Web Release Date: January 31,
Determination of Internuclear Distances in Uniformly Labeled Molecules by Rotational-Resonance Solid-State NMR
Contribution from the Laboratory for Physical Chemistry, ETH-Zürich, ETH-Hönggerberg, 8093-Zürich, Switzerland
Received August 20, 2002
Abstract:
Rotational-resonance magic-angle spinning NMR experiments are frequently used to measure dipolar couplings and to determine internuclear distances. So far most measurements were performed on samples containing isolated spin pairs. Thus, extensive structure elucidation, for example in biomolecules, requires the preparation of a whole set of doubly labeled samples. Here, we describe the analysis of the rotational-resonance polarization-exchange curves obtained from a single, uniformly labeled sample. It is shown experimentally that, at a magnetic field of 14.09 T, the rotational-resonance conditions in uniformly 13C-labeled threonine are sufficiently narrow to permit the measurement of five distances between the four carbon spins with an accuracy of better than 10%. The polarization-exchange curves are analyzed using a modified two-spin model consisting of the two active spins. The modified model includes an additional offset in the final polarization, which comes from the coupling to the additional, passive, spins. The validity of this approach is experimentally verified for uniformly 13C-labeled threonine. The broader applicability of such a model is demonstrated by numerical simulations which quantify the errors as a function of the most relevant parameters in the spin system.
High-resolution solid-state NMR is a useful tool to study both
micro- and noncrystalline materials. To achieve the spectral
resolution required for measuring specific distance constraints,
the application of magic-angle spinning (MAS) is usually
mandatory. However, the improved spectral resolution comes
at the expense of loosing the information contained in the
anisotropic interactions. One example of such an averaged
anisotropic interaction is the dipolar-coupling tensor, which is
proportional to the inverse cube of the distance between two
nuclei. To reintroduce the anisotropic dipolar interaction into
MAS NMR spectra, dipolar recoupling methods are usually
applied.1,2
Recoupling methods may be classified into broad-banded and selective methods. In broad-banded methods, all the spins are recoupled simultaneously. The polarization-transfer dynamics in such multispin experiments contain, possibly among other contributions, the information about all dipolar couplings in the system. It is, however, difficult to extract this information quantitatively3 and the contribution of the small couplings tends to be negligible. Selective methods ideally recouple only two spins at a time and determine accurately a single internuclear distance per experiment. The pair of "active" spins is selected by its spectral properties, for example, the isotropic chemical-shift difference.
The rotational-resonance experiment (RR)4-8r =
-
. Despite this
inherent selectivity of rotational-resonance recoupling,6 most
practical applications have been to systems where relatively
isolated spin pairs have been introduced chemically by selective
labeling. This leads to particularly simple polarization-exchange
dynamics. The drawback, however, is that producing the
selectively labeled compounds is both expensive and time-consuming.
In rotational-resonance experiments, weak dipolar couplings are characterized through the measurement of polarization exchange between the two recoupled spins. These polarization-exchange curves are analyzed in terms of the internuclear distance and the zero-quantum relaxation time with other parameters such as chemical-shielding tensors playing a minor role at the n = 1 rotational resonance condition. This approach is known to work well in selectively labeled systems with isolated spin pairs because there are no additional dipolar couplings to "passive" spins which could influence the polarization-exchange curves.
In this paper we demonstrate that, under certain circumstances, the polarization exchange at the rotational-resonance condition in uniformly labeled systems can be modeled by a modified two-spin system. In the presence of the additional passive spins the difference polarization evolves towards a finite value, while it evolves toward zero in an isolated two-spin system. Similar effects were observed in systems with inhomogeneously broadened lines.9 The modified long-time behavior of the difference polarization can be included phenomenologically into the analysis of the polarization-exchange curves. Such a simple modified two-spin model of rotational-resonance polarization exchange gives accurate results if the isotropic chemical-shift differences between the passive and active spins are far enough from any rotational-resonance conditions.
In this paper we will show experimental rotational-resonance measurements on fully labeled L-threonine that will be analyzed using the modified two-spin model. In addition, synthetic data generated by numerical simulations of model three-spin systems will show in which range of parameters such a model gives accurate results.
