The Ramachandran number: an order parameter for protein geometry

TL;DR

The paper introduces the Ramachandran number, a novel single parameter derived from dihedral angles, to effectively describe and analyze protein secondary structures and their geometric content.

Contribution

It proposes a new scalar parameter, the Ramachandran number, that compactly encodes regions of the Ramachandran plot for improved structural analysis.

Findings

The Ramachandran number effectively summarizes secondary structure regions.

It enables visualization of structural content in large protein datasets.

Potential as an order parameter for protein geometry is demonstrated.

Abstract

Three-dimensional protein structures usually contain regions of local order, called secondary structure, such as $\alpha$-helices and $\beta$-sheets. Secondary structure is characterized by the local rotational state of the protein backbone, quantified by two dihedral angles called $\phi$ and $\psi$. 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 $\phi$ and $\psi$, 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, ${\mathcal{R}}$, which is a structurally meaningful combination of $\phi$ and $\psi$. We show that the potential applications of ${\mathcal{R}}$ 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 ${\mathcal{R}}$ might be used as an order parameter for protein geometry for a wide range of applications.