# Multiple scales and phases in discrete chains with application to folded proteins

## TL;DR

This paper introduces a new method to identify and analyze multiple length scales and phase behaviors in chiral heteropolymers like proteins, enhancing understanding of their structure and dynamics.

## Contribution

It proposes a systematic approach using a novel order parameter and a variant of Kadanoff's block-spin transformation to distinguish scales and phases in proteins.

## Key findings

- The order parameter reveals multiple length scales in protein structures.
- Different phases can coexist at various scales within the same heteropolymer.
- The method correlates scales and phases with folding pathway complexity.

## Abstract

Chiral heteropolymers such as larger globular proteins can simultaneously support multiple length scales. The interplay between different scales brings about conformational diversity, and governs the structure of the energy landscape. Multiple scales produces also complex dynamics, which in the case of proteins sustains live matter. However, thus far no clear understanding exist, how to distinguish the various scales that determine the structure and dynamics of a complex protein. Here we propose a systematic method to identify the scales in chiral heteropolymers such as a protein. For this we introduce a novel order parameter, that not only reveals the scales but also probes the phase structure. In particular, we argue that a chiral heteropolymer can simultaneously display traits of several different phases, contingent on the length scale at which it is scrutinized. Our approach builds on a variant of Kadanoff's block-spin transformation that we employ to coarse grain piecewise linear chains such as the C$\alpha$ backbone of a protein. We derive analytically and then verify numerically a number of properties that the order parameter can display. We demonstrate how, in the case of crystallographic protein structures in Protein Data Bank, the order parameter reveals the presence of different length scales, and we propose that a relation must exist between the scales, phases, and the complexity of folding pathways.