Multichain Systems Research Articles

Fast off-lattice Monte Carlo simulations with “soft” repulsive potentials

The basic idea of fast off-lattice Monte Carlo (FOMC) simulations is to use "soft" repulsive potentials that allow particle overlapping in continuum Monte Carlo (MC) simulations. For multichain systems, this gives much faster chain relaxation and better sampling of the configurational space than conventional molecular simulations using "hard" excluded-volume interactions that prevent particle overlapping. Such coarse-grained models are particularly suitable for the study of equilibrium properties of soft materials. Since soft potentials are commonly used in polymer field theories, it is another advantage of FOMC simulations that using the same Hamiltonian in both FOMC simulations and the theories enables quantitative comparisons between them without any parameter fitting to unambiguously reveal the consequences of approximations in the theories. Moreover, FOMC simulations can be performed with various chain models and in any statistical ensemble, and all the advanced off-lattice MC techniques proposed to date can be implemented to further improve the sampling efficiency. We have performed canonical-ensemble FOMC simulations with an isotropic soft pair potential for three systems: we first used (small-molecule) soft spheres to demonstrate the improved sampling of FOMC simulations over conventional molecular simulations; we then used single-chain simulations to show that the effects of excluded-volume interactions can be well captured by the soft repulsive potential; finally, for compressible homopolymer melts, we compared FOMC results with those under the random-phase approximation to demonstrate that FOMC simulations can be used to unambiguously and quantitatively reveal the fluctuation/correlation effects in the system. In addition, we examined in detail in our single-chain simulations the spatial discretization scheme used in all previous FOMC simulations.

Computer simulations of amorphous polymers: From quantum mechanical calculations to mesoscopic models

The development of computational methods for predicting the structure and properties of complex macromolecular systems in condensed phases, e.g. amorphous and melted states, has provided the basis of the hierarchical modeling strategies currently used in polymer science. This paper reviews some methods, many of them developed or extensively applied by the researchers in the groups of Madrid and Barcelona, that appear particularly promising. Specifically, after a brief discussion about the applicability of quantum mechanical methods to the development of reliable force-field parameters for condensed-phase studies, this work describes: (a) strategies recently reported for atomistic simulations of hetereogeneously ordered multichain denses systems, i.e. a procedure based on a random search of microstructures with minimum energy and advanced Monte Carlo methods; and (b) methodologies to study the mesoscopic properties or amorphous and melted polymer, i.e. pseudoparticle-based algorithms and the primitive path network. The examples chosen to illustrate the performance of the different methods have been focused on polyethylene to illustrate the different stages within the hierarchical approach.

Properties of polymer sandwich brushes

We designed and developed a simple model of multi-chain polymer systems. The macromolecules consisted of identical beads were restricted to a simple cubic lattice with the excluded volume interactions only (an athermal system). The polymers were confined between two parallel impenetrable walls and were grafted with one-to-one end of the walls. The properties of the model sandwich brush were studied using Monte Carlo simulations. A Metropolis-like sampling algorithm with local changes of chain's conformation was used. The interactions between the brushes were analyzed and discussed. It was found that the interaction between two brushes depend mostly on the length of the chains and weaker on the number of chains.

Linear Viscoelasticity from Molecular Dynamics Simulation of Entangled Polymers

The linear viscoelastic (LVE) spectrum is one of the primary fingerprints of polymer solutions and melts, carrying information about most relaxation processes in the system. Many single chain theories and models start with predicting the LVE spectrum to validate their assumptions. However, until now, no reliable linear stress relaxation data were available from simulations of multichain systems. In this work, we propose a new efficient way to calculate a wide variety of correlation functions and mean-square displacements during simulations without significant additional CPU cost. Using this method, we calculate stress−stress autocorrelation functions for a simple bead−spring model of polymer melt for a wide range of chain lengths, densities, temperatures, and chain stiffnesses. The obtained stress−stress autocorrelation functions were compared with the single chain slip−spring model in order to obtain entanglement related parameters, such as the plateau modulus or the molecular weight between entanglements. Then, the dependence of the plateau modulus on the packing length is discussed. We have also identified three different contributions to the stress relaxation: bond length relaxation, colloidal and polymeric. Their dependence on the density and the temperature is demonstrated for short unentangled systems without inertia.

