Michaelis-Menten calculator
Fits the Michaelis-Menten equation to substrate concentration and initial velocity pairs.
Fits the Michaelis-Menten equation to substrate concentration and initial velocity pairs.
Paste substrate concentrations with the initial velocity measured at each one and the page fits the Michaelis-Menten equation by nonlinear least squares, then reports Km andVmax with the fitted curve through your points.
Vmax is the velocity the enzyme approaches when substrate is saturating; Km is the substrate concentration at which the velocity is half of Vmax. Both come from the same fit, and neither is read off a single measurement. The velocities must be initial rates, measured while less than about 10 % of the substrate has been consumed and the product is not yet inhibiting, because the equation describes the reaction at time zero.
For any Km the model is linear in Vmax, so the best Vmax is written down directly and only Km is searched: the tool scans Km over a wide logarithmic range, then refines the minimum of the sum of squared residuals by golden-section search. That makes the fit deterministic, with no starting guess to get wrong and no local minimum to fall into. Standard errors come from the curvature of the residual surface at the minimum, and R² compares the residuals with the spread of the velocities.
The double-reciprocal plot was how Km and Vmax were extracted before computers, and it is still the clearest way to show inhibition patterns, which is why it is under Details here. As a fitting method it is poor. Rewriting the equation as
puts the low-substrate points, which are the slowest and noisiest, far out on the right of the plot where they dominate the regression, while the high-substrate points that actually define Vmax bunch up near the intercept. Ordinary least squares then assumes an equal error on every 1/v, which is exactly what taking reciprocals destroys. Eadie-Hofstee and Hanes-Woolf plots rearrange the same problem differently and are also biased. Fit the curve itself and use the straight lines for display.
"Try an example" loads eight substrate concentrations from 2 to 320 µM with the measured velocities. The nonlinear fit gives Km = 44.9 µM and Vmax = 2.2 in the units of v, with R² = 0.9995 and standard errors of 1.34 on Km and 0.0216 on Vmax. The points were generated from Km = 45 and Vmax = 2.2 with a few percent of noise, so the fit recovers them.
The same points put through the double-reciprocal plot give Km = 37.2 andVmax = 1.91, low by 17 % and 13 %. That is the cost of the reciprocal transform on data with ordinary noise, and it is why the nonlinear estimate is the one reported above. If the assay held 5 nM enzyme and v is in µM/min, entering 0.005 as the enzyme concentration gives kcat = 440 min⁻¹ and kcat/Km = 1.63 × 10⁵ M⁻¹ s⁻¹.
Vmax depends on how much enzyme you happened to add; kcat does not, which is why turnover numbers are what papers compare. Enter the total enzyme concentration in the same concentration unit as the velocity and choose the time unit v was measured per. kcat/Km is reported in M⁻¹ s⁻¹, the usual units of the specificity constant, using Km converted to mol/L. To work out the enzyme concentration from an absorbance reading, use theA280 protein concentration calculator or theprotein concentration calculator, and for the extinction coefficient of your enzyme the protein properties calculator.
If the highest concentration tested is still below Km, the curve never levels off, Vmax is an extrapolation and Km follows it upwards. The tool says so when that happens, and the fitted numbers should then be treated as a lower bound.
One substrate concentration per line, followed by one or more velocities separated by spaces, tabs or commas, so a paste from a spreadsheet works. Several velocities on a line are treated as replicates, and repeating a concentration on further lines does the same. A header row is ignored. Concentrations of zero are allowed, negative ones are not, and the unit is set on the result so Km can be reported in it.
For a sigmoid dose-response curve rather than a hyperbola, for example an inhibitor titration, use the dose-response curve fitter or theIC50 calculator, and convert an IC50 to Ki with theCheng-Prusoff converter. To prepare the substrate series itself, see the serial dilution planner.
Vmax is the rate the enzyme reaches when the substrate is saturating, so every active site is occupied. Km is the substrate concentration that gives half of Vmax. A low Km means the enzyme reaches half speed at a low substrate concentration, which is often described as high apparent affinity, although Km is only a true dissociation constant when the chemical step is slow compared with substrate release.
Taking reciprocals turns the smallest velocities into the largest numbers on the plot, and their errors grow with them. A point at low substrate with 10 percent noise can sit far from the line and drag the regression, while the well-measured points near saturation crowd towards the origin and hardly influence it. The nonlinear fit weights every measurement as it was made, so it is the one to report; the double-reciprocal plot is a display, not a fitting method.
Five to eight, spread roughly evenly on a log scale from about one fifth of Km to five or ten times Km. That range covers both the rising part of the curve, which sets Km, and the plateau, which sets Vmax. Three points is the mathematical minimum for two parameters and leaves no way to check the model.
kcat is Vmax divided by the total enzyme concentration, the number of substrate molecules one active site turns over per unit time. kcat/Km is the specificity constant, the second-order rate constant for the reaction of free enzyme with free substrate, and it is the number to compare when asking which of two substrates an enzyme prefers. Diffusion sets a ceiling around 10⁸ to 10⁹ M⁻¹ s⁻¹.
That is substrate inhibition, and the Michaelis-Menten equation cannot describe it: the fit will report a Km and a Vmax that are both too high. Either drop the concentrations beyond the maximum and fit only the rising part, or use the substrate inhibition model v = Vmax·[S] / (Km + [S]·(1 + [S]/Ki)). A dip only in the last point is more often a depleted substrate or a detection limit than real inhibition.