The GraphPad Guide to Analyzing Radioligand Binding Data
Dr. Harvey Motulsky, President GraphPad Software
Copyright © 199596 by GraphPad Software, Inc. All rights reserved.
Preface
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 accessibility for various drugs. Radioligand binding experiments
are easy to perform, and provide useful data in many fields. But
many find it difficult to analyze radioligand binding data. That's
why I wrote this booklet.
If you want more detailed information beyond the
scope of this booklet, consult one of these general references:
 LE Limbird, Cell surface receptors: A short
course on theory and methods, Kluwer Academic Publishers,
second edition, 1996.
 HI Yamamura, et. al. Methods in neurotransmitter
receptor analysis, Raven Press, 1990.
 T Kenakin, Pharmacologic analysis of drugreceptor
interaction, second edition, Raven Press, 1993.
You have permission to duplicate this booklet for
use in teaching and research, provided that you duplicate the
entire document (including the copyright page) and don't charge
for copies. You may also download this document from the internet
(http://www.graphpad.com) or order additional copies from GraphPad
Software.
I thank Drs. Lee Limbird and Richard Neubig for making
very helpful comments!
The law of mass action
Most analyses of radioligand binding experiments
are based on a simple model, called the law of mass action:
The model is based on these simple ideas:
At equilibrium, ligandreceptor complexes form at
the same rate that they dissociate:
[Ligand][Receptor]k_{on} = [LigandReceptor]k_{off
}
Rearrange to define the equilibrium dissociation
constant K_{d}.
The K_{d}, expressed in units of moles/liter
or molar, is the concentration of ligand which occupies half of
the receptors at equilibrium. A small K_{d} means that
the receptor has a high affinity for the ligand. A large K_{d}
means that the receptor has a low affinity for the ligand. Don't
mix up K_{d}, the equilibrium dissociation constant, with
k_{off}, the dissociation rate constant. They are not
the same, and aren't even expressed in the same units.
The law of mass action predicts the fractional receptor
occupancy at equilibrium as a function of ligand concentration.
Fractional occupancy is the fraction of all receptors that are
bound to ligand.
A bit of algebra creates a useful equation. Multiply
both numerator and denominator by [Ligand] and divide both by
[LigandReceptor]. Then substitute the definition of K_{d}.
When [Ligand]=0, the occupancy equals zero. When
[Ligand] is very high (many times K_{d}) , the fractional
occupancy approaches 1.00. When [Ligand]=K_{d}, fractional
occupancy is 0.50. The approach to saturation as [ligand] increases
is slower than many people appreciate. When the ligand concentration
equals four times its K_{d}, it will only occupy 80% of
the receptors at equilibrium. The occupancy rises to 90% when
the ligand concentration equals 9 times the K_{d}. It
takes a concentration equal to 99 times the K_{d} to occupy
99% of the receptors at equilibrium.
Although termed a "law", the law of mass
action is simply a model based on these assumptions:
If these assumptions are not met, you have two choices.
One choice is to develop a more complicated model, beyond the
scope of this booklet. The other choice is to analyze your data
in the usual way, but interpret the results as an empirical description
of the data without attributing rigorous thermodynamic meaning
to the K_{d} values and rate constants.
Saturation binding experiments
Saturation binding experiments measure specific binding
at equilibrium at various concentrations (often 612) of the radioligand
to determine receptor number and affinity. Because this kind of
experiment can be graphed as a Scatchard plot (more accurately
attributed to Rosenthal), they are sometimes called "Scatchard
experiments".
The analyses depend on the assumption that the incubation
has reached equilibrium. This can take anywhere from a few minutes
to many hours, depending on the ligand, receptor, temperature,
and other experimental conditions. Since lower concentrations
of radioligand take longer to equilibrate, use a low concentration
of radioligand (perhaps 1020% of the Kd) when measuring how long
it takes the incubation to reach equilibrium.