NMR Experiments. All experiments were performed on uniformly
13C and 15N labeled L-threonine (Cambridge Isotope Laboratories) and
diluted in natural-abundance material at a ratio of 1:10 to minimize
intermolecular dipolar contacts. Subsequently the sample was recrystallized from a hot aqueous solution. Narrow resonances (full width at
half-height 0.15 ppm) observed in the CP-MAS spectra indicated a
uniform crystal form throughout the sample. NMR experiments were
recorded at a static magnetic field of 14.09 T using a Bruker Avance
600 spectrometer. All data were acquired with a Bruker 2.5 mm o.d.
triple-resonance CP-MAS probe. The spinning frequency was stabilized
to ±5 Hz. Rotational-resonance experiments were performed following
adiabatic cross polarization from 1H to 13C.10 Afterward, the polarization
of one of the resonances was selectively inverted using a DANTE pulse
train.11,12s with an effective 180
pulse of 25
s. After a variable mixing time, the polarization was
converted to single-quantum coherence by a hard 90
pulse and
detected. During the mixing time and data acquisition, TPPM13
decoupling was applied using a 10
phase angle and a pulse length of
5
s. Each point on the polarization-exchange curve was the result of
the summation of 128 transients. An example of a typical spectrum of
L-threonine used to determine the difference polarization used for the
calculation of the experimental exchange curves is shown in Figure 1
together with its assignment. The data were processed and integrated
in Felix 97.0 (Accelrys, CA).
Simulations. Numerical Liouville-space simulations of a hypothetical
three-spin system were performed using the GAMMA spin-simulation
environment14 extended by a block-diagonalization package to speed
up the diagonalization of the Liouvillian matrixes. The input to these
simulations included both the size and the orientation of the chemical-shift anisotropy tensors (CSA), the dipolar-coupling tensors, and the
isotropic J-couplings (see Table 1). The relaxation was implemented
as an uncorrelated random field fluctuation along the z-direction, with
an identical rate constant for every spin (kz = 100 s-1). In an isolated
two-spin system, this leads to a zero-quantum relaxation time of T2ZQ
= 5 ms.
To obtain the apparent distances and relaxation times for the active
spin pair from the exchange curves obtained by the numerically exact
three- or four-spin simulations or from experiments on fully labeled
samples, the data were fitted by the model of a single homonuclear
spin pair at rotational resonance5 which includes the following
parameters: the chemical-shift anisotropy of both spins, the dipolar
coupling, a phenomenological zero-quantum relaxation-rate constant,
and the initial difference polarization. In addition to these standard
parameters, our modified model includes the value of the final difference
polarization. Under the influence of T2ZQ-relaxation, the difference
polarization approaches this final difference polarization for t
.
Nonlinear least-squares fits of the data were carried out using the
MINUIT routines.15 The dipolar-coupling constant, T2ZQ, the initial
difference polarization, and the final difference polarization were free
parameters for the fit. The remaining parameters were kept fixed. The
isotropic chemical shifts of L-threonine were measured in a spectrum
recorded at an MAS frequency of 22.5 kHz as follows: C' 173.0 ppm;
C 62.4 ppm; C
68.0 ppm; C
21.6 ppm. The magnitude and the
orientation of the chemical-shielding tensors were taken from James
et al.16
Six experimental polarization-transfer curves for uniformly
13C and 15N labeled threonine are shown in Figure 2. The
experimental data were fitted by the model as described in the
previous section. The fits yield quite good agreement with the
measured data with some systematic deviations. The parameters
obtained from the analysis of these six rotational-resonance
experiments are given in Table 2. The experimental data were
normalized by setting the magnitude of the initial experimental
data point to 1. The internuclear distances obtained from these
fits compare favorably with the distances obtained from crystal-structure data (see Table 2), with a maximum deviation of 0.18
Å (<10%). The good agreement between the fitted distances
and the distances obtained from X-ray diffraction data suggest
that under the given experimental conditions (B0 = 14.09 T)
the selectivity of the rotational-resonance condition is sufficiently narrow to permit an analysis of rotational-resonance
data from uniformly labeled threonine in terms of a simple
modified two-spin model.
