NMR relaxometry of paramagnetic molecules Giacomo Parigi CERM - - PowerPoint PPT Presentation

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NMR relaxometry of paramagnetic molecules Giacomo Parigi CERM - - PowerPoint PPT Presentation

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NMR relaxometry of paramagnetic molecules Giacomo Parigi CERM University of Florence The paramagnetic contribution to relaxation t M t M H t R H H t M H H t s O M e O O O H H + H H H H 2 O O t fast t M Bulk water H NH

slide-1
SLIDE 1

NMR relaxometry of paramagnetic molecules

Giacomo Parigi CERM

University of Florence

slide-2
SLIDE 2

O H H tM Bulk water O H H NH tM H2O tM tfast tR H t’M Bulk water O H H M O H H tM ts e O H H tD tR

R1=R1dia+R1para

r1, relaxivity = paramagnetic relaxation rate due to 1 mmol/dm3 paramagnetic centers

+

mS mI1 mI2

mS=658.2 mI R1para=[Me]r1 r1=(R1-R1dia)/[Me]

The paramagnetic contribution to relaxation

slide-3
SLIDE 3

The paramagnetic contribution to relaxation

R1=R1dia+R1para

mS mI1 mI2

mS=658.2 mI R1para=[Me]r1

0.01 0.1 1 10 100 1 2 3 4 5 6 7 8 9 10

Proton relaxivity (s

  • 1 mM
  • 1)

Proton Larmor frequency (MHz)

0.01 0.1 1 10 100 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Proton relaxation rate (s

  • 1)

Proton Larmor frequency (MHz)

Diamagnetic molecule 2 mM

0.01 0.1 1 10 100 2 4 6 8 10 12 14 16 18 20

Proton relaxation rate (s

  • 1)

Proton Larmor frequency (MHz)

Paramagnetic molecule 2 mM Relaxivity (1 mM)

  • =

r1, relaxivity = paramagnetic relaxation rate due to 1 mmol/dm3 paramagnetic centers

r1=(R1-R1dia)/[Me]

slide-4
SLIDE 4

N S m

m

 r

through space Fluctuations in the Hamiltonian couple each spin to the external world (lattice), thus allowing for energy exchanges

Nucleus-electron dipole-dipole coupling

  • +

+

wI wS

  • -

+ + mS,mI

B

S S

 w = B

I I

 w =

slide-5
SLIDE 5

The transition probabilities per unit time (for stochastic, stationary perturbations) are:

  • +

w0 w1 w1 w2

I I

+

MAGNETICALLY COUPLED TWO-SPIN SYSTEM (DIPOLE-DIPOLE COUPLING)

If tc is the correlation time of the relaxation mechanism, for t < tc, there is a large correlation and G is large for t > tc, the correlation goes to zero

R1M = w0+2w1

I+w2

Nucleus-electron dipole-dipole coupling

R1  Edip

2f(tc, w)

wI wS - -

+ +

c c

m H n n H m G G

mn mn t t t t

t

/ * 1 1 /

e | | | | (0)e ) (

  = =

= =

 

  • t

w

t t t w

d e G J

c

i mn mn /

) ( ) (

 

2 2 1

1 ) (

c mn c mn

H t w t   

n m mn

E E - = w

slide-6
SLIDE 6

For stationary perturbations: and

c c

m H n n H m G G

mn mn t t t t

t

/ * 1 1 /

e | | | | (0)e ) (

  = =

=

  • =

=

 

c i mn i mn mn

i e G d e G J

c c

t w t w

t t t w t t t w

/ 1 ) ( ) ( ) (

/ /

=  =        

  • =

1 ) ( 2 / 1 1 ) ( 2

c c mn c mn

i G i G wt t t w

    

           =

2 2 * 1 1

1 1 | | | | 2

c c c c

i m H n n H m wt wt wt t

2

slide-7
SLIDE 7

t t w

t w d

e G J

i mn mn

= ) ( ) (

    

           =

2 2 * 1 1

1 1 | | | | 2 ) (

c c c c mn

i m H n n H m J wt wt wt t w

  

