Drug resistance, which arises in biological systems ranging from bacterial infections to cancer, represents a growing public health threat that is often combated with multi-drug therapies. However, a potent drug combination designed to target a particular infection, for example, may be rendered ineffective by the rapid evolution of drug-resistant mutants. In addition, each new mutant may respond to the treatment in a unique way. Hence, the rate of increasing biological diversity outpaces our ability to adapt the treatment. To address this problem, we hypothesized that drug-resistance modifies how cells interpret extracellular drug concentrations but preserves all other features of the original multi-drug response. To formalize this hypothesis, we first developed a single class of phenomenological models for two-drug interactions that allowed us to compare the multi-drug response in wild type and mutant cells using a common framework. Then, we experimentally showed that resistance mutations act as a nonlinear transformation of drug concentration space but leave undisturbed all other features of the original two-drug growth response. This transformation is difficult to identify from traditional pharmaceutical methods but can be uncovered when drug interactions are cast as modifications of effective drug concentrations. The framework reduces the two-drug response surface to three simpler, one-dimensional "basis" functions: the single drug dose response curves for each drug alone and a single drug-drug coupling function. We found that, surprisingly, drug-resistant and drug-sensitive cells—even those with drastically different response surfaces—are characterized by basis functions related by simple scaling parameters. We verified these scaling laws using combinations of antibiotics targeting both gram-positive (S. aureus, E faecalis) and gram negative (E. coli) drug-resistant bacteria, combinations of antifungal drugs targeting drug resistant yeast (S. cerevisiae), and combinations of anti-cancer drugs targeting drug-resistant cancers, including melanoma, non small cell lung cancer (NSCLC), breast cancer, and cancer stem cells. Finally, we exploited these scaling relationships to develop a method for rapidly characterizing the response surfaces of drug resistant mutants using only a small number of data points. These results identify a scaling law connecting drug-sensitive and drug-resistant cells that can be exploited to provide rapid, systematic adaptation of multi-drug therapies targeting emergent drug- resistance.
Single drug toxicity functions and coupling functions (D2,eff/D2) for drug- resistant mutants can be rescaled to match those in parental E. coli.
A. Two drug response of resistant mutants are described the 3 "basis" functions: single drug toxicity functions (top left, top center) and D2,eff/D2 functions (top right) for 18 mutant strains isolated by selection in chloramphenicol (Cm) and ciprofloxacin (Cip) at various doses. Each color / marker combination represents a single mutant. Drug concentrations are in units of μg/ml for chloramphenicol and ng/ml for ciprofloxacin. D2,eff/D2 functions are constructed point-by-point from raw growth data. Error bars can be further reduced by using alternate parameterizations, which minimize overfitting.
B. Single drug toxicity functions (bottom left, bottom center) and D2,eff/D2 functions (bottom right) for all mutant strains are related by simple rescaling to the corresponding function in the wild type cells . A set of three scaling parameters, (a1, a2, a3), provide a set of coordinates that define each mutant. The parameters a1, a2, and a3 describe the change in resistance to drug 1, the change in resistance to drug 2, and the change in the drug interaction in the resistant mutant. Solid line, bottom panels: single toxicity functions (left and center panel) and D2,eff/D2 function (right panel) that best describe rescaled data.