Rotational-resonance distance measurements in multi-spin systems

Article abstract of DOI:10.1016/j.jmr.2004.03.009

It is demonstrated that internuclear distances can be evaluated from rotational-resonance (RR) experiments in uniformly ¹³C-labelled compounds. The errors in the obtained distances are less than 10% without the need to know any parameters of th

Full text of DOI:10.1016/j.jmr.2004.03.009

Journal of Magnetic Resonance 168 (2004) 314–326
www.elsevier.com/locate/jmr
Rotational-resonance distance measurements in multi-spin systemsq
Aswin Verhoeven,a Philip T.F. Williamson,a Herbert Zimmermann,b
Matthias Ernst,a and Beat H. Meiera,*
a
Physical Chemistry, ETH Zurich, CH-8093 Zurich, Switzerland
Max-Planck-Institut f€ur medizinische Forschung, Abteilung Biophysik, Jahnstrasse 29, D-69120 Heidelberg, Germany
b
Received 14 November 2003; revised 8 March 2004
Available online 14 April 2004
Abstract
It is demonstrated that internuclear distances can be evaluated from rotational-resonance (RR) experiments in uniformly 13C-
labelled compounds. The errors in the obtained distances are less than 10% without the need to know any parameters of the spin
system except the isotropic chemical shifts of all spins. We describe the multi-spin system with a simple fictitious spin-1/2 model. The
influence of the couplings to the passive spins (J and dipolar coupling) is described by an empirical constant offset from the ro-
tational-resonance condition. Using simulated data for a three-spin system, we show that the two-spin model describes the rota-
tional-resonance transfer curves well as long as none of the passive spins is close to a rotational-resonance condition with one of the
active spins. The usability of the two-spin model is demonstrated experimentally using a sample of acetylcholine perchlorate with
labelling schemes of various levels of complexity. Doubly-, triply-, and fully labelled compounds lead to strongly varying RR
polarization-transfer curves but the evaluated distances using the two-spin model are identical within the expected error limits and
coincide with the distance from the X-ray structure. Rotational-resonance distance measurements in fully labelled compounds allow,
in particular, the measurement of weak couplings in the presence of strong couplings.
Ó 2004 Elsevier Inc. All rights reserved.
1. Introduction
and largely complementary, the former giving a large
number of relatively inaccurate distance constraints, the
latter only a few but more accurate distances. Selective
methods can also be used to measure smaller dipolar
couplings, corresponding to larger distances, in the
presence of larger couplings [3,4], thereby, overcoming
the problems of dipolar truncation [5].
The rotational-resonance experiment (RR) is a robust
and widely applied method for selective homonuclear
recoupling [6–8]. It requires the MAS frequency to
match an integer submultiple of the isotropic chemical-
shift difference between the two selected active spins, i.e.,
nxr ¼ jX1iso ꢀ X2isoj. The RR condition must be met to an
accuracy prescribed by the magnitude of the active di-
polar coupling [7,8]. Despite this inherent spectral se-
lectivity of RR recoupling, most practical applications
have been to spin systems where relatively isolated pairs
have been introduced by selective labelling. Such
chemically isolated spin pairs lead to particularly simple
spin dynamics which has been extensively characterized
[8]. The drawback of this procedure is the necessity of
The measurement of internuclear distances by high-
resolution solid-state magic-angle spinning (MAS)
NMR is an important tool for structure determination in
micro-crystalline and non-crystalline samples in analogy
to similar protocols available in liquid-state NMR
spectroscopy [1]. Because MAS is indispensable for ob-
taining high spectral resolution, recoupling methods [2]
must be employed to recover distance information.
Recoupling methods can be broadly divided into
broadband and selective methods depending on the
question whether all dipolar couplings in the sample are
recovered simultaneously or whether spin pairs are re-
coupled selectively based on their spectral properties,
e.g., the chemical shifts. Both approaches are valuable
q Supplementary data associated with this article can be found, in
the online version, at doi: 10.1016/j.jmr.2004.03.009.
* Corresponding author. Fax: +41-1-632-1621.
E-mail address: beme@ethz.ch (B.H. Meier).
1090-7807/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved.
doi:10.1016/j.jmr.2004.03.009

A. Verhoeven et al. / Journal of Magnetic Resonance 168 (2004) 314–326
315
preparing a number of selectively labelled samples which
is expensive and time consuming.
either one of the active spins. For simplicity, we neglect
the chemical-shift anisotropy and consider the following
simple model Hamiltonian in the usual rotating frame:
iso
We have recently shown experimentally and by nu-
merical calculations that relatively accurate distance
information (with a precision of a few percent) can in-
deed be obtained from uniformly 13C-labelled com-
pounds [3]. In principle, the RR polarization-transfer
curves for small spin systems (less than 10 spins) can
readily be calculated numerically [9]. However, such
calculations require either pre-existing knowledge about
the spin-system parameters (in particular J couplings,
dipolar-coupling tensors, and chemical-shift tensor), or
the fitting of these parameters. The strategy described in
this publication aims at identifying conditions under
which the RR curves can be fitted by a modified two-
spin system (and ultimately a single fictitious spin 1/2)
with the desired internuclear distance and an effective
offset as the only relevant fit parameter. In this article,
we expand our earlier phenomenological description [3]
of the effects of the passive spins on the polarization-
transfer curve and provide a theoretical basis for the
reduction of a three-spin to a two-spin problem. Gen-
eralization to more spins is not difficult.
HðtÞ ¼ H þ HddðtÞ þ HJ
ð1Þ
cs
with the isotropic chemical-shift Hamiltonian
iso
cs
H
¼ X1I1z þ X2I2z þ X3I3z;
ð2Þ
the J-coupling Hamiltonian
*
*
*
HJ ¼ 2pJ12 I 1 ꢁ *I 2 þ 2pJ13 I 1 ꢁ *I 3 þ 2pJ23 I 2 ꢁ *I 3;
and the dipolar-coupling Hamiltonian
ð3Þ
HddðtÞ¼b12ðtÞð3I1zI2z ꢀ*I 1 ꢁ*I 2Þþb13ðtÞ
ꢂð3I1zI3z ꢀ*I 1 ꢁ*I 3Þþb23ðtÞð3I2zI3z ꢀ*I 2 ꢁ*I 3Þ; ð4Þ
with
2
X
n
r
bijðtÞ ¼
ðbijeinx tÞ:
ð5Þ
n¼ꢀ2
n¼0
The constants bnij are the Fourier coefficients of the
dipolar coupling whose explicit form is given in Ap-
pendix A.
To obtain a time-independent Hamiltonian, we
transform the time-dependent Hamiltonian of Eq. (1)
into the appropriate interaction frame and invoke the
secular approximation. The active spin pair is close to a
rotational-resonance condition with a small offset
d12 ¼ X2 ꢀ X1 ꢀ n12xr. Spins 1 and 3 are offset by
d13 ¼ X3 ꢀ X1 ꢀ n13xr from the nearest rotational-res-
onance condition and the corresponding rotational-res-
onance condition for the spin pair 2–3 is then given by
n23 ¼ n13 ꢀ n12 leading to X3 ꢀ X2 ¼ d13 ꢀ d12 þ n23xr.
2. Theoretical description
We consider a three-spin system under MAS with one
of the spin pairs on or close to rotational resonance. We
call this spin pair the ÔactiveÕ spin pair and denote these
two spins with labels 1 and 2 (Fig. 1A). The active spin
pair has dipolar and J couplings to an additional Ôpas-
siveÕ spin 3 which is not on rotational resonance with
A
B
Fig. 1. (A) Topology of the spin system used in the simulations. Spins 1 and 2 form the ‘‘active’’ spin pair which is on rotational resonance while spin
3 is the passive spin. (B) Schematic representation of the block diagonalization of the three-spin Hamiltonian of Eq. (8). The 2 ꢂ 2 subspaces of the
active spin pair are shown in black, the elements outside these subspaces in a lighter shading.

