Michaelis-Menten Km and Vmax Nonlinear Kinetics Fitter
Fit initial reaction velocities across a substrate concentration series to estimate the Michaelis constant and maximal velocity by direct nonlinear least squares. This reference explains why nonlinear regression with profile intervals supersedes double-reciprocal linearisation and how substrate range governs parameter identifiability.
The initial-rates worked example: duplicate synthetic rates generated from Vmax 10 µM/min and Km 2 µM with fixed noise, with 10 nM active sites for kcat. This page opens with the initial-rates fit already run on these readings, kcat in per minute as the rates are; the engine itself is compared with a separate SciPy nonlinear least-squares fixture.
Verify initial-rate linearity before fitting the hyperbola
Every point on a Michaelis–Menten curve represents an initial velocity measured from the linear early phase of a reaction progress curve. If substrate depletion, product inhibition or enzyme inactivation causes the progress curve to bend before the readout is taken, the apparent rates at low substrate concentrations will be systematically underestimated. That distortion artificially inflates the fitted Km and can mimic cooperativity.
Span a wide range of substrate concentrations that brackets the expected Km, ideally extending from roughly one fifth of Km to at least five times Km. Observations below Km anchor the initial slope, observations near Km locate the curvature, and observations well above Km establish the upper asymptote. Without points on the upper shoulder, the optimiser must extrapolate Vmax from a weakly curved segment.
Read parameter correlation and turnover constants carefully
Because Vmax and Km appear together in the rational function, an upward shift in the trial Vmax can be partially compensated by an upward shift in Km over a finite substrate range. Profile likelihood intervals trace that banana-shaped objective surface by refitting the nuisance parameter at each trial value, revealing asymmetric uncertainty that symmetric standard errors conceal.
When reporting catalytic efficiency or kcat, include the calibration factor that converted raw instrument signal per unit time into molar product formation rate, together with the active-site concentration basis. Keeping those unit conversions explicit prevents arbitrary instrument units from being mistaken for absolute molecular turnover rates.
Equations and variables
Michaelis–Menten initial velocity
v0 = (Vmax × [S]) / (Km + [S])
- v0
- Measured initial reaction velocity before significant substrate depletion (rate units)
- Vmax
- Asymptotic maximal velocity at saturating substrate concentration (rate units)
- [S]
- Initial free substrate concentration (mol/L)
- Km
- Substrate concentration yielding half of the maximal velocity (mol/L)
Catalytic turnover number
kcat = Vmax / [E]t
- kcat
- First-order catalytic rate constant per active site, in the rates' time unit (per minute here) (min⁻¹ or s⁻¹)
- Vmax
- Maximal velocity expressed in molar product per second (mol L⁻¹ s⁻¹)
- [E]t
- Active-site molar concentration in the reaction (mol/L)
Scope and limitations
- The hyperbolic model assumes steady-state initial velocities with negligible product accumulation, no substrate inhibition and excess substrate over enzyme.
- When the highest tested substrate concentration remains below Km, Vmax and Km become strongly correlated and cannot be resolved independently.
- Converting Vmax into kcat requires velocity and active-site concentration to share compatible molar units rather than arbitrary fluorescence or absorbance units per minute.
Sources and provenance
- Published numerical verification inputs and independent referencesOriginal synthetic inputs; see individual fixture notices
The generated inputs are demonstrations, not experimental measurements. Existing versioned fixtures are retained unchanged.
- Independent Michaelis–Menten initial-rate verification setMIT; SciPy curve_fit oracle reference
Engine verification cases; the worked example on this page is separate synthetic data.
- Johnson and Goody: the original Michaelis constant and translation of the 1913 paperArticle citation; historical method context
Technical questions
Why avoid Lineweaver–Burk double-reciprocal plots for fitting Km and Vmax?
Taking the reciprocal of both substrate concentration and velocity severely distorts experimental error, placing the heaviest leverage on the lowest substrate concentrations where signal-to-noise is worst. Nonlinear least squares fits the untransformed observations directly on their measurement scale.
What happens if substrate solubility prevents testing above Km?
Below Km, the hyperbola is nearly a straight line whose slope equals the specificity ratio Vmax divided by Km. Under that design, the ratio is well determined by the data, but the individual Km and Vmax parameters trade off against one another and will show wide or open profile bounds.
How do I detect substrate inhibition in an initial-rate series?
If initial velocity rises to a peak and then declines at high substrate concentrations, the monotone hyperbola is misspecified. Inspect the residuals at the top concentrations and compare a Haldane substrate-inhibition model only when the downturn is supported by multiple concentrations beyond the optimum.
Connected workflows
- EC50 and Hill slope fitting
Compare empirical sigmoidal activation fits with hyperbolic saturation kinetics.
- Protein A280 concentration calculation
Calculate the molar enzyme concentration needed to convert Vmax into kcat.
- Monod fed-batch growth simulation
Contrast isolated enzyme saturation with whole-cell Monod substrate uptake.