BIOCHEMICAL AND MOLECULAR TOXICOLOGY

RECEPTOR THEORY AND PRACTICE At the end of this section, you should be able to:

1. Know the fundamentals of characterizing a receptor. 2. Know the fundamentals of characterizing a drug or toxicant. 3. Be able to use small ligands properly in toxicological studies as guided by basic principles.

1 INTRODUCTION.............................................................................................................................................................. 1 1.1 SOME TERMS AND DEFINITIONS ................................................................................................................................ 1 1.2 THEORIES OF DRUG-RECEPTOR INTERACTION........................................................................................................... 2

2 RADIOACTIVITY, RADIOLIGANDS, AND BINDING ASSAYS................................................................................. 6 2.1 HOW TO SEPARATE BOUND FROM FREE ..................................................................................................................... 7 2.2 TOTAL VS. FREE CONCENTRATIONS OF LIGAND. LIGAND DEPLETION. .......................................................................... 7 2.3 RADIOACTIVITY ...................................................................................................................................................... 8 2.4 SELECTING THE RADIOLIGAND ................................................................................................................................. 9

3 CHARACTERIZATION OF A RECEPTOR USING A RADIOLIGAND ................................................................... 12 3.1 ANALYSIS OF SATURATION RADIOLIGAND BINDING DATA......................................................................................... 12

4 DETERMINATION OF KINETIC PARAMETERS ..................................................................................................... 14 4.1 DISSOCIATION BINDING DATA ................................................................................................................................ 14 4.2 ASSOCIATION BINDING DATA ................................................................................................................................. 15

5 COMPETITIVE BINDING DATA WITH ONE CLASS OF RECEPTORS ................................................................ 17 5.2 SHALLOW COMPETITIVE BINDING CURVES............................................................................................................... 20 5.3 LIGAND, PRISM AND OTHER CURVE FITTING PROGRAMS ........................................................................................... 25

6 ANTAGONISM ............................................................................................................................................................... 25 6.1 COMPETITIVE ANTAGONISM. ................................................................................................................................. 25 6.2 SCHILD REGRESSION ............................................................................................................................................. 27 6.3 NONCOMPETITIVE ANTAGONISM. ........................................................................................................................... 28

7 PARTIAL AGONISTS .................................................................................................................................................... 28 7.1 WHAT DOES A PARTIAL AGONIST LOOK LIKE?.......................................................................................................... 29 7.2 ESTIMATING THE POTENCY OF PARTIAL AGONISTS................................................................................................... 30 7.3 DECREASING RECEPTOR CONCENTRATION .............................................................................................................. 31

Receptor Theory and Practice: Page 1

1 INTRODUCTION Note: Gluttons for punishment can refer to the following for more in dpeth information. • LE Limbird. Cell surface receptors: A Short Course on Theory and Methods, Second Edition.

Kluwer Academic Publishers, 1996. • HI Yamamura et al. Methods in Neurotransmitter Receptor Analysis, Raven Press, 1990. • T Kenakin. Pharmacologic Analysis of Drug Receptor Interaction (2nd ed.) Raven Press,

1993.

1.1 Some terms and definitions • Affinity: the “tenacity” by which a drugs binds to its receptor (Discussion: the solution

of a lipid containing drug in the bilayer may be essentially irreversible: is this a high affinity event?)

• Intrinsic activity (=efficacy?): the relative maximal response caused by a drug in a tissue preparation. A full agonist causes a maximal effect equal to that of the endogenous ligand (or sometimes another reference compound if the endogenous ligand is not known); a partial agonist causes less than a maximal response.

• Intrinsic efficacy: a drug’s ability to affect a receptor and cause a biological response (hence a property of a drug). (Discussion: can a drug have “negative efficacy”?).

• Potency: broad definition: wide the ability of a drug to cause a measured biological change; narrow definition: wide the ability of a drug to cause a measured functional change.

1.1.1 The biologist’s dream: From a biological perspective, one might simplify drug, hormone, or neurotransmitter

action via the following scheme:

Drug + ReceptorDrug-Receptor

ComplexResponse(s)

Ideally, we would wish to understand, and then predict, all of the properties of how a drug can cause a response in any tissue. Unfortunately, there are many factors that make achieving this goal at best difficult, and at worst, impossible. For example, events that affect the equilibrium of the drug at the receptor (limited diffusion on a macro or micro scale, metabolism, entrapment, etc.) can cause experimental results to deviate from theory. Even more importantly, the production of a stimulus often does not have a one-to-one correspondence to the measured response. The response caused by an activated receptor can involve a variety of different mechanisms (see cartoon in Figure 1). Some receptors directly effect the response of interest (e.g., the ionotropic receptor in A). Even in this case, other factors (including allosteric modulators, cofactors, etc.) that can influence the observed response. In the case of G protein coupled receptors (B), the receptor may start a cascade of biochemical events due to actions at several effectors, and in some cases, also may be phosphorylated itself. In other cases, like a

Receptor Theory and Practice: Page 2 tyrosine kinase (C), the receptor may itself be modified (e.g., phosphorylated) in the process of catalyzing a reaction (phosphorylation in this example). Finally, nuclear receptors are examples of a class (D) that are actually translocated as part of their normal function.

A Ion

R R

ligand

A Ion

R R

ligandB

αβ γ

α

βE

2

R E1

ligand

γ

B

αβ γ

α

βE

2

R E1

ligand

γligand

nucleus

R

R

D ligand

nucleus

R

R

D ligand

E

R R

C

P

P

P

P

R R

ligand

E

R R

C

P

P

PP

PP

P

P

PP

PP

R R

1.2 Theories of Drug-Receptor Interaction There have been several major theories that have been proposed to provide a theoretical

basis for understanding, modeling, and thereby predicting, drug response. Three of the most widely known of these schemes are described as follows:

• Occupation Theory: the idea that a response emanates from a receptor only when it is occupied by an appropriate ligand (drug).

• Rate Theory: the idea that a response emanates from a receptor in proportion to the kinetic rate of onset and offset of drug binding to the receptor.

• Operational Model: a modified, semi-empirical approach that is based on occupancy theory modified to incorporate a factor relating agonist-receptor complex and response. All of these theories have specific strengths and appeal, and all have significant failings.

In general, rate theory is now considered to be the one of least utility. Conversely, since the advent of radioreceptor methods in the mid-1970’s, it is now possible to measure drug-receptor interactions directly, making occupation theory of particular interest because receptor occupation can be measured directly for the first time. As molecular tools begin to provide ways of studying several sequential molecular events, operational theory or direct multi-step models can certainly be applied.

Receptor Theory and Practice: Page 3 1.2.1 Law of mass action

Although there are a plethora of complexities that arise, it is true that the large majority of experiments [especially those using radioreceptor methods (radioligand binding methods)] are based on very simple application of the law of mass action. In the case of a drug (ligand) interacting with a homogeneous population of receptors, this relationship can be expressed:

Ligand + Receptor Ligand-Receptorkon

koff Binding occurs when ligand and receptor collide (due to diffusion) in the correct

orientation and with enough energy. The rate of association (number of binding events per unit of time) equals [Ligand]⋅[Receptor]⋅kon.

Once binding has occurred, the ligand and receptor remain bound together for a random amount of time. The rate of dissociation (number of dissociation events per unit time) equals [ligand⋅receptor]⋅koff. The probability of dissociation is the same at every instant of time. The receptor doesn't “know” how long it has been bound to the ligand. After dissociation, the ligand and receptor are the same as at they were before binding. If either the ligand or receptor are chemically modified, then the binding does not follow the law of mass action.

Equilibrium is reached when the rate of formation of new ligand-receptor complexes equals the rate at which existing ligand⋅receptor complexes dissociate. By definition, at equilibrium this means that:

[Ligand] [Receptor] k [Ligand Receptor] kon off⋅ ⋅ = ⋅ ⋅

Rearrange that equation to define the equilibrium dissociation constant KD.

