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PLoS Comput Biol. Jun 2010; 6(6): e1000796.
Published online Jun 3, 2010. doi:  10.1371/journal.pcbi.1000796
PMCID: PMC2880566

Optimal Drug Synergy in Antimicrobial Treatments

Philip E. Bourne, Editor

Abstract

The rapid proliferation of antibiotic-resistant pathogens has spurred the use of drug combinations to maintain clinical efficacy and combat the evolution of resistance. Drug pairs can interact synergistically or antagonistically, yielding inhibitory effects larger or smaller than expected from the drugs' individual potencies. Clinical strategies often favor synergistic interactions because they maximize the rate at which the infection is cleared from an individual, but it is unclear how such interactions affect the evolution of multi-drug resistance. We used a mathematical model of in vivo infection dynamics to determine the optimal treatment strategy for preventing the evolution of multi-drug resistance. We found that synergy has two conflicting effects: it clears the infection faster and thereby decreases the time during which resistant mutants can arise, but increases the selective advantage of these mutants over wild-type cells. When competition for resources is weak, the former effect is dominant and greater synergy more effectively prevents multi-drug resistance. However, under conditions of strong resource competition, a tradeoff emerges in which greater synergy increases the rate of infection clearance, but also increases the risk of multi-drug resistance. This tradeoff breaks down at a critical level of drug interaction, above which greater synergy has no effect on infection clearance, but still increases the risk of multi-drug resistance. These results suggest that the optimal strategy for suppressing multi-drug resistance is not always to maximize synergy, and that in some cases drug antagonism, despite its weaker efficacy, may better suppress the evolution of multi-drug resistance.

Author Summary

The use of antibiotics against bacterial infections has led to the emergence of multi-drug resistant pathogens such as tuberculosis and MRSA. In order to control resistance, clinicians have increasingly turned to multi-antibiotic therapies. The common wisdom is to use combinations of drugs that act synergistically to kill the infection, but the impact of drug synergy on the evolution of resistance is unclear. Using mathematical simulations of an in vivo infection model, we asked what level of drug synergy would minimize the risk of multi-drug resistance while preserving the efficacy of treatment. We found that synergy may increase or decrease the risk of multi-drug resistance in a given treatment, depending on infection properties such as mutation rate and the availability of resources. Surprisingly, under conditions of strong competition for resources within the host, we found that maximal synergy—currently favored in clinical settings—can actually increase the risk of multi-drug resistance. Our results identify conditions under which drug synergy exacerbates the problem of multi-drug resistance, and offer guidelines for the selection of drug pairs that suppress it.

Introduction

As antibiotic-resistant pathogens become more common, clinicians increasingly turn to multi-drug treatment to control infection [1][5]. The inhibitory effect of two drugs in combination can be larger or smaller than expected from their individual effects, corresponding to synergistic or antagonistic interactions between the drugs respectively [6][9]. Synergistic interactions are usually thought of as advantageous since, for a given amount of drug, they more effectively inhibit the growth of drug-sensitive pathogens. However, in vitro studies have suggested that, for the same level of inhibition, more synergistic drug pairs may foster antibiotic resistance [10][12]. Antagonistic drug combinations, on the other hand, are less effective at inhibiting drug-sensitive pathogens, but can reduce and even invert the selective advantage of single-drug resistant mutants, causing selection against resistance [13].

These recent observations point to a possible tradeoff in the choice of synergistic versus antagonistic drug combinations with respect to their effects on treating infection and suppressing antibiotic resistance. However, while antagonistic drug combinations increase selection against resistance, and should therefore minimize resistance, they also kill the infection more slowly, giving resistance more time to emerge. Antagonism therefore has two contradicting effects on the evolution of resistance: on one hand, it increases the risk of resistance by decreasing antibiotic inhibition and allowing more time for resistance to evolve; on the other hand, it decreases the risk of resistance by decreasing the selective advantage of single drug resistant mutants. We ask which of these opposing effects is stronger, and therefore which type of drug interaction – synergistic or antagonistic – best prevents the overall chance of emergence of multi-drug resistance.

We frame this problem in the context of a clinical infection, formalizing the two main factors in the success of an antibiotic treatment as “treatment efficacy” and “prevention of multi-drug resistance.” Treatment efficacy is the rate at which the infection is cleared by the treatment, and can be defined as the reciprocal of the time, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e001.jpg, at which the total infection is eliminated, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e002.jpg. Prevention of multi-drug resistance is defined as the reciprocal of the number of double-drug resistant mutants expected to arise during the course of treatment, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e003.jpg. In real infections, multi-drug resistance can arise either through a single mutation conferring cross-resistance to both drugs simultaneously, or through the sequential acquisition of mutations conferring resistance to each drug individually [10], [14][16]. Furthermore, resistance to a single drug can develop in several small steps or in one large step [15], [17][19]. For simplicity, and to emphasize the role of drug interactions, we concentrate here on an idealized case in which resistance to the two-drug combination evolves through sequential acquisition of two spontaneous mutations, each conferring strong resistance specific to one of the two antibiotics (Fig. 1).

Figure 1
Model of the evolution of resistance in synergistic and antagonistic drug treatments.

