The standard methods of analyzing saturation binding assume that a tiny fraction of the radioligand binds to receptors. This means that the concentration of radioligand added is very similar to the concentration of radioligand free in solution.

In some experimental situations, where the receptors are present in high concentration and have a high affinity for the ligand, that assumption is not true. A large fraction of the radioligand binds to receptors, so the free concentration added is quite a bit lower than the concentration you added. The free ligand concentration is depleted by binding to receptors.

If possible you should avoid experimental situations where the free ligand concentration is far from the total concentration. You can do this by increasing the volume of the assay without changing the amount of tissue. The drawback is that you'll need more radioligand, which is usually expensive or difficult to synthesize.

If you can't avoid radioligand depletion, you need to account for the depletion in your analyses. The obvious way to do this is to subtract the number of cpm (counts per minute) of total binding from the cpm of added ligand to calculate the number of cpm free in solution. This can then be converted to the free concentration in molar. There are four problems with this approach:

If you used this method, experimental error in determining specific binding would affect the free ligand concentration you calculate. Error in Y would affect X, which violates an assumption of nonlinear regression.

	•	Since the free concentration in the nonspecific tubes is not the same as the free concentration in the total tubes, it is difficult to deal with nonspecific binding using this approach. You can not calculate specific binding as the difference between the total binding and nonspecific binding.

	•	This method works only for saturation binding experiments, and cannot be extended to analysis of competition curves.

	•	You cannot implement this method with Prism, which does not let you subtract Y from X. (Since Prism allows for many Y data sets per table, but only one X column, subtracting Y from X would be ambiguous).

S. Swillens (Molecular Pharmacology, 47: 1197-1203, 1995) developed an equation that defines total binding as a function of added ligand, accounting  for nonspecific binding and ligand depletion. By analyzing simulated experiments, that paper shows that fitting total binding gives more reliable results than you would get by calculating free ligand by subtraction. The equations shown below are not exactly the same as in Swillens' paper, but the ideas are the same.

From the law of mass action, total binding follows this equation.

The first term is the specific binding, which equals fractional occupancy times Bmax, the total number of binding sites. The second term is nonspecific binding, which is assumed to be proportional to free ligand concentration. The variable NS is the fraction of the free ligand that binds to nonspecific sites.

This equation is not useful, because you don't know the concentration of free ligand. What you know is that the free concentration of ligand equals the concentration you added minus the concentration that bound (specific and nonspecific). Defining X to be the amount of ligand added and Y to be total binding, the system is defined by two equations:

Combining the two equations:

X, Y and Bmax are expressed in units of cpm. To keep the equation consistent, therefore, Kd must also be converted to cpm units (the number of cpm added to each tube when the total concentration equals the Kd).

You cannot enter that equation into Prism for nonlinear regression because Y appears on both sides of the equal sign. But simple algebra rearranges it into a quadratic equation. The solution is shown below as a user defined Prism equation.

;X is total ligand added in cpm. Y is total binding in cpm

;SpecAct is specific radioactivity in cpm/fmol
;Vol is reaction volume in ml
;Both must be set to be CONSTANTS

;Calc KD in cpm from nM
KdCPM=KdnM * Vol * 1000 * SpecAct
; (nm/L * mL * 0.001 L/ml * 1000000 fmol/nmol * cpm/fmol)

a = -1-NS
b = KdCPM + NS*KdCPM + X + 2*X*NS + Bmax
c= -1*X*(NS*KdCPM + X*NS+Bmax)
Y= -b + SQRT(b*b - 4*a*c))/(2*a)

Determining nonspecific binding experimentally in saturation binding experiments with ligand depletion

The method described above fits total binding data to an equation that includes both specific and nonspecific components. It does not require that you experimentally determine nonspecific binding. While this is convenient, many investigators would feel uneasy trusting those results without determining nonspecific binding experimentally.

You can experimentally determine nonspecific binding by including a large concentration of an unlabeled ligand in your incubations. This will bind to virtually all the receptors, leaving only nonspecific sites free to bind radioligand. The conventional approach is to measure total and nonspecific binding at each ligand concentration, and to define specific binding as the difference. This approach cannot be used when a high fraction of ligand binds, because the free concentration of ligand in the total tubes is not the same as the free concentration of ligand in the nonspecific tubes.

We assume that nonspecific binding is a constant fraction of the concentration of free ligand.

We also assume that the free concentration of ligand equals the added concentration (X) minus the amount that bound (Y).

Combining the two equations:

To experimentally determine nonspecific binding:

1. Enter the data with X equal to the added concentration in cpm and Y equal to the nonspecific binding in cpm.

2. Fit by nonlinear regression to this user-defined equation: Y=X*NS/(NS+1)

3. Since this is really a linear equation, you'll get the same fit no matter what initial value you enter. Set the initial value of NS equal to 0.01.

4. Compare the value of NS determined here with the NS determined from analysis of the total binding. The two values should be similar.

If you think the value of NS determined here is more accurate than the NS determined from analysis of the total binding, refit the total binding data holding NS constant (equal to the value determined from the nonspecific binding analysis).