An Exponentially$epsilon$-Convergent Control Algorithm for Chained Systems and Its Application to Automatic Parking Systems

In this brief, a simple control approach is proposed for the multichain systems based on a switching algorithm of two sets of steering laws. Global exponential convergence to any prescribed norm bound (epsi-convergence) is guaranteed and the convergence rate is explicitly given. This approach has a distinctive feature that one of the inputs (driving input) is free. This feature plays a key role in achieving realistic parking tasks. For the design of steering laws, a method based on backstepping is proposed and analyzed in detail. Further, this control algorithm is applied to the automatic parking systems. The corresponding system package is developed which is composed of control laws for the steering angle and the driving velocity, algorithms for various kinds of parking tasks under steering angle and parking space limitations. A prototype system is built to validate this package, and the system works well in experiments subject to various parking situations

Recent advances in the modeling and simulation of metallocene catalysis, sequence distribution, chain conformations, and crystallization of polymers

AbstractThis article is a review describing the latest advances in modeling and simulation of polymers. Sequence distributions in stereoblock polymers and copolymers greatly affect their chain conformations and thermal transitions. While no theory was able to fully predict the influence of chain conformations on polymer crystallization, modeling techniques have shown good reliance in simulating the conformational behavior of polymeric chains and the related physical properties. This is done by generating representative sequences, and studying the chain conformation and packing of such sequences into crystalline regions in multichain systems. Metallocene catalysts, a class of single site catalysts, also showed unprecedented performance in the polymerization of olefins, most notably their activity, copolymerization capabilities, and potential for precise control of stereostructures. These attributes are among the most important issues in the manufacture of polyolefins and olefin copolymers, and are too good to be ignored. These polymers consist of alternating atactic sequences, which are amorphous and act as elastomeric chains, and ordered isotactic or syndiotactic sequences which, if long enough, will crystallize and act as physical reinforcing crosslinks. Mesoscopic investigation utilizing computer modeling and simulation into the effects of the sequence length and sequence length distribution on the reinforcement of stereoblock and stereoregular polyolefins has also been reviewed. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 2524–2541, 2006

Mean-field theory of ordering in polymer systems with orientational-deformation interactions

On the basis of the mean-field method (single-chain approximation), expressions were derived for the free energy of a chain with dipolar orientation-deformation interactions of Gaussian subchains with a fixed mean-square length. In terms of this approach, the systems experience a second-order phase transition from the isotropic state to an ordered state. The Curie point and the dipole and quadrupole order parameters were related to the thermodynamic chain bending rigidity. The types and characteristics of transitions in anisotropic lattice models with dipolar orientational interactions were compared with those in continuous models with quadrupole interactions in terms of the mean-field approximation (Maier-Saupe theory). The short-and long-range orientational order in single-chain models (mean molecular field) and the corresponding multichain models (with local intrachain interactions of subchains) were compared. A comparison of the orientational correlation functions in single-chain and multichain models composed of flexible segments of a fixed length and rigid elements showed their qualitative equivalence as applied to the description of the statistical properties of chains in polymer systems with dipolar orientational-deformational interchain interactions, especially in the case of high chain ordering. Owing to the long-range fluctuation instability in two-dimensional multichain systems, the results of the mean-field theory are applicable only to parts of systems with finite sizes (domains).

Protein-folding landscapes in multichain systems

Computational studies of proteins have significantly improved our understanding of protein folding. These studies are normally carried out by using chains in isolation. However, in many systems of practical interest, proteins fold in the presence of other molecules. To obtain insight into folding in such situations, we compare the thermodynamics of folding for a Miyazawa-Jernigan model 64-mer in isolation to results obtained in the presence of additional chains. The melting temperature falls as the chain concentration increases. In multichain systems, free-energy landscapes for folding show an increased preference for misfolded states. Misfolding is accompanied by an increase in interprotein interactions; however, near the folding temperature, the transition from folded chains to misfolded and associated chains is entropically driven. A majority of the most probable interprotein contacts are also native contacts, suggesting that native topology plays a role in early stages of aggregation.

Phase separation in binary mixtures containing polymers: A quantitative comparison of single‐chain‐in‐mean‐field simulations and computer simulations of the corresponding multichain systems

