These experiments provide information about the concentration of a receptor. They are solicited to compare the concentrations of different receptors in a given tissue and to monitor variations in receptor concentration as a result of normal physiological regulation, medication and pathophysiological conditions.

For saturation binding experiments, constant amounts of membrane suspension are incubated with increasing concentrations of radioligand. Obviously, both total and non-specific binding should be measured at each concentration of radioligand (Figure 31). In the example shown, binding is expressed as a function of the free concentration of radioligand by a saturation binding plot. Obviously, only the specific binding is of interest.

To analyze these saturation binding data, it is necessary to advance a relevant molecular model for the radioligand-receptor interaction. In the simple (and fortunately the most common) situation, the interaction of the radioligand (L) with the

Figure 31 Saturation binding of the a2-adrenergic antagonist [3H]RX 821002 to a2 adrenergic receptors in membranes from the human frontal cortex. Reprinted from Neurochemistry International, 17, Vauquelin G., De Vos H., De Backer J.-P. and Ebinger G., Identification of a2 adrenergic receptors in human frontal cortex membranes by binding of [3H]RX 821002, the 2-methoxy analog of [3H]idazoxan, 537-546. Copyright (1990), with permission from Elsevier.

Figure 31 Saturation binding of the a2-adrenergic antagonist [3H]RX 821002 to a2 adrenergic receptors in membranes from the human frontal cortex. Reprinted from Neurochemistry International, 17, Vauquelin G., De Vos H., De Backer J.-P. and Ebinger G., Identification of a2 adrenergic receptors in human frontal cortex membranes by binding of [3H]RX 821002, the 2-methoxy analog of [3H]idazoxan, 537-546. Copyright (1990), with permission from Elsevier.

receptor (R) can be expressed as a reversible bimolecular reaction that obeys the law of mass action: i.e.

Where k1 and are the association and dissociation rate constants, respectively. The equilibrium dissociation constant (KD) is given by:

Where [R] is the amount of free receptors, [L] the amount of free ligand and [L-R] the amount of bound ligand/receptors.

The relationship between the amount of occupied receptors and the free radioligand concentration (i.e. the saturation binding plot) is as follows:

Where [Rtot] is the total number of receptors.

'B' and 'Bmax' (which stand for binding and maximum binding and are often expressed in fmol/mg protein) usually replace [L - R] and [Rtot]. Equation (3) then becomes:

This equation is analogous to the Michaelis-Menten equation of enzyme kinetics and describes a rectangular hyperbola. Initially, B increases almost linearly with L. Then

B tends to level off when L is further increased. The limit value is Bmax (Figure 32). It is important to notice that this that Bmax will be attained only at infinite concentrations of L. Thus, one will never observe Bmax experimentally; Bmax may be approached but never attained. Half-maximal binding is obtained when L = KD (since Equation (4) becomes B = Bmax/2). In other words, the KD of a radioligand corresponds to its concentration for which half of the receptors are occupied. The KD value is thus an 'inverse' measure of the radioligand's affinity for the receptor: a low KD corresponds to high affinity and a high KD to low affinity.

Bmax and KD cannot be easily determined by graphical analysis of the saturation binding plot (Figure 32) since Equation (4) is a non-linear relationship and since Bmax is only reached when L = This equation can, however, be transformed mathematically to yield a linear 'Scatchardplot (Figure 33) corresponding to the following equation:

SCATCHARD PLOT

SCATCHARD PLOT

C : positive cooperativity Analysis: computer-assisted

A : 1 site or > 1 site with the same affinity

Analysis: linear regression

B : > 1 site or negative cooperativity Analysis: computer-assisted bound

Figure 34 Scatchard plots: different possibilities.

C : positive cooperativity Analysis: computer-assisted

A : 1 site or > 1 site with the same affinity

Analysis: linear regression

B : > 1 site or negative cooperativity Analysis: computer-assisted bound

Figure 34 Scatchard plots: different possibilities.

