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Four pulse recoupling
Accepted Manuscript Four Pulse Recoupling Navin Khaneja, Ashutosh Kumar PII: DOI: Reference:
S1090-7807(16)30189-6 http://dx.doi.org/10.1016/j.jmr.2016.09.019 YJMRE 5961
To appear in:
Journal of Magnetic Resonance
Received Date: Accepted Date:
16 July 2016 25 September 2016
Please cite this article as: N. Khaneja, A. Kumar, Four Pulse Recoupling, Journal of Magnetic Resonance (2016), doi: http://dx.doi.org/10.1016/j.jmr.2016.09.019
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Four Pulse Recoupling Navin Khaneja
∗†
‡
Ashutosh Kumar
September 26, 2016
Abstract
The paper describes a family of novel recoupling pulse sequences in magic angle spinning (MAS) solid state NMR, called four pulse recoupling. These pulse sequences can be employed for both homonuclear and heteronuclear recoupling experiments and are robust to dispersion in chemical shifts and rf-inhomogeneity.
The homonuclear pulse sequence consists of a building block
π 3π ◦ ◦ ◦ ◦ ( π2 )0◦ ( 3π 2 )φ ( 2 )180 +φ ( 2 )180 where φ =
π n
( φ◦ =
180◦ n ),
and n is number of blocks in a two rotor
π 3π ◦ ◦ ◦ ◦ period. The heteronuclear recoupling pulse sequence consists of a building block ( π2 )0◦ ( 3π 2 )φ1 ( 2 )180 +φ1 ( 2 )180 π 3π ◦ ◦ ◦ ◦ and ( π2 )0◦ ( 3π 2 )φ2 ( 2 )180 +φ2 ( 2 )180 on channel I and S, where φ1 =
3π 2n ,
φ2 =
π 2n
and n is number
of blocks in a two rotor period. The recoupling pulse sequences mix the y magnetization. We show that four pulse recoupling is more broadband compared to three pulse recoupling [1]. Experimental quantification of this method is shown for 13 Cα -13 CO, homonuclear recoupling in a sample of Glycine and
15
N-13 Cα , heteronuclear recoupling in Alanine. Application of this method is demonstrated on
a sample of tripeptide N-formyl-[U-13 C,15 N]-Met-Leu-Phe-OH (MLF). ∗ To
whom correspondence may be addressed. Email:[email protected] of Electrical Engineering, IIT Bombay, Powai - 400076, India. ‡ Department of Biosciences and Bioengineering , IIT Bombay, Powai- 400076, India. † Department
1
1
Introduction
Solid State nuclear magnetic resonance (NMR) spectroscopy opens up the possibility of studying insoluble protein structures such as membrane proteins, fibrils, and extracellular matrix proteins which are otherwise difficult to analyze using conventional atomic-resolution structure determination methods, including liquid-state NMR and X-ray crystallography [4, 5, 6, 7, 8, 16]. Recoupling pulse sequences that enable transfer of magnetization between coupled spins is the workhorse of all these experiments, either as a means to obtain structural information (e.g., internuclear distances) or as a means to improve resolution, as building blocks in multiple-dimensional correlation experiments. The present paper describes some new methodology development for design of recoupling pulse sequences and demonstration of their use in correlation experiments. The paper is organized as follows. In section 2, we describe a novel approach to homonuclear recoupling that recouple dipolar coupled spins under Magic angle spinning (MAS) experiments. We call this four pulse recoupling. These experiments are broadband and robust to rf-inhomogeneity. This work extends recently developed techniques for broadband homonuclear recoupling as reported in the [1, 2, 3, 9, 11, 12, 13]. In section 3, we describe these methods in the context of heteronuclear experiments. In the context of heteronuclear spins, the recoupling is achieved by matching the syncronized phases on the two rf-channels (analogous to Hartmann-Hahn matching of the rf-power commonly seen in heteronuclear recoupling experiments [15] ). Section 4, describes experimental verification of the proposed techniques.
2
Four pulse recoupling in homonuclear spins
Consider, two homonuclear spins I and S under magic angle spinning condition [14]. In a rotating frame, rotating with both the spins at their common Larmor frequency, the Hamiltonian of the spin system takes the form H(t) = ωI (t)Iz + ωS (t)Sz + ωIS (t)(3Iz Sz − I · S) + 2πA(t)(cos φ(t)Fx + sin φ(t)Fy ),
(1)
where the operator Fx = Ix +Sx , and ωI (t) and ωS (t) represent the chemical shift for the spins I and S respectively and ωIS (t) represents the time varying couplings between the spins under magic-angle
2
spinning. These interactions may be expressed as a Fourier series ωλ (t) =
2
ωλm exp(imωr t),
(2)
m=−2
where ωr is the spinning frequency (in angular units), while the coefficients ωλ (λ = I, S, IS) reflect the dependence on the physical parameters like the isotropic chemical shift, anisotropic chemical shift, the dipole-dipole coupling constant and through this the internuclear distance [16]. The term I · S in (1), commutes with the rf-field Hamiltonian, and in the absence of the chemical shifts, it averages to zero under MAS. Consider the rf irradiation on homonuclear spin pair, whose amplitude is chosen as A(t) = C 2π ,
where C = nωr and phase φ(t), such that the pulse sequence consists of the building block
( π2 )0◦ (3π)φ◦ ( π2 )0◦ , which takes 2τc units of time where τc =
2π C .
There are n building blocks in 2
rotor periods. We can think of phase as starting from zero and jumping to value φ = and then returning to 0 at t =
7τc 4
π n
τc 4
at t =
and this cycle continues. See Fig. 1A.
Over one cycle, the rate of change of phase φ˙ takes the form (See Fig. 1B.)
˙ = Cπ (δ(Ct − π ) − δ(Ct − 7π )), (3) φ(t) n 2 2 t where δ(t) is a delta function ( −t f (τ )δ(τ )dτ = f (0) for t > 0). The instantaneous change of phase of φ is modelled by a delta function in its rate of change. In the modulation frame of the phase φ(t), the rf-field Hamiltonian takes the form ˙ z, H rf (t) = CFx − φF
(4)
where C is in the angular frequency units and we choose C ωI (t), ωS (t), ωIS (t), ωr . In the interaction frame of the irradiation along x axis, with the strength C, the chemical shifts of the spins are averaged out. The rf-field Hamiltonian of the spin system transforms to HIrf (t) = −
π 7π Cπ (δ(Ct − ) − δ(Ct − ))(Fz cos(Ct) + Fy sin(Ct)), 2 2 n
(5)
˙ φ(t)
which accumulates an integral only when delta function peaks at Ct =
π 2
and Ct =
7π 2 .
The net
rf rotation generated by HIrf (t) is therefore − 2π n Fy , which corresponds to a phase advance of at Ct =
π 2
and phase decrement of
π n
at Ct =
7π 2
3
π n
( a negative phase decrement is multiplied by
B
A
φ
φ 7 τ 4 c 1 τ 4 c
t
7 τ 2τ 4 c c
1 τ 4 c
Fz
1 0 0 1
0 1 1 0 0 1
t
c
Fz
C
2τ
1 0 0 1
D 0 1 1 0 0 1
Fy
0 1 1 0 0 1
Figure 1: Fig. A shows how phase changes for the pulse sequence ( π2 )0◦ (3π)φ◦ ( π2 )0◦ . Fig. B shows the rate of change of phase. Area under this curve gives the total phase change. Fig. C shows the accumulation of phase area at the dotted points when the Fz is rotated in the interaction frame of CFx π 3π ◦ ◦ ◦ ◦ as in Eq. 5. Fig. D shows the same for the compensated sequence ( π2 )0◦ ( 3π 2 )φ ( 2 )180 +φ ( 2 )180 , where we alternate between the interaction frame of CFx and −CFx . See Eq. 12 and 13.
4
0 1 1 0 0 1
Fy
− sin( π2 ), which makes it a positive accumulation). See Fig. 1C. A net rotation of 2τc =
2τr n
corresponds to a net effective field of strength
ωr 2
2π n
in time
along −y direction. This effective field
recouples. As an example if we choose C = 6ωr , then n = 6. The above may be implemented with a pulse sequence
π π ( )0◦ (3π)30◦ ( )0◦ 2 2 where π is the flip angle which takes
τr 2n
(6)
units of time with pulse amplitude of C = nωr , so that in
two rotor periods of duration 2τr , we have n of these units. When pulse amplitude has inhomogeneity, then in the modulation frame of the phase φ(t), the rf-field Hamiltonian takes the form ˙ z, H rf (t) = C(1 + )Fx − φF
(7)
In the interaction frame of the irradiation along x axis, with the strength C, the rf-field Hamiltonian of the spin system transforms to HIrf (t) = CFx −
π 7π Cπ (δ(Ct − ) − δ(Ct − ))(Fz cos(Ct) + Fy sin(Ct)), 2 2 n
(8)
˙ φ(t)
which has an additional factor of CFx , which accumulates (in time 2τc ) to first order an evolution 4πFx − π 5 Fx
2π n Fy ,
which for an inhomogeneity of = .05 and n = 6, corresponds to an evolution
− π3 Fy which limits the transfer efficiency in presence of inhomogeneity.
