c) RDC Theory

Residual dipolar couplings 

NMR workshop, Rosario November 2008 

Markus Zweckstetter 

Department for NMR-based Structural Biology, 

Göttingen, Germany 

15 N 

Max Planck Institute 

For Biophysical Chemistry 

1 H 

1 

B 0


Background material 

Saupe A, Englert G (1963) Phys. Rev. Lett. 11: 462-464 

BothnerBy AA. (1996) In Grant DM, Harris RK (eds.), Encyclopedia of Nuclear Magnetic Resonance. 

Wiley, Chichester: pp. 2932-2938 

Tjandra N, Bax A (1997) Direct measurement of distances and angles in biomolecules by NMR in a 

dilute liquid crystalline medium. Science 278: 1111-1114 

Bax A, Kontaxis G, Tjandra N. (2001) Dipolar couplings in macromolecular structure determination. In 

Nuclear Magnetic Resonance of Biological Macromolecules, Pt B, Vol. 339, pp. 127-174 

Prestegard JH, Al-Hashimi HM, Tolman JR (2000) NMR structures of biomolecules using field 

oriented media and residual dipolar couplings. Q. Rev. Biophys. 33: 371-424 

Kramer F, Deshmukh MV, Kessler H, Glaser SJ (2004) Residual dipolar coupling constants: An 

elementary derivation of key equations. Concepts Magn. Reson. Part A 21A: 10-21 

2

NOE/ROE~1/r 

(NOESY, ROESY) 

N 

H N 

φ 

H α k 

C α k 

R 

C' 

ψ ω 

O 

Distance Restraints 

Bacteria 

13C,15N-Source 


N 

H k+1 

Nk+1 

NMR structure determination 

J 

ω 

J − Coupling 

(E.COSY, DQ/ZQ, FIDS, quant.J) 

N 

H N 

Η α k 

C α k 

R 

C' 

φ ψ ω 

O 

Dihedral Restraints 

Sample 

N 

Ηk+1 

Nk+1 

Multi-D-NMR 

Structure Calculation 

(Simulated Annealing, Distance Geometry, 

Molecular Dynamics) 

N 

H N 

Assignment 

Γ c NH,CH 

Cross Correlated Relaxation 

φ 

H α k 

C α k 

R 

ω 

θ 

C' 

ψ ω 

O 

1) Dipole-Dipole 

2) Dipole-CSA 

3) CSA-CSA 

N 

H k+1 

Nk+1 

Projection Restraints 

1) Residual Dipolar Couplings 

2) N-H Relaxation Rates 

R 

H N 

N 

φ ψ 

C α k 

Η α k 

C' k+1 

1) Dipole-Susceptibility Tensor 

2) Dipole-Mass Tensor 

3D-Structure 

N 

H N 

Η α k 

φ 

C α k 

R 

C' 

ψ ω 

N 

Ηk+1 Nk+1 

Chemical Shift Restraints 

O 

3

Why do we want to use dipolar couplings in 

solution NMR? 

• all these parameters give essentially local information: 

distances


RDCs today 

• Residual dipolar couplings (RDCs) can be observed in solution when a molecule 

is aligned with the magnetic field 

• When alignment can be kept sufficiently weak 

• NMR spectra remain simple as in isotropic solution 

• quantitative measurement of a wide variety of RDCs 

• Several dilute liquid crystalline media are now available 

• RDC measurements and analysis are highly efficient 

Residual dipolar couplings are a generally applicable tool for NMR 

structure determination 

5


The dipolar coupling 

6

Theclassicaldipolarinteraction 

Nuclear spin magnetic dipole μ magnetic field 


0 ( ) 5 ( ) ( ) 