We asked what level of drug interaction (ranging from strong synergy to strong antagonism) maximizes treatment efficacy (An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e021.jpg) and prevention of multi-drug resistance (An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e022.jpg). Maximizing An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e023.jpg is straightforward: as more synergistic drug pairs have increased killing potency and clear the infection more quickly, maximally synergistic drug pairs should maximize An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e024.jpg [4], [20]. In attempting to maximize An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e025.jpg, however, the best choice of drug interaction is less clear. Assuming sequential acquisition of resistance to each drug, the rate at which multi-drug resistance arises will depend on the size of the single-drug resistant mutant population. The size of this single-mutant population, in turn, depends on two factors: the rate at which such mutants arise, and their selective advantage over the wild-type. Synergistic drug pairs decrease the first factor because they more quickly kill the source wild-type population from which single mutants arise. However, synergistic drug pairs also increase the second factor: single-drug resistant mutants will have a strong selective advantage in a synergistic treatment because resistance removes both the burden of one drug, and its enhancing effect on the other drug [13] (Fig. 1B). Synergistic drug pairs therefore decrease the rate at which single-drug resistant mutants appear, but increase their selective advantage. Antagonistic drug pairs do the opposite: though they allow a larger number of single-drug resistant mutants to arise, they also diminish the selective advantage of these mutants. The net effect of a given drug interaction on the evolution of multi-drug resistance is therefore not obvious, and requires a quantitative model to determine the overall impact of mutation and selection's countervailing effects.

To better understand how drug interactions affect the risk of multi-drug resistance, we used a population genetic model of microbial infection previously applied to predict single-drug resistance in vivo in mice [21], and modified it to account for the sequential acquisition of mutations leading to multi-drug resistance. We used this model to ask what level of drug interaction maximizes treatment efficacy (An external file that holds a picture, illustration, etc.
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Results

Simple model for the evolution of resistance in multi-drug treatment

We based our model on work by Jumbe et al. [21], which investigated a mouse-thigh P. aeruginosa infection model [22][24] and provided a mathematical model that quantitatively described the relationship between exposure to the fluoroquinolone antibiotic levofloxacin, and changes in drug-susceptible and drug-resistant bacterial subpopulations over time. This mathematical model was successful both in reproducing the observed changes in drug-susceptible and –resistant subpopulations over time, and in predicting the dose of levofloxacin needed to suppress amplification of levofloxacin-resistant (efflux-pump-expressing) mutants. To investigate the effect of antibiotic interactions on treatment efficacy and the prevention of multi-drug resistance in a simple scenario, we modified the Jumbe et al. model in four ways: we include a second antibiotic in our model; we assume a constant antibiotic dose; we assume a low hill coefficient, consistent with the mechanisms of a range of antibiotics [25]; and we assume no cost for antibiotic resistance. The consequences of these assumptions are discussed throughout the text.

Our model incorporates treatment with two antibiotics, A and B. It uses a set of ordinary differential equations (ODEs) to follow the population sizes of the drug-sensitive wild-type strain (An external file that holds a picture, illustration, etc.
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Object name is pcbi.1000796.e031.jpg) arising over time (Fig. 1A; Methods):

equation image
(1)
equation image
(2)
equation image
(3)

Populations are affected by growth, antibiotic killing and mutation, where An external file that holds a picture, illustration, etc.
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Object name is pcbi.1000796.e036.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e037.jpg are the growth rate, antibiotic killing rate and frequency of resistance mutations per generation, respectively, and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e038.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e039.jpg are the effective doses of antibiotic felt by the wild-type and single-drug resistant mutant populations. We assume for simplicity that antibiotic resistance imposes no fitness cost, so that the growth rates of the sensitive and resistant populations are the same. To account for competition for resources, we assume this growth rate is given by the logistic equation,

equation image
(4)

where An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e041.jpg is the maximal growth rate, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e042.jpg is the total population size, and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e043.jpg is the maximal carrying capacity (Fig. S1B). This competition for resources was included in the in vivo murine infection model [21], and has been observed in or inferred from a range of infections [26], including S. pneumoniae [27], and Methicillin-Resistant S. aureus (MRSA) [28].

While we assume the growth rates of the wild-type and single-drug resistant mutants are the same, the rates An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e044.jpg at which they are killed by antibiotic are different and depend on the effective antibiotic dose, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e045.jpg, felt by each population:

equation image
(5)