[Ligand] [Receptor][Ligand Receptor]

kk

Koff

onD

⋅⋅

= =

This equation gives you a feel for what KD means. When the ligand occupies half the receptors, the concentration of unoccupied receptors equals the concentration of occupied receptors: [Receptor] = [Ligand⋅Receptor].This can only be true when KD equals [Ligand]. In other words, the KD is the concentration of ligand that, at equilibrium, will cause binding to half the receptors.

REMEMBER, the KD is the equilibrium dissociation constant, whereas the koff is the dissociation rate constant. The two measure different things and are expressed in different units.

Variable Name Units kon Association rate constant or “on” rate constant M-1min-1

koff Dissociation rate constant or “off” rate constant min-1

KD Equilibrium dissociation constant M

KA Equilibrium association constant (=1/KD) M-1

A term that is sometimes useful to pharmacologists is fractional occupancy. Based on the law of mass action, this term describes receptor occupancy at equilibrium as a function of ligand concentration. Specifically:

Receptor Theory and Practice: Page 4

Fractional occupancy [Ligand Receptor][Total Receptor]

[Ligand Receptor][Receptor] Ligand Receptor]

=⋅

=⋅

+ ⋅[

From the equation for KD derived above, it is seen that:

[Receptor] K [Ligand Receptor]D=⋅ ⋅

[ ]Ligand

One can substitute this value for [Receptor] in the denominator of the equation for fractional occupancy and after simplifying, obtain the following (Do this yourself to see if it’s correct):

Fractional occupancy [Ligand][Ligand] K D

=+

This equation assumes equilibrium. To make sense of it, think about a few different values for [Ligand]. When [Ligand]=0, the occupancy equals zero. When [Ligand] is very high (many times KD) , the fractional occupancy approaches 100%. When [Ligand]=KD, fractional occupancy is 50%.

Although termed a “law”, the law of mass action is simply a model that can be used to explain some experimental data. As noted above, this model is not useful in all situations.

• The model assumes that all receptors are equally accessible to ligands.

• The model ignores any states of partial binding. According to the model, receptors are either free or bound to ligand. It also doesn’t allow for more than one affinity state (although this can be accommodated in some ways).

• The model assumes that the ligand is not altered by binding.

• The model assumes that binding is reversible. Despite is simplicity, the law of mass action has proven to be very useful in describing

many aspects of receptor pharmacology.

Just because binding is constant over time does not mean the system is in equilibrium. Other reactions could be happening as well, especially when agonists are used. Many investigators use the term "steady state" to describe binding that has plateaued, and reserve the term "equilibrium" to describe the ideal model.

1.2.2 The theoretical basis for characterizing receptors using saturation radioligand assays Radioreceptor assays were first developed in the early 1970’s. They were based on two

very simple, but very elegant concepts. 1. If a ligand had high affinity for a macromolecular target (as had been shown by classical

pharmacological studies over many decades), it should be thermodynamically possible to measure the binding of the ligand to the receptor without the need to perform equilibrium dialysis (the only method then used) as long as one could separate the ligand-receptor complex from the free ligand.

Receptor Theory and Practice: Page 5 2. By labeling ligands with appropriate radioactive atoms, one could detect the ligand-

receptor sensitively and rapidly. (This was the key point, since chemical methods were neither sufficiently sensitive nor inexpensive for this use.) The sections that follow will discuss many of the actual considerations for such assays,

but it is first important to understand the conceptual basis for this approach. We begin with the simple law of mass action derived earlier:

Ligand + Receptor Ligand-Receptorkon

koff which led to the following equation:

[Ligand] [Receptor][Ligand Receptor]

kk

Koff

onD

⋅⋅

= =

The receptor that has radioligand bound to it is called, in lab jargon, “Bound” or “B”. The ligand that is not bound to receptor is called “Free” or “F”. Both of these are measurable experimentally, as will be discussed below. This allows us to substitute these more common terms in the previous equation.

F [Receptor]B

K D⋅

=

Since F and B are independent variables, and we wish to solve for KD, it is necessary to be able to quantify the unbound receptor. This is technically impossible at present. Yet we now that:

[Total Receptor] = [Receptor] + [Ligand-Receptor] In fact, the total amount of receptor present (i.e., the maximal number of binding sites) is

termed “Bmax” in lab jargon, and is another desired experimental parameter. Thus, with more common lab terms:

Bmax = [Receptor] + B or by rearranging:

[Receptor] = Bmax - B If we take the equation from above:

F [Receptor]B

K D⋅

=

and substitute for [Receptor], we get:

F Bmax - B)B

K D⋅

=(

We can simplify this by multiplying both sides by B and expanding the left-hand parenthetical expression to yield:

F ⋅ Bmax - F ⋅ B = KD*B

Receptor Theory and Practice: Page 6 With simple rearrangement, we get:

KD ⋅ B + FB = F ⋅ Bmax This can be factored:

B(KD + F) = F ⋅ Bmax ..... resulting in the following equation:

B =⋅F B

K + Fmax

D

This equation should make clear the experimental design. One has an independent variable (F) and a dependent variable (B), and a successful experiment should allow one to arrive at estimates of two biologically meaningful constants: KD and Bmax. Hopefully this equation will look somewhat familiar, as it is functionally identical to the Michaelis-Menten equation of enzyme kinetics.

Discussion: Why are the KD and Bmax of interest to pharmacologists, neurobiologists, molecular biologists, etc.?. What information can they convey, and in what types of experiments?

We are now ready to begin to perform studies, and then determine how we solve this equation in practice.

2 RADIOACTIVITY, RADIOLIGANDS, AND BINDING ASSAYS As we discussed, a radioligand is a radioactively labeled drug that can associate with a

receptor, transporter, enzyme, or any protein of interest. Measuring the rate and extent of binding provides information on the number of binding sites, and their affinity and pharmacological (biological) characteristics.

There are three commonly used experimental protocols:

• Saturation binding experiments. These are the experiments whose theoretical basis we just derived. In these experiments the extent of binding is measured in the presence of different concentrations of the radioligand. From an analysis of the relationship between binding and ligand concentration, we can determine the number of binding sites, Bmax, and ligand affinity, KD.

• Kinetic experiments. Saturation and competition experiments are allowed to incubate until binding has reached equilibrium. Kinetic experiments measure the time course of binding and dissociation to determine the rate constants for radioligand binding and dissociation. Together, these values also permit a calculation of the KD.

• Competitive binding experiments. Measure the binding of a single concentration of radioligand at various concentrations of an unlabeled competitor. Analyze these data to learn the affinity of the receptor for the competitor.

Receptor Theory and Practice: Page 7 2.1 How to separate Bound from Free

For any of these approaches to work, one must be able to determine how much radioligand is associated with the receptor, and how much is unbound. It is fortunate that the huge majority of target macromolecules are either insoluble (e.g., membranous) or can be made insoluble with simple biochemical tricks (e.g., using polyethylene glycol). There are three general approaches of which you should be aware.

2.1.1 Equilibrium dialysis: Technical problems prevent its use except in rare circumstances. Some issues include:

degradation or sticking of receptor or ligand; cumbersome nature of assays when large numbers of samples are needed; time to obtain equilibrium; etc. Moreover, this technique cannot be used for kinetic analysis.

2.1.2 Centrifugation: The [R] changes during pelleting and

trapped radioactivity increases non-specific binding (NSB; see later sections). One can calculate time allowable for separation based on KD and the derived rate constants; the relationship will be logarithmic. The table to the left makes the assumption that separation must be complete in ca. 0.15 t0.5 if one is to avoid losing more than 10% of DR complex, thus introducing unacceptable error. As one can see, assays with low affinity ligands introduce very specific experimental problems.

KD (nM) 0.15(t0.5)(sec) 1000 0.1 100 1. 10 10. 1 100. 0.1 1000.

2.1.3 Filtration: With sufficiently high KD, several washes allow very low NSB. Cold wash buffer (like a cold

centrifuge above) will increase further the separation time [see values for centrifugation (above)].