316
A. Verhoeven et al. / Journal of Magnetic Resonance 168 (2004) 314–326
We can rewrite the isotropic chemical-shift Hamiltonian
of Eq. (2) using the rotational-resonance conditions
mentioned above and obtain
tions jnijj ¼ 1; 2 while the presence of shift anisotropy
can lead to nonzero values for all integer values of n,
including 0. Each ZQ subspace supports three zero-
quantum transitions and the difference polarization of
the two active spins SðtÞ ¼ hI1z ꢀ I2ziðtÞ can be written in
the general form
iso
cs
H
¼ X1Fz þ ðn12xr þ d12ÞI2z þ ðn13xr þ d13ÞI3z
ð6Þ
with the total spin operator Fz ¼ I1z þ I2z þ I3z. Except
for the offsets from the rotational-resonance condition
this Hamiltonian has the required form for the rotating-
frame transformation operator, namely:
iso
þ1=2
SðtÞ ¼ C þ A
cosðxþ121=2tÞ þ Aþ1=2 cosðx1þ31=2tÞ
12
13
þ1=2
þ A
þ A
cosðxþ231=2tÞ þ Aꢀ1=2 cosðx1ꢀ21=2tÞ
23
12
ꢀ1=2
cosðx1ꢀ31=2tÞ þ Aꢀ1=2 cosðx2ꢀ31=2tÞ:
ð14Þ
and the six fre-
R ¼ H ꢀ ðd12I2z þ d13I3zÞ
¼ X1Fz þ n12xrI2z þ n13xrI3z:
13
23
cs
ꢃ1=2
ð7Þ
The seven amplitudes C and A
ij
can be evaluated analytically from the
ꢃ1=2
quencies x
ij
Neglecting all time-dependent terms in the interac-
tion-frame Hamiltonian, expðꢀiRtÞH expðRtÞ, we obtain
a time-independent zeroth-order average Hamiltonian
iso
matrices given above. The resulting general equations are,
however, lengthy and do not provide much insight into
the problem. Furthermore, they are usually not suited to
analyze experimental data because of the large number of
parameters that have either to be known beforehand,
assumed, or fitted.
H ¼ H þ Hdd þ HJ ;
with the chemical-shift Hamiltonian
ð8Þ
cs
iso
We, therefore, introduce a simple perturbation treat-
ment which reduces the three-spin problem to an effective
two-spin problem for the case that the two active spins are
close to rotational resonance and the passive spins are far
from any rotational-resonance condition. Reduction to
an effective 2-spin system is equivalent to block-diago-
nalizing the two 3 ꢂ 3 subblocks (see Fig. 1B) into a 1 ꢂ 1
and a 2 ꢂ 2 subblock. The two 2 ꢂ 2 subblocks represent
the fictitious spin-1/2 systems which describe, approxi-
mately, the ZQ subspace of the active spin pair. In this
approximation, the two 2 ꢂ 2 subblocks provide all the
information needed to calculate SðtÞ. In the present
treatment, a single passive spin is taken into account but
the generalization to a larger number of passive spins is
H
¼ d12I2z þ d13I3z;
ð9Þ
cs
the J-coupling Hamiltonian
HJ ¼ 2pJ12I1zI2z þ 2pJ13I1zI3z þ 2pJ23I2zI3z;
and the dipolar-coupling Hamiltonian
ð10Þ
ꢀ
ꢁ
ꢀ
ꢁ
Hdd ¼ bn12 ðI1 I2 þ I1 I2 Þ þ b13 ðI1 I3 þ I1 I3 Þ
þ
ꢀ
ꢀ
þ
n
þ
ꢀ
ꢀ þ
1
2
1
2
ꢀ
ꢁ
n
þ
ꢀ
ꢀ
þ
1
2
þ b23 ðI2 I3 þ I2 I3 Þ :
ð11Þ
It is well known that the matrix representation of the
Hamiltonian of Eq. (8) is block diagonal if the basis
functions are ordered according to the total magnetic
quantum number fz (see Fig. 1B). Because the initial
density operator can also be block diagonalized in the
same way, the time evolution of the spin system can be
evaluated for each subspace separately. We can restrict
the discussion to the two 3 ꢂ 3 zero-quantum (ZQ)
subspaces with the matrix representation
possible. We separate the Hamiltonian into an unper-
ð0Þ
turbed part H which contains the diagonal elements
and the active dipolar coupling and into a perturbation
ꢃ1=2
V
ꢃ1=2 which contains the passive couplings and is defined
2
3
pðꢀJ12 ꢀ J13 þ J23Þ þ d12 þ d13
ꢀbn12
ꢀbn13
1
ZQ
ꢀðbn12Þꢄꢄ
ꢀðbn13Þ
pðꢀJ12 þ J13 ꢀ J23Þ ꢀ d12 þ d13
ꢀðbn23Þꢄ
ꢀbn23
4
5
H
¼
;
ð12Þ
ð13Þ
ð15Þ
1=2
2
pðJ12 ꢀ J13 ꢀ J23Þ þ d12 ꢀ d13
and
2
4
3
5
pðꢀJ12 ꢀ J13 þ J23Þ ꢀ d12 ꢀ d13
ꢀðbn12Þꢄ
pðꢀJ12 þ J13 ꢀ J23Þ þ d12 ꢀ d13
ꢀbn23
ꢀðbn13Þꢄꢄ
ꢀðbn23Þ
1
ZQ
H
¼
ꢀbn12
ꢀbn13
:
ꢀ1=2
2
pðJ12 ꢀ J13 ꢀ J23Þ ꢀ d12 þ d13
Note that the coupling elements bnij in the two ma-
(for the +1/2 subspace) by
trices given above are the Fourier components corre-
sponding to the nij rotational-resonance condition for
the transition involved. In the absence of chemical-shift
anisotropy, they are only nonzero for the four condi-
2
3
5
0
0
0
0
ꢀbn13
1
2
n
4
ꢀb23
V1=2
¼
:
ꢀðbn13Þꢄ ꢀðbn23Þꢄ
0

