The problem
A transformation that fixes the geometry breaks the errors.
The Michaelis–Menten model, v = Vmax·[S] / (Km + [S]), is a curve. Before nonlinear regression was routine, the standard move was to plot 1/v against 1/[S], read Vmax from the intercept and Km from the slope, and fit by eye or by least squares. The algebra is exact. The statistics are not: taking reciprocals compresses the high-concentration points and stretches the low-concentration ones, so the fit is dominated by the measurements with the least information and the largest relative error.
The measurements
| [Substrate] (µM) | 1 | 2 | 5 | 10 | 20 | 50 |
|---|---|---|---|---|---|---|
| Velocity | 12 | 20 | 38 | 55 | 72 | 90 |
Six concentrations spanning a fiftyfold range, one reading each. The velocity is still climbing at 50 µM, which matters later.
The same six points, two fitting routes
| Route | Vmax | Km (µM) | Fit quality |
|---|---|---|---|
| Nonlinear regression on v vs [S] | 105.106 (SE 1.548) | 8.872 (SE 0.369) | R² = 0.99913 |
| Lineweaver–Burk (1/v vs 1/[S]) | 92.459 | 6.820 | r² = 0.99766 |
The double-reciprocal route reports a Vmax 12.0% lower and a Km 23.1% lower than the direct fit — and it reports them with an r² that would pass any informal quality check. r² is computed in reciprocal space, where it describes how well a line fits transformed numbers, not how well the model fits the measurements.
Residuals from the direct fit
In velocity units, in concentration order: +1.354, +0.666, +0.117, −0.692, −0.807, +0.735. They change sign, stay small relative to velocities of 12 to 90, and show no systematic sweep — which is what supports the model here, far more than R² = 0.99913 does on its own.
What Km and Vmax actually claim
Km is the substrate concentration at which velocity reaches half of Vmax; it is a concentration, reported in the units of [S], not a rate and not an affinity constant in general. Vmax is an extrapolated ceiling: the velocity the model predicts at infinite substrate. Neither is measured directly here. With the highest concentration at 50 µM and a fitted Km of 8.872 µM the series does reach roughly 5.6 × Km, so Vmax is reasonably constrained — but a design whose top concentration sits near or below Km makes Vmax an extrapolation the data cannot support, and Km inherits that uncertainty.
A defensible kinetics workflow
- Choose concentrations that bracket the expected Km, ideally spanning well below to several multiples above it.
- Replicate each concentration; single readings cannot separate model error from measurement error.
- Fit v against [S] directly by nonlinear regression. Do not fit a transformed variable.
- Inspect residuals against [S] for systematic curvature before reading either constant.
- Report Km with its units and Vmax with its uncertainty, and state the concentration range the estimates rest on.
What Plotwright checks
Both fits above come from the engine's regression routes — the same nonlinear machinery used for dose–response curves, which carries reference checks on the validation page. Every value on this page is reproducible from the six visible measurements in the linked editable project.
Sources and reproducibility
- Johnson, K. A. & Goody, R. S. (2011), “The Original Michaelis Constant: Translation of the 1913 Michaelis–Menten Paper”, Biochemistry—the original derivation with modern commentary on how the constants are defined and estimated.
- IC50 and EC50 dose response—the same nonlinear-fitting discipline applied to sigmoidal potency curves.
- Standard curves and interpolation—the related trap of judging a fit by r² instead of its residuals.
- Plotwright statistical validation—published expected values, numerical tolerances and evidence limits.