Pharmacological Assay Formats

Chapter 4

Pharmacological Assay Formats: Binding

The yeoman work in any science . . . is done by the experimentalist who must keep the theoreticians honest. —Michio Kaku (1995).

4.1 4.2 4.3

The Structure of This Chapter Binding Theory and Experiment Complex Binding Phenomena: Agonist Affinity from Binding Curves

71 71

4.4

4.5

Experimental Prerequisites for Correct Application of Binding Techniques 84 Binding in Allosteric Systems 88

4.6

Chapter Summary and Conclusions 4.7 Derivations References

91 92 96

80

4.1 THE STRUCTURE OF THIS CHAPTER This chapter discusses the application of binding techniques to the study of drug
receptor interaction. It will be seen that the theory of binding and the methods used to quantify drug effect are discussed before the experimental prerequisites for good binding experiments are given. This may appear to be placing the cart before the horse in concept. However, the methods used to detect and rectify nonequilibrium experimental conditions utilize the very methods used to quantify drug effect. Therefore, they must be understood before their application to optimize experimental conditions can be discussed. This chapter first presents what the experiments strive to achieve and then explores the possible pitfalls of experimental design that may cause the execution to fall short of the intent.

4.2 BINDING THEORY AND EXPERIMENT A direct measure of the binding of a molecule to a protein target can be made if there is some means to distinguish the bound molecule from the unbound and a means to quantify the amount bound. Historically, the first widely used technique to do this was radioligand binding. Radioactive molecules can be detected by observation of radioactive decay, and their amount quantified through A Pharmacology Primer. DOI: https://doi.org/10.1016/B978-0-12-813957-8.00004-7 © 2019 Elsevier Inc. All rights reserved.

calibration curves relating the amount of molecule to the amount of radioactivity detected. An essential part of this process is the ability to separate the bound from the unbound molecule. This can be done by taking advantage of the size of the protein versus the soluble small molecule. The protein can be separated by centrifugation, equilibrium dialysis, or filtration. Alternatively, the physical proximity of the molecule to the protein can be used. For example, in scintillation proximity assays, the receptor protein adheres to a bead containing scintillant, a chemical that produces light when close to radioactivity. Thus, when radioactive molecules are bound to the receptor (and therefore are near the scintillant), a light signal is produced, heralding the binding of the molecule. Other methods of detecting molecules such as fluorescence are increasingly being utilized in binding experiments. For example, molecules that produce different qualities of fluorescence, depending on their proximity to protein, can be used to quantify binding. Similarly, in fluorescence polarization experiments, fluorescent ligands (when not bound to protein) reduce the degree of light polarization of light passing through the medium through free rotation. When these same ligands are bound, their rotation is reduced, thereby concomitantly reducing the effect on polarization. Thus, binding can be quantified in terms of the degree of light polarization in the medium. In general, there are emerging technologies available to discern bound from unbound molecules, and many of 71

72

these can be applied to receptor studies. It will be assumed from this point that the technological problems associated with determining bound species are not an experimental factor, and subsequent discussions will focus on the interpretation of the resulting binding data. Several excellent sources of information on the technology and practical aspects of binding are available [1
3]. It is important to note that pharmacological binding versus functional studies measure different protein species in the assay. Binding measures the amount of protein bound to a radioactive tracer while function measures the effects of an activated receptor species as sensed by the cell (see Fig. 4.1). Therefore, there are numerous cases where binding versus functional studies yield different data (see Fig 8.25 for an example) and in terms of therapeutic drug activity, functional data are preferred. However, binding may give insights not obvious from functional studies and thus it can still be a useful endeavor. Binding experiments can be done in three modes: saturation, displacement, and kinetic. Saturation binding directly observes the binding of a tracer ligand (radioactive, fluorescent, or otherwise detectable) to the receptor. The method quantifies the maximal number of binding sites and the affinity of the ligand for the site (equilibrium dissociation constant of the ligand
receptor complex). This is a direct measure of binding using the Langmuir adsorption isotherm model. A major limitation of this technique is the obvious need for the ligand to be traceable (i.e., it can be done only for radioactive or fluorescent molecules). Displacement studies overcome this limitation by allowing measurement of the affinity of nontraceable ligands through their interference with the binding of tracer ligands. Thus, molecules are used to displace or otherwise prevent the binding of tracer ligands and the reduction in signal is used to quantify the affinity of the displacing ligands. Finally, kinetic studies follow the binding of a tracer ligand with time. This can yield first-order rate constants for the onset and offset of

Chapter | 4 Pharmacological Assay Formats: Binding

binding, which can be used to calculate equilibrium binding constants to assess the temporal approach to equilibrium or to determine binding reversibility or to detect allosteric interactions. Each of these is considered separately. The first step is to discuss some methodological points common to all these types of binding experiments. The aim of a binding experiment is to define and quantify the relationship between the concentration of ligand in the receptor compartment and the portion of the concentration that is bound to the receptor at any one instant. A first prerequisite is to know that the amount of bound ligand that is measured is bound only to the receptor and not to other sites in the sample tube or well (i.e., cell membrane, wall of the vessel containing the experimental solution, and so on). The amount of ligand bound to these auxiliary sites but not specifically to the target is referred to as nonspecific binding (denoted nsb). The amount bound only to the pharmacological target of interest is termed the specific binding. The amount of specific binding is defined operationally as the bound ligand that can be displaced by an excess concentration of a specific antagonist for the receptor that is not radioactive (or otherwise does not interfere with the signals). Therefore, another prerequisite of binding experiments is the availability of a nontracer ligand (for the specific target defined as one that does not interfere with the signal, whether it be radioactivity, fluorescence, or polarized light). Optimally, the chemical structure of the ligand used to define nsb should be different from the binding tracer ligand. This is because the tracer may bind to nonreceptor sites (i.e., adsorption sites, other nonspecific proteins), and if a nonradioactive version of the same molecular structure is used to define specific binding, it may protect those very same nonspecific sites (which erroneously define specific binding). A ligand with different chemical structure may not bind to the same nonspecific sites and thus lessen the potential of defining nsb sites as biologically relevant receptors. The nsb of low concentrations of biologically active ligands is essentially linear and nonsaturable within the ranges used in pharmacological binding experiments. For a traceable ligand (radioactive, fluorescent, and so on), nsb is given as nsb 5 kU½AT

(4.1)

where k is a constant defining the concentration relationship for nsb and [A*] is the concentration of the traceable molecule. The specific binding is saturable and defined by the Langmuir adsorption isotherm: FIGURE 4.1 Model depicting receptor that can exist in an active (Ra) and inactive (Ri) state formed by binding of agonist A and allosteric ligand B. When the radioligand is agonist A, binding experiments detect the species in red (panel labeled binding). In functional experiments (panel labeled function), the cellular response is produced by agonist bound species and any constitutively active receptor species (in the form of Ra or BRa) shown in red.

Specific binding 5

½AT ½AT 1 Kd

(4.2)

where Kd is the equilibrium dissociation constant of the ligand
receptor complex. The total binding is the sum of these and is given as

4.2 BINDING THEORY AND EXPERIMENT

Total binding 5

73

½ATUBmax 1 kU½AT ½AT 1 Kd

(4.3)

The two experimentally derived variables are nsb and total binding. These can be obtained by measuring the relationship between the ligand concentration and the amount of ligand bound (total binding) and the amount bound in the presence of a protecting concentration of receptorspecific antagonist. This latter procedure defines the nsb. Theoretically, specific binding can be obtained by subtracting these values for each concentration of ligand, but a more powerful method is to fit the two data sets (total binding and nsb) to Eqs. (4.1) and (4.3) simultaneously. One reason that this is preferable is that more data points are used to define specific binding. A second reason is that a better estimate of the maximal binding (Bmax) can be made by simultaneously fitting two functions. Since Bmax is defined at theoretically infinite ligand concentrations, it is difficult to obtain data in this concentration region. When there is a paucity of data points, nonlinear fitting procedures tend to overestimate the maximal asymptote. The additional experimental data (total plus nsb) reduce this effect and yields more accurate Bmax estimates. In binding, a good first experiment is to determine the time required for the binding reaction to come to equilibrium with the receptor. This is essential to know, since most binding reactions are made in stop-time mode, and real-time observation of the approach to equilibrium is not possible (this is not true of more recent fluorescent techniques where visualization of binding in real time can be achieved). A useful experiment is to observe the approach to equilibrium of a given concentration of tracer ligand and then to observe reversal of binding by addition of a competitive antagonist of the receptor. An example of this experiment is shown in Fig. 4.2. Valuable data are obtained with this approach, since it indicates the time needed to reach equilibrium and confirms the fact that the binding is reversible. Reversibility is essential to the attainment of steady states and equilibria (i.e., irreversible binding reactions do not come to equilibrium).

4.2.1 Saturation Binding A saturation binding experiment consists of the equilibration of the receptor with a range of concentrations of traceable ligand in the absence (total binding) and presence of a high concentration (approximately 100 3 Kd) of antagonist to protect the receptors (and thus determine the nsb). Simultaneous fitting of the total binding curve [Eq. (4.3)] and nsb line [Eq. (4.1)] yields the specific binding with parameters of maximal number of binding sites (Bmax) and equilibrium dissociation constant of the ligand
receptor complex (Kd) [see Eq. (4.2)]. An example of this procedure for the human calcitonin receptor is shown in Fig. 4.3 [4]. Before the widespread use of nonlinear fitting programs, the Langmuir equation was linearized for ease of fitting graphically. Thus, specific binding ([A*R]) according to mass action is represented as ½ATR ½AT 5 Bmax ½AT 1 Kd which yields a straight line with the transforms ½ATR Bmax ½ATR 5 2 ½AT Kd Kd

1 1 Kd 1 5 U 1 ½ATR ½AT Bmax Bmax

½AT ½AT Kd 5 1 ½ATR Bmax Bmax

Fractional binding

0.9 A

0.5 0.3 0.1 –0.1 0

50

100

150

200 250 Time (min)

300

(4.6)

This is referred to as a double reciprocal or Lineweaver
Burk plot. From this linear plot, Kd 5 slope/ intercept and the 1/intercept 5 Bmax. Finally, a linear plot can be achieved with

1.1

–50

(4.5)

referred to alternatively as a Scatchard, Eadie, or Eadie
Hofstee plot. From this linear plot, Kd 5 21/slope and the x intercept equals Bmax. Alternatively, another method of linearizing the data points is by using

B

0.7

(4.4)

350

400

450

(4.7)

FIGURE 4.2 Time course for the onset of a radioligand onto the receptor and the reversal of radioligand binding upon addition of a high concentration of a nonradioactive antagonist ligand. The object of the experiment is to determine the times required for steady-state receptor occupation by the radioligand and confirmation of reversibility of binding. The radioligand is added at point A, and an excess competitive antagonist of the receptor at point B.

74

Chapter | 4 Pharmacological Assay Formats: Binding

0.20

Scatchard

[A*R]/[A*]

0.16 0.12 0.06 0.04 0.00 0

1

2

3

16 14

14 (1/[A*R]) x 1011

12 [A*R] : pM

4

5

6

7

8

[A*R] : pM

10 8 6 4

Double recip.

