§ C-28 Enzyme Catalysis, Michaelis–Menten Kinetics & Active-Site Inhibition (Grade 10–11 · Enzyme Kinetics)
Enzyme–substrate complex ES, lock-and-key active site, Vₘax, K_M, and competitive inhibition. An enzyme is a molecular glove whose palm, the active site, is shaped to embrace the transition state of its substrate.
Theory Summary & Scientific Mechanism
Enzymes achieve extraordinary catalytic rate enhancements by binding substrate S within a complementary active-site pocket to form an enzyme–substrate complex ES: E + S ⇄ ES xrightarrowk_cat E + P.
Under steady-state conditions (d[ES]/dt = 0), reaction velocity obeys the Michaelis–Menten equation v = (Vₘax[S])/(K_M + [S]), transitioning from 1st-order (v ≈ (Vₘax)/(K_M)[S]) at low [S] to 0th-order saturation (v → Vₘax = k_cat[E]₀) when all active sites are occupied ([S] ≫ K_M).
A competitive inhibitor I competes reversibly for the active site (E + I ⇄ EI), increasing the apparent Michaelis constant to K_M^app = K_M(1 + [I]/K_I) while leaving the asymptotic Vₘax unchanged.
Everyday Analogy
An enzyme active site is a lock and the substrate is the key. When keys flood the room, every lock is busy working at top speed Vₘax. A competitive inhibitor is a decoy key that jams the lock, requiring more real keys (higher K_M^app) to reach Vₘax!
Interactive Question & Derivation
Why does the reaction rate v([S]) of an enzyme-catalyzed reaction plateau at Vₘax = k_cat[E]₀ at high substrate concentrations, and how does a competitive inhibitor [I] alter K_M and Vₘax?
- A: At [S] ≫ K_M all enzyme active sites are saturated in the [ES] complex (v → Vₘax, 0th order in [S]); a competitive inhibitor increases the apparent Michaelis constant K_M^app = K_M(1 + [I]/K_I) without changing Vₘax
- B: Reaction rate increases linearly with [S] without bound, and a competitive inhibitor permanently destroys Vₘax
- C: The Michaelis constant K_M is the maximum possible velocity of the enzyme
Study Hint: When active sites saturate ([S] ≫ K_M), velocity plateaus at Vₘax = k_cat[E]₀; competitive inhibition increases K_M^app without changing Vₘax.
Proof & Explanation: In the Michaelis–Menten mechanism (E + S ⇄ ES xrightarrowk_cat E + P), the steady-state rate is v = (Vₘax[S])/(K_M + [S]), where Vₘax = k_cat[E]₀ and K_M is the substrate concentration at which v = (1)/(2)Vₘax. At low [S] ≪ K_M, v ≈ (Vₘax)/(K_M)[S] (1st order); at high [S] ≫ K_M, active sites are fully saturated and v → Vₘax (0th order). A competitive inhibitor I competes with S for the active site, raising K_M^app = K_M(1 + [I]/K_I) while leaving Vₘax unchanged at high [S].
Key Equations & 3D Simulation Parameters
E + S undersetk₋₁oversetk₁⇄ ES oversetk_catlongrightarrow E + P, v = (Vₘax[S])/(K_M≤ft(1 + ([I])/(K_I) ) + [S])
- Substrate concentration [S]: 0.25 … 12 mmol/L (default: 4 mmol/L)
- Michaelis constant K_M (inverse affinity): 0.5 … 5 mmol/L (default: 1.5 mmol/L)
- Catalytic turnover number k_cat (Vₘax): 2 … 14 s⁻¹ (default: 8 s⁻¹)
- Competitive inhibitor concentration [I]: 0 … 6 mmol/L (default: 0 mmol/L)