Residual dipolar coupling

Traditionally, NMR structure elucidation has relied upon analysis of chemical shifts, coupling constants, and NOEs. In most cases this is sufficient but occasionally mistakes have been made. Recently, additional NMR observables have been introduced to the structure elucidation toolkit that reduce the chances of getting things wrong. One of these observables is residual dipolar coupling or RDC.

Most users of solution NMR are familiar with the concept of scalar coupling; the splitting of an NMR signal of one nucleus due to another nearby nucleus connected by covalent bonds. This scalar coupling is mediated by the electrons in the bond or bonds that link the two nuclei. Dipolar coupling, however, is direct coupling of the two nuclei through space.

The figure below shows two nuclei, I and S, in a static magnetic field, B₀. the distance between the two nuclei is r. The dipolar coupling, D, between these two nuclei is given by the formula on the right, where μ₀ is the permeability of free space, γ is the gyromagnetic ratio, ℏ is Planck's constant divided by 2π, and θ is the angle the vector between I and S makes with the magnetic field. The angle brackets indicate that the term inside is averaged over space and time.

Entering some reasonable values into the equation gives values for D of ±100,000 Hz! Much larger than the roughly 8 Hz seen for ¹H-¹H scalar coupling or 150 Hz for ¹H-¹³C. But if dipolar coupling is so large then why is it not normally observed? The reason is the averaging term in the equation. In solution, molecules are free to move in any direction. This means that while the position of the two nuclei I and S may be fixed relative to each other, their orientation to the magnetic field can change over time, or be different from one molecule to the next. The result is that cos²θ adopts a range of values that averages to zero, rendering the dipolar coupling zero as well.

To observe dipolar coupling in solution the motion of the molecules needs to be modified so that it is no longer random and cos²θ does not average to zero. However, we don't want to restrict motion completely because then the dipolar couplings would be so large that the spectra would be uninterpretable. Instead, a partial alignment of the molecule with the magnetic field is desired so that a "residual" dipolar coupling can be measured.

Several different methods of inducing partial alignment have been developed. Most involve adding large dipolar molecules or molecular assemblies to the sample. Phage¹, liquid crystals², and bicelles³ have all been used. Another method, more amenable to small molecule work, is to soak the sample into a stretched or compressed gel⁴. Deformation of the gel elongates the pores in a controlled manner. This biases the motion of the sample within the pores, inducing partial alignment, and enabling the measurement of RDCs. Most alignment methods allow the solution conditions to be adjusted so that the RDCs can be scaled up or down to give values that can be easily and accurately measured.

This seems like a lot of trouble to go to to measure some obscure NMR parameter. Why would anyone want to measure RDCs? Most NMR parameters provide local, short-range information. For example, chemical shifts report on the nuclei within two or three bonds, scalar coupling is used to infer orientation about three bonds, and NOEs are only observed out to 5 or 6 Å. RDCs, however, through their proportionality to θ, report on the orientation of a pair of nuclei with respect to the orientation of the entire molecule. If multiple RDCs can be measured then the orientation of bonds, in distant parts of the molecule, can be fixed. This is very useful information that can be used to define stereochemistry or identify the correct structure from a list of possibilities^5,6.

References

1. Hansen MR, Mueller L, Pardi A.
Tunable alignment of macromolecules by filamentous phage yields dipolar coupling interactions.
Nat Struct Bio 1998 Dec;5(12):1065-74

2. Tjandra N and Bax A
Direct measurement of distances and angles in biomolecules by NMR in a dilute liquid crystalline medium.
Science 1997 Nov 7;278(5340):1111-4

3. Sanders CR and Prestegard JH.
Magnetically orientable phospholipid bilayers containing small amounts of a bile salt analogue, CHAPSO.
Biophys J. 1990 58:447–46

4. Luy, B., Kobzar, K. and Kessler, H
An Easy and Scalable Method for the Partial Alignment of Organic Molecules for Measuring Residual Dipolar Couplings.
Angewandte Chemie International Edition, 2004 43:1092-1094

5. Thiele CM
Residual Dipolar Couplings (RDCs) in Organic Structure Determination.
Eur. J. Org. Chem., 2008: 5673-5685.

6. Liu Y, Saurí J, Mevers E, Peczuh MW, Hiemstra H, Clardy J, Martin GE, Williamson RT.
Unequivocal determination of complex molecular structures using anisotropic NMR measurements.
Science. 2017 Apr 7;356(6333). pii: eaam5349.