Introduction

The standard dose-response curve is sometimes called the four-parameter logistic equation. It fits the bottom and top plateaus of the curve, the EC50, and the slope factor (Hill slope). This curve is symmetrical around its midpoint. To extend the model to handle curves that are not symmetrical, the Richards equation adds an additional parameter, S, which quantifies the asymmetry. This equation is sometimes referred to as a five-parameter logistic equation, abbreviated 5PL.

Step by step

Create an XY data table. Enter the logarithm of the concentration of the agonist into X. Enter response into Y in any convenient units.

From the data table, click Analyze, choose nonlinear regression, and choose the panel of equations:  Dose-Response -- Special. Then choose Asymmetrical (five parameter).

Consider constraining the Hill Slope to a constant value of 1.0 (stimulation) or -1 (inhibition).

Also consider whether Bottom or Top should be fixed to constant values, or shared between data sets.

Model

LogXb = LogEC50 + (1/HillSlope)*Log((2^(1/S))-1)

Numerator = Top - Bottom

Denominator = (1+10^((LogXb-X)*HillSlope))^S

Y = Bottom + (Numerator/Denominator)

Interpret the parameters

Bottom and Top are the plateaus at the left and right ends of the curve, in the same units as Y.

LogEC50 is the concentrations that give half-maximal effects, in the same units as X. Note that the logEC50 is not the same as the inflection point Xb (see below).

HillSlope is the unitless slope factor or Hill slope. Consider constraining it to equal 1.0 (stimulation) or -1 (inhibition).

S is the unitless symmetry parameter. If S=1, the curve is symmetrical and identical to the standard dose-response equation. If S is distinct than 1.0, then the curve is asymmetric as shown below.

References                                                                          

Giraldo, J., Vivas, N. M., Vila, E. & Badia, A. Assessing the (a)symmetry of concentration-effect curves: empirical versus mechanistic models. Pharmacol Ther 95, 21–45 (2002).

Gottschalk, P. G. & Dunn, J. R. The five-parameter logistic: a characterization and comparison with the four-parameter logistic. Anal Biochem 343, 54–65 (2005).