Chemical shift titrations

NMR is one of the most useful techniques for monitoring molecular interactions. It is particularly useful for characterizing weakly interacting systems. Typically, weakly interacting systems produce spectra that are in fast exchange, i.e. the NMR observables are detected as population weighted averages. By manipulating the populations, through changing concentration for example, a series of spectra can be obtained from which information like the dissociation constant (K_D) and the dissociation rate (k_off) may be determined. Titrating one molecule with another and monitoring the chemical shift changes is a common method for determining K_D.

The stackplot below shows a chemical shift titration. The blue spectrum at the bottom shows the pure macromolecule, a derivatised β-cyclodextrin, whose resonances all appear below 2.9 ppm. In each successive spectrum an aliquot of a ligand, rimantadine, has been added. Its resonances appear above 2.9 ppm and increase in intensity as more is added for each spectrum. The ratio of ligand to macromolecule is indicated on the left of the stackplot.

As the ligand concentration increases, resonances from both the ligand and the macromolecule move, indicating that the molecules interact and that the system is in fast exchange. Under fast exchange the position of each of the averaged resonances (δ_ave) is weighted by the populations of each of the states present. Assuming the presence of only two states, free and bound, simplifies analysis of the system and is likely to be the case here. Thus,

δ_ave = δ_b⋅P_b + δ_f⋅P_f

Since the population of the bound state (P_b) depends on the dissociation constant (K_D) we can determine K_D by fitting the equation above to a graph of the change in chemical shift versus P_b. The population of the free state (P_f) is calculated using P_b + P_f = 1, while P_b is calculated from K_D and the total concentration of macromolecule (M) and ligand (L) in both free and bound states, using the equation below.  

P_b = (M+L+K_D - ((M+L+K_D)² - 4⋅M⋅L)^½ ) / (2⋅L)

Simulated data calculated using these equations is shown in the graph below. The lines show the chemical shift of a macromolecule resonance as the ligand concentration is increased. For this graph the macromolecule concentration (M) was fixed at 1 mM, δ_b was set to 100 Hz and δ_f to 0 Hz. The lines show the behavior of the macromolecule resonances for K_D=1.0x10^-8 M (red), 1.0x10^-6 M (purple), 1.0x10^-5 M (blue), and 1.0x10^-4 M (green).

The plot shows the behavior of the macromolecule peaks as the ligand concentration is changed, but it is also possible to fit the ligand peaks as the macromolecule concentration is changed. In that case, the P_b equation is modified to replace L with M in the denominator. Note that to accurately determine K_D, the titration must sample points above and below the 1:1 equivalence point. Also, as K_D becomes much larger than the ligand and macromolecule concentrations it becomes increasingly difficult to determine K_D accurately. One way to increase the accuracy is to simultaneously fit data from multiple resonances.

Acknowledgments
Thanks to Katy Kellet in the Gilson Lab for use of her titration spectra.