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 Systems Biology
Drug interactions modulate the potential for evolution of resistance

Edited by Richard P. Novick, New York University School of Medicine, New York, NY, and approved July 1, 2008

↵^{‡}J.B.M. and P.J.Y. contributed equally to this work. (received for review January 29, 2008)
Related Article
 In This Issue Sep 30, 2008
Abstract
Antimicrobial treatments increasingly rely on multidrug combinations, in part because of the emergence and spread of antibiotic resistance. The continued effectiveness of combination treatments depends crucially on the frequency with which multidrug resistance arises. Yet, it is unknown how this propensity for resistance depends on crossresistance and on epistatic interactions—ranging from synergy to antagonism—between the drugs. Here, we analyzed how interactions between pairs of drugs affect the spontaneous emergence of resistance in the medically important pathogen Staphylococcus aureus. Resistance is selected for within a window of drug concentrations high enough to inhibit wildtype growth but low enough for some resistant mutants to grow. Introducing an experimental method for highthroughput colony imaging, we counted resistant colonies arising across a twodimensional matrix of drug concentrations for each of three drug pairs. Our data show that these different drug combinations have significantly different impacts on the size of the window of drug concentrations where resistance is selected for. We framed these results in a mathematical model in which the frequencies of resistance to single drugs, crossresistance, and epistasis combine to determine the propensity for multidrug resistance. The theory suggests that drug pairs which interact synergistically, preferred for their immediate efficacy, may in fact favor the future evolution of resistance. This framework reveals the central role of drug epistasis in the evolution of resistance and points to new strategies for combating the emergence of drugresistant bacteria.
The widespread use of antibiotics pits clinical need against the reality of evolution (1–3). The clinical goal is to kill as many pathogenic bacteria as possible, or inhibit their growth to allow the immune system to gain the upper hand; but a drug that kills or inhibits the growth of susceptible pathogens confers a dramatic selective advantage to resistant lineages, eventually making the drug ineffective. Although major advances have been made in describing the impact of single drugs on bacterial resistance (3), it is still unclear how drugs in combination affect the evolution of resistance. Combinations of drugs may inhibit bacterial growth in complex ways, deviating from the neutral situation expected when the drugs do not interact (4–6). Compared with this null situation, drug combinations that interact to increase each other's effects are termed “synergistic”; drugs whose combined effect is smaller than expected are termed “antagonistic” (4–7, 39) (Fig. 1D). We have previously shown that these epistatic drug interactions profoundly affect the selective advantage of a single horizontally transferred resistance allele (8). Here, we focus on the more complex scenario of the evolution of multidrug resistance by spontaneously occurring mutations.
In many infectious and noninfectious diseases, including HIV (9), tuberculosis (10), malaria (11, 12), and cancer (13), high rates of mutation confer resistance to individual drugs. Combination therapies are therefore used to increase the killing of singledrugresistant strains or mutants. Unfortunately, multidrug resistance still arises: multiple mutations conferring resistance may accumulate, or a single mutation may confer resistance to several drugs (crossresistance). We address here the frequency of such spontaneous resistant mutations in Staphylococcus aureus, one of the most worrisome multidrugresistant bacteria (14). Although a major mode of resistance in S. aureus is horizontal gene transfer, resistance acquired vertically by spontaneous mutations is another concern and combination therapies aimed at preventing their emergence are frequently used (15, 16). The approach we develop using S. aureus as a model system is general in scope and can be applied to pathogens such as Mycobacterium tuberculosis, where resistance acquired during treatment by spontaneous mutations is critical.
Antibiotics impose a strong selection pressure on bacterial populations (17, 18): susceptible cells do not grow, and resistant cells already present in the population are selectively enriched. A commonly used measure of the potential to evolve resistance by spontaneous mutations is the size of the mutant selection window (MSW)—the range of drug concentrations where resistance is selectively enriched (19, 20). The MSW ranges from the minimum inhibitory concentration (MIC) that inhibits wildtype growth, to the mutant prevention concentration (MPC) where even very rare mutants are unlikely to grow (Fig. 1A). The frequency of resistance and the MSW of several clinically important individual drugs have been well characterized (21–26), and evidence suggests that the MSW of drug combinations can be smaller than the MSW of any of their individual drug constituents (27). Yet, a general relationship between the frequencies of cells resistant to combinations of drugs and the frequencies of resistance to each drug alone has not been established, and the effect of drug–drug interactions (epistasis and crossresistance) on this relationship is not known.
