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Two pulse recoupling
Journal of Magnetic Resonance 281 (2017) 162–171
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Two pulse recoupling Navin Khaneja a,⇑, Ashutosh Kumar b a b
Department of Electrical Engineering, IIT Bombay, Powai 400076, India Department of Biosciences and Bioengineering, IIT Bombay, Powai 400076, India
a r t i c l e
i n f o
Article history: Received 21 April 2017 Revised 2 June 2017 Accepted 3 June 2017
Keywords: Recoupling Chemical shifts Broadband rf-inhomogeneity Hartmann-Hahn
a b s t r a c t The paper describes a family of novel recoupling pulse sequences in magic angle spinning (MAS) solid state NMR, called two pulse recoupling. These pulse sequences can be employed for both homonuclear and heteronuclear recoupling experiments and are robust to dispersion in chemical shifts and rfp, inhomogeneity. The homonuclear pulse sequence consists of a building block ðpÞ/ ðpÞ/ where / ¼ 4n and n is number of blocks in a rotor period. The recoupling block is made robust to rf-inhomogeneity by extending it to ðpÞ/ ðpÞ/ ðpÞpþ/ ðpÞp/ . The heteronuclear recoupling pulse sequence consists of a p and n is number building block ðpÞ/1 ðpÞ/1 and ðpÞ/2 ðpÞ/2 on channel I and S, where /1 ¼ 38np ; /2 ¼ 8n of blocks in a rotor period. The recoupling block is made robust to rf-inhomogeneity by extending it to ðpÞ/1 ðpÞ/1 ðpÞpþ/1 ðpÞp/1 and ðpÞ/2 ðpÞ/2 ðpÞpþ/2 ðpÞp/2 on two channels respectively. The recoupling pulse sequences mix the z magnetization. Experimental quantification of this method is shown for 13Ca-13CO homonuclear recoupling in a sample of Glycine and 15N-13Ca heteronuclear recoupling in Alanine. Application of this method is demonstrated on a sample of tripeptide N-formyl-[U-13C,15N]Met-Leu-Phe-OH (MLF). Compared to R-sequences (Levitt, 2002), these sequences are more robust to rf-inhomogeneity and give better sensitivity, as shown in Fig. 3. Ó 2017 Elsevier Inc. All rights reserved.
1. Introduction 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 difficult to analyze using conventional atomic-resolution structure determination methods, including liquid-state NMR and X-ray crystallography [1–6]. 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. To put this work in proper context, and motivate the proposed new methodology, we look at development of dipolar recoupling. The work of Tycko [7–9] on DRAMA initiated homonuclear dipolar recoupling methods in solids. This was followed by methods like Rotational Resonance [10,11] and RFDR [12]. Later came gamma encoded recoupling in HORROR [13], and its adiabatic version ⇑ Corresponding author. E-mail address: [email protected] (N. Khaneja). http://dx.doi.org/10.1016/j.jmr.2017.06.004 1090-7807/Ó 2017 Elsevier Inc. All rights reserved.
DREAM [14]. Further developments include, DRAWS [15] and MELODRAMA [16]. Subsequently, there was development of C7 [17], POSTC7 [19], SPC5 [18] and CMR7 [20]. Recently Levitt [24] and co-workers have further developed symmetry based pulse sequences. Some new work in recoupling includes, CMRR [28,29], phase alternating recoupling [25] and most recently TPR and FPR recoupling [21,22]. If we study sequences like C7, POSTC7, SPC5, CMR7, Symmetry sequences, CMRR and TPR and FPR recoupling, we find they have a common design principle. A strong rf-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. The second oscillating field comes about by principled phase changes in these sequences. To add to this family, we propose in this paper, a new two phase modulated recoupling, we call TOPR. The sequence is interesting from simplicity of its design for broadband recoupling and robustness to rf-inhomogeneity. Compared to symmetry based R-sequences, [24] these sequences are more robust to rfinhomogeneity and give better sensitivity, as shown in Fig. 3. Furthermore, TOPR adds to our repertoire of recoupling sequences and to our understanding of recoupling which is fundamental to solid state NMR. The paper is organized as follows. In Section 2, we describe TOPR, a novel approach to homonuclear recoupling that recouple
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dipolar coupled spins under Magic angle spinning (MAS) experiments. TOPR experiments are broadband and robust to rfinhomogeneity. This work extends recently developed techniques for broadband homonuclear recoupling as reported in the [21,25,26,28–30]. 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 [31]). Section 4 describes experimental verification of the proposed techniques. We conclude in Section 5 by comparing TOPR with state of the art pulse sequences. 2. TOPR in homonuclear spins Consider two homonuclear spins, I and S, under magic angle spinning condition [13]. In a rotating frame, rotating with both the spins at their common Larmor frequency, the Hamiltonian of the spin system takes the form
HðtÞ ¼ xI ðtÞIz þ xS ðtÞSz þ xIS ðtÞð3Iz Sz I SÞ þ 2pAðtÞ ðcos /ðtÞF x þ sin /ðtÞF y Þ;
ð1Þ
where the operator F x ¼ Ix þ Sx , and xI ðtÞ and xS ðtÞ represent the chemical shift for the spins I and S respectively and xIS ðtÞ represents the time varying couplings between the spins under magicangle spinning. These interactions may be expressed as a Fourier series,
xk ðtÞ ¼
2 X
xmk expðimxr tÞ;
ð2Þ
m¼2
ðpÞ/ ðpÞ/ ; p . This means ampliwith n such blocks in a rotor period and / ¼ 4n tude A of pulses is 2pA ¼ C ¼ nxr and phase /ðtÞ alternates p and / every sc ¼ p units of time as shown in between / ¼ 4n 2 C Fig. 1, where sc is duration of building block. We can think of phase as starting from zero and jumping to value / at t ¼ 0 and then jumping to / at t ¼ s2c and then returning to 0 at t ¼ sc and this cycle continues as shown in Fig. 1A. Over one building block, the rate of change of phase /_ takes the
form,
Cp ðdðCtÞ dðCt pÞ þ dðCt 2pÞÞðF z cosðCtÞ þ F y 4n |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} _ /ðtÞ
sinðCtÞÞ;
ð5Þ
which accumulates an integral only when delta function peaks at Ct ¼ 0; Ct ¼ p and Ct ¼ 2p. The net rf rotation generated by HrfI ðtÞ p at is therefore pn F z , which corresponds to a phase advance of 4n p Ct ¼ 0 and Ct ¼ 2p and phase decrement of 2n at Ct ¼ p (a negative phase decrement is multiplied by cosðpÞ, which makes it a positive accumulation). See Fig. 1C. A net rotation of pn in time sc ¼ snr corresponds to a net effective field of x2r along z direction. This field recouples, as field of strength x2r is a recoupling field. As an example, if we choose C ¼ 6xr , with n ¼ 6, TOPR takes the form
ðpÞ7:5 ðpÞ7:5 : When pulse amplitude has inhomogeneity, such that rf-field strength instead of C is Cð1 þ Þ, then, in the modulation frame of the phase /ðtÞ, the rf-field Hamiltonian takes the form
_ z: Hrf ðtÞ ¼ Cð1 þ ÞF x /F
ð6Þ
In the interaction frame of the irradiation along x axis, with the strength C, the rf-field Hamiltonian of the spin system transforms to
HrfIa ðtÞ ¼ CF x
Cp ðdðCtÞ dðCt pÞ þ dðCt 2pÞÞðF z cosðCtÞ 4n |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
þ F y sinðCtÞÞ;
ð7Þ
which has an additional factor of CF x coming from rfinhomogeneity, which over a building block, accumulates to first order an evolution 2pF x pn F z , which for an inhomogeneity of ¼ :05 and n ¼ 6, corresponds to an evolution 10p F x p6 F z . Presence p F , limits the transfer efficiency. This inhomoof additional factor 10 x geneity factor can be canceled by following this building block with a building block of amplitude C (adding 180 to all phases). In the interaction frame of the irradiation along x axis, with the strength C, the rf-field Hamiltonian of the spin system transforms to
HrfIb ðtÞ ¼ CF x
Cp ðdðCtÞ dðCt pÞ þ dðCt 2pÞÞ 4n |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} _ /ðtÞ
ðF z cosðCtÞ F y sinðCtÞÞ;
ð8Þ
Which accumulates an integral only when delta function peaks at Ct ¼ 0; 2p and Ct ¼ p. The net rf rotation is therefore pn F z 2pF x . The CF x in HrfIb cancels CF x in HrfIa to first order.