In addition to binding to the receptors of physiological
interest, radioligands bind to nonreceptor sites. When performing
radioligand binding experiments, you need to measure both total
and nonspecific binding, and calculate specific (receptor) binding
as the difference.
Assess nonspecific binding by measuring radioligand
binding in the presence of a concentration of an unlabeled compound
that binds to essentially all the receptors. Since all the receptors
are occupied by the unlabeled drug, the radioligand only binds
nonspecifically.
Which unlabeled drug should you use? The obvious
answer is to use the same compound as the radioligand, but unlabeled.
In many cases, this is necessary as no other drug is known to
bind to the receptors. But most investigators avoid using the
same compound as the hot and cold ligand for routine work, and
prefer to define nonspecific binding with a standard drug that
is known to bind to that particular receptor.
What concentration of unlabeled drug should you use?
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. If
you are studying a wellcharacterized receptor, a useful ruleofthumb
is to use the unlabeled compound at a concentration equal to 100
times its K_{d} for the receptors.
Ideally, you should get the same results defining
nonspecific binding with a range of concentrations of several
drugs.
Nonspecific binding is generally proportional to
the concentration of radioligand (within the range it is used).
The left figure shows total and nonspecific binding. The dotted
curve shows the difference between total and nonspecific binding
 the specific binding.
The right panel shows the specific binding again on a graph with
a logarithmic X axis. Notice that the saturation binding curve
plotted on a log axis looks like the familiar sigmoidal doseresponse
curve. The dotted curves in the two panels represent the same
range of radioligand concentrations. The solid portion of the
curve on the right shows binding at higher radioligand concentrations.
These high concentrations are rarely used because radioligands
are expensive and nonspecific binding would be too high a fraction
of total binding.
Equilibrium specific binding at a particular radioligand concentration
equals fractional occupancy times the total receptor number (Bmax):
This equation describes a rectangular hyperbola or a binding isotherm.
[L] is the concentration of free radioligand, the value plotted
on the X axis. Bmax is the total number of receptors expressed
in the same units as the Y values (i.e., cpm, sites/cell or fmol/mg
protein) and Kd is the equilibrium dissociation constant (expressed
in the same units as [L], usually nM). Typical values might be
a Bmax of 101000 fmol binding sites per milligram of protein
and a Kd between 10 pM and 100 nM.
To determine the B_{max} and K_{d},
fit data to the equation using nonlinear regression.
This analysis is based on these assumptions:
 Binding follows the law of mass action and has
equilibrated.
 There is only one population of receptors.
Scatchard plots
Before nonlinear regression programs were widely
available, scientists transformed data to make a linear graph,
and then analyzed the transformed data with linear regression.
There are several ways to linearize binding data, but Scatchard
plots (more accurately attributed to Rosenthal) are used most
often. In this plot, the X axis is specific binding (usually labeled
"bound") and the Y axis is the ratio of specific binding
to concentration of free radioligand (usually labeled "bound/free").
Bmax is the X intercept; Kd is the negative reciprocal of the
slope.
When making a Scatchard plot, you have to choose
units for the Y axis. One choice is to express both free ligand
and specific binding in cpm so the ratio bound/free is a unitless
fraction. The advantage of this choice is that you can interpret
Y values as the fraction of radioligand bound to receptors. If
the highest Y value is large (greater than 0.10), then the free
concentration will be substantially less than the added concentration
of radioligand, and the standard analyses won't work. You should
either revise your experimental protocol or use special analysis
methods that deal with ligand depletion (see page 29). The disadvantage
is that you cannot interpret the slope of the line without performing
unit conversions.
An alternative is to express the Y axis as sites/cell/nM
or fmol/mg/nM. While these values are hard to interpret, they
simplify calculation of the K_{d} which equals the reciprocal
of the slope. The specific binding units cancel when you calculate
the slope. The negative reciprocal of the slope is expressed in
units of concentration (nM) which equals the K_{d}.
