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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.)