The “long tail” of the protein tumbling correlation function: observation by 1H NMR relaxometry in a wide frequency and concentration range

Inter-protein interactions in solution affect the auto-correlation function of Brownian tumbling not only in terms of a simple increase of the correlation time, they also lead to the appearance of a weak slow component (“long tail”) of the correlation function due to a slowly changing local anisotropy of the microenvironment. The conventional protocol of correlation time estimation from the relaxation rate ratio R1/R2 assumes a single-component tumbling correlation function, and thus can provide incorrect results as soon as the “long tail” is of relevance. This effect, however, has been underestimated in many instances. In this work we present a detailed systematic study of the tumbling correlation function of two proteins, lysozyme and bovine serum albumin, at different concentrations and temperatures using proton field-cycling relaxometry combined with R1ρ and R2 measurements. Unlike high-field NMR relaxation methods, these techniques enable a detailed study of dynamics on a time scale longer than the normal protein tumbling correlation time and, thus, a reliable estimate of the parameters of the “long tail”. In this work we analyze the concentration dependence of the intensity and correlation time of the slow component and perform simulations of high-field 15N NMR relaxation data demonstrating the importance of taking the “long tail” in the analysis into account. Electronic supplementary material The online version of this article (doi:10.1007/s10858-015-0001-1) contains supplementary material, which is available to authorized users.

Thus, the two quantities that map out J() can be compared visually in the same plot.
In contrast, our R 2 measurements were performed at a Larmor frequency of 20 MHz, which does not allow for safely neglecting the contribution of 0 (2 ) J  to R 2 . A correct rescaling of R 2 thus requires knowledge on both 0 () J  and 0 (2 ) J  , which is impossible without further data analysis. Of course, data fitting was performed using the exact equations for R 1, R 1 and R 2 .

Reliability of the FC-NMR R 1 measurements at low fields
The shortest relaxation delay in our field-cycling measurements was 0.7 ms ("dead time"). In the presence of a distribution of R 1 s, the initial slope of the field-cycling relaxation decay may thus not reflect the true initial slope at t=0, as illustrated by Fig. S1.  Figure S1. BSA (c=213 mg/ml, T=20 ºC) R 1 and R 1 relaxation decays after the water component subtraction. As seen from the R 1 decay, the experimentally observed relaxation rate (solid and dashed line) may significantly change during the dead time of the FC NMR measurements. This issue is of relevance as soon as the relaxation time of the fast relaxing protein component becomes comparable to the dead time.
The relaxation decays were only considered as being reliable if a bi-exponential fit to the data provided -within the fitting uncertainty -the same initial slope as the fitting result based upon a log-normal distribution, see Fig. S2. In the case of bi-exponential analysis, the relaxation decay was fitted according to the equation where P a,b and R a,b are the intensities and relaxation rates of the components a and b. Then, the mean relaxation rate was defined as If the two evaluation methods gave different results, the distribution of R 1 s within the undetected part of the decay (initial 0.7 ms), and thus the estimate of <R 1 >, was considered uncertain. This situation occurs in the case of BSA at low relaxation fields, potentially due to the presence of oligomers, as these cause increased relaxation rates and an increased curvature of the relaxation decays.
For the sake of clarity in notation, the brackets of <R 1 > will henceforth be skipped, as is also the case in the main paper.  Figure S2. Relaxation rates of (a) LYZ and (b) BSA as provided by bi-exponential (open symbols) and lognormal-distribution fitting to the raw data. If not indicated by error bars, the fitting error was of the same order of magnitude as or smaller than the symbol size. Data points marked with a cross were not used for further data analysis.
An additional argument supporting the accuracy of the FC-NMR measurements at low field in LYZ solution is provided by the comparison of R 1 with R 1 measurements, as the latter provide a safe, dead-time free estimate of the relaxation rate at low frequencies in absence of any relevant delays during the pulse sequence. At 65 mg/ml and 130 mg/ml protein concentration, the FC-NMR measurements provide a plateau value of R 1 that at all temperatures matches the corresponding value of 10/3 R 1 see Fig. S3)  The same holds for the samples of higher protein concentrations (213 mg/ml, 257 mg/ml) at temperatures above 15°C. At lower temperatures of the highly concentrated samples, however, no plateau of R 1 is observed, which we address to the increased relevance of the slow component of rotational diffusion. However, to be safe with respect to the influence of potentially biased data points in highly concentrated LYZ solutions, we excluded FC-NMR values of a mean relaxation rate above 0.5 ms -1 (at and below this value, R 1 reaches a plateau, in agreement with R 1 ) from data fitting, and observed very similar results compared to the fit of all data without exceptions, see Table S1.

