Pharmacology

1. Law of Mass Action (Drug-Receptor Binding)

A drug (D) reversibly binds a receptor (R) to form a drug-receptor complex (DR). The forward rate constant is k₁ (association) and the reverse rate constant is k₋₁ (dissociation):

The rate of complex formation is governed by:

At equilibrium, d[DR]/dt = 0, so the rate of association equals the rate of dissociation: k₁[D][R] = k₋₁[DR]. This is the starting point for deriving K_D.

2. Dissociation Constant (K_D)

From the equilibrium condition k₁[D][R] = k₋₁[DR], rearrange to isolate the ratio of rate constants:

K_D has units of concentration (e.g., nM, μM). When [D] = K_D, exactly half of all receptors are occupied ([DR] = [R]). Therefore:

• Lower K_D = higher affinity (less drug needed for 50% occupancy)
• Higher K_D = lower affinity (more drug needed)
• K_D is the reciprocal of the association constant: K_D = 1/K_A

3. Fractional Receptor Occupancy

Let total receptors [R_T] = [R] + [DR]. From the K_D expression, [R] = K_D·[DR]/[D]. Substituting:

Solving for fractional occupancy f = [DR]/[R_T]:

This is a rectangular hyperbola (Langmuir isotherm). At [D] = K_D, f = 0.5. As [D] → ∞, f → 1 (saturation). This forms the basis of the Clark equation.

4. Hill Equation (Dose-Response with Cooperativity)

The simple occupancy model assumes independent binding sites. The Hill equation generalizes this by introducing a cooperativity coefficient n (Hill coefficient):

Interpretation of the Hill coefficient n:

• n = 1: No cooperativity (simple hyperbolic curve, equivalent to Langmuir)
• n > 1: Positive cooperativity (sigmoidal curve, steeper — binding of one molecule facilitates the next)
• n < 1: Negative cooperativity (shallower curve — binding of one molecule hinders the next)

EC₅₀ is the drug concentration producing 50% of maximum effect. When n = 1 and effect is proportional to occupancy, EC₅₀ = K_D.

5. Schild Equation (Competitive Antagonism)

When a competitive antagonist B is present, it competes with agonist D for the same binding site. The dose ratio r is the factor by which the agonist concentration must increase to achieve the same effect in the presence of the antagonist:

where K_B is the equilibrium dissociation constant of the antagonist. Taking logarithms:

A Schild plot of log(r-1) vs log[B] yields a straight line with slope = 1 for a true competitive antagonist. The x-intercept gives pA₂ = -log K_B, a measure of antagonist potency.

6. Clark Equation (Classical Receptor Theory)

A.J. Clark proposed that the biological effect is directly proportional to receptor occupancy. Combining fractional occupancy with this assumption:

This assumes a linear stimulus-response coupling: maximum effect occurs only when all receptors are occupied. Modern receptor theory (Stephenson, Furchgott) introduces efficacy and receptor reserve, allowing maximum effect at sub-maximal occupancy. Partial agonists have lower intrinsic efficacy even at full occupancy.

7. First-Order Elimination Kinetics

Most drugs follow first-order elimination, where the rate of elimination is proportional to the current plasma concentration. Starting from the differential equation:

Separating variables and integrating: ∫dC/C = -k_e ∫dt → ln(C) = ln(C₀) - k_e·t:

The half-life t₁/₂ is the time for concentration to fall by half (C = C₀/2):

After 5 half-lives, ~97% of the drug is eliminated. The half-life is a key determinant of dosing frequency.

8. Michaelis-Menten Enzyme Kinetics

Many drug targets are enzymes. The enzyme (E) binds substrate (S) to form an enzyme-substrate complex (ES), which then converts to product (P):

Applying the steady-state assumption (d[ES]/dt = 0) and defining K_m = (k₋₁ + k_cat)/k₁:

K_m is the substrate concentration at half-maximal velocity. Competitive enzyme inhibitors increase the apparent K_m, while non-competitive inhibitors decrease V_max. This equation is analogous to receptor occupancy — K_m plays a role similar to K_D.

9. Volume of Distribution (V_d)

V_d is a theoretical volume that relates the total amount of drug in the body to its plasma concentration. After an IV bolus dose D:

Interpretation of V_d values:

• V_d ~ 3-5 L: Drug confined to plasma (e.g., warfarin, highly protein-bound)
• V_d ~ 14 L: Drug distributes into extracellular fluid
• V_d ~ 42 L: Drug distributes into total body water
• V_d >> 42 L: Extensive tissue binding/sequestration (e.g., chloroquine, V_d ~ 13,000 L)

10. Clearance (CL)

Clearance is the volume of plasma completely cleared of drug per unit time. It connects the elimination rate constant and the volume of distribution:

Substituting k_e = 0.693/t₁/₂:

Clearance can also be determined from the area under the concentration-time curve (AUC): CL = Dose/AUC (for IV administration). Total clearance is the sum of renal, hepatic, and other routes: CL_total = CL_renal + CL_hepatic + CL_other.

11. Bioavailability (F)

Bioavailability is the fraction of an administered dose that reaches systemic circulation unchanged. It is determined by comparing the AUC of an oral dose to an IV dose (100% bioavailable):

Factors reducing oral bioavailability:

• First-pass metabolism: Hepatic extraction before reaching systemic circulation
• Incomplete absorption: Poor solubility, degradation in GI tract
• Efflux transporters: P-glycoprotein pumps drug back into gut lumen

IV drugs have F = 1 by definition. Oral drugs typically have F between 0.05 and 1.0.

12. Steady-State Concentration & Loading Dose

For repeated dosing at a fixed interval τ, drug accumulates until the rate of administration equals the rate of elimination. The average steady-state concentration is:

Steady state is reached after approximately 4-5 half-lives. To achieve therapeutic levels immediately, a loading dose can be administered:

The loading dose is independent of clearance or dosing interval — it depends only on the target concentration, volume of distribution, and bioavailability. After the loading dose, maintenance doses of D = CL·C_ss·τ/F sustain the steady state.