         =

2 * 1 1 2

1 | | | | 2

c mn c mn

m H n n H m W t w t 

t t

t w

d e G W

mn

i t t mn mn

= ) ( 1

2

slide-8
SLIDE 8

A B C D E F

  • +

w0 w1 w1 w2

I I

+

     

* 2 2 * 1 1 dip

4 1 F S I F S I F S I S I F S I S I F S I S I S I H

z z z z z z

 

            

  • =

16 | | | |

2 * 1 1

F H H w =  -



-   4 | | | |

2 1 * 1 1 1

F H H w =   -  -   

2 2 * 1 1 2

| | | | F H H w =  

  • 
  • 

    =  | 2 1 |

z

I

  • -

+ +

slide-9
SLIDE 9

)) ( cos 3 1 ( ) (

2

t k t F 

  • =

) ( 1

e ) ( )cos ( sin 2 3 ) (

t

  • i

t t k t F

 

  • =

) ( 2 2 2

e ) ( sin 4 3 ) (

t i

  • t

k t F

  • =

2 1 1 4 2 2 2 2 2 2

5 4 ) (cos d ) cos 9 cos 6 1 ( 2 1 | cos 3 1 | | | k k k F = 

  • =

= 

  

2 2 2 2 1

10 3 | | | | k F F = = 

3 2

4 r k

S I

  m  =

2 2 2 2 2 2 2 2

) ( 1 10 ) ( 1 5 4 16 1 2

c I S c c I S c

k k w t w w t t w w t

=

=  

          

        =

  • 2

2 2 2 2 2 6 2 2 2 2 1 1

) ( 1 6 ) ( 1 1 3 4 10 1

c S I c c S I c c I c S I

r T t w w t t w w t t w t    m 

slide-10
SLIDE 10

0.01 0.1 1 10 100 1000 1 2 3 4 5 6 7 8 9 10

Spectral density (tc units) Proton Larmor frequency (MHz)

Nuclear relaxation due to the electron-nucleus dipolar coupling Solomon equation

tc (s)

Solomon, Phys. Rev. 99 (1955) 559 Bertini, Luchinat, Parigi, Ravera, NMR of paramagnetic molecules, Elsevier, 2016

7J(wS) 3J(wI)

10-9 10-10 10-8

 

                =

2 2 2 2 6 2 2 2 2 1

1 3 1 7 1 4 15 2

c I c c S c B e I M

r S S g R t w t t w t m   m

B

S S

 w = B

I I

 w =

slide-11
SLIDE 11

Three times modulate the dipolar Hamiltonian: 1) Electron relaxation ts 2) Rotation tr 3) Chemical exchange tM

e e N N

B0

N e N e N e N e

B0 B0

A B C

kT a

r

3 4

3

 t =

slide-12
SLIDE 12

tr ts tM s

10-13 10-11 10-9 10-7 10-5

Each time contributes to the decay of the correlation function:

] ) ( exp[ ) / exp( ) / exp( ) / exp(

1 1 1

t t t t

M r s M r s

  • =
  • t

t t t t t

1 1 1 1

 = 

M r s c

t t t t

slide-13
SLIDE 13

The paramagnetic contribution to solvent relaxation

Bulk water O H H M O H H tM ts e tR

If tM << 1/R1M fM = mole fraction of ligand nuclei, in water: R1M q = number of coordinated water molecules

r1, relaxivity = paramagnetic relaxation rate due to 1 mmol/dm3 paramagnetic centers

M M R

f r

1 1 =

6 . 55 001 . q fM =

slide-14
SLIDE 14

0.01 0.1 1 10 100 0.0 0.5 1.0 1.5 2.0 2.5 3.0

Proton relaxivity (s

  • 1 mM
  • 1)

Proton Larmor frequency (MHz)

Copper(II) aqua ion

298 K Lorentzian dispersion Rl.f. Rh.f. Rl.f.=10/3 Rh.f.

slide-15
SLIDE 15

Best fit (with q=6) r = 0.27 nm tc = 2.610-11 s (ts = 310-10 s )

