Random loop model for long polymers

Remarkably little is known about the higher-order folding motifs of the chromatin fiber inside the cell nucleus. Folding depends among others on local gene density and transcriptional activity and plays an important role in gene regulation. Strikingly, at fiber lengths above 5 to 10 Mb the measured mean square distance < R-2 > between any two points on the chromatin fiber is independent of polymer length. We propose a polymer model that can explain this leveling-off by means of random looping. We derive an analytical expression for the mean square displacement between two arbitrary beads. Here 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.