2.2 Total vs. free concentrations of ligand. Ligand depletion. The equations that describe the law of mass action include the variable F ([Ligand]), the

free radioligand. In many experimental situations, you can assume that only a very small fraction of the ligand ever binds to receptors. In these situations, you can assume that the free concentration of ligand is approximately equal to the concentration you added. This assumption vastly simplifies the analysis of binding experiments, and the standard analysis methods depend on this assumption. In other situations, a large fraction of the ligand binds to the receptors. This means that the concentration of ligand free in solution does not equal the concentration you added, and the discrepancy is not the same in all tubes or at all times. The free ligand concentration is depleted by binding. Many investigators use this rule of thumb. If less than 10% of the ligand binds to receptors, don't worry about ligand depletion. If more than 10% of the ligand binds, you have three choices:

• Change the experimental conditions. The simplest approach is to decrease the amount of receptor in the incubation by using less tissue or fewer cells. The problem is that this will decrease the number of radioactive counts. The way around this is to increase the

Receptor Theory and Practice: Page 8 reaction volume without changing the amount of tissue. The problem with this approach is that it requires more radioligand, which is usually very expensive.

• Measure the free concentration of ligand in every tube. This is fairly straightforward if you use centrifugation or equilibrium dialysis, but is difficult if you use vacuum filtration. One can also estimate the free concentration by subtracting the total bound from the total added.

• Use analysis techniques that adjust for the difference between the concentration of added ligand and the concentration of free ligand.

2.3 Radioactivity

2.3.1 Specific radioactivity When you buy radioligands, the packaging usually states the specific radioactivity as

Curies per millimole (Ci/mmol). Since you measure counts per minute (cpm) , the specific radioactivity is more useful when stated in terms of cpm. Often the specific radioactivity is expressed as cpm/fmol (1 fmol = 10-15 mole). To convert from Ci/mol to cpm/fmol, you need to know the efficiency of your counter. Efficiency is the fraction of the radioactive disintegration that are detected by the counter.

Radionuclides that decay with high energy can be counted more efficiently than those with low energies modes. For example, 125I can be counted at very high efficiencies, usually 70-90+% depending on the geometry of the gamma counter (e.g., if the detector doesn’t entirely surround the tube, some gamma rays miss the detector).

With 3H, the efficiency of counting is much lower (maximally ca. 60%). The low efficiency is mostly a consequence of the physics of decay, and can not be improved by better instrumentation. When a tritium atom decays, a neutron converts to a proton and the reaction shoots off an electron and neutrino. The energy released is always the same, but it is randomly partitioned between the neutrino (not detected) and an electron (that we try to detect). When the electron has sufficient energy, it will travel far enough to encounter a fluor molecule in the scintillation fluid. This fluid amplifies the signal and gives off a flash of light detected by the scintillation counter. The intensity of the flash (number of photons) is proportional to the energy of the electron. If the electron has insufficient energy, it is not captured by the fluor and is not detected. If it has low energy, it is captured but the light flash has few photons and is not detected by the instrument. Since the decay of many tritium atoms does not lead to a detectable number of photons, the efficiency of counting is less than 100%.

To convert from Ci/mmol to cpm/fmol, you need to know that 1 Ci equals 2.22 x 1012 dpm (disintegrations per minute). Another unit of radioactivity that is becoming more common is the Becquerel (Bq). 1 µCi = 27.0270 MBq (that’s mega, not milli). The same exercise applies if you receive your radioactive sample in units of Bq.

Use this equation to convert Z Ci/mmol to Y cpm/fmol when the counter has an efficiency (expressed as a fraction) equal to E.

Y cpmfmol

= Z Cimmole

2.22x10 dpmCi

10 mmolefmole

E cpmdpm

= Z 2.22 E

Y cpmfmol

= Z Cimmole

2.22 E

12 12⋅ ⋅ ⋅ ⋅

⋅ ⋅

− ⋅

Receptor Theory and Practice: Page 9 2.3.2 Calculating the concentration of the radioligand

Rather than trust your dilutions, you can accurately calculate the concentration of radioligand in a stock solution. Measure the number of counts per minute in a small volume of solution and use this equation. C is cpm counted, V is volume of the solution you counted in ml, and Y is the specific activity of the radioligand in cpm/fmol (calculated in the previous section).

Concentration in pM =

C cpmY cpm / fmolV ml

pmol / fmol0.001 liter / ml

⋅ =0 001. /C Y

V

2.3.3 Radioactive decay Radioactive decay is entirely random. A particular atom has no idea how old it is, and

can decay at any time. The probability of decay at any particular interval is the same as the probability of decay at any other interval. If you start with N0 radioactive molecules, the number remaining at time t is:

N N etK tdecay= ⋅ − ⋅

0

KDecay is the rate constant of decay expressed in units of inverse time. Each radioactive isotope has a different value of KDecay. The half-life (t½) is the time it takes for half the isotope to decay. Half-life and decay rate constant are related by this equation:

tK Kdecay decay

1 22 0 693

/ln( ) .

= =

It is this relationship that allowed us to formulate the equation presented earlier (and shown below) that uses the t0.5 rather than the less commonly seen Kdecay.

NN

= Fraction Remaining = et

0

-0.693 t

t0.5

⋅

2.4 Selecting the radioligand

2.4.1 What chemical structure: agonist vs. antagonist In general antagonists are used much more widely, in large measure because they often

have much higher affinity than available agonists. (Discussion: why is this?) Resulting technical problems (degradation or sticking of receptor or ligand; cumbersome nature of assays when large numbers of samples are need; time to obtain equilibrium; etc.) have limited use of agonists with most receptors.

2.4.2 Choice of isotope One can predict what type of radionuclide can be used successfully based on the density

of receptors in the preparation being studied. This is either known from direct experimental evidence, or can be estimated by analogy to well characterized systems. Usually, one finds densities of from 10-500 fmol receptor/mg protein in “normal neural tissue”, and from 200-3,000 fmol/mg in transfected cells, depending on the promoter used and other factors.

One can decide what radionuclides are suitable based on the estimated density of receptors and on elementary principles of detection of radioactivity. These calculations needed

Receptor Theory and Practice: Page 10 to determine feasibility for use of a radioligand in radioreceptor assays are identical to those for the use of radioisotopes in any biology problem. If you are not familiar with such calculations, please ask in class or see me before class.

The factors to consider include: •environmental and experimental background •efficiency of counting •amount of radiolabel incorporation into radioligand •specific activity of radionuclide The specific activity of the radionuclide is based solely on its half-life, and is

independent on the mode or energy of decay. The following table shows the half-lives for commonly used radioisotopes. The table also

shows the specific activity assuming that each molecule is labeled with one atom of an isotope (as is often the case with 125I and 32P). Tritiated molecules often incorporate several tritium atoms, resulting in increased the specific radioactivity of the molecule.

Radionuclide Half life Specific Activity (Ci/mmol)

Decays to: β Energy (keV)

3H 12.43 y 28.8 3He 18 125I 59.6 d

2176 125Te -

32P 14.3 d 9131 32S 1710 35S 87.4 d 1494 35Cl 167 14C 5730 y 0.062 14N 156

You can calculate radioactive decay from a date where you knew the concentration and specific radioactivity using this equation (see later sections for more detail).

Fraction Remaining = e- 0.693 t

t 0.5

⋅

It turns out that the decay of most isotopes of biological interest result in either destruction of the molecule in which the atom is contained or in significant chemical change (see table; for example 32P ⇒ 32S). Thus, rather than changing the specific radioactivity of the ligand (as is commonly - and mistakenly - done), the concentration. of radioligand is reduced.

2.4.3 The Poisson distribution The decay of a population radioactive atoms is random, and therefore subject to a

sampling error. (This sampling error has nothing to do with other experimental factors, such as the differences in efficiency of counting between samples.) For example, the radioactive atoms in a tube containing 1000 cpm of radioactivity won’t give off exactly 1000 counts in every minute. There will be more counts in some minutes and fewer in others, with the distribution of counts following a Poisson distribution.

