1

NMR (CHEM8028)

Solid-state NMR: Anisotropic interactions and how we use them

Dr Philip WilliamsonJanuary 2015

2

NMR: From Molecular to Cellular Level

!

SolutionMembraneMitochondrionCell

Increasing complexity

Key proteinsLipid/proteinassemblies

Tissues Organelles

Solid State NMR Liquid NMR

3

Solid-state NMR spectra

3

4

Solid-state NMR

• Anisotropic Interactions

• What are they, what do they do (to our spectra)

• How can we manipulate them

• Oriented samples

• Magic angle spinning

• How can we exploit them

• Cross polarization

• Dipolar recoupling

• How can we use them to probe structure/dynamics (2nd

series of lectures)4

5

Outline (1)

• What is anisotropy

• How does it effect NMR spectra

• What interactions give rise to anisotropic properties?

• Describing interactions: tensors

• Chemical Shielding Anisotropy

– Orientational dependence of resonance frequency

– Powder spectra

• Dipolar interactions

• Quadrupolar interactions5

6

What is anisotropy

• Something whose properties depend on its orientation

e.g. stress

6

7

How does it effect the NMR spectrum

• Each molecular orientation gives rise to a difference resonance frequency

• In powder we have the sum of all distributions

• In the liquid state these anisotropic properties are averaged on the NMR timescale 7

8

Which interactions in NMR

8QDipolarCSAJCS HHHHHH

Isotropic Anisotropic

JCS HHH

DipolarCSAJCS HHHHH

CSAJCS HHHH

CSHH

9

Describing interactions: tensors (1)

We are concerned with 3 flavours

• Zero rank tensors

– Physical property independent of coordinate system in which it is described (scalar, distance)

• First rank tensors

– Coordinate, depends on frame of reference (vector, has size and direction)

• Second rank tensors

– Multiple first rank tensors e.g. stress (matrix)

• Higher rank exist – but we will not be considering9

10

Describing interactions: tensors (2)

Rank zero tensor Rank one tensor

10

r

B0

(0,0,Bz)Isotropic chemical shift,

J-coupling

11

Describing interactions: tensors (3)

• Second rank tensors

11

zz

yy

xx

PAS

000000

zzzyzx

yzyyyx

xzxyxx

i j k

i

j

k

x

y

z

zz

yy

xx

12

Parameterizing 2nd rank tensors

• In cartesian notation tensors defined by principle components, Axx, Ayy andAzz

• Frequently parameterized with

• This assumes

• Thus the asymmetry 0.0<η<1.0 and anisotropy can be both positive and negative 12

xxyy

zz

zzyyxx

AA

aA

AAAATra

3

333aAaAaA yyxxzz

13

Chemical Shielding Anisotropy (1)

• Perturbation of the magnetic field due to interaction with surrounding electrons

• Inherently asymmetric (e.g. electron distribution surrounding carbonyl group)

13

14

Chemical Shielding Anisotropy (2)

• We can describe the perturbation of the main field (B0), by the second rank tensor,

• The Hamiltonian which describes the interaction with the modified field is:

Which can be written in a simplified form as:

14

0

00

ˆ,ˆ,ˆ

BIIIH

zzzyzx

yzyyyx

xzxyxx

kkzkykxkCSA

0ˆ BIH k

kkkCSA

0

00

BB

zzzyzx

yzyyyx

xzxyxx

S

15

Chemical Shielding Anisotropy (3)

Thus the chemical shielding Hamiltonian simplifies to:

and the resonance frequency of the line is:

Thus the resonance frequency is proportional to zz in the laboratory frame.

However, is usually defined in the principle axis system (PAS) not in the lab frame (LF). Therefore, we need to transform from the PAS to LF.

15

0)(ˆ BIH k

zzk

kzkCSA

0)(

12 kzz

16

Transformations

16

z

x

y

zz

yyxx

z

x

y

Principle Axis System

Lab Frame

• Rotation characterized by the three Euler angles (α,β,γ)

• Multiple by rotation matrix R

17

Transformation matrix

Can derive a rotation matrix which bring about the rotation described above:

To determine in the laboratory frame, need to apply to the chemical shielding tensor in the principle axis system:

This can be simplified to give general Hamiltonian for CSA in lab frame of:

17

cossinsinsincos

sinsincoscossincossinsincoscoscossincossinsincoscoscossinsinsincoscoscos

,,R

),,(),,( 1 RR PASLAB

kzisokCS IBH ˆ2cossin1cos32

220

18

Effect on resonance position

18

z

x

y

zz =3000Hz

yy=-1500Hzxx =-1500Hz

σiso = 1/3(σxx+σyy+σzz) = 0Hz

δ = σzz-σiso = 3000 Hz

= (σyy-σxx)/

kzisokCS IBH ˆ2cossin1cos32

220

/2

19

Powder Patterns

• In powders we have a random distribution of molecular orientations.

