Helix parameters and protein structure using quaternions

The mathematics of helices is important for understanding protein secondary and super-secondary structure, since every regular protein backbone struct ure can be considered as a helix. This paper presents a mathematical approa ch to helices using geometric algebra in the form of quaternions. The motiv ation is to make it more convienient to compute a solid-state NMR picture o f the protein using orientational constraints. In terms of two parameters s pecifying a helix, formulas are given for the various helical parameters of interest in considering protein structure: residues per turn, pitch, radiu s and helix axis direction. Helices with period more than one residue are a lso considered, depending on more parameters and giving more complicated fo rmulas. Applications to determining protein structure using solid-state NMR are considered. (C) 1999 Elsevier Science B.V. All rights reserved.