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Spatial organization of chromatin plays a critical role in genome regulation. Previously, various types of affinity mediators and enzymes have been attributed to regulate spatial organization of chromatin from a thermodynamics perspective. However, at the mechanistic level, enzymes act in their unique ways and perturb the chromatin. Here, we construct a polymer physics model following the mechanistic scheme of Topoisomerase-II, an enzyme resolving topological constraints of chromatin, and investigate how it affects interphase chromatin organization. Our computer simulations demonstrate Topoisomerase-II’s ability to phase separate chromatin into eu- and heterochromatic regions with a characteristic wall-like organization of the euchromatic regions. We realized that the ability of the euchromatic regions to cross each other due to enzymatic activity of Topoisomerase-II induces this phase separation. This realization is based on the physical fact that partial absence of self-avoiding interaction can induce phase separation of a system into its self-avoiding and non-self-avoiding parts, which we reveal using a mean-field argument. Furthermore, motivated from recent experimental observations, we extend our model to a bidisperse setting and show that the characteristic features of the enzymatic activity-driven phase separation survive there. The existence of these robust characteristic features, even under the non-localized action of the enzyme, highlights the critical role of enzymatic activity in chromatin organization.

A non-Markovian process, in which stochastic evolution of the system depends on its past history, often shows up in soft matter, living cells, and other complex systems. Despite its importance, however, the statistics of first passage time in such systems is not well understood. This is largely due to the fact that most theoretical frameworks are based on Markovian description, and incorporation of the memory effect into the problem remains a challenge. Here, we argue that a key quantity in the problem, i.e., the average behavior of a non-Markovian walker after the first passage, can be linked to its dynamical response, and propose a simple framework to compute important observables in the first passage problem. We perform a mean-field analysis and demonstrate semiquantitative description of the one-dimensional fractional Brownian motion in the presence of an absorbing boundary.

Topological constraints (TCs) between polymers determine the behavior of complex fluids such as creams, oils, and plastics. Most of the polymer solutions used in everyday life employ linear chains; their behavior is accurately captured by the reptation and tube theories which connect microscopic TCs to macroscopic viscoelasticity. On the other hand, polymers with nontrivial topology, such as rings, hold great promise for new technology but pose a challenging problem as they do not obey standard theories; additionally, topological invariance, i.e., the fact that rings must remain unknotted and unlinked if prepared so, precludes any serious analytical treatment. Here we propose an unambiguous, parameter-free algorithm to characterize TCs in polymeric solutions and show its power in characterizing TCs of entangled rings. We analyze large-scale molecular dynamics simulations via persistent homology, a key mathematical tool to extract robust topological information from large datasets. This method allows us to identify ring-specific TCs which we call “homological threadings” (H-threadings) and to connect them to polymer behavior. It also allows us to identify, in a physically appealing and unambiguous way, scale-dependent loops which have eluded precise quantification so far. We discover that while threaded neighbors slowly grow with ring length, the ensuing TCs are extensive also in the asymptotic limit. Our proposed method is not restricted to ring polymers and can find broader applications for the study of TCs in generic polymeric materials.

[This corrects the article DOI: 10.1371/journal.pone.0060938.].

Some biochemical systems show oscillation. They often consist of feedback loops with repressive transcription regulation. Such biochemical systems have distinctive characteristics in comparison with ordinary chemical systems: i) numbers of molecules involved are small, ii) there are typically only a couple of genes in a cell with a finite regulation time. Due to the fluctuations caused by these features, the system behavior can be quite different from the one by deterministic rate equations, because the rate equations ignore molecular fluctuations and thus are exact only in the infinite molecular number limit. The molecular fluctuations on a free-running circadian system have been studied by Gonze et al. (2002) by introducing a scale parameter [Formula: see text] for the system size. They consider, however, only the first effect, assuming that the gene process is fast enough for the second effect to be ignored, but this has not been examined systematically yet. Here we study fluctuation effects due to the finite gene regulation time by introducing a new scale parameter [Formula: see text], which we take as the unbinding time of a nuclear protein from the gene. We focus on the case where the fluctuations due to small molecular numbers are negligible. In simulations on the same system studied by Gonze et al., we find the system is unexpectedly sensitive to the fluctuation in the transcription regulation; the period of oscillation fluctuates about 30 min even when the regulation time scale [Formula: see text] is around 30 s, that is even smaller than 1/1000 of its circadian period. We also demonstrate that the distribution width for the oscillation period and amplitude scales with [Formula: see text], and the correlation time scales with [Formula: see text] in the small [Formula: see text] regime. The relative fluctuations for the period are about half of that for the amplitude, namely, the periodicity is more stable than the amplitude.

