Full relaxation matrix analysis of apparent cross-correlated relaxation rates in four-spin systems

Journal of Magnetic Resonance 226 (2013) 52–63

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Full relaxation matrix analysis of apparent cross-correlated relaxation rates in four-spin systems Beat Vögeli ⇑ Laboratory of Physical Chemistry, Swiss Federal Institute of Technology, ETH-Hönggerberg, CH-8093 Zürich, Switzerland

a r t i c l e

i n f o

Article history: Received 8 September 2012 Revised 15 October 2012 Available online 12 November 2012 Keywords: Full relaxation matrix Cross-correlated relaxation Secular approximation

a b s t r a c t Cross-correlated relaxation (CCR) rates are an established tool for the extraction of relative bond orientations in biomolecules in solution. CCR between dipolar interactions in four-spin systems is a particularly well-suited mechanism. In this paper, a simple approach to analyze systematic experimental errors is formulated in a subspace of the complete four-spin Hilbert space. It is shown that, contrary to the common assumption, the secular approximation of the relaxation matrix is marginal for the most prominent spin systems. With the main focus on the model protein GB3 at room temperature, it is shown that the apparent experimental CCR rates have errors between 12% and +4% for molecules with a molecular tumbling time of 3.5 ns. Although depending on the specific pulse sequence used, the following rule-of-thumb can be established: Judged by absolute values, the errors for Ha–Ca/Ha–Ca, HN–N/Ca–C0 , HN–N/Cc–Cb and HN–N/Hb–Cb CCR rates can safely be neglected. However, errors for HN–N/HN–N and HN–N/Ha–Ca CCR rates are on the order of 0.1–0.3 s1 and must be considered. Tabulated correction factors may be used for their extraction. If larger systems are studied, in most cases the errors cannot be neglected anymore. On the other hand, well-calibrated pulses can safely be assumed to be perfect. Ó 2012 Elsevier Inc. All rights reserved.

1. Introduction Dipolar cross-correlated relaxation (CCR) effects are an excellent tool to analyze relative bond orientations in molecules [1–6]. In particular, multiple-quantum measurements in four-spin systems offer the possibility to correlate two interspin vectors that do not share a common spin. In most cases, u, w, or v1 angles in proteins have been assessed [7–14]. As the CCR rate is a timeaveraged observable, it has been proposed to inform on dynamics [1,4,15–21]. A prerequisite of such a study is sufficient measuring precision and accuracy. It has been shown that different mechanisms affect the extraction of angles from CCR rates. Pulse sequences that make use of decoupling sequences to conserve inphase magnetization are prone to the creation of unwanted antiphase magnetization caused by imperfections in decoupling sequences [22]. Angle calculation from a CCR rate is usually based on the assumption of isotropic molecular tumbling. Obviously, an error is introduced depending on the degree of the anisotropy [4,21,23]. For a four-spin ½ system, effects due to longitudinal relaxation rates of the passively coupled spins have been examined using perturbation theory [24]. In addition, the nuclear Overhauser enhancement between the two passively coupled spins has different effects on the double- and ⇑ Fax: +41 44 632 10 21. E-mail address: [email protected] 1090-7807/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jmr.2012.11.002

zero-quantum coherences which are difficult to account for if their scalar couplings to the MQ spins are similar (that is, JI1S1  JI2S2) [25]. Apparent measured CCR rates are also affected by scalar couplings, and interference between them and longitudinal relaxation [26,27]. For example, Hu et al. demonstrated the effect on CCR of longitudinal relaxation and J couplings in a two-spin ½ system [27]. This work pursues two goals. First, it is geared towards the specific case of CCR rates in the model protein GB3 [28]. In an attempt to characterize the dynamic behavior of GB3 (molecular tumbling time 3.4 ns at 298 K) more holistically, a large body of CCR rates, mainly from the protein backbone, is currently collected. Some of them, namely those between HN–N/Ha–Ca, HN–N/Cb–Cc and HN– N/Ca–C0 have previously been published [5,21,29]. Further spin systems of interest in this investigation are HN–N/HN–N, Ha–Ca/ Ha–Ca and HN–N/Hb–Cb. For this purpose, it is crucial to understand all sources of systematic errors at every stage of the procedure. The measurements are described with four-spin ½ systems. All experiments are based on multiple-quantum coherences scalar coupled to two longitudinal spins. Errors of the apparent CCR rates are calculated which depend specifically on the pulse sequences, pulse imperfections, the method of CCR rate calculation and spin systems used in those studies. The calculations follow exactly the experimental procedures by evolving the magnetization operators numerically in a full evolution- and relaxation-matrix approach. In doing so, the usual assumption of the secular approximation of the

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B. Vögeli / Journal of Magnetic Resonance 226 (2013) 52–63

matrix is circumvented and off-diagonal elements that alter the apparent CCR rates are explicitly included. Ultimately, correction factors to the apparent CCR rates for specific cases are provided. It is shown that, contrary to the common assumption, the secular approximation is marginal. In some cases, it provides false results while in others it may be safely used. A second aim of this work is to make the approach seizable for a more general use. It can easily be adapted to any spin system and pulse sequence, as only the basic elements of the pulse sequence have to be brought in the correct order. In addition, results for larger molecules with three and six times the tumbling time of GB3 are provided.

~ u;v ðTÞ ¼ eFtN M MQ

N1 Y

~ u;v ð0Þ Pi eFti M MQ

ð3Þ

i¼1

Free evolution is computed upon diagonalization: Ft i

e

¼ UeDti U 1

ð4Þ

D is the diagonal matrix containing the eigenvalues of F and U is the transformation matrix containing the eigenvectors. The intensity of an observed signal selected by the observation ~ obs is given by the projection operator M

   ~  T ~ u;v Iu;v MQ ¼ ðM obs Þ  M MQ ðTÞ

ð5Þ

2. Theory 2.2. Construction of pulse matrices 2.1. Full matrix description of cross-correlated relaxation in a four-spin ½ system In the following, a matrix including spectral frequencies and contributions to relaxation is constructed. In practice, the spin system consists of two spins involved in a multiple-quantum coherence MQ, I1 and S1, and two spins, I2 and S2, which are passively coupled by the scalar couplings JI1I2 and JS1S2. This condition restricts the calculation to a subspace of a general four-spin Hilbert space. The MQ subspace is only spanned by transverse operators which are chosen to be the raising and lowering operators. I2 and S2 can be expressed by the identity (E) and longitudinal operators. These are transformed into the basis formed by a and b states, {aa, ab, ba, bb}. Any magnetization in the subspace under consideration can then be expressed by a 16-dimensional magnetization vector:

