Dynamics of star branched polymers in a matrix of linear chains — a Monte Carlo study

A simple cubic lattice model of the melt of 3-arm star-branched polymers of various length dissolved in a matrix of long linear chains (n₁ = 800 beads) is studied using a dynamic Monte Carlo method. The total polymer volume fraction is equal to 0,5, while the volume fraction of the star polymers is about ten times smaller. The static and dynamic properties of these systems are compared with the corresponding model systems of isolated star-branched polymers and with the melt of linear chains. It has been found that the number of dynamic entanglements for the star polymers with arm length up to 400 segments is too small for the onset of the arm retraction mechanism of polymer relaxation. In this regime dynamics of star-branched polymers is close to the dynamics of linear polymers at corresponding concentration and with equivalent chain length. The entanglement length for star polymers appears to be somewhat larger compared with linear chains.     

Monte Carlo simulation of tetrahedral chains, 5. The pivot-algorithm for non-athermal linear and star-branched polymers

By use of the pivot algorithm, star-branched chains with F = 3–12 arms of length n, nF = 480, and linear chains (F = 2) are generated on a tetrahedral lattice. In order to simulate different qualities of the solvent, specific short-range interactions are taken into account. Whereas in athermal systems a new configuration — which is obtained by rotating that part of an arm which contains the chain end around a randomly selected bond by ± 120° — is accepted if it is self-avoiding, for non-athermal systems the Metropolis-Rosenbluth criterion with respect to energy must be satisfied in addition. Calculating the energy of the configuration, nearest-neighbour interactions (each contact contributes an energy Φ·kT, Φ < 0 characterizing endothermal solutions and Φ > 0 exothermal ones) are considered only; no energy is introduced to distinguish between trans and gauche bonds. A rather quick response of chain properties to the variation of thermodynamic conditions is demonstrated. The average acceptance fraction f̄ has its maximum value for athermal conditions, slightly decreases for exothermal conditions and strongly decreases with increasingly negative Φ-values. However, for Φ = −0,5 — representing a solution near theta-conditions — still 25% of attempted moves are accepted for all F-values examined. The dependence of global properties (characterized by the mean-square radius of gyration) on Φ and F is in full accordance with most Monte Carlo results.     

Motion of star‐branched chains in polymer networks: a Monte Carlo study

A simple model of branched polymers in confined space is developed. Star‐branched polymer molecules are built on a simple cubic lattice with excluded volume and no attractive interactions (good solvent conditions). A single star molecule is trapped in a network of linear polymer chains of restricted mobility. The simulations are carried out using the classical Metropolis algorithm. Static and dynamic properties of the star‐branched polymer are determined using various networks. The dependence of the longest relaxation time and the self‐diffusion coefficient on chain length and network properties are discussed and the proper scaling laws formulated. The possible mechanism of motion is discussed. The differences between the motion of star‐branched polymers in such a network are compared with the cases of a dense matrix of linear chains and regular rod‐like obstacles.     

Histogram Monte Carlo simulation on phase transitions in single‐polymer systems

Applying the histogram Monte Carlo simulation method and the bond‐fluctuation model, various phase transitions in single‐polymer systems were investigated. The critical transition temperature (Θ point) in the coil‐globule collapse transition of a macromolecular chain is accurately determined. Finite‐size scaling results near the transition point are verified. The first‐order transition associated with the freezing/crystallization of a polymer at a temperature below the Θ point is also observed. The free energy profiles associated with these two transitions are explicitly computed. Furthermore, the unfolding phase transition associated with stretching a collapsed polymer chain is investigated. The free energy profile associated with the transition