CH 7

Kinetics and Regulation

7.1 Kinetics Is the Study of Reaction Rates

• Reaction velocity (or rate) is determined by measuring:
  • How much reactant A disappears as a function of time.
  • How much product P appears as a function of time.

7.2 The Michaelis–Menten Model Describes the Kinetics of Many Enzymes

• Consider a simple reaction where an enzyme E catalyzes the conversion of substrate S to product P: E + S \rightleftharpoons ES \rightarrow E + P, with k1, k2, and k_{-1} being the rate constants for the indicated reaction steps.
• To ignore the reverse reaction of P to S, activity is measured when [P] ≈ 0.
• Under these conditions, the velocity is called the initial velocity or V_0.

Determining Initial Velocity

• The initial velocity is determined by measuring product formation as a function of time shortly after the reaction has started.
• Conditions for determining initial velocity:
  • Constant enzyme concentration [E].
  • Varying substrate concentration [S].
• These conditions can be used to create a Michaelis-Menten graph.

Michaelis–Menten Kinetics

• Leonor Michaelis and Maud Menten derived an equation to describe the initial reaction velocity as a function of substrate concentration:
  • V0 = \frac{V{max}[S]}{K_M + [S]}
  • Where KM = [S] at V{max}/2

What is K_M?

• Michaelis Constant (K_M) relates to the nature of the Enzyme-Substrate interaction:
  • KM = \frac{k2 + k{-1}}{k1}
  • Can be estimated as the [S] at \frac{1}{2} V_{max}.
• K_M is an inverse measure of the affinity of the enzyme for its substrate.
  • Higher K_M means lower affinity.
  • Lower K_M means higher affinity.
• An enzyme with a higher KM will require more substrate to reach V{max}.

Clinical Insight: Variations in K_M Can Have Physiological Consequences

• Two enzymes play a key role in the metabolism of alcohol.
• Some people respond to alcohol consumption with facial flushing and rapid heartbeat, symptoms caused by excessive amounts of acetaldehyde in the blood.
• There are two different aldehyde dehydrogenases in most people, one with a low KM and one with a high KM.
• The low K_M enzyme is inactivated in susceptible individuals.
• The enzyme with the high K_M cannot process all of the acetaldehyde, so some acetaldehyde appears in the blood.

Determining KM and V{max} Values

• The Michaelis–Menten equation can be manipulated into one that yields a straight-line plot by taking the reciprocal of both sides of the equation.
• This double-reciprocal equation is the Lineweaver–Burk equation.
  • \frac{1}{V0} = (\frac{KM}{V{max}})(\frac{1}{[S]}) + \frac{1}{V{max}}
  • This follows the form y = m * x + b.

Lineweaver-Burk Plot

• Y-intercept = 1/V_{max}.
• X-intercept = -1/K_M.
• Slope = KM/V{max}.

Significance of KM and V{max} Values

• The K_M value is approximately the concentration of the substrate found in vivo.
• K_M values for enzymes vary widely.
  • For Example:
    • Chymotrypsin (Acetyl-L-tryptophanamide) K_M = 5000 µM
    • Lysozyme (Hexa-N-acetylglucosamine) K_M = 6 µM
    • β-Galactosidase (Lactose) K_M = 4000 µM
    • Carbonic anhydrase (CO2) KM = 8000 µM
    • Penicillinase (Benzylpenicillin) K_M = 50 µM
• If the enzyme concentration, [E]T, is known, then V{max} can be used to calculate the turnover number (k_{cat}).
  • k_{cat} is the number of substrate molecules converted into product per second by a single enzyme molecule.

Calculating k_{cat}

• k{cat} = \frac{V{max}}{[E]_T}

Catalytic Efficiency: k{cat}/KM

• k{cat}/KM is a measure of catalytic efficiency because it considers both the rate of catalysis (k{cat}) and the enzyme-substrate interaction (KM).

• Catalytic efficiency represents the number of molecules of substrate that an enzyme can convert to product per liter per second.

• Examination of the k{cat}/KM ratio reveals that some enzymes approach catalytic perfection.

• Expanding the equation for k{cat}/KM shows that k_1, the rate of formation of the enzyme-substrate complex, is the rate-limiting step.

• The rate at which enzyme and substrate diffuse together (\approx 10^8 – 10^9 s^{-1} M^{-1}) is the maximum possible catalytic efficiency.

• If k{-1} is very small, the k{cat}/K_M of some enzymes approaches the rate of diffusion!

• Enzymes for which k{cat}/KM is Close to the Diffusion-controlled Rate of Encounter

  • Acetylcholinesterase 1.6 × 10^8 s^{−1} M^{−1}
  • Carbonic anhydrase 8.3 × 10^7 s^{−1} M^{−1}
  • Catalase 4 × 10^7 s^{−1} M^{−1}
  • Crotonase 2.8 × 10^8 s^{−1} M^{−1}
  • Fumarase 1.6 × 10^8 s^{−1} M^{−1}
  • Triose phosphate isomerase 2.4 × 10^8 s^{−1} M^{−1}
  • β-Lactamase 1 × 10^8 s^{−1} M^{−1}
  • Superoxide dismutase 7 × 10^9 s^{−1} M^{−1}