While Scatchard plots are very useful for visualizing
data, they are not the most accurate way to analyze data. The
problem is that the linear transformation distorts the experimental
error. Linear regression assumes that the scatter of points around
the line follows a Gaussian distribution and that the standard
deviation is the same at every value of X. These assumptions are
not true with the transformed data. A second problem is that the
Scatchard transformation alters the relationship between X and
Y. The value of X (bound) is used to calculate Y (bound/free),
and this violates the assumptions of linear regression.
Since the assumptions of linear regression are violated,
the B_{max} and K_{d} you determine by linear
regression of Scatchard transformed data are likely to be further
from their true values than the B_{max} and K_{d}
determined by nonlinear regression. Considering all the time and
effort you put into collecting data, you want to use the best
possible analysis technique. Nonlinear regression produces the
most accurate results. Scatchard plots produce approximate results.
The figure below shows the problem of transforming
data. The left panel shows data that follows a rectangular hyperbola
(binding isotherm). The right panel is a Scatchard plot of the
same data. The solid curve on the left was determined by nonlinear
regression. The solid line on the right shows how that same curve
would look after a Scatchard transformation. The dotted line shows
the linear regression fit of the transformed data. The transformation
amplified and distorted the scatter, and thus the linear regression
fit does not yield the most accurate values for B_{max}
and K_{d}. In this example, the B_{max} determined
by the Scatchard plot is about 25% too large and the K_{d}
determined by the Scatchard plot is too high. The errors could
just as easily have gone in the other direction.
The experiment in the figure was designed to determine
the B_{max} and the experimenter didn't care too much
about the value of the K_{d}. So it was appropriate to
obtain only a few data points at the beginning of the curve and
many in the plateau region. Note however how the Scatchard transformation
gives undo weight to the data point collected at the lowest concentration
of radioligand (the lower left point in the left panel, the upper
left point in the right panel). This point dominates the linear
regression calculations on the Scatchard graph. It has "pulled"
the regression line to become shallower, resulting in an overestimate
of the B_{max}.
Although it is inappropriate to analyze data
by performing linear regression on a Scatchard plot, it is often
helpful to display data as a Scatchard plot. Many people
find it easier to visually interpret Scatchard plots than binding
curves, especially when comparing results from different experimental
treatments.
Competitive binding experiments
Competitive binding experiments measure the binding
of a single concentration of labeled ligand in the presence of
various concentrations of unlabeled ligand.
Competitive binding experiments are used to:
The experiment is done with a single concentration
of radioligand. How much should you use? There is no clear answer.
Higher concentrations of radioligand are more expensive and result
in higher nonspecific binding, but also result in higher numbers
of cpm bound and thus lower counting error. Lower concentrations
save money and reduce nonspecific binding, but result in fewer
counts of specific binding and thus more counting error. Many
investigators choose a concentration approximately equal to about
the K_{d} of the radioligand for binding to the receptor,
but this is not universal.
You need to let the incubation occur until equilibrium
has been reached. How long does that take? Your first thought
might be: "as long as it takes the radioligand to reach equilibrium
in the absence of competitor." It turns out that this may
not be long enough. You should incubate for 45 times the halflife
for receptor dissociation as determined in an offrate experiment
(see page 19).
Typically, investigators use 1224 concentrations
of unlabeled compound spanning about six orders of magnitude.
The top of the curve is a plateau at a value equal
to radioligand binding in the absence of the competing unlabeled
drug. This is total binding. The bottom of the curve is a plateau
equal to nonspecific binding (NS). The difference between the
top and bottom plateaus is the specific binding. Note that this
not the same as B_{max}. When you use a low concentration
of radioligand (to save money and avoid nonspecific binding),
you have not reached saturation so specific binding will be much
lower than the B_{max}.
The Y axis can be expressed as cpm or converted to
more useful units like fmol bound per milligram protein or number
of binding sites per cell. Some investigators like to normalize
the data from 100% (no competitor) to 0% (nonspecific binding
at maximal concentrations of competitor).
