Heteronuclear Decoupling and Recoupling Christopher Jaroniec, Ohio State University 1. Brief review of nuclear spin interactions, MAS, etc. 2. Heteronuclear decoupling (average Hamiltonian analysis of CW decoupling, intro to improved decoupling schemes) 3. Heteronuclear recoupling (R 3 , REDOR) 4. AHT analysis of finite pulse REDOR 5. 13 C- 15 N distance measurements in multispin systems (frequency selective REDOR, 3D TEDOR methods) 6. Intro to dipole tensor correlation experiments for measuring torsion angles
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analysis of CW decoupling, intro to improved decoupling schemes)
3. Heteronuclear recoupling (R3, REDOR)4. AHT analysis of finite pulse REDOR5. 13C-15N distance measurements in multispin systems
(frequency selective REDOR, 3D TEDOR methods)6. Intro to dipole tensor correlation experiments for
measuring torsion angles
Isolated Spin-1/2 (I-S) SystemCS CS J D
int I S IS ISH H H H H= + + +
2
CSI IDIS ISJIS IS
H
H
H
γ
π
= ⋅ ⋅
= ⋅ ⋅
= ⋅ ⋅
I σ B
I D S
I J S
• Relevant interactions expressed in general as couplingof two vectors by a 2nd rank Cartesian tensor (3 x 3 matrix)
Rotate Tensors: PAS Lab
1( ) ( ) ; , ,LAB PAS α β γ−= Ω ⋅ ⋅ Ω Ω =σ R σ R
• SSNMR spectra determined by interactions in lab frame
• Rotate tensors from their principal axis systems(matrices diagonal) into lab frame (B0 || z-axis)
• In general, a rotation is accomplished using a setof 3 Euler angles α,β,γ
Multiple Interactions
• In case of multiple interactions first transform all tensors into common frame (molecular- or crystallite-fixed frame)
• Powder samples: rotate each crystallite into lab frame
High Field Truncation: HCS( )2 2 2 2 2
0 0
2 20 0
sin cos sin sin cos
1 3cos 1 sin cos(2 )2
CS LABI I zz z I xx yy zz z
I iso I z
H B I B I
B B I
γ σ γ σ θ φ σ θ φ σ θ
γ σ γ δ θ η θ φ
= = + +
⎧ ⎫⎡ ⎤= + − −⎨ ⎬⎣ ⎦⎩ ⎭
• Retain only parts of HCS that commute with Iz
( )13iso xx yy zz
zz iso
yy xx
σ σ σ σ
δ σ σσ σ
ηδ
= + +
= −−
=
High Field Truncation: HJ and HD
( )2
03
21 3cos 1 22
4
JIS IS z z
DIS IS z z
I SIS
IS
H J I S
H b I S
br
π
θ
µ γ γπ
=
= −
= −
• J-anisotropy negligiblein most cases
• bIS in rad/s; directly related to I-S distance
Dipolar Couplings in Proteins
~504.015N13C
~2002.515N13C
~9001.515N13C
~2,2001.513C13C
~10,8001.0415N1H
~21,5001.1213C1H
b12/2π (Hz)r12 (Å)Spin 2Spin 1
Static Powder Spectra: HCS & HD
