Mathematical Formulations Used in Solid-State NMR for Structural Biology

Recent advances in the field of structural biology, with relatively new biophysical techniques such as solid-state nuclear magnetic resonance that holds promise for determining the structures of peptides and proteins located within the cell membrane, facilitate the collection of data on the atomic structures of the biological molecules. Classical methods such as X-ray diffraction (XRD) and liquid-state NMR spectroscopy suffer from difficulties in crystallizing functional proteins in membrane environments (XRD) or too slow molecular motion for averaging of anisotropic nuclear spin interactions (liquid-state NMR) [1]. These problems have motivated the search for alternative methods such as solid-state NMR (SS-NMR). In the solid state NMR, the nuclear spin interactions are typically governed by anisotropic (orientation dependent) components in addition to the isotropic (orientation independent) components known from liquid-state NMR [2-4]. The method (SS-NMR) elucidates molecular structure and dynamics in systems not amenable to characterization by any other way. The importance of the technique stands in its ability to determine accurately intermolecular distances and molecular torsion angles. The technique has been used in systems including both microscopically ordered preparations such as membrane proteins, nanocrystalline proteins, amyloid fibrils, and also disordered or amorphous systems such as glasses. This chapter presents a view of algorithm formulation of structural biology with the mathematical foundation of the determination of protein structure from orientation constraints which highlight the solid-state NMR used as probe for the determination of peptide and protein structures. We also review the continuity conditions and torsion angles from solid-state NMR orientational restraints. Tools such as vector algebra, Gram matrices, and determinants, are used.

Keywords: Distance constraints, gram matrices, orientational constraints, structural biology, solid-state NMR, tensors, torsion angles.