The inhomogeneity factor can be cancelled by applying the following compensated pulse sequence called FPR, Four Pulse Recoupling. π 3π π 3π ( )0◦ ( )φ◦ ( )180◦ +φ◦ ( )180◦ 2 2 2 2 N
(9)
Then in the modulation frame of the phase φ(t), the rf-field Hamiltonian takes the form H rf (t) = C(1 + )Fx −
π Cπ δ(Ct − ) Fz , n 2 ˙ φ(t)
for t ∈ [0, τc ] followed by 5
(10)
H rf (t) = −C(1 + )Fx +
π Cπ δ(Ct − ) Fz . n 2
(11)
˙ φ(t)
In the interaction frame of the irradiation along x axis, with the strength C for time τc and −C for next τc , the rf-field Hamiltonian of the spin system transforms to
(t) = CFx − HIrf a
π Cπ δ(Ct − )(Fz cos(Ct) + Fy sin(Ct)), n 2
(12)
˙ φ(t)
followed by
HIrf (t) = −CFx + b
π Cπ δ(Ct − )(Fz cos(Ct) − Fy sin(Ct)). n 2
(13)
˙ φ(t)
Eq. (12, 13) accumulate an integral only when delta function peaks at Ct =
π 2.
The net rf
rotation is therefore − 2π n Fz and ±CFx cancel to first order. See, figure 1D. For n = 6, the compensated FPR pulse sequence takes the form
π 3π π 3π ( )0 ( )30◦ ( )210◦ ( )180◦ 2 2 2 2
(14)
as opposed to uncompensated FPR in Eq. (6). We have calculated
τc 0
HIrf (t)dt and a
τc 0
HIrf (t)dt, which are first order contributions to generb
ated rotation by time varying Hamiltonian HIrf (t) and HIrf (t). We can calculate the second order a b terms by writing evolution of HIrf (t) and HIrf (t) as a b
Ua (t) = exp(−i
Ub (t) = exp(i
3π π π Fx ) exp(i Fy ) exp(−i Fx ), 2 n 2
3π π π Fx ) exp(i Fy ) exp(i Fx ). 2 n 2
ωr Therefore, we prepare to first order, effective Hamiltonian − 2τ1c 2π n Fy = − 2 Fy . The second
order Hamiltonian is of order π2 ωr Fz . We have a recoupling field along − ω2r Fy + π2 ωr Fz . 6
We evaluate the effect of recoupling field on the coupling Hamiltonian. By transforming into the interaction frame of irradiation along x axis which alternates between CFx and −CFx , we prepare HIrf (τ ) and HIrf (τ ) every τc units of time. The coupling Hamiltonian alternates between a b DD HI+ (t) =
3 3 ωIS (t)(Iz Sz + Iy Sy ) + ωIS (t)((Iz Sz − Iy Sy ) cos(2Cτ ) + (Iz Sy + Iy Sz ) sin(2Cτ )), (15) 2 2
and
DD HI− (t) =
3 3 ωIS (t)(Iz Sz + Iy Sy ) + ωIS (t)((Iz Sz − Iy Sy ) cos(2C(−τ )) + (Iz Sy + Iy Sz ) sin(2C(−τ )). 2 2 (16)
Now, transforming the coupling Hamiltonian in Eq. 15 and Eq. 16, into the interaction frame of the rf-field Hamiltonian HIrf , we act with an effective recoupling field − ω2r Fy + π2 ωr Fz , that points in unit direction −m, ˆ where m ˆ = cos θˆ y − sin θˆ z and θ is of order π. We only retain terms that give static contribution to the effective Hamiltonian, i.e., terms oscillating with frequency 2C are dropped and the residual Hamiltonian takes the form (neglecting the terms oscillating at frequency 2ωr as they are not recoupled) HII (t) = κh {cos(ωr t) cos(ωr t + γ)(In Sn − Ix Sx ) + sin(ωr t) cos(ωr t + γ)(In Sx + Ix Sn )},
(17)
which averages to ¯ II = κh {cos(γ)(In Sn − Ix Sx ) − sin(γ)(In Sx + Ix Sn )}, H 2 where κh =
3 √ b 4 2 IS
(18)
sin(2β), where bIS is the dipole coupling constant and n ˆ is the effective direction n ˆ = cos θ zˆ + sin θ yˆ,
and m ˆ ×n ˆ =x ˆ. For small , n ˆ = zˆ and m ˆ = yˆ. It takes time
2π κh
to transfer Iy → Sy , under the
above coupling Hamiltonian, where β is chosen at a nominal value. The rf-interaction frame, HIrf prepares an effective field − ω2r Fy , every τc units of time. Transformation of the coupling Hamiltonian can be evaluated every τc units of time [9], by realizing that T
kτc U (t)JU (t)dt = k k−1τc U ((k − 1)τc )Uk (t)JUk (t)U ((k − 1)τc )dt, where U ((k − 1)τc ) is the 0 interaction frame propagator at time (k − 1)τc and is same as evolution under effective field − ω2r Fz at this time, and Uk (t) evolves in time [(k − 1)τc , kτc ] with Uk (t) = I + o( ω2r τc ). For ωr C, we can 7
neglect the second factor and assume we evolve the coupling Hamiltonian under an effective field, − ω2r Fy . We can add more details to the above analysis, and consider the coupling Hamiltonian cos(ωr t + γ)2Iz Sz = cos(ωr t + γ){(Iz Sz − Ix Sx ) + (Iz Sz + Ix Sx )}. The zero quantum term (Iz Sz + Ix Sx ) doesn’t evolve under effective field and averages out. We can write
cos(ωr t + γ)(Iz Sz − Ix Sx ) = +
exp(j(ωr t + γ)) + exp(−j(ωr t + γ)) {(Iz Sz − Ix Sx ) + j(Iz Sx + Ix Sz )} 4 exp(j(ωr t + γ)) + exp(−j(ωr t + γ)) {(Iz Sz − Ix Sx ) − j(Iz Sx + Ix Sz )}. 4
In the rf-interaction frame, HIrf , we have a pulse at time {(Iz Sz − Ix Sx ) ± j(Iz Sx + Ix Sz )} → exp(∓j
π 2n
which transforms,
2π ){(Iz Sz − Ix Sx ) ± j(Iz Sx + Ix Sz )}. n
Then over τc , J cos(ωr t + γ)(Iz Sz − Ix Sx ) averages to Ix Sz ) sin γ } where γ = γ −
τc 4 ,
π Jτc sin( n ) {(Iz Sz π 2 n
− Ix Sx ) cos γ − (Iz Sx +
and this is the average Hamiltonian every τc units of time.
In Fig. 2, we simulated the performance of compensated and uncompensated FPR for C = 60 kHz and ωr = 10 kHz which gives n = 6 and φ = 30◦ .
2.1
Offset performance of FPR
The FPR pulse sequence is broadband as a large value of C averages out the chemical shift [11]. We can understand the effect of chemical shift offset on the recoupling field. In presence of a chemical shift Δω on spin S, we have in the interaction frame of the rf phase the Hamiltonians ˙ z and ΔωSz − CSx + φS ˙ z . We write these as ω ˙ z and ω ˙ z where ΔωSz + CSx − φS ˜ Sx − φS ˜ Sx + φS √ x = x cos θ + z sin θ and x = −x cos θ + z sin θ and ω ˜ = C 2 + Δω 2 , for cos θ = C/˜ ω , see Fig. 3. By proceeding in frame CSx and CSx , we create Hamiltonians HIrf (t) and HIrf (t) whose a b evolution is Ua (t)
=
exp(−i
3τc π τc Δ˜ ω Sx ) exp(i (cos θSy + sin θSx )) exp(−i Δ˜ ω Sx ) 4 n 4
and
8
Build up curve for homonuclear uncompensated FPR 0.8 0.7
transfer efficiency
A
0.6 0.5 a
0.4 0.3 0.2 b 0.1 0 c −0.1
0
0.5
1 time (ms)
1.5
2
Build up curve for homonuclear compensated FPR 0.8 0.7
B
transfer efficiency
0.6 0.5 b c
0.4
a
0.3 0.2 0.1 0 −0.1
0
0.5
1 time (ms)
1.5
2
offset performace of FPR 0.8 0.7
C
transfer efficiency
0.6 0.5
a b
0.4
c
0.3
d
0.2 0.1 0 −0.1
0
0.5
1 time (ms)
1.5
2
Figure 2: Fig. A shows the build up of transfer of magnetization for a ideal two spin system 13 C-13 C on a 750 MHz (proton frequency) static field, using the FPR pulse unit with C = 6ωr , n = 6 with φ = 30◦ and no compensation. Simulation uses internuclear distance of 1.52A◦ and powder avaraging. a, b, c corresponds to inhomogeneity value of 0, .02, .05 respectively. Fig. B shows the build up of basic 13 C-13 C correlation experiment using the compensated FPR pulse unit with φ = 30◦ . a, b, c corresponds to inhomogeneity value of 0, .02 and .05 respectively. In practice, the rf-inhomogeneity results in a weighted sum of these different . Offset of spin pair is assumed to be on resonance. Fig. C shows the build up of basic 13 C-13 C correlation experiment using the compensated FPR pulse unit with φ = 30◦ with different chemical shifts of the spin pair. a, b, c, d, corresponds to chemical shift difference of 0, 8, 16 and 24 kHz with carrier in the center. In Fig. C, we include a CSA value of 11 and 80 ppm for the two spins respectively with anisotropy parameter .3 and .9, representing the Cα -CO spin pair in Glycine respectively. Simulation were performed with spinevolution [10] software.
9
z z
z θ
θ x
x θ
θ x
Figure 3: Fig. shows effective directions x , z , x and z in presence of chemical shift offset Δω.
