homi bhabha centenary school on relaxation in …nmr/workshops/ws_nmr_09/madhu/pkmadhu...nmr...

Welcome to all the participants

Homi Bhabha Centenary School on Relaxation in NMR and Related Aspects

February 16-20, 2009National Facility for High-Field NMR, TIFR

Relaxation References

A. Abragam: Principles of Nuclear Spin Magnetism

C. P. Slichter: Principles of Magnetic Resonance

A. G. Redfied: Adv. Magn. Reson., 1, 1, 1961

L. Werbelow and D. M. Grant: Adv. Magn. Reson., 9, 189, 1977

N. Murali and V. V. Krishnan, Conc. Magn. Reson., 17, 86, 2003 (A primer innuclear magnetic relaxation in liquids)

Paramagnetic Relaxation

P. K. P. K. MadhuMadhuDept. of Chemical SciencesDept. of Chemical Sciences

TIFRTIFR

February 18, 2009

How complicated it all is

An example (without eqution) of the hyperfine relaxation

enrotationτ

er

er

ττττ

><IS-dipolar

In solution

complex rler

rler

ττττττττ

<>><

;;

lifetimeτ

IS-dipolarIS-scalar

The solvent itself

naltranslatioτ

e-e interactions

Curie spins

Relaxation Mechanisms in NMR

•Chemical–shift anisotropy

•Scalar coupling

•Dipole-dipole coupling

•Quadrupole coupling

•Spin rotation

A X

Relaxation pathways are treated independentlySimultaneous presence of them is not considered

M

RelA=CSAM+DDAX+DDAM

Relaxation pathwaysIndependentexistence

Autocorrelation Crosscorrelation

Simulatneous presence

Affects T1, T2, and NOE

Causes differential effect/multipleteffect among the various transitions

A X

M

RelA=CSAM+DDAX+DDAM+CSAMXDDAX+ CSAMXDDAM+DDAXXDDAM

What are Cross-Correlations?

Cross-Correlations: Are They Useful?

The differential effects coming from cross-terms of CSA with DD orDD with DD may be exploited to get local structural information, suchas distances, bond angles, and torsion angles

The differential effects may also be used to increase the apparentresolution of a multi-dimensional spectrum of large molecules, TROSY

Anil Kumar et al. Prog. NMR Specy. 37, 191, 319, 2000

Manifestation of Cross-Correlations

Auto-correlation

Single-spin magnetisation modes(total magnetisation of a spin), Az, Mz, Xz

Cross-correlation

Multi-spin magnetisation modesAz, Mz, Xz, 2AzMz, 2AzXz, 2MzXz,4AzMzXzThey reflect the difference in the intensity of various transitions of a spin

A X

MCSAXDD, ΔA

AM, ΔAAX………..

DDXDD, δAMAX........................

Applications of Cross-Correlations

• DLB and longitudinal multi-spin orders

– Determine the absolute sign of the various spin couplings

– Position the principal axes of the spin interaction

– Evolution of multi-spin orders being sensitive to motional anisotropies can be used to study highly anisotropic systems where conventional NMR relaxation studies normally would not work

– Detailed description of molecular dynamics and anisotropic interactions at the molecular level

– Measurement of various structural parameters

– Increasing the resolution of multi-dimensional spectra

Cross-Correlations and Geometry

Cross-correlation spectral densities

Relative orientation of the anisotropic Interactions and their motions

Rotational correlation time of the molecule

The order parameter, S2, hence, dynamics

21( ) ( )( ) (3cos 1) ( )2ij i j ij cJ fω σ σ θ τ∝ Δ Δ −CSAXCSA cross-correlation

CSAXDD cross-correlation 2, , 3

1 1( ) ( ) (3cos 1) ( )2i ij i i ij c

ij

J fr

ω σ θ τ∝ Δ −

DDXDD cross-correlation2

, , 3 3

1 1 1( ) (3cos 1) ( )2ij il ij il c

ij il

J fr r

ω θ τ∝ −

ril

rij θij,il

Relaxation Mechanisms in Paramagnetic Systems

Nuclear relaxation processes of paramagnetic complexes: The slow-motioncase, A. J. Vega, D. Fiat, Mol. Phys., 31, 347, 1976

Nuclear relaxation in macromolecules by paramagnetic ions: A novel mechanism,M. Gueron, J. Magn. Reson., 19, 58, 1975

Solution NMR of paramagnetic molecules, I. Bertini, C. Luchinat, G. Parigi, Currentmethods in inorganic Chemistry, Vol. 2, Elsevier, 2002

Papers by La Mar….

Paramagnetic Effects

• Quantitative expressions available for paramagnetic relaxation

• Paramagnetic metal ions cause – shifts of the resonances (could be used to resolve overlapping

NMR signals)– Enhanced nuclear spin relaxation, PRE– Molecular alignment in the magnetic field– Various cross-correlation effects

• Paramagnetic shifts convey useful long-range structural information and can be measured with high accuracy– Structure determination of paramagnetic metalloproteins

Paramagnetism

Paramagnetism associated with unpaired electrons

A paramagnetic centre may be also introduced exogenously, spin labels

The magnetic moment due to unpaired electrons

1/2[ ( 1)]S Bg S Sμ μ= +

2.0023 for free electrons

Essentially the electron-nuclear interaction similar to nuclear spin-spin interaction, except that the electronic magnetic moment is much higher

Electron Around the Nuclear Spin

Fast molecular reorientation with respect to the g-anisotropy

Electron Around the Paired Electron Spins

How Does Nuclear Spin Sense Electron Spin?

Electron spins relax faster than the nuclear spin

Nuclear spins in each MI level see one electron rapidly changingits orientation among its MS levels and they see anaverage electron magnetic moment

The average electron moment can influence the nuclear energy levels:Local field then fluctuates with a correlation time T1e

The dipolar interaction between electron and nuclear spinsis zero when the electron induced magnetic moment is orientationindependent and non-zero otherwise

A fraction of the electron spins sit on the resonating nucleus leading toa direct shielding constant: Contact term

Relaxation can be caused by the fluctuation of the interaction energy, T1e

Relaxation also by the fluctuation of the average electron moment by the molecular rotation: Curie spin relaxation (CSR)

Nuclear-Electron Interaction

The interaction between a nuclear spin and an unpaired electronis termed as hyperfine coupling

A is the hyperfine coupling tensor

. .H I A S=

(2) (2) (2)0

, ,( ) ( )z

lH I Q K Dμ ν μν

μ ν

ωΩ = + Ω∑

Chemical Shifts in Paramagnetic Molecules

Direct delocalisation of the electron spin at various nuclear spin sites

Effect arising from the anisotropy ofthe g-tensor of the electron spin

Contact shift (Fermi contact term) Pseudo-contact shift (dipolar term)

The average induced electron magnetic moment gives a contribution to the chemical shift called the hyperfine shift

Due to the spin densityon the resonating nucleus

Due to the spin densityoutside of the nucleus

(more useful)

Fermi Contact Coupling and Pseudo-Contact Shift

The contact shift arises from the additional magnetic field generated atthe site of the nuclear spin by the electron magnetic moment locatedat the nuclear spin itself.

This magnetic moment arises from the electron spin density at the nucleus,weighted by <Sz>.

The PCS arises from the spin density distribution all over the space aroundthe nuclear spin.

This is dipolar in nature, and has useful structural information.

The unpaired electron is not localised on a single point, but delocalised onthe entire molecule.

Hence, in every point of space, where the molecular orbital containing theunpaired electron has a non-zero value, the average electron magneticmoment sensed by the nuclear spin is non-zero and proportional to <Sz>times the fraction of unpaired electron present at that point.

Such a fraction is called spin density which for a single electron is given bythe square of its wave function at that point.

Electron Spin Density

Expressions

con pcδ δ+

0

( 1)3

e Bcon z

I I

g S SA A SkT B

μδγ γ

+= = − < >

23

1 ( )(3cos 1)12pc r

δ χ χ θπ ⊥= − −

(Isotropic dipolar shift- The contribution to this shift is isotropic

Pseudo-contact shift has a square dependence on B0, hence the shifts may belarger in higher magnetic fields

Isotropic electron magnetic moment, isotropic susceptibility, pseudocontactshift goes to zero

If the induced magnetic field changes intensity with the molecular orientationsue to the susceptibility being anisotropic, PCS is not zero

Pseudo-Contact Shift

Expressions

Spectra

Separation of Contact and PC Shifts

The sole presence of an unpaired electron spin causes nuclear spin relaxation

The correlation times for the electron-nucleus interaction?

Equations for dipolar and contact interaction mediated nuclear spin relaxation?

Can any of these relaxation pathways cross-correlate with the other nuclearrelaxation mechanisms?

Electron Induced Nuclear Spin Relaxation

Electron Induced Nuclear Spin Relaxation

•The magnetic nucleus does not see unpaired electrons as localised but as spin density distributed throughout the space

•The spin density, in every unit volume, will spend more time in thelow Zeeman energy level(s) and less time in the upper levels

•Changes in the Ms values involve changes in the orientation of the electron magnetic moment

•The time sharing of the levels occurs through electron relaxation

•Electron relaxation thus provides fluctuating magnetic fields causingnuclear spin relaxation

Nuclear spin relaxation due to electrons is contact in origin if reference is madeto spin density at the resonating nucleus

The rest of the electron density and associated electron relaxation is sensed by the nucleus through dipolar coupling- This relaxation is dipolar in origin

Paramagnetic Relaxation Enhancements: PRE

Fluctuation of the electron dipolar field at the nucleus due to the electron relaxation-Dipolar in origin

There are other mechanisms for nuclear spin relaxation besides electron relaxation

Nuclear spin relaxationdue to electron relaxation

Paramagnetic Relaxation Enhancements: PRE

Rotation of the molecular frame causes the nucleusto see the electron in different positions. If the rotation is fasterthan the electron relaxation time, on the rotational time scalethe nucleus sees the electron with the same MS value buton different positions in space. This random motion of the electronaround the nucleus can be again seen as a fluctuating magneticfield that causes nuclear relaxation via dipolar coupling.

In fact, the nucleus sees the induced electronic magnetic moment <m>aligned along the magnetic field. Upon rotation, this average momentcan cause fluctuating magnetic fields sensed by the nucleus through space.This is also dipolar, called Curie spin relaxation (CSR).

Nuclear spin relaxationdue to molecular rotation

Paramagnetic Relaxation Enhancements: PRE

Nuclear spin relaxationdue to chemical exchange

The binding and detachment of a moiety containing the resonatingnucleus and unpaired electron (chemical exchange) can cause fluctuating magnetic fields at the nucleus through both contact and dipolar mechanisms

Relaxation measurements- Information on the interaction between nuclei andunpaired electrons and time dependence ofthe interaction parameters

Dipolar coupling: Electron-nucleus distance, and structural information

Contact interaction: Unpaired electron spin density on the various resonatingnuclei and hence to the topology (via chemical bonds)and the electronic structure of the molecule

We need to decouple these mechanisms to get useful information

Paramagnetic Relaxation Enhancements: PRE

Correlation Times

The longitudinal relaxation enhancement of the nuclear spin due toa coupling with the unpaired electron

Time-independentZeeman fields Time dependent electronic field

seen by the nucleus

The fluctuation of this causes nuclear spin relaxation

Paramagnetic Relaxation Enhancements: PRE

Dipolar term Contact term

Modulation of Hyperfine Interaction

Nuclear spin relaxation through the modulation of the hyperfine coupling

Consider only the dipolar relaxation (due to relevant structural information content)

Even here, there are two sources for the fluctuations:

The relaxation of the electron spin S, dipolar/SolomonThe motion of r

We can write S=SC+swhere SC is the thermal average of S, called the Curie spinSC is aligned along the magnetic field B0 such that

0 ( 1)3

e Bz C

I

g B S SS SkT

μγ

+< >= = (high-temperature approx.)

