Finding potent multidrug combinations against cancer and infections is a pressing therapeutic challenge; however, screening all combinations is difficult because the number of experiments grows exponentially with the number of drugs and doses. To address this, we present a mathematical model that predicts the effects of three or more antibiotics or anticancer drugs at all doses based only on measurements of drug pairs at a few doses, without need for mechanistic information. The model provides accurate predictions on available data for antibiotic combinations, and on experiments presented here on the response matrix of three cancer drugs at eight doses per drug. This approach offers a way to search for effective multidrug combinations using a small number of experiments.drug combinations | drug cocktails | cancer treatment | mechanism-free formula | predictive formula T o kill cancer cells or bacteria, combination therapy can be more effective than individual drugs (1-6). Combination therapy is thought to allow increased efficacy at low doses, thus reducing side effects and toxicity; it is also believed to minimize the chances of resistance (7-9), a pressing problem in treating cancer and infectious diseases.Much work has been devoted to classifying how pairs of drugs interact (10)(11)(12)(13)(14). Across systems, a good first approximation is the Bliss independence model (15,16), in which the pair effect is the product of the individual drug effects: If the effect of the drugs are g 1 and g 2 , the effect of the combination is g 12 = g 1 · g 2 . The Bliss model ignores interactions in which drugs enhance each other effectssynergism-or inhibit each other's effects-antagonism. Some drugs even inhibit each other so much that the combined effect is lower than either drug alone, an effect called hyperantagonism (17)(18)(19)(20)(21).Going beyond drug pairs has been difficult. Experimentally testing high-order combinations beyond pairs in a systematic way is challenging because it requires an exponentially large number of experiments (22-25): For N drugs at D doses, one needs D N experiments. For N = 10 drugs and D = 8 doses, this means ∼10 9 measurements. The combinatorial explosion makes exhaustive testing of drug dose combinations unfeasible. This problem is especially acute in cases where material is scarce, such as testing of patient-derived samples (26)(27)(28)(29). Hence, models for predicting high-order effects are essential.Apart from detailed simulations of particular systems (22,30,31), there has been little study of general mechanism-independent models for multiple drugs. An exception is the elegant study by Wood et al. (32) that showed that combinations of antibiotics can be predicted by an Iserliss-like formula that uses pair effects to predict the effects of higher-order mixtures. For example, the effect of a drug triplet is modeled as g 123 = g 12 g 3 + g 13 g 2 + g 23 g 1 − 2g 1 g 2 g 3 . This formula has not been tested on cancer drug mixtures, to the best of our knowledge.Another line of research u...