12 10 8 6 4 2 0

2 0

0

0

50

100

150 200 [A*] : pM

250

5

300

10

15

20

25

30

250

300

1/[A*] x 1010

[A*]/[A*R]

35

Hanes plot

30 25 20 15 10 5 0 0

50

100

150

200

[A*] : pM FIGURE 4.3 Saturation binding. Left panel: Curves showing total binding (filled circles), nonspecific binding (filled squares), and specific binding (open circles) of the calcitonin receptor antagonist radiolabel 125I AC512 (Bmax 5 6.63 pM; Kd 5 26.8 pM). Panels to the right show linear variants of the specific binding curve: Scatchard [Eq. (4.5)], double reciprocal [Eq. (4.6)], and Hanes plots [Eq. (4.7)] cause distortion and compression of data. Nonlinear curve-fitting techniques are preferred. Left panel: Data redrawn from W.-J. Chen, S. Armour, J. Way, G. Chen, C. Watson, P. Irving, et al., Expression cloning and receptor pharmacology of human calcitonin receptors from MCF-7 cells and their relationship to amylin receptors, Mol. Pharmacol. 52 (1997) 1164
1175.

This is referred to as a Hanes, Hildebrand
Benesi, or Scott plot. From this linear plot, Kd 5 intercept/slope and 1/slope 5 Bmax. Examples of these are shown for the saturation data in Fig. 4.3. At first glance, these transformations may seem like ideal methods for analyzing saturation data. However, transformation of binding data is not generally recommended. This is because transformed plots can distort experimental uncertainty, produce compression of data, and cause large differences in data placement. Also, these transformations violate the assumptions of linear regression and can be curvilinear simply because of statistical factors (e.g., Scatchard plots combine dependent and independent variables). These transformations are valid only for ideal data and are extremely sensitive to different types of experimental errors. They should not be used for

estimation of binding parameters. Scatchard plots compress data to the point where a linear plot can be obtained. Fig. 4.4 shows a curve with an estimate of Bmax that falls far short of being able to furnish an experimental estimate of the Bmax, yet the Scatchard plot is linear with an apparently valid estimate from the abscissal intercept. In general, nonlinear fitting of the data is essential for parameter estimation. Linear transformations, however, are useful for visualization of trends in data. Variances from a straight edge are more discernible to the human eye than are differences from curvilinear shapes, so linear transformations can be a useful diagnostic tool. An example of where the Scatchard transformation shows significant deviation from a rectangular hyperbola is shown in Fig. 4.5. The direct presentation of the data shows little

75

1.4 1.2 1.0 0.8

[A*R]/[A*]

Fraction max.

4.2 BINDING THEORY AND EXPERIMENT

0.6 0.4 0.2 0.0 –4

–2

0 Log ([A*]/Kd)

2

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0

4

0.2

0.4

0.6 0.8 [A*R]/Kd

1

1.2

FIGURE 4.4 Erroneous estimation of maximal binding with Scatchard plots. The saturation binding curve shown to the left has no data points available to estimate the true Bmax. The Scatchard transformation to the right linearizes the existing points, allowing an estimate of the maximum to be made from the x-axis intercept. However, this intercept in no way estimates the true Bmax since there are no data to define this parameter.

(A)

(B) 30 40 [A*R]/[A*] x 104

Bo (pmol/mg)

25 20 15 10

30

20

10

5 0

0 –9

–8

–7

–6

–5

–4

Log [A*]

0

5

10

15

20

25

[A*R] : pM

FIGURE 4.5 Saturation binding expressed directly and with a Scatchard plot. (A) Direct representation of a saturation binding plot (Bmax 5 25 pmol/mg, Kd 5 50 nM). Data points are slightly deviated from ideal behavior (lower two concentrations yield slightly lower values for binding, as is common when slightly too much receptor protein is used in the assay, vide infra). (B) Scatchard plot of the data shown in panel (A). It can be seen that the slight deviations in the data lead to considerable deviations from linearity on the Scatchard plot.

deviation from the saturation binding curve as defined by the Langmuir adsorption isotherm. The data at 10 and 30 nM yield slightly underestimated levels of binding, a common finding if slightly too much protein is used in the binding assay (see Section 4.4.1). While this difference is nearly undetectable when the data are presented as a direct binding curve, it does produce a deviation from linearity in the Scatchard curve (see Fig. 4.5B). Estimating the Bmax value is technically difficult since it basically is an exercise in estimating an effect at infinite drug concentration. Therefore, the accuracy of the estimate of Bmax is proportional to the maximal levels of radioligand that can be used in the experiment. The attainment of saturation binding can be deceiving when the ordinates are plotted on a linear scale, as they are in Fig. 4.3. Fig. 4.6 shows a saturation curve for calcitonin binding that appears to reach a maximal asymptote on a

linear scale. However, replotting the graph on a semilogarithmic scale illustrates the illusion of maximal binding on the linear scale and, in this case, how far short of true maxima a linear scale can present a saturation binding curve. An example of how to measure the affinity of a radioligand and obtain an estimate of Bmax (maximal number of binding sites for that radioligand) is given in Section 13.1.1.

4.2.2 Displacement Binding In practice, there will be a limited number of ligands available that are chemically traceable (i.e., radioactive, fluorescent). Therefore, the bulk of radioligand experiments designed to quantify ligand affinity are done in a displacement mode whereby a ligand is used to displace

76

Chapter | 4 Pharmacological Assay Formats: Binding

[Bound] : nM

0.18 0.12 0.06 0.00 –11

–10

–9

–8

–7

–6

125I-hCal]

Log [ 125I-hCal

[Bound] : pM

80 60 40 20 0 0

4

6 nM

10

8

FIGURE 4.6 Saturation binding of the radioligand human 125I-human calcitonin to human calcitonin receptors in a recombinant cell system in human embryonic kidney cells. Left-hand panel shows total binding (open circles), nonspecific binding (open squares), and specific receptor binding (open triangles). The specific binding appears to reach a maximal asymptotic value. The specific binding is plotted on a semilogarithmic scale (shown in the right-hand panel). The solid line on this curve indicates an estimate of the maximal receptor binding. The data points (open circles) on this curve show that the data define less than half the computer-estimated total saturation curve. Data redrawn from W.-J. Chen, S. Armour, J. Way, G. Chen, C. Watson, P. Irving, et al., Expression cloning and receptor pharmacology of human calcitonin receptors from MCF-7 cells and their relationship to amylin receptors, Mol. Pharmacol. 52 (1997) 1164
1175.

or otherwise affect the binding of a traceable ligand. In general, an inverse sigmoidal curve is obtained with reduction in radioligand binding upon addition of nonradioactive antagonist. An example of how to measure the affinity of a displacing ligand is given in Section 13.1.2. The equations describing the amount of bound radioligand observed in the presence of a range of concentrations of nontraceable ligand vary with the model used for the molecular antagonism. These are provided in material following, with brief descriptions. More detailed discussions of these mechanisms can be found in Chapter 6, Orthosteric Drug Antagonism. If the binding is competitive (both ligands compete for the same binding domain on the receptor), the amount of tracer ligand
receptor complex (ρ*) is given as (see Section 4.7.1) ρT 5

½AT=Kd ½AT=Kd 1 ½B=KB 1 1

(4.8)

where the concentration of tracer ligand is [A*], the nontraceable displacing ligand is [B], and Kd and KB are respective equilibrium dissociation constants. If the binding is noncompetitive (binding of the antagonist precludes the binding of the tracer ligand), the signal is given by (see Section 4.7.2)

ρT 5

½AT=Kd ½AT=Kd ð½B=KB 1 1Þ 1 ½B=KB 1 1

(4.9)

If the ligand allosterically affects the affinity of the receptor (antagonist binds to a site separate from that for the tracer ligand) to produce a change in receptor conformation to affect the affinity of the tracer (vide infra) for the tracer ligand (see Chapter 7: Allosteric Modulation for more detail), the displacement curve is given by (see Section 4.7.3) ρT 5

½AT=Kd ð1 1 α½B=KB Þ ½AT=Kd ð1 1 α½B=KB Þ 1 ½B=KB 1 1

(4.10)

where α is the multiple factor by which the nontracer ligand affects the affinity of the tracer ligand (i.e., α 5 0.1 indicates that the allosteric displacing ligand produces a tenfold decrease in the affinity of the receptor for the tracer ligand). As noted previously, in all cases these various functions describe an inverse sigmoidal curve between the displacing ligand and the signal. Therefore, the mechanism of interaction cannot be determined from a single displacement curve. However, observation of a pattern of such curves obtained at different tracer ligand

4.2 BINDING THEORY AND EXPERIMENT

77

concentrations (range of [A*] values) may indicate whether the displacements are due to a competitive, noncompetitive, or allosteric mechanism. Competitive displacement for a range of [A*] values [Eq. (4.8)] yields the pattern of curves shown in Fig. 4.7A. A useful way to quantify the displacement is to determine the concentration of displacing ligand that produces a diminution of the signal to 50% of the original value. This concentration of displacing ligand will be referred to as the IC50 (inhibitory concentration for 50% decrease). For competitive antagonists, it can be shown that the IC50 is related to the concentration of tracer ligand [A*] by (see Section 4.7.4) IC50 5 KB Uð½AT=Kd 1 1Þ

(4.11)

This is a linear relation often referred to as the Cheng
Prusoff relationship [5]. It is characteristic of competitive ligand
receptor interactions. An example is shown in Fig. 4.7B. In most conventional biochemical binding studies, the concentration of receptor protein is well below that of the ligands and thus the binding process does not significantly deplete the ligands. However, there are certain procedures such as fluorescent binding assays which require high concentrations of receptor to maximize the window for observing a response. Under these circumstances, the fluorescent probe concentration is kept below the Kd value (where Kd is the equilibrium dissociation constant of the fluorescent probe
receptor complex) and the receptor concentration is maximized (above the Kd value) [6]. In these types of assays, the standard correction for

(A)

IC50 to Ki values is not valid and a revised procedure utilizing the following equation must be used [6]: Ki 5

(B) 12

1.0

10 300 100 30 10

8 [IC50]/KB

0.8 0.6 1

0.4

6 4

[A*]/Kd = 0.03 2

0.2

0

0.0 –3

–1

1 3 Log ([B]/KB)

5

(4.12)

where [I]50 is the free antagonist concentration at 50% inhibition, Kd is the equilibrium dissociation constant of the fluorescent probe
receptor complex, [A*]50 is the free concentration of fluorescent probe at 50% inhibition, and [R]0 is the free concentration of receptor at 0% inhibition. The practical application of this equation is discussed in detail in Section 4.7.4. The displacement of a tracer ligand, for a range of tracer ligand concentrations, by a noncompetitive antagonist is shown in Fig. 4.8. In contrast to the pattern shown for competitive antagonists, the IC50 for inhibition of tracer binding does not change with increasing tracer ligand concentrations. In fact, it can be shown that the IC50 for inhibition is equal to the equilibrium dissociation constant of the noncompetitive antagonist
receptor complex (see Section 4.7.2). Allosteric antagonist effects can be an amalgam of competitive and noncompetitive profiles in terms of the relationship between IC50 and [A*]. This relates to the magnitude of the term α, specifically the multiple ratio of the affinity of the receptor for [A*] imposed by the binding of the allosteric antagonist. A hallmark of allosteric inhibition is that it is saturable (i.e., the antagonism maximizes upon saturation of the allosteric binding site). Therefore, if a given antagonist has a value of α of 0.1, this means that the saturation binding curve will shift to

1.2

Fraction max.