We use both experimental and theoretical tools to explore how interactions between antibiotics impact the landscape on which selection can act. We develop a highthroughput system to measure frequencies of resistant S. aureus mutants over a matrix of concentrations of pairwise drug combinations, and apply this tool to three different types of drug pairs. Motivated by the diversity of results observed, we develop a quantitative theoretical model that makes clear the central role of drug epistasis and the impact of crossresistance on the potential for evolution of resistance in multidrug environments. This model yields direct predictions for the impact of drug synergy on the emergence of resistance.
Results
Extending the MPC to Antibiotic Combinations Necessitates Consideration of Drug Epistasis.
Resistant mutants may appear spontaneously because of replication errors at frequencies typically lower than 1 in 10^{6} cells. The frequency of those mutants and the drug window within which they survive characterize the potential of the population to evolve resistance and is represented by the curve F_{X}(C_{X})—the frequency of mutants that resist concentration C_{X} of drug X (Fig. 1A). This curve typically presents plateaus with sharp drops, indicating the existence of subpopulations of resistant cells. The drug concentration at which the frequency drops to nondetectable levels is the MPC, defined here as F_{X} < 10^{−9} (see Materials and Methods). The mutant selection window (MSW) of a drug extends from the MIC to the MPC. To allow comparison between different antibiotics, we normalize drug concentrations to their respective MICs (note, though, that the absolute values of the MPCs in μg/ml are also of direct clinical importance due to drug toxicity). The size of the MSW is then (MPC − MIC)/MIC (for instance, if the MIC is 1 μg/ml and MPC = 100 μg/ml, MSW ≅ 100).
By analogy with resistance to a single drug, we introduce the surface F_{XY}(C_{X}, C_{Y}), the frequency of cells that can grow in an environment containing a combination of the two drugs X and Y at the concentrations C_{X} and C_{Y} (Fig. 1B). As in the singledrug case, F_{XY} is likely to exhibit plateaus: the first drop in frequency occurs as wildtype growth is inhibited, and defines the MIC line. The MPC line bounds the region where resistant mutants are unlikely to occur (F_{XY} < 10^{−9}). The drugs can be combined in different proportions to effectively produce new “single” drugs, represented geometrically by linear lines extending from the origin in the drug concentrations plane (Fig. 1B, black line at angle θ corresponding to the drug ratio). These effective drugs have their own MSW, according to the geometric points of intersection with the MIC and MPC line. The smallest MSW obtained over all of the combinations of X and Y characterizes the potential of the drug pair X–Y for limiting the evolution of resistance (Fig. 1C, arrow).
The simplest approach in the absence of information on how the drugs interact would assume the frequency of resistance to the drug combination to be the product of the frequencies of resistance to each of the individual drugs, F_{XY}(C_{X}, C_{Y}) = F_{X}(C_{X})F_{Y}(C_{Y}) (12, 28). It is known that this expectation breaks down in the presence of mutations that confer resistance to X and Y simultaneously (crossresistance), but we highlight here that F_{XY} may also significantly depend on epistatic interactions between drugs. The effect of drug combinations on growth inhibition can deviate from the null situation expected by Loewe additivity (Fig. 1D Left) (5), defining synergistic or antagonistic epistasis [Fig. 1D and supporting information (SI) Fig. S1]. Consider, for example, an environment containing 0.75 MIC of drug X and of 0.75 MIC of Y (Fig. 1D, ). In the antagonistic case, the wild type is able to grow and therefore the frequency of resistance is effectively 1 [F_{XY}(0.75, 0.75) = 1]. In contrast, if the drugs interact synergistically, the wild type cannot grow and therefore F_{XY}(0.75, 0.75) ≪ 1. The frequency of resistance to each drug alone is the same in both cases [F_{X}(0.75) = F_{Y}(0.75) = 1], but epistatic interactions dramatically affect resistance to the combination. This simple example illustrates that even in the absence of crossresistance, the frequency of cells resistant to the combination is not trivially the product of the frequencies of cells resistant to each of the single drugs.