(i) LYZ
. Dispersion profiles of LYZ at several temperatures. The shown data points include the full FC-R 1 data set, a part of it being shown in Fig. 5 of the main paper. 10/3R 1 data points are shown as solid symbols. Additionally, we here include data points for R 1 measured with a saturation recovery sequence on a Bruker minispec mq20 at the resonance frequency 20 MHz. In order not to overload the figure, error bars were skipped, the measuring uncertainty of the relaxation rates is comparable to the symbol size. A part of these data plotted as a function of temperature is shown in Fig. S5a.

(ii) BSA
Commercially available BSA contains a significant fraction of oligomers in solution. If these are neglected upon fitting the data, the procedure will give a mean rotational correlation time that incorporates an uncertain weighting of monomers vs. oligomers. However, since  S >> rot a spread of the rotational correlation does not challenge the sensitivity of the fit to the slow component: R 2 and R 1 values are most sensitive to the latter, whereas relaxation at higher Larmor frequencies serves to fix  rot .
Taking into account the oligomer fraction increases the number of fitting parameters, but provides more reasonable results. To reduce the complexity of the fitting model (otherwise, the fitting is too uncertain and ambiguous), we assumed the oligomer fraction P M to stay the same over the entire temperature range ( 4 C 30 C T     ), and fixed the rotational activation energy of oligomers and monomers at all concentrations to 20 kJ/mol, that is, the activation energy of water flow that governs the protein rotational motion (in the case of LYZ and Bcrystallin (Roos et al. 2015) we obtained similar values of E rot ). We divided the rotational correlation time into two components, rot  and rot2  , that correspond to the monomer and oligomer fractions, respectively. In practice, the second term of J(), see Eq. (7)  (1 ) 1 1 1 Figs. S4 and S5 displays the corresponding fitting results. According to the data analysis, oligomers comprise more than half of the protein mass. This outcome is in accordance with our size-exclusion chromatography results obtained for dilute BSA solutions (cf. Materials and Methods), and further agrees well with the findings of ref. (Atmeh et al. 2007). Table S2 presents the fitting results for the two options of the data analysis (single  rot and rot  / rot2  components of the overall tumbling correlation function). Fitting curves (solid red lines) correspond to the analysis that takes into account oligomers (Eq. S8). Crossed symbols are uncertain data points that were not included in data fitting (see explanation in the text). Other BSA data are shown as function of temperature in Fig. S5b. To stabilize these fits, the activation energy of rot  was fixed to 20 kJ/mol at all concentrations.
(iii) Relaxation rates as a function of temperature

Fitting LYZ data assuming no "long tail" component
The necessity of inclusion of the "long tail" in the fitting model was justified in our previous papers, see the Introduction. Here, by the example of LYZ data, we demonstrate once more that neglecting the "long tail" provides a systematic inconsistency between experimental points and fitting curves. Fig. S6 presents the LYZ relaxation data in frequency and temperature domains along with the fitting curves assuming 2 rot S =0.   The regions of systematic inconsistency between the experimental and fitting relaxation rates values are indicated by red circles/ovals.
As seen in that graph, at low concentration R 2 values cannot be reasonably fitted. The correlation time  S at this concentration is large, and thus it affects only R 2 ; all other relaxation data can be described by the simplest model quite well. Upon increasing concentration,  S becomes shorter (see the discussion of this phenomenon in the main paper), and thus the region of the fitting inconsistency shifts towards FC R 1 's in the frequency range 0.5 -1 MHz. The higher the concentration, the larger the order parameter 2 rot S and, hence, the larger the fitting discrepancy, which is seen in Fig. S6. Finally, we note that the root mean square deviation of the fitting (see Eq. 11 of the main paper) with and without the "long tail" is 0.12 and 0.2, respectively.

Simulation of high-field 15 N NMR relaxation data.
The 15 N relaxation parameters were simulated according to the standard formulae for the heteronuclear dipole-dipole and CSA relaxation mechanisms with a typical values for the NH pair in the protein backbone (see e.g. Daragan and Mayo 1997). Below we show the simulated R 1 , R 2 and NOE's for three frequencies as a function of the product 2 rot S  S for the "rigid" and "mobile" residues:  "mobile" "rigid" Figure S7. Simulated R 1 , R 2 and NOE's as a function of the product 2 rot S S  for "rigid" and "mobile" residues.