ð3Þ
Rt R0 where dðtÞ is a delta function ( 0 f ðsÞdðsÞds ¼ t f ðsÞdðsÞds ¼ R 1 t f ðsÞdðsÞds ¼ f ð0Þ for t > 0). See Fig. 1B. 2 t In the modulation frame of the phase /ðtÞ, the rf-field Hamiltonian takes the form
_ z; Hrf ðtÞ ¼ CF x /F
HrfI ðtÞ ¼
_ /ðtÞ
where xr is the spinning frequency (in angular units), while the coefficients xk ðk ¼ I; S; ISÞ 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 [6]. 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 pulse sequence, made of building block
Cp _ /ðtÞ ¼ ½dðCtÞ dðCt pÞ þ dðCt 2pÞ; 4n
spins are averaged out. The rf-field Hamiltonian of the spin system transforms to
ð4Þ
where C is in the angular frequency units and we choose C xI ðtÞ; xS ðtÞ; xIS ðtÞ; xr . In the interaction frame of the irradiation along x axis, with the strength C, the chemical shifts of the
The C amplitude may be implemented with a two pulse element (for n ¼ 6)
ðpÞ187:5 ðpÞ172:5 : The total compensated pulse sequence takes the form
ðpÞ7:5 ðpÞ7:5 ðpÞ187:5 ðpÞ172:5 : We have calculated
R sc 0
HrfIa ðtÞdt and
R sc 0
HrfIb ðtÞdt, which are first
order contributions to generated rotation by time varying Hamiltonian HrfIa ðtÞ and HrfIb ðtÞ. We can calculate the second order terms, by writing evolution of HrfIa ðtÞ and HrfIb ðtÞ as,
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Fig. 1. Fig. A shows how phase of homonuclear recoupling sequence changes with time between / and /. Fig. B shows time derivative of phase in Eq. (3). Area under this curve gives the total phase change. Fig. C shows the dotted points (bold dots), where in Eq. (5), delta functions peak and net rotation happens, as F z is rotated in the interaction frame of CF x .
U 1 ðtÞ ¼ exp i
p F z expðipF x Þ exp i F z expðipF x Þ 2n p p expði F z Þ exp i F z 2pF x ; 4n n
U 2 ðtÞ ¼ exp i
p
Now transforming the coupling Hamiltonian in Eqs. (10) and
4n
(11) into the interaction frame of the rf-field Hamiltonian HrfI , we act on coupling Hamiltonian with an effective recoupling field x2r F z (see following paragraph), and we only retain terms that give static contribution to the coupling Hamiltonian, i.e., terms oscillating with frequency 2C are dropped and the coupling Hamiltonian takes the form (neglecting the terms oscillating at frequency 2xr as they are not recoupled)
p F z expðipF x Þ exp i F z expðipF x Þ 4n 2n p p exp i F z exp i F z þ 2pF x : 4n n
p
Therefore,
we
prepare
effective
Hamiltonians
CF x nps F z ¼ CF x sp F z ¼ CF x x2 F z , followed by the Hamiltonian CF x x2 F z . Toggling between them eliminates CF x . The second order Hamiltonian is of the order p2 xr F y . We have a recoupling field along x2 F z p2 xr F y x2 F z . r
c
HII ðtÞ ¼ jh fcosðxr tÞ cosðxr t þ cÞðIy Sy Ix Sx Þ sinðxr tÞ
r
cosðxr t þ cÞðIy Sx þ Ix Sy Þg;
r
r
r
To evaluate the effect of recoupling field on the coupling Hamiltonian, consider the compensated pulse sequence
h
ðpÞ/ ðpÞ/ ðpÞpþ/ ðpÞp/
i
N
:
ð9Þ
In the modulation frame of the rf-phase, rf-Hamiltonian alter_ z and CF x /F _ z every sc units of time, nates between CF x /F _ where / is as in Fig. 1A. By transforming into the interaction frame of irradiation along x axis, which alternates between CF x and CF x , we prepare HrfIa ðsÞ and HrfIb ðsÞ every
sc units of time. The coupling
HDD Ia ðtÞ ¼
HDD Ib ðtÞ ¼
which averages to
HII ¼
3 3 xIS ðtÞðIz Sz þ Iy Sy Þ þ xIS ðtÞððIz Sz Iy Sy Þ cosð2C sÞ 2 2 þ ðIz Sy þ Iy Sz Þ sinð2C sÞÞ;
ð10Þ
3 3 xIS ðtÞðIz Sz þ Iy Sy Þ þ xIS ðtÞððIz Sz Iy Sy Þ 2 2 cosð2CðsÞÞ þ ðIz Sy þ Iy Sz Þ sinð2CðsÞÞ:
ð11Þ
jh 2
fcosðcÞðIy Sy Ix Sx Þ þ sinðcÞðIy Sx þ Ix Sy Þg:
ð13Þ
jh ¼ 4p3 ffiffi2 bIS sinð2bÞ and bIS is the dipole coupling constant. Thus TOPR has a scaling factor of 12 compared to a standard cwhere
encoded experiment like HORROR [13]. The rf-interaction frame, HrfI prepares an effective field x2r F z , every sc units of time. Transformation of the coupling (denoted J below) under rf-propagator UðtÞ can be evaluated every sc units of time [26], by realizing that
Z
Hamiltonian alternates between
ð12Þ
0
T
U 0 ðtÞJUðtÞ dt ¼
X Z ksc k
ðk1Þsc
U 0 ððk 1Þsc Þ U 0k ðtÞJU k ðtÞ Uððk 1Þsc Þ dt:
Uððk 1Þsc Þ is the interaction frame propagator at time ðk 1Þsc and is same as evolution under effective field x2r F z at this time, and U k ðtÞ evolves in time ½ðk 1Þsc ; ksc with U k ðtÞ ¼ I þ oðx2r sc Þ. For xr C, we can neglect the second factor and assume we evolve the coupling Hamiltonian under an effective field x2r F z .
