Hong Life CalculatorsUnderstanding the four-parameter logistic (IC50/EC50) calculator — concept and principle
The four-parameter logistic (4PL) model is the standard sigmoid curve for dose-response data, immunoassays such as ELISA. Given the curve's bottom and top plateaus, its midpoint (IC50 or EC50) and its Hill slope, this calculator returns the expected response at any concentration x. Use it to predict a response from fitted parameters or to draw a theoretical curve.
How it is calculated
y = Bottom + (Top − Bottom) ÷ (1 + (IC50 ÷ x)^Hill)
- Bottom and Top: the response plateaus at very low and very high concentration.
- IC50/EC50: the concentration giving a response halfway between Bottom and Top, in the same unit as x.
- Hill slope: steepness. With this form, a positive Hill gives a rising curve (activation, EC50); a negative Hill gives a falling curve (inhibition, IC50).
x must be greater than zero.
The principle
Plotted against log concentration, many biological responses follow an S-shape: nothing happens at very low doses, the response rises steeply over roughly two orders of magnitude, then saturates. When x equals IC50, the ratio IC50/x is 1, so the denominator is 2 and the response sits exactly halfway between the plateaus. The Hill exponent comes from A. V. Hill's work on oxygen binding to haemoglobin; a slope of 1 corresponds to simple one-site binding, and values above 1 give a steeper switch.
Worked example
- Inputs: Bottom = 0, Top = 100, IC50 = 10, Hill = 1, x = 5.
- IC50 ÷ x = 10 ÷ 5 = 2; 2¹ = 2.
- Denominator: 1 + 2 = 3.
- y = 0 + (100 − 0) ÷ 3 = 33.33.
At half the EC50, a curve with slope 1 reaches one third of the full response.
| Top | Response at x |
|---|---|
| 25 | 8.333333333 |
| 50 | 16.66666667 |
| 75 | 25 |
| 100 | 33.33333333 |
| 125 | 41.66666667 |
| 150 | 50 |
| 200 | 66.66666667 |
How to read the result
With Hill = 1, the response is 10% of the span at x = EC50/9 and 90% at x = 9 × EC50, so going from 10% to 90% takes an 81-fold change in concentration. A lower IC50 means a more potent compound. When comparing compounds, compare IC50 values measured in the same assay, because IC50 depends on conditions such as substrate concentration.
Common mistakes
- Fitting on a linear concentration axis with too few points around the midpoint.
- Using the sign convention of another program; some write the formula with (x/IC50) and a positive slope for inhibition.
- Treating relative IC50 (midpoint between fitted plateaus) as absolute IC50 (concentration giving 50% of control) when plateaus are not 0 and 100.
Frequently asked questions
What is the difference between IC50 and EC50?
IC50 is the concentration that inhibits a response by half; EC50 is the concentration that produces half the maximal effect. The maths is the same.
How do I convert IC50 to Ki?
For competitive inhibition, the Cheng-Prusoff equation Ki = IC50 ÷ (1 + [S]/Km) is commonly used.
Why plot concentration on a log axis?
The 4PL curve then becomes a symmetric S, so the midpoint and slope are easy to see.
Sources: A. V. Hill, "The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves", Journal of Physiology 40 (1910) · Y. Cheng and W. H. Prusoff, Biochemical Pharmacology 22, 3099–3108 (1973)