The concentration of unlabeled drug that results
in radioligand binding halfway between the upper and lower plateaus
is called the IC_{50} (inhibitory concentration 50%) also
called the EC_{50} (effective concentration 50%). The
IC_{50} is the concentration of unlabeled drug that blocks
half the specific binding.
If the labeled and unlabeled ligand compete for a
single binding site, the steepness of the competitive binding
curve is determined by the law of mass action. The curve descends
from 90% specific binding to 10% specific binding with an 81fold
increase in the concentration of the unlabeled drug. More simply,
nearly the entire curve will cover two log units (100fold change
in concentration).
Competitive binding curves are described by this
equation:
Y is the total binding you measure in the presence
of various concentrations of the unlabeled drug, and log[D] is
the logarithm of the concentration of competitor plotted on the
X axis. Nonspecific is binding in the presence of a saturating
concentration of D, and Total is the binding in the absence of
competitor. Y, Total and Nonspecific are all expressed in the
same units, such as cpm, fmol/mg, or sites/cell.
Use nonlinear regression to fit your competitive
binding curve to determine the log(IC_{50}).
In order to determine the bestfit value of IC_{50}
(the concentration of unlabeled drug that blocks 50% of the specific
binding of the radioligand), the nonlinear regression problem
must be able to determine the 100% (total) and 0% (nonspecific)
plateaus. If you have collected data over wide range of concentrations
of unlabeled drug, the curve will have clearly defined bottom
and top plateaus and the program should have no trouble fitting
all three values (both plateaus and the IC_{50}).
With some experiments, the competition data may not
define a clear bottom plateau. If you fit the data the usual way,
the program might stop with an error message. Or it might find
a nonsense value for the nonspecific plateau (it might even be
negative). If the bottom plateau (0%) is incorrect, the IC_{50}
will also be incorrect. To solve this problem, you should define
the nonspecific binding from other data. All drugs that bind to
the same receptor should compete for all specific radioligand
binding and reach the same bottom plateau value. When running
the curve fitting program, set the bottom plateau of the curve
to a constant equal to binding in the presence of a standard drug
known to block all specific binding.
Similarly, if the curve doesn't have a clear top
plateau, you should set the total binding to be a constant equal
to binding in the absence of any competitor.
The value of the IC_{50} is determined by
three factors:
Calculate the K_{i} from the IC_{50},
using the equation of Cheng and Prusoff (Cheng Y., Prusoff W.
H., Biochem. Pharmacol. 22: 30993108, 1973).
In thinking about this equation, remember that K_{i
}is a property of the receptor and unlabeled drug, while
IC_{50} is a property of the experiment. By changing your
experimental conditions (changing the radioligand used or changing
its concentration), you'll change the IC_{50} without
affecting the K_{i}.
This equation is based on these assumptions:
Why determine log(IC_{50}) rather than IC_{50}?
The equation for a competitive binding curve (page
14) looks a bit strange since it combines logarithms and antilogarithms
(10 to the power). A bit of algebra simplifies it :
If you fit the data to this equation, you'll get
the same curve and the same IC_{50}. Since the equation
is simpler, why not use it? The difference appears only when you
look at how nonlinear regression programs assess
the accuracy of the fit as a confidence interval. Even after converting
from a log scale to a linear scale, you'll end up with different
confidence intervals for the IC_{50}.
Which confidence interval is correct? With nonlinear
regression, the standard error of the fit variables are only approximately
correct. Since the confidence intervals are calculated from the
standard errors, they too are only approximately correct. The
problem is that the real confidence interval may not be symmetrical
around the best fit value. It may extend further in one direction
than the other. However, nonlinear regression programs always
calculate symmetrical confidence intervals (unless you use advanced
techniques). When writing the equation for nonlinear regression,
therefore, you want to arrange the variables so the uncertainty
is as symmetrical as possible. Because data are collected at concentrations
of D equally spaced on a log axis, the uncertainty is symmetrical
when the equation is written in terms of the log of IC_{50},
but is not symmetrical when written in terms of IC_{50}.
