Question

In Michaelis-Menten kinetics to increase the reaction velocity by 9 fold from 10% of $V_{max}$ to 90% of $V_{max}$ the substrate concentration has to increase by:

• A. 3 fold
• B. 9 fold
• C. 27 fold
• D. 81 fold

Hint:

For 10% to 90% $V_{max}$, $[S]$ always increases 81-fold.

Answer 1

The Correct Option is D

Step 1: Concept
The relationship between velocity and substrate is $v = \frac{V_{max}[S]}{K_{m} + [S]}$.

Step 2: Meaning
We need to find $[S]$ at $v = 0.1 V_{max}$ and $v = 0.9 V_{max}$.

Step 3: Analysis
- At $v = 0.1 V_{max}$: $0.1 = \frac{[S_{1}]}{K_{m} + [S_{1}]} \implies [S_{1}] = \frac{1}{9} K_{m}$. - At $v = 0.9 V_{max}$: $0.9 = \frac{[S_{2}]}{K_{m} + [S_{2}]} \implies [S_{2}] = 9 K_{m}$. - Increase factor = $[S_{2}] / [S_{1}] = 9 / (1/9) = 81$.

Step 4: Conclusion
The substrate concentration must increase by 81-fold to shift the velocity from 10% to 90% of $V_{max}$. Final Answer: (D)