nmr relaxation and cross-relaxation to study protein...
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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 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)
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 filtering
Stockman & Dalvit, 2002
A single FKBP binding ligand is identified in a mixture of 9 compounds by transverse relaxation editing.
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
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.
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