BioDeviceHub

Enzyme Kinetics Calculator (Km, Vmax Fit)

Fit Km and Vmax to initial-rate data by nonlinear least squares, with standard errors, kcat, the fitted curve, residuals and the linearised plots for comparison.

Formula

v=Vmax⁡ [S]Km+[S],fitted by nonlinear least squaresv = \dfrac{V_{\max}\,[S]}{K_{\mathrm{m}} + [S]},\quad \text{fitted by nonlinear least squares}
kcat=Vmax⁡[E]k_{\mathrm{cat}} = \dfrac{V_{\max}}{[E]}
Lineweaver-Burk:1v=KmVmax⁡⋅1[S]+1Vmax⁡\text{Lineweaver-Burk:}\quad \dfrac{1}{v} = \dfrac{K_{\mathrm{m}}}{V_{\max}}\cdot\dfrac{1}{[S]} + \dfrac{1}{V_{\max}}
kcatk_{\mathrm{cat}}
turnover number: substrate molecules converted per enzyme molecule per unit time
kcat/Kmk_{\mathrm{cat}}/K_{\mathrm{m}}
the catalytic efficiency, which limits the rate at low substrate

How it works

Km and Vmax are estimated by fitting the Michaelis-Menten equation directly to the initial rates, using a Levenberg-Marquardt nonlinear least-squares fit. This is the recommended method, because it weights all the data as they were measured.

The classical linear transformations (Lineweaver-Burk, Eadie-Hofstee and Hanes-Woolf) are shown alongside for comparison, because they are still widely used. They distort the error structure, especially Lineweaver-Burk, which gives the most weight to the least reliable points at the lowest substrate concentration, so their estimates can differ from the nonlinear fit. If the total enzyme concentration is known, kcat and kcat/Km follow.

Worked example

Seven substrate concentrations from 0.5 to 32 with rates 1.3, 2.2, 3.6, 5.1, 6.2, 7.0 and 7.3 (illustrative).

  1. The nonlinear fit gives Vmax = 7.997 ± 0.106 and Km = 2.43 ± 0.115.
  2. Lineweaver-Burk gives Km = 2.68 and Vmax = 8.24; Hanes-Woolf gives Km = 2.35.

Km = 2.43 and Vmax = 8.00, with the linearised methods within about 10%, as expected for data with some scatter.

These are the values the calculator opens with, so you can check its output against this example.

Assumptions

  • Rates are initial rates, with product formation small and the substrate nearly undepleted.
  • Errors in the rate are similar in size at all substrate concentrations; if they grow with the rate, a weighted fit is better.
  • A single-substrate enzyme with hyperbolic kinetics.

Common mistakes

  • Using Lineweaver-Burk as the primary estimate.
  • Choosing substrate concentrations that all lie above or below Km, so the curve's shape is not sampled.
  • Fitting progress curves rather than initial rates without a model for substrate depletion.

Related tools

Michaelis-Menten CalculatorRate at a given [S], or [S] for a target rate, with the curveOpen calculatorEnzyme Activity Calculator (Units, Specific Activity)U/mL from an absorbance rate, specific activity and unit conversionOpen calculatorIC50 CalculatorFour-parameter logistic fit, with Ki by Cheng-PrusoffOpen calculatorProtein Binding Affinity (Kd) CalculatorKd and Bmax from a saturation curve, and fraction boundOpen calculator

Related equipment

Service documentation, failure modes and parts for the instruments this calculation is used with.