Self-avoiding Polymer Research Articles

A generalization of the bond fluctuation model to viscoelastic environments

A lattice-based simulation method for polymer diffusion in a viscoelastic medium is presented. This method combines the eight-site bond fluctuation model with an algorithm for the simulation of fractional Brownian motion on the lattice. The method applies to unentangled self-avoiding chains and is probed for anomalous diffusion exponents α between 0.7 and 1.0. The simulation results are in very good agreement with the predictions of the generalized Rouse model of a self-avoiding chain polymer in a viscoelastic medium.

Dynamics of the reaction between the free end of a tethered self-avoiding polymer and a flat penetrable surface: a renormalization group study.

The average time τr for one end of a long, self-avoiding polymer to interact for the first time with a flat penetrable surface to which it is attached at the other end is shown here to scale essentially as the square of the chain's contour length N. This result is obtained within the framework of the Wilemski-Fixman approximation to diffusion-limited reactions, in which the reaction time is expressed as a time correlation function of a "sink" term. In the present work, this sink-sink correlation function is calculated using perturbation expansions in the excluded volume and the polymer-surface interactions, with renormalization group methods being used to resum the expansion into a power law form. The quadratic dependence of τr on N mirrors the behavior of the average time τc of a free random walk to cyclize, but contrasts with the cyclization time of a free self-avoiding walk (SAW), for which τr ∼ N(2.2). A simulation study by Cheng and Makarov [J. Phys. Chem. B 114, 3321 (2010)] of the chain-end reaction time of an SAW on a flat impenetrable surface leads to the same N(2.2) behavior, which is surprising given the reduced conformational space a tethered polymer has to explore in order to react.

Structure of Microgels with Debye–Hückel Interactions

The structural properties of model microgel particles are investigated by molecular dynamics simulations applying a coarse-grained model. A microgel is comprised of a regular network of polymers internally connected by tetra-functional cross-links and with dangling ends at its surface. The self-avoiding polymers are modeled as bead-spring linear chains. Electrostatic interactions are taken into account by the Debye–Hückel potential. The microgels exhibit a quite uniform density under bad solvent conditions with a rather sharp surface. With increasing Debye length, structural inhomogeneities appear, their surface becomes fuzzy and, at very large Debye lengths, well defined again. Similarly, the polymer conformations change from a self-avoiding walk to a rod-like behavior. Thereby, the average polymer radius of gyration follows a scaling curve in terms of polymer length and persistence length, with an asymptotic rod-like behavior for swollen microgels and self-avoiding walk behavior for weakly swollen gel particles.

Investigation of the Chromosome Regions with Significant Affinity for the Nuclear Envelope in Fruit Fly – A Model Based Approach

Three dimensional nuclear architecture is important for genome function, but is still poorly understood. In particular, little is known about the role of the “boundary conditions” – points of attachment between chromosomes and the nuclear envelope. We describe a method for modeling the 3D organization of the interphase nucleus, and its application to analysis of chromosome-nuclear envelope (Chr-NE) attachments of polytene (giant) chromosomes in Drosophila melanogaster salivary glands. The model represents chromosomes as self-avoiding polymer chains confined within the nucleus; parameters of the model are taken directly from experiment, no fitting parameters are introduced. Methods are developed to objectively quantify chromosome territories and intertwining, which are discussed in the context of corresponding experimental observations. In particular, a mathematically rigorous definition of a territory based on convex hull is proposed. The self-avoiding polymer model is used to re-analyze previous experimental data; the analysis suggests 33 additional Chr-NE attachments in addition to the 15 already explored Chr-NE attachments. Most of these new Chr-NE attachments correspond to intercalary heterochromatin – gene poor, dark staining, late replicating regions of the genome; however, three correspond to euchromatin – gene rich, light staining, early replicating regions of the genome. The analysis also suggests 5 regions of anti-contact, characterized by aversion for the NE, only two of these correspond to euchromatin. This composition of chromatin suggests that heterochromatin may not be necessary or sufficient for the formation of a Chr-NE attachment. To the extent that the proposed model represents reality, the confinement of the polytene chromosomes in a spherical nucleus alone does not favor the positioning of specific chromosome regions at the NE as seen in experiment; consequently, the 15 experimentally known Chr-NE attachment positions do not appear to arise due to non-specific (entropic) forces. Robustness of the key conclusions to model assumptions is thoroughly checked.

