PMC:2728246 / 21919-66646 JSONTXT

{"target":"https://pubannotation.org/docs/sourcedb/PMC/sourceid/2728246","sourcedb":"PMC","sourceid":"2728246","source_url":"https://www.ncbi.nlm.nih.gov/pmc/2728246","text":"Results\n\nMF-derived side chain dynamics\nThe order parameters derived from the average spherical harmonics obtained from methyl group RDCs measured in 13 different alignment conditions according to the MFA are plotted in Fig. 2. Of the 50 methyl groups in ubiquitin, 37 gave rise to order parameters and MF parameters. Residues for which RDC data was too sparse, as indicated by a small 5th eigenvalue (\u003c0.35) of the F-matrix, were rejected. This affected the following residues: Thr7(Cγ1), L15(Cδ1), V17(Cγ1), I23(Cδ1), V26(Cγ2), I30(Cδ1), L43(Cδ1 + Cδ2), Thr55(Cγ1), L56(Cδ1), Thr66(Cγ1), L69(Cδ1), and L71(Cδ1). Dynamic Q-values have been calculated (see Supplementary Information, Table S1). The average dynamic Q-value is \u003cQdyn\u003e = 0.13.\nFig. 2 Methyl group a order parameters, and b amplitude of anisotropy, ηrdc measured from RDCs plotted against residue number in human ubiquitin. Also encoded in the graph are residue type (letter), their solvent exposure (grey core exposed; white solvent exposed) and the distance of the methyl group to the backbone (blueCβ; greenCγ; orangeCδ; redCε). RDC-based order parameters have a mean value of 0.43 with a standard deviation of 0.25\nThe range spanned by the methyl axial order parameters shows striking heterogeneity in methyl group dynamics. They vary from very rigid (e.g.,  ~ 1 for A28 (Cβ), I44(Cγ2) and A46(Cβ) to very mobile [e.g.,  ~ 0.05 and 0.1 for L69(Cδ1) and L8(Cδ1)]. The mean order parameter over all measured methyl groups in ubiquitin is with a standard deviation of σ = 0.25. This value is much lower than the average order parameter measured on the more rigid backbone NH groups on the same protein and under the same conditions (\u003c(NH)\u003e = 0.72 ± 0.02; Lakomek et al. 2008a).\nThe magnitude of errors is larger for methyl groups than for NH backbone amide groups (errors for DNH were estimated to be 0.3 Hz or less). The first reason is a limited resolution of the RDC measurements using a constant-time HSQC (Kontaxis and Bax 2001) with the limited maximum evolution time corresponding to approximately 15 Hz resolution compared to better than 2 Hz resolution for NH groups measured on a normal HSQC (without zero filling). Secondly, the coverage of the five-dimensional RDC space is less complete than for the NH measurements, as indicated by a lower 5th eigenvalue of the F-matrix. Although the combined effect of both errors propagates, a quantitative analysis of the results was still possible.\nThree notable order parameters have values larger than 1, based on the MF theory. Those correspond to methyl groups of A28 (Cβ), I44(Cγ2) and A46(Cβ). Whereas such values are impossible in reality, they are still within experimental error in the allowed range for order parameters (0–1). For statistical purposes, these three order parameters were corrected to the maximum physical value of 1.\nIn Fig. 2b, the MF parameter ηrdc is shown for each methyl group in ubiquitin. This parameter represents the anisotropy of motion, assuming values from 0 (perfectly axially symmetric motion) to √¾ (fully anisotropic). This parameter, in combination with the order parameter, is useful for differentiating complex types of motions; for example, deciphering those dominated by two or three rotameric jumps or by libration motion. Also for the amplitude of anisotropy ηrdc, a large heterogeneity is observed, with variations seen from 0.04 ± 0.05 (V17(Cγ2)) to 0.83 ± 0.15 (L71(Cδ1)).\n\nValidation of MFA\nAs shown in Fig. 2a, methyl groups located at positions distant from the rigid backbone are more mobile and show smaller . This is expected since their orientational fluctuation will depend on the combined rotameric jumps of all intervening torsion angles between the methyl group and the backbone. It is also known that the first χ1 angle (N–Cα–Cβ–Cγ) has a preference for the gauche− position (−60°) due to steric hindrance with the backbone (Janin et al. 1978), whereas similar effects are less pronounced for all subsequent torsional angles (χ2,χ3,…). Similar properties have been seen already from relaxation studies of protein side chains, in which there is a general tendency for decreasing order parameters with increasing distance from the backbone (LeMaster 1999; LeMaster and Kushlan 1996; Houben and Boelens 2004; Mittermaier et al. 1999). The extent of anisotropy (Fig. 2b) can also be quite revealing; for instance, it is expected that geminal methyl groups of leucine and valine side chains not only have similar order parameters but also similar motional anisotropy in the simplistic model where their mobility depends on a single torsion angle. However, exceptions would be expected for cases where adjacent side chain and backbone dihedral angles undergo substantial concerted motion. In the following, these and other MF parameters for methyl groups are examined in the context of their distance to the backbone, of residue type and of solvent accessibility.\n\nIsoleucines\nFor the isoleucine residues, a tendency is observed for the Cδ1 methyl groups attached to Cγ1 to be more mobile than those at Cγ2. Since the Cδ1 methyl groups are further away from the backbone than the Cγ2 methyl groups this increased mobility does not come as a surprise. It can be observed for I3 in the first β-strand with (Cγ2) = 0.67 ± 0.15 \u003e (Cδ1) = 0.30 ± 0.19, I44 in the third β-strand with (Cγ2) = 1.25 ± 0.44 \u003e (Cδ1) = 0.13 ± 0.06 and I61 in loop region with (Cγ2) = 0.72 ± 0.12 \u003e (Cδ1) = 0.37 ± 0.18.