( ) ( ) ( ) sin Tr exp expCS x CSI t d d I iH t I iH tφ θ θ+ +∝ ⋅ −∫ ∫
σzz
σxx
σyy σzz
σxx
σyy
φ
θ
θB0
NMR of Rotating Samples
2
( )
2
( )
( )2
2
( ) exp
CSI I zDIS IS z z
JIS IS z z
mr
m
H t I
H t I S
H J I S
t im tλ λ
ω
ω
π
ω ω ω=−
=
=
=
= ∑
HD for Rotation at Magic Angle (θm = 54.74o)
( )
( ) ( )
( )
2( )
2
2 2(0)
( 1)
( 2) 2
exp 2
3cos 1 3cos 10
2 2
sin 2 exp2 2
sin exp 24
D mIS IS r z z
m
PR mIS IS
ISIS PR PR
ISIS PR PR
H im t I S
b
b i
b i
ω ω
β θω
ω β γ
ω β γ
=−
±
±
⎧ ⎫= ⎨ ⎬
⎩ ⎭
− −= =
= − ±
= ±
∑
• For spinning at the magic angle the time-independent dipolar (and CSA) components vanish; terms modulated at ωr and 2ωr vanish when averaged over the rotor cycle
Frequency (Hz)-30000 -20000 -10000 0 10000 20000 30000
Static
νr = 2 kHz
νr = 20 kHz
13C-1H DipolarSpectra underMAS
Incr
easi
ngR
esol
utio
n
13C SSNMR Spectra atHigh MAS Rates
M. Ernst, JMR 2003
13C-1H 13C-1H2
• Presence of many strong 1H-1H couplings leads to anincomplete averaging of 13C-1H dipolar coupling by MAS
13C Frequency
( )rr
af bνν
= +
High-Power CW Decoupling
• Average 13C-1H couplings by simultaneously usingMAS and high-power 1H RF irradiation
• Traditionally for efficient decoupling 1H RF fieldsof ~50-200 kHz were used (i.e., ω1H >> bHH, bHX)
CW off
150 kHz
13C-VF, 28 kHz MAS
CW Decoupling: AHT Analysis
1
( ) ( )2 ( )2
iso isotot S z S z I z I z
IS z z IS z z x
H S t S I t IJ I S t I S I
ω ω ω ωπ ω ω
= + + ++ + +
(I)
(S)
• Average Hamiltonian analysis: RF and MAS modulations must be synchronized to obtain cyclic propagator
AHT: Summary
1
0
1
0
0
(0) (1) (2)
(0)
( ) ( ) (0) ( ) ; ( ) exp ( )
( ) ( ) ( ) ( ) exp ;
( ) ( ) exp exp (for ( ) 1)c
t
tot tot tot RF
t
tot RF RF RF RF
t
tot c c c RF c
t U t U t U t T i dt H H
U t U t U t U t T i dt H H U HU
U t U t T i dt H iHt U t
H H H H
H
ρ ρ −
−
′= = − +
′= = ⋅ − =
′= = − = − =
= + + +
∫
∫
∫
…
(1)
0 0 0
1 1; ( ), ( ) ; 2
c ct t t
c c
dt H H dt dt H t H tt it
′′⎡ ⎤′ ′′ ′ ′′ ′= = ⎣ ⎦∫ ∫ ∫ …
Haeberlen & Waugh, Phys. Rev. 1968
Interaction Frame Hamiltonian
( ) ( )
11 1
1 1
1 1
exp exp
( )
2 cos( ) sin( )
( )2 cos( ) sin( )
( )
r r r r
RF RF x x
J DS I IS IS
isoS S z S z
JIS IS z z y
in t in t in t in tIS z z y
DIS IS z z y
inIS z z
H U HU i tI H i tI
H H H H H
H S t S
H J S I t I t
J S I e e iI e e
H t S I t I t
t S I e
ω ω ω ω
ω
ω ω
ω ω
π ω ω
π
ω ω ω
ω
−
− −
= = −
= + + +
= +
= +
= + − −
= +
= ( ) ( ) r r r rt in t in t in tye iI e eω ω ω− −+ − −
Interaction Frame Cont.
