Enzyme Kinetics and Michaelis-Menten Equation

Enzyme Kinetics: Basics - Speedy Catalysts

• Catalysts (proteins/RNA) that ↑ reaction rates.
• Mechanism: Lower activation energy ($E_a$).
  • Provide alternative reaction pathway.
  • Stabilize transition state.
  • Crucially, enzymes do NOT alter $\Delta G$ (Gibbs free energy) or $K_{eq}$ (equilibrium constant).
• Reaction Velocity (V): Rate of [product] formation or [substrate] consumption.
• Factors Affecting Velocity:
  • Enzyme concentration [E]: V ∝ [E] (if [S] saturating).
  • Substrate concentration [S]: V ↑ with [S] until $V_{max}$.
  • Temperature: Bell-shaped curve (optimum T); high T denatures.
  • pH: Bell-shaped curve (optimum pH); extreme pH denatures.

⭐ While enzymes don't change the reaction's endpoint (equilibrium), they drastically reduce the time taken to reach it.

Enzyme Kinetics: Michaelis-Menten - The Speed Formula

• Describes how initial reaction velocity (V) changes with substrate concentration ([S]).
• The fundamental equation: $V = \frac{V_{max}[S]}{K_m + [S]}$
• $V_{max}$ (Maximum Velocity):
  • The theoretical maximum rate when all enzyme active sites are saturated.
  • Directly proportional to total enzyme concentration [E]$_{total}$; units: rate (e.g., µmol/min).
• $K_m$ (Michaelis Constant):
  • Substrate concentration ([S]) at which V = ½ $V_{max}$.
  • An inverse measure of enzyme's affinity for its substrate: ↓ $K_m$ implies ↑ affinity.
  • Units: concentration (e.g., M, mM).
• Key Assumptions:
  • [S] >> [E] (substrate in large excess).
  • Steady-state: d[ES]/dt = 0 (ES complex concentration is constant).
  • Initial velocities measured (product P ≈ 0, reverse reaction is negligible).

⭐ Low $K_m$ indicates the enzyme is highly efficient at low substrate concentrations.

Enzyme Kinetics: Lineweaver-Burk - Straight Plot Story

• A double reciprocal plot of Michaelis-Menten kinetics: linearizes data.
• Equation: $\frac{1}{V_0} = \left(\frac{K_m}{V_{max}}\right) \frac{1}{[S]} + \frac{1}{V_{max}}$ (form $y=mx+c$)
• Plot parameters:
  • Y-axis: $\frac{1}{V_0}$
  • X-axis: $\frac{1}{[S]}$
  • Y-intercept: $\frac{1}{V_{max}}$
  • X-intercept: $-\frac{1}{K_m}$
  • Slope: $\frac{K_m}{V_{max}}$
• 📌 Mnemonic for intercepts: Y-intercept is $\frac{1}{V_{max}}$ (V for Vertical). X-intercept is $-\frac{1}{K_m}$ (negative value on X-axis).
• 
• Key for determining $K_m$, $V_{max}$, and analyzing enzyme inhibition types.
• Limitation: Distorts errors at low [S] (high $\frac{1}{[S]}$ values), less accurate.

⭐ Lineweaver-Burk plots clearly distinguish competitive, non-competitive, and uncompetitive inhibition patterns by their effects on $V_{max}$ and $K_m$.

Enzyme Kinetics: Inhibition - Kinetic Blockers

• Competitive Inhibition:
  • Inhibitor (I) mimics substrate; binds active site.
  • Effect: $K_m$ ↑ (affinity ↓), $V_{max}$ unchanged.
  • Reversible by ↑ [S].
  • E.g., Statins, Methotrexate.
• Non-competitive Inhibition:
  • I binds E or ES at an allosteric site.
  • Effect: $K_m$ unchanged, $V_{max}$ ↓.
  • Not reversible by ↑ [S].
  • E.g., Lead, some forms of Cyanide.
• Uncompetitive Inhibition:
  • I binds only to ES complex (allosteric site).
  • Effect: Both $K_m$ ↓ and $V_{max}$ ↓ (proportionally).
  • Lineweaver-Burk: Parallel lines.
  • E.g., Lithium.

⭐ Competitive inhibitors increase the Michaelis constant (apparent $K_m$) but do not affect the maximum velocity ($V_{max}$). More substrate is needed to reach half $V_{max}$, but $V_{max}$ itself is still achievable with sufficient substrate concentration.

High‑Yield Points - ⚡ Biggest Takeaways

• Michaelis-Menten equation: Relates initial velocity (v₀) to substrate concentration [S].
• K_m_ (Michaelis constant): [S] at ½ V_max_. Inverse measure of substrate affinity (↓K_m_ = ↑affinity).
• V_max_ (maximum velocity): Maximum velocity at enzyme saturation. Proportional to enzyme concentration.
• Lineweaver-Burk plot: Double reciprocal (1/v vs 1/[S]), linearizes kinetics.
• Competitive inhibitors: ↑K_m_, V_max_ unchanged. Overcome by ↑[S].
• Non-competitive inhibitors: ↓V_max_, K_m_ unchanged. Not overcome by ↑[S].
• Uncompetitive inhibitors: ↓V_max_ and ↓K_m_. Bind ES complex only.