Materials and methods Sample and alignment media Wild-type 15N,13C-labeled human ubiquitin was expressed according to a previous protocol (Johnson et al. 1999) and dispersed in 13 different alignment conditions for RDC measurements. Sample conditions for which datasets 1 through 10 were collected have already been described by (Lakomek et al. 2008a). Datasets 12 and 13 are obtained from additional alignments (positive gel and Pf1 phage) not previously reported. Sample for dataset 12: A 7% positively charged gel sample was prepared according to (Cierpicki and Bushweller 2004). A stock solution of 40% acrylamide and N,N′-methylenebisacrylamide in a 19:1 ratio was mixed 1:1 with a stock solution of 40% (3-acrylamidopropyl)-trimethylammonium chloride (APTMAC) and N,N′-methylenbisacrylamide in a 19:1 ratio. The mixtures were diluted with 10× TBE buffer (0.9 M TRIS, 0.9 M borate, 0.02 M EDTA, pH = 8.2) to a final concentration of 7%. Polymerization was initiated by the addition of 0.15% ammonium peroxide sulphate and 1% tetramethylethylenediamine. Polymerization of the mixture was carried out overnight in plastic tubes of 3.5 mm in diameter. Gels were extensively washed in deionized water where they swelled significantly to a diameter of 8 mm. Gels were then cut to 4 cm in length to match a ratio between length and diameter of 5:1. Gels were dried for several days on a plastic support wrapped with polyvinylidene chloride foil. To prepare the sample, the dried gel was transferred to a 5-mm microcell tube (Shigemi Inc., Allison Park, PA), ~2 mg of protein were dissolved in 300 μl of buffer (50 mM NaPO4, 100 mM NaCl, pH 6.5, 0.05% NaN3) and added to the gel. The plunger was positioned to limit the final length of the gel to 12 mm. Sample for dataset 13: The equivalent of 5 mg Pf1 phage (ASLA Ltd., Riga, Lativa) was suspended in buffer (50 mM NaPO4, 100 mM NaCl, pH = 6.5, 0.05% NaN3) and pelleted three times at 140,000 g for 1 h. The pellet was then carefully suspended in a 15N,13C-labeled human ubiquitin sample (~2 mg) in the same buffer. NaCl salt was later added to a final concentration of about 300 mM. The water deuterium quadrupolar splitting was 15.8 Hz. An additional alignment condition and corresponding RDC measurements (dataset 11) were taken from the literature (Ottiger and Bax 1998, 1999). Table 1 shows the corresponding dataset nomenclature. Table 1 Summary of sample conditions Dataset Corresponding SCRM alignment Short description References 1 A1 7% Positively charged gel (APTMAC/acryl. 1:3) Lakomek et al. (2008a, b) 2 A2 7% Positively charged gel (APTMAC/acryl. 1:1) Lakomek et al. (2008a, b) 3 A3 5% Negatively charged gel (acrylic cid/acryl. 1:1) Lakomek et al. (2008a, b) 4 A4 PEG/hexanol Lakomek et al. (2008a, b) 5 A7 Bicelles: DMPC/DHPC (3:1) Lakomek et al. (2008a, b) 6 A8 Bicelles: DMPC/DHPC/SDS (30:10:2) Lakomek et al. (2008a, b) 7 A9 Bicelles: DLPC/DHPC/SDS (30:10:2) Lakomek et al. (2008a, b) 8 A10 Bicelles: DMPC/DHPC/C14PC (30:10:1) Lakomek et al. (2008a, b) 9 A12 Bicelles: DMPC/CHAPSO/CTAB (50:10:1) Lakomek et al. (2008a, b) 10 A13 Bicelles: DMPC/DHPC/CTAB (30:10:1) Lakomek et al. (2008a, b) 11 A20 Bicelles: DMPC/DHPC/CTAB (30:10:1) Ottiger and Bax (1998) 12 – 7% positively charged gel (APTMAC/acryl. 1:1) This work 13 – Pf1 phages: 15 mg/ml + 300 mM NaCl This work NMR measurements Backbone amide group NH RDCs have been recorded and evaluated as described in (Lakomek et al. 2008a). For methyl group CH couplings, four constant-time 13C,1H-HSQC sub-spectra with varying 1JCH(+1DCH) evolution delays (Δ1 = 0, 1/6J, 1/3J, 1/2J) were recorded in an interleaved manner using the pulse sequence described by (Kontaxis and Bax 2001). Typically, each experiment included 89 complex increments in the indirect 13C dimension for the sampling of a 16-ppm sweep width, centered at 20 ppm. All experiments were recorded either on a 600 MHz DRX Bruker instrument equipped with a 5-mm H/C/N-probehead or on a 700 MHz Avance Bruker instrument equipped with a 5 mm H/C/N-probehead. Sub-spectra were separated, processed and co-added to yield four spectra with single peak components corresponding to the ααα, ααβ/αβα/βαα, αββ/βαβ/ββα, or βββ coherence of each methyl group. Couplings (1JCH and 1JCH + 1DCH) were evaluated from the separation between the four components; for precise measurement, a dedicated FELIX script was designed to perform the spectral shifts necessary to simultaneously overlay the four components. Uncertainties on the couplings have been estimated as the range of shifts for which 4-peak overlays was acceptable. Assuming fast axial reorientation, each