~ M

aa

a aa ab ba bb aa ¼ ðZQ þ ; ZQ þ ; ZQ bþ ; ZQ bb þ ; ZQ þ ; ZQ þ ; ZQ þ ; ZQ þ ; DQ þþ ; ab a aa ab ba bb T ; DQ bþþ ; DQ bb DQ þþ þþ ; DQ  ; DQ  ; DQ  ; DQ  Þ

~ AP;AP M ZQ ~ IP;IP M DQ

1 0

0

0; 0; 0; 0; 0; 0; 0; 0ÞT pffiffiffi ¼ 2ð1=4; 1=4; 1=4; 1=4; 1=4; 1=4; 1=4;

1=4; 1=4; 1=4ÞT pffiffiffi ¼ 2ð0; 0; 0; 0; 0; 0; 0; 0; 1=4; 1=4; 1=4; 1=4; 1=4; 1=4; 1=4; 1=4ÞT

The normalization factor is chosen such that the scalar product ~ xP;xP ÞT  M ~ yP;yP ¼ du;v;x;y , where du,v;x,y is Kronecker’s delta ðM uQ vQ function. Manipulation of the initial magnetization vector is simulated by matrices describing pulses, relaxation, scalar coupling and chemical shift evolution. The total time T is divided up into N periods of free evolution ti and (N  1) (possibly multispin-) pulses which are assumed to be executed instantaneously. The pulse operations are given by matrices Pi and free evolution is given by the matrix F, which is identical for all periods. In integrated form the magnetization evolution is given as:

1 ð6Þ

The tensor product (direct product) of matrices is used. P is e Inversion pulses composed of four identical 4  4 submatrices P. only applied to I2 or S2 are given as:

1 0 0 1 0 B0 0 0 1C C B ¼B C; @1 0 0 0A 0

e 180 P I2

1 0 1 0 0 B1 0 0 0C C B ¼B C @0 0 0 1A 0

e 180 P S2

0

0

0

ð7:1 — 2Þ

1 0

If the pulses are imperfect, three- or four-spin coherences are generated which are assumed to be undetectable (this can be shown to be true for all pulse sequences analyzed in this work). e is replaced by linear combinations of P e and the idenTherefore, P tity matrix:

0

1

0

j 0 k j B0 k 0 jC Bj k C e B eI ¼ B P B C; P S2 ¼ B 2 @j 0 k 0 A @0 0 0 j 0 k 0 0 k¼

ð2:1 — 4Þ

0

B0 1 0 0C C e B P I ¼B C  P I2 2 @0 0 1 0A S2 S2 0 0 0 1 I 2 S2 I 2 S2

0 1 0 ð1Þ

pffiffiffi 2ð1=4; 1=4; 1=4; 1=4; 1=4; 1=4; 1=4; 1=4;

1=4; 0; 0; 0; 0; 0; 0; 0; 0ÞT pffiffiffi ¼ 2ð0; 0; 0; 0; 0; 0; 0; 0; 1=4; 1=4; 1=4; 1=4; 1=4;

~ AP;AP M DQ

0

ab

ZQ and DQ are the zero- and double-quantum coherences obtained by the respective combinations of the raising (+) and lowering operators () of I1 and S1 as indicated in the subscript and the superscripts define the states of I2 and S2. Double-inphase (IP, IP) and double-antiphase (AP, AP) zero- and double-quantum coherences with respect to the two passively coupled spins are then expressed as:

~ IP;IP ¼ M ZQ

The P matrices executing pulses on I2, S2 or both have the following form since transitions occur only within the ZQ and DQ subspaces:

k

0

1

0

0

0 k

0C C C jA

ð8:1-2Þ

j k

1 þ Cosð/Þ 1  Cosð/Þ ;j ¼ 2 2

/ is the effective angle rotation caused by the pulse. Simultaneous execution of pulses on both I2 and S2 is described by:

eI S ¼ P eS eI P P 2 2 2 2

ð9Þ

Inversion pulses on I1 and S1 induce transitions between ZQ and DQ coherences:

0

0

0

0 1

1

0

1

0

0

0

1

B0 0 1 0C B0 1 0 0C C C B B P180 C; CB I1 ¼ B @0 1 0 0A @0 0 1 0A 1 0 B0 B ¼B @1 0

P180 S1

0 0 0 1 0 0 0 1 1 0 1 0 0 0 0 1 0 C B 0 0 1C C B0 1 0 0C C CB 0 0 0A @0 0 1 0A

0 1 0

0

0

0

ð10:1 — 2Þ

0 1

Here no modification for imperfect pulses is used since it is assumed that they lead to dephasing of the MQ magnetization by

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B. Vögeli / Journal of Magnetic Resonance 226 (2013) 52–63

RZQ and RDQ are the autorelaxation rates of the ZQ and DQ coherS2 ences, and RI2 z and Rz are the longitudinal autorelaxation rates of spins I2 and S2. Rn (with 1 6 n 6 15) are the 15 relevant cross-correlated relaxation rates associated with the multiple-quantum coherences between spins I1 and S1. If the rates cannot be separated due to symmetry properties, they are summarized into one term which is the sum over the indices. In addition, there are two contributions from dipolar interaction between I2 and S2. The convention is taken from Ref. [25] and listed in Table 1. XX is the Larmor frequency of spin X, and JXY the J coupling between spins X and Y. The explicit expressions for the dipole/dipole, dipole/CSA and CSA/CSA CCR rates are [31]

means of gradients (again, this can be shown to be true for all pulse sequences analyzed here). 2.3. Construction of free evolution matrix As is well known from the Redfield kite, no transitions occur between ZQ and DQ coherences during free evolution [30]. As a consequence, the ZQ and DQ conditions can be treated in 8dimensional subspaces. In the following, the matrices are given as 8  8 matrices which relate to the full 16  16 matrix as:

F¼

e F ZQ ~ 0

~ 0 e F DQ

! ð11Þ

RI1 I2 =S1 S2 ¼

~ is the zero 8  8 matrix and e F DQ are the according matri0 F ZQ and e ces in the ZQ and DQ subspaces. These matrices are composed of the real R and the imaginary part C which contain the relaxation and the chemical shift evolution/scalar coupling terms, respectively:

0

e e e e þC R ZQ B ZQ B DQ B DQ e e e F ZQ ¼ R ZQ þ C ZQ ¼ B B ~~ @ 0 DQ DQ DQ

RI1 I2 =S1 ¼ 

1

~~ 0

C C C C C e e e ~ R ZQ  C ZQ A DQ DQ

 l  2 c c c c h2 1 I1 I2 S1 S2 0  3  3 J I1 I2 =S1 S2 4p 10p2 r eff r eff I1 I2 S1 S2

ð15Þ

 l  2c c c hB0 i 1 h eff xx I1 I2 S 1 yy eff 0  3 DrS1 ;xx JI1 I2 =S1 þ DrS1 ;yy J I1 I2 =S1 4p 15p reff I1 I2 ð16Þ

ð12Þ RI1 =S1 ¼

e ZQ and R e DQ are constructed by two identical 4  4 submatrices: R

X 8cI1 cS1 B20 X ll;l0 l0 eff Dreff I1 ;ll DrS1 ;l0 l0 J I1 =S1 45 0 0 0 0 0 0 ll¼xx;yyl l ¼x x ;y y

ð17Þ

cX is the gyromagnetic ratio of nucleus X, B0 the polarizing magnetic 0

e e ZQ R

RZQ  R1;2  R3;4;5 þ R6;7 þ R8;9 B þR B 10;11  R12;13  R14;15 þ W 2 B B B RSz2 =2 B B ¼ B B B RIz2 =2 B B B @ W 2

1

RSz2 =2

RIz2 =2

W 2

RZQ þ R1;2  R3;4;5  R6;7 þ R8;9 R10;11  R12;13 þ R14;15 þ W 0

W 0

RIz2 =2

RZQ þ R1;2  R3;4;5  R6;7  R8;9

W 0

RSz2 =2

þR10;11 þ R12;13  R14;15 þ W 0

RIz2 =2

RZQ  R1;2  R3;4;5 þ R6;7  R8;9

RSz2 =2

C C C C C C C C C C C C C A

R10;11 þ R12;13 þ R14;15 þ W 2 ð13:1Þ

0

e e DQ R

RDQ þ R1;2 þ R3;4;5 þ R6;7 þ R8;9

B þR B 10;11 þ R12;13 þ R14;15 þ W 2 B B B RSz2 =2 B B ¼ B B B RIz2 =2 B B B @ W 2

RSz2 =2 RDQ  R1;2 þ R3;4;5  R6;7 þ R8;9 R10;11 þ R12;13  R14;15 þ W 0 W 0

1

RIz2 =2

W 2

W 0

RIz2 =2

RDQ  R1;2 þ R3;4;5  R6;7  R8;9

RSz2 =2

þR10;11  R12;13 þ R14;15 þ W 0

RIz2 =2

RSz2 =2

RDQ þ R1;2 þ R3;4;5 þ R6;7  R8;9

C C C C C C C C C C C C C A

R10;11  R12;13  R14;15 þ W 2 ð13:2Þ

e DQ are constructed by two identical 4  4 submatrices: e ZQ and C C

0 B e e ZQ ¼ iB C B @ 0 B e e DQ ¼ iB C B @

ðXI1  XS1 Þ þ pðJ I1 I2  J S1 S2 Þ

0

0

0

0

ðXI1  XS1 Þ þ pðJ I1 I2 þ J S1 S2 Þ

0

0

0 0

0 0

ðXI1 þ XS1 Þ þ pðJ I1 I2 þ J S1 S2 Þ

0

0

0

0

ðXI1 þ XS1 Þ þ pðJ I1 I2  J S1 S2 Þ

0

0

0

0

ðXI1 þ XS1 Þ þ pðJI1 I2 þ J S1 S2 Þ

0

0

0

0

ðXI1 þ XS1 Þ þ pðJ I1 I2  J S1 S2 Þ

ðXI1  XS1 Þ þ pðJ I1 I2  J S1 S2 Þ 0 0 ðXI1  XS1 Þ þ pðJ I1 I2 þ JS1 S2 Þ

1 C C C A

ð14:1Þ

1 C C C A

ð14:2Þ

55

B. Vögeli / Journal of Magnetic Resonance 226 (2013) 52–63 Table 1 Cross-correlated relaxation rates and cross-relaxation rates.

W2 ¼

Code

Interactions

Spectral density function

R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 R13 R14 R15 W0 W2

Dipole I1–I2/dipole S1–S2 Dipole I1–S2/dipole S1–I2 CSA I1/CSA S1 Dipole I1–I2/dipole S1–I2 Dipole I1–S2/dipole S1–S2 Dipole I1–I2/dipole I1–S2 Dipole S1–I2/dipole S1–S2 CSA I1/dipole I1–I2 CSA S1/dipole S1–I2 CSA I1/dipole I1–S2 CSA S1/dipole S1–S2 CSA I1/dipole S1–I2 CSA S1/dipole I1–I2 CSA I1/dipole S1–S2 CSA S1/dipole I1–S2 Dipole I2–S2/dipole I2–S2 Dipole I2–S2/dipole I2–S2

J(0) J(0) J(0) J(0) + 3=4 J(XI2) J(0) + 3=4 J(XS2) J(0) + 3=4 J(XI1) J(0) + 3=4 J(XS1) J(0) + 3=4 J(XI1) J(0) + 3=4 J(XS1) J(0) + 3=4 J(XI1) J(0) + 3=4 J(XS1) J(0) J(0) J(0) J(0) J(XI2  XS2) J(XI2 + XS2)

 l 2 c2 c2 h2 1 I2 S 2 0  6 J I2 S2 ðXI2  XS2 Þ 4p 40p2 eff r I2 S2

4p

40p2

1  6 J I2 S2 ðXI2 þ XS2 Þ eff r I2 S2

ð19Þ

The matrices are constructed with experimentally known relaxation rates if they are available. Otherwise, they are estimated using formulae (17)–(19) and inclusion of all spins exerting a relevant influence. In general, very precise values are used for protein backbone spins and less well-defined values for side-chain atoms (in particular, CSA tensors are ill-defined in some cases).

3. Results and discussion

field, r eff XY is the effective distance between nuclei X and Y (under the assumption that radial and spherical motion of the bond vector and the CSA tensor are not correlated) [32,33], l0 is the permeability of free space and h denotes Planck’s constant. reff S1ll denote the principal components of the effective chemical shielding anisotropy teneff eff eff sor reff S1 of spin S1 along the l dimension and DrS1;ll  rS1;zz  rS1;ll (convention used in Ref. [34]). The spectral density function J is listed in Table 1 and consists of one term at zero frequency and a second term at the Larmor frequency XX if spin X is involved in both interactions. The spectral density function depends on the orientation and dynamics of reff Xll and the X1–X2 bond vectors. In the simplest case of isotropic molecular tumbling with correlation time sc, no internal dynamics and only zero frequency, the spectral density function is J = P2(cos h)sc, where P2 is the Legendre polynomial of second order and h the projection angle between the two interaction axes. The cross-relaxation rates between spins I2 and S2 are:

W0 ¼

 l 2 6c2 c2 h2 I2 S2 0

ð18Þ

Cross-correlated cross-relaxation rates in four spin-½ systems can be measured by three different approaches, here referred to as DIAI, ACE and MMQ methods. The relevant periods of the pulse sequences are shown in Figs. 1 and 2. For convenience, the periods will be referred to according to their figure code (that is, 1A, etc.). For a specific spin system, only one or more approaches may be applicable and as a consequence, the apparent CCR rate may be very different. In the following, analyses for each approach are presented with a focus on practically relevant spin systems. For all these spin systems, three representative geometries are assumed: If all involved atoms are located in the backbone, the secondary structure dominates the results and the categories are a helix, b sheet and loop. While the former two have highly defined geometries, the latter exhibits large variation and the most distinct geometry from the former ones is chosen. If the side-chain conformation is relevant as well, a complex interplay of backbone angles and (in this study) the v1 angle is active. Instead of choosing specific secondary elements or side-chain conformers, three groups as distinct as possible are chosen by inspection of experimental CCR rates obtained from GB3 (referred to as types 1–3). The selection is clearly non-exhaustive and can easily be extended to further cases with the Mathematica [35] template provided in the Supporting Information. In the following, the discussion refers to the CCR rates simulated for the tumbling time of GB3 (here sc = 3.5 ns), if not explicitly stated otherwise. CCR rates between the I2–I1 and S2–S1 dipoles are designated as I2–I1/S2–S1, where I1 and S1 form the measured multiple-quantum coherence.

Fig. 1. Relevant periods of pulse sequences for CCR measurement with the DIAI approach. A and B are applied if JI1I2 – JS1S2, and C if JI1I2 = JS1S2. Narrow and wide bars indicate 90° and 180° pulses. The delays have the following values: DI = 1/(8JI1I2), DS = 1/(8JS1S2), DIS = DI  DS. For the reference experiments, the black pulses are applied. For the cross-relaxation experiments, the pulses marked with arrows are shifted to the gray positions. During the period in the box the magnetization evolves as single transition (effectively, selective evolution under JI1I2) rather than as MQ coherences. In pulse sequence A, the effective evolution time is T’ = T – 1/(2JI1I2) + 1/(2JS1S2).

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B. Vögeli / Journal of Magnetic Resonance 226 (2013) 52–63

Fig. 2. Relevant periods of pulse sequences for CCR measurement with the ACE and MMQ approaches. The sequence with the minimal number of pulses for ACE is shown in A and one example of MMQ in B. The bars indicate 180° pulses. t is the incremented delay for frequency labeling of the multiple-quantum coherence. The relaxation mechanisms are active during the constant period T.

3.1. DIAI method The DIAI method is based on double inphase–antiphase conversion [36,37]. The initial magnetization is 2I1,xS1,x or 4I1,xS1,yI2,z (x and y may be interchanged) which are partially converted into 8I1,yS1,yI2,zS2,z or 4I1,yS1,xS2,z, respectively, during the course of T. After time T, the initial magnetization operator is detected. Two experiments are carried out. In the first one (‘‘ref’’), the cosine term of both scalar couplings (=1) and cross-relaxation rate is detected, and in the second one (‘‘cross’’), the sine term (=1) of both scalar couplings and cross-relaxation rate is detected. Then the signal intensities are related to the wanted CCR rate R = RI1I2/S1S2 + RI1S2/ S1I2 by

R¼

  arctanh Icross I ref

T

ð20Þ

Two cases of the DIAI method must be considered depending on whether JI1I2 = JS1S2 or JI1I2 – JS1S2. 3.1.1. DIAI method, JI1I2 = JS1S2 Here, it is assumed that only I1/S1, and I2/S2 can be excited separately. The measurement is most easily achieved by the following MQ-evolution element (Fig. 1C) [25,38]. In the center of the period, an inversion pulse refocuses the MQ chemical-shift evolution (theoretically, two separate pulses on I1 and S1 are possible). The observed operator is the initial magnetization operator 2I1,xS1,x = pffiffiffi ~ IP;IP Þ= 2, which is partially converted into 8I1,yS1,yI2,z~ IP;IP þ M ðM ZQ DQ pffiffiffi ~ AP;AP  M ~ AP;AP Þ= 2 during T. In the reference measurement, S2,z = ðM ZQ

DQ

two inversion pulses on I2 and S2 are applied after T/4 and 3T/4, and in the ‘‘cross’’ experiment, these two pulses are shifted by 1/ (8JI1I2) = 1/(8JS1S2). The most important examples are the measurements of dipolar HN–N/HN–N (T = 43 ms) or dipolar Ha–Ca/Ha–Ca (T = 28 ms) CCR rates. Note that they can only be measured with the DIAI approach. Simulated true and apparent values as typically observed in an a helix, a b sheet or loop are listed in Table S1 in the Supporting Information. Apparent CCR rates normalized to the true ones are shown in panels A and B in Figs. 3–5. 3.1.1.1. HN–N/HN–N CCR rates. Under the assumption that the sample is deuterated, the calculated errors are in the range of 12% to +4%, which become relevant if the measuring error is 0.1–0.2 s1. Indeed, this is the error range obtained from new experiments (to be published elsewhere). In an a helix, an apparent CCR rate of 1 1 Rapp is obtained for a true rate of Rtrue 1;2 ¼ 5 s . Setting 1;2 ¼ 5:186 s

selectively one of the relaxation rates to zero reveals that the largest impacts are obtained from W0, R6,7, R3,4,5 and (RDQ  RZQ) 1 (Rapp , 5.123 s1, 5.139 s1 and 5.147 s1). Primary effects 1;2 = 5.065 s can be identified if all relaxation parameters are set to zero except for the one under consideration (note, these differential equations 1 can also be solved analytically). Again, for Rtrue 1;2 ¼ 5 s , changes app 1 are only obtained for W0, yielding R1;2 ¼ 5:100 s . This result is in accordance with the fact that W0 only contributes to ZQ but not to DQ coherence if JI1I2 = JS1S2, thereby increasing Rapp 1;2 by W0/4 [25]. All other impacts are interference effects summing up to 0.065 s1, 0.086 s1, and 0.565 s1, if W0 = 0 s1, 0.4 s1, and 2 s1, respectively. This can be further examined by repeating all the calculations with W0 = 0 and all other rates kept at the initial values. 1 The largest influence on Rapp is obtained for R6,7, R3,4,5, 1;2 ¼ 5:065 s 1 1 (RDQ  RZQ) (Rapp = 5.000 s , 5.030 s and 5.036 s1). This 1;2 demonstrates that W0 and R6,7 must not be zero to obtain changes due to interference effects of all other parameters. If W0 is equal to zero, R6,7 has only an effect if R3,4,5 or (RDQ  RZQ) are not equal to zero. Two values have a relatively large uncertainty in the above assumptions that must be checked: First, JI1I2 = JS1S2 is usually not strictly correct; second, (RDQ  RZQ) may vary substantially if exchange mechanisms are active [20,39]. As mentioned above, W0 only contributes to ZQ but not to DQ coherence if JI1I2 = JS1S2. This condition, however, is unlikely to be exactly fulfilled. Typically, JHN,N is between 96 and 91 Hz. In that case, there is a negligibly small contribution to DQ as well. Fig. 4, panel A, shows that the variation of JI1I2 has virtually no impact on the apparent CCR rate. In general, the impact from variation in (RDQ  RZQ) is also negligibly small. A variation of 5 Hz for some loop conformations (small CCR rates), however, can alter the apparent rate by 3% as demonstrated in Fig. 5, panel A, but this change is much smaller than the random measuring error. It has been proposed to measure only DQ instead of (ZQ + DQ) to remove W0 effects [19,20,25]. This comes at the cost of adding R6,7 to the observed CCR rate. In addition, half of the initial magnetization is sacrificed and DQ relaxes faster than ZQ effectively leaving 1 less than 50% of the (ZQ + DQ) signal. Indeed, if Rtrue 1;2 ¼ 5 s , 1 W0 = 0.4 s1 and all other values are set to zero, Rapp ¼ 5:2 s 1;2 and 5.0 s1 for ZQ and DQ, respectively, as expected (see Table S1). If all values are set to default, Rapp 1;2 obtained from DQ coherence is 4.85 s1 yielding a similar error as (ZQ + DQ) but with opposite trend. The deviation is exactly given by R6,7, but in general, there is a large spread of R6,7. For example, for GB3 it can be as positive as 0.21 s1 and as negative as 0.37 s1, which is much larger than

B. Vögeli / Journal of Magnetic Resonance 226 (2013) 52–63

57

Fig. 3. Apparent experimental CCR rates normalized to the true ones versus molecular tumbling time sc. The parameters used for the simulations are listed in Tables S1–S11 in the Supporting Information. Spin systems, specific geometry and pulse sequences are indicated.

the maximal W0/4 value. In conclusion, it may be recommended to record only DQ for proteins with sc larger than 10 ns (compare apparent rates for DQ and (ZQ + DQ) in Table S1).

S2 Due to the weak dependence on RI2 z and Rz , all calculations are virtually the same for a protonated protein or for a protein whose Rz are affected by exchange (see Table S1).

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Fig. 4. Apparent experimental CCR rates normalized to the true ones versus JI1I2, while JS1S2 is kept constant (93 and 145 Hz in A and B, respectively). DIAI pulse sequence 1C is used. The molecular tumbling time sc is 3.5 ns. Spin system and specific geometry are indicated.

Fig. 5. Apparent experimental CCR rates normalized to the true ones versus DRMQ = (RDQ  RZQ). The molecular tumbling time sc is 3.5 ns. Spin systems, specific geometry and pulse sequences are indicated.

3.1.1.2. Ha–Ca/Ha–Ca CCR rates. The errors are in the per-mil range (see Table S2) and are beyond the detectability of the current tech1 niques. In an a helix, an apparent CCR rate of Rapp is 1;2 ¼ 9:986 s trie 1 obtained for a true rate of R1;2 ¼ 10 s . Setting selectively one of the relaxation rates to zero reveals that the largest influence is obS2 app 1 tained for W0 and RI2 and  9:990 s1 Þ. z or Rz ðR1;2 ¼ 9:991 s Again, primary effects are identified. For Rtrue ¼ 10 s1 , changes 1;2 S2 1 are obtained for W0 yielding Rapp ¼ 9:995 s and for RI2 z or Rz 1;2

1 yielding Rapp 1;2 ¼ 9:996 s . In addition to the sum of these three effects, there are no interference effects. Simple summation fully explains the overall deviation. The two values which have a relatively large uncertainty in the assumption made above ((JI1I2  JS1S2) and (RDQ  RZQ)) must be checked: As mentioned previously, W0 only affects ZQ but not DQ coherence if JI1I2 = JS1S2. Typically, JHa,Ca is between 138 and 150 Hz. As demonstrated in Fig. 4B, deviations are negligibly small in that range. Only for small CCR rates

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59

I2 Fig. 6. Apparent experimental CCR rates normalized to the true ones versus an exchange-induced additional contribution to RI2 z ; DRZ ¼ ðRDQ  RZQ Þ. The molecular tumbling time sc is 3.5 ns. Spin systems, specific geometry and pulse sequences are indicated.

1 (Rtrue 1;2 ¼ 1 s ) there is an almost linear dependence on (RDQ  RZQ): every 1 s1 contributes 1% error as shown in Fig. 5B. This value, however, is usually very small. As discussed above, it has been proposed to measure only DQ instead of (ZQ + DQ) to remove W0 effects [19,20,25]. In the present case, W0  0.02 s1 is not large enough to have a considerable effect, but R6,7 has a dominant impact on the rate. There is no reason to record DQ spectra.

3.1.2. DIAI method, JI1I2 – JS1S2 Transitions from inphase to antiphase coherences with respect to I2 and S2 (and vice versa) cannot be achieved simultaneously. Two solutions have been proposed in the literature: First, the synchronization of the transitions can be achieved during the MQ evolution at the cost of a short intermixing of the ZQ and DQ coherences during which the effect of RI1I2/S1S2 is refocused (Fig. 1A) [36]. Second, the MQ evolution may be preceded or followed by a single-quantum evolution to complete both transitions (Fig. 1B) [14,24,37]. The most prominent example is the measurement of the dipolar HN–N/Ha–Ca CCR rate. 3.1.2.1. HN–N/Ha–Ca CCR rates. First, pulse sequence 1A is examined. The pulse sequence features a temporary interchange of the ZQ and DQ coherences during which some unwanted pathways are active [36]. To compensate for the partial reversal of the impact of R1,2, the evolution time T in Eq. (20) is replaced by an effective time T’ = T  1/(2JI1I2) + 1/(2JS1S2). Results are shown in Tables S3 and S4 and panel C and D of Fig. 3 for intraresidual and sequential CCR rates, respectively. The errors are in the range of 3% to +3%, which become relevant if the measuring error is 0.1–0.2 s1. The experimental random error obtained in experiment 1A is circa 0.3 s1 and the error of averages between 1A and other approaches is also circa 0.3 s1 (to be published elsewhere). Here, the discussion is focused on the sequential rates because only these have been measured with pulse sequence 1A [21,36]. The intraresidual analysis yields very similar results. In the b sheet, 1 an apparent CCR rate of Rapp is obtained for a true 1;2 ¼ 11:223 s true 1 rate of R1;2 ¼ 11 s . Setting selectively one of the relaxation rates to zero reveals that the largest impacts are obtained for W0, S2 1 1 RI2 ðRapp , z , (RDQ  RZQ), R3,4,5 and Rz 1;2 ¼ 11:037 s , 11.105 s 11.207 s1, 11.232 s1 and 11.230 s1). Interestingly, setting 1 all contributions to zero yields Rapp rather than 1;2 ¼ 10:929s 1 11 s . This is caused by a second approximation used in Eq. (20) when applied to pulse sequence 1A: While the ‘‘cross’’ evolution is indeed T’ in the reference experiment, it is actually T in the