Experiments Show That Multidrug Resistance Varies Dramatically Between Drug Combinations.
To explore how resistance to combinations of drugs depends on resistance to each drug alone and interactions between the drugs (epistasis and crossresistance), we designed an experimental setup to systematically measure the frequencies of resistance to more than a hundred different concentrations of a given pair of drugs (see Materials and Methods and Fig. 2). We sampled the resistance surfaces of three drug pairs: fusidic acid–erythromycin (FUSERY), ciprofloxacin–ampicillin (CPRAMP), and fusidic acid–amikacin (FUSAMI). The drugs were chosen for their clinical relevance, diversity of mechanisms of action, and potential to evolve spontaneous resistance. The assayed drug pairs cover the range of epistatic interactions: the MIC lines, measured independently in liquid media on clonal wildtype populations, show synergy between FUS and ERY (epistasis parameter ε = −0.1; see Materials and Methods), antagonism between FUS and AMI (ε = 0.3), and no epistasis between CPR and AMP (ε = 0.06) (Fig. 2 C–E Insets). We measured resistance to each drug pair at 11 × 11 combined concentrations. Agar plates containing the relevant concentrations of drugs were prepared, inoculated with S. aureus at a range of inoculum sizes (from 10^{1.5} to 10^{9} cells per plate), and placed on scanners taking timelapse highresolution pictures every hour for 5 days (Fig. 2A). The images obtained were analyzed by an automated imageprocessing platform designed to count visible colonies on each plate (Fig. 2B).
Our results show qualitatively different patterns of resistance to the three pairwise drug combinations. The frequency of cells resistant to combinations of FUS and AMI is comparable with that of cells resistant to AMI alone (Fig. 2E). By contrast, the frequency of cells resistant to combinations of CPR and AMP is significantly smaller than the frequencies of resistance to the same concentration of CPR alone, or AMP alone (Fig. 2D). The same is observed for FUSERY (Fig. 2C). Furthermore, mixing together FUS and ERY leads to effective drugs whose MSWs can be up to one order of magnitude smaller than the MSW of FUS or ERY alone (Fig. 4 Insets). Thus, combinations of FUS and ERY can be found that significantly narrow the drug regime that selects for resistance. This effect is not observed for FUSAMI or CPRAMP. We find that the multiplicative model (i.e., F_{XY} = F_{X}F_{Y}) is unable to capture such diverse behaviors (Fig. S2), underscoring the need for a predictive model of resistance frequencies inclusive of epistasis and crossresistance.
Resistance Frequencies to Individual Drugs, CrossResistance, and Epistasis Determine the Frequency of Resistance to Multidrug Combinations.
The frequency of resistance F_{X} to a single antibiotic X derives from the makeup of the bacterial population at the time the antibiotic is introduced. We define p_{X}(x), the probabilistic density of cells whose MIC of drug X is exactly x (Fig. 3A): it derives from the frequency of resistance as p_{X}(x) = −dF_{X}(x)/dx. Because the frequency of resistance to C_{X} of X is, by definition, the frequency of cells whose MIC is greater than C_{X} (Fig. 3B), we have where η_{X}(z) = 1 if z < 1 and is 0 otherwise. In this equation, η_{X} characterizes the growth of one individual cell: for simplicity we consider only step functions, but η_{X} could also be smooth and account for natural variations in the response of isogenic cells.
In the case of two drugs X and Y, the growth region of a bacterial cell is a function of the twodrug concentration, and is delimited by its MIC line [η(C_{X}, C_{Y}) = 1 if the cell can grow in (C_{X}, C_{Y}), zero otherwise]. Earlier work (8) has shown that growth regions of wild type (η_{XY}) and resistant mutants (η) tend to have a common shape, which characterizes the epistatic interactions between the drugs. We therefore approximate the region of growth of mutant cells in the population by a linear scaling of the wild type's growth region (Fig. 3C Inset; blue, wild type; red, a mutant). This approximation cannot be tested directly within our experimental data, but the fit of the resulting model to measured resistance frequencies supports its validity. The growth region of a bacterial cell whose MIC of drug X is x and whose MIC of Y is y is approximated by η(C_{X}, C_{Y}) = η_{XY}(C_{X}/x, C_{Y}/y).
