Subtopic: Fundamentals of Biomolecules, enzyme kinetics
Question Statement:
If the function
f(x) = {sin 2x , for x > 0,
a + bx, for x ≤ 0,
where a and b are constants, is differentiable at x = 0, then a + b is equal to
Options:
(A) 0
(B) 1
(C) 2
(D) 3
Official Answer Key:
C
Step-by-Step Solution & Explanation
### **Official Solution & Concept Analysis (GATE BT 2025 Q.46)**
**1. Governing Concept**:
- **Topic**: Biochemistry & Allosteric Enzyme Kinetics (Hill Equation)
- Cooperative binding in multimeric allosteric enzymes (such as hemoglobin or ATCase) is described by the **Hill equation**:
$\theta = \frac{[L]^n}{K_d + [L]^n}$
where $n$ is the Hill coefficient:
- If **$n > 1$**: **Positive cooperativity** (binding of one ligand increases the affinity of remaining vacant sites for subsequent ligands; yields a sigmoidal binding curve).
- If **$n = 1$**: **Non-cooperative binding** (independent binding sites; hyperbolic curve).
- If **$n < 1$**: **Negative cooperativity** (binding of one ligand decreases the affinity of subsequent sites).
**2. Option Analysis**:
- **Option (A) is INCORRECT**: $n = 1$ indicates completely independent, non-cooperative binding.
- **Option (B) is INCORRECT**: $n < 1$ indicates negative cooperativity.
- **Option (C) is CORRECT**: A Hill coefficient greater than 1 ($n > 1$) mathematically and physically defines positive cooperativity.
- **Option (D) is INCORRECT**: The Hill coefficient cannot be negative for binding sites.
**3. Final Answer**:
- **Correct Choice**: **C**
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