Bead-spring Model Research Articles

Simulating Dynamic Chromosome Compaction: Methods for Bridging In Silico to In Vivo.

The application of polymer models to chromosome structure and dynamics is a powerful approach for dissecting functional properties of the chromosome. The models are based on well-established bead-spring models of polymers and are distinct from molecular dynamics studies used in structural biology. In this work, we outline a polymer dynamics model that simulates budding yeast chromatin fibers in a viscous environment inside the nucleus using DataTank as a user interface for the C++ simulation. We highlight features for creating the nucleolus, a dynamic region of chromatin with protein-mediated, transient chromosomal cross-links, providing a predictive, stochastic polymer-physics model for versatile analyses of chromosome spatiotemporal organization. DataTank provides real-time visualization and data analytics methods during simulation. The simulation pipeline provides insights into the entangled chromosome milieu in the nucleus and creates simulated chromosome data, both structural and dynamic, that can be directly compared to experimental observations of live cells in interphase and mitosis.

Brownian simulations for tetra-gel-type phantom networks composed of prepolymers with bidisperse arm length.

We studied the effect of arm length contrast of prepolymers on the mechanical properties of tetra-branched networks via Brownian dynamics simulations. We employed a bead-spring model without the excluded volume interactions, and we did not consider the solvent explicitly. Each examined 4-arm star branch prepolymer has uneven arm lengths to attain two-against-two (2a2) or one-against-three (1a3) configurations. The arm length contrast was varied from 38-2 to 20-20 for 2a2, and from 5-25 to 65-5 for 1a3, with the fixed total bead number of 81, including the single bead located at the branch point for prepolymers. We distributed 400 molecules in the simulation box with periodic boundary conditions, and the bead number density was fixed at 4. We created polymer networks by cross-end-coupling of equilibrated tetra-branched prepolymers. To mimic the experiments of tetra gels, we discriminated the molecules into two types and allowed the reaction only between different types of molecules at their end beads. The final conversion ratio was more than 99%, at which unreacted dangling ends are negligible. We found that the fraction of double linkage, in which two of the four arms connect a pair of branch points, increases from 3% to 15% by increasing the arm length contrast. We stretched the resultant tetra-type networks to obtain the ratio of mechanically effective strands. We found that the ratio is 96% for the monodisperse system, decreasing to 90% for high arm length contrast. We introduced bond scission according to the bond stretching to observe the network fracture under sufficiently slow elongation. The fracture behavior was not correlated with the fraction of double linkage because the scission occurs at single linkages.

Poisson’s ratio of two-dimensional hexagonal materials under finite strains

Herein, we developed a bead-spring model to predict the Poisson’s ratios of two-dimensional hexagonal materials under finite strains. The predicted strain-dependent Poisson’s ratios were supported by simulations. The bounds on the Poisson’s ratios of two-dimensional hexagonal materials under finite strains (-1/3, +∞) were established. The underlying mechanism for the strain-dependent Poisson’s ratios was uncovered as the interplay between bond stretching and angle bending during the stretching. These findings indicate that, even for the simplified material system under finite strains, there is no upper bound on the Poisson’s ratios, and that there is plenty of room for strain-engineering the Poisson’s ratios.

Role of chain crossing prohibition on chain penetration in ring-linear blends through dissipative particle dynamics simulations

In this study, we investigated the topological effect caused by chain crossing prohibition in ring-linear blends through dissipative particle dynamics (DPD) simulations. Multipoint segmental repulsive potential (MP-SRP) was used to ensure thermodynamic consistency between the systems that permitted and prohibited chain crossing in DPD polymer simulations. We successfully extracted the topological effect of ring polymers by observing the distribution of penetration in ring–linear blends and the radius of gyration of rings. The topological effect prohibits ring–ring crossing, resulting in a small radius of gyration of the rings. In the DPD polymer system with MP-SRP, the minimum size (number of particles) of a ring with penetration was found to be 30, which is less than half of the value of 80 observed in previous work using the Kremer–Grest bead-spring model.

FRET Measurement of Polymer Response under Shear.

Polymer solutions under shear flow are often observed in manufacturing processes. Classically, polymer behavior is represented by Kuhn’s bead-spring model, in which only the elongation of polymer chains is assumed. In recent years, the compression of polymer chains under shear flow has been reported. In this study, we investigated the behavior of polymer chains dissolved in various concentrations under shear flow. We measured the time variation of the fluorescence intensity emitted from a FRET (fluorescence resonance energy transfer) polymer, which enabled us to study the change in the distance between both ends of a polymer chain. The polymer chains appeared to stretch and compress depending on the concentration of the polymer solution. The results showed that the deformation of polymer chains was different from the classical theory. The FRET measurement is a promising diagnostic method for understanding the dynamics of polymer chains.

