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Lineweaver–Burk &
Alternative Linear Plots

Transform Michaelis–Menten kinetics into straight lines to extract \( V_{\max} \) and \( K_m \) with confidence.

Biochemistry

Why Linearize Enzyme Kinetics?

The Michaelis–Menten equation describes a rectangular hyperbola: \( v_0 = \dfrac{V_{\max}[\text{S}]}{K_m + [\text{S}]} \). Fitting a curve to this hyperbola is hard by eye: \( V_{\max} \) is approached asymptotically and small errors in high-\([\text{S}]\) data distort the estimate.

The Michaelis–Menten Curve

\([\text{S}]\) \(v_0\) \(V_{\max}\) \(K_m\) \(\frac{V_{\max}}{2}\)

Three Linear Transformations

Lineweaver–Burk
Plot \( \frac{1}{v_0} \) vs \( \frac{1}{[\text{S}]} \). The "double-reciprocal" plot.
Hanes–Woolf
Plot \( \frac{[\text{S}]}{v_0} \) vs \( [\text{S}] \). Better statistical weighting.
Eadie–Hofstee
Plot \( v_0 \) vs \( \frac{v_0}{[\text{S}]} \). Spreads data evenly.

The Lineweaver–Burk Equation

Start with Michaelis–Menten and take the reciprocal of both sides:

y-axis
\( \frac{1}{v_0} \)
x-axis
\( \frac{1}{[\text{S}]} \)
Slope
\( \frac{K_m}{V_{\max}} \)
y-intercept
\( \frac{1}{V_{\max}} \)

Reading the Double-Reciprocal Plot

\(\frac{1}{[\text{S}]}\) \(\frac{1}{v_0}\) \(\frac{1}{V_{\max}}\) \(-\frac{1}{K_m}\) \(\text{slope} = \frac{K_m}{V_{\max}}\) 0

Lineweaver–Burk: Weaknesses

The double-reciprocal plot is the most famous, but it has serious statistical drawbacks that can mislead you.

1
Unequal error magnification: taking \( \frac{1}{v_0} \) amplifies errors at low \( [\text{S}] \) (small \( v_0 \)), compressing reliable high-\([\text{S}]\) data near the origin.
2
Clustering of points: most data points crowd near the y-axis, while a few distant points at low \( [\text{S}] \) dominate the regression.
3
Intercept extrapolation: both intercepts require extrapolating beyond the measured data range, increasing uncertainty.

The Hanes–Woolf Equation

Multiply both sides of the Lineweaver–Burk equation by \( [\text{S}] \):

y-axis
\( \frac{[\text{S}]}{v_0} \)
x-axis
\( [\text{S}] \)
Slope
\( \frac{1}{V_{\max}} \)
y-intercept
\( \frac{K_m}{V_{\max}} \)

Hanes–Woolf Plot

\([\text{S}]\) \(\frac{[\text{S}]}{v_0}\) \(\frac{K_m}{V_{\max}}\) \(-K_m\) \(\text{slope} = \frac{1}{V_{\max}}\) 0

The Eadie–Hofstee Equation

Rearrange Michaelis–Menten by isolating \( v_0 \):

y-axis
\( v_0 \)
x-axis
\( \frac{v_0}{[\text{S}]} \)
Slope
\( -K_m \)
y-intercept
\( V_{\max} \)

Eadie–Hofstee Plot

\(\frac{v_0}{[\text{S}]}\) \(v_0\) \(V_{\max}\) \(\frac{V_{\max}}{K_m}\) \(\text{slope} = -K_m\)

Step-by-Step: Building a Linear Plot

1
Collect data: measure \( v_0 \) at 5+ different \([\text{S}]\) values, keeping \([\text{E}]_{\text{total}}\) constant.
2
Choose your plot: calculate the transformed variables (e.g. \( \frac{1}{v_0} \) and \( \frac{1}{[\text{S}]} \) for Lineweaver–Burk).
3
Plot and fit: graph the data, draw the best-fit line using linear regression.
4
Extract parameters: read intercepts and slope; invert or rearrange to get \( V_{\max} \) and \( K_m \).
5
Cross-check: compare values from two different linear plots to confirm consistency.