Ub (t) = exp(−i
3τc π τc Δ˜ ω Sx ) exp(i (cos θSy − sin θSx )) exp(−i Δ˜ ω Sx ) 4 n 4
where Δ˜ ω=ω ˜ − C. Using the approximation x ∼ −x , we get Then Ub Ua ∼ exp(i
2π (cos θSy + sin θSx )) n
Therefore we have a recoupling field of strength
ωr 2
even in presence of offset. Therefore FPR
is quite broadband pulse sequence. It is worthwhile to compare the effective field of FPR with effective field of TPR [1]. The TPR pulse sequence takes the form
π π π π ( )0◦ (π) φ◦ ( )0◦ ( )180◦ (π)180◦ − φ◦ ( )180◦ 2 2 2 2 2 2 N where φ =
π n.
Proceeding analogously to above we find
10
Ua (t)
=
τc π τc Δ˜ ω Sx ) exp(i (cos θSy − sin θSx )) exp(−i Δ˜ ω Sx ) 4 2n 2 τc π ω Sx ) exp(i (cos θSy + sin θSx )) exp(−i Δ˜ 2n 4 exp(−i
and
τc π τc Δ˜ ω Sx ) exp(i (cos θSy + sin θSx )) exp(−i Δ˜ ω Sx ) 4 2n 4 τc π ω Sx ) exp(i (cos θSy − sin θSx )) exp(−i Δ˜ 2n 4
Ub (t) = exp(−i
Then Ub Ua ∼ exp(i
2π cos θSy ) n
We prepare an recoupling field − cos θ ω2r Sy . Due to presence of offset, we deviate from the normal recoupling field of
ωr 2
by an amount
Δω 2 2C 2 (ωr /2).
Hence FPR pulse sequence performs better
than TPR in presence of chemical shift offsets. See Fig. 4. We can evaluate the effect of FPR pulse sequence on the chemical shift anisotropy in the interaction frame of irradiation along x axis which alternates between CFx and −CFx . The chemical shift anisotropy of spin I can be expressed as (ωI±1 exp(±iωr t) + ωI±2 exp(±i2ωr t))Iz , we can decompose the operator Iz = I + + I − , where I ± = Iz ± iIy , where exp(−iCIx t)I ± exp(+iCIx t) = exp(±iCt)(I ± ). Using this, For n = 6, the CSA averages every two rotor periods, where TPR toggles between ±C every τc . Proceeding to interaction frame, HIrf , by acting with an effective field − ω2r Fy , we find that for example n = 6, the CSA averages to zero in first order.
3
Phase Matching and Heteronuclear Recoupling
Consider two coupled heteronuclear spins I and S under magic angle spinning condition [15]. The spins are irradiated with rf fields at their Larmor frequencies along say the x direction. In a doublerotating Zeeman frame, rotating with both the spins at their Larmor frequency, the Hamiltonian of the system takes the form H(t) = ωI (t)Iz + ωS (t)Sz + ωIS (t)2Iz Sz + H rf (t), 11
(19)
Offset performace of FPR vs TPR 0.7 0.6
transfer efficiency
0.5 0.4
a
0.3 0.2
b
0.1 0 0
0.5
1 time (ms)
1.5
2
Figure 4: Fig. shows the build up curve for transfer of magnetization I → S, using homonuclear FPR (a) and TPR (b) for ωr = 10 kHz and n = 6, C = 60 kHz, and φ = 30◦ , with offset ΔωI = −15 kHz, and ΔωS = 15 kHz.
where ωI (t), ωS (t), and ωIS (t) represent time-varying chemical shifts for the two spins I and S and the coupling between them, respectively. These interactions can be expressed as a Fourier
2 series ωλ (t) = m=−2 ωλm exp(imωr t), where ωr is the spinning frequency (in angular units), while the coefficients ωλ , (λ = I, S) reflect the dependence on the physical parameters like the isotropic chemical shift, anisotropic chemical shift, the dipole-dipole coupling constant and through this the internuclear distance [16]. Consider the rf irradiation on heteronuclear spin pair, where amplitude on spin I and S is chosen as Aλ (t) =
C 2π
(λ = I, S) such that C = nωr , and phase modulation, given by pulse sequence
( π2 )0◦ (3π)φ◦1 ( π2 )0◦ and ( π2 )0◦ (3π)φ◦2 ( π2 )0◦ respectively on two channels. Here we choose φ1 = φ2 =
π 2n .
3π 2n
and
This gives φ˙1 (t)
=
φ˙2 (t)
=
π 7π 3Cπ (δ(Ct − ) − δ(Ct − )) 2n 2 2 π 7π Cπ (δ(Ct − ) − δ(Ct − )) 2n 2 2
(20) (21)
where phase jump is three halves and half of the modulation considered in previous section. As a result, the effective fields prepared on two channels will be 12
3ωr 4
and
ωr 4 ,
with average
ωr 2 ,
which will
Buildup curve for heteronuclear uncompensated FPR 0.7 0.6 0.5 b transfer efficiency
A
a
0.4 0.3 0.2 c
0.1 0 −0.1
0
2
4
6
8
10
time (ms) build up curve for compensated heteronuclear FPR 0.8 0.7 0.6 transfer efficiency
B
0.5 a
0.4 b
0.3
c 0.2 0.1 0 −0.1
0
2
4
6
8
10
time (ms) offset performance of hetronuclear FPR 0.8 a b
0.7
C
transfer efficiency
0.6
c
0.5 0.4 0.3 0.2 d 0.1 0 0
2
4
6
8
10
time (ms)
Figure 5: Fig. A shows the build up of 15 N to 13 C magnetization transfer on a 750 MHz (proton frequency) static field, using the FPR pulse unit with C = 40 kHz and ωr = 10 kHz and n = 4 with φ1 = 67.5◦ and φ2 = 22.5◦ on 13 C and 15 N respectively with no compensation. a, b, c corresponds to inhomogeneity value of 0, .02 and .05 on Carbon channel. Fig. B shows the build up using the FPR pulse unit with compensation. Fig. C shows the build up of basic -15 N-13 C correlation experiment using the FPR pulse unit, with different chemical shift of the 13 C spin. a, b, c, d corresponds to chemical shift of 0, 4, 8, 12 kHz of the 13 C spin respectively.
13
recouple a double quantum Hamiltonian 1 . To mitigate the effect of inhomogeneity of rf-field, we use a compensating pulse sequence. In nutshell the pulse sequence on channels I and S (say C 13 and N 15 ) is following respectively.
3π π 3π π ( )0◦ ( )φ◦1 ( )180◦ +φ◦1 ( )180◦ 2 2 2 2
(22)
3π π 3π π ( )0◦ ( )φ◦2 ( )180◦ +φ◦2 ( )180◦ 2 2 2 2
(23)
In the frame of the rf-field along x axis, that toggles between ±C(Ix + Sx ), the coupling Hamiltonian is averaged to
DD HI± (t) = ωIS (t)(Iz Sz + Iy Sy ) + ωIS (t)((Iz Sz − Iy Sy ) cos(2Cτ ) ± (Iz Sy + Iy Sz ) sin(2Cτ )), (24)
With effective rf-field − 3ω4 r Iy −
ωr 4 Sy
written as −ωr
(Iy +Sy ) 2
−
ωr (Iy −Sy ) , 2 2
the coupling Hamilto-
nian is further averaged to
¯ II = κd {(Iz Sz − Ix Sx ) cos(γ) − (Iz Sx + Ix Sz ) sin(γ)} . H
(25)
We call the above experiments heteronuclear FPR. Fig. 5 shows simulation results for a heteronuclear magnetization transfer in
15
N-13 C heteronu-
clear system for C = 40 kHz and ωr = 10 kHz with n = 4. This gives φ1 = 67.5◦ and φ2 = 22.5◦ .
4
Experimental Results
All experiments were performed on a 750 MHz spectrometer (1 H Larmor frequency of 750 MHz) equipped with a triple resonance 3.2 mm MAS probe. Uniformly
13
C labeled sample of Glycine and
uniformly 13 C, 15 N -labeled sample of Alanine were used in the full volume of standard 3.2 mm rotor at ambient temperature for homonuclear and heteronuclear experiments respectively. Uniformly 13
C,
15
N -labeled sample of MLF was used for both homonuclear and heteronuclear recoupling
1 When
gyromagnetic ratio of two spins are of different sign, we recouple a zero quantum Hamiltonian.
14
experiments. The experiments used 2s recycling delay. In all experiments, CW decoupling of 120 kHz is used on protons. All 2D experiments were processed using TPPI processing. Fig. 6A shows the build up curve for transfer of magnetization, for
13
Cα to
13
CO every rotor
period, N = 3 (N as in Eq. 9), with FPR as the recoupling element. The experiment uses an initial ramped CP for 1 H to
13
C cross polarization. The FPR recoupling block is designed for a
nominal power of 60 kHz as described in the text and spinning speed of 10 kHz. This gives n = 6 and φ = 30◦ . Fig. 6B shows a
13
Cα -13 CO 2D correlation spectrum obtained using the FPR as the
recoupling element. Mixing time is 10 rotor periods. The experiment used 1024 points in direct and 512 in indirect dimension with spectral width of 220 ppm in both dimensions. The carrier is placed in center of the two resonances at 119 ppm during the transfer. 4 scans were collected for every t1 increment. Fig. 7A shows the build up of transfer of magnetization, shown every two rotor period, N = 4, for the
15
N →13 Cα experiment with FPR as the recoupling element. The FPR recoupling is done
at spinning speed of 10 kHz and nominal power of 40 kHz on carbon and nitrogen as described in the text. This gives n = 4 and φ1 = 67.5◦ (on initial ramped CP for 1 H to for
13
15
13
C) and φ2 = 22.5◦ (on
15
N). The experiment uses an
N cross polarization. Bottom panel of Fig. 7, shows the 2D spectrum
Cα -15 N experiment with FPR as the recoupling element. The magntization precesses on
15
N
during indirect evolution. Mixing time is 28 rotor periods. The experiment used 1024 points in direct and 256 in indirect dimension with spectral width of 220 ppm in dimension. The carrier is placed on Fig. 8 shows the 2D
13
15
N and
13
13
Cα and 100 ppm in
15
N
Cα resonance during transfer.