Dipolar Relaxation Mechanisms

Curie Spin Relaxation (CSR)

CSR in the interaction between the nuclear spin and the staticmagnetic moment <Sz> (the time-averaged electron spin moment)

This interaction cannot be modulated by the electron spin relaxation,since <Sz> is already an average over the electron spin states

The correlation time for this coupling is only determined by τρ

This relaxation mechanism is called the magnetic susceptibility orCSR (to reflect its relationship with the magnetic susceptibility ofa sample via Curie law)

I-S Spin System: Energy Levels, W’s, Transition Frequencies

Relaxation Expressions: Solomon Term

Nuclear relaxation times, 1/T1 and 1/T2

Here, 1 1

2 2

1 1 1

1 1 1s c e

s c e

T

T

τ τ

τ τ

= +

= +

I Sω ω<<and

In the fast motion limit, ωS and ωI <<1/τc), absence of chemical exchange,and T1e=T2e

This is the qualitative statement that the shorter the electronic relaxation times, smaller the paramagnetic effects on nuclear relaxation

Paramagnetic Relaxation Enhancements: PRE

Fast relaxing electrons: PRE is smaller, nuclear spins can be studied with NMR

Slow relaxing electrons: PRE is larger, severe line broadeningfor the nuclear resonances

Ideally we need fast relaxing electrons, PRE negligible, and T1e , T2e effects can be neglected leading to only CSR

Relaxation Expressions: Contact Term

Only flip-flop terms are active

In the fast motion limit,

Relaxation Expressions: CSR

Curie Spin Relaxation (CSR)

CSR is significant when the dipolar coupling described by the Solomon’sequations is governed by the electronic relaxation times, Te<<τc

1 1

1 1 1

s c eTτ τ= +

Efficient relaxation-long correlation times-large T1e-Slow electronic relaxationUse of ions like Mn2+ or Gd3+

CSR may be neglectedBut one should know T1e and this is difficult

On the other hand use ions with short T1e, fast electronic relaxation, thenCSR broadening is overwhelming, and only rotational correlation timesneed to be consideredCSR increases with magnetic field and the size of moleculesUse of ions such as high-spin Fe2+ (S=2) as in hemoglobin or rare earthIons which have short T1e.

Relaxation Mechanisms in NMR

•Chemical–shift anisotropy

•Scalar coupling

•Dipole-dipole coupling

•Quadrupole coupling

•Spin rotation

•Curie spin relaxation

A

M

X

RelA=CSAM+DDAX+DDAM+CSRAe

Relaxation pathways are treated independentlySimultaneous presence of them is not considered

electron

Geometry Relative to Paramagnetic Centre

S

I

re−N

re−H

ΘH

ΘN

rN−HrIS

reS

reI

eSI

IeIS

Geometric Dependence of the Dipolar EffectsInvolving an Electronic Spin

Pseudocontact shift

Relaxation

Interference with nucleus- nucleus dipolar relaxation

θ

ϕχx

χy

χz

re

I

re

I

θr

e

I K

Differential Narrowing in a 2D Multiplet

TROSY (ββ)

anti-TROSY (αα)

semi-TROSY 1 (βα)

semi-TROSY 2 (αβ)

TROSY in Paramagnetic Proteins

TROSY has enabled NMR of very large proteins ofca. 500-800 amino acid residues long

Higher magnetic fields of 700-800 MHz are required tosee the manifestation of TROSY effects

TROSY effects may be more evident in paramagnetic proteins due to additional cross-correlations, CSA*DD+DD*DD+CSR*DD

TROSY in paramagnetic proteins will have directional informationas CSR is geometry dependent with respect to the electron

For all practical purposes, CSRXDD cross correlation Is like CSAXDD cross correlation

