Theory for nuclear magnetic relaxation of probes in anisotropic systems: application of cholesterol in phospholipid vesicles.

The nuclear magnetic relaxation of a nucleus in a cylindrical probe embedded in a bilayer vesicle is considered. The probe is assumed to diffuse freely about its unique (C infinity) symmetry axis with an effective correlation time tau parallel, and the C infinity axis moves in a potential which is azimuthally symmetric about a director, with an effective correlation time tau perpendicular. The overall isotropic rotational correlation time of the membrane is tau M. Dipolar relaxation and, in the special case that the relevant tensors are axially symmetric, quadrupolar and chemical shift anisotropy relaxation are treated. An expression for the appropriate correlation function is derived which depends on the above effective correlation times, on the order parameter of the C infinity axis of the probe, and on the angle (beta) which in the case of dipolar relaxation of a protonated 13 C nucleus is between the 13 C-H vector and the C infinity axis of the probe. A significant feature of this formulation of the dynamics is that no assumptions need be made concerning the relative order of magnitudes of the effective correlation times and the Larmor frequencies (e.g., such as the extreme narrowing limit). The model not only can be used to describe the dynamics of probes such as cholesterol in membranes but also is applicable to certain anisotropic internal motions in proteins. In addition, by allowing the angle beta to fluctuate, a simple model for segmental motion in lipid molecules can be obtained. As an application, 13 C nuclear magnetic relaxation experiments on cholesterol in sonicated egg yolk phosphatidylcholine vesicles are interpreted within the framework of the model. The remarkable observation that the protonated C6 carbon has a line width which is much narrower than those of other methine carbons and is in fact comparable to the line width of the nonprotonated C5 carbon is shown to be the consequence of (1) the anisotropic nature of the cholesterol motion in the bilayer and (2) the fact that angle between the 13 C6-H vector and the long axis of cholesterol is very close to the "magic" value of 54.7 degrees.