Plotwright

Worked nonlinear-regression example

Calculate IC50 or EC50 by fitting the curve—not a line.

A fitted midpoint is only as defensible as the model, concentration range and experimental replication behind it. Work through a four-parameter logistic fit and the checks that should accompany the number.

7 concentrations4PL variable slopeProfile-likelihood CI9-minute read

The short answer

Fit a sigmoidal model that matches how concentration was entered.

EC50 is the concentration producing half of the specified maximal effect. IC50 is the concentration producing half-maximal inhibition under the stated assay conditions. Neither is an assay-independent property: name the response, system, exposure conditions and model when you report it.

Input rule: if X contains log10 molar concentrations, use an equation whose X variable is log concentration. If X contains concentrations, use a concentration-form equation and display the axis logarithmically. Do not log zero.

The four-parameter logistic model

For an increasing response with X entered as log10 concentration, Plotwright fits:

Y = Bottom + (Top − Bottom) / (1 + 10(LogEC50 − X) × HillSlope)

Bottom and Top are the fitted plateaus. LogEC50 is the midpoint between those plateaus on the log10 concentration scale. HillSlope controls steepness. This is an empirical description of the observed curve; the slope is not automatically a binding stoichiometry or molecular mechanism.

An exact worked EC50 example

Plotwright includes this seven-point synthetic dataset as a teaching sample. X is log10 agonist concentration in mol/L and Y is response percent:

log10 concentration (M) −9 −8 −7 −6 −5 −4 −3
Response (%) 1 4 11 50 89 98 100
Four-parameter logistic fit to seven dose-response points Seven observations rise sigmoidally from one to one hundred percent. The fitted EC50 is 1.04 micromolar with a profile-likelihood 95 percent confidence interval from 0.90 to 1.20 micromolar. −9 log10 M: 1% −8 log10 M: 4% −7 log10 M: 11% −6 log10 M: 50% −5 log10 M: 89% −4 log10 M: 98% −3 log10 M: 100% 0 50 100 −9 −7 −5 −3 log10 concentration (M) Response (%) EC50 = 1.04 µM
Points are the seven synthetic observations; the line is the variable-slope 4PL fit. The dashed marker shows the fitted midpoint, not an additional observation.
Fitted midpoint EC50 = 1.04 µM

Profile-likelihood 95% CI: 0.90 to 1.20 µM; LogEC50 = −5.983.

Fit diagnostics HillSlope = 0.933

R² = 0.9998; Sy.x = 0.913; 3 residual degrees of freedom; fit converged.

Scope: these are exact results for a clean synthetic teaching sample, not evidence that an assay is biologically valid. With only seven points and no independent runs shown, the example demonstrates the calculation—not the reproducibility of a real experiment.

Five checks before reporting IC50 or EC50

  1. Match equation to X. Concentration and log-concentration equations are not interchangeable merely because the graph uses a logarithmic axis.
  2. Support both plateaus. If the data do not approach a bottom and a top, all four parameters may be weakly identified. Fixing either plateau makes a strong assumption; normalization alone does not prove that the true values are 0 and 100.
  3. Interpolate rather than extrapolate. A reported midpoint should fall inside the tested concentration range, with observations on both sides. If it does not, report the range limitation instead of a precise unsupported number.
  4. Distinguish technical wells from independent runs. Replicate wells on one plate can share preparation and plate effects. Averaging wells does not create new independent experiments; preserve the run or plate structure in the analysis.
  5. Inspect uncertainty and residuals. Confirm convergence, examine residual patterns and report a confidence interval. A large R² can coexist with an inappropriate model or underestimated uncertainty.

What changes for an inhibition curve?

Use the decreasing-response form, whose midpoint parameter is LogIC50. The same design checks apply. State what was inhibited and how the response was normalized; “IC50” alone can refer to different assay quantities. Do not convert IC50 into a binding constant unless the experimental system and an appropriate mechanistic relationship justify that conversion.

A compact reporting template

“Response was fitted against log10 molar concentration using a four-parameter logistic model with variable slope. The fitted EC50 was 1.04 µM (profile-likelihood 95% CI 0.90–1.20 µM), with HillSlope 0.933. Both plateaus and concentrations on both sides of the midpoint were observed; convergence and residual diagnostics were inspected.”

Replace every value and design statement with the actual experiment. Report the number of independent runs, treatment of technical replicates, parameter constraints and any prespecified exclusion or weighting rule.

What Plotwright checks

Plotwright fits increasing and inhibition 4PL models, concentration-form 4PL and 5PL models, fixed-slope variants, residual diagnostics, confidence and prediction bands, and profile-likelihood parameter intervals. Release tests compare defined nonlinear-regression cases with independent SciPy calculations and checked-in reference result files, using explicit parameter and interval tolerances.

That evidence checks numerical agreement for defined cases. It cannot choose the biological model, establish independence, rescue an unsupported concentration range or interpret the fitted potency for the researcher.

Sources and reproducibility

  1. Assay Guidance Manual: Assay Operations for SAR Support—definitions, four-parameter fitting, interpolation and replicate-design guidance.
  2. Assay Guidance Manual: In Vivo Assay Guidelines—nonlinear regression, 4PL parameters and the assumptions introduced by fixed plateaus.
  3. IUPHAR/BPS Guide to Pharmacology help—pharmacological term definitions and the need to state IC50 context.
  4. Plotwright statistical validation—the public evidence model, numerical tolerances and limits of software validation.

Keep the curve, diagnostics and reported value together.

Plotwright turns an XY concentration table into an inspectable nonlinear result, confidence interval and editable scientific graph in one project.

Try the built-in dose-response sample Next guide: Kaplan-Meier curves