# Question fe76a

Apr 6, 2015

The slope represents K_"m"/V_("max").

The starting curve for your double reciprocal plot is actually the Michaelis-Menten Plot, which looks like this The above plot shows the rate of an enzyme-catalyzed reaction, $V$, as a function of the substrate concentration, $\left[S\right]$.

The equation plotted looks like this

$V = \frac{{V}_{\text{max}} \cdot \left[S\right]}{{K}_{m} + \left[S\right]}$, where

${V}_{\text{max}}$ - the maximum rate of the reaction;
${K}_{m}$ - the substrate concentration when the rate is equal to ${V}_{\text{max}} / 2$.

Now, you need to get from $V$ and $\left[S\right]$ to $\frac{1}{V}$ and $\frac{1}{\left[S\right]}$, so just use the multiplicative inverse of the above equation to get

$\frac{1}{V} = \frac{{K}_{m} + \left[S\right]}{{V}_{\text{max}} \cdot \left[S\right]}$

$\frac{1}{V} = {K}_{m} / \left({V}_{\text{max") * [S]) + cancel([S])/(cancel([S]) * V_("max}}\right)$

1/V = K_m/(V_("max") * [S]) + 1/V_("max")

Now, to get to slope, think of the $y = m x + b \text{ }$ equation for a line

underbrace(1/V)_("y") = underbrace(K_m/V_("max"))_(color(red)(m)) * underbrace(1/([S]))_text(x) + underbrace(1/(V_max))_(b)

This plot is called the Lineweaver-Burk plot and it looks like this Apr 6, 2015

The slope is equal to ${K}_{\text{m""/"V_"max}}$.

A double reciprocal plot is used for analyzing enzyme kinetics.

It is a useful method for analyzing the Michaelis-Menten equation:

$V = \left({V}_{\text{max"[S])/(K_"m}} + \left[S\right]\right)$

where $V$ is the reaction velocity, ${V}_{\text{max}}$ is the maximum reaction velocity, $\left[S\right]$ is the substrate concentration, and ${K}_{\text{m}}$ is the Michaelis-Menten constant.

Taking the reciprocal gives

1/V = (K_"m" + [S])/(V_"max"[S]) = K_"m"/V_"max" 1/([S]) + 1/V_"max"#

The equation is of the form $y = m x + b$

where $y = \frac{1}{V}$, $m = {K}_{\text{m"/V_"max}}$, $x = \frac{1}{\left[S\right]}$, and $b = \frac{1}{V} _ \text{max}$.

A plot of $y$ vs $x$ or $\frac{1}{V}$ vs $\frac{1}{\left[S\right]}$ is called a double reciprocal plot or a Lineweaver-Burk plot. The $x$-intercept is $- \frac{1}{K} _ \text{m}$.

The $y$-intercept $b = \frac{1}{V} _ \text{max}$.

And the slope $m = {K}_{\text{m"/V_"max}}$.