where An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e047.jpg is the maximal killing rate and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e048.jpg is the Hill coefficient, which determines the steepness of the killing rate as a function of drug dose. In contrast to Jumbe et al., we set An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e049.jpg, which is representative of many common antibiotics [25], although different values of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e050.jpg give rise to similar overall model behavior (Fig. S2). The effective drug dose, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e051.jpg, depends on the dosage of the two drugs and on their interaction (Fig. 1B, Text S1, Fig. S1A). For simplicity we assume both drugs are administered at the same dose, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e052.jpg, defined in units of their minimum inhibitory concentration (An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e053.jpg), the single-drug dose at which the wild-type death rate equals its growth rate at resource-unlimited conditions: An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e054.jpg. For the wild-type, the effective dose is the sum of the dosage of the two drugs plus their level of interaction An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e055.jpg: An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e056.jpg. Values of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e057.jpg are positive, zero, or negative for synergy, additivity and antagonism, respectively. While in practice the value of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e058.jpg is specific to a given drug pair [29], we treat it as continuous in order to investigate all potential treatment strategies. We assume that single-drug resistant mutants are affected by only one of the drugs, which is reasonable in the case of resistance mechanisms that decrease the intracellular concentration of antibiotic, such as efflux pump expression or enzymatic degradation [11], [13], [30]. The effective dose of single-drug resistant mutants is therefore An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e059.jpg, and is independent of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e060.jpg (Fig. 1B). Except where indicated, we set An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e061.jpg (general model behavior is robust to changes in drug dosage, Fig. S2A), which for an additive drug pair is consistent with the drug dosage used in Jumbe et al. [21].

Mutations from wild-type to single-drug resistance, or from single- to double-drug resistance, arise at a rate An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e062.jpg per individual per replication, or An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e063.jpg per individual per unit time. Since in any effective treatment the number of double mutants arising is smaller than 1, we do not account for the growth or death of this fractional population, but rather define An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e064.jpg as the integrated number of double mutants generated via mutation during treatment (Eq. 3). Prevention of multi-drug resistance is then defined as An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e065.jpg.

The model therefore consists of Eqs. 1–5. Parameter values, following Jumbe et al. [21], are given in Table S1. Initial conditions for the model are An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e066.jpg, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e067.jpg, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e068.jpg - the population sizes at the onset of treatment. We assume that prior to treatment, the infections have grown from a single cell to the initial population size An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e069.jpg while mutating; while the overall mutation rate is a function of model parameters An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e070.jpg, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e071.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e072.jpg (Eqs. 2, 3), An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e073.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e074.jpg are functions of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e075.jpg alone: An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e076.jpg, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e077.jpg. No double-drug resistant mutants are present: An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e078.jpg. We integrate the ODEs with these initial conditions (Methods) and define An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e079.jpg as the time at which the total population size drops below one (An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e080.jpg is defined as infinity if the population reaches a non-zero steady state).

Antibiotic interactions create a saturable tradeoff between treatment efficacy and prevention of multi-drug resistance

To determine the impact of drug interaction on treatment outcome, we first looked at the differences in treatment efficacy (An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e081.jpg) and prevention of multi-drug resistance (An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e082.jpg) over a range of drug interaction values (An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e083.jpg) while holding drug dosage fixed (Fig. 2, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e084.jpg). We observed two distinct and robust (Fig. S2) behaviors, depending on whether An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e085.jpg falls above or below a critical value, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e086.jpg (Fig. 2; for the parameters used, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e087.jpg).

Figure 2
Choice of drug interaction presents a tradeoff between treatment efficacy and prevention of multi-drug resistance.

For An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e093.jpg, we observed a tradeoff between treatment efficacy and prevention of resistance. In this regime, increasing synergy yields greater An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e094.jpg (Fig. 2, unshaded region); this is expected, as increasing the synergistic interaction between the drugs kills the wild-type more quickly. Despite faster infection clearance, however, greater synergy actually decreases An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e095.jpg; namely, it increases the risk of multi-drug resistance. Conversely, more antagonistic drug pairs increase An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e096.jpg, albeit at the expense of reduced efficacy.

This tradeoff between efficacy and prevention of resistance breaks down at a critical threshold, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e097.jpg, above which increasing synergy no longer increases An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e098.jpg, but still decreases An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e099.jpg (Fig. 2, shaded region). Above this “synergy ceiling,” further increasing synergy therefore has only undesirable effects, since it increases the risk of multi-drug resistance without increasing efficacy. Optimal drug pairs for treating the given infection must therefore have a level of drug interaction lower than An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e100.jpg and, due to the tradeoff between An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e101.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e102.jpg, the optimal value of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e103.jpg will depend on the relative importance assigned to these two conflicting goals.

The frequency of resistance mutations determines the “synergy ceiling” An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e104.jpg

To understand what determines the level of the synergy ceiling, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e105.jpg, we asked what causes the transition from tradeoff behavior at An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e106.jpg, to plateau behavior at An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e107.jpg (Fig. 2). Due to the sharp biphasic behavior of efficacy (An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e108.jpg) around An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e109.jpg, we looked to population time courses to determine how the time of clearance, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e110.jpg, was affected by drug interactions below, at or above An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e111.jpg (An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e112.jpg, respectively; Fig. 3A). For An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e113.jpg (Fig. 3A, top), the wild-type subpopulation outlives the single-drug resistant mutants and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e114.jpg. Since wild-type killing is stronger for more synergistic drug pairs, increasing An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e115.jpg decreases An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e116.jpg, explaining why efficacy increases with An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e117.jpg in this region (Fig. 2, unshaded region). For An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e118.jpg (Fig. 3A; bottom), however, the wild-type is eliminated before the single-mutant population, and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e119.jpg. Because the killing rate of the single-drug resistant mutants is independent of An external file that holds a picture, illustration, etc.
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Object name is pcbi.1000796.e126.jpg; Fig. 3A, middle).