Liquid-liquid phase separation (LLPS) has increasingly been found to play pivotal roles in a number of intracellular events and reactions, and has introduced a new paradigm in cell biology to explain protein-protein and enzyme-ligand interactions beyond conventional molecular and biochemical theories. LLPS is driven by the cumulative effects of weak and promiscuous interactions, including electrostatic, hydrophobic and cation-π interactions, among polypeptides containing intrinsically disordered regions (IDRs) and describes the macroscopic behaviours of IDR-containing proteins in an intracellular milieu. Recent studies have revealed that interactions between 'charge blocks' - clusters of like charges along the polypeptide chain - strongly induce LLPS and play fundamental roles in its spatiotemporal regulation. Introducing a new parameter, termed 'charge blockiness', into physicochemical models of disordered polypeptides has yielded a better understanding of how the intrinsic amino acid sequence of a polypeptide determines the spatiotemporal occurrence of LLPS within a cell. Charge blockiness might also explain why some post-translational modifications segregate within IDRs and how they regulate LLPS. In this Review, we summarise recent progress towards understanding the mechanism and biological roles of charge block-driven LLPS and discuss how this new characteristic parameter of polypeptides offers new possibilities in the fields of structural biology and cell biology. [ABSTRACT FROM AUTHOR]

We investigated the collapsed structure of a weakly charged wormlike chain under a moderate concentration of 1:1 electrolyte solution. By assuming a torus as a grand state, we found that the size of a torus is determined by the balance between surface energy and electrostatic energy, which leads to a finite torus thickness almost independent of the chain contour length. Owing to this unique characteristic, a long charged wormlike chain forms multiple tori structure as a collapsed product, which is never seen with a neutral wormlike chain. These features were confirmed by a Monte Carlo simulation. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

The way tension propagates along a chain is a key to govern many anomalous dynamics in macromolecular systems. After introducing the weak and the strong force regimes of the tension propagation, we focus on the latter, in which the dynamical fluctuations of a segment in a long polymer during its stretching process is investigated. We show that the response, i.e., average drift, is anomalous, which is characterized by the nonlinear memory kernel, and its relation to the fluctuation is nontrivial. These features are discussed on the basis of the generalized Langevin equation, in which the role of the temporal change in spring constant due to the stress hardening is pinpointed. We carried out the molecular dynamics simulation, which supports our theory. [ABSTRACT FROM AUTHOR]

We report the spontaneous patterning of polymer microgels by confining a polymer blend within microspheres. A poly(ethylene glycol) (PEG) and gelatin solution was confined inside water-in-oil (W/O) microdroplets coated with a layer of zwitterionic lipids: dioleoylphosphatidylethanolamine (PE) and dioleoylphosphatidylcholine (PC). The droplet confinement affected the kinetics of the phase separation, wetting, and gelation after a temperature quench, which determined the final microgel pattern. The gelatin-rich phase completely wetted to the PE membrane and formed a hollow microcapsule as a stable state in the PE droplets. Gelation during phase separation varied the relation between the droplet size and thickness of the capsule wall. In the case of the PC droplets, phase separation was completed only for the smaller droplets, wherein the microgel partially wetted the PC membrane and had a hemisphere shape. In addition, the temperature decrease below the gelation point increased the interfacial tension between the PEG/gelatin phases and triggered a dewetting transition. Interestingly, the accompanying shape deformation to minimize the interfacial area was only observed for the smaller PC droplets. The critical size decreased as the gelatin concentration increased, indicating the role of the gel elasticity as an inhibitor of the deformation. Furthermore, variously patterned microgels with spherically asymmetric shapes, such as discs and stars, were produced as kinetically trapped states by regulating the incubation time, polymer composition, and droplet size. These findings demonstrate a way to regulate the complex shapes of microgels using the interplay among phase separation, wetting, and gelation of confined polymer blends in microdroplets. [ABSTRACT FROM AUTHOR]

We analyze the electrorheological effect in immiscible fluid mixtures with dielectric mismatch. By taking the electric field effect into account, which couples to the dynamics of domain morphology under flow, we propose a set of electrorheological constitutive equations valid under the condition where the relative magnitude of the flow field is stronger than that of the electric field. Through comparison with recent experiments, we point out a unique dynamical stress response inherent in situations where the cross-coupling between different fields is essential. [ABSTRACT FROM AUTHOR]