Worked Example: Lineweaver–Burk

An enzyme gives a y-intercept of \(0.02\;\mu\text{M}^{-1}\text{s}\) and a slope of \(0.05\;\text{s}\) on a Lineweaver–Burk plot. Find \( V_{\max} \) and \( K_m \).

1
y-intercept \( = \frac{1}{V_{\max}} = 0.02 \), so \( V_{\max} = \frac{1}{0.02} = \) \( 50\;\mu\text{M}\,\text{s}^{-1} \).
2
Slope \( = \frac{K_m}{V_{\max}} = 0.05 \), so \( K_m = 0.05 \times 50 = \) \( 2.5\;\mu\text{M} \).

Worked Example: Hanes–Woolf

The same enzyme is plotted on a Hanes–Woolf graph. The slope is \( 0.02\;\text{s}\cdot\mu\text{M}^{-1} \) and the y-intercept is \( 0.05\;\text{s} \). Confirm \( V_{\max} \) and \( K_m \).

1
Slope \( = \frac{1}{V_{\max}} = 0.02 \), so \( V_{\max} = \frac{1}{0.02} = \) \( 50\;\mu\text{M}\,\text{s}^{-1} \).
2
y-intercept \( = \frac{K_m}{V_{\max}} = 0.05 \), so \( K_m = 0.05 \times 50 = \) \( 2.5\;\mu\text{M} \).

Worked Example: Eadie–Hofstee

On an Eadie–Hofstee plot the y-intercept is \( 50\;\mu\text{M}\,\text{s}^{-1} \) and the slope is \( -2.5\;\mu\text{M} \). Extract \( V_{\max} \) and \( K_m \).

1
y-intercept \( = V_{\max} = \) \( 50\;\mu\text{M}\,\text{s}^{-1} \): read directly.
2
Slope \( = -K_m = -2.5 \), so \( K_m = \) \( 2.5\;\mu\text{M} \).
3
x-intercept \( = \frac{V_{\max}}{K_m} = \frac{50}{2.5} = \) \( 20\;\text{s}^{-1} \). Verified.

Common Mistakes

✕ Forgetting the Negative
The x-intercept of Lineweaver–Burk is \( -\frac{1}{K_m} \). Dropping the minus sign gives a nonsensical negative \( K_m \).
✕ Swapping Slope Meanings
The slope formula differs in each plot. Confusing \( \frac{K_m}{V_{\max}} \) (LB) with \( \frac{1}{V_{\max}} \) (HW) gives wrong answers.
✕ Trusting Outliers on LB
A single noisy low-\([\text{S}]\) point can tilt the entire regression line. Always inspect residuals or use an alternative plot.

Summary Comparison

Lineweaver–Burk Hanes–Woolf Eadie–Hofstee
y-axis \( \frac{1}{v_0} \) \( \frac{[\text{S}]}{v_0} \) \( v_0 \)
x-axis \( \frac{1}{[\text{S}]} \) \( [\text{S}] \) \( \frac{v_0}{[\text{S}]} \)
Slope \( \frac{K_m}{V_{\max}} \) \( \frac{1}{V_{\max}} \) \( -K_m \)
y-intercept \( \frac{1}{V_{\max}} \) \( \frac{K_m}{V_{\max}} \) \( V_{\max} \)
Best for Visualizing inhibition Reliable regression Spotting outliers

Question 1 of 5

On a Lineweaver–Burk plot, the y-intercept equals:

Question 2 of 5

In the Eadie–Hofstee plot, the slope of the line equals:

Question 3 of 5

A Lineweaver–Burk plot has a y-intercept of \( 0.01\;\mu\text{M}^{-1}\text{s} \) and an x-intercept of \( -0.5\;\mu\text{M}^{-1} \). What is \( K_m \)?

Question 4 of 5

Which linear plot has the best statistical reliability for determining \( V_{\max} \)?

Question 5 of 5

On a Hanes–Woolf plot, the y-intercept equals: