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Residual Dipolar Coupling

Figure 1. Depiction of interaction between a protein and bicelle at their interface.

Residual dipolar couplings (RDCs) had been observed as early as 1963 [1] in nematic environments. A number of recent applications [2-7] have reignited their wide use in application to a broad spectrum of biomolecules. RDCs have been used in studies of carbohydrates [8-10], nucleic acids [11-13] and proteins [14-16] to mention a few. Residual dipolar couplings can be acquired very rapidly and accurately by a number of techniques including direct measurement of splittings in coupled heteronuclear single quantum coherence spectra (HSQC) [17-19] and provide simultaneous structural [4;15;20] and motional [5;7;13;21-23] information.
RDCs arise from the interaction of two magnetically active nuclei in the presence of the external magnetic field of a NMR instrument [3;4]. This interaction is normally reduced to zero due to the isotropic tumbling of molecules in their aqueous environment. The introduction of a partial order to the molecular alignment by minutely limiting their isotropic tumbling will resurrect the RDC observable. This partial order can be introduced by either magnetic anisotropy of the molecule [3], a crystalline aqueous solution [24] as illustrated in Figure1 or incorporation of artificial tags with magnetic anisotropy susceptibility such as Lanthanide [25]. Equation 1 describes the time average observable of the RDC interaction between a pair of spin ½ nuclei.

Here, Dij denotes the residual dipolar coupling in units of Hz between nuclei i and j, γi and γj are nuclear magnetogyric ratios, rij is the internuclear distance (assumed fixed for directly bonded atoms) and θij(t) is the time dependent angle of the internuclear vector with respect to the external magnetic field. The angle brackets signify the time average of the quantity.
Simple algebraic manipulation of equation 1 produces the more familiar formulation of RDC interaction shown in Equation 2.

$D_{\mathit{ij}}=D_{\mathit{max}}\underset{k,l}{\sum}{s_{\mathit{kl}}\cos \theta _{k}^{\mathit{ij}}\cos \theta_{l}^{\mathit{ij}}}$ (2)
The indexes k and l in Eq 2 denote the orthonormal basis sets that span the Cartesian space $(\hat{i}, \hat{j}, \hat{k})$ and $D_{\mathit{max}}=-{\mu _{0}\gamma _{i}\gamma _{j}h}/{(2\pi r)^{3}}$. Entities skl in this equation represent various components of anisotropy, are referred to as the elements of the Saupe order tensor matrix, and are defined by Eq 3. Note that the entities $\cos \theta_{k}^{\mathit{ij}}$ represent the direction cosine of the vector connecting nuclei i and j to the k-th axis of the molecular frame. Equation 4 illustrates a more familiar form of this equation after expansion of the two embedded summations in Eq 2.
$s_{\mathit{kl}}=\left\langle \frac{3\cos \left(\theta _{k}\right)\cos\left(\theta _{l}\right)-\delta _{\mathit{kl}}}{2}\right\rangle$ (3)

$D_{\mathit{ij}}=D_{\mathit{max}}\left(\cos ^{2}\theta_{x}^{\mathit{ij}}s_{\mathit{xx}}+\cos ^{2}\theta_{y}^{\mathit{ij}}s_{\mathit{yy}}+\cos ^{2}\theta_{z}^{\mathit{ij}}s_{\mathit{zz}}+2\cos \theta _{x}^{\mathit{ij}}\cos\theta _{y}^{\mathit{ij}}s_{\mathit{xy}}+2\cos \theta_{x}^{\mathit{ij}}\cos \theta _{z}^{\mathit{ij}}s_{\mathit{xz}}+2\cos\theta _{y}^{\mathit{ij}}\cos \theta_{z}^{\mathit{ij}}s_{\mathit{yz}}\right)$ (4)

Bibliography

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Written by homayoun

October 9th, 2009 at 1:37 pm