We analyze static properties of a strongly confined semiflexible polymer, i.e., either trapped in a closed space or compressed by external forces, in an athermal solvent. Like a flexible polymer case, we can resort to an analogy with the semidilute solution, but a complication due to the additional length scale arising from the chain rigidity results in more diverse behaviors depending on system parameters. For each regime, scaling forms of the excess free energy of the confinement are derived. Effects of the confinement geometry and the solvent quality are briefly discussed. [ABSTRACT FROM AUTHOR]

The compaction of long duplex DNA by cationic nanoparticles (NP) used as a primary model of histone core particles has been investigated. We have systematically studied the effect of salt concentration, particle size, and particle charge by means of single-molecule observationsfluorescence microscopy (FM) and transmission electron microscopy (TEM)and molecular dynamics (MD) simulations. We have found that the large-scale DNA compaction is progressive and proceeds through the formation of beads-on-a-string structures of various morphologies. The DNA adsorbed amount per particle depends weakly on NP concentration but increases significantly with an increase in particle size and is optimal at an intermediate salt concentration. Three different complexation mechanisms have been identified depending on the correlation between DNA and NPs in terms of geometry, chain rigidity, and electrostatic interactions:  free DNA adsorption onto NP surface, DNA wrapping around NP, and NP collection on DNA chain. [ABSTRACT FROM AUTHOR]

It has been observed that the direction in which a sand dune extends its crest line depends on seasonal variation of wind direction; when the variation is small, the crest line develops more or less perpendicularly to the mean wind direction to form a transverse dune with some undulation. In the case of bimodal wind with a large relative angle, however, the dune extends its crest along the mean wind direction and evolves into an almost straight longitudinal dune. Motivated by these observations, we investigate the dynamical stability of isolated dunes using the crest line model, where the dune dynamics is represented by its crest line motion. First, we extend the previous linear stability analysis under the unidirectional wind to the case with nonzero slant angle between the wind direction and the normal direction of the crest line, and show that the stability diagram does not depend on the slant angle. Second, we examine how the linear stability is affected by the seasonal changes of wind direction in the case of bimodal wind with equal strength and duration. For the transverse dune, we find that the stability is virtually the same with that for the unidirectional wind as long as the dune evolution during a season is small. On the other hand, in the case of the longitudinal dune, the dispersions of the growth rates for the perturbation are drastically different from those of the unidirectional wind, and we find that the largest growth rate is always located at k=0. This is because the growth of the perturbation with k≠0 is canceled by the alternating wind from opposite sides of the crest line even though it grows during each duration period of the bimodal wind. For a realistic parameter set, the system is in the wavy unstable regime of the stability diagram for the unidirectional wind, thus the straight transverse dune is unstable to develop undulation and eventually evolves into a string of barchans when the seasonal variation of wind direction is small, but the straight longitudinal dune is stabilized under the large variation of bimodal wind direction. We also perform numerical simulations on the crest line model, and find that the results are consistent with our linear analysis and the previous reports that show that the longitudinal dunes tend to have a straight ridge elongating over time. [ABSTRACT FROM AUTHOR]

The loop extrusion theory predicts that the loops of chromosomes are produced by cohesin molecules that uni-directionally extrude a chromatin fiber. We here use an extension of the Rouse model to predict the chain conformational dynamics driven by the loop extrusion process. Our theory predicts that in a bulk solution, the mean square distance between the starting and ending sites of the loop extrusion process decreases with a constant rate. This is because the tension generated by the loop extrusion process drives the displacement of the starting site towards the ending site. In contrast, when the cohesin is entrapped at an interface, the mean square distance does not decrease until the tension generated by the loop extrusion process arrives at the ending site. This theory highlights the fact that the chain dynamics strongly depends on the mobility of the chain segments bound by cohesin. [ABSTRACT FROM AUTHOR]

Inspired by recent experiments on chromosomal dynamics, we introduce an exactly solvable model for the interaction between a flexible polymer and a set of motorlike enzymes. The enzymes can bind and unbind to specific sites of the polymer and produce a dipolar force on two neighboring monomers when bound. We study the resulting nonequilibrium dynamics of the polymer and find that the motion of the monomers has several properties that were observed experimentally for chromosomal loci: a subdiffusive mean-square displacement and the appearance of regions of correlated motion. We also determine the velocity autocorrelation of the monomers and find that the underlying stochastic process is not fractional Brownian motion. Finally, we show that the active forces swell the polymer by an amount that becomes constant for large polymers. [ABSTRACT FROM AUTHOR]

Configurations of a single semiflexible polymer is studied when it is pushed into a nanochannel in the case where the polymer persistence length lp is much longer than the channel diameter D:lp/D≫1. Using numerical simulations, we show that the polymer undergoes a sequence of recurring structural transitions upon longitudinal compression: random deflection along the channel, a helix going around the channel wall, double-fold random deflection, double-fold helix, etc. We find that the helix transition can be understood as buckling of deflection segments, and the initial helix formation takes place at very small compression with no appreciable weak compression regime of the random deflection polymer. [ABSTRACT FROM AUTHOR]

Several subdiffusive stochastic processes in nature, e.g., the motion of a tagged monomer in polymers, the height fluctuation of interfaces, particle dynamics in single-file diffusion, etc., can be described rigorously or approximately by the superposition of various modes whose relaxation times are broadly distributed. In this paper, we propose a mode analysis generating superdiffusion, which is paired with or complementary to subdiffusion. The key point in our discussion lies in the identification of a pair of conjugated variables, which undergo sub- and superdiffusion, respectively. We provide a simple interpretation for the sub- and superdiffusion duality for these variables using the language of polymer physics. The analysis also suggests the usefulness of looking at the force fluctuation in experiments, where a polymer is driven by a constant velocity. [ABSTRACT FROM AUTHOR]

Some immiscible blends under a strong electric field often exhibit periodic structures, bridging the gap between two electrodes. Upon shear, the structures tilt, and exhibit an elastic response which is mostly governed by the electric energy. Assuming a two-dimensional stripe structure, we calculate the Maxwell stress, and derive an expression for the shear modulus, demonstrating how it depends on the external electric field, the composition, and the dielectric properties of the blend. We also suggest the notion of effective interfacial tension, which renormalizes the effect of the electric field. This leads to a simple derivation of the scaling law for the selection of the wavelength of the structure formed under an electric field. [ABSTRACT FROM AUTHOR]

The miscibility of polymer blends, a classical problem in polymer science, may be altered, if one or both of the component do not have chain ends. Based on the idea of topological volume, we propose a mean-field theory to clarify how the topological constraints in ring polymers affect the phase behavior of the blends. While the large enhancement of the miscibility is expected for ring-linear polymer blends, the opposite trend toward demixing, albeit comparatively weak, is predicted for ring-ring polymer blends. Scaling formulas for the shift of critical point for both cases are derived. We discuss the valid range of the present theory, and the crossover to the linear polymer blends behaviors, which is expected for short chains. These analyses put forward a view that the topological constraints could be represented as an effective excluded-volume effects, in which the topological length plays a role of the screening factor. [ABSTRACT FROM AUTHOR]

We study inelastic collapse in a one-dimensional N-particle system when the system is driven from below under gravity. We investigate the hard-sphere limit of inelastic soft-sphere systems by numerical simulations to find how the collision rate per particle noeu increases as a function of the elastic constant of the sphere k when the restitution coefficient e is kept constant. For systems with large enough N > 20, we find three regimes in e depending on the behavior of ncon in the hard-sphere limit: (i) an uncollapsing regime for 1 ≥ e > ecl, where «cou converges to a finite value, (ii) a logarithmically collapsing regime for ecl > e > ec2, where ncon diverges as «cou ~ log k, and (iii) a power-law collapsing regime for eci> e > 0, where nco)| diverges as ncon ~ ka with an exponent a that depends on N. The power-law collapsing regime shrinks as N decreases and seems not to exist for the system with N = 3, while, for large N, the size of the uncollapsing and the logarithmically collapsing regime decreases as ec\ ~ 1-2.6/N and eci ~ 1-3.0/N. We demonstrate that this difference between large and small systems exists already in the inelastic collapse without external drive and gravity. [ABSTRACT FROM AUTHOR]

We show that a single DNA molecule confined and extended in a nanochannel can be dynamically compressed by sliding a permeable gasket at a fixed velocity relative to the stationary polymer. The gasket is realized experimentally by optically trapping a nanosphere inside a nanochannel. The trapped bead acts like a "nanodozer," directly applying compressive forces to the molecule without requirement of chemical attachment. Remarkably, these strongly nonequilibrium measurements can be quantified via a simple nonlinear convective-diffusion formalism and yield insights into the local blob statistics, allowing us to conclude that the compressed nanochannel-confined chain exhibits mean-field behavior. [ABSTRACT FROM AUTHOR]