\nAll three residues, I3, I44 and I61, also show substantial differences in amplitudes of anisotropy: η(Cγ2) = 0.73 ± 0.13 and η(Cδ1) = 0.27 ± 0.10 for I13, η(Cγ2) = 0.18 ± 0.08 and η(Cδ1) = 0.56 ± 0.11 for I44 and η(Cγ2) = 0.08 ± 0.06 and η(Cδ1) = 0.26 ± 0.12 for I61.\n\nAlanines\nOnly two alanine residues are present in ubiquitin, at positions 28 and 46. Alanine residues have a short side chain with the methyl groups directly attached to the backbone. The Cα–Cβ axis motion does not depend on any side chain dihedral angle and only fluctuates along with backbone motions. Both A28 and A46 show the highest order parameters with (Cβ) = 1.38 ± 0.47 and (Cβ) = 1.09 ± 0.26, respectively. Whereas A28 is located in the rigid α-helix ((NH) = 0.85 ± 0.06), A46 is at the slightly less rigid interface between the β4-strand and the following loop.\n\nValines\nOf the four valines present, only two (V5 and V70) yield both Cγ1 and Cγ2 data. Both for V5 and V70, the (Cγ1) and (Cγ2) order parameters have almost identical values. While for V70, the amplitudes of anisotropy ηrdc are very similar, those for V5 were somewhat different with η(Cγ1) = 0.28 ± 0.19 and η(Cγ1) = 0.05 ± 0.09, although they are equivalent within the experimental error. However, since V5 is involved in the packing of the core, the presence of concerted motions cannot be ruled out. V70 is also the only valine of the four to exhibit some solvent exposure; accordingly, it has the lowest order parameters of the four ((Cγ1/Cγ2) = 0.27/0.22 vs. 0.41/0.39 (V5), –/0.82 (V17), 0.62/– (V26)) and the highest level of anisotropy (ηrdc(Cγ1/Cγ2) = 0.48/0.45 vs. 0.28/0.05 (V5), –/0.04 (V17), 0.10/– (V26)), suggesting the sampling of more than one rotameric states about the χ1 torsion angle.\n\nLeucines\nFor four leucines, L8, L50, L67, and L73, RDC-based order parameters were available both for Cδ1 and Cδ2. For L8, L50, and L67, (Cδ1) and (Cδ2) were identical within the error. The same also holds for the amplitudes of anisotropy. This indicates isotropic behavior for the leucines apart from L73 in the very flexible C-terminal part of ubiquitin. L73, being very mobile, shows a statistically significant difference both in order parameter and amplitude of anisotropy with (Cδ1) = 0.05 ± 0.04 and (Cδ2) = 0.18 ± 0.06 and η(Cδ1) = 0.83 ± 0.15 and η(Cδ2) = 0.14 ± 0.11. The difference in η looks very dramatic, however, it must be realized that in this case, big differences in η can occur even if the motion of the two methyl groups is not vastly different, given the very low order parameter.\n\nGeminal methyl groups\nThe (CC) measurements do not show large significant differences for geminal methyl groups in the two valines and four leucines for which both order parameters were available, except for the very flexible L73. The question whether geminal methyl groups exhibit similar mobility in proteins has been addressed by several different methods, but has led to conflicting interpretations. A study on 13C–13C cross-relaxation on serine-protease PB92 (Houben and Boelens 2004) reports consistently higher order parameters for the methyl group that is trans in the main chain as compared to the methyl group that is gauche. In support of this, relaxation studies on thioredoxin (LeMaster and Kushlan 1996; LeMaster 1999) have led to the observation of important differences in S2, which were interpreted as concerted motions. This effect has however been observed to a much smaller extent from 2H-relaxation measurements for various other proteins (Millet et al. 2003; Mittermaier et al. 1999; Flynn et al. 2001). For ubiquitin, there is no indication of this behaviour from 2H-relaxation measurements either. Interestingly, the anisotropy of motion η (Fig. 2b) show some differences for geminal methyl groups and could also underline the presence of different dynamic properties. This point is addressed again in the rotameric analysis (vide infra).\n\nSolvent accessibility\nNeglecting the two alanines and the Cγ for I44 which have unusually large order parameters, a correlation between the side chain solvent-accessibility and the side chain mobility can be observed (Fig. 3a). The correlation coefficient between solvent-accessibility and (CC) order parameters is r = −0.501. The side chains which are more solvent-accessible show a tendency to be more mobile.\nFig. 3 a Methyl group order parameters measured from RDCs in ubiquitin are plotted against side chain solvent-exposure parameters calculated with Molmol (Koradi et al. 1996) on pdb:1ubi structure. The straight line represents the correlation between the two parameters (r = −0.501). b Influence of the number of torsion angles from the backbone to the methyl carbon on its measured order parameter. Mean values are 1.24, 0.52, and 0.25 for χ1-, χ1χ2- and χ1χ2χ3-dependent methyl groups. c Order parameters as a function of the angle between the average axial orientation predicted from MFA and from X-ray structure pdb:1ubi for methyl groups\n\nDistance to backbone\nLooking at the methyl group order parameters for all residues, there is a very clear correlation between mobility and distance to the backbone as shown in Fig. 3c. Methyl groups separated by two torsion angles (Cδ1 and Cδ2, depending on χ1 and χ2) have a mean order parameter of only 0.25 compared to 0.52 for those separated by only one torsion angle (Cγ1 and Cγ2, depending only on χ1). As seen above, the alanines Cβ have much higher order parameters with a mean value of 1.