reference experiment [36]. Primary effects are then obtained for 1 1 W0, RI2 and RS2 yielding Rapp and z z 1;2 ¼ 11:102 s , 11.048 s 10.921 s1. The individual effects have the same size but go the opposite way. This demonstrates that the effects observed for (RDQ  RZQ) and R3,4,5 are pure interference effects. Again, (RDQ  RZQ) has a relatively large uncertainty in the above assumption that must be checked. The plot of Rapp 1;2 versus (RDQ  RZQ) in panel C of Fig. 5 shows that for small CCR rates there is a strong relative dependence. However, these changes are beyond detectN ability. In addition, the increase of RI2 z due to exchange of H may alter the apparent rate. However, Fig. 6 demonstrates that the impact remains also beyond detectability for a realistic range of RI2 z . Second, an alternative pulse sequence, 1B, is examined. The pulse sequence does not feature a temporary interchange of ZQ and DQ coherences, but involves an evolution of single-quantum coherence on I1 to complete the coupling period under JI1I2 [24,37]. This evolution cannot be strictly represented in the chosen space. For simplicity, a modified F matrix was used to mimic the combined effect of the 90° pulse and the single-quantum evolution. The following parameters were changed in F: RMQ is replaced by RSQ,I; because all CCR rates involving S1 vanish, R1,2 = R3,4,5 = R12,13 = R14,15 = 0 (all others are kept as in MQ evolution, since contributions from the terms involving S1 are small except that R10,11 is replaced by R10); XS1 = 0; JS1S2 = 0. Because the SQ evolution is described as a MQ evolution, a 180° pulse on I1 at the end is added to reverse the ZQ–DQ interchange. Results are shown in Tables S3 and S4 and panels C and D of Fig. 3, respectively, for intraresidual and sequential rates. The CCR rates are overestimated by 0–4%, which becomes relevant if the measuring error is 0.1– 0.3 s1. The experimental random error obtained in experiment 1B is circa 0.5 s1 and the error of averages between 1B and other approaches is also circa 0.3 s1 (to be published elsewhere). Again, the discussion is focused on the sequential rates, but both have been measured with the pulse sequence 1B [14,21,24]. The intraresidual analysis yields very similar results. In the b sheet, 1 an apparent CCR rate of Rapp is obtained for a true 1;2 ¼ 11:068 s 1 rate of Rtrue ¼ 11 s . Setting selectively one of the relaxation 1;2 rates to zero, it is found that the largest influences are obtained S2 1 for RI2 ðRapp z , W0, (RDQ  RZQ), R3,4,5 and Rz 1;2 ¼ 10:876 s , 1 1 1 11.181 s , 11.045 s , 11.082 s and 11.075 s1). Primary S2 app 1 effects are only obtained for RI2 z , W0 and Rz ðR1;2 ¼ 11:192 s , 10.875 s1 and 10.992 s1). As for pulse sequence 1A, the effects have the same size but go the opposite way. This shows that the effects observed for (RDQ  RZQ) and R3,4,5 are pure interference effects. Since (RDQ  RZQ) has a relatively large uncertainty in the

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above assumption, the impact of the variation must be checked: The plot of Rapp 1;2 versus (RDQ  RZQ) in panel C of Fig. 5 shows that for small CCR rates there is a strong dependence. However, these changes are beyond detectability. The same result is found for a possible increase of RI2 z due to exchange (see Fig. 6). although the effect is somewhat stronger than for 1A. In general, pulse sequence 1B introduces less systematic error than 1A. 3.2. ACE/MMQ methods The ACE (all components evolution) and MMQ (mixed multiple quantum) methods are based on the individual evolution of all components with respect to I2 and S2 {aa, ab, ba, bb} of the ZQ/ DQ and MQ coherences, respectively. Here, we assume the initial magnetization to be 2I1,xS1,x. Theoretically, it could also be 4I1,yS1,xI2,z, 4I1,xS1,yS2,z, or 8I1,yS1,yI2,zS2,z (or x and y interchanged). One experiment is carried out. All components show up in multiplet patterns of the multiple-quantum coherences. Their signal intensities I are related to the wanted CCR rate R = RI1I2/S1S2 + RI1S2/S1I2 by ZQ DQ DQ IZQ 1 aa Ibb Iab Iba ln ZQ ZQ DQ DQ 8T Iab Iba Iaa Ibb

! ¼ RdðI1S1Þ=dðI2S2Þ þ RdðI1S2Þ=dðI2S1Þ

ð21Þ

for the ACE approach [2,26] and ZQ ;DQ IZQ;DQ 1 aa;ab Ibb;ba ln ZQ;DQ ZQ ;DQ 4T Iab;aa Iba;bb

! ¼ RdðI1S1Þ=dðI2S2Þ þ RdðI1S2Þ=dðI2S1Þ

ð22Þ

for the MMQ approach [7]. For ACE, the simplest pulse sequence does not apply any pulses on the passively coupled spins during the relevant period (see Fig. 2A). Since the multiple-quantum frequency labeling is also implemented during that period, the pair of pulses on I1 and S1 must be shifted to achieve constant-time t incrementation. For MMQ, the simplest sequences employ three pairs of pulses. The minimal number of pulses on the passively coupled spins is one. An example is shown in Fig. 2B. All simulations are carried out for the incremented delay t set to zero. Importantly, as opposed to DIAI, the apparent CCR rates are not sensitive to (RDQ  RZQ) (see Fig. 5, panels C and D for some examples). In the following, the apparent CCR rates obtained from the two methods are discussed separately. 3.2.1. ACE method 3.2.1.1. HN–N/Ha–Ca CCR rates. The pulse sequence 2A is examined. Note that some pulse sequences create 4I1,xS1,xKz with K = C’ rather than 2I1,xS1,x at the beginning of the MQ period [2,21,26], but K is not included here. Results are shown in Tables S5 and S6 and panels C and D in Fig. 3, respectively, for intraresidual and sequential CCR rates. The apparent rates are 1–2% smaller than the true ones, which becomes relevant if the measuring error is 0.1–0.2 s1. The experimental random error obtained in experiment 2A is circa 0.4 s1 and the error of averages between 2A and other approaches is similar (to be published elsewhere). Focusing on the sequential CCR rates in the b sheet, an apparent 1 rate of Rapp is obtained for a true rate of 1;2 ¼ 10:766 s 1 Rtrue ¼ 11 s . Setting selectively one of the relaxation rates to 1;2 zero, it is found that the largest impacts are obtained for W0, RI2 z , app 1 1 RS2 , 10.778 s1, z , R8,9 and R10,11 ðR1;2 ¼ 10:896 s , 10.857 s 10.768 s1 and 10.768 s1). Primary effects are then obtained S2 app 1 1 for W0, RI2 and z and Rz yielding R1;2 ¼ 10:870 s , 10.909 s 10.988 s1. This shows that the effects observed for R8,9 and R10,11 are pure interference effects.