3 

μ 

Bμr = ⎡ − ⎤ 

4π r 

⎣ μr r rr μ⎦ 

2nd nuclear spin magnetic moment in B μ interaction energy 

E 

μ0 

1 ⎡ 3 

⎤ 

=− B ( r12 ) μ2= 3 ⎢μ1μ2− 2 ( μ1r12 )( μ2r12 ) ⎥ 

4π 

r12 ⎣ r12 

⎦ 

μμ 1 2 μ1 

Within strong external magnetic field B 0 alignment 

μ r / | r | = μ cosθ 

1 12 12 1 

E 

μμ 

1 2 

μ 

1 

⎡ 

⎣ 

⎤ 

⎦ 

0 

2 

= μ 3 1μ2 1−3cos θ 

4π 

r12 

7

Theclassicaldipolarinteraction 

E 

molecular 

reorientation 

μμ 

1 2 

μ 


1 

⎡ ⎤ 

0 

2 

= μ 3 1μ2 1 3cos θ 

4π 

r ⎣ − ⎦ 

12 

15 N 

1 H 

B 0 

isotropic 

π 

2 

∫ ( ) 

0 

magic angle ~ 54° 

1− 3cos θ sinθdθ = 0 

8

spin I 


Quantum mechanical picture 

μ = γ hI 

I 

H 

μ 1 ⎛ x x ⎞ γ γ hμ 

⎛ 

I 

x x ⎞ 

S 

⎝ ⎠ ⎝ ⎠ 

3 3 

0 i j I S 0 

i j 

D =− 3∑μ1i⎜3 − δij⎟μ2 j =− 3 

3 ∑ i⎜ −δij⎟ 

j 

4π r i, j= 1 r r 4πr 

i, j= 

1 r r 

2 

z 

IS z z :3 −1 

2 

r 

xz 

IS x z + IS z x :3 2 

r 

yz 

IS y z + IS z y :3 2 

r 

xy 

IS x y + IS y x :3 2 

r 

IS 

x x 

IS 

y y 

2 

x ⎫ 

:3 −1 2 

r ⎪⎬⎪ 

2 

y 

:3 −1 2 

r ⎪⎭ 

2 2 2 2 2 2 2 2 2 

3x + 3y −2x −2y −2z − 2z + x + y ⎛3z ⎞1 

IS x x + IS y y :3 = − 1 

2 2 2 

2r 2r ⎜ − 

r 

⎟ 

⎝ ⎠2 

9


Quantum mechanical picture 

⎧ 2 

⎫ 

⎪⎡ 1 

⎤⎛ 

z ⎞ 

IS z z − ( IS x x + IS y y) 

3 − 1 + ⎪ 

⎪⎢ ⎜ 2 ⎟ 

⎣ 2 

⎥ 

⎦⎝ r ⎠ 

⎪ 

⎪ ⎪ 

γγ I Shμ0⎪⎡ 

xz yz ⎤ ⎪ 

HD =− 3 ⎨ ( IxSz IzSx) 3 2 ( IySz IzSy) 3 2 

4π 

r ⎢ 

+ + + 

r r ⎥ 

+ ⎬ 

⎪⎣ ⎦ ⎪ 

⎪ 2 2 

⎡ xy 

3( 

x − y ) ⎤⎪ 

⎪⎢( IS x y + IS y x) 3 + 2 ( IS x x + IS y y) 

⎥⎪ 

2 

⎪⎢r 2r 

⎥⎪ 

⎩⎣⎦⎭ spherical coordinates 

D 

x y z 

= sinθ cos φ; = sinθsin φ; = cosθ 

r r r 

m m −m 

∑( 1 ) 2 ( θφ , ) 2 

1 

H = − F T 

m 

coordinates spin operators 

where 

γγ hμ 

⎛24π ⎞ 

F =− ⎜ ⎟ Y 

4πr⎝ 5 ⎠ 

m I S 0 

2 

m 

2 3 

2 

10


Spherical harmonics 

m 

component ( , ) 

m = 0 ⎛ 5 ⎞21 

2 

⎜ ⎟ ( 3cos θ −1) 

⎝4π⎠ 2 

1 

m =± 1 ⎛15 ⎞2 

− ⎜ ⎟ sinθ cosθ exp( 

± iφ) 

⎝8π⎠ 1 

m =± 2 ⎛15 ⎞21 

2 

⎜ ⎟ sin θ exp( ± i2φ) 

⎝2π⎠ 4 

m 

Y2 θφ T2 − 

1 

⎛ 1 ⎞⎡ 1 

⎤ 

⎜ ⎟⎢ 

IS z z − ( IS x x + IS y y) 

6 ⎣ 2 

⎥ 

⎝ ⎠ 

⎦ 

1 

± + 

2 

[ IS IS] 

z ± ± z 

1 

2 IS 

± ± 

+ ( x y ) = + I− = ( Ix−iIy) I I iI 

( ) 

exp iφ= cosφ + 

isinφ 

11


Secular approximation 

γγ I Shμ0 

2 ⎡ 1 

⎤ 

H D =− 3 ( 3cos θ −1) IS z z − ( IS + − + IS − + ) 

4πr⎢ ⎣ 4 

⎥ 

⎦ 

heteronuclear case 

γ γ hμ 

2 

H ( θ ) I S 

4π 

r 

I S 0 

D =− 3cos −1 

3 z z 

γ Iγ Shμ0 

3 

4π 

r 

1 

2 

( ) 

IS + IS = IS+ IS 

+ − − + 

x x y y 

r = 1.04 1 D(N,H) = 21.7 kHz 

12

molecular tumbling/internal motion 


PQ γ γ hμ 

2 

D =− ∫ P( β , γ) F ( θ, φ) d( cosβ) 

dγ 

P Q 0 0 

3 

4πr4π 2 

PQ 

D 

γ γ hμ 

1 

3cos 1 

4πr2 ( θ ) 

P Q 0 2 

=− − 

3 

sampled orientations 

13

Molecular alignment and RDCs 


14



( ) ( () () () ) 2 

3 1 

P2 cosθ = cos βx t cosα x + cos βy 

t cosαy + cos βz 

t cosαz 

− 

2 2 

P 

2 

( ) ( ) 

Ci t = cos βi 

t and 

ci = cosα 

i 

( cosθ 

) 

2 2 

2 

⎡ 2 2 2 

3 Cx() t cx + Cy() t cy + Cz() t cz 

+ 

⎤ 

1 

= ⎢ ⎥− 

2⎢2 C () () 2 () () 2 () () 2 

x t Cy t cxcy Cx t Cz t cxcz Cy t Cz t cyc ⎥ 

⎣ 

+ + 

z⎦ 

B 0 

θ 

β x (t) 

βz (t) 

βy (t) 

P2( 

cosθ ) = 

3 T 

2 

( br 0 0) 

−1 

2 

⎧ ⎛ 

⎪ ⎜ 

3⎪ 0 0 0 ⎜ 

= ⎨( rx , ry , rz ) 

2 ⎜ 

⎪ ⎜ 

⎪ ⎜ 

⎩ ⎝ 

2 

Cx() t 

Cx() t Cy() t 

Cx() t Cz() t 

Cx() t Cy() t 

2 

Cy() t 

Cy() t Cz() t 

Cx() t Cz() t ⎞ ⎫ 

0 

⎟⎜⎛r⎞ x ⎪ 

⎟ 0 ⎟⎪ 

1 

Cy() t Cz() t 

⎟⎜ry 

⎟⎬− 

0 2 

2 ⎟⎜r⎟⎪ z 

Cz() t ⎟⎝ 

⎠⎪ 

⎠ ⎭ 

α x 

x 

P 

z 

α z 

Q 

α y 

15 

orientational probability distribution 

y



( ) ( ) ( ) ( ) 