\n\nAverage orientation\nAnother validation comes from the MF parameters (ϕav,θav) which represent the polar angles of the average orientation of the C–C bond within the frame of the ubiquitin reference structure. It is expected that for those methyl groups with low mobility (high order parameter), the average orientation lies close to the corresponding orientation defined from the electron density in the reference crystal structure, especially if those are well-defined and unaffected by the crystal packing effects. It would also be expected that for more mobile methyl groups, this correlation would weaken due to the nature of the averaging effect of second order tensors explained earlier in the Materials and Methods section. Figure 3b shows the relationship between the order parameter and the angle between the average orientation defined from the X-ray structure (pdb code:1ubi). There is indeed a strong tendency for the MFA to yield a similar orientation of C–C bonds relative to the crystal structure for the more rigid methyl groups. Except for A28 and L69, all methyl groups with  \u003e 0.45 are predicted within 20° of the reference. Conversely, for all other methyl groups with lower order parameter, an even distribution of angles is observed within the allowed range from 0° to 90°, due to the averaging properties of second order tensors.\n\nMFA detects important supra-τc motion in the side chains\nPrevious studies have reported on the dynamics of methyl groups in ubiquitin using different techniques. It is of interest to compare these results with the RDC-based order parameters reported here. It is important, however to highlight the differences between the various approaches with regards to the time scales as well as the conformational space sampled by the methyl group motion.\nOrder parameters were measured (Lee et al. 1999) from 2H relaxation on selectively 2H-labeled ubiquitin under conditions very similar to ours in concentration, composition and temperature. As for the RDC-based order parameters, the relaxation-based order parameters, (CC) [subscript LS for Lipari-Szabo, in reference to (Lipari and Szabo 1982)], reflect motional fluctuations of the methyl axial CC bond relative to an external reference frame, but with the important difference that the time scale sampled by these motions is strictly faster than the correlation time (τc) of ubiquitin corresponding to about 4 ns. Since RDC-based order parameters are additionally susceptible to the motions over time scales 5–6 orders of magnitude slower than τc, it is generally expected that the condition:10 will be respected for every methyl group, in a similar fashion than what has been described for backbone amide groups when comparing relaxation and RDC-dynamics data (Lakomek et al. 2005, 2006, 2008a, 2008b; Lange et al. 2008; Meiler et al. 2000, 2001). It is however, noteworthy to mention that, whereas the overall scaling of amide order parameters, (NH), strictly depends on the corresponding condition ( within the experimental error), no additional scaling is used here for the methyl groups, since the alignment tensor is predetermined from the amide order parameters (which already include this condition).\nThe relaxation-based order parameters are represented in a correlation plot against the corresponding model-free RDC-based order parameters from our analysis and on a per-residue context in Fig. 4a and b. In both figures, for all but a single residue, the condition described by (10) is respected within the error ranges. It can also be appreciated from those figures that the is much smaller than for a large majority of residues. Not surprisingly, this indicates that there are very important additional motions present in the ns–μs time scale, which are invisible to relaxation. The mean and standard deviation for corresponding sets of methyl groups are  ± σ = 0.66 ± 0.25 and  ± σ = 0.43 ± 0.25. An estimate of the extent of the supra-τc order parameters gives indicating that there is almost as much mobility present in the supra-τc as in the sub-τc ranges for methyl groups This supra-τc mobility of the methyl groups has much larger amplitude than what was observed for amide groups in the same time window. From a similar comparison, the respective amount of backbone motion in the supra-τc region was observed to be on the order of  = 0.93 (Lakomek et al. 2008a). Despite these large differences, the order parameters on both scales show a correlation coefficient of r = 0.72, suggesting that a fair portion of the additional mobility in the supra-τc region can be interpreted mostly as a “broadening” of the amplitude of motion.\nFig. 4 Methyl order parameters measured by other methods are plotted against order parameters obtained from RDCs in ubiquitin (a, c, g, e) and against residue position (b, d, f, h) in color ( are in black). a,b are obtained from 2H relaxation experiments (Lee et al. 1999) and sample motions faster than the correlation time. c,d are obtained from the CxH dipolar splitting reduction for ubiquitin in a microcrystalline form using separated local field experiments by solid-state NMR under magic-angle spinning condition. of CxHx are plotted with the of CxCmet. These values are obtained by squaring the values reported by (Lorieau and McDermott 2006). The time scale sampled also includes the ns–ms range. e,f obtained from rotameric fitting of χ1 based on averaged 3JNCγ- and 3JC′Cγ-couplings and on CH RDCs according to (Chou et al. 2003). These order parameters are based on a finite rotameric jump model and sample dynamics over similar time scales. g,h The same order parameters are multiplied by S2(NH)LS for the same residue (i, green) and for the following residue (i + 1, red) to include rapid small-scale fluctuations. The Pearson correlation r (with p value) and rmsd fit are also given\nA simple analytical method was proposed to back-predict methyl group Lipari-Szabo-type sub-τc order parameters from static protein structures based on the distribution of local contacts around methyl carbons and the number of χ angles to the backbone (Ming and Brüschweiler 2004). Interestingly, since a similar correlation coefficient is obtained from the back-predicted order parameters, (individual data not shown, correlation coefficient r = 0.66), it is reasonable to suppose that this simple local contact model would also apply in the supra-τc time window.