Copper(II) aqua ion

0.01 0.1 1 10 100 0.0 0.5 1.0 1.5 2.0 2.5 3.0

Proton relaxivity (s

  • 1 mM
  • 1)

Proton Larmor frequency (MHz)

298 K

M

R q r

1 1

6 . 55 001 . =

slide-16
SLIDE 16

0.01 0.1 1 10 100 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Proton relaxivity (s

  • 1 mM
  • 1)

Proton Larmor frequency (MHz)

298 K 338 K 278 K Best fit r = 0.27 nm tc (278)= 4.010-11 s tc (298)= 2.610-11 s tc (338)= 0.910-11 s

N e N e

B0

B

tM<1/R1M10-5 s

Copper(II) aqua ion

kT a

r

3 4

3

 t =

slide-17
SLIDE 17

The paramagnetic contribution to solvent relaxation

Bulk water O H H M O H H tM ts e tR

R1M

r1=fm(1/R1M+tM)-1

If temperature , times are faster: tr and tM Since R1M  tc, R1M

  • tM <<1/R1M
  • tM >>1/R1M

r1 r1

0.01 0.1 1 10 100 5 10 15 20 25 30 35

Proton relaxivity (s

  • 1 mM
  • 1)

Proton Larmor frequency (MHz)

278 K 288 K 298 K

slide-18
SLIDE 18

 

                                      =

2 , 2 , 2 , 2 , 6 2 2 2 2 6 2 2 2 2 1

1 3 1 7 1 3 1 7 15 1 2 4 6 . 55 001 .

ss M I ss M ss M S ss M ss ss c I c c S c is is B e I

r q r q S S g r t w t t w t t w t t w t m   m

 

                = 

2 2 2 2 2 2 2 2 6 1

1 3 1 7 15 1 2 4 6 . 55 001 .

c I c c S c B e I i i i

S S g r q r t w t t w t m   m

The paramagnetic contribution to solvent relaxation

q = number of coordinated water molecules

H t’M Bulk water O H H M O H H tM ts e tR

r1, relaxivity = paramagnetic relaxation rate due to 1 mmol/dm3 paramagnetic centers

  • and tc,i = tc

First and second-sphere contributions:

  • and tM,ss < tc,in

) / 1 ( 6 . 55 001 .

, , 1 1 i M i M i i

R q r t  =

slide-19
SLIDE 19

Effect of tM

tM= 1 ns Proton Larmor frequency (MHz) Proton Relaxivity (s-1 mM-1) 0.1 ns 0.01 ns 0.001 ns 0.0001 ns 0.00001 ns

slide-20
SLIDE 20

ts0 for paramagnetic metal ions

Cu(II) 300 ps VO(IV) 500 Ti(III) 40 Mn(II) 3500 Fe(III) 90 Fe(II) 1 Cr(III) 400 Co(II) 3 Ni(II) 4 Gd(III) 120 Ln(III) 0.1-1

ts < tr (30 ps) in aqua ions (Low lying excited states make Orbach process very efficient)

slide-21
SLIDE 21

No dispersion is detected

Fast relaxing metal ions

0.01 0.1 1 10 100 0.0 0.2 0.4 0.6 0.8 1.0

Ni

2+

Co

2+

Proton relaxivity (s

  • 1 mM
  • 1)

Proton Larmor frequency (MHz)

Fe

2+

at 1 T

 

                =

2 2 2 2 6 2 2 2 2 1

1 3 1 7 1 4 15 2

s I s s S s B e I M

r S S g R t w t t w t m   m

ps 5 MHz 50 2 658 1 =     t s 1 

s St

w

s c

t t =

slide-22
SLIDE 22

Proton Larmor Frequency (MHz)

0.01 0.1 1 10 100

Proton Relaxivity (s-1mM-1)

0.0 0.2 0.4 0.6 0.8 1.0 1.2

Co(II) carbonic anhydrase

nitrate adduct OH Co His His His r = 0.27 nm ts = 10-11 s D  10 cm-1 No ZFS with ZFS

A fifth ligand reduces ts of one order of magn.

