Model Of Polymer Chain Research Articles

Langevin Dynamics Study on the Driven Translocation of Polymer Chains with a Hairpin Structure.

The hairpin structure is a common and fundamental secondary structure in macromolecules. In this work, the process of the translocation of a model polymer chain with a hairpin structure is studied using Langevin dynamics simulations. The simulation results show that the dynamics of hairpin polymer translocation through a nanopore are influenced by the hairpin structure. Hairpin polymers can be classified into three categories, namely, linear-like, unsteady hairpin, and steady hairpin, according to the interaction with the stem structure. The translocation behavior of linear-like polymers is similar to that of a linear polymer chain. The time taken for the translocation of unsteady hairpin polymers is longer than that for a linear chain because it takes a long time to unfold the hairpin structure, and this time increases with stem interaction and decreases with the driving force. The translocation of steady hairpin polymers is distinct, especially under a weak driving force; the difficulty of unfolding the hairpin structure leads to a low translocation probability and a short translocation time. The translocation behavior of hairpin polymers can be explained by the theory of the free-energy landscape.

A Chain Scission-Induced Anisotropic Damage Constitutive Model for Double Network Hydrogels

This work develops a chain scission-induced anisotropic damage constitutive model for double network (DN) hydrogels while considering the damage cross effect. A free energy of a single polymer chain comprising the configuration entropy of the polymer chain and the internal energy of persistent segments is proposed to avoid the force singularity caused by chain stretch limit. Then, a distribution function of the initial stretch ratio of polymer chains in DN hydrogels is derived to consider the randomness of chain length. Based on the micro-sphere full-network model and an energy-form fracture criterion of polymer chains, the damage evolution model of polymer chains in a certain orientation is formulated by the critical initial stretch ratio of the polymer chain, showing that only polymer chains with the initial stretch ratio lower than the critical initial stretch ratio survive in a certain chain stretch. The critical initial stretch ratio is determined by the maximum chain stretch and the maximum first invariant of the deformation state by considering the damage cross effect. Using the developed damage constitutive model, we describe the stress softening, strain hardening behavior and the nonmonotonic stress–strain curve of DN hydrogels. It also reveals that the residual strain of DN hydrogels in the loading cycles is caused by the anisotropic damage of polymer chains in DN hydrogels.

Statistics of Gaussian polymer chains in harmonic applied fields.

The model of an ideal polymer chain in a harmonic applied field has broad applicability in situations involving polymer confinement and deformation due to applied stress. In this work we (1) formulate a general analytical model for a continuous Gaussian chain under a harmonic applied potential and (2) evaluate the statistical mechanics of this model given the potential, obtaining partition functions and moment generating functions (MGFs) that describe the chain configurations. Closed-form expressions for the squared radius of gyration, potential energy, partition function, and MGF for the center of mass are obtained for a general and multidimensional harmonic field. The expressions are compared with results of Monte Carlo simulations of a discrete Gaussian chain as well as results for related systems obtained from the literature. The theory derived here is used to test the applicability of the current model assumptions to relations from the literature describing polymer confinement and deformation in experiment, theory, and simulations.

Effects of chain resolution on the configurational and rheological predictions of dilute polymer solutions in flow fields with hydrodynamic interactions

In a recent study, the resolution of a polymer chain model was shown to significantly affect rheological predictions from Brownian dynamics (BD) simulations [Kumar and Dalal, “Effects of chain resolution on the configurational and rheological predictions from Brownian dynamics simulations of an isolated polymer chain in flow,” J. Non-Newtonian Fluid Mech. 315, 105017 (2023)], even in the absence of hydrodynamic interactions (HI) and excluded volume. In this study, we investigate the effects of chain resolution in the presence of HI. Toward this, we perform BD simulations of a long polymer chain, with the discretization level varying from a single Kuhn step (bead–rod model) to several tens of Kuhn-steps (bead–spring model). The chain models were subjected to flow fields of uniaxial extension (purely stretching) and steady shear (equal rates of stretching and rotation). Broadly, our results indicate an amplification of the differences observed between the differently resolved bead–rod and bead–spring models, in the presence of HI. Interestingly, all rheological predictions qualitatively fall in two groups for extensional flow, with the predictions from the bead–spring model with HI being close to those of the bead–rod model without HI. This indicates significantly reduced sensitivity of coarser bead–spring models to HI, relative to the one resolved to a single Kuhn step. However, in shear flow, the bead–spring rheological predictions fall between those of the bead–rod model with and without HI, forming a third group. This is linked to the presence of stretched and coiled states in the ensemble for shear flow. HI effects are large for the coiled states and weak for the stretched states, thereby yielding predictions that are intermediate between those for no HI and dominant HI. Thus, quite surprisingly, the quality of predictions of the bead–spring models is strongly affected by the physics of the flow field, irrespective of the parameterization.

Polymer Modeling Reveals Interplay between Physical Properties of Chromosomal DNA and the Size and Distribution of Condensin-Based Chromatin Loops.

Transient DNA loops occur throughout the genome due to thermal fluctuations of DNA and the function of SMC complex proteins such as condensin and cohesin. Transient crosslinking within and between chromosomes and loop extrusion by SMCs have profound effects on high-order chromatin organization and exhibit specificity in cell type, cell cycle stage, and cellular environment. SMC complexes anchor one end to DNA with the other extending some distance and retracting to form a loop. How cells regulate loop sizes and how loops distribute along chromatin are emerging questions. To understand loop size regulation, we employed bead-spring polymer chain models of chromatin and the activity of an SMC complex on chromatin. Our study shows that (1) the stiffness of the chromatin polymer chain, (2) the tensile stiffness of chromatin crosslinking complexes such as condensin, and (3) the strength of the internal or external tethering of chromatin chains cooperatively dictate the loop size distribution and compaction volume of induced chromatin domains. When strong DNA tethers are invoked, loop size distributions are tuned by condensin stiffness. When DNA tethers are released, loop size distributions are tuned by chromatin stiffness. In this three-way interaction, the presence and strength of tethering unexpectedly dictates chromatin conformation within a topological domain.

Toward a 3D physical model of diffusive polymer chains.

Recent studies in polymer physics have created macro-scale analogs to solute microscopic polymer chains like DNA by inducing diffusive motion on a chain of beads. These bead chains have persistence lengths of O(10) links and undergo diffusive motion under random fluctuations like vibration. We present a bead chain model within a new stochastic forcing system: an air fluidizing bed of granular media. A chain of spherical 6 mm resin beads crimped onto silk thread are buffeted randomly by the multiphase flow of grains and low density rising air "bubbles". We "thermalize" bead chains of various lengths at different fluidizing airflow rates, while X-ray imaging captures a projection of the chains' dynamics within the media. With modern 3D printing techniques, we can better represent complex polymers by geometrically varying bead connections and their relative strength, e.g., mimicking the variable stiffness between adjacent nucleotide pairs of DNA. We also develop Discrete Element Method (DEM) simulations to study the 3D motion of the bead chain, where the bead chain is represented by simulated spherical particles connected by linear and angular spring-like bonds. In experiment, we find that the velocity distributions of the beads follow exponential distributions rather than the Gaussian distributions expected from polymers in solution. Through use of the DEM simulation, we find that this difference can likely be attributed to the distributions of the forces imparted onto the chain from the fluidized bed environment. We anticipate expanding this study in the future to explore a wide range of chain composition and confinement geometry, which will provide insights into the physics of large biopolymers.

Mapping scheme as key element in coarse-graining of methacrylate-based polymers

Atomistic molecular modelling of polymeric hybrid materials becomes more relevant with continually increasing computing power. Currently, the simulations are still limited to medium size atomistic models, although polymer dynamics demand large polymer chain models and time scales. Coarse-graining of atomistic polymer models is aiming at the transfer of structure related properties from fully atomistic models to bead representations having less degrees of freedom but allowing the increase of model size and time scales. Key for a successful bead representation is the mapping scheme. Although there are a lot of meaningful approaches for polymer mapping schemes available (e. g. mapping one repeat unit to one bead), the successful development is not always a trivial task and every approach has advantages and shortcomings. Even for the rather “simple” polystyrene at least seven different and meaningful mapping scheme approaches are known (H. A. Karimi-Varzaneh, N. F. A. van der Vegt, F. Müller-Plathe, P. Carbone, ChemPhysChem2012, 13, 3428.).Here, we show the importance of the mapping scheme for a more sophisticated hybrid material substance and evaluate its quality, i. e. speedup and atomistic model representation by the coarse-grained models. The degree of coarse-graining of different mapping schemes is discussed. The coarse-grained models are evaluated according to their glass transition analysis in comparison to the full atomistic glass transition analysis. It is shown, that the choice of the mapping scheme becomes more crucial with increasing monomer complexity. Finally, the trade-off between efficiency, i. e. time-saving with speed-up during simulation vs. time-loss for coarse-grained force field parameter development, and quality of results, i. e. comparability with fully atomistic models, has to be considered carefully.

Complex Monte Carlo Light-Driven Dynamics of Monomers in Functionalized Bond Fluctuation Model Polymer Chains.

We study Monte Carlo dynamics of the monomers and center of mass of a model polymer chain functionalized with azobenzene molecules in the presence of an inhomogeneous linearly polarized laser light. The simulations use a generalized Bond Fluctuation Model. The mean squared displacements of the monomers and the center of mass are analyzed in a period of Monte Carlo time typical for a build-up of Surface Relief Grating. Approximate scaling laws for mean squared displacements are found and interpreted in terms of sub- and superdiffusive dynamics for the monomers and center of mass. A counterintuitive effect is observed, where the monomers perform subdiffusive motion but the resulting motion of the center of mass is superdiffusive. This result disparages theoretical approaches based on an assumption that the dynamics of single monomers in a chain can be characterized in terms of independent identically distributed random variables.

Solvent-Induced Negative Energetic Elasticity in a Lattice Polymer Chain.

The negative internal energetic contribution to the elastic modulus (negative energetic elasticity) has been recently observed in polymer gels. This finding challenges the conventional notion that the elastic moduli of rubberlike materials are determined mainly by entropic elasticity. However, the microscopic origin of negative energetic elasticity has not yet been clarified. Here, we consider the n-step interacting self-avoiding walk on a cubic lattice as a model of a single polymer chain (a subchain of a network in a polymer gel) in a solvent. We theoretically demonstrate the emergence of negative energetic elasticity based on an exact enumeration up to n=20 and analytic expressions for arbitrary n in special cases. Furthermore, we demonstrate that the negative energetic elasticity of this model originates from the attractive polymer-solvent interaction, which locally stiffens the chain and conversely softens the stiffness of the entire chain. This model qualitatively reproduces the temperature dependence of negative energetic elasticity observed in the polymer-gel experiments, indicating that the analysis of a single chain can explain the properties of negative energetic elasticity in polymer gels.

A statistical mechanics framework for polymer chain scission, based on the concepts of distorted bond potential and asymptotic matching

To design increasingly tough, resilient, and fatigue-resistant elastomers and hydrogels, the relationship between controllable network parameters at the molecular level (bond type, non-uniform chain length, entanglement density, etc.) to macroscopic quantities that govern damage and failure must be established. Many of the most successful constitutive models for elastomers have been rooted in statistical mechanical treatments of polymer chains. Typically, such constitutive models have used variants of the freely jointed chain model with rigid links. However, since the free energy state of a polymer chain is dominated by enthalpic bond distortion effects as the chain approaches its rupture point, bond extensibility ought to be accounted for if the model is intended to capture chain rupture. To that end, a new bond potential is supplemented to the freely jointed chain model (as derived in the uFJC framework of Buche and Silberstein (2021) and Buche et al. (2022)), which we have extended to yield a tractable, closed-form model of single chain behavior that should be amenable to continuum-level constitutive model development. Inspired by the asymptotically matched uFJC model response in both the low/intermediate chain force and high chain force regimes, a simple, quasi-polynomial bond potential energy function is derived. This bond potential exhibits harmonic behavior near the equilibrium state and anharmonic behavior for large bond stretches tending to a characteristic energy plateau (akin to the Lennard-Jones and Morse bond potentials). Using this bond potential, approximate yet highly-accurate analytical functions for bond stretch and chain force dependent upon chain stretch are established. Then, using this polymer chain model, a stochastic thermal fluctuation-driven chain rupture framework is developed. This framework is based upon a force-modified tilted bond potential that accounts for distortional bond potential energy, allowing for the derivation and subsequent calculation of the dissipated chain scission energy. The cases of rate-dependent and rate-independent scission are accounted for throughout the rupture framework. The impact of Kuhn segment number on chain rupture behavior is also investigated. The model is fit to single chain mechanical response data collected from atomic force microscopy tensile tests for validation and to glean deeper insight into the molecular physics taking place. Due to their analytical nature, this polymer chain model and the associated rupture framework can in the future be implemented in finite element models accounting for fracture and fatigue in polydisperse elastomer networks.