( ) ( )
1 1
2( ) ( )( )
2
( ) ( )( )
( )2 cos( ) sin( )
( )
r r r r
r r
r r
DIS IS z z y
in t in t in t in tIS z z y
i m n t i m n tmIS z z
m
i m n t i m n tmIS y z
H t S I t I t
t S I e e iI e e
e e I S
i e e I S
ω ω ω ω
ω ω
ω ω
ω ω ω
ω
ω
ω
− −
+ −
=−
+ −
= +
= + − −
⎡ ⎤= +⎣ ⎦
⎡ ⎤− −⎣ ⎦
∑
Lowest-Order Average Hamiltonian
( ) ( )
(0) (0) (0) (0), ,
2(0) ( )
0 02
(0), 0
0
r rr
rr r r r
S J IS D IS
isoim tm isoS z z
S S S zmr r
in t in t in t in tIS zJ IS z y
r
H H H H
S SH dt dt e S
J SH dt I e e iI e e
τ τ ω
τ ω ω ω ω
ω ω ωτ τ
πτ
=−
− −
= + +
⎧ ⎫= + =⎨ ⎬
⎩ ⎭
= + − − =
∑∫ ∫
∫
• S-spin CSA refocused by MAS, I-S J-coupling eliminatedby I-spin decoupling RF field
Lowest-Order Average HD
2( ) ( )(0) ( )
02
( ) ( )( )
( )( )( )0
1
0 if 01 if 0
rr r
r r
rr
i m n t i m n tmS IS z z
mr
i m n t i m n tmIS y z
i m n tmIS n
ISr
H dt e e I S
i e e I S
m ndt e
m n
τ ω ω
ω ω
τ ω
ωτ
ω
ωωτ
+ −
=−
+ −
±
⎡ ⎤= +⎣ ⎦
⎡ ⎤− −⎣ ⎦
± ≠⎧= ⎨ ± =⎩
∑∫
∫ ∓
Lowest-Order Average HD
( ) ( )
(0),
(0) ( ) ( ) ( ) ( ),
0 D IS
n n n nD IS IS IS z z IS IS y z
H
H I S i I Sω ω ω ω− −
=
= + − −
n π 1,2 ô I-S Decoupling
n = 1,2 ô I-S Dipolar Recoupling!
• Rotary resonance recoupling (R3) arises from the interference of MAS and I-spin RF (when ω1 = ωr or 2ωr)
• Additional (much-weaker) resonances (n = 3,4,…) are also possible due to higher order average Hamiltonian terms involving the I-spin CSA
• Apply a series of rotor-synchronized π pulses (2 per τr) to 15N spins(this is usually called a dephasing or S experiment)
• Typically a reference (or S0) experiment with pulses turned off is also,acquired – normally report S/S0 ratio (or ∆S/S0 = 1-S/S0)
Gullion & Schaefer, JMR 1989
REDOR: AHT Summary
˜ S z =
Sz 0 < t ≤ τ−Sz τ < t ≤τ +τ r / 2Sz τ + τr / 2 < t ≤ τr
⎧
⎨ ⎪
⎩ ⎪ S
212( ) ( )2 sin ( )cos[2( )] 2 sin(2 )cos( )2IS IS z z IS r r z zH t t I S b t t I Sω β γ ω β γ ω= = − + − +
(0) 2 sin(2 )sin( ) 2 ; (sequence phase)IS IS z z rH b I Sβ γ ψ ψ ω τπ
= + ⋅ =
(0)
2 sin(2 )sin( ) 2 for /2
2 sin(2 )sin( ) 2 for 0
IS z z r
IS
IS z z
b I SH
b I S
β γ τ τπ
β γ τπ
⎧− ⋅ =⎪⎪= ⎨
⎪ ⋅ =⎪⎩
• Effective Hamiltonian changes sign as a function of position ofpulses within the rotor cycle (must be careful about this in some implementations of REDOR)
REDOR: Typical Implementation
• Rotor synchronized spin-echo on 13C channel refocuses 13C isotropic chemical shift and CSA evolution
• 2nd group of pulses moved by -τr/2 relative to 1st group to change sign of HD and avoid refocusing the 13C-15N dipolar coupling
