Michaelis-Menten calculator

What is the Michaelis–Menten equation?

The equation v = Vmax × [S] ÷ (Km + [S]) gives the initial rate v of a simple enzyme reaction at substrate concentration [S]. Vmax is the rate when the enzyme is saturated, and Km, the Michaelis constant, is the substrate concentration at which the rate is half of Vmax. It goes back to Leonor Michaelis and Maud Menten’s 1913 study of the enzyme invertase. The modern derivation assumes a steady state for the enzyme–substrate complex and an enzyme concentration much lower than the substrate. With Vmax = 100 and Km = 5, a substrate concentration equal to Km gives v = 50, nine times Km gives 90, and a tenth of Km gives only 9.09.

What do the three modes do?

“Rate from [S]” evaluates the equation. “[S] from rate” inverts it: [S] = Km × v ÷ (Vmax − v), which is only possible for a rate below Vmax, because no amount of substrate pushes the enzyme past saturation. For example, to reach v = 75 with Vmax = 100 and Km = 5 you need [S] = 15, that is three times Km. “Fit my data” takes a table of substrate concentrations and measured initial rates, one pair per line, and estimates Vmax and Km. Concentration and time units are labels for the results; the math does not change.

Worked example: fitting a table of rates

Suppose you measured these (substrate, rate) pairs: (0.5, 8), (1, 15), (2, 22), (5, 40), (10, 52), (20, 60), (50, 68). These numbers are illustrative, not from a real experiment. The nonlinear fit finds Vmax = 73.59 and Km = 4.28, with standard errors of 0.95 and 0.189. A Hanes–Woolf line gives Vmax = 73.64, Km = 4.25, and the Lineweaver–Burk line gives Vmax = 70.58, Km = 3.89. All three come out different because each method weights the points differently; with perfectly error-free data they all agree exactly.

Why not just use a Lineweaver–Burk plot?

Lineweaver and Burk (1934) showed that plotting 1/v against 1/[S] gives a straight line with intercept 1/Vmax and slope Km/Vmax, which made kinetic constants easy to read off before computers. The catch is the error distortion: the points at low substrate concentration, where v is small and relatively noisy, become the largest 1/v values and sit far to the right, so they dominate the line. Common sources on enzyme kinetics therefore recommend a properly weighted nonlinear fit, with the Hanes–Woolf plot (S/v against S) as the best of the linear forms when the rate has a constant error. This calculator shows all three so you can see how much your data depend on the method.

How does the nonlinear fit work?

It searches for the Vmax and Km that minimize the sum of squared differences between your measured rates and the curve, working on the rates themselves instead of on reciprocals. The search is a Gauss–Newton iteration with Levenberg–Marquardt damping, started from the Hanes–Woolf estimate. The standard errors come from the curvature at the best fit and the residual variance, which needs at least three points. The quality of the fit depends on having concentrations that span from well below Km to several times Km; if the highest [S] is under about twice Km, Vmax is largely an extrapolation, and the calculator says so.

Common mistakes

Using rates that are not initial rates is the most frequent problem: when product builds up or substrate is used up during the measurement, the apparent rate falls and Km looks wrong. Another is mixing units, for example substrate in µM and Km in mM. Vmax also depends on how much enzyme you used, so it is not a property of the enzyme alone; Km, unlike Vmax, does not change with enzyme amount, but it does depend on temperature, pH and the substrate. Finally, a Michaelis–Menten curve cannot describe cooperative enzymes, which give an S-shaped plot, or substrate inhibition, where the rate falls at high [S].

Limits of this method

The model describes one substrate and one active site per enzyme in a steady state; it assumes the enzyme is far less concentrated than the substrate and that product does not rebind. It does not cover inhibitors, which need an extra term (competitive, noncompetitive or uncompetitive models), nor reactions with two substrates. With only three or four points the standard errors are large and the numbers are rough. For a serious analysis, replicate the measurements and use dedicated curve-fitting software with weighting that matches your assay’s error.

For study and lab-bench use; confirm results with your own analysis software. Sources (secondary summaries; the original papers were not opened): Wikipedia, “Michaelis–Menten kinetics” (L. Michaelis and M. L. Menten, Biochemische Zeitschrift 49, 1913, 333–369) and “Lineweaver–Burk plot” (H. Lineweaver and D. Burk, Journal of the American Chemical Society 56, 1934, 658–666). The fitting algorithm is standard nonlinear least squares. Related: half-life calculator, linear regression calculator, activation energy calculator, molarity calculator and dilution calculator.