Theory of Ligand-Receptor Interactions Enzyme Kinetics. Drug-receptor binding has parallels with enzyme kinetics. Enzymes produce a product by catalysis, but the intermediate is a substrate-enzyme complex analogous to the ligand-receptor complex. Most of the methods of analysis are similar. In enzyme kinetics, are interested in the velocity of the reaction, which is proportional to the concentration of the high-energy intermediate ES. So most of the receptor analysis is simply a substitution of [LR] for Velocity.

ProductEnzyme Enzyme)(Substrate EnzymeSubstrate +→↔+Response eptor)(LigandRec Receptor Ligand →→↔+

Drug Receptor Binding. From the laws of mass action, binding of a ligand (L) to its receptor (R) to form a ligand-receptor complex (LR) leads to the equations:

LRk

kRL

⎯⎯ ⎯←⎯⎯→⎯

+

− 1

1

where k1 = association rate constant, or ON rate (units: sec-1 M-1) k-1 = dissociation rate constant, or OFF rate (units: sec-1) [L], [R], [LR] = concentrations of free ligand, free receptor and ligand-receptor complex The rate of formation of LR = k1 [L] [R] The rate of breakdown of LR = k-1 [LR] At equilibrium, rate of formation of LR = rate of breakdown of LR k1 [L] [R] = k-1 [LR]

][]][[

LRRL

kkKd ==

−

1

1

Kd = equilibrium dissociation constant = dissociation rate association rate Units of Kd: Molar We can measure the concentration of occupied receptor [LR] (e.g. with radioactive ligand), and the total number of receptors (with saturating concentration of ligand), but cannot measure the unliganded receptor, so would like to express the binding in terms of liganded receptors and total receptor. Since the total number receptors Rt is fixed in a given experiment, we can separate them into those that are free [R] and those bound with ligand [LR]: [Rt] = [R] + [LR] or [R] = [Rt] – [LR]

Therefore, substituting for [R]

[LR]) [LR] [Rt] ( [L]Kd −

=

Then, by combining the terms with [LR],

[LR] [L] [Rt] [L]Kd [LR] −=

[Rt] [L]Kd) ([L] [LR] =+

Kd) ([L] [Rt] [L] [LR]+

=

There are several ways of expressing ligand/receptor binding data in graphical form:

• Dose-Occupation curve (& Log Dose-Occupation curve) • Double reciprocal plot • Scatchard Plot

Dose-Occupation Curve, or Dose-Response Curve. This produces the usual Langmuir adsorption isotherm when one plots [LR] vs. [L], giving a retangular hyperbola.

1. What happens when [L] = Kd?

2

[Rt]Kd) (Kd

[Rt] Kd [LR] =+

=

2. What happens when [L] >> Kd?

[Rt][L]

[L][Rt]Kd) ([L]

[Rt] [L] [LR] ==+

=

Problem with this graph: it doesn’t saturate fast enough, and can’t estimate Rt unless go to very high [L]. Often the binding exhibits a lower affinity non-saturable component that is subtracted to obtain the dose-occupation curve for the receptor of interest.

Log Dose-Occupation Plot. Dose occupation curves are often plotted on a semilog plot because the affinities of drugs vary over several orders of magnitude. A sigmoid curve results. At half saturation of receptor occupation by ligand, Log[L] gives the Kd.

Double Reciprocal Plot. For many experiments, it may be preferable to look at the data in a linear plot so that the Kd and number of total receptors may be more readily extrapolated. This is particularly useful when the binding data do not extend to high ligand concentrations and the receptor occupation by ligand has not yet saturated. The double reciprocal plot has the following formulation (obtained by taking the reciprocal of the basic binding equation):

[Rt]1

[L]1

[Rt]Kd

[LR]1

+⎟⎠

⎞⎜⎝

⎛=

Thus a plot of 1/[LR] vs. 1/[L] has

• a slope of Kd/[Rt] • a y-intercept of 1/[Rt] • an x-intercept of –1/Kd

(This plot is similar to the Lineweaver-Burk plot of enzyme kinetics: 1/V vs. 1/[S] )

Problems: the error bars are greatest at the right side of the graph for low [L], and therefore these data points are weighted too heavily by linear curve-fitting routines, and single outliers can distort the interpretation. Scatchard Plot. This representation of binding data also is used widely in Pharmacology papers to determine the Kd and number of receptors present. Rearranging the equation above yields the following:

Now if [LR]/[L] is plotted against [LR], a straight line will result with

• slope of –1/Kd • y-intecept of [Rt]/Kd • x-intercept is the number of binding sites [Rt]

Deviations from linearity suggest more than one binding site: the steeper slope having the higher affinity, and the shallower slope the lower affinity binding which is often nonspecific) (Similar to Eadie-Hofstee plot, but axis are reversed: V vs. V/[S], slope –Km)