Hi everybody,
I would be grateful if someone could give me some suggestions on how to properly fit binding data, in particular using the Hill equation.
I often use biophysical techniques (e.g. SPR, FP, etc.) to quantify protein-protein interactions. In several cases, it happened to me that the data could not be easily fitted by the classical "1:1 binding" model. In particular, when experimental points lie on a steep sigmoidal curve, I used the Hill equation to fit my data, but this raised some doubts to me.
- First, could the Hill equation account for biological phenomena other that binding cooperativity (e.g. ligand heterogeneity)? Otherwise, are there better models to be used in those cases in which no cooperativity is expected?
- Second, in the Hill equation F=(ch)/(ch + kAh) [F is fractional occupancy of binding sites; c is the concentration of ligand, kA is the half-maximal ligand concentration and h is the Hill coefficient], the dissociation constant (Kd) corresponds to kAh . Since this is a power function of concentration, the unit by which concentration is expressed heavily impacts on the fitting value of Kd. How should concentrations be expressed?
Thanks in advance to anyone that could give me some hints.
Best,
Filippo
Remember that equilibrium constants are dimensionless (i.e., no units). That is achieved by dividing all concentrations or activities in the expressions of the equilibrium constants by the concentration of the standard state (e.g., c0 = 1 M). This will fix the scale for the Gibbs energy. Then, selecting the unit for the concentration, the Kd will be estimated.
If there is no possibility of cooperativity, the Hill coefficient h should be set equal to 1. If you find that the dat are not well fit by the Hill equation, and you expect a 1:1 binding stoichiometry, and you need a Hill coefficient greater than 1 to fit the data, a probable reason is that you are in tight-binding mode.
In the binding isotherm equation (the Hill equation with h=1) , the concentration of the titrant is the free concentration, not the total concentration. If the binding is tight (i.e., the Kd is similar to the receptor concentration or lower), the total and free concentrations will differ significantly. If the total titrant concentration is used for the curve fitting, the data won't be well-fit by the Hill equation.
Instead, you should use the tight binding (Morrison) equation, which has a quadratic form, and in which the total titrant concentration is used.
Binding heterogeneity can be modelled by the sum of 2 binding equations. I have done that in the case of ATP binding to Hsc70 (DOI:10.1042/bj3180923). If there are more than 2 different binding sites, the statistics becomes difficult because of the large number of parameters.
The number of binding sites can be determined by a Job-plot (Ann.
Chim. (Paris) 9 (1928), 113–203)
In normal binding, the free ligand concentration is F = T - B, and if you replace F in the binding equation you get a quadratic equation, the Langmuir-isotherm (DOI:10.1021/ja02242a004) that can be used for fitting B as a function of T if F is significantly different from T.
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