½I50 ½AT50 =Kd 1 ½R0 =Kd 1 1

0

2

4

6 [A*]/Kd

8

10

FIGURE 4.7 Displacement of a radioligand by a competitive nonradioactive ligand. (A) Displacement of radioactivity (ordinate scale) as curves shown for a range of concentrations of displacing ligand (abscissae as log scale). Curves shown for a range of radioligand concentrations denoted on the graph in units of [A*]/Kd. Curved line shows the path of the IC50 for the displacement curves along the antagonist concentration axis. (B) Multiple values of the Ki for the competitive displacing ligand (ordinate scale) as a function of the concentration of radioligand being displaced (abscissae as linear scale). Linear relationship shows the increase in observed IC50 of the antagonist with increasing concentrations of radioligand to be displaced [according to Eq. (4.11)].

78

Chapter | 4 Pharmacological Assay Formats: Binding

the right by a factor of 10 in the presence of an infinite concentration of allosteric antagonist. Depending on the initial concentration of radioligand, this may cause the displacement binding curve to fail to reach nsb levels. This effect is illustrated in Fig. 4.9. Therefore, in contrast to competitive antagonists, where displacement curves all take binding of the radioligand to nsb values, an allosteric ligand will displace only to a maximum value determined by the initial concentration of radioligand and the value of α for the allosteric antagonist. In fact, if a displacement curve is observed where the radioligand binding is not displaced to nsb values, this is presumptive evidence that the antagonist is operating through an allosteric mechanism. The maximum displacement of a given concentration of radioligand [A*] by an allosteric antagonist with given values of α is (see Section 4.7.5) Maximal Fractional Inhibition 5

½ATKd 1 1 (4.13) ½AT=Kd 1 1=α

1.2

0.8 0.6

IC50 5 KB

0.4 0.2 0.0 –3

–1

1 Log ([B]/KB)

3

5

FIGURE 4.8 Displacement curves for a noncompetitive antagonist. Displacement curve according to Eq. (4.9) for values of radioligand [A*]/Kd 5 0.3 (curve with lowest ordinate scale beginning at 0.25), 1, 3, 10, 30, and 100. While the ordinate scale on these curves increases with increasing [A*]/Kd values, the location parameter along the x-axis does not change.

(A)

(4.14)

(B) 1.2

Fraction max.

ð1 1 ð½AT=Kd ÞÞ ð1 1 αð½AT=Kd Þ

It can be seen from this equation that the maximum of the hyperbola defined by a given antagonist (with ordinate values expressed as the ratio of IC50 to KB) will have a maximum asymptote of 1/α. Therefore, observation of a range of IC50 values needed to block a range of radioligand concentrations can be used to estimate the value of α for a given allosteric antagonist. Fig. 4.12 shows the relationship between the IC50 for allosteric antagonism and the concentration of radioligand used in the assay, as

α = 0.01

1.2

1.0 0.8 [A*]/Kd = 100

0.6 0.4 0.3

0.2

1

3

10

30

Fraction max.

Fraction max.

1.0

where Kd is the equilibrium dissociation constant of the radioligand
receptor complex (obtained from saturation binding studies). The observed displacement for a range of allosteric antagonists for two concentrations of radioligands is shown in Fig. 4.10. The effects shown in Fig. 4.10 indicate a practical test for the detection of allosteric versus competitive antagonism in displacement binding studies. If the value of the maximal displacement varies with different concentrations of radioligand, this would suggest that an allosteric mechanism is operative. Fig. 4.11 shows the displacement of the radioactive peptide ligand 125I-MIP-1α from chemokine CCR1 receptors by nonradioactive peptide MIP-1α and by the allosteric small molecule modulator UCB35625. Clearly, the nonpeptide ligand does not reduce binding to nsb levels, indicating an allosteric mechanism for this effect [7]. Another, more rigorous, method to detect allosteric mechanisms (and one that may furnish a value of α for the antagonist) is to formally observe the relationship between the concentration of radioligand and the observed antagonism by displacement with the IC50 of the antagonist. As shown with Eq. (4.11), for a competitive antagonist, this relationship is linear (Cheng
Prusoff correction). For an allosteric antagonist, the relationship is hyperbolic and given by (see Section 4.7.6)

α = 0.1 [A*]/Kd = 100

1.0 30

0.8 10

0.6

3 1

0.4 0.3

0.2

0.0

0.0 –3

–1

1 Log ([B]/KB)

3

5

–3

–1

1 Log ([B]/KB)

3

5

FIGURE 4.9 Displacement curves according to Eq. (4.10) for an allosteric antagonist with different cooperativity factors [panel (A), α 5 0.01; panel (B), α 5 0.1]. Curves shown for varying values of radioligand ([A*]/Kd). It can be seen that the curves do not reach nsb values for high values of radioligand and that this effect occurs at lower concentrations of radioligand for antagonists of higher values of α. nsb, nonspecific binding.

4.2 BINDING THEORY AND EXPERIMENT

79

(A)

(B) 0.35

1.0

0.30

0.8

α 0.5

0.6

0.20

α

0.15

0.5

0.10

0.3

0.05 0.00 0.001 0.01

0.1 0.03

0.1

1

10

100

100

Bo

Bo

0.25

0.3

0.4 0.1

0.2 0.0 0.001 0.01

[B]/KB : log scale

0.03 0.01

0.1

1

10

100

100

[B]/KB : log scale

FIGURE 4.10 Displacement curves for allosteric antagonists with varying values of α (shown on figure). Ordinates: bound radioligand. (A) Concentration of radioligand [A*]/Kd 5 0.1. (B) Displacement of higher concentration of radioligand [A*]/Kd 5 3.

100

70 α = 0.003

60 50 [IC50]/KB

%Bo

80 60

40

0.01

30 0.03 0.1

20

40

10 0

20

0

10

20

30

40

50

60

70

[A*]/Kd

0 –11

–9 –7 Log [antagonist]

–5

FIGURE 4.11 Displacement of bound 125I-MIP-1α from chemokine CCR1 by MIP-1α (filled circles) and the allosteric ligand UCB35625 (open circles). Note how the displacement by the allosteric ligand is incomplete. CCR1, C receptors type 1. Data redrawn from I. Sabroe, M. J. Peck, B.J. Van Keulen, A. Jorritsma, G. Simmons, P.R. Clapham, A small molecule antagonist of chemokine receptors CCR1 and CCR3, J. Biol. Chem. 275 (2000) 25985
25992.

a function of α. It can be seen that unlike the linear relationship predicted by Eq. (4.11) (see Fig. 4.7B), the curves are hyperbolic in nature. This is another hallmark of allosteric versus simple competitive antagonist behavior. An allosteric ligand changes the shape of the receptor, and in so doing will necessarily alter the rate of association and dissociation of some trace ligands. This means that allosterism is tracer dependent (i.e., an allosteric change detected by one radioligand may not be detected in the same way, or even detected at all, by another). For example, Fig. 4.13 shows the displacement binding of two radioligand antagonists, [3H]-methyl-quinuclidinyl benzilate (QNB) and [3H]-atropine, on muscarinic receptors by the allosteric ligand alcuronium. It can be seen that quite different effects are observed. In the case of [3H]-methyl-QNB, the allosteric ligand displaces the

FIGURE 4.12 Relationship between the observed IC50 for allosteric antagonists and the amount of radioligand present in the assay according to Eq. (4.14). Dotted line shows relationship for a competitive antagonist.

radioligand and reduces binding to the nsb level. In the case of [3H]-atropine, the allosteric ligand actually enhances binding of the radioligand [8]. There are numerous cases of probe dependence for allosteric effects. For example, the allosteric ligand strychnine has little effect on the affinity of the agonist methylfurmethide (twofold enhanced binding) but a much greater effect on the agonist bethanechol (49-fold enhancement of binding [9]). An example of the striking variation of allosteric effects on different probes by the allosteric modulator alcuronium is shown in Table 4.1 [8,10,11].

4.2.3 Kinetic Binding Studies A more sensitive and rigorous method of detecting and quantifying allosteric effects is through observation of the kinetics of binding. In general, the kinetics of most allosteric modulators has been shown to be faster than the kinetics of binding of the tracer ligand. This is an initial assumption for this experimental approach. Under these circumstances, the

80

Chapter | 4 Pharmacological Assay Formats: Binding

250

TABLE 4.1 Differential Effects of the Allosteric Modulator Alcuronium on Various Probes for the m2 Muscarinic Receptor

[3H]Atropine binding

% Basal

200 150 100 50

[3H]methyl-QNB binding

0 –9

–8

–7

–5 –6 Log [alcuronium]

–4

–3

FIGURE 4.13 Effect of alcuronium on the binding of [3H]-methylQNB (filled circles) and [3H]-atropine (open circles) on muscarinic receptors. Ordinates are percentage of initial radioligand binding. Alcuronium decreases the binding of [3H]-methyl-QNB and increases the binding of [3H]-atropine. Data redrawn from L. Hejnova, S. Tucek, E.E. El-Fakahany, Positive and negative allosteric interactions on muscarinic receptors, Eur. J. Pharmacol. 291 (1995) 427
430.

