Biomedical Engineering Reference
In-Depth Information
8.2 Residual Dipolar Couplings as Probes of Protein
Conformation
The coupling of a given magnetic moment with any other magnetic moment in
its surroundings gives rise to a dipole-dipole interaction which we can be
expressed in the following way for two spins I and S:
12
c
I
c
S
m
0
h
16p
3
r
IS
D
i
~{
SP
2
cos h
I
ð
T
ð
8
:
1
Þ
where c is the gyromagnetic ratio for the two spins, r is the distance between
the spins, m
0
is the permeability of free space, and h is Planck's constant. When
the interaction vector connecting two spins samples all orientations with equal
probability (as is the case in isotropic solution), the measured value is averaged
to zero, whereas in the solid-state case, the near-complete absence of
macroscopic motion leaves the dipolar interaction (tens of kHz for covalently
bound
15
N-
1
H spins) essentially undiminished, leading to spectra which are
dominated by extensive dipolar couplings. The idea of using a mesomorphic or
liquid-crystalline phase is to establish a regime whereby the geometric
information content inherent to dipolar couplings (either radial, or more
commonly orientational) in a residual dipolar coupling can be obtained while
retaining the simplicity, and spectral resolution of liquid-state NMR.
13
The demonstration that precise structural information can be acquired using
relatively straightforward experimental and analytical procedures,
9,14,15
was
followed by intense development of alignment media that are compatible with
the study of proteins in standard aqueous buffer conditions. This in turn led to
the proposal of a range of dilute liquid crystals that do not interact specifically
with the protein of interest, but induce sufficient alignment while retaining the
high-resolution characteristics of solution NMR. Commonly used alignment
media are based on, for example: bicelles of diverse compositions,
14
mixtures
of polyethylene glycol and alcohol,
16
bacteriophage,
17
and stretched or
compressed polyacrylamide gels.
18
Under conditions of weak alignment, the
dipolar coupling can be expressed in the following form (in this chapter we will
consider the case of covalently bound spins, whose internuclear distance (r
IS
)is
either known, or can be determined):
"
#
r
4p
5
r
R SY
2
h
i
,w
i
3
8
c
I
c
S
m
0
h
8p
3
r
IS
D
i
~A
j
a
SY
2
h
i
,w
i
Þ
TzSY
0
ð
Þ
Tz
ð
{2
h
i
,w
i
ð
Þ
T
ð
8
:
2
Þ
where A
a
is the amplitude and R is the rhombicity of the alignment tensor and
the average spherical harmonics SY
2
h
i
,w
ð Þ
T define the orientational sampling
of each vector relative to a common molecular alignment frame (h and w are
the polar angles relative to the alignment tensor axes). RDCs are first and
foremost highly sensitive structural constraints, due to the angular dependence
of eqn (8.2), which describes the orientation of internuclear vectors with