• xy-type phase cycling of 15N pulses is critical (Gullion, JMR 1990)
REDOR Dipolar Evolution
( ) exp 2 exp 2 cos( ) 2 sin( )
2 sin(2 )sin( )
z z x z z
x y z
IS
t i C N t C i C N tC t C N t
b
ρ ω ωω ω
ω β γπ
= −
= +
= −
REDOR: Example
1.51 ± 0.01 Å
• Experiment highly robust toward 15N CSA, experimental imperfections, resonance offset and finite pulse effects(xy-4/xy-8 phase cycling is critical for this)
• REDOR is used routinely to measure distances up to ~5-6 Å(D ~ 25 Hz) in isolated 13C-15N spin pairs
• Simpler schemes (xy-4, xy-8) seem to perform better with respect to 15N CSA compensation than the longer xy-16
• Since [HD(0),HCSA
(0)] = 0 the behavior is likely due to finite pulses and higher order terms in the average Hamiltonian expansion
REDOR at High MAS Rates
2 p
r
τϕ
τ=
0.42010
0.82020
0.21010
0.1510
ϕνr (kHz)τp (µs)
REDOR (xy-4) at High MAS: AHT ( ) ( ) ( )2 ( )2 ( )2IS IS z z z x z yH t t f t I S g t I S h t I Sω= + +
1
2 2
3
4
(0) 1 ( ) 2
( ) cos[ ( )] 2 ( )sin[ ( )] 2
( ) 2
( ) cos[ ( )] 2 (
IS z zr t
z z z yt t
z zt
z zt
H ac bc dt I S
ac bc t dt I S ac bc t dt I S
ac bc dt I S
ac bc t dt I S a
τ
θ θ
θ
′′ ′∝ + ⋅
′′ ′ ′′ ′+ + ⋅ + + ⋅
′′ ′− + ⋅
′′ ′ ′− + ⋅ −
∫
∫ ∫
∫
∫4
5
6 6
7
) sin[ ( )] 2
( ) 2
( ) cos[ ( )] 2 ( )sin[ ( )] 2
( ) 2
(
z xt
z zt
z z z yt t
z zt
c bc t dt I S
ac bc dt I S
ac bc t dt I S ac bc t dt I S
ac bc dt I S
a
θ
θ θ
′ ′+ ⋅
′′ ′+ + ⋅
′′ ′ ′′ ′+ + ⋅ − + ⋅
′′ ′− + ⋅
−
∫
∫
∫ ∫
∫
8 8
) cos[ ( )] 2 ( )sin[ ( )] 2 z z z xt t
c bc t dt I S ac bc t dt I Sθ θ′′ ′ ′′ ′+ ⋅ + + ⋅∫ ∫
Jaroniec et al, JMR 2000
REDOR (xy-4) at High MAS: AHT
( )22
(0)
cos2 sin(2 )sin( )2 ; finite pulses1
2 sin(2 )sin( )2 ; ideal pulses
IS z z
IS
IS z z
b I SH
b I S
π ϕβ γ
π ϕ
β γπ
⎧−⎪ −⎪= ⎨
⎪ −⎪⎩
• For xy-4 phase cycling, finite π pulses result only in a simple scaling of the dipolar coupling constant by an additional factor, κ, between π/4 and 1
• For xx-4 spin dynamics are more complicated and converge to R3 dynamics in the limit of ϕ = 1
( )22
cos; / 4 1
1
effIS
IS
bb
π ϕκ π κ
ϕ≡ = ≤ ≤
−
REDOR (xy-4) at High MAS:AHT vs. Numerical Simulations
10 µs 15N pulsesno 15N CSA
REDOR (xy-4) Experiments
REDOR in Multispin Systems
( ) ( )1 1 2 2
1 2
2 2
( ) cos cosIS z z z z
x
H I S I S
I t t t
ω ω
ω ω
= +
=
• Strong 13C-15N couplings dominate REDOR dipolar dephasing;weak couplings become effectively ‘invisible’
I
S
S
200 Hz(2.5 Å) 50 Hz
(4 Å)45o
