The Ramachandran Number: An Order Parameter for Protein Geometry

Abstract

Three-dimensional protein structures usually contain regions of local order, called secondary structure, such as α-helices and β-sheets. Secondary structure is characterized by the local rotational state of the protein backbone, quantified by two dihedral angles called Øand Ψ. Particular types of secondary structure can generally be described by a single (diffuse) location on a two-dimensional plot drawn in the space of the angles Ø andΨ, called a Ramachandran plot. By contrast, a recently-discovered nanomaterial made from peptoids, structural isomers of peptides, displays a secondary-structure motif corresponding to two regions on the Ramachandran plot [Mannige et al., Nature 526, 415 (2015)]. In order to describe such 'higher-order' secondary structure in a compact way we introduce here a means of describing regions on the Ramachandran plot in terms of a single Ramachandran number, R, which is a structurally meaningful combination of Ø andΨ. We show that the potential applications of R are numerous: it can be used to describe the geometric content of protein structures, and can be used to draw diagrams that reveal, at a glance, the frequency of occurrence of regular secondary structures and disordered regions in large protein datasets. We propose that R might be used as an order parameter for protein geometry for a wide range of applications.