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165
TOPR pulse sequence is broadband, as a large value of C averages out the chemical shift [28]. The high power pulse sequence is made robust to rf-inhomogeneity by incrementing the phase by p every sc units of time. In Fig. 2, we simulated the performance of compensated and uncompensated TOPR for C ¼ 60 kHz and xr ¼ 10 kHz which gives n ¼ 6 and / ¼ 7:5 . We observe that compensated TOPR is much robust to rf-inhomogeneity as compared to uncompensated TOPR. It is worth pointing out that for the above choice of parameters, the uncompensated TOPR is simply the symmetry based p. R-sequence, R2412 , that has / ¼ 24 In Fig. 3A, we compare experimentally, the sensitivity of com-
pensated TOPR to symmetry based R-sequence, R2412 , for transfer of magnetization, from 13Ca to 13CO in a Glycine, at 10 kHz spinning, and rf-power C ¼ 60 kHz. We find that since TOPR is much robust to rf-inhomogeneity as compared to R2412 , we get almost twice the senstivity experimentally. To further quantify the senstivity in Fig. 3A, we provide a simulation for this 13C-13C transfer with a 10% Lorentzian rf-inhomogeneity in Fig. 3B. We find TOPR gives 50% transfer, while R2412 gives 25% transfer for the same mixing time of 10 rotor periods. 2.1. Chemical shifts We can understand the effect of chemical shift offset on the recoupling field. In the presence of chemical shift, we have in the interaction frame of the rf phase the Hamiltonians (for spin I), _ z followed by DxIz CIx /I _ z . We write these as DxIz þ CIx /I _ z ~ Ix0 /I x
_ z where x0 ¼ x cos h þ z sin h and ~ Ix00 /I x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ , see ~ ¼ C 2 þ Dx2 , for cos h ¼ C=x x ¼ x cos h þ z sin h and x and
00
Fig. 4. By proceeding in frame CIx0 and CIx00 , we create Hamiltonians HrfIa ðtÞ and HrfIb ðtÞ whose evolution is
U 1 ðtÞ ¼ exp i
s c ~ I x0 ðcos hIz0 þ sin hIx0 Þ exp i Dx 4n 2 p exp i ðcos hIz0 sin hIx0 Þ 2n p s c ~ Ix0 exp i ðcos hIz0 þ sin hIx0 Þ ; exp i Dx 2 4n
p
and
U 2 ðtÞ ¼ exp i
s p c ~ Ix00 ðcos hIz00 þ sin hIx00 Þ exp i Dx 4n 2 p exp i ðcos hIz00 sin hIx00 Þ 2n p s c ~ Ix00 exp i ðcos hIz00 þ sin hIx00 Þ : exp i Dx 2 4n
~ ¼x ~ C. where Dx Then
U 2 U 1 exp i2sc
p ~ sin hIz cos2 hIz þ Dx nsc
:
Therefore we have a recoupling field of
x x Dx2 Dx2 r ~ sin h Iz r 1 þ Dx I z : cos2 h þ Dx 2 2 2 2 DC 2C The ideal recoupling field is x2r Iz . Due to presence of offset, we deviate from it by an amount
D x2 2C 2
ðDx þ xr Þ. When Dx is negative,
then ðDx þ xr Þ is smaller, hence the pulse sequence performs better as compared to positive offsets. See Fig. 5. If instead we use the pulse sequence ðpÞ/ ðpÞ/ ðpÞ180/ ðpÞ180þ/ , we prepare an recoupling field x2r Iz . Due to presence of offset, we deviate from it by an amount
Dx2 2C 2
ðDx xr Þ. When Dx is positive,
Fig. 2. Fig. A shows the build up of transfer of magnetization for a ideal two spin system, 13C-13C, on a 750 MHz (proton frequency) static field, using the uncom pensated TOPR pulse unit with C ¼ 6xr ; n ¼ 6 with / ¼ 7:5 . Simulation uses internuclear distance of 1.52 A and powder avaraging. a; b; c; d, corresponds to inhomogeneity value of 0; :02; :05 and :1 respectively. Fig. B shows the build up basic 13C-13C correlation experiment using the compensated TOPR pulse unit, with / ¼ 7:5 . a; b; c; d, corresponds to inhomogeneity value of 0; :02 and :05 and :1 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 13C-13C correlation experiment using the compensated TOPR pulse unit with / ¼ 7:5 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 Ca-CO spin pair in Glycine respectively. Simulation were performed with spinevolution [27] software.
then ðDx xr Þ is smaller, as compared to when Dx is negative, hence the pulse sequence performs better for positive offsets. See Fig. 5.
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Fig. 3. Fig. A shows the transfer amplitude using (a) TOPR vs (b) R2412 sequence for transfer of magnetization, from 13Ca to 13CO in a Glycine at 10 kHz spinning, and rf power C ¼ 60 kHz. This gives / ¼ 7:5 . Mixing time is 10 rotor periods. Fig. B shows simulation of transfer amplitude of (a) TOPR vs (b) R2412 sequence for transfer of magnetization, from 13Ca to 13CO in a Glycine at 10 kHz spinning, and rf-power C ¼ 60 kHz. We use 10% Lorentzian for simulating rf-inhomogeneity. The chemical shift difference between spins in taken as 24 kHz, with carrier in the center. We include a CSA value of 11 and 80 ppm for the two spins respectively, with anisotropy parameter :3 and :9, representing the Ca-CO spin pair in Glycine respectively. Fig. 5. Left panel shows the build up curve for transfer of magnetization I ! S, using homonuclear TOPR ðpÞ/ ðpÞ/ ðpÞpþ/ ðpÞp/ for xr ¼ 10 kHz and n ¼ 6; C ¼ 60 kHz, and / ¼ 7:5 , with negative offset DxS ¼ 12 kHz (curve a) and positive offset DxS ¼ 12 kHz (curve b). Left panel compares the performace of ðpÞ/ ðpÞ/ ðpÞp/ ðpÞpþ/ (curve a), with ðpÞ/ ðpÞ/ ðpÞpþ/ ðpÞp/ (curve b) for positive offset DxS ¼ 12 kHz.
where
I ¼ Iz iIy ,
where
expðiCIx tÞI expðþiCIx tÞ ¼
expð iCtÞðI Þ. Using this for n ¼ 6, the CSA averages every two rotor periods, where TOPR toggles between C every sc . Proceeding to interaction frame of HrfI , by acting with an effective field x2r F z , we find that for example n ¼ 6, the CSA averages to zero in the first order. 3. Phase matching and heteronuclear recoupling Consider two coupled heteronuclear spins I and S under magic angle spinning condition [31]. 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 Fig. 4. Above Fig. shows the direction x0 of effective field, resulting from a x-phase rf-field of strength C and resonance offset Dx, and its orthogonal direction z0 . Similarly directions x00 and z00 result from a x phase rf-field of strength C and resonance offset Dx.
We can evaluate the effect of TOPR pulse sequence on the chemical shift anisotropy in the interaction frame of irradiation along x axis, which alternates between CF x and CF x . The chemical shift anisotropy of spin I can be expressed as ðx 1 I expð ixr tÞþ þ x 2 I expð i2xr tÞÞIz . We can decompose the operator Iz ¼ I þ I ,
HðtÞ ¼ xI ðtÞIz þ xS ðtÞSz þ xIS ðtÞ2Iz Sz þ Hrf ðtÞ;
ð14Þ
where xI ðtÞ; xS ðtÞ, and xIS ðtÞ 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 series P xk ðtÞ ¼ 2m¼2 xmk expðimxr tÞ, where xr is the spinning frequency (in angular units), while the coefficients xk ; ðk ¼ I; SÞ 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 [6]. Consider the rf irradiation on heteronuclear spin pair, where amplitude on spin I and S is chosen as Ak ðtÞ ¼ 2Cp (k ¼ I; S) such that
N. Khaneja, A. Kumar / Journal of Magnetic Resonance 281 (2017) 162–171
167
C ¼ nxr , and phase modulation, given by pulse sequence ðpÞ/1 ðpÞ/1 and ðpÞ/2 ðpÞ/2 respectively on two channels. Here p . This gives we choose /1 ¼ 38np and /2 ¼ 8n
3C p /_1 ðtÞ ¼ ðdðCtÞ dðCt pÞ þ dðCt 2pÞÞ; 8n Cp /_2 ðtÞ ¼ ðdðCtÞ dðCt pÞ þ dðCt 2pÞÞ; 8n
ð15Þ ð16Þ
where phase jump is three halves and half of the modulation considered in previous section. As a result, the effective fields prepared r and x4r , with average x2r , which will on two channels will be 3x 4 recouple a double quantum Hamiltonian. To mitigate the effect of inhomogeneity of rf-field, the basic pulse unit is accompanied with a compensating unit. In nutshell, the pulse sequence on channels I and S (say C 13 and N 15 ) is following respectively.