AbstractWe discuss a phenomenological, coarse‐grained simulation scheme, single‐chain‐in‐mean‐field (SCMF) simulation, for investigating the kinetics of phase separation in dense polymer blends and mixtures of polymers and solvents. In the spirit of self‐consistent‐field calculations, we approximate the interacting multichain problem by that of a single chain in an external field, which, in turn, depends on the local densities of the components. To study the time evolution of the mixture, we perform an explicit Monte Carlo (MC) simulation of an ensemble of independent chains in the external field and periodically calculate the average densities and update the external field. Unlike dynamic self‐consistent‐field theory, these SCMF simulations do not assume that the chain conformations relax much more quickly than the density and incorporate the single‐chain dynamics explicitly rather than via an Onsager coefficient. This allows us to study systems with large spatial inhomogeneities and dynamic asymmetries. To assess the accuracy and limitations of the simulation scheme, we compare the results of SCMF simulations using a discretized Edwards Hamiltonian with computer simulations of the corresponding multichain system for (1) the early stages of spinodal decomposition of a symmetric binary polymer blend in response to a quench from χN = 0.314 to χN = 5 (where χ is the Flory–Huggins parameter and N is the number of segments), for which the growth rate of composition fluctuations is compared with MC simulations of the bond fluctuation model and alternative dynamic self‐consistent‐field calculations, and (2) the evaporation of a solvent from a low‐molecular‐weight thin polymer film, for which a comparison is made with molecular dynamics (MD) simulations of a bead‐necklace model with a monomeric solvent. In the latter case, the polymer conformations are extracted from MD simulations and modeled in the SCMF simulations by a discretized Edwards Hamiltonian augmented by a chain‐bending potential. From the MD simulations of thin polymer films in equilibrium with its vapor, phase coexistence has been determined, and the second‐ and third‐order virial coefficients in the SCMF simulations have been adjusted accordingly. Finally, MD simulations of bulk solutions of a polymer and a solvent over a range of compositions, as well as the pure solvent at various densities, have been performed to determine self‐diffusion coefficients that enter the SCMF simulations in the form of density‐dependent segmental mobilities. A comparison of the polymer and solvent profiles in a thin film as a function of time and the fraction of the solvent evaporating from a solvent‐swollen film, as obtained from MD simulations and parameterized SCMF simulations, shows satisfactory agreement for this simple mapping procedure. © 2005 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 934–958, 2005

Kinetics of Fibril Formation by Polyalanine Peptides

Ordered beta-sheet complexes, termed amyloid fibrils, are the underlying structural components of the intra- and extracellular fibrillar protein deposits that are associated with a variety of human diseases, including Alzheimer's, Parkinson's, and the prion diseases. In this work, we investigated the kinetics of fibril formation using our newly developed off-lattice intermediate resolution model, PRIME. The model is simple enough to allow the treatment of large multichain systems while maintaining a fairly realistic description of protein dynamics without built-in bias toward any conformation when used in conjunction with constant temperature discontinuous molecular dynamics, a fast alternative to conventional molecular dynamics. Simulations were performed on systems containing 48-96 model Ac-KA14K-NH2 peptides. We found that fibril formation for polyalanines incorporate features that are characteristic of three models, the templated assembly, nucleated polymerization, and nucleated conformational conversion models, but that none of them gave a completely satisfactory description of the simulation kinetics. Fibril formation was nucleation-dependent, occurring after a lag time that decreased with increasing peptide concentration and increased with increasing temperature. Fibril formation appeared to be a conformational conversion process in which small amorphous aggregates --> beta-sheets --> ordered nucleus --> subsequent rapid growth of a small stable fibril or protofilament. Fibril growth in our simulations involved both beta-sheet elongation, in which the fibril grew by adding individual peptides to the end of each beta-sheet, and lateral addition, in which the fibril grew by adding already formed beta-sheets to its side. The initial rate of fibril formation increased with increasing concentration and decreased with increasing temperature.

Phase Diagrams Describing Fibrillization by Polyalanine Peptides

Amyloid fibrils are the structural components underlying the intra- and extracellular protein deposits that are associated with a variety of human diseases, including Alzheimer’s, Parkinson’s, and the prion diseases. In this work, we examine the thermodynamics of fibril formation using our newly-developed off-lattice intermediate-resolution protein model, PRIME. The model is simple enough to allow the treatment of large multichain systems while maintaining a fairly realistic description of protein dynamics when used in conjunction with constant-temperature discontinuous molecular dynamics, a fast alternative to conventional molecular dynamics. We conduct equilibrium simulations on systems containing 96 Ac-KA 14K-NH 2 peptides over a wide range of temperatures and peptide concentrations using the replica-exchange method. Based on measured values of the heat capacity, radius of gyration, and percentage of peptides that form the various structures, a phase diagram in the temperature-concentration plane is constructed delineating the regions where each structure is stable. There are four distinct single-phase regions: α-helices, fibrils, nonfibrillar β-sheets, and random coils; and four two-phase regions: random coils/nonfibrillar β-sheets, random coils/fibrils, fibrils/nonfibrillar β-sheets, and α-helices/nonfibrillar β-sheets. The α-helical region is at low temperature and low concentration. The nonfibrillar β-sheet region is at intermediate temperatures and low concentrations and expands to higher temperatures as concentration is increased. The fibril region occurs at intermediate temperatures and intermediate concentrations and expands to lower as the peptide concentration is increased. The random-coil region is at high temperatures and all concentrations; this region shifts to higher temperatures as the concentration is increased.

