Protein Structure from Solid-State NMR

Abstract

This article deals with mathematical questions arising from the determination of protein structure from data obtained by solid-state nuclear magnetic resonance (NMR). Solid-state NMR holds the promise of revealing the structure of membrane proteins in a lipid bilayer. The derivation of protein structure from NMR data has most often been done using proteins in liquid state, and the mathematical analysis has been done using distance geometry and distance matrices. The mathematical analysis for solid state NMR uses orientational constraints rather than distance constraints, and matrices of inner products rather than distance matrices. Solving the structure from the data requires supplying a sequence of signs, a situation somewhat analogous to the necessity to supply the phases to solve a structure from x-ray crystallographic data. Other problems in solving for the structure arise from the condition that the gram determinants be non-negative, and this is analogous problem in distance geometry that the distance matrix must satisfy the conditions of the Cayley-Menger theorem.