\nA second set of order parameters, , comes from solid-state NMR measurements of the dynamic averaging of the CxH dipolar couplings, adjacent to a CxCmet bond, using separated local field (SLF) methodologies on ubiquitin in microcrystalline form undergoing magic-angle spinning near ambient temperatures (Lorieau and McDermott 2006). It is reasonable to expect that the adjacent CxH and CxCmet bonds would have similar mobility, assuming that concerted motions are not dominant. Despite the important difference in the sample configuration, the averaging of the dipolar couplings in solids or of the residual dipolar couplings in solution is susceptible to motions including the supra-τc (faster than μs for SLF, faster than ms for RDCs). The SLF generalized order parameters reported as |S| = |\u003cP2(cosθ)\u003e| represents the coefficient of reduction of the corresponding static CH dipolar coupling. For comparison with our order parameters, these coefficient were squared, giving |S|2 = .\nFigure 4c and d show the correlation between methyl groups and the corresponding methine and methylene groups (in the solid state) for which both dynamic parameters were measured using the two different methods. Whereas agreement for individual residues cannot be expected, we find remarkable concurrence of the overall scale of order parameters for the ensemble of methyl groups. Indeed, we notice a much less pronounced tendency for these order parameters to be either larger or smaller. For comparison, whereas the average order parameters measured using relaxation methods is generally higher the average order parameters from SS-NMR are much closer to that of the RDC-based order parameters  = 0.48 vs.  = 0.43. Furthermore, we see agreement within error ranges for over 61% of the order parameters using SLF. The correlation coefficient r in this analysis is 0.38 which indicates that there is quite a bit of variation in the differences between the solid-state derived order parameters and the ones derived from our MFA. The rmsd is with 0.24, however, relatively low.\nAnother relevant set of order parameters, , available in the literature are those obtained indirectly for methyl groups at position Cγ using 3-bond J-couplings and a set of RDCs from two alignment media by (Chou et al. 2003). From the rotameric angular distributions obtained from grid search protocols, the corresponding order parameters ranges for each residue were back-calculated. represents the intersecting range between J-coupling- and RDC-based ensembles of rotameric solutions. Since these order parameters sample the same time scales as those presented here, they are expected to fit in the optimal way to the model-free order parameters derived here. Of course, there are differences in the sensitivity of the involved observables to certain motions. For example, the J-coupling averaging is, to a first approximation, only susceptible to axial fluctuation about the Cα–Cβ bond, such as rotameric jumps of the χ1 torsion angle. They are insensitive to the orientational fluctuation of the Cα–Cβ axis itself and because of the nature of the Karplus curves, especially around the gauche positions, they are rather insensitive to fast fluctuations around each rotameric equilibrium position. In contrast, RDCs are sensitive to all types of fluctuations as long as the 5-dimensional space is properly represented, owing to sufficiently linearly independent alignment media. Despite these important differences, the overall correlation between these order parameters and our model-free is rather good with a correlation coefficient of r = 0.58 and an rmsd of 0.28.\nMost striking, however, is the observation of in all available cases as seen in Fig. 4e, f. The obvious interpretation is that the rotameric jump model only explains part of the mobility experienced by the Cβ–Cγ bond. Just like the ratio defines the temporally distinct supra-τc order parameters, the ratio could define a geometrically distinct libration order parameter around the equilibrium positions. For instance, I30(Cγ1) is predicted to mostly sample a single rotamer with the gauche− rotamer dominating the χ1 distribution, based on . This same methyl group is however, quite mobile based on (0.48 ± 0.20). Interestingly, the backbone at that position over the same time scale is relatively flexible based on (NH) = 0.78, which would suggest that some of the mobility would be due to “wobbling” of the Cα–Cβ bond with the backbone. Nevertheless, for other residues where the Cγ undergo significant rotameric interconversions, is exquisitely sensitive to supra-τc motions. Notably, the were significantly lower than the (CC) for several residues in the first pair of ubiquitin’s anti-parallel β-strand, indicating that rotameric averaging on a time scale slower than the rotational correlation time is taking place.