A. Verhoeven et al. / Journal of Magnetic Resonance 168 (2004) 314–326
317
ð0Þ
The correction to the diagonal elements of H can
be obtained in second-order perturbation theory by
and is exact in the absence of J couplings. The well-known
equations for rotational-resonance polarization transfer
in an isolated two-spin system are then recovered:
1=2
2
2
jbn13j
jbn13j
E1ð2Þ
E2ð2Þ
¼
¼
ffi
;
SðtÞ
4pðꢀJ12 þ J23Þ þ 4d13
4d13
¼ C þ A ꢁ cosðxeff tÞ
ð23Þ
Sð0Þ
with
2
2
jbn23j
jbn23j
ð16Þ
ffi
;
4pðꢀJ12 þ J13Þ ꢀ 4d12 þ 4d13
4d13
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
E3ð2Þ ¼ ꢀE1ð2Þ ꢀ E2ð2Þ
:
ðD12Þ
jbn12j
2
2
C ¼
;
A ¼
;
2
xeff
¼
jbn12j þ ðD12Þ :
2
ðxeff Þ
ðxeff Þ
Neglecting the part proportional to the unity opera-
tor, the 2 ꢂ 2 zero-quantum subspaces of the active spin
pair can now be written as
ð24Þ
Such a description of the time evolution of the difference
polarization does not include relaxation. In many cases,
however, the relaxation-rate constant (in the zero-
quantum subspace) is in the same order of magnitude as
the dipolar coupling of the active spin pair and can,
therefore, not be neglected. The Liouville-space de-
scription of the time evolution, introduced by Levitt
et al. [8], includes relaxation according to
H
¼ Dꢃ1=2SzZQ þ jb1n2jSuZQ
;
ð17Þ
ꢃ1=2
12
with
2
D
¼ ꢀpJ13 þ pJ23 ꢃ d12
ꢃ
2 Ç
:
ð18Þ
jbn13j
jbn23j
ꢃ1=2
12
4d13
4d13
The operators SZQ and SuZQ are fictitious spin-1/2 oper-
ators in the 2 ꢂz2 zero-quantum subspace of the active
spin pair and SuZQ is a generalized transverse spin oper-
ator in the xy plane as defined by
^
^
^
^
^
L
^
¼ ꢀiHꢃ1=2 ꢀ C
;
ꢃ1=2
ð25Þ
ꢃ1=2
where the relaxation operator describes a random field
along the z axis, as defined by
SuZQ ¼ cosðuÞSxZQ þ sinðuÞSyZQ
;
ð19Þ
^
^
where u depends on the phase of the complex Fourier
component bn12. By a simple (crystal orientation depen-
dent) z-rotation, the dipolar field can be aligned with the
x-axis of the ZQ subspace:
¼ Dꢃ1=2SzZQ þ jb1n2jSxZQ
Eq. (20) is formally identical to the Hamiltonian of a
two-spin system under off-RR conditions. The ꢃc1r=y2stal-
orientation-dependent effective RR mismatch D12 de-
pends on the passive dipolar and J couplings. In the
absence of J couplings, the offsets in the two subblocks
are the same, except of the sign and the time evolution in
C ¼ k1½I1z; ½I1z; ŠŠ þ k2½I2z; ½I2z; ŠŠ:
ð26Þ
Here, k1 and k2 are the transverse relaxation-rate con-
stants of the spins 1 and 2. Assuming the random fields
on the two spins are uncorrelated, they constitute an
effective transverse relaxation rate in the zero-quantum
subspace with R2 ¼ k1 þ k2.
H
:
ð20Þ
ꢃ1=2
12
In a basis spanned by the fictitious spin-1/2 operators
ðSxZQ; SyZQ; SzZQÞ, the matrix representation of the Liou-
villian can be written as
2
3
ꢃ1=2
ꢀR2 ꢀD
0
12
ꢃ1=2
^
ꢃ1=2
^
L12
4
5
¼
:
ð27Þ
D12
ꢀR2
ꢀjbn12j
0
jbn12j
0
the two subspaces is identical. We furthermore note that
in the absence of J couplings the offset D
ꢃ1=2
for a single
12
ꢃ1=2
As pointed out above, the offset terms D
can often be
crystallite, but not in the powder average, can be re-
moved by setting the MAS frequency slightly offset
(d12 ¼ 0) from the exact RR condition.
12
assumed to be identical and the superscript can be
omitted. Under this approximation, the density operator
in the zero-quantum subspace evolves according to
The time evolution of the difference polarization in
such a two-spin model is given by
^
^
rðtÞ ¼ expðLtÞrð0Þ
ð28Þ
SðtÞ
¼ ½C þ Aꢀ1=2 cosðxeff tÞ þ Aþ1=2 cosðxeþff1=2tފ;
ꢀ1=2
1
2
with the initial density operator, in vector representa-
tion, given by rð0Þ ¼ ½0; 0; Sð0ފT. This equation is iso-
morphic with the Bloch equations with transverse but
no longitudinal relaxation. Starting with an initial den-
sity operator proportional to SzZQ, it leads to a damped
oscillation of the difference polarization SðtÞ with an
effective frequency xeff (see Eq. (22)). The non-oscillat-
ing part (term C in Eq. (21)) will also decay to zero
under the action of R2 according to the coupled system
of equations defined by the Liouvillian.
Sð0Þ
ð21Þ
with
2
þ1=2
12
2
2
ꢀ1=2
12
2
2
jbn12j
ðD
ðx
Þ
Þ
ðD
ðx
Þ
Aꢃ1=2
¼
¼
;
C ¼
þ
;
ꢃ1=2
2
þ1=2
ꢀ1=2
ðx
Þ
Þ
ð22Þ
eff
eff
eff
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ꢃ1=2
2
2
x
jbn12j þ ðDꢃ121=2Þ :
eff
þ1=2
ꢀ1=2
The condition jD
j ¼ jD
j is often fulfilled to a
To account for a possible pedestal of signal-compo-
nents not decaying under the zero-quantum relaxation
12
12
good approximation even in the presence of J couplings

