Fitting binding of fluorescent ligands

Fitting binding of fluorescent ligands presents two challenges usually not seen with radioactive ligands:

• To get sufficient signal, the reactions are usually run so a substantial portion of the added ligand binds to receptors. This invalidates the conventional (in radioligand binding) assumption that the total (added) concentration of ligand is essentially equal to the free concentration. 
• The molar fluorescence of the ligand differs when the ligand is bound. Binding may increase or decrease the flourescence.

The equations shown here were developed by Dr. Richard Neubig at the University of Michigan.  These two equations are pretty versatile, but cover only analyses of fluorescent intensity, so don't cover all experimental situations (such as fluorescence polarization). The equations cover two experimental designs. In the first, you use a single concentration of ligand and vary receptor concentration. In the other, you use a single concentration of receptors and vary ligand concentration.

Determining the molar fluorescence

Fitting each equation below requires that you first determine the molar fluorescence of free ligand. To determine this, measure fluorescence in the absence of receptor with various concentrations of ligand. Use linear regression to find the slope, which is the molar fluorescence. You'll need to enter this value as a constant when using either equation shown below. 

Molar fluorescence can be strongly dependent on experimental conditions such as buffers, pH, and temperature. So it is critical that the value of molar fluorescence be determined under exactly the same conditions (and in the same experiment) as the binding reaction.

Vary amount of protein (receptors)

In this experimental design, you use a single concentration of ligand, and vary the concentration of receptor. The equation is:

LR= ((X+Ltot+KD)-SQRT((X+Ltot+KD)^2-4*X*Ltot))/2
L= Ltot - LR
Y= BKG + MF*L + FR*MF*LR

CONSTANTS
Ltot: Total ligand concentration (Same units as X).
BKG: Background fluorescence w/o receptors  (Same units as Y). 
MF: Molar fluorescence of free Ligand (Y units divided by X units)

PARAMETERS TO FIT
Kd: Dissociation constant (X units)
FR: Fluoresence ratio. MF of bound ligand=MF*FR (unitless ratio)
    so FR > 1 means binding increases fluorescence
       FR < 1 means binding quenches fluorescence 

Vary concentration of ligand

In this experimental design, you use a single concentration of receptor, and vary the concentration of ligand. The equation is:

LR= ((X+Rtot+KD)-SQRT((X+Rtot+KD)^2-4*X*Rtot))/2
L= X - LR
Y= BKG + MF*L + FR*MF*LR

 

CONSTANTS
Rtot: Total receptor concentration (Same units as X).
BKG: Background fluorescence w/o ligand (Same units as Y).
MF: Molar fluorescence of free Ligand (Y units divided by X units)

 

PARAMETERS TO FIT
Kd: Dissociation constant (X units)
FR: Fluorescence ratio. MF of bound ligand=MF*FR (unitless ratio)
     so FR > 1 means binding increases fluorescence
          FR < 1 means binding quenches fluorescence 

 

Vary concentration of ligand. Determine Kd and molar fluorescence in one analysis.

In this experiment, place the actual data (same as previous one) into column B. In column A, omit the receptors, so it can assess MF. An alternative is to include the receptors, but also include a huge concentration of unlabeled ligand that will block all receptor binding sites.

 

<A>Y=BKG + MF*X

<B>LR= ((X+Rtot+KD)-SQRT((X+Rtot+KD)^2-4*X*Rtot))/2

<B>L= X - LR

<B>Y= BKG + MF*L + FR*MF*LR

 

CONSTANTS

Rtot: Total receptor concentration (Same units as X).

BKG: Background fluorescence w/o ligand (Same units as Y).

 

PARAMETERS TO FIT

Kd: Dissociation constant (X units)

MF: Molar fluorescence of free Ligand (Y units divided by X units)

FR: Fluoresence ratio. MF of bound ligand=MF*FR (unitless ratio)

     so FR > 1 means binding increases fluorescence

          FR < 1 means binding quenches fluorescence 

 

Note that the equation line preceded by <A> only applies to the data in column A, and the lines preceded by <B> apply to data set B. Global fitting is used to determine one value of MF (which is shared) for both data sets. In the example below, the FR is greater than 1.0, so the signal with receptors is brighter than the signal without. Some ligands are quenched when bound to receptors, and in those cases the signal with receptors would be lower than the signal without receptors.

 

 

This Prism file contains these equations and the associated Prism graphs. Note that each equation has one or two constants, and these are included in the Prism analses. You'll need to open the parameters dialog and change that constant to the appropriate value for your experiment.