S = 3/2 cosβ t cosβ t − 1/2δ = 3/2 C t C t −1/2δ 

ij i j ij i j ij 

2 2 2 

x + y + z = 1 CC i j = CjCi C C C 

P2( cosθ ) = ∑ Sij 

cosαicosα j 

{ } 

i, j= x, y, z 

( θPQ = αz; cz= cos θPQ; cx= sinθPQ cos φPQ; cy= 

sinθ PQ sinφPQ 

) 

S is real, traceless, symmetric 

5 independent elements 

principal alignment frame, i.e. diagonalization of S S d 

3 

( ) { } 

α , α , α = ⎡ + + ⎤−1 

PQ PQ 

2 2 2 2 2 2 

x y z max x x y y z z 

D D C c C c C c 

2 ⎢⎣ ⎥⎦ 

2 

i 1/3 ij 

C A 

P 

x; S xx d 

φ PQ 

z; S zz d 

PQ PQ 2 2 2 2 2 

= + ( ) 

θ PQ 

Q 

y; S yy d 

3 

D αx, αy, αz = D ⎡ max cos θPQAzz + sin θPQ cos φPQ Axx+ sin θPQsin φPQ 

A ⎤ yy 

2 ⎣ ⎦ 

PQ 3 PQ ⎡ 1 2 

⎤ 

D ( θPQ, φPQ) = Dmax P2( cosθPQ ) Azz sin θPQcos 2φPQ 

( Axx 16Ayy 

) 

2 ⎢ 

+ − 

⎣ 2 

⎥ 

⎦

γ PγQhμ PQ 

⎡ 3 

⎤ 

D ( θ φ ) S 

⎢( θ ) R θ φ 

⎣ 

⎥ 

⎦ 


A a =3/2A zz =S zz d , Ar =(A xx -A yy )=2/3 (S xx d -Syy d ) 

PQ PQ ⎡ 3 2 

⎤ 

D ( θPQ, φPQ) = Dmax ⎢ 

P2( cosθPQ) Aa + Arsin 

θPQcos2φPQ ⎣ 4 

⎥ 

⎦ 

PQ PQ ⎡ 2 3 2 

⎤ 

D ( θPQ, φPQ ) = Da⎢( 3cos θPQ − 1) + Rsin 

θPQcos2φPQ ⎣ 2 

⎥ 

⎦ 

0 2 2 

PQ, PQ =− LS 3cos 1 sin cos2 

2 3 

PQ − + 

PQ PQ 

4π 

r 

2 

PQ 

D a PQ = ½ D PQ max A a : magnitude of alignment tensor (A a =10 -3 D a NH ~ 10 Hz) 

R = A a /A r : rhombicity of alignment tensor; R ∈ [0; 2/3] 

use only a very slight orientational preference ("partial alignment"), i.e., 

only ca. 1 out of 1000 solute molecules 

Generalized degree of order (GDO): 

Euclidean norm ϑ = ( 2 / 3 4 /5 π ∑ i,j S ij 2 ) 1/2 

17

How to Get Alignment 


18


Partial Alignment 

• dipolar couplings are LARGE in solids (~22 kHz for 1 D HN !), but 

• fortunately average out for isotropically fast tumbling (solution NMR) 

⇒ full dipolar couplings are not desirable in high-resolution NMR 

Can we get orientational information WITHOUT messing up our nice NMR spectra ? 

YES! 

• use only a very slight orientational preference ("partial alignment"), i.e., 

only ca. 1 out of 1000 solute molecules 

result: dipolar coupling from the (0.1%) non-isotropic fraction is scaled down to 

residual dipolar couplings (RDCs) of ± 20 Hz max.! (0.1% of ±20 kHz) 

19

Alignment – Anisotropic Tumbling 

To extract dipolar coupling data, the molecule must behave anisotropically! 

