The pEC50

The pEC50 is defined as the negative logarithm of the EC50. If the EC50 equals 1 micromolar (10-6 molar), the log(EC50) is -6 and the pEC50 is 6. There is no particular advantage to expressing potency this way, but it is the custom in some fields.

If you fit a dose response curve using one of the classic equations, Prism will report the logEC50. Multiply by -1 to obtain the pEC50.

If you want to fit the pEC50 directly, perhaps to embed into a table on a Prism graph, use the following equation.Y=Bottom + (Top-Bottom)/(1+10^((X - pEC50)*HillSlope))

Calculating any EC value from the EC50 and Hill slope

The potency of a drug is commonly quantified as the EC50, the concentration that leads to 50% maximal response (or the logarithm of the EC50).  But in some systems you might be more interested in the EC80 or the EC90 or some other value. You can compute the EC80 or EC90 (or any other EC value) from the EC50 and Hill slope. Or you can fit data to determine any EC value directly. If you express response as a percentage, a standard dose response curve is described by this equation:

L is the ligand concentration, EC50 is the concentration that gives half-maximal effect, and H is the Hill constant or slope factor that defines the steepness of the curve. L and EC50 are expressed in the same units of concentration, so the units cancel out.  F is the fractional response (or fractional occupancy for binding data), expressed as a percentage.

Set F to any percentage you want (80 if you want to obtain the EC80) and then solve for L. We call that value of L the ECF, as that is the quantity of drug needed to elicit an F percentage response (or with binding data, F is the concentration of ligand needed to occupy fraction F of the receptors). A bit of algebra yields this equation:

If you know the EC50 and Hill slope (H), you can easily compute the EC80 or EC10 or any other value you want. For example, if the Hill slope equals 1, the EC90 equals the EC50 times nine. If H equals 0.5, the curve is shallower and the EC90 equals the EC50 times 81.

Determining any EC value directly

You can also fit data directly to an equation written in terms of the ECF. The advantage of this approach is that Prism will report the 95% confidence value for ECF. Use the equation below, where X is the log of concentration and Y is response, which ranges from Bottom to Top.

F=80
logEC50=logECF - (1/HillSlope)*log(F/(100-F))
Y=Bottom + (Top-Bottom)/(1+10^((LogEC50-X)*HillSlope))

To fit data to this equation, you'll need to enter rules for computing initial values. Set Top equal to 1*Ymax, Bottom equal to 1*Ymin. For the Hill slope, simply pick a value, probably +1.0 or -1.0. For the log EC value, enter 1*XMID as a crude initial value, or enter a value based on the range of concentrations you use.

Here is a simplified equation, for fitting the EC90. Here the response is expressed as a percent ranging from zero to one hundred, so we dispense with the variables Top and Bottom.

logEC50=logEC90 - (1/HillSlope)*log(9)
Y=100/(1+10^((LogEC50-X)*HillSlope))