Lineweaver-Burk | Equation, Derivation and Rearrangements

Core idea

Overview

The Lineweaver-Burk equation is a linear transformation of the Michaelis-Menten equation achieved by taking the reciprocal of both sides. This double-reciprocal plot allows biochemists to easily determine the maximum reaction velocity (Vmax) and the Michaelis constant (Km) by fitting experimental data to a straight line.

When to use: Apply this equation when you need to calculate kinetic parameters from experimental reaction rates at varying substrate concentrations. It is particularly useful for identifying the mechanism of enzyme inhibition, such as distinguishing between competitive and non-competitive inhibitors based on changes in the intercept and slope.

Why it matters: This linear model simplifies the analysis of enzyme kinetics, which is vital for drug discovery and understanding metabolic pathways. It allows for the visual diagnosis of how a molecule affects an enzyme's affinity and catalytic power, facilitating the development of therapeutic inhibitors.

Symbols

Variables

y = 1/v, m = Gradient (Km/Vmax), x = 1/[S], c = Y-intercept (1/Vmax)

y

1/v

s/u

m

Gradient (Km/Vmax)

s

x

1/[S]

1/mM

c

Y-intercept (1/Vmax)

s/u

Walkthrough

Derivation

Derivation of Lineweaver-Burk Plot Equation

A linear form of Michaelis–Menten used to estimate Vmax and Km from a straight-line graph.

Michaelis–Menten kinetics applies to the enzyme under the conditions used.

1

Start with Michaelis–Menten:

Begin with the standard saturation equation.

V = \frac{V _{m a x} [ S ]}{K _{m} + [ S ]}

2

Take Reciprocals:

Invert both sides to move toward a linear relationship.

\frac{1}{V} = \frac{K _{m} + [ S ]}{V _{m a x} [ S ]}

3

Split the Fraction:

Separate into a term involving $1/ [S]$ plus a constant.

\frac{1}{V} = \frac{K _{m}}{V _{m a x} [ S ]} + \frac{1}{V _{m a x}}

4

Write in y=mx+c Form:

Plotting $1/ V$ against $1/ [S]$ gives a straight line with gradient $K_{m} / V_{m a x}$ and y-intercept $1/ V_{m a x}$ .

\frac{1}{V} = (\frac{K _{m}}{V _{m a x}}) \frac{1}{[ S ]} + \frac{1}{V _{m a x}}

Result

\frac{1}{V} = (\frac{K _{m}}{V _{m a x}}) \frac{1}{[ S ]} + \frac{1}{V _{m a x}}

Source: AQA A-Level Biology — Biological Molecules (Enzymes)

Free formulas

Rearrangements

Solve for $y$

Make y the subject

y = m x + c

Transform the Lineweaver-Burk equation into the standard straight-line form, y = mx + c, by identifying y as 1/v and its corresponding gradient (m), independent variable (x), and y-intercept (c) terms.

Difficulty: 2/5

Solve for $m$

Make m the subject

m = \frac{y - c}{x}

Start from the Lineweaver-Burk equation, which is in the form y = mx + c. To make m the subject, subtract c from both sides, then divide by x.

Difficulty: 2/5

Solve for $x$

Lineweaver-Burk: Make x the subject

[S] = \frac{v K _{m}}{V _{ma x} - v}

Rearrange the Lineweaver-Burk equation to solve for the substrate concentration [S].

Difficulty: 3/5

Solve for $c$

Make c the subject

c = y - m x

Rearrange the Lineweaver-Burk equation to make `c` (the Y-intercept, `\frac{1}{V_{max}}`) the subject, expressing it in the form `c = y - mx`.

Difficulty: 2/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

The graph is a straight line representing a direct linear relationship between y and x, where m is the slope and c is the y-axis intercept. For a biology student, this shape simplifies the analysis of enzyme kinetics because small x-values represent high substrate concentrations while large x-values represent low concentrations. The most important feature is that the linear relationship allows for a constant rate of change, meaning that equal increments in x result in consistent changes in y across the entire line.

Graph type: linear

Why it behaves this way

Intuition

The Lineweaver-Burk equation transforms the hyperbolic enzyme kinetics into a straight line, where the slope, y-intercept, and x-intercept directly correspond to key kinetic parameters Vmax and Km.

v

Initial reaction velocity

The rate at which product is formed at the beginning of an enzyme-catalyzed reaction, before substrate depletion becomes significant.

[S]

Substrate concentration

The amount of reactant molecules available for the enzyme to act upon; higher concentration typically leads to more frequent enzyme-substrate encounters.

