Intro to Pharmacology and Toxicology
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A ligand is any ion or molecule that reacts to form a complex with another molecule, usually a macromolecule. A receptor is a molecule or other cellular component than when activated by a ligand or physical change produces and effect. An agonist a a chemical that, when bound to a receptor, stimulates the receptor to produce its appropriate response. The action of the agonist may be recognized as stimulation or inhibition of the cellular activity, depending on the type of receptor and the type of cell. A partial agonist acts as an agonist on a receptor but fails to elicit the full response of the receptor.
The affinity of a ligand for a receptor relates to the frequency with with which the ligand, when in proximity to the receptor by diffusion, will reside at a position of minimum free energy within the force field of the receptor.
Intrinsic efficacy is a dimensionless proportionality factor that represents the stimulus per receptor molecule produced by an agonist. For example, one agonist may produce an effect of a specific magnitude by occupying 10% of the available receptors, while another agonist may need to occupy 50% of the receptors to produce an effect of the same magnitude. An inverse agonist can be thought of as having negative efficacy. When the inverse agonist binds to the receptor, the basal level of activity produced by the receptor in the absence of agonist diminishes.
A simple bimolecular association model involving a receptor and a ligand can be describe by:
[R] + [L] 
[RL]
The velocity of the forward reaction is proportional to the concentration of ligand and receptor, and the rate constant k1 allows that proportionality to be converted to an equation:
vf µ [R] [L] Þ VF = k1 [R] [L]
k1 is the association (on-rate) rate constant of the receptor-ligand association. [R][L] is related to the probability that a ligand molecule and a receptor will associate. In a similar way, the velocity of the reverse reaction is proportional to the concentration of the drug-ligand complex:
vr µ [RL] Þ VF = k-1 [RL]
k-1 is the dissociation (off-rate) rate constant of the receptor-ligand complex. [RL] is related to the probability that a ligand-receptor complex will dissociate. If experimental conditions are held constant (temperature, pH, etc.), the change in ligand-receptor complex concentration over time is the difference between the forward and reverse velocities:
d[RL] / dt = k1 [R] [L] - k-1 [RL]
If we stipulate several assumptions, a mathematical model can be developed:
Given the conditions above, the forward and reverse velocities at equilibrium are equal (because there is no net change in the concentration of reactants), meaning that:
0 = d[RL] / DT = k1 [R] [L] - k-1 [RL]
k1
[R] [L] = k-1 [RL]
Applying the mass conservation equations above to put everything in terms of [R]total and [L]total and setting k-1/k1 = Kd we can obtain the simple Langmuir isotherm:
k1 ([R]total - [RL]) [L]total = k-1 [RL]
([R]total - [RL]) [L]total = (k-1/k1) [RL]
[R]total [L]total - [RL] [L]total = Kd [RL]
[R]total [L]total = Kd [RL] + [RL] [L]total
[R]total [L]total = [RL] (Kd + [L]total)
[R]total
[L]total = [RL]
Kd
+ [L]total
To determine Kd experimentally, the total receptor concentration is set constant during a series of experiment while varying the total ligand concentration for each run and measuring [RL] at equilibrium.
The Langmuir isotherm describes a hyperbolic curve, i.e. there is a saturable increase in ligand-receptor concentration with increases in free ligand concentration. The curve approaches an asymptote equal to the total receptor concentration. Kd is equal to the ligand concentration that produces half maximal binding (i.e. half or the receptors are occupied).
The data from receptor binding experiments is most often plotted on a semilogaritmic scale to yield what is commonly called a log-dose response curve, a regular sigmoid approaching a maximal value equal to the total concentration of receptors and with an inflection point corresponding to Kd.
Other
common transformations of the receptor binding data are the double-reciprocal
(Lineweaver-Burk) and Scatchard plot (Eadie-Hofstee).
1 =
Kd x
1 + 1
[RL] [R]total
[L]total
[R]total
[RL]
= Kd x
1 + 1
[L]total
[R]total
Kd
[R]total
The Scatchard plot expresses data
as bound/free vs. bound. Although widely used, it is fundamentally because the
dependent variable [RL] is on both axes, magnifying statistical errors. Furthermore,
the Scatchard plot for a system involving complex receptors may be curved instead
of linear. One of several possible explanations is that binding occurs at two
separate and independent sites. Further experiments will be needed in such a
case 
to
resolve the mechanism. The mechanism of interaction cannot be proven by kinetics
experiments alone. Kinetics data may suggest mechanisms but proof must come
from experiments designed to test the hypothetical mechanism.
The analysis of receptor binding may include experiments carried out under conditions where equilibrium does not apply. In such cases, time course studies are used to establish time to equilibrium and for directly determining the rate constants.
At the instant free receptor and free ligand are mixed, association to form the complex begins as described by the following equation:
[RL]
= [RL]eq (1-e^{-(k1[L]+k-1)t})
where [RL]eq is the concentration of complex at equilibrium for any given [L]. This equation can also be used after linear transformation by plotting ln (1- [RL]/[RL]eq) vs. time.
ln (1- [RL]/[RL]eq) = -(k1 [L] + k-1) t
The dissociation constant can be measured directly if receptor and ligand are mixed and allowed to produce complex in a preincubation, followed by a dilution step in which free ligand is suddenly removed and the complex allowed to dissociate. The dissociation is described by:
[RL] = [RL]0 e^-k-1t OR ln [RL] = ln [RL]0 - k-1 t
where [RL]0 is the concentration of complex at the instant free ligand was removed and dissociation started. The dissociation rate is dependent only on the dissociation rate constant k-1. If time is plotted against ln[RL], k-1 is the slope.
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