NMR relaxation and cross-relaxation

to study protein interactions

Pavia, September 7th 2007

Protein interactions withLarge molecules

Small ligands

NMR: more easily applicable to

protein-small ligand interactions

Protein-small ligand interactions: relevant for drug discovery

1) Lead generation

2) Lead optimization

3) Preclinical development

Identification of suitable ligands with specific activity and structure activity relationship (SAR)

Improving in vitro performance by design-synthesis-assay cycles based on 3D structure

Improving in vivo performance and bioavailability - ADMET: absorption, distribuition, metabolic stability, excretion, toxicity

Initially NMR appeared ideally suited for stage 2, i.e. lead optimization, but NMR 3D structure determinations are time-consuming and routinely feasible for molecules below 10-12 kDa, which does not fit with the requirements of pharmaceutical industry research.

In mid 1990s it was shown that NMR could be profitably used for lead generation with the SAR-by-NMR approach.

NMR was successfully applied, for specific issues, also to stage 3.

SAR by NMR

Ligand library screening by NMR. Additional nearby binding site. NMR assay of double-binding-site ligands.

Shuker, Hajduk, Meadows & Fesik, Science, 1996

Movie: Brian HargreavesEquilibrium

Movie: Brian HargreavesExcitation: laboratory frame

Movie: Brian HargreavesExcitation: rotating frame

Movie: Brian HargreavesExcitation: laboratory frame

Movie: Brian Hargreaves

Relaxation = restoring equilibrium

Loss of x-y coherence = transverse relaxation T2

Recovery of z magnetization = longitudinal ralaxation, T1

Relaxation

This frequency can be reached by fluctuations, due to molecular motions, of the local magnetic field.

The event that ultimately determines relaxation is the transition of the nuclear magnetic moment that occurs at a precise frequency value.

2

v Bγπ

=

One of the most relevant sources of fluctuating local magnetic fields are the dipole-dipole interactions among different spins.

In isotropic liquids, the longitudinal and trasverse components of Bdip with respect to the static field are modulated by the molecular motions and thus generate local fluctuating magnetic fields.

B0

µS

µI

r

µS

µIr

µS

µI

r

If the molecular motion frequency will be appropriate, the ensuing fluctuations of the local magnetic fields will be able to induce transitions, that is inducing relaxation:

2dipB

γν

π=

Dipole-Dipole interaction

D-D interactions, cross-relaxation and nuclear Overhauser effect - NOE

The effects of the dipolar interaction between two nuclei in a magnetic field depend on the internuclear separation and on the reorientation speed of the internuclear vector with respect to the external magnetic field, i.e. the frequency of θ change.

I

S

r B0

θ

θt

θ

τc >> ω0−1

τc << ω0−1

( )6,c ISf rη τ −=NOE, molecular motion and internuclear distance

Relaxation parameters

( ) ( )

( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

2 2

6

2 2

1 6

2 2

1 6

2 2

2 6

2 2

1 16

6 2 020

13 12 2

20

10 3 6 2

20

15 0 9 6 2

40

1 5 2 9 6 240

ij ij ijij

nsij ij

j i ij

selij ij ij

j i ij

ij ij ijj i ij

ij ij ijj i ij

J Jr

R J Jr

R J J Jr

R J J Jr

R J J Jrρ

γσ ω

γω ω

γω ω

γω ω

γ ω ω ω

≠

≠

≠

≠

= −

= +

= + +

= + +

= + +

∑

∑

∑

∑

h

h

h

h

h

( )( )2

2

1c

c

J nn

τω

ωτ=

+

For homonuclear D-D relaxation (I = 1/2)

Molecular mobility and relaxation

The dependence of relaxation on molecular reorientation translates into the measured values of T1 and T2, that are equal for small molecules (fast motions) − extreme narrowing limit − and divergent (with T1 > T2) for large molecules (slow motions) − spin diffusion limit.

Harris, 1983

θt

θ

τc >> ω0−1

τc << ω0−1

I

S

r B0

θ

Macura & Ernst, 1980

( ) ( )2 2

6 6 2 020ij ij ij

ij

J Jrγ

σ ω = − h

Receptor-ligand binding equilibria

8 1 1

10

1 10 (diffusion limited)T

on

E M

k M s

µ− −

=

= ×

[ ][ ][ ]

[ ] [ ] [ ] [ ] [ ][ ]

[ ][ ]

[ ][ ]

[ ][ ]

[ ][ ]

; ;

on

off

k offDk

on

TT T

T

E LB B

T T

E LB B

D D

kE LE L EL K

EL k

LE EL E L EL L

E

EL ELP P

E L

L EP P

L K E K

ε

+ = =

= + = + =

= =

= = + +

ˆ ˆ ˆ†‡ ˆ ˆ ˆ

2 4 15 10 5 10offk s−= × − ×[ ]0.5E

B DP L K= ⇒ =

Receptor-ligand binding equilibria

8 1 1

10

1 10 (diffusion limited)T

on

E M

k M s

µ− −

=

= ×

[ ][ ][ ]

