Lineweaver–Burk plot (double-reciprocal) — basic theory
The Lineweaver–Burk method is a classic way to analyze enzyme kinetics by turning the
Michaelis–Menten curve into a straight line using a reciprocal transformation.
It is commonly taught because it makes the parameters appear through the line’s slope and intercept.
Important: because it uses reciprocals, this plot tends to overweight measurements at low substrate concentration
(low [S]) and can amplify noise. Use it mainly for learning/diagnostic checks, and compare with a nonlinear fit
of the original Michaelis–Menten form when possible.
1) Starting point: Michaelis–Menten model
The Michaelis–Menten rate law (for initial rates) is:
\[
v=\frac{V_{\max}[S]}{K_m+[S]}
\]
- v — initial reaction rate
- [S] — substrate concentration
- Vmax — maximum rate (when enzyme is saturated)
- Km — substrate concentration where \(v = \frac{1}{2}V_{\max}\)
2) Double-reciprocal transformation
Take the reciprocal of both sides and rearrange to get a line in the variables \(x=\frac{1}{[S]}\) and \(y=\frac{1}{v}\):
\[
\frac{1}{v}=\frac{K_m}{V_{\max}}\cdot\frac{1}{[S]}+\frac{1}{V_{\max}}
\]
This matches the straight-line form:
\[
y = m\,x + b
\]
where:
3) Computing Vmax and Km from the fitted line
Once you fit a line \(y = m x + b\) to the reciprocal points \(\left(\frac{1}{[S]},\frac{1}{v}\right)\),
you can recover the kinetic parameters as:
\[
V_{\max}=\frac{1}{b},\qquad K_m=\frac{m}{b}
\]
The calculator also displays the x-intercept, which ideally equals \(-\frac{1}{K_m}\).
4) Why the method is sensitive (and what “exclude lowest [S]” does)
-
The transformation uses \(\frac{1}{v}\) and \(\frac{1}{[S]}\). Small errors in \(v\) at low rates become
large errors in \(\frac{1}{v}\).
-
Points at low [S] often have large \(\frac{1}{[S]}\) values, which can have high leverage on the fitted slope.
-
As a result, Lineweaver–Burk fits can produce biased Vmax and Km, especially if the low-[S]
measurements are noisy.
The calculator includes an option to exclude the lowest [S] point(s). This is a practical robustness trick:
removing the lowest [S] group(s) can reduce the leverage of extreme reciprocal-x values and sometimes stabilizes the line.
It is not a substitute for proper weighting or nonlinear regression; it is mainly a learning aid.
5) Fit quality (R² and RMSE) — what it means here
The calculator reports R² and RMSE for the straight-line regression in reciprocal space
(i.e., for \(y=\frac{1}{v}\) vs \(x=\frac{1}{[S]}\)).
- R² is descriptive: it tells you how well a line explains variation in \(\frac{1}{v}\), not how good the enzyme model is in original units.
- RMSE is also computed in reciprocal units (1/v).
A visually “good” line in the Lineweaver–Burk plot does not guarantee an accurate Vmax, Km.
Always check the original-scale curve.
6) Connecting back to the original curve
After estimating Vmax and Km from the Lineweaver–Burk line, the calculator draws an
inset plot of \(v\) vs \([S]\) using:
\[
v_{\text{fit}}=\frac{V_{\max}[S]}{K_m+[S]}
\]
This helps you see how the reciprocal-line estimates behave on the real (nonlinear) Michaelis–Menten scale.
Practical checklist
- Use only valid data: \([S] > 0\) and \(v > 0\). (Reciprocals require it.)
- If your lowest [S] values look noisy, try excluding 1 point and compare results.
- Interpret R² as “how linear the reciprocal plot is,” not as final model validation.
- Whenever possible, prefer a nonlinear Michaelis–Menten fit for parameter estimation.