318
A. Verhoeven et al. / Journal of Magnetic Resonance 168 (2004) 314–326
mechanism (e.g., caused by natural-abundance signals
and overlap with other resonances not coupled to the
active spin system), we add a phenomenological term
rð1Þ ¼ ½0; 0; Sð1ފ and obtain the equation of motion
3. Analysis of the equations in the absence of relaxation
To obtain an estimate how well the approximate two-
spin model developed in the previous chapter describes a
multi-spin system, numerically exact Hilbert-space cal-
culations of an example three-spin system were per-
formed using the spin-simulation environment
GAMMA [9]. The active spin pair was chosen to have
T
^
^
rðtÞ ¼ expðLtÞrð0Þ þ rð1Þ:
ð29Þ
This equation will be used to describe the spin system.
The observable SðtÞ in the RR experiment is the third
vector component of rðtÞ. Relevant model parameters
are the active dipolar coupling bn12, the offset D1ꢃ21=2, the
zero-quantum relaxation rate constant R2 and the phe-
nomenological offset Sð1Þ.
ꢀ
an internuclear distance r12 ¼ 4:0 A (corresponding to
the anisotropy of the dipolar coupling dðD1;3Þ ¼ 118:5 Hz)
and no J coupling. All calculations were performed on
the exact rotational-resonance condition of the active
spin pair with xr=ð2pÞ ¼ 20 kHz, and d12 ¼ 0. To sim-
plify the situation the coupling between spins 2 and 3 of
our model system was set to zero, i.e., J23 ¼ b23 ¼ 0. The
relative orientation of the two dipolar couplings (1,2)
and (1,3) was chosen to be b ¼ 90°.
It should be noted that both the effective offset and
the dipolar coupling are functions of the crystallite ori-
entation with rꢃes1=p2ect to a rotor-fixed coordinate system,
ꢃ1=2
i.e., D
fore, a powder average must be performed to obtain the
time evolution.
¼ D12 ða; b; cÞ and bn12 ¼ b1n2ða; b; c). There-
12
The exact simulations (Fig. 2, black lines) are com-
pared to approximate solutions of Eqs. (23) and (24)
(Fig. 2, red lines) describing an effective two-spin system
and an approximate analytical description of the three-
spin system in secular approximation as described in
Appendix B (Fig. 2, blue lines) for four different situa-
tions: (i) For an isolated two-spin system, i.e., J13 ¼ 0 Hz
and b13 ¼ 0 Hz. The three curves are, as expected, in-
distinguishable showing that for the present case the
secular approximation of Eq. (8) is very good; (ii)
Adding a passive J coupling, J13 ¼ 50 Hz, which corre-
sponds to a spin system with a heteronuclear passive
spin, leads to the behaviour shown in Fig. 2B with a
modified precession frequency and a pedestal which are
excellently described by the two-spin model of Eq. (23);
(iii) In the presence of a strong passive dipolar coupling
For a numerical treatment, it is also possible to avoid
the secular approximation employed to obtain Eq. (8)
and to keep the time-dependent components of the di-
polar coupling tensors. Then a time-dependent model
Liouvillian is used
2
3
ꢀR2
Çn12xr ꢀ Dꢃ1=2
0
12
^
ꢃ1=2
^
L
4
5
¼
ꢃ1=2
ꢃn12xr þ D12
ꢀR2
ꢀb12ðtÞ
0
b12ðtÞ
0
ð30Þ
and the Liouville–von-Neumann equation is solved by
the method of time slicing. In this case, the presence of
chemical-shift anisotropy for the active spins can easily
be integrated.
Fig. 2. Polarization-transfer curves of a powder sample calculated using Eq. (21) (red), Eq. (B.1) (blue), and by performing an exact Hilbert-space
simulation of the spin system (black). For the blue curves the frequency-domain data are shown below the time-domain data. All data were calculated
ꢀ
at exact rotational resonance for the active spin pair. In (A) the curves show a two-spin system with r12 ¼ 4:0 A and xr=ð2pÞ ¼ 20 kHz. In (B) a
passive spin was added, with J13 ¼ 50 Hz but no dipolar coupling equivalent to a situation where the passive spin pair is far from any RR condition.
ꢀ
In (C) the passive spin is close to the n ¼ 1 RR condition with spin 1, with r13 ¼ 1:5 A and d13=ð2pÞ ¼ 3 kHz. In (D) the passive spin is even closer to
the n ¼ 1 RR condition with spin 1, i.e., d13=ð2pÞ ¼ 1 kHz.