[ ][ ]

[ ][ ]

on

off

k offDk

on

E EB B

T D

kE LE L EL K

EL k

EL LP P

E L K

+ = =

= =+

ˆ ˆ ˆ†‡ ˆ ˆ ˆ

[ ][ ][ ]

[ ][ ][ ]

[ ] ( ) ( )21 14

2 2

T TD

T T D T T D T T

E L E EL L ELK

EL EL

EL L E K L E K L E

− −= =

= + − − + − −

[ ][ ]

EB T

T

ELP vs L

E=

The fraction of ligand-bound receptor can be calculated in terms of known quantities.

The NMR spectrum of a system undergoing binding exchange equilibria exhibits changes of the observable NMR parameters, Q, i.e. chemical shifts (Ω), relaxation rates (R), cross-relaxation rates (σ) diffusion constants (D).

This is accounted for by the modified Bloch equations:

( ) ( )

( )

ddtd

idt + +

= − +

= − + −

z z?M R K ?M

M R K O M

The solutions of the modified Bloch equations give the time course of free and receptor-bound ligand magnetizations.

2

2

0 0

0 0B ex F exF F F F

B ex F exB B B B

P k P kM R Mdi

P k P kM R Mdt+ +

+ +

− Ω = − + − − Ω

[ ][ ] [ ] [ ]

12 2with , , ; ;

; ; ;

ex on off

B F ex B on ex F offT T

F B free bound R T k k E k

EL LP P k P k E k P k

L L

−= = = +

= = = =

By symmetrization and diagonalization of R, Ω and K matrices, one obtains the exchange-modulated relaxation rates and precession frequencies.

Peng et al., 2004

Analytical solutions are available (Hahn, Maxwell, McConnel solution or Swift-Connick formula).

Slow-exchange limit

The exchange matrix K introduces only a small perturbation to (R − iΩ): ( ) ( )2 2 ,F B ex F B F BP P k R R− Ω − Ω=

The free and bound ligand signals retain their precession frequency and have modified transverse relaxation rates:

2 , 2 2 , 2 F sl F B ex B sl B F exR R P k R R P k= + = +

Typically 2 2 and B F B FR R P P? =

Hence2 , 2 F sl FR R ⇒; it may be hard to distinguish slow

exchange and lack of binding.

Fast-exchange limit

Fast exchange on the chemical shift and relaxation time scales means that the term (R − iΩ) becomes a small perturbation to the exchange matrix K.

A single averaged value is observed for chemical shift and transverse relaxation relaxation rate:

( )22, 2 2 with

avg F F B B

F Bavg F F B B ex ex F B

ex

P P

P PR P R P R R R

k

Ω = Ω + Ω

= + + = Ω −Ω

For very fast exchange Rex = 0.

The information on the bound state is encoded by the averaged relaxation rate of a single resonance.

Fast-exchange limit

The conditions of fast-exchange limit are quite usual and convenient.

[ ][ ][ ]

on

off

k offDk

on

kE LE L EL K

EL k+ = =ˆ ˆ ˆ†‡ ˆ ˆ ˆ

With a typical value of KD = 100 µM, if kon is in the range 107-109 M−1s−1, then 103 < koff < 105 s−1.

This value of exceeds most differences of transverse relaxationrates, rotating frame relaxation rates, cross-relaxation rates, chemical shifts, diffusion coefficients i.e. ∆Q.

Fast-exchange limitA single averaged value of a generic NMR parameter may be observed as:

avg F F B B

avg F F B B ex

Q P Q P Q

Q P Q P Q Q

= +

= + +

and compared with the corresponding value in the absence of binding, i.e. QF.

Observing Qavg is convenient when QB >> QF because typically screening conditions imply:

1 TB F

T

LP P

Eε = ⇒? =

Under these conditions it is convenient to consider ( )avg FQ Q−

Fast-exchange limit [ ][ ][ ]

[ ][ ]

[ ][ ]

[ ][ ]

[ ][ ]

; ; + =1

; ; + =1

on

off

k offDk

on

E E E EB F B F

T T

B F B FT T

kE LE L EL K

EL k

EL EP P P P

E E

EL LP P P P

L L

+ = =

= =

= =

ˆ ˆ ˆ†‡ ˆ ˆ ˆ

E

T BB

T

L PP

Eε

ε= =

( )

( ) ( )

( ) ( ) [ ][ ] [ ] ( )

( ) [ ][ ] ( )

1avg B B F F B B B F

EB

avg F B B F B F

Eavg F B B F B F

avg F B FD

Q P Q P Q P Q P Q

PQ Q P Q Q Q Q

ELQ Q P Q Q Q Q

E EL

LQ Q Q Q

L K

ε

ε

ε

= + = + −

− = − = −

− = − = −+

− = −+

This is a dose-response hyperbolic curve.