ðpÞ/1 ðpÞ/1 ðpÞpþ/1 ðpÞp/1 :
ð17Þ
ðpÞ/2 ðpÞ/2 ðpÞpþ/2 ðpÞp/2 :
ð18Þ
In the frame of the rf-field along x axis, that toggles between
CðIx þ Sx Þ, the coupling Hamiltonian is averaged to
HDD I ðtÞ ¼ xIS ðtÞðI z Sz þ Iy Sy Þ þ xIS ðtÞððIz Sz Iy Sy Þ cosð2C sÞ
ðIz Sy þ Iy Sz Þ sinð2C sÞÞ; With ðIz þSz Þ r 2
x to
effective
rf-field
ð19Þ r 3x I x4r Sz 4 z
written
as
xr ðIz Sz Þ 2
2
, the coupling Hamiltonian is further averaged
HII ¼ jd fðIy Sy Ix Sx Þ cosðcÞ þ ðIy Sx þ Ix Sy Þ sinðcÞg : |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
ð20Þ
We call the above experiments heteronuclear TOPR. Fig. 6 shows simulation results for a heteronuclear magnetization transfer in 15N-13C heteronuclear system for C ¼ 40 kHz and
xr ¼ 10 kHz with n ¼ 4. This gives /1 ¼ 16:875 and /2 ¼ 5:625 . 4. Experimental results All experiments were performed on a 750 MHz spectrometer (1H Larmor frequency of 750 MHz) equipped with a triple resonance 3:2 mm probe. Uniformly 13C labeled sample of Glycine and uniformly 13C, 15N-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 13C, 15N-labeled sample of MLF was used for both homonuclear and heteronuclear recoupling experiments. The experiments used 2 s recycling delay. In all experiments, CW decoupling of 120 kHz is used on protons. All 2D experiments were acquired using TPPI. Fig. 7A shows the build up curve for transfer of magnetization, for 13Ca to 13CO every rotor period, N ¼ 2 (N as in Eq. (9)), with TOPR as the recoupling element. The experiment uses an initial ramped CP for 1H to 13C cross polarization. The TOPR recoupling block is designed for a nominal power of 60 kHz as described in the text and spinning speed of 15 kHz. This gives n ¼ 4 and
/ ¼ 11:25 . Fig. 7B shows a 13Ca-13CO 2D correlation spectrum obtained using the TOPR as the recoupling element. Mixing time is 8 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 two resonances at 119 ppm during transfer. 4 scans were collected for every t 1 increment. Fig. 8A shows the build up of transfer of magnetization, shown every rotor period, N ¼ 2, for the 15N !13 Ca experiment with TOPR as the recoupling element. The TOPR recoupling is done at spin-
Fig. 6. Fig. A shows the build up of 15N to 13C magnetization transfer on a 750 MHz (proton frequency) static field, using the TOPR pulse unit with C ¼ 40 kHz and xr ¼ 10 kHz and n ¼ 4 with /1 ¼ 16:875 and /2 ¼ 5:625 on 13 C and 15N 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 TOPR pulse unit with compensation. Fig. C shows the build up of basic -15N-13C correlation experiment using the TOPR pulse unit, with different chemical shift of the 13C spin. a; b; c, corresponds to chemical shift of 0; 4; 8 kHz of the 13C spin respectively.
ning speed of 10 kHz and nominal power of 40 kHz on carbon and nitrogen as described in the text. This gives n ¼ 4 and /1 ¼ 16:875 (on 13C) and /2 ¼ 5:625 (on 15N). The experiment uses an initial ramped CP for 1H to 15N cross polarization. Bottom panel of Fig. 8 shows the 2D spectrum for 13Ca-15N experiment with
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Fig. 7. Fig. A shows the build up curve for transfer of magnetization from 13Ca to 13CO in a Glycine, after every 2N ¼ 4 blocks, at 15 kHz spinning, and rf-power C ¼ 60 kHz. This gives n ¼ 4 and / ¼ 11:25 . Fig. B shows corresponding 13Ca-13CO 2D correlation spectrum obtained using the TOPR as the recoupling element. Mixing time is 8 rotor periods. The carrier is placed in center of 13Ca-13CO resonances. Fig. C shows the pulse sequence for the 2D experiment. The phase of CP pulse on 13C is TPPI phase cycled.
TOPR as the recoupling element. The magntization precesses on 15 N during indirect evolution. Mixing time is 17 rotor periods. The experiment used 1024 points in direct and 256 in indirect dimension with spectral width of 220 ppm in 13Ca and 100 ppm in 15N dimension. The carrier is placed on 15N and 13Ca resonance during transfer. Fig. 9 shows the 2D 13C correlation in tripeptide MLF obtained using TOPR as recoupling element. Experiment is done at 15 kHz spinning, and rf-power C ¼ 60 kHz. This gives n ¼ 4 and
/ ¼ 11:25 . 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 ¼ 16 or 8 rotor periods. The experiment uses an initial ramped CP for 1H to 13C 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 t 1 increment. Fig. 10 shows the 2D 15N-13C correlation in tripeptide MLF obtained using TOPR as recoupling element. Experiment is done at 10 kHz spinning, and rf-power C ¼ 40 kHz on carbon and nitrogen channels. This gives n ¼ 4 and /1 ¼ 16:875 (on 13C) and
N. Khaneja, A. Kumar / Journal of Magnetic Resonance 281 (2017) 162–171
169
Fig. 8. Top panel shows the build up of transfer of magnetization, shown every 2N ¼ 4 blocks, at 10 kHz spinning, for the 15 N ! 13 C a experiment, in uniformly labeled sample of Alanine, with TOPR as the recoupling element as described in text. The rf power used is C ¼ 40 kHz, giving n ¼ 4 and /1 ¼ 16:875 (on 13C) and /2 ¼ 5:625 (on 15N). Panel B shows the 2D spectrum for 15N-13Ca experiment with TOPR as the recoupling element. Mixing time is 17 rotor periods. The magntization precesses on 15N during indirect evolution. The carrier is placed on 15N and 13Ca resonance.
Fig. 9. Top panel shows the 2D 13C correlation in tripeptide MLF obtained using TOPR as recoupling element. Experiment is done at 15 kHz spinning, and rf-power C ¼ 60 kHz. This gives n ¼ 4 and / ¼ 11:25 . 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 ¼ 16 or 8 rotor periods.
/2 ¼ 5:625 (on 15N). The carbon carrier is placed near CO resonances in A and Ca resonances in B. Mixing time corresponds to N ¼ 34 or 17 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 t 1 increment.
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Fig. 10. Top panel shows the 2D 15N-13C correlation in tripeptide MLF obtained using TOPR as recoupling element. Experiment is done at 10 kHz spinning, and rf-power C ¼ 40 kHz on carbon and nitrogen channels. This gives n ¼ 4 and /1 ¼ 16:875 and /2 ¼ 5:625. The carbon carrier is placed near CO resonances in A and Ca resonances in B. CW decoupling of 120 kHz is used on protons. Mixing time corresponds to N ¼ 34 or 17 rotor periods.
5. Conclusion In this paper we introduced a class of recoupling pulse sequences, which rest on the principle of second oscillating field [21,25,26,28,29,32]. 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. If we compare TOPR with TPR and FPR recoupling [21,22], we find that here, we produce a recoupling effective field along z direction, as opposed to an effective recoupling field along y direction in [21,22]. Pulse sequences presented in this paper do mixing of the z magnetization. If we compare TOPR to HORROR [13], we find the sequence in this paper is much more broadband. When we compare the sequence to C7, POSTC7, SPC5 [17,19,18], and the symmetry based sequences CNmn , [24], we find these pulse sequences also produce an efffective recoupling field along z direction, however these sequences have constant phase increments [23] unlike TOPR, where phase switches back and forth between two values. In that respect TOPR is similar to symmetry based RNmn pulse sequences [24]. If we compare the pulse sequence ðpÞ/ ðpÞ/ as appearing in
symmetry based RNmn pulse sequences [24], we point out that we have a completely different design philosophy, where we produce an effective recoupling field of strength x2r . If we consider a symmep and has n ¼ 6 blocks in try based pulse sequence R2412 , it has / ¼ 24 a rotor period and hence is our uncompensated TOPR, discussed in main text. We emphasize that our design principle is based on constructing a recoupling field of strength x2r and doesn’t rely on any symmetry principles. The work in this paper shows how symmetry based RNmn pulse sequences can be seen as synthesizing a effective recoupling field. Furthermore we present compensated pulse sequences, which make them robust to rf-inhomogeneity and gives better sensitivity. This is shown in Fig. 3. If we compare TOPR to CMRR, we find CMRR has continuous phase modulation while TOPR has only two phases. It is worthwhile to note that the square modulation of the phase as presented
in the Fig. 1 and Eq. (3) can also seen when CMRR is implemeted in r a TPPM mode [29], by synthesizing Hamiltonians CF x þ px F y and 4 sc r F , each for units of time. This will create an effective CF x px y 4 2 recoupling field of strength x2r F z . For C ¼ nxr , this is a pulse sequence with a square waveform phase modulation with phase qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 p and amplitude x n2 þ p16. Therefore this / such that tan / ¼ 4n r qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p2 , which is a similar pulse sequence is ðhÞ/ ðhÞ/ with h ¼ p 1 þ 16n 2 pulse sequence to TOPR but not the same. Furthermore TOPR is robust to rf-inhomogeneity.
Acknowledgement The authors would like to thank the HFNMR lab facility at IIT Bombay where the data was collected.