You'll get more accurate confidence intervals from fits of competitive
binding data when the equation is written in terms of the log(IC_{50}).
A competitive binding experiment is termed homologous
when the same compound is used as the hot and cold ligand.
The term heterologous is used when the hot and cold ligands
differ. Homologous competitive binding experiments can be used
to determine the affinity of a ligand for the receptor and the
receptor number. In other words, the experiment has the same goals
as a saturation binding curve. Because homologous competitive
binding experiments use a single concentration of radioligand
(which can be low), they consume less radioligand and thus are
more practical when radioligands are expensive or difficult to
synthesize.
To analyze a homologous competitive binding curve,
you need to accept these assumptions:
Analyze a homologous competitive binding curve using
the same equation used for a onesite heterologous competitive
binding to determine the top and bottom plateaus and the IC_{50}.
The Cheng and Prussoff equation lets you calculate
the K_{i} from the IC_{50} (see page 15). In the
case of a homologous competitive binding experiment, you assume
that the hot and cold ligand have identical affinities so that
K_{d} and K_{i} are the same. Knowing that, simple
algebra converts the equation to:
You set the concentration of radioligand in the experimental
design, and determine the IC_{50} from nonlinear regression.
The difference between the two is the K_{d} of the ligand
(assuming hot and cold ligands bind the same).
The difference between the top and bottom plateaus
of the curve represents the specific binding of radioligand at
the concentration you used. Depending on how much radioligand
you used, this value may be close to the B_{max} or far
from it. To determine the B_{max}, divide the specific
binding by the fractional occupancy, calculated from the K_{d}
and the concentration of radioligand.
Kinetic binding experiments
A dissociation binding experiment measures the "off
rate" for radioligand dissociating from the receptor. Perform
dissociation experiments to:
To perform an offrate experiment, first allow ligand
and receptor to bind, perhaps to equilibrium. At that point, block
further binding of radioligand to receptor using one of these
methods:
After initiating dissociation, measure binding over
time (typically 1020 measurements) to determine how rapidly the
ligand dissociates from the receptors.
Fit the data to this equation using nonlinear regression
to determine the rate constant. This is called an exponential
decay equation.
Total binding and nonspecific binding (NS) are expressed
in cpm, fmol/mg protein, or sites/cell. Time (t) is usually expressed
in minutes. The dissociation rate constant (k_{off}) is
expressed in units of inverse time, usually min^{1}.
Since it is hard to think in those units, it helps to calculate
the halflife for dissociation which equals ln(2)/k_{off}
or 0.6931/k_{off}. In one halflife, half the radioligand
will have dissociated. In two halflives, three quarters the radioligand
will have dissociated, etc.
Typically the dissociation rate constant of useful
radioligands is between 0.001 and 0.1 min^{1}. If the
dissociation rate constant is any faster, it would be difficult
to perform radioligand binding experiments as the radioligand
would dissociate from the receptors while you wash the filters.
This analysis assumes that the law of mass action
applies to your experimental situation. Dissociation binding experiments
also let you test that assumption. If the law of mass action applies
to your system, the answer to all these questions is yes:
If you plot ln(specific binding) vs. time, the graph
of a dissociation experiment will be linear if the system follows
the law of mass action with a single affinity state. The slope
of this line will equal k_{off}.
Notes:
 The log plot will only be linear if you take
the logarithm of specific binding. A graph of log(total binding)
vs. time will not be linear.
 You must use the natural logarithm, not the log
base ten in order for the slope to equal k_{off}.
 Use the log plot to display data, not to analyze
data. You'll get a more accurate rate constant by fitting the
raw data using nonlinear regression.
Association binding xperiments are used to determine
the association rate constant. This is useful to characterize
the interaction of the ligand with the receptor. It also is important
as it lets you determine how long it takes to reach equilibrium
in saturation and competition experiments.
You add radioligand and measure specific binding
at various times thereafter.
