Section5:Binding Parameters

From Assay Guidance Wiki

1 Binding Paramters
2 Saturation Binding
3 Considerations/Assumptions for Saturation Binding Experiments
4 Scatchard Plots
5 Homologous Competition
6 Association Rate at Various Radioligand Concentrations (Optional)
7 Heterologous Competition
8 Pharmacological Profile
9 OLO Guidelines for Nonlinear Regression Curve Fitting

Binding Paramters

The determination of the equilibrium dissociation constant for the radioligand (Kd) or equilibrium dissociation constants for unlabeled compounds (Ki) should be performed after the SPA receptor binding assay has been fully optimized for the conditions outlined in the prior sections.

Three methods are described for the determination of the receptor affinity for the radioligand, Kd:

Saturation analysis
Homologous competition
Association rate at various radioligand concentrations

A heterologous competition binding assay is used to determine the affinity of the receptor for an unlabeled compound, Ki.

Saturation Binding

An equilibrium saturation binding experiment measures total and nonspecific binding at various radioligand concentrations. The equilibrium dissociation constant or affinity for the radioligand, Kd, and the maximal number of receptor binding sites, Bmax, can be calculated from specific binding (total - NSB) using non-linear regression analysis.

Requirements:

Steady state for low concentrations of radioligand (i.e. 1/10 estimated Kd) has been
Ensure that <10% of the added radioligand is bound (at all radioligand concentrations tested) to prevent ligand depletion - if this is not met, lower the receptor concentration.

The range of radioligand concentrations tested in a saturation binding experiment is typically from 1/10 Kd to 10x Kd to yield an appropriate curve for nonlinear regression analysis methods. Radioligand specific activity, concentration or expense may prevent a wide range of concentrations from being used.

A high concentration of unlabeled compound (1000 x Ki or Kd value) is used to determine nonspecific binding. Ideally, the unlabeled compound should be structurally different than the radioligand. Nonspecific binding should be less than 50% of the total binding at the highest concentration of [L] tested.

Calculate Kd for specific binding using a one-site binding hyperbola nonlinear regression analysis (i.e. GraphPad Prism) as shown in the equation below:

Bmax is the maximal number of binding sites (pmol/mg), and Kd (nM, pM, etc.) is the concentration of radioligand required to reach half-maximal binding.

Setup: Measure total binding (receptor + radioligand + SPA beads) and nonspecific binding (receptor + radioligand + excess unlabeled competitor + SPA beads) at various concentrations of radioligand using the determined optimum incubation time, optimum receptor concentration and optimum SPA bead amount. Include the expected concentration of DMSO or other solvent for compound testing. To assess non-proximity effects (NPE), a condition without receptor can be included (radioligand + SPA beads).

Results Analysis: Plot total, NSB and specific binding (total - NSB) versus free concentration of radioligand. Plot NPE if no receptor condition was performed.

A representative saturation binding experiment is shown below. Y-axis data has been expressed in pmol/mg, using conversion methods shown in the Appendix.

A listing of the calculations required for analysis of saturation binding data is shown below. Details for each of these calculations are shown in the Appendix.

Determine total radioactivity added per well by counting an aliquot of each radioligand mix in a gamma counter or a liquid scintillation counter. Convert to DPM if necessary using the equation below: DPM = CPM/Efficiency
Convert binding data (total bound, NSB) from CPM to DPM data using above equation.
Calculate specific binding in DPM: Specific bound = Total Bound - NSB
Calculate unbound (free) DPM: Free = Total Added - Total Bound
Convert free DPM to concentration units (i.e. nM) using the radioligand specific activity (expressed as DPM/fmol) and the volume of sample used.
Convert Total bound, NSB and Specific bound DPM to pmol/mg units using specific activity expressed as DPM/fmol and the amount of receptor added per assay well in mg units.
Plot Bound (in pmol/mg) on Y-axis versus Free concentration of radioligand (in nM) on X-axis.
Determine Kd and Bmax using a non-linear regression analysis for a single site binding (hyperbola).

Considerations/Assumptions for Saturation Binding Experiments

The binding reaction must be at equilibrium for all concentrations of radioligand. Lower concentrations of radioligand require longer times to reach equilibrium.

Less than 10% of the total added radioligand should be bound at all concentrations of radioligand tested. At lower concentrations of radioligand, it is more likely that greater than 10% of the added radioligand will be bound (if this is the case, receptor concentration should be lowered).

If reagents and the assay system allow, radioactive concentrations of at least 10 times the Kd should be tested to provide suitable data for a nonlinear regression analysis and accurate determination of the binding parameters. The Kd and Bmax values can be calculated from less than complete data sets, but the statistical reliability of the returned values may be lower.

Ideally, nonspecific binding should be less than 50% of the total binding.