Agonistsa

(1/α)

Arecoline

1.7

Acetylcholine

10

Bethanechol

10

Carbachol

9.5

Furmethide

8.4

Methylfurmethide

7.3

Antagonists Atropineb

0.26 b

0.54

Methyl-N-piperidinyl benzilate

c

Methyl-N-quinuclidinyl benzilate

63

Methyl-N-scopolamine

0.24

a

From [10]. From [11]. c From [8]. b

rate of dissociation of the tracer ligand (ρA*t) in the presence of the allosteric ligand is given by [12,13] ρATt 5 ρAT Ue2koff2obs Ut

(4.15)

where ρA* is the tracer ligand receptor occupancy at equilibrium and koff-obs is given by koff2obs 5

α½Bkoff2ATB =KB 1 koff2AT 1 1 α½B=KB

(4.16)

Therefore, the rate of offset of the tracer ligand in the presence of various concentrations of allosteric ligand can be used to detect allosterism (change in rates with allosteric ligand presence) and to quantify both the affinity (1/ KB) and α value for the allosteric ligand. Allosteric modulators (antagonists) will generally decrease the rate of association and/or increase the rate of dissociation of the tracer ligand. Fig. 4.14 shows the effect of the allosteric ligand 5-(N-ethyl-N-isopropyl)-amyloride (EPA) on the kinetics of binding (rate of offset) of the tracer ligand [3H]-yohimbine to α2-adrenoceptors [14]. It can be seen from this figure that EPA produces a concentrationdependent increase in the rate of offset of the tracer ligand, thereby indicating an allosteric effect on the receptor.

change in the receptor (i.e., analogous to Langmuir’s binding of molecules to an inert surface). The conclusions drawn from a system where the binding of the ligand changes the receptor are different. One such process is agonist binding, in which, due to the molecular property of efficacy, the agonist produces a change in the receptor upon binding to elicit a response. Under these circumstances, the simple schemes for binding discussed for antagonists may not apply. Specifically, if the binding of the ligand changes the receptor (produces an isomerization to another form), the system can be described as A+R

Ka

The foregoing discussion has been restricted to the simple Langmuirian system of the binding of a ligand to a receptor. The assumption is that this process produces no

χ σ

AR *

ð4:17Þ

Under these circumstances, the observed affinity of the ligand for the receptor will not be described by KA (where KA 5 1/Ka) but rather by that microaffinity modified by a term describing the avidity of the isomerization reaction. The observed affinity will be given by (see Section 4.7.7) Kobs 5

4.3 COMPLEX BINDING PHENOMENA: AGONIST AFFINITY FROM BINDING CURVES

AR

KA Uχ=σ 1 1 χ=σ

(4.18)

One target type for which the molecular mechanism of efficacy has been partly elucidated is the G-proteincoupled receptor (GPCR). It is known that activation of GPCRs leads to an interaction of the receptor with separate membrane G-proteins to cause dissociation of the Gprotein subunits and subsequent activation of effectors (see Chapter 2: How Different Tissues Process Drug

4.3 COMPLEX BINDING PHENOMENA: AGONIST AFFINITY FROM BINDING CURVES

(A)

81

(B) 0.0

0.40 0.30

–0.4 Koffobs

Log (Bt/Bo)

–0.2

–0.6 –0.8 –1.0

0.20 0.10

–1.2 –1.4

0.00 0

10

20

30

40

–2

Time (min)

–1 0 Log [EPA]

1

FIGURE 4.14 Effect of the allosteric modulator EPA on the kinetics dissociation of [3H] yohimbine from α2-adrenoceptors. (A) Receptor occupancy of [3H] yohimbine with time in the absence (filled circles) and presence (open circles) of EPA 0.03, 0.1 (filled triangles), 0.3 (open squares), 1 (filled squares), and 3 mM (open triangles). (B) Regression of observed rate constant for offset of concentration of [3H] yohimbine in the presence of various concentrations of EPA on concentrations of EPA (abscissae in mM on a logarithmic scale). EPA, 5-(N-ethyl-N-isopropyl)-amyloride. Data redrawn from R.A. Leppick, S. Lazareno, A. Mynett, N.J. Birdsall, Characterization of the allosteric interactions between antagonists and amiloride at the human α2A-adrenergic receptor, Mol. Pharmacol. 53 (1998) 916
925.

A+R

Ka

AR + [G]

Kg

ARG

ð4:19Þ

In the absence of two-stage binding, the relative quantities of [AR] and [R] are controlled by the magnitude of Ka in the presence of ligand [A]. This, in turn, defines the affinity of the ligand for R (affinity 5 [AR]/([A] [R])). Therefore, if an outside influence alters the quantity of [AR], the observed affinity of the ligand for the receptor R will change. If a ligand predisposes the receptor to bind to G-protein, then the presence of G-protein will drive the binding reaction to the right (i.e., [AR] complex will be removed from the equilibrium defined by Ka). Under these circumstances, more [AR] complex will be produced than that governed by Ka. The observed affinity will be higher than it would be in the absence of Gprotein. Therefore, the property of the ligand that causes the formation of the ternary ligand/receptor/G-protein complex (in this case, efficacy) will cause the ligand to have a higher affinity than it would have if the receptor were present in isolation (no G-protein present). Fig. 4.15 shows the effect of adding a G-protein to a receptor system on the affinity of an agonist [15]. As shown in this figure, the muscarinic agonist oxotremorine has a receptor equilibrium dissociation constant of 6 μM in a reconstituted phospholipid vesicle devoid of G-proteins.

120 100 80 % Bo

Response). For the purposes of binding, this process can lead to an aberration in the binding reaction as perceived in experimental binding studies. Specifically, the activation of the receptor with subsequent binding of that receptor to another protein (to form a ternary complex of receptor, ligand, and G-protein) can lead to the apparent observation of a “high-affinity” site—a ghost site that has no physical counterpart but appears to be a separate binding site on the receptor. This is caused by two-stage binding reactions, represented as

60 40 20 0 –11

–7

–9

–5

–3

Log [oxotremorine] FIGURE 4.15 Effects of G-protein on the displacement of the muscarinic antagonist radioligand [3H]-L-quinuclidinyl benzylate by the agonist oxotremorine. Displacement in reconstituted phospholipid vesicles (devoid of Gprotein subunits) shown in open circles. Addition of G-protein (G0 5.9 nM βγ-subunit/3.4 nM α0-IDP subunit) shifts the displacement curve to the left (higher affinity; see filled circles) by a factor of 600. Data redrawn from V.A. Florio, P.C. Sternweis, Mechanism of muscarinic receptor action on Go in reconstituted phospholipid vesicles, J. Biol. Chem. 264 (1989) 3909
3915.

However, upon addition of G0 protein, the affinity increases by a factor of 600 (10 nM). This effect can actually be used to estimate the efficacy of an agonist (i.e., the propensity of a ligand to demonstrate high affinity in the presence of G-protein, vide infra). The observed affinity of such a ligand is given by (see Section 4.7.8) Kobs 5

KA 1 1 ½G=KG

(4.20)

where KG is the equilibrium dissociation constant of the receptor/G-protein complex. A low value for KG indicates

82

Chapter | 4 Pharmacological Assay Formats: Binding

tight binding between receptors and G-proteins (i.e., high efficacy). It can be seen that the observed affinity of the ligand will be increased (decrease in the equilibrium dissociation constant of the ligand
receptor complex) with increasing quantities of G-protein [G] and/or very efficient binding of the ligand-bound receptor to the Gprotein (low value of KG, the equilibrium dissociation constant for the ternary complex of ligand/receptor/G-protein). The effects of various concentrations of G-protein on the binding saturation curve to an agonist ligand are shown in Fig. 4.16A. It can be seen from this figure that increasing concentrations of G-protein in this system cause a progressive shift to the left of the saturation dose
response curve. Similarly, the same effect is observed in displacement experiments. Fig. 4.16B shows the effect of different concentrations of G-protein on the displacement of a radioligand by a nonradioactive agonist.

(A)

(B)

120

120

100

100

80

80 0 = [G]/[KG]

60 100 30

40

% Bo

% Bo

FIGURE 4.16 Complex binding curves for agonists in G-protein unlimited receptor systems. (A) Saturation binding curves for an agonist where there is highaffinity binding due to G-protein complexation. Numbers next to curves refer to the amount of Gprotein in the system. (B) Displacement of antagonist radioligand by same agonist in Gprotein unlimited system.

The previous discussion assumes that there is no limitation on the stoichiometry relating receptors and Gproteins. In recombinant systems, where receptors are expressed in surrogate cells (often in large quantities), it is possible that there may be limited quantities of Gprotein available for complexation with receptors. Under these circumstances, complex saturation and/or displacement curves can be observed in binding studies. Fig. 4.17A shows the effect of different submaximal effects of G-protein on the saturation binding curve to an agonist radioligand. It can be seen that clear two-phase curves can be obtained. Similarly, two-phase displacement curves also can be seen with agonist ligands displacing a radioligand in binding experiments with subsaturating quantities of G-protein (Fig. 4.17B). Fig. 4.18 shows an experimental displacement curve of the antagonist radioligand for human calcitonin receptors [125I]-AC512 by the agonist amylin in a recombinant

0 = [G]/[KG] 60 100 30

40 1 10

20

1 10

20

0

0 –5

–3

–1

1

–5

3

–3

Log ([A*]/Kd)

(A)

(B) 120

120

100

100 0

80

80 [G]/[Rtot] = 1 60

% Bo

% Bo

–1 Log ([B]/KB)

0.75 0.50

40

0.50 0.75

40

0.15

[G]/[Rtot] = 1

0

20

0.15 60

20

0

0 –5

–3

–1 Log ([A*]/Kd)

1

3

–5

–3

–1 Log ([B]/KB)

1

3

FIGURE 4.17 Complex binding curves for agonists in G-protein limited receptor systems. (A) Saturation binding curves for an agonist where the high-affinity binding due to G-protein complexation 5 100 3 Kd (i.e., Kobs 5 Kd/100). Numbers next to curves refer to ratio of G-protein to receptor. (B) Displacement of antagonist radioligand by same agonist in G-protein limited system.

1

3

100

10

80

9

83

8

60

7

40

6 pKI

% Bo

4.3 COMPLEX BINDING PHENOMENA: AGONIST AFFINITY FROM BINDING CURVES

20

5 4

0 –12

–11

–10

–9

–8

–7

–6

–5

3 2

Log [amylin] 125

FIGURE 4.18 Displacement of antagonist radioligand I-AC512 by the agonist amylin. Ordinates: percentage of initial binding value for AC512. Abscissae: logarithms of molar concentrations of rat amylin. Open circles are data points, solid line fit to two-site model for binding. Dotted line indicates a single phase displacement binding curve with a slope of unity. Data redrawn from W.-J. Chen, S. Armour, J. Way, G. Chen, C. Watson, P. Irving, et al., Expression cloning and receptor pharmacology of human calcitonin receptors from MCF-7 cells and their relationship to amylin receptors, Mol. Pharmacol. 52 (1997) 1164
1175.

system where the number of receptors exceeds the amount of G-protein available for complexation to the ternary complex state. It can be seen that the displacement curve has two distinct phases: a high-affinity (presumably due to coupling to G-protein) binding process followed by a lower affinity binding (no benefit of Gprotein coupling). While high-affinity binding due to ternary complex formation (ligand binding to the receptor followed by binding to a G-protein) can be observed in isolated systems where the ternary complex can accumulate and be quantified, this effect is canceled in systems where the ternary complex is not allowed to accumulate. Specifically, in the presence of high concentrations of GTP (or a chemically stable analog of GTP such as GTPγS), the formation of the ternary complex [ARG] is followed immediately by hydrolysis of GTP and the Gprotein and dissociation of the G-protein into α- and γβ-subunits (see Chapter 2: How Different Tissues Process Drug Response for further details). This causes subsequent dissolution of the ternary complex. Under these conditions, the G-protein complex does not accumulate, and the coupling reaction promoted by agonists is essentially nullified (with respect to the observable radioactive species in the binding reaction). When this occurs, the high-affinity state is not observed in the binding experiment. This has a practical consequence in binding experiments. In broken-cell preparations for binding, the concentration of GTP can be depleted and thus the twostage binding reaction is observed (i.e., the ternary complex accumulates). However, in whole-cell experiments, the intracellular concentration of GTP is high and the ternary complex [ARG] species does not accumulate. Under these circumstances, the high-affinity binding of

