Pharmacodynamics 3

Quantitative Principles of Receptor Occupancy and Parameters

• Equilibrium Dissociation Constant ($K_D$): Defined as the molar concentration of agonist that occupies $50\%$ of total receptors at equilibrium.

• Agonist Fractional Occupancy ($p_D$): Modeled using the Law of Mass Action as:

$\frac{[RD]}{R_T} = \frac{[D]}{K_D + [D]}$

• Potency Index ($pD_2$): Defined as the negative logarithm of the agonist molar concentration ($EC_{50}$) that produces $50\%$ of the maximal biological response:

$pD_2 = -\log_{10}(EC_{50})$

Classification and Mechanisms of Receptor Antagonism

 Receptor antagonism is broadly categorized into pharmacological and physiological mechanisms:

• Pharmacological Antagonism: Occurs when an agonist and an antagonist interact through the same receptor macromolecule.

• Physiological Antagonism: Occurs when two agonists act through completely independent receptor pathways to produce opposing physiological effects within a biological tissue (e.g., histamine causing bronchoconstriction via $H_1$ receptors versus epinephrine causing bronchodilations via $\beta_2$ adrenergic receptors).

• Competitive Antagonists: The agonist and antagonist compete directly for the same binding site on the receptor (orthosteric site). Blockade can be surmounted by increasing the concentration of the agonist.

• Non-Competitive Antagonists: The antagonist binds to a distinct regulatory site on the receptor (allosteric site) or binds covalently / irreversibly to the orthosteric site. Blockade cannot be surmounted by increasing the concentration of the agonist.

Competitive Antagonism

An established example of competitive antagonism is atropine acting against acetylcholine in the guinea pig ileum.

Increasing the concentration of a competitive antagonist ($[A]$ or [atropine]) results in specific changes to the agonist concentration-response relationship:

• Concentration-response curves shift to the right in a parallel fashion along the log-concentration axis.

• The apparent potency ($pD_2$) of the agonist decreases ($EC_{50}$ increases).

• The maximal attainable effect ($E_{max}$) remains unchanged because high agonist concentrations completely displace the competitive antagonist from orthosteric binding sites.

Non-Competitive Antagonism

An established example of non-competitive antagonism is benzilycholine mustard acting against acetylcholine.

Increasing the concentration of a non-competitive antagonist causes characteristic modifications to the agonist concentration-response relationship:

• The apparent $pD_2$ ($EC_{50}$) of the agonist is usually unchanged (little to no horizontal shift along the log-concentration axis).

• The maximal biological response ($E_{max}$) decreases progressively as antagonist concentration increases.

• Concentration-response curves are not parallel to the control curve.

Comparative Summary of Antagonist Types

Mathematical Modeling of Competitive Antagonism

The ability of an antagonist to block a receptor-mediated response depends on:

1. The relative affinities of the agonist ($K_D$) and antagonist ($K_A$) for the receptor.

2. The relative concentrations of the agonist ($[D]$) and antagonist ($[A]$).

Applying the Law of Mass Action for the antagonist ($A$):

$R + A \rightleftharpoons RA$

$K_A = \frac{[R][A]}{[RA]}$

Applying the Law of Mass Action for the agonist ($D$):

$R + D \rightleftharpoons RD$

$K_D = \frac{[R][D]}{[RD]}$

Conservation of total receptors ($R_T$):

$R_T = [R] + [RD] + [RA]$

$[R] = R_T - [RD] - [RA]$

Derivation yields fractional occupancy ($p_D$) of agonist in the presence of competitive antagonist:

$\frac{[RD]}{R_T} = \frac{\frac{[D]}{K_D}}{1 + \frac{[D]}{K_D} + \frac{[A]}{K_A}}$

When $[A] = 0$, this equation simplifies back to standard fractional occupancy:

$\frac{[RD]}{R_T} = \frac{[D]}{K_D + [D]}$

Concept and Calculation of the Dose Ratio

The Dose Ratio ($DR$) is defined as the ratio of agonist concentrations that produce the same biological response in the absence ($[D_0]$) versus the presence ($[D_A]$) of a given concentration of antagonist:

$DR = \frac{[D_A]}{[D_0]}$

Worked Example

• If $1\,\mu\text{M}$ of agonist produces a $50\%$ response in the absence of antagonist ($[D_0] = 1\,\mu\text{M}$).

