NMR experiment
The pulse sequence of the experiment for \({\text {H}^{\text{N}}_{i}}{\text {H}^{\alpha }_{i-1}}\)dd – C\({^\prime }_{i-1}\) CSA CCR rate measurement is shown in Fig. 1. It includes three indirectly-detected dimensions (\(\text {C}^\alpha _{i-1}\), C\({^\prime }_{i-1}\) and \(\text {N}_i\)) and one directly-detected \(\text {H}^{\text{N}}_{i}\) dimension. The employed method of CCR rate quantification is called quantitative spectroscopy, which means that two independent data sets (later referred to as reference and cross) are acquired. The coherence entering the CCR block of the pulse sequence is partially preserved and partially converted to another one. The conversion is CCR-mediated, thus the measurement of the intensities of signals originating from both components allows for quantification of the CCR effect. In the cross spectrum the observable magnetization originates from the converted coherence, while in the reference spectrum it originates from the preserved coherence. Crucial is the fact that the peak intensities in the reference spectrum are proportional to \(\cosh {(\varGamma T_c)}\) where \(\varGamma\) is the corresponding CCR rate and \(T_c\) is the time of the CCR evolution, while in the cross spectrum they are proportional to \(\sinh {(\varGamma T_c)}\) which is why their values (\(I_{ref}\) and \(I_{cross}\)) are not identical (see Fig. 2). Therefore the CCR rates are calculated (separately for each residue of the protein under investigation) using the following formula:
$$\begin{aligned} \varGamma =\frac{1}{T_c} {{\,\text{arctanh}\,}}\left( \frac{I_{cross}}{I_{ref}} \right) \end{aligned}$$
(1)
Importantly, the formula should be modified if the reference and cross experiments were acquired with different number of scans, which is a good practice due to significantly lower sensitivity of the cross experiment. In such a case, the intensity ratio should be divided by the ratio of number of scans acquired in cross and reference experiments.
The coherence transfer pathway is shown below. The beginnings of reference and cross experiments are identical, up to the CCR block:
$$\begin{aligned}&{\text {H}^{\text{N}}_{i} \,{}_{\text{z }}}\rightarrow 2{\text {H}^{\text{N}}_{i} \,{}_{\text{z }}}\text {N}_i \,{}_{\text{z }}\rightarrow 2\text {N}_i \,{}_{\text{z }}\text {C}{^\prime}_{i-1} \,{}_{\text{z }}\\&\quad \rightarrow 4\text {N}_i \,{}_{\text{z }}\text {C}{^\prime}_{i-1} \,{}_{\text{z }}\text {C}^{\alpha }_{i-1} \,{}_{\text{z }}\xrightarrow []{C^{\alpha }_{i-1} evolution} 4\text {N}_i \,{}_{\text{z }}\text {C}{^\prime}_{i-1} \,{}_{\text{z }}\text {C}^{\alpha }_{i-1} \,{}_{\text{z }}\\&\quad \rightarrow 8{\text {H}^{\text{N}}_{i} \,{}_{\text{z }}}\text {N}_i \,{}_{\text{z }}\text {C}{^\prime}_{i-1} \,{}_{\text{z }}\text {C}^{\alpha }_{i-1} \,{}_{\text{z }}\end{aligned}$$
Then, during the CCR block, the product operator 8\({\text {H}^{\text{N}}_{i} \,{}_{\text{z }}}\)\(\text {N}_i \,{}_{\text{z }}\)\(\text {C}{^\prime}_{i-1} \,{}_{\text{z }}\)\(\text {C}^{\alpha }_{i-1} \,{}_{\text{z }}\)is partially preserved and partially converted into 16\({\text {H}^{\text{N}}_{i} \,{}_{\text{z }}}\)\(\text {N}_i \,{}_{\text{z }}\)\(\text {C}{^\prime}_{i-1} \,{}_{\text{z }}\)\(\text {C}^{\alpha }_{i-1} \,{}_{\text{z }}\)\(\text {H}^{\alpha }_{i-1} \,{}_{\text{z }}\). This conversion occurs for total time \(T_c\). Also, the evolution of \(\text {C}{^\prime }_{i-1}\) nuclei occurs here. Importantly, coherence of no other CCR rate occurs during the CCR block, thus providing a clean