CH 7

Kinetics and Regulation

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.

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.

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.

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

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}.

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.

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.

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.

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}