Figure 3
The synergy ceiling An external file that holds a picture, illustration, etc.
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Since An external file that holds a picture, illustration, etc.
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Object name is pcbi.1000796.e161.jpg and An external file that holds a picture, illustration, etc.
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Object name is pcbi.1000796.e164.jpg (Fig. S2), the strongest effect was due to changes in the frequency of resistance mutations, An external file that holds a picture, illustration, etc.
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Object name is pcbi.1000796.e180.jpg falls below zero (representing an antagonistic drug interaction); in this case mildly synergistic and even additive interactions fall in the undesirable regime where the risk of multi-drug resistance increases without any corresponding gain in treatment efficacy.

Competition for resources underlies the tradeoff between treatment efficacy and prevention of multi-drug resistance

Why does synergy, despite clearing the infection faster, increase the risk of multi-drug resistance (Fig. 2)? Since synergistic drug pairs clear the infection more quickly than antagonistic drug pairs, the rate at which they generate double mutants must also be higher. The overall rate at which double mutants arise, An external file that holds a picture, illustration, etc.
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Object name is pcbi.1000796.e183.jpg, due to the inhibitory effect of resource limitation on growth and mutation (Fig. 4A). The total number of double mutants expected to arise is simply the integral of this instantaneous rate over the treatment course. In order to determine why synergistic treatments increase An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e184.jpg, we therefore analyzed the trajectories of synergistic and antagonistic treatments through the space of An external file that holds a picture, illustration, etc.
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Object name is pcbi.1000796.e186.jpg (Fig. 4A; An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e187.jpg, solid line, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e188.jpg, dashed line). The initial slopes of these trajectories (Fig. 4A, arrows) are determined by the relative fitness of the wild-type and single-drug resistant populations under antibiotic treatment. The synergistic treatment selects strongly against the wild-type population, producing a trajectory with a steep slope that drives treatment into a region of high An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e189.jpg (Fig. 4A, red region); this is because the rapid decrease in wild-type population size relieves competition for resources, creating a window of opportunity in which the still-large single-mutant population can rapidly grow and mutate. Conversely, the antagonistic treatment selects only weakly against the wild-type, producing a trajectory with a shallow slope that skirts the high An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e190.jpg region. Antagonistic drug pairs therefore decrease An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e191.jpg in a competition-dependent fashion: weak killing of the wild-type maintains competition for resources, limiting growth and mutation of the single-drug resistant population until it is eliminated.

Figure 4
Prevention of multi-drug resistance by drug antagonism depends on resource competition.

It is important to note that resource competition is significant only at the beginning of treatment, when An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e209.jpg. If competition for resources is required for the advantage of antagonism over synergy in preventing resistance, then we should expect a decrease in initial population size to decrease this advantage. To test this prediction, we looked at the relative ability of our representative synergistic (An external file that holds a picture, illustration, etc.
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Object name is pcbi.1000796.e213.jpg (Fig. 4C, circles; sensitivity to other model parameters is minimal, Fig. S2). Indeed, we found that the advantage of antagonistic drug pairs in preventing resistance (An external file that holds a picture, illustration, etc.
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Object name is pcbi.1000796.e218.jpg, the trend reverses and synergy better prevents resistance (An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e219.jpg, above dashed line). This is because, for low population sizes, resource competition effects are negligible; An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e220.jpg therefore no longer depends on An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e221.jpg, and becomes a function of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e222.jpg alone. Because synergistic drug pairs better limit An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e223.jpg by quickly killing the wild-type population from which single mutants arise, they therefore also better limit multi-drug resistance in cases of weak competition. Indeed, this advantage of synergy disappeared entirely when we artificially turned off wild-type to single-mutant mutation, allowing only those single mutants present at the start of treatment to contribute to An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e224.jpg (Fig. 4C, triangles).

Importantly, when competition for resources is weak (An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e225.jpg significantly less than An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e226.jpg, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e227.jpg), the tradeoff between treatment efficacy and prevention of multi-drug resistance no longer exists (Fig. 4D; compare with Fig. 2). As a result, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e228.jpg is no longer useful as a “synergy ceiling” because, although drug pairs with An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e229.jpg do not further improve An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e230.jpg, they do improve An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e231.jpg. For infections with weak competition, the use of maximally synergistic drug pairs therefore represents the best possible treatment strategy.

Discussion

We used a population dynamic model of bacterial infection to determine what drug interactions best suppress the emergence of multi-drug resistance. Whereas antagonistic drug pairs kill bacterial populations more slowly, and therefore allow more time for resistance to emerge, they also decrease the selective advantage of resistant mutants. Which of these two opposing effects of antagonism dominates in determining its overall impact on the chance of evolving multi-drug resistance? Framing this problem in the context of a clinical infection, we asked how two measures of treatment outcome, treatment efficacy and prevention of multi-drug resistance, depend on drug interaction.

We found that the optimal drug interaction can be determined primarily as a function of two infection parameters: population size at the outset of treatment, and the frequency of resistance mutations (see summary of our results in Fig. S3). For clinically relevant scenarios where initial population sizes are well below the carrying capacity, competition for resources is weak and synergy, which is typically preferred in clinical settings for its superior treatment efficacy [3], [4], [20], is also expected to best prevent the emergence of multi-drug resistance.

