A Random Loop Model for Long Polymers

TL;DR

This paper introduces a random loop model to explain the higher-order folding of chromatin, accounting for experimental observations of mean square displacement leveling-off at large distances.

Contribution

The paper develops an analytical and numerical model of chromatin folding based on random loops, providing insights into the role of loops at multiple scales.

Findings

Loops on all scales are necessary to fit experimental data

The model explains the leveling-off of mean square displacement in chromatin

Analytical expression derived for mean square displacement in the presence of loops

Abstract

While the structure of chromatin has been studied in great detail on length scales below 30 nm, amazingly little is known about the higher-order folding motifs of chromatin in interphase. Recent experiments give evidence that the folding may depend locally on gene density and transcriptional activity and show a leveling-off at long distances where approximately $<R^2> \sim O(1)$. We propose a new model that can explain this leveling-off by the formation of random loops. We derive an analytical expression for the mean square displacement between two beads where the average is taken over the thermal ensemble with a fixed but random loop configuration, while quenched averaging over the ensemble of different loop configurations -- which turns out to be equivalent to averaging over an ensemble of random matrices -- is performed numerically. A detailed investigation of this model shows that loops on all scales are necessary to fit experimental data.