Biomedical Engineering Reference
In-Depth Information
principle more sensitive spatial probe of structure and dynamics. Methodsto
fully
harness
this
sensitivity
in
studies
of
dynamics
remain
to
be
fully
established.
9.5
Dynamic Interpretation of RDCs Measured in RNA
9.5.1 Dynamics Information Contained Within RDCs
To appreciate the full angular dynamic information contained within RDCs,it
is useful to use a spherical tensor representation to express the measured time-
averaged dipolar tensor element SD
0
T in terms of the overall alignment tensor,
O
2
m
, of the molecule and 5 out of 25 time-averaged Wigner rotation elements,
SD
n0
(bc)T [Figure 9.4(A)]. These elements are functions of the Euler angles
(bc) describing the orientation of the bond vector relative to the molecular
frame:
27,71,121
SD
0
T
l
~
X
X
2
2
O
2
m
(PAS)
l
D
2
mn
(h
l
)SD
n0
(bc)T
ð
9
:
6
Þ
m~{2
n~{2
Here, O
2
m
(PAS)
l
are elements of the l
th
overall order/alignment tensor
describing averaging of the dipolar interaction due to overall motions
expressed in the PAS of the tensor. D
2
mn
(h
l
) are elements of a time-independent
Wigner rotation matrix that transform the PAS of the l
th
overall tensor into a
common molecular frame. Importantly, eqn (9.6) assumes that the internal and
overall motions of the molecule are uncorrelated to one another.
The information regarding internal motions is contained within the five
time-average Wigner elements SD
nk
(abc)T ({n}522, 21, 0, 1, 2) which are
trigonometric functions of two Euler angles describing the orientation of the
bond vector relative to the chiral frame (Table 9.3). The five time-averaged
Wigner elements can be determined experimentally for each bond vector
provided the measurement of RDCs under five linearly independent alignment
conditions, as shown elegantly by Griesinger and Tolman.
26,28
Like the overall
order tensor, these five Wigner elements—like the overall order tensor, can be
parameterised into an alignment/order tensor, except in this case, the tensor
describes the internal dynamics of the axially symmetric RDC bond vector (as
opposed to the magnetic field direction) relative to a chiral molecular
frame.
26,28,71
The five parameters specify the average orientation of the bond
vector relative to the chiral frame, the amplitude of any internal motions,as
well as the extent and direction of motional asymmetry. Note that due to the
inherent axial symmetry of the dipolar interaction, there is no sensitivity to
internal motions that lead to rotations about the bond vector itself (a),
therefore limiting sensitivity to only two of the three Euler angles [eqn (9.6)].
The bond-vector-type analysis of RDCs has been successfully applied to
proteins
24,26,28,50,51,80
but has yet to be applied to nucleic acids. Such
applications are challenging because of the difficulty in varying the overall
