A Random Loop Model for Long Polymers

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.