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Coherent and stochastic averaging in solid-state NMR
Accepted Manuscript Coherent and Stochastic Averaging in Solid-State NMR Alexander A. Nevzorov PII: DOI: Reference:
S1090-7807(14)00268-7 http://dx.doi.org/10.1016/j.jmr.2014.09.023 YJMRE 5524
To appear in:
Journal of Magnetic Resonance
Received Date: Revised Date:
5 August 2014 16 September 2014
Please cite this article as: A.A. Nevzorov, Coherent and Stochastic Averaging in Solid-State NMR, Journal of Magnetic Resonance (2014), doi: http://dx.doi.org/10.1016/j.jmr.2014.09.023
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09/15/14
Coherent and Stochastic Averaging in Solid-State NMR Alexander A. Nevzorov Department of Chemistry, North Carolina State University 2620 Yarbrough Drive, Raleigh, North Carolina, 27695-8204 Phone: (919) 515-3199 E-mail: [email protected] Keywords: Stochastic Liouville equation, uniaxial diffusion, motional averaging, dipolar recoupling, membrane proteins
Abstract
A new approach for calculating solid-state NMR lineshapes of uniaxially rotating membrane proteins under the magic-angle spinning conditions is presented. The use of stochastic Liouville equation (SLE) allows one to account for both coherent sample rotation and stochastic motional averaging of the spherical dipolar powder patterns by uniaxial diffusion of the spin-bearing molecules. The method is illustrated via simulations of the dipolar powder patterns of rigid samples under the MAS conditions, as well as the recent method of rotational alignment in the presence of both MAS and rotational diffusion under the conditions of dipolar recoupling. It has been found that it is computationally more advantageous to employ direct integration over a spherical grid rather than to use a full angular basis set for the SLE solution. Accuracy estimates for the bond angles measured from the recoupled amide 1H-15N dipolar powder patterns have been obtained at various rotational diffusion coefficients. It has been shown that the rotational alignment method is applicable to membrane proteins approximated as cylinders with radii of approximately 20 Å, for which uniaxial rotational diffusion within the bilayer is sufficiently fast and exceeds the rate 2×105 s-1.
2
1. Introduction The explicit time-dependence of the relevant Hamiltonians in MAS NMR makes the calculation of spectral lineshapes especially challenging. Notable approaches to this problem include the Average Hamiltonian Theory [1-3], which is based on the Magnus expansion, and more elaborate treatments employing the multi-modal Floquet theories [4-7], which involve expansions of the density matrix into a set of operators with periodic time dependences at integral rotor frequencies. At the same time, especially in biological samples, the simultaneous sample rotation and molecular dynamics (both local and global, slow and fast) can profoundly influence the observed NMR lineshapes. A detailed understanding of the arising lineshape patterns can allow one to extract structural and dynamic information about the sample. Of particular interest here is the uniaxial rotational diffusion of a membrane protein embedded in a fluid-like lipid bilayer [8-12]. Recently, a new solid-state NMR method for obtaining structural information for uniaxially diffusing membrane proteins has been proposed [13,14]. It is based on obtaining the bond angles from measuring the scaled splittings of the recoupled spherical powder patterns partially averaged out by anisotropic uniaxial rotational diffusion that the protein molecules undergo about the membrane normal. The idea behind this clever method is that the splitting between the singularities (relative to the rigid limit) is proportional to the factor (3cos2θB -1), where θB is the angle that the bond connecting the two interacting spin dipoles forms with the axis of diffusional rotation. Similar effects have been known for a long time, such as the scaling of the deuterium powder patterns for the rapidly rotating methyl CD3 groups [15]. However, without a priori knowledge of the diffusion rates, it is rather difficult to predict whether the spectra of uniaxially diffusing protein molecules would fall into a particular motional regime. Moreover, even in the intermediate motional regime, a description of static
3
lineshapes in the presence of stochastic uniaxial rotation modeled as three-site jumps [16] becomes quite involved. Suffice it to say that a rigorous description of this effect under the MAS conditions at arbitrary diffusion rates and, in addition, in the presence of radiofrequency irradiation, is challenging [17]. To address this problem for the general motional case, we utilize here the method of Stochastic Liouville equation (SLE). The SLE approach allows one to simulate the spectra at arbitrary rotation and diffusion coefficients, from fully static to fastmotional limits, and has been introduced in ESR by Freed and co-workers [18,19] following the seminal work of Kubo [20,21]. Over the past decades, the SLE approach has also been applied to NMR [22-28] to describe the effect of intermediate motional time scales on NMR relaxation and lineshapes. Recently, the SLE method has been applied to simulate solid-state NMR spectra in the presence of uniaxial rotational diffusion of protein molecules in macroscopically aligned bilayers, [29,30] and also to treat the time-dependent rotation of the sample in terms of a generalized MAS superoperator [31]. In the present work, we provide an alternative derivation of the relaxation superoperator for the case of MAS NMR of randomly oriented molecules without using the Fokker Planck equation [31]. Furthermore, we illustrate the utility of SLE for calculating MAS lineshapes in simple and more complex cases, such as the recoupling of the dipolar powder patterns under the conditions of ‘rotational alignment’ in the presence of both coherent and stochastic rotations of the spin-bearing molecules.
4
2. Derivation of the SLE under the MAS conditions As in virtually every NMR experiment, the formal solution for the Liouville-von Neumann equation for the density matrix ρ(t) under the evolution governed by a time-dependent Hamiltonian, H(t), viz.
∂ ρ (t) = −i[H (t), ρ (t)] ∂t
(1)
can be written compactly as:
⎤ ⎡ t g(t) = expO ⎢i ∫ L(t ')dt '⎥ g(0) ⎦ ⎣ 0
(2)
Where the Liouvillian superoperator is given by:
L(t) = H T (t) ⊗ 12 N −12 N ⊗ H (t)
(3)
Here 12N is a 2N-by-2N identity matrix (for spins ½) and ⊗ stands for the Kronecker product, the symbol “O” denotes the Dyson time-ordering, and we omit for the reason of compactness the usual hat and double hat notations for the operators and superoperators, respectively. The explicit time dependence is contained in the rotation angles that the anisotropic interactions (e.g. chemical shift tensor and dipolar) depend on. Furthermore, in Eq. 2, g(0) = vec[ρ(0)] denotes the vectorization [32] of the initial density matrix (i.e., a stacking of its columns to form a density vector, g). We also note that the plus sign in front of the Liouvillian is in accord with the above definition of the Liouvillian, Eq. (3), and the vectorization operation of the density matrix as defined in ref. [32]. If the time dependence of the Liouvillian is solely contained in a single angle, φ(t), the formal solution of Eq. 2 can be expanded as a time-ordered exponential (t1>t2>t3>…) [33,34]:
5
t
t
exp O i ∫ L[ϕ (t ')]dt ' = 1+ i ∫ L[ϕ (t1 )]dt1 + 0
0
t
t1
0
0
t
t1
t2
0
0
0
(i)2 ∫ dt1 ∫ dt 2 L[ϕ (t1 )]L[ϕ (t 2 )]+
(4)
(i)3 ∫ dt1 ∫ dt 2 ∫ dt 3L[ϕ (t1 )]L[ϕ (t 2 )]L[ϕ (t 3 )]+ ... When the molecules are randomly distributed around the axis of MAS rotation, the resulting NMR signal can be calculated as an average over the distribution of the molecules. However, instead of summing over deterministic trajectories, the average can also be treated as an average over a stochastic Markov process, thus yielding a time-average propagator: t
t
expO i ∫ L[ϕ (t ')]dt ' = 1+ i ∫ dt1 ∫ d ϕ1L(ϕ1 )p(ϕ1 ,t1 )+ 0
0
t1
t
(i)2 ∫ dt1 ∫ dt 2 ∫ d ϕ1 ∫ d ϕ 2 L(ϕ1 )L(ϕ 2 )p(ϕ1 ,t1; ϕ 2 ,t 2 )+ 0
0
t
t1
t2
0
0
0
(i)3 ∫ dt1 ∫ dt 2 ∫ dt 3 ∫ d ϕ1 ∫ d ϕ 2 ∫ d ϕ 3L(ϕ1 )L(ϕ 2 )L(ϕ 3 )p(ϕ1 ,t1; ϕ 2 ,t 2 ; ϕ 3 ,t 3 )+ ...
