Chemistry Reference
In-Depth Information
Fig. 1 Illustrations of relationships between RDC internuclei vector
AB
and an arbitrary molecu-
lar frame (a) and the alignment tensor frame (b). A protein molecule, carrying spin nuclei
A
and
B
,
is represented using an ellipsoid.
B
0
is the external field.
Y
is the instantaneous angle between
the internuclei vector
AB
and
B
0
.
b
x
,
y
,
z
specify the projection angles of
B
0
onto each axis of
a molecular frame. Polar angle
y
and azimuth angle
f
are spherical coordinates of the vector
AB
in
the alignment tensor frame
for
D
will be zero if Eq. (
2
) is integrated over the variable
that covers the surface of a
sphere, corresponding to all possible orientation for an isotropic diffusing molecule.
However, due to a weak alignment causing a nonisotropic sampling of orientations,
the average of
D
is not zero. In practice, adjusting the concentration of the alignment
media can allow between an equivalent of 0.1-1% fractions of protein molecules
being aligned and it results in a value for
D
to be within
Y
20 Hz for
1
H-
15
Nvectors,
referred to as RDC.
D
max
¼
m
0
h
g
A
g
B
;
(1)
3
r
AB
8
p
3cos
2
Y
1
D
¼
D
max
;
(2)
2
0
@
1
A
0
@
1
A
;
S
xx
S
xy
S
xz
x
y
z
D
¼
D
max
ð
xyz
Þ
S
yx
S
yy
S
yz
(3)
S
zx
S
zy
S
zz
3 cos
b
i
cos
b
j
d
ij
S
ij
¼
;
ij
¼f
x
;
y
;
z
g:
(4)
2
The equation for RDC, Eq. (
2
), could be rewritten in the form of the expectation
value for a vector in a Saupe order matrix, i.e., Eqs. (
3
,
4
)[
9
]. The
x
,
y
,
z
in Eq. (
3
)are
directional cosines of a bond vector in an arbitrary molecular frame (Fig.
1a
), the
most convenient would be an existing PDB coordinate frame; Saupe element
S
ij
is
the averaged projection of axes of the molecular frame onto the direction of
B
0
with
b
x
,
y
,
z
(Fig.
1a
) specifying the projection angle for each axis;
d
ij
is Kronecker delta.
The Saupe matrix is symmetric and traceless and contains five unknown variables,
