The topological glass in ring polymers

We study the dynamics of concentrated, long, semi-flexible, unknotted and unlinked ring polymers embedded in a gel by Monte Carlo simulation of a coarse-grained model. This involves the ansatz that the rings compactify into a duplex structure where they can be modelled as linear polymers. The classical polymer glass transition involves a rapid loss of microscopic freedom within the polymer molecule as the temperature is reduced toward Tg. Here we are interested in temperatures well above Tg where the polymers retain high microscopic mobility. We analyse the slowing of stress relaxation originating from inter-ring penetrations (threadings). For long polymers an extended network of quasi-topological penetrations forms. The longest relaxation time appears to depend exponentially on the ring polymer contour length, reminiscent of the usual exponential slowing (e.g., with temperature) in classical glasses. Finally, we discuss how this represents a universality class for glassy dynamics

Static Rouse Modes and Related Quantities: Corrections to Chain Ideality in Polymer Melts

Following the Flory ideality hypothesis intrachain and interchain excluded volume interactions are supposed to compensate each other in dense polymer systems. Multi-chain effects should thus be neglected and polymer conformations may be understood from simple phantom chain models. Here we provide evidence against this phantom chain, mean-field picture. We analyze numerically and theoretically the static correlation function of the Rouse modes. Our numerical results are obtained from computer simulations of two coarse-grained polymer models for which the strength of the monomer repulsion can be varied, from full excluded volume (`hard monomers') to no excluded volume (`phantom chains'). For nonvanishing excluded volume we find the simulated correlation function of the Rouse modes to deviate markedly from the predictions of phantom chain models. This demonstrates that there are nonnegligible correlations along the chains in a melt. These correlations can be taken into account by perturbation theory. Our simulation results are in good agreement with these new theoretical predictions.Comment: 9 pages, 7 figures, accepted for publication in EPJ

Topological effects in the thermal properties of knotted polymer rings

The topological effects on the thermal properties of several knot configurations are investigated using Monte Carlo simulations. In order to check if the topology of the knots is preserved during the thermal fluctuations we propose a method that allows very fast calculations and can be easily applied to arbitrarily complex knots. As an application, the specific energy and heat capacity of the trefoil, the figure-eight and the $8_1$ knots are calculated at different temperatures and for different lengths. Short-range repulsive interactions between the monomers are assumed. The knots configurations are generated on a three-dimensional cubic lattice and sampled by means of the Wang-Landau algorithm and of the pivot method. The obtained results show that the topological effects play a key role for short-length polymers. Three temperature regimes of the growth rate of the internal energy of the system are distinguished.Comment: 7 pages, 12 figures, LaTeX + RevTeX. With respect to the first version, in the second version the text has been improved and all figures are now in black and whit

Tensile Fracture of Welded Polymer Interfaces: Miscibility, Entanglements and Crazing

Large-scale molecular simulations are performed to investigate tensile failure of polymer interfaces as a function of welding time $t$. Changes in the tensile stress, mode of failure and interfacial fracture energy $G_I$ are correlated to changes in the interfacial entanglements as determined from Primitive Path Analysis. Bulk polymers fail through craze formation, followed by craze breakdown through chain scission. At small $t$ welded interfaces are not strong enough to support craze formation and fail at small strains through chain pullout at the interface. Once chains have formed an average of about one entanglement across the interface, a stable craze is formed throughout the sample. The failure stress of the craze rises with welding time and the mode of craze breakdown changes from chain pullout to chain scission as the interface approaches bulk strength. The interfacial fracture energy $G_I$ is calculated by coupling the simulation results to a continuum fracture mechanics model. As in experiment, $G_I$ increases as $t^{1/2}$ before saturating at the average bulk fracture energy $G_b$. As in previous simulations of shear strength, saturation coincides with the recovery of the bulk entanglement density. Before saturation, $G_I$ is proportional to the areal density of interfacial entanglements. Immiscibiltiy limits interdiffusion and thus suppresses entanglements at the interface. Even small degrees of immisciblity reduce interfacial entanglements enough that failure occurs by chain pullout and $G_I \ll G_b$

Structure and dynamics of ring polymers: entanglement effects because of solution density and ring topology

The effects of entanglement in solutions and melts of unknotted ring polymers have been addressed by several theoretical and numerical studies. The system properties have been typically profiled as a function of ring contour length at fixed solution density. Here, we use a different approach to investigate numerically the equilibrium and kinetic properties of solutions of model ring polymers. Specifically, the ring contour length is maintained fixed, while the interplay of inter- and intra-chain entanglement is modulated by varying both solution density (from infinite dilution up to \approx 40 % volume occupancy) and ring topology (by considering unknotted and trefoil-knotted chains). The equilibrium metric properties of rings with either topology are found to be only weakly affected by the increase of solution density. Even at the highest density, the average ring size, shape anisotropy and length of the knotted region differ at most by 40% from those of isolated rings. Conversely, kinetics are strongly affected by the degree of inter-chain entanglement: for both unknots and trefoils the characteristic times of ring size relaxation, reorientation and diffusion change by one order of magnitude across the considered range of concentrations. Yet, significant topology-dependent differences in kinetics are observed only for very dilute solutions (much below the ring overlap threshold). For knotted rings, the slowest kinetic process is found to correspond to the diffusion of the knotted region along the ring backbone.Comment: 17 pages, 11 figure

Scale-free static and dynamical correlations in melts of monodisperse and Flory-distributed homopolymers: A review of recent bond-fluctuation model studies

It has been assumed until very recently that all long-range correlations are screened in three-dimensional melts of linear homopolymers on distances beyond the correlation length $\xi$ characterizing the decay of the density fluctuations. Summarizing simulation results obtained by means of a variant of the bond-fluctuation model with finite monomer excluded volume interactions and topology violating local and global Monte Carlo moves, we show that due to an interplay of the chain connectivity and the incompressibility constraint, both static and dynamical correlations arise on distances $r \gg \xi$. These correlations are scale-free and, surprisingly, do not depend explicitly on the compressibility of the solution. Both monodisperse and (essentially) Flory-distributed equilibrium polymers are considered.Comment: 60 pages, 49 figure

Colocalization of coregulated genes: a steered molecular dynamics study of human chromosome 19

The connection between chromatin nuclear organization and gene activity is vividly illustrated by the observation that transcriptional coregulation of certain genes appears to be directly influenced by their spatial proximity. This fact poses the more general question of whether it is at all feasible that the numerous genes that are coregulated on a given chromosome, especially those at large genomic distances, might become proximate inside the nucleus. This problem is studied here using steered molecular dynamics simulations in order to enforce the colocalization of thousands of knowledge-based gene sequences on a model for the gene-rich human chromosome 19. Remarkably, it is found that most, ~80% gene pairs can be brought simultaneously into contact. This is made possible by the low degree of intra-chromosome entanglement and the large number of cliques in the gene coregulatory network. A clique is a set of genes coregulated all together as a group. The constrained conformations for the model chromosome 19 are further shown to be organised in spatial macrodomains that are similar to those inferred from recent HiC measurements. The findings indicate that gene coregulation and colocalization are largely compatible and that this relationship can be exploited to draft the overall spatial organization of the chromosome in vivo. The more general validity and implications of these findings could be investigated by applying to other eukaryotic chromosomes the general and transferable computational strategy introduced here

Hi-C-constrained physical models of human chromosomes recover functionally-related properties of genome organization

Combining genome-wide structural models with phenomenological data is at the forefront of efforts to understand the organizational principles regulating the human genome. Here, we use chromosome-chromosome contact data as knowledge-based constraints for large-scale three-dimensional models of the human diploid genome. The resulting models remain minimally entangled and acquire several functional features that are observed in vivo and that were never used as input for the model. We find, for instance, that gene-rich, active regions are drawn towards the nuclear center, while gene poor and lamina associated domains are pushed to the periphery. These and other properties persist upon adding local contact constraints, suggesting their compatibility with non-local constraints for the genome organization. The results show that suitable combinations of data analysis and physical modelling can expose the unexpectedly rich functionally-related properties implicit in chromosome-chromosome contact data. Specific directions are suggested for further developments based on combining experimental data analysis and genomic structural modelling

The fractal globule as a model of chromatin architecture in the cell

The fractal globule is a compact polymer state that emerges during polymer condensation as a result of topological constraints which prevent one region of the chain from passing across another one. This long-lived intermediate state was introduced in 1988 (Grosberg et al. 1988) and has not been observed in experiments or simulations until recently (Lieberman-Aiden et al. 2009). Recent characterization of human chromatin using a novel chromosome conformational capture technique brought the fractal globule into the spotlight as a structural model of human chromosome on the scale of up to 10 Mb (Lieberman-Aiden et al. 2009). Here, we present the concept of the fractal globule, comparing it to other states of a polymer and focusing on its properties relevant for the biophysics of chromatin. We then discuss properties of the fractal globule that make it an attractive model for chromatin organization inside a cell. Next, we connect the fractal globule to recent studies that emphasize topological constraints as a primary factor driving formation of chromosomal territories. We discuss how theoretical predictions, made on the basis of the fractal globule model, can be tested experimentally. Finally, we discuss whether fractal globule architecture can be relevant for chromatin packing in other organisms such as yeast and bacteria