3.2.1.2. HN–N/Ca –C0 CCR rates. It is assumed that the sample is deuterated. This rate has previously been measured with pulse sequence 2A [5]. Results are shown in Table S7 and panel E of Fig. 3. Here, the discussion is focused on the sequential rates which are determined by a nearly conserved geometry. The rate is very small because the projection angle is close to the magic angle [5]. The error of the apparent rate is less than 0.0004 s1, with the largest true 1 impact coming from RI2 z , which is 0.2% for R1;2 ¼ 0:2 s . Even if I2 1 Rz is in the range of 10 s due to exchange effects the impact does not exceed 1%. The error can safely be neglected in practice. 3.2.1.3. HN–N/Cc–Cb CCR rates. It is assumed that the sample is deuterated. Pulse sequence 2A has been used for this spin system [29]. This pulse sequence creates 4I1,xS1,xKz with K = Ca rather than 2I1,xS1,x at the beginning of the MQ period [29], but K is omitted here. Results are shown in Table S8 and panel F of Fig. 3, and Table S9 and Fig. S1, respectively, for intraresidual and sequential CCR rates. An error of 0.5% is not exceeded in any case. This corresponds to an error no more than 0.001 s1, which is well below the measuring precision. Note that the individual contributions may be not as accurate as those obtained for the backbone due to large variations in side-chain composition and conformation and the less well-determined CSA tensor of Cb. 3.2.1.4. HN–N/Hb–Cb CCR rates. The pulse sequence 2A, which has not been reported so far, is examined. Note that some pulse sequences create 4I1,xS1,xKz with K = Ca or C’ rather than 2I1,xS1,x at the beginning of the MQ period, but K is omitted here. Results are shown in Tables S10 and S11 and panels G and H of Fig. 3, respectively, for intraresidual and sequential CCR rates. The analysis of the intraresidual and sequential rates yields very similar results. Due to the similarity of the spin system to the one of HN–N/ Ha–Ca, all influences are structured analogously. The error does not exceed 0.5% for any geometry type. This corresponds to an error of no more than 0.1 s1, which is below the measuring precision. 3.2.2. MMQ method 3.2.2.1. HN–N/Ca–C0 CCR rates. It is assumed that the sample is deuterated. The sequential CCR rate has been recorded with pulse sequence 2B [5]. Results are shown in Table S7 and panel E of Fig. 3. The error of the rate is less than 0.0002 s1, with the largest influtrue I2 1 ence coming from RI2 z , which is 0.15% for R1;2 ¼ 0:2 s . Even if Rz is in the range of 10 s1 due to exchange effects the impact does not exceed 0.5%. It is smaller than the one obtained in 2A pulse sequence. 3.2.2.2. HN–N/Cc–Cb CCR rates. It is assumed that the sample is deuterated. Although never reported so far, this CCR rate can be recorded with pulse sequence 2B. This pulse sequence creates 4I1,xS1,xKz with K = Ca rather than 2I1,xS1,x at the beginning of the MQ period, but K is not considered here. Results are shown in Tables S8 and S9, panel F of Fig. 3 and in Fig. S1, respectively, for intraresidual and sequential rates. Although larger than in the ACE approach, the errors are still very small. With a rather uniform value of circa 0.006 s1, it is well below the measuring precision. Note that the individual contribution may not be as accurate as those calculated for the backbone due to large variations in sidechain composition and conformation and the less well-determined CSA tensor of Cb. 3.2.2.3. HN–N/Hb–Cb CCR rates. The pulse sequence 2B, which has not been reported to date, is examined. Analysis of the intraresidual and sequential CCR rates yields very similar results as shown in Tables S10 and S11 and panels G and H in Fig. 3. Although the performance is slightly worse than for 2A with a maximal error of 4%,

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Fig. 7. Apparent experimental CCR rates normalized to the true ones versus the pulse calibration error DP. The molecular tumbling time sc is 3.5 ns. Spin systems, specific geometry and pulse sequences are indicated.

it is still below the measuring precision. If RI2 z is in the range of 10 s1 due to exchange effects the CCR rates change by 3% which may be larger than the measuring precision for 15 s1 (see Fig. 6). If a protonated sample is used for the measurement of the HN– N/Ca–C0 and HN–N/Cc–Cb rates with the ACE or MMQ methods, all errors are still within measuring uncertainty (Tables S1 and S7– S9). 3.3. Impact from pulse imperfections So far, all simulations were carried out under the assumption of perfect pulses. In practice, however, it is impossible to operate under such conditions due to uncertainties in pulse-length calibration, offset effects, etc. Imperfections of the inversion pulses on the spin involved in the multiple-quantum coherences lead to a reduction of the signal rather than errors in the extraction of the CCR rates, since gradients are applied which embrace the pulses. Here, the impact of imperfect pulses on the passively coupled spins are examined by replacing the pulse matrices given in Eqs. (7.1 and 2) by those in (8.1 and 2). Some cases are plotted in Fig. 7. All cases investigated here feature only inversion pulses on 1H, which are easier to calibrate than those on heteronuclei. It can safely be assumed that the errors are smaller than 3%, especially if pulse trains such as 90°–180°–90° are employed. For a 3% miscalibration, the maximal impact on the apparent CCR rate is about 0.5%. Even for large rates such as Ha–Ca/Ha–Ca, this corresponds to 0.1 s1. It can be concluded that pulse calibration is not a critical aspect in CCR measurements. More detailed inspection reveals that the ACE method 2A is completely independent of calibration because no pulses are

