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 betasheets. In this paper, we show that, under reasonable assumptions, these second order surfaces are indeed the best approximating family for beta-sheets. Introduction. Proteins are biological polymers that perform most of the life’s function. A single chain polymer (protein) is folded in such a way that forms local substructures called secondary structure elements. In order to study the structure and function of proteins it is extremely important to have a good geometrical description of the proteins structure. There are two important secondary structure elements: alpha helices and beta-sheets. A part of the protein structure where different fragments of the polypeptide align next to each other in extended conformation forming a surface-like feature defines a secondary structure called a beta pleated sheet, or, for short, a beta-sheet; see, e.g., (Branden et al. 1999). Beta-sheets are coming in many forms and shapes. In some cases, we have a cylinder-like structure called a beta-barrel that is “closed” in one dimension and “open” in the other, but in most cases, we have a surface that is open in both directions. The actual shapes of the beta-sheets can be very complicated, so we would like to approximate them by simpler 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; see, e.g., (Novotny et val. 1984). However, it is not clear whether they are indeed a good approximating family. Of course, the more parameters we allow, the better the approximation. So, the question can be reformulated as follows: for a given number of parameters (i.e., for a given dimension of approximating family), which is the best family? In this paper, we formalize and solve this problem. Specifically, we show that, under reasonable assumptions, these second order surfaces are indeed the best low-parameter approximating family for