\nDespite already good agreement, we could improve the correlation between the two independently determined order parameters even further in the following way: We multiplied the rotamer-based order parameter (CCi) with a local backbone order parameter, (NHi) or (NHi+1), representing small-scale rapid fluctuation (compare Fig. 4g, h). Slightly better agreement is obtained using the following NHi+1 order parameters (Fig. 4g, h, residue i + 1, red, rmsd = 0.18 and r = 0.63) compared to those from the previous NHi (Fig. 4g, h, residue i, green, rmsd = 0.22 and r = 0.47). This result indicates that both rotameric sampling and fast small-scale libration around equilibrium positions generally have an important contribution to the overall mobility of the CCmet bond. With this correction of the (CCi) order parameters, they fit with the largest correlation coefficient r and the smallest rmsd to the order parameters derived from our MFA.\n\nCorrelation between (CC) and (NH)\nExtending the investigation of side chains dynamics to the supra-τc time windows reveals substantial additional mobility slower than τc, which is likely to be composed of rotameric jumps around torsion angles and of small-scale rapid fluctuations around equilibrium positions. Whereas the mechanism underlying these motions is too complex to decipher with the approach used in this study, it would be intriguing to study whether these substantial movements might have an effect on the motions at the level of the backbone, especially considering that the important rearrangements associated with rotameric jumps may require local structural accommodations even from the backbone. Mechanisms of such transfer of motion have already been suggested and applied to improve side chain conformational variability models (Davis et al. 2006; Friedland et al. 2008).\nRecently, a tendency was observed that residues with solvent exposed side chains appear more mobile in protein backbone. A transfer of motion from the side chains to the protein backbone had already been hypothesized (Lakomek et al. 2005). Due to the lack of dynamic information for the side chains, conclusions were based on the assumption that solvent-exposed side chains were more mobile than those in the core of the protein. Now that direct experimental information about side chain dynamics in a time window from pico- to milliseconds becomes available, it is of interest to investigate whether there is a direct correlation between the order parameters of the methyl groups and those of the backbone amides to which they are directly bonded.\nIn Figure 5a and b, the order parameters of methyl groups are plotted against the order parameters of the same (i) and of the following (i + 1) NH group over different time scales and the linear correlation coefficients are summarized in Table 2. For pairs of values where the correlation is deemed “significant” based on p-values smaller than 0.1 (probability that this correlation would occur by chance, in brackets), a few observation can be made. On all time scales, S2(CCi) show better correlation with S2(NHi) than with S2(NHi+1). This is the case for fs-ms dynamics with Pearson coefficients of ri = 0.49 and ri+1 = 0.34, as well as for fs-ns dynamics  = 0.60 and  = 0.41. These numbers show that a possible mechanism of transfer of motion from the side chain to the backbone is more prevalent from the side chain to the amide group of the same residue than to the next one as has been postulated earlier (Lakomek et al. 2005).\nFig. 5 Correlation of RDC-based order parameters from side chains at residue position i from methyl Cγ (full circles) and Cδ (open circles) with RDC-based backbone NH order parameters at the corresponding i and i + 1 positions taken from (Lakomek et al. 2008a). Short and long dashed lines correspond to the linear function fitting for Cγ and Cδ order parameters\nTable 2 Pearson correlation coefficient of side chain and backbone order parameters over various time scales\nTime scale Order parameters S2(NHi)a,b No. of data S2(NHi+1)a,b No. of data\nfs–ms (CCi) 0.49 (0.006) 30 0.34 (0.07) 29\n(CCγi) 0.50 (0.06) 15 0.19 (0.52) 13\n(CCδi) 0.46 (0.11) 13 0.26 (0.37) 14\nfs–ns (CCi) 0.60 (0.003) 23 0.41 (0.06) 22\n(CCγi) −0.17 (0.58) 13 0.36 (0.33) 9\n(CCδi) 0.76 (0.02) 9 0.48 (0.10) 13\naNumber in brackets is the null-hypothesis p-value\nbOnly correlations in bold are considered significant. All the others (in italic) are not significant\nWhen classifying the data according to the distance of the methyl group to the backbone, it seems that the order parameters at the Cγ methyl show practically the same correlations to the amide motion on the same residue as the Cδ methyl groups (e.g., ri(γ) = 0.50 vs. ri(δ) = 0.46). The slope of the linear correlation is, however, roughly twice for Cδ compared to Cγ, indicating that the methyl group motion transfer are significantly dampened through the degrees of freedom of the linear side chain.\nWhen comparing the correlation of motion between side chains and backbone for the fast (ps to ns) to the full (ps to ms) time window detected by RDCs a better correlation from faster motions is obtained. This suggests that the major contribution of this transfer already occurs at time-scales faster than the correlation time (τc). These findings encourage further investigations of a possible transfer mechanism of motion from the side chains to the protein backbone.