tetracoordinated

Koenig SH, Brown RD III, Bertini I, Luchinat C, Biophys. J. 1983; 41: 179

slide-23
SLIDE 23

ts0 for paramagnetic metal ions

Cu(II) 300 ps VO(IV) 500 Ti(III) 40 Mn(II) 3500 Fe(III) 90 Fe(II) 1 Cr(III) 400 Co(II) 3 Ni(II) 4 Gd(III) 120 Ln(III) 0.1-1

tr << ts in aqua ions

0.01 0.1 1 10 100 0.0 0.5 1.0 1.5 2.0 2.5 3.0

Proton relaxivity (s

  • 1 mM
  • 1)

Proton Larmor frequency (MHz)

tc = tr  30 ps

slide-24
SLIDE 24

0.01 0.1 1 10 100 5 10 15 20 25 30 35 40 45 50

Proton Relaxivity (s

  • 1mM
  • 1)

Proton Larmor frequency (MHz)

298 K

Manganese(II) aqua ion

Ri.f. Rh.f. Ri.f. = 10/3 Rh.f. tc1  30 ps tc2  3000 ps

An additional dispersion is present at low fields

slide-25
SLIDE 25

The hyperfine coupling

N S m

m

 r

through bonds

M e- H

through space

= unpaired electron spin density on the nucleus

i i

r

) ( 

slide-26
SLIDE 26

Contact relaxation

Bloembergen, J. Chem. Phys. 27 (1957) 572 Bertini, Luchinat, Parigi, Ravera, NMR of paramagnetic molecules, Elsevier, 2016

R1M  E2 f(tc,w)  Ac

2 f(tc,w)

Ac = contact coupling constant, proportional to the unpaired electron spin density at the nucleus

1 1 1

=

M s c

t t t

slide-27
SLIDE 27

0.01 0.1 1 10 100 5 10 15 20 25 30 35 40 45 50

Proton Relaxivity (s

  • 1mM
  • 1)

Proton Larmor frequency (MHz)

298 K 313 °C

Manganese(II) aqua ion

Dispersion due to contact

M

R q r

1 1

6 . 55 001 . =

con 1 dip 1 1 M M M

R R R  =

Dispersion due to dipolar q=6 S=5/2

Best fit parameters (298 K) r = 0.28 nm Ac/h = 0.8 MHz tr = 310-11 s ts0 = 310-9 s

slide-28
SLIDE 28

298 K

Gadolinium(III) aqua ion

0.01 0.1 1 10 100 5 10 15 20 25 30

Proton Relaxivity (s

  • 1mM
  • 1)

Proton Larmor frequency (MHz)

Rl.f. Rh.f.

  • ne single dispersion
  • Rl.f.  10/3 Rh.f.

Since in fast exchange (as seen from temperature dependence), tc must be field dependent

field dependent ts

slide-29
SLIDE 29

0.01 0.1 1 10 100 5 10 15 20 25 30 35

Proton Relaxivity (s

  • 1mM
  • 1)

Proton Larmor frequency (MHz)

Gadolinium(III) complex bound to a protein

field dependent ts

tr very long tc =ts

slide-30
SLIDE 30

Electron relaxation due to modulation of ZFS

  • Fluctuations in the spin-orbit coupling and in the spin-spin

coupling (for S>1/2 systems) may induce electron relaxation

  • Such fluctuations are induced by deformations of the

coordination polyhedron by collision with solvent molecules.

  • They cause a transient ZFS (Dt) which allows the coupling of

rotation with spin transitions.

slide-31
SLIDE 31

Electron relaxation (pseudorotational model)

due to modulation of ZFS transient ZFS correlation time for electron relaxation

10 100 1000 1E-10 1E-9 1E-8 1E-7

Electron relaxation time (s) Proton Larmor frequency (MHz) Bloembergen, Morgan, J. Chem. Phys. 1961, 34, 842

          

2 2 2 2

4 1 4 1

v s v v s v

t w t t w t

 

3 ) 1 ( 4 50 2

2 1

D = S S R

t e

e s

R1 / 1 = t

tv Dt

slide-32
SLIDE 32

Nuclear relaxation due to the electron-nucleus dipolar coupling SBM model

transient ZFS correlation time for electron relaxation

 