(5)
where pn(φ1, t1 ; φ2, t2 ;... φn, tn) is the n’th order joint probability. For a Markovian path average, the Chapman-Kolmogorov equation employing the conditional probabilities, P(φn-1 , tn-1 | φn , tn) can be written as:
pn (ϕ1 ,t1; ϕ 2 ,t 2 ;...ϕ n ,t n ) = P(ϕ1,t1 | ϕ 2 ,t 2 )...P(ϕ n−1 ,t n−1 | ϕ n ,t n )p(ϕ n ,t n )
(6)
To evaluate the expansion, Eq. 5, let’s introduce an auxiliary function g(φ, t) such that:
g(t) =
2π
∫ dϕ g(ϕ ,t)
(7)
0
Following ref. [34] it can be easily verified that the auxiliary function is given by the expansion:
6
t
g(ϕ ,t) = [1 peq (ϕ )+ i ∫ dt1 ∫ d ϕ1L(ϕ1 )P(ϕ ,t | ϕ1 ,t1 )peq (ϕ1 )+ 0
t1
t
(i)2 ∫ dt1 ∫ dt 2 ∫ d ϕ1 ∫ d ϕ 2 L(ϕ1 )L(ϕ 2 )P(ϕ ,t | ϕ1 ,t1 )P(ϕ1 ,t1 | ϕ 2 ,t 2 )peq (ϕ 2 )+ (i)
3
0
0
t
t1
t2
∫ dt ∫ dt ∫ dt ∫ dϕ ∫ dϕ ∫ dϕ L(ϕ )L(ϕ 1
0
2
0
3
1
2
3
1
2
)L(ϕ 3 )P(ϕ ,t | ϕ1 ,t1 )P(ϕ1 ,t1 | ϕ 2 ,t 2 )P(ϕ 2 ,t 2 | ϕ 3 ,t 3 )peq (ϕ 3 )+ ...]g(0)
0
(8) where, for a stationary Markov process: p(φ, t) = peq(φ) =1/(2π). Moreover, if the averaging process is coherent (such as the MAS rotation about the angle φ with the frequency ωr), the conditional probability can be written as:
P(ϕ n−1 ,t n−1 | ϕ n ,t n ) = δ[ϕ n−1 − ϕ n − ωr (t n−1 − t n )]
(9)
Which has the following formal property:
1 ∂P(ϕ n−1,t n−1 | ϕ n ,t n ) ∂P(ϕ n−1,t n−1 | ϕ n ,t n ) = ωr ∂t n−1 ∂ϕ n−1
(10)
By differentiating g(φ, t) with respect to time, t and using the chain rule for the integrals involved, one obtains for the second term of the expansion:
∂ ∂t
t
∫ dt ∫ dϕ L(ϕ )P(ϕ ,t | ϕ ,t )p 1
1
1
1
1
eq
(ϕ1 ) =
0
∫ dϕ1L(ϕ1 )P(ϕ ,t | ϕ1,t)peq (ϕ1 )+ L(ϕ )peq (ϕ )+ ω r
∂ ∂ϕ
t
∫ ∫ dϕ L(ϕ ) 1
1
0
∂P(ϕ ,t | ϕ1 ,t1 ) peq (ϕ1 ) = ∂t
t
∫ ∫ dϕ L(ϕ )P(ϕ ,t | ϕ ,t )p 1
1
1
1
eq
(ϕ1 )
0
(11)
All other terms in the expansion for g(φ, t) can be evaluated in a similar manner and re-combined into two main terms: one having the superoperator iL in front of it, and the other containing the operator ωr∂/∂ϕ. We, therefore, obtain that g(φ, t) satisfies the corresponding Stochastic Liouville Equation in the presence of coherent rotation of the sample about the angle φ:
7
∂g(ϕ ,t) ∂g(ϕ ,t) = iLg(ϕ ,t)+ ω r ∂t ∂ϕ
(12)
Which can be written more compactly as:
∂g(ϕ ,t) = (iL + Γ)g(ϕ ,t) ∂t where the superoperator Γ is defined as: Γ = ω r
(13)
∂ . If, at the same time, the molecule undergoes ∂ϕ
stochastic rotational diffusion about another angle ψ with diffusion coefficient D||, by assuming stochastic independence of the two processes one can write for the two simultaneous rotations that:
Γ = ωr
∂ ∂2 + D|| 2 ∂ϕ ∂ψ
(14)
while Eq. 13 is still generally valid. Thus, Eq. (14) allows one to treat both coherent and stochastic averaging in a single formalism. Also, it should be noted in the derivation of Eq. (12) the non-intuitive differentiation of the probability density with respect to the density matrix employed by the previous authors [31,35] was not necessary.
3. Numerical solutions to the SLE To obtain the solution of the SLE, we shall use a minimalistic geometrical framework [36] to describe basic NMR experiments performed on a membrane protein molecule incorporated within spherical liposomes, Fig. 1.
In general, the Hamiltonian contains the
orientational dependence of the magnetic field B0 relative to some arbitrary molecular frame as given by the angles α and β, which are contained in the chemical shift anisotropy (CSA) and dipolar terms (written in the angular frequency units):
8
H = ΔΩI z + ω S S x + ω I I x + ν CSA ( β ,α )S z + ν DIP ( β ,α )I z S z
(15)
Here ΔΩ accounts for the frequency offset of the proton (I spins), ωS and ωI are the radiofrequency amplitudes for the low (or S) spin (e.g. nitrogen or carbon) and the proton, respectively. Only the radiofrequency irradiation along the x axis in the rotating frame is included here; other phases can be accounted for in a similar manner. The explicit dependences of the CSA, νCSA , and dipolar frequencies, νDIP , on the angles α and β are given in ref. [30]. The decomposition of the angular functions involving the angles α and β into the individual angles describing the overall sample orientation (θ), sample rotation (φ) spherical distribution or mosaic spread (Δθ) and, finally, uniaxial diffusion (ψ) is depicted in Fig. 1. Mathematically, it can be readily accomplished by using the irreducible spherical basis of spherical harmonics of rank 2 [30,36]: →
→
Y (β , α ) = Y (θ ,0)× D(ϕ ,Δθ , ψ )× D(0,θ B , ϕ B )
(16)
Here Y ’s are the row vectors of spherical harmonics of rank 2 (ordered according to their index m as: 2, 1, 0 -1, -2), D’s are the second-rank Wigner rotation matrices, θB and φB are the angles that the axis of uniaxial diffusion a forms with respect to given bond (or a principal axis system, in the case of CSA tensor). For calculating the dipolar patterns, as is done in the present paper, only θB becomes of interest; the MAS patterns due to CSA averaging have been calculated elsewhere [31]. To look for the solution of Eq. 13, one can use the expansion in terms of the Wigner rotation matrix coefficients (denoted compactly as a ket, |l, m, m’ >) involving the indices l (for the angle Δθ), m (for φ), m’ (for ψ), viz:
g(ϕ ,Δθ , ψ ,t) =
∑g
l,m,m '
(t) ⊗ l, m, m '
(17)
l,m,m '
9
Here the index l runs from 0 to a certain cut-off value, and the indices m and m’ run from –l to l. The Hilbert-space representation of the Liouvillian marix, L involves the integrals of three Wigner rotation matrices containing products of the Clebsch-Gordan coefficients [31,37]. Alternatively, one can co-add various spectra calculated for different values of Δθ by using either the Monte-Carlo method [36] or a uniform grid with the weighting function ½×sinΔθ, and utilize the Fourier expansions in just m and m’. In the Hilbert space, this expansion can be written in terms of the Kronecker products in a vector form as:
g(ϕ ,Δθ , ψ ,t) = ∑ g m,m ' (Δθ ,t) ⊗ m ⊗ m' m,m '
(18)
This way of expanding the solution significantly reduces the size of the superoperator matrices and makes the calculation of the superoperator matrices especially straightforward (they become tridiagonal in each angular subspace). In Eqs. 17-18 we have used again the Kronecker product,
⊗ to factorize the spin and the angular subspaces in the Hilbert space of the total superoperator, iL+Γ. The final time-averaged density matrix simply corresponds to the expansion functions with l=0 or m=m’=0, depending on whether Eq. 17 or Eq. 18 is used, respectively. The solution for the observed NMR signal is then given by the quadratic form:
G(t) = g 0T exp[(iL + Γ)t]g(0)
(19)
where g0 represents the vectorization of the detection operator, usually S- or Sx . We note that the SLE approach renders the superoperator matrices time-independent, which requires only one (or a few, depending on the pulse sequence modeled) diagonalization steps in simulating the spectra.
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4. Results All simulations in this work have been performed on a desktop computer having two sixcore processors and operating at 2.66 GHz using a Matlab (Mathworks, Inc.) script. Figure 2 illustrates the well-known result of averaging of the NH amide dipolar couplings (with the constant χ = 10 kHz) by the magic angle spinning calculated using the SLE, Eq. 13. In Figs 2AD we use the expansion of Eq. 17 (with the the spin space factored out and no rotational diffusion, D|| = 0, m’ = 0) with the cutoff value l = 50. Various spinning speeds are used, from the nearly static case of ωr = 100 Hz to the spinning rate of ωr = 5 kHz illustrating the appearance of the spinning side bands. By contrast, in Figs. 2E-H we utilized the expansion of Eq. 18 with the absolute-value cutoff for the index m of 100, (still with D|| = 0) with co-adding of 1000 spectra at various values of Δθ randomly chosen by the Monte-Carlo method. We note that the cutoff indices are determined by the spinning rate, and much lower values <5 are sufficient to obtain convergent spectra for faster speeds, (>5 kHz) which makes the calculations considerably faster. It is clear that the direct integration combined with expansion of Eq. 19 is computationally superior to the full grid-free approach [31] yielding at least a factor of six gain in speed with a similar degree of convergence for the spectra. Figure 3 represents the solution of Eq. 13 in the presence of both MAS spinning and rotational diffusion given by the superoperator of Eq. 14 at various bond angles, ranging from θB = 0o to 90o. Using the Stokes-Einstein Equation and approximating the protein as a cylinder [8] having the radius of 20 Å (which would be a rough estimate for the transmembrane core of a GPCR [38]), we obtain for the value of D|| = 2×105 s-1. Here a membrane bilayer is assumed to have hydrophobic thickness of 30 Å and viscosity of ~100 cPoise [8]. Similar rates have been reported elsewhere [9-12]. This value serves as a rough estimate for the actual diffusion rate of a
11
membrane protein within the bilayer. More recent studies [39], however, suggest a considerably faster diffusion coefficient of D|| = 5×106 s-1 for GPCRs, presumably due to the perturbed local bilayer viscosity at the protein interface, which would be even more beneficial for the rotational alignment method. The recently developed CP-PI pulse sequence [40] was used to achieve the recoupling of the dipolar Hamiltonian. The spin Hamiltonian of Eq. 15 was used and the spinning speed was chosen to be 20 kHz with 60 kHz rf amplitude on the low side. The starting vector was taken as:
g(0) = vec(Sx − I x ) ⊗ (0,..., 0,1,0,...0) 0,1,0,.., 0) ⊗ (0,..., T
2 m +1
T
(20)
2 m ' +1
The chemical shifts at these relatively high rf amplitudes have been neglected. To yield first-order rotary dipolar recoupling, the rf amplitudes on the proton side were additionally toggled between 80 and 40 kHz for the even and odd dwells, and the dwell time was chosen to be one-half of the rotor period, π/ωr . The cutoffs for the index m was chosen to be ±4 and ±6 for m’, thus making the total matrix size of 16(2|m|+1)(2|m’|+1)=1872 by 1872, and 400 spectra were co-added by using a uniform distribution in the angle Δθ with the weighting function ½×sinΔθ. It can be seen from Fig. 3 that at these parameter values a good convergence of line shapes was achieved. It can also be seen that, depending on the value of θB the recoupled powder patterns are scaled differently depending on the angle that the chemical bond connecting two interacting spins forms with the instantaneous axis of rotation. This provides the source for the structural information that can be directly extracted from the spectra [13,14]. In Fig. 4, the diffusion coefficient was decreased down to D|| = 2×104 s-1 to illustrate the effect of slow motions on the resolution of the rotationally averaged recoupled powder patterns. While the slower rotational diffusion still yields the scaling of the recoupled powder patterns, resolution becomes
12
