Hydrogen bond geometry in protein helices

Protein backbone helical structures have the same values of dihedral angles φ and ψ from unit to unit and thus, can be uniquely determined by a pair (φ,ψ). We focus our attention on two types of protein secondary structure αand 310-helices stabilized by 1-5 and 1-4 hydrogen bonding pattern respectively. Even though both αand 310-helices each represent a group of helices, various studies show that dihedral angle pairs (−63◦,−43◦) and (−60◦,−25◦) characterize the most stable helices within each group respectively. Hydrogen bond has geometrical constraints with respect to linearity, length and various associated angles, that must influence the formation of most stable helices. The relationship between hydrogen bond geometrical requirements and stability of various protein secondary structures has been extensively studied by applying statistical analysis to protein experimental data. Here, we study this relationship via mathematical optimization. Previously, mathematical optimization was only applied to α-helix and considered just the linearity of a hydrogen bond. We express other various geometrical parameters of a hydrogen bond as functions of dihedral angles (φ,ψ) and study their influence on helical stability. In particular, we take two major hydrogen bond requirements: linearity and length constraints and ask whether the most stable αand 310-helix result from optimization with respect to a linear combination of these two criteria. We show that these criteria are not sufficient to explain the peaks. Moreover we show that another hydrogen bond parameter plays an important role in the formation of peaks.