measured methyl CH RDC (1DCH3) was rescaled to the methyl CC RDC (1DCC), corresponding to the CC bond co-linear to the rotation axis, using a universal factor of −0.3155 according to (Ottiger and Bax 1999). RDC-based model-free analysis The MFA applied to the methyl group data was similar to the one applied to backbone NH (Lakomek et al. 2008a, 2006). However, important modifications were required to account for the different properties of the methyl groups and are laid out here. A summary of the entire approach is illustrated in Fig. 1. Fig. 1 Schematics of the approach used to obtain model-free parameters for methyl groups in ubiquitin. A total of m methyl group CH RDCs are measured in i alignment conditions. The homogeneity of these alignment conditions is analyzed by SECONDA (Hus and Brüschweiler 2002; Hus et al. 2003). Since methyl groups undergo rapid axial reorientation ( for each residue. All reported order parameters represent the axial motion of the CC bond The object of the MFA is to extract the five average spherical harmonics components by inverting the equation:1 where Di,m represents the measured RDC for methyl groups i in alignment media m, while the Fi,M term is a i-by-5 matrix representing the ensemble of second order alignment tensors in which the RDCs were measured. The five symmetric complex terms collectively describe the dynamic averaging of the RDC, and thus the spatial fluctuation of the methyl CC vector in 5-dimensional space, and occurring on time scales faster than the ms range. Methyl groups are too mobile to reliably determine the alignment tensors of the F-matrix from their RDCs. It is also not possible to calculate the alignment tensor exclusively from the more rigid alanine side chains, since ubiquitin contains only two of them. However, for each alignment condition, a set of NH RDCs (1DNH) from the much more rigid backbone were also available. Thus, we have used the alignment tensor estimated from the backbone NH RDCs and applied a linear scaling to the principal component in alignment medium i to obtain . This conversion factor was estimated to:2 assuming NH and CCmet bond lengths corresponding to rNH = 1.04 Å and rCC = 1.53 Å. For the NH couplings, alignment tensors have been calculated using the software PALES (Zweckstetter and Bax 2000) applied to the dynamic average NH orientations obtained after four SCRM cycles for the NH RDC data set D36M, as described in (Lakomek et al. 2008a). Rhombicity and Eulerian angles were assumed to remain unchanged. The 20% most mobile NH residues as derived for datasets D36M were excluded from alignment tensor calculation in order to minimize the effect of alignment tensor fluctuations. Since only a dynamically reduced principal component of the alignment tensor can be derived from the dynamically averaged , the static value of Di,zz has to be estimated using a robust statistical scaling method that is explained in detail in the supporting information of (Lakomek et al. 2008a). The same overall scaling factor has been used. We want to emphasize that only the order parameter is affected by the scaling factor. All other MF parameters (see below) are entirely unaffected by a change in the overall scaling. Also, the rotameric analysis (last section) is largely unaffected by a change in the scaling factor as long as this change can be compensated by adjusting the local order parameter for the fast fluctuations within a minimum. In order to quantify the similarity of structure and dynamics in the different alignment media (homogeneity of RDC data), a SECONDA analysis was applied (Hus and Brüschweiler 2002; Hus et al. 2003) on the methyl RDCs. The SECONDA ratio of the 5th to 6th eigenvalues gave a reasonable value of 2.83. The fifth eigenvalue of the diagonalised F-matrix was 0.53. The five methyl group average spherical harmonics components were obtained from the inversion of (1). In previous reports, the information contained in the cumbersome five spherical harmonics parameters was translated to the so-called model-free (MF) parameters, which are more intuitive and can easily be related to a starting structure. These MF parameters are the order parameter, the amplitude (ηrdc) and direction (ϕ′rdc) of motional anisotropy and the average orientation polar angles (ϕav,θav), and are obtained according to transformations derived in (Lakomek et al. 2006, 2008a).3 4 5 6 For (4) and (6), the corresponding orientations are obtained by applying a second order Wigner rotation defined by the three Eulerian angles. However, despite the fact that the MF parameters are just linear transformations