A. Verhoeven et al. / Journal of Magnetic Resonance 168 (2004) 314–326
319
(dDð1;3Þ ¼ 2248 Hz) and a J coupling (J13 ¼ 50 Hz) to a
passive spin off rotational resonance by d13=ð2pÞ ¼ 3 kHz
the simple two-spin approximation (Eq. (23)) is still
quite good as can be seen in Fig. 2C. However, some
additional low-amplitude high-frequency oscillations
become apparent which cannot be described by the
pseudo two-spin model but by the additional frequencies
contained in Eq. (14); (iv) If the offset from the rota-
tional-resonance condition d13=ð2pÞ is reduced further,
these high-frequency oscillations become stronger. This
is shown in Fig. 2D, where the offset is smaller than
1 kHz, which is less than the passive dipolar coupling.
These components decrease in frequency but increase in
amplitude for decreasing values of d13. The simplified
model does not only fail to reproduce the high frequency
oscillations, it also gives the wrong magnitude for the
amplitude of the time-independent term. Despite the
small offset from the rotational-resonance condition for
the passive spin pair, the approximate description from
Appendix B still leads to good results. The shape of the
exchange curves also depends somewhat on the relative
orientation of the active and passive dipolar coupling
tensors (see Additional material).
Larger deviations can be observed at lower MAS
frequencies. If the calculation of Fig. 2C is repeated
with an MAS frequency of xr=ð2pÞ ¼ 5 kHz, while
adjusting the chemical-shift difference of the active spin
pair to remain on the n ¼ 1 rotational resonance, larger
deviations of the approximate solution from the exact
simulations are observed (Fig. 3). This is due to the fact
that the spin pair 1–3 is now simultaneously close to
two rotational-resonance conditions since the offset
from the n13 ¼ 1 condition is 3 kHz and the offset from
the n13 ¼ 2 condition is 2 kHz. In this case, the secular
approximation, which takes into account only the
closest RR condition of the spin pair 1–3, becomes
inadequate.
4. Analysis of the equations including relaxation effects
To interpret the experimental RR polarization-
transfer curves from multiply labelled samples, three
simple models all based on the Liouvillian of Eq. (27)
were employed:
(A) In the simplest approach, the multi-spin system
was treated like a two-spin system on rotational
ꢃ1=2
resonance and the offsets D
were set to zero. A
12
powder average was calculated using the Liouvillian of
Eq. (27) with the dipolar-coupling constant b12 and the
zero-quantum relaxation-rate constant R2 as well as the
initial polarization difference Sð0Þ as free parameters
(see also Eq. (28)).
(B) Based on the observation that the dipolar oscil-
lation is offset by a value C, the model A was extended
by a pedestal for the dipolar oscillation (or a base-line
offset for the transfer curves). This is the model em-
ployed in [3]. Free fit parameters are Sð0Þ, b12, R2, and
the pedestal Sð1Þ (see also Eq. (29)).
ꢃ1=2
(C) The full Liouvillian of Eq. (27) with D
¼ 0,
12
according to the model sketched in Chapter 2 was used
in this model. Since the orientation-depenꢃd1e=n2t offset
from the rotational-resonance condition D12 ða; b; c)
depends on the usually unknown relative orientation of
the two dipolar-ꢀco1=u2pling tensors, the approximation
þ1=2
ꢁ
ꢁ
D
ða; b; cÞ ¼ D12 ða; b; cÞ ¼ D was used, where D is
12
constant and independent of the crystal orientation.
Free parameters of the fit (according to Eq. (29)) are the
dipolar-coupling constant b12, the zero-quantum relax-
ation-rate constant R2, the initial polarization difference
Sð0Þ, and the average effective offset from the RR con-
ꢁ
dition D. Sð1Þ was held constant at the value expected
from the natural-abundance contributions.
To estimate the errors associated with the three models,
we have fitted the numerically exact polarization-transfer
curves for a three-spin model system with each of the
three two-spin models described above. The three-spin
simulations were performed in the full Liouville space
using the GAMMA spin-simulation environment [9]
complemented by block-diagonalization code to speed
up the matrix diagonalization. Relaxation was imple-
mented as described by Eqs. (25) and (26). The relax-
ation-rate constants were chosen to be identical for
every spin (k1 ¼ k2 ¼ 100sꢀ1). The maximum mixing
time in each simulation was chosen as a function of the
3
distance according to smmax ¼ 5ms ꢁ ðr12=ð1:5AÞÞ or
ꢀ
smmax ¼ 100 ms whichever is shorter. In this way we
account for the decreasing oscillation frequency of the
transfer process for longer distances. The difference
polarization was sampled every rotor period. Gaussian
noise with a standard deviation of 0.005 was added to
Fig. 3. Polarization transfer curves generated in a similar way as in Fig.
2. In these simulations the MAS frequency is reduced to
xr=ð2pÞ ¼ 5 kHz. The passive spin is now close to both the n ¼ 1
(d13=ð2pÞ ¼ 3 kHz) and the n ¼ 2 (d13=ð2pÞ ¼ 2 kHz) RR conditions.

320
A. Verhoeven et al. / Journal of Magnetic Resonance 168 (2004) 314–326
ꢁ
Table 1
Parameters for the numerical simulations in the three-spin model
an Euler-angle independent offset value D. We have,
therefore, repeated the fit with model C with a fit by a
model D which uses the dipolar-coupling constant b12,
the zero-quantum relaxation-rate constant R2, and the
initial polarization difference Sð0Þ as free parameters.
The pedestal Sð1Þ is set to zero. The orientation-de-
4A–C
simulations
5B simulations
(X1 (Hz), d1 (Hz), g1)
(a1, b1, c1)
(X2 (Hz), d2 (Hz), g2)
(0, 0, 0)
(), ), ))
(variable, 0, 0)
(0, 6083, 0.2466)
(36°, 134°, 127° )
(variable, 1542,
0.2432)
pendent offset from the rotational-resonance condition
ꢃ1=2
D
ða; b; cÞ is calculated from the spin-system data
12
(a2, b2, c2)
(X3 (Hz), d3 (Hz), g3)
(), ), ))
(18750, 0, 0)
(160°, 137°, 82° )
(18750, 9250,
0.8919)
which involves knowledge of the parameters of the
passive spins. The resulting fit is shown in Fig. 5A and is
very similar to the one of Fig. 4C indicating that this
approximation of model C is justified.
(a3, b3, c3)
d12 (Hz) (r12 (A))
(), ), ))
Variable
(42°, 109°, 87°)
Variable
ꢀ
(a12, b12, c12
d13 (Hz) (r13 (A))
(a13, b13, c13
)
(0°, 0°, 0°)
2248 (1.54)
(0°, 109.5°, 0°)
0 (1Þ
(0°, 0°, 0°)
2248 (1.54)
Our models ignore the effects of chemical-shift an-
isotropy. Formally, this additional interaction is easy to
include into the theoretical description [7,8]. Since the
magnitude and orientation of the chemical-shift tensors
are often unknown their effects are usually ignored in
the analysis at the expense of some systematic errors.
The situation is similar in our case. The simulated
spectra that were analyzed by our models A–C in Fig. 4
were generated without anisotropic chemical-shift in-
teractions. Anisotropic chemical-shift terms were in-
cluded in the simulations used for Fig. 5B (see Table 2)
but the analysis was still performed in the framework of
our simple model C ignoring chemical-shift anisotropy
effects. Clearly, the inclusion of chemical-shift aniso-
tropies in the simulated data reduces the fidelity with
which internuclear distances can be determined. How-
ever, this happens primarily in an area where the MAS
frequency becomes so slow that it is exceeded by the
sizes of the CSA tensors. This finding is in accordance
with observations on isolated two-spin systems [8].
ꢀ
)
(0°, 109.5°, 0°)
0 (1Þ
ꢀ
d23 (Hz) (r23 (A))
the curves after normalizing the first point to a value of
1. The v2 of the fit was normalized according to the
noise, i.e., a perfect fit would result in a v2 ¼ 1. The
parameters for the spin systems used in these simula-
tions and fits are summarized in Table 1. The fitting
program uses the optimization routines from the
package MINUIT [10].
The simulated data for the three-spin system was
fitted with models A, B, and C, respectively which all
employ an effective two-spin system. The results of this
analysis are presented in Figs. 4A–C as the relative de-
viation of the fitted distance from the actual distance
used in the simulations. As expected, strong deviations
(more than 15%) in the resulting distances are observed
near the n ¼ 1 and n ¼ 2 rotational-resonance condi-
tions for the passive spin, i.e., at X3 ꢀ X1 ꢅ nxr using
model A (Fig. 4A). The deviations in the distance are
reduced for model B (Fig. 4B) and even more so when
using model C with Sð1Þ ¼ 0 (Fig. 4C). This is also il-
lustrated by a general decrease in the v2 of the fits, as
shown in Figs. 4D–F. For a large range of spin-system
parameters, the fit yields the ‘‘true’’ distance within 10%
accuracy. Regions where the fitted distance does not
agree with the true distance are often (in particular for
model C) but not always flagged by bad fits as expressed
by large v2 values (compare Figs. 4A–C with D–F).
Using our simulated data, we can check the system-
5. Fits of experimental RR polarization-transfer curves
In the previous two sections we have illustrated by
way of representative examples that the effect of the
passive spins on the polarization-transfer curves can be
described by an effective two-spin system. Models B and
C have been shown to yield improved accuracy com-
pared to simply neglecting the passive spins (model A).
In the following we use models B and C to analyze the
experimental RR polarization-transfer curves of four
differently labelled acetylcholine perchlorate samples
representing chemically isolated two-spin, three-spin, or
atic errors introduced by neglecting the angular depen-
dence of D
ꢃ1=2
ða; b; c) and using the approximation of
12
c
Fig. 4. Relative difference between the true internuclear distance of the active spin pair in a three spin system and the value obtained by a least square
analysis using: (A) model A, (B) model B, and (C) model C as described in the text. The plots are given as a function of the internuclear distance of
the active spins r12 and the chemical-shift difference of the active spins X1 ꢀ X2. The chemical-shift difference between the active and the passive spins
was kept constant at (X3 ꢀ X1Þ=ð2pÞ ¼ 18:75 kHz. The MAS frequency xr was set to X1 ꢀ X2 in order to always fulfill the RR condition for the
active spin pair. (D–F) show the least square deviations of the simulated data and the best fit by models A–C.
Fig. 5. (A) Relative difference between the true internuclear distance between the active spins in a three-spin system and the value obtained by a least-
square analysis using model D described in the text. (B) Analysis of a simulated three-spin spectrum including chemical-shift anisotropy (see Table 1)
by model C as described in the text.