result. After the CCR block, the coherence (8\({\text {H}^{\text{N}}_{i} \,{}_{\text{z }}}\)\(\text {N}_i \,{}_{\text{z }}\)\(\text {C}{^\prime}_{i-1} \,{}_{\text{z }}\)\(\text {C}^{\alpha }_{i-1} \,{}_{\text{z }}\)for the reference experiment and 16\({\text {H}^{\text{N}}_{i} \,{}_{\text{z }}}\)\(\text {N}_i \,{}_{\text{z }}\)\(\text {C}{^\prime}_{i-1} \,{}_{\text{z }}\)\(\text {C}^{\alpha }_{i-1} \,{}_{\text{z }}\)\(\text {H}^{\alpha }_{i-1} \,{}_{\text{z }}\)for the cross experiment) is gradually back-converted into an observable \(\text {H}^{\text{N}}_{i}\) transverse magnetization. In particular, in the second INEPT after the CCR block, 8\(\text {N}_i \,{}_{\text{z }}\)\(\text {C}{^\prime}_{i-1} \,{}_{\text{z }}\)\(\text {C}^{\alpha }_{i-1} \,{}_{\text{z }}\)\(\text {H}^{\alpha }_{i-1} \,{}_{\text{z }}\)operator in cross version is converted into 4\(\text {N}_i \,{}_{\text{z }}\)\(\text {C}{^\prime}_{i-1} \,{}_{\text{z }}\)\(\text {C}^{\alpha }_{i-1} \,{}_{\text{z }}\)one using the \(J_{CA{-}HA}\) scalar coupling. Notably, the operator involving a glycine residue contains two alpha protons and thus will not be converted into an observable magnetization using this pulse scheme. This is the reason why the presented experiment does not allow to determine CCR rates for glycine residues. The last indirect evolution (of \(\text {N}_{i}\) nuclei) occurs during the 2\(\text {N}_i \,{}_{\text{z }}\)\(\text {C}{^\prime}_{i-1} \,{}_{\text{z }}\)\(\rightarrow\) 2\({\text {H}^{\text{N}}_{i} \,{}_{\text{z }}}\)\(\text {N}_i \,{}_{\text{z }}\)INEPT block.
The proposed pulse sequence was designed in a way which precludes the evolution of any other CCR rate. Nonetheless, the results still may be perturbed by another factor, namely the dispersion of the scalar coupling constants between alpha carbon and alpha proton \(J_{CA{-}HA}\) throughout the protein. After the CCR block, in the cross version of the experiment this coupling is evolved. It is however not evolved in the reference experiment. Therefore deviation from this assumed coupling constant will cause perturbation of the obtained \(\varGamma\) value. For deviations of ± 5 % of the 146 Hz value of the J-coupling, the \(\varGamma\) perturbation is not big and typically does not exceed 1.5 %.
An important issue is parameters of amide-proton selective pulses in the CCR block of the pulse sequence. The excitation range, defined using ’offset’ and ’bandwidth’ parameters of the pulse, should cover the whole amide-proton region, but not overlap with the alpha-proton region. For residues with \(H^N\) outside or \(H^{\alpha }\) inside the excitation range, the measured ratio of peak intensities from reference and cross spectra will not provide a correct value of the CCR rate. As the resonance assignment is already known at the stage of CCR rates measurements, it is possible to adjust the offset and/or bandwidth to match the particular protein. In general, the problem is less pronounced for IDPs, which feature narrower chemical shift ranges, than for folded proteins. The values used in the experiments shown in the present study (offset of 8.3 ppm and bandwidth of 3.5 ppm) matched well both proteins used, Ubiquitin and \(\alpha\)-Synuclein. The \(H^N\) of one of the Ubiquitin residues (Ile36) was outside of the excitation range, but the peak involving this nucleus provides information on the CCR rate of the preceding Gly35 residue, which—being a glycine—is not useable anyway.
The pulse sequence of the presented experiment can be obtained from the authors upon request.