\n\nCorrelation between MFA and dynamic ensembles\nIn recent years, a number of independent computational ensembles have been calculated to fulfill both structural restraints and dynamical information (Clore and Schwieters 2004, 2006; Markwick et al. 2007; Lindorff-Larsen et al. 2005; Richter et al. 2007; Lange et al. 2008; Friedland et al. 2009). From a number of available ubiquitin ensembles, we have back-calculated the average spherical harmonics and related MF parameters and compared them to the new data presented in this study. Thus, further insight into the conformations adopted by the side chains, especially in the ns-ms time scale can be obtained. Data for the ensembles of structures used here are presented in the Table 3.\nTable 3 Conformational heterogeneity for methyl groups from the average structure and dynamic ensembles of ubiquitin and correlation with RDC-derived average spherical harmonics\nPDB Number of conformers Positional rmsd of all methyl group carbons Av ± st dev order parameter Pearson correlation (\u003cY2,M\u003e) rmsd (\u003cY2,M\u003e) References\n1d3z 10 0.54 ± 0.23 Å 0.87 ± 0.19 0.8379 0.0758 Cornilescu et al. (1998)\n1xqq 128 1.76 ± 0.32 Å 0.59 ± 0.27 0.8497 0.0597 Lindorff-Larsen et al. (2005)\n2nr2 144 1.43 ± 0.24 Å 0.66 ± 0.21 0.8748 0.0588 Richter et al. (2007)\n2k39 116 2.28 ± 0.43 Å 0.45 ± 0.29 0.8568 0.0496 Lange et al. (2008)\n«brub» 50 1.77 ± 0.28 Å 0.59 ± 0.26 0.8092 0.0648 Friedland et al. (2009)\nFigure 6 displays a per-residue comparison of the order parameters from the highly restrained lowest-energy average NMR structures (pdb:1d3z), from the three dynamically-refined ensembles (pdb:1xqq, 2nr2 and 2k39) and from the “Backrub”-generated ensembles (“brub”) with those obtained from the MFA. While the pdb:1d3z structure can be considered as a good approximation for the average structure in solution for the backbone (Cornilescu et al. 1998), our results indicate that a single structure representation is not appropriate to describe the large amplitude side chain motions observed for ubiquitin. Not surprisingly, all the order parameters for this ensemble are larger than those from the MFA. Accordingly, the fit of the back-calculated order parameters with the experimental ones is the worst, both in terms of correlation factor (r) as well as rmsd as shown in Table 3.\nFig. 6 Order parameters plotted against residue position as back-calculated from different structural ensembles (blue:1d3z, green: 2nr2, yellow: 1xqq, red: 2k39, cyan: “backrub”-ensemble (brub), black: MFA)\nIn comparison, the dynamic ensembles pdb:1xqq and pdb:2nr2 show a much higher level of conformational heterogeneity, by virtue of their refinement against not only backbone but also side chain (methyl group) relaxation-based order parameters. However, as discussed previously, these order parameters are insensitive to supra-τc motion. Predictably, the back-calculated order parameters from these ensembles do not reproduce very well those acquired from RDCs, which are smaller for the large majority of residues. The average order parameters (σ = 0.27) and (σ = 0.27) are both significantly larger than the average from the MFA Agreement within error ranges only occur in 33% of the cases.\nThe EROS ensemble (pdb:2k39) is also a dynamically refined structure, with the important difference that the ensemble refinement is not performed against order parameters but rather against orientational restraints provided from RDCs (including backbone DNH and methyl group DCC). As a consequence, it is anticipated that the conformational heterogeneity present in the ensemble reflect fluctuations over a wider time scale (fs-ms).\nOf the five ensembles, the pdb:2k39 boasts together with pdb:1xqq the best correlation (Pearson correlation r = 0.857) and by quite a big margin the best overall agreement (rmsd = 0.0496) with the MFA results. The average order parameter is almost identical to the average from the MFA. Agreement is present in 66% of the residues measured. Despite the high fluidity reproduced in the Lipari-Szabo-based dynamic ensembles, this supports that there is a large amount of motion that is not sampled by relaxation techniques because it occurs in the ns-ms time scale.\nThe “Backrub”-generated ensemble (brub) is an ensemble of ubiquitin structures in solution that is created by sampling conformational space without experimental information using the “Backrub” motions, inspired by alternative conformations observed in sub-Angstrom resolution crystal structures (Davis et al. 2006). An ensemble of 50 Backrub-generated structures was selected to optimize agreement with 23 datasets of NH backbone RDCs for ubiquitin (PDB entry pending; Friedland et al. 2009). As opposed to the other ensembles, the methyl group RDCs were not used as restraints to generate the brub ensemble. The ensemble represents therefore an intermediate between 1d3z which has not been optimized to match conformational averaging seen by RDCs and 2k39 where backbone and side chain RDCs were utilized. Accordingly, the brub ensemble demonstrates an average methyl group order parameter of 0.59 that resides between the ones reported for 1d3z and 2k39 (Table 4). Interestingly, the rmsd to the RDC-derived average spherical harmonics is also reduced with respect to 1d3z but elevated when compared to 2k39.