1) large magnetic susceptibility anisotropy 

• diamagnetic systems such as DNA (small anisotropy in each base) 

• metalloproteins with paramagnetic centers 

• lanthanide-binding tags 

2) anisotropic environment 

• oriented liquid-crystalline phase 

• anisotropically compresed gel 


field-dependent alignment of molecules 

field-independent alignment 

20

Requirements 


Alignment media 

• liquid crystalline at < 10% w/v order of biomolecules: ~ 0.002 

• aqueous 

• uniform anisotropy over the whole sample volume, 

• stable at different ionic strength, pH, temperature 

• not too strongly charged < 0.5 e/nm 2 , 

• solute should not bind (significantly) to medium 

21


Bicelles 

• Diskshaped particles made from DMPC and DHPC (q = 3:1) 

• concentration usually 5% (w/v) 

• degree of protein alignment can be “tuned” by adjusting the 

bicelle concentration 

• alignment is temperature dependent (liquid crystal > 37°C) 

• aligning with their normal perpendicular to the direction of the 

magnetic field 

• degree of alignment can be determined by measuring the 2H 

quadrupolar splitting in the HDO resonance. 

22


Bicelles (2) 

• Isotropic bicelles (q ~ 0.5) for solubilization of integral 

membrane proteins solution-state NMR 

• Anisotropic bicelles for solid-state NMR and X-ray 

crystallography 

Disadvantages: 

• unstable in the presence of certain proteins 

• offers only a limited temperature and pH range 

23

• bicelles 


Alignment media - Toolbox 

• alkyl poly(ethylene glycol) based media 

• filamentous phage (Pf1,fd; -0.47 e/nm 2 ) 

• polyacrylamide gel (charged, uncharged) 

• cellulose crystallites 

• purple membrane fragments 

• cetylpyrimidinium-based media, ... 

24

Alignment of Membrane Proteins 

Acrylamide gels DNA nanotubes 

G-tetrad DNA 


Douglas et al. 

PNAS, 2007 

Lorieau et al. 

JACS, 2008 

25

Modulation of alignment tensor 


26

The problem: Orientational degeneracy 


D PQ = D a PQ [(3 cos 2 θ –1) + 3/2 R sin 2 θ cos(2φ)] 

Ramirez & Bax 

JACS, 1998 

27

Strategies for Modulation of Alignment 

• Alignment media with different properties (steric/electrostatic) 

• Charged bicelles: doping with small charged amphiphiles to alter 

their charge 

Positive: CTAB / Negative: SDS 

•Chargedgels 

• variation of pH, ionic strength 

• mutation (introduction of charged residues) 

• paramagnetic alignment (different tags/lanthanides) 


28

Attenuation of alignment strength by 

increasing the ionic strength 

450 mM NaCl 

20 mg/ml Pf1 

150 mM NaCl 

9.0 8.0 

7.0 

ppm 


ubiquitin at 450 mM NaCl 

in 20 mg/ml Pf1 

29

Modulation of alignment tensor orientation 

by ionic strength changes 


GB1 

30


Liquid crystal theory 

31



Onsager (1949): concentration at which a solution undergoes a spontaneous first 

order phase transition from an isotropic to a chiral nematic phase 

25 Hz 

B 2 c p > c a 

B 2 = π D eff L 2 /4 

first-order transition coexistence region c p ∈ [c i , c a ] 

for semi-flexible rods 

c i = 0.3588 [(1-√x)B 2 ] -1 c a = 0.3588 [(x-√x)B 2 ] -1 

32

B 2 c p > c a 

B 2 = π D eff L 2 /4 


Onsager theory 

effective increase in rod diameter 

D eff = D + κ -1 (ln ω + 0.7704) 

contact potential ω = 2π Z2 [βγx0 K1 (x0 )] -2 Q κ-1 exp(-2 x0 ) 

contribution of the polyions to the ionic strength κ = 8πQ(cs + Γzpcp ) 

33



density of most nematogens is higher than for the solvent 

average density of the nematic region is higher than for the isotropic region gravity 

causes it to occupy the bottom region of the sample 

Pf1 (12 mg/ml) 

