The morphology of cell-based dose-matrix responses to chemical combinations differs between mechanisms. Data for two synergistic antibacterial combinations using a methicillin-resistant Staphylococcus aureus strain are shown in three-dimensional projections. In Augmentin^® (A), clavulanate disables a bacterial defence against penicillin drugs, whereas both agents in Bactrim^® (B) inhibit enzymes in folate metabolism. The mechanisms of action differ between the two combinations, as do the shapes of their response surfaces. Dose-matrix response surfaces show the inhibition of growth (1−treated/untreated) for each pairwise permutation of the serially diluted single agent doses. The 2D maps (C, D) show a top-down view of the surface, using the same colours for inhibition levels, to better display mathematical descriptions of the shape without obscuring any data, and permit many matrices to be shown together with clarity.

The best-fit shape models for our simulated pathway (Figure 3) consistently depend on target connectivity. Combination effects between competitive inhibitors are shown as symbols whose area is proportional to the HSA volume and whose colour indicates the best-fit model shape: black for HSA, blue for Loewe additivity, green for Bliss boosting, red for potentiation, and grey when two or more models could not be distinguished (see Supplementary Figure S4). The Bliss boosting model is subdivided using shape to indicate the boosting level (square for saturating, diamond for multiplicative, circle for masking, and open circle for suppressing). The bold lines separate Pathway A from Pathway B combinations, and the thin lines partition Pathway A into its component sections. All inhibitor-with-self pairings fit Loewe additivity, as expected. Other combinations within Pathway A consistently fit the Bliss model with distinct boosting levels. Cross-target combinations within the negative feedback in Pathway B consistently show potentiation, and cross-pathway combinations always have masking responses (close to Bliss masking, additivity, or HSA).

Simulations of a multiply inhibited network yield distinct response shapes that depend on target connectivity. Substrates (black nodes) are metabolized through a series of Michaelis–Menten reactions (grey circles) from sources with constant reaction velocities to a limitless sink at the end-point. We calculated dose-matrix inhibitions of the end point velocity at R_E to paired competitive inhibitors at various enzymes. The response surfaces for some combinations are shown, with the inhibitors indicated by joined markers. For Pathway A, same-target pairs produced Loewe additivity, and separated targets led to various Bliss boosts depending on where the targets were placed. Pathway B was added to investigate combination effects across pathways, with a bypass to ensure clear distinctions between cross-pathway boosting levels. Because both inputs are required for R_AB, the junction is equivalent to a logical AND function. The negative feedback in Pathway B was introduced to represent processes like sterol biosynthesis in yeast. Inhibiting across pathways produced HSA-like masking effects, and inhibiting within the negative feedback loop yielded potentiation effects.

Combination effect statistics from the HCT116 screen show expected trends, and the exceptionally strong synergies between inhibitors of EGFR signalling (Table II) reflect known connections. The distributions of HSA volumes (A) and synergy profile correlations (B) are shown for drug pairs in the HCT116 tumour cell screen (Supplementary 6), grouped by target similarity (‘Identical' for drug-with-self combinations; ‘Same' for distinct drugs sharing a target; ‘Related' for differing targets sharing a common functional group; ‘Different' for pairs with different functions; and a separate group ‘Kinases' for combinations between kinase inhibitors). There is a significant trend towards more variable synergy and lower profile correlations for more distantly related targets (Supplementary 7), and the most synergistic and highly correlated pairs within the ‘Kinases' set are dominated by those targeting the EGFR signalling pathways, which are major drivers of HCT116 proliferation. (C) Combinations between MAPK/PI3K, MAPK/PKC, and PI3K/PKC targets show different levels of response that are consistent across mechanistic replicates, despite limited concentration sampling (only TUBC and ROTT were optimal) and evidence of off-target effects (synergy between PI3K inhibitors and discrepant ROTT effects compared to the other PKC inhibitors). Shape model degeneracies are shown in Supplementary Figure S6.

Combination response shape models that describe many of the observed response morphologies. Each model (shown using the same colour scale as Figures 1 and 3) is used to calculate an expected combination effect I_model at any concentration X,Y, based on the single agent response curves. (A) HSA is a superposition of the X and Y single agent responses, calculated from the inhibitions I_X at X and I_Y at Y. (B) Loewe additivity (Loewe, 1928) is the drug-with-itself reference for synergy, where I_Loewe at X,Y yields additive doses relative to the components' effective concentrations X_I,Y_I at I_Loewe. (C) Bliss boosting describes combinations with a variable boost β above E_max (the greater of the single agent limiting efficacies E_X,E_Y), at high combined concentrations. Useful reference levels for Bliss boosting are ‘cancelling', ‘suppressing', ‘masking', ‘multiplicative' corresponding to Bliss independence (Bliss, 1939), and ‘saturating' (see Materials and methods). Finally, (D) potentiation can characterize responses similar to those of Bactrim^® (Figure 1), where one single agent's curve I_X(C) is shifted with a power-law slope p above an enhancer concentration Y_pot, and superposed on the enhancer's own activity. These models can be extended to higher-order combinations, and used in the same form (with adjustments to Bliss boosting) for any type of measurement, provided that the activities of both agents vary monotonically with concentration. Of these models, only Loewe additivity has an a priori mechanistic basis.

Validation of expected combination effects in a yeast experiment focused on sterol biosynthesis. In yeast, the sterol pathway (left) produces ergosterol, an essential cell wall component, through a series of metabolic enzymes (grey symbols) acting on intermediate substrates (black symbols), and regulated by negative feedback (dotted connections). The experiment tested C. glabrata proliferation responses to all combinations of 10 antifungal drugs (Table I), six with known targets on the sterol pathway (inhibitor markers). The results are shown (centre) with concentrations increasing from the bottom left of each drug pair's dose matrix. The combination effect symbols (right) summarize the observed response for each pair in terms of the models presented in Figure 2. Size shows synergy as measured by V_HSA, the volume between the data and HSA surface. The best-fit shape model is indicated by colour: black for HSA, blue for Loewe additivity, green for Bliss boosting, red for potentiation, and grey when two or more models could not be distinguished (see Supplementary Figure S5). Drug pairs sharing a target mostly produced weak responses (V_HSA<1) consistent with Loewe additivity, and cross-target sterol combinations produced potentiation, as expected for a linear pathway under negative feedback (∼70% prediction accuracy). Combinations involving non-sterol drugs showed fewer strong combination effects and their synergy profiles across all the drugs were more variable.