significantly worse, especially at the values of the bond angle that are close to the magic angle (i.e. θB ≈ 54.7o). Figure 5 shows correlation plots of the ratio of the recoupled NH dipolar splittings as determined from the simulations relative to their static value, χ/√2 as a function of the actual bond-angle factor, |3cos2θB -1|/2. It can be seen that at the diffusion rate of D|| = 2×105 s-1 there is a very good correlation between the relative splittings measured from the singularities and the actual bond factor. The relative errors vary from 1-5% for larger splittings up to 14% for smaller splittings. However, a systematic underestimation of the true bond factors has been observed for nearly all bond angles. We also note that for measuring the aliphatic carbon 1H-13C couplings for the same molecule at the same degree of precision this lower limit would need to be approximately doubled due to the linear scaling property of the SLE, Eq. 13, at the relatively fast MAS speeds and high rf amplitudes. When the diffusional rate is decreased down to D|| = 2×104 s-1 in the simulations (which corresponds to intermediate-to-slow motional regime at the given dipolar coupling constant of χ = 10 kHz; πχ/D|| = 1.57 at these values), the agreement becomes worse, cf. Fig. 4B, thus potentially interfering with the accuracy of the experimentally derived angular factors. However, if the recently reported [39] value of D|| = 5×106 s-1 is used instead, the maximum deviation of the measured splittings from the true bond factors considered in the simulations is reduced to less than 10%. Even though the correlation plots of Fig. 5 look quite good, they do not reflect individual errors for the specific bond angles that may ultimately affect the final structure calculation. Figure 6 shows plots of relative errors in determining the bond-angle factors from the rotationally averaged recoupled dipolar splittings, Δ, as a function of the bond angle, θB and the diffusion coefficient, D|| . The relative error has been calculated as: 13
Error(%) =
2 2Δ
χ 3cos 2 θ B −1
−1 ×100%
(21)
It is clear that the largest error arises near the value θB = 54.7, where the splitting becomes small and on the order of the spectral linewidth. Also, the faster is the diffusion, the greater accuracy in determining the bond angle can be achieved. However, markedly smaller errors are associated with the splittings that correspond to smaller bond angles (θB < 30o) at all diffusion rates considered. 5. Conclusions The presented SLE approach provides an alternative to the existing numerical methods for simulating MAS spectra that involve direct time-step integration [41,42]. Importantly, the presented formalism allows one to describe more complicated time dependencies, both periodic and stochastic, and allows one to simulate MAS NMR spectra at arbitrary rotational and diffusional rates. It suffers, however, from the relatively large size of the superoperators matrices involved, which limits its applicability to only a few spins, at least at present computational powers. This problem, however, can be circumvented by using space-restriction techniques as implemented in SPINACH [43], and, more recently, by the “tensor train” formalism [44]. In addition, it was determined that an upper-size limit for obtaining reliable bond-angle factors from ‘rotational alignment’ corresponds to proteins with an average radius of 20 Å when roughly treated as cylinders. It is worthwhile to mention that, according to the previous work [30,36] the rate D|| = 2×105 s-1 would be insufficient to obtain sharp uniform linewidths for membrane proteins incorporated in magnetically aligned bicelles in oriented-sample NMR. By contrast, the spherical powder patterns as recoupled in MAS NMR appear to be less sensitive to the
14
diffusional rate since it is the overall distribution and singularities that are averaged out by the motions, not the individual bond orientations. We also note that using the SLE-based model one may be able to interpret the scaled dipolar couplings even at slower motional rates if the appropriate correction factors or a pre-defined measure of uncertainty is used in structure calculations. This, however, would require a more accurate knowledge of the radius of gyration for the protein and the value for the viscosity of the bilayer environment, which still remains somewhat controversial. The SLE treatment presented herein is potentially applicable to other solid-state NMR applications including simulations of more complex multidimensional pulse sequences.
Acknowledgement The author wishes to thank Dr. Ilya Kuprov for stimulating discussions. Supported by NSF MRI 1229547.
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Nevzorov, A. A.; De Angelis, A. A.; Park, S. H.; Opella, S. J. In NMR
Spectroscopy of Biological Solids; Ramamoorthy, A., Ed.; Marcel Dekker: New York, 2005, p 177. [30]
Nevzorov, A. A. Orientational and Motional Narrowing of Solid-State NMR
Lineshapes of Uniaxially Aligned Membrane Proteins, Journal of Physical Chemistry B 115 (2011), 15406. [31]
Edwards, L. J.; Savostyanov, D. V.; Nevzorov, A. A.; Concistrè, M.; Pileio, G.;
Kuprov, I. Grid-free powder averages: On the applications of the Fokker–Planck equation to solid state NMR, Journal of Magnetic Resonance 235 (2013), 121. [32]
Steeb, W. H. Matrix Calculus and Kronecker Product with Applications and C++
Programs; World Scientific: Singapore, 1997. [33]
Pedersen, J. B. In Electron spin relaxation in liquids; Muus, L. T., Atkins, P. W.,
Eds.; Plenum Press: New York, 1972, p 25. [34]
Nevzorov, A. A.; Freed, J. H. Spin relaxation by dipolar coupling: from motional
narrowing to the rigid limit, J. Chem. Phys. 112 (2000), 1413. [35]
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equilibrium, Pure and Applied Chemistry 40 (1974), 181. [36]
Nevzorov, A. A.; Moltke, S.; Heyn, M. P.; Brown, M. F. Solid-State NMR
Lineshapes of Uniaxially-Oriented Immobile Systems, J. Am. Chem. Soc. 121 (1999), 7636. [37]
Abramowitz, M.; Stegun, I. Handbook of mathematical functions with formulas,
graphs, and mathematical tables; U. S. Govt. Print. Off.: Washington, 1972. [38]
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kinds, Trends in Neurosciences 31 (2008), 74.
19
[39]
Spille, J. H.; Zurn, A.; Hoffmann, C.; Lohse, M. J.; Harms, G. S. Rotational
diffusion of the alpha(2a) adrenergic receptor revealed by FlAsH labeling in living cells, Biophysical Journal 100 (2011), 1139. [40]
Paluch, P.; Pawlak, T.; Amoureux, J.-P.; Potrzebowski, M. J. Simple and accurate
determination of X–H distances under ultra-fast MAS NMR Journal of Magnetic Resonance 233 (2013), 56. [41]
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simulation of solid and liquid state NMR experiments, Journal of Magnetic Resonance 178 (2006), 248. [42]
Tošner, Z.; Andersen, R.; Stevensson, B.; Edén, M.; Nielsen, N. C.; Vosegaard, T.
Computer-intensive simulation of solid-state NMR experiments using SIMPSON, J. Magn. Reson. in press (2014). [43]
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densely coupled spin systems with applications to spin chemistry, Journal of Chemical Physics 132 (2010), 174101. [44]
Savostyanov, D. V.; Dolgov, S. V.; Werner, J. M.; Kuprov, I. Exact NMR
simulation of protein-size spin systems using tensor train formalism, Phys. Rev. B 90 (2014), 085139.
20
Figure Captions
Figure 1. A series of angular transformations used to describe a uniaxially rotating membrane protein reconstituted in spherical liposomes under the conditions of MAS. Here the overall sample rotor orientation is given by the angle θ, sample rotation about the angle φ occurs with the frequency ωr , and the spin bearing molecules are distributed over a sphere with the mosaic spread angle Δθ . Uniaxial diffusion of the molecules about their local alignment axes occurs in the angle ψ with diffusional rate D|| . A bond connecting two interacting spin dipoles forms an angle θB with respect to the diffusion axis.
Figure 2. A comparison between grid-free and direct integration approaches for powder averaging of 1H-15N dipolar couplings under MAS at the spinning speeds as indicated. The average simulation time is ca. 460 sec in the grid-free case (A-D) and ca. 60-80 sec for the direct integration method (E-H). When direct integration over the angle Δθ is performed, the size of the Liouvillian supermatrix is considerably decreased thus yielding faster computational time, even though 1000 spectra were co-added. For further details of the simulations cf. the text.
Figure 3. CP-PI Recoupling pulse sequence (from ref. [40]) simulated for protein rotational diffusion with coefficient D|| = 2×105 s-1 at ω = 20 kHz MAS for the various bond angles θ as r
B
indicated. The powder patterns are averaged out differently by uniaxial rotational diffusion depending on the angle θ that the bond connecting the two interacting spins forms with the axis B
of diffusion.
