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The dipole-dipole interaction
The dipolar Hamiltonian is given by H ij
D=�0 �i � j h
8�2 r ij3
[ I i I j�31
r ij2�I i r ij ��I j r ij�]
for a heteronuclear spin system in the high field approximation the following simplification can be obtained
H ISD �t �=�
�I �S �0 h
8�3 � r IS
3 �t ����3 cos
2IS �t ��1��
2I z S z
in analogy to the scalar coupling (relevant spin operator is also IzS
z ) the dipolar coupling results in a
dublett splitting with
D IS=��I �S �0 h
8�3 � r IS
3 �t ����3cos
2IS �t ��1��
2
B0
I
S
rIS
�IS
but coupling is a function of orientation relative to the external magnetic field !
Averaging of Dipolar Coupling by molecular tumpling in isotropic solution
molecular tumpling modulates orientation of internuclear vector relative to the static magnetic field in a random mannertime scale for modulation of D
ij is significant faster (�
c 1-50 ns) than the time scale for signal detection ( s � ms)� �
All orientations have the same probabilityAverage value of D
ij is obtained by integrating orientational dependence over surface of a sphere
�3 cos2�1�=
14�
�0
2�
d��0
�3 cos2�1�sind
=12�0
�3cos2�1�sin d
=12[3�
0
cos2sind ��0
sind ]
=12
[ [�3 cos3]0� [cos]0
� ]=0
random molecular tumpling averages Dij to zero
Dij=�� i � j�0 h
8�3� r ij
3 ���3 cos2ij�1��
2
x
z
yαx
αy
αz
βx
βy
βz
B0
n
m
Description in an arbitrary frame
cosmn=�cos�x
cos� y
cos� z��
cos�x
cos � y
cos �z�=cos�x cos �x cos� y cos � y cos�z cos� z
� 3cos2mn�1
2 �=32��cos�x cos �x cos� y cos� y cos�z cos �z�
2��12
=32[ �cx
2 �C x2 �c y
2 �C y2 �c z
2�C z2
2 �cx c y �C x C y 2�c x cz �C x C z 2 �c y cz �C y C z ]
C i=cos�i ; ci=cos�i
fixed orientation of (m,n) vector relative to molecular frame,averaging only occurs by overall molecular tumbling
The average orientation of the molecule relative to the magnetic field can then be described by a symmetric order matrix A(alignment tensor), which is traceless (sum of diagonal elements = 0):
Aij=32�cos�i cos� j��
12�ij
The dipolar coupling between m, n is then Dmn=��m �n �0 h
8�3 �rmn3 ��
kl
Akl cos�k cos�l=Dmn
max�kl
Akl cos�k cos�l
if the alignment tensor become collinear with the molecular frame, the order matrix will be of diagonal form:
A=�Axx 0 0
0 Ayy 0
0 0 Azz� �A zz���A yy���A xx�
Dmn
max �� x ,� y ,�z �= �i= x , y , z
Aii cos2�iThe dipolar coupling becomes then
transforming into polar coordinates results in
cos� z=cos
cos� x=sincos�
cos� y=sin sin�
Dmn � ,��=Dmn
max[ Azz cos
2 A xx sin
2cos
2� A yy sin
2 sin
2�]
Da=12
Azz Dmnmax Dr=
13� Axx�A yy�Dmn
max R=Dr
Da
axial component rhombic component rhombicity
with
Dnm � ,��=Da [�3 cos2�1� 32
R �sin2cos 2��]
Description of dipolar coupling in the principal frame of the alignment tensor
D = V-1 A V and A = V D V-1
Diagonalization of the Alignment tensor
V is rotation matrix, describing the rotation of the molecular frame to coincide with the alignment tensor
Diagonalization Solving the Eigenvalue problemEigenvalues are D
ii (i = x, y, z)
Eigenvectors describe the rotation matrix V
Rotation of frames is often given by three Euler angles ( , , ):� � �
1. Rotation around z-axis with angle �2. Rotation around y-axis with angle �3. Rotation around z-axis with angle �
Warning: Check convention, whether rotation of the Alignment tensor or the rotation of the molecular frame is meant
A v=� v
cos�i n=c in
i= x , y , z
Dnm ��x ,� y ,�z �=Dnm
max �i , j= x , y , z
Aij cos�i cos� j
�c y1
2 �c x1
2cz1
2 �c x1
2 2cx1 c y1 2c x1 c z1 2c y1c z1
c y22 �c x2