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Contents lists available at ScienceDirect
Journal of Magnetic Resonance journal homepage: www.elsevier.com/locate/jmr
Two pulse recoupling Navin Khaneja a,⇑, Ashutosh Kumar b a b
Department of Electrical Engineering, IIT Bombay, Powai 400076, India Department of Biosciences and Bioengineering, IIT Bombay, Powai 400076, India
a r t i c l e
i n f o
Article history: Received 21 April 2017 Revised 2 June 2017 Accepted 3 June 2017
Keywords: Recoupling Chemical shifts Broadband rf-inhomogeneity Hartmann-Hahn
a b s t r a c t The paper describes a family of novel recoupling pulse sequences in magic angle spinning (MAS) solid state NMR, called two pulse recoupling. These pulse sequences can be employed for both homonuclear and heteronuclear recoupling experiments and are robust to dispersion in chemical shifts and rfp, inhomogeneity. The homonuclear pulse sequence consists of a building block ðpÞ/ ðpÞ/ where / ¼ 4n and n is number of blocks in a rotor period. The recoupling block is made robust to rf-inhomogeneity by extending it to ðpÞ/ ðpÞ/ ðpÞpþ/ ðpÞp/ . The heteronuclear recoupling pulse sequence consists of a p and n is number building block ðpÞ/1 ðpÞ/1 and ðpÞ/2 ðpÞ/2 on channel I and S, where /1 ¼ 38np ; /2 ¼ 8n of blocks in a rotor period. The recoupling block is made robust to rf-inhomogeneity by extending it to ðpÞ/1 ðpÞ/1 ðpÞpþ/1 ðpÞp/1 and ðpÞ/2 ðpÞ/2 ðpÞpþ/2 ðpÞp/2 on two channels respectively. The recoupling pulse sequences mix the z magnetization. Experimental quantification of this method is shown for 13Ca-13CO homonuclear recoupling in a sample of Glycine and 15N-13Ca heteronuclear recoupling in Alanine. Application of this method is demonstrated on a sample of tripeptide N-formyl-[U-13C,15N]Met-Leu-Phe-OH (MLF). Compared to R-sequences (Levitt, 2002), these sequences are more robust to rf-inhomogeneity and give better sensitivity, as shown in Fig. 3. Ó 2017 Elsevier Inc. All rights reserved.
1. Introduction 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 difficult to analyze using conventional atomic-resolution structure determination methods, including liquid-state NMR and X-ray crystallography [1–6]. 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. To put this work in proper context, and motivate the proposed new methodology, we look at development of dipolar recoupling. The work of Tycko [7–9] on DRAMA initiated homonuclear dipolar recoupling methods in solids. This was followed by methods like Rotational Resonance [10,11] and RFDR [12]. Later came gamma encoded recoupling in HORROR [13], and its adiabatic version ⇑ Corresponding author. E-mail address: [email protected] (N. Khaneja). http://dx.doi.org/10.1016/j.jmr.2017.06.004 1090-7807/Ó 2017 Elsevier Inc. All rights reserved.
DREAM [14]. Further developments include, DRAWS [15] and MELODRAMA [16]. Subsequently, there was development of C7 [17], POSTC7 [19], SPC5 [18] and CMR7 [20]. Recently Levitt [24] and co-workers have further developed symmetry based pulse sequences. Some new work in recoupling includes, CMRR [28,29], phase alternating recoupling [25] and most recently TPR and FPR recoupling [21,22]. If we study sequences like C7, POSTC7, SPC5, CMR7, Symmetry sequences, CMRR and TPR and FPR recoupling, we find they have a common design principle. A strong rf-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. The second oscillating field comes about by principled phase changes in these sequences. To add to this family, we propose in this paper, a new two phase modulated recoupling, we call TOPR. The sequence is interesting from simplicity of its design for broadband recoupling and robustness to rf-inhomogeneity. Compared to symmetry based R-sequences, [24] these sequences are more robust to rfinhomogeneity and give better sensitivity, as shown in Fig. 3. Furthermore, TOPR adds to our repertoire of recoupling sequences and to our understanding of recoupling which is fundamental to solid state NMR. The paper is organized as follows. In Section 2, we describe TOPR, a novel approach to homonuclear recoupling that recouple
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dipolar coupled spins under Magic angle spinning (MAS) experiments. TOPR experiments are broadband and robust to rfinhomogeneity. This work extends recently developed techniques for broadband homonuclear recoupling as reported in the [21,25,26,28–30]. 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 [31]). Section 4 describes experimental verification of the proposed techniques. We conclude in Section 5 by comparing TOPR with state of the art pulse sequences. 2. TOPR in homonuclear spins Consider two homonuclear spins, I and S, under magic angle spinning condition [13]. In a rotating frame, rotating with both the spins at their common Larmor frequency, the Hamiltonian of the spin system takes the form
HðtÞ ¼ xI ðtÞIz þ xS ðtÞSz þ xIS ðtÞð3Iz Sz I SÞ þ 2pAðtÞ ðcos /ðtÞF x þ sin /ðtÞF y Þ;
ð1Þ
where the operator F x ¼ Ix þ Sx , and xI ðtÞ and xS ðtÞ represent the chemical shift for the spins I and S respectively and xIS ðtÞ represents the time varying couplings between the spins under magicangle spinning. These interactions may be expressed as a Fourier series,
xk ðtÞ ¼
2 X
xmk expðimxr tÞ;
ð2Þ
m¼2
ðpÞ/ ðpÞ/ ; p . This means ampliwith n such blocks in a rotor period and / ¼ 4n tude A of pulses is 2pA ¼ C ¼ nxr and phase /ðtÞ alternates p and / every sc ¼ p units of time as shown in between / ¼ 4n 2 C Fig. 1, where sc is duration of building block. We can think of phase as starting from zero and jumping to value / at t ¼ 0 and then jumping to / at t ¼ s2c and then returning to 0 at t ¼ sc and this cycle continues as shown in Fig. 1A. Over one building block, the rate of change of phase /_ takes the
form,
Cp ðdðCtÞ dðCt pÞ þ dðCt 2pÞÞðF z cosðCtÞ þ F y 4n |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} _ /ðtÞ
sinðCtÞÞ;
ð5Þ
which accumulates an integral only when delta function peaks at Ct ¼ 0; Ct ¼ p and Ct ¼ 2p. The net rf rotation generated by HrfI ðtÞ p at is therefore pn F z , which corresponds to a phase advance of 4n p Ct ¼ 0 and Ct ¼ 2p and phase decrement of 2n at Ct ¼ p (a negative phase decrement is multiplied by cosðpÞ, which makes it a positive accumulation). See Fig. 1C. A net rotation of pn in time sc ¼ snr corresponds to a net effective field of x2r along z direction. This field recouples, as field of strength x2r is a recoupling field. As an example, if we choose C ¼ 6xr , with n ¼ 6, TOPR takes the form
ðpÞ7:5 ðpÞ7:5 : When pulse amplitude has inhomogeneity, such that rf-field strength instead of C is Cð1 þ Þ, then, in the modulation frame of the phase /ðtÞ, the rf-field Hamiltonian takes the form
_ z: Hrf ðtÞ ¼ Cð1 þ ÞF x /F
ð6Þ
In the interaction frame of the irradiation along x axis, with the strength C, the rf-field Hamiltonian of the spin system transforms to
HrfIa ðtÞ ¼ CF x
Cp ðdðCtÞ dðCt pÞ þ dðCt 2pÞÞðF z cosðCtÞ 4n |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
þ F y sinðCtÞÞ;
ð7Þ
which has an additional factor of CF x coming from rfinhomogeneity, which over a building block, accumulates to first order an evolution 2pF x pn F z , which for an inhomogeneity of ¼ :05 and n ¼ 6, corresponds to an evolution 10p F x p6 F z . Presence p F , limits the transfer efficiency. This inhomoof additional factor 10 x geneity factor can be canceled by following this building block with a building block of amplitude C (adding 180 to all phases). In the interaction frame of the irradiation along x axis, with the strength C, the rf-field Hamiltonian of the spin system transforms to
HrfIb ðtÞ ¼ CF x
Cp ðdðCtÞ dðCt pÞ þ dðCt 2pÞÞ 4n |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} _ /ðtÞ
ðF z cosðCtÞ F y sinðCtÞÞ;
ð8Þ
Which accumulates an integral only when delta function peaks at Ct ¼ 0; 2p and Ct ¼ p. The net rf rotation is therefore pn F z 2pF x . The CF x in HrfIb cancels CF x in HrfIa to first order.
