The power-law "potentiation" model describes strong shifts in potency1. For such combinations, an active single agent's response curve shows an increase in potency as the enhancer is titrated in, which appears as linear iso-effect contours in logarithmic concentration space. The two free parameters for this model are the threshold concentration Ypot above which potentiation takes effect, and the potentiation slope p (synergy for positive and antagonism for negative p). There is no potentiation for p = 0, where this model reduces to an HSA surface. Just as for HSA and Loewe additivity, the form of this model is identical for any type of effect measure.

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 more effective single agent.


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 inhibition IPotent = max( IX(C) , IY ), where IX(C) is the single agent response curve of the potentiated compound, at a shifted concentration C = X [1+(Y/Ypot)|p|]sign(p), where sign(p) is a unit sign function evaluated at (-1,0,+1) corresponding to the sign of its argument. The free parameters are determined using an iterative Nelder-Mead optimization algorithm. First the model is optimized first assuming that the dose matrix's X-agent is being potentiated by the Y-agent, and second assuming the X-agent as the potentiator. The solution with the lowest chi-squared is taken as the best fit. The best fit parameters are reported along with their standard errors, in each case estimated from the corresponding parameter's 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 POT model's calculation.

Limitations and Constraints

This model is appropriate for all 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 algorithm may get confused about when the potentiation begins. Also, because there are two parameters to optimize, the searching can be computationally expensive.

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