Overlap Concentration in Salt-Free Polyelectrolyte Solutions

For strongly charged polyelectrolytes in salt-free solutions, we use molecular dynamics simulations of a coarse-grained bead-spring model to calculate overlap concentrations c* and chain structure ...

Observation of elastic heterogeneity and phase evolution in 2D layered perovskites using coherent acoustic phonons

Abstract Two-dimensional (2D) organic–inorganic perovskites have shown interesting optical properties due to the natural quantum-well structures. The repetition of soft organic and hard inorganic intercalations also renders 2D perovskites rich phonon dynamics. Here, we investigated the coherent acoustic phonons in (PEA)2PbI4 perovskite films by time-resolved Brillouin spectroscopy. The coherent acoustic phonons were launched indirectly in perovskite films by exciting Au nanoplates which were used as optoacoustic transducers. A longitudinal sound velocity ν = 1937 ± 31 m/s, and an elastic modulus E = 9.84 GPa along the cross-plane direction of perovskites were obtained from analysis of the Brillouin oscillation frequency. Following a bead-spring model, we calculated a spring constant k ≈ 1.709 N m−1 for PEA cations which is comparably small for perovskites. We also demonstrated that coherent acoustic phonons are sensitive to differentiate structural heterogeneity and monitor dynamic phase evolution in perovskite films. Domains of PbI2-rich and PbI2-poor phases were identified. Under light stimulus, PbI2-poor phases were gradually disappearing and PbI2-rich phases became crystallized. The observations of structural and elastic heterogeneity and dynamic phase evolution using coherent acoustic phonons provide a toolbox for submicroscale elastic characterization of perovskites.

Effects of chain length on Rouse modes and non-Gaussianity in linear and ring polymer melts.

The dynamics of ring polymer melts are studied via molecular dynamics simulations of the Kremer-Grest bead-spring model. Rouse mode analysis is performed in comparison with linear polymers by changing the chain length. Rouse-like behavior is observed in ring polymers by quantifying the chain length dependence of the Rouse relaxation time, whereas a crossover from Rouse to reptation behavior is observed in linear polymers. Furthermore, the non-Gaussian parameters of the monomer bead displacement and chain center-of-mass displacement are analyzed. It is found that the non-Gaussianity of ring polymers is remarkably suppressed with slight growth for the center-of-mass dynamics at long chain length, which is in contrast to the growth in linear polymers for both the monomer bead and center-of-mass dynamics.

Emergence of quasiequilibrium state and energy distribution for the beads-spring molecule interacting with a solvent.

We study the energy distribution during the emergence of a quasiequilibrium (QE) state in the course of relaxation to equipartition in slow-fast Hamiltonian systems. A bead-spring model where beads (masses) are connected by springs is considered. The QE lasts for a long time because the energy exchange between the high-frequency vibrational and other motions is prevented when springs in the molecule become stiff. We numerically calculated the time-averaged kinetic energy and found that the kinetic energy of the solvent particles was always higher than that of the bead in a molecule. This is explained by adopting the equipartition theorem in QE, and it agrees well with the numerical results. The energy difference can help determine how far the system is from achieving equilibrium, and it can be used as an indicator of the number of frozen or inactive degrees existing in the molecule.

Friction on incommensurate substrates: Role of anharmonicity and defects.

We present molecular dynamics simulations of one- and two-dimensional bead-spring models sliding on incommensurate substrates after an initial kick, in the case where the coupling to the underlying substrate is weak, i.e., energy can dissipate only into the internal degrees of freedom of the sliding object, but not into the substrate below. We investigate how sliding friction is affected by structural defects and interaction anharmonicity. In their absence, we confirm earlier findings, namely, that at special resonance sliding velocities, friction is maximal. When sliding off-resonance, partially thermalized states are possible, whereby only a small number of vibrational modes becomes excited, but whose kinetic energies are already Maxwell-Boltzmann distributed. Anharmonicity and defects typically destroy partial thermalization and instead lead to full thermalization, implying much higher friction. For sliders with periodic boundaries, thermalization begins with vibrational modes whose spatial modulation is compatible with the incommensurate lattice. For a disk-shaped slider, modes corresponding to modulations compatible with the slider radius are initially the most dominant. By tuning the mechanical properties of the slider's edge, this effect can be controlled, resulting in significant changes in the sliding distance covered.

