On the Influence of Topological Catenation and Bonding Constraints on Ring Polymers

Abstract

The ring structure of certain polymers in nature, like proteins and DNA indicates a benefit compared with the linear form. Transcriptional regulation in higher eukaryotes is maintained among others by the formation of chromatin loops. Experimental studies revealed that different chromosomes as well as chromatin regions on one single chromosome tend to be segregated into distinct territories. Here we study a system of two rings in both catenane and bonded topology as a toy model for the influence of loops and topological constraints on the polymers' conformational properties. Athermal Monte Carlo simulations reveal that the mean square radius of gyration AER gyr 2 ae of catenated or bonded rings follows a similar scaling exponent as isolated rings, which in turn is close to the value of ν ≈ 0.588 for a self-avoiding walk. However, the effective segment length is larger for catenated rings, reflecting the swelling of the polymers. The shape of catenated and bonded rings, in contrast, shows pronounced differences even in the limit of infinite chain length. We observe a strong tendency toward segregation for the bonded topology in comparison with a similar ring-linear and linear-linear system. The orientation of the rings' gyration ellipsoids is slightly perpendicular, trying to minimize the overlap area. These findings indicate that loops might play an important role in the entropy-driven segregation of chromatin.

Topological interactions between ring polymers: Implications for chromatin loops

“…We know from several studies that isolated ring polymers follow a similar scaling law as linear polymers R 2 g ∼ N 2ν where ν ≈ 0.59 14,22,41 . However, non-catenated rings in a melt behave like compact polymers with a scaling exponent ν = 1/3 11,17 .…”

“…For a random walk, the ratio between the mean squared radius of gyration and the end-to-end distances is 1/6. For isolated ring polymers, the ratio R 2 g / d 2 extrapolates to 0.3053(2) in the asymptotic limit, while catenated rings display a ratio of 0.2995(3) 14,22 . For the case studied here, the ratio between the radius of gyration and ring diameter R 2 g / d 2 changes significantly for different CM separations (see Fig.…”

“…The interaction between non-linked ring polymers without excluded volume has been investigated for chains of length up to N = 80 21 . In a recent study 14 out that while the dimensions of catenated rings show the same scaling behavior than their linear counterparts, the shape changes significantly. The topological states of circular DNA have been analyzed in various publications 37,38 .…”

“…As only local moves are applied, consecutive conformations are highly correlated. To determine the relaxation time of a ring polymer, we evaluated the autocorrelation function C(t) 28 of the squared radius of gyration A(t) = R 2 g (t) and from C(t) we estimated the integrated autocorrelation time τ using Sokal's windowing procedure 14,29 . Two subsequent conformations are considered independent after 5τ Monte Carlo steps (MCS).…”

“…14 ). Figure 6A shows the root mean squared radius of gyration R g (r) = R g (r) 2 for self-avoiding walks (SAW), ring polymers without mutual topological interactions (arbitrary linking number Φ) and non-catenated rings (linking number Φ = 0) in dependence of the scaled center-of-mass separation s = r/R g .…”