                =

2 2 2 2 6 2 2 2 2 1

1 3 1 7 1 4 15 2

c I c c S c B e I M

r S S g R t w t t w t m   m

r s

t t 

1 1 1 1

 =

M r e c

R t t t

t

D

v

t

slide-33
SLIDE 33

Effect of electron relaxation

Proton Larmor Frequency (MHz)

0.01 0.1 1 10 100 1000

T1M

  • 1(s-1mM-1)

5 2 20

Proton Larmor Frequency (MHz)

0.01 0.1 1 10 100 1000

T1M

  • 1(s-1mM-1)

0.05 0.03 0.1

tv (ps) Dt (cm-1)

slide-34
SLIDE 34

S=7/2

Gadolinium(III) aqua ion

0.01 0.1 1 10 100 5 10 15 20 25 30 35 313 K

Proton Relaxivity (s

  • 1mM
  • 1)

Proton Larmor frequency (MHz)

298 K

Best fit parameters (298 K) r = 0.30 nm tr = 3810-12 s ts0 = 1.110-10 s tv = 1610-12 s (Dtr = 0.036 cm-1; Dt

2 = (2cDtr)2 = 4.61019 s-2)

q=8 ts =110 ps  tr ts =3300 ps >> tr tc = 28 ps tc = 38 ps

) (

1 1 1 1

 =

M r s c

t t t t

dip 1 1

6 . 55 001 .

M

R q r =

slide-35
SLIDE 35

Outer-sphere relaxation

Bulk water M ts e O H H tD tR

Mn(II)

Mn(DTPA)

H t’M Bulk water O H H M O H H tM ts e O H H tD tR

In general, all contributions should be considered:

  • s

i M i M i i

r R q r

, 1 , , 1 1

) / 1 ( 6 . 55 001 .   = t

slide-36
SLIDE 36

Outer-sphere relaxation due to diffusing water protons

Hwang and Freed, J Chem Phys 1975, 63:4017 Polnaszek and Bryant, J Chem Phys 1984, 81:4038

Solomon

   

) ( 3 ) ( 7 ) ( ) 1 ( 1000 4 405 32

2 2 2 2 1p I S L M B e I A

J J D D d S S g M N R w w m   m           =

648 / 81 / 81 / 4 6 / 2 / 1 8 / 8 / 5 1 ) (

6 5 4 3 2 2

z z z z z z z z J         = w

         =

s D D

z t t wt 2

L M D

D D d  =

2

t

slide-37
SLIDE 37

Outer-sphere relaxation

slide-38
SLIDE 38

A zinc-sensing contrast agent for MRI

with Zn2+ without Zn2+

1 Gd-coordinated water molecule + Outer-sphere water molecules Outer sphere water molecules Major, Parigi, Luchinat, Meade,

  • Proc. Natl. Acad. Sci. USA, 2007
slide-39
SLIDE 39

Proton Larmor Frequency (MHz)

0.01 0.1 1 10 100

Proton Relaxivity (s-1mM-1)

1 2 3 4 5 6 7 8 9 10

  • uter-sphere

inner-sphere

Gd-DTPA Best fit parameters r = 0.31 nm – q = 1 tr = 7510-12 s ts0 = 9010-12 s (Dtr = 0.036 cm-1) tv = 2010-12 s d = 0.36 nm D = 2.610-9 m2s-1

Gd(III) H2O

Gd(III) complexes

slide-40
SLIDE 40

Gd(III) macromolecules

The effect of increasing tr

Other parameters: r = 0.31 nm – q = 1 Dtr = 0.03 cm-1 tv = 2010-12 s (ts0 = 13010-12 s) tM = 1010-9 s (fast exchange) 0.01 0.1 1 10 100 0.0002 0.0023 0.0235 0.2349 2.3485 10 20 30 40 50 60

0.1 1 10 100

Magnetic field (T) Proton relaxivity (s

  • 1 mM
  • 1)

Proton Larmor frequency (MHz) tr (ns)

slide-41
SLIDE 41

0.01 0.1 1 10 100 5 10 15 20 25 30

Relaxivity (s

  • 1 mM
  • 1)