Data analysis
The expected angular dependence was modelled in accordance with Yang et al. (1997) assuming model-free dynamics (Lipari and Szabo 1982),
$$\begin{aligned} \varGamma ^{DD,CSA}_{AB,C}(\psi ,\theta (\psi )) = \frac{4}{15} \frac{\mu _0 \hbar }{4\pi } \frac{\gamma _A\gamma _B}{r^3_{AB}(\psi )} B_0 \gamma _{C} \times f_C \times \tau _c S^2, \end{aligned}$$
(2)
where
$$\begin{aligned} f_C= & {} \frac{1}{2}[\sigma _{xx}(3\cos ^2\theta _{AB,X}-1)+\sigma _{yy}(3\cos ^2\theta _{AB,Y}-1)\nonumber \\&+\sigma _{zz}(3\cos ^2\theta _{AB,Z}-1)], \end{aligned}$$
(3)
with \(A=H^N_i\), \(B=H^\alpha _{i-1}\) and \(C = C{^\prime }_{i-1}\). \(\gamma\) is the gyromagnetic ratio, \(\mu _0\) is the vacuum permeability, \(\hbar\) is the reduced Planck constant, \(B_0\) is the magnetic field strength, \(\tau _c\) is the global correlation time, \(S^2\) is the local order parameter, \(\sigma _{xx,yy,zz}\) are the tensor components of the diagonal CSA tensor (in ppm), \(r_{AB}\) is the internuclear distance between A and B, \(\theta\) denotes the projection angles between the dipolar unit vector AB and the principal axes X, Y, Z of the CSA tensor coordinate system.
\(\varGamma\) as a function of \(\psi\) was calculated numerically using an Avogadro-generated (Hanwell et al. 2012) backbone geometry with \(\psi = -180^{\circ }\) (Table 1) and rotating around the \(\text {C}^\alpha\)–\(\text {C}{^\prime }\) bond in \(1^{\circ }\) steps.
Table 1 The model protein backbone in x,y,z-coordinates with \(\psi = \omega = -180^{\circ }\) Parameters were adapted primarily from Engh and Huber (2006), angles involving hydrogens were taken from Momany et al. (1975). The principal axes of the carbonyl CSA tensor were set in accordance with Teng et al. (1992): The Z-axis was defined as the cross product of the \(\text {C}{^\prime }\)–\(\text {O}\) and the \(\text {C}{^\prime }\)–\(\text {C}^{\alpha }\) bond unit vectors, the X- and Y-axis as clockwise rotations of the \(\text {C}{^\prime }\)–\(\text {O}\) bond unit vector around the Z-axis by 82\(^{\circ }\) and − 8\(^{\circ }\), approximating the \(\text {O}\)–\(\text {C}{^\prime }\)–\(\text {N}\) angle with 120\(^{\circ }\).
The tensor components of Ubiquitin were adapted from Cisnetti et al. (2004). \(\sigma _{xx}\) and \(\sigma _{zz}\) were set according to the reported averages as 249.4 ppm and 87.9 ppm. Using the suggested calibration, the average \(\sigma _{yy}\) was derived from the chemical shifts (BMRB ID 17769, Cornilescu et al. (1998)) as 191.1 ppm. Uncertainties were estimated by allowing the tensor components to vary within their reported standard deviations (x, y, z = 6.1, 6.1, 5.4 ppm) while still matching the chemical shift range, yielding a lower limit of \(\sigma _{xx},\sigma _{yy},\sigma _{zz}\) = 243.3, 211.7, 87.4 ppm and an upper limit of \(\sigma _{xx},\sigma _{yy},\sigma _{zz}\) = 255.5, 172.9, 93.3 ppm.
A correlation time \(\tau _c\) of 4.1 ns was assumed (Schneider et al. 1992), order parameters were taken from Tjandra et al. (1995) with an average \(S^2\) of 0.84, a lower limit of 0.70 and an upper limit of 0.91 (excluding the reported outlier of 0.565 at L73).
Neighbour-corrected Structural Propensity Calulator (ncSPC) values for Ubiquitin and \(\alpha\)-Synuclein (BMRB ID 17769 and 6968, respectively) were calculated using the tool of Tamiola and Mulder (2012) with default settings and the Tamiola et al. (2010) library.
CCR rates expected for random-coil-like residues were estimated using the random coil library of Mantsyzov et al. (2015) with a total of 152870 \(\psi\)-angles (excluding glycine and proline residues). Rates were calculated according to Equation 2 and averaged. The effective correlation time was estimated from the experimentally observed values: The range of Ubiquitin (0.67 to 10.39 \(\text {s}^{-1}\)) was normalized to the mean correlation time (\(3.44 \text { ns} = \tau _c S^2\)). Dividing the observed range of \(\alpha\)-Synculein (1.38 to 8.16 \(\text {s}^{-1}\)) by this factor yields an estimate for the effective correlation time of 2.40 ns. Thus, the effective correlation time was estimated to lie within 2 and 3 ns, equating to an expected \(\varGamma\) for random-coil-like residues between 3.10 and 4.66 \(\text {s}^{-1}\).