Transient Electric Birefringence of Linear and Circular DNA: A Comparison of Kinetic Theory Predictions.

The use of monodisperse DNA and restriction enzyme modification allows the preparation of model samples of polymer rings and linear chains with a precision that is not possible by conventional synthetic routes. These samples allow direct comparisons of the predictions of polymer kinetic theories. Transient electric birefringence allows rapid imposition or forces (∼100 ns) and monitoring of conformational changes (∼0.7 μs). We determined the relaxation spectra of once-cut linear ΦX-174 (5386 bp), twice-cut linear ΦX-174 (2693 bp), and relaxed ring (5386 bp) ΦX-174 with transient electric birefringence. This allows comparison of the relaxation dynamics of a linear chain and a polymer loop with exactly the same contour length. The relaxed loop and the twice-cut DNA had the longest relaxation times of 1039 and 1076 μs, respectively. The loop was measured both in the native supercoiled state and in the relaxed state. We found the birefringence decayed in agreement with bead-spring (Rouse/Zimm) models for polymer dynamics determined by both a model-independent average relaxation time and deconvolution of the decay into a sum of exponentials. Deconvolution was performed by CONTIN and by the Padé-Laplace method. For both the relaxed ring and the linear fragments, we found the longer time constants τ1 and τ2 were discrete and had a spacing that was in agreement with the Rouse model without hydrodynamic interaction (τ1/τ2 = 0.25), which would be expected for a chain with 20 statistical segments. The excitation of higher relaxation modes could be observed as the electric field pulse length increased.

Geometrical nonlinear elasticity of axon under tension: A coarse-grained computational study

Axon bundles cross-linked by microtubule (MT) associate proteins and bounded by a shell skeleton are critical for normal function of neurons. Understanding effects of the complexly geometrical parameters on their mechanical properties can help gain a biomechanical perspective on the neurological functions of axons and thus brain disorders caused by the structural failure of axons. Here, the tensile mechanical properties of MT bundles cross-linked by tau proteins are investigated by systematically tuning MT length, axonal cross-section radius, and tau protein spacing in a bead-spring coarse-grained model. Our results indicate that the stress-strain curves of axons can be divided into two regimes, a nonlinear elastic regime dominated by rigid-body like inter-MT sliding, and a linear elastic regime dominated by affine deformation of both tau proteins and MTs. From the energetic analyses, first, the tau proteins dominate the mechanical performance of axons under tension. In the nonlinear regime, tau proteins undergo a rigid-body like rotating motion rather than elongating, whereas in the nonlinear elastic regime, tau proteins undergo a flexible elongating deformation along the MT axis. Second, as the average spacing between adjacent tau proteins along the MT axial direction increases from 25 to 125 nm, the Young’s modulus of axon experiences a linear decrease whereas with the average space varying from 125 to 175 nm, and later reaches a plateau value with a stable fluctuation. Third, the increment of the cross-section radius of the MT bundle leads to a decrease in Young’s modulus of axon, which is possibly attributed to the decrease in MT numbers per cross section. Overall, our research findings offer a new perspective into understanding the effects of geometrical parameters on the mechanics of MT bundles as well as serving as a theoretical basis for the development of artificial MT complexes potentially toward medical applications.

Increasing the efficiency and maneuverability of one-hinge swimmer

Understanding the dynamics of micro-organisms will help in developing artificial swimmers for applications like drug delivery. In the present study, a two-dimensional one-hinge swimmer resembling a scallop in Newtonian fluid is explored. To model the one-hinge swimmer, we use bead-spring model and the fluid is simulated using multi-particle collision dynamics with Anderson thermostat. We consider a non-uniform distribution of the bending rigidity along the arms of the swimmer, where we reduce the bending rigidity progressively from the hinge to the end of the arms. The non-uniform arms show higher swimming speed for the same average bending rigidity, thereby enhancing the efficiency of the swimmer. It was observed that the bending rigidity variation along the arm of the swimmer following a geometric sequence was more efficient than linear or quadratic for the same average bending rigidity. We also study the maneuverability of the one-hinge swimmer having asymmetrical bending rigidity for the arms, thereby the swimmer undergoes curved path. We find that depending upon the stiffness of the arm, the swimmer undergoes clockwise or anticlockwise rotation. We also find that the angular and transnational velocities of the swimmer are maximum at approximately the same sperm number ∼1.8. The angular velocity of the swimmer scaled linearly with the amplitude of actuation as predicted by resistive force theory. Finally, we show that in the case of a two-dimensional one-hinge swimmer angular velocity, curvature and the direction of rotation can be controlled by just changing the relative bending rigidity of the arms.