C correlation in tripeptide MLF obtained using FPR as recoupling
element. Experiment is done at 10 kHz spinning, and rf-power C = 60 kHz. This gives n = 6 and φ = 30◦ . The carrier is placed in center of carbon spectrum at 119 ppm. CW decoupling of 120 kHz is used on protons. Mixing time corresponds to N = 30 or 10 rotor periods. The experiment uses an initial ramped CP for 1 H to
13
C cross polarization. The experiment used 1024 points in direct
and 1024 in indirect dimension with spectral width of 220 ppm in both dimensions. 16 scans were collected for every t1 increment. Fig. 9 shows the 2D
15
N-13 C correlation in tripeptide MLF obtained using TPR as recoupling
element. Experiment is done at 10 kHz spinning, and rf-power C = 40 kHz on carbon and nitrogen channels. This gives n = 4 and φ1 = 67.5◦ (on
13
C) and φ2 = 22.5◦ (on
15
15
N). The carbon carrier
is placed in center at 119 ppm. Mixing time corresponds to N = 56 or 28 rotor periods. The experiment used 1024 points in direct and 512 in indirect dimension with spectral width of 220 ppm in carbon and 100 ppm in nitrogen dimension. 8 scans were collected for every t1 increment.
5
Conclusion
In this paper we introduced a class of recoupling pulse sequences, which rest on the principle of second oscillating field [1, 3, 9, 11, 12, 17]. A strong field is used to eliminate chemical shifts and make the sequence broadband. Furthermore this strong rf-field is used to demodulate a second oscillating field which performs recoupling. In our design, the second oscillating field comes about by principled phase changes which are described in the paper. The recoupling sequences presented in the paper for homonuclear and heteronuclear spin systems are broadband and robust to rf-inhomogeneity. We compared the work in this paper with TPR pulse sequence in [1], where we also produce an effective
ωr 2
field along y axis. We found that the effective recoupling field in FPR is more robust
to resonance offsets as compared to TPR, which makes the sequence more broadband.
6
Acknowledgement
The authors would like to thank the HFNMR lab facility at IIT Bombay , funded by RIFC, IRCC, where the data was collected.
References [1] J. Lin, R.G. Griffin, N.C. Nielsen and N. Khaneja, Three pulse recoupling and phase jump matching, J. Magn. Reson. 263, 172-183 (2016). [2] M.H. Levitt, Symmetry-Based Pulse Sequences in Magic-Angle Spinning Solid-State NMR, Encyclopedia of Nuclear Magnetic Resonance, Volume 9, pp 165-196, Edited by David M. Grant and Robin K. Harris (2002). [3] J. Lin, R.G. Griffin, and N. Khaneja, Recoupling in solid state NMR using γ prepared states and phase matching, J Magn Reson. 2011 Oct; 212(2): 402411. [4] S. J. Opella, NMR and membrane proteins, Nat. Struct. Biol. 4, 845-848 (1997). 16
[5] R. G. Griffin, Dipolar recoupling in MAS spectra of biological solids, Nat. Struct. Biol. 5, 508-512 (1998). [6] F. Castellani, B. van Rossum, A. Diehl, M. Schubert, K. Rehbein, and H. Oschkinat, Structure of a protein determined by solid-state magic-angle-spinning NMR spectroscopy, Nature 420, 98-102 (2002). [7] A. T. Petkova, Y. Ishii, J. J. Balbach, O. N. Antzutkin, R. D. Leapman, F. Deglaglio, and R. Tycko, A structural model for alzheimer’s β-amyloid fibrils based on experimental constraints from solid state NMR, Proc. Natl. Acad. Sci. 99, 16742-16747 (2002). [8] C. P. Jaroniec, C. E. MacPhee, V.S. Baja, M.T. McMahon, C.M. Dobson, and R. G. Griffin, High-resolution molecular structure of a peptide in an amyloid fibril determined by magic angle spinning NMR spectroscopy, Proc. Natl. Acad. Sci. 101, 711-716 (2004). [9] J. Lin, “Solid State NMR Experiments with powder dephased states and phase matching.” Phd Thesis, School of Engineering and Applied Sciences, Harvard (2010). [10] M. Veshtort and R. G. Griffin, SPINEVOLUTION: A powerful tool for the simulation of solid and liquid state NMR experiments, J. Magn. Reson., 178, 248-282 (2006). [11] G. De Pa¨epe, M. J. Bayro, J. Lewandowski, R. G. Griffin, Broadband homonuclear correlation spectroscopy at high magnetic fields and MAS frequencies, J. Amer. Chem. Soc. 128, 17761777(2006). [12] G. De Pa¨epe, J. Lewandowski, R. G. Griffin, Spin dynamics in the modulation frame: Application to homonuclear recoupling in magic angle spinning solid state NMR, J. Chem. Phys. 128, 124503 (2008). [13] J. Lin, M. Bayro, R. G. Griffin, and N. Khaneja, Dipolar recoupling in solid state NMR by phase alternating pulse sequences, J. Mag. Reson 197, 145-152 (2009). [14] N. C. Nielsen, H. Bildsøe, H. J. Jakobsen, and M. H. Levitt, Double-quantum homonuclear rotary resonance: Efficient dipolar recovery in magic-angle spinning nuclear magnetic resonance, J. Chem. Phys. 101, 1805-1812 (1994). [15] J. Schaefer, R. A. McKay, and E. O. Stejskal, Double-cross-polarization NMR of solids, J. Magn. Reson. 34, 443-447 (1979). [16] M. Duer, Solid State NMR Spectroscopy, Blackwell Publishing, (2000). 17
[17] A.B. Nielsen, L.A. Straaso, A.J. Nieuwkoop, C.M. Rienstra, M. Bjerring, and N.C. Nielsen, Broadband Heteronuclear Solid-State NMR Experiments by Exponentially Modulated Dipolar Recoupling without Decoupling J. Phys. Chem. Lett., 2010, 1 (13), pp 1952-1956.
18
Figure 6: Fig. A shows the build up curve for transfer of magnetization, from 13 Cα to 13 CO in a Glycine, after every N = 3 blocks, at 10 kHz spinning, and rf-power C = 60 kHz. This gives n = 6 and φ = 30◦ . Fig. B shows corresponding 13 Cα -13 CO 2D correlation spectrum obtained using the FPR as the recoupling element. Mixing time is 10 rotor periods. The carrier is placed in center of two resonances.
19
Figure 7: Fig. A shows the build up of transfer of magnetization, shown every N = 4 blocks, at 10 kHz spinning, for the 15 N →13 Cα experiment, in uniformly labelled sample of Alanine, with FPR as the recoupling element as described in text. The rf power used is C = 40 kHz, giving n = 4 and φ1 = 67.5◦ (on 13 C) and φ2 = 22.5◦ (on 15 N) . Fig. B shows the 2D spectrum for 15 N-13 Cα experiment with FPR as the recoupling element. Mixing time is 28 rotor periods. The magntization precesses on 15 N during indirect evolution. The carrier is placed on 15 N and 13 Cα resonance.
20
Figure 8: Fig. shows the 2D 13 C correlation in tripeptide MLF obtained using FPR as recoupling element. Experiment is done at 10 kHz spinning, and rf-power C = 60 kHz. This gives n = 6 and φ = 30◦ . The carrier is placed in center of carbon spectrum at 119 ppm. CW decoupling of 120 kHz is used on protons. Mixing time corresponds to N = 30 or 10 rotor periods.
Figure 9: Fig. shows the 2D 15 N-13 C correlation in tripeptide MLF obtained using FPR as recoupling element. Experiment is done at 10 kHz spinning, and rf-power C = 40 kHz on carbon and nitrogen channels. This gives n = 4 and φ1 = 67.5◦ and φ2 = 22.5◦ . The carbon carrier is placed in center of carbon spectrum at 119 ppm. CW decoupling of 120 kHz is used on protons. Mixing time corresponds to N = 56 or 28 rotor periods.
21
1. We present a family of broadband recoupling pulse sequences, called four pulse recoupling. 2. These pulse sequences can be employed for both homonuclear and heteronuclear recoupling experiments. 3. We show these sequences are robust to rf-inhomogeneity and chemical shift dispersion. 4. We present solid state NMR experiments that use four pulse recoupling.