No positive or negative binding cooperativity

Binding is reversible and obeys the Law of Mass Action:

Scatchard Plots

In the past, nonlinear saturation binding data was transformed into linear data followed by analysis using linear regression, resulting in a Scatchard (or Rosenthal) plot. Although perhaps useful for the display of data, the Scatchard plot is not used anymore for the determination of Kd or Bmax values. These values are determined using nonlinear regression analysis as described above. Scatchard plots distort the experimental error (X value is used to calculate Y), hence the assumptions of linear regression are violated and the resulting values are not accurate.

It is inappropriate to analyze transformed data for the determination of Kd and Bmax.

Homologous Competition

A homologous competition is a concentration response curve with an unlabeled compound that is identical to the radioligand being used. Radioligand concentration is constant in the experiment. Homologous competition experiments can be used as an alternative to saturation experiments to determine receptor affinity (Kd) and density (Bmax), provided the criteria shown below are met. When using [125I]-ligands, a non-radioactive iodo-ligand should be used if possible.

Assumptions:

The receptor has identical affinity for the radioligand and unlabeled ligand.
There is no cooperativity.
No ligand depletion (<10% of the total added radioligand is bound)
Nonspecific binding is proportional to the concentration of labeled ligand.

The concentration-response curve should ideally descend from 90% specific binding to 10% specific binding over an 81-fold (or approximately 2 log scales) increase in concentration of the unlabeled ligand.

A homologous competition experiment has been designed correctly if the IC50 is between 2 and 10 times the concentration of radioligand.

Two methods can be used to analyze data from a homologous competition experiment and determine the Kd and Bmax. They are described below as Results Analysis 1 and Results Analysis 2. The experimental setup is identical for both types of analysis.

Setup: Measure binding (receptor + radioligand + SPA beads) at various concentrations of unlabeled competitor using a single concentration of radioligand (≤Kd) and the determined optimum incubation time, optimum receptor concentration and optimum SPA bead amount. In some cases (3H-label with low specific activity), concentrations above the Kd may be required. Total binding is determined in the absence of any added competitor. Nonspecific binding (receptor + radioligand + excess unlabeled competitor + SPA beads) is included for calculation of specific binding.

Results Analysis 1: Plot specific bound (Bound - NSB) at each concentration of unlabeled competitor. Conversion to percent specific bound is performed using the following equation:

Step 1. Determine the IC50 using a sigmoidal dose-response (variable slope), which is also known as a four-parameter logistic nonlinear regression analysis (i.e. using GraphPad Prism) as shown in the equation below (use log concentration values for proper analysis):

Representative data for a homologous competition:

Step 2. Determine the Kd and Bmax using nonlinear regression analysis (i.e. GraphPad Prism) for the following equation:

[Hot] is the concentration of radioligand used in the assay (in nM) [Cold] is the concentration of competitor, which varies, (in nM)

In GraphPad Prism, enter Y in CPM or DPM and X in log concentration of competitor.

Calculate Kd and Bmax with above curve fit analysis. Use instrument counting efficiency and specific activity of radioligand to convert the calculated maximum signal units (CPM or DPM) to pmol/mg units.

The calculated IC50 for this homologous competition experiment is 1.5 x 10-10 M. The concentration of [L] used in this homologous competition is 3.8 x 10-11 M.

The calculated IC50 is between 2 and 10 times the concentration of added [L].

The concentration-response curve descends from 90% specific binding to 10% specific binding over an 86-fold increase in concentration of the unlabeled ligand.

Inputting X (log concentration) and Y (Total DPM) into the homologous binding analysis equation above yields (in parentheses are the values obtained from saturation binding analysis): Log Kd = -9.962 → Antilog of -9.962 = 1.09 x 10-10 M = 109 pM (79 pM) Bmax = 27773 DPM → Convert using specific activity, Bmax = 12.5 pmol/mg (5 pmol/mg)

Results Analysis 2:

Alternatively, convert specific DPM bound to molar units (i.e. nM) bound.

The molar concentration of labeled ligand ([L]) is calculated using the DPM added per well, the specific activity and the conversion factor, 1 μCi = 2.2 x 106 DPM.

The formula is [L] nM = (specific counts) * (1/2200000) * (1 / Specific Activity) * 10000
This concentration is added to all concentrations of unlabeled ligand to determine the final added ligand concentration.
As the added ligand concentration increases (due to increase added unlabeled ligand), the specific activity of the labeled ligand is decreased.
For each specific DPM bound determine the specific molar units bound by using the corresponding specific activity in that condition.

The formula is [RL] = (added ligand) * (specific DPM) / (DPM added per well).
Use a one-site binding (hyperbola) similar to the saturation binding data to calculate Kd and Bmax.

Example: Labeled ligand specific activity is 90 Ci/mmol and 66398 dpm are added per well (100 μl final volume). The concentration of labeled ligand in all wells is 3.3 nM. At unlabeled ligand concentration of 125 nM, the final added ligand (unlabeled + labeled) is 128.3 nM.