An energy minimization strategy based on an improved nonlinear conjugate gradient method for accelerating the charged polymer dynamics simulation.

Using a hybrid simulation method that combines a non-linear conjugate gradient (NCG) method for solving large-scale unconstrained optimization problems with a Brownian dynamics (BD) model for polymer chains, we investigate the pre-equilibrium simulation of charged polymers in different dielectric systems from an energy minimization perspective. We propose an improved NCG coefficient (βLPRPk) that satisfies a sufficient descent condition with a greater parameter under a strong Wolfe line search and converges globally for nonconvex minimization. Furthermore, preliminary numerical results show that the βLPRPk coefficient is more efficient than many existing NCG coefficients for a large number of practical test problems from our model. We further compare the performance of the improved NCG method with that of other mainstream numerical methods in energy minimization, and the simulation results suggest that the NCG method is more competitive in terms of cost-effectiveness. Importantly, we apply the geometrically optimized configuration obtained by performing the NCG method to the pre-equilibrium simulation, and the numerical results show that it increases the computational efficiency of a pure solvent and biomolecule-solution systems at most by about 32 and 70 times, respectively, with the relative energy errors being controlled below 1 × 10-2 and 4.5 × 10-3, respectively. More importantly, the final pre-equilibrium configuration of the BD simulation that performs energy minimization and the traditional BD simulation matched closely.

Lattice conformation of theta-curves accompanied with Brunnian property

A theta-curve is an embedding of the Greek letter Θ shaped graph in three-dimensional space. This is a useful physical model for polymer chains since theta-curve motifs are often present in many circular proteins with internal bridges. A Brunnian theta-curve is a nontrivial theta-curve with the property that if we remove any one among three edges, then the remaining knot can be laid in the plane without crossings. We focus on the rigidity of polymer chains with the Brunnian theta-curve shape by using the lattice stick number which is the minimal number of sticks glued end-to-end that are necessary to construct the theta-curve in the cubic lattice. The authors have already shown in a previous research that at least 15 lattice sticks are needed to construct Brunnian theta-curves. In this paper, we improve the lower bound of the lattice stick number for Brunnian theta-curves to 16.

Self-propulsion with speed and orientation fluctuation: Exact computation of moments and dynamical bistabilities in displacement.

We consider the influence of active speed fluctuations on the dynamics of a d-dimensional active Brownian particle performing a persistent stochastic motion. The speed fluctuation brings about a dynamical anisotropy even in the absence of shape anisotropy. We use the Laplace transform of the Fokker-Planck equationto obtain exact expressions for time-dependent dynamical moments. Our results agree with direct numerical simulations and show several dynamical crossovers determined by the activity, persistence, and speed fluctuation. The dynamical anisotropy leads to a subdiffusive scaling in the parallel component of displacement fluctuation at intermediate times. The kurtosis remains positive at short times determined by the speed fluctuation, crossing over to a negative minimum at intermediate times governed by the persistence before vanishing asymptotically. The probability distribution of particle displacement obtained from numerical simulations in two dimensions shows two crossovers between compact and extended trajectories via two bimodal distributions at intervening times. While the speed fluctuation dominates the first crossover, the second crossover is controlled by persistence like in the wormlike chain model of semiflexible polymers.