Proton Larmor Frequency (MHz)

N N N CONH SO2NH2 COO- COO-

  • OOC
  • OOC

Na+ Gd3+

Gd(III) DTPA-SA + Carbonic Anhydrase

Best fit parameters: 2 protons at 3.0 Å Dtr = 0.017 cm-1 tv = 18·10-12 s tM = 560·10-9 s D=0.01 cm-1 (tr = 12·10-9 s)

Anelli, Bertini, Fragai, Lattuada, Luchinat & Parigi, Eur.J.Inorg.Chem. 2000

Gd(III) macromolecules

ZFS

SBM model

slide-42
SLIDE 42

Degeneracy in fit parameters can be limited by analyzing NMRD profiles at several temperatures: d, r, Ac (and possibly Dt) should not change

0.01 0.1 1 10 100 1000 1 2 3 4 5 283 K 298 K Proton relaxivity (s

  • 1 mM
  • 1)

Proton Larmor frequency (MHz) 318 K 0.01 0.1 1 10 100 1000 2.5 5.0 7.5 10.0 12.5 283 K 298 K Proton relaxivity (s

  • 1 mM
  • 1)

Proton Larmor frequency (MHz) 318 K

Temperature dependence

Slow exchange Fast exchange

Other parameters: r = 0.31 nm – q = 1 Dtr = 0.02 cm-1 tv = 16, 12, 910-12 s tr = 150, 94, 5410-12 s

tM = 5.8, 3.0, 1.410-6 s tM = 58, 30, 1410-9 s

) / exp( T B T A

M M M =

t ) / exp( T B A

v v v =

t ) / exp( T B A

r r r =

t

slide-43
SLIDE 43

0.01 0.1 1 10 100 5 10 15 20 25 30 35 40 45

Inner-sphere relaxivity (mM

  • 1s
  • 1)

Proton Larmor Frequency (MHz)

Gd(III) MS-325 + HSA

Caravan, Parigi, Chasse, Cloutier, Ellison, Lauffer, Luchinat, McDermid, Spiller, McMurry, Inorg. Chem, 2007

Best fit parameters: r =3.1 Å, q=1 Dtr = 0.015 cm-1 tv = 0.130·10-10 exp(135/T) s tM = 0.510·10-12 exp(3963/T) s D=0.024 cm-1 308 K 278 K 298 K 288 K

slide-44
SLIDE 44

Approximations in the SBM model

  • the electron is supposed to reside in a single point and to behave as

a magnetic point-dipole, thus neglecting electron spin density delocalization (point-dipole approximation);

  • the perturbation Hamiltonian changes stochastically and is

stationary;

  • molecular reorientation is isotropic;
  • molecular reorientation and electron spin relaxation (i.e. dynamics of

the electron magnetic moment) are uncorrelated (decomposition approximation);

  • electron spin relaxation is a single exponential process (as if all

electron spin transitions have the same relaxation rates);

  • the correlation time for nuclear relaxation is shorter than the nuclear

relaxation time, as well as the correlation time for electron relaxation is shorter than the electron relaxation time (Redfield limit);

  • the electron g tensor is isotropic, and the static Hamiltonian is

dominated by the electron Zeeman interaction.

slide-45
SLIDE 45

Internal motions

Large tr Low relaxivity at high fields

0.01 0.1 1 10 100 0.0002 0.0023 0.0235 0.2349 2.3485 10 20 30 40 50 60

0.1 1 10 100

Magnetic field (T) Proton relaxivity (s

  • 1 mM
  • 1)

Proton Larmor frequency (MHz) tr (ns)

If the decay is not as steep as expected, fast local motions:

 

                  

             =

2 2 2 2 2 2 2 2 2 2 6 2 2 2 1

1 3 1 7 1 1 3 1 7 ) 1 ( 15 2

f H f f S f LS c H c c S c LS B e H M

S S r S S g R t w t t w t t w t t w t m 

slide-46
SLIDE 46

0.01 0.1 1 10 100 5 10 15 20 25 30 35 40 45

Inner-sphere relaxivity (mM

  • 1s
  • 1)