Where resource competition is significant, however, strong synergy may not always be the optimal treatment strategy. Real infections frequently exhibit competition, due either to a scarcity of carbon or iron [26], [31], [32], or saturation of available adhesion sites (e.g. in biofilm formation [33], [34]). In our model, such competition is predicted to give rise to a tradeoff between treatment efficacy and resistance prevention: increased synergy leads to greater efficacy, but at the expense of an increased risk of multi-drug resistance. Importantly, this tradeoff saturates for levels of synergy greater than a critical value An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e232.jpg, above which greater synergy does not further increase efficacy, but still increases the risk of multi-drug resistance. If the goal is to minimize multi-drug resistance, then choosing drug interactions above this “synergy ceiling” may be counterproductive. This is especially important given the dependence of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e233.jpg on the frequency of resistance mutations: our model predicts that for infections where resistance rates are high (An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e234.jpg) An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e235.jpg may be negative (antagonistic), favoring the use of antagonistic drug pairs over mildly synergistic or even purely additive antibiotic combinations. Indeed, for the modified Jumbe et al. model that we study, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e236.jpg is nearly additive; and while the resistance frequency we use may be an overestimate (Jumbe et al. determined this as the rate of all mutations conferring only a 3-fold increase in the MIC), these and higher mutation rates have been identified in human pathogens [35], [36]. Together, the potential for strong competition and high mutation rates in infection suggest that the tradeoff and synergy ceiling behaviors observed in our model – as well as the ability of antagonistic drug pairs to minimize multi-drug resistance – may describe the properties of some clinical infections.

We emphasize that drawing concrete therapeutic conclusions from this study would be beyond its scope. Our model incorporates many simplifying assumptions: we assume An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e237.jpg to be a fixed value, although it has been observed to change with both the absolute and relative doses of the antibiotics administered [37], [38]; drug administration and pharmacokinetics are not considered, although they may significantly impact the evolution of resistance [23], [39][42]; resistance mutation rates per generation are assumed to be independent of growth and antibiotic-killing rates; and while we consider an idealized case in which multi-drug resistance arises from strong, sequential mutations conferring resistance to each antibiotic, real mutations may confer cross-resistance to both drugs simultaneously, or only partial resistance to a single drug [10], [14], [16]. One consequence of partial resistance is antibiotic killing of drug-resistant mutants for drug interactions above An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e238.jpg; while for strong resistance this killing would be minimal, weak resistance may allow enough killing to undermine synergy ceiling behavior (Fig. S4). Finally, we note that this model does not consider the impact of host immune defenses, which may substantially impact microbial growth and death rates in clinical infections [43], [44]; whether the influence of host defenses favors the use of some drug combinations over others, however, remains to be seen.

While these caveats indicate the limitations of this simple model and suggest important avenues for future study, our results make a number of novel predictions about the relationship between drug interaction and multi-drug resistance: that there exist conditions under which antagonistic drug pairs may better prevent multi-drug resistance despite their weaker efficacy; that there is a synergy ceiling to how much efficacy can be achieved by modulating drug interaction; and that, below this ceiling, changes in drug interaction may produce a tradeoff between inhibition and multi-drug resistance. By basing our model on a previous experimental model of infection [21], we have identified regions of parameter space in which such behaviors may be relevant in a clinical scenario, and which could be tested in future experimental models of infection. Finally, our model highlights the idea that the optimal choice of drug pair in treating an infection may be contextual: while strongly synergistic drug pairs seem the preferred strategy in scenarios where resource limitation and other forms of competition are negligible, antagonistic drug pairs may best prevent resistance in cases of high mutation rates and strong intra-infection competition. While present therapeutic knowledge generally favors synergistic drug pairs, our work motivates further research into the impact and potential utility of antagonistic interactions both in clinical and in ecological settings.

Methods

Model details

Our model consists of 3 ODEs (Eq. 1–3) describing the population sizes of the wild-type and single-drug resistant mutants (An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e239.jpg, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e240.jpg), as well as the number of double mutants expected to arise during a treatment course (An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e241.jpg). Parameter values for this model include first-order maximal growth (An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e242.jpg) and death rate (An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e243.jpg) constants, carrying capacity An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e244.jpg and mutation rate An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e245.jpg (per individual per generation), which were taken from the in vivo murine model investigated in Jumbe et al. [21] (Table S1). Initial population sizes (An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e246.jpg) were determined by assuming that, prior to treatment, the infections grew from a single cell to the initial population size An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e247.jpg while mutating, such that An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e248.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e249.jpg; unless otherwise indicated, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e250.jpg. ODEs were solved in MATLAB (Version 7.1, MathWorks, Natick, MA) using a built-in, numerical ODE solver (ODE45). To avoid artifacts associated with using continuous ODEs to describe finite populations, each step was modified with the assumption that the wild-type or single-drug resistant population is eliminated (size decreases to zero) if its size drops below one.

