Antagonists II: Quantifying Reversible Competitive Antagonist Action

BS2013: Physiology and Pharmacology: Topic 1 Lecture 1.5: Antagonists II: Quantifying Reversible Competitive Antagonist Action

Introduction

• Key Figures:

  • Sir John Henry Gaddum FRS (1900 - 1965)

  • Heinz Otto Schild, FRS (1906 - 1984)

Modeling of Competitive Antagonism

• **Law of Mass Action:

  • Agonist (A) + Antagonist (B) present leads to two reactions:**

  • In the presence of only the agonist (A): $ext{Occupancy (agonist)} = X<em>A = rac{[A]}{[A] + K</em>A}$

    • Where:

    • $X_A$ = agonist concentration

    • $K_A$ = concentration of agonist A at 50% occupancy (Agonist dissociation constant)

  • In similar fashion, for binding of antagonist (B) to receptor: $ext{Occupancy (antagonist)} = X<em>B = rac{[B]}{[B] + K</em>B}$

    • Where:

    • $X_B$ = antagonist concentration

    • $K_B$ = concentration of antagonist B at 50% occupancy (Antagonist dissociation constant)

• Combined Equation for Agonist Occupancy:

  • The expressions can be combined to derive:
    $ext{Occupancy (agonist)} = X<em>A imes rac{[A]}{[A] + K</em>A[1 + rac{X<em>B}{K</em>B}]}$

Equilibrium Between Competing Ligands

• Predicts that in the presence of a competitive antagonist, the fractional occupancy by a certain [agonist] will be lower than in its absence.

• Mutually Exclusive Binding:

  • Agonist and antagonist binding occurs in a mutually exclusive manner.

• The Gaddum Equation:

  • A formal representation of the relationship between agonist and antagonist presence.

Quantification of Parallel Shifts

• Dose Ratio (DR):

  • Represents how many fold greater agonist concentration is needed in the presence of an antagonist to stimulate the same response.

  • Defined as:
    $ext{[Agonist EC}<em>{50} ext{ in presence of antagonist]} = ext{Dose Ratio (DR)} imes ext{[Agonist EC}</em>{50} ext{ in absence of antagonist]}$

• Graphs:

  • Graphical representation showing the increasing antagonist concentration against agonist concentration (μM).

pA2 Values for Antagonists

• Definition and Calculation of pA2:

  • A measure of antagonist potency. Defined as:
    $pA<em>2 = - ext{log}</em>{10}(A_2)$

  • Higher pA2 values denote more potent antagonists compared to lower values.

  • Example:

    • Antagonist X: $pA_2 = 10$, requires $1 imes 10^{-10} ext{ M}$ of X to achieve a dose ratio of 2.

    • Antagonist Z: $pA_2 = 4$, requires $1 imes 10^{-4} ext{ M}$ of Z to achieve the same dose ratio.

• pH Relation:

  • $pH = - ext{log}_{10}[H^+]$

• A2 Value Explanation:

  • Represents molar concentration of antagonist that results in a dose ratio of 2.

Calculation of pA2 Using Schild Regression Analysis

• Occupancy and Dose Ratio:

  • $ext{Occupancy (agonist)} = X<em>A imes rac{1}{[A] + K</em>A[1 + rac{X<em>B}{K</em>B}]}$

  • $ext{Dose Ratio (DR)} = [X_B] + 1$

  • Where $X<em>B$ = concentration of antagonist and $K</em>B$ = antagonist equilibrium constant.

• Schild Equation Simplification:

  • Rearranging gives:
    $ext{DR}- 1 = [X_B]$

• Logarithmic Relationship:

  • Taking the logarithm of both sides leads to:
    $ext{log}<em>{10}( ext{DR}-1) = ext{log}</em>{10}[X<em>B] - ext{log}</em>{10}K_B$

Schild Plot

• Graphical Representation:

  • $ext{log}<em>{10}[antagonist] ext{ versus } ext{log}</em>{10}( ext{DR}-1) = ext{log}<em>{10}[X</em>B] - ext{log}<em>{10}K</em>B$

• Determination of A2 Value:

  • The inverse log of the X-axis intercept yields the A2 value.

  • Example Calculation:

    • Inverse log of -8.5 results in $3.16 imes 10^{-9} ext{ M}$, leading to $pA_2 = 8.5$.

    • When $ ext{log}_{10}( ext{DR}-1) = 0$, results in a calculated X-axis intercept.

Summary Points

• Antagonist Receptor Interactions:

  • Definition: An antagonist is a drug that hinders a functional response.

  • Various types:

  • Competitive antagonists: Block agonist binding; can be reversible (surmountable) or irreversible (insurmountable).

  • Mechanisms: Antagonists may act via different mechanisms, primarily at receptor level.

  • Potency Measurement: Reversible competitive antagonists' potency quantified via $pA_2$:

  • $pA<em>2 = - ext{log}</em>{10}( ext{[antagonist]} ext{ required to double EC}_{50} ext{ value})$

  • Resulting in a dose ratio of 2.

  • Shifts in Curves:

  • Reversible antagonists cause a rightward parallel shift in the agonist concentration:response curve without affecting Emax.

    • Irreversible antagonists also induce a rightward shift but reduce Emax, particularly in the absence of spare receptors.

  • Schild Plot Intercept: The pA2 value corresponds to the negative value of the X-axis intercept on a Schild plot.