A. Verhoeven et al. / Journal of Magnetic Resonance 168 (2004) 314–326
321

322
A. Verhoeven et al. / Journal of Magnetic Resonance 168 (2004) 314–326
Table 2
Results from the fits on acetylcholine perchlorate
X-ray data
Model Ca
Model Ba
Sample a
Samples b/c
Sample d
Sample a
Samples b/c
Sample d
ꢀ
C1–C3
C1–C4
C2–C3
C2–C4
r (A)
3.69
4.73
2.37
3.63
3.690(16)
8.1
9.5
3.889(24)
9.8
34.8
3.84(4)
6.3
40.8
3.671(14)
7.5
[0]
3.55(3)
4.6
[0]
3.43(7)
2.0
[0]
T2ZQ (ms)
D/2p (Hz)
Sð1Þ
[0.09]
[0.09]
[0.09]
0.10
0.32
0.25
ꢀ
r (A)
4.55(7)
6.3
31.9
4.60(10)
5.6
42.6
4.13(11)
2.5
[0]
3.9(3)
1.2
[0]
T2ZQ (ms)
D/2p (Hz)
Sð1Þ
[0.09]
[0.09]
0.28
0.32
ꢀ
r (A)
2.383(3)
15.0
22.0
2.388(4)
10.9
42.7
2.378(3)
17.5
[0]
2.371(3)
16.7
[0]
T2ZQ (ms)
D/2p (Hz)
Sð1Þ
[0.09]
[0.09]
0.12
0.18
ꢀ
r (A)
3.643(15)
10.6
30.1
3.740(20)
10.6
39.0
3.476(16)
7.9
[0]
3.42(3)
5.3
[0]
T2ZQ (ms)
D/2p (Hz)
Sð1Þ
[0.09]
[0.09]
0.26
0.35
a Values in [ ] were kept constant and fixed in the fits.
The experimental transfer curves are shown in dark
blue in Fig. 7, the fitted curves using model C in red. The
resulting fit parameters are given in Table 2. Overall,
good agreement between the different measurements as
well as with the X-ray structure is observed.
It is illustrating to compare the three transfer curves
between C1 and C3 measured in almost isolated two-
spin, three-spin, and five-spin systems. Experimentally
we observe that the presence of the additional passive
spins causes a damping of the oscillations and slightly
increases the oscillation frequency. Despite the obvious
differences between the three transfer curves, the inter-
nuclear distances obtained with model C, namely 3.69,
Fig. 6. The labelling schemes of the acetylcholine perchlorate samples
used in the acquisition of the experimental RR curves.
five-spin systems, respectively. The compounds used in
this study were synthesized as described in Appendix C
and diluted to 10% in natural-abundance material. The
labelling schemes are shown in Fig. 6. The chemical
shifts of the five carbon atoms were determined as 19.4,
171.7, 58.8, 64.2, and 53.6 ppm for C1–C5. All experi-
ments were performed at the nominal n ¼ 1 RR condi-
tion for the active spin pair on a Bruker Avance 600
spectrometer equipped with a Bruker 2.5 mm MAS
probe. The spinning frequency was stabilized to about
ꢃ5 Hz. Rotational-resonance experiments were per-
formed following adiabatic cross polarization from
protons [11]. After cross polarization, the polarization
of one of the 13C resonances was selectively inverted
using a DANTE pulse train [12] empirically optimized
for each of the transfer curves. After a variable mixing
time, the polarization was converted to single-quantum
coherence by a 90° pulse and detected. During the
mixing time and data acquisition, TPPM [13] decoupling
was applied using an RF-field strength of about
120 kHz, a 10° phase angle, and a pulse length of 5 ls.
Each point of the polarization-transfer curve was the
result of the summation of 128 transients. The data were
processed and integrated into Felix 97.0 (Accelrys, CA).
ꢀ
3.88, and 3.84 A, do not deviate from each other by
more than about 5%. As expected, the result from the
RR experiment on an isolated two-spin system is very
ꢀ
close to the X-ray distance of 3.69 A [14]. Model B yields
ꢀ
distances of 3.67, 3.55, and 3.43 A, respectively and
underestimates the distances measured in multi-spin
systems somewhat because the faster oscillation fre-
quency is interpreted in terms of a stronger dipolar
coupling only (see Eq. (24) with D12 ¼ 0). Using model B,
the maximum error in the distance amounts to about 8%.
For the longest distance measured in acetylcholine
ꢀ
perchlorate, namely C1–C4 with a distance of 4.73 A, an
error of less than 4% was encountered by analyzing the
multi-spin RR experiments with model C, which clearly
performs much better than model B which shows an
error of about 20%. Additional simulations (not shown)
indicate that the systematic deviations between mea-
sured and modelled curves in Fig. 7 which occur at
longer mixing times even for the isolated two-spin sys-
tem, should be attributed to the effect of the chemical-
shielding anisotropy which is not accounted for in the

