Crumpled globule formation during collapse of a long flexible and semiflexible polymer in poor solvent

Coil-globule transition in two-dimensional polymer chains in an explicit solvent

The structure of two-dimensional polymer chains in a solvent at different temperatures is still far from being fully understood. Computer simulations of high-density macromolecular systems require the use of appropriate algorithms, and therefore the simulations were carried out using the Cooperative Motion Algorithm. The polymer model studied was exactly two-dimensional, coarse-grained and based on a triangular lattice. The theta temperature and temperature of coil-to-globule transition, and critical exponents were determined. The differences between the structure of such a disk and that of a chain in a dense polymer liquid were shown.

Phase behaviour of semiflexible lattice polymers in poor-solvent solution: Mean-field theory and Monte Carlo simulations

We study a solution of interacting semiflexible polymers with curvature energy in poor-solvent conditions on the d-dimensional cubic lattice using mean-field theory and Monte Carlo computer simulations. Building upon past studies on a single chain, we construct a field-theory representation of the system and solve it within a mean-field approximation supported by Monte Carlo simulations in d = 3. A gas-liquid transition is found in the temperature-density plane that is then interpreted in terms of real systems. Interestingly, we find this transition to be independent of the bending rigidity. Past classical Flory-Huggins and Flory mean-field results are shown to be particular cases of this more general framework. Perspectives in terms of guiding experimental results towards optimal conditions are also proposed.

Structure of Strongly Adsorbed Polymer Systems: A Computer Simulation Study

The structure of very thin polymer films formed by strongly adsorbed macromolecules was studied by computer simulation. A coarse-grained model of strictly two-dimensional polymer systems was built, and its properties determined by an efficient Monte Carlo simulation algorithm. Properties of the model system were determined by means of Monte Carlo simulations with a sampling algorithm that combines Verdier-Stockmayer, pivot and reputation moves. The effects of temperature, chain length and polymer concentration on the macromolecular structure were investigated. It was shown that at low temperatures, the chain size increases with the concentration, that is, inversely with high temperatures. This behavior should be explained by the influence of inter-chain interactions.

Microstructural Dynamics of Polymer Melts during Stretching: Radial Size Distribution

The transient elongational viscosity ηe(t) of the polymer melt is known to exhibit strain hardening, which depends on the strain rate ε˙. This phenomenon was elucidated by the difference of chain stretching in the entanglement network between extension and shear. However, to date, the microscopic evolution of polymer melt has not been fully statistically analyzed. In this work, the radial size distributions P(Rg,t) of linear polymers are explored by dissipative particle dynamics during the stretching processes. In uniaxial extensional flow, it is observed that the mean radius of gyration R¯g(t) and standard deviation σ(t) remain unchanged until the onset of strain hardening, corresponding to linear viscoelasticity. Both R¯g and σ rise rapidly in the non-linear regime, and bimodal size distribution can emerge. Moreover, the onset of strain hardening is found to be insensitive to the Hencky strain (ε˙Ht) and chain length (N).

Expansion of Single Chains Released from a Spherical Cavity

A two-stage model is developed to explain the phenomena of chain expansion, released from a confining cavity. In the first stage, the chain is assumed to expand as a sphere, while in the second stage it expands like a coil. The kinetic equations for the variation of chain size are derived in the two stages by balancing the rate of the free energy change with the rate of the energy dissipation. Langevin dynamics simulations are then performed to examine the theory. We find that the expansion process is dominated by the second stage and the evolution of chain size follows, mainly, the predicted curve for coil expansion, which depends on the chain length and is not sensitive to the confining volume fraction. It permits to define the expansion time for the process. Further study reveals that the chain does undergo a spherical expansion in the first stage with the characteristic time much shorter than the one for the second stage. As a consequence, the first-stage variation of chain size can be regarded as an add-on to the principal curve of expansion designated by the second stage. The scaling behaviors and the associated scaling exponents are analyzed in details. The simulation results well support the theory.

Topological Constraints with Optimal Length Promote the Formation of Chromosomal Territories at Weakened Degree of Phase Separation

It is generally agreed that the nuclei of eukaryotic cells at interphase are partitioned into disjointed territories, with distinct regions occupied by certain chromosomes. However, the underlying mechanism for such territorialization is still under debate. Here we model chromosomes as coarse-grained block copolymers and to investigate the effect of loop domains (LDs) on the formation of compartments and territories based on dissipative particle dynamics. A critical length of LDs, which depends sensitively on the length of polymeric blocks, is obtained to minimize the degree of phase separation. This also applies to the two-polymer system: The critical length not only maximizes the degree of territorialization but also minimizes the degree of phase separation. Interestingly, by comparing with experimental data, we find the critical length for LDs and the corresponding length of blocks to be respectively very close to the mean length of topologically associating domains (TADs) and chromosomal segments with different densities of CpG islands for human chromosomes. The results indicate that topological constraints with optimal length can contribute to the formation of territories by weakening the degree of phase separation, which likely promotes the chromosomal flexibility in response to genetic regulations.