of the average spherical harmonics, their interpretation might lead to problems for methyl groups, for the following reasons: (1) The side chain positions in the reference structure may sometimes be less well defined because of mobility or crystal packing, making the structure generally less reliable to compare to. (2) The fourfold degeneracy of valid RDC-derived orientations combined with the high mobility may make the interpretation of their average orientation more uncertain. (3) The dynamic average orientation is the result of the averaging of the dipolar second order tensor, for which large amplitude motions may lead to “swapping” of principal components, and can thus not always be compared to the starting structure in which the C–C orientation is more akin to vectorial averaging. As simple examples of the last two cases, a C–C vector undergoing a 1:1 jump between a trans and a gauche+ would have an average orientation (ϕav,θav) within the plane formed by the trans and gauche+ position but outside their intervening angle (rather than exactly halfway between them). Also, a C–C vector equally sampling all three rotamers (or freely rotating axially) would have its average orientation perpendicular rather then parallel to the rotation axis, according to the definition of the MF parameters. Due to this very non-linear behaviour of averaged orientations, care must be taken when interpreting these MF parameters for comparison and validation purposes. For these reasons, we also invoke the average spherical harmonic terms for modeling the motional heterogeneity of side chains. For the quality estimation of the MFA fit to the measurements, the RDCs were back-calculated from the average spherical harmonics. As quality criteria, for residue i, the root mean square deviation over all alignment media between the back-calculated and the measured RDCs () were determined. In addition, for each alignment medium m, the dynamic Q-values () were determined to compare the RDCs back-calculated from the MFA spherical harmonics to the experimental RDCs (Lakomek et al. 2008a; Table S1, Supplementary Information) and are defined analogously to the standard Q-value (Bax and Grishaev 2005). Errors for the five MF parameters (, ϕav, θav, ηrdc, ϕ′rdc) were calculated by adding Gaussian noise to the measured RDCs. The input RDCs with added noise Di,m,noise were generated by drawing N = 1,000 random samples Di,m,noise = randomp(Di,m) from a Gaussian distribution p(D) with standard deviation σi. Both the experimental error and the residue-specific σirmsd = rmsd(rdc,i) values were used to assess the combined effect of experimental error and additional systematic errors introduced by the MFA. Rotameric analysis A Matlab7 (Mathworks, MA) script was designed to perform a grid search in order to model the full side chain rotameric sampling of valines (V5, V70), isoleucines (I3, I36, I44, and I61) and leucines (L8, L67) over the fs-ms time range based on two average spherical harmonics sets (ten parameters) from their Cγ and/or Cδ RDCs. The protocol searches populations of ideal rotamers in trans (tr), gauche+ (g+) and gauche− (g−) of χ1 and χ2 angles and computes the average spherical harmonics based on the following expressions:7 8 where the terms p1xx represent populations between 0 and 1 for torsion angle χ1 in ideal rotamer xx = [tr, g+, g−] and the terms p2yy|1xx represent populations of χ2 in ideal rotamer yy when χ1 is in ideal rotamer xx. The isotropic general order parameters Sγ and Sδ are terms representing local small-scale fluctuations specific to Cγ and Cδ, and all spherical harmonics, and are generated from the N–Cα–Cβ bond of the reference crystal structure pdb:1ubi and represent the idealised rotameric geometries for methyls at Cγ and Cδ positions. Due to the fact that the  = 1 − ptr −  for all χ angle, the grid search was performed for valines on three variables (p1tr, Sγ) in increment steps of (0.02, 0.02, 0.01) and on isoleucines on 10 variables (p1tr,p1tr|2tr, , , , , , Sγ, Sδ) in increments steps of (0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.05, 0.05). The grid search on leucines was performed as for isoleucines, except that the two isotropic order parameters were combined into a single one. In certain cases, where relaxation-based order parameters were available from (Lee et al. 1999) and did not suggest any rotameric exchange , these values were used directly in the result of the grid search as . The minimized target function is the squared sum of differences (ssd) of the real and imaginary terms of the average spherical harmonics, defined as:9