The residual dipolar coupling between two spins in a molecule occurs if the molecules in solution exhibit a partial alignment leading to an incomplete averaging of spatially anisotropic dipolar couplings. Partial molecular alignment leads to an incomplete averaging of anisotropic magnetic interactions such as the magnetic dipole-dipole interaction (also called dipolar coupling), the chemical shift anisotropy, or the electric quadrupole interaction. The resulting so-called residual anisotropic magnetic interactions are useful in biomolecular NMR spectroscopy.
History and pioneering works NMR spectroscopy in partially oriented media was reported by Alfred Saupe. After this initiation, several NMR spectra in various liquid crystalline phases were reported (see e.g. ). A second technique for partial alignment that is not limited by a minimum anisotropy is strain-induced alignment in a gel (SAG). The technique was extensively used to study the properties of polymer gels by means of high-resolution deuterium NMR, but only lately gel alignment was used to induce RDCs in molecules dissolved into the gel. SAG allows the unrestricted scaling of alignment over a wide range and can be used for aqueous as well as organic solvents, depending on the polymer used. As a first example in organic solvents, RDC measurements in stretched polystyrene (PS) gels swollen in CDCl3 were reported as a promising alignment method. In 1995, NMR spectra were reported for cyanometmyoglobin, which has a very highly anisotropic paramagnetic susceptibility. When taken at very high field, these spectra may contain data that can usefully complement NOEs in determining a tertiary fold. In 1996 and 1997, the RDCs of a diamagnetic protein ubiquitin were reported. The results were in good agreement with the crystal structures.
Physics
The secular dipolar coupling Hamiltonian of two spins, I {\displaystyle I} and S , {\displaystyle S,} is given by:
H D = ℏ 2 γ I γ S 4 π r I S 3 [ 1 − 3 cos 2 θ ] ( 3 I z S z − I → ⋅ S → ) {\displaystyle H_{\mathrm {D} }={\frac {\hbar ^{2}\gamma _{I}\gamma _{S}}{4\pi r_{IS}^{3}}}[1-3\cos ^{2}\theta ](3I_{z}S_{z}-{\vec {I}}\cdot {\vec {S}})}
where
ℏ {\displaystyle \hbar } is the reduced Planck constant.
γ I {\displaystyle \gamma _{I}} and γ S {\displaystyle \gamma _{S}} are the gyromagnetic ratios of spin I {\displaystyle I} and spin S {\displaystyle S} respectively.
r I S {\displaystyle r_{IS}} is the inter-spin distance.
θ {\displaystyle \theta } is the angle between the inter-spin vector and the external magnetic field.
I → {\displaystyle {\vec {I}}} and S → {\displaystyle {\vec {S}}} are vectors of spin operators. The above equation can be rewritten in the following form:
H D = D I S ( θ ) [ 2 I z S z − ( I x S x + I y S y ) ] {\displaystyle H_{\mathrm {D} }=D_{IS}(\theta )[2I_{z}S_{z}-(I_{x}S_{x}+I_{y}S_{y})]\!}
where
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