An analytical solution to the problem of the orientation of rigid particles by planar obstacles. Application to membrane systems and to the calculation of dipolar couplings in protein NMR spectroscopy

Nonspherical particles or molecules experience an ordering effect in the pr esence of obstacles due to the restrictions they place on the orientation o f those molecules that are in their proximity. Obstacles may be the limits of a membrane in which the molecule is embedded, oriented mesoscopic system s such as bicelles, or membrane fragments used to induce weak protein align ment in a magnetic field. The overall shape of most proteins can be describ ed to a good approximation by an ellipsoidal particle. Here we describe and solve analytically the problem of the orientation of ellipsoidal particles by planar obstacles. Simple expressions are derived for the orientational distribution function and the order parameter. These expressions al-low the analytical calculation of the residual dipolar couplings for a protein of known three-dimensional structure oriented by steric effects. The results a re in good agreement with experiment and with the results of previously des cribed simulations. However, they are obtained analytically in a fraction o f the time and therefore open the possibility to include the optimization o f the overall shape in the determination of three-dimensional structures us ing residual dipolar coupling constraints. The equations derived are genera l and can also be applied to problems of a completely different nature. In particular, previous equations describing the orientation of particles embe dded in membranes are verified and generalized here.