Introducing homologous competition

The most common way to determine receptor number and affinity is to perform a saturation binding experiment where you vary the concentration of radioligand. An alternative is to keep the radioligand concentration constant, and compete for binding with the same chemical, but not radioactively labeled. Since the hot (radiolabeled) and cold (unlabeled) ligands are chemically identical, this is called a homologous competitive binding experiment.

Most analyses of homologous competition data are based on these assumptions:

Theory of homologous competition binding

Start with the equation for equilibrium binding to a single class of receptors.

Set [Ligand] equal to the sum of the labeled (hot) and unlabeled ligand (cold). Specific binding you measure (specific binding of the labeled ligand) equals specific binding of all ligand times the fraction of the ligand that is labeled. This fraction, hot/(hot+cold), varies from tube to tube.  Therefore specific binding of labeled ligand follows this equation:

Specific binding and Bmax are in the same units, usually cpm, sites/cell or fmol/mg. [Hot], [Cold] and Kd are in concentration units. Those units cancel so it doesn't matter if you use molar, nM, or some other unit, so long as you are consistent.

Maximum binding of labeled ligand occurs when the concentration of cold ligand equals zero. This is not the same as Bmax, because the concentration of hot ligand will not saturate all the receptors. In the absence of cold ligand (set [cold]=0), the binding equals

The IC50 in a homologous binding experiment is the concentration of [Cold] that reduces specific binding of labeled ligand by 50%. So the IC50 is the concentration of cold that solves the equation below. The left side of the equation is half the maximum binding with no cold ligand. The right side is binding in the presence of a particular concentration of cold ligand. We want to solve for [Cold].

Solve this equation for [Cold], and you'll find that you achieve half-maximal binding when [Cold] = [Hot] + Kd .  In other words,

Why homologous binding data can be ambiguous

Since the IC50 equals [Hot] + Kd , the value of the Kd doesn't affect the IC50 very much when you use a high concentration of radioligand. This means that you'll see the same IC50 with a large range of Kd values. For example if you use a Hot ligand concentration of 10 nM, the IC50 will equal 10.1 nM if the Kd is 0.1 nM (dashed curve below), and the IC50 will equal 11 nM if the Kd is 1 nM (solid curve below). These two IC50 values are almost identical, and cannot be distinguished in the presence of experimental error.

If the concentration of hot (radiolabeled) ligand greatly exceeds the Kd , the curve is ambiguous. There are an infinite number of curves, defined by different Kd and Bmax values, which are almost identical. The data simply don't define the Kd and Bmax . No curve-fitting program can determine the Kd and Bmax from this type of experiment - the data are consistent with many Kd and Bmax values.

Fitting homologous competition data (one site)

We recommend that you follow a two-step procedure for fitting homologous competition data, at least with new systems. Once you have a routine assay, you may want to skip step 1.

Step 1. Determine the IC50

This first step is to check that you used a reasonable concentration of radioligand.

Fit your data to the built-in equation, One-site competition. If your competition curve doesn't have clearly defined top and bottom plateaus, you should set one or both of these to constant values based on control experiments.

Compare the best-fit value of the EC50  (same as IC50) to the concentration of hot ligand you used. Homologous competition experiments only lead to useful results when the concentration of hot ligand is less than half the IC50.

If	Then do this
IC50 is more than ten times [Hot]	The concentration of [Hot] is lower than it needs to be. If you have quality data, you can continue with step 2. If you have too few cpm bound or if more than 10% of the added radioactivity binds to tissue, rerun the experiment with more radioligand
IC50 is between two and ten times [Hot]	You've designed the experiment appropriately. Continue to step 2 to determine the Kd and Bmax with confidence limits.
IC50 is greater than [Hot] but not twice as high	You'll get better results if you repeat the experiment using less hot ligand.
IC50 is less than [Hot]	If binding follows the assumptions of the analysis, it is impossible for the IC50 to be less than the concentration of hot ligand. Since Kd  = IC50 -[Hot], these data would lead to a negative value for Kd , which is impossible. One possibility is that your system does not follow the assumptions of the analysis. Perhaps the hot and cold ligands do not bind with identical affinities. Another possibility is that you simply used too high a concentration of hot ligand, so the true IC50 is very close to (but greater than) [Hot]. Experimental scatter may lead to a best-fit IC50 that is too low, leading to the impossible results. Repeat the experiment using a lot less hot ligand.