Proton Larmor Frequency (MHz)

Gd(III) MS-325 + HSA

Caravan, Parigi, Chasse, Cloutier, Ellison, Lauffer, Luchinat, McDermid, Spiller, McMurry, Inorg. Chem, 2007

Best fit parameters: r =3.1 Å, q=1 Dtr = 0.013 cm-1 tv = 16·10-12 exp(180(1/T-1/308)) s tR = 4.9·10-9 exp(1020(1/T-1/308)) s tM = 67·10-9 exp(6620(1/T-1/308)) s D=0.024 cm-1 308 K 278 K 298 K 288 K

SLS

2 ≈ 0.63

tfast < 300 ps

slide-47
SLIDE 47

0.01 0.1 1 10 100 20 40 60 80 100 NSs - 25 °C NSs - 37 °C

Proton Relaxivity (mM

  • 1s
  • 1)

Proton Larmor Frequency (MHz)

37 °C 25 °C Dt 0.020 cm-1 tv 8.7 ps 10 ps ZFS 0.06 cm-1  67° q1 1 r1 3.05 Å tM1 25 ns 30 ns q2 4a r2 3.5 Å tM2 0.30 ns 0.55 ns d 3.6 Å Ddiff 3.310-9 2.310-9 m2/s tR > 1 ms

a q2 = 3, r2 = 3.3 Å; q2 = 6, r2 = 3.7 Å; etc.

Transmission electron microscope image

NMRD of Gd(III)-DNA-Gold NanoStars

Rotz, Culver, Parigi, MacRenaris, Luchinat, Odom, Meade, ACS Nano (2015) 9, 3385-3396

slide-48
SLIDE 48

r

Gd3+ te tr tM O H H O H H O t'M

O H H O H H O H H Bulk water r' O H H r"

t"M O H H tD

NMRD of Gd(III)-DNA-Gold NanoStars

 37 °C  25 °C

slide-49
SLIDE 49

Approximations in the SBM model

  • the electron is supposed to reside in a single point and to behave as

a magnetic point-dipole, thus neglecting electron spin density delocalization (point-dipole approximation);

  • the perturbation Hamiltonian changes stochastically and is

stationary;

  • molecular reorientation is isotropic;
  • molecular reorientation and electron spin relaxation (i.e. dynamics of

the electron magnetic moment) are uncorrelated (decomposition approximation);

  • electron spin relaxation is a single exponential process (as if all

electron spin transitions have the same relaxation rates);

  • the correlation time for nuclear relaxation is shorter than the nuclear

relaxation time, as well as the correlation time for electron relaxation is shorter than the electron relaxation time (Redfield limit);

  • the electron g tensor is isotropic, and the static Hamiltonian is

dominated by the electron Zeeman interaction.

slide-50
SLIDE 50

+ IAS hyperfine coupling + SDS zero field splitting + mBSgB0 g-anisotropy

Other terms in static Hamiltonian

important when the energies related to these effects are not negligible with respect to the Zeeman energy (i.e., at low fields)

Bertini, Galas, Luchinat & Parigi, J.Magn.Reson. 1995

slide-51
SLIDE 51
  • 1/2

2D B0 E +1

B0 E +3/2 B0 E +1

D +1/2

  • 3/2

2D B0 E +3/2 D +1/2

  • 3/2
  • 1/2

4D

  • 5/2

+5/2 S=1 S=3/2 S=2 S=5/2 3D +2

  • 2

The effect of ZFS

E B0 +7/2

  • 7/2

+5/2

  • 5/2

+3/2

  • 3/2

+1/2

  • 1/2

Zeeman energy DE=gem0B0

S=7/2

slide-52
SLIDE 52

Effect of ZFS

D = 0.5 cm-1

S=1 S=3/2

D = 0 (cm-1) D = 0 (cm-1)

slide-53
SLIDE 53

Static and transient ZFS

Bertini, Kowalewski, Luchinat, Nilsson & Parigi, J.Chem.Phys. 1999 Kruk, Nilsson & Kowalewski, Phys.Chem.Chem.Phys. 2001