A. Verhoeven et al. / Journal of Magnetic Resonance 168 (2004) 314–326
323
Fig. 7. Experimental RR polarization-transfer curves for acetylcholine perchlorate with labelling schemes as indicated by model C. The parameter
values obtained by the fits are collected in Table 2.
model. Finally, we note that, on careful inspections of
the initial behaviour of some experimental polarization-
transfer curves (e.g., for the C2–C4 curve), the appear-
ance of fast oscillations, as discussed in the context of
the simulated data in Fig. 2D can be observed.
Our experimental data on acetylcholine perchlorate
show, in agreement with numerical simulations, that it is
possible to determine internuclear distances of up to
ꢀ
about 5 A with an accuracy better than 5% using rota-
tional-resonance measurements in many uniformly la-
beled samples. The experimental data on acetylcholine,
a neurotransmitter, suggest that measurements on a
single sample with acetylcholine bound to the receptor,
can provide information about the structure of a small
molecule in the bound state. Similar experiments with
other drugs or hormones are conceivable. We also
foresee that the method is applicable to macromolecules
because spectral crowding does, by itself, not compro-
mise the power of the analysis presented here as only
passive spins with a sizeable coupling need to be taken
into account in the analysis.
While more complex models which take into account
explicitly the presence of passive spins have the potential
to lead to more accurate distance constraints, the model
presented here has the advantage to be very simple and
not to require any previous knowledge of the spin sys-
tem except for the isotropic chemical shifts of the cou-
pled spins. This information is easily obtainable by a
broadband chemical-shift correlation experiment. The
accuracy of the distance constraints is not quite as pre-
cise as those obtained from selectively doubly labelled
compounds, but, in our hands, much higher than con-
straints from broadband experiments of the driven
6. Conclusions
An improved but still very simple model is introduced
which allows the evaluation of internuclear distances
from multiply labelled compounds by rotational-reso-
nance experiments obviating the need to synthesize a
large number of doubly labelled compounds. The model
describes the oscillation of the difference polarization
between the two active spins by an effective two-spin
system with a dipolar-coupling constant bn12 (the pa-
ꢁ
rameter one is usually interested in), an effective offset D
from the nominal RR condition, and a zero-quantum
relaxation-rate constant R2. The parameters of the ad-
ditional passive spins need not be known except that the
isotropic chemical shifts should be checked to make sure
that the offset of each passive spin from the nearest ro-
tational-resonance condition is larger than the dipolar
coupling to that spin. If such an undesired resonance
appears, it will, however, become often apparent
through the failure of the proposed models to describe
the polarization-transfer curves accurately.

324
A. Verhoeven et al. / Journal of Magnetic Resonance 168 (2004) 314–326
spin-diffusion type. Finally, it should be mentioned that
the accuracy of the methods described here improves
with increasing magnetic field strength. Our example
describes the situation at 14.1 T.
(bn23 ¼ 0). We first diagonalize the 2 ꢂ 2 subspace of spin
pair (1,3) and subsequently the 2 ꢂ 2 subspace of the
active spin pair (1,2). The off-diagonal terms still re-
maining after these two Jacobi steps were neglected.
This procedure leads us to the following approxima-
tion for Eq. (14):
Acknowledgments
SðtÞ
¼ Cꢀ1=2 þ Cþ1=2 þ Aꢀ121=2 cosðxeff;12
ꢀ1=2 tÞ þ Aþ1=2
12
Sð0Þ
Financial support from the Swiss National Science
Foundation and the ETH Zurich is gratefully ac-
knowledged.
ꢂ cosðxeff;12
þ1=2 tÞ þ Aꢀ1=2 cosðxꢀeff1;=123tÞ
13
þ1=2
þ A
þ A
cosðxeff;13
þ1=2 tÞ þ Aꢀ1=2 cosðxꢀeff1;=223tÞ
13
23
þ1=2
cosðxþef1f;=223tÞ
ðB:1Þ
Appendix A. Fourier components
23
with the offset is given by
The Fourier components for the dipolar interaction
are given by
ꢃ1=2
ꢃ1=2
cꢃ1=2
¼
ðP1þ2 ꢀ 2P3Þð1 ꢀ cꢃ1=2Þð5 þ 7c
þ ð3 þ c
Þ
1
64
13
13
13
pffiffi
ði;jÞ
ði;jÞ
bij ¼ 2dD eꢀic
e
Þ ꢂ ðsinðbpðim;jÞÞ sinðbmrÞ
1
ꢀ2iðamrþc
mr
pm
P1ꢀ2
128
ꢃ1=2
12
ꢃ1=2
2
ꢃ1=2
2
ꢃ1=2
16
ꢂ ððc
Þ ꢀ ðs
Þ ÞÞ þ
ð37 ꢀ 12c
12
12
13
mrþcði;jÞ
ꢀ 2eiða
Þ cosðbpðim;jÞÞ cosðbmrÞ
Þ sinðbðpim;jÞÞ sinðbmrÞÞ ꢂ ðsinðbði;jÞÞ
pm
ꢃ1=2
ꢃ1=2
2
2
þ 2ð3 þ c
Þððc
Þ ꢀ ðs
Þ ÞÞ
Þ Þ
13
12
mrþcði;jÞ
þ e2iða
pm
ꢃ1=2
2
ꢃ1=2
2
pm
þ 7ððc
and the amplitudes by
Þ ꢀ ðs
ðB:2Þ
12
12
mrþcði;jÞ
þ cosðbmrÞ sinðbðpim;jÞÞ þ 2eiða
Þ cosðbðpim;jÞÞ
pm
mrþcði;jÞ
ꢂ sinðbmrÞ þ e2iða
Þðꢀ sinðbðpim;jÞÞ
pm
P1ꢀ2
32
ðP1þ2 ꢀ 2P3Þ
ꢃ1=2
ꢃ1=2
ꢃ1=2
12
2
A
¼
ð3 þ c
Þðs
Þ þ
12
13
þ cosðbmrÞ sinðbði;jÞÞÞÞ
ðA:1Þ
ðA:2Þ
32
2
pm
ꢃ1=2
ꢃ1=2
ꢃ1=2
ꢂ ð1 ꢀ c
Þð3 þ c
Þðs
Þ ;
ðB:3Þ
ðB:4Þ
13
13
12
and
ði;jÞ
ði;jÞ
bij ¼ 1 dD eꢀ2ic
e
Þðsinðbpðim;jÞÞ cosðbmrÞ
ðP1ꢀ2 ꢀ P1þ2 þ 2P3Þ
2
ꢀ2iðamrþc
mr
ꢃ1=2
ꢃ1=2
ꢃ1=2
2
pm
A
A
¼
¼
ð1 þ c
Þðs
Þ ;
32
13
12
13
16
mrþcði;jÞ
ði;jÞ
ꢂ sinðbpm Þ þ 2eiða
Þ cosðbpðim;jÞÞ sinðbmrÞ
pm
ðP1ꢀ2 ꢀ P1þ2 þ 2P3Þ
ꢃ1=2
ꢃ1=2
ꢃ1=2
2
mrþcði;jÞ
ð1 ꢀ c
ÞðS
Þ :
ðB:5Þ
þ e2iða
Þðꢀ sinðbpðim;jÞÞ þ cosðbmrÞ
23
12
13
pm
16
2
ꢂ sinðbðpim;jÞÞÞÞ
Here, P1þ2; P1ꢀ2, and P3 represent the initial sum and
difference polarization of the active spin pair and the
initial polarization on spin 3, respectively, while
ckij ¼ cosðhkijÞ and sikj ¼ sinðhikjÞ represent the cosines and
sines of the tilt of the effective field with respect to the z-
axis in the fictitious spin-1/2 zero-quantum subspaces of
spins i and j. They are defined by
with dDði;jÞ ¼ ꢀ2l0cicjꢁh=ð4pr3 Þ and bn ¼ ðbꢀnÞꢄ. The Euler
ij
ij
ij
angles bpðim;jÞ; cði;jÞ describe the orientation of the principal-
axis system of the dipolar-coupling tensors with respect
pm
to the molecular frame which is most conveniently
chosen to be the principal axis systemð1o;2fÞ the dipolar
ð1;2Þ
tensor of the active spin pair such that bpm ¼ c
¼ 0.
pm
ꢃ1=2
ꢃ1=2
ꢃ1=2
ꢃ1=2
c
¼ D
for the (1,3) subspace, and by
=X
and
S
¼ jb13j=Xꢃ1=2
ðB:6Þ
The other set of Euler angles, (amr; bmr; cmr), gives the
orientation of the molecular frame with respect to the
rotor-fixed frame. The latter set of Euler angles will be
used to perform the powder average.
13
13
eff;13
13
eff;13
ꢃ1=2
ꢃ1=2
ꢃ1=2
eff;12
c
¼ D
=x
and
12
12
ꢂ
ꢃ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðB:7Þ
ꢃ1=2
ꢃ1=2
ꢃ1=2
1
2
S
¼
jb12j ð1 þ c13 Þ =xeff;12
12
Appendix B. Theoretical description of rotational reso-
nance with higher accuracy
for the (1,2) subspace. The effective offsets from the RR
condition in the two subspaces are given by
Here we shortly discuss an approximate analytical
solutions for the diagonalization of the two 3 ꢂ 3 sub-
spaces of Eqs. (12) and (13) based on the Jacobi method
of matrix diagonalization [15]. These formulas have
been used to calculate the blue curves in Fig. 2. One of
the passive dipolar couplings was assumed to be zero
J23
ꢃ1=2
D
D
¼
¼
ꢃ d13 and
13
2
!
ꢃ1=2
eff;13
ðB:8Þ
X
J13 J23
d13
ꢃ1=2
ꢀ
þ
ꢃ d12
Ç
þ
12
2
4
2
2

