pharmacodynamics 2

Law of Mass Action and Quantitative Drug-Receptor Interactions

To allow comparison to be made between drugs, their biological and physiological effects must be quantified. Quantifying drug-receptor interactions relies on fundamental kinetic principles describing how ligands interact with biological targets.

Fundamental Assumptions

The standard quantitative framework for drug-receptor interactions is built upon three primary assumptions:

1. Law of Mass Action: The binding reaction between a drug ($[D]$) and a receptor ($[R]$) is reversible and governed by the Law of Mass Action:

$[D] + [R] \rightleftharpoons [DR]$

The rate of the forward reaction (association) is proportional to the concentration of free drug and free receptor:

$\text{Rate}_{\text{forward}} = k_{+1} [D] [R]$

Where $k_{+1}$ is the association rate constant.

The rate of the reverse reaction (dissociation) is proportional to the concentration of the drug-receptor complex:

$\text{Rate}_{\text{reverse}} = k_{-1} [DR]$

Where $k_{-1}$ is the dissociation rate constant.

2. Negligible Bound Drug Assumption: It is assumed that only a negligible amount of the total drug in the system is bound to receptors at any given time. Consequently, the concentration of free drug ($[D]$) is treated as equal to the total drug concentration added to the system:

$\text{free drug in system} = \text{total drug concentration} = [D]$

3. Equilibrium State: At equilibrium, the rate of the forward reaction equals the rate of the reverse reaction:

$k_{+1} [D] [R] = k_{-1} [DR]$

Derivation of the Equilibrium Dissociation Constant ($K_D$)

Rearranging the equilibrium equation $k_{+1} [D] [R] = k_{-1} [DR]$ gives:

$\frac{k_{-1}}{k_{+1}} = \frac{[D][R]}{[DR]}$

The ratio of the dissociation rate constant $k_{-1}$ to the association rate constant $k_{+1}$ is defined as the equilibrium dissociation constant ($K_D$):

$K_D = \frac{k_{-1}}{k_{+1}} = \frac{[D][R]}{[DR]}$

The constant $K_D$ serves as a quantitative measure of the affinity of a drug for its receptor. A lower $K_D$ indicates higher affinity, meaning less drug is required to occupy a given fraction of receptors.

To express $K_D$ in terms of total receptors, define total receptor concentration ($[R_T]$) as the sum of free receptors ($[R]$) and bound receptors ($[DR]$):

$[R_T] = [R] + [DR]$

Solving for free receptor concentration ($[R]$):

$[R] = [R_T] - [DR]$

Substituting $[R] = [R_T] - [DR]$ into the $K_D$ expression yields:

$K_D = \frac{[D] ([R_T] - [DR])}{[DR]}$

Derivation of the Langmuir Equation and Fractional Occupancy

To quantify receptor occupancy, parameters are defined as follows:

• $[R_T]$: Total number or concentration of receptors in the system.

• $[D]$: Total drug concentration (assumed equal to free drug concentration).

• $[DR]$: Concentration of the bound drug-receptor complex.

• $r$: Fractional occupancy of the receptors, defined as the ratio of bound receptors to total receptors:

$r = \frac{[DR]}{[R_T]}$

• $K_D$: Equilibrium dissociation constant ($K_D = \frac{[D][R]}{[DR]}$).

Step-by-Step Algebraic Derivation

1. Starting from the rearranged dissociation constant equation:

$K_D = \frac{[D] ([R_T] - [DR])}{[DR]}$

2. Multiply both sides by $[DR]$:

$K_D [DR] = [D] ([R_T] - [DR])$

3. Expand the right side of the equation:

$K_D [DR] = [D] [R_T] - [D] [DR]$

4. Add $[D] [DR]$ to both sides to group all $[DR]$ terms together:

$K_D [DR] + [D] [DR] = [D] [R_T]$

5. Factor out $[DR]$ on the left side:

$[DR] (K_D + [D]) = [D] [R_T]$

6. Divide both sides by $[R_T]$:

$\frac{[DR]}{[R_T]} (K_D + [D]) = [D]$

7. Divide both sides by $(K_D + [D])$ to isolate the fractional occupancy $r = \frac{[DR]}{[R_T]}$:

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

This equation is known as the Langmuir equation (or Langmuir adsorption isotherm).

