The Bliss independence model1 corresponds to a multiplicative effect in growth measures, and is the preferred reference for synergy in some contexts2, especially genetics3. This model is a special case of the Bliss Boosting model, with boost parameter β = Emin(1 's Emax).

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 differs from both agents' high-dose levels by an amount determined by the boosting parameter β.


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 IBLISS = IX + IY + IX IY. For fold measures the expression is IBLISS = IX + IY, provided that fold increase is calculated as I = ln(Treated/Vehicle).

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 (b) 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 fitted curve values (option b) are used for the Bliss model's calculation.

Limitations and Constraints

This model is appropriate for most effect measures, including raw values, inhibition, and fold increases. However, the calculation only works well when both agents have effects that are positive at all concentrations (excluding noise) and monotonically increasing. When there are negative effect levels, the high-dose combination effects no longer vary in a way that makes sense for the intuition underlying the model. If the measurement scale differs from inhibition, the saturating and multiplicative reference levels need to be adjusted accordingly. For example, if we consider fitness ratios (treated over untreated growth ratios) with f = ln(T/U) in place of inhibitions, there is no saturating level and β = Emin indicates multiplicative. Note that Bliss masking is similar to both HSA and Loewe additivity for very high or very low concentrations where the single agent effects are almost flat.

  1. Bliss (1939), Ann. Appl. Biol. 26:585-615.
  2. Greco, Bravo, Parsons (1995), Pharmacol Rev 47:331-85.
  3. Mani et al. (2008), PNAS 105:3461-6.