Fit dose-response and kinetic curves, find peaks, and tame instrument signals.
Lab and instrument data has its own shapes: sigmoidal dose-response curves, saturating kinetics, exponential decays, and noisy traces with peaks to quantify. This guide covers the curve-fitting and signal-processing tools that turn raw readings into meaningful parameters like EC50, Km, or a rate constant.
For any non-linear fit, judge it by more than R²: check that the parameters are physically sensible and that the residuals scatter randomly around zero rather than curving.
For regulated method validation, ICH Q2(R2) names the parameters to characterize — specificity, linearity, range, accuracy, precision, and detection/quantitation limits. Two fitting habits pay off: weight the fit when scatter grows with the signal (constant relative error), and choose between nested models (one- vs two-phase decay, 4PL vs 5PL) with an extra-sum-of-squares F-test or an information criterion rather than R² alone, which almost always rewards the more complex model.
Dose-response curves (Motulsky & Christopoulos)
The four-parameter logistic (4PL) is the workhorse for sigmoidal concentration-response, returning EC50/IC50, the top and bottom plateaus, and the slope. Use 5PL when the curve is asymmetric, the Hill equation when you want the Hill coefficient explicitly, and probit analysis for quantal (all-or-nothing) responses like LD50.
▸ — One shared fit across plates/datasets, with parameters held in common.
▸ — Hill equation — cooperativity via the Hill coefficient.
▸ — Quantal dose-response (LD50/ED50) by probit.
Enzyme kinetics & binding (Michaelis–Menten)
Michaelis-Menten fits Vmax and Km directly from rate-versus-substrate data — no Lineweaver-Burk linearization, which distorts the error structure and biases the estimates.
▸ — Type your own model formula when no preset matches the mechanism.
▸ — Is the extra parameter earning its keep? F-test plus AICc weights.
▸ — Not sure of the shape? Fit nine standard curves and rank them.
▸ — Vmax and Km from initial-rate data.
Growth & decay over time (Motulsky & Christopoulos)
Time-course data follows characteristic shapes: one- or two-phase exponential decay for clearance/washout, association-to-plateau for uptake, and Gompertz or logistic growth for populations, cultures, and tumors.
▸ — Single-exponential decay (one rate constant).
▸ — Fast + slow decay components.
▸ — Exponential association up to a plateau.
▸ — Asymmetric (Gompertz) growth.
▸ — Logistic (S-shaped) growth to a carrying capacity.
Signals, peaks & spectra (Savitzky–Golay 1964)
Noisy traces need smoothing before you can read them — Savitzky-Golay removes noise while preserving peak shape. Peak analysis locates and measures peaks; deconvolution separates overlapping ones; spectral analysis exposes periodicities.
▸ — Periodogram — dominant frequencies in a signal.
Interpolation, area & calibration (ICH Q2(R2))
Read a value between measured points with spline interpolation, integrate a signal (drug exposure, chromatographic area) with the trapezoidal rule or numerical integration, and build calibration curves with linear or polynomial regression.
▸ — A curved calibration when a straight line will not do.
▸ — Spline / linear read-off between measured points.
▸ — Area under a paired X-Y curve (exposure / chromatogram).
▸ — Numerical integration & differentiation of a series.
▸ — Calibration line + inverse prediction.
▸ — Do two calibration slopes differ?
Distributions & outliers (ISO 5725 · Grubbs)
Identify the distribution your measurements follow, and screen replicates for a bad point before fitting a curve to them.
▸ — A response surface over two inputs, as iso-value contours.
▸ — Three-component mixtures and formulations.
▸ — Overlay a theoretical curve on your data.
▸ — MLE distribution fit + goodness-of-fit.
▸ — Test for a single outlier.
▸ — Robust outlier removal designed for curve fits.
References & further reading
Motulsky & Christopoulos (2004). Fitting Models to Biological Data using Linear and Nonlinear Regression. Oxford University Press.