# Michaelis-Menten Calculator - Km, Vmax & kcat Nonlinear Fit, Inhibition Ki

> Paste [S] and rate data to fit Michaelis–Menten by nonlinear regression: Km, Vmax with 95% CI, kcat, Lineweaver–Burk comparison and inhibition Ki.

- URL: https://qilingtools.com/en/chemistry/enzyme-kinetics
- Category: Chemistry Tools (https://qilingtools.com/en/chemistry)
- Tags: [Chemistry](https://qilingtools.com/en/tags/chemistry)
- Last updated: 2026-10-29
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

The Michaelis–Menten equation v = Vmax[S]/(Km + [S]) relates the initial rate of an enzyme reaction to substrate concentration: Km is the substrate concentration at half-maximal rate, Vmax the rate when the enzyme is saturated, kcat = Vmax/[E]t the turnover number per active site, and kcat/Km the catalytic efficiency. Paste [S] and v (extra columns are treated as replicates) and this calculator fits the equation by nonlinear least squares with a hand-written Levenberg–Marquardt solver, reporting Vmax and Km with standard errors, 95% confidence intervals, R² and a residual plot. It also draws the Lineweaver–Burk (double reciprocal), Eadie–Hofstee and Hanes–Woolf plots with their own estimates. Exact data generated with Vmax = 10 and Km = 2 is recovered exactly; for the noisy simulated triplicate example the nonlinear fit gives Vmax 10.01 ± 0.15 and Km 1.94 ± 0.09 μM, while the double-reciprocal plot gives 9.50 and 1.75, because 1/v magnifies the error of the low-[S] points — which is why nonlinear regression is preferred. With 10 nM enzyme, kcat ≈ 16.7 s⁻¹ and kcat/Km ≈ 8.6 × 10⁶ M⁻¹s⁻¹. The inhibition tab fits data at several inhibitor concentrations globally to competitive, uncompetitive, noncompetitive and mixed models, compares them by AICc, reports Ki and αKi and shows the characteristic Lineweaver–Burk pattern. The Hill equation (n_H, K₀.₅), a substrate-inhibition model and an interactive simulator are included.

## How to use

1. Pick an example dataset or paste [S] and v from a spreadsheet, set the concentration and time units, and enter the enzyme concentration if you want kcat
2. Read the Michaelis–Menten fit with confidence intervals and residuals, the AICc comparison with Hill and substrate-inhibition models, and the three linearized plots
3. Paste rates at several [I] in the inhibition tab to identify the inhibition type and Ki, or drag the sliders in the simulator

## Features

- Levenberg–Marquardt nonlinear least squares with standard errors and t-based 95% confidence intervals
- Replicate columns and header lines handled automatically — paste straight from Excel
- Lineweaver–Burk, Eadie–Hofstee and Hanes–Woolf plots compared side by side with the nonlinear fit
- kcat and kcat/Km in s⁻¹ and M⁻¹s⁻¹ with covariance-based error propagation
- Hill equation (n_H, K₀.₅) and substrate inhibition fits, ranked by AICc and Akaike weights
- Global fits for competitive, uncompetitive, noncompetitive and mixed inhibition with Ki, αKi and apparent parameters at each [I]
- Simulator: drag Vmax, Km, [I], Ki and α to see v vs [S] curves and Lineweaver–Burk patterns
- Export charts as PNG or SVG and data with fitted values as CSV; everything runs in your browser

## FAQ

### Why is the Lineweaver–Burk plot not recommended for estimating Km and Vmax?

Taking reciprocals distorts the errors. At low substrate concentrations v is small and its relative error large, and 1/v magnifies that error; those points also lie at the far right of the plot where they have the most leverage on the slope. Km and Vmax from a double-reciprocal fit are therefore often biased. Fitting the Michaelis–Menten equation directly by nonlinear regression keeps the original error structure. The double-reciprocal plot is still useful for displaying data and recognising inhibition patterns.

### What do Km, Vmax, kcat and kcat/Km mean?

Km is the substrate concentration at which the rate is half of Vmax; a lower Km usually means higher apparent affinity. Vmax is the rate at saturating substrate and is proportional to the amount of enzyme. kcat = Vmax/[E]t is the turnover number — substrate molecules converted per active site per second. kcat/Km is a second-order rate constant describing efficiency at low substrate concentration; diffusion-limited enzymes reach 10⁸–10⁹ M⁻¹s⁻¹.

### How do I tell competitive, uncompetitive and noncompetitive inhibition apart?

A competitive inhibitor raises the apparent Km but leaves Vmax unchanged, so Lineweaver–Burk lines meet on the y-axis. An uncompetitive inhibitor lowers Vmax and Km by the same factor, giving parallel lines. A pure noncompetitive inhibitor lowers Vmax only, so the lines meet on the x-axis at −1/Km, and mixed inhibition gives an intersection left of the y-axis. This tool fits all inhibitor concentrations globally and compares the models by AICc, which is more objective than judging intersections by eye.

### When should I use the Hill equation instead of Michaelis–Menten?

Use it when the v vs [S] curve is sigmoidal — slow at low [S], then steep — which indicates cooperative binding, as in allosteric enzymes. A Michaelis–Menten fit then shows systematic residuals. In v = Vmax[S]ⁿ/(K₀.₅ⁿ + [S]ⁿ), n_H > 1 means positive cooperativity, n_H < 1 negative cooperativity, and K₀.₅ is the half-saturating concentration. The tool fits both models and uses AICc to judge whether the extra parameter is justified.

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