Frequency Selective REDOR
Jaroniec et al, JACS 1999, 2001
• Use a pair of weak frequency-selective pulses to ‘isolate’ the 13C-15N dipolar coupling of interest; all other couplings refocused
• This trick is possible because all relevant interactions commute
FS-REDOR
H = ωkl2IkzSlz
H = ω ij2Iiz S jzi, j∑
+ πJij2I izI jzi< j∑
FS-REDOR Evolution
0 2 1
0 2
( /2) ( /2)
( /2) ( /2) ( /2)( /2)
( ) kx lx lx kx kx lx
kx lx
kx lx
i I i S i S i I i I i SiH t iH t
iH t i I i SiH t iH tiH t
iH t i I i SiH t
U t e e e e e e e ee e e e e ee e e e
π π π π π π
π π
π π
− − − −− −
− − −−−
− − −−
=
=
=
[ ] [ ] [ ][ ] [ ]
0 1 2 0 1 0 2 1 2
0 0 0
1
; , , , 0
2 2 ; , , 0
2 2 2 ;
ij iz jz ij iz jz kx lxi k j l i j k
ki kz iz il iz lz ki kz izi l i k i
H H H H H H H H H H
H I S J I I H I H S
H I S I S J I I
ω π
ω ω π≠ ≠ < ≠
≠ ≠ ≠
= + + = = =
= + = =
= + +
∑∑ ∑
∑ ∑ [ ] [ ]
[ ] [ ]
1
2 2 2
for each term in , 0 or , 0
2 ; , 0 and , 0
kx lxk
kl kz lz kx lx
H I S
H I S H I H Sω
≠ ≠
= ≠ ≠
∑
0
2 2
(0) ; (0), (0), (0), 0
( ) cos( ) 2 sin( )
lx kxi S i I iH tkx
iH t iH tkx kx kl ky lz kl
I e e e
t e I e I t I S t
π πρ ρ ρ ρ
ρ ω ω
− − −
−
⎡ ⎤ ⎡ ⎤ ⎡ ⎤= = = =⎣ ⎦ ⎣ ⎦ ⎣ ⎦= = +
1
Dipolar evolution under only bkl
13C Selective Pulses: U-13C Thr
FS-REDOR: U-13C,15N Asn
1 ms
2 ms
4 ms
6 ms
8 ms
10 ms
FS-REDOR: U-13C,15N Asn
FS-REDOR: U-13C,15N-f-MLF
FS-REDOR: U-13C,15N-f-MLFMet Cβ-Phe NLeu Cβ-Phe NLeu Cβ-Leu N
X-ray: 2.50 ÅNMR: 2.46 ± 0.02 Å
X-ray: 3.12 ÅNMR: 3.24 ± 0.12 Å
X-ray: 4.06 ÅNMR: 4.12 ± 0.15 Å
FS-REDOR: U-13C,15N-f-MLF
• 16 13C-15N distances could be measured in MLF tripeptide
• Selectivity of 15N pulse + need of prior knowledge of which distances to probe is a major limitation to U-13C,15N proteins
Simultaneous 13C-15N Distance Measurements in U-13C,15N Molecules
General Pseudo-3D HETCOR(Heteronuclear Correlation) Scheme
• I-S coherence transfer as function of tmix via DIS
• Identify coupled I and S spins by chemical shift labeling in t1, t2
Transferred Echo Double Resonance
Hing, Vega & Schaefer, JMR 1992
• Similar idea to INEPT experiment in solution NMR
• Cross-peak intensities depend on all 13C-15N dipolar couplings
• Experiment not directly applicable to U-13C-labeled samples
3D TEDOR Pulse Sequence
( /2) ( /2)
2
2 sin( / 2)
2 sin( / 2) sin ( / 2)
x xI SREDORx z y IS mix
REDORy z IS mix x IS mix
S I S t
I S t I t
π πω
ω ω
+⎯⎯⎯→ ⎯⎯⎯⎯⎯⎯→
− ⎯⎯⎯→
3D TEDOR: U-13C,15N Molecules
Michal & Jelinski, JACS 1997
