The Highest Single Agent (HSA) model, or Gaddum's non-interaction reference (Berenbaum 1989), is based on the intuition that if a combination's effect exceeds those of its constituents, there must be some interaction. Mathematically, the HSA model is a superposition of the single agent curves where, at any combined concentration (X,Y), the model inhibition IHSA = max(IX, IY), where IX and IY, are effects produced by the single agents at (X,0) and (0,Y) respectively. The model values are calculated at each dose matrix point, where IX and IY were determined using sigmoidal fits to the single agent response data.

This model is the expectation for combinations of chemicals with only positive effect values and whose targets have no functional connection. The HSA model predominates in experimental combination screens, and is the product of simulated chemical combinations of inhibitors with unrelated targets1. Combination responses above this reference can arise either from effect boosts or potency shifts.

Model Response Surface

Single agent responses at varying concentrations X and Y are shown along the bottom and left edges of the dose matrix, using colors that run from black (no effect) through the rainbow to pink/white (total effect). Here, the X-agent reaches ~60% and the Y-agent ~40% effect.

The response follows the closest single agent until concentrations where both are active, at which point the combined effect matches that of the stronger agent at corresponding concentrations.


In Chalice, the model values are calculated at each dose matrix point based on the single agent response curves. For a combination point at concentrations X,Y, the corresponding single agent effect levels IX and IY are determined and the model value is calculated as IHSA = max(IX, IY). The single agent effect levels are set as a user-controlled option to be either: (a) a linear interpolation between single agent values, returning blank if data do not exist both for higher and lower non-zero single agent concentrations; or (c) the calculated response value for a sigmoidal fit to the single agent curve data, returning blank only if the model evaluation fails (see SigmoidCurve description). By default, the linear interpolations (option a) are used for the HSA model's calculation.

Limitations and Constraints

This reference model is appropriate for all effect measures, including raw values, inhibition, and fold increases. However, the HSA calculation only makes sense when both agents have effects that are positive at all concentrations (excluding noise). When there are negative effect levels, it becomes unclear which effect is "higher" when the single agents go in different directions.

  1. Lehar et al. (2007), Mol Syst Biol 3:80.