Heterologous competitive binding with ligand depletion

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Heterologous competitive binding with ligand depletion

The standard sigmoidal equations used to fit competitive binding data assume that a small fraction of the radioligand binds. This means that the free concentration of radioligand is almost equal to the concentration you added, and that the free concentration is the same in all tubes in the assay.

If a large (say greater than 10%) fraction of the radioligand binds to receptors, then the free concentration will be less than the added concentration of radioligand. The discrepancy between free and added radioligand concentration depends on the concentration of the unlabeled drug. The standard equation for competitive binding, shown below, needs two corrections to account for ligand depletion.


	•	The free concentration of labeled ligand equals the amount you added minus the amount that bound.


	•	The nonspecific binding is not the same for all tubes. As you increase the concentration of cold ligand, less radioligand binds to receptors so the free concentration of radioligand increases. Since nonspecific binding is assumed to be proportional to the free concentration of radioligand, there will be more nonspecific binding in the tubes with higher concentrations of unlabeled drug.

Y, [Free ligand], and [Added ligand] are expressed in units of cpm. To be consistent, therefore the Kd also needs to be expressed in cpm units. [Cold ligand] and Ki are expressed in the same units (molar), so the ratio is unitless.

Combine these equations, and you end up with a complicated quadratic equation whose solution is shown here:

KdCPM=KdnM*SpAct*vol*1000
; nmol/L *(cpm/fmol * ml * .001L/ml * 1000000fmol/nmol) = cpm

R=NS+1

S=[1+10^(X-LogKi)]*KdCPM+Hot

a=-1*R

b=R*S+NS*Hot + Bmax

c= -1*Hot*(S*NS + Bmax)

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

Select this equation from the Advanced Radioligand Binding equation library.

Variable

Units

Comments

log(Molar)

CPM

Hot

CPM

Amount of labeled ligand added to each tube. Set to a constant value.

SpAct

cpm/fmol

Specific radioactivity of labeled ligand. Set to constant value.

Vol

Incubation volume. Set to a constant value.

KdnM

Kd of labeled ligand. Set to a constant value.

LogKi

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.)

Unitless fraction

Initial value =0.01

You need to set four of the parameters to constant values. Hot is the number of cpm of labeled ligand added to each tube. SpAct is the specific activity of the radioligand in cpm/fmol, Vol is the incubation volume in ml, and Kd is the KdnM of the radioligand in nM (determined from other experiments). The program fits this equation to your data to determine the logKI. It also fits two other variables which are of less interest: Bmax which is the maximum binding of radioligand (if present at a saturating concentration) in cpm, and NS which is the fraction of the free ligand that binds nonspecifically.Notes:


	•	This equation accounts for ligand depletion when a large fraction of the radioligand binds to receptors. It does not adjust for depletion of the unlabeled compound. It assumes that the concentration of unlabeled compound that you added (antilog of X) equals the free concentration. If your unlabeled compound binds with high affinity, this assumption may not be true.


	•	You may use this equation for any competitive binding curve, even if only a small fraction of the radioligand binds. The results will be identical to the results from the more conventional equations.


	•	This equation is not easily extended to a situation with two binding sites.