21
Figure 4. CP-PI Recoupling pulse sequence [40] simulated for slow rotational diffusion with the coefficient D|| = 2×104 s-1 at ω = 20 kHz MAS for the various bond angles θ as indicated. As in r
B
Fig. 3, the powder patterns are averaged out differently depending on the angle θ ; however, B
spectral resolution is significantly affected by slower diffusion.
Figure 5. Correlation between the absolute values of the observed splittings, Δ, and the bond angle-factors, |3cos2θB -1|/2. A. Rotational diffusion coefficient D|| = 2×105 s-1; B. Slow rotational diffusion coefficient D|| = 2×104 s-1. A very good correlation is obtained in part A; whereas the relatively slow rotational diffusion yields larger discrepancies between the measured and the actual bond-angle factors.
Figure 6. Relative errors in determining the bond-angle factors, |3cos2θB -1|/2 from the recoupled powder patterns as a function of the bond angle, θB , and the rotational diffusion coefficient, D|| , at ωr = 20 kHz MAS spinning frequency. Plots are calculated for the values of the diffusion coefficient D|| = 2×104 s-1 (dotted line); D|| = 2×105 s-1 (dashed line); and D|| = 5×106 s-1 (solid line). Greater diffusional rates provide better accuracy in determining the bond angle due to the more complete motional averaging of the powder patterns.
22
Figgurre 1.
23
Figgurre 2.
24
Figgurre 3.
25
Figgurre 4.
26
Figgurre 5.
27
Figgurre 6.
28
MAS superoperator is derived by assuming stochastic averaging Recoupled dipolar spectra were simulated in the presence of uniaxial diffusion Accuracy for deriving angular restraints from the dipolar spectra was assessed
29
30
S1090-7807(14)00268-7 http://dx.doi.org/10.1016/j.jmr.2014.09.023 YJMRE 5524
To appear in:
Journal of Magnetic Resonance
Received Date: Revised Date:
5 August 2014 16 September 2014
Please cite this article as: A.A. Nevzorov, Coherent and Stochastic Averaging in Solid-State NMR, Journal of Magnetic Resonance (2014), doi: http://dx.doi.org/10.1016/j.jmr.2014.09.023
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
09/15/14
Coherent and Stochastic Averaging in Solid-State NMR Alexander A. Nevzorov Department of Chemistry, North Carolina State University 2620 Yarbrough Drive, Raleigh, North Carolina, 27695-8204 Phone: (919) 515-3199 E-mail: [email protected] Keywords: Stochastic Liouville equation, uniaxial diffusion, motional averaging, dipolar recoupling, membrane proteins
Abstract
A new approach for calculating solid-state NMR lineshapes of uniaxially rotating membrane proteins under the magic-angle spinning conditions is presented. The use of stochastic Liouville equation (SLE) allows one to account for both coherent sample rotation and stochastic motional averaging of the spherical dipolar powder patterns by uniaxial diffusion of the spin-bearing molecules. The method is illustrated via simulations of the dipolar powder patterns of rigid samples under the MAS conditions, as well as the recent method of rotational alignment in the presence of both MAS and rotational diffusion under the conditions of dipolar recoupling. It has been found that it is computationally more advantageous to employ direct integration over a spherical grid rather than to use a full angular basis set for the SLE solution. Accuracy estimates for the bond angles measured from the recoupled amide 1H-15N dipolar powder patterns have been obtained at various rotational diffusion coefficients. It has been shown that the rotational alignment method is applicable to membrane proteins approximated as cylinders with radii of approximately 20 Å, for which uniaxial rotational diffusion within the bilayer is sufficiently fast and exceeds the rate 2×105 s-1.
2
1. Introduction The explicit time-dependence of the relevant Hamiltonians in MAS NMR makes the calculation of spectral lineshapes especially challenging. Notable approaches to this problem include the Average Hamiltonian Theory [1-3], which is based on the Magnus expansion, and more elaborate treatments employing the multi-modal Floquet theories [4-7], which involve expansions of the density matrix into a set of operators with periodic time dependences at integral rotor frequencies. At the same time, especially in biological samples, the simultaneous sample rotation and molecular dynamics (both local and global, slow and fast) can profoundly influence the observed NMR lineshapes. A detailed understanding of the arising lineshape patterns can allow one to extract structural and dynamic information about the sample. Of particular interest here is the uniaxial rotational diffusion of a membrane protein embedded in a fluid-like lipid bilayer [8-12]. Recently, a new solid-state NMR method for obtaining structural information for uniaxially diffusing membrane proteins has been proposed [13,14]. It is based on obtaining the bond angles from measuring the scaled splittings of the recoupled spherical powder patterns partially averaged out by anisotropic uniaxial rotational diffusion that the protein molecules undergo about the membrane normal. The idea behind this clever method is that the splitting between the singularities (relative to the rigid limit) is proportional to the factor (3cos2θB -1), where θB is the angle that the bond connecting the two interacting spin dipoles forms with the axis of diffusional rotation. Similar effects have been known for a long time, such as the scaling of the deuterium powder patterns for the rapidly rotating methyl CD3 groups [15]. However, without a priori knowledge of the diffusion rates, it is rather difficult to predict whether the spectra of uniaxially diffusing protein molecules would fall into a particular motional regime. Moreover, even in the intermediate motional regime, a description of static
3
lineshapes in the presence of stochastic uniaxial rotation modeled as three-site jumps [16] becomes quite involved. Suffice it to say that a rigorous description of this effect under the MAS conditions at arbitrary diffusion rates and, in addition, in the presence of radiofrequency irradiation, is challenging [17]. To address this problem for the general motional case, we utilize here the method of Stochastic Liouville equation (SLE). The SLE approach allows one to simulate the spectra at arbitrary rotation and diffusion coefficients, from fully static to fastmotional limits, and has been introduced in ESR by Freed and co-workers [18,19] following the seminal work of Kubo [20,21]. Over the past decades, the SLE approach has also been applied to NMR [22-28] to describe the effect of intermediate motional time scales on NMR relaxation and lineshapes. Recently, the SLE method has been applied to simulate solid-state NMR spectra in the presence of uniaxial rotational diffusion of protein molecules in macroscopically aligned bilayers, [29,30] and also to treat the time-dependent rotation of the sample in terms of a generalized MAS superoperator [31]. In the present work, we provide an alternative derivation of the relaxation superoperator for the case of MAS NMR of randomly oriented molecules without using the Fokker Planck equation [31]. Furthermore, we illustrate the utility of SLE for calculating MAS lineshapes in simple and more complex cases, such as the recoupling of the dipolar powder patterns under the conditions of ‘rotational alignment’ in the presence of both coherent and stochastic rotations of the spin-bearing molecules.
4
2. Derivation of the SLE under the MAS conditions As in virtually every NMR experiment, the formal solution for the Liouville-von Neumann equation for the density matrix ρ(t) under the evolution governed by a time-dependent Hamiltonian, H(t), viz.