• And $10\,\mu\text{M}$ of agonist is required to produce that same $50\%$ response in the presence of antagonist ($[D_A] = 10\,\mu\text{M}$):

$DR = \frac{10\,\mu\text{M}}{1\,\mu\text{M}} = 10$

This calculated Dose Ratio indicates that the antagonist caused a $10$-fold rightward shift of the agonist concentration-response curve.

Step-by-Step Determination of Dose Ratio

1. Select any target response level (typically $50\%$ of maximal response).

2. Determine the agonist concentrations required to achieve this specific response in the absence ($[D_0]$) and presence ($[D_1], [D_2], \dots$) of antagonist concentrations ($[A_1], [A_2], \dots$).

3. Calculate the Dose Ratio ($DR_1 = \frac{[D_1]}{[D_0]}$, $DR_2 = \frac{[D_2]}{[D_0]}$, etc.) for each antagonist concentration.

The Gaddum-Schild Equation

If the measured biological response in the presence and absence of antagonist is identical, it is assumed that agonist receptor occupancy ($\frac{[RD]}{R_T}$) is also identical, regardless of spare receptor reserves or post-receptor transduction mechanisms:

$\frac{[D_0]}{[D_0] + K_D} = \frac{\frac{[D_A]}{K_D}}{1 + \frac{[D_A]}{K_D} + \frac{[A]}{K_A}}$

Simplifying this expression yields the Gaddum-Schild Equation:

$\frac{[D_A]}{[D_0]} - 1 = \frac{[A]}{K_A}$

$DR - 1 = \frac{[A]}{K_A}$

Properties of the Gaddum-Schild Analysis

• Founded on the Law of Mass Action and assumes simple competitive equilibrium antagonism.

• Makes no assumptions about the functional relationship between receptor occupancy and downstream tissue response.

• Is independent of the specific agonist used, provided the agonist competes with the antagonist for the exact same receptor binding site.

Quantitative Definition of pA2

When the Dose Ratio $DR = 2$:

$DR - 1 = \frac{[A]}{K_A}$

$2 - 1 = \frac{[A]}{K_A} \implies 1 = \frac{[A]}{K_A} \implies [A] = K_A$

Taking the negative logarithm of both sides:

$-\log_{10}([A]) = -\log_{10}(K_A) = pA_2$

Definitions

• $pA_2$ is defined as the negative logarithm ($-\log_{10}$) of the molar concentration of competitive antagonist that requires doubling the agonist concentration to produce the same biological response (i.e., causing $DR = 2$).

• The concentration of antagonist that doubles the agonist $EC_{50}$ (causing $DR = 2$) is equal to its equilibrium dissociation constant ($K_A$).

The Schild Plot and Experimental Determination of Antagonist Affinity

Logarithmic transformation of the Gaddum-Schild equation gives:

$\log_{10}(DR - 1) = \log_{10}\left(\frac{[A]}{K_A}\right)$

$\log_{10}(DR - 1) = \log_{10}([A]) - \log_{10}(K_A)$

Since $pA_2 = -\log_{10}(K_A)$, this rearranges into linear form ($y = mx + c$):

$\log_{10}(DR - 1) = \log_{10}([A]) + pA_2$

Key Analysis Features

• Plot Variables: $\log_{10}(DR - 1)$ on the y-axis against $\log_{10}([A])$ on the x-axis.

• Slope $m = 1$: A regression slope equal to $1$ confirms simple competitive reversible equilibrium antagonism.

• x-Intercept ($y = 0$): Setting $\log_{10}(DR - 1) = 0$ gives:

$0 = \log_{10}([A]) + pA_2 \implies \log_{10}([A]) = -pA_2$

The x-intercept directly equals $-pA_2$.

• Value Notation: $pA_2$ is expressed as a positive value.

!Schild plot showing log(dose ratio - 1) versus log[A] with x-intercept at -pA2 and a theoretical slope equal to 1