0.5M 

B 2 c p > 4.19 and B 2 c p > 5.51 

34


Paranematic phase of Pf1 phage 

For a fully nematic phase, the degree of alignment is independent of field 

strength above a typically very low threshold 

paranematic 

Q cc ( 2 H) = 0.886 c Pf1 

c Pf1 [mg/ml] 

• 600 MHz 

□ 800 MHz 

35

Measurement of RDCs 


36


NOESY HSQC 

37

Accuracy of measured splitting: ΔJ = LW/SN 

1 JHN [1]: IPAP-HSQC, DSSE-HSQC, 3D HNCO 

1 JC‘Cα [5]: 3D HNCO (CSA(C‘) ~ 500 MHz optimum) 

1 JC‘N & 2 J C‘HN [8.3]: 2D HSQC, 3D TROSY-HNCO 

1 JCαHα [0.5]: 2D J CH -modulated HSQC, (HA)CANH, HN(CO)CA 

1 JCH (side-chain): 2D J CH -mod. HSQC, CCH-COSY, SPITZE-HSQC 

1 H- 1 H: COSY, CT-COSY, HNHA, 3D SS-HMQC2 (long-range) 


required accuracy < 5% * Da 

Bax, Kontaxis & Tjandra Method Enzymol. 339, 127-174, 2001; 

Chou & Bax JBNMR, 2001; Delaglio et al. JMR 2001; Wu & Bax, JACS, 2002; 

38

RDC measurement: J splitting ( 1 J HN) 

IPAP-HSQC 

Ottiger et al. JMR, 1998 


39

RDC measurement: Quantitative J 

correlation ( 1 J C‘N) 


Chou & Bax 

JBNMR, 2001 

40

Determination of a Molecular 


Alignment Tensor 

41

1) RDC distribution analysis 


Four Methods 

2) Back-calculation of alignment tensor 

3) Prediction of alignment from structure 

4) Prediction of alignment from structure and charge distribution 

42

Count 

1) Estimate for alignment tensor 

30 

20 

10 

D zz PQ = 2 Da PQ 

D yy PQ = –Da PQ (1 + 1.5 R) 

D xx PQ = –Da PQ (1 – 1.5 R) 

with D ii PQ = D PQ max S ii d no structure necessary ! 


0 

D 

A xx B 

D zz 

-40 -20 0 20 

D yy 

R 

0.2 

0.1 

0 

1d(NH) [Hz] 

-20 -15 

D aNH [Hz] 

-20 0 20 

log( L( d 1…n PQ | Da PQ , R)) = ∑i=1,..,N log (P(d i PQ )) 

43

2) Back-calculation of alignment tensor 

if well-defined structure available 

• singular value decomposition (SVD) 

very stable & with a minimum 

of five RDCs possible 

• iterative least squares procedure (Levenberg-Marquardt minimization) 

χ 2 = ∑ i=1,..,N [d i PQ (exp) – di PQ (calc)] 2 /(σi PQ ) 2 

fixing of alignment parameters (e.g. rhombic component zero due to 

three-fold or higher symmetry) 


2 GLN HN 2 GLN N -8.170 1.000 1.00 

3 ILE HN 3 ILE N 8.271 1.000 1.00 

4 PHE HN 4 PHE N 10.489 1.000 1.00 

5 VAL HN 5 VAL N 9.871 1.000 1.00 

6 LYS HN 6 LYS N 9.152 1.000 1.00 

7 THR HN 7 THR N 3.700 1.000 1.00 

8 LEU HN 8 LEU N 6.461 1.000 1.00 

10 GLY HN 10 GLY N 7.634 1.000 1.00 

11 LYS HN 11 LYS N -7.528 1.000 1.00 

44

Evaluation of uncertainty in backcalculated 

alignment tensors (I) 

Can you trust a back-calculated alignment tensor? 