applied on the passively coupled spins. Although not strictly comparable (because of the different spin systems), the following ranking may be used as a rule-of-thumb: The ACE method 2A is the most insensitive to pulse imperfections, followed by MMQ 2B, DIAI 1B, DIAI 1A, and DIAI 1C, which is the most sensitive. This is not surprising, as it corresponds to the number of the relevant inversion pulses. The smaller rates are more likely to develop larger errors as the calibration becomes worse. Of a special note, some rates show a decreasing trend with small certain miscalibrations and pass through a trend inversion for larger miscalibrations. For example, the apparent Ha–Ca/Ha–Ca CCR rate in the loop increases from 10% miscalibration on. As this is a small rate, some unwanted pathways that are not well suppressed anymore start to dominate the desired one. One may ask if the impact of pulse miss-calibration is sensitive to the chosen evolution time T. For illustration, the effects of pulse miss-calibration in pulse sequence 1C for the HN–N/HN–N spin system are compared for the two evolution times 43 and 21.5 ms (see Fig. S2 in the Supporting Information). For the loop, 21.5 ms is favorable, whereas for the a helix and the b sheet, 43 ms provides more accurate values. Interestingly, for the former, the trend is similar, and for the latter, the trends are opposite. Obviously, the dependence of pulse miss-calibration effects on T is complex and no general rule can be deduced. 3.4. Larger systems (sc > 3.5 ns) The analysis up to this point focused on the model case of GB3 at room temperature (sc = 3.5 ns). It is instructive to examine the behavior for larger molecules as it depends strongly on the

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Fig. 8. Apparent experimental CCR rates normalized to the true ones versus the molecular tumbling time sc. The evolution delay T is the values used above divided by sc such that the product T  sc is kept constant. Spin systems, specific geometry and pulse sequences are indicated.

molecular tumbling time (see Fig. 3). All apparent CCR rates deviate more strongly from the true values with increasing sc. With sc = 21 ns and pulse sequence 1C, the apparent HN–N/HN–N rates are between 10% too large and 20% too small, and the apparent Ha–Ca/Ha–Ca rates is no more than 3% off target. In the case of HN–N/Ha–Ca, the error depends strongly on the method. While DIAI 1A and 1B deviate up to +15%/30% and +15%/20%, respectively, ACE 2A only deviates by +2%/1%. It is noteworthy that for DIAI, the CCR rate is strongly underestimated for b sheets, although it has the largest absolute values (for example, for sc of 21 ns, as much as 30%). MMQ 2B is dramatically more advantageous than ACE 2A for the HN–N/Ca–C0 and HN–N/Hb–Cb rates (+50%/50% and +5%/15% for ACE, improve to errors smaller than 2% for MMQ), while both of them are small for all HN–N/Cc–Cb rates with errors smaller than 0.5%. If a protonated instead of a deuterated sample is used, all errors are virtually the same for the HN–N/HN–N rates due to the weak S2 N dependence on RI2 z and Rz (see Table S1). The error for the H –N/ a 0 C –C rates is still negligible for MMQ 2B (Table S7). However, while it is also negligible for 3.5 and 10.5 ns molecular tumbling times as obtained from ACE 2A, the apparent rate is clearly erroneous for 21 ns. The error for the HN–N/Cc–Cb rates are negligible for ACE 2A (Tables S8 and S9). Method MMQ 2B yields somewhat larger errors, in particular for 3.5 ns tumbling time with up to 20%, but the absolute errors are still within the measuring uncertainty. If exchange events effect RI2 z , appreciable changes in the CCR rates in a protein with a 21 ns overall correlation time are expected for the HN–N/Ha–Ca CCR rates (weak for 1A and 1B, strong for 2A),

HN–N/Ca–C0 (only for 2A, absent for 2B), HN–N/Cc–Cb C0 (yet small for 2B, absent for 2A) and HN–N/Hb–Cb (for 2A and 2B). Of course, the pulse sequences may be adjusted to larger-size systems. Since the CCR rates are proportional to the molecular tumbling time, it may be sufficient to use much shorter evolution times. In Fig. 8, relative errors are shown for pulse sequences where the product of the molecular tumbling time and the evolution time is kept constant (for simplicity, the evolution times are not adjusted to multiples of refocusing periods for the various scalar couplings). Although it is obvious that proteins with smaller tumbling times still yield smaller errors, the errors are in almost all cases smaller than those obtained with longer evolution times for a specific tumbling time.

4. Conclusions Generally, the secular approximations of the relaxation matrix are valid for systems of the size of GB3. Nevertheless, if highly accurate and precise CCR rates are required, it is recommended to apply corrections to the apparent experimental rates by using the tabulated values. If larger systems are studied, the error cannot be neglected anymore. The approach which produces a smaller error may be used in those cases where different methods are applicable. Note, however, that the DIAI approach has a smaller random error than the ACE and MMQ approaches. The pulse sequences may be adjusted to larger-size systems. Since the CCR rates are proportional to the molecular tumbling time, it may be sufficient to use much shorter evolution times if they are not fixed

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by decoupling conditions (such as 30 ms for Ca to decouple Cb). For example, the DIAI 1C method to measure HN–N/HN–N CCR rate has been used with an evolution time of 43/2 ms = 21.5 ms [25]. The error for all spin systems are then reduced by one quarter. Well-calibrated pulses can safely be assumed to be perfect. Acknowledgment Prof. Roland Riek is thanked for critical reading of the manuscript. This work was supported by the SNF. Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.jmr.2012.11.002. References [1] V.A. Daragan, K.H. Mayo, Prog. Nucl. Magn. Reson. Spectrosc. 31 (1997) 63. [2] B. Reif, M. Hennig, C. Griesinger, Science 276 (1997) 1230. [3] A. Kumar, C.R.R. Grace, P.K. Madhu, Prog. Nucl. Magn. Reson. Spectrosc. 37 (2000) 191. [4] B. Vögeli, J. Chem. Phys. 133 (2010). 014501-1. [5] B. Vögeli, J. Biomol. NMR 50 (2011) 315. [6] R.B. Fenwick, S. Esteban-Martin, B. Richter, D. Lee, K.F.A. Walter, D. Milanovic, S. Becker, N.A. Lakomek, C. Griesinger, X. Salvatella, J. Am. Chem. Soc. 133 (2011) 10336. [7] D.W. Yang, L.E. Kay, J. Am. Chem. Soc. 120 (1998) 9880. [8] T. Carlomagno, M. Maurer, M. Hennig, C. Griesinger, J. Am. Chem. Soc. 122 (2000) 5105. [9] D. Früh, J. Tolman, G. Bodenhausen, C. Zwahlen, J. Am. Chem. Soc. 123 (2001) 4810. [10] K. Kloiber, W. Schüler, R. Konrat, J. Biomol. NMR 22 (2002) 349. [11] B. Vögeli, K. Pervushin, J. Biomol. NMR 24 (2002) 291.

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