Biomedical Engineering Reference
In-Depth Information
field for half of the carbon nuclei and subtract for the other half.
51
Consequently, the carbon resonance frequency splits into a doublet, reflecting
the addition and subtraction of the average proton field. The magnitude of this
splitting is referred to as a 'residual dipolar coupling'.
20,52
Through-space
dipolar couplings (D) and through-bond scalar couplings (J) both effectively
increase or decrease the average magnetic field at a given nucleus, which
manifests in a splitting of resonances. This makes it possible to readily measure
RDCs as new contributions to splittings when a molecule is partially aligned
[Figure 9.1(C)].
At high magnetic fields, the dipolar interaction can be simplified to a
truncated dipolar Hamiltonian,
53,54
resulting in the following expression (in
Hz) describing the local field contribution between nuclei i and j:
c
i
c
j
h
2p
2
r
ij,ef f
S
3 cos
2
h{1
2
D
ij
~{
m
0
4p
T,
ð
9
:
1
Þ
where m
0
is the magnetic permittivity of a vacuum, h is Planck's constant, r
ij
is
the inter-nuclear distance between the spins, and c is the gyromagnetic ratio.
The angular bracket denotes a time average over all orientations sampled,
while distance averaging of the inter-nuclear distance is represented by the
effective bond length r
ij,eff
.
55-57
The utility of RDCs in studies of dynamics arises chiefly from the angular
dependence of eqn (9.1),
22
rendering RDCs sensitive to internal motions that
re-orient bond vectors at timescales faster than the inverse of the dipolar
interaction. For typical levels of alignment, this encompasses a wide range of
timescales spanning picoseconds to y10 milliseconds.
22
Although RDCs do
not provide information about motional timescales, they are exquisitely
sensitive to the orientation distribution sampled by the bond vector and,
therefore, the 3D choreography of the motion.
23,50,58
In addition, a wide
variety of RDCs can be measured in nucleic acids (C-H, C-C, C-N, N-H, H-
H,
P-H,
etc.),
providing
the
basis
for
comprehensively
mapping
out
nucleobase, sugar, and phosphodiester backbone dynamics.
9.2.2 The Alignment Tensor
Central to the dynamic interpretation of RDCs is a description of the overall
alignment of a molecule and specifically, the contributions to the angular term,
S
3cos
2
h
{
2
T, arising due to overall re-orientation. In general, overall re-
orientation dominates the averaging of this angular term, scaling its value
down by a factor 10
24
compared to typically only 10
21
due to internal
motions. The overall alignment of an internally rigid molecule relative to the
magnetic field and any observed RDCs can be fully accounted for by
specifying five elements of a traceless and symmetric overall order or alignment
tensor.
21,41
The order tensor describes the orientation distribution of the
axially symmetric magnetic field direction relative to the chiral molecular
