Additivity of inhibitory effects in multidrug combinations


From natural ecology1,2,3,4 to clinical therapy5,6,7,8, 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)9,10,11. Here, noting that these models diverge with increased number of drugs, we contrast their predictions with growth measurements of four phylogenetically distant microorganisms including Escherichia coli, Staphylococcus aureus, Enterococcus faecalis and Saccharomyces cerevisiae, under combinations of up to ten different drugs. In all species, 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 R. A. Fisher’s classical geometric model12, dosages of independent drugs add up as orthogonal vectors rather than linearly. This dose-orthogonality approximation provides results similar to Bliss, yet uses the dosage language as in Loewe and is hence easier to implement and intuit. The rejection of dosage additivity in favour of effect additivity and dosage orthogonality provides a framework for understanding how multiple drugs and stressors add up in nature and the clinic.

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Fig. 1: Schematic depiction of effect additivity (Bliss) and dosage additivity (Loewe).
Fig. 2: Pairwise measurements do not resolve the Bliss and Loewe models of additivity.
Fig. 3: The Loewe model of additivity loses its predictive power with increased number of combined drugs.
Fig. 4: A square-root scaling law of inhibitory total dosage with effective number of drugs is explained by a simple dosage-orthogonality model.

Data availability

The data sets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.


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We thank U. Alon and I. Katzir for thoughtful discussions and suggestions, Y. Arava for strains, and members of the Kishony lab, especially M. Datta, E. Tamar, I. Yelin and N. Yin, for experimental support and comments on the manuscript. This work was supported in part by the US National Institutes of Health grant R01-GM081617, the Israeli Centers of Research Excellence I-CORE Program ISF grant 152/11 and the European Research Council FP7 ERC Grant 281891 (to R.K.).

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Correspondence to R. Kishony.

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Supplementary Information

Supplementary Information

Supplementary Notes, Supplementary Figures 1–13 and Supplementary Tables 1–8.

Reporting Summary

Supplementary Table 9

Dose response of all drug combinations.

Supplementary Table 10

Measured and predicted potencies of drug combinations.

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Russ, D., Kishony, R. Additivity of inhibitory effects in multidrug combinations. Nat Microbiol 3, 1339–1345 (2018).

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