∂ ρ (t) = −i[H (t), ρ (t)] ∂t
(1)
can be written compactly as:
⎤ ⎡ t g(t) = expO ⎢i ∫ L(t ')dt '⎥ g(0) ⎦ ⎣ 0
(2)
Where the Liouvillian superoperator is given by:
L(t) = H T (t) ⊗ 12 N −12 N ⊗ H (t)
(3)
Here 12N is a 2N-by-2N identity matrix (for spins ½) and ⊗ stands for the Kronecker product, the symbol “O” denotes the Dyson time-ordering, and we omit for the reason of compactness the usual hat and double hat notations for the operators and superoperators, respectively. The explicit time dependence is contained in the rotation angles that the anisotropic interactions (e.g. chemical shift tensor and dipolar) depend on. Furthermore, in Eq. 2, g(0) = vec[ρ(0)] denotes the vectorization [32] of the initial density matrix (i.e., a stacking of its columns to form a density vector, g). We also note that the plus sign in front of the Liouvillian is in accord with the above definition of the Liouvillian, Eq. (3), and the vectorization operation of the density matrix as defined in ref. [32]. If the time dependence of the Liouvillian is solely contained in a single angle, φ(t), the formal solution of Eq. 2 can be expanded as a time-ordered exponential (t1>t2>t3>…) [33,34]:
5
t
t
exp O i ∫ L[ϕ (t ')]dt ' = 1+ i ∫ L[ϕ (t1 )]dt1 + 0
0
t
t1
0
0
t
t1
t2
0
0
0
(i)2 ∫ dt1 ∫ dt 2 L[ϕ (t1 )]L[ϕ (t 2 )]+
(4)
(i)3 ∫ dt1 ∫ dt 2 ∫ dt 3L[ϕ (t1 )]L[ϕ (t 2 )]L[ϕ (t 3 )]+ ... When the molecules are randomly distributed around the axis of MAS rotation, the resulting NMR signal can be calculated as an average over the distribution of the molecules. However, instead of summing over deterministic trajectories, the average can also be treated as an average over a stochastic Markov process, thus yielding a time-average propagator: t
t
expO i ∫ L[ϕ (t ')]dt ' = 1+ i ∫ dt1 ∫ d ϕ1L(ϕ1 )p(ϕ1 ,t1 )+ 0
0
t1
t
(i)2 ∫ dt1 ∫ dt 2 ∫ d ϕ1 ∫ d ϕ 2 L(ϕ1 )L(ϕ 2 )p(ϕ1 ,t1; ϕ 2 ,t 2 )+ 0
0
t
t1
t2
0
0
0
(i)3 ∫ dt1 ∫ dt 2 ∫ dt 3 ∫ d ϕ1 ∫ d ϕ 2 ∫ d ϕ 3L(ϕ1 )L(ϕ 2 )L(ϕ 3 )p(ϕ1 ,t1; ϕ 2 ,t 2 ; ϕ 3 ,t 3 )+ ...
(5)
where pn(φ1, t1 ; φ2, t2 ;... φn, tn) is the n’th order joint probability. For a Markovian path average, the Chapman-Kolmogorov equation employing the conditional probabilities, P(φn-1 , tn-1 | φn , tn) can be written as:
pn (ϕ1 ,t1; ϕ 2 ,t 2 ;...ϕ n ,t n ) = P(ϕ1,t1 | ϕ 2 ,t 2 )...P(ϕ n−1 ,t n−1 | ϕ n ,t n )p(ϕ n ,t n )
(6)
To evaluate the expansion, Eq. 5, let’s introduce an auxiliary function g(φ, t) such that:
g(t) =
2π
∫ dϕ g(ϕ ,t)
(7)
0
Following ref. [34] it can be easily verified that the auxiliary function is given by the expansion:
6
t
g(ϕ ,t) = [1 peq (ϕ )+ i ∫ dt1 ∫ d ϕ1L(ϕ1 )P(ϕ ,t | ϕ1 ,t1 )peq (ϕ1 )+ 0
t1
t
(i)2 ∫ dt1 ∫ dt 2 ∫ d ϕ1 ∫ d ϕ 2 L(ϕ1 )L(ϕ 2 )P(ϕ ,t | ϕ1 ,t1 )P(ϕ1 ,t1 | ϕ 2 ,t 2 )peq (ϕ 2 )+ (i)
3
0
0
t
t1
t2
∫ dt ∫ dt ∫ dt ∫ dϕ ∫ dϕ ∫ dϕ L(ϕ )L(ϕ 1
0
2
0
3
1
2
3
1
2
)L(ϕ 3 )P(ϕ ,t | ϕ1 ,t1 )P(ϕ1 ,t1 | ϕ 2 ,t 2 )P(ϕ 2 ,t 2 | ϕ 3 ,t 3 )peq (ϕ 3 )+ ...]g(0)
0
(8) where, for a stationary Markov process: p(φ, t) = peq(φ) =1/(2π). Moreover, if the averaging process is coherent (such as the MAS rotation about the angle φ with the frequency ωr), the conditional probability can be written as:
P(ϕ n−1 ,t n−1 | ϕ n ,t n ) = δ[ϕ n−1 − ϕ n − ωr (t n−1 − t n )]
(9)
Which has the following formal property:
1 ∂P(ϕ n−1,t n−1 | ϕ n ,t n ) ∂P(ϕ n−1,t n−1 | ϕ n ,t n ) = ωr ∂t n−1 ∂ϕ n−1
(10)
By differentiating g(φ, t) with respect to time, t and using the chain rule for the integrals involved, one obtains for the second term of the expansion:
∂ ∂t
t
∫ dt ∫ dϕ L(ϕ )P(ϕ ,t | ϕ ,t )p 1
1
1
1
1
eq
(ϕ1 ) =
0
∫ dϕ1L(ϕ1 )P(ϕ ,t | ϕ1,t)peq (ϕ1 )+ L(ϕ )peq (ϕ )+ ω r
∂ ∂ϕ
t
∫ ∫ dϕ L(ϕ ) 1
1
0
∂P(ϕ ,t | ϕ1 ,t1 ) peq (ϕ1 ) = ∂t
t
∫ ∫ dϕ L(ϕ )P(ϕ ,t | ϕ ,t )p 1
1
1
1
eq
(ϕ1 )
0
(11)
All other terms in the expansion for g(φ, t) can be evaluated in a similar manner and re-combined into two main terms: one having the superoperator iL in front of it, and the other containing the operator ωr∂/∂ϕ. We, therefore, obtain that g(φ, t) satisfies the corresponding Stochastic Liouville Equation in the presence of coherent rotation of the sample about the angle φ:
7
∂g(ϕ ,t) ∂g(ϕ ,t) = iLg(ϕ ,t)+ ω r ∂t ∂ϕ
(12)
Which can be written more compactly as:
∂g(ϕ ,t) = (iL + Γ)g(ϕ ,t) ∂t where the superoperator Γ is defined as: Γ = ω r
(13)
∂ . If, at the same time, the molecule undergoes ∂ϕ
stochastic rotational diffusion about another angle ψ with diffusion coefficient D||, by assuming stochastic independence of the two processes one can write for the two simultaneous rotations that:
Γ = ωr
∂ ∂2 + D|| 2 ∂ϕ ∂ψ
(14)
while Eq. 13 is still generally valid. Thus, Eq. (14) allows one to treat both coherent and stochastic averaging in a single formalism. Also, it should be noted in the derivation of Eq. (12) the non-intuitive differentiation of the probability density with respect to the density matrix employed by the previous authors [31,35] was not necessary.
3. Numerical solutions to the SLE To obtain the solution of the SLE, we shall use a minimalistic geometrical framework [36] to describe basic NMR experiments performed on a membrane protein molecule incorporated within spherical liposomes, Fig. 1.