Supporting Information

Figure S1

Models of drug interaction and logistic growth. (A) Model of drug interaction. The effective drug dose for the wild-type strain, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e251.jpg, is a function of three variables: the doses of drugs A and B (An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e252.jpg, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e253.jpg) and the interaction parameter An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e254.jpg (Text S1). Isoboles of the wild-type effective dose (An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e255.jpg), are shown for additive (An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e256.jpg, black), synergistic (An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e257.jpg, red) and antagonistic (An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e258.jpg, blue) drug pairs. While for additive drug pairs the effective dose is a simple sum of the drugs' individual doses, synergistic or antagonistic drug pairs achieve the same effective dose with smaller or larger drug doses, respectively. All model simulations fall on the dashed line, where drug doses are equal: An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e259.jpg. (B) Logistic growth model. As the population size, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e260.jpg, increases, competition causes the growth rate, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e261.jpg, to fall from its maximal value, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e262.jpg, to 0 at the carrying capacity, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e263.jpg (Eq. 4). Unless otherwise indicated, in model simulations An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e264.jpg at the outset of treatment (black circle).

(0.11 MB TIF)

Figure S2

Prevention of resistance, and the synergy ceiling An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e265.jpg, are robust to changes in model parameters. To test the robustness of the model to parameter changes, we varied each parameter independently and measured its effect on both the relative ability of strongly synergistic and antagonistic drug pairs (An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e266.jpg) to prevent multi-drug resistance, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e267.jpg (solid black line), and the level of the synergy ceiling An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e268.jpg (solid blue line). All lines have undergone 5-point smoothing. Dashed lines indicate the points at which synergistic and antagonistic drug pairs prevent resistance equally well (An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e269.jpg, black), or the synergy ceiling is additive (An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e270.jpg, blue). In each panel, those points corresponding to the original set of model parameters are indicated by circles. (A) An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e271.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e272.jpg vary little with changes in drug dose D, (B) carrying capacity An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e273.jpg, or (C) maximal growth rate An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e274.jpg. (D) As previously discussed, increases in the frequency of resistance mutations An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e275.jpg decrease An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e276.jpg substantially (Fig. 3), while having no significant effect on An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e277.jpg. (E) Likewise, increases in the initial population size, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e278.jpg, significantly decrease An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e279.jpg (Fig. 4), but also decrease An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e280.jpg. (F) Changes in the Hill coefficient, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e281.jpg, of antibiotic killing had a more complex effect on An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e282.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e283.jpg: while An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e284.jpg was consistently less than 1 over a wide range of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e285.jpg (antagonism better prevents resistance), its magnitude was parabolic with An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e286.jpg, with antagonistic drug pairs having the greatest advantage for An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e287.jpg. An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e288.jpg also appeared parabolic with An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e289.jpg and was lowest (most antagonistic) at An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e290.jpg. In the limit of large An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e291.jpg, maximal antibiotic killing rates are achieved for both wild-type and single-drug resistant populations, regardless of drug interaction. Synergistic and antagonistic drug pairs therefore fail to differentially impact wild-type killing rates, and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e292.jpg at high An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e293.jpg (black line). Furthermore, saturation of killing rates causes An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e294.jpg to be greater than An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e295.jpg for all values of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e296.jpg; An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e297.jpg is therefore undefined for An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e298.jpg (blue line), though saturation of the wild-type killing rate still causes efficacy to effectively plateau for low values of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e299.jpg.

(0.37 MB TIF)

Figure S3

Optimal drug interactions as a function of resistance frequency and initial population size. The contour map shows the synergy ceiling, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e300.jpg, for a given combination of resistance mutation frequency, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e301.jpg, and population size at the start of treatment, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e302.jpg. An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e303.jpg decreases monotonically with increasing An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e304.jpg (as in Fig. 3), but is nearly unaffected by An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e305.jpg. The black line is a single contour above which antagonistic drug pairs prevent multi-drug resistance better than synergistic drug pairs (An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e306.jpg). Above this contour, greater synergy increases the chance of multi-drug resistance; the optimal drug interaction must therefore fall below An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e307.jpg, with its specific value depending on the priority assigned to treatment efficacy versus prevention of multi-drug resistance. Below the contour, however (An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e308.jpg), greater synergy decreases the chance of multi-drug resistance; the optimal drug interaction is therefore maximal synergy (region below the black contour is colored dark red), regardless of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e309.jpg. The magenta circle indicates the combination of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e310.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e311.jpg corresponding to the original set of model parameters.

(0.10 MB TIF)

Figure S4

Partial antibiotic resistance weakens synergy ceiling behavior. In the model we assume strong antibiotic resistance, such that the antibiotic-resistant subpopulation feels the effect of only a single drug; effectively, this makes the MIC of the drug to which it is resistant infinite and produces the familiar synergy ceiling, in which efficacy increases with An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e312.jpg up to a critical level An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e313.jpg and plateaus above it (blue line, Fig. 2; all lines have been shifted on the vertical axis for clarity). As previously discussed (Fig. 3), this plateau is due to the resistant subpopulation dying after the wild-type when An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e314.jpg. When we weaken the assumption of strong resistance, however (MIC<∞), drug interactions still affect drug-resistant mutants, even when such mutants are killed after the wild-type. This results in efficacy increasing over all An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e315.jpg, but retaining its characteristic biphasic profile (red, green, orange lines). This biphasic behavior is due to the stronger killing of the wild-type than the resistant mutant, which persists even with only partial resistance. While stronger resistance produces behavior similar to the typical synergy ceiling (MIC increases 100-fold, red line), weaker resistance (MIC increases 4-fold, orange line) yields a still-biphasic curve, but one in which increases in An external file that holds a picture, illustration, etc.
Object name is pcbi.1000796.e316.jpg improve efficacy substantially in all cases. The synergy ceiling behavior is therefore most relevant in cases of strong antibiotic resistance.

