Rotameric analysis: evidence for internal concerted motions
Following 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.
To 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.
Figure 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.
Fig. 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
In 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.
Different 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 <6%/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.
Distributions 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.
According 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 > g+> 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 > g− > g+ when χ1 is in the g− state, but has a preference for g+>g− > 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.
Evidence 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.
The 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 χ1.