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 presence of obstacles due to the restrictions they place on the orientation of those molecules that are in their proximity. Obstacles may be the limits of a membrane in which the molecule is embedded, oriented mesoscopic systems such as bicelles, or membrane fragments used to induce weak protein alignment in a magnetic field. The overall shape of most proteins can be described 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 allow the analytical calculation of the residual dipolar couplings for a protein of known three-dimensional structure oriented by steric effects. The results are in good agreement with experiment and with the results of previously described simulations. However, they are obtained analytically in a fraction of the time and therefore open the possibility to include the optimization of the overall shape in the determination of three-dimensional structures using residual dipolar coupling constraints. The equations derived are general and can also be applied to problems of a completely different nature. In particular, previous equations describing the orientation of particles embedded in membranes are verified and generalized here.