Monte-Carlo type approach (RDC noise) 

´ 

repeat SVD calculation many times (~1000 times) 

each time add different Gaussian noise to experimental RDCs 

accept only those solutions for which all back-calculated RDCs are within 

a given margin of the original experimental dipolar couplings 

error in the data is dominated by the random measurement error in the dipolar couplings 

indirectly take into account uncertainties in the structure 

set the amplitude of the added noise two to three times higher than the measurement uncertainty 


45

Evaluation of uncertainty in backcalculated 

alignment tensors (II) 

Structural noise Monte-Carlo type approach 

repeat SVD calculation many times (~1000 times) 

each time add different Gaussian noise to the original structure (match the 

RMSD between the experimental and back-calculated RDCs) 

spread in alignment parameters obtained for these noise-corrupted structures, 

when using the coupling constants calculated for the original structure (i.e., 

yielding a perfect fit if no structural noise were added) 


46



no RDCs necessary ! 

47


Computer experiment: PALES 

S ij =1/2 (i,j=x,y,z) 

S mol linear average over all non-excluded S matrices 

Periodic boundary conditions 

r < d/(2 V f ) (wall model), or r < d/(4V f ) 1/2 (cylinder) 

48

Shape prediction of magnitude and 

orientation of alignment 

1 DNH predicted [Hz] 


1 DNH measured [Hz] 

Protein alignment in bicelles is sterically induced 

49

Weak alignment in Pf1 bacteriophage 

http://www.asla-biotech.com/asla-phage.htm 


50


and charge distribution 

B 0 


p B = exp[ -ΔG el (r,Ω)/k B T] 

ΔG el (r,Ω) = ∑ i q i φ[r i (r,Ω)] 

S ij mol = ∫ Sij p B (r,Ω) dr dΩ / ∫ p B (r,Ω) dr dΩ 

steric 

p = 0 p = 1 p = 1 

51

How to calculate the electrostatic energy? 

• Continuum electrostatic theory (Debye and Hueckel 1923): protein embedded in 

a dielectric medium containing excess ions 

• non-linear Poisson-Boltzmann equation (Chapman 1913; Gouy 1910) 

• Further simplification: Protein = a particle in the external field of the liquid 

crystal many approximations ! 

• protein = charges of their ionizable residues 

• static dielectric constant of water ε = 78.29 

• average surface charge of phages: -0.47 e/nm2 NMR workshop, Rosario November 2008 

52

elative G 

φ [kT/e] 

0.5 

0 

-2 

-4 

0 


Electrostatic potential 

10 20 

distance from Pf1 [nm] 

B 

53

ubiquitin 

DinI 

GB3 

GB1 

DNA 

Experiment versus PALES prediction 

steric 


electrostatic + steric 

Zweckstetter et al., 

Biophys. J. 2004 

54

ubiquitin 

DinI 

GB3 

GB1 

DNA 


Ionic strength dependence 

magnitude 

orientation 

55

10-20 nm 

Weak alignment in surfactant liquid 

crystalline phases 

3 nm 

PALES 


+ 

+ + 

+ 

+ 

+ 

+ 

B 0 

Lα 

56

Residual dipolar couplings in 

cetylpyridinium bromide/hexanol/sodium 

bromide 


Zweckstetter, 

submitted 

steric and electrostatic interactions 

dominate weak alignment of biomolecules 

in polar liquid crystalline media 

57

Barrientos et al., 

JMR 2001 

pH 7 

pH 3 


Partial alignment at pH 3 

pH 3 

pH 7 

Zweckstetter, Eur. Biophys. J. 2005 

58


PH dependence of alignment 

59

RDCs are a sensitive probe of protein 

electrostatics 

DinI (PDB code: 1GHH) 

Zweckstetter et al., Biophys. J. 2004 


60

Charge/Shape prediction: applications 

• differentiation of monomeric and homodimeric states 

(Zweckstetter and Bax 2000) 

• conformational analysis of dynamic systems such as oligosaccharides 

(Azurmendi and Bush 2002) 