1 0

1

2

3

4

5

6

FIGURE 4.19 Affinity of adenosine receptor agonists in whole cells (red bars) and membranes (blue bars, high-affinity binding site). Data shown for (1) 2-phenylaminoadenosine, (2) 2-chloroadenosine, (3) 50 -N-8ethylcarboxamidoadenosine, (4) N6-cyclohexyladenosine, (5) (2)-(R)-N6-phenylisopropyladenosine, and (6) N6-cyclopentyladenosine. Data redrawn from P. Gerwins, C. Nordstedt, B.B. Fredholm, Characterization of adenosine A1 receptors in intact DDT1 MF-2 smooth muscle cells, Mol. Pharmacol. 38 (1990) 660
666.

agonists is not observed, only the so-called low-affinity state of agonist binding to the receptor. Fig. 4.19 shows the binding (by displacement experiments) of a series of adenosine receptor agonists to a broken-cell membrane preparation (where high-affinity binding can be observed) and the same agonists in a whole-cell preparation (where the results of G-protein coupling are not observed) [16]. It can be seen from this figure that a phase shift for the affinity of the agonists under these two binding experiment conditions is observed. The broken-cell preparation reveals the effects of the ability of the agonists to promote G-protein coupling of the receptor. This latter property, in effect, is the efficacy of the agonist. Thus, ligands that have a high observed affinity in broken-cell systems often have a high efficacy. A measure of this efficacy can be obtained by observing the magnitude of the phase shift of the affinities measured in broken-cell and whole-cell systems. A more controlled experiment to measure the ability of agonists to induce the high-affinity state, in effect a measure of efficacy, can be done in broken-cell preparations in the presence and absence of saturating cNoncentrations of GTP (or GTPγS). Thus, the ratio of the affinity in the absence and presence of GTP (ratio of the high-affinity and low-affinity states) yields an estimate of the efficacy of the agonist. This type of experiment is termed the “GTP shift” after the shift to the right of the displacement curve for agonist ligands after cancellation of G-protein coupling. Fig. 4.20 shows the effects of saturating concentrations of GTPγS on the affinity of β-adrenoceptor agonists in turkey erythrocytes [17]. As can be seen from this figure, a correlation of the magnitude of GTP shifts for a series of agonists and their

84

Chapter | 4 Pharmacological Assay Formats: Binding

KL/KH

intrinsic activities as measured in functional studies (a more direct measure of agonist efficacy; see Chapter 5: Agonists: The Measurement of Affinity and Efficacy in Functional Assays). The GTP shift experiment is a method to estimate the efficacy of an agonist in binding studies. The previous discussions indicate how binding experiments can be useful in characterizing and quantifying the activity of drugs (provided the effects are detectable as changes in ligand affinity). As for any experimental procedure, there are certain prerequisite conditions that must be attained for the correct application of this technique to the study of drugs and receptors. A short list of required and optimal experimental conditions for successful binding experiments is given in Table 4.2. Some special experimental procedures for determining equilibrium conditions involve the adjustment of biological material (i.e., membrane or cells) for maximal signal-to-noise ratios and/or temporal approach to equilibrium. These are outlined in the material following.

180 160 140 120 100 80 60 40 20 0 0.0

0.2

0.4

0.6 0.8 Intrinsic activity

1.0

1.2

FIGURE 4.20 Correlation of the GTP shift for β-adrenoceptor agonists in turkey erythrocytes (ordinates) and intrinsic activity of the agonists in functional studies (abscissae). Data redrawn from R.J. Lefkowitz, M.G. Caron, T. Michel, J.M. Stadel, Mechanisms of hormone-effector coupling: the β-adrenergic receptor and adenylate cyclase, Fed. Proc. 41 (1982) 2664
2670.

4.4 EXPERIMENTAL PREREQUISITES FOR CORRECT APPLICATION OF BINDING TECHNIQUES 4.4.1 The Effect of Protein Concentration on Binding Curves In the quest for optimal conditions for binding experiments, there are two mutually exclusive factors with regard to the amount of receptor used for the binding reaction. On the one hand, increasing receptor (Bmax) also increases the signal strength and usually the signal-tonoise ratio. This is a useful variable to manipulate. On the other hand, a very important prerequisite to the use of the Langmuirian type kinetics for binding curves is that the binding reaction does not change the concentration of tracer ligand being bound. If this is violated (i.e., if the binding is high enough to deplete the ligand), then distortion of the binding curves will result. The amount of tracer ligand
receptor complex as a function of the amount of receptor protein present is given as (see Section 4.7.9) ( 1   ½ATR 5 AT 1 Kd 1 Bmax 2 ) (4.21) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ð½AT 1Kd 1Bmax Þ2 2 4½AT Bmax where the radioligand
receptor complex is [A*R] and ½AT  is the total concentration of radioligand. Ideally, the amount of receptor (magnitude of Bmax) should not limit the amount of [A*R] complex formed and there should be a linear relationship between [A*R] and Bmax. However, Eq. (4.21) indicates that the amount of [A*R] complex formed for a given [A*] indeed can be limited by the amount of receptor present (magnitude of Bmax) as Bmax values exceed Kd. A graph of [A*R] for a concentration of [A*] 5 3 3 Kd as a function of Bmax is shown in

TABLE 4.2 Minimal Criteria and Optimal Conditions for Binding Experiments G

G G G G G G G

Minimal criteria and optimal conditions for binding experiments: The means of making the ligand chemically detectable (i.e., addition of radioisotope label, fluorescent probe) does not significantly alter the receptor biology of the molecule. The binding is saturable The binding is reversible and able to be displaced by other ligands There is a ligand available to determine nonspecific binding There is sufficient biological binding material to yield a good signal-to-noise ratio but not too much so as to cause depletion of the tracer ligand For optimum binding experiments, the following conditions should be met: There is a high degree of specific binding and a concomitantly low degree of nonspecific binding Agonist and antagonist tracer ligands are available The kinetics of binding are rapid The ligand used for determination of nonspecific binding has a different molecular structure from the tracer ligand

4.4 EXPERIMENTAL PREREQUISITES FOR CORRECT APPLICATION OF BINDING TECHNIQUES

Fig. 4.21. It can be seen that as Bmax increases, the relationship changes from linear to curvilinear as the receptor begins to deplete the tracer ligand. The degree of curvature varies with the initial amount of [A*] present. Lower concentrations are affected at lower Bmax values than are higher concentrations. The relationship between [AR] and Bmax for a range of concentrations of [A*] is shown in Fig. 4.22A. When Bmax levels are exceeded (beyond the linear range), saturation curves shift to the right and do not come to an observable maximal asymptotic value. The effect of excess receptor concentrations on a saturation curve is shown in Fig. 4.22B. For displacement curves, a similar error occurs with excess protein concentrations. The concentration of [A*R] in the presence of a nontracer-displacing ligand [B] as a function of Bmax is given by (see Section 4.7.10)

Bmax = 130 pM

140

[A*R] : nM

[A*R] : pM

Bmax (mol/L) x 109

3 2 1 0 0

1

2

3 [AR] nM

4

5

6

FIGURE 4.21 Effect of increasing protein concentration on the binding of a tracer ligand present at a concentration of 3 3 Kd. Ordinates: [A*R] in moles/L calculated with Eq. (4.21). Abscissae: Bmax in moles/ L 3 109. Values of Bmax greater than the vertical solid line indicate region where the relationship between Bmax and [A*R] begins to be nonlinear and where aberrations in the binding curves will be expected to occur.

Bmax = 60 nM

50

100 80 60 40

40 30 20 10

20 0 –12

4

60

120

85

0 –11

–10 –9 Log [A*]

–8

–6

–12

–11

–10 –9 Log [A*]

5

–8

100 Bmax = 130 pM

4 3

% max.

Log [A*R] : pM

–6

2

50 60 nM

1 0 1 (A)

2

3

4

5

6

0 –11

Log (Bmax) : pM

–10

–9

–8

–7

–6

Log [A*] (B)

FIGURE 4.22 Effects of excess protein on saturation curves. (A) Bound ligand for a range of concentrations of radioligand, as a function of pM of receptor (Fig. 4.21 is one example of these types of curves). The binding of the range of concentrations of radioligands are taken at two values of Bmax (shown by the dotted lines, namely, 130 pM and 60 nM) and plotted as saturation curves for both Bmax values on the top panels (note the difference in the ordinate scales). (B) The saturation curves shown on the top panels are replotted as a percentage of the maximal binding for each level of Bmax. These comparable scales allow comparison of the saturation curves and show the dextral displacement of the curves with increasing protein concentration.

86

Chapter | 4 Pharmacological Assay Formats: Binding

( 1   ½ATR 5 AT 1Kd ð11 ½B=KB Þ1Bmax 2 ) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2   2 ð½AT 1Kd1ð11½B=KB Þ1Bmax Þ 24½AT Bmax

where [B] is the competitor and k3 and k4 the respective rates of onset and offset from the receptor. As described by Motulsky and Mahan [18], the following differential equations describe the binding of the radioligand and competitor with time:

(4.22) where the concentration of the displacing ligand is [B] and KB is the equilibrium dissociation constant of the displacing ligand
receptor complex. A shift to the right of displacement curves, with a resulting error in the IC50 values, occurs with excess protein concentration (see Fig. 4.23).

4.4.2 The Importance of Equilibration Time for Equilibrium Between Two Ligands In terms of ensuring that adequate time is allowed for the attainment of equilibrium between a single ligand and receptors, the experiment shown in Fig. 4.2 is useful. However, in displacement experiments, there are two ligands (tracer and nontraceable ligand) present and they must compete for the receptor. This competition can take considerably longer than the time required for just a single ligand. This is because the free ligands can bind only to free unbound receptors (except in the case of allosteric mechanisms, vide infra). Therefore, the likelihood of a receptor being free to accept a ligand depends on the reversibility of the other ligand, and vice versa. Assuming mass action kinetics describes the binding of the radioligand [A*] and competitive antagonist [B] A*+ R

k1

ð4:23Þ

A*R

k2

d½ATR 5 ½AT½Rk1 2 ½ARk2 dt d½BR 5 ½B½Rk3 2 ½BRk4 dt

(4.25) (4.26)

The solution to the differential equations leads to an expression that describes the amount of radioligand bound to receptors with time in the presence of the competitor [18]:   k1 ½AT k4 ðΩ 2 ΨÞ ðk4 2 ΩÞ 2Ωt ðk4 2 ΨÞ 2Ψt 1 e 2 e ρATt 5 Ω2ψ ΩΨ Ω Ψ (4.27) where Ω

5

1 ½k3 ½B 1 k4 1 k1 ½AT 1 k2 2

1 ððk3 ½B1k4 2k1 ½AT2k2 Þ2 1 4k3 k1 ½AT½B1=2  and Ψ

5

1 ½k3 ½B 1 k4 1 k1 ½AT 1 k2 2

2 ððk3 ½B1k4 2k1 ½AT2k2 Þ2 1 4k3 k1 ½AT½B1=2  Fig. 4.24 shows the kinetics of binding of a radioligand in the absence and presence of a competitor with comparatively rapid binding kinetics; it can be seen that it takes longer to reach equilibrium for the radioligand in the presence of the competitor. This should be considered when designing binding experiments, i.e., measurement of radioligand kinetics to determine when the

where [A] is the radioligand and k1 and k2 the respective rates of onset and offset from the receptor. k3 k4

ð4:24Þ

BR

Fraction radioligand occupancy

B+R

% Bo

100

50

–Log Bmax

9

8

7

6

0

Radioligand alone

1 0.8 0.6 0.4

Radioligand + competitor 0.2 0 50

–3

–1

1

3

5

Log [antagonist]

FIGURE 4.23 Effect of excess protein concentration on displacement curves [as predicted by Eq. (4.22)]. As the Bmax increases (2log Bmax values shown next to curves), the displacement curves shift to the right.