ð3Þ
Rt R0 where dðtÞ is a delta function ( 0 f ðsÞdðsÞds ¼ t f ðsÞdðsÞds ¼ R 1 t f ðsÞdðsÞds ¼ f ð0Þ for t > 0). See Fig. 1B. 2 t In the modulation frame of the phase /ðtÞ, the rf-field Hamiltonian takes the form
_ z; Hrf ðtÞ ¼ CF x /F
HrfI ðtÞ ¼
_ /ðtÞ
where xr is the spinning frequency (in angular units), while the coefficients xk ðk ¼ I; S; ISÞ 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 [6]. 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 pulse sequence, made of building block
Cp _ /ðtÞ ¼ ½dðCtÞ dðCt pÞ þ dðCt 2pÞ; 4n
spins are averaged out. The rf-field Hamiltonian of the spin system transforms to
ð4Þ
where C is in the angular frequency units and we choose C xI ðtÞ; xS ðtÞ; xIS ðtÞ; xr . In the interaction frame of the irradiation along x axis, with the strength C, the chemical shifts of the
The C amplitude may be implemented with a two pulse element (for n ¼ 6)
ðpÞ187:5 ðpÞ172:5 : The total compensated pulse sequence takes the form
ðpÞ7:5 ðpÞ7:5 ðpÞ187:5 ðpÞ172:5 : We have calculated
R sc 0
HrfIa ðtÞdt and
R sc 0
HrfIb ðtÞdt, which are first
order contributions to generated rotation by time varying Hamiltonian HrfIa ðtÞ and HrfIb ðtÞ. We can calculate the second order terms, by writing evolution of HrfIa ðtÞ and HrfIb ðtÞ as,
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Fig. 1. Fig. A shows how phase of homonuclear recoupling sequence changes with time between / and /. Fig. B shows time derivative of phase in Eq. (3). Area under this curve gives the total phase change. Fig. C shows the dotted points (bold dots), where in Eq. (5), delta functions peak and net rotation happens, as F z is rotated in the interaction frame of CF x .
U 1 ðtÞ ¼ exp i
p F z expðipF x Þ exp i F z expðipF x Þ 2n p p expði F z Þ exp i F z 2pF x ; 4n n
U 2 ðtÞ ¼ exp i
p
Now transforming the coupling Hamiltonian in Eqs. (10) and
4n
(11) into the interaction frame of the rf-field Hamiltonian HrfI , we act on coupling Hamiltonian with an effective recoupling field x2r F z (see following paragraph), and we only retain terms that give static contribution to the coupling Hamiltonian, i.e., terms oscillating with frequency 2C are dropped and the coupling Hamiltonian takes the form (neglecting the terms oscillating at frequency 2xr as they are not recoupled)
p F z expðipF x Þ exp i F z expðipF x Þ 4n 2n p p exp i F z exp i F z þ 2pF x : 4n n
p
Therefore,
we
prepare
effective
Hamiltonians
CF x nps F z ¼ CF x sp F z ¼ CF x x2 F z , followed by the Hamiltonian CF x x2 F z . Toggling between them eliminates CF x . The second order Hamiltonian is of the order p2 xr F y . We have a recoupling field along x2 F z p2 xr F y x2 F z . r
c
HII ðtÞ ¼ jh fcosðxr tÞ cosðxr t þ cÞðIy Sy Ix Sx Þ sinðxr tÞ
r
cosðxr t þ cÞðIy Sx þ Ix Sy Þg;
r
r
r
To evaluate the effect of recoupling field on the coupling Hamiltonian, consider the compensated pulse sequence
h
ðpÞ/ ðpÞ/ ðpÞpþ/ ðpÞp/
i
N
:
ð9Þ
In the modulation frame of the rf-phase, rf-Hamiltonian alter_ z and CF x /F _ z every sc units of time, nates between CF x /F _ where / is as in Fig. 1A. By transforming into the interaction frame of irradiation along x axis, which alternates between CF x and CF x , we prepare HrfIa ðsÞ and HrfIb ðsÞ every
sc units of time. The coupling
HDD Ia ðtÞ ¼
HDD Ib ðtÞ ¼
which averages to
HII ¼
3 3 xIS ðtÞðIz Sz þ Iy Sy Þ þ xIS ðtÞððIz Sz Iy Sy Þ cosð2C sÞ 2 2 þ ðIz Sy þ Iy Sz Þ sinð2C sÞÞ;
ð10Þ
3 3 xIS ðtÞðIz Sz þ Iy Sy Þ þ xIS ðtÞððIz Sz Iy Sy Þ 2 2 cosð2CðsÞÞ þ ðIz Sy þ Iy Sz Þ sinð2CðsÞÞ:
ð11Þ
jh 2
fcosðcÞðIy Sy Ix Sx Þ þ sinðcÞðIy Sx þ Ix Sy Þg:
ð13Þ
jh ¼ 4p3 ffiffi2 bIS sinð2bÞ and bIS is the dipole coupling constant. Thus TOPR has a scaling factor of 12 compared to a standard cwhere
encoded experiment like HORROR [13]. The rf-interaction frame, HrfI prepares an effective field x2r F z , every sc units of time. Transformation of the coupling (denoted J below) under rf-propagator UðtÞ can be evaluated every sc units of time [26], by realizing that
Z
Hamiltonian alternates between
ð12Þ
0
T
U 0 ðtÞJUðtÞ dt ¼
X Z ksc k
ðk1Þsc
U 0 ððk 1Þsc Þ U 0k ðtÞJU k ðtÞ Uððk 1Þsc Þ dt:
Uððk 1Þsc Þ is the interaction frame propagator at time ðk 1Þsc and is same as evolution under effective field x2r F z at this time, and U k ðtÞ evolves in time ½ðk 1Þsc ; ksc with U k ðtÞ ¼ I þ oðx2r sc Þ. For xr C, we can neglect the second factor and assume we evolve the coupling Hamiltonian under an effective field x2r F z .