(0.08 MB TIF)

Table S1

Parameters used in this study.

(0.03 MB DOC)

Text S1

(0.02 MB DOC)

Acknowledgments

For helpful comments and discussion, we thank R. Jajoo, J.-B. Michel, R.C. Moellering, and R. Ward.

Footnotes

The authors have declared that no competing interests exist.

This work was supported in part by US National Institutes of Health grant R01 GM081617 (to RK) and a Herchel Smith Graduate Fellowship (to JPT). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

References

1. Levy SB, Marshall B. Antibacterial resistance worldwide: causes, challenges and responses. Nat Med. 2004;10:S122–S129. [PubMed]
2. Golan DE, Tashjian AH, Armstrong EJ, Armstrong AW, editors. Principles of Pharmacology: The Pathophysiologic Basis of Drug Therapy. Philadelphia: Lippincott Williams & Wilkins; 2005.
3. Pillai SK, Moellering RC, Eliopoulos GM. In: Antibiotics in Laboratory Medicine. Lorian V, editor. Philadelphia: Lippincott Williams & Wilkins; 2005. pp. 365–440.
4. Rybak MJ, McGrath BJ. Combination antimicrobial therapy for bacterial infections - Guidelines for the clinician. Drugs. 1996;52:390–405. [PubMed]
5. Fitzgerald JB, Schoeberl B, Nielsen UB, Sorger PK. Systems biology and combination therapy in the quest for clinical efficacy. Nat Chem Biol. 2006;2:458–466. [PubMed]
6. Loewe S. The problem of synergism and antagonism of combined drugs. Arzneimittel-Forschung-Drug Research. 1953;3:285–290. [PubMed]
7. Berenbaum MC. What is synergy? Pharmacol Rev. 1989;41:93–141. [PubMed]
8. Greco WR, Bravo G, Parsons JC. The search for synergy - a critical review from a response-surface perspective. Pharmacol Rev. 1995;47:331–385. [PubMed]
9. Keith CT, Borisy AA, Stockwell BR. Multicomponent therapeutics for networked systems. Nat Rev Drug Discov. 2005;4:71–78. [PubMed]
10. Michel JB, Yeh PJ, Chait R, Moellering RC, Kishony R. Drug interactions modulate the potential for evolution of resistance. Proc Natl Acad Sci U S A. 2008;105:14918–14923. [PMC free article] [PubMed]
11. Hegreness M, Shoresh N, Damian D, Hartl D, Kishony R. Accelerated evolution of resistance in multidrug environments. Proc Natl Acad Sci U S A. 2008;105:13977–13981. [PMC free article] [PubMed]
12. Klein M, Schorr SE. The role of bacterial resistance in antibiotic synergism and antagonism. J Bacteriol. 1953;65:454–465. [PMC free article] [PubMed]
13. Chait R, Craney A, Kishony R. Antibiotic interactions that select against resistance. Nature. 2007;446:668–671. [PubMed]
14. Sanders CC, Sanders WE, Goering RV, Werner V. Selection of multiple antibiotic-resistance by quinolones, beta-lactams, and aminoglycosides with special reference to cross-resistance between unrelated drug classes. Antimicrob Agents Chemother. 1984;26:797–801. [PMC free article] [PubMed]
15. Mwangi MM, Wu SW, Zhou YJ, Sieradzki K, de Lencastre H, et al. Tracking the in vivo evolution of multidrug resistance in Staphylococcus aureus by whole-genome sequencing. Proc Natl Acad Sci U S A. 2007;104:9451–9456. [PMC free article] [PubMed]
16. Cohen SP, McMurry LM, Hooper DC, Wolfson JS, Levy SB. Cross-resistance to fluoroquinolones in Multiple-Antibiotic-Resistant (MAR) Escherichia coli selected by tetracycline or chloramphenicol - decreased drug accumulation associated with membrane changes in addition to OmpF reduction. Antimicrob Agents Chemother. 1989;33:1318–1325. [PMC free article] [PubMed]
17. Normark BH, Normark S. Evolution and spread of antibiotic resistance. J Intern Med. 2002;252:91–106. [PubMed]
18. Larder BA, Kemp SD. Multiple Mutations in HIV-1 Reverse Transcriptase Confer High-Level Resistance to Zidovudine (AZT). Science. 1989;246:1155–1158. [PubMed]
19. Handel A, Regoes RR, Antia R. The Role of Compensatory Mutations in the Emergence of Drug Resistance. PLoS Comput Biol. 2006;2 [PMC free article] [PubMed]
20. Acar JF. Antibiotic synergy and antagonism. Med Clin North Am. 2000;84:1391–1406. [PubMed]
21. Jumbe N, Louie A, Leary R, Liu WG, Deziel MR, et al. Application of a mathematical model to prevent in vivo amplification of antibiotic-resistant bacterial populations during therapy. J Clin Invest. 2003;112:275–285. [PMC free article] [PubMed]
