Coherent and stochastic averaging in solid-state NMR.

A new approach for calculating solid-state NMR lineshapes of uniaxially rotating membrane proteins under the magic-angle spinning conditions is presented. The use of stochastic Liouville equation (SLE) allows one to account for both coherent sample rotation and stochastic motional averaging of the spherical dipolar powder patterns by uniaxial diffusion of the spin-bearing molecules. The method is illustrated via simulations of the dipolar powder patterns of rigid samples under the MAS conditions, as well as the recent method of rotational alignment in the presence of both MAS and rotational diffusion under the conditions of dipolar recoupling. It has been found that it is computationally more advantageous to employ direct integration over a spherical grid rather than to use a full angular basis set for the SLE solution. Accuracy estimates for the bond angles measured from the recoupled amide 1H-15N dipolar powder patterns have been obtained at various rotational diffusion coefficients. It has been shown that the rotational alignment method is applicable to membrane proteins approximated as cylinders with radii of approximately 20Å, for which uniaxial rotational diffusion within the bilayer is sufficiently fast and exceeds the rate 2×105s-1.