If the specific binding at 125 nM unlabeled ligand condition is 3283 DPM, then the specific molar unit bound would be (3283 x128.3)/66398 = 6.34 nM.

Representative results for homologous competition analyzed using the Results Analysis 2 method is shown on the following page.

Homologous Competition for Determination of Kd

In this example data, the Kd determined from a homologous competition experiment is 12.6 nM.

Association Rate at Various Radioligand Concentrations (Optional)

An optional method, which can be used early in development for both determination of optimum incubation time and provide an estimate for the Kd, is to perform an association rate experiment at various radioligand concentrations.

Setup: Measure total binding (receptor + radioligand + SPA beads) and nonspecific binding (receptor + radioligand + excess unlabeled competitor + SPA beads) at various times and at various concentrations of added radioligand.

Results Analysis: Plot specific binding (total binding - NSB) versus time at each radioligand concentration tested.

Calculate the observed association rate constant (kobs) by fitting the signal versus time data to a one-phase exponential association nonlinear regression analysis for each concentration of radioligand tested. The kobs value is returned as one of the resulting curve fit parameters. There will be different kobs for each radioligand concentration.

Plot the observed association rate constant (kobs) versus concentration of [L].

This should result in a linear function with a slope equal to the association rate constant (kon) and the Y-intercept equal to the dissociation rate constant (koff). An estimate for the equilibrium dissociation constant (Kd) can be calculated using the equation below with the kinetically determined rate constants:

Kd = koff/kon

Example Data: SPA Method: Reaction mix was read at different time points

The Kd calculated from saturation binding for this receptor was 35 pM.

Heterologous Competition

Experimentally, a heterologous competition is identical to a homologous competition. Heterologous competition assays measure concentration-response binding with unlabeled ligands that are structurally different than the radioligand. The IC50 for the unlabeled compound is determined from the experimental data and the equilibrium dissociation constant, Ki, can be calculated using a mathematical formula (Cheng-Prusoff equation).

Setup: Measure binding (receptor + radioligand + SPA beads) at various concentrations of unlabeled competitor using a single concentration of radioligand (≤Kd) and the determined optimum incubation time, optimum receptor concentration and optimum SPA bead amount. Total binding is determined in the absence of any added competitor. Nonspecific binding (receptor + radioligand + excess unlabeled competitor + SPA beads) is included for calculation of specific binding.

Results Analysis: Plot specific bound (Bound - NSB) at each concentration of unlabeled competitor. Conversion to percent specific bound is performed using the following equation:

Determine IC50 using a sigmoidal dose-response (variable slope), which is also known as a four-parameter logistic nonlinear regression analysis (i.e. using GraphPad Prism) as shown in the equation below:

where Y is the specific binding and X is the log concentration of competitor.

Calculate the equilibrium dissociation constant for the unlabeled compound (Ki) using the Cheng-Prusoff equation (valid when Hill Slope is near unity):

Ki = IC50/[1 + ([L]/Kd)]

where

Ki is the equilibrium dissociation constant for the unlabeled compound
IC50 is the concentration causing 50% inhibition of binding
[L] is the concentration of radioligand
Kd is the equilibrium dissociation constant for the radioligand
Further calculation details on the Cheng-Prusoff equation can be found in the Appendix.

A representative heterologous competition curve is similar to the one shown in the homologous competition section.

Several assumptions, based on specific criteria, are made to allow calculations using the Cheng-Prusoff equation to be reliable:

Law of Mass Action applied (10-90% of displacement occurs over 81-fold concentration range)
A single class of receptor binding sites
No ligand depletion
Receptor concentration < Kd
Assay is at equilibrium or steady state
The concentration of the added inhibitor is equal to the free concentration of the inhibitor

For special cases associated with high affinity compounds, where ligand depletion must be accounted for, see page 37 of this section.

Pharmacological Profile

A pharmacological profile is a heterologous competition testing several unlabeled compounds simultaneously. The Ki for each compound can be computed and compared to each other. A rank affinity can also be calculated. The data below demonstrates a typical pharmacological profile with representative IC50, Ki and rank affinity data shown in the table.

Notice that different concentration ranges may be required for each drug to fully define top and bottom portions of the curves

The IC50 is determined from experimental data, the Ki is calculated using the Cheng-Prusoff equation and the Relative Affinity is relative to a particular compound of interest (Drug 1 in this example).

Drug	IC₅₀, nM	Ki, nM	Relative Affinity
1	103	53	1.00
2	190	95	0.56
3	0.25	0.13	424
4	8.9	4.5	11.9
5	30.9	15.5	3.4

[L] = 0.025

Kd = 0.025

Relative Affinity = IC50 for Drug 1/IC50 for Drug

A typical plate setup for competitive binding is shown in the Plate Layout section of the Appendix. A control compound is tested on each plate and can be used for determination of the relative affinity. This process aids in analyzing the statistical significance of differences between the individual compounds.