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.