Observing the overall rocking motion of a protein in a crystal

The large majority of three-dimensional structures of biological macromolecules have been determined by X-ray diffraction of crystalline samples. High-resolution structure determination crucially depends on the homogeneity of the protein crystal. Overall ‘rocking' motion of molecules in the crystal is expected to influence diffraction quality, and such motion may therefore affect the process of solving crystal structures. Yet, so far overall molecular motion has not directly been observed in protein crystals, and the timescale of such dynamics remains unclear. Here we use solid-state NMR, X-ray diffraction methods and μs-long molecular dynamics simulations to directly characterize the rigid-body motion of a protein in different crystal forms. For ubiquitin crystals investigated in this study we determine the range of possible correlation times of rocking motion, 0.1–100 μs. The amplitude of rocking varies from one crystal form to another and is correlated with the resolution obtainable in X-ray diffraction experiments.

The R 1 rates have been computed using the model-free formalism with a single motional correlation time. [5][6][7] The protonnitrogen distance was assumed to be 1.02 Å, and the anisotropy of the (presumed axially symmetric) nitrogen CSA tensor was taken to be -172 ppm. The 1 H Larmor frequency was set to 600 MHz, mimicking our experimental setup. The 15 N R 1 rate constants are highest for correlation times of approximately 1-10 ns. Measurable rate constants R 1 >0.01 s -1 are produced by motions occurring on time scales from tens of picoseconds to ~100 ns, depending also on the motional amplitude. The calculations were performed ignoring the inherent multi-exponential nature of the R 1 decay that arises from orientation-dependent relaxation. We have shown previously that the error associated with this approximation is very small, generally well below the precision of experimental measurements.

Supplementary Figure 5. The dependence of the 15 N R1ρ relaxation rate constant on the amplitude and time scale of reorientational motion.
The R 1ρ rates have been computed using the model-free formalism with a single motional correlation time, similar to Supplementary Figure 4. Additionally, it has been assumed that ω 1,15N /2π = 15 kHz and ω MAS /2π = 39.5 kHz, in line with our experimental setup. The calculations were conducted using the formula by Kurbanov et al., which accounts for the entry of ω 1,15N and ω MAS frequencies into spectral densities. 8 This formula shows an appreciable difference from the standard solution-type expression for the correlation times τ exceeding ca. 1μs. Dashed lines in the plot show the relaxation rate constants close to the detection limit (determined by the maximum duration of the spin-lock period that is dictated by hardware limitations) and thus delineate the range of motional parameters to which R 1ρ measurements are sensitive.
One should bear in mind that the formula by Kurbanov, as well as other similar results, 9 are derived from the Redfield theory. In principle, the range of validity of this formula is given by the following relationship, 2π(1-S)d NH τ < 0.1, where d NH is the strength of the proton-nitrogen dipolar coupling (11.5 kHz). The region where this condition is violated is shown as a grey hatched area in the plot. However, our numeric simulations suggest that Kurbanov's results remain sufficiently accurate over the broad (S 2 , τ) region, see Figure 5. To explain this observation, one needs to re-analyze the conditions of validity of the Redfield treatment in the rotating frame under fast MAS conditions, as appropriate for the spin-lock experiment under consideration. Such analysis is beyond the scope of this work.
The behavior of 15 N R 1ρ rate constants, as seen in the contour plot Supplementary Figure 5, can be easily rationalized. Generally, the 15 N R 1ρ rate constant increases for larger motional amplitudes (1-S 2 ) and longer correlation times τ. However, when the motion becomes sufficiently slow (with τ in microsecond range) the dipolar interactions, as well as CSA interactions, are efficiently refocused by the fast magic angle spinning, as well as strong 15 N spin lock field. As a consequence, the R 1ρ rate constant declines toward the upper edge of the graph.

Supplementary Figure 6. Comparison of order parameters S 2 in MPD-ub (black) and cubic-PEG-ub (red).
The data in panel (a) are the experimental data shown in Figure 2 of the main text. In panel (b), the order parameters of cubic-PEG-ub have been scaled by a factor 1.04. This factor minimizes the difference between the two data sets, MPD-ub and cubic-PEG-ub, excluding residues G10 and Q62 which have clear differences in local dynamics in the two crystal forms. N R 1ρ data, suggesting that the MD simulations suffer from a lack of convergence and/or from "structural drift" (see main text). We have verified that the internal coordinates of ubiquitin molecules are well preserved during the simulations; therefore, it is the dynamics of the crystal lattice that is problematic. It is also instructive to discuss MD results in terms of the mean amplitude of the rocking motion. In the case of MPD-ub, the mean amplitude of the rocking motion is 4.6° if crystal structure is used as a reference. Alternatively, if one uses the average MD coordinates as a reference, the amplitude is 3.3°. The corresponding numbers for chain A in the cubic-PEG-ub trajectory are 11.5° and 6.3°. The substantial difference between the two values also points toward the lack of convergence and/or "structural drift". 10 Ultimately, our analysis confirms that MD simulations provide only a qualitative, rather than a quantitative, picture of the rocking motion in protein crystals.