1
B
A
φ
φ 7 τ 4 c 1 τ 4 c
C
7 τ 2τ 4 c c
0 1 1 0 0 1
00 11 11 00 00 11
1 τ 4 c
t
2τ
c
Fz
Fz 1 0 1 0
D 00 11 11 00 00 11
Fy
1
00 11 11 00 00 11
00 11 11 00 00 11
F
S1090-7807(16)30189-6 http://dx.doi.org/10.1016/j.jmr.2016.09.019 YJMRE 5961
To appear in:
Journal of Magnetic Resonance
Received Date: Accepted Date:
16 July 2016 25 September 2016
Please cite this article as: N. Khaneja, A. Kumar, Four Pulse Recoupling, Journal of Magnetic Resonance (2016), doi: http://dx.doi.org/10.1016/j.jmr.2016.09.019
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Four Pulse Recoupling Navin Khaneja
∗†
‡
Ashutosh Kumar
September 26, 2016
Abstract
The paper describes a family of novel recoupling pulse sequences in magic angle spinning (MAS) solid state NMR, called four pulse recoupling. These pulse sequences can be employed for both homonuclear and heteronuclear recoupling experiments and are robust to dispersion in chemical shifts and rf-inhomogeneity.
The homonuclear pulse sequence consists of a building block
π 3π ◦ ◦ ◦ ◦ ( π2 )0◦ ( 3π 2 )φ ( 2 )180 +φ ( 2 )180 where φ =
π n
( φ◦ =
180◦ n ),
and n is number of blocks in a two rotor
π 3π ◦ ◦ ◦ ◦ period. The heteronuclear recoupling pulse sequence consists of a building block ( π2 )0◦ ( 3π 2 )φ1 ( 2 )180 +φ1 ( 2 )180 π 3π ◦ ◦ ◦ ◦ and ( π2 )0◦ ( 3π 2 )φ2 ( 2 )180 +φ2 ( 2 )180 on channel I and S, where φ1 =
3π 2n ,
φ2 =
π 2n
and n is number
of blocks in a two rotor period. The recoupling pulse sequences mix the y magnetization. We show that four pulse recoupling is more broadband compared to three pulse recoupling [1]. Experimental quantification of this method is shown for 13 Cα -13 CO, homonuclear recoupling in a sample of Glycine and
15
N-13 Cα , heteronuclear recoupling in Alanine. Application of this method is demonstrated on
a sample of tripeptide N-formyl-[U-13 C,15 N]-Met-Leu-Phe-OH (MLF). ∗ To
whom correspondence may be addressed. Email:[email protected] of Electrical Engineering, IIT Bombay, Powai - 400076, India. ‡ Department of Biosciences and Bioengineering , IIT Bombay, Powai- 400076, India. † Department
1
1
Introduction
Solid State nuclear magnetic resonance (NMR) spectroscopy opens up the possibility of studying insoluble protein structures such as membrane proteins, fibrils, and extracellular matrix proteins which are otherwise difficult to analyze using conventional atomic-resolution structure determination methods, including liquid-state NMR and X-ray crystallography [4, 5, 6, 7, 8, 16]. Recoupling pulse sequences that enable transfer of magnetization between coupled spins is the workhorse of all these experiments, either as a means to obtain structural information (e.g., internuclear distances) or as a means to improve resolution, as building blocks in multiple-dimensional correlation experiments. The present paper describes some new methodology development for design of recoupling pulse sequences and demonstration of their use in correlation experiments. The paper is organized as follows. In section 2, we describe a novel approach to homonuclear recoupling that recouple dipolar coupled spins under Magic angle spinning (MAS) experiments. We call this four pulse recoupling. These experiments are broadband and robust to rf-inhomogeneity. This work extends recently developed techniques for broadband homonuclear recoupling as reported in the [1, 2, 3, 9, 11, 12, 13]. In section 3, we describe these methods in the context of heteronuclear experiments. In the context of heteronuclear spins, the recoupling is achieved by matching the syncronized phases on the two rf-channels (analogous to Hartmann-Hahn matching of the rf-power commonly seen in heteronuclear recoupling experiments [15] ). Section 4, describes experimental verification of the proposed techniques.
2
Four pulse recoupling in homonuclear spins
Consider, two homonuclear spins I and S under magic angle spinning condition [14]. In a rotating frame, rotating with both the spins at their common Larmor frequency, the Hamiltonian of the spin system takes the form H(t) = ωI (t)Iz + ωS (t)Sz + ωIS (t)(3Iz Sz − I · S) + 2πA(t)(cos φ(t)Fx + sin φ(t)Fy ),
(1)
where the operator Fx = Ix +Sx , and ωI (t) and ωS (t) represent the chemical shift for the spins I and S respectively and ωIS (t) represents the time varying couplings between the spins under magic-angle
2
spinning. These interactions may be expressed as a Fourier series ωλ (t) =
2
ωλm exp(imωr t),
(2)
m=−2
where ωr is the spinning frequency (in angular units), while the coefficients ωλ (λ = I, S, IS) reflect the dependence on the physical parameters like the isotropic chemical shift, anisotropic chemical shift, the dipole-dipole coupling constant and through this the internuclear distance [16]. The term I · S in (1), commutes with the rf-field Hamiltonian, and in the absence of the chemical shifts, it averages to zero under MAS. Consider the rf irradiation on homonuclear spin pair, whose amplitude is chosen as A(t) = C 2π ,
where C = nωr and phase φ(t), such that the pulse sequence consists of the building block
( π2 )0◦ (3π)φ◦ ( π2 )0◦ , which takes 2τc units of time where τc =
2π C .
There are n building blocks in 2
rotor periods. We can think of phase as starting from zero and jumping to value φ = and then returning to 0 at t =
7τc 4
π n
τc 4
at t =
and this cycle continues. See Fig. 1A.
Over one cycle, the rate of change of phase φ˙ takes the form (See Fig. 1B.)
˙ = Cπ (δ(Ct − π ) − δ(Ct − 7π )), (3) φ(t) n 2 2 t where δ(t) is a delta function ( −t f (τ )δ(τ )dτ = f (0) for t > 0). The instantaneous change of phase of φ is modelled by a delta function in its rate of change. In the modulation frame of the phase φ(t), the rf-field Hamiltonian takes the form ˙ z, H rf (t) = CFx − φF
(4)
where C is in the angular frequency units and we choose C ωI (t), ωS (t), ωIS (t), ωr . In the interaction frame of the irradiation along x axis, with the strength C, the chemical shifts of the spins are averaged out. The rf-field Hamiltonian of the spin system transforms to HIrf (t) = −
π 7π Cπ (δ(Ct − ) − δ(Ct − ))(Fz cos(Ct) + Fy sin(Ct)), 2 2 n
(5)
˙ φ(t)
which accumulates an integral only when delta function peaks at Ct =
π 2
and Ct =
7π 2 .
The net
rf rotation generated by HIrf (t) is therefore − 2π n Fy , which corresponds to a phase advance of at Ct =
π 2
and phase decrement of
π n
at Ct =
7π 2
3
π n
( a negative phase decrement is multiplied by
B
A
φ
φ 7 τ 4 c 1 τ 4 c
t
7 τ 2τ 4 c c
1 τ 4 c
Fz
1 0 0 1
0 1 1 0 0 1
t
c
Fz
C
2τ
1 0 0 1
D 0 1 1 0 0 1
Fy
0 1 1 0 0 1
Figure 1: Fig. A shows how phase changes for the pulse sequence ( π2 )0◦ (3π)φ◦ ( π2 )0◦ . Fig. B shows the rate of change of phase. Area under this curve gives the total phase change. Fig. C shows the accumulation of phase area at the dotted points when the Fz is rotated in the interaction frame of CFx π 3π ◦ ◦ ◦ ◦ as in Eq. 5. Fig. D shows the same for the compensated sequence ( π2 )0◦ ( 3π 2 )φ ( 2 )180 +φ ( 2 )180 , where we alternate between the interaction frame of CFx and −CFx . See Eq. 12 and 13.
4
0 1 1 0 0 1
Fy
− sin( π2 ), which makes it a positive accumulation). See Fig. 1C. A net rotation of 2τc =
2τr n
corresponds to a net effective field of strength
ωr 2
2π n
in time
along −y direction. This effective field
recouples. As an example if we choose C = 6ωr , then n = 6. The above may be implemented with a pulse sequence
π π ( )0◦ (3π)30◦ ( )0◦ 2 2 where π is the flip angle which takes
τr 2n
(6)
units of time with pulse amplitude of C = nωr , so that in
two rotor periods of duration 2τr , we have n of these units. When pulse amplitude has inhomogeneity, then in the modulation frame of the phase φ(t), the rf-field Hamiltonian takes the form ˙ z, H rf (t) = C(1 + )Fx − φF
(7)
In the interaction frame of the irradiation along x axis, with the strength C, the rf-field Hamiltonian of the spin system transforms to HIrf (t) = CFx −
π 7π Cπ (δ(Ct − ) − δ(Ct − ))(Fz cos(Ct) + Fy sin(Ct)), 2 2 n
(8)
˙ φ(t)
which has an additional factor of CFx , which accumulates (in time 2τc ) to first order an evolution 4πFx − π 5 Fx
2π n Fy ,
which for an inhomogeneity of = .05 and n = 6, corresponds to an evolution
− π3 Fy which limits the transfer efficiency in presence of inhomogeneity.
