Abstract
From natural ecology 1–4 to clinical therapy 5–8, cells are often exposed to mixtures of multiple drugs. Two competing null models are used to predict the combined effect of drugs: response additivity (Bliss) and dosage additivity (Loewe) 9–11. Here, noting that these models diverge with increased number of drugs, we contrast their predictions with measurements of Escherichia coli growth under combinations of up to 10 different antibiotics. As the number of drugs increases, Bliss maintains accuracy while Loewe systematically loses its predictive power. The total dosage required for growth inhibition, which Loewe predicts should be fixed, steadily increases with the number of drugs, following a square root scaling. This scaling is explained by an approximation to Bliss where, inspired by RA Fisher’s classical geometric model 12, dosages of independent drugs adds up as orthogonal vectors rather than linearly. This dose-orthogonality approximation provides results similar to Bliss, yet uses the dosage language as Loewe and is hence easier to implement and intuit. The rejection of dosage additivity in favor of effect additivity and dosage orthogonality provides a framework for understanding how multiple drugs and stressors add up in nature and the clinic.
Main Text
In both nature and the clinic, cells are often exposed to combinations of multiple stresses and drugs. In natural ecosystems, such as the soil, dozens of microbial species capable of producing different antimicrobial compounds coexist in very close proximity, thus exposing each-others to a mixture of multiple stressors 1–4 In clinical settings, drug combinations, aimed at reducing side effects and counteracting resistance 13–17, are becoming increasingly important in treatment for infectious diseases and cancer 5–8, 18, 19. It is therefore of wide importance to understand how cell growth is affected by combinations of multitude of stressors and what are thereby the general rules of high-dimensionality drug arithmetics.
When combined together, drugs can interact to synergize or antagonize each other effects relative to a null additive model. Synergy occurs when the combined effect of drugs is larger than expected based on their individual effects. Conversely, drugs can also antagonize each other, leading to a combined effect that is smaller than expected. These interactions are important in clinical settings as a way of increasing treatment potency and selectivity 20–22 or slowing selection for resistance 14,17,23. Importantly, the definition of both synergy and antagonism relies on comparing the effect of drug combinations with a null model of “additive expectation” 6,24–27.
There are two primary models for the null effect of drug combinations 6,28: the Bliss model 10,29, which assumes response additivity and the Loewe model 9,11, which assumes dose additivity. According to Bliss, the combined effect of two drugs E1+2 is simply the sum of their individual effects 30, E1+2 = E1 + E2, where E, = (g0−gi)/g0 is the effect of drug i on the normalized growth rate g/g0 (Fig. 1; when effects are measured based on total yield rather than growth rate, Bliss additivity becomes multiplicativity; Supplementary Note 1). In contrast, according to Loewe additivity, the effect of drugs in combination is determined not by the sum of their normalized effects, but rather by the sum of their normalized dosages, such that their combined effect is the same across all combinations that have the same total normalized dosage. Namely, according to Loewe, lines of equal combination effect in drug-dosage space (isoboloes), are linear 26 (Fig. 1). For example, if two drugs are additive with respect to Loewe, their 50% inhibition isobole is a straight line satisfying d1/d150 + d2/d250 = 1, where di is the dose of drug i and is the dose at which drug i alone causes 50% growth inhibition (IC50). Though conceptually different from one another, mechanistic support is available for both the Bliss and the Loewe models 31 and there is no agreement on which model should generally be used 32–35. Models that implement pairwise interaction data as well as higher order interactions can improve multi-drug predictions of either Bliss or Loewe 21,36–39. Yet, regardless of pairwise interactions, it remains unknown which of these two null models best predict the combined effect of multiple drugs.
Here, measuring bacterial response to antibiotic combinations, we contrast the Bliss and Loewe models for an increasing number of drugs, where we show that expectations of these models increasingly diverge. Quantitating bacterial response to combinations of up to 10 different drugs, we find that the Bliss model maintains accuracy with increased number of drugs, while the Loewe model loses its predictive power. Indeed, in contrast to Loewe, which predicts that the total drug dosage required for inhibition is constant, we find that total dosage increases monotonically with the number of drugs. Interestingly, our data show that this increase follows a square-root scaling, inspiring a simple model for orthogonality of dose additivity which follows a classical evolutionary optimization principle developed by R. A. Fisher 12.