Note that the maximum binding (Y_{max}) is
not the same as B_{max}. The maximum binding achieved
in an association experiment depends on the concentration of radioligand.
Low to moderate concentrations of radioligand will bind to only
a small fraction of all the receptors no matter how long you wait.
To analyze the data, use nonlinear regression to
fit the specific binding data to the one phase exponential association
equation.
The observed rate constant, k_{ob} is expressed
in units of inverse time, usually min^{1}. It is a measure
of how quickly the incubation reaches equilibrium, and is determined
by three factors:
To calculate the association rate constant usually
expressed in units of Molar^{1} min^{1}, use
the following equation. Typically ligands have association rate
constants of about 10^{8 }M^{1 }min^{1}.
Analyses of association experiments assume:
Once you have separately determined k_{on}
and k_{off}, you can combine them to calculate the K_{d}
of receptor binding:
The units are consistent: k_{off} is in units
of min^{1}; k_{on} is in units of M^{1}min^{1},
so K_{d} is in units of M.
If binding follows the law of mass action, the K_{d}
calculated this way should be the same as the K_{d} calculated
from a saturation binding curve.
Two binding sites
Competitive binding experiments are often used in
systems where the tissue contains two classes of binding sites,
perhaps two subtypes of a receptor. Analysis of these data are
straightforward when you accept these assumptions:
If you accept those assumptions, binding follows
this equation:
This equation has five variables: the total and nonspecific
binding (the top and bottom binding plateaus), the fraction of
binding to receptors of the first type of receptor (F), and the
IC_{50} of the unlabeled ligand for each type of receptors.
If you know the K_{d} of the labeled ligand and its concentration,
you can convert the IC_{50} values to K_{i} values
(see page 15).
Since there are two different kinds of receptors
with different affinities, you might expect to see a biphasic
competitive binding curve. In fact, you will only see a biphasic
curve only in unusual cases where the affinities are extremely
different. More often you will see a shallow curve with the two
components blurred together. For example, the following graph
shows competition for two equally abundant sites with a ten fold
(one log unit) difference in IC_{50}. 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.
When the radioligand binds to two classes of receptors,
analyze the data by using this equation.
The left panel of the figure below shows specific
binding to two classes of receptors present in equal quantities,
whose K_{d} values differ tenfold. The right panel shows
the transformation to a Scatchard plot. In both graphs the dotted
and dashed lines show binding to the two individual receptors
that sum to the solid curves.
Note the following:
A twosite model almost always fits your data better
than a onesite model. A threesite model fits even better, and
a foursite model better yet! As you add more variables (sites)
to the equation, the curve becomes "more flexible" and
gets closer to the points. You need to use statistical calculations
to see if the improvement in fit between twosite and onesite
models is more than you'd expect to see by chance.
Before thinking about statistical comparisons, you
should look at whether the results make sense. Sometimes the twosite
fit gives results that are clearly nonsense. Disregard a twosite
fit when:
If the twosite fit seems reasonable, then you should
test whether the improvement is statistically significant.
Even if the simpler onesite model is correct, you
expect it to fit worse (have the higher sumofsquares) because
it has fewer inflection points (more degrees of freedom). In fact,
statisticians have proven that the relative increase in the sum
of squares (SS) is expected to equal the relative increase in
degrees of freedom (DF). In other words, if the onesite model
is correct you expect that:
If the more complicated twosite model is correct,
then you expect the relative increase in sumofsquares (going
from twosites to onesite) to be greater than the relative increase
in degrees of freedom:
The F ratio quantifies the relationship between the
relative increase in sumofsquares and the relative increase
in degrees of freedom.
If the onesite model is correct you expect to get
an F ratio near 1.0. If the ratio is much greater than 1.0, there
are two possibilities:
Many programs calculate the P value for you. If not,
you can find it using a table of F statistics. You need to know
that the numerator has DF1DF2 degrees of freedom and the denominator
has DF2 degrees of freedom. To use statistical tables you also
have to remember that a larger value of F corresponds to a lower
P value.