After counting a certain number of counts in your tube, you want to know what the “real” number of counts is. Obviously, there is no way to know that. But you can calculate a range of counts that is 95% certain to contain the true average value. So long as the number of counts, C,

Receptor Theory and Practice: Page 11 is greater than about 50 you can calculate the confidence interval using this approximate equation:

( ) ( )95% Confidence Interval: C - 1.96 C to C + 1.96 C

When calculating the confidence interval, you must set C equal to the total number of counts you measured experimentally, not the number of counts per minute. Example: You placed a radioactive sample into a scintillation counter and counted for 10 min. The counter tells you that there were 225 cpm. What is the 95% confidence interval? Since you counted for 10 min, the instrument must have detected 2250 cpm. The 95% confidence interval of this number extends from 2157 to 2343. This is the confidence interval for the number of counts in 10 min, so the 95% confidence interval for the average number of cpm extends from 216 to 234. If you had attempted to calculate the confidence interval using the number 225 cpm rather than 2250 (actual counts detected), you would have calculated a wider (incorrect) interval.

The Poisson distribution explains why it is helpful to counts your samples longer when the number of counts is small. For example, this table shows the confidence interval for 100 cpm counted for various times. When you count for longer times, the confidence interval will be narrower.

Counting Time 1 minute 10 minutes 100 minutes Counts per minute (cpm) 100 100 100

Total counts 100 1,000 10,000

95% CI (in counts) 81.4 to 121.6 938 to 1,062 9,804 to 10,196

95% CI (in cpm) 81.4 to 121.6 93.8 to 106.2 98.0 to 102.0 This graph shows percent error as a function of number of counts (C). Percent error is

defined from the width of the confidence interval divided by the number of counts:

Percent Error = 1001.96 C

C⋅

⋅

Number of counts

Perc

ent E

rror

102 103 104 105 1060.1

1.0

10.0

100.0

3 CHARACTERIZATION OF A RECEPTOR USING A RADIOLIGAND

3.1 Analysis of saturation radioligand binding data Saturation radioligand binding experiments measure specific radioligand binding at

equilibrium at various concentrations of the radioligand. Analyze these data to determine

Receptor Theory and Practice: Page 12 receptor number and affinity. Because this kind of experiment used to be analyzed with Scatchard plots (more accurately attributed to Rosenthal), they are sometimes called “Scatchard experiments”.

The analyses depend on the assumption that you have allowed the incubation to proceed to equilibrium. This can take anywhere from a few minutes to many hours, depending on the ligand, receptor, temperature, and other experimental conditions. The lowest concentration of radioligand will take the longest to equilibrate. When testing equilibration time, therefore, use a low concentration of radioligand (perhaps 10-20% of the KD).

3.1.1 Nonspecific binding (NSB) In addition to binding to the receptors, radioligand also binds to other sites termed

nonspecific sites. Nonspecific binding is assessed by measuring radioligand binding in the presence of a saturating concentration of an unlabeled drug that binds to the receptor(s) of interest. The theory is that under those conditions, virtually all the receptors are occupied by the unlabeled drug, so the radioligand can only bind to nonspecific sites. Subtract the nonspecific binding at a particular concentration of radioligand from the total binding at that concentration to calculate the specific radioligand binding to receptors. [Discussion: what causes NSB and what can one do to minimize it.]

Two questions should be obvious: 1) what unlabeled drug should you use, and 2) at what concentration? In characterizing an assay system, the rule of thumb is to use a different drug than the radioligand, ideally in a different chemical class. (Once a system is well characterized, it may be acceptable to use the same drug.) You want to use enough to block virtually all the specific radioligand binding, but not so much that you cause more general physical changes to the membrane that might alter specific binding. A useful rule-of-thumb is to use the unlabeled compound at a concentration equal to 100-1000 times its KD for the receptors.

Nonspecific binding is usually linear with the concentration of radioligand (within the range it is used). Add twice as much radioligand, and you'll see twice as much nonspecific binding. The left figure shows a schematic of total and nonspecific binding. The figure on the right shows the difference between total and nonspecific binding – the specific binding.

0 2 4 6 8 10 12 14 16 18

Radioligand Added (cpm x 1000)

Am

ount

Bo u

n d

800

600

400

200

0

Total Bound

Non-Specific

0 2 4 6 8 10 12 14 16 18

Radioligand Added (cpm x 1000)

800

600

400

200

0

Specific

Experimental Calculated

If only a small fraction of the ligand binds to the receptor, then the free concentration of

ligand equals the added concentration in both the tubes used to measure total binding and the tubes used to measure nonspecific binding. In this case, you can subtract the nonspecific from the total to get specific binding. If the assumption is not valid, then the free concentration of ligand will differ in the two sets of tubes. In this case subtracting the two values makes no sense, and determining specific binding is difficult, and other strategies are needed.

Receptor Theory and Practice: Page 13 3.1.2 Analysis of data to determine Bmax and KD

As derived earlier, such experiments often are based on the One Site Binding Equation that we derived earlier ⇒ ⋅F BAgain, this analysis is based on several assumptions:

• Only a small fraction of the radioligand binds. The free concentrathe concentration you added.

• There is no cooperativity. Binding of a ligand to one binding site of another binding site.

• Your experiment has reached equilibrium.

• Binding is reversible and follows the law of mass action.

3.1.3 Scatchard plots In the days before nonlinear regression programs were wi

transformed data into a linear form, and then analyzed the data by linevariety of algebraically equivalent ways to linearize such data, includiEadie-Hofstee, Wolff, and Scatchard-Rosenthal plots. With “perfect”answers, yet each is affected more by different types of experimental eused of these methods in the pre-computer era was the Scatchard pspecific binding (B) and the Y axis is specific binding divided by free (B/F).

It is possible to estimate the Bmax and KD from a Scatchard ploKD is the negative reciprocal of the slope). This transformationexperimental error, and thus violates the assumptions of linear regression. The Bmax and KD values you determine by linear regression of Scatchard transformed data are likely to be far from their true values. You should analyze your data with nonlinear regression. Do not analyze your data with Scatchard plots.

Spe

cific

Bin

ding

Fr

ee R

adio

ligan

d

After analyzing your data with nonlinear regression, however, it is often useful (traditional?) to display data as a Scatchard plot. The human retina and visual cortex are wired to detect edges (straight lines), not rectangular hyperbolas. Scatchard plots are often shown as insets to the saturation binding curves. They are especially useful when you want to show a change in Bmax or KD.

When making a Scatchard plot, you have to choose what units yothe Y axis. Some investigators express both free ligand and specifso the ratio bound/free is a unit-less fraction. While this is easy tofraction of radioligand bound to receptors), a more rigorousexpress specific binding in sites/cell or fmol/mg protein and theconcentration in nM. While this makes the Y axis hard to intprovides correct units for the slope (which is -1/KD).

B =K + F

max

D

tion is almost identical to

does not alter the affinity

dely available, scientists ar regression. There are a ng the Lineweaver-Burk, data, all yield identical rror. The most commonly lot, where the X axis is radioligand concentration

t (Bmax is the X intercept; , however, distorts the

Specific Binding

Scatchard Plot

Bmax

slope = -1/ KD

u want to use for ic binding in cpm interpret (it is the alternative is to free radioligand erpret visually, it

Receptor Theory and Practice: Page 14

4 DETERMINATION OF KINETIC PARAMETERS In general, when a new receptor or radioligand is characterized, both equilibrium and

kinetic techniques are used. In addition, kinetic methods also are used to deal with specific mechanistic questions.

4.1 Dissociation binding data

4.1.1 How "off rate" experiments work? A dissociation binding experiment measures the “off rate” for radioligand dissociating

from the receptor. Initially ligand and receptor are allowed to bind, perhaps to equilibrium. At that point, you need to block further binding of radioligand to receptor so you can measure the rate of dissociation. There are several ways to do this:

• If the tissue is attached to a surface, you can remove the buffer containing radioligand and replace with fresh buffer without radioligand.

• You can spin the suspension and resuspend in fresh buffer.

• Add a very high concentration of an unlabeled ligand. If the concentration is high enough it will instantly bind to nearly all the unoccupied receptors and thus block binding of the radioligand.