In general, the Hamiltonian contains the
orientational dependence of the magnetic field B0 relative to some arbitrary molecular frame as given by the angles α and β, which are contained in the chemical shift anisotropy (CSA) and dipolar terms (written in the angular frequency units):
8
H = ΔΩI z + ω S S x + ω I I x + ν CSA ( β ,α )S z + ν DIP ( β ,α )I z S z
(15)
Here ΔΩ accounts for the frequency offset of the proton (I spins), ωS and ωI are the radiofrequency amplitudes for the low (or S) spin (e.g. nitrogen or carbon) and the proton, respectively. Only the radiofrequency irradiation along the x axis in the rotating frame is included here; other phases can be accounted for in a similar manner. The explicit dependences of the CSA, νCSA , and dipolar frequencies, νDIP , on the angles α and β are given in ref. [30]. The decomposition of the angular functions involving the angles α and β into the individual angles describing the overall sample orientation (θ), sample rotation (φ) spherical distribution or mosaic spread (Δθ) and, finally, uniaxial diffusion (ψ) is depicted in Fig. 1. Mathematically, it can be readily accomplished by using the irreducible spherical basis of spherical harmonics of rank 2 [30,36]: →
→
Y (β , α ) = Y (θ ,0)× D(ϕ ,Δθ , ψ )× D(0,θ B , ϕ B )
(16)
Here Y ’s are the row vectors of spherical harmonics of rank 2 (ordered according to their index m as: 2, 1, 0 -1, -2), D’s are the second-rank Wigner rotation matrices, θB and φB are the angles that the axis of uniaxial diffusion a forms with respect to given bond (or a principal axis system, in the case of CSA tensor). For calculating the dipolar patterns, as is done in the present paper, only θB becomes of interest; the MAS patterns due to CSA averaging have been calculated elsewhere [31]. To look for the solution of Eq. 13, one can use the expansion in terms of the Wigner rotation matrix coefficients (denoted compactly as a ket, |l, m, m’ >) involving the indices l (for the angle Δθ), m (for φ), m’ (for ψ), viz:
g(ϕ ,Δθ , ψ ,t) =
∑g
l,m,m '
(t) ⊗ l, m, m '
(17)
l,m,m '
9
Here the index l runs from 0 to a certain cut-off value, and the indices m and m’ run from –l to l. The Hilbert-space representation of the Liouvillian marix, L involves the integrals of three Wigner rotation matrices containing products of the Clebsch-Gordan coefficients [31,37]. Alternatively, one can co-add various spectra calculated for different values of Δθ by using either the Monte-Carlo method [36] or a uniform grid with the weighting function ½×sinΔθ, and utilize the Fourier expansions in just m and m’. In the Hilbert space, this expansion can be written in terms of the Kronecker products in a vector form as:
g(ϕ ,Δθ , ψ ,t) = ∑ g m,m ' (Δθ ,t) ⊗ m ⊗ m' m,m '
(18)
This way of expanding the solution significantly reduces the size of the superoperator matrices and makes the calculation of the superoperator matrices especially straightforward (they become tridiagonal in each angular subspace). In Eqs. 17-18 we have used again the Kronecker product,
⊗ to factorize the spin and the angular subspaces in the Hilbert space of the total superoperator, iL+Γ. The final time-averaged density matrix simply corresponds to the expansion functions with l=0 or m=m’=0, depending on whether Eq. 17 or Eq. 18 is used, respectively. The solution for the observed NMR signal is then given by the quadratic form:
G(t) = g 0T exp[(iL + Γ)t]g(0)
(19)
where g0 represents the vectorization of the detection operator, usually S- or Sx . We note that the SLE approach renders the superoperator matrices time-independent, which requires only one (or a few, depending on the pulse sequence modeled) diagonalization steps in simulating the spectra.
10
4. Results All simulations in this work have been performed on a desktop computer having two sixcore processors and operating at 2.66 GHz using a Matlab (Mathworks, Inc.) script. Figure 2 illustrates the well-known result of averaging of the NH amide dipolar couplings (with the constant χ = 10 kHz) by the magic angle spinning calculated using the SLE, Eq. 13. In Figs 2AD we use the expansion of Eq. 17 (with the the spin space factored out and no rotational diffusion, D|| = 0, m’ = 0) with the cutoff value l = 50. Various spinning speeds are used, from the nearly static case of ωr = 100 Hz to the spinning rate of ωr = 5 kHz illustrating the appearance of the spinning side bands. By contrast, in Figs. 2E-H we utilized the expansion of Eq. 18 with the absolute-value cutoff for the index m of 100, (still with D|| = 0) with co-adding of 1000 spectra at various values of Δθ randomly chosen by the Monte-Carlo method. We note that the cutoff indices are determined by the spinning rate, and much lower values <5 are sufficient to obtain convergent spectra for faster speeds, (>5 kHz) which makes the calculations considerably faster. It is clear that the direct integration combined with expansion of Eq. 19 is computationally superior to the full grid-free approach [31] yielding at least a factor of six gain in speed with a similar degree of convergence for the spectra. Figure 3 represents the solution of Eq. 13 in the presence of both MAS spinning and rotational diffusion given by the superoperator of Eq. 14 at various bond angles, ranging from θB = 0o to 90o. Using the Stokes-Einstein Equation and approximating the protein as a cylinder [8] having the radius of 20 Å (which would be a rough estimate for the transmembrane core of a GPCR [38]), we obtain for the value of D|| = 2×105 s-1. Here a membrane bilayer is assumed to have hydrophobic thickness of 30 Å and viscosity of ~100 cPoise [8]. Similar rates have been reported elsewhere [9-12]. This value serves as a rough estimate for the actual diffusion rate of a
11
membrane protein within the bilayer. More recent studies [39], however, suggest a considerably faster diffusion coefficient of D|| = 5×106 s-1 for GPCRs, presumably due to the perturbed local bilayer viscosity at the protein interface, which would be even more beneficial for the rotational alignment method. The recently developed CP-PI pulse sequence [40] was used to achieve the recoupling of the dipolar Hamiltonian. The spin Hamiltonian of Eq. 15 was used and the spinning speed was chosen to be 20 kHz with 60 kHz rf amplitude on the low side. The starting vector was taken as:
g(0) = vec(Sx − I x ) ⊗ (0,..., 0,1,0,...0) 0,1,0,.., 0) ⊗ (0,..., T
2 m +1
T
(20)
2 m ' +1
The chemical shifts at these relatively high rf amplitudes have been neglected. To yield first-order rotary dipolar recoupling, the rf amplitudes on the proton side were additionally toggled between 80 and 40 kHz for the even and odd dwells, and the dwell time was chosen to be one-half of the rotor period, π/ωr . The cutoffs for the index m was chosen to be ±4 and ±6 for m’, thus making the total matrix size of 16(2|m|+1)(2|m’|+1)=1872 by 1872, and 400 spectra were co-added by using a uniform distribution in the angle Δθ with the weighting function ½×sinΔθ. It can be seen from Fig. 3 that at these parameter values a good convergence of line shapes was achieved. It can also be seen that, depending on the value of θB the recoupled powder patterns are scaled differently depending on the angle that the chemical bond connecting two interacting spins forms with the instantaneous axis of rotation. This provides the source for the structural information that can be directly extracted from the spectra [13,14]. In Fig. 4, the diffusion coefficient was decreased down to D|| = 2×104 s-1 to illustrate the effect of slow motions on the resolution of the rotationally averaged recoupled powder patterns. While the slower rotational diffusion still yields the scaling of the recoupled powder patterns, resolution becomes
12