Next, by analogy to the single drug case, p_{XY}(x, y) is the density of cells in the population whose MIC of drug X alone is x and whose MIC of drug Y alone is y (Fig. 3C)—note that it does not contain information on the ability of cells to grow in combinations of these drugs, which is carried by η_{XY}. With these definitions, the frequency of resistance to a given combination (C_{XY},C_{Y}) of drugs X and Y is given by (Fig. 3D) Unlike the MIC line (η_{XY}), which is experimentally measured in liquid media, the density of population p_{XY} cannot easily be measured. We have built a simple model where p_{XY} depends only on p_{X} and p_{Y} (measured as the derivatives of F_{X} and F_{Y}), and on crossresistance between the drugs. When crossresistance is absent, mutations conferring resistance to X or Y are independent: the density of population is p_{XY}^{Indep}(x, y) ≡ p_{X}(x)p_{Y}(y). Conversely, the mechanisms of resistance could be extremely correlated, as would be the case if the “two” drugs were in fact the same: this corresponds to a density p_{XY}^{Correl} that depends only on p_{X} and p_{Y} (mathematically defined as the density that maximizes the correlation between resistance to X and Y under the constraints p_{X}(x) = ∫_{y>0} p_{XY}(x, y)dy and p_{Y}(y) = ∫_{x>0} p_{XY}(x, y)dx; see SI Text.). In general, some mutations confer resistance to only one of the drugs, and others confer resistance to both drugs at once: we model p_{XY} as a linear combination between the two extreme cases, p_{XY} = ξp_{XY}^{Correl} + (1 − ξ)p_{XY}^{Indep}. The parameter ξ reflects crossresistance, absent when ξ = 0 and maximal when ξ = 1.
This constitutes a model that quantifies exactly how drug epistasis, crossresistance between drugs, and singledrug resistance determine the frequency of cells resistant to any combination of the drugs X and Y. The densities p_{X} and p_{Y} are estimated by the derivatives of the measured frequencies of resistance to the individual drugs F_{X}, F_{Y} (Eq. 1). They enable the construction of p_{XY}^{Indep} and p_{XY}^{Correl}, which, tuned by the crossresistance ξ, produce p_{XY}. The MIC line of the wildtype is a direct measurement of η_{XY}. Finally, p_{XY} and η_{XY} predict the frequency of resistance F_{XY} to any combined concentration of the drugs X and Y (Eq. 2).
The Theoretical Model Captures the Experimental Results.
We next compared the predictions of this model with our experimental results. Frequencies of resistance to the single drugs alone (F_{X}, F_{Y}) were measured together with the whole surface. The MIC line of each drug pair (η_{XY}) was directly measured in liquid media. Crossresistance (ξ) is the only free parameter and was estimated for each drug pair by a leastsquares fitting of predicted to experimental frequencies of resistance. The drug pair FUSAMI shows strong crossresistance (ξ = 0.3), whereas the drug pairs CPRAMP and FUSERY show almost no crossresistance (ξ < 10^{−2}). Our experimental results on resistance to combinations of drugs are well captured by the model (Fig. 4). This framework successfully accounts for very different behaviors in terms of epistatic interactions and crossresistance.
Theory Predicts That Synergistic Epistasis Favors Resistance.
We use this framework to weigh the impact of drug interactions on the evolution of resistance. Crossresistance increases the MSW of drug combinations; with increased crossresistance, mutants resistant to both drugs individually occur at frequencies high enough to appear in the population and contribute to the MSW of drug combinations (Fig. S3). The impact of epistasis, however, is less obvious. We present here the simplest model exhibiting the mechanisms by which epistasis affects the potential to evolve resistance to a combinations of drugs. For simplicity, we assume no crossresistance; the combined impact of crossresistance and epistasis is presented in Fig. S3.
