Enzymes that saturate
Enzymes speed up chemical reactions in living cells. At low substrate concentration, doubling the substrate nearly doubles the rate. At high concentration, every enzyme molecule is already busy, so adding more substrate barely helps: the rate levels off at a maximum. Leonor Michaelis and Maud Menten described this behavior in 1913 with a simple equation that is still the starting point of enzyme kinetics.
The equation
v = Vmax × [S] ÷ (Km + [S])
- v is the initial reaction rate.
- Vmax is the maximum rate when the enzyme is saturated.
- [S] is the substrate concentration.
- Km, the Michaelis constant, is the substrate concentration at which v = Vmax ÷ 2.
When [S] is far below Km, the rate is nearly proportional to [S]. When [S] is far above Km, the rate is nearly constant. Reaching 90% of Vmax takes a substrate concentration of 9 × Km, and 99% takes 99 × Km.
Example
An enzyme with Vmax = 100 µmol/min and Km = 2 mmol/L, at [S] = 5 mmol/L:
- v = 100 × 5 ÷ (2 + 5) = 71.4 µmol/min, or 71.4% of Vmax
- Lineweaver-Burk slope: 2 ÷ 100 = 0.02, y-intercept: 0.01
With a competitive inhibitor at 1 mmol/L and Ki = 0.5 mmol/L, the factor 1 + [I]/Ki equals 3. The apparent Km triples to 6 mmol/L, and the rate falls to 100 × 5 ÷ 11 = 45.5 µmol/min.
Types of reversible inhibition
| Inhibitor | Binds to | Apparent Km | Apparent Vmax |
|---|---|---|---|
| Competitive | Free enzyme, at the active site | × (1 + [I]/Ki) | Unchanged |
| Noncompetitive (pure) | Enzyme with or without substrate | Unchanged | ÷ (1 + [I]/Ki) |
| Uncompetitive | Enzyme–substrate complex only | ÷ (1 + [I]/Ki) | ÷ (1 + [I]/Ki) |
Many drugs are competitive inhibitors: statins compete with the natural substrate of HMG-CoA reductase to lower cholesterol production. A competitive inhibitor can be overcome by adding more substrate; a noncompetitive one cannot.
Linear plots
Before computers, biochemists turned the curve into a straight line. The Lineweaver-Burk plot graphs 1/v against 1/[S], with slope Km/Vmax and intercept 1/Vmax. It is easy to read but gives too much weight to measurements at low substrate concentration, which are the least precise. Today, fitting the curve directly with nonlinear regression gives better estimates of Km and Vmax.
Assumptions
The equation assumes a single substrate, initial rates measured before much product forms, an enzyme concentration much lower than the substrate concentration, and a steady state. Allosteric enzymes, which show S-shaped curves, follow the Hill equation instead.