Once you've determined that the IC50 is quite a bit larger than the concentration of hot ligand, continue with this step to determine the Bmax and Kd.

Total binding equals specific binding plus nonspecific binding. Nonspecific binding is the same for all tubes since it only depends on the concentration of hot ligand, which is constant. The equation for specific binding is derived in Theory of homologous competition binding. Add a nonspecific binding term to define total binding:

Since this equation is not built-in to Prism, you'll need to enter it as a user-defined equation as follows:

ColdNM=10^(x+9) ;Cold concentration in nM

KdNM=10^(logKD+9) ;Kd  in nM

Y=(Bmax*HotnM)/(HotnM + ColdNM + KdNM) + NS

With any user-defined equation, you need to define initial values. The best way to do this is to enter rules, so Prism can compute initial values for any set of data. After entering the equation, press the button to define rules for initial values. Here are some suggested rules.

This equation assumes that you have entered X values as the logarithm of the concentrations of the unlabeled ligand in molar, so 1nM (10-9 molar) is entered as -9. The first line in the equation adds 9 to make it the logarithm of the concentration in nM, and then takes the antilog to get concentration in nM.

Since the experiment is performed with the concentrations of unlabeled ligand equally spaced on a log scale, the confidence intervals will be most accurate when the Kd is fit as the log(Kd). The second line converts the log of the Kd  in moles/liter to nM.

Set HotnM equal to the concentration of labeled ligand in nM, and set it to be a constant value. Prism cannot fit this value; you must make it a constant and enter its value.

Homologous competitive binding with ligand depletion

If a large fraction of the radioligand is bound to the receptors, homologous binding will be affected. Although you add the same amount of labeled ligand to each tube, the free concentration will not be the same. High concentrations of unlabeled drug compete for the binding of the labeled ligand, and thus increase the free concentration of the labeled ligand.

Nonspecific binding is also affected by ligand depletion. Since nonspecific binding is proportional to the free concentration of labeled ligand, the amount of nonspecific binding will not be the same in all tubes. The tubes with the highest concentration of unlabeled drug have the highest concentration of free radioligand, so will have the most nonspecific binding.

Because the free concentration varies among tubes, as does the nonspecific binding, there is no simple relationship between IC50 and Kd. The IC50 is nearly meaningless in a homologous binding curve with ligand depletion.

The equations for homologous binding with ligand depletion are quite a bit more complicated than for homologous binding without depletion. The math that follows is adapted from S. Swillens (Molecular Pharmacology, 47: 1197-1203, 1995).

Start with the equation for total binding in homologous competition as a function of the free concentration of radioligand.

This equation defines total binding as specific binding plus nonspecific binding. Nonspecific binding equals a constant fraction of free radioligand, and we define this fraction to be NS. To keep units consistent, the radioligand concentration is expressed in nM in the left half of the equation (to be consistent with Kd and the concentration of cold ligand) and is expressed in cpm on the right half of the equation (to be consistent with Y).

The problem with this equation is that you don't know the concentrations of free radioligand or free cold ligand. What you know is the concentrations of labeled and unlabeled ligand you added. Since a high fraction of ligand binds to the receptors, you cannot assume that the concentration of free ligand equals the concentration of added ligand.

Defining the free concentration of hot ligand is easy. You added the same number of cpm of hot ligand to each tube, which we'll call HotCPM. The concentration of free radioligand equals the concentration added minus the total concentration bound, or HotCPM-Y (both HotCPM and Y are expressed in cpm).

Defining the free concentration of cold ligand is harder, so it is done indirectly. The fraction of hot radioligand that is free equals (HotCPM - Y)/HotCPM. This fraction will be different in different tubes. Since the hot and cold ligands are chemically identical, the fraction of cold ligand that is free in each tube is identical to the fraction of hot ligand that is free. Define X to be the logarithm of the total concentration of cold ligand, the variable you vary in a homologous competitive binding experiment. Therefore, the total concentration of cold ligand is 10X, and the free concentration of cold ligand is 10X(HotCPM - Y)/HotCPM.

Substitute these definitions of the free concentrations of hot and cold ligand into the equation above, and the equation is still unusable. The problem is that the variable Y appears on both sides of the equal sign.  Some simple, but messy, algebra puts Y on the left side of a quadratic equation, shown below as a user-defined Prism equation.