22. Craig WA. Pharmacokinetic/pharmacodynamic parameters: rationale for antibacterial dosing of mice and men. Clin Infect Dis. 1998;26:1–10; quiz 11–12. [PubMed]
23. Craig WA. Does the dose matter?; 2001. University of Chicago Press; pp. S233–S237.
24. Eagle H, Fleischman R, Levy M. “Continuous” vs. “discontinuous” therapy with penicillin; the effect of the interval between injections on therapeutic efficacy. N Engl J Med. 1953;248:481–488. [PubMed]
25. Regoes RR, Wiuff C, Zappala RM, Garner KN, Baquero F, et al. Pharmacodynamic functions: A multiparameter approach to the design of antibiotic treatment regimens. Antimicrob Agents Chemother. 2004;48:3670–3676. [PMC free article] [PubMed]
26. Smith VH, Holt RD. Resource competition and within-host disease dynamics. Trends Ecol Evol. 1996;11:386–389. [PubMed]
27. Lipsitch M, Dykes JK, Johnson SE, Ades EW, King J, et al. Competition among Streptococcus pneumoniae for intranasal colonization in a mouse model. Vaccine. 2000;18:2895–2901. [PubMed]
28. Dall'Antonia M, Coen PG, Wilks M, Whiley A, Millar M. Competition between methicillin-sensitive and -resistant Staphylococcus aureus in the anterior nares. J Hosp Infect. 2005;61:62–67. [PubMed]
29. Yeh P, Tschumi AI, Kishony R. Functional classification of drugs by properties of their pairwise interactions. Nat Genet. 2006;38:489–494. [PubMed]
30. Yeh PJ, Hegreness MJ, Aiden AP, Kishony R. Drug interactions and the evolution of antibiotic resistance. Nat Rev Microbiol. 2009;7:460–466. [PMC free article] [PubMed]
31. Freter R, Brickner H, Botney M, Cleven D, Aranki A. Mechanisms that control bacterial populations in continuous-flow culture models of mouse large intestinal flora. Infect Immun. 1983;39:676–685. [PMC free article] [PubMed]
32. Weinberg ED, Weinberg GA. The role of iron in infection. Curr Opin Infect Dis. 1995;8:164–169.
33. Nadell CD, Xavier JB, Levin SA, Foster KR. The evolution of quorum sensing in bacterial biofilms. PLoS Biol. 2008;6:171–179. [PMC free article] [PubMed]
34. Zhang TC, Fu YC, Bishop PL. Competition for substrate and space in biofilms. Water Environ Res. 1995;67:992–1003.
35. Martinez JL, Baquero F. Mutation frequencies and antibiotic resistance. Antimicrob Agents Chemother. 2000;44:1771–1777. [PMC free article] [PubMed]
36. Tucker SP, Stiebel TR, Potts KE, Smidt ML, Bryant ML. Estimate of the frequency of human immunodeficiency virus type 1 protease inhibitor resistance within unselected virus populations in vitro. Antimicrob Agents Chemother. 1998;42:478–480. [PMC free article] [PubMed]
37. Meletiadis J, Stergiopoulou T, O'Shaughnessy EM, Peter J, Walsh TJ. Concentration-dependent synergy and antagonism within a triple antifungal drug combination against Aspergillus species: Analysis by a new response surface model. Antimicrob Agents Chemother. 2007;51:2053–2064. [PMC free article] [PubMed]
38. Rand KH, Brown P. Concentration-dependent synergy and antagonism between cefoperazone and imipenem against methicillin-resistant Staphylococcus aureus. Antimicrob Agents Chemother. 1995;39:1173–1177. [PMC free article] [PubMed]
39. Lipsitch M, Levin BR. The population dynamics of antimicrobial chemotherapy. Antimicrob Agents Chemother. 1997;41:363–373. [PMC free article] [PubMed]
40. Roberts JA, Kruger P, Paterson DL, Lipman J. Antibiotic resistance - What's dosing got to do with it? Crit Care Med. 2008;36:2433–2440. [PubMed]
41. Knudsen JD, Odenholt I, Erlendsdottir H, Gottfredsson M, Cars O, et al. Selection of resistant Streptococcus pneumoniae during penicillin treatment in vitro and in three animal models. Antimicrob Agents Chemother. 2003;47:2499–2506. [PMC free article] [PubMed]
42. Foo J, Michor F. Evolution of Resistance to Targeted Anti-Cancer Therapies during Continuous and Pulsed Administration Strategies. PLoS Comput Biol. 2009;5 [PMC free article] [PubMed]
43. Grant AJ, Restif O, McKinley TJ, Sheppard M, Maskell DJ, et al. Modelling within-host spatiotemporal dynamics of invasive bacterial disease. PLoS Biol. 2008;6:757–770. [PMC free article] [PubMed]
44. Wodarz D. Ecological and evolutionary principles in immunology. Ecol Lett. 2006;9:694–705. [PubMed]

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