Conditions for Validity

• The Langmuir equation is valid for a simple bimolecular interaction between a drug molecule and its target receptor.

• The equation holds only at equilibrium.

Plotting Receptor Occupancy: Rectangular Hyperbola vs Sigmoidal Curve

Linear Concentration Scale ($r$ vs $[D]$)

When fractional occupancy $r$ is plotted on the y-axis against drug concentration $[D]$ on a linear x-axis, the resulting curve is a rectangular hyperbola.

Key features of this plot include:

• Zero Occupancy: At $[D] = 0$, $r = 0$

• Fractional Saturation ($f\text{-Saturation}$): As $[D] \to \infty$, $r \to 1$. At saturation, all available receptors are occupied by drug molecules ($[DR] = [R_T]$).

Special Case: Half-Maximal Receptor Occupancy ($r = 0.5$)

To determine the drug concentration required to occupy exactly half of the total receptor population ($r = 0.5$ or $50\%$ occupancy):

$r = \frac{1}{2} = \frac{[D]}{[D] + K_D}$

Cross-multiplying yields:

$[D] + K_D = 2 [D]$

Subtracting $[D]$ from both sides gives:

$[D] = K_D$

Definition of $K_D$: The equilibrium dissociation constant $K_D$ is equal to the concentration of free drug ($[D]$) required to occupy exactly $50\%$ (half) of the available receptors at equilibrium.

Semi-Logarithmic Scale ($r$ vs $\log[D]$)

Because pharmacological drug concentrations span many orders of magnitude (e.g., from $10^{-10}\,M$ to $10^{-5}\,M$), fractional occupancy $r$ is usually plotted against the logarithm of drug concentration ($\log[D]$).

Key features of the semi-logarithmic plot:

• The transformation converts the rectangular hyperbola into a sigmoidal (S-shaped) curve.

• Linear Region: The sigmoidal curve is approximately linear in the range between $20\%$ and $80\%$ fractional occupancy ($r = 0.2$ to $0.8$).

• Inflection Point: The midpoint of the sigmoidal curve occurs at $r = 0.5$, corresponding directly to $\log(K_D)$.

Complex Drug-Receptor Interactions and the Hill Equation

While simple bimolecular interactions conform to the standard Langmuir equation, many drug-receptor interactions exhibit greater complexity.

Examples of Complex Interactions

1. Receptors Binding Multiple Drug Molecules Simultaneously:

   • An example is the nicotinic acetylcholine receptor ($nAChR$), an ionotropic receptor that binds two acetylcholine ($ACh$) molecules simultaneously to trigger the opening of a $Na^+$ ion channel.

2. Receptors Interconverting Between Conformational States:

   • Receptors that dynamically transition between high-affinity and low-affinity functional states.

The Hill Equation and Hill Coefficient ($n$)

To account for cooperativity and non-simple stoichiometry, the interaction can be represented as:

$n [D] + [R] \rightleftharpoons [D_n R]$

Modifying the Langmuir equation with an exponent $n$ yields the Hill equation:

$r = \frac{[D]^n}{[D]^n + K_D^n}$

Where $n$ is the Hill coefficient, representing the slope of the interaction curve.

Interpretation of the Hill Coefficient ($n$)

• $n = 1$: Indicates a simple, non-cooperative bimolecular interaction (1:1 binding stoichiometry conforming to standard Langmuir kinetics).

• $n > 1$: Indicates positive cooperativity, where the binding of the first drug molecule increases the receptor's binding affinity for subsequent drug molecules.

