When a macromolecule has n independent and identical binding sites for a ligand (if they are not identical or not independent, the model gets more complicated), the binding polynomial is given by:

Z = (1+K[L])n

from which the number of ligand molecules bound per macromolecule is given by:

nLB = dlnZ/dln[L] = nK[L]/(1+K[L])

which is the expression for a single binding site with a multiplicative factor n. The expression for the total ligand becomes:

[L]T = [L] + [L]B = [L] + [M]TnLB = [L] + [M]TnK[L]/(1+K[L])

The analysis of the binding isotherm is based on solving the previous equation for [L] as unknown, and calculating the expected heat effect at each of the ligand additions (injections or experimental points). The parameter n accounts for the fact that n ligand molecules can interact simultaneously with the macromolecule (stoichiometry 1:n), although, below saturating conditions (ideally at infinite or very high ligand concentration) all different type of complexes (MLm, m

When a macromolecule has n independent and identical binding sites for a ligand (if they are not identical or not independent, the model gets more complicated), the binding polynomial is given by:

Z = (1+K[L])n

from which the number of ligand molecules bound per macromolecule is given by:

nLB = dlnZ/dln[L] = nK[L]/(1+K[L])

which is the expression for a single binding site with a multiplicative factor n. The expression for the total ligand becomes:

[L]T = [L] + [L]B = [L] + [M]TnLB = [L] + [M]TnK[L]/(1+K[L])

The analysis of the binding isotherm is based on solving the previous equation for [L] as unknown, and calculating the expected heat effect at each of the ligand additions (injections or experimental points). The parameter n accounts for the fact that n ligand molecules can interact simultaneously with the macromolecule (stoichiometry 1:n), although, below saturating conditions (ideally at infinite or very high ligand concentration) all different type of complexes (MLm, m

I see. So if there is a misinterpretation of the n value as a number of ligand binding site, is that mostly caused by fitting the data to a wrong binding polynomial? What could be some sources that could yield a wrong n as a number of ligand binding site? Thanks!

It is not really a problem of using a wrong binding polynomial, since there are not many possibilities.

Either you have one or more than one binding site. Then, in case there are more than one binding site (e.g., more than one inflection point, or asymmetric isotherm), then you must decide if the sites are identical or not, independent or not.

If the binding sites appear to be identical and independent, the binding polynomial is the one I wrote before. In that case the parameter n will represent the stoichiometry (number of binding sites), but, because it appears multiplying the total concentration of protein in cell, it may also represent the fraction of active protein in the cell. Due to this coupling, the parameter "n" might be difficult to interpret. For example, if you get a value n = 1.5, what can you infer? Well, maybe there is a single binding site, but there is more protein in the cell than expected, or less ligand in the syringe than expected. Or there are two identical, independent binding sites but there is less protein in the cell than expected, or more ligand in the syringe than expected. I fact, getting a value n = 1 does not guarantee your estimated parameters are correct, because the two concentrations, in the cell and in the syringe, may have some errors canceling out, but the binding affinity and the binding enthalpy will be erroneous. Do not forget that the parameter n reflects the position of the inflection point (if there is an inflection point) on an x-axis that is the ratio of two concentrations (molar ratio).

In other binding polynomials the number of binding sites and the fraction of active protein appear decoupled. For example, in a macromolecule with two different and independent binding sites, the stoichiometry is explicitly considered (two factors in the binding polynomial), while the fraction of active protein is accounted for a parameter "n" that in this case does not represent the stoichiometry.

Dear all

i m also confused about the value of n.should i fix the value of n=1 in fitting the data ?or let it be vary to get different value like 1.5 /0.025 /65 (generaly i get this type of n value in different experiment by fitting it one site of binding)

waiting for reply

Thanks Adrian Velazquez-Campoy for  sharing your insight.  Could you also elaborate on what Z and K are in your answer?

In all previous comments Z is the binding polynomial, K is the association constant (its inverse is the dissociation constant Kd=1/K), [L] is the concentration of free ligand, [L]T is the total concentration of ligand, [L]B is the concentration of bound ligand, [M]T is the total concentratioin of macromolecule.

The binding polynomial is the partition fucntion of the system (macromolecule interacting with ligand) that is defined by the characteristics of the interaction: number of binding sites, type of binding sites (identical or non-identical, independent or cooperative), and the energetics of the interaction (association constants). Once you define the binding polynomial, all other quantities (average number of ligand molecules bound per macromolecule, average interaction enthalpy) derive from the binding polynomial through derivatives.

@Samima Khatun, care should be taken with the parameter n, even if you know the stoichiometry. In general n should be an adjustable parameter. For example, let's assume c= 100 and the stoichiometry is 1:1. Then, n equal to 1 is expected, but there might be some errors in the reagents concentrations that result in an inflection point located in 0.75, and if you fix n to 1, there is no way to get a good fit. In general, for c>1 you should not fix n, because it will be determined by the location of the inflection point. For c