significantly worse, especially at the values of the bond angle that are close to the magic angle (i.e. θB ≈ 54.7o). Figure 5 shows correlation plots of the ratio of the recoupled NH dipolar splittings as determined from the simulations relative to their static value, χ/√2 as a function of the actual bond-angle factor, |3cos2θB -1|/2. It can be seen that at the diffusion rate of D|| = 2×105 s-1 there is a very good correlation between the relative splittings measured from the singularities and the actual bond factor. The relative errors vary from 1-5% for larger splittings up to 14% for smaller splittings. However, a systematic underestimation of the true bond factors has been observed for nearly all bond angles. We also note that for measuring the aliphatic carbon 1H-13C couplings for the same molecule at the same degree of precision this lower limit would need to be approximately doubled due to the linear scaling property of the SLE, Eq. 13, at the relatively fast MAS speeds and high rf amplitudes. When the diffusional rate is decreased down to D|| = 2×104 s-1 in the simulations (which corresponds to intermediate-to-slow motional regime at the given dipolar coupling constant of χ = 10 kHz; πχ/D|| = 1.57 at these values), the agreement becomes worse, cf. Fig. 4B, thus potentially interfering with the accuracy of the experimentally derived angular factors. However, if the recently reported [39] value of D|| = 5×106 s-1 is used instead, the maximum deviation of the measured splittings from the true bond factors considered in the simulations is reduced to less than 10%. Even though the correlation plots of Fig. 5 look quite good, they do not reflect individual errors for the specific bond angles that may ultimately affect the final structure calculation. Figure 6 shows plots of relative errors in determining the bond-angle factors from the rotationally averaged recoupled dipolar splittings, Δ, as a function of the bond angle, θB and the diffusion coefficient, D|| . The relative error has been calculated as: 13
Error(%) =
2 2Δ
χ 3cos 2 θ B −1
−1 ×100%
(21)
It is clear that the largest error arises near the value θB = 54.7, where the splitting becomes small and on the order of the spectral linewidth. Also, the faster is the diffusion, the greater accuracy in determining the bond angle can be achieved. However, markedly smaller errors are associated with the splittings that correspond to smaller bond angles (θB < 30o) at all diffusion rates considered. 5. Conclusions The presented SLE approach provides an alternative to the existing numerical methods for simulating MAS spectra that involve direct time-step integration [41,42]. Importantly, the presented formalism allows one to describe more complicated time dependencies, both periodic and stochastic, and allows one to simulate MAS NMR spectra at arbitrary rotational and diffusional rates. It suffers, however, from the relatively large size of the superoperators matrices involved, which limits its applicability to only a few spins, at least at present computational powers. This problem, however, can be circumvented by using space-restriction techniques as implemented in SPINACH [43], and, more recently, by the “tensor train” formalism [44]. In addition, it was determined that an upper-size limit for obtaining reliable bond-angle factors from ‘rotational alignment’ corresponds to proteins with an average radius of 20 Å when roughly treated as cylinders. It is worthwhile to mention that, according to the previous work [30,36] the rate D|| = 2×105 s-1 would be insufficient to obtain sharp uniform linewidths for membrane proteins incorporated in magnetically aligned bicelles in oriented-sample NMR. By contrast, the spherical powder patterns as recoupled in MAS NMR appear to be less sensitive to the
14
diffusional rate since it is the overall distribution and singularities that are averaged out by the motions, not the individual bond orientations. We also note that using the SLE-based model one may be able to interpret the scaled dipolar couplings even at slower motional rates if the appropriate correction factors or a pre-defined measure of uncertainty is used in structure calculations. This, however, would require a more accurate knowledge of the radius of gyration for the protein and the value for the viscosity of the bilayer environment, which still remains somewhat controversial. The SLE treatment presented herein is potentially applicable to other solid-state NMR applications including simulations of more complex multidimensional pulse sequences.
Acknowledgement The author wishes to thank Dr. Ilya Kuprov for stimulating discussions. Supported by NSF MRI 1229547.
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Figure Captions
Figure 1. A series of angular transformations used to describe a uniaxially rotating membrane protein reconstituted in spherical liposomes under the conditions of MAS. Here the overall sample rotor orientation is given by the angle θ, sample rotation about the angle φ occurs with the frequency ωr , and the spin bearing molecules are distributed over a sphere with the mosaic spread angle Δθ . Uniaxial diffusion of the molecules about their local alignment axes occurs in the angle ψ with diffusional rate D|| . A bond connecting two interacting spin dipoles forms an angle θB with respect to the diffusion axis.
Figure 2. A comparison between grid-free and direct integration approaches for powder averaging of 1H-15N dipolar couplings under MAS at the spinning speeds as indicated. The average simulation time is ca. 460 sec in the grid-free case (A-D) and ca. 60-80 sec for the direct integration method (E-H). When direct integration over the angle Δθ is performed, the size of the Liouvillian supermatrix is considerably decreased thus yielding faster computational time, even though 1000 spectra were co-added. For further details of the simulations cf. the text.
Figure 3. CP-PI Recoupling pulse sequence (from ref. [40]) simulated for protein rotational diffusion with coefficient D|| = 2×105 s-1 at ω = 20 kHz MAS for the various bond angles θ as r
B
indicated. The powder patterns are averaged out differently by uniaxial rotational diffusion depending on the angle θ that the bond connecting the two interacting spins forms with the axis B
of diffusion.
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Figure 4. CP-PI Recoupling pulse sequence [40] simulated for slow rotational diffusion with the coefficient D|| = 2×104 s-1 at ω = 20 kHz MAS for the various bond angles θ as indicated. As in r
B
Fig. 3, the powder patterns are averaged out differently depending on the angle θ ; however, B
spectral resolution is significantly affected by slower diffusion.
Figure 5. Correlation between the absolute values of the observed splittings, Δ, and the bond angle-factors, |3cos2θB -1|/2. A. Rotational diffusion coefficient D|| = 2×105 s-1; B. Slow rotational diffusion coefficient D|| = 2×104 s-1. A very good correlation is obtained in part A; whereas the relatively slow rotational diffusion yields larger discrepancies between the measured and the actual bond-angle factors.
Figure 6. Relative errors in determining the bond-angle factors, |3cos2θB -1|/2 from the recoupled powder patterns as a function of the bond angle, θB , and the rotational diffusion coefficient, D|| , at ωr = 20 kHz MAS spinning frequency. Plots are calculated for the values of the diffusion coefficient D|| = 2×104 s-1 (dotted line); D|| = 2×105 s-1 (dashed line); and D|| = 5×106 s-1 (solid line). Greater diffusional rates provide better accuracy in determining the bond angle due to the more complete motional averaging of the powder patterns.
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Figgurre 1.
23
Figgurre 2.
24
Figgurre 3.
25
Figgurre 4.
26
Figgurre 5.
27
Figgurre 6.
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MAS superoperator is derived by assuming stochastic averaging Recoupled dipolar spectra were simulated in the presence of uniaxial diffusion Accuracy for deriving angular restraints from the dipolar spectra was assessed
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