The inhomogeneity factor can be cancelled by applying the following compensated pulse sequence called FPR, Four Pulse Recoupling. π 3π π 3π ( )0◦ ( )φ◦ ( )180◦ +φ◦ ( )180◦ 2 2 2 2 N
(9)
Then in the modulation frame of the phase φ(t), the rf-field Hamiltonian takes the form H rf (t) = C(1 + )Fx −
π Cπ δ(Ct − ) Fz , n 2 ˙ φ(t)
for t ∈ [0, τc ] followed by 5
(10)
H rf (t) = −C(1 + )Fx +
π Cπ δ(Ct − ) Fz . n 2
(11)
˙ φ(t)
In the interaction frame of the irradiation along x axis, with the strength C for time τc and −C for next τc , the rf-field Hamiltonian of the spin system transforms to
(t) = CFx − HIrf a
π Cπ δ(Ct − )(Fz cos(Ct) + Fy sin(Ct)), n 2
(12)
˙ φ(t)
followed by
HIrf (t) = −CFx + b
π Cπ δ(Ct − )(Fz cos(Ct) − Fy sin(Ct)). n 2
(13)
˙ φ(t)
Eq. (12, 13) accumulate an integral only when delta function peaks at Ct =
π 2.
The net rf
rotation is therefore − 2π n Fz and ±CFx cancel to first order. See, figure 1D. For n = 6, the compensated FPR pulse sequence takes the form
π 3π π 3π ( )0 ( )30◦ ( )210◦ ( )180◦ 2 2 2 2
(14)
as opposed to uncompensated FPR in Eq. (6). We have calculated
τc 0
HIrf (t)dt and a
τc 0
HIrf (t)dt, which are first order contributions to generb
ated rotation by time varying Hamiltonian HIrf (t) and HIrf (t). We can calculate the second order a b terms by writing evolution of HIrf (t) and HIrf (t) as a b
Ua (t) = exp(−i
Ub (t) = exp(i
3π π π Fx ) exp(i Fy ) exp(−i Fx ), 2 n 2
3π π π Fx ) exp(i Fy ) exp(i Fx ). 2 n 2
ωr Therefore, we prepare to first order, effective Hamiltonian − 2τ1c 2π n Fy = − 2 Fy . The second
order Hamiltonian is of order π2 ωr Fz . We have a recoupling field along − ω2r Fy + π2 ωr Fz . 6
We evaluate the effect of recoupling field on the coupling Hamiltonian. By transforming into the interaction frame of irradiation along x axis which alternates between CFx and −CFx , we prepare HIrf (τ ) and HIrf (τ ) every τc units of time. The coupling Hamiltonian alternates between a b DD HI+ (t) =
3 3 ωIS (t)(Iz Sz + Iy Sy ) + ωIS (t)((Iz Sz − Iy Sy ) cos(2Cτ ) + (Iz Sy + Iy Sz ) sin(2Cτ )), (15) 2 2
and
DD HI− (t) =
3 3 ωIS (t)(Iz Sz + Iy Sy ) + ωIS (t)((Iz Sz − Iy Sy ) cos(2C(−τ )) + (Iz Sy + Iy Sz ) sin(2C(−τ )). 2 2 (16)
Now, transforming the coupling Hamiltonian in Eq. 15 and Eq. 16, into the interaction frame of the rf-field Hamiltonian HIrf , we act with an effective recoupling field − ω2r Fy + π2 ωr Fz , that points in unit direction −m, ˆ where m ˆ = cos θˆ y − sin θˆ z and θ is of order π. We only retain terms that give static contribution to the effective Hamiltonian, i.e., terms oscillating with frequency 2C are dropped and the residual Hamiltonian takes the form (neglecting the terms oscillating at frequency 2ωr as they are not recoupled) HII (t) = κh {cos(ωr t) cos(ωr t + γ)(In Sn − Ix Sx ) + sin(ωr t) cos(ωr t + γ)(In Sx + Ix Sn )},
(17)
which averages to ¯ II = κh {cos(γ)(In Sn − Ix Sx ) − sin(γ)(In Sx + Ix Sn )}, H 2 where κh =
3 √ b 4 2 IS
(18)
sin(2β), where bIS is the dipole coupling constant and n ˆ is the effective direction n ˆ = cos θ zˆ + sin θ yˆ,
and m ˆ ×n ˆ =x ˆ. For small , n ˆ = zˆ and m ˆ = yˆ. It takes time
2π κh
to transfer Iy → Sy , under the
above coupling Hamiltonian, where β is chosen at a nominal value. The rf-interaction frame, HIrf prepares an effective field − ω2r Fy , every τc units of time. Transformation of the coupling Hamiltonian can be evaluated every τc units of time [9], by realizing that T
kτc U (t)JU (t)dt = k k−1τc U ((k − 1)τc )Uk (t)JUk (t)U ((k − 1)τc )dt, where U ((k − 1)τc ) is the 0 interaction frame propagator at time (k − 1)τc and is same as evolution under effective field − ω2r Fz at this time, and Uk (t) evolves in time [(k − 1)τc , kτc ] with Uk (t) = I + o( ω2r τc ). For ωr C, we can 7
neglect the second factor and assume we evolve the coupling Hamiltonian under an effective field, − ω2r Fy . We can add more details to the above analysis, and consider the coupling Hamiltonian cos(ωr t + γ)2Iz Sz = cos(ωr t + γ){(Iz Sz − Ix Sx ) + (Iz Sz + Ix Sx )}. The zero quantum term (Iz Sz + Ix Sx ) doesn’t evolve under effective field and averages out. We can write
cos(ωr t + γ)(Iz Sz − Ix Sx ) = +
exp(j(ωr t + γ)) + exp(−j(ωr t + γ)) {(Iz Sz − Ix Sx ) + j(Iz Sx + Ix Sz )} 4 exp(j(ωr t + γ)) + exp(−j(ωr t + γ)) {(Iz Sz − Ix Sx ) − j(Iz Sx + Ix Sz )}. 4
In the rf-interaction frame, HIrf , we have a pulse at time {(Iz Sz − Ix Sx ) ± j(Iz Sx + Ix Sz )} → exp(∓j
π 2n
which transforms,
2π ){(Iz Sz − Ix Sx ) ± j(Iz Sx + Ix Sz )}. n
Then over τc , J cos(ωr t + γ)(Iz Sz − Ix Sx ) averages to Ix Sz ) sin γ } where γ = γ −
τc 4 ,
π Jτc sin( n ) {(Iz Sz π 2 n
− Ix Sx ) cos γ − (Iz Sx +
and this is the average Hamiltonian every τc units of time.
In Fig. 2, we simulated the performance of compensated and uncompensated FPR for C = 60 kHz and ωr = 10 kHz which gives n = 6 and φ = 30◦ .
2.1
Offset performance of FPR
The FPR pulse sequence is broadband as a large value of C averages out the chemical shift [11]. We can understand the effect of chemical shift offset on the recoupling field. In presence of a chemical shift Δω on spin S, we have in the interaction frame of the rf phase the Hamiltonians ˙ z and ΔωSz − CSx + φS ˙ z . We write these as ω ˙ z and ω ˙ z where ΔωSz + CSx − φS ˜ Sx − φS ˜ Sx + φS √ x = x cos θ + z sin θ and x = −x cos θ + z sin θ and ω ˜ = C 2 + Δω 2 , for cos θ = C/˜ ω , see Fig. 3. By proceeding in frame CSx and CSx , we create Hamiltonians HIrf (t) and HIrf (t) whose a b evolution is Ua (t)
=
exp(−i
3τc π τc Δ˜ ω Sx ) exp(i (cos θSy + sin θSx )) exp(−i Δ˜ ω Sx ) 4 n 4
and
8
Build up curve for homonuclear uncompensated FPR 0.8 0.7
transfer efficiency
A
0.6 0.5 a
0.4 0.3 0.2 b 0.1 0 c −0.1
0
0.5
1 time (ms)
1.5
2
Build up curve for homonuclear compensated FPR 0.8 0.7
B
transfer efficiency
0.6 0.5 b c
0.4
a
0.3 0.2 0.1 0 −0.1
0
0.5
1 time (ms)
1.5
2
offset performace of FPR 0.8 0.7
C
transfer efficiency
0.6 0.5
a b
0.4
c
0.3
d
0.2 0.1 0 −0.1
0
0.5
1 time (ms)
1.5
2
Figure 2: Fig. A shows the build up of transfer of magnetization for a ideal two spin system 13 C-13 C on a 750 MHz (proton frequency) static field, using the FPR pulse unit with C = 6ωr , n = 6 with φ = 30◦ and no compensation. Simulation uses internuclear distance of 1.52A◦ and powder avaraging. a, b, c corresponds to inhomogeneity value of 0, .02, .05 respectively. Fig. B shows the build up of basic 13 C-13 C correlation experiment using the compensated FPR pulse unit with φ = 30◦ . a, b, c corresponds to inhomogeneity value of 0, .02 and .05 respectively. In practice, the rf-inhomogeneity results in a weighted sum of these different . Offset of spin pair is assumed to be on resonance. Fig. C shows the build up of basic 13 C-13 C correlation experiment using the compensated FPR pulse unit with φ = 30◦ with different chemical shifts of the spin pair. a, b, c, d, corresponds to chemical shift difference of 0, 8, 16 and 24 kHz with carrier in the center. In Fig. C, we include a CSA value of 11 and 80 ppm for the two spins respectively with anisotropy parameter .3 and .9, representing the Cα -CO spin pair in Glycine respectively. Simulation were performed with spinevolution [10] software.
9
z z
z θ
θ x
x θ
θ x
Figure 3: Fig. shows effective directions x , z , x and z in presence of chemical shift offset Δω.