• Dilute the incubation by a large factor, perhaps a 20-fold dilution. This will reduce the concentration of radioligand by that factor. At such a low concentration, new binding of radioligand will be negligible. For this method to be useful, you need to use a low radioligand concentration to start with.

You then measure binding at various times after that to determine how rapidly the ligand falls off the receptors.

Plateau+ span

Plateau

t1/2 Time

Y

Binding follows this equation:

Y Span e PlateaK X= • +− • u

Receptor Theory and Practice: Page 15 Variable Meaning Comment

X Time

Y Specific Binding

Plateau Binding that doesn't dissociate. Nonspecific binding.

K Dissociation rate constant often called koff.

Expressed In units of inverse time

t0.5 Half-life 0.6902/koff

An analysis of dissociation binding data assumes that the law of mass action applies to your experimental situation; dissociation binding experiments also let you test that assumption. Ask yourself these questions:

• Does all the specific binding dissociate? Is the binding truly reversible?

• Is the dissociation rate constant the same no matter how long you incubated the cells before initiating dissociation?

• Is the dissociation rate constant the same when you initiate dissociation by diluting and by adding unlabeled drug? If not, consider the possibility of cooperativity (binding sites are clustered, and binding of ligand to one binding site changes the affinity of the others).

• After dissociation, is the ligand chemically intact? Or has the ligand degraded?

4.2 Association binding data

4.2.1 How "on rate experiments" work Association binding experiments are used to determine the association rate constant. You

add radioligand and measure specific binding at various times thereafter.

Ymax

Time

Y

The graph shows you the rate at which the binding approaches equilibrium. This is

determined by four factors:

• The association rate constant, kon or k1. This is what you are trying to determine.

• The concentration of radioligand. If you use more radioligand, the system equilibrates faster.

Receptor Theory and Practice: Page 16 • The dissociation rate constant, koff or k-1. Some people are surprised to find that the

observed rate of association depends in part on the dissociation rate constant. During the incubation, radioligand is binding to the receptors and radioligand is dissociating from receptors. The system reaches equilibrium when the two rates are equal. So the observed rate of association measures how long it takes to reach equilibrium. If the radioligand dissociates quickly from the receptor, equilibrium will be reached faster.

• Temperature affects the values of kon and koff. The next section explains how to calculate kon from the rate of equilibration.

Variable Units Comment kon M-1 min-1 The association rate constant (i.e., what you

want to know).

kob min-1 The value determined by fitting an exponential association equation to your data.

koff min-1 The dissociation rate constant. See previous section.

[radioligand] M Set by the experimenter. Assumed to be constant during the experiment (a small fraction binds).

4.2.2 To determine kon if you don't know koff: 1. Perform the association binding experiment at several different concentrations of

radioligand.

2. Use nonlinear regression to find kob at each concentration.

3. Create a graph with [radioligand] on the X axis and kob on the Y axis.

4. Fit linear regression to this graph. The slope equals kon and the Y-intercept is koff.

Receptor Theory and Practice: Page 17

5 COMPETITIVE BINDING DATA WITH ONE CLASS OF RECEPTORS

5.1.1 Present and future uses of competition binding assays It is too expensive and too technically challenging to characterize receptors only by direct

radioligand binding assays. Moreover, when studying a new drug, it is usually not feasible to radiolabel the drug prior to understanding its receptor properties. For these reasons, competition assays are used most widely in receptor studies. One of the interesting new approaches in drug design and development is an elegant brute force approach called combinatorial chemistry. In this approach, one starts with several structural backbones that are felt likely to be components of a receptor pharmacophore (the structural features of a drug that cause binding and/or activation or blockade) and does chemical reactions that allow several different “pieces” to react in a way that hundreds of products may be formed. Rather than separating this gemisch and testing each component, the mixture is tested, and if the mixture is found to be “positive”, the active species are purified and studied more conventionally. Such an approach dramatically increases the throughput of tested compounds. The most commonly used test system in combinatorial chemistry is the radioreceptor assay!

5.1.2 What is a competitive binding curve? Competitive binding experiments measure the binding of a single concentration of

labeled ligand in the presence of various concentrations of unlabeled ligand. Typically, the concentration of unlabeled ligand varies over at least six orders of magnitude.

log[Unlabeled drug]

Tota

l Rad

iolig

and

Bin

ding

IC50

The top of the curve is a plateau at a value equal to radioligand binding in the absence of

the competing unlabeled drug. The bottom of the curve is a plateau equal to nonspecific binding. The concentration of unlabeled drug that produces radioligand binding half way between the upper and lower plateaus is called the IC50 (inhibitory concentration 50%) or EC50 (effective concentration 50%).

If the radioligand and competitor both bind reversibly to a single binding site, binding at equilibrium follows this equation (where Top and Bottom are the Y values at the top and bottom plateau of the curve).

( )Y Bottom

Top BottomX LogEC= +

−+ −1 10 50

Competitive binding experiments are often used in more complicated situations where this equation doesn't apply.

Receptor Theory and Practice: Page 18 5.1.3 Entering data

If you performed an experiment with triplicate samples and you are using software for analysis, let the software automatically plot the error bars. Some investigators transform their data to percent specific binding. The problem with this approach is that you need to define how many cpm equal 100% binding and how many equal 0% specific binding. Deciding on these values is usually somewhat arbitrary. It is usually better to enter the data as cpm.

5.1.4 Decisions to make before fitting the data

5.1.4.1 Weighting When analyzing the data, you need to decide whether to minimize the sum of the square

of the absolute distances of the points from the curve or to minimize the sum of the square of the relative distances. The choice depends on the source of the experimental error. Follow these guidelines:

• If the bulk of the error comes from pipetting, the standard deviation of replicate measurements will be, on average, a constant fraction of the amount of binding. In a typical experiment, for example, the highest amount of binding might be 2000 cpm with a SD of 100 cpm. The lowest binding might be 400 cpm with a SD of 20 cpm. With data like this, you should evaluate goodness-of-fit with relative distances. The details on how to do this are in the next section.

• In other experiments, there are many contributions to the scatter and the standard deviation is not related to the amount of binding. With this kind of data, you should evaluate goodness-of-fit using absolute distances, which is the default choice.

• You should only consider weighting by relative distances when you are analyzing total binding data. When analyzing specific binding (or data normalized to percent inhibition), you should evaluate goodness-of-fit using absolute distances, as there is no clear relationship between the amount of scatter and the amount of specific binding.

5.1.4.2 Constants With experience, you will notice that even with the use of the best software, data that is

non-ideal (i.e., most data) requires the experimenter to make decisions during the course of the analysis. Many times, your data will span a wide range of concentrations of unlabeled drug, and clearly define the bottom or top plateaus of the curve. If this is the case, software can estimate with accuracy the 0% and 100% values from the plateaus of the curve.

In some cases, your competition data may not define a clear bottom plateau, but you can define the plateau from other data. All drugs that bind to the same receptor should compete all specific radioligand binding and reach the same bottom plateau value. This means that you can define the 0% value (the bottom plateau of the curve) by measuring radioligand binding in the presence of a standard drug at a concentration known to block all specific binding. If you do this, make sure that you use plenty of replicates to determine this value accurately. If your definition of the lower plateau is wrong, the values for the EC50 will be wrong as well. You can also define the top plateau as binding in the absence of any competitor.

To summarize, if you have collected enough data to clearly define the entire curve, let the software fit all the variables and fit the top and bottom plateaus based on the overall shape of your data. If your data do not define a clear top or bottom plateau, you should define one or both of these values to be constants fixed to values determined from other data. Because of factors such as intra-assay drift, this often provides a challenge to the investigator. In the class exercises,

Receptor Theory and Practice: Page 19 change the top and bottom values, and observe the effects on the derived variables and the fitted curve.

5.1.5 Interpreting the results

5.1.5.1 Assumptions To interpret the results, you must make these assumptions:

• Only a small fraction of your radiolabeled ligand binds to the tissue so that the free concentration is virtually the same as the added concentration. (There are alternative methods that do not make this assumption.)

• There is no cooperativity.

