RÉSUMÉ
We demonstrate that Fokker-Planck equations in which spatial coordinates are treated on the same conceptual level as spin coordinates yield a convenient formalism for treating magic angle spinning NMR experiments. In particular, time dependence disappears from the background Hamiltonian (sample spinning is treated as an interaction), spherical quadrature grids are avoided completely (coordinate distributions are a part of the formalism) and relaxation theory with any linear diffusion operator is easily adopted from the Stochastic Liouville Equation theory. The proposed formalism contains Floquet theory as a special case. The elimination of the spherical averaging grid comes at the cost of increased matrix dimensions, but we show that this can be mitigated by the use of state space restriction and tensor train techniques. It is also demonstrated that low correlation order basis sets apparently give accurate answers in powder-averaged MAS simulations, meaning that polynomially scaling simulation algorithms do exist for a large class of solid state NMR experiments.
RÉSUMÉ
The orientation of prosthetic groups in membrane proteins is of considerable importance in understanding their functional role in energy conversion, signal transduction, and ion transport. In this work, the orientation of the retinylidene chromophore of bacteriorhodopsin (bR) was investigated using 2H NMR spectroscopy. Bacteriorhodopsin was regenerated with all-trans-retinal stereospecifically deuterated in one of the geminal methyl groups on C1 of the cyclohexene ring. A highly oriented sample, which is needed to obtain individual bond orientations from 2H NMR, was prepared by forming hydrated lamellar films of purple membranes on glass slides. A Monte Carlo method was developed to accurately simulate the 2H NMR line shape due to the distribution of bond angles and the orientational disorder of the membranes. The number of free parameters in the line shape simulation was reduced by independent measurements of the intrinsic line width (1.6 kHz from T2e experiments) and the effective quadrupolar coupling constant (38. 8-39.8 kHz from analysis of the line shape of a powder-type sample). The angle between the C1-(1R)-1-CD3 bond and the purple membrane normal was determined with high accuracy from the simultaneous analysis of a series of 2H NMR spectra recorded at different inclinations of the uniaxially oriented sample in the magnetic field at 20 and -50 degrees C. The value of 68.7 +/- 2.0 degrees in dark-adapted bR was used, together with the previously determined angle of the C5-CD3 bond, to calculate the possible orientations of the cyclohexene ring in the membrane. The solutions obtained from 2H NMR were then combined with additional constraints from linear dichroism and electron cryomicroscopy to obtain the allowed orientations of retinal in the noncentrosymmetric membrane structure. The combined data indicate that the methyl groups on the polyene chain point toward the cytoplasmic side of the membrane and the N-H bond of the Schiff base to the extracellular side, i.e., toward the side of proton release in the pump pathway.