To assess if this approach works for spin systems other than
threonine and in order to estimate the magnitude of the
systematic errors that may be incurred in the analysis, polarization-exchange curves were simulated in Liouville-space for
different three-spin systems with two active spins (spins 1 and
2) and one passive spin (spin 3). The passive spin is dipolar
coupled to one of the active spins (spin 1) (see schematic in
Figure 3). For the simulations used in Figure 3A the dipolar
coupling to the passive spin (d13) was chosen such as to represent
a one-bond C-C coupling and the chemical-shift difference
(13) corresponds to the one between C' and C
in an amino
acid at 14.09 T. The detailed parameters used in these simulations are summarized in Table 1. The parameters of the two
active spins, i.e., the dipolar-coupling constant d12 and the
isotropic chemical-shift difference
12 were varied in an
attempt to mimic the range of parameters found in organic
molecules and peptides. The active spin pair was always kept
at the n = 1 rotational-resonance condition by setting
r =
12
. The resulting simulated polarization-transfer curves were
analyzed in the same manner as the experimental data and the
parameters of the fits compared with the input values of the
simulation. The values of the parameters where chosen for the
simulations shown in Figure 3A such that they do not match
directly to the experimentally measured threonine spin system
to demonstrate the generality of the observations. However,
additional simulations (given as Supporting Information) using
a three-spin approximation of the threonine four-spin system
show the same qualitative results and confirm that the general
features of this analysis do not depend on the precise values of
the spin-system parameters.
The relative deviation of the distances extracted with our two-spin model from the "true" distance, ( - r12)/r12 is plotted in
Figure 3A as a function of the distance r12 and the isotropic
chemical-shift difference,
12. As expected, accurate results
(indicated in green) for the internuclear distance r12 were
obtained if the passive spin is far from rotational resonance,
while the largest deviations (indicated in red) occur when the
passive spin is also close to a rotational-resonance condition,
i.e., n
r
13
. The width of these unfavorable recoupling
conditions is of the size of the dipolar coupling to the passive
spin (d13) and becomes smaller for higher-order rotational-resonance conditions. In addition, the J-couplings to the passive
spins can contribute to the deviations in the fitted distance. For
Figure 3A, we have chosen the worst case scenario with the
passive coupling being about d13/(2
) = 2.25 kHz, corresponding to a one-bond carbon-carbon coupling. The errors in Figure
3A, therefore, correspond to the errors in the value of the small
coupling, measured in the presence of a one-bond coupling. It
is noteworthy that the relative size of the unfavorable regions
with large errors will be reduced at higher B0-fields as the dipolar
coupling is independent of the static magnetic field. In the
regions with large errors, the fitted distances are shorter than
the theoretical ones because the oscillation frequency increases
as the passive spin (spin 3) approaches a rotational-resonance
condition. For some parameter values good agreement for the
distance was observed even when there was poor agreement
between the fit and the simulated spectra, corresponding to a
high value of
2. A contour plot of
2 as a function of the
chemical-shift difference,
12, is given Figure 3B. Poor fits
with high
2 (areas indicated in red) are again obtained when a
resonance condition of an active/passive spin pair is approached.
For larger distances, the values of
2 decrease, because
polarization-exchange curves without large oscillations can be
fitted quite well even in the proximity of a rotational-resonance
condition with a passive spin. Thus, from Figure 3 we conclude
that internuclear distances up to 4.5 Å can be measured, in the
presence of a one-bond coupling, with an error of less than 10%
for spin pairs with favorable chemical-shift values (areas
indicated in green/yellow in Figure 3).
The data represented in Figure 3 correspond to a special,
although practically important, situation because 13 was kept
constant at a value typical for a C'-C
spin pair. The influence
of variations in
13 is illustrated in Figure 4 where both
12
and
13 are varied. In these simulations, both internuclear
distances r12 and r13 were fixed at 4 and 1.5 Å, respectively.
Again, large discrepancies between the apparent distance
obtained in our two-spin analysis and the true distance occur
only when the spinning frequency (
r =
12
) is close to a
rotational-resonance condition involving the passive spin (n
r
13
). Simulations of a selection of four-spin systems (data
not shown) gave results similar to those described for a three-spin system.
The chemical-shielding tensor orientations are known to
influence the rotational-resonance polarization-exchange curves
obtained for an isolated two-spin system. The effects on the
fitted distance are, however, usually small at the n = 1
rotational-resonance condition and are often neglected.4 Additional simulations (data not shown) indicate that the same
assumption is justified in the context of the analysis of a three-spin system.
These simulations on three- and four-spin systems support
also the empirical findings in threonine and indicate that similar
accuracies can be expected for other compounds. We have
observed for threonine (see Table 2) that the C'-C distance
of 2.9 Å corresponding to a dipolar coupling constant of 309
Hz was measured relatively accurately in the presence of two
one-bond C'-C
coupling of more than 2 kHz. The largest
deviation between the X-ray and the NMR distances measured
in L-threonine (see Table 2) is found for the spin pair C
-C
.
