Bliss boosting1, which models boosts in efficacy at high concentrations different from what the single agents can achieve, is adapted the Bliss independence model2 that corresponds to a multiplicative effect in growth measures. The one free parameter β, in units of effect, determines the amount of boosting above Emax, the greater of the single agent efficacies. This model is used to quantify the strength of a combination effect in terms of its high-dose boost above the single agents.

For inhibition effects, there are a number of useful reference levels: β = 'Emax produces "canceling", with zero effect at high concentration; β = Emin 'Emax is "suppressing", where the less effective agent prevails; β = 0 yields "masking", where the more effective agent prevails; β = Emin(1 'Emax) is multiplicative (Bliss independence); and β = (1 'Emax) produces "saturating" at 100% inhibition. At the high concentration limit, the suppressing, masking, multiplicative, and saturating levels correspond to "suppression", "buffering", "no epistasis" and "synthetic lethal" in recent classifications for epistasis3,4>\.

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 IBOOST = IX + IY + (β 'Emin) (IX IY/EX EY), where Emin is the lesser of EX and EY, the limiting single agent efficacies. For fold measures the expression is the same provided that fold increase is calculated as I = ln(Treated/Vehicle). The boosting parameter is found using an iterative root finding algorithm which identifies the value of β with the lowest chi-squared difference between the model and the data. The best fit boost parameter is reported along with its standard error, estimated from the range providing a unit change in reduced chi-squared.

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 Boost 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. Lehar et al. (2007), Mol Syst Biol 3:80.
  2. Bliss (1939), Ann. Appl. Biol. 26:585-615.
  3. Segra et al. (2005), Nat Genet 37:77-83.
  4. Mani et al. (2008), PNAS 105:3461-6.