Chapter 13, Problem 1P

Answers to all problems are at the end of this book. Detailed solutions are available in the Student Solutions Manual, Study Guide, and Problems Book.

Exploring the Michaelis-Menten Equation - I According to the Michaelis-Menten equation, what is the v/V_max ratio when [S] = 4 K_m?

Interpretation Introduction

To propose:

With the use of Michaelis-Menten equation I, the value of $\upsilon$ / $V_{\max }$ ratio when [S] = 4 $K_m$ .

Introduction:

The Michaelis-Menten Equation relates reaction velocity with substrate concentration where they form an enzyme-substrate complex which further reacts irreversibly to form a product and free enzyme and it can be represented mathematically as below:

  $E+S\rightleftharpoons ES\to E+P$

  $\upsilon =\frac{d\left[P\right]}{dt}=\frac{V_{\max }\left[S\right]}{K_m+\left[S\right]}$

Where,

• $V_{\max }$= maximum velocity
• $K_m$= Michaelis-Menten constant
• $\left[S\right]$= substrate concentration

Explanation of Solution

For enzyme kinetics, Leonor Michaelis, a German biochemist had made a very best-understood and known model as Michaelis-Menten equation. The rate in enzyme reactions, in relation to the rate of reaction $\upsilon$

  $\upsilon =\frac{d\left[P\right]}{dt}=\frac{V_{\max }\left[S\right]}{K_m+\left[S\right]}$

Where,

• $V_{\max }$= at the time of saturation, the maximum rate achieved by any system
• $K_m$ = Michaelis- Menten constant and at the time of this, the value of $V_{\max }$becomes half.

Now, we have the value [S] = 4 $K_m$ and we can determine the value of $\upsilon$ / $V_{\max }$

ratio as below:

  $\begin{array}{l}\upsilon =\frac{d\left[P\right]}{dt}=\frac{V_{\max }\left[S\right]}{K_m+\left[S\right]}\\ \upsilon =\frac{V_{\max }\times 4K_m}{K_m+4K_m}\\ \upsilon =\frac{4V_{\max }}{5}\\ \frac{\upsilon }{V_{\max }}=0.8\end{array}$

Therefore, the value of $\upsilon$ / $V_{\max }$ ratio when [S] = 4 $K_m$ is 0.8.