Enzymes and Kinetics

Michaelis-Menten Kinetics

Enzymes accelerate reactions by lowering activation energy without changing $\Delta G$ or equilibrium. Initial velocity follows $v = \frac{V_{max}[S]}{K_m + [S]}$ where $K_m$ is the substrate concentration giving half of $V_{max}$: a low $K_m$ means high affinity. $V_{max} = k_{cat}[E]_T$ scales with enzyme amount, and $k_{cat}/K_m$ measures catalytic efficiency at low substrate.

Lineweaver-Burk Analysis

The double-reciprocal plot linearizes the hyperbola: $\frac{1}{v} = \frac{K_m}{V_{max}}\cdot\frac{1}{[S]} + \frac{1}{V_{max}}$ The y-intercept is $1/V_{max}$, the x-intercept is $-1/K_m$. Inhibitors move these intercepts diagnostically.

Inhibition and Regulation

Competitive inhibitors bind the active site: apparent $K_m$ rises, $V_{max}$ unchanged, overcome by excess substrate. Noncompetitive inhibitors bind elsewhere on free enzyme or ES: $V_{max}$ falls, $K_m$ unchanged. Uncompetitive inhibitors bind only the ES complex: both fall (parallel Lineweaver-Burk lines). Allosteric enzymes show sigmoidal kinetics and cooperativity; phosphorylation and feedback inhibition tune pathway flux.

Key Takeaways

• $K_m$ is inverse to affinity; $V_{max}$ reflects enzyme quantity and turnover.
• Competitive: $K_m$ up. Noncompetitive: $V_{max}$ down. Uncompetitive: both down.
• Enzymes change rates, never equilibria.
• Sigmoidal curves signal cooperativity, not Michaelis-Menten behavior.