Supplementary Figure 8. Simulated B factors for C α atoms in ubiquitin.
Two different protocols have been used to compute B factors on the basis of 1-μs-long crystal trajectories. In the first protocol, all protein molecules are first superimposed via the crystal symmetry transformations and then transferred to origin (through their respective centers of mass). The B factors for i-th atom are then evaluated as follows, B = (8π , where x i is the vector of atomic coordinates and angular brackets denote averaging over all copies of the protein and all frames in the trajectory. The second protocol is different in that the copies of the protein are superimposed via the least-square fitting of the C α atoms belonging to the secondary structure of the protein. Importantly, the first definition (non-aligned, blue symbols) includes the effect of re-orientational rocking dynamics, i.e. rotational fluctuations of the molecule as a whole, alongside with internal protein dynamics. In contrast, the second definition (aligned, green symbols) is confined to the motions representing internal protein dynamics. The inspection of the plot shows that rocking motion is relatively insignificant in the case of MPD-ub trajectory, but has a pronounced effect in the case of cubic-PEG simulation (especially for chain B); the results demonstrate the extent to which x-ray diffraction data deteriorate as a result of rocking dynamics in the crystal lattice. Of interest, the plot also highlights differences between chains A and B with respect to the dynamic status of the β1-β2 loop.

Supplementary Figure 9. TLS analysis of the three different crystal forms of ubiquitin.
The translation-libration-screw (TLS) parameters have been determined by means of the TLSMD algorithm 11,12 for three crystal forms of ubiquitin using the crystal structures available in the Protein Data Bank or obtained in this study. The PDB identifiers are marked in the plot; letters A, B and C refer to non-equivalent protein molecules in the crystal unit cells. In the TLSMD analyses each protein molecule was treated as a rigid body and has not been partitioned into segments (groups). This minimal model involves 20 unique fitting parameters, which underscores the risk of overfitting and the difficulty in interpreting the results. Note that the TLS model takes a formalized view of protein rigid-body dynamics. For instance, it is straightforward to show that a sequence of rotations with different pivot points can be described as a combination of a single rotation and a translation. It is this latter (minimalistic) description that is implemented in the TLS model, as well as other similar models such as vGNM. 13 As a consequence, the absolute values of angular fluctuations as seen by NMR (sensitive exclusively to rotation) and TLS (possibly entangled rotation/translation) are not expected to be identical. MPD-ub crystals grew in the form of sea urchins composed of thousands of extremely thin rods (~100-200x5x5 µm), impossible to isolate and loop individually. To check whether these crystals had the same space group as those previously reported (3ONS), we looped a large number of these thin rods and collected 9 frames of 20° oscillation (180° total -20 min exposure time per frame) using our in-house X-ray source. To generate a powder diffraction of these crystals, all frames have been summed up (shown in the left half of the figure). A simulated pattern obtained from the deposited ubiquitin model 3ONS was generated using the software powder 0.9.1 (https://pypi.python.org/pypi/powder). To compare the experimental and simulated patterns, 1D azimuthal integrations were performed as a function of 2theta using the program Fit-2D. 14 The spectral peaks obtained in this manner are represented by vertical orange bars; the simulated peaks are represented by blue bars (right half of the figure). For 2theta values ranging from 0 to 31°, 70 and 81 peaks were observed in the experimental and simulated spectra, respectively. The overall standard deviation between the 2theta values from each experimental peak and the closest simulated peak is 0.07°. The 2theta difference in peak position ranges from 0 to 0.33°. Each bar in the figure is centered at its respective 2theta value and plotted with a width corresponding to two standard deviations (0.14°). The simulated spectrum (blue) is superimposed on top of the experimental spectrum (orange) so that the agreement between the two can be judged by the number of orange bars that remain visible (out of 70).

Supplementary Figure 12: Stereo view images of electron density maps.
Representative portions of the electron density maps (2mFO-DFc map, plotted at 1 σ) of (a) cubic-PEG-ub, (b) rod-PEGub, and (c) rod-PEG-ub-II, as determined in this study.