Geometry of Protein Structures. I. Why Hyperbolic Surfaces are a Good Approximation for Beta-Sheets

Protein structure is invariably connected to protein function. To analyze the structural changes of proteins, we should have a good description of basic geometry of proteins' secondary structure. A beta-sheet is one of important elements of protein secondary structure that is formed by several fragments of the protein that form a surface-like feature. The actual shapes of the beta-sheets can be very complicated, so we would like to approximate them by simpler geometrical shapes from an approximating family. Which family should we choose? Traditionally, hyperbolic (second order) surfaces have been used as a reasonable approximation to the shape of beta-sheets. In this paper, we show that, under reasonable assumptions, these second order surfaces are indeed the best approximating family for beta-sheets.