Modified 3D TEDOR Pulse Sequence
( /2) ( /2)
( /2) ( /2)
2
2 sin( / 2)
2 sin( / 2)
2 sin( / 2) sin ( / 2)
x x
x x
S IREDORx y z IS mix
S Iz y IS mix
REDORy z IS mix x IS mix
I I S t
I S t
I S t I t
π π
π π
ω
ω
ω ω
+
−
⎯⎯⎯→ ⎯⎯⎯⎯⎯⎯→
− ⎯⎯⎯⎯⎯⎯→
− ⎯⎯⎯→
3D TEDOR: U-13C,15N Molecules
N-acetyl-Val-Leu
• Cross-peak intensities roughly proportional to 13C-15N dipolar
couplings
3D TEDOR: U-13C,15N N-ac-VL
• Spectral artifacts (spurious cross-peaks, phase twisted lineshapes) appear at longer mixing times as result of 13C-13C J-evolution
Improved Scheme:3D Z-Filtered TEDOR
• Unwanted anti-phase and mulitple-quantum coherences
responsible for artifacts eliminated using two z-filter periods
3D ZF TEDOR Pulse Sequence
Jaroniec, Filip & Griffin, JACS 2002
Results in N-ac-VL
• 3D ZF TEDOR generates purely absorptive 2D spectra
• Cross-peak intensities give qualitative distance information
• 13C-15N cross-peak intensities reduced 2- to 5-fold
C
N
50 Hz
J1 J2C
C
13C Band-Selective 3D TEDOR Scheme
• 13C-13C J-couplings refocused using band-selective 13C pulses
(no z-filters required)
• Most useful for strongly J-coupled sites (e.g., C’) but requires
resolution in 13C dimension
ZF-TEDOR vs. BASE-TEDORGly 13Cα-15N
Thr 13Cγ-15N
Gly 13C’-15N
Cross-Peak Trajectories in TEDOR Expts.
• Intensities depend on all spin-spin couplings to particular 13C• Use approximate models based on Bessel expansions of REDOR signals to describe cross-peak trajectories (Mueller, JMR 1995)
3D ZF TEDOR
IiSj
( )
( ) ( )
2
2 2
(0) cos
sin cos
i
i
m
ij i ill i
N
ij ikk j
V V Jπ τ
ω τ ω τ
≠
≠
= ×∏
∏
3D BASE TEDOR
IiSj
( ) ( )2 2(0) sin cosiN
ij i ij ikk j
V V ω τ ω τ≠
= ∏
3D ZF-TEDOR: N-ac-VL
X-ray: 2.95 ÅNMR: 3.1 Å
X-ray: 4.69 ÅNMR: 4.7 Å
Val(N)
Leu(N)
X-ray: 3.81 ÅNMR: 4.0 Å
X-ray: 3.38 ÅNMR: 3.5 Å
Val(N)
Leu(N)
3D BASE TEDOR: N-ac-VL
X-ray: 2.39 ÅNMR: 2.4 Å
X-ray: 1.33 ÅNMR: 1.3 Å
Val(N)
Leu(N)
X-ray: 1.33 ÅNMR: 1.4 Å
X-ray: 4.28 ÅNMR: 4.1 Å
Val(N)
Leu(N)
Summary of DistanceMeasurements in N-ac-VL
Application to TTR(105-115)Amyloid Fibrils
• ~70 13C-15N distances measured by 3D ZF TEDOR in several U-13C,15N labeled fibril samples (30+ between 3-6 Å)
Slice from 3D ZF TEDOR Expt.
Cross-Peak Trajectories (T106)Y105
T106
I107
A108
Y105
T106
I107
A108
REDOR in Multispin Systems
( ) ( )1 1 2 2
1 2
2 2
( ) cos cosIS z z z z
x
H I S I S
I t t t
ω ω
ω ω
= +
=
• Strong 13C-15N couplings dominate REDOR dipolar dephasing;weak couplings become effectively ‘invisible’