HA (Hydrophobic/Amphiphilic) Copolymer Model: Coil−Globule Transition versus Aggregation

For simulating hydrophobic−amphiphilic (HA) copolymers, we have developed a “side-chain” HA model in which hydrophilic (P) interaction sites are attached to hydrophobic (H) main chain, thereby forming amphiphilic (A) monomer units, each with dualistic (hydrophobic/hydrophilic) properties. Using this coarse-grained model, we performed molecular dynamics simulations of the hydrophobically driven self-assembly in a selective solvent, for both single-chain and multichain systems. The focus is on the regime in which H and P interaction sites are strongly segregated. Single-chain simulations are performed for copolymers with the same HA composition but with different distribution of H and A monomer units along the hydrophobic backbone, including regular copolymers comprising H and A units in alternating sequence, (HA)x, regular multiblock copolymers (HLAL)x composed of H and A blocks of equal lengths L = 3, and quasi-random proteinlike copolymers having quenched primary structure. In a solvent selectively poor for H sites, the proteinlike polyamphiphiles can readily adopt spherical-shaped compact conformations with the hydrophobic chain sections clustered at the globular core and the hydrophilic groups forming the envelope of this core and buffering it from solvent. Because of the fact that these globules are size- and shape-persistent objects, they maintain their morphological integrity even in rather concentrated solutions where no large-scale aggregation is observed. Moreover, we find that the population of aggregates generally decreases with worsening solvent quality. The compact conformations of long regular copolymers tend to be strongly elongated in one direction.

The competition between protein folding and aggregation: Off-lattice minimalist model studies

Protein aggregation has been associated with a number of human diseases, and is a serious problem in the manufacture of recombinant proteins. Of particular interest to the biotechnology industry is deleterious aggregation that occurs during the refolding of proteins from inclusion bodies. As a complement to experimental efforts, computer simulations of multi-chain systems have emerged as a powerful tool to investigate the competition between folding and aggregation. Here we report results from Langevin dynamics simulations of minimalist model proteins. Order parameters are developed to follow both folding and aggregation. By mapping natural units to real units, the simulations are shown to be carried out under experimentally relevant conditions. Data pertaining to the contacts formed during the association process show that multiple mechanisms for aggregation exist, but certain pathways are statistically preferred. Kinetic data show that there are multiple time scales for aggregation, although most association events take place at times much shorter than those required for folding. Last, we discuss results presented here as a basis for future work aimed at rational design of mutations to reduce aggregation propensity, as well as for development of small-molecular weight refolding enhancers.

Structure and properties of poly(methyl methacrylate) particles prepared by a modified microemulsion polymerization

AbstractNanoscale poly(methyl methacrylate) (PMMA) particles were prepared by modified microemulsion polymerization. Different from particles made by traditional microemulsion polymerization, the particles prepared by modified microemulsion polymerization were multichain systems. PMMA samples, whether prepared by the traditional procedure or the modified procedure, had glass‐transition temperatures (Tg's) greater than 120 °C and were rich in syndiotactic content (55–61% rr). After the samples were dissolved in CHCl3, there were decreases in the Tg values for the polymers prepared by the traditional procedure and those prepared by the modified process. However, a more evident Tg decrease was observed in the former than in the latter; still, for both, Tg was greater than 120 °C. Polarizing optical microscopy and wide‐angle X‐ray diffraction indicated that some ordered regions formed in the particles prepared by modified microemulsion polymerization. The addition of a chain‐transfer agent resulted in a decrease in both the syndiotacticity and Tg through decreasing polymer molecular weight. © 2003 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 42: 733–741, 2004

Intramolecular Nucleation Model for Polymer Crystallization

We report a numerical study of the free energy barrier for crystallization and melting of a single homopolymer chain. The simulations show that the free energy barrier separating the crystalline and molten states at the same free energy level strongly depends on the chain length. However, at a fixed temperature the barrier for single-chain crystallization is independent of chain length. This observation is in agreement with recent experiments on multichain bulk-polymer systems and can be understood theoretically if we assume that the primary nucleation of polymer crystals is determined by intramolecular nucleation. If we further assume that the subsequent growth of polymer crystals is controlled by two-dimensional intramolecular nucleation on the growth front, we can even account for the experimentally observed molecular segregation during crystal growth as well as the chain-length independence of the free energy barrier for secondary nucleation.