100 Time (min)

150

200

FIGURE 4.24 Binding kinetics of a radioligand with: k1 5 3.0 3 105 min21/mol, k2 5 3.0 3 1023 min21, [A]/KA 5 3.0 in the absence (solid line) and presence (dotted line) of a competitor for receptor binding (k3 5 106 min21/mol, k4 5 0.03 min21, [B]/KB 5 10).

4.4 EXPERIMENTAL PREREQUISITES FOR CORRECT APPLICATION OF BINDING TECHNIQUES

(A)

87

(B) 0.6

0.6

0.4

Fract. Bo

Fract. Bo

[B]/KB = 0

10 0.2

0.4 240 min

30 min

0.2

30 0.0

0.0 –10

40

90 140 Time (min)

190

–10

240

–9

–8 –7 Log [antagonist]

–6

–5

FIGURE 4.25 Time course for equilibration of two ligands for a single receptor. (A) Time course for displacement of a radioligand present at a concentration of [A*]/Kd 5 1. Kinetic parameter for the radioligand k1 5 105 s21/mol, k2 5 0.05 s21. Equilibrium is attained within 30 min in the absence of a second ligand ([B]/KB 5 0). Addition of an antagonist (kinetic parameters 5 k1 5 106 s21/mol, k2 5 0.001 s21) at concentrations of [B]/KB 5 10 and 30, as shown in panel (A). (B) Displacement of radioligand [A*] by the antagonist B measured at 30 and 240 min. It can be seen that a tenfold error in the potency of the displacing ligand [B] is introduced into the experiment by inadequate equilibration time. O

0.06

H N

H N

N S

O

O

Ki20227

O O

FRET

experiment should be terminated and measurements taken by observation of radioligand binding alone may underestimate the time needed for attainment of equilibrium. Radioligand binding experiments are usually initiated by addition of the membrane to a premade mixture of radioactive and nonradioactive ligand. After a period of time thought adequate to achieve equilibrium (guided by experiments like that shown in Fig. 4.2), the binding reaction is halted and the amount of bound radioligand is quantified. Fig. 4.25 shows the potential hazard of using kinetics observed for a single ligand (i.e., the radioligand) as being indicative of a two-ligand system. In the absence of another ligand, Fig. 4.25A shows that the radioligand comes to equilibrium binding within 30 minutes. However, in the presence of a receptor antagonist (at two concentrations [B]/KB 5 10 and 30), a clearly biphasic receptor occupancy pattern by the radioligand can be observed, in which the radioligand binds to free receptors quickly (before occupancy by the slower acting antagonist) and then a reequilibration occurs as the radioligand and antagonist redistribute according to the rate constants for receptor occupancy of each. The equilibrium for the two ligands does not occur until .240 minutes. Fig. 4.25B shows the difference in the measured affinity of the antagonist at times of 30 and 240 minutes. Fig. 4.26 shows this effect with fluorescence resonance energy transfer (FRET) binding where the tracer and antagonist are added simultaneously and the FRET signal is monitored in real time. This figure shows that a biphasic binding curve is seen for the slowly dissociating antagonist Ki20227 and not for the rapidly dissociating antagonist sunitinib [20]. It can

N

0.04 Sunitinib H N

0.02 F N H

0

250

500

750

1000

O

H N

N

O

1250

1500

1750

Time (s) FIGURE 4.26 FRET signal for a labeled antibody for colonystimulating factor 1 receptors and small molecule kinase inhibitor tracer conjugated to a label. Antibody, tracer, and antagonist were added simultaneously and the FRET signal monitored in real time; data shown for Ki20227 (316 nM, t1/2 5 330 min) and sunitinib (316 nM, t1/2 5 1 min). FRET, fluorescence resonance energy transfer. Data redrawn from C.M. Uitdhaag, C.M. Sunnen, A.M. van Doornmalen, N. de Rouw, A. Oubrie, R. Azevedo, Multidimensional profiling of CSF1R screening hits and inhibitors: assessing cellular activity, target residence time, and selectivity in a higher throughput way. J. Biomol. Screen. 16 (2011) 1007
1017.

also be seen from these data that the times thought adequate from the observation of a single ligand to the receptor (as that shown in Fig. 4.2) may be quite inadequate compared to the time needed for two ligands to come to temporal equilibrium with the receptor. Therefore, in the case of displacement experiments utilizing more than one ligand, temporal experiments should be carried out to ensure that adequate times are allowed for complete equilibrium to be achieved for two ligands.

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Chapter | 4 Pharmacological Assay Formats: Binding

FIGURE 4.27 Allosteric models showing receptor species present in system through formation of an active receptor state (R*)—panel (A), Hall Allosteric model [20] or through formation of an active state and allowing the receptor to couple to G proteins [panel (B)] [21]. Agonist is A, allosteric modulator is B, receptor is R and G protein is G.

4.5 BINDING IN ALLOSTERIC SYSTEMS As noted earlier in this chapter, there can be dissimulations between the protein species binding ligands and those producing pharmacological response (see Fig. 4.1). While pharmacological function is the main activity monitored in drug discovery, it can sometimes also be useful to know the receptor species binding allosteric and orthosteric ligands. There are various models that have been published to represent the receptor species present in allosteric systems; one of the first is the Hall model [20,21] which represents active-state and inactive-state receptors bound to agonist [A] and allosteric modulator [B]—see Fig. 4.27A. This model can be extended to include binding of signaling protein (such as a G protein)—see Fig. 4.27B. A practical problem with extended models to account for all protein species and activation states is that they become heuristic, i.e., they require many parameters that cannot be independently verified— see Fig 3.5 for an example. However, allosteric binding models can be useful to account for unique behaviors. A minimal model to account for the protein binding species in an allosteric system is shown in Fig. 4.28. The radioligand binding species are [AR], [ARG], [ARBG], and [ABR]. The factors α, β, γ, σ denote the influence of the various ligands on the receptor (R) and signaling protein species (i.e., G protein, G) to associate. Specifically, α is the cooperativity of B imposed on binding of A, γ the cooperativity of G protein binding imposed by A (efficacy of A), σ the cooperativity of B to G protein interaction (efficacy of B), and β the dual cooperativity of G protein imposed by binding of A and B (quaternary complex formation). This model can be used to determine the effects of a modulator on the binding of an orthosteric radioligand with the following equation (derived in Section 4.7.11):

FIGURE 4.28 Simple allosteric binding model showing the relationship between (R), radioligand (A), allosteric modulator (B) and G protein (G).

½B=KB ðα½A=KA ð1 1 γβσ½G=KG ÞÞ 1 ½A=KA ð1 1 γ½G=KG Þ ρAT 5 ½B=KB ðα½A=KA ð1 1 γβσ½G=KG Þ (4.28) 1 σ½G=KG 1 1Þ 1 ½A=KA ð1 1 γ½G=KG Þ 1 ½G=KG 1 1 Binding can provide models of drug effects in receptor systems and ligand properties. For example, Fig. 4.29 shows the effect of the allosteric modulator Sch527123 (α 5 0.1, β 5 0.05, γ 5 30, σ 5 0.05) on [125I]-CXCL8 binding to CXCR1 receptors [22]; the 125I-CXCL8 binds to form the [ARG] species and the data show an incomplete blockade which the model shows to be residual [ARBG], [ABR], and [ARG] receptors species binding 125 I-CXCL8 The same allosteric parameters for Sch527123 can be used to describe the displacement of radioactive Sch527123 by nonradioactive CXCL8. In this case, a modified equation from Eq. (4.28) is used to denote the allosteric molecule as the radioligand ([B*]) and the orthosteric ligand as the nonradioactive species ([A]) (see

4.5 BINDING IN ALLOSTERIC SYSTEMS

89

Section 4.7.11). Thus, the fraction of receptor bound to the radioactive allosteric modulator (ρB) is α½A=KA ½B=KB ð1 1 βγσ½G=KG Þ 1 σ½B=KB ½G=KG 1 ½B=KB ρB 5 ½A=KA ð1 1 α½B=KB 1 γ½G=KG ð1 1 αβσ½B=KB ÞÞ 1 ½B=KB ð1 1 σ½G=KG Þ 1 ½G=KG 1 1 (4.29)

Fig. 4.30 shows experimental data indicating incomplete blockade by CXCL8 fit to the binding model [22]. The incomplete blockade is caused by the fact that CXCL8 forms a large proportion of [ARBG] which, because Sch527123 is an allosteric ligand that can bind to the receptor when CXCL8 is also bound, the [ARBG] species registers as bound radioligand and the blockade is “incomplete.” Fitting to the binding model enables a conceptual scheme for the antagonist activity of Sch527123

FIGURE 4.29 Interaction of a nonradioactive allosteric modulator Sch527123 (B) and bound radioactive 125I-CXCL8 (A) with CXCR2 receptors according to Eq. (4.28). Model shown in Fig. 4.28 used to fit data from [22]. Parameters are α 5 0.1, β 5 0.05, σ 5 0.1, γ 5 50, [A]/KA 5 1, [G]/KG 5 3, and KB 5 50 pM. In the absence of Sch527123, 97% of the receptor is in the ARG form; in the presence of Sch527123, the receptor species distribute to BR (33%), BRG (52%) with small amounts of ABR (3%), ABRG (7%), and ARG (3%). The fact that these small amounts of radioligand binding species (containing A) exist is indicated by the failure of Sch527123 to completely suppress the 125ICXCL8 signal in the assay (see red arrow).

FIGURE 4.30 Interaction of nonradioactive CXCL8 (A) and bound radioactive allosteric modulator Sch527123 (B) with CXCR2 receptors according to Eq. (4.29). Model parameters (Fig. 4.28) are α 5 0.17, β 5 0.1, σ 5 300, γ 5 1, [B]/KB 5 1, KA 5 1 nM to fit data from [22]. In the absence of CXCL8, 100% of the radioactive species is in the BRG form. CXCL8 is an agonist which promotes complexation of the receptor with G protein. In the presence of CXCL8, the radioactive species formed are ARG (31%), AR (10%), and ARBG (58%). This latter species contains radioactive Sch527123; therefore, there is a large residual radioactive signal in the assay (note arrow in red).