• $n < 1$: Indicates negative cooperativity, where the binding of the first drug molecule decreases the receptor's binding affinity for subsequent drug molecules.

Relating Receptor Occupancy to Biological Effect: Occupation Theory vs Rate Theory

A central problem in quantitative pharmacology is understanding how physical receptor occupancy ($r$) translates into a measurable biological or tissue effect ($E$). Two classical theories address this relationship:

1. Occupation Theory

• Core Premise: The biological response ($E$) produced by a drug is directly proportional to the number of receptors occupied ($[DR]$).

$\text{Response } [E] \propto \text{number of receptors occupied}$

• Maximal Effect: The maximum possible biological response ($E_{max}$) occurs when all available receptors are fully occupied ($r = 1$).

• Fractional Response Equation: Under Occupation Theory, the fractional response ($\frac{E}{E_{max}}$ or $R$) is strictly equal to the fractional occupancy ($r$):

$\frac{E}{E_{max}} = r = \frac{[D]}{[D] + K_D}$

• Graphical Shape: Graphs of fractional biological response ($\frac{E}{E_{max}}$) plotted against drug concentration ($[D]$ or $\log[D]$) have the exact same shape (hyperbolic on linear scale, sigmoidal on logarithmic scale) as plots of fractional occupancy ($r$) against concentration.

2. Rate Theory

• Core Premise: The biological response ($E$) produced by a drug is directly proportional to the rate of receptor occupation (the number of drug-receptor association events taking place per unit time), rather than the absolute number of occupied receptors.

$\text{Response } [E] \propto \text{rate of receptor occupation}$

Agonist Potency Quantification ($pD_2$) and Limitations of Occupation Theory

Quantification of Agonist Potency ($pD_2$)

The parameter $pD_2$ is a widely used pharmacological index that quantifies the affinity and potency of an agonist for its receptor.

Definition and Formula

$pD_2$ is defined as the negative logarithm (base 10) of the molar concentration of agonist that produces a half-maximal biological response ($50\%$ of $E_{max}$ or $EC_{50}$):

$pD_2 = -\log_{10}([D]_{50})$

Alternatively expressed as:

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

Where $[D]_{50}$ (or $EC_{50}$) is the drug concentration yielding \frac{E}{E_{max}} = 0.5$.\n\n### Characteristics of pD_2\n- **Always Positive**: Because effective agonist concentrations are typically sub-molar (e.g., 10^{-6}\,M$or$10^{-9}\,M$), taking the negative logarithm yields positive values (e.g.,$6$or$9).\n- **Potency Indicator**: Drugs with **higher pD_2 values** act at **lower concentrations**, reflecting greater potency and receptor affinity.\n\n## Theoretical Relationship Between pD_2$and$K_D\n\nIf all assumptions of Occupation Theory hold true, then at half-maximal response:\n\n\frac{E}{E_{max}} = 0.5 = \frac{[D]}{[D] + K_D}\n\nThis requires that:\n\n[D]{50} = K_D\n\nSubstituting [D]{50} = K_D$into the$pD_2 formula gives:\n\npD_2 = -\log_{10}(K_D)\n\nUnder strict Occupation Theory, pD_2$is equal to the negative logarithm of the equilibrium dissociation constant$K_D$.

Limitations and Breakdown of Occupation Theory

In real biological systems, the core assumptions of Occupation Theory frequently do not hold true:

• For many receptor systems, experimental determinations reveal that $pD_2$ overestimates $K_D$ (i.e., $pD_2 > -\log_{10}(K_D)$).

• This means the drug appears to produce biological effects at concentrations much lower than those required to physically occupy $50\%$ of the receptors (the drug appears to bind more tightly than it actually does).

• This discrepancy occurs because full biological responses can often be elicited when only a small fraction of receptors are occupied, a phenomenon attributed to spare receptors (receptor reserve) and efficient intracellular signal amplification cascades.