Biomedical Engineering Reference
In-Depth Information
frame. The physical significance of the order tensor can be best understood
using a Cartesian representation, S
kl
.
41
Two angular terms define the
orientation of a principal direction of order, S
zz
, relative to the chiral
molecular fragment. The S
zz
axis defines the average orientation of the
magnetic field relative to the fragment; it is oriented on average along and
perpendicular to the magnetic field direction for S
zz
.0 and S
zz
,0,
respectively. A third angular term specifies the orientation of S
xx
and S
yy
,
axes with S
yy
pointing along the direction of asymmetry (i.e., the direction
about which the magnetic field is most likely to rotate about compared to
other axes perpendicular to S
zz
). An order parameter, referred to as the
generalised degree of order (q)
59
describes the degree of alignment (q 5
r
2
3
) and the extent to which the magnetic field direction is
ordered relative to the molecule. Finally, an asymmetry parameter (g 5
S
xx
{S
yy
S
xx
zS
yy
zS
zz
S
zz
)
describes the extent of asymmetry in the distribution of the magnetic field
direction relative to the chiral frame [Figure 9.2(A)]. When in the principal axis
system (PAS) of the order tensor, the two order parameters are frequently
expressed in terms of a magnitude, D
a
and rhombicity, R
21
:
2
S
zz
;
c
i
c
j
h
2p
2
r
ij
m
0
4p
1
2
3
g
D
i
a
~{
R~
ð
9
:
2
Þ
A similar order/alignment tensor type description can be used to describe
internal motions within the molecule; however, in this case one describes the
average orientation of an axially symmetric RDC bond vector or an axially
asymmetric fragment relative to the chiral molecular frame. The dynamics
interpretation of RDCs is discussed in greater detail in Section 9.5.
For a rigid object, the time-averaged angular term in eqn (9.1) can be
expressed in terms of a time-independent orientation of the internuclear vector
relative to an arbitrary frame and the five-order tensor elements (S
kl
):
41,60
T~
X
kl~xyz
S
3cos
2
h{1
2
S
kl
cos (a
k
) cos (a
l
),
ð
9
:
3
Þ
where a
n
is the angle between the ij
th
internuclear vector and the n
th
axis of
arbitrarily defined coordinates. In practice, the overall order/alignment tensor
can be determined for a solute molecule provided the measurement of five or
more spatially independent RDCs for bond vectors that do not undergo
internal motions and whose relative orientation (but not necessarily transla-
tion) within the structure is known.
9.3 Partial Alignment of Nucleic Acids
The measurement of RDCs under solution conditions hinges on being able to
introduce a particular level of alignment,
61
either by dissolving the solute in an