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TOPR pulse sequence is broadband, as a large value of C averages out the chemical shift [28]. The high power pulse sequence is made robust to rf-inhomogeneity by incrementing the phase by p every sc units of time. In Fig. 2, we simulated the performance of compensated and uncompensated TOPR for C ¼ 60 kHz and xr ¼ 10 kHz which gives n ¼ 6 and / ¼ 7:5 . We observe that compensated TOPR is much robust to rf-inhomogeneity as compared to uncompensated TOPR. It is worth pointing out that for the above choice of parameters, the uncompensated TOPR is simply the symmetry based p. R-sequence, R2412 , that has / ¼ 24 In Fig. 3A, we compare experimentally, the sensitivity of com-
pensated TOPR to symmetry based R-sequence, R2412 , for transfer of magnetization, from 13Ca to 13CO in a Glycine, at 10 kHz spinning, and rf-power C ¼ 60 kHz. We find that since TOPR is much robust to rf-inhomogeneity as compared to R2412 , we get almost twice the senstivity experimentally. To further quantify the senstivity in Fig. 3A, we provide a simulation for this 13C-13C transfer with a 10% Lorentzian rf-inhomogeneity in Fig. 3B. We find TOPR gives 50% transfer, while R2412 gives 25% transfer for the same mixing time of 10 rotor periods. 2.1. Chemical shifts We can understand the effect of chemical shift offset on the recoupling field. In the presence of chemical shift, we have in the interaction frame of the rf phase the Hamiltonians (for spin I), _ z followed by DxIz CIx /I _ z . We write these as DxIz þ CIx /I _ z ~ Ix0 /I x
_ z where x0 ¼ x cos h þ z sin h and ~ Ix00 /I x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ , see ~ ¼ C 2 þ Dx2 , for cos h ¼ C=x x ¼ x cos h þ z sin h and x and
00
Fig. 4. By proceeding in frame CIx0 and CIx00 , we create Hamiltonians HrfIa ðtÞ and HrfIb ðtÞ whose evolution is
U 1 ðtÞ ¼ exp i
s c ~ I x0 ðcos hIz0 þ sin hIx0 Þ exp i Dx 4n 2 p exp i ðcos hIz0 sin hIx0 Þ 2n p s c ~ Ix0 exp i ðcos hIz0 þ sin hIx0 Þ ; exp i Dx 2 4n
p
and
U 2 ðtÞ ¼ exp i
s p c ~ Ix00 ðcos hIz00 þ sin hIx00 Þ exp i Dx 4n 2 p exp i ðcos hIz00 sin hIx00 Þ 2n p s c ~ Ix00 exp i ðcos hIz00 þ sin hIx00 Þ : exp i Dx 2 4n
~ ¼x ~ C. where Dx Then
U 2 U 1 exp i2sc
p ~ sin hIz cos2 hIz þ Dx nsc
:
Therefore we have a recoupling field of
x x Dx2 Dx2 r ~ sin h Iz r 1 þ Dx I z : cos2 h þ Dx 2 2 2 2 DC 2C The ideal recoupling field is x2r Iz . Due to presence of offset, we deviate from it by an amount
D x2 2C 2
ðDx þ xr Þ. When Dx is negative,
then ðDx þ xr Þ is smaller, hence the pulse sequence performs better as compared to positive offsets. See Fig. 5. If instead we use the pulse sequence ðpÞ/ ðpÞ/ ðpÞ180/ ðpÞ180þ/ , we prepare an recoupling field x2r Iz . Due to presence of offset, we deviate from it by an amount
Dx2 2C 2
ðDx xr Þ. When Dx is positive,
Fig. 2. Fig. A shows the build up of transfer of magnetization for a ideal two spin system, 13C-13C, on a 750 MHz (proton frequency) static field, using the uncom pensated TOPR pulse unit with C ¼ 6xr ; n ¼ 6 with / ¼ 7:5 . Simulation uses internuclear distance of 1.52 A and powder avaraging. a; b; c; d, corresponds to inhomogeneity value of 0; :02; :05 and :1 respectively. Fig. B shows the build up basic 13C-13C correlation experiment using the compensated TOPR pulse unit, with / ¼ 7:5 . a; b; c; d, corresponds to inhomogeneity value of 0; :02 and :05 and :1 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 13C-13C correlation experiment using the compensated TOPR pulse unit with / ¼ 7:5 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 Ca-CO spin pair in Glycine respectively. Simulation were performed with spinevolution [27] software.
then ðDx xr Þ is smaller, as compared to when Dx is negative, hence the pulse sequence performs better for positive offsets. See Fig. 5.
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Fig. 3. Fig. A shows the transfer amplitude using (a) TOPR vs (b) R2412 sequence for transfer of magnetization, from 13Ca to 13CO in a Glycine at 10 kHz spinning, and rf power C ¼ 60 kHz. This gives / ¼ 7:5 . Mixing time is 10 rotor periods. Fig. B shows simulation of transfer amplitude of (a) TOPR vs (b) R2412 sequence for transfer of magnetization, from 13Ca to 13CO in a Glycine at 10 kHz spinning, and rf-power C ¼ 60 kHz. We use 10% Lorentzian for simulating rf-inhomogeneity. The chemical shift difference between spins in taken as 24 kHz, with carrier in the center. We include a CSA value of 11 and 80 ppm for the two spins respectively, with anisotropy parameter :3 and :9, representing the Ca-CO spin pair in Glycine respectively. Fig. 5. Left panel shows the build up curve for transfer of magnetization I ! S, using homonuclear TOPR ðpÞ/ ðpÞ/ ðpÞpþ/ ðpÞp/ for xr ¼ 10 kHz and n ¼ 6; C ¼ 60 kHz, and / ¼ 7:5 , with negative offset DxS ¼ 12 kHz (curve a) and positive offset DxS ¼ 12 kHz (curve b). Left panel compares the performace of ðpÞ/ ðpÞ/ ðpÞp/ ðpÞpþ/ (curve a), with ðpÞ/ ðpÞ/ ðpÞpþ/ ðpÞp/ (curve b) for positive offset DxS ¼ 12 kHz.
where
I ¼ Iz iIy ,
where
expðiCIx tÞI expðþiCIx tÞ ¼
expð iCtÞðI Þ. Using this for n ¼ 6, the CSA averages every two rotor periods, where TOPR toggles between C every sc . Proceeding to interaction frame of HrfI , by acting with an effective field x2r F z , we find that for example n ¼ 6, the CSA averages to zero in the first order. 3. Phase matching and heteronuclear recoupling Consider two coupled heteronuclear spins I and S under magic angle spinning condition [31]. 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 Fig. 4. Above Fig. shows the direction x0 of effective field, resulting from a x-phase rf-field of strength C and resonance offset Dx, and its orthogonal direction z0 . Similarly directions x00 and z00 result from a x phase rf-field of strength C and resonance offset Dx.
We can evaluate the effect of TOPR pulse sequence on the chemical shift anisotropy in the interaction frame of irradiation along x axis, which alternates between CF x and CF x . The chemical shift anisotropy of spin I can be expressed as ðx 1 I expð ixr tÞþ þ x 2 I expð i2xr tÞÞIz . We can decompose the operator Iz ¼ I þ I ,
HðtÞ ¼ xI ðtÞIz þ xS ðtÞSz þ xIS ðtÞ2Iz Sz þ Hrf ðtÞ;
ð14Þ
where xI ðtÞ; xS ðtÞ, and xIS ðtÞ 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 series P xk ðtÞ ¼ 2m¼2 xmk expðimxr tÞ, where xr is the spinning frequency (in angular units), while the coefficients xk ; ðk ¼ I; SÞ 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 [6]. Consider the rf irradiation on heteronuclear spin pair, where amplitude on spin I and S is chosen as Ak ðtÞ ¼ 2Cp (k ¼ I; S) such that
N. Khaneja, A. Kumar / Journal of Magnetic Resonance 281 (2017) 162–171
167
C ¼ nxr , and phase modulation, given by pulse sequence ðpÞ/1 ðpÞ/1 and ðpÞ/2 ðpÞ/2 respectively on two channels. Here p . This gives we choose /1 ¼ 38np and /2 ¼ 8n
3C p /_1 ðtÞ ¼ ðdðCtÞ dðCt pÞ þ dðCt 2pÞÞ; 8n Cp /_2 ðtÞ ¼ ðdðCtÞ dðCt pÞ þ dðCt 2pÞÞ; 8n
ð15Þ ð16Þ
where phase jump is three halves and half of the modulation considered in previous section. As a result, the effective fields prepared r and x4r , with average x2r , which will on two channels will be 3x 4 recouple a double quantum Hamiltonian. To mitigate the effect of inhomogeneity of rf-field, the basic pulse unit is accompanied with a compensating unit. In nutshell, the pulse sequence on channels I and S (say C 13 and N 15 ) is following respectively.