\nTable 4 Ideal rotamer populations derived from the grid search against RDC-based average spherical harmonics and comparison to rotamer statistics from three dynamic ensembles, the backrub ensemble and the rotameric fitting to J-couplings\n1xqqa 2nr2b 2k39c brubd Je MFAf\nV5\n    0 0 0.04 0 0.02 0.09 (0.04)\n    1 1 0.88 1 0.92 0.70 (0.03)\n    0 0 0.08 0 0.06 0.21 (0.04)\n    0.88 (0)g\n   ssd 0.037\nI3\n    1 1 1 0.78 0.47 0.84 (0.05)\n    0 0 0 0.22 0.21 0 (0)\n    0 0 0 0 0.32 0.16 (0.05)\n    0.06 0.01 0.06 0.05 – 0.38 (0.06)\n    0.94 0.99 0.94 0.95 – 0.48 (0.06)\n    0 0 0 0 – 0.14 (0.05)\n    – – – 0 – –\n    – – – 1 – –\n    – – – 0 – –\n    – – – – – 0.08 (0.09)\n    – – – – – 0.06 (0.09)\n    – – – – – 0.86 (0.10)\n    0.96 (0.03)g\n    0.75 (0.03)g\n   ssd 0.072\nI44\n    0.05 0 0.05 0 0.03 0\n    0.02 0 0.07 0 0.06 0.10 (0.0)\n    0.94 1 0.88 1 0.91 0.90 (0.0)\n    0 – 0 – – –\n    1 – 1 – – –\n    0 – 0 – – –\n    0.5 – 0 – – 0.18 (0.17)\n    0.5 – 0.63 – – 0.16 (0.17)\n    0 – 0.38 – – 0.66 (0.20)\n    0 0.02 0.08 0.04 – 0.03 (0.04)\n    0.45 0.43 0.59 0.60 – 0.46 (0.05)\n    0.55 0.55 0.33 0.36 – 0.51 (0.05)\n    0.76 (0.01)g\n    0.70 (0)\n   ssd 0.047\nL8\n    0 0 0.18 0 – 0.15 (0.06)\n    0.66 0.4 0.28 0 – 0.40 (0.08)\n    0.34 0.6 0.54 1 – 0.44 (0.06)\n    – – 0.33 – – 0.37 (0.19)\n    – – 0.52 – – 0.12 (0.12)\n    – – 0.14 – – 0.52 (0.15)\n    0.92 0.98 0.5 – – 0.42 (0.06)\n    0.08 0.02 0.16 – – 0.03 (0.05)\n    0 0 0.34 – – 0.54 (0.06)\n    0.07 0 0.17 0.24 – 0.03 (0.04)\n    0.93 0.99 0.68 0.76 – 0.68 (0.06)\n    0 0.01 0.14 0 – 0.30 (0.04)\n    1h\n    0.78 (0.14)\n   ssd 0.021\nV70\n    0 0 0.04 0 0.02 0.09 (0.04)\n    1 1 0.88 1 0.92 0.70 (0.03)\n    0 0 0.08 0 0.06 0.21 (0.04)\n    0.88 (0)g\n   ssd 0.037\nI36\n    0 0 0.04 0 0 0\n    0.05 0 0.03 0 0.1 0.2 (0.0)\n    0.95 1 0.93 1 0.9 0.8 (0.0)\n    – – 0 – – –\n    – – 0.8 – – –\n    – – 0.2 – – –\n    0 – 0.67 – – 0.53 (0.21)\n    1 – 0.33 – – 0.18 (0.17)\n    0 – 0 – – 0.28 (0.13)\n    0 0.03 0.02 0 – 0 (0)\n    0.87 0.92 0.81 0.58 – 0.80 (0.05)\n    0.13 0.05 0.18 0.42 – 0.20 (0.05)\n    0.83 (0.05)g\n    0.78 (0.07)g\n   ssd 0.013\nI61\n    0 0 0.01 0 0.01 0.15 (0.05)\n    0 0 0 0 0.01 0.01 (0.03)\n    1 1 0.99 1 0.98 0.84 (0.05)\n    – – – – – 0.57 (0.18)\n    – – – – – 0.11 (0.11)\n    – – – – – 0.32 (0.17)\n    – – – – – –\n    – – – – – –\n    – – – – – –\n    0 0.01 0 0 – 0.01 (0.02)\n    0.85 0.9 0.88 0.70 – 0.87 (0.05)\n    0.15 0.09 0.12 0.30 – 0.12 (0.04)\n    0.95 (0.04)g\n    0.81 (0.09)\n   ssd 0.025\nL67\n    0 0 0.08 0 – 0.17 (0.02)\n    0.01 0 0.07 0.1 – 0.08 (0.04)\n    0.99 1 0.85 0.9 – 0.72 (0.04)\n    – – 0.89 – – 0.69 (0.11)\n    – – 0 – – 0.23 (0.13)\n    – – 0.11 – – 0.07 (0.07)\n    1 – 0.5 1 – 0.15 (0.10)\n    0 – 0.38 0 – 0.33 (0.21)\n    0 – 0.13 0 – 0.57 (0.25)\n    0.52 0.58 0.64 0.26 – 0.54 (0.04)\n    0.48 0.42 0.3 0.74 – 0.45 (0.04)\n    0.01 0.01 0.06 0 – 0.00 (0.01)\n    1h\n    0.87 (0.09)\n   ssd 0.015\naLindorff-Larsen et al. (2005)\nbRichter et al. (2007)\ncLange et al. (2008)\ndFriedland et al. (2009)\neValues from Chou et al. (2003)\nfAverage values (with standard deviation) among grid search solutions within 10% of overall minimum\ngWithin allowed range of corresponding value Lee et al. (1999)\nhFixed at 1 (no variation in grid search)\nSide chain coordinates within the brub ensemble are constructed using a rotamer library (Dunbrack and Karplus 1993). The improved agreement of RDC-derived average spherical harmonics when compared to 1d3z is therefore triggered by accurate representation of backbone conformational averaging that is propagated to the side chains. Thus, an accurate representation of backbone dynamics induces an improved but imperfect representation of side chain conformational averaging in the brub ensemble. This result supports the notion that backbone and side chain motions are coupled and therefore correlated. The accuracy of conformational averaging in side chains of the brub ensemble would benefit from a rotamer selection protocol that optimizes agreement with methyl group RDCs.\n\nRotameric analysis: evidence for internal concerted motions\nFollowing in the steps of (Chou and Bax 2001; Mittermaier and Kay 2001), we address the question whether it is possible to model the mobility of side chain methyl groups based on RDC data. The MFA can provide in principle a detailed description of the side chain mobility with the five independent terms contained in the average spherical harmonics. Side chains with 2 methyl groups (Val, Ile, Leu) contain ten unique parameters to describe the intricate mobility occurring on a time scale faster than ms. For individual side chains pointing inside the hydrophobic core of the protein where motion is restricted, it would be interesting to try to describe the complex side chain fluctuations and decipher internal concerted motions, i.e., correlated rearrangements around consecutive torsion angles.\nTo this end, we chose several residues, for which MFA data were available with comparatively small experimental errors for both their methyl groups (Cγ1 and Cγ2 for valines, Cγ2 and Cδ1 for isoleucines and Cδ1 and Cδ2 for leucines), each acting as individual probes directly on the χ1 or χ2 angles. To model these side chain motions, we assume that their mechanism is dominated by rotameric jumps and fast libration fluctuation around equilibrium positions. As described in the Materials and Methods section, a grid search was performed on all population factors p representing the occupancy of each of the three (Val) or nine (Ile, Leu) possible combinations of ideal [gauche+ (60°), trans (180°), gauche− (-60°)] rotameric pairs for χ1 (V, I, L) and χ2 (I, L). In addition, we include isotropic order parameter terms, and , to account for small scale fluctuations around each χ angle. Table 4 lists the average results for all solutions within 10% of the overall minimum of the squared sum of difference [ssd, (9)] of the grid search and compares them to the statistics from different ensembles.