A topology framework for macromolecular complexes and condensates

Macromolecular assemblies such as protein complexes and protein/RNA condensates are involved in most fundamental cellular processes. The arrangement of subunits within these nano-assemblies is critical for their biological function and is determined by the topology of physical contacts within and between the subunits forming the complex. Describing the spatial arrangement of these interactions is of central importance to understand their functional and stability consequences. In this concept article, we propose a circuit topology-based formalism to define the topology of a complex consisting of linear polymeric chains with inter- and intrachain interactions. We apply our method to a system of model polymer chains as well as protein assemblies. We show that circuit topology can categorize different forms of chain assemblies. Our multi-chain circuit topology should aid analysis and predictions of mechanistic and evolutionary principles in the design of macromolecular assemblies.

Fraenkel Springs as an Efficient Approximation to Rods for Brownian Dynamics Simulations and Modeling of Polymer Chains

AbstractRecent studies indicate the importance of using bead‐rod models for polymer chains, resolving to a single Kuhn step, especially in strong flows. Earlier, researchers suggested the Fene‐Fraenkel spring as a computationally efficient approximation to the rod for Brownian dynamics (BD) simulations. This analysis reveals that the Fraenkel spring is an even better alternative. The predictions from BD simulations with stiff Fraenkel springs are nearly identical to those using Fene‐Fraenkel springs and rigid rods. Significantly, such excellent agreement is obtained when the spring lengths at every time step are updated once, without any further iterations for refinement. The computational speeds for single‐step update of the spring lengths, using Fraenkel springs, is an order of magnitude faster than the usual procedure of iterating until convergence. Even when multiple iterations are allowed within each time step, the computational time remains about 20–25% lower for longer chains with Fraenkel springs than Fene‐Fraenkel springs. The merits of using highly resolved chains, using Fraenkel springs, in preaveraged models are also highlighted. The predictions of such preaveraged models agree reasonably well with BD simulations. This is particularly advantageous since the predictions for shear flows are poor, especially at higher shear rates, for well‐known models like FENE‐P.

The Regulatory Roles of Intrinsically Disordered Linker in VRN1-DNA Phase Separation.

Biomacromolecules often form condensates to function in cells. VRN1 is a transcriptional repressor that plays a key role in plant vernalization. Containing two DNA-binding domains connected by an intrinsically disordered linker (IDL), VRN1 was shown to undergo liquid-like phase separation with DNA, and the length and charge pattern of IDL play major regulatory roles. However, the underlying mechanism remains elusive. Using a polymer chain model and lattice-based Monte-Carlo simulations, we comprehensively investigated how the IDL regulates VRN1 and DNA phase separation. Using a worm-like chain model, we showed that the IDL controls the binding affinity of VRN1 to DNA, by modulating the effective local concentration of the VRN1 DNA-binding domains. The predicted binding affinities, under different IDL lengths, were in good agreement with previously reported experimental results. Our simulation of the phase diagrams of the VRN1 variants with neutral IDLs and DNA revealed that the ability of phase separation first increased and then decreased, along with the increase in the linker length. The strongest phase separation ability was achieved when the linker length was between 40 and 80 residues long. Adding charged patches to the IDL resulted in robust phase separation that changed little with IDL length variations. Our study provides mechanism insights on how IDL regulates VRN1 and DNA phase separation, and why naturally occurring VRN1-like proteins evolve to contain the charge segregated IDL sequences, which may also shed light on the molecular mechanisms of other IDL-regulated phase separation processes in living cells.

Rheo-Dielectrics and Diffusion of Type-A Rouse Chain under Fast Shear Flow: Method of Evaluation of Non-Equilibrium Parameters for FENE, Friction-Reduction, and Brownian Force Intensity Variation