• Thus the lineshape is the weighted superposition of all the different orientations:

19

..sin),,,(8

1)(2

0 0

2

02 tsts

20

Empirical relation between PAS and MF1) Methyl carbons axially symmetric, axis along threefold symmetry

axis

2) Ring carbons three distinct tensor elements, most shielded perpendicular to plane, least shielded bisecting C-C-C angle of ring

3) Most shielded direction:1) Perpendicular to ring in aromatic carbons2) Along C3 axis for methyl carbons3) Perpendicular to the sp2 plane for carbonyl/carboxylic acids

4) Least shielded direction:1) In the ring plane, bisecting C-C-C angle2) Perpendicular to C3 axis for methyl groups3) In the sp2 place for carbonyl/carboxylic acids

5) Intermediate shielding1) Tangential to ring for aromatic systems2) In the sp2 plane and perpendicular to the C-C bond for COOH

20

21

Dipolar Interaction

Classical interpretation

Classical interaction energy between two magnetic (dipole) moments when both are aligned with the magnetic field:

Quantum mechanical

where:

• Symmetric second rank axially symmetric tensor.

• Again we need to rotate from the PAS to LF to obtain resonance frequency.

21

B0

1

2

)cos31(14

2213

12

0

r

E

21

122121212

21312

210

ˆˆ

.ˆ.ˆ3ˆˆ4

IDI

rIrIr

IIr

HD

200010001

4 312

210

rD

22

Orientation dependence of dipolar interactionHomo-nuclear Dipolar Hamiltonian: Hetero-nuclear Dipolar Hamiltonian:

22

2121

2

312

210,

ˆˆˆˆ32

)cos31(4

IIIIr

H zzIID

zzISD II

rH 21

2

312

210,

ˆˆ22

)cos31(4

dip=20 kHz

dipdip

23

Quadrupolar Interaction (1)If spin>1/2, nucleus contains an electronic quadrupole moment (Q).

Electronic quadrupole moment interacts with surrounding electron cloud (electric field gradient(EFG), V).

where:

Again we can define the anisotropy and asymmetry:

23

kkQ IQIHˆˆ

zz

yy

xx

VV

V

IIeQQ

000000

)12(2

zz

xxyyQ

ZZQ

VVV

IIqQeQ

)12(2

2

24

Quadrupolar Interaction (2)To calculate the resonance frequency, we must transform from the PAS of the EFG to the laboratory frame.

Retaining only the “secular terms” gives the following Hamiltonian in the LF:

24

)1(ˆ32cossin1cos32

222 IIIH ZQQ

Q

Orientation dependence of a single crystal of Ala-d3

Powder spectrum of Ala-d3

Q

25

Exploitation of anisotropic interaction

• Oriented samples

– Single Crystal studies

– Oriented Biological Membranes

• Dynamics

– Averaging of anisotropic interaction

• Local electronic environment

– Perturbation in chemical shielding anisotropy

25

26

Dynamics: averaging of anisotropy

26

Gel Phase

Liquid Crystalline Phase

Axis of rotational averaging

Rotational diffusion: Scaling of interaction by where is the angle between axis of motional averaging and the PAS of the interaction

1cos321 2

27

Oriented samples

27

Cys192/193

Field (B0)

C3

C3’

Orientation

0°

90°

Necessary to introduce macroscopic alignment:1) Crystallization2) Oriented membranes3) Fibres (Silk/DNA)

28

Oriented samples – ligand orientations

B0 B0

Orientation±5° Mosaic Spread±5° Orientation±5° Mosaic Spread±5°

29

Protein Backbone Orientation

Opella et al. 1998

15N chemicalshielding anisotropy

15N-1H hetero-nucleardipolar coupling

Bo

30

Local electronic environment

30

HCl

As we shall see next week, typically these parameters are obtained under conditions of magic-angle spinning to enhance signal to noise.

