On the tree-like structure of rings in dense solutions

Abstract

One of the most challenging problems in polymer physics is providing a theoretical description for the behaviour of rings in dense solutions and melts. Although it is nowadays well established that the overall size of a ring in these conditions scales like that of a collapsed globule, there is compelling evidence that rings may exhibit ramified and tree-like conformations. In this work I show how to characterise these local tree-like structures by measuring the local writhing of the rings' segments and by identifying the patterns of intra-chain contacts. These quantities reveal two major topological structures: loops and terminal branches which strongly suggest that the strictly double-folded "lattice animal" picture for rings in the melt may be replaced by a more relaxed tree-like structure accommodating loops. In particular, I show that one can identify hierarchically looped structures whose degree increases linearly with the size of a ring, and that terminal branches are found to store about 30% of the whole ring mass, irrespectively of its length. Finally, I draw an analogy between rings in the melt and slip-linked chains, where contact points are enforced by mobile slip-links and for which a field-theoretic treatment can be employed to get some insight into their typical conformations. These findings are ultimately discussed in the light of recent works on the static structure of rings and on the existence of inter-ring threadings.

Nonequilibrium Chromosome Looping via Molecular Slip Links

“…S3 and S4), i.e. an effective potential which accounts for the entropy, or probability of formation, of a network of loops [14,15]. This effective potential has a quantitative effect on our results, but it does not modify the qualitative trends; in this section we report findings from 1D stochastic simulations without looping weight, as this simpler version is simpler to analyse theoretically.…”

“…1a), or as a single ring that embraces two duplexes [13]. In both cases, the dimer/ring acts as a sliding bridge or molecular slip-link [14,15], and we will use the latter term to describe both cases. In vitro and in vivo experiments show that cohesin does indeed topologically link to DNA (with binding 2 mediated by "loader proteins" such as Scc2 or NIPBL [1,16]), that it can slide along DNA diffusively, and that it remains bound for τ ∼ 20 minutes before dissociating [16][17][18][19][20].…”

“…1b), with a power law correction for slip-links, p slip (l) ∼ e −αl l −c/2 . This is markedly different from the power laws which determine the looping probability of an equilibrium polymer [14,15]. The decay length is v/k off for the loop extrusion model [22], and (typical loop size) equal to ∼ 500 − 1000 kbp (Fig.…”

“…In this formula, c is a universal exponent, which in 3D is equal to 1.5 for loops made by infinitesimally thin random walks [28], ∼ 2.1 for internal looping within self-avoiding chains [28,29], and 1 for contacts within a "fractal globule" [25]. We refer to this third model, with a logarithmic looping potential, as the "sliplink" model, as it more closely resembles the dynamics of slip-links on polymers [4,14,15].As detailed in the SI, we can analytically find the probability that a cohesin dimer binding at t = 0 will, at some point, form a CTCF-mediated loop before detaching. Denoting this probability by p(l), the three models predict the following dependence on loop size l (Fig.…”

Non-equilibrium chromosome looping via molecular slip-links

“…1a), or as a single ring that embraces two duplexes [13]. In both cases, the dimer/ring acts as a sliding bridge or molecular slip-link [14,15], and we will use the latter term to describe both cases. In vitro and in vivo experiments show that cohesin does indeed topologically link to DNA (with binding 2 .…”