The measurement of the distance between these two spins
represents the determination of a relatively weak two-bond
coupling (d12/(2
)
569 Hz) in the presence of two one-bond
couplings of more than 2 kHz. In addition, one of the passive
spins (C
) is separated by only 850 Hz from a rotational-resonance condition with one of the active spins (C
). Despite
these unfavorable conditions, the distance measured by rotational-resonance NMR and the distance measured by X-ray diffraction
differ only by about 7%.
The experimental data and our numerical simulations show that, in many practical cases, it is possible to determine internuclear distances of up to 4.5 Å with an accuracy better than 10% using rotational-resonance measurements in uniformly labeled samples. The experiments were performed on a small molecule. They can be translated directly to biologically interesting compounds like neurotransmitters (e.g., acetylcholine where preliminary measurements have been performed) and hormones whose structure could be determined bound to proteins.17 We also foresee that the method is applicable to larger molecules like peptides. Spectral crowding does, by itself, not compromise the power of the analysis presented here as only passive spins with a sizeable coupling need to be taken into account in the analysis.
The analysis introduced here describes a many-spin system as a pair of active spins and a set of passive spins. If the chemical-shift differences between the active and the passive spins are not closer to a rotational-resonance condition than the dipolar coupling constant between the active/passive spin pair, the presence of the passive spins only leads to a finite final polarization approached in the dynamics of the active spin pair and to a moderate systematic error in the internuclear distance. We have quantified the systematic errors for simple model systems and conclude that it is possible to identify critical cases based on the assignment of the resonances in the molecule under study.
The increase in uncertainty in the distance evaluated from the exchange curves in uniformly labeled samples using our modified two-spin model, compared to the one obtained in selectively labeled samples, is often more than offset by the increase in the number of distances which can be measured from a single, uniformly labeled sample. The method presented can be looked at as a homonuclear version of the recently introduced selective heteronuclear recoupling experiments in uniformly labeled molecules (REDOR20).
A table listing additional simulations using a three-spin approximation of the threonine four-spin system. This information is available free of charge via the Internet at http://pubs.acs.org.
* In papers with more than one author, the asterisk indicates the name of the author to whom inquiries about the paper should be addressed.
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|
spin 2 |
|
spin 1 |
spin 3 |
|
1542 Hz |
|
6083 Hz |
9250 Hz |
|
0.2432 |
|
0.2466 |
0.8919 |
( |
(160 |
|
(0 |
(42 |
|
variable |
|
0 |
variable |
dij/(2 |
variable 2248 Hz (1.54 Å) |
|||
( |
(0 |
|||
Jij |
0 Hz 50 Hz |
|
r12/Åb |
T2ZQ/msb |
|||||||
|
|||||||||
sitese |
|
fitted |
fitted |
X-rayc |
fitted |
est.d |
init. pop. |
final pop. |
|
A |
C'-C |
16703 |
2011 |
1.556(4) |
1.54 |
10(8) |
2.1 |
0.94(4) |
0.16(1) |
B |
C'-C |
15854 |
501 |
2.47(3) |
2.55 |
4(1) |
1.7 |
0.93(5) |
0.29(2) |
C |
C'-C |
22831 |
300 |
2.93(3) |
3.09 |
5(1) |
2.9 |
0.94(4) |
0.19(2) |
D |
C |
22831 |
309 |
2.91(4) |
3.09 |
5(1) |
2.9 |
0.91(4) |
0.19(2) |
E |
C |
6128 |
569 |
2.37(4) |
2.55 |
2(1) |
2.7 |
0.95(5) |
0.19(2) |
F |
C |
6977 |
2022 |
1.554(6) |
1.52 |
4(2) |
2.1 |
1.00(5) |
0.17(1) |
a Parameters obtained after fitting the data shown in Figure 2 to theoretical
polarization-exchange curves for a homonuclear two-spin system as
described in the text.b Values in parentheses represent the error (1 standard
deviation) calculated during the fitting of the experimental data.c Values
obtained from the crystal structure of L-threonine.21 d T2ZQ calculated
from the single quantum line widths, assuming 1/T2ZQ = 1/(2(
+
)).22,23 e The first site listed is the one whose resonance line was
inverted in the experiments.