Florence NMRD program

Bertini, Galas, Luchinat & Parigi, J.Magn.Reson. 1995 Includes the effects of static ZFS, hyperfine coupling between unpaired electron and metal nucleus, g-anisotropy The effect of static ZFS is included in the calculation of the field dependent electron relaxation

Modified Florence NMRD program

Programs available at http://www.cerm.unifi.it/softwares/software-nmrd

If tr >> ts (slow rotation limit) and ts > tv (Redfield limit)

No field dependent electron relaxation In the presence of both static and transient ZFS

slide-54
SLIDE 54

Static and transient ZFS

S=1 Analogous corrections are necessary for all S values

Bertini, Kowalewski, Luchinat, Nilsson & Parigi, J.Chem.Phys. 1999 Kruk, Nilsson & Kowalewski, Phys.Chem.Chem.Phys. 2001

Conditions: Dttv<<1, tr>>ts

Kowalewski, Luchinat, Nilsson & Parigi J.Phys.Chem. 2002 Kruk & Kowalewski, J.Magn.Reson. 2003

 

av sp s

j j j j R ) ( ) ( ) ( ) (

3 4 2 3 1 2 1 2 , 1

w  w  w      =

2 2 2

1 5 ) (

v v t

j t w t w   D  =

) , , , , (   w   E D

s k k =

slide-55
SLIDE 55

B0 Dz Dx Dy r Me H  R = Wigner rotation matrix, which contains the Euler angles , b and , indicating the magnetic field direction in the molecular frame

              =

ij c I ij c M

j H i R conjugate complex ) ( 1 | | Re 1

2 2 2 dip 2 dip 1

t w w t 

 (rotational average)

slide-56
SLIDE 56

Kowalewski, Luchinat, Nilsson & Parigi J.Phys.Chem. A 2002, 106, 7376

Other approaches for taking into account the static ZFS are:

  • Slow motion theory, developed by J. Kowalewski in Stockholm
  • Grenoble approach, developed by P. Fries

valid beyond the Redfield limit Good agreement between the modified Florence NMRD program and slow-motion theory, in the Redfield limit and slow rotation conditions: Dttv<<1, tr>>ts

Modified NMRD vs. slow-motion theory

slide-57
SLIDE 57

2A B0 E +1/2,+1/2 B0 E A I=1/2 S=1/2 I=3/2 S=1/2

  • 1/2,+1/2
  • 1/2,-1/2

+1/2,-1/2 +1/2,+3/2 +1/2,+1/2 +1/2,-1/2 +1/2,-3/2

  • 1/2,-3/2
  • 1/2,-1/2
  • 1/2,+3/2
  • 1/2,+3/2

Hyperfine coupling with the metal nucleus

A|| = 0.016 cm-1, A = 0 A|| = 0.016 cm-1, A = 0.005 cm-1

A|| = A

slide-58
SLIDE 58

e e N N

B0

A

tR=9 ns

Best fit parameters (298 K) r = 0.34 nm tc (273)= 4.610-9 s tc (298)= 1.810-9 s A|| = 137  10-4 cm-1 A = 0

Cu(II) superoxide dismutase

Bertini I, Briganti F, Luchinat C, Mancini M, Spina G .

  • J. Magn. Reson. 1985; 63: 41
slide-59
SLIDE 59

Bertini, Luchinat, Parigi, Ravera, NMR of paramagnetic molecules, Elsevier, 2017

  • I. Bertini, C. Luchinat, G. Parigi, “1H NMRD profiles of paramagnetic

aquo-complexes and metalloproteins”, Adv. Inorg. Chem. (2005) 57, 105-

  • 172. In R. van Eldik and I. Bertini, “Relaxometry of water-metal ion

interactions”.

  • J. Kowalewski, D. Kruk, G. Parigi, “NMR relaxation in solution of

paramagnetic complexes: recent theoretical progress for S  1”, Adv.

  • Inorg. Chem. (2005) 57, 41-104. In R. van Eldik and I. Bertini,

“Relaxometry of water-metal ion interactions”.