Polymer escape from a confining potential

The rate of escape of polymers from a two-dimensionally confining potential well has been evaluated using self-avoiding as well as ideal chain representations of varying length, up to 80 beads. Long timescale Langevin trajectories were calculated using the path integral hyperdynamics method to evaluate the escape rate. A minimum is found in the rate for self-avoiding polymers of intermediate length while the escape rate decreases monotonically with polymer length for ideal polymers. The increase in the rate for long, self-avoiding polymers is ascribed to crowding in the potential well which reduces the free energy escape barrier. An effective potential curve obtained using the centroid as an independent variable was evaluated by thermodynamic averaging and Kramers rate theory then applied to estimate the escape rate. While the qualitative features are well reproduced by this approach, it significantly overestimates the rate, especially for the longer polymers. The reason for this is illustrated by constructing a two-dimensional effective energy surface using the radius of gyration as well as the centroid as controlled variables. This shows that the description of a transition state dividing surface using only the centroid fails to confine the system to the region corresponding to the free energy barrier and this problem becomes more pronounced the longer the polymer is. A proper definition of a transition state for polymer escape needs to take into account the shape as well as the location of the polymer.

Circular Kardar-Parisi-Zhang equation as an inflating, self-avoiding ring polymer

We consider the Kardar-Parisi-Zhang equation for a circular interface in two dimensions, unconstrained by the standard small-slope and no-overhang approximations. Numerical simulations using an adaptive scheme allow us to elucidate the complete time evolution as a crossover between a short-time regime with the interface fluctuations of a self-avoiding ring or two-dimensional vesicle, and a long-time regime governed by the Tracy-Widom distribution expected for this geometry. For small-noise amplitudes, scaling behavior is only of the latter type. Large noise is also seen to renormalize the bare physical parameters of the ring, akin to analogous parameter renormalization for equilibrium three-dimensional membranes. Our results bear particular importance on the relation between relevant universality classes of scale-invariant systems in two dimensions.

Dynamics of a polymer adsorbed to an attractive homogeneous flat surface.

The adsorption of a bond fluctuation self-avoiding walk polymer on an attractive homogeneous flat surface at temperature below the critical adsorption point is studied using dynamic Monte Carlo simulation. Results show that the apparent size Rg,xy(2) of the polymer parallel to the surface increases exponentially with time during the adsorption process. The relaxation time for Rg,xy(2) reaching its asymptotic value σeq decreases with the increase in the polymer-surface attraction strength Eps, whereas σeq increases with Eps, indicating that the polymer is adsorbed faster and becomes more extended at stronger adsorption. The polymer's asphericity Axy parallel to the surface is sensitive to intra-polymer interaction and its behavior is different from that of Rg,xy(2). Simulation results also show that the two-dimensional behaviors of Rg,xy(2) and Axy are different from that of the three-dimensional conformational size Rg(2) and asphericity A during the adsorption process. During the adsorption, the surface contacted monomer number M increases with time, but Rg(2) and A show novel behavior as they first increase with M at small M, then decrease with M at moderate M and finally increase with M again at large M. Whereas Rg,xy(2) and Axy first decrease with M and then increase with M during the adsorption.

Expansion dynamics of a self-avoiding polymer in a cylindrical pore

Inspired by recent bacterial chromosome experiments in narrow channels, we simulate the expansion (and internal) dynamics of a self-avoiding polymer under cylindrical confinement. The chain is trapped in a piston, compressed up to of its equilibrium length, and released unidirectionally from the right end of the piston. Our results suggest that the chain initially expands like a concentrated hard-sphere system, enters a subdiffusive regime at an intermediate time, and eventually relaxes globally to its equilibrium size. Using our results, we test a few theoretical models (e.g., a Flory-type approach), in which the blob-blob or monomer-monomer interaction determines “expansion forces,” clarifying their applicability. Our results can be used for exploring further the polymer aspect of bacterial chromosomes.

Monomer Distributions and Intrachain Collisions of a Polymer Confined to a Channel

We study the conformations of a self-avoiding polymer confined to a channel by computing the cross-sectional distributions of the positions of its monomers. By means of Monte-Carlo simulations for a self-avoiding, freely-jointed chain we determine how the cross-sectional distribution for a given monomer depends on its location in the polymer, and how strongly this distribution is affected by self avoidance. To this end we analyze how the frequency of intra-chain collisions between monomers depends on their spatial position in the channel and on their location within the polymer. We show that most collisions occur between closely neighboring monomers. As a consequence the collision probability depends only weakly on the spatial position of the monomers. Our results explain why the effect of self-avoidance on the monomer distributions is weaker than predicted by mean-field theory. We discuss the relevance of our results for studies of DNA conformations in nanofluidic channels.

