Chemistry Reference
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i.e.,
S
xx
,
S
yy
,
S
xy
,
S
xz
, and
S
yz
. The product on the right side of Eq. (
3
) is a scalar, the
value of which equals the average of the second order Legendre polynomials in
Eq. (
2
). The aforementioned 10
3
to 10
4
(0.1-1%) scaling in the dipolar coupling
is contained within all Saupe order elements. Since the five unknowns in the Saupe
matrix are common to all bond vectors or RDC measurements in one aligned
protein sample, in theory with more than five RDC bond vectors pointing in
different directions one could solve the Saupe matrix. Prestegard and coworkers
have written a protocol for solving the five Saupe unknowns using the singular-
value-decomposition (SVD) method to obtain alignment parameters [
10
].
0
1
0
1
A
xx
00
0
A
yy
0
00
A
zz
x
y
z
3
2
D
max
ð
T
ða; b; gÞ
@
A
T
@
A
;
D
¼
xyz
Þ
ða; b; gÞ
(5)
0
1
0
1
x
0
y
0
z
0
x
y
z
@
A
¼
@
A
;
T
ða; b; gÞ
(6)
3
2
R
sin
2
D
a
3cos
2
D
¼
y
1
þ
y
cos 2
f
:
(7)
One can diagonalize the Saupe matrix to obtain the alignment parameters.
Diagonalization of the Saupe matrix results in the principal order matrix and
Euler rotation matrices
T
(
a
,
b
,
g
) and
T*
(
a
,
b
,
g
), where
a
,
b
, and
g
are Euler angles
and * indicates conjugate transpose (Eq. (
5
)). The Euler rotation of Cartesian
coordinates in the molecular frame
xyz
of Eq. (
3
) results in a set of new coordinates
x
0
y
0
z
0
Eq. (
6
) for the bond vector within the principle order frame (Fig.
1b
). The
principal order matrix equals the alignment tensor matrix with its Eigenvalues,
A
xx
,
A
yy
,
A
zz
in Eq. (
5
), representing the alignment order in the corresponding tensor
direction. There are different conventions in describing the alignment order
parameters. For instance, one can keep the
S
representation for Eigenvalues in the
alignment tensor matrix, i.e.,
S
zz
, etc. Alternatively, the
A
zz
, notation which is
equivalent to
2S
zz
/3 in the diagonalized Saupe matrix can be used [
11
].
Following the convention in Eq. (
5
), and in analogy to
S
xx
and
S
yy
of the
diagonalized Saupe matrix, there are the following relationships:
A
xx
¼
A
zz
(
1/
2+3
R
/4) and
A
yy
¼
1/2-3
R
/4), where
R
is the rhombicity which can be in
the range of 0-2/3, with the convention that |
A
zz
|
A
zz
(
|
A
xx
|. Because
A
zz
,on
the order of 10
3
to 10
4
, is not a convenient number,
A
zz
can be replaced with a
more convenient
D
a
(
|
A
yy
|
3
D
max
A
zz
/4) representation, which allows for an easier
comparison among different alignment conditions. Here is an alternative to using
the five Saupe unknowns. We use
D
a
to specify the alignment order of a sample,
rhombicity
R
to describe asymmetry of the alignment tensor, and three Euler angles
to define the tensor directions instead. With Euler angles one could conveniently
visualize the tensor within a molecular frame. Further simplifications can be made
by using spherical coordinates, i.e., polar angle
¼
y
and azimuth angle
f
, to replace