( ) [ ][ ] ( )avg F B F

D

LQ Q Q Q

L Kε − = −

+

Fast-exchange limit

Analogy with Michaelis-Menten steady state kinetics equation:

[ ][ ]0 max

M

SV V

S K= ⋅

+

[S]

0V

K M

maxV

max2

V

[L]K D

( )B FQ Q−

( )avg FQ Qε −

( )2

B FQ Q−

By measuring Qavg and correcting for QF, if ε >> 1, [L] ~ [LT] and the curve yields estimates for KD and the plateau value.

27

Binding and line broadening

Broadening, i.e. increased R2, can reveal binding. Mixture of small ligands (A) in the absence and (B) in the presence of p38 MAP kinase receptor. Solid arrows highlight broadened signals.

Peng et al., 2004

Relaxation parameters

( ) ( )

( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

2 2

6

2 2

1 6

2 2

1 6

2 2

2 6

2 2

1 16

6 2 020

13 12 2

20

10 3 6 2

20

15 0 9 6 2

40

1 5 2 9 6 240

ij ij ijij

nsij ij

j i ij

selij ij ij

j i ij

ij ij ijj i ij

ij ij ijj i ij

J Jr

R J Jr

R J J Jr

R J J Jr

R J J Jrρ

γσ ω

γω ω

γω ω

γω ω

γ ω ω ω

≠

≠

≠

≠

= −

= +

= + +

= + +

= + +

∑

∑

∑

∑

h

h

h

h

h

( )( )2

2

1c

c

J nn

τω

ωτ=

+

For homonuclear D-D relaxation (I = 1/2)

Binding and line broadening

β2m

β2m + αBcrystallin

Binding and line broadening

β2m β2m + αBcrystallin

Binding and line broadening

Binding and line broadening

Trp60 occurs in αL backbone conformation in β2m

Transverse relaxation and aggregation

Transverse relaxation and aggregation

NMR: 15N relaxation measurements

Transverse relaxation editing by paramagnetic labeling

A paramagnetic labelled ligand (34) magnifies the tranverse relaxation enhancement of a adjacent ligand (35).

T2-edited spectra of a ligand mixture without and with FKBP protein and with spin-labeled FKBP.

Stockman & Dalvit, 2002

Transverse relaxation editing by paramagnetic labeling

Niccolai et al., 2001

Transverse relaxation filtering

Stockman & Dalvit, 2002

A single FKBP binding ligand is identified in a mixture of 9 compounds by transverse relaxation editing.

Movie: Brian Hargreaves

Transverse relaxation filtering

Stockman & Dalvit, 2002

a) transverse-relaxation-edited spectrum of 9 compounds without FKBP.

b) transverse-relaxation-edited spectrum of 9 compounds with FKBP minus the same editing for FKBP alone.

c) difference spectrum from a) minus b).

d) reference spectrum of the ligand.

e) same difference spectrum as in c) obtained with a mixture of the 8 non-binding ligands

Diffusion filtering

Johnson, 1999

Diffusion filtering

Johnson, 1999

Diffusion filtering - DOSY

Johnson, 1999

Peng et al., 2004

Saturation transfer difference

Exchange driven spin diffusion

The rotation of Tyr or Phe aromatic ring of can be as slow as 100 s−1 and enable the observation of different resonance frequencies for the ε and δ hydrogens.

For a typical protein with τc = 6 ns, σmax ≈ 10 sec−1 for geminal protons (r = 0.17 nm). k >> σmax

Exchange driven spin diffusion

ε1

ε2

δ1

δ2

1 1 1

1 2 2

1 1 1

2 1 2

00

00

kk

kk

ε εδ

ε εδ

εδ

δε δ

δε δ

ρ σρ σ

σ ρσ ρ

−

−

=

L

ε1-δ2 = ε2-δ1 ≈ 0.5 nm σ ≈ 0

Non-zero antidiagonal values build up by spin diffusion.

L14 = 2k1+σ2 +σ1 , L23 = 2k−1+σ1+σ2 , ecc.

Saturation transfer difference

Peng et al., 2004

Very advantageous with large receptor molecules.

[ ]STD STDI ELα∝

(αSTD = amplification factor)

0 TI L∝

[ ][ ]

[ ][ ]

0

0

ESTD B

STD STD B STDT

STDSTD

D

ELI PP

I L

LII L K

α α αε

ε α

= = =

= +

Saturation transfer difference

Peng et al., 2004

A) Mixture of ligands in the absence of receptor.

B) STD spectrum of the ligand mixture in the presence of p38 MAP kinase (42 kDa).

Only the signals of the binding ligandare seen because the receptor is filetered through relaxation filtering.

Saturation transfer versus broadening

Peng et al., 2004

Cross-relaxation highlights binding effects more effectively than R2 increase.

Peng et al., 2004