Schematic illustration of the MIC and MPC, and their extensions to multidrug environments. (A) Selection for resistance occurs primarily within the mutant selection window (MSW)—drug concentrations ranging from the MIC (dashed blue) inhibiting wild-type growth to the MPC (dashed red) above which the frequency of resistance (F_X) drops to nondetectable values. (B) In drug combinations, these notions extend to the MIC line (blue) and the MPC line (red). Resistance frequency [F_XY(C_X, C_Y), gray surface] is a function of the two-drug dosage and depends on the frequencies of resistance to the individual drugs, cross-resistance, and epistatic interactions. “Effective drugs,” obtained by combining X and Y at fixed proportions, are geometrically represented by lines extending from the origin (thick black line at angle θ), from which their MSW is defined (double-headed arrow). (C) The MSW of these effective drugs is plotted against the ratio of the drug combination (represented by θ); the smallest of these windows characterizes the drug combination's potential to limit the emergence of resistance. (D) The shape of the MIC line of two drugs defines their epistatic interactions: a linear line signifies no epistasis (Left), deviation below linearity signifies synergy (Center), deviation above linearity signifies antagonism (Right). The drug concentration marked by allows wild-type growth in the antagonistic case (therefore, F_XY = 1) but not in the synergistic case (F_XY ≪ 1). In both cases, frequencies of resistance to the individual drugs alone are the same (F_X = 1, F_Y = 1), illustrating that F_XY is not in general equal to F_XF_Y but rather depends critically on epistatic interactions.

A mathematical model for multidrug resistance that incorporates resistance to the individual drugs, epistatic interactions, and cross-resistance. We demonstrate the model by considering a same hypothetical bacterial population subjected to a single drug X (A and B) or to two drugs X and Y combined (C and D). (A) The population is composed of wild-type (WT) and mutant cells that differ in their MICs of drug X. The positions and heights of the bars represent the MIC and the frequency of the corresponding subpopulation. (Inset) Each subpopulation can grow at any drug concentration below its own MIC. (B) The frequency of resistance to any given concentration C_X is the sum of the frequency of the phenotypes that resist concentrations greater than C_X. (C) The same population also analyzed with respect to another drug Y—the bacterial cells' MICs of X and MICs of Y define different subpopulations. Here, the population is made up of five phenotypes represented by 3D bars (blue, wild type; shades of red, mutants) positioned at their MICs of drugs X and MICs of drug Y. The height of each bar represents the frequency of the corresponding subpopulation. Note the projections of these bars onto the x–z plane (gray flat bars) are the three subpopulations obtained in A; similarly, the projections onto the y–z plane are the two subpopulations which differ by their MIC of Y. (Inset) The regions of drug space where each phenotype can grow (e.g., one of the mutants, red) are approximated by scaling the wild-type growth region (blue), which characterizes epistasis. (D) The frequency of resistance at any given concentration (C_X, C_Y) is the sum of the frequencies of the phenotypes whose growth region contains the point (C_X, C_Y). The MPC line (red) is the line above which no mutant can grow.

Synergy is predicted to be less efficient than antagonism at reducing the potential for evolving resistance. We consider for simplicity a bacterial population containing only three subpopulations: the wild type, a mutant population resistant only to drug X, and a mutant population resistant only to drug Y (no double mutants or cross-resistance). (A and B) The frequency of resistance to combinations of X and Y strongly depends on their epistatic interactions (synergy in A, and antagonism in B). The MSW of the drug pair is defined by the smallest MSW of the “effective” drugs obtained by mixing X and Y at constant proportions. Here, in each case, the effective drug that possesses the smallest MSW is obtained for the proportion 1X:1Y (black lines with arrows). The MPC of that effective drug (solid line) is smaller relative to the MIC (dashed line) when the drug pair is antagonistic: the MSW of the combination is smaller for antagonistic than for synergistic epistasis. (C) The size of the MSW decreases as epistasis changes from synergy to antagonism. (Inset) This reduction in the size of the MSW (red) is a result of the MPC of the “best” effective drug (solid line) growing more slowly than its MIC (dashed line). As the epistatic interactions vary from synergy to antagonism, the MSW shrinks and eventually vanishes.

Measurement of resistance frequencies to pairwise drug combinations. (A) Frequencies of resistance of S. aureus to each of 11 × 11 concentrations of a drug pair measured by counting colonies arising on agar plates (see Materials and Methods). Example colony images are shown for the pair of antibiotics ERY-FUS at the final time point (5-day incubation) for the most concentrated cell inoculum (10⁹ cells per well). (B) Individual colonies detected by a custom image-processing platform—here, in false colors, on the plate labeled by an asterisk in A. (C–E) Measured frequencies of resistance (filled circles, or open circles if no resistant colonies appeared) plotted against the two-drug concentrations for FUS-ERY, CPR-AMP, and FUS-AMI. A standard polynomial interpolation surface is shown together with the points (gray surface). The MIC lines, measured independently in liquid media, define the nature of the epistatic interactions between the drugs (blue line, Insets). The MPC lines (red) represent the regions of drug concentrations above which no mutants appear.

Good agreement between experimental measures and theoretical predictions. Shown are comparisons of experimental (Left) and theoretical (Right) frequencies of resistance for FUS-ERY (A), CPR-AMP (B), and FUS-AMI (C) (blue, dark red, and light red as indicated). Theoretical surfaces for each drug pair were computed based on measured frequencies of resistance to each individual drug, measured epistasis between the drugs (MIC line of the wild type), and fitted cross-resistance (see Materials and Methods). Although the behaviors of the three pairs of drugs in terms of epistasis and cross-resistance are very different, the model accounts remarkably well for the frequencies of resistance and MPC lines (red). (Insets) MSW of “effective drugs” obtained by mixing the two drugs at different proportions; experimental points are plotted with their conservative error bars, and the lines correspond to the model prediction. FUS and ERY can be combined to significantly reduce the MSW compared with that of FUS or ERY alone, whereas this is not possible for combinations of CPR and AMP, or FUS and AMI (arrows).