A. Verhoeven et al. / Journal of Magnetic Resonance 168 (2004) 314–326
325
and the effective dipolar couplings are defined by
Appendix C. Synthesis of 13C labelled acetylcholine
compounds
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ꢃ1=2
eff;13
ꢃ1=2
2
2
jb13j þ ðDꢃ1=2Þ ;
ðB:9Þ
A strategy for the synthesis of acetylcholine perchlo-
rate was developed which permitted the introduction of
13C labels in any combination of positions from a range
of commercially available 13C-labelled precursors. An
overview is presented in Fig. C.1. The key intermediate in
the synthesis of 13C labelled acetylcholine compounds
was the synthesis of 13C-labelled 2-aminoethanol and is
illustrated in detail below for [1,2-13C2]2-aminoethanol.
Triethylamine (40 ml) was added to a cooled stirred
suspension (0 °C) of 13C-labelled glycine ethylester hy-
drochloride (18.3 g), CH2Cl2 (260 ml), MgSO4 (13.1 g),
and benzaldehyde (14.1 g). The mixture slowly reached
room temperature and stirring was continued for 24 h.
X
x
¼ sgnðD
Þ
Þ
13
13
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ꢃ1=2
eff;12
ꢃ1=2
ꢃ1=2
2
2
ꢃ1=2
1
2
¼ sgnðD
ðD
Þ þ jb12j ð1 þ c13 Þ;
12
12
ðB:10Þ
ðB:11Þ
!
ꢃ1=2
ꢃ1=2
x
D
ꢃ1=2
eff;13
ꢃ1=2
eff;13
eff;12
12
x
x
¼
¼
ꢀ X
ꢀ X
þ
ꢀ
;
2
2
!
ꢃ1=2
ꢃ1=2
x
D
ꢃ1=2
eff;23
ꢃ1=2
eff;13
eff;12
12
:
ðB:12Þ
2
2
Fig. C.1. Reaction scheme for the synthesis of fully or specufically labelled acetylcholine.

326
A. Verhoeven et al. / Journal of Magnetic Resonance 168 (2004) 314–326
The solid was filtered, the filtrate evaporated at low
temperature versus a liquid nitrogen trap. The oily res-
idue was cooled to 0 °C, diluted with water (60 ml) and
extracted extensively with ether. The combined etheral
extracts were dried with magnesium sulfate and again
evaporated at low temperature, resulting in N-benzy-
lidene-[13C2]glycine ethylester (25 g).
The N-benzylidene-[13C2]glycine ethylester was dis-
solved in dry ether (200 ml) and slowly added to LiAlH4
(10 g) in ether (400 ml). After stirring at room tempera-
ture for 1 h the mixture was refluxed for 24 h. After
hydrolysis with H2O and 10% NaOH in H2O, the sus-
pension was filtrated. The filtrate was evaporated to
yield N-benzylamino-[1,2-13C2]glycinol as a colorless oil
(21 g).
The N-benzylamino-[1,2-13C2]glycinol was dissolved
in CH3OH (300 ml), Pd/C 20% (Degussa, 6 g) was added
and under shaking the mixture was catalytically hydro-
genated at room temperature during 20 h. The catalyst
was filtered off and the filtrate was evaporated, which
resulted in [1,2-13C2]2-aminoethanol (8.7 g).
The 2-aminoethanol was subsequently alkylated with
excess methyliodide with sodium methoxide as base. The
obtained choline iodide was acetylated by 13C-labelled
acetic anhydride in pyridine with DMAP as catalyst.
The iodide counter ion was replaced by a perchlorate
ion by adding the acetylcholine iodide to a saturated
ammonium-perchlorate solution in water at room tem-
perature. The compound was dissolved by heating the
mixture to the boiling point. After slow cooling to room
temperature, acetylcholine perchlorate crystals are ob-
tained as needles.
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