Linear forms of michaelis-menten equation

 The Michaelis-Menten equation describes the rate of enzymatic reactions by relating reaction rate 

$v$ to substrate concentration $[S]$, maximum reaction rate $V_{max}$, and the Michaelis constant $K_m$. The equation is given by:

$v=\frac{V_{max}[S]}{K_m+[S]}$

To determine $K_m$ and $V_{max}$, the Michaelis-Menten equation can be linearized in a few different ways. The two most common linear forms are the Lineweaver-Burk plot and the Eadie-Hofstee plot.

Lineweaver-Burk Plot (Double Reciprocal Plot)

The Lineweaver-Burk plot is obtained by taking the reciprocal of both sides of the Michaelis-Menten equation:

$\frac{1}{v}=\frac{K_m+[S]}{V_{max}[S]}$

This can be rearranged to:

$\frac{1}{v}=\frac{K_m}{V_{max}}\cdot \frac{1}{[S]}+\frac{1}{V_{max}}$

This equation is in the form of a straight line $y=mx+b$ where:

• $y=\frac{1}{v}$
• $x=\frac{1}{[S]}$
• Slope $m=\frac{K_m}{V_{max}}$
• Y-intercept $b=\frac{1}{V_{max}}$

By plotting $\frac{1}{v}$ against $\frac{1}{[S]}$, $V_{max}$ can be determined from the y-intercept (which is $\frac{1}{V_{max}}$) and $K_m$ can be determined from the slope (which is $\frac{K_m}{V_{max}}$).

Eadie-Hofstee Plot

The Eadie-Hofstee plot is obtained by rearranging the Michaelis-Menten equation to express $v$ as a function of $\frac{v}{[S]}$:

$v=V_{max}-K_m\left(\frac{v}{[S]}\right)$

This is also in the form of a straight line $y=mx+b$ where:

• $y=v$
• $x=\frac{v}{[S]}$
• Slope $m=-K_m$
• Y-intercept $b=V_{max}$

By plotting $v$ against $\frac{v}{[S]}$, $V_{max}$ can be determined from the y-intercept and $K_m$ can be determined from the negative slope.

Summary

1. Lineweaver-Burk Plot: $\frac{1}{v}=\frac{K_m}{V_{max}}\cdot \frac{1}{[S]}+\frac{1}{V_{max}}$

   • Plot $\frac{1}{v}$ vs. $\frac{1}{[S]}$
   • Slope = $\frac{K_m}{V_{max}}$
   • Y-intercept = $\frac{1}{V_{max}}$

2. Eadie-Hofstee Plot: $v=V_{max}-K_m\left(\frac{v}{[S]}\right)$

   • Plot $v$ vs. $\frac{v}{[S]}$
   • Slope = $-K_m$
   • Y-intercept = $V_{max}$

Using these linear transformations, $K_m$ and $V_{max}$ can be determined from experimental data by fitting the appropriate straight lines and extracting the parameters from the slope and intercept.