Crumpled Polymer with Loops Recapitulates Key Features of Chromosome Organization

Abstract

Chromosomes are exceedingly long topologically constrained polymers compacted in a cell nucleus. We recently suggested that chromosomes are organized into loops by an active process of loop extrusion. Yet loops remain elusive to direct observations in living cells; detection and characterization of myriads of such loops is a major challenge. The lack of a tractable physical model of a polymer folded into loops limits our ability to interpret experimental data and detect loops. Here, we introduce a new physical model - a polymer folded into a sequence of loops - and solve it analytically. Our model and a simple geometrical argument show how loops affect the statistics of contacts in a polymer across different scales, explaining universally observed shapes of the contact probability. Moreover, we reveal that folding into loops reduces the density of topological entanglements, a novel phenomenon we refer to as "the dilution of entanglements."Supported by simulations, this finding suggests that up to approximately 1-2-Mb chromosomes with loops are not topologically constrained, yet become crumpled at larger scales. Our theoretical framework allows inference of loop characteristics, draws a new picture of chromosome organization, and shows how folding into loops affects topological properties of crumpled polymers.