Enzymatic Catalysis
Enzymes are protein (or ribozyme) biocatalysts of remarkable efficiency and specificity. They accelerate biological reactions by factors of 10⁶–10¹⁴ relative to the uncatalyzed rate. Their kinetics are formalized by the Michaelis-Menten model (1913).
The Michaelis-Menten Model
The simplified mechanism is:
E + S ⇌ ES → E + P
- k₁ (ES complex formation); k₋₁ (dissociation); k_cat (reaction → product).
Applying the steady-state approximation to intermediate ES:
d[ES]/dt = k₁[E][S] − k₋₁[ES] − k_cat[ES] = 0
Define the Michaelis constant: Km = (k₋₁ + k_cat) / k₁
Total enzyme concentration: [E]_T = [E] + [ES]
The Michaelis-Menten equation:
v = Vmax · [S] / (Km + [S])
with Vmax = k_cat · [E]_T
![v = f([S]) curve — Michaelis-Menten hyperbola](/courses/figures/enzymatic-catalysis-mm-curve.png)
Interpreting Km and Vmax
| Parameter | Meaning | Units |
|---|---|---|
| Vmax | Maximum rate at substrate saturation | mol·L⁻¹·s⁻¹ |
| Km | [S] at which v = Vmax/2; measures E/S affinity | mol·L⁻¹ |
| k_cat | Turnover number | s⁻¹ |
| k_cat/Km | Catalytic efficiency; diffusion limit ≈ 10⁸–10⁹ M⁻¹s⁻¹ | M⁻¹s⁻¹ |
- Low Km → high enzyme/substrate affinity.
- High k_cat/Km → kinetically perfect enzyme (near diffusion limit).
The Lineweaver-Burk plot (double-reciprocal) linearizes the curve: 1/v = (Km/Vmax) · 1/[S] + 1/Vmax, allowing graphical determination of Km and Vmax.
Enzyme Inhibition
Inhibition modifies enzymatic activity through binding of an inhibitor molecule I. Two main types:
Competitive inhibition: I binds to the active site (competes with S).
- [ES] decreases, but at very high [S], the inhibitor is displaced.
- Vmax unchanged; apparent Km increases: Km^app = Km(1 + [I]/Ki)
- On Lineweaver-Burk: lines converge on the y-axis.
Non-competitive inhibition: I binds to an allosteric site (independent of active site).
- ES complex forms normally, but the ES → P step is slowed.
- Km unchanged; apparent Vmax decreases: Vmax^app = Vmax / (1 + [I]/Ki)
- On Lineweaver-Burk: lines converge on the x-axis.
| Inhibition type | Km^app | Vmax^app | L-B convergence |
|---|---|---|---|
| Competitive | increases | unchanged | y-axis |
| Non-competitive | unchanged | decreases | x-axis |
| Mixed | increases | decreases | 2nd quadrant |
Zinc (Zn) is an essential cofactor in many enzymes (carbonic anhydrase, carboxypeptidase); Iron (Fe) is central to hemoproteins and oxidoreductases.