. The first mutant, ΔtolC _{A} smaller than in wild type), but the cost of salicylate remains unchanged (K_{S} approximately same as wild type). β_{max} was measured to be −0.15 +/− 0.01. According to the phase diagram (Figure

Since the suppressive interaction between salicylate and chloramphenicol is partially associated with the synthesis of AcrAB-TolC efflux pumps, we also hypothesized that cells with constitutive _{A} is larger than in wild-type) and a near-additive drug interaction (β_{max} = 0.19 +/− 0.03) between salicylate and chloramphenicol (Figure _{max} is approximately 17% of that of the wild type, and the antagonism between the drugs is markedly decreased).

Discussion

Efflux is a widespread cellular defense mechanism with clinical consequences for antimicrobial treatments. Cost-benefit principles provide a convenient tool for analyzing these ubiquitous systems. We have shown that such a framework provides a quantitative description of drug interactions associated with the classic MAR system, which mediates an inducible resistance to many different toxic agents. The model allows us to quantify the relative importance of the induction mediating reversible resistance and the toxicity of the inducing agent. In particular, we observe a transition between two regimes, one where salicylate is harmful and one where it is beneficial, as the concentration of antibiotic crosses a threshold A_{crit}. By disentangling the contributions of inducer cost and benefit, the analysis also provides a quantitative measure (β_{max}) of the reversible (MAR-mediated) resistance to each antibiotic used. This measure complements previous MIC measurements

This framework may be applicable to other systems where environmental signals can be toxic but also induce transient resistance. For example, cancer cells can activate pro-survival stress responses upon treatment with chemotherapies or radiation**→** A_{eff} for the effective concentration of drug A, and this form could be used to describe other common mechanisms of resistance, including enzymatic decay and target modification. As a consequence, the cost-benefit framework developed here may prove useful for modeling and understanding a wide range of systems governed by the inherent trade-offs of inducible resistance.

Conclusions

Our analysis provides a quantitative description of the MAR system and demonstrates that optimal cell growth in the presence of an inducer and an antibiotic does not coincide with maximum induction of the

Methods

Strains

We used the

Drugs

Drug solutions were made from solid stocks (sodium salicylate, Sigma-Aldrich; chloramphenicol, MP Biomedicals; tetracycline hydrochloride, Acros Organics; doxycycline hyclate, Sigma- Alderich; kanamycin, Fisher; ciprofloxacin, Sigma-Alderich). All antibiotic stock solutions were stored in the dark at −20°C. All drugs were thawed (when necessary) and diluted in sterilized Lennox LB broth (Fisher) for experimental use.

Growth conditions and drug treatments

We inoculated LB media with a single colony and grew the cells overnight (12 h at 30°C with shaking at 200 rpm). Following overnight growth, stationary phase cells were diluted (600–1000 fold) in LB media containing 0.2% glucose and grown for an additional 3 h (30°C with shaking at 200 rpm). Following the initial dilution period, we transferred cells to 96 well microplates (round bottom, polystyrene, Corning) and set up a two-dimensional matrix of drug concentrations in each 96-well microplate (165 μl media per well). For the remainder of the experiment (4–8 h), cells were grown at 30°C with shaking at 1200 rpm on a Heidolph Titramax vibrating microplate shaker (Brinkmann). Plates were wrapped in aluminum foil to minimize evaporation. A600 (absorbance at 600 nm, proportional to optical density OD) and YFP fluorescence were measured at 15–25 min intervals for 4–8 h using a Wallac Victor-2 1420 Multilabel Counter (PerkinElmer).

Growth rate estimation and MIC determination

Growth rates were determined by fitting multiple regions of growth curves (A_{600} vs. time, Additional file

Drug resistant mutants

We diluted liquid cultures of wild-type cells (in stationary phase) 1000-fold into 96 individual 150 μL cultures on a single microplate, with each well supplemented with 1.0 μg/mL tetracycline. After approximately 48 h of growth at 30°C with shaking, optical density and fluorescence were measured from each well. Samples from randomly selected cultures exhibiting high OD and fluorescence (Additional file

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

KW and PC designed research, analyzed data, and wrote the paper. KW performed all experiments. Both authors read and approved the final manuscript.

Acknowledgements

We thank L. Bruneaux, E. Balleza and A. Subramaniam for technical guidance, C. Guet for advice and for providing the Frag-1B pZS*2 MAR-venus strain, and all members of the Cluzel lab for many helpful discussions. We also thank K. Dave for editorial advice and assistance and Kris Wood for comments on the manuscript. This work was supported in part by the NIH P50 award P50GM081892-02 to the University of Chicago and the NSF Postdoctoral Fellowship 0805462 (to K.W.).