V_{ma x}

Maximum reaction velocity

The highest possible rate an enzyme can achieve when it is fully saturated with substrate, representing its maximum catalytic capacity.

K_{m}

Michaelis constant

The substrate concentration at which the reaction velocity is half of Vmax; it reflects the enzyme's apparent affinity for its substrate (lower Km means higher affinity).

Signs and relationships

1/v: Taking the reciprocal of the initial velocity linearizes the hyperbolic Michaelis-Menten relationship, transforming it into a straight line for easier graphical analysis.
1/[S]: Taking the reciprocal of the substrate concentration similarly linearizes the independent variable, allowing for a straight-line plot against 1/v.

Free study cues

Insight

Canonical usage

The Lineweaver-Burk equation requires consistent units for concentration (e.g., M, mM) and time (e.g., s, min) across all terms to ensure dimensional homogeneity.

Common confusion

A common mistake is using inconsistent units for concentration (e.g., M for [S] but mM for Km) or for time (e.g., seconds for v but minutes for Vmax). This will lead to incorrect kinetic parameter calculations.

Unit systems

$v$ M s^-1 - Initial reaction velocity, typically expressed as change in substrate or product concentration per unit time. Units like μM min^-1 are also common.

KmM - Michaelis constant, representing the substrate concentration at half Vmax. Its units must be consistent with the substrate concentration [S].

VmaxM s^-1 - Maximum reaction velocity. Its units must be consistent with the initial reaction velocity v.

[S]M - Substrate concentration. Units like mM or μM are also common.

Ballpark figures

Quantity:

One free problem

Practice Problem

Given an enzyme-catalyzed reaction where the slope of the Lineweaver-Burk plot (m) is 0.5 min and the y-intercept (c) is 0.2 min/µmol, calculate the reciprocal velocity (y) when the reciprocal of the substrate concentration (x) is 4 L/µmol.

Gradient (Km/Vmax)0.5 s

1/[S]4 1/mM

Y-intercept (1/Vmax)0.2 s/u

Solve for: $y$

Hint: Use the linear form y = mx + c to find the total reciprocal velocity.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

When finding Km from a straight-line fit, Lineweaver-Burk is used to calculate 1/v from Gradient (Km/Vmax), 1/[S], and Y-intercept (1/Vmax). The result matters because it helps compare enzyme activity, saturation, or inhibitor strength in an assay or drug-response setting.

Study smarter

Tips

The y-intercept (c) represents 1/Vmax.
The x-intercept is -1/Km, though the x-variable itself is 1/[S].
The slope (m) represents the ratio Km/Vmax.
Be cautious with low substrate concentrations, as small measurement errors are magnified when taking reciprocals.

Avoid these traps

Common Mistakes

Plotting v instead of 1/v.
Forgetting the y-intercept term.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

A linear form of Michaelis–Menten used to estimate Vmax and Km from a straight-line graph.

Apply this equation when you need to calculate kinetic parameters from experimental reaction rates at varying substrate concentrations. It is particularly useful for identifying the mechanism of enzyme inhibition, such as distinguishing between competitive and non-competitive inhibitors based on changes in the intercept and slope.

This linear model simplifies the analysis of enzyme kinetics, which is vital for drug discovery and understanding metabolic pathways. It allows for the visual diagnosis of how a molecule affects an enzyme's affinity and catalytic power, facilitating the development of therapeutic inhibitors.

Plotting v instead of 1/v. Forgetting the y-intercept term.

When finding Km from a straight-line fit, Lineweaver-Burk is used to calculate 1/v from Gradient (Km/Vmax), 1/[S], and Y-intercept (1/Vmax). The result matters because it helps compare enzyme activity, saturation, or inhibitor strength in an assay or drug-response setting.

The y-intercept (c) represents 1/Vmax. The x-intercept is -1/Km, though the x-variable itself is 1/[S]. The slope (m) represents the ratio Km/Vmax. Be cautious with low substrate concentrations, as small measurement errors are magnified when taking reciprocals.

References

Sources

IUPAC Gold Book: Michaelis constant, Km
IUPAC Gold Book: Maximum velocity, Vmax
Lehninger Principles of Biochemistry, 7th Edition, Nelson, D.L. and Cox, M.M.
Wikipedia: Lineweaver-Burk plot
Lehninger Principles of Biochemistry, 7th Edition, by Nelson and Cox
Berg, Tymoczko, and Stryer Biochemistry
Nelson and Cox Lehninger Principles of Biochemistry
Atkins Physical Chemistry