The Scatchard plot of the above saturation binding data reveals a linear relationship between B/[L] (the ordinate) and B (the abscissa). KD corresponds to the negative reciprocal of the line. The intercept of the line with the abscissa (i.e. when B/[L] = 0) is Bmax. Thus, it is relatively easy to calculate KD and Bmax values by linear regression analysis of the Scatchard plot.

The relationship described by Equation (5) is for the simplest case; i.e. a single class of non-interacting receptor sites. However, it is possible that the radioligand binds to two different receptors with different affinities or even that one receptor is present in two or more (non-interconverting) affinity states for the radioligand. This situation will result in a non-linear Scatchard plot: i.e. showing downward concavity (Figure 34 curve B).

Moreover, certain receptors (e.g. ion channel-gating receptors which make part of a larger structure) possess multiple binding which influence each other's binding characteristics. This may result in either negative or positive co-operative interactions among the binding sites. In other words, binding of the radioligand to one site decreases (negative co-operativity) or increases (positive co-operativity) the affinity of the radioligand for other sites. This will also result in non-linear Scatchard plots with, respectively, downward concavity (negative co-operativity, Figure 34 curve B) or upward concavity (positive co-operativity, Figure 34 curve C).

A more sensitive method to ascertain whether radioligand binding obeys the law of mass action is to analyse the 'Hill plot' of the saturation binding data (Figure 35). The Hill equation is, in fact, a logarithmic transformation of Equation (4).

Log(B/(Bmax — B)) is the ordinate and Log([L]) is the abscissa of the Hill plot. The slope corresponds to the Hill coefficient: 'nH'. The law of mass action is obeyed if nH = 1 (in practice, values between 0.8 and 1.2 will do). This means that the radioligand binds with the same affinity to all the sites. nH < 1 is indicative of either negative co-opera-tivity or of the existence of binding sites with different affinity. nH > 1 is indicative of positive co-operativity, i.e. where radioligand binding to one site increases the affinity of the radioligand for other sites.

HILL PLOT

HILL PLOT

[3H]RX821002 concentration (Log)

Figure 35 Hill plot of the saturation binding data from Figure 31 (B in fmol/mg protein, F in nM).

[3H]RX821002 concentration (Log)

Figure 35 Hill plot of the saturation binding data from Figure 31 (B in fmol/mg protein, F in nM).

A disadvantage of a Scatchard plot is that the extrapolation of data obtained over an insufficient concentration range of L may give an artificial impression of the lack of complexity of the radioligand-receptor interactions. This may result in inaccurate Hill plots since they rely on a correct estimation of the Bmax value. Although Scatchard and Hill plots are still sometimes shown in publications for the sake of clarity (e.g. it is easy to visualize differences in KD and Bmax values of one or more radioligands with a Scatchard plot), radioligand binding parameters are now almost always calculated by computer programmes which are based on non-linear regression analysis of the saturation binding data. In the case of two non-interconverting binding sites, these programmes even allow the calculation of the concentration of each site and its respective affinity for the radioligand.

Finally, certain important considerations need to be taken into account before correctly analyzing saturation binding data; they include:

• The data must represent an equilibrium situation. In practice, this means that the incubation must have occurred long enough for equilibrium to be reached. Investigating binding of a given concentration of radioligand as a function of the incubation time can check this. This binding will increase time-wise until a plateau value (corresponding to the equilibrium situation) is reached. Equilibrium binding is often obtained within minutes at usual incubation temperatures (20-37 °C), but that it may become considerably longer when the temperature is lowered to (0-4 °C).

• Binding is expressed as a function of the free concentration of radioligand. This concentration may be set equal to the concentration of radioligand added (i.e. [L] = [Linit]) if only a small fraction of the added radioligand is bound (in other words, if most of the added radioligand still remains free). If a more substantial amount of radioligand is bound (e.g. >5%), then [L] is smaller than [Linit], and its correct value should be calculated by the equation: [L] = [Linit] — [L — R].

• The ligand must not aggregate, at higher concentrations, to a dimer or multimer.

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