To contrast the Bliss and Loewe models, we calculated how their predictions scale with increased number of drugs. As a natural measure of the combined potency of multiple drugs, we considered the total drug dosage needed to achieve a given level of inhibition. Defining “total dosage” (D) as the sum of the concentration of the individual drugs weighted by their corresponding IC50’s, , the “combination potency” (D50) is the total dosage D that yields 50% growth inhibition. As the number of drugs N increases, Loewe additivity predicts that the combination potency remains fixed, . The prediction of Bliss, on the other hand, depends on the individual dose response curves of the different drugs. Assuming a Hill equation 40 for the single drug dose response , where h is the Hill coefficient, and equating the Bliss prediction of the combined effect to 50%, yields the Bliss predicted scaling of combination potency with the number of drugs: . Thus, while Loewe predicts that the total dosage required for inhibition is constant, Bliss predicts that it increases with the number of drugs. The two models can therefore best be contrasted by measuring the combined action of increased number of drugs.
We considered 10 mechanistically different antibiotics and measured their effects on the growth rate of Escherichia coli populations, individually as well as in combinations of increased number of drugs. We chose antibiotics acting on a range of cellular functions, including cell wall synthesis, DNA replication, transcription and translation (Table 1). Measuring optical density (OD) versus time of bacterial growth on gradients of each of the individual drugs, we determined the dose response curve g(di) for each of the drugs (Fig. 2a, Extended Data Figs 1–4, Extended Data Tables 1 and 2). These dose response curves are well fit by Hill functions, with Hill coefficients of the individual drugs ranging from 1 to 6.9 (Extended Data Figs 5 and 6). These fits also define the concentrations of 50% inhibition for each of the drugs in isolation.
Moving to drug pairs, we measured their combination potency and compared it to Bliss and Loewe predictions. We first measured the full response surface across 2-D dose gradients for two drug pairs: Tetracycline and Ciprofloxacin (TET-CIP) and Tetracycline and Erythromycin (TET-ERY), representing well-known examples of antibiotic antagonism and synergy (Figs 2b and 2c, response surface and IC50 isoboles) 41,42. Using the growth measurements of the individual drug gradients Ei(di), we derived the response surface and the IC50 isobole predictions of Loewe (straight line connecting the points and ) and Bliss (the set of all points [d1,d2] satisfying E1(d1)+ E2(d2) = 50%, Methods). As expected, the measured IC50 isobole lie above these predictions for the TET-CIP pair (indicating antagonism) and below for the TET-ERY pair (indicating synergy). While these two-dimensional gradients allow clear definition of synergy and antagonism, they require many growth measurements and become combinatorially prohibitive in a high-dimensional multi-drug space.
To effectively sample the concentrations space of multiple drugs, we performed growth measurements along a “co-potent” line 38, a curve in concentration space where the individual drugs have equal potencies in isolation (E1(d1) = E2(d2) = … = EN (dN), Figs 2b-e). This co-potent line sampling method vastly reduces the dimensionality of required measurements while guaranteeing that null models are evaluated in a region in drug concentration space where all drugs are active, rather than one in which the combined effect is dominated by a subset of drugs. Identifying the point P =(d1, d2,…, dN) on the co-potent line where growth is inhibited by 50% yields the combination potency, . This measured combination potency was contrasted with the expected potencies and , defined as the points along the co-potent line where the single-drug based calculations of Bliss and Loewe predict 50% inhibition (Supplementary Methods). The interaction between drugs was then defined as the deviation between the observed and predicted potencies of the combination , which captures the extent of antagonism (ε>0) and synergy (ε<0) (Figs 2d and 2e).
Measuring combination effects for all drug pairs, we find that their joint potencies are similarly well-predicted by both the Bliss and the Loewe models. For each of the 45 drug pairs, we measured their dose response along co-potent concentration gradient and determined their combination potency, (supplementary methods, Fig. 2f, Extended Data Fig. 4). Comparing these combination potencies with predictions of the Bliss and Loewe null models, we find that both positive (ε > 0, antagonism) and negative (ε < 0, synergism) deviations are prevalent with respect to either model (Figs 2f and 2g). This prevalence of both antagonism and synergy among drug pairs overwhelms any small deviations between the two models (Fig. 2g; σ(εBliss)= 0.40; σ(sLoewe) = 0.41; < εBliss >−< εLoewe > =0.06, t-test: P=0.49). Further, clustering drugs based on these pairwise interactions, defined with respect to either Bliss or Loewe, leads to similar grouping by mechanism of action (Extended Data Fig. 7; possible small advantage to Bliss in resolving fine functionality differences) 43. The similarity of pairwise null predictions, the prevalence and magnitude of pairwise interactions with respect to both models, and their similar correlation with cellular function, prohibit distinction of the Loewe and Bliss null models based on drug pairs.
