Hi Gary,

To get the binding stoichiometry from ITC data you need to work at protein concentrations where the value of c, defined as  Kb ×[P] × n, was  ≥ 5. In fact, five is rather in the limit. Unless you can work in such conditions, the stoichiometry should be fixed based on additional information. The risk of doing so, is that usually we do not know if all the protein molecules in the sample are functional or not.

Whether your ligand is small molecule or another protein (which probably is being titrated) you need to start with 5-10 fold higher stock concentration than the Kd.

Another experiment is worth performing with new stock concentrations.

Calorimetric titrations are usually plotted employing the Wiseman representation (normalized heat per peak, Q, as a function of molar ratio, XR). This binding isotherm has an inflection point located at a molar ratio XR = 1 - 1/c, where c = Ka x Pt (product of the association constant and the total protein concentration in the cell for a 1:1 interaction). For a 1:n interaction a factor n should be added, but let’s assume we are dealing with a 1:1 interaction for now on. Therefore, the curve will exhibit an inflection point if c is equal or larger than 1. By the way, the deflection of the isotherm at the inflection point is always equal to Q = dH/2 (where dH is the binding enthalpy, and considering the background injection heat is subtracted).

The indication from Margarita ensures the inflection point is located close enough to the equivalence point (XR = 1), because at c = 1 the inflection point will coincide with the y-axis intercept (XR = 0), but at c = 5 the inflection point will be located at XR = 0.8.

When the binding affinity is high (c much larger than 1) the location of the inflection point is well defined and it may be close to XR = 1 or not. If it is close to 1 we may think that the stoichiometry is right and both concentrations (titrant and titrand) are well determined. If it is not close to XR = 1 we could think that: 1) the concentration of the titrant or the titrant (or both) are not well determined; or 2) the stoichiometry is not right. Regarding the first possibility, when studying protein-ligand interactions it is common to observe that the active fraction of protein is not 1, but a fractional value because not all protein molecules are well folded in their binding-competent conformation, or there are some contaminants. That problem usually does not arise with the ligand (unless it is a molecule prone to easy degradation). Thus, even if we observe an inflection point located close to XR = 1, there is a possibility to have errors in both titrant and titrand concentrations canceling out to give an apparent right location of the inflection point. The message is: if you can measure the “active” concentration of one of the two reactants, you can use that for calibrating the concentration of the other reactant (that is the purpose of a “titration”).

When the binding affinity is low (c smaller than 1) there is a problem associated with the fitting analysis because the absence of inflection point increases the correlation between the three binding parameters (association constant, Ka, enthalpy, dH, and stoichiometry, n). The good news is that usually Ka is weakly dependent on n; the bad news is that dH is strongly dependent on n. Therefore, if we are only interested on the binding affinity we may fix the value of n.

If we are interested on dH we should either fix n to a reasonable value or let it adjustable. The usual problem is that the correlation between n and dH results in no convergence or unexpectedly low or high dH values. Thus, very often we must fix n through the fitting. Which is the appropriate value for fixing n? We need additional information (e.g. another technique or another titration with a calibrating ligand). But, in the absence of that additional information we must select a reasonable value. For a 1:1 titration that value should be n = 1.

This constraint may seem very strong for many people, but I would say that this is actually what everybody does when analyzing spectroscopic titrations. The parameter n is not considered in spectroscopic titrations. On the contrary, it is assumed that the concentrations are well determined and there is 100% of active titrant and titrand. The reason is that spectroscopic titrations are not as sensitive to n as calorimetric titrations (the calorimetric titration is a finite-difference plot, whereas the spectroscopic titration is a cumulative plot). Because the calorimetric titration is more sensitive and provides that information, the problem arises. But nothing is questioned ever in spectroscopic titrations.

Summarizing, when the binding affinity is low and there is no inflection point, you have some ways to overcome the difficulties associated with data analysis: 1) obtain additional information on the “active” concentration of reactants (e.g., titrate the protein with a stronger ligand, change the experimental conditions to increase the binding affinity, increase the concentration of reactants) in order to employ the right concentrations through the data analysis; 2) fix n to the expected value (as commented above, this will provide a good estimation of the association constant, but the estimation of the enthalpy might be questionable).

I would second what Adrian has said and just add a couple of points.

First, for c > 1 in 1:1 binding, n and DH are estimated with greater precision than K, but dropping below c = 1, n and DH become increasingly correlated and imprecise, such that at some point you would HAVE to freeze one of these in the fit; clearly n is the one to freeze.

Fortunately, K is essentially decoupled from n and DH at low c, so you get good estimates of it regardless.  If you can do temperature-dependent experiments over a range where the mechanism remains 1:1 binding, you can then get DH from the T-dependence of K (van't Hoff method) and accordingly, you can also get n.  I discussed this in a short note on low-c work published in Anal. Biochem. in 2008.  Note that the precisely determined quantity in ITC is the product n x DH.  The van't Hoff DH is typically less precise than the calorimetric estimate where the latter can be reliably obtained (c > 1), but it will be better for smaller c, since K remains well determined.

With enough precision, your resulting estimates of n may still differ from what you expect for your system by up to ~10%.  That is because n absorbs errors in the active cell volume, and some widely used instruments appear to have actual cell volumes that differ by 5% or more from the values stated by the manufacturer.

Hi Gary,

ITC  practically measures dH and the fits are made to give you Ka (or Kd), dH and "n". Said that, the issues come when you give the values of the concentrations of your receptor (protein) and ligand (small molecule). Its very common that the actual concentration of your ligand/receptor is far from what you think is in the experiment. Generally in case of protein and ligand, its the ligand thats tricky to handle. some times its not soluble, some times you weigh some thing on a balance but you don't have it in the solution as much as you think you weighed. So, the first and foremost thing you should pay attention is to get the concentrations of your Ligand and Receptor right before you fix the "n" (stoichiometry).

Hope this helps

Good Luck with your analysis

Venu 

Dear Gary,

I recommend to read this publication from Turnbull and Daranas. They outlined the importance of the c-value for low affinity systems.

Just be careful with the interpreatation ot the deltaH values.

Regards

Michael