Rouse model is the basic molecular model for unentangled linear polymer chains in melts. Recent studies suggest that the Rouse parameters characterizing the unentangled melts, the spring strength κ , the segmental friction coefficient ζ, and the mean-square Brownian force intensity B, change under fast flow thereby providing significant rheological nonlinearities with those melts (Matsumiya et al., Macromolecules, 51, 9710 (2018)). This article theoretically analyzes effects of the flow-induced changes of these parameters on the dielectric and diffusion behavior of the Rouse chain having so-called type-A dipoles parallel along the chain backbone. It turned out that the dielectric behavior under steady shear is determined by non-equilibrium parameters, rκ = κ/κeq and rζ,ij = ζij/ζeq but irrelevant to rB,ij = B ij/Beq, where the subscript “eq” stands for the quantities at equilibrium and the subscripts “ij” denote the spatial direction under shear flow: i and/or j = x, y, and z for the velocity, velocity gradient, and vorticity directions, respectively. Specifically, the complex dielectric permittivity ε* is analytically expressed in terms of the parameters rκ and rζ,ij, which in turn enables us to evaluate these parameters from data of the dielectric relaxation time and intensity under steady shear (obtained in future experiments). In contrast, the advective diffusion behavior under steady shear is determined by rζ,ij and rB,ij but irrelevant to rκ. The mean-square displacement ‹Δr2CM› of the center of mass defined with respect to the known non-diffusive advection point is analytically expressed in terms of rζ,ij and rB,ij, thereby allowing us to evaluate rB,ij from the diffusion data under steady shear and the dielectrically determined rζ,ij. The method of experimental evaluation of the non-equilibrium parameters presented in this article provides us with a complementary basis for analyzing nonlinear rheological behavior of unentangled melts.

Evaluation of loop formation dynamics in a chromatin fiber during chromosome condensation

Abstract Chromatin fibers composed of DNA and proteins fold into consecutive loops to form rod-shaped chromosomes in mitosis. Although the loop growth dynamics has been investigated in several studies, its detailed processes are unclear. Here, we describe the time evolution of the loop length for thermal-driven loop growth processes as an iterative map by calculating the physical quantities involved in the processes. We quantify the energy during the chromatin loop formation by calculating the free energies of unlooped and looped chromatins using the Domb–Joyce model of a lattice polymer chain incorporating the bending elasticity for thermal-driven loop growth processes. The excluded volume interaction among loops is integrated by employing the mean-field theory. We compare the loop formation energy with the thermal energy and evaluate the growth of the loop length via thermal fluctuation. By assuming the dependence of the excluded volume parameter on the loop length, we construct an iterative map for the loop growth dynamics. The map demonstrates that the growth length of the loop for a single reaction cycle decreases with time to reach the condensin size, where the loop growth dynamics can be less stochastic and be regarded as the direct power stroke of condensin as a kind of motor protein.

A multiscale analysis framework for formation and failure of the thermoplastic interface

Formation and failure of the interface determine the quality of thermoplastic composite structures. However, accurate prediction of interface strength evolution during advanced manufacture is still challenging due to the multiscale feature of thermoplastics, which hinders the design of composites processing and analysis of structure reliability. Here, we develop a multiscale model for formation and failure of thermoplastic interface. The reptation model of polymer chains is used to describe the chain density evolution at the interface during manufacture. Then, by releasing the rigid-bond assumption in the traditional freely-jointed-chain model, the Lennard-Jones potential function is incorporated to model polymer chain scission. Finally, chain-length dispersity, individual chain scission, and chain density reduction are incorporated to model the interface failure. The model predictions are in good agreement with the experimental results of composites welding, indicating that the developed model can be used to analyze the interface strength evolution of thermoplastics in the advanced manufacturing processes.

Critical and geometric properties of magnetic polymers across the globule-coil transition.

We study a lattice model of a single magnetic polymer chain, where Ising spins are located on the sites of a lattice self-avoiding walk in d=2. We consider the regime where both conformations and magnetic degrees of freedom are dynamic, thus the Ising model is defined on a dynamic lattice and conformations generate an annealed disorder. Using Monte Carlo simulations, we characterize the globule-coil and ferromaget-to-paramagnet transitions, which occur simultaneously at a critical value of the spin-spin coupling. We argue that the transition is continuous-in contrast to d=3 where it is first order. Our results suggest that at the transition the metric exponent takes the θ-polymer value ν=4/7 but the crossover exponent ϕ≈0.7, which differs from the expected value for a θ polymer.