• Your experiment has reached equilibrium.

• Binding is reversible and follows the law of mass action.

• You use a valid value for the KD of the radioligand (e.g., under similar conditions with a similar receptor).

5.1.5.2 IC50 The IC50 is influenced by three theoretical factors (in addition to the factors controlled,

or affected, by the experimenter :

• The affinity of the receptor for the competing drug (as reflected by its theoretical KD). If the affinity is high (i.e., small KD) the IC50 will be low.

• The concentration of the radioligand. If you choose to use a higher concentration of radioligand, it will take a larger concentration of unlabeled drug to compete for the binding.

• The affinity of the radioligand for the receptor (KD). It takes more unlabeled drug to compete for a tightly bound radioligand (small KD) than for a loosely bound radioligand (high KD).

5.1.5.3 KI

• The KI is an equilibrium dissociation constant. It is the concentration of the competing ligand that would bind to half the binding sites at equilibrium in the absence of radioligand or other competitors.

• The subscript “I” is used to indicate that the competitor inhibited radioligand binding. Thus, it is determined indirectly by measuring competition with a known radioligand.

• The Ki can be calculated once the IC50 is determined, and the competition curve has been analyzed. Commonly, we use one of the equations from Cheng and Prusoff (Biochem. Pharmacol. 22: 3099-3108, 1973). Note that there is not one Cheng-Prusoff relationship, but several. The one below is for pure competition at a single site:

K ICligand

K

i

d

=+

50

1 [ ]

Receptor Theory and Practice: Page 20 5.2 Shallow competitive binding curves

5.2.1 The slope factor or Hill slope If the labeled and unlabeled ligand compete for a single binding site, the competitive

binding curve will have a shape determined by the law of mass action. In this case, the curve will descend from 90% specific binding to 10% specific binding over an 81-fold increase in the concentration of the unlabeled drug. More simply, virtually the entire curve will cover two log units (100-fold change in concentration).

[Unlabeled drug, M]

Perc

ent S

peci

fic B

indi

ng

10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -3

0

25

50

75

10090%

10%81 Fold

To quantify the steepness of a competitive binding curve, one determines a slope factor

(also called Hill slope). A standard competitive binding curve that follows the law of mass action has a slope of -1.0. If the slope is more shallow, the slope factor will be a negative fraction (i.e. -0.85 or -0.60).

5.2.2 The Hill Equation The Hill Equation provides information about the nature of ligand-receptor interactions.

Widely used in the past, and often demanded by reviewers, its utility has been decreased by the wide availability of nonlinear regression analysis programs (like Prism, a program designed for receptor nalysis). Thus, while less useful, it is important to understand the Hill equation since it has been a standard way of describing the slope factor (also called Hill slope or Hill number). It is derived from the binding equation modified to include the situation whereby the ligand may have multiple sites for interaction “n”.

B B DK D

n

Dn=

⋅

+max [ ]

' [ ]

This can then be rearranged in several steps as follows:

B K B D B DDn n⋅ + ⋅ = ⋅' [ ] [ ]max

B K B D B DDn n⋅ = ⋅ − ⋅' [ ] [max ]

BB B

K DDn

max' [ ]

−⋅ =

Receptor Theory and Practice: Page 21

BB B

DK

n

Dmax

[ ]'−

=

To yield the “Hill Equation”.

log log[ ] log 'max

BB B

n D K D−= ⋅ −

Thus, a plot of the log [Drug] vs. the logit function on the left hand side of the equation results in a straight-line with a slope = nH. (the Hill slope or Hill number)

The slope factor describes the steepness of a curve. In most situations, there is no way to interpret the value in terms of chemistry or biology. If the slope factor is far from 1.0, then the binding does not follow the law of mass action with a single site. Often investigators transform the data to create a linear Hill plot. The slope of this plot equals the slope factor, and the intercept with an ordinate value of “0” equals the IC50.

Concentration (nM)

0

10

20

30

40

50

60

70

80

90

100

0.10.01 1.0 10 100 1000

Spec

ific

Bin

ding

(%)

B

A C

Competition Curves

Log Concentration

Hill

num

b er

2

1

0

-1

C

-2 0 2-1 1

Hill Plot

B

A

In these figures, A is a curve of normal steepness, with a Hill slope of 1. B is a curve of shallow steepness, with a Hill slope < 1. C is a steep curve with a Hill slope > 1

Explanations for shallow binding curves include:

• Experimental problems. If the serial dilution of the unlabeled drug concentrations was done incorrectly, the slope factor is not meaningful.

• Curve fitting problems. If the top and bottom plateaus are not correct, then the slope factor is not meaningful. Don't try to interpret the slope factor unless the curve has clear top and bottom plateaus.

• Negative cooperativity. You will observe a shallow binding curve if the binding sites are clustered (perhaps several binding sites per molecule) and binding of the unlabeled ligand to one site causes the remaining site(s) to bind the unlabeled ligand with lower affinity.

• Heterogeneous receptors. The receptors do not all bind the unlabeled drug with the same affinity.

• Ternary complex model with limiting availability of G protein. This is a common situation that results with heterogeneous receptors. See below.

Receptor Theory and Practice: Page 22 5.2.3 Competitive binding with two sites

Often, data does not fit one site models well, and a two site analysis may be useful. (Normal experimental variance usually does not permit discrimination of two from three high affinity sites). Some assumptions needed to do such analyses include:

• There are two distinct classes of receptors. For example, a tissue could contain a mixture of β1 and β2 adrenergic receptors.

• The unlabeled ligand has distinct affinities for the two sites.

• The labeled ligand has equal affinity for both sites.

• Binding has reached equilibrium.

• A small fraction of both labeled and unlabeled ligand bind. This means that the concentration of labeled ligand added is very close to the free concentration in all tubes.

When you look at the competitive binding curve, you will only see a biphasic curve only in the rare cases where the affinities are very different. More often you will see a shallow curve with the two components blurred together. For example, this graph shows competition for two equally abundant sites with a ten fold (one log unit) difference in EC50. If you look carefully, you can see that the curve is shallow (it takes more than two log units to go from 90% to 10% competition), but you cannot see two distinct components.

log[competitor]

Spec

ific

Bin

ding

%

-12 -11 -10 -9 -8 -7 -6 -5 -4

0

25

50

75

100

5.2.4 Comparing one- and two-site models Some software (like Prism) can simultaneously fit your data to two equations and

compare the two fits. This feature is commonly used to compare a one-site competitive binding model and a two-site competitive binding model. Since the model has an extra parameter and thus the curve has an extra inflection point, the two-site model almost always fits the data better than the one site model. And a three site model would fit even better. Before accepting the more complicated models, you need to ask whether the improvement in goodness of fit is more than you would expect by chance. Prism, for example, answers this question with an F test that yields a P value that answers this question: If the one site model were really correct, what is the chance that randomly chosen data points would fit to a two-site model this much better (or more so) than to a one-site model?

Before looking at Prism’s comparison of the two equations, you should look at both fits yourself. Sometimes the two-site fit gives results that are clearly nonsense. Here are some examples to when to disregard a two-site fit:

• The two IC50 values are almost identical.

Receptor Theory and Practice: Page 23 • One of the IC50 values is outside the range of your data.

• The fraction of the total binding represented by any of the sites is either close to 1.0, or to 0.0. In this case, most of the receptors have the same affinity, and the IC50 value for the low occurring site cannot be reliable.

• The calculated values for either the BOTTOM or TOP of your curves are far from the range of real values in your experiment.

Basic Principle: If the results don’t make sense, don’t believe them (don’t ever say “The computer said ….”). Only pay attention to the results of the comparison when both fits are reasonable.

This equation has five variables: the top and bottom plateau binding, the fraction of the receptors of the first class, and the IC50 of competition of the unlabeled ligand for both classes of receptors. If you know the KD of the labeled ligand and its concentration, you can convert the IC50 values to Ki values.

