Determination of quantitative distance and angle restraints for protein-

To quantitatively describe the experimental I[¹H]/I[²H] intensity ratios, we carried out spin simulations, employing the SIMPSON software package [Bak et al., 2000]. To construct an adequate spin system, we considered a spin geometry consisting of four nuclei, resembling the protein¹H^N,¹⁵N nuclei and the RNA¹H^C,¹³C nuclei (Figure 3.70), respectively. Since the signal attenuation in the¹H,¹⁵N correlation of the protein in close proximity to the RNA is induced by dipole-mediated line broadening, the simulatedI[¹H]/I[²H]ratio was expected to be dependent on the distance, as well as on the orientation of all considered dipoles. Therefore, we introduced two angles,β₁andβ₂, to allow sampling of all orientations of the¹H^N,¹⁵N and¹H^C,¹³C dipole

tensors with respect to each other. Besides theβ₁andβ₂angles, the only other variable was the

1H^N,¹H^Cdistance. All other dipolar couplings and orientations were calculated accordingly.

As expected, we found that the simulatedI[¹H]/I[²H]ratios depend on the relative orientation of the two dipole-dipole vectors. This dependency decreases with increasing distance. Further-more, the simulation revealed an up to fourfold degeneracy in the angular space (Figure 3.70).

To reduce the degeneracy, we determined theβ₁ and β₂ angle distribution for the L7Ae-box C/D RNA complex as well as for two additional protein-RNA complexes and used their com-mon angle interval for constraining the angular space during the fitting procedure (Figure 3.71).

The simulated data matrix, I[¹H]/I[²H], as a function of the ¹H^N,¹H^C distance,β₁ andβ₂ is represented in Figure 3.72A.

The fit of the experimental data did not assume a prior knowledge of the structure of the system. For the fitting of the experimentally determinedI[¹H]/I[²H]intensity ratios, first the peak volumes were determined by integration, employing in-house Python scripts using the I/O routines of nmrglue v0.2 [Helmus and Jaroniec, 2013]. The error of the peak volume was set to two times the noise RMSD. The noise level was determined individually for every peak by averaging over 1,000 equally sized random boxes in the noise region of the spectrum. A grid of simulated data was created, using the interval[0^◦,180^◦](5^◦ steps) for β₁,[0^◦,355^◦](5^◦steps) forβ2, and [2.5 Å, 102.9 Å] for the¹H^N,¹H^Cdistancer₂(nomenclature given in Figure 3.70, page 151). The peak height for the¹H^N resonance was determined and stored into the matrix I_H(β1,β2,r2) andI_D(β1,β2,r2), employing the¹H- or²H-RNA spin system, respectively. The ratio matrixR(β₁,β₂,r₂)was calculated byIH(β₁,β₂,r₂)/ID(β₁,β₂,r₂). As can be seen in Figure 3.70, the contour plots show an up to four fold degeneracy for theβ1/β2angle space.

We quantified the occurrence ofβ₁andβ₂for different protein-RNA complexes (Figure 3.71).

In the analysis, protein¹H^Nand RNA¹H^C pairs only within 6 Å were considered. We found, that theβ₂ angle adopts values in the range[70^◦,100^◦], whereas the full range of values was found forβ1. To reduce the degeneracy and improve the fitting, we thus constrained the angular space to two times the standard deviation from each expectation value.

As a first attempt, we assumed a linear arrangement of the¹H^N,¹⁵N and¹H^C,¹³C dipole ten-sors, which yielded a poor correlation (Figure 3.72B, cyan). However, introducing the anglesβ₁

Figure 3.70: Numerical simulation of the illustrated four-spin system using SIMPSON [Bak et al., 2000]. The respective tensor magnitudes and orientations were calculated as a function of the¹H1,¹H2 distancer2and the angles β1andβ2. All possible dipolar couplings were taken into account, as well as the spatial orientation of the planar spin system, which was defined only by theβ Euler angle (Ω= (0,β,0)). The isotropic chemical shift values were set according to the average shift found in the BMRB. The chemical shift anisotropies (asymmetries) in [ppm] for

1H2,¹³C,¹H1,¹⁵N were as follows: -4.55 (0.9) [Aravamudhan et al.,1979], -118.5 (0.9) [Stueber and Grant,2002], 7.67 (0.65) [Wu et al.,1995], 170 (0.2) [Bak et al.,2002,Schanda et al.,2010]. The¹H Larmor frequency was set to 700 MHz and the MAS frequency to 20 kHz. Here, only the¹H1 resonance was detected and the FID apodized with a line broadening of 5 Hz. After Fourier transformation, the peak height for¹H1 was extracted. For simulating the spin system of the²H-RNA sample, the¹H2 proton was replaced by a deuteron (CQ/2π= 150 kHz). In the contour plots, the intensity ratioI[¹H]/I[²H]was plotted as a function of the anglesβ1andβ2for three different¹H1,¹H2 distances (2.50 Å, 4.77 Å and 6.02 Å). The ratios were normalized with respect to the maximal value obtained at r₂→∞.Reproduced with kind permission from Asami, S., et al., Angew. Chem. Int. Ed. 2013, 52 (8), pp 2345-2349.

Copyright 2013 Wiley-VCH Verlag GmbH Co. KGaA. DOI: 10.1002/anie.201208024.

Figure 3.71:Distribution of anglesβ1andβ2for different protein-RNA complexes. Here,(180^◦β1)corresponds to

∠(N,H1,H2) and(180^◦β2)to∠(C,H2,H1), respectively, as illustrated in Figure 3.70. The coordinate of the carbon nucleus C was replaced by its projection onto the N,H1,H2 plane. The employed protein-RNA complexes were the L7Ae-box C/D RNA complex fromA. fulgidus(PDB: 1RLG) [Moore et al.,2004], the Cse3-RNA complex from T. thermophilus(PDB: 2Y8Y) [Sashital et al.,2011] and the Dbp5-RNA complex fromS. cerevisiae(PDB: 3PEY) [Montpetit et al.,2011], respectively. Assuming a normal distribution for 1RLG, 2Y8Y and 3PEY, the following expectation values/standard deviations forβ1(β2) were extracted: 43/21 (86/7), 63/44 (84/6) and 64/47 (88/7) [^◦].

Reproduced with kind permission from Asami, S., et al., Angew. Chem. Int. Ed. 2013, 52 (8), pp 2345-2349.

Copyright 2013 Wiley-VCH Verlag GmbH Co. KGaA. DOI: 10.1002/anie.201208024.

andβ2, improved the correlation significantly (Figure 3.72B, black). As expected, restricting the β₂angular space resulted in smaller errors; however, it did not significantly change the absolute determined distances (Figure 3.72B, red). The determined parameters are summarized in Table 3.4 and the obtained distances were plotted in Figure 3.69A, together with the distances from the crystal structure. The estimated distances and angles were within the error margins. For the experiments, we used only about 2 mg of the²H-RNA sample. Obviously, a higher fitting accuracy can be achieved by increasing the experimental sensitivity for the two samples.

It should be noted, that the proton density of RNA is naturally much lower than that of a protein. Here, the low proton density simplified the determination of structural parameters, since, to first approximation, every protein¹H^N was in proximity to only very few RNA¹H^C protons.