Mathematical Aspects of Protein Structure Determination with NMR Orientational Restraints

J. R. Quine, Timothy A. Cross, Michael S. Chapman, and Richard Bertram

The field of structural biology is becoming increasingly important as new technological developments facilitate the collection of data on the atomic structures of proteins and nucleic acids. Solid-state NMR is a relatively new biophysical technique that holds particular promise for determining the structures of peptides and proteins that are located within the cell membrane. This method provides information on the orientation of the peptide planes relative to an external magnetic field. In this article, we discuss some of the mathematical methods and tools that are useful in deriving the atomic structure from these orientational data. We first discuss how the data are viewed as tensors, and how these tensors can be used to construct an initial atomic model, assuming ideal stereochemistry. We then discuss methods for refining the models using global optimization, with stereochemistry constraints treated as penalty functions. These two processes, initial model building followed by refinement, are the two crucial steps between data collection and the final atomic model.