ðpÞ/1 ðpÞ/1 ðpÞpþ/1 ðpÞp/1 :
ð17Þ
ðpÞ/2 ðpÞ/2 ðpÞpþ/2 ðpÞp/2 :
ð18Þ
In the frame of the rf-field along x axis, that toggles between
CðIx þ Sx Þ, the coupling Hamiltonian is averaged to
HDD I ðtÞ ¼ xIS ðtÞðI z Sz þ Iy Sy Þ þ xIS ðtÞððIz Sz Iy Sy Þ cosð2C sÞ
ðIz Sy þ Iy Sz Þ sinð2C sÞÞ; With ðIz þSz Þ r 2
x to
effective
rf-field
ð19Þ r 3x I x4r Sz 4 z
written
as
xr ðIz Sz Þ 2
2
, the coupling Hamiltonian is further averaged
HII ¼ jd fðIy Sy Ix Sx Þ cosðcÞ þ ðIy Sx þ Ix Sy Þ sinðcÞg : |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
ð20Þ
We call the above experiments heteronuclear TOPR. Fig. 6 shows simulation results for a heteronuclear magnetization transfer in 15N-13C heteronuclear system for C ¼ 40 kHz and
xr ¼ 10 kHz with n ¼ 4. This gives /1 ¼ 16:875 and /2 ¼ 5:625 . 4. Experimental results All experiments were performed on a 750 MHz spectrometer (1H Larmor frequency of 750 MHz) equipped with a triple resonance 3:2 mm probe. Uniformly 13C labeled sample of Glycine and uniformly 13C, 15N-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 13C, 15N-labeled sample of MLF was used for both homonuclear and heteronuclear recoupling experiments. The experiments used 2 s recycling delay. In all experiments, CW decoupling of 120 kHz is used on protons. All 2D experiments were acquired using TPPI. Fig. 7A shows the build up curve for transfer of magnetization, for 13Ca to 13CO every rotor period, N ¼ 2 (N as in Eq. (9)), with TOPR as the recoupling element. The experiment uses an initial ramped CP for 1H to 13C cross polarization. The TOPR recoupling block is designed for a nominal power of 60 kHz as described in the text and spinning speed of 15 kHz. This gives n ¼ 4 and
/ ¼ 11:25 . Fig. 7B shows a 13Ca-13CO 2D correlation spectrum obtained using the TOPR as the recoupling element. Mixing time is 8 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 two resonances at 119 ppm during transfer. 4 scans were collected for every t 1 increment. Fig. 8A shows the build up of transfer of magnetization, shown every rotor period, N ¼ 2, for the 15N !13 Ca experiment with TOPR as the recoupling element. The TOPR recoupling is done at spin-
Fig. 6. Fig. A shows the build up of 15N to 13C magnetization transfer on a 750 MHz (proton frequency) static field, using the TOPR pulse unit with C ¼ 40 kHz and xr ¼ 10 kHz and n ¼ 4 with /1 ¼ 16:875 and /2 ¼ 5:625 on 13 C and 15N 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 TOPR pulse unit with compensation. Fig. C shows the build up of basic -15N-13C correlation experiment using the TOPR pulse unit, with different chemical shift of the 13C spin. a; b; c, corresponds to chemical shift of 0; 4; 8 kHz of the 13C spin respectively.
ning speed of 10 kHz and nominal power of 40 kHz on carbon and nitrogen as described in the text. This gives n ¼ 4 and /1 ¼ 16:875 (on 13C) and /2 ¼ 5:625 (on 15N). The experiment uses an initial ramped CP for 1H to 15N cross polarization. Bottom panel of Fig. 8 shows the 2D spectrum for 13Ca-15N experiment with
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Fig. 7. Fig. A shows the build up curve for transfer of magnetization from 13Ca to 13CO in a Glycine, after every 2N ¼ 4 blocks, at 15 kHz spinning, and rf-power C ¼ 60 kHz. This gives n ¼ 4 and / ¼ 11:25 . Fig. B shows corresponding 13Ca-13CO 2D correlation spectrum obtained using the TOPR as the recoupling element. Mixing time is 8 rotor periods. The carrier is placed in center of 13Ca-13CO resonances. Fig. C shows the pulse sequence for the 2D experiment. The phase of CP pulse on 13C is TPPI phase cycled.
TOPR as the recoupling element. The magntization precesses on 15 N during indirect evolution. Mixing time is 17 rotor periods. The experiment used 1024 points in direct and 256 in indirect dimension with spectral width of 220 ppm in 13Ca and 100 ppm in 15N dimension. The carrier is placed on 15N and 13Ca resonance during transfer. Fig. 9 shows the 2D 13C correlation in tripeptide MLF obtained using TOPR as recoupling element. Experiment is done at 15 kHz spinning, and rf-power C ¼ 60 kHz. This gives n ¼ 4 and
/ ¼ 11:25 . 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 ¼ 16 or 8 rotor periods. The experiment uses an initial ramped CP for 1H to 13C 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 t 1 increment. Fig. 10 shows the 2D 15N-13C correlation in tripeptide MLF obtained using TOPR as recoupling element. Experiment is done at 10 kHz spinning, and rf-power C ¼ 40 kHz on carbon and nitrogen channels. This gives n ¼ 4 and /1 ¼ 16:875 (on 13C) and
N. Khaneja, A. Kumar / Journal of Magnetic Resonance 281 (2017) 162–171
169
Fig. 8. Top panel shows the build up of transfer of magnetization, shown every 2N ¼ 4 blocks, at 10 kHz spinning, for the 15 N ! 13 C a experiment, in uniformly labeled sample of Alanine, with TOPR as the recoupling element as described in text. The rf power used is C ¼ 40 kHz, giving n ¼ 4 and /1 ¼ 16:875 (on 13C) and /2 ¼ 5:625 (on 15N). Panel B shows the 2D spectrum for 15N-13Ca experiment with TOPR as the recoupling element. Mixing time is 17 rotor periods. The magntization precesses on 15N during indirect evolution. The carrier is placed on 15N and 13Ca resonance.
Fig. 9. Top panel shows the 2D 13C correlation in tripeptide MLF obtained using TOPR as recoupling element. Experiment is done at 15 kHz spinning, and rf-power C ¼ 60 kHz. This gives n ¼ 4 and / ¼ 11:25 . 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 ¼ 16 or 8 rotor periods.
/2 ¼ 5:625 (on 15N). The carbon carrier is placed near CO resonances in A and Ca resonances in B. Mixing time corresponds to N ¼ 34 or 17 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 t 1 increment.
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Fig. 10. Top panel shows the 2D 15N-13C correlation in tripeptide MLF obtained using TOPR as recoupling element. Experiment is done at 10 kHz spinning, and rf-power C ¼ 40 kHz on carbon and nitrogen channels. This gives n ¼ 4 and /1 ¼ 16:875 and /2 ¼ 5:625. The carbon carrier is placed near CO resonances in A and Ca resonances in B. CW decoupling of 120 kHz is used on protons. Mixing time corresponds to N ¼ 34 or 17 rotor periods.
5. Conclusion In this paper we introduced a class of recoupling pulse sequences, which rest on the principle of second oscillating field [21,25,26,28,29,32]. 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. If we compare TOPR with TPR and FPR recoupling [21,22], we find that here, we produce a recoupling effective field along z direction, as opposed to an effective recoupling field along y direction in [21,22]. Pulse sequences presented in this paper do mixing of the z magnetization. If we compare TOPR to HORROR [13], we find the sequence in this paper is much more broadband. When we compare the sequence to C7, POSTC7, SPC5 [17,19,18], and the symmetry based sequences CNmn , [24], we find these pulse sequences also produce an efffective recoupling field along z direction, however these sequences have constant phase increments [23] unlike TOPR, where phase switches back and forth between two values. In that respect TOPR is similar to symmetry based RNmn pulse sequences [24]. If we compare the pulse sequence ðpÞ/ ðpÞ/ as appearing in
symmetry based RNmn pulse sequences [24], we point out that we have a completely different design philosophy, where we produce an effective recoupling field of strength x2r . If we consider a symmep and has n ¼ 6 blocks in try based pulse sequence R2412 , it has / ¼ 24 a rotor period and hence is our uncompensated TOPR, discussed in main text. We emphasize that our design principle is based on constructing a recoupling field of strength x2r and doesn’t rely on any symmetry principles. The work in this paper shows how symmetry based RNmn pulse sequences can be seen as synthesizing a effective recoupling field. Furthermore we present compensated pulse sequences, which make them robust to rf-inhomogeneity and gives better sensitivity. This is shown in Fig. 3. If we compare TOPR to CMRR, we find CMRR has continuous phase modulation while TOPR has only two phases. It is worthwhile to note that the square modulation of the phase as presented
in the Fig. 1 and Eq. (3) can also seen when CMRR is implemeted in r a TPPM mode [29], by synthesizing Hamiltonians CF x þ px F y and 4 sc r F , each for units of time. This will create an effective CF x px y 4 2 recoupling field of strength x2r F z . For C ¼ nxr , this is a pulse sequence with a square waveform phase modulation with phase qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 p and amplitude x n2 þ p16. Therefore this / such that tan / ¼ 4n r qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p2 , which is a similar pulse sequence is ðhÞ/ ðhÞ/ with h ¼ p 1 þ 16n 2 pulse sequence to TOPR but not the same. Furthermore TOPR is robust to rf-inhomogeneity.
Acknowledgement The authors would like to thank the HFNMR lab facility at IIT Bombay where the data was collected.
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