Example data table
Paste these pairs to see a typical cooperative binding curve.
| Concentration (µM) | Fractional saturation (Y) |
|---|---|
| 0.1 | 0.12 |
| 0.3 | 0.28 |
| 1.0 | 0.55 |
| 3.0 | 0.78 |
| 10.0 | 0.92 |
Formula used
Note: Hill analysis is an approximation. For mechanistic binding, consider a full nonlinear fit to your experimental model.
How to use this calculator
- Collect binding pairs: concentration and fractional saturation (Y).
- Choose Regression for multiple data points.
- Paste your dataset: one pair per line.
- Pick a log base and your unit label for display.
- Press Calculate to show results above the form.
- Use Download CSV or Download PDF for reporting.
Interpretation tips
R² close to 1 suggests a cleaner Hill-plot linear region, but always inspect the transformed table for outliers and saturation limits.
What the Hill coefficient quantifies
The Hill coefficient (n) summarizes how sharply fractional saturation rises with ligand concentration. When n≈1, binding behaves roughly independently. Values below 1 often reflect site heterogeneity or negative cooperativity, while n>1 indicates positive cooperativity and a steeper transition near the midpoint. In practice, n is an empirical slope taken from transformed data, so it describes the response curve shape rather than the exact number of binding sites. Always interpret n alongside raw curves and experimental design constraints for confidence today.
Using a Hill plot for dataset-driven estimates
This calculator linearizes measurements by plotting log(Y/(1−Y)) versus log([L]). With multiple points spanning low to high saturation, ordinary least squares returns the slope (n) and intercept. The intercept maps to an apparent Kd through log(Kd)=−intercept/n. Because the transformation amplifies noise near Y→0 or Y→1, trimming extreme points can improve stability and produce a more meaningful linear region.
Fit quality, R², and practical diagnostics
R² reports how well the transformed points align to a straight line. A high R² can indicate a consistent cooperative region, but it does not guarantee the underlying mechanism is correct. Review the computed table to spot outliers, repeated concentrations, or saturation plateaus. If data cluster narrowly in concentration, n becomes poorly constrained, and the regression can be dominated by measurement error.
Interpreting apparent Kd in context
The Kd reported here is an apparent constant derived from the Hill relationship, useful for comparing conditions such as pH, ionic strength, or mutations. It is not always identical to a microscopic dissociation constant for multi-step binding. Report Kd with the unit label you used for concentration inputs, and compare only when the same response definition for Y is applied across experiments.
Recommended reporting for reproducible results
For lab notes or manuscripts, report the method, log base, n, Kd, R², and number of points included. Exporting CSV preserves the transformed values so others can verify the linear region. PDF output is best for quick sharing. When possible, keep at least five well-spaced concentrations and replicate measurements to reduce uncertainty in n and improve interpretability.
FAQs
1) What does an n value greater than 1 mean?
n>1 suggests positive cooperativity, meaning binding becomes easier as occupancy increases. It also reflects a steeper response near the midpoint. It does not automatically equal the number of binding sites.
2) Why must Y be between 0 and 1?
The Hill transform uses Y/(1−Y). At Y=0 or Y=1 the ratio becomes 0 or infinite, which breaks the logarithm and inflates noise. Use values strictly inside the open interval.
3) Should I use the two-point estimate?
Use it only for quick checks when you have limited measurements. Two points cannot reveal curvature or outliers, so n and Kd may be unstable. A regression on multiple concentrations is preferred.
4) How many concentrations are recommended for regression?
Five or more well-spaced concentrations typically give a more reliable slope and intercept. Include points across low, mid, and high saturation, and avoid clustering too tightly around a single concentration range.
5) What does a low R² indicate here?
A low R² means the transformed points do not align well to a straight line. This can happen from experimental noise, mixed binding modes, or using points near saturation limits. Inspect the table and consider filtering extremes.
6) Is the reported Kd always a true dissociation constant?
Not always. The calculator reports an apparent Kd from the Hill relationship, which is useful for comparisons under consistent definitions of Y. Complex mechanisms, multiple states, or allosteric models can shift apparent values.