| This dissertation addresses the geometry of protein structures and methods of determining global structural information for proteins from solid-state nuclear magnetic resonance data. Geometrically, models of protein backbone structures can be viewed as discrete curves with vertices representing atoms and edges representing bonds. A discrete version of the Frenet frame is defined for discrete curves and developed in relation to elementary differential geometry concepts. Using the discrete Frenet frame formalism, regular protein backbone structures are viewed as the orbit of a set of points under an orientation-preserving Euclidean motion computed as a product of Frenet frames, and Chasles' formula is used to compute the axes of such structures. These concepts are implemented in a Maple package which contains procedures for computing with protein structures, displaying protein backbones, and exporting Protein DataBank files.;To understand the nature of coiled proteins, the discrete Frenet frame concept is applied to examine the coiling problem. Given an Euclidean motion that generates a coiled protein structure, this problem seeks to find sets of torsion angles so that the generating Euclidean motion can be written as a product of Euclidean motions whose rotational parts are discrete Frenet frames. This problem is examined as a generalization of the ring closure problem. By reducing the number of parameters, and by simplification of the bond angle sequence, solutions can be given in a number of special cases and give insight into the various conformations of the gramicidin A peptide.;The Polarization Inversion Spin Exchange at Magic Angle (PISEMA) SSNMR experiment has recently become a powerful tool for studying membrane protein structure. Mathematical analysis shows that boundary of PISEMA powder patterns consists of an ellipse and a wedge-shaped curve. Further, PISEMA spectra of oriented membrane alpha-helices show an interesting feature called a "Polarity Index Slant Angle (PISA) wheel" which is closely related to the helical wheel of the protein. In modeling work, PISA wheels are seen to lie on fourth degree curves, while the center of a PISA wheel determines the helical tilt without using resonance assignments. The distribution of resonances from amino acid specific labels around the PISA wheel defines the rotational orientation of a helix and yields preliminary site specific assignments. The envelope of the PISA wheels for a membrane helix is the boundary of the powder pattern. The usefulness of this analysis to the assignment of resonances and to the determination of protein structure is demonstrated. |