5.2.5 Agonist binding and the ternary complex If you use an antagonist radioligand and compete with an unlabeled agonist you are likely

to observe a shallow competitive binding curve. Receptors can interact with intracellular molecules in a way that alters agonist binding. One situation that has been well studied is the competition of unlabeled agonist ligand for antagonist radioligands for binding to receptors linked to G proteins in membrane preparations lacking GTP.

The ternary complex model explains some of these data. This model shows binding of an agonist (A) to receptor (R) coupled to G protein (G):

A + R + G AR + G

A + RG ARG Some features of the model:

• The left half of the picture shows the equilibrium between receptor (R) and G protein (G) in the absence of agonist (A).

• The right half shows the agonist-receptor complex binding to G to form the ternary complex (ARG).

• The top part shows the binding of agonist to receptors not linked to G.

• The bottom part shows the binding of agonist to receptors linked to G.

• Not shown is the radioligand L that binds with equal affinity to R and RG. • This model can explain the shape of agonist competition curves if the following are

true:

• Agonist binds more tightly to the receptor-G complex than to receptor alone. In other words, A binds with low affinity to R but to high affinity to RG.

Receptor Theory and Practice: Page 24 • The total concentration of G is limiting. If G is present in higher concentrations than R, this

model does not predict shallow binding curves (Neubig et al. Mol. Pharmacol. 28:475, 1985).

• GTP is omitted from the incubation. When GTP is present, as it is inside cells, the model is more complicated. Soon after the ARG complex forms, GTP binds to the G protein, and activates it. The activated G then dissociates from AR, and its α subunit dissociates from the βγ subunits. Because the ARG complex is so transient, all you see in binding experiments is low affinity AR binding.

A few investigators have fit data to equations describing the ternary complex. Most do not, for these reasons:

• The model is too complicated to fit well. There are too many parameters to fit. You need to fit all four equilibrium constants, plus the relative concentration of receptor to G.

• The model is too simple to be useful. The ternary complex model only predicts a shallow competitive binding curve when the total concentration of G is less than or equal to the total concentration of receptors. But biochemical evidence in many systems demonstrates that G is present in vastly higher concentrations than receptors. Yet these systems demonstrate shallow competitive binding curves. To resolve this discrepancy, you must make the model still more complicated (either many of the G proteins are not available to bind to receptors, or many of the receptors are not available to bind to G proteins). For a review of these issues, see RR Neubig, Faseb J. 8:939-946, 1994.

Rather than fit data to the ternary complex model, most investigators fit data to a two-site binding model. The data fit well to the two site model, even though we know it is too simple. The two-site model is shown below – it is not the same as the ternary complex model.

A + R AR

A + RG ARG

Klo

Khi

The two-site model, as its name implies, assumes that there are two distinct kinds of

receptors. In this context, one kind of receptor is coupled to G, and the other is not. This simpler model provides values for Klo, Khi, and relative numbers of the two kinds of receptors, and these can be compared after different treatments. Since we know that the two-site model is too simple (the ternary complex is a better model for many systems) you shouldn't interpret Khi and Klo strictly. Nonetheless, the ratio of the Khi and Klo is a useful empirical measure which often correlates with agonist efficacy.

This short section is hardly adequate to explain the complexities of the ternary complex model, but should be sufficient to keep you from making common mistakes in interpretation.

5.3 Ligand, Prism and other curve fitting programs The first widely used program in the field was “Ligand”, a freely available, special-

purpose nonlinear regression program for analyzing equilibrium radioligand binding data written

Receptor Theory and Practice: Page 25 by Munson, Rodbard, and collaborators (Methods in Enzymology 92:543, 1983). Today, there are many other alternatives to Ligand (for many different computer platforms) that may be easier to use or more facile with certain types of analyses.

6 ANTAGONISM Antagonism is the process of inhibiting or preventing an agonist-induced receptor

response. Agents that produce this effect are called antagonists. The availability of selective antagonists has provided an important mechanistic tool. On the one hand, the classification of receptor subtypes was accomplished largely because of the availability of selective antagonists. Even with the proliferation of receptors resulting from molecular cloning, selective antagonists are an essential element of understanding the functional role of such receptors. Based on the kinetics of interaction of the antagonist with the receptor, antagonism is classified as competitive and non-competitive.

6.1 Competitive Antagonism. Competitive antagonism is based on the principle that an agonist or antagonist can bind

to the same recognition site(s) on the receptor, and when both agonist and antagonist are present concomitantly, they can compete for such sites. Reversible competitive antagonism occurs either when the binding of the antagonist can be eliminated by increasing the concentration of agonist, or when the antagonist dissociates as the free concentration decreases.

The ability of an antagonist to influence the receptor occupancy by an agonist (and therefore to elicit a response) is determined by the relative affinity of the antagonist for the receptor, and by the relative concentration of the antagonist. Antagonism can be viewed as two independent equilibria occurring for agonist (A) and antagonist (B):

A + R A-R

B + R B-R Based on the concentrations and equilibrium dissociation constants of both agonist and

antagonist, Gaddum derived the equation that predicts the effects of antagonists on the fraction of receptors occupied by the agonist:

[ ][ ]

[ ']

[ '] ( [ ])

A RR

A

A K BK

tA

B

⋅=

+ +1

Where [A] is the concentration of agonist; [A'], the concentration of agonist that in the presence of the antagonist, [B] is necessary to produce the same response as [A]; RT is the total receptor population, and KA and KB, the equilibrium dissociation constants of the agonist and antagonist, respectively.

The effects of competitive antagonists on agonist induced responses is studied by determining the dose-response relationships for the agonists in the absence or in the presence of increasing concentrations of antagonists. This system assumes that the response of the test

Receptor Theory and Practice: Page 26 system depends only on the agonist occupancy, and therefore, that equal agonist occupancies in the absence or in the presence of antagonist will produce equal responses. The multiple increase in the agonist concentration required to achieve a given response is called the dose ratio. This can be seen in the following Figure.

Log Agonist Concentration (nM)

0

0.2

0.4

0.6

0.8

1.0

0.10.01 1.0 10 100 1000

Resp

onse

(% o

f ma x

imal

)Control (agonist

with no antagonist)

+ Increasing concentrationsof antagonist B

Raw Data

The effects of two concentrations of antagonist on agonist response. For clarity, this figure shows the effects of two concentration of antagonist on agonist response. This analysis requires a minimum of five concentrations for an accurate analysis.

Schild simplified Gaddum’s equation to the following:

[ '][ ]

[ ]AA

BKB

= +1

As you can see this formulation permits the assessment of KB without making assumptions regarding the relationship between fractional occupancy and the ultimate response.

An empirical scale for antagonist potencies based on dose ratios termed the pAX scale was introduced by Schild in 1949.

log (dose ratio - 1) = log KB - pAX pAx is defined as the negative logarithm of the molar concentration of antagonist that

produces an equiactive dose ratio of magnitude x. For example, the pA10 is the concentration of antagonist that produces a 10-fold shift to the right in the agonist dose response curve. Notice that the pAX is an empirical value with no theoretical relevance to molecular mechanisms.

6.2 Schild Regression The dose ratios calculated for a competitive antagonist can be used in a quantitative

model of competitive antagonism derived by Arunlakshana and Schild.

log (x - 1) = log [B] - log KB

Receptor Theory and Practice: Page 27

0

1

2

3

4

5

21 3 4 5 6

log

(dos

e ra

tio-1

)

-log [B]=pA x

pA = -log KDB2

pA = -log KDB2Schild Plot

0

1

2

3

4

-5-6 -4 -3 -2

log

(dos

e ra

tio-1

)

log [antagonist]

KDB

Schild Regression

Notice that the dose ratio is referred to as X in the Schild regression. This equation is a powerful tool in the drug and receptor classification process. If a

regression of log (x-1) vs. log [B] is linear and has a slope of unity, it indicates that the antagonism is competitive. This relationship is independent of the characteristics of the agonist, and should be the same for all agonists that act on the same population of receptors.

Notice that the dose ratio depends only on the concentration and on the equilibrium dissociation constant of the antagonist. The dose ratio is independent of the magnitude of the response chosen as a reference point for the measurements and of the equilibrium dissociation constant of the agonist.