Ub (t) = exp(−i
3τc π τc Δ˜ ω Sx ) exp(i (cos θSy − sin θSx )) exp(−i Δ˜ ω Sx ) 4 n 4
where Δ˜ ω=ω ˜ − C. Using the approximation x ∼ −x , we get Then Ub Ua ∼ exp(i
2π (cos θSy + sin θSx )) n
Therefore we have a recoupling field of strength
ωr 2
even in presence of offset. Therefore FPR
is quite broadband pulse sequence. It is worthwhile to compare the effective field of FPR with effective field of TPR [1]. The TPR pulse sequence takes the form
π π π π ( )0◦ (π) φ◦ ( )0◦ ( )180◦ (π)180◦ − φ◦ ( )180◦ 2 2 2 2 2 2 N where φ =
π n.
Proceeding analogously to above we find
10
Ua (t)
=
τc π τc Δ˜ ω Sx ) exp(i (cos θSy − sin θSx )) exp(−i Δ˜ ω Sx ) 4 2n 2 τc π ω Sx ) exp(i (cos θSy + sin θSx )) exp(−i Δ˜ 2n 4 exp(−i
and
τc π τc Δ˜ ω Sx ) exp(i (cos θSy + sin θSx )) exp(−i Δ˜ ω Sx ) 4 2n 4 τc π ω Sx ) exp(i (cos θSy − sin θSx )) exp(−i Δ˜ 2n 4
Ub (t) = exp(−i
Then Ub Ua ∼ exp(i
2π cos θSy ) n
We prepare an recoupling field − cos θ ω2r Sy . Due to presence of offset, we deviate from the normal recoupling field of
ωr 2
by an amount
Δω 2 2C 2 (ωr /2).
Hence FPR pulse sequence performs better
than TPR in presence of chemical shift offsets. See Fig. 4. We can evaluate the effect of FPR pulse sequence on the chemical shift anisotropy in the interaction frame of irradiation along x axis which alternates between CFx and −CFx . The chemical shift anisotropy of spin I can be expressed as (ωI±1 exp(±iωr t) + ωI±2 exp(±i2ωr t))Iz , we can decompose the operator Iz = I + + I − , where I ± = Iz ± iIy , where exp(−iCIx t)I ± exp(+iCIx t) = exp(±iCt)(I ± ). Using this, For n = 6, the CSA averages every two rotor periods, where TPR toggles between ±C every τc . Proceeding to interaction frame, HIrf , by acting with an effective field − ω2r Fy , we find that for example n = 6, the CSA averages to zero in first order.
3
Phase Matching and Heteronuclear Recoupling
Consider two coupled heteronuclear spins I and S under magic angle spinning condition [15]. The spins are irradiated with rf fields at their Larmor frequencies along say the x direction. In a doublerotating Zeeman frame, rotating with both the spins at their Larmor frequency, the Hamiltonian of the system takes the form H(t) = ωI (t)Iz + ωS (t)Sz + ωIS (t)2Iz Sz + H rf (t), 11
(19)
Offset performace of FPR vs TPR 0.7 0.6
transfer efficiency
0.5 0.4
a
0.3 0.2
b
0.1 0 0
0.5
1 time (ms)
1.5
2
Figure 4: Fig. shows the build up curve for transfer of magnetization I → S, using homonuclear FPR (a) and TPR (b) for ωr = 10 kHz and n = 6, C = 60 kHz, and φ = 30◦ , with offset ΔωI = −15 kHz, and ΔωS = 15 kHz.
where ωI (t), ωS (t), and ωIS (t) represent time-varying chemical shifts for the two spins I and S and the coupling between them, respectively. These interactions can be expressed as a Fourier
2 series ωλ (t) = m=−2 ωλm exp(imωr t), where ωr is the spinning frequency (in angular units), while the coefficients ωλ , (λ = I, S) reflect the dependence on the physical parameters like the isotropic chemical shift, anisotropic chemical shift, the dipole-dipole coupling constant and through this the internuclear distance [16]. Consider the rf irradiation on heteronuclear spin pair, where amplitude on spin I and S is chosen as Aλ (t) =
C 2π
(λ = I, S) such that C = nωr , and phase modulation, given by pulse sequence
( π2 )0◦ (3π)φ◦1 ( π2 )0◦ and ( π2 )0◦ (3π)φ◦2 ( π2 )0◦ respectively on two channels. Here we choose φ1 = φ2 =
π 2n .
3π 2n
and
This gives φ˙1 (t)
=
φ˙2 (t)
=
π 7π 3Cπ (δ(Ct − ) − δ(Ct − )) 2n 2 2 π 7π Cπ (δ(Ct − ) − δ(Ct − )) 2n 2 2
(20) (21)
where phase jump is three halves and half of the modulation considered in previous section. As a result, the effective fields prepared on two channels will be 12
3ωr 4
and
ωr 4 ,
with average
ωr 2 ,
which will
Buildup curve for heteronuclear uncompensated FPR 0.7 0.6 0.5 b transfer efficiency
A
a
0.4 0.3 0.2 c
0.1 0 −0.1
0
2
4
6
8
10
time (ms) build up curve for compensated heteronuclear FPR 0.8 0.7 0.6 transfer efficiency
B
0.5 a
0.4 b
0.3
c 0.2 0.1 0 −0.1
0
2
4
6
8
10
time (ms) offset performance of hetronuclear FPR 0.8 a b
0.7
C
transfer efficiency
0.6
c
0.5 0.4 0.3 0.2 d 0.1 0 0
2
4
6
8
10
time (ms)
Figure 5: Fig. A shows the build up of 15 N to 13 C magnetization transfer on a 750 MHz (proton frequency) static field, using the FPR pulse unit with C = 40 kHz and ωr = 10 kHz and n = 4 with φ1 = 67.5◦ and φ2 = 22.5◦ on 13 C and 15 N respectively with no compensation. a, b, c corresponds to inhomogeneity value of 0, .02 and .05 on Carbon channel. Fig. B shows the build up using the FPR pulse unit with compensation. Fig. C shows the build up of basic -15 N-13 C correlation experiment using the FPR pulse unit, with different chemical shift of the 13 C spin. a, b, c, d corresponds to chemical shift of 0, 4, 8, 12 kHz of the 13 C spin respectively.
13
recouple a double quantum Hamiltonian 1 . To mitigate the effect of inhomogeneity of rf-field, we use a compensating pulse sequence. In nutshell the pulse sequence on channels I and S (say C 13 and N 15 ) is following respectively.
3π π 3π π ( )0◦ ( )φ◦1 ( )180◦ +φ◦1 ( )180◦ 2 2 2 2
(22)
3π π 3π π ( )0◦ ( )φ◦2 ( )180◦ +φ◦2 ( )180◦ 2 2 2 2
(23)
In the frame of the rf-field along x axis, that toggles between ±C(Ix + Sx ), the coupling Hamiltonian is averaged to
DD HI± (t) = ωIS (t)(Iz Sz + Iy Sy ) + ωIS (t)((Iz Sz − Iy Sy ) cos(2Cτ ) ± (Iz Sy + Iy Sz ) sin(2Cτ )), (24)
With effective rf-field − 3ω4 r Iy −
ωr 4 Sy
written as −ωr
(Iy +Sy ) 2
−
ωr (Iy −Sy ) , 2 2
the coupling Hamilto-
nian is further averaged to
¯ II = κd {(Iz Sz − Ix Sx ) cos(γ) − (Iz Sx + Ix Sz ) sin(γ)} . H
(25)
We call the above experiments heteronuclear FPR. Fig. 5 shows simulation results for a heteronuclear magnetization transfer in
15
N-13 C heteronu-
clear system for C = 40 kHz and ωr = 10 kHz with n = 4. This gives φ1 = 67.5◦ and φ2 = 22.5◦ .
4
Experimental Results
All experiments were performed on a 750 MHz spectrometer (1 H Larmor frequency of 750 MHz) equipped with a triple resonance 3.2 mm MAS probe. Uniformly
13
C labeled sample of Glycine and
uniformly 13 C, 15 N -labeled sample of Alanine were used in the full volume of standard 3.2 mm rotor at ambient temperature for homonuclear and heteronuclear experiments respectively. Uniformly 13
C,
15
N -labeled sample of MLF was used for both homonuclear and heteronuclear recoupling
1 When
gyromagnetic ratio of two spins are of different sign, we recouple a zero quantum Hamiltonian.
14
experiments. The experiments used 2s recycling delay. In all experiments, CW decoupling of 120 kHz is used on protons. All 2D experiments were processed using TPPI processing. Fig. 6A shows the build up curve for transfer of magnetization, for
13
Cα to
13
CO every rotor
period, N = 3 (N as in Eq. 9), with FPR as the recoupling element. The experiment uses an initial ramped CP for 1 H to
13
C cross polarization. The FPR recoupling block is designed for a
nominal power of 60 kHz as described in the text and spinning speed of 10 kHz. This gives n = 6 and φ = 30◦ . Fig. 6B shows a
13
Cα -13 CO 2D correlation spectrum obtained using the FPR as the
recoupling element. Mixing time is 10 rotor periods. The experiment used 1024 points in direct and 512 in indirect dimension with spectral width of 220 ppm in both dimensions. The carrier is placed in center of the two resonances at 119 ppm during the transfer. 4 scans were collected for every t1 increment. Fig. 7A shows the build up of transfer of magnetization, shown every two rotor period, N = 4, for the
15
N →13 Cα experiment with FPR as the recoupling element. The FPR recoupling is done
at spinning speed of 10 kHz and nominal power of 40 kHz on carbon and nitrogen as described in the text. This gives n = 4 and φ1 = 67.5◦ (on initial ramped CP for 1 H to for
13
15
13
C) and φ2 = 22.5◦ (on
15
N). The experiment uses an
N cross polarization. Bottom panel of Fig. 7, shows the 2D spectrum
Cα -15 N experiment with FPR as the recoupling element. The magntization precesses on
15
N
during indirect evolution. Mixing time is 28 rotor periods. The experiment used 1024 points in direct and 256 in indirect dimension with spectral width of 220 ppm in dimension. The carrier is placed on Fig. 8 shows the 2D
13
15
N and
13
13
Cα and 100 ppm in
15
N
Cα resonance during transfer.