However, for increasing numbers of drugs, we find that their combined effect is well predicted by Bliss, while the Loewe prediction systematically diverges. Given that predictions of the two models should diverge with increased number of drugs, we measured the combined effect of multiple combinations with a varying number of antibiotics. We chose 35 combinations of three to ten antibiotics, including 8 randomly chosen sets from each combination size of N = 3, 5 and 7 drugs, all 10 sets of 9 drugs, and the whole 10-drug set (Figs 3a and 3b, Extended Data Fig. 8). Following the procedure used for the drug pairs, we measured the combined effect of each multi-drug set as a function of total dosage along co-potent lines and identified their combination potencies D50. Contrasting these measured potencies with the predicted potencies of Bliss and Loewe based on the single-drug measurements, we find that the Bliss model maintain good accuracy regardless of the number of drugs, while the accuracy of Loewe model declines as the number of drugs increases (Fig. 3c). The multiplicative form of the Bliss model (more suitable for yield measurements, Supplementary Note 1) is less predictive than the additive form, yet still much better compared to Loewe (Extended Data Fig. 9). We conclude that the Loewe model, predicting that the total dosage required for inhibition is fixed regardless of the number of drugs, can be rejected as a general predictor for the potency of multi-drug combinations.
Next, we tested how the potency of drug combinations, namely the total dosage required for inhibition, varies with the number of drugs. To account for any slight experimental deviations from the ideal co-potent line, we use a natural entropy-like definition of an effective number of drugs Neff which is based on the uniformity of the individual drug effects (Neff equals N if all drugs have the same effect; is slightly smaller than N when these effects are uneven; and converges to 1 at the extreme case when a single drug dominates and all the other have no effect; see definition of Neff in Fig. 4 caption). Contrary to the Loewe prediction, we find that the total dosage required for inhibition increases with the effective number of drugs (Fig. 4a). Moreover, this inhibitory total dosage seems to obey a simple scaling law: it increases as the square root of the effective number of drugs (D50 = (Neff)α, least square fit yields: α = 0.48 ±0.03).
The square root scaling of the inhibitory dosage with number of drugs can be explained by an approximation to Bliss, inspired by the classical optimization principle of Fisher’s geometric model of adaptation 12. Fisher’s model describes the fitness f in a space of N independent orthogonal phenotypes and assumes that it declines as a function of the euclidean distance from an optimal point (, where and xi are phenotypic distances from the optimal point). For a given fitness value, the phenotypic distances xi therefore decline as and their sum, , increases as . The analogy of drug dosages with Fisher’s phenotypes explains the square root scaling of total inhibitory dosage with number of drugs and underscores that drug concentrations should be summed not linearly by simple addition as in Loewe, but rather as the geometric sum of orthogonal vectors (Fig. 4b; Of course, orthogonality is a null idealization from which drug combinations can deviate due to synergy or antagonism, or when similar drugs act along the same axis). This Fisher’s inspired “dose-orthogonality” model can also be derived as an approximation of Bliss additivity at the limit of small dosages (Supplementary Note 2). Indeed, we find that even strongly interacting drug pairs assume more circular isoboles for small fitness effects (Figs 2b, 2c, 4c). Similarly to Bliss and in contrast to Loewe, combination potency predictions of the dose-othethognality model (derived by intersecting the co-potent lines with spherical IC50 isoboloes , Methods), are consistent with the drug combination measurements (Fig. 4d). Yet, unlike Bliss these predictions do not require fine measurements of the minute individual drug effects Ei(di), but rather depend on the more robust measurements of their individual IC50 dosages, . Using Loewe’s dose language, the dose-orthogonality model provides an intuitive and robust approximation of Bliss (Extended Data Table 3), which predict the potency of drug combinations similarly well and explain the square-root scaling law of potency with number of drugs.
Our measurements reject the Loewe model of dosage-additivity for predicting combination of multiple diverse drugs, favoring the Bliss effect-additivity and motivating a simple model of dosage-orthogonality. In contrast to Loewe additivity, which predicts that the total dosage required for inhibition is fixed, we find that the total inhibitory dosage increases with the number of drugs, following a square root scaling law. This general reduction in potency with increased number of drugs implies that bacterial inhibition by multi-drug combinations may be require higher total drug dosages than classically anticipated. The square root scaling supports a model for drug additivity where dosages of independent drugs add up orthogonally rather than linearly. This dosage-orthogonality model provides an approximation to Bliss, yet it uses dosage arithmetics which allows a more robust implementation and simple intuition. It will be interesting to explore the generality of these results and the limit on the number of orthogonal pharmacological axes as more antibiotics and stresses are added, as well as beyond the minimal inhibitory concentration and in more complex systems such as in cancer therapy. Throughout such clinical systems and natural ecologies, our findings provide a uniform framework for understanding the null arithmetics of many-drug combinations.