Michaelis–Menten calculator
Paste the substrate concentrations and the initial rates you measured. The calculator fits v = Vmax·[S] ÷ (Km + [S]) directly to the data by nonlinear least squares and gives Vmax, Km, how well the curve fits and the rate at any concentration you ask about.
Runs in your browser: nothing you enter is sent anywhere.
How to use it
- Enter the substrate concentrations, then the initial rates in the same order.
- Optionally enter a concentration to predict the rate at.
- Read Vmax and Km. R² close to 1 means the curve follows the points closely.
The equation
v = Vmax × [S] ÷ (Km + [S])
At [S] = Km the rate is exactly Vmax ÷ 2; at [S] = 9 × Km it is 90 % of Vmax. The units of Vmax follow the rates you enter (µmol/min, mM/s, absorbance per minute) and the units of Km follow the concentrations.
R² is 1 − (sum of squared residuals) ÷ (total sum of squares) of the rates around their mean.
Worked example
The default data come from an enzyme with Vmax near 10 and Km near 2, measured with a little scatter:
| [S] |
0.5 |
1 |
2 |
4 |
8 |
16 |
| v |
2.1 |
3.4 |
5.0 |
6.6 |
8.0 |
8.9 |
The fit gives Vmax = 9.9436 and Km = 1.9544, with R² = 0.9996. At [S] = 3 the predicted rate is 9.9436 × 3 ÷ (1.9544 + 3) = 6.0211, or 60.6 % of Vmax.
Good practice
- Measure initial rates, before the substrate is used up or the product inhibits the enzyme.
- Spread the concentrations from about 0.2 × Km to 5 × Km or more.
- Look at the points around the curve: a pattern in the misses suggests inhibition, cooperativity or a wrong blank.
Frequently asked questions
What do Vmax and Km mean?
Vmax is the rate the enzyme approaches when the substrate saturates it. Km, the Michaelis constant, is the substrate concentration at which the rate is half of Vmax; a lower Km means the enzyme reaches half speed at a lower concentration. Both come out in the units of your data.
How is the curve fitted?
By nonlinear least squares: the calculator looks for the Vmax and Km that make the sum of squared differences between measured and predicted rates as small as possible (the Levenberg–Marquardt method), starting from the Hanes–Woolf straight line.
Why not use the Lineweaver–Burk plot?
Taking reciprocals magnifies the error of the slowest rates at the lowest concentrations, so a double-reciprocal line can pull Km and Vmax off. A direct fit weights every point equally. For the default data the direct fit gives Vmax 9.9436 and Km 1.9544, the Lineweaver–Burk line 9.659 and 1.8104.
How many points do I need?
At least three, but a useful fit needs more, spread from well below Km to several times above it. If every concentration is far below Km the curve looks like a straight line and Vmax cannot be pinned down.