A constitutive equation for Rouse model modified for variations of spring stiffness, bead friction, and Brownian force intensity under flow

We derived a constitutive equation for the Rouse model (the most frequently utilized bead-spring model) with its spring constant κ, bead friction coefficient ζ, and the (squared) Brownian force intensity B being allowed to change under flow. Specifically, we modified the Langevin equation of the original Rouse model by introducing time (t)-dependent κ, ζ, and B (of arbitrary t dependence), which corresponded to the decoupling and preaveraging approximations often made in bead-spring models. From this modified Langevin equation, we calculated time evolution of second-moment averages of the Rouse eigenmode amplitudes and further converted this evolution into a constitutive equation. It turned out that the equation has a functional form, σ(t)=∫−∞tdt′{κ(t)/κ(t′)}M(t,t′)C−1(t,t′), where σ(t) and C−1(t,t′) are the stress and Finger strain tensors, and M(t,t′) is the memory function depending on κ(t′), ζ(t′), and B(t′) defined under flow. This equation, serving as a basis for analysis of nonlinear rheological behavior of unentangled melts, reproduces previous theoretical results under specific conditions, the Lodge–Wu constitutive equation for the case of t-independent κ, ζ, and B [A. S. Lodge and Y. Wu, “Constitutive equations for polymer solutions derived from the bead/spring model of Rouse and Zimm,” Rheol. Acta 10, 539 (1971)], the finite extensible nonlinear elastic (FENE)-Peterlin mean-Rouse formulation for the case of t-dependent changes of the only κ reported by Wedgewood and co-workers [L. E. Wedgewood et al., “A finitely extensible bead-spring chain model for dilute polymer solutions,” J. Non-Newtonian Fluid Mech. 40, 119 (1991)], and analytical expression of steady state properties for arbitrary κ(t), ζ(t), and B(t) reported by ourselves [H. Watanabe et al., “Revisiting nonlinear flow behavior of Rouse chain: Roles of FENE, friction reduction, and Brownian force intensity variation,” Macromolecules 54, 3700 (2021)]. It is to be added that a constitutive equation reported by Narimissa and Wagner [E. Narimissa and M. H. Wagner, “Modeling nonlinear rheology of unentangled polymer melts based on a single integral constitutive equation,” J. Rheol. 64, 129 (2020)] has a significantly different functional form and cannot be derived from the Rouse model with any simple modification of the Rouse–Langevin equation.

Knots are Generic Stable Phases in Semiflexible Polymers

Semiflexible polymer models are widely used as a paradigm to understand structural phases in biomolecules including folding of proteins. Since stable knots are not so common in real proteins, the existence of stable knots in semiflexible polymers has not been explored much. Here, via extensive replica exchange Monte Carlo simulation we investigate the same for a bead-stick and a bead-spring homopolymer model that covers the whole range from flexible to stiff. We establish the fact that the presence of stable knotted phases in the phase diagram is dependent on the ratio $r_b/r_{\rm{min}}$ where $r_b$ is the equilibrium bond length and $r_{\rm{min}}$ is the distance for the strongest nonbonded contacts. Our results provide evidence for both models that if the ratio $r_b/r_{\rm{min}}$ is outside a small window around unity then depending on the bending stiffness one always encounters stable knotted phases along with the usual frozen and bent-like structures at low temperatures. These findings prompt us to conclude that knots are generic stable phases in semiflexible polymers.

