Michaelis-Menten Kinetics and Steady-State

Jun 18, 2024

Michaelis-Menten Kinetics and the Steady-State

Enzyme Catalysis

  • Enzymes accelerate reactions and can be divided into two steps:
    1. Binding of enzyme to substrate
    2. Formation of products
  • Each step has its own rate
  • At high substrate concentrations, the reaction hits maximum speed (Vmax) with constant enzyme concentration

Steady-State Assumption

  • Steady-State: The concentration of enzyme-substrate complex (ES) is constant
    • Formation of ES = Loss/Dissociation of ES
  • Reactions can go forward or backward (equilibrium)
    • Reverse reactions (
    • Some exceptions due to thermodynamic stability
    • Rate (k_{-2}) -> negligible compared to (k_1)
  • Replace the second double-headed arrow with a single-headed arrow

Deriving the Michaelis-Menten Equation

  1. Sequence Reactions: Formation (
  • Rates forming ES = Rates taking away ES
  1. Swap out rates with rate constants multiplied by reactants
  • (k_1[E][S]) for rate 1, etc.
  1. Introduce total enzyme concentration (ET)
  • (ET = E + ES)
  • Rewrite E as (ET - ES)
  1. Expand the left side of the equation, simplify by dividing by (k_1)
  2. Introduce Michaelis constant (KM)
  • (k_{-1} + k_2 / k_1)
  1. Derive final form to get Michaelis-Menten Equation
  • (v_0 = \frac{V_{max}[S]}{K_M + [S]})
  1. Discuss Vmax at high substrate concentration where all enzymes are saturated
  • Sub in for Vo and Vmax accordingly

Michaelis Constant (KM)

  • Define KM as substrate concentration at which reaction speed is half of Vmax (Vo = 1/2 Vmax)
  • KM specific to circumstances, lower KM = better enzyme efficiency at low substrate concentrations

Catalytic Efficiency

  • KM in units of molar (M)
  • Kcat: Maximum speed of reaction divided by total enzyme available (turnover number)
    • Units: seconds⁻¹ (reactions per second)
  • Catalytic Efficiency: (\frac{Kcat}{KM})
    • Higher Kcat or lower KM = higher catalytic efficiency
    • Varies for different enzymes under different conditions

Summary

  • Steady-State Assumption: ES concentration remains constant
  • Michaelis-Menten Equation: (v_0 = \frac{V_{max}[S]}{K_M + [S]})
  • Catalytic Efficiency: Combines KM and Kcat to score enzyme efficiency