The intercept on the x axis of a plot of log (x -1) versus log [B] yields directly KB, the equilibrium dissociation constant for the antagonist. The term pKB is the value of -log (x-1) when the dose ratio (x) = 2. This quantity therefore corresponds to pA2, the empirical parameter defined as the negative logarithm of the molar concentration of the antagonist that produce a dose ratio = 2. Therefore, the pKB always correspond to the pA2. The opposite is only true, however, when the Schild regression has a slope of unity.

An important practical aspect of experimental Schild regressions is that sufficient equilibration time should be allowed to achieve a thermodynamic equilibrium between the antagonist and the receptor before the challenge with the agonist.

Schild regressions with slopes different from unity can be originated by:

The antagonist is non competitive. i.e. interacts with the receptor at non overlapping sites with agonist binding sites.

A drug disposition mechanism or other nonequilibrium steady state obscures the competitive nature of the antagonism

The competitive antagonism of a heterogeneous receptor population subserving the same response is observed; or

Multiple drug properties are expressed in the concentrations used to make the measurements (such as effects on agonist degradation or uptake).

Receptor Theory and Practice: Page 28 6.3 Noncompetitive Antagonism.

In this type of antagonism, the blockade of agonist response is produced by the interaction of the antagonist with binding sites intimately associated with the receptor, but distinct from the agonist binding site. It is assumed that the binding of the antagonist to this site precludes the activation of the receptor by the agonist. This could be conceptualized as if the noncompetitive antagonist removes the receptor, or eliminates the system's capacity to respond. Therefore, the maximal capacity of the receptor to respond is decreased due to a progressive decline in agonist fractional receptor occupancy. The agonist, however, operates normally at receptor-effector units not influenced by the antagonist. The affinity of the remaining receptors for the agonist, and the potency of the agonist are not altered. This type of antagonism is not surmountable by increasing the concentration of agonist. Note:

In a situation where "spare receptors" are present, the presence of a noncompetitive antagonist will first shift to the right the dose response of the agonist without a change in the maximal response (pseudo competitive antagonism). Increasing the concentration of the non competitive antagonist will eventually reduce the number of available receptors so that the maximal response can not be obtained.

7 PARTIAL AGONISTS In the classical receptor theory developed by Clark, it was assumed that the effect of a

drug is proportional to the fraction of receptors occupied by the drug and that maximal effect results when all receptors are occupied. This assumption may only be true for a limited number of cases. To account for the discrepancies, Ariëns introduced the term intrinsic activity to describe the relationship between the effect elicited by a drug and the concentration of drug-receptor complexes.

E= α[DR]

Where E is the effect; α is a proportionality constant or the intrinsic activity of the drug; and [DR] the concentration of drug-receptor complex. This relationship addressed the observation that some drugs do not elicit a maximal response even at apparently maximal receptor occupancies. These drugs are called partial agonists.

7.1 What does a partial agonist look like? The figure below on the left shows the concentration effect curve for a full agonist and a

partial agonist, the binding affinity of the receptor for both drugs is the same. Notice the shift to the right of the concentration effect curve for the partial agonist relative to the full agonist. On the right are plotted the occupancy response curves for the data on the left. Note that the partial agonist in spite of achieving maximal occupancy, elicit only 45 % of the maximal response.

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Stephenson introduced the concept of efficacy as a property of the drug to explain the

nonlinear relationships between occupancy and response, and for the ability of a group of drugs (partial agonists) to produce equal responses while occupying different proportions of the receptor population. According to Stephenson, the response, R, of a tissue is a function of the stimulus, S, given to the tissue: R = f (S). S was defined as the product of the efficacy of an agonist, e , multiplied by the fraction of receptors occupied by the agonist.

R f S f eRAR tot

= = ⋅ ⋅( )[ ][ ]

The differences in the effects of a drug in different tissues are a reflection of the contribution of properties of the drug, properties of the receptor, and properties of a tissue in terms of receptor density, and the coupling of receptor occupancy to the ultimate response.

Partial agonists by virtue of the occupation of a large number of receptors without eliciting a response, competitively block the effects of agonists of higher intrinsic efficacies or full agonists. Since both effects of the partial agonist are the result of the interaction with the same site, antagonism from partial agonists should be observed at the same concentrations that produce the agonist effect. Therefore, the agonistic response to a drug that occurs at concentrations different than those that produce antagonism, cannot be ascribed to interaction of the drug with a single receptor.

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The figure above shows a series of concentration-effect curves for a partial agonist in the

absence and in the presence of different concentrations of a full agonist. The partial agonist produced 20 % of the maximal response. Note that the effects of the partial agonist in the presence of low concentrations of full agonist (producing a response lower than the maximal response of the partial agonist i.e. 20 %) are additive to those of the full agonist up to the maximal response of the partial agonist (20%). In contrast, the partial agonist in the presence of concentration of the full agonist that produce responses higher than those of the partial agonist (more than 20%) has antagonistic effects, decreasing the response of the full agonist to the level of the maximal response of the partial agonist (20%). Note also that the agonistic and antagonistic effects of the partial agonist are produced at the same concentration range at low concentrations of the full agonist. The increase in concentrations of the partial agonist required to antagonize the effects of increasingly higher concentrations of full agonist are indicative of the competitive nature of the effect.

7.2 Estimating the potency of partial agonists Several approaches have been used to estimate the potency (equilibrium dissociation

constants) of partial agonists (see Chapter 2 in Limbird).

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The figure above shows the effects of various concentrations of a partial agonist on the

concentration response curves for a full agonist. Note the dual effects of the partial agonist in this plot. The increase in the response produced by the partial agonist in the absence of the full agonist, and the shifts to the right in the concentration effect curves for the full agonist which can provide dose-ratios for Schild analysis as indicated in the previous section.

This approach nonetheless has some caveats, first, which region of the concentration effect curve of the full agonist should be used to measure the dose ratios? It has been observed that the dose ratio taken from the half-maximal concentration of the full agonist dose response curves produce the most unambiguous approach; second, the intrinsic efficacy of the partial agonist could introduce an error in the estimation of the KB. This error is proportional to the magnitude of the intrinsic efficacy of the partial agonist and to the efficiency of the coupling of the tissue stimulus-response mechanism. In general, for low efficacy partial agonist this method produces accurate estimations of the KB. A most unambiguous method to determine the KB for a partial agonist consist in to irreversibly block or inactivate a sufficient fraction of the receptor population so that the partial agonist no longer produces a response but that still permit to observe the effects of a full agonist. The interaction of the partial agonist with its receptor can then be studied as a competitive antagonist of the full agonist as indicated in the previous section.

7.3 Decreasing receptor concentration Several approaches have been used successfully to decrease the number of the receptor

population. The primary criterion for the validity of any of these approaches is that the treatment will eliminate a fraction of the receptor population, yet should not affect coupling of the remaining receptors to any of the response system (e.g., an equal number of agonist-receptor

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complexes elicit equal responses before and after the irreversible receptor blockade). Some possible approaches include: • Receptor-selective irreversible antagonists (unfortunately such drugs may not be available

for most receptor subtypes). • Non-selective irreversible receptor antagonists. (Question: how can you use what you

have learned about receptror binding to minimize the artifacts induced by non-selective irreversible antagonists?)

• Non-ligand inactivators that cross-linking the ligand to the receptor. For example, a bifunctional reagent, such as succinimidyl suberimidate may achieve a specific irreversible blockade.

• Non-selective biochemical inactivation. Reagents such as N-ethylmaleimide or other residue-directed reagents, or limited proteolysis, have sometimes been used.

• Immunotitration. An excellent method if a selective deactivating antibody is available. • Molecular control. With a cloned, expressed receptor, one can often manipulate (or at least

select) for varying levels of receptor expression. The primary criterion for the validity of any of these approaches is that the treatment will

only eliminate a fraction of the receptor population and must not affect in any way the receptor coupling of the remaining receptors, or the response system, so that equal number of agonist-receptor complexes elicit equal responses before and after the irreversible receptor blockade.