The P value answers this question: If the onesite
model is really correct, what is the chance that you'd randomly
obtain data that fits the twosite model so much better? If the
P value is small, you conclude that the twosite model is significantly
better than the onesite model. Most scientists reach this conclusion
when P<0.05, but this threshold P value is arbitrary.
Advanced topics
Many competitive binding curves are shallower than
predicted by the law of mass action for binding to a single site.
The steepness of a binding curve can be quantified with a slope
factor, often called a Hill slope. A onesite competitive binding
curve that follows the law of mass action has a slope of 1.0.
If the curve is more shallow, the slope factor will be a negative
fraction (i.e. 0.85 or 0.60). The slope factor is negative because
the curve goes downhill.
To quantify the steepness of a competitive binding
curve (or a doseresponse curve), fit the data to this equation:
The slope factor is a number that describes the steepness
of the curve. In most situations, there is no way to interpret
the value in terms of chemistry or biology. If the slope factor
differs significantly from 1.0, then the binding does not follow
the law of mass action with a single site.
Explanations for shallow binding curves include:
All of the analyses presented in this booklet are
based on the law of mass action. This assumes that the receptor
and ligand reversibly bind, but nothing else happens. The law
of mass action may not be true when the ligand or competitor is
an agonist. By definition, something happens when an agonist binds
to the receptor. For example, the agonist may alter the interaction
of the receptor with a G protein, which then changes the affinity
of the receptor for the agonist. These complexities mean that
the law of mass action may be too simple when agonists are involved,
and that the results of analyses based on the law of mass action
can't be rigorously interpreted.
Some investigators have attempted to fit data to
more complicated models, but these analyses are beyond the scope
of this booklet. Despite the theoretical complexities, agonist
binding curves often turn out to fit rectangular hyperbolas or
one or twosite competitive binding curves. Most investigators
analyze their data using these standard analyses. The bestfit
curves often fit the data nicely, and provide K_{d} or
K_{i} values that can be compared between conditions.
It is important to realize that these analyses are based on a
model that is too simplistic. Treat the bestfit values of K_{d}
or K_{i} as empirical descriptions of the data, and are
not as true equilibrium dissociation constants.
The equations that describe the law of mass action
include the variable [Ligand] which is the free concentration
of ligand. All the analyses presented so far depend on the assumption
that a very small fraction of the ligand binds to receptors (or
to nonspecific sites) so you can assume that the free concentration
of ligand is approximately equal to the concentration added. This
is sometimes called "zone A".
In some experimental situations, the receptors are
present in high concentration and have a high affinity for the
ligand, so that assumption is not valid. A large fraction of the
radioligand binds to receptors so the free concentration of radioligand
is quite a bit lower than the concentration added. The system
is not in "zone A". The discrepancy is not the same
in all tubes or at all times.
Many investigators use this rule of thumb. If less
than 10% of the ligand binds, don't worry about ligand depletion.
If possible you should design your experimental protocol
to avoid situations where more than 10% of the ligand binds. You
can do this by using less tissue in your assays. The problem is
that this will also decrease the number of counts. An alternative
is to increase the volume of the assay without changing the amount
of tissue. The problem with this approach is that you'll need
more radioligand.
If you can't avoid radioligand depletion, you need
to account for the depletion in your analyses. You may use several
approaches.
GraphPad Prism
All the graphs in this booklet were created using
GraphPad Prism, a generalpurpose
curve fitting and scientific graphics program for Windows.
Although GraphPad Prism was designed to be a general
purpose program, it is particularly wellsuited for analyses of
radioligand binding data.
Please visit our web site at http://www.graphpad.com
to read about Prism and download a free demo. Or contact GraphPad
Software to request a brochure and demo disk by phone (6194573909),
fax (6194578141) or email sales@graphpad.com. The demo
is not a slide show  it is a functional version of Prism with
no limitations in data analysis. Try it out with your own data,
and see for yourself why Prism is the best solution for analyzing
and graphing scientific data.