Hence similar effects, such as in TROSY, can be expected

IS

IS

IS

IS

IS

Angular Dependence in 2D Multiplet Effect

PIN Geometry Dial

Paramagnetic Induced Narrowing Muller, Otting, Brutscher

TROSY vs HSQC, Myoglobin

TROSY HSQC

Coupled HN-HSQC: Myoglobin

Madhu et al., J. Biomol. NMR 20, 31-37 (2001)

Cross-Correlation Effects:Overlay of 4 αβ-HSQC-αβ

CSR: Structure Elucidation

Ferrocytochrome, S=2

Boisbouvier….Brutscher..,JACS, 121, 7700, 1999

Long-range information possible

S=0Fe (II)

S=1/2Fe (III)

S=5/2Fe (III)

Sperm Whale Myoglobin (153 Residues)

Two Distance Shells from the Metal can be Studied in the Two Spin Samples

CN-Mb7 Å < rHFe < 11 Å 10 Å < rHFe < 25 Å

F-Mb

PCS and Susceptibility Anisotropy

Line Shapes and Pulse Sequence

Quantitative Evaluation

isotropic case

Anisotropy of the Magnetic Susceptibility Tensor

Isotropic χ-tensor -Axially symmetric dipolar shift tensor

High spin Low spin

σDSA σ DSA

Anisotropic χ-tensor -rhombic dipolar shift tensor

isotropic susceptibility axial DSA

non-axial DSAanisotropic susceptibility

Anisotropic Magnetic Susceptibility

Quantitative Evaluation

anisotropic caseθZ’HN

θY’HNθX’HN

z’

x’y’

H N

Angular Dependence of the Cross-Correlation Rates

Isotropic magnetic susceptibility

Anisotropic magnetic susceptibility

θzHFe= 0θzHFe= 90

HN within xz plane

HN normal to xz plane

CSR Cross-Correlation Data

Complete assignment of myoglobin in the low-spin and high- spin states

Distance measurements possible upto 11 A in low- spinstate and 25 A in high- spin state

Systematic study of the contribution of the g- tensor madepossible

Both diamagnetic and paramagnetic samples needed for getting pure cross-correlation contribution and quantification of model

Assignment of paramagnetic proteins will be possible by monitoring only cross-correlated relaxation without any NOE constraints

Identification of solvent exposed regions of a protein: Usefulfor the detection of intermolecular contact sites in protein-ligandcomplexes and protein-multimers

TEMPOLGd(DTPA- BMA)

Omniscan(Gadolinium diethylentriamine pentaacetic acid

bismethylamide)

Gadolinium complexes feature a STRONGER paramagnetism

Gd(DTPA-BMA) is less hydrophobicthan TEMPOL

lower concentrations are required

the binding potential to proteins is minimized

1.

2.

Effect of Paramagnetic Additives on the NMR Parameters of the Protein (in Particular its 1H Relaxation Times)

3. Gd(DTPA-BMA) stable over a widerange of pH and against redox-active

compounds in solution

Ideal binding agent

Both are uncharged and highly water soluble

Effect of Paramagnetic Additives on the NMR Parameters of the Protein (in Particular its 1H Relaxation Times)

The absence of binding of Gd(DTPA-BMA) was confirmed using:

Resonances were attenuated, but none disappeared

No significant chemical shift changes observed upon the addition of the relaxation agent

In the absence of specific binding, the paramagnetic agent is expectedto enhance the relaxation rate of the protein protons as a function of theirsolvent exposure and distance from the surface

Effect on Ubiquitin Proton Resonances

largely affected resonances

slightly affected protons

Relaxation enhancements are reliably big for highly surface exposed protons and small for the deeply buried ones

0 mM Omniscan

4 mM Omniscan

Quantitative Description of the Relaxation Enhancements

1. Second-sphere relaxation

position independent relaxation time τc

Bertini, I.. et al. Nuclear and electron relaxation;VCH, Weinheim, 1991.

