Kinetics of Bi-Substrate Enzymes and Allosteric Enzymes

KWETICS OF BI-SUBSTRATE ENZYMES

General Overview

• Enzyme Reaction Mechanism: The bi-substrate enzyme kinetics involves the interaction of an enzyme (E) with two substrates (A and B) to form products (P and Q).
• General Reaction Formula:
    - $E + A + B \rightleftharpoons P + Q$

Equations and Constants

1. Formation of Enzyme-Substrate Complexes
       - The complexes formed during the reaction include:
         - First substrate forming a complex:
           - $E + A \rightleftharpoons EA$
             where $K_a$ is the association constant for substrate A.
         - Second substrate forming a complex:
           - $E + B \rightleftharpoons EB$
             where $K_b$ is the association constant for substrate B.
       - Combined complex for both substrates:
         - $E + A + B \rightleftharpoons EAB$
           - $[EAB] = [E][A][B]$
           - $K_{ab} = K_a K_b$

2. Substitution of Values
       - From earlier equations, substituting gives:
           - $K_a' = \frac{[EA][B]}{[E]}$
           - $K_b' = \frac{[EB][A]}{[E]}$
       - Equating constants:
         - $K_a K_b = \frac{[EAB]}{[E][A][B]}$

3. Total Enzyme Concentration (ET)
       - Total enzyme concentration is defined as:
           - $E_T = E + EA + EB + EAB$

4. Making EAB Subject of the Formula
       - When rearranging leads to:
           - $EAB = \frac{ET}{E} \cdot (TEA + TEA + 1)$

Sequential Order Mechanism

• Stepwise formation:
    1. The enzyme (E) binds to substrate A or B.
    2. Formation of complexes and transitions as follows:
        - Sequential binding, where:
          - $EIB = \frac{ET}{E} \cdots$
          - Factors such as $K_a$ and $K_b$ contribute to the binding affinity.

Michaelis-Menten Kinetics

• Rate of Reaction:
    1. At the initial stage, the relationship between reaction rate and substrate concentration is direct and linear. As concentration increases, the rate reaches a plateau when the enzyme is saturated.
    2. The general form of the Michaelis-Menten equation is:
        - $V = V_{max} \cdot \frac{[S]}{K_m + [S]}$
            - where:
              - $V$ = reaction velocity,
              - $V_{max}$ = maximum reaction velocity,
              - $[S]$ = substrate concentration,
              - $K_m$ = Michaelis constant.

Allosteric Enzymes

Definition and Mechanism

• Allosteric Enzymes: Enzymes possessing an extra binding site (allosteric site) that can modulate the activity of the active site.
    - Example: Isocitrate dehydrogenase, a heterodimer enzyme that binds with both substrates and regulators.

Characteristics of Allosteric Enzymes

1. Multi-Subunit Structure: Allosteric enzymes typically consist of multiple subunits which encompass both catalytic and regulatory sites.
2. Regulatory Sites: Binding of effector molecules at the allosteric site leads to conformational changes that influence catalytic activity.
3. Types of Regulation: - Heterotrophic: When binding of a ligand affects substrate binding at a different site.
     - Homotropic: When binding at the regulatory site affects substrate binding at an identical site.

Kinetics of Allosteric Enzymes

• Allosteric enzyme reactions do not follow classical Michaelis-Menten kinetics; instead, they generally exhibit a sigmoid (S-shaped) curve in kinetic plots. Initially, the rate of reaction increases linearly then plateaus as all active sites become saturated.

Hill's Equation and Plot

• Hill Equation:
    - $[Y] = \frac{[L]^n}{K_a + [L]^n}$
      - where $Y$ = observed saturation, $[L]$ = ligand concentration, and $n$ = Hill coefficient indicating cooperative binding.

Steady-State Kinetics

• Steady-State Hypothesis: Suggested by E. Bridger and J.B.S. Haldane, stating that once the transient phase has passed, the concentration of the enzyme-substrate complex remains approximately constant, balancing the formation and breakdown rates.