2 cz22 �c x2
2 2c x2 c y2 2c x2 c z2 2c y2 c z2
� � � � �� � � � �� � � � �
c yn
2�c xn
2czn
2�c xn
22 cxn c yn 2c xn c zn 2c ync zn
� �A yy
A zz
Axy
Axx
A yz
�=�D1
D2
��Dn
�
Determination of the Alignment tensor with given dipolar couplings, Di, and a known structure
the direction cosini, cos �in, for the different vectors can be calculated from the structure
the couplings Di are measured:
overdetermined system of linear equations (n > 5), can be solved by numerical methods (e.g. Singular Value Decomposition SVD)
−15 −15−10 −10−5 −50 05 510 1015 15
−15 −15
−10 −10
−5 −5
0 0
5 5
10 10
15 15
10 2010 3020 4030 5040−20
50
−10
−20
0
−10
10
0
20
10
20
D [Hz]calc
D
[H
z]ob
sD
−
D
[Hz]
calc
obs
Sequenz Sequenz
NMR Struktur (1DX8, keine RDC’s)
rmsd = 4.99 Hz
Kristallstruktur (1RDG, 1.4 A)
rmsd = 1.69 Hz
NMR structure of Guillardia theta rubredoxin
1.4 Å crystal structure of Desulfovibrio gigas rubredoxin
overlay of bothbackbone rmsd : 1.0 Å
Quality parameter for least-square fit of the Alignment tensor
root mean square deviation (rmsd) of observed and calculated couplingsProblem: no normalization to the absolute values of measured values
Q-factor: Normalization of the rmsd to the absolute value of measured couplings
for isotropic distribution of the couplings
rmsd =� 1N�i=1
N
�Dmeas
i �Dcalc
i �2
Q=rmsd
Dmeas
rms
Dmeas
rms={Da
2�4 3 R
2�/5}
1/2
Dmeas
rms =� 1N�i=1
N
�Dmeas
i �2
Remark: If couplings are used for structure calculation, the 'quality factors' do not serve for an independentmeasure of quality !
O
C
NC
R
Hα HN
C
1D(N,HN)
1D(N,Cα)
1D(N,C')
2D(HN,C')
1D(Cα,C')
1D(Cα,Hα)
1D(N,HN)1D(Cα,Hα)
= -0.5
1D(N,HN)= 8.2
1D(N,C')
1D(N,HN)= -3.2
2D(HN,C')
DOPr =DOP
�N �H rOP
3
�O �P r NH
3
Residual dipolar couplings in the protein backbonepeptide bond geometry results in rigid distances !
normalization of RDC's to 1D(N, HN)
−20 −10 0 10 200
−20 −10 0 10 200
D = 10 Hz, R = 0aD = 10 Hz, R = 0.3a
zzS
zzS
xxS
yyS
xxS
yyS
axialsymmetrischer Tensor rhombischer Tensor
Szz = 2 Da Syy = -Da(1 + 1.5R) Sxx = -Da(1-1.5R)
isotropic distribution of RDC's
powder pattern like distribution
−20 −10 0 100
2
4
6
8
norm. RDC(DNH) (Hz)
Anz
ahl
−20 −10 0 100
2
4
6
8
norm. RDC (DNCO) (Hz)
Anz
ahl
−20 −10 0 100
2
4
6
8
norm. RDC (DHNCO) (Hz)
Anz
ahl
−20 −10 0 100
2
4
6
8
norm. RDC (DCAHA) (Hz)
Anz
ahl
−20 −10 0 100
2
4
6
8
norm. RDC (DCACO) (Hz)
Anz
ahl
−20 −10 0 100
5
10
15
20
25
norm. RDC gesamt
Anz
ahl
Distribution of couplings in a real proteinhostrogram methods for estimating the alignment tensor
step width in histogram : 1 Hz
−20 −10 0 100
2
4
6
8
norm. RDC(DNH) (Hz)
Anz
ahl
−20 −10 0 100
2
4
6
8
norm. RDC (DNCO) (Hz)
Anz
ahl
−20 −10 0 100
2
4
6
8
norm. RDC (DHNCO) (Hz)
Anz
ahl
−20 −10 0 100
2
4
6
8
norm. RDC (DCAHA) (Hz)
Anz
ahl
−20 −10 0 100
2
4
6
8
norm. RDC (DCACO) (Hz)
Anz
ahl
−20 −10 0 100
5
10
15
20
25
norm. RDC gesamt
Anz
ahl
step width in histogram: 2 Hz
6.66.76.86.97.07.17.27.37.47.5
122.7
123.2
123.7
124.2
124.7
125.2
125.7
126.2
126.7
127.2
127.7
128.2
128.7
129.2
129.7
J J + D
unorientiert orientiert
6.66.76.86.97.07.17.27.37.47.51
9.09.19.29.39.49.59.69.79.89.9
121
122
123
124
125
126
127
128
1H [ppm]
15N
[ppm
]
8.99.09.19.29.39.49.59.69.79.8
121
122
123
124
125
126
127
128
Y17
F34E5
E9
−126.92 Hz
−123.69 Hz
−82.06 Hz
E9
Y17
F34E5
−109.43 Hz
−108.54 Hz
−86.90 Hz
G. theta rubredoxin in Pf1 phage solution (25 mg/ml) G. theta rubredoxin in Pf1 phage solution (18 mg/ml)
Practical aspects for determination of RDC's
orientation to large inefficient coherence transfer because of large variation of (1J + 1D)passive nD (e.g. between protons) increase line width