ColdnM=10^(X+9)

KDnM=10^(LogKD+9)

HotnM=HotCPM/(SpAct*vol*1000)

; cpm/(cpm/fmol * ml * .001L/ml * 1000000fmol/nmol)

TotalnM=HotnM+ColdnM

Q=HotCPM*(TotalnM + KDnM)

a=(Ns+1)*TotalnM*-1

b=Q*(NS+1)+TotalnM*HotCPM*NS + Bmax*HotnM

c=-1*Q*HotCPM*NS - HotCPM*Bmax*HotnM

Y= (-1*b + sqrt(b*b-4*a*c))/(2*a)

Select this equation from the Advanced Radioligand Binding equation library.

Variable	Units	Comments
X	Log(Molar)
Y	Cpm
HotCPM	Cpm	Amount of labeled ligand added to each tube. Set to a constant value.
SpAct	Cpm/fmol	Specific radioactivity of labeled ligand. Set to a constant value.
Vol	ml	Incubation volume. Set to a constant value.
logKd	Log(Molar)	Initial value = 1.0*XMID
Bmax	Units of Y axis, usually cpm	Initial value = 10*YMAX (This assumes that you've used a concentration of radioligand that binds to one tenth of the receptors. You may wish to adjust this.)
NS	Unitless fraction	This is the fraction of free ligand that binds nonspecifically. Initial value =0.01

Fitting homologous competition data (two sites)

With some systems it is possible to determine Kd  and Bmax values for two independent sites using homologous competition data. With most systems, however, you won't get reliable results.

You can only determine Bmax and Kd from homologous binding data if you use a concentration of hot ligand that is much lower than the Kd value. If your system has two binding sites, you must choose a concentration much lower than the Kd of the high affinity site. Using such a low concentration of radioligand, you'll bind only a small fraction of low-affinity sites. You only be able to detect the presence of the second, low-affinity, site if they are far more abundant than the high-affinity sites.

For example, imagine that the low affinity site (Kd=10 nM) is ten times as abundant as the high affinity site (kd=0.1 nM). You need to use a concentration of hot ligand less than 0.1 nM, say 0.05 nM. At this concentration you bind to 33.33% of the high affinity sites, but only to 0.0049% of the low affinity sites. Even though the low affinity sites are ten times as abundant, you won't find them in your assay (low affinity binding will be only 0.15% of the binding).

To attempt to determine the two Kd  and Bmax   values from a homologous competition curve, fit the data to the equation below. Assuming no cooperativity and no ligand depletion, the binding to each site is independent and depends on the Bmax and Kd values of each site. The binding that you measure, Y, is the sum of the binding to the two receptor sites plus nonspecific binding.  

Site1=(Bmax1*HotnM)/(HotnM + 10^(X+9)+ 10^(logKd1+9))

Site2=(Bmax2*HotnM)/(HotnM + 10^(X+9)+ 10^(logKd2+9))

Y= site1 + site2 + NS

Define rules for initial values using the suggestions in the table below. This is a difficult equation to fit, and you will almost certainly have to try many sets of initial values to converge on a reasonable solution. It is especially important to adjust the initial values of the two Bmax values.

Consider this approach for analyzing homologous competitive binding data to determine the characteristics of two sites. First use a very low concentration of radioligand and fit to a single site. This will determine the Bmax and Kd  of the high affinity site. Then repeat the experiment with a higher concentration of radioligand. Fit these data to the two-site equation, but set the Kd  and Bmax  of the high affinity site to constant values determined from the first experiment.

Advantages and disadvantages of homologous binding experiments

Determining receptor number with homologous binding has one clear advantage: You need far less radioligand than you would need if you performed a saturation binding experiment. This reason can be compelling for ligands that are particularly expensive or difficult to synthesize.

The disadvantage of determining receptor number and affinity from a homologous competitive binding experiment is that it can be hard to pick an appropriate concentration of radioligand. If you use too little radioligand, you'll observe little binding and will obtain poor quality data. If you use too much radioligand, the curve will be ambiguous and you won't be able to determine Bmax and Kd.

Using homologous binding to determine the Kd and Bmax of two binding sites with homologous binding is difficult. You are probably better off using a saturation binding experiment.