C correlation in tripeptide MLF obtained using FPR as recoupling
element. Experiment is done at 10 kHz spinning, and rf-power C = 60 kHz. This gives n = 6 and φ = 30◦ . The carrier is placed in center of carbon spectrum at 119 ppm. CW decoupling of 120 kHz is used on protons. Mixing time corresponds to N = 30 or 10 rotor periods. The experiment uses an initial ramped CP for 1 H to
13
C cross polarization. The experiment used 1024 points in direct
and 1024 in indirect dimension with spectral width of 220 ppm in both dimensions. 16 scans were collected for every t1 increment. Fig. 9 shows the 2D
15
N-13 C correlation in tripeptide MLF obtained using TPR as recoupling
element. Experiment is done at 10 kHz spinning, and rf-power C = 40 kHz on carbon and nitrogen channels. This gives n = 4 and φ1 = 67.5◦ (on
13
C) and φ2 = 22.5◦ (on
15
15
N). The carbon carrier
is placed in center at 119 ppm. Mixing time corresponds to N = 56 or 28 rotor periods. The experiment used 1024 points in direct and 512 in indirect dimension with spectral width of 220 ppm in carbon and 100 ppm in nitrogen dimension. 8 scans were collected for every t1 increment.
5
Conclusion
In this paper we introduced a class of recoupling pulse sequences, which rest on the principle of second oscillating field [1, 3, 9, 11, 12, 17]. A strong field is used to eliminate chemical shifts and make the sequence broadband. Furthermore this strong rf-field is used to demodulate a second oscillating field which performs recoupling. In our design, the second oscillating field comes about by principled phase changes which are described in the paper. The recoupling sequences presented in the paper for homonuclear and heteronuclear spin systems are broadband and robust to rf-inhomogeneity. We compared the work in this paper with TPR pulse sequence in [1], where we also produce an effective
ωr 2
field along y axis. We found that the effective recoupling field in FPR is more robust
to resonance offsets as compared to TPR, which makes the sequence more broadband.
6
Acknowledgement
The authors would like to thank the HFNMR lab facility at IIT Bombay , funded by RIFC, IRCC, where the data was collected.
References [1] J. Lin, R.G. Griffin, N.C. Nielsen and N. Khaneja, Three pulse recoupling and phase jump matching, J. Magn. Reson. 263, 172-183 (2016). [2] M.H. Levitt, Symmetry-Based Pulse Sequences in Magic-Angle Spinning Solid-State NMR, Encyclopedia of Nuclear Magnetic Resonance, Volume 9, pp 165-196, Edited by David M. Grant and Robin K. Harris (2002). [3] J. Lin, R.G. Griffin, and N. Khaneja, Recoupling in solid state NMR using γ prepared states and phase matching, J Magn Reson. 2011 Oct; 212(2): 402411. [4] S. J. Opella, NMR and membrane proteins, Nat. Struct. Biol. 4, 845-848 (1997). 16
[5] R. G. Griffin, Dipolar recoupling in MAS spectra of biological solids, Nat. Struct. Biol. 5, 508-512 (1998). [6] F. Castellani, B. van Rossum, A. Diehl, M. Schubert, K. Rehbein, and H. Oschkinat, Structure of a protein determined by solid-state magic-angle-spinning NMR spectroscopy, Nature 420, 98-102 (2002). [7] A. T. Petkova, Y. Ishii, J. J. Balbach, O. N. Antzutkin, R. D. Leapman, F. Deglaglio, and R. Tycko, A structural model for alzheimer’s β-amyloid fibrils based on experimental constraints from solid state NMR, Proc. Natl. Acad. Sci. 99, 16742-16747 (2002). [8] C. P. Jaroniec, C. E. MacPhee, V.S. Baja, M.T. McMahon, C.M. Dobson, and R. G. Griffin, High-resolution molecular structure of a peptide in an amyloid fibril determined by magic angle spinning NMR spectroscopy, Proc. Natl. Acad. Sci. 101, 711-716 (2004). [9] J. Lin, “Solid State NMR Experiments with powder dephased states and phase matching.” Phd Thesis, School of Engineering and Applied Sciences, Harvard (2010). [10] M. Veshtort and R. G. Griffin, SPINEVOLUTION: A powerful tool for the simulation of solid and liquid state NMR experiments, J. Magn. Reson., 178, 248-282 (2006). [11] G. De Pa¨epe, M. J. Bayro, J. Lewandowski, R. G. Griffin, Broadband homonuclear correlation spectroscopy at high magnetic fields and MAS frequencies, J. Amer. Chem. Soc. 128, 17761777(2006). [12] G. De Pa¨epe, J. Lewandowski, R. G. Griffin, Spin dynamics in the modulation frame: Application to homonuclear recoupling in magic angle spinning solid state NMR, J. Chem. Phys. 128, 124503 (2008). [13] J. Lin, M. Bayro, R. G. Griffin, and N. Khaneja, Dipolar recoupling in solid state NMR by phase alternating pulse sequences, J. Mag. Reson 197, 145-152 (2009). [14] N. C. Nielsen, H. Bildsøe, H. J. Jakobsen, and M. H. Levitt, Double-quantum homonuclear rotary resonance: Efficient dipolar recovery in magic-angle spinning nuclear magnetic resonance, J. Chem. Phys. 101, 1805-1812 (1994). [15] J. Schaefer, R. A. McKay, and E. O. Stejskal, Double-cross-polarization NMR of solids, J. Magn. Reson. 34, 443-447 (1979). [16] M. Duer, Solid State NMR Spectroscopy, Blackwell Publishing, (2000). 17
[17] A.B. Nielsen, L.A. Straaso, A.J. Nieuwkoop, C.M. Rienstra, M. Bjerring, and N.C. Nielsen, Broadband Heteronuclear Solid-State NMR Experiments by Exponentially Modulated Dipolar Recoupling without Decoupling J. Phys. Chem. Lett., 2010, 1 (13), pp 1952-1956.
18
Figure 6: Fig. A shows the build up curve for transfer of magnetization, from 13 Cα to 13 CO in a Glycine, after every N = 3 blocks, at 10 kHz spinning, and rf-power C = 60 kHz. This gives n = 6 and φ = 30◦ . Fig. B shows corresponding 13 Cα -13 CO 2D correlation spectrum obtained using the FPR as the recoupling element. Mixing time is 10 rotor periods. The carrier is placed in center of two resonances.
19
Figure 7: Fig. A shows the build up of transfer of magnetization, shown every N = 4 blocks, at 10 kHz spinning, for the 15 N →13 Cα experiment, in uniformly labelled sample of Alanine, with FPR as the recoupling element as described in text. The rf power used is C = 40 kHz, giving n = 4 and φ1 = 67.5◦ (on 13 C) and φ2 = 22.5◦ (on 15 N) . Fig. B shows the 2D spectrum for 15 N-13 Cα experiment with FPR as the recoupling element. Mixing time is 28 rotor periods. The magntization precesses on 15 N during indirect evolution. The carrier is placed on 15 N and 13 Cα resonance.
20
Figure 8: Fig. shows the 2D 13 C correlation in tripeptide MLF obtained using FPR as recoupling element. Experiment is done at 10 kHz spinning, and rf-power C = 60 kHz. This gives n = 6 and φ = 30◦ . The carrier is placed in center of carbon spectrum at 119 ppm. CW decoupling of 120 kHz is used on protons. Mixing time corresponds to N = 30 or 10 rotor periods.
Figure 9: Fig. shows the 2D 15 N-13 C correlation in tripeptide MLF obtained using FPR as recoupling element. Experiment is done at 10 kHz spinning, and rf-power C = 40 kHz on carbon and nitrogen channels. This gives n = 4 and φ1 = 67.5◦ and φ2 = 22.5◦ . The carbon carrier is placed in center of carbon spectrum at 119 ppm. CW decoupling of 120 kHz is used on protons. Mixing time corresponds to N = 56 or 28 rotor periods.
21
1. We present a family of broadband recoupling pulse sequences, called four pulse recoupling. 2. These pulse sequences can be employed for both homonuclear and heteronuclear recoupling experiments. 3. We show these sequences are robust to rf-inhomogeneity and chemical shift dispersion. 4. We present solid state NMR experiments that use four pulse recoupling.
1
B
A
φ
φ 7 τ 4 c 1 τ 4 c
C
7 τ 2τ 4 c c
0 1 1 0 0 1
00 11 11 00 00 11
1 τ 4 c
t
2τ
c
Fz
Fz 1 0 1 0
D 00 11 11 00 00 11
Fy
1
00 11 11 00 00 11
00 11 11 00 00 11
F