We review pro and contra of the hypothesis that generic polymer properties of topological constraints are behind many aspects of chromatin folding in eukaryotic cells. For that purpose, we review, first, recent theoretical and computational findings in polymer physics related to concentrated, topologically simple (unknotted and unlinked) chains or a system of chains. Second, we review recent experimental discoveries related to genome folding. Understanding in these fields is far from complete, but we show how looking at them in parallel sheds new light on both.

In order to quantify the effect of mutual threading on conformations and dynamics of unconcatenated and unknotted rings in the melt we computationally examine their minimal surfaces. We found a linear scaling of the surface area with the ring length. Minimal surfaces allow for an unambiguous algorithmic definition of mutual threading between rings. Based on it, we found that, although ring threading is frequent, majority of cases correspond to short loops. These findings explain why approximate theories that neglect threading are so unexpectedly successful despite having no small parameter justification. We also examine threading dynamics and identify the threading order parameter that reflects the ring diffusivity.

Dynamical properties of a long polymer ring in a melt of unknotted and unconcatenated rings are calculated. We re-examine and generalize the well known model of a ring confined to a lattice of topological obstacles in the light of the recently developed Flory theory of unentangled rings which maps every ring on an annealed branched polymer and establishes that the backbone associated with each ring follows self-avoiding rather than Gaussian random walk statistics. We find the scaling of ring relaxation time and diffusion coefficient with ring length, as well as time dependence of stress relaxation modulus, zero shear viscosity and mean square averaged displacements of both individual monomers and ring's mass center. Our results agree within error bars with all available experimental and simulations data of the ring melt, although the quality of the data so far is insufficient to make a definitive judgment for or against the annealed tree theory. In the end we review briefly the relation between our findings and experimental data on chromatin dynamics.

Recent theoretical studies found that mixtures of active and passive colloidal particles phase separate but only at very high activity ratio. The high value poses serious obstacles for experimental exploration of this phenomenon. Here we show using simulations that when the active and passive particles are polymers, the critical activity ratio decreases with the polymer length. This not only facilitates the experiments but also has implications on the DNA organization in living cell nuclei. Entropy production can be used as an accurate indicator of this non-equilibrium phase transition.PACS numbers: 64.75. Va, 87.15.Zg, 05.70.Ln Active matter consists of microscopic constituents, such as bacteria, micro-swimmers or molecular motors that transform energy from the surroundings or their own sources into mechanical work. The energy is fed into the system on the particle scale producing local spatiotemporal gradients which give rise to a number of interesting macroscopic non-equilibrium phenomena such as emergence of spatial structures through dynamical phase transitions [1], directed rotational motion [2], propagating waves [3,4] and phase separation of mixtures of active and passive particles with purely repulsive interactions [5][6][7][8][9][10].The latter phenomenon, that we focus on here, is particularly interesting as often the active particles reside in a passive environment or the level of their activity is heterogeneous. Moreover, the mixture of active and passive polymers was hypothesised to play a role in the DNA organization within the cell nucleus during the interphase (metabolic phase of the cells life) [11][12][13]. Indeed, DNA transcription or the hypothesised active loop extrusion [14] involve continual energy influx and dissipation on microscopic length scales. Observations of living cells [15,16] and the "C-experiments" [17] confirm that euchromatin -the active DNA, is colocalized and separated from the inactive denser heterochromatin by an unknown mechanism. The chemical difference of the hetero-and euchromatin [16] could play a role in the chromatin separation [18], and, as first shown by Ganai et al [11], the active process that we are focusing on here too, could bring an additional contribution to the separation.Recently, two works [6, 10] modelled active colloidal particles as having higher diffusivity as their passive counterparts, by simply connecting them to a higher temperature thermostat. Both, simulations and analytical theory predict phase separation at a temperature ratio of about 30. Similarly, high critical activity contrast is observed also for other, "vectorial" activity models such as active Brownian particles (ABP) [5] (Péclet number over 50) and Vicsek model [9,19] where the generated force or velocity acting on the particle has a specific direction that randomises on longer time scales. These models are more complex due to other effects like pressure dependence on the interaction details of the particles with the container [20][21][22]. Thus we here restrict ourselves to the...