Chemistry Reference
In-Depth Information
RDCs within a protein that correspond to bond directions within the alignment tensor
frame, providing orientation restraints for protein structure determination.
There are at least two approaches to create weak alignment conditions for
measuring RDCs. One is to take advantage of the large magnetic susceptibility of
a protein where its interaction with the magnetic field could produce a weak
alignment [
5
,
6
]. The other is to mix the protein sample with a medium that can
be mechanically manipulated to create an anisotropic matrix or one containing
supramolecules with substantially large susceptibility anisotropy that can be
aligned under an external magnetic field. The interaction between the proteins
and the media in turn could induce weak alignment of the proteins [
7
]. The latter
approach creates a degree of alignment that is roughly one order of magnitude
stronger than the former one, significantly larger than the experimental error, and
thus is more practical for general applications.
A common usage of RDCs is to include themwith other NMR restraints in refining
a protein structure. For studies of large or membrane associated proteins, where high
level of deuteration is required to achieve narrower line-widths, the number of
observed NOEs will be reduced greatly. Therefore RDC restraints are necessary [
8
].
In this chapter we will focus on a short description on how RDC was developed, its
practical mathematical expressions, and novel methods used in creating different
alignment conditions that would allow more proteins to be studied using RDCs. We
will describe RDC data interpretations and some common approximations. Finally we
will discuss the most recent RDC applications in ensemble structure refinement,
protein-ligand, and protein high-order structure determinations.
2 Theoretical Expressions
Dipolar coupling measures the interaction between two magnetic nuclei in an
external field
B
0
. If the vector connecting nuclei
A
and
B
is parallel to the field
B
0
, the coupling is at its strongest with a magnitude
D
max
, which is given by Eq. (
1
),
where
m
0
is the vacuum permeability,
h
is Planck's constant,
g
A
and
g
B
are the
gyromagnetic ratios of nuclei
A
and
B
, respectively, and
r
AB
is the distance between
nuclei
A
and
B
. In some expressions vacuum permeability
m
0
was assumed and
therefore omitted in the
D
max
expression, and there will be a factor of 4
difference
in the denominator of Eq. (
1
).
D
max
is bond type dependent and usually on the order
of 10
3
Hz, e.g.,
D
max
for protein backbone amide
1
H-
15
N spin nuclei pair is
21.66 kHz with an assumed bond length of 1.04
˚
. Because of diffusive motion
the direction of the internuclei vector fluctuates relative to the
B
0
direction and
therefore the dipolar coupling has to be evaluated with respect to every possible
orientation. This orientation dependency follows the second order Legendre
polynomials. Shown in Eq. (
2
) is the expression for the dipolar coupling
D
of an
internuclei vector with a fixed distance (which is the case for bonded nuclei), where
Y
p
is the instantaneous angle between the dipole-dipole or bond vector and
B
0
(Fig.
1
), and the angular bracket indicates time or population average. The average
