Biomolecules, enzyme kinetics

### **Step-by-Step Solution & Mathematical Derivation (GATE BT 2025 Q.32)** **1. Governing Concept & Definition**: - **Topic**: Enzyme Kinetics & Michaelis-Menten Equation - The Michaelis-Menten rate equation is: $v_0 = \frac{V_{\max} [S]}{K_m + [S]}$ - Dividing both sides by $V_{\max}$: $\frac{v_0}{V_{\max}} = \frac{[S]}{K_m + [S]}$ **2. Detailed Step-by-Step Calculation**: - **Given**: - Substrate concentration: $[S] = 20 K_m$ - **Substitution**: $\frac{v_0}{V_{\max}} = \frac{20 K_m}{K_m + 20 K_m} = \frac{20 K_m}{21 K_m} = \frac{20}{21}$ - **Numerical Evaluation**: $\frac{20}{21} \approx 0.95238 \approx 0.95$ **3. Final Answer**: - **Calculated Value**: **0.95** - **Official GATE Accepted Range**: **0.94 to 0.96**