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Chapter | 4 Pharmacological Assay Formats: Binding

to emerge as a negative allosteric modulator (NAM) for CXCR1 effects of the agonist CXCL8. Binding studies reveal that SCH527123 decreases the binding affinity of the receptor for CXCL8 but actually promotes receptor coupling to G proteins by CXCL8. However, the resulting CXCL8 ternary complex becomes devoid of signaling properties, i.e., the resulting ternary complex is sterile from the standpoint of signaling. Binding can yield insight into allosteric ligand behaviors that may not otherwise be evident. For example, the NAM for the CXCR2 receptors SB265610 blocks the binding of the orthosteric CXCR2 agonist 125I-IL-8 [23]. However, binding studies with nonradioactive IL-8 show that IL-8 is unable to displace radioactive 3H-SB265610 (see Fig. 4.31A); this effect was shown not to be due to pseudoirreversible binding of SB265610. Experimental data have shown that in general, SB265610 does not interfere with orthosteric agonist binding but blocks response either through negative effects on agonist-receptor signaling protein coupling or simply by shutting off the ability of the agonist-bound receptor to signal. Thus, binding in a low G-protein experimental system would still allow 3HSB265610 binding to the receptor even in the presence of SB265610 (formation of the [ABR] complex); this would not change the bound radioactivity ([BR] complex)—see

Fig. 4.31B. In a high G protein system, IL-8 would have the power to create the alternative [ARG] species thus reducing the radioactivity due to bound 3H-SB265610, i.e., IL-8 will block the binding of 3H-SB265610 in a high G protein system—see Fig. 4.31. This is important in the light of the fact that SB265610 blocks the response to chemokines in functional systems. However, this is also problematic in that there is a disparity in the potency of SB265610 as a blocker of chemokine binding versus function; specifically, SB265610 is considerably more potent in reversing chemokine binding than it is in reversing chemokine function. This question can be addressed with the allosteric binding model as well. Fig. 4.32 shows the effect of SB265610 displacement of 125I-IL-8 binding in a low G protein and high G protein system. In a low G protein system (as was utilized in the binding study [23]), SB265610 readily forms the [BR] complex thereby reducing the bound radioactivity ([AR] and [ARG] complex). However, in a high G protein containing system, bound 125 I-IL-8 is in the form of a ternary complex ([ARG]) which now requires SB265610 to uncouple the receptor from the G protein to form the [BR] complex. This causes a 30
100-fold decrease in SB265610 potency as observed in binding and functional studies—see Fig. 4.32.

FIGURE 4.31 Blockade of allosteric [3H]-SB265610 binding by nonradioactive orthosteric ligand IL-8 on CXCR2 receptors. Panel (A): Experimental data from [23] showing the inability of IL-8 to affect binding of [3H]-SB265610 in a low G protein system. Nonradioactive SB265610 affects binding of [3H]-SB265610 eliminating the possibility of irreversible [3H]-SB-265610 binding. Panel B: Model shown in Fig. 4.28 used to fit data from [23]; parameters are α 5 1, β 5 1, σ 5 0.001, γ 5 10, [B]/KB 5 1, KA 5 1 nM. The agonist IL-8 promotes receptor coupling to G protein for a high-affinity binding. In the presence of low concentrations of G protein ([G]/KG 5 0.01), IL-8 is unable to affect the binding of [3H]SB265610 because there is insufficient G protein to create the high-affinity species ARG; the receptor species are AR (35%), ARG (28%), and R (36%) [panel (C)]. In the presence of SB265610, the species revert to BR (88%) and ABR (12%); the affinity of SB265610 for this conversion is high [dotted line curve panel (B)]. Panel (D): In the presence of high amounts of G protein ([G]/KG 5 1), IL-8 promotes a high level of ARG (91%) and can affect the binding of [3H]SB265610 [solid line curve in panel (B)].

4.6 CHAPTER SUMMARY AND CONCLUSIONS

91

FIGURE 4.32 Interaction of allosteric modulator SB265610 (B) receptor bound to orthosteric radioligand 125 I-IL-8 (A) with CXCR2 receptors. Model shown in Fig. 4.28 used to fit data from [23]; parameters are α 5 0.1, β 5 0.1, σ 5 0.001, γ 5 10, [A]/KA 5 1, KB 5 1 nM. The agonist IL-8 promotes receptor coupling to G protein for a high-affinity binding. In the presence of low concentrations of G protein ([G]/KG 5 0.1), the affinity of 125I-IL-8 for the receptor is low and the receptor species are AR (35%), ARG (28%), and R (36%) [panel A]. In the presence of SB265610, the species revert to BR (88%) and ABR (12%); the affinity of SB265610 for this conversion is high (dotted line curve). In the presence of high amounts of G protein ([G]/ KG 5 10), 125I-IL-8 promotes a high level of ARG (91%) [panel (B)]. SB265610 must overcome this G protein complexation to reverse radioligand binding; thus, there is fall in the observed affinity of SB265610 (solid line curve).

4.6 CHAPTER SUMMARY AND CONCLUSIONS G

G

G

G

G

G

If there is a means to detect (i.e., radioactivity, fluorescence) and differentiate between proteinbound and free ligand in solution, then binding can directly quantify the interaction between ligands and receptors. Binding experiments are done in three general modes: saturation, displacement, and kinetic binding. Saturation binding requires a traceable ligand but directly measures the interaction between a ligand and a receptor. Displacement binding can be done with any molecule and measures the interference of the molecule with a bound tracer. Displacement experiments yield an inverse sigmoidal curve for nearly all modes of antagonism. Competitive, noncompetitive, and allosteric antagonism can be discerned from the pattern of multiple displacement curves. Allosteric antagonism is characterized by the fact that it attains a maximal value. A sensitive method for the detection of allosteric effects is through studying the kinetics of binding.

G

G

G

G

G

G

Kinetic experiments are also useful to determine the time needed for attainment of equilibria and to confirm reversibility of binding. Agonists can produce complex binding profiles due to the formation of different protein species (i.e., ternary complexes with G-proteins). The extent of this phenomenon is related to the magnitude of agonist efficacy and can be used to quantify efficacy. While the signal-to-noise ratio can be improved with increasing the amount of membrane used in binding studies, too much membrane can lead to depletion of radioligand with a concomitant introduction of errors in the estimates of ligand affinity. The time to reach equilibrium for two ligands and a receptor can be much greater than that required for a single receptor and a single ligand. Allosteric binding models can be more complex because allosteric ligands may still allow binding of radioligand (if the allosteric ligand is a radioligand); thus, the presence of radioligand does not necessarily indicate ligand-free receptor. Allosteric binding models also can be extremely useful to determine mode of action of modulators and identify receptor species with varying functions.

92

Chapter | 4 Pharmacological Assay Formats: Binding

4.7 DERIVATIONS G

G

G

G

G

G

G G

G

G

Converting to equilibrium dissociation constants (i.e., Kd 5 1/Ka) leads to the following equation:

Displacement binding: competitive interaction (Section 4.7.1). Displacement binding: noncompetitive interaction (Section 4.7.2). Displacement of a radioligand by an allosteric antagonist (Section 4.7.3). Relationship between IC50 and KI for competitive antagonists (Section 4.7.4). Maximal inhibition of binding by an allosteric antagonist (Section 4.7.5). Relationship between IC50 and KI for allosteric antagonists (Section 4.7.6). Two-stage binding reactions (Section 4.7.7). Effect of G-protein coupling on observed agonist affinity (Section 4.7.8). Effect of excess receptor in binding experiments: saturation binding curve (Section 4.7.9). Effect of excess receptor in binding experiments: displacement experiments (Section 4.7.10).

4.7.1 Displacement Binding: Competitive Interaction The effect of a nonradioactive ligand [B] displacing a radioligand [A*] by a competitive interaction is shown schematically as

ð4:30Þ

where Ka and Kb are the respective ligand
receptor association constants for radioligand and nonradioactive ligand. The following equilibrium constants are defined: ½R 5

½ATR ½ATKa

½BR 5 Kb ½B½R 5

Kb ½B½ATR ½ATKa

(4.31) (4.32)

Total receptor concentration ½Rtot  5 ½R 1 ½ATR 1 ½BR (4.33) This leads to the expression for the radioactive species [A*R]/[Rtot] (denoted as ρ*): ρT 5

½ATKa ½ATKa 1 ½BKb 1 1

(4.34)

ρT 5

½ATKd ½ATKd 1 ½B=KB 1 1

(4.35)

4.7.2 Displacement Binding: Noncompetitive Interaction It is assumed that mass action defines the binding of the radioligand to the receptor and that the nonradioactive ligand precludes binding of the radioligand [A*] to receptor. There is no interaction between the radioligand and displacing ligand. Therefore, the receptor occupancy by the radioligand is defined by mass action times the fraction q of receptor not occupied by noncompetitive antagonist: ρT 5

½AT=Kd Uq ½AT=Kd 1 1

(4.36)

where Kd is the equilibrium dissociation constant of the radioligand
receptor complex. The fraction of receptor bound by the noncompetitive antagonist is given as (1 2 q). This yields the following expression for q: q 5 ð11½B=KB Þ21

(4.37)

Combining Eqs. (4.36) and (4.37) and rearranging yield the following expression for radioligand bound in the presence of a noncompetitive antagonist: ρT 5

½AT=Kd ½AT=Kd ð½B=KB 1 1Þ 1 ½B=KB 1 1

(4.38)

The concentration that reduces binding by 50% is denoted as the IC50. The following relation can be defined: ½AT=Kd 0:5½AT=Kd 5 ½AT=Kd ðIC50 =KB 1 1Þ 1 IC50 =KB 1 1 ½AT=Kd 1 1 (4.39) It can be seen that the equality defined in Eq. (4.39) is true only when IC50 5 KB (i.e., the concentration of a noncompetitive antagonist that reduces the binding of a tracer ligand by 50% is equal to the equilibrium dissociation constant of the antagonist
receptor complex).