31

An aside: spherical tensors

• Make the calculations a lot easier to handle

• Frequently used in papers

31

32

Sensitivity and resolution enhancement in solid-state NMR

32

33

Resume

33QDipolarCSAJCS HHHHHH

Isotropic Anisotropic

JCS HHH

DipolarCSAJCS HHHHH

CSAJCS HHHH

CSHH

34

Oriented samples

• Increase resolution by orienting interactions, therefore all spins resonate at the same frequency

• As all spins resonate with the same frequency the sensitivity of the measurements is higher

34

35

Magic-angle spinning

35

36

Magic Angle Spinning

Seeks to reintroduce averaging process through mechanical rotation

36

Magic Angle Spinning Probehead (Doty)

Sample rotors (Varian)

37

Averaging of anisotropic interactions

37

38

Averaging of anisotropic interactions

38

The Hamiltonian becomes time dependent:

We can deconvolute this into the iso- and an-isotropic contributions:

where

and

Where C1, C2, S1 and S2 relate the anisotropic interaction to magnetic field (Appendix 1).

ZCS IttH ˆ),,,()(ˆ

),,(),,,( CSAisot

0iso

)22sin()22cos()sin()cos(),,(

22

11

tStC

tStCCSA

39

Analysis of MAS spectra

• All anisotropic interactions become time dependent

• To analyze spectra need to treat these time dependencies

• Several mathematical descriptions that allow us to do this

– Average Hamiltonian Treatment

– Floquet Theory

– Piece wise integration

39

40

Slow speed spinning• Rotational echoes apparent in

fid which characterise the anisotropy of the interaction

• At lower spinning speed the intensity of the sidebands characterises the anisotropic interaction ( and )

40

41

Herzfeld-Berger Analysis

Expression exist to calculate the intensity of sidebands for a given anisotropic interaction:

where

and

411) Herzfeld and Berger, J.Chem.Phys 73 (1980) 6021

2

0 0

2),,(sin41)( rr NFNI

)exp(),,( CSAr iNF

)2sin(2

)2sin(2

)cos()sin(),,,(

)(),,,(

22

11

tStC

tStCt

ttt

rr

rr

rr

rr

CSA

isoiso

CSAisoCSA

42

CSA analysis in reality

Several programs now available that now facilitate this task:

1) Tables – Paper by Herzfeld and Berger

2) matNMR (routines for analysis of both CSA and quadrupolar interactions in bothe static and MAS spectra) http://matnmr.sourceforge.net/ (requires matlab)

3) MAS sideband analysis (Levitt group homepage) http://www.mhl.soton.ac.uk/public/Main/index.html(requires mathematica)

42

43

Effect of off-angle MAS

• Anisotropic interaction scaled by ½(3cos2-1)

• Useful for characterizing anisotropy whilst gaining some sensitivity

• Indicates why magic angle should be carefully set!

43

44

When does MAS not work?

• Homogeneous interactions

– e.g. Homonuclear dipolar interactions

• Heterogeneous line-broadening

– e.g. Samples with conformational heterogeneity (lyophilized solids)

• Nuclei with large quadrupolar interactions

• When samples are not ‘solid’

44

45

Applications of MAS

• Resolution/Sensitivity Enhancement

• Low speed spinning – characterisation of anisotropy

45

Isotropic chemical shifts in the protein backbone are sensitive to secondary structure

Analysis of the principle components of the chemical shielding tensor reveals that larger changes are seen in 22making it a sensitive probe of protein secondary structure.

Wei et al. 2001 JACS 123: 6118-26

46

Applications of MAS

• Low speed spinning

– anisotropymobility

Amyloid precursor protein in differing lipid environments has different propensity to oligomerise. Sideband analysis reveals changes in peptide mobility

Marenchino et al. Biophysical Journal 2008 46

47

Magic-angle spinning and metabolomics

47

48

Tale of a hungry worm

4kHz MAS spectrum of C.elegans (400MHz)

49

Carbohydrate metabolism

50

Fatty acid metabolism

51

MAS-NMR and metabolism

Spectroscopically:• Simple markers for metabolites• Observe changes in metabolite levels• Labelling possible to aid in assignment

Biologically:• Genetics of C.elegans well characterized• Large library of mutations• Development well understood• Genes linked to phenotype• Behavioral differences

52

A molecular view of biological systems

Structure of high affinity ligands

Membrane proteincomplexes Regulation of

intra-cellular trafficking

Whole organisms