Calculator
Example data table
Example parameters: EC50=10, n=2, Bottom=0, Top=1, unit µM.
| x (µM) | θ | y |
|---|---|---|
| 0.1 | 0.0001 | 0.0001 |
| 0.3 | 0.000899 | 0.000899 |
| 1 | 0.009901 | 0.009901 |
| 3 | 0.082569 | 0.082569 |
| 10 | 0.5 | 0.5 |
| 30 | 0.9 | 0.9 |
| 100 | 0.990099 | 0.990099 |
Formula used
Fractional saturation
θ = xn / (EC50n + xn)
Response form
y = Bottom + (Top − Bottom) · xn / (EC50n + xn)
- x is the ligand, stimulus, or input level.
- EC50 is the half‑max constant in x units.
- n controls steepness and cooperativity.
- Top/Bottom scale the output range in response mode.
How to use this calculator
- Select the output mode: fractional saturation or scaled response.
- Enter x units for display, then set the Hill coefficient n.
- Provide EC50 in the same units as x, unless using log10.
- Set Top and Bottom if you want a calibrated response range.
- Enter one x value for the summary, plus optional extra values.
- Press Calculate to display results above the form instantly.
- Use the download buttons to export CSV or a PDF report.
Professional article
1) Where the Hill equation fits
The Hill equation summarizes saturating behavior seen in cooperative occupancy, adsorption, and thresholded activation. It maps an input level x to fractional saturation θ (0–1) or to a scaled signal y between Bottom and Top. This calculator lets you test parameters and generate exportable tables for reports cleanly.
2) Core saturation relation
Fractional saturation is θ = xn/(EC50n + xn). The midpoint is fixed: when x = EC50, θ = 0.5 for any n>0. At x ≪ EC50, θ ≈ (x/EC50)n; at x ≫ EC50, θ approaches 1.
3) Turning θ into a measured signal
For instruments, use y = Bottom + (Top − Bottom)·θ. If a detector spans 120 to 980 a.u., set Bottom = 120 and Top = 980 to predict signals directly. Keeping Bottom and Top explicit helps separate baseline drift from real sensitivity changes.
4) Meaning of the Hill coefficient
The Hill coefficient n sets steepness near EC50. n = 1 gives a simple hyperbola. n > 1 sharpens the transition (often interpreted as positive cooperativity or multi‑site gating), while n < 1 broadens it (heterogeneity, negative cooperativity, or mixed populations). Many fitted datasets land around n ≈ 0.5–4.
5) How EC50 shifts the curve
EC50 sets the x scale. With EC50 = 10 µM and n = 2, x = 3 µM gives θ ≈ 0.083 and x = 30 µM gives θ ≈ 0.900. Changing EC50 by a factor of 10 shifts the curve by one decade on a log axis, without changing its shape.
6) Linear vs log10 entry
Experiments often use decade spacing (0.1, 0.3, 1, 3, 10, 30, 100). The log10 options accept −1, −0.5, 0, 0.5, 1, 1.5, 2 and convert internally to linear units. This reduces transcription errors and preserves intended spacing.
7) Checks that prevent nonsense
Require EC50 > 0 and n > 0. Negative x is physically meaningless for concentration‑style inputs and is skipped. If Top equals Bottom, the response becomes constant and no longer carries information about θ. Always report units, EC50, n, Top, and Bottom alongside exported results.
8) A practical reporting workflow
Estimate EC50 from the visual midpoint of your curve, then tune n to match steepness. Confirm θ(EC50)=0.5 and that θ approaches 0 and 1 at your lowest and highest x values. Export CSV for plotting and the PDF for sharing assumptions and parameters.
FAQs
1) What does EC50 represent here?
EC50 is the x value where the fractional saturation θ equals 0.5. In binding language it often matches an effective Kd; in activation curves it is the half‑max stimulus level.
2) Can I use this for pressure, voltage, or temperature inputs?
Yes. x is any non‑negative driving variable. Enter a unit label for display, keep EC50 in the same units, and interpret θ as a normalized activation fraction rather than a chemical occupancy.
3) Why is n sometimes not an integer?
n is an empirical slope parameter. Non‑integer values commonly arise from heterogeneity, partial cooperativity, or mixed populations, even when the underlying mechanism has discrete sites.
4) What is the difference between θ mode and response mode?
θ mode outputs the pure normalized saturation from 0 to 1. Response mode multiplies that saturation into a real measurement range using Bottom and Top, producing y in your chosen signal units.
5) When should I use the log10 checkboxes?
Use them when your inputs are stored as logarithms, such as pEC50 or log‑spaced dose lists. The calculator converts each value with 10^x so computations still occur in linear space.
6) Why are negative x values ignored?
The Hill forms use x^n and assume x is non‑negative. For concentration‑style inputs, negative values have no physical meaning and would also cause issues for non‑integer n.
7) How many x values can I evaluate at once?
You can compute a combined set of up to 200 x values per run. Extra entries beyond that limit are truncated to keep the table, exports, and browser performance stable.