Pressing soft membrane on a self-avoiding polymer against a flat wall

A polymer-membrane interacting system can produce a variety of structures. Here, we theoretically study a model system in which a membrane pushes a polymer against a hard surface; we show that a first-order structural phase transition can occur. Using a Monte Carlo simulation, we reveal that the system undergoes a transition from a confined (bump) state to a strongly confined (flattened-out) state as the pressure increases. A scaling argument is also made to understand the physical mechanism behind the phase transition and the properties of each state.

Critical adsorption of a flexible polymer confined between two parallel interacting surfaces

Critical adsorption of a lattice self-avoiding bond fluctuation polymer chain confined between two parallel impenetrable surfaces is studied using the Monte Carlo method. The dependence of the mean contact number <M> on the temperature T and on the chain length N is simulated for a polymer-surface interaction E=-1. A critical adsorption of the polymer is found at T(c)=1.65 for large surface separation distance D>N(ν)b, whereas no critical adsorption is observed for small distance D<N(ν)b, where ν ≈ 0.58 is the Flory exponent and b is the mean bond length. The critical adsorption point T(c)=1.65 is the same as that of a grafted polymer. Normal diffusion is observed for the confined polymer; however, the diffusion rate is dependent on the temperature and surface separation distance.

Population Properties of Self-Avoiding Polymer Chain Models of Chromosomes in a Confined Space of Nucleus

Global architecture of cell nucleus and the spatial organization of chromatin play important roles in gene expression and nuclear function. Single-cell imaging and chromosome conformation capture-based techniques provide information on chromosome conformations and their spatial organizations. Here we propose a polymer model to study the higher order chromatin organization in the nucleus. This model is based on generating self-avoiding chains with appropriate diameters of chromatin fiber in a confined space where the excluded volume effect is taken into consideration. using a sequential importance sampling technique, we are able to generate 10,000 independent conformations explicitly and study their statistical properties, such as spatial distance vs. genomic length relationship and loop forming probability. This model can capture many scaling properties of chromatin folding reported experimentally and can also explain many biological properties of chromatin fibers.

Budding transition of a self-avoiding polymer confined by a soft membrane adhering onto a flat wall

Monte Carlo simulation is used to study the structural properties of the system consisting of a self-avoiding polymer chain confined between a fluid membrane and a flat hard surface. As the adhesion between the soft membrane and the hard-wall surface increases, the polymer is subject to a strong confinement; a pancake-shaped polymer conformation state eventually yields to a bud state through an abrupt, first-order phase transition. We explore the scaling behavior of the physical properties of the system as functions of the polymer's size, the membrane's surface tension, and the adhesion energy, for both pancake and bud states, in terms of Monte Carlo data and analytic scaling theories.

Polymer-mediated entropic forces between scale-free objects

The number of configurations of a polymer is reduced in the presence of a barrier or an obstacle. The resulting loss of entropy adds a repulsive component to other forces generated by interaction potentials. When the obstructions are scale invariant shapes (such as cones, wedges, lines, or planes) the only relevant length scales are the polymer size R(0) and characteristic separations, severely constraining the functional form of entropic forces. Specifically, we consider a polymer (single strand or star) attached to the tip of a cone, at a separation h from a surface (or another cone). At close proximity, such that h << R(0), separation is the only remaining relevant scale and the entropic force must take the form F = Ak(B)T/h. The amplitude A is universal and can be related to exponents η governing the anomalous scaling of polymer correlations in the presence of obstacles. We use analytical, numerical, and ε-expansion techniques to compute the exponent η for a polymer attached to the tip of the cone (with or without an additional plate or cone) for ideal and self-avoiding polymers. The entropic force is of the order of 0.1 pN at 0.1 μm for a single polymer and can be increased for a star polymer.

Variation of the folding and dynamics of the Escherichia coli chromosome with growth conditions

We examine whether the Escherichia coli chromosome is folded into a self-adherent nucleoprotein complex, or alternately is a confined but otherwise unconstrained self-avoiding polymer. We address this through in vivo visualization, using an inducible GFP fusion to the nucleoid-associated protein Fis to non-specifically decorate the entire chromosome. For a range of different growth conditions, the chromosome is a compact structure that does not fill the volume of the cell, and which moves from the new pole to the cell centre. During rapid growth, chromosome segregation occurs well before cell division, with daughter chromosomes coupled by a thin inter-daughter filament before complete segregation, whereas during slow growth chromosomes stay adjacent until cell division occurs. Image correlation analysis indicates that sub-nucleoid structure is stable on a 1 min timescale, comparable to the timescale for redistribution time measured for GFP-Fis after photobleaching. Optical deconvolution and writhe calculation analysis indicate that the nucleoid has a large-scale coiled organization rather than being an amorphous mass. Our observations are consistent with the chromosome having a self-adherent filament organization.