Life time of the intermolecular adduct

Relaxation agent is non-specific, yet rotationally correlated, complexwith the protein, and the dipolar coupling between the electron spinand 1H spin is modualted by τr, T1e, and τM

Second-Sphere Interaction Mode

Omniscan

TEMPOL

criteria for the identification of the dimerization site

locations protected from access to Omniscan

although significantly solvent exposed in the NMR structure

Conclusions

correlations between spectral and structural features

Omniscan

Myoglobin

signals of the HSQC are affected proportionally to their solvent exposure

assignment can be performed with only the 3D NOESY-HSQC using :crystal structure

diamagnetic assignmentestimates on the χ tensor

depending on the metal spin, CSR-DD cross-correlation Is measurable at different distance shells from the metal

the effect of anisotropic magnetic susceptibility can deeply affect the use of the cross-correlated relaxation rates

• The paramagnetism of Lanthanide ions useful for fast 3D structure determination protein-ligand complexes

• Combination of PCS induced by a site-specifically bound lanthanide ion and prior knowledge of the 3D strucutre of the La-labelled protein can be used to achieve

– Rapid assignment of the NMR spectra

– Structure determinations of protein-protein complexes

– Identification of the binding mode of low-molecular weight compounds in complexes with proteins

Lanthanide Ions for Structure Determination

Lanthanide ions are attractive due to their relatively largeparamagnetic effects which are also varied

These could be nice targets in the study of large protein-proteincomplexes which are other difficult to study with NMR or X-ray

Lanthanides also have no known essential role in biology

Ca2+ ions, for instance, in calcium binding proteins may be replacedwith lanthanide ions embedding the lanthanide ion in a rigid andextended molecular framework of defined 3D structure (the ionicradii of Ca and La ions are nearly the same)

Lanthanide Ions for Structure Determination

Pseudocontact Shift

θ

ϕχx

χy

χz

re

I

( ) ⎥⎦⎤

⎢⎣⎡ Δ+−Δ= ϕϑχϑχ

πδ 2cossin

231cos3

121 22

3 rhaxpc

r

Orientation dependenceLong-range information

Pseudocontact Shift

Protein complex with La3+ and Tb3+

PCS Isosurfaces

Most Common Paramagnetic Metals in Biomolecular NMRc

Most Common Paramagnetic Metals From the d-Transition

Attachment of a Lanthanide Binding Site in a Protein

(b) site-specific derivatization of a free Cys thiol with a lanthanide-complexing agent (Dvoretsky et al., FEBS Lett., 2002, 528, 189; Ikegami et al., J. Biomol. NMR, 2004, 29, 339), Purdy et al., Acta Crystallogr. D, 2002, 58, 1111.

(c) design of fusion proteins with lanthanide-binding motifs

EF-hands (Ma and Opella, J. Magn. Reson., 2000, 146, 381)calmodulin (Feeney et al., J. Biomol. NMR, 2001, 21, 41)combinatorially screened peptides (Wohnert et al., J. Am. Chem. Soc.,

2003, 125, 13338)

(a) Replacement of a pre-existing calcium binding site (Bertini et al., J. Am. Chem. Soc., 2001, 123, 4181.

Fast Determination of the Binding Mode of Small Ligand

One example:

Determination of the 3D structure of a small ligand moleculebound to its protein target in solutionSimultaneously, the location and orientation of the ligand molecule with respect to the protein

An eg: thymidine bound to ε186/θ loaded with Dy3+, Tb3+, or Er3+

M. John, G. Pintacuda, A.-Y. Park, N. E. Dixon and G. Otting J. Am. Chem. Soc., 2006, 128, 12910-12916.

Fast Determination of the Binding Mode of Small Ligands

Lanthanide Labeling: Advantages

Lanthanide Labeling: Advantages

Fast Determination of the Binding Mode of Small Ligands

Concentration-dependent relaxation enhancements and PCS lead to binding affinity and the fraction of bound thymidine

From the PCS data of thymidine, the position of 1H and 13Ccan be positioned with respect to the Δχ tensor and the ε186 molecule

The structure and the location of the thymidine molecule was foundto be similar to that determined in single crystal studies

Further CSRXDD Cross-Correlation Effects