Linewidth titrations

NMR is one of the most powerful techniques to monitor molecular interactions because of the wealth of information it provides. The previous post described monitoring chemical shift changes during titrations, but chemical shifts are not the only NMR parameter that can change during titrations. Often linewidths change as well. By measuring linewidth changes the dissociation rate, k_off, may be determined.

Under conditions of fast exchange, linewidths, like chemical shifts, are observed as the population weighted average of the linewidths in all the states present. Linewidth changes, however, are typically smaller than chemical shift changes and so are less likely to fall into the fast exchange regime. Often they appear under "intermediate exchange" where behavior is more complex.

In the panel below the stackplot on the left shows intermediate exchange spectra simulated with the software LineShapeKin¹. Over the course of the titration the peak at zero frequency shifts downfield, broadens, then sharpens. Note that the size of the chemical shift change is not constant. Also, the area of the peak, or integral, is the same in each 1D spectrum, but as the peak broadens its intensity decreases. The 2D spectrum on the right of the panel shows an experimental titration of an ¹⁵N labeled protein, PDZ1, with an unlabeled protein, PBM². Only the ¹⁵N labeled PDZ1 is detected. As the titration proceeds many peaks move, and some disappear then reappear. The peaks that disappear then reappear (e.g. Leu34 at 9.3,126.0 ppm) are in intermediate exchange and have broadened enough that their intensity drops below the lowest threshold of the contour plot. Other peaks that are in fast exchange (e.g. Arg85 at 9.1,126.0 ppm) show constant intensity throughout the titration. This demonstrates that for one molecular interaction different resonances can fall into different exchange regimes. Since K_D and k_off are the same for all resonances, the exchange regime for each resonance is determined by the size of the chemical shift or linewidth change for that particular resonance.

Under fast exchange, the linewidth (λ) observed is the population weighted average of the linewidths from each state present in the exchanging system, just like chemical shifts (δ). Under intermediate exchange, however, things are more complex. To describe these linewidth changes the following expression for the observed linewidth (λ_obs) in a two-state system undergoing intermediate exchange has been derived³.

λ_obs = P_f⋅λ_f + P_b⋅λ_b + (P_b⋅P_f²⋅4π(δ_b-δ_f)²)/k_off

The final term in this equation, the "exchange contribution to linewidth", includes k_off, so measuring changes in linewidth with concentration will enable one to determine k_off. The graph below shows data simulated using the equation above. The macromolecule concentration was fixed at 1 mM, the change in chemical shift was set to 100 Hz and the change in linewidth was set to 10 Hz. Each line shows the lineshape behavior for different combinations of K_D and k_off. At lower values of k_off (blue and orange lines) the linewidth increases, then decreases. For larger k_off values (yellow and green lines) the observed linewidth assymptotes to the bound linewidth but never gets larger than it. In these latter two cases the exchange contribution to linewidth becomes small and the system approaches the fast exchange regime. For small K_D values (blue and yellow lines) the linewidth approaches the bound linewidth when the ligand concentration equals the macromolecule concentration, while for larger values of K_D (orange and green lines) it continues to change past this point.

Under intermediate exchange, linewidth and chemical shift behavior can be quite complex. To extract reliable information it is best to simultaneously fit all the spectra acquired during a titration. Software for performing these types of analyses are available. For analysis of 1D spectra LineShapeKin¹ seems to be the best package, while for 2D spectra TITAN⁴ is popular.

References
1. Evgenii L. Kovrigin
"NMR line shapes and multi-state binding equilibria"
J. Biomol. NMR 2012 53(3), 257-270

2. Fabian A. Renschler, Susanne R. Bruekner, Paulin L. Salomon, Amrita Mukherjee, Lars Kullmann, Mira C. Schütz-Stoffregen, Christine Henzler, Tony Pawson, Michael P. Krahn, Silke Wiesner
"Structural basis for the interaction between the cell polarity proteins Par3 and Par6"
Sci. Signal. 2018 11(517), eaam9899

3. James Feeney, J.G. Batchelor, J.P. Albrand, Gordon C.K. Roberts
"The effects of intermediate exchange processes on the estimation of equilibrium constants by NMR"
J. Magn. Reson. 1979 33(3), 519-529

4. Christopher A. Waudby, Andres Ramos, Lisa D. Cabrita, John Christodoulou
"Two-Dimensional NMR Lineshape Analysis"
Scientific Reports 2016 6, 24826