\nFigure 7 models the result of the grid search. For the MFA-based model, we generated 200 structures satisfying the relative rotamer populations and represented the overall isotropic order parameter as a normal distribution of deviations around each of the χ angles. The figure gives a sense of how the RDC-based MFA data can roughly reproduce the same dominant rotameric conformations as those present in the different ensembles for all types of residues. Furthermore, the majority of missing rotamers in the ensembles are also excluded or very rare in the results of the grid search. This result is quite remarkable considering that the ensemble conformations are largely driven by distance and J-coupling restraints as well as hydrophobic packing energy during minimization, whereas the MFA models are purely and uniquely based on orientation restraints.\nFig. 7 Eight representative examples of methyl bearing ubiquitin side chain heterogeneity from average structure (pdb:1d3z), from backrub-ensemble (brub), from dynamic ensembles (pdb:1xqq, 2nr2, 2k39) and from ensembles generated by rotamer model-fitting of the average spherical harmonics resulting from the Model-free Analysis (MFA). Backbone is shown in black, Cβ–Cγ1 (Cβ–Cγ2), Cγ–Cδ (Cβ–Cγ2) and Cγ–Cδ1 (Cγ–Cδ2) bonds are in red (green) for valines, isoleucines, and leucines, respectively. In the MFA rotamer construction of leucines, the small scale fluctuation order parameter is entirely represented in χ2. Side chains are aligned based on N, Cα,Cβ and C′ positions\nIn all dynamic ensembles (1xqq, 2nr2, 2k39) as well as in the backrub-ensemble (brub), residues I3, V5, I36, I44, I61, and L67 have a very strong preference for a single rotameric state of χ1. In contrast, the MFA results predict prevalence of the same rotamer but with substantial additional excursions into a second rotameric equilibrium position in up to 20% of the cases. These few odd rotamers are sufficient to account for part of the lowering of the (Cγ) order parameters as well as increasing their anisotropy (ηrdc(Cγ)) (see Fig. 2). For the solvent-exposed residues L8 and V70, χ1 samples more than one state in all ensembles. For V70, the same 2-state distribution (tr/g−) of χ1 rotamers is reproduced by all ensembles as well as by the MFA. However, the distribution of L8 predicted from the MFA or from the EROS ensemble includes significant representation of all three rotamers whereas the relaxation-based ensembles include only the tr and g− rotamers. This result suggests the existence of an additional mode of motion in the supra-τc for residue L8.\nDifferent levels of agreement are also observed in distributions of χ2, when χ1 is in its dominant rotameric state. Whereas the MFA rotameric predictions for I36, I44, I61, and L67 reproduce the distribution of rotamers for χ2 of the ensembles, I3 adopts more frequent occurrences in the g+/g− positions than in the ensembles (38%/14% compared to \u003c6%/0%, respectively). This distribution for I3 χ2 is imposed by the very high anisotropy value for Cδ1(ηrdc = 0.7), not observed for other residues.\nDistributions for χ1 in ubiquitin have also been previously reported based on averaging of 3JNCγ and 3JCCγ scalar couplings by (Chou et al. 2003) and are also compared in Table 4. In general, these are in good agreement with those presented with the exception of I3. For this residue, the J-couplings-based analysis predicts important populations for all three rotamers (g+/tr/g− = 0.47/0.21/0.32), in sharp contrast to the dynamic ensembles dominated with only gauche+ rotamers (g+/tr/g− = 1/0/0). The authors explain their result by a “rotameric averaging on a time scale slower than the rotational correlation time (4 ns)”. Our results for this residue rather suggest that the distribution of χ1 rotamers (g+/tr/g− = 0.84/0/0.16) are only slightly modified when taking the supra-τc motions into account.\nAccording to the motional heterogeneities predicted for isoleucines I3, I36, I44, and I61, there is evidence for some level of internal concerted motion along the side chain, as the χ2 distribution clearly depends on the value of χ1. For example, in I3, the χ2 angle has decreasing preference for rotamers in tr \u003e g+\u003e g− when χ1 is in the g+ state, but has a preference for rotamers in g− ≫ g+,tr when χ1 is g−. In I61, the χ2 has decreasing preference for rotamers in tr \u003e g− \u003e g+ when χ1 is in the g− state, but has a preference for g+\u003eg− \u003e tr when χ1 is g+. As appreciated from Fig. 7, these combinations of χ angles indicate that I3 adopts a more narrow and elongated conformation whereas I61 is wider and shorter. Similar correlations between χ angles can be made for I36 and I44.\nEvidence for concerted motions is also strong for the two leucines, L8 and L67. While we do not probe directly the motion around the Cα–Cβ torsional angle from methyl group RDCs, this angle is still reflected in both of the δ-methyl groups. From the minimum search, we find that, in L8 where all three χ1 rotamers are present, χ2 prefers being in the g+ and g− positions when χ1 is in g+ (populations: 0.54 and 0.44) or tr (populations: 0.41 and 0.57), but prefers being in tr when χ1 is in g− (population: 0.66). Similarly in L67, three different distributions of tr, g+ and g− rotamers of χ2 are observed depending on the state of χ1.\nThe results of the rotamer analysis indicate that side chains may adopt more than one conformation. Based on a number of residues both solvent and core exposed, there is important evidence for internal correlated concerted motion. This may well be due to the fact that the rather tight packing restriction only allow for certain side chain conformations and therefore the switch from one rotamer state of χ2 requires compensation by a concerted jump from one rotamer state to another for 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