Molecular Dynamics Simulations of Polyelectrolyte Solutions: Osmotic Coefficient and Counterion Condensation

Osmotic coefficients and counterion distribution functions of rodlike and flexible polyelectrolyte chains have been studied using molecular dynamics simulations of multichain systems with explicit counterions in salt-free solutions. Using the counterion density profile, we have verified different regimes in the phase diagram of rodlike and flexible chains predicted by the two-zone model of Deshkovski et al. The agreement between our simulation results and predictions of the two-zone model is reasonably good for weakly charged rodlike chains. However, for flexible chains our results for the distance dependence of the counterion density profile in different regions of the phase diagram are only in qualitative agreement with the predictions of the two-zone model. The osmotic coefficient changes nonmonotonically with polymer concentration, in agreement with predictions of the two-zone model. The osmotic coefficient decreases with increasing polymer concentration in dilute solutions of both rodlike and flexible polyelectrolytes. In semidilute solutions of flexible chains the osmotic coefficient is an increasing function of polymer concentration. We have found that position of the minimum in the osmotic coefficient is close to the overlap concentration.

Effect of secondary structure on protein aggregation: A replica exchange simulation study

The ability to control or reverse protein aggregation is vital to the production and formulation of therapeutic proteins and may be the key to the prevention of a number of neurodegenerative diseases. In recent years, laboratory studies of the phenomenon have been accompanied by a growing number of computational treatments aimed at elucidating the molecular mechanisms of aggregation. The present article is a continuation of our simulation studies of coarse-grained model oligopeptides that mimic aggregating proteins. The potential function of a multichain system is expressed in terms of a generalized Go model for a set of sequences with varying contents of secondary-structural motifs akin to α-helices and β-sheets. Conformational evolution is considered by conventional Monte Carlo simulation, and by a variation of the Replica Monte Carlo technique that facilitates barrier-crossing in glasslike aggregated systems. The foldability and aggregation propensity are monitored as functions of the extent of different secondary structures and the length of the chains. Our results indicate that an increased proportion of sheetlike structures facilitates folding of isolated chains, but strongly favors the formation of misfolded aggregates in multichain systems, in agreement with experimental observations. This behavior is interpreted in terms of cooperativity effects associated with the formation of multiple residue–residue bonds involving adjacent monomers in interacting segments, which enhance both intramolecular binding and interprotein association.

Off-lattice Monte Carlo methods for coarse-grained models of polymeric materials and selected applications

This brief review presents a general introduction to the Monte Carlo simulation of coarse-grained models for flexible macromolecules under various conditions: from the solution to the melt, in the bulk as well as in geometries confined by walls etc. In the first part, the general concepts for the construction of such coarse-grained models for the simulation of cooperative phenomena in multi-chain systems and `technical' aspects of Monte Carlo simulations are discussed, as well as the limitations that this methodology still has to face. In the second part, typical applications are briefly reviewed, including the dynamics of chains adsorbed on surfaces or confined between plates, the scattering function from semidilute polymer solutions, and the formation of micelles from solutions of short block copolymers in the bulk and at the surface. Finally, a brief comparison to related but somewhat different approaches (Monte Carlo simulation of lattice models, Molecular Dynamics of bead-spring models) is given.

Integrability and exact solution of correlated hopping multi-chain electron systems

Exact quantum integrability is established for a class of multi-chain electron models with correlated hopping and spin models with interchain interactions, by constructing the related Lax operators and R-matrices through twisting and gauge transformations. Exact solution of the eigenvalue problem for commuting conserved quantities of such systems is achieved through algebraic Bethe ansatz, on the examples of Hubbard and t–J models with correlated hopping. Our systematic construction identifies the integrable subclass of such known solvable models and also generates new systems including the generalized t–J models. At the same time it makes proper correction to a well known model and resolves recent controversies regarding the equivalence and solvability of some known models.

Preparation of bulk melt chain configurations of polycyclic polymers

The configurations of oligomers of polyimide and polyetherketone polycyclic polymers in the melt are predicted by a new hybrid pivot Monte Carlo (PMC)/molecular dynamics (MD) single-chain sampling technique restricted to a limited number of near-neighbor interactions. These are then compared to configurations obtained for the same models by running MD simulations on the corresponding multichain systems in the bulk melt. A new phantom-atom technique is introduced which avoids interlocking rings during construction of the bulk melt samples. Similar to earlier work carried out on polyethylene, polyvinylchloride and uncharged polyethylene oxide, both theoretical and bulk melt sampled conformational and configurational properties are found to be in very good agreement. This confirms that the new hybrid PMC/MD sampling is a promising and cost-effective technique for preparing polymer samples prior to subsequent MD simulations of the bulk amorphous phase.