We consider the case where only two mutations exist, one conferring resistance to drug X, the other to drug Y; small mutation frequencies preclude the appearance of double mutants, and the MSWs for drug X and for drug Y have the same size M. Only epistasis between the drugs X and Y varies, and is quantified by the real number ε, which parameterizes the MIC line as x^{10ε} + y^{10ε} = 1. Absence of epistatic interactions is represented by ε = 0, while antagonism and synergy correspond to ε > 0 and ε < 0, respectively. Fig. 5 A and B shows the frequencies of resistance to combinations of X and Y for synergistic or antagonistic epistasis. In both cases, 1:1 (θ = 45°) is the ratio of X and Y for which the effective drug has the smallest MSW, defining MSW_{XY}. Analytical expressions for the corresponding MIC_{XY} (dashed line) and MPC_{XY} (solid line) of this “best” effective drug are geometrically derived and a closed form of the smallest MSW, MSW_{XY} ≡ (MPC_{XY} − MIC_{XY})/MIC_{XY}, is obtained: Although the MSW of the combination increases with the MSW of the single drugs, it decreases as the drugs become less synergistic (Fig. 5C). In other words, the more antagonistic the epistasis, the smaller the potential to evolve resistance to the drug combination. This signifies that antagonistic drug combinations can be more efficient than synergistic combinations at reducing the range of drug concentrations that select for resistance. Although the MPC_{XY} and the MIC_{XY} of the “best” effective drug both increase in absolute value with the degree of epistasis, the MPC_{XY} increases at a slower pace than the MIC_{XY}: relative to the MIC_{XY}, the value of the MPC_{XY} decreases (Fig. 5C Inset). Therefore, as epistasis goes from synergy to antagonism, the MSW of the combination shrinks and eventually vanishes when the drugs completely buffer or even suppress one another (Fig. S4).
Discussion
Current clinical practice emphasizes the use of multidrug treatments primarily to increase the spectrum of activity (29–33), to increase efficacy (34), and, in some pathogens, to decrease the likelihood of the emergence of resistance (29). Clinicians generally prefer synergistic drug pairs when prescribing combination treatments to broaden the spectrum but usually do not consider the effect of drug epistasis on resistance (29). Our results imply that synergistic drug pairs may favor the evolution of resistance. In contrast, largely overlooked antagonistic drug combinations may suppress the emergence of resistance. This study, designed to explore the mostly uncharted territory of resistance to combinations of drugs, comprises drug pairs with different epistasis, different degrees of crossresistance, and different frequencies of resistance to single drugs. Our theoretical prediction above can be validated experimentally by comparing pairs of drugs with different epistasis but with similar degree of crossresistance, and similar singledrug MSW. Future screens focusing on finding such pairs could be designed in light of this model.
We focused on the frequency of spontaneous mutations conferring resistance in multidrug environments and emphasized the drug window selective for resistance. We note that this measure of propensity for evolution of resistance does not include other factors that may affect the emergence and spread of drugresistant pathogens, such as variation in the drug concentration, natural variability in the response of isogenic cells, or the pharmacodynamics of the particular antibiotics (35). The dynamics of antibiotic treatment can lead to rounds of selection and adaptation: resistance may be acquired during the course of treatment. Temporal variations in drug dosage and nongenetic phenotypic tolerance may substantially increase the likelihood of emergence and rate of evolution of resistance. For instance, persister cells may remain dormant long enough for the antibiotic to decay, enabling adaptation and subsequent selection (36, 40). Furthermore, in natural and clinical settings, resistance acquired through horizontal transfer may play a major role in the evolution of resistance. Examining the impact of drug–drug interactions on these factors central to the development of drug resistance is a promising avenue for future research.
In conclusion, we present an experimental–theoretical framework that offers a quantitative, unified understanding of how resistance to individual drugs, crossresistance, and epistatic interactions affect the propensity for resistance in multidrug combinations. Importantly, our results suggest that antagonistic combinations may narrow the range of drug concentrations where resistant is selected for. In contrast, synergistic drug combinations, typically preferred in clinical settings, may in fact favor the evolution of resistance even though they increase killing efficiency. Our results indicate that drug interactions could be central to a tradeoff between immediate efficacy and the future prevention of resistance.
Materials and Methods
Bacteria and Antibiotics.