Event-Chain Monte-Carlo Simulations of Dense Soft Matter Systems

We discuss the rejection-free event-chain Monte-Carlo algorithm and several applications to dense soft matter systems. Event-chain Monte-Carlo is an alternative to standard local Markov-chain Monte-Carlo schemes, which are based on detailed balance, for example the well-known Metropolis-Hastings algorithm. Event-chain Monte-Carlo is a Markov chain Monte-Carlo scheme that uses so-called lifting moves to achieve global balance without rejections (maximal global balance). It has been originally developed for hard sphere systems but is applicable to many soft matter systems and particularly suited for dense soft matter systems with hard core interactions, where it gives significant performance gains compared to a local Monte-Carlo simulation. The algorithm can be generalized to deal with soft interactions and with three-particle interactions, as they naturally arise, for example, in bead-spring models of polymers with bending rigidity. We present results for polymer melts, where the event-chain algorithm can be used for an efficient initialization. We then move on to large systems of semiflexible polymers that form bundles by attractive interactions and can serve as model systems for actin filaments in the cytoskeleton. The event chain algorithm shows that these systems form networks of bundles which coarsen similar to a foam. Finally, we present results on liquid crystal systems, where the event-chain algorithm can equilibrate large systems containing additional colloidal disks very efficiently, which reveals the parallel chaining of disks.

Effects of interaction strength of associating groups on linear and star polymer dynamics.

A small number of associating groups incorporated onto a polymer backbone have dramatic effects on the mobility and viscoelastic response of the macromolecules in melts. These associating groups assemble, driving the formation of clusters, whose lifetime affects the properties of the polymers. Here, we probe the effects of the interaction strength on the structure and dynamics of two topologies, linear and star polymer melts, and further investigate blends of associative and non-associating polymers using molecular dynamics simulations. Polymer chains of approximately one entanglement length are described by a bead-spring model, and the associating groups are incorporated in the form of interacting beads with an interaction strength between them that is varied from 1 to 20 kBT. We find that, for all melts and blends, interaction of a few kBT between the associating groups drives cluster formation, where the size of the clusters increases with increasing interaction strength. These clusters act as physical crosslinkers, which slow the chain mobility. Blends of chains with and without associating groups macroscopically phase separate for interaction strength between the associating groups of a few kBT and above. For weakly interacting associating groups, the static structure function S(q) is well fit by functional form predicted by the random phase approximation where a clear deviation occurs as phase segregation takes place, providing a quantitative assessment of phase segregation.

Molecular Dynamics Simulations of Ion-Containing Polymers Using Generic Coarse-Grained Models

Molecular dynamics simulations with generic bead–spring models have been instrumental in revealing the molecular-scale behavior that underlies structure–property relationships of various types of p...

Multiscale modeling framework to predict the effective stiffness of a crystalline-matrix nanocomposite

Recently, multiscale modeling frameworks combining micromechanics-based homogenization methods and atomistic simulations have been widely applied to predict the effective stiffness of particulate-reinforced composites. Although most studies demonstrated that theoretical predictions incorporating interfacial damage are necessary to explain atomistic simulation results, the microscopic origin of the interfacial damage has not been systematically analyzed in terms of interatomic potential and interfacial structure. In this study, first, we conduct a series of particle simulations of two fictitious model crystalline composites with coherent interfaces: one has a two-dimensional triangular structure described by a bead-spring model and the other has a face-centered cubic structure described by the artificial Lennard-Jones potential. By comparing the simulation results with micromechanics theory, we obtain the interfacial bonding (damage) parameter used in the homogenization method in terms of parameters at the atomistic level. Second, we study the effects of the interfacial structures (coherent/incoherent) because of lattice or crystallographic orientation mismatch on the effective properties of composites. We obtain the elastic stiffness of Si(matrix)-Ge(nanoparticle) nanocomposites with different interfacial structures (coherent/incoherent structures) using atomistic simulations and observe that nanoparticle-size-dependency occurs only for the composite with incoherent interfaces. We propose a homogenization scheme considering the pre-stress (or residual stress) and interfacial imperfection, and explain the results from Si-Ge nanocomposite simulations.

Effect of chain-penetration on ring shape for mixtures of rings and linear polymers

Coarse-grained molecular dynamics simulations of Kremer–Grest-type bead-spring models were performed on mixtures of ring and linear polymers to investigate the effects of chain-penetration on polymer ring shape. We found that the number of linear chains penetrating through a ring increased with ring size, while penetration was inhibited for smaller rings composed of less than 80 beads. From the eigenvalue analyses of the gyration tensor of each ring, the ring shape was characterized by asphericity and prolateness. The number of chains penetrating through a ring (nP) was determined from the Gauss linking numbers of all ring and linear chain pairs. The radius of gyration of the rings were dependent on the nP value, suggesting that the linear chains expanded the ring upon penetration. The obtained results provide important insights in ring-linear blends for theoretical research as well as simulations and experiments especially for rheological measurements.