• refinement of nucleic acid structures 

(Warren and Moore, 2001) 

• determination of the relative orientation of protein domains 

(Bewley and Clore 2000) 

• validation of structures of protein complexes 

(Bewley 2001) 

• classify protein fold families on the basis of unassigned NMR data 

(Valafar and Prestegard 2003) 

• Probing surface electrostatics of proteins and nucleic acids, 

refinement of side chain orientations in proteins, … 


61

RDCs in Structure Calculation 


62

RDC-based refinement of structures 

PROBLEM: 

Potential energy surface is very rough many local/false minima 

convergence problem 


RDC-based refinement of starting structures 

k dip force constant adjust such that dipolar RMS is equal to 

measurement error) 

63


Crititical Points 

How to use the structural information obtained from molecular alignment 

1. In order to use the information one needs to know the direction and the size of the 

tensor (susceptibility, alignment, etc). 

2. Minimization of the deviation between the measured quantity and its calculated 

value from a given structure and set of tensor parameters. 

3. One has to be able to evaluate the outcome of the minimization. 

Minimization procedure 

1. Get a good estimate on the tensor parameters (magnitude and rhombicity). 

2. Define structural constraints with respect to the arbitrary tensor coordinate system. 

3. Turn on the force constant for a particular alignment potential such that the final 

RMS between the measured and calculated values reflect the experimental error. 

4. During the calculation the tensor orientation will be automatically determined 

through global minimization with respect to the current structure. 

5. Refine the initial estimate of the tensor parameters. This can be achieved through 

either grid search method or built into the minimization itself. 

64

Define axis system representing the 

alignment tensor 


65


From PALES to XPLOR 

rDC(NH) measured aligned - isotropic (PALES convention) 

(min(DC)=-16.2 Hz; max(DC)=12.7 Hz 

==> estimated Da(NH)=-8.1*(-1) as correct measurement isotropic-aligned) 

==> PALES gives: DATA Da -4.177638e-04 

==> get correct Da by multiplying -4.177638e-04*-21585.19 = 9.0 Hz 

(for XPLOR however use -9.0 Hz, i.e. sign of Szz/2) 

rDC(NH) measured isotropic - aligned (SSIA convention) 

(min(DC)=-12.7 Hz; max(DC)=16.2 Hz 

===> estimated Da(NH)=16.2/2=8.1 

==> PALES gives: DATA Da 4.177638e-04 

==> get correct Da by multiplying 4.177638e-04*21585.19 = 9.0 Hz 

(for XPLOR however use -9.0 Hz, i.e. (-1)*sign of Szz/2) 

66

How to evaluate structures produced by 

minimizing against alignment data 

1. The final dipolar RMS between measured .vs. calculated values 

2. Consistency between dipolar coupling data and NOE data 

(decrease in non-RDC energy terms) 

3. If one has more than one class of dipolar couplings: cross validation with 

quality factor 

4. Use programs such as Procheck to look at the overall quality of the structure 

(distribution in Ramchandran plot should improve) 

5. In most cases where one obtains a high degree of consistency one would also 

gain in the overall RMSD of the family of calculated structures. 


Q = 

rms (D obs -D calc ) 

rms (D obs ) 

67

1 DNH predicted [Hz] 

Quality measure of calculated/simulated 

RDCs 


1 DNH measured [Hz] 

Q = 

rms (D obs -D calc ) 

rms (D obs ) 

rms(Dobs norm 2 2 1/2 

)= [2 (D ) (4 + 3R )/5] a 

Q ~ 17% ≈ 1.8 Å X-ray 

O ~ 11% ≈ 1.1 Å X-ray 

• use only for RDCs not included in 

structure determination ! 

• no translational validation 

68

Impact of RDC refinement 

a) without rdc 

b) with rdc 


Zhou et al. Biopolymers (1999-2000) 52, 168. 

69

c) RDC Theory

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