orientation to small RDC's are small, especially 1D(N, C) cannot be determined
optimal orientation Da (15N, 1HN) approx. 8-15 Hz, that means maximal 1D(HN, N) approx. 16-30 Hz
Media for orienting biological macromolecules
general: The degree of alignment of the protein (nucleic acid) should be ca. 0.001 (-> Da
HN ca. 10 Hz)
liquid crystalline media have a degree of alignment of ca. 0.5 -0.85
very weak interaction between protein (nucleic acid) and alignment medium
typical alignment media very large particles (> 500 Å)
large free spaces between the particles allow weak orientation
together with rotational freedom comparable to isotropic solution
only small contributions to transverse relaxation
Bicelles
first liquid crystalline medium for the weak alignment of proteins and DNA
consists of usual phospholipids, they form rod-like micelles that orient in the magnetic field
examples: mixtures of DMPC (Dimyristoyl-phosphatidylcholine)
and DHPC (Dihexanoylphosphatidylcholine)
molar ratio DMPC:DHPC 3:1, 5% w/v lipids in water (5 g / 100 ml)
liquid crystalline phase in temperature range 30° - 45° C
bicelles can be stabilized with ionic detergents (SDS or CTAB)
filamentous phages (fd, Pf1) contour length of 2 m, diameter 6.5 nm�
liquid crystalline at low concentrations of few mg / ml
ionic strength reduces alignment at concentrations < 15 mg / ml
in praxis: interaction between protein and phage is electrostatic !
further media ternary mixtures of cetylpyridimiumchloride (bromide),
hexanol and NaCl (NaBr) in water
purple membrane fragments (cellular membrane from Halobacterium salinarum)
Alkyl-polyethylene glycol based media
e.g. C10E5 : hexanol (molar ratio 0.95), 3% (wt) C10E5 in water
−40 −30 −20 −10 0 10 20 30 40 50 60
Hz
2H quadrupolar splitting in 25 mg/ml Pf1 phage solution
Structure calculation with RDC's
XPLOR refinement 1. Calculation of a structure ensemble with 'classical' restraints
(NOE's, J-couplings, hydrogen bonds)
2. Determination of the alignment tensor (Da, R)
either by analysis of RDC-distribution (histograms)
or least square fit to preliminary structures
then ' restrained molecular dynamics' refinement with additional potential Edip
= kdip
(Dobs
� Dcalc
)2
The orientation relative to a external coordinate system will be varied.
These coordinate system represents the alignment tensor.
general problems: additional potential term creates an energy surface with a large number of local minima,
this results in bad convergence during structure calculation.
Improvements: several independent repetitions of individual refinements
increasing simulation time (very slow cooling)
−15 −15−10 −10−5 −50 05 510 1015 15
−15 −15
−10 −10
−5 −5
0 0
5 5
10 10
15 15
10 1020 2030 3040 4050 50−20 −20
−10 −10
0 0
10 10
20 20
D [Hz]calc
D
[H
z]ob
sD
−
D
[Hz]
calc
obs
Sequenz Sequenz
NMR Struktur (1DX8, keine RDC’s)
rmsd = 4.99 Hz
NMR Struktur nach RDC Refinement
rmsd = 0.3 Hz
Inclusion of RDC's into structure calculation
A
B
SA SB
Domain orientation
only few NOE's define the domain interface
therefore ususally bad definition of domain orientation
with 'classical' NMR restraints
rigid orientation: SA = S
B
one alignment tensor describes the complete alignment
of the complete molecule (both domains)
relative flexibility: SA S�
B
two different alignment tensors (each for one domain)
are necessary for describing the alignment
relative flexibilityt in multidomain proteins(Braddock et al. JACS 123 (2001), 8634.)