We used a streptomycinresistant S. aureus strain Newman NCTC 8178 (37). Growth media was liquid or agar Luria broth (LB) supplemented, as indicated, with one or two of five different antibiotics (see Table S1). A single colony (starting from a single cell) was inoculated in LB liquid and grown overnight: frozen aliquots of this culture were kept at −80°C. All experiments were initiated from a freshly thawed aliquot from this single batch.
MIC Line.
The MIC line of a drug pair was measured by a standard overnight growth assay in liquid media, inoculating ≈10^{3} wildtype cells in each of 96 wells (Costar plate; 150 μl per well) forming a 12 × 8 gradient of drug concentration (dilutions of 2/3 to 9/10). The MIC line was defined as the line separating regions of growth and no growth (practically, the contour line of optical density 0.1). The shape of the MIC line was used to define the function η for each specific drug pair (Fig. 2 C–E Insets). The sign of the epistatic interactions was determined by fitting η with the function x^{10ε} + y^{10ε} = 1; synergy, additivity and antagonism correspond respectively to negative, null, and positive values of ε.
Frequency of Resistance.
We measured the frequency of resistance with a resolution spanning nine orders of magnitude, across an 11 × 11 grid of drug concentrations. For each twodrug concentration we used one sixwell plate (Becton Dickinson Multiwell), and poured 7 ml of agar supplemented with the same concentration of drugs in all six wells (e.g., Fig. S5). After agar solidification, each of the six wells was inoculated with a different number of bacterial cells (approximately 10^{1.5}, 10^{3}, 10^{4.5}, 10^{6}, 10^{7.5}, and 10^{9}). The plates were then incubated on scanners (see below) in a controlled environmental room at 30°C and 70% relative humidity for 5 days, a duration optimized for the detection of resistant colonies by our custom software (Fig. S6: effect of the incubation time). The frequency of resistance to a given concentration of the drug pair was defined as the total number of mutants in all wells of the relevant drug concentrations with countable colonies (typically, <500 cfu per well) divided by the total number of cells plated on these specific wells. The MPC (respectively MPC line) bounds the drug concentrations where at least one growing colony was observed. This corresponds here to frequencies of resistance greater than 10^{−9}, while the standard definition of the MPC is usually F_{X} > 10^{−10} (20, 38): our measurements of the MSW may differ from those obtained by the standard method. The use of frozen cell aliquots prepared from the same single culture eliminates much of the Luria–Delbrück fluctuations. Some additional fluctuations due to the last 100fold amplification step are still present (Fig. S7).
Scanner and Imaging Platform.
We built an array of 30 office scanners (Epson Perfection 3170/3490) controlled by one computer. Five plates were placed in each scanner. The scanners were programmed to take timelapse pictures of the plates at 600 dpi every hour for 5 days. We built an imageanalysis platform in MATLAB (MathWorks) to count the number of colonies arising in each plate. The platform detects single colonies larger than onetenth of a millimeter and tracks their growth by using a custom contourdetection algorithm based on contrast gradients (Fig. 2B).
Acknowledgments
For comments and discussion, we thank K. Vetsigian, N. Shoresh, T. Bollenbach, A. DeLuna, M. Hegreness, M. Ernebjerg, M. Elowitz, M. Ackermann, P. Bordalo, R. Ward, D. Andersson, J. Skerker, and R. Milo. For strains and advice, we thank G. RegevYochay and M. Lipsitch. This work was supported in part by National Institutes of Health Grant R01 GM081617 (to R.K.) and a National Institutes of Health National Research Service Award (to P.J.Y.).
Footnotes
 ^{‖}To whom correspondence should be addressed at: Department of Systems Biology, 200 Longwood Avenue, Boston, MA 02115. Email: roy_kishony{at}hms.harvard.edu

Author contributions: J.B.M., P.J.Y., R.C.M., and R.K. designed research; J.B.M. and P.J.Y. performed research; J.B.M., P.J.Y., and R.C. contributed new analytical tools; J.B.M., P.J.Y., and R.K. analyzed data; and J.B.M., P.J.Y., and R.K. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/cgi/content/full/0800944105/DCSupplemental.
 © 2008 by The National Academy of Sciences of the USA
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