4.7.3 Displacement of a Radioligand by an Allosteric Antagonist It is assumed that the radioligand [A*] binds to a site separate from the one binding an allosteric antagonist [B]. Both ligands have equilibrium association constants for

4.7 DERIVATIONS

93

receptor complexes of Ka and Kb, respectively. The binding of either ligand to the receptor modifies the affinity of the receptor for the other ligand by a factor α. There can be three ligand-bound receptor species, namely, [A*R], [BR], and [BA*R]:

ð4:40Þ

From this, the relationship between the IC50 and the amount of tracer ligand [A*] is defined as [2] IC50 5 KB Uð½AT=Kd 1 1Þ

If it cannot be assumed that the free concentration of binding probe molecule (in most cases a fluorescent) and/ or competing ligand does not change with receptor binding, then the calculation of Ki values from IC50 values requires a different procedure [6]. The base equation for the conversion is Ki 5

The resulting equilibrium equations are Ka 5

½ATR ½AT½R

(4.41)

Kb 5

½BR ½B½R

(4.42)

αKa 5

½ATRB ½BR½AT

(4.43)

αKb 5

½ATRB ½ATR½B

(4.44)

Solving for the radioligand-bound receptor species [A*R] and [A*RB] as a function of the total receptor species:

½I50 ½AT50 =kd 1 ½R0 =Kd 1 1

½R20 1 ½R0 ðKd 1 ½ATT Þ 2 ½RT 5 0

where [A*]T and [R]T are the total concentration of fluorescent probe and receptor, respectively. The positive root of Eq. (4.48) is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðKd 1½ATT Þ2 1 4½RT 2 Kd 2 ½ATT ½R0 5 0:5

The conservation equation for total receptor in the absence of antagonist [B] is

ðð1=α½BKb Þ 1 1Þ ðð1=α½BKb Þ 1 ð1=αKa Þ 1 ð1=α½ATKa Kb Þ 1 1Þ (4.45)

Simplifying and changing association to dissociation constants (i.e., Kd 5 1/Ka) yield (as defined by Ehlert [19a]): (4.46)

4.7.4 Relationship Between IC50 and KI for Competitive Antagonists A concentration of displacing ligand that produces a 50% decrease in ρ* is defined as the IC50. The following relation can be defined: ½AT=Kd 0:5½AT=Kd 5 ½AT=Kd 1 1 ½AT=Kd 1 IC50 =KB 1 1

(4.50)

(4.51)

½ATR 1 ½ATRB ½Rtot 

½AT=Kd ð1 1 α½B=KB Þ ρT 5 ½AT=Kd ð1 1 α½B=KB Þ 1 ½B=KB 1 1

(4.49)

where [I]50 is the free antagonist concentration at 50% inhibition, Kd is the equilibrium dissociation constant of the fluorescent probe
receptor complex, [A*]50 is the free concentration of fluorescent probe at 50% inhibition and [R]0 is the free concentration of receptor at 0% inhibition. The value for [R]0 is obtained from calculating the positive root of

ð½Rtot  5 ½R 1 ½ATR 1 ½BR 1 ½ATRBÞ yields

5

(4.48)

(4.47)

½RT 5 ½R0 1 ½ATR0

(4.52)

A value for [A*R]0 can be calculated. For the total fluorescent probe concentration, the following relation holds: ½ATT 5 ½AT0 1 ½ATR0

(4.53)

From Eq. (4.53) and obtaining [A*R]0 from Eq. (4.52), a value for [A*]0 is obtained. A value of [A*R]50 is then defined as the concentration of tracer
receptor complex present at 50% inhibition of binding ([A*R]50 5 [A*R]0/2). By analogy to Eq. (4.53), [A*]50 5 [A*R]T 2 [A*R]50 5 [A*]T 2 [A*R]0/2. The conservation equation for receptor in the presence of a concentration of antagonist that produces 50% reduction in binding is ½RT ½R50 1 ½ATR50 1 ½BR50

(4.54)

The value for free receptor at the 50% inhibition point (defined as [R]50) is given by the mass action equation for free tracer ligand concentration at 50% inhibition:

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Chapter | 4 Pharmacological Assay Formats: Binding

Kd 5

½R50 ½AT50 ½AT50

(4.55)

4.7.7 Two-Stage Binding Reactions

(4.56)

Assume that the ligand [A] binds to receptor [R] to produce a complex [AR], and by that, reaction changes the receptor from [R] to [R*]:

By analogy to Eq. (4.53), I50 5 IC50 2 ½BR50

where IC50 is the concentration of antagonist found to reduce the binding by 50% under experimental conditions. Substituting for [BR]50 from Eq. (4.54) with [A*R]50 as [A*R]0/2 from Eq. (4.55) yields Kd ½ATR50 1 ½ATR50 ½I50 5 IC50 2 ½RT 1 ½AT50

4.7.5 Maximal Inhibition of Binding by an Allosteric Antagonist From Eq. (4.41), the ratio of bound radioligand [A*] in the absence and presence of an allosteric antagonist [B], denoted by ρA*/ρA*B, is given by (4.58)

The fractional inhibition is the reciprocal, namely, ρA*/ρA*B. The maximal fractional inhibition occurs as [B]/KB-N. Under these circumstances, maximal inhibition is given by Maximal Inhibition 5

½AT=Kd 1 1 ½AT=Kd 1 1=α

AR

χ

AR *

σ

ð4:62Þ

The equilibrium equations are ½A½R ½AR

(4.63)

χ ½AR 5 σ ½ART

(4.64)

Ka 5

(4.57)

Thus the procedure begins with the determination of [R]0 [Eq. (4.51)], then [A*R]0 [Eq. (4.52)], and obtaining [A*]0 [Eq. (4.53)]. This is followed by dividing [A*R]0 by 2 to yield [A*R]50, calculating [R]50 [Eq. (4.55)] which then allows calculation of [BR]50 [Eq. (4.54)]. The I50 value then is calculated [Eq. (4.56)] Substituting for I50, [A*]50, and [R]0 into Eq. (4.49) allows calculation of the true Ki value for the antagonist.

ρATB ½AT=Kd ð1 1 α½B=KB Þ 1 ½B=KB 1 1 5 ð½AT=Kd 1 1ÞUð1 1 α½B=KB Þ ρAT

Ka

A+R

The receptor conservation equation is ½Rtot  5 ½R 1 ½AR 1 ½ART

(4.65)

Therefore, the quantity of end product [AR*] formed for various concentrations of [A] is given as ½ART ½A=KA 5 ½Rtot  ½A=KA ð1 1 χ=σÞ 1 χ=σ

(4.66)

where KA 5 1/Ka. The observed equilibrium dissociation constant (Kobs) of the complete two-stage process is given as KA Uχ=σ 1 1 χ=σ

Kobs 5

(4.67)

It can be seen that for nonzero positive values of χ/σ (binding promotes formation of R*), Kobs , KA.

4.7.8 Effect of G-Protein Coupling on Observed Agonist Affinity Receptor [R] binds to agonist [A] and goes on to form a ternary complex with G-protein [G]:

(4.59) A+R

Ka

AR + [G]

Kg

ARG

ð4:68Þ

The equilibrium equations are

4.7.6 Relationship Between IC50 and KI for Allosteric Antagonists The concentration of allosteric antagonist [B] that reduces a signal from a bound amount [A*] of radioligand by 50% is defined as the IC50: ð1 1 ½AT=Kd Þ 5 0:5 ½AT=Kd ð1 1 αIC50 =KB Þ 1 IC50 =KB 1 1

(4.60)

ð1 1 ð½AT=Kd ÞÞ ð1 1 αð½AT=Kd ÞÞ

(4.69)

Kg 5

½AR½G ½AR

(4.70)

The receptor conservation equation is ½Rtot  5 ½R 1 ½AR 1 ½ARG

(4.71)

Converting association to dissociation constants (i.e., 1/Ka 5 KA):

This equation reduces to IC50 5 KB

½A½R ½AR

Ka 5

(4.61)

½ARG ð½A=KA Þð½G=KG Þ 5 ½Rtot  ½A=KA ð1 1 ½G=KG Þ 1 1

(4.72)

4.7 DERIVATIONS

95

The observed affinity according to Eq. (4.72) is Kobs 5

KA 1 1 ð½G=KG Þ

(4.73)

4.7.9 Effect of Excess Receptor in Binding Experiments: Saturation Binding Curve The Langmuir adsorption isotherm for radioligand binding [A*] to a receptor to form a radioligand
receptor complex [A*R] can be rewritten in terms of one where it is not assumed that receptor binding produces a negligible effect on the free concentration of ligand: ½A  R 5

ð½AT  2 ½A  RÞBmax ½AT  2 ½A  R 1 Kd

(4.74)

where Bmax reflects the maximal binding (in this case, the maximal amount of radioligand
receptor complex). Under these circumstances, analogous to the derivation shown in Section 2.11.4, the concentration of radioligand bound is ½A  R2 2 ½A  RðBmax 1 ½AT  1 Kd Þ 1 ½AT Bmax 5 0 (4.75) One solution to Eq. (4.75) is ( 1 ½A  R 5 ½AT  1 Kd 1 Bmax 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ð½AT 1Kd 1Bmax Þ2 2 4½AT Bmax g (4.76)

4.7.10 Effect of Excess Receptor in Binding Experiments: Displacement Experiments The equation for the displacement of a radioligand [A*] by a nonradioactive ligand [B] can be rewritten in terms of one where binding does deplete the amount of radioligand in the medium (no change in ½Afree ): ð½AT  2 ½A  RÞBmax ½A  R 5  ½AT  2 ½A  R 1 Kd 1 ½B=KB

(4.77)

where Bmax reflects the maximal formation of radioligand
receptor complex. Under these circumstances, the concentration of radioligand bound in the presence of a nonradioactive ligand displacement is ½A  R2

2 ½A  RðBmax 1 ½AT  1 Kd ð1 1 ½B=KBÞÞ 1 ½AT Bmax 5 0: (4.78)

One solution to Eq. (4.78) is ( 1 ½AT  1 Kd ð1 1 ½B=KB Þ 1 Bmax ½A  R 5 2 ) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2   2 ð½AT 1Kd ð11½BKB Þ1Bmax Þ 2 4½AT Bmax (4.79)

4.7.11 Derivation of an Allosteric Binding Model Referring to Fig. 4.28, the following receptor species can be identified: ½ARG 5 ½R 5

½ARBG αβδ½BKb

½ARBG αβγσ½BKb ½AKa ½GKg

(4.80) (4.81)

½RG 5

½ARBG αβγσ½BKb ½AKa

(4.82)

½AR 5

½ARBG αβγσ½BKb ½GKg

(4.83)

½BR 5

½ARBG αβγσ½AKa ½GKg

(4.84)

½ABR 5

½ARBG βγσ½GKg

(4.85)

½BRG 5

½ARBG αβγ½AKa

(4.86)

The receptor conservation equation is ½Rtot  5 ½ARBG 1 ARG 1 ½½AR 1 ½ABR 1 ½RG 1 ½BRG 1 ½BR 1 ½R (4.87) Converting equilibrium association constants to equilibrium dissociation constants (i.e., Ka 5 KA), the receptor conservation equation can be reexpressed as ½Rtot  5 ½B=KBð1 1 1 σ½G=KG 1 α½A=KAð1 1 βγσ½G=KGÞÞ 1 ½A=KAð1 1 γ½G=KGÞ 1 1

(4.88)

When the radioligand is [A], then the radioligand bound species are [ARBG], [ARG], [ABR], [AR]. The fraction of receptors bound by radioligand (ρA*) is this sum divided by [Rtot] which is

96

Chapter | 4 Pharmacological Assay Formats: Binding

½B=KB ðα½A=KA ð1 1 γβσ½G=KG ÞÞ ρA 5

1 ½A=KA ð1 1 γ½G=KG Þ ½B=KB ðα½A=KA ð1 1 γβσ½G=KG Þ 1 σ½G=KG 1 1Þ

[11]

1 ½A=KA ð1 1 γ½G=KG Þ 1 ½G=KG 1 1 (4.89)

[12]

This equation calculates the effect of a nonradioactive allosteric modulator on the binding of a radioactive orthosteric ligand. If the radioactive species is the allosteric modulator ([ARBG], [ABR], [BRG], [BR]), then Eq. (4.86) can be rewritten to yield (ρB*) as

[13]

α½A=KA ½B=KB ð1 1 βγσ½G=KG Þ ρB 5

1 σ½B=KB ½G=KG 1 ½B=KB ½A=KA ð1 1 α½B=KB 1 γ½G=KG ð1 1 αβσ½B=KB ÞÞ 1 ½B=KB ð1 1 σ½G=KG Þ 1 ½G=KG 1 1 (4.90)

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