Multiparticle collision dynamics simulation of a polymer in a viscous fluid

We perform a hybrid molecular dynamics simulation of a polymer immersed in a viscous liquid. Using the multiparticle collision dynamics algorithm to incorporate the hydrodynamic effects, we examine static and dynamic properties of a self-avoiding flexible polymer. The diffusion coefficient of the polymer is found to be proportional to N −ν , where N is the molecular weight of the polymer and ν = 3/5 is the swelling exponent, as predicted by the Zimm model. The velocity autocorrelation function of the center of mass of the polymer is obtained as a function of time t, which shows an algebraic decay as t −3/2 at long-time scales. Our results confirm the theoretical prediction for the dynamical properties of a polymer and show the validity of the hybrid simulation scheme in studying the dynamics of a polymer in a viscous fluid.

Coarse graining of star-polymer – colloid nanocomposites

We consider mixtures of self-avoiding multiarm star polymers with hard colloids that are smaller than the star polymer size. By employing computer simulations, and by extending previous theoretical approaches, developed for the opposite limit of small star polymers [A. Jusufi et al., J. Phys.: Condens. Matter 13, 6177 (2001)], we coarse-grain the mixture by deriving an effective cross-interaction between the unlike species. The excellent agreement between theory and simulation for all size ratios examined demonstrates that the theoretical approaches developed for the colloidal limit can be successfully modified to maintain their validity also for the present case of the protein limit, in contrast to the situation for mixtures of colloids and linear polymers. We further analyze, on the basis of the derived interactions, the non-additivity parameter of the mixture as a function of size ratio and star functionality and delineate the regions in which we expect mixing as opposed to demixing behavior. Our results are relevant for the study of star-colloid nanocomposites and pave the way for further investigations of the structure and thermodynamics of the same.

The Impact of Entropy on the Spatial Organization of Synaptonemal Complexes within the Cell Nucleus

We employ 4Pi-microscopy to study SC organization in mouse spermatocyte nuclei allowing for the three-dimensional reconstruction of the SC's backbone arrangement. Additionally, we model the SCs in the cell nucleus by confined, self-avoiding polymers, whose chain ends are attached to the envelope of the confining cavity and diffuse along it. This work helps to elucidate the role of entropy in shaping pachytene SC organization. The framework provided by the complex interplay between SC polymer rigidity, tethering and confinement is able to qualitatively explain features of SC organization, such as mean squared end-to-end distances, mean squared center-of-mass distances, or SC density distributions. However, it fails in correctly assessing SC entanglement within the nucleus. In fact, our analysis of the 4Pi-microscopy images reveals a higher ordering of SCs within the nuclear volume than what is expected by our numerical model. This suggests that while effects of entropy impact SC organization, the dedicated action of proteins or actin cables is required to fine-tune the spatial ordering of SCs within the cell nucleus.

Thermodynamic behaviour of two-dimensional vesicles revisited

We study pressurised self-avoiding ring polymers in two dimensions using Monte Carlo simulations, scaling arguments and Flory-type theories, through models which generalise the model of Leibler, Singh and Fisher (Phys. Rev. Lett. 59, 1989 (1987)). We demonstrate the existence of a thermodynamic phase transition at a non-zero scaled pressure [Formula: see text] , where [Formula: see text] = Np/4[Formula: see text] , with the number of monomers N [Formula: see text] ∞ and the pressure p [Formula: see text] 0 , keeping [Formula: see text] constant, in a class of such models. This transition is driven by bond energetics and can be either continuous or discontinuous. It can be interpreted as a shape transition in which the ring polymer takes the shape, above the critical pressure, of a regular N -gon whose sides scale smoothly with pressure, while staying unfaceted below this critical pressure. Away from these limits, we argue that the transition is replaced by a sharp crossover. The area, however, scales with N(2) for all positive p in all such models, consistent with earlier scaling theories.

Simulation of a Flexible Polymer Confined between Two Parallel Surfaces

The conformational properties of a lattice self-avoiding polymer chain confined between two parallel impenetrable surfaces are simulated using Monte Carlo method. The statistical conformation size of polymer can be scaled by polymer chain length N and distance D between two surfaces. Chain is nearly free at large scaled distance Δ = D/Nνb with ν = 0.58 the scaling exponent and b = 1.42 mean bond length. Then it is squeezed along normal direction to surface at intermediate Δ, and at last it spreads out along surface at small Δ. For attractive surfaces, the chain is adsorbed at low temperature. For small D, we find an abnormal temperature dependence of chain size due to chain adsorbed to both surfaces simultaneously.