3.3 Agonists & Antagonists

Drugs interact with receptors as agonists (activators) or antagonists (blockers). The spectrum ranges from full agonists through partial agonists to inverse agonists, and antagonists can be competitive, noncompetitive, or irreversible. Understanding these classifications through their mathematical models is essential for predicting drug effects and designing therapeutics.

Historical Development

Dale (1906) first classified drug actions at the neuromuscular junction as "nicotinic" and "muscarinic." Ahlquist (1948) proposed alpha and beta adrenergic receptor subtypes based on agonist potency orders. Black developed propranolol (1964, first beta-blocker) and cimetidine (1976, first H2-antagonist), earning the Nobel Prize in 1988 for demonstrating that rational antagonist design could revolutionize therapeutics.

Derivation 1: Two-State Receptor Model & Agonist Classification

The two-state model provides a unified framework for understanding the full spectrum of drug efficacies, from full agonists to inverse agonists.

Step 1: Receptor Equilibrium

Receptors exist in an equilibrium between inactive (R) and active (R*) states:

\( R \underset{}{\overset{L_0}{\rightleftharpoons}} R^* \quad \text{where } L_0 = \frac{[R^*]}{[R]} \)

L_0 is the isomerization constant. In most systems L_0 is small (less than 0.1), meaning few receptors are spontaneously active.

Step 2: Drug Binding to Both States

A drug A can bind both R and R* with different affinities. Define selectivity ratio alpha:

\( \alpha = \frac{K_R}{K_{R^*}} \)

where K_R = dissociation constant for R and K_R* = dissociation constant for R*.

If alpha > 1: drug prefers R* (agonist). If alpha = 1: no preference (neutral antagonist). If alpha < 1: drug prefers R (inverse agonist).

Step 3: Fraction in Active State

The fraction of total receptors in the active state (R* + AR*) in the presence of drug:

\( f_{active} = \frac{L_0(1 + \alpha[A]/K_R)}{1 + L_0 + [A]/K_R(1 + \alpha L_0)} \)

Full Agonist

alpha is much greater than 1

Isoproterenol

Partial Agonist

alpha > 1 (moderate)

Pindolol

Neutral Antag.

alpha = 1

Blocks all

Inverse Agonist

alpha < 1

Some beta-carbolines

Derivation 2: Gaddum Equation for Competitive Antagonism

Gaddum (1937) derived the occupancy equation when both an agonist A and competitive antagonist B are present at the same receptor.

Step 1: Three-Species Equilibrium

With both A and B competing for receptor R:

\( [R_T] = [R] + [AR] + [BR] \)

From mass action:

\( [AR] = \frac{[A][R]}{K_A}, \quad [BR] = \frac{[B][R]}{K_B} \)

Step 2: Solve for Agonist Occupancy

Substituting and solving for [AR]/[R_T]:

\( [R_T] = [R]\left(1 + \frac{[A]}{K_A} + \frac{[B]}{K_B}\right) \)

\( [R] = \frac{[R_T]}{1 + [A]/K_A + [B]/K_B} \)

Gaddum Equation

\( \frac{[AR]}{[R_T]} = \frac{[A]/K_A}{1 + [A]/K_A + [B]/K_B} = \frac{[A]}{[A] + K_A(1 + [B]/K_B)} \)

The antagonist effectively increases the apparent K_A by a factor of (1 + [B]/K_B). This is the Gaddum equation, from which the Schild equation follows directly.

Derivation 3: Noncompetitive & Irreversible Antagonism

Noncompetitive antagonists bind at a different site from the agonist, while irreversible antagonists form covalent bonds. Both reduce the functional receptor pool.

Noncompetitive: Reduced E_max

If the antagonist binds at an allosteric site and prevents receptor activation regardless of agonist binding, the effective receptor pool is reduced by a fraction q:

\( q = \frac{1}{1 + [B]/K_B} \)

The dose-response becomes:

\( E = q \cdot E_{max} \cdot \frac{[A]}{K_A + [A]} \)

E_max is reduced proportionally while EC_50 remains unchanged (no rightward shift).

Irreversible: Progressive Receptor Loss

Irreversible antagonists permanently inactivate receptors. If a fraction q of receptors remain functional:

\( q = e^{-k_{inact} \cdot [B] \cdot t} \)

In tissues with spare receptors, initial receptor loss shifts the curve rightward without reducing E_max (because fewer receptors still suffice for maximal response). Only when receptor reserve is exhausted does E_max decline. Recovery requires new receptor synthesis (days to weeks).

Derivation 4: Allosteric Modulators (alpha-beta Model)

Allosteric modulators bind at a distinct site from the orthosteric (agonist) site and can alter both binding affinity and efficacy of the orthosteric ligand.

The Allosteric Ternary Complex Model

When modulator M is bound, the agonist's affinity changes by factor alpha and its efficacy changes by factor beta:

\( K_{A,apparent} = \frac{K_A}{\alpha} \quad \text{(alpha > 1: increased affinity)} \)

\( e_{apparent} = \beta \cdot e \quad \text{(beta > 1: increased efficacy)} \)

Full Allosteric Equation

The fraction of receptors occupied by agonist A in the presence of allosteric modulator M (at concentration [M] with dissociation constant K_M):

\( \frac{[AR]_{total}}{[R_T]} = \frac{[A](1 + \alpha[M]/K_M)}{[A](1 + \alpha[M]/K_M) + K_A(1 + [M]/K_M)} \)

PAM

Positive Allosteric Modulator

alpha > 1 and/or beta > 1

Benzodiazepines at GABA_A

NAM

Negative Allosteric Modulator

alpha < 1 and/or beta < 1

Maraviroc at CCR5

SAM

Silent Allosteric Modulator

alpha = beta = 1

Blocks other modulators

Derivation 5: Partial Agonist as Functional Antagonist

When a partial agonist is combined with a full agonist, it can act as a functional antagonist by competing for receptors while producing a lower maximal response.

Combined Occupancy

When full agonist A (efficacy e_A, affinity K_A) and partial agonist P (efficacy e_P < e_A, affinity K_P) compete:

\( E = E_{max}\left( e_A \cdot \frac{[A]/K_A}{1 + [A]/K_A + [P]/K_P} + e_P \cdot \frac{[P]/K_P}{1 + [A]/K_A + [P]/K_P} \right) \)

As [P] increases, it displaces A from receptors. The net effect decreases because e_P < e_A. At very high [P], the effect approaches e_P * E_max (the partial agonist ceiling).

Clinical example: Buprenorphine (partial mu agonist) displaces morphine and reduces overall opioid effect, while providing enough activation to prevent withdrawal. This dual agonist/antagonist behavior is the basis for its use in opioid use disorder treatment.

Antagonist Types: Dose-Response Comparison

Competitive Antagonist

Noncompetitive Antagonist

Python Simulation: Agonists & Antagonists

Agonists & Antagonists — Two-State Model, Gaddum, Allosteric Modulation

Python

script.py99 lines

import numpy as np

log_conc = np.linspace(-3, 3, 200)
conc = 10**log_conc
KA = 1.0

# ── Panel 1: Agonist Spectrum (Two-State Model) ──
L0 = 0.01  # basal isomerization
alphas_2state = [1000, 50, 10, 1, 0.1, 0.01]  # selectivity ratio
KR = 1.0

print("CHART1_TITLE: Two-State Model: Agonist Spectrum")
print("CHART1_XLABEL: log[Drug]")
print("CHART1_YLABEL: Fraction Active Receptors")
for i in range(0, len(conc), 5):
    vals = []
    for a2 in alphas_2state:
        f_active = L0 * (1 + a2 * conc[i] / KR) / (1 + L0 + conc[i] / KR * (1 + a2 * L0))
        label = f"a_{a2:.0f}" if a2 >= 1 else f"a_{a2:.2f}"
        vals.append(f"{label}={f_active:.5f}")
    print(f"logA={log_conc[i]:.2f} " + " ".join(vals))

# ── Panel 2: Competitive Antagonism (Gaddum) ──
KB = 10.0
antag_concs_gaddum = [0, 10, 50, 200]
print("\nCHART2_TITLE: Competitive Antagonism (Gaddum Equation)")
print("CHART2_XLABEL: log[Agonist]")
print("CHART2_YLABEL: Fractional Occupancy")
for i in range(0, len(conc), 5):
    vals = []
    for B in antag_concs_gaddum:
        occ = conc[i] / (conc[i] + KA * (1 + B / KB))
        label = f"B_{B}" if B > 0 else "control"
        vals.append(f"{label}={occ:.4f}")
    print(f"logA={log_conc[i]:.2f} " + " ".join(vals))

# ── Panel 3: Noncompetitive Antagonism ──
print("\nCHART3_TITLE: Noncompetitive Antagonism (Reduced E_max)")
print("CHART3_XLABEL: log[Agonist]")
print("CHART3_YLABEL: Response (% E_max)")
nc_concs = [0, 5, 20, 100]
for i in range(0, len(conc), 5):
    vals = []
    for B in nc_concs:
        q = 1 / (1 + B / KB)
        resp = 100 * q * conc[i] / (KA + conc[i])
        label = f"B_{B}" if B > 0 else "control"
        vals.append(f"{label}={resp:.2f}")
    print(f"logA={log_conc[i]:.2f} " + " ".join(vals))

# ── Panel 4: Allosteric Modulation ──
KM = 1.0
mod_conc = 10.0
alpha_vals = [0.1, 0.5, 1.0, 2.0, 10.0]
beta = 1.0  # efficacy unchanged for this panel

print("\nCHART4_TITLE: Allosteric Modulation (varying alpha, beta=1)")
print("CHART4_XLABEL: log[Agonist]")
print("CHART4_YLABEL: Fractional Occupancy")
for i in range(0, len(conc), 5):
    vals = []
    for alph in alpha_vals:
        num = conc[i] * (1 + alph * mod_conc / KM)
        den = conc[i] * (1 + alph * mod_conc / KM) + KA * (1 + mod_conc / KM)
        occ = num / den
        vals.append(f"alpha_{alph:.1f}={occ:.4f}")
    print(f"logA={log_conc[i]:.2f} " + " ".join(vals))

# ── Panel 5: Partial Agonist + Full Agonist ──
eA = 1.0   # full agonist efficacy
eP = 0.3   # partial agonist efficacy
KP = 0.5   # partial agonist affinity
full_agonist_conc = 10.0

print("\nCHART5_TITLE: Partial Agonist Displacing Full Agonist")
print("CHART5_XLABEL: log[Partial Agonist]")
print("CHART5_YLABEL: Response (% E_max)")
for i in range(0, len(conc), 5):
    denom = 1 + full_agonist_conc / KA + conc[i] / KP
    fracA = (full_agonist_conc / KA) / denom
    fracP = (conc[i] / KP) / denom
    E_total = 100 * (eA * fracA + eP * fracP)
    print(f"logP={log_conc[i]:.2f} response={E_total:.2f}")

# ── Panel 6: Irreversible Antagonist with Spare Receptors ──
print("\nCHART6_TITLE: Irreversible Antagonist (with receptor reserve)")
print("CHART6_XLABEL: log[Agonist]")
print("CHART6_YLABEL: Response (% E_max)")
tau = 20.0  # high receptor reserve
q_values = [1.0, 0.5, 0.2, 0.05]
for i in range(0, len(conc), 5):
    vals = []
    for q in q_values:
        tau_eff = tau * q
        e = 100 * (tau_eff * conc[i]) / (conc[i] * (1 + tau_eff) + KA)
        vals.append(f"q_{q:.2f}={e:.2f}")
    print(f"logA={log_conc[i]:.2f} " + " ".join(vals))

print("\nSimulation complete: 6 panels generated")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Clinical Applications

Naloxone for Opioid Reversal

Competitive mu-opioid antagonist. Rapidly displaces morphine/fentanyl from receptors. Short half-life (30-90 min) means opioid effects may return (renarcotization), requiring repeated dosing or continuous infusion.

Phenoxybenzamine (Irreversible)

Irreversible alpha-adrenergic antagonist. Used preoperatively for pheochromocytoma. Effects last 24-48 hours until new receptors are synthesized. Initial rightward shift becomes E_max depression as receptor reserve is depleted.

Benzodiazepines (PAMs)

Positive allosteric modulators at GABA_A receptors. They increase GABA's affinity (alpha > 1) and efficacy (beta > 1) without directly activating the channel, providing a built-in ceiling effect that limits toxicity compared to barbiturates.

Aripiprazole (Partial Agonist)

Partial agonist at D2 dopamine receptors. In hyperdopaminergic states (psychosis), it acts as a functional antagonist. In hypodopaminergic states (negative symptoms), it provides modest agonism. This "dopamine stabilizer" concept reduces extrapyramidal side effects.

Key Takeaways

1.
The two-state model classifies drugs by their selectivity for R* (active) vs R (inactive) receptor conformations: alpha > 1 = agonist, alpha = 1 = neutral antagonist, alpha < 1 = inverse agonist.
2.
Gaddum equation: competitive antagonists increase apparent K_A by factor (1 + [B]/K_B), producing rightward shift without E_max reduction.
3.
Noncompetitive antagonists reduce E_max (q = 1/(1+[B]/K_B)) without shifting EC_50.
4.
Allosteric modulators alter agonist affinity (alpha factor) and efficacy (beta factor) without competing at the orthosteric site.
5.
Partial agonists can function as antagonists in the presence of full agonists, reducing the overall response toward their own submaximal ceiling.

← 3.2 Dose-Response 3.4 Receptor Theory →

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