Crystallization of semiflexible polymers in melts and solutions

We studied the crystallization of semiflexible polymer chains in melts and poor-solvent solutions with different concentrations using dissipative particle dynamics (DPD) computer simulation techniques. We used the coarse-grained polymer model to reveal the general principles and microscopic scenario of crystallization in such systems at large time and length scales. It covers both primary and secondary nucleation as well as crystallites' merging. The parameters of the DPD model were chosen appropriately to reproduce the entanglements of polymer chains. We started from an initial homogeneous disordered solution of Gaussian chains and observed the initial stages of crystallization process caused in our model by orientational ordering of polymer chains and polymer-solvent phase separation. We found that the overall crystalline fraction at the end of the crystallization process decreases with the increasing polymer volume fraction while the steady-state crystallization speed at later stages does not depend on the polymer volume fraction. The average crystallite size has a maximal value in the systems with a polymer volume fraction from 0.7 to 0.95. In our model, these polymer concentrations represent an optimal value in the sense of balance between the amount of polymer material available to increase the crystallite size and chain entanglements, that prevent crystallites' growth and merging. On large time scales, our model allows us to observe lamellar thickening linear in logarithmic time scale.

Order and stochasticity in the folding of individual Drosophila genomes

Mammalian and Drosophila genomes are partitioned into topologically associating domains (TADs). Although this partitioning has been reported to be functionally relevant, it is unclear whether TADs represent true physical units located at the same genomic positions in each cell nucleus or emerge as an average of numerous alternative chromatin folding patterns in a cell population. Here, we use a single-nucleus Hi-C technique to construct high-resolution Hi-C maps in individual Drosophila genomes. These maps demonstrate chromatin compartmentalization at the megabase scale and partitioning of the genome into non-hierarchical TADs at the scale of 100 kb, which closely resembles the TAD profile in the bulk in situ Hi-C data. Over 40% of TAD boundaries are conserved between individual nuclei and possess a high level of active epigenetic marks. Polymer simulations demonstrate that chromatin folding is best described by the random walk model within TADs and is most suitably approximated by a crumpled globule build of Gaussian blobs at longer distances. We observe prominent cell-to-cell variability in the long-range contacts between either active genome loci or between Polycomb-bound regions, suggesting an important contribution of stochastic processes to the formation of the Drosophila 3D genome.

Genomes are partitioned into topologically associating domains (TADs). Here the authors present single-nucleus Hi-C maps in Drosophila at 10 kb resolution, demonstrating the presence of chromatin compartments in individual nuclei, and partitioning of the genome into non-hierarchical TADs at the scale of 100 kb, which resembles population TAD profiles.

Kinetic mechanisms of crumpled globule formation

A homopolymer chain with beads forming pairwise reversible bonds is a well-known model in polymer physics. We studied the kinetics of homopolymer chain collapse, which was induced by pairwise reversible bond formation. We compared the kinetic mechanism of this coil-globule transition with the mechanism of collapse in a poor solvent. We discovered that if collapse of a spatially confined chain is induced by formation of long-living pairwise reversible bonds, the transient structures do not resemble classical pearl-necklace conformations formed during collapse in a poor solvent. However, both types of collapse lead to formation of a metastable state of a crumpled globule, which is one of the well-known models of interphase chromatin structure in different organisms. Moreover, we found that the stability and dynamics of this state can be controlled by the fraction of reversible bonds and bond lifetime.

Kinetic Signature of Cooperativity in the Irreversible Collapse of a Polymer

We investigate the kinetics of a polymer collapse due to the formation of irreversible cross-links between its monomers. Using the contact probability P(s) as a scale-dependent order parameter depending on the chemical distance s, our simulations show the emergence of a cooperative pearling instability. Namely, the polymer undergoes a sharp conformational transition to a set of absorbing states characterized by a length scale ξ corresponding to the mean pearl size. This length and the transition time depend on the polymer equilibrium dynamics and the cross-linking rate. We confirm experimentally this transition using a DNA conformation capture experiment in yeast.

The biology and polymer physics underlying large-scale chromosome organization

Chromosome large-scale organization is a beautiful example of the interplay between physics and biology. DNA molecules are polymers and thus belong to the class of molecules for which physicists have developed models and formulated testable hypotheses to understand their arrangement and dynamic properties in solution, based on the principles of polymer physics. Biologists documented and discovered the biochemical basis for the structure, function and dynamic spatial organization of chromosomes in cells. The underlying principles of chromosome organization have recently been revealed in unprecedented detail using high-resolution chromosome capture technology that can simultaneously detect chromosome contact sites throughout the genome. These independent lines of investigation have now converged on a model in which DNA loops, generated by the loop extrusion mechanism, are the basic organizational and functional units of the chromosome.

Active chromatin and transcription play a key role in chromosome partitioning into topologically associating domains

Recent advances enabled by the Hi-C technique have unraveled many principles of chromosomal folding that were subsequently linked to disease and gene regulation. In particular, Hi-C revealed that chromosomes of animals are organized into topologically associating domains (TADs), evolutionary conserved compact chromatin domains that influence gene expression. Mechanisms that underlie partitioning of the genome into TADs remain poorly understood. To explore principles of TAD folding in Drosophila melanogaster, we performed Hi-C and poly(A)⁺ RNA-seq in four cell lines of various origins (S2, Kc167, DmBG3-c2, and OSC). Contrary to previous studies, we find that regions between TADs (i.e., the inter-TADs and TAD boundaries) in Drosophila are only weakly enriched with the insulator protein dCTCF, while another insulator protein Su(Hw) is preferentially present within TADs. However, Drosophila inter-TADs harbor active chromatin and constitutively transcribed (housekeeping) genes. Accordingly, we find that binding of insulator proteins dCTCF and Su(Hw) predicts TAD boundaries much worse than active chromatin marks do. Interestingly, inter-TADs correspond to decompacted inter-bands of polytene chromosomes, whereas TADs mostly correspond to densely packed bands. Collectively, our results suggest that TADs are condensed chromatin domains depleted in active chromatin marks, separated by regions of active chromatin. We propose the mechanism of TAD self-assembly based on the ability of nucleosomes from inactive chromatin to aggregate, and lack of this ability in acetylated nucleosomal arrays. Finally, we test this hypothesis by polymer simulations and find that TAD partitioning may be explained by different modes of inter-nucleosomal interactions for active and inactive chromatin.

Coalescence Model for Crumpled Globules Formed in Polymer Collapse

The rapid collapse of a polymer, due to external forces or changes in solvent, yields a long-lived "crumpled globule." The conjectured fractal structure shaped by hierarchical collapse dynamics has proved difficult to establish, even with large simulations. To unravel this puzzle, we study a coarse-grained model of in-falling spherical blobs that coalesce upon contact. Distances between pairs of monomers are assigned upon their initial coalescence, and do not "equilibrate" subsequently. Surprisingly, the model reproduces quantitatively the dependence of distance on segment length, suggesting that the slow approach to scaling is related to the wide distribution of blob sizes.

Anomalous diffusion in fractal globules

The fractal globule state is a popular model for describing chromatin packing in eukaryotic nuclei. Here we provide a scaling theory and dissipative particle dynamics computer simulation for the thermal motion of monomers in the fractal globule state. Simulations starting from different entanglement-free initial states show good convergence which provides evidence supporting the existence of a unique metastable fractal globule state. We show monomer motion in this state to be subdiffusive described by ⟨X(2)(t)⟩∼t(αF) with αF close to 0.4. This result is in good agreement with existing experimental data on the chromatin dynamics, which makes an additional argument in support of the fractal globule model of chromatin packing.

Effects of topological constraints on globular polymers

Topological constraints can affect both equilibrium and dynamic properties of polymer systems and can play a role in the organization of chromosomes. Despite many theoretical studies, the effects of topological constraints on the equilibrium state of a single compact polymer have not been systematically studied. Here we use simulations to address this longstanding problem. We find that sufficiently long unknotted polymers differ from knotted ones in the spatial and topological states of their subchains. The unknotted globule has subchains that are mostly unknotted and form asymptotically compact RG(s)∼s1/3 crumples. However, crumples display a high fractal dimension of the surface db=2.8, forming excessive contacts and interpenetrating each other. We conclude that this topologically constrained equilibrium state resembles a conjectured crumpled globule [Grosberg et al., Journal de Physique, 1988, 49, 2095], but differs from its idealized hierarchy of self-similar, isolated and compact crumples.