Subtopic: Fundamentals of Biomolecules, enzyme kinetics
Question Statement:
An enzymatic reaction exhibits Michaelis-Menten kinetics. For this reaction, on doubling the
concentration of enzyme while maintaining [S] >> [Eo].
Options:
(A) both Km and Vmax will remain the same.
(B) Km will remain the same but max will increase.
(C) Km will increase but max will remain the same.
(D) both Km and max will increase.
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE BT 2017 Q.21)**
**1. Governing Concept & Question Classification**:
- **Section**: BT
- **Question Type**: **MCQ** (1 Mark)
- **Topic**: Biochemistry / Enzyme Kinetics (Michaelis-Menten Parameters: $K_m$ and $V_{\max}$)
**2. Detailed Step-by-Step Derivation**:
- In Michaelis-Menten enzyme kinetics:
$v = \frac{V_{\max}[S]}{K_m + [S]}$
- The maximum velocity is directly proportional to the total active enzyme concentration $[E_0]$:
$V_{\max} = k_{\text{cat}} [E_0]$
where $k_{\text{cat}}$ is the turnover number (a constant intrinsic to the enzyme).
- If the concentration of enzyme is doubled ($[E_0]' = 2[E_0]$):
$V_{\max}' = k_{\text{cat}} [E_0]' = 2 k_{\text{cat}} [E_0] = 2 V_{\max}$
Therefore, $V_{\max}$ doubles (increases).
- The Michaelis constant, $K_m = \frac{k_{-1} + k_{\text{cat}}}{k_1}$, represents the substrate concentration at which $v = V_{\max}/2$.
- $K_m$ is an **intrinsic characteristic constant** of the enzyme-substrate pair reflecting the affinity under defined temperature, pH, and ionic strength, and is **completely independent of total enzyme concentration $[E_0]$**.
- Thus, doubling the enzyme concentration causes **$K_m$ to remain the same while $V_{\max}$ increases**.
- **Option-by-Option Evaluation**:
- **Option (A) [INCORRECT]**: $V_{\max}$ is directly proportional to $[E_0]$ and therefore cannot remain the same when $[E_0]$ doubles.
- **Option (B) [CORRECT]**: $K_m$ is an intrinsic kinetic constant and remains unchanged, whereas $V_{\max}$ doubles with doubled $[E_0]$.
- **Option (C) [INCORRECT]**: $K_m$ does not depend on enzyme concentration.
- **Option (D) [INCORRECT]**: $K_m$ does not increase with enzyme concentration.
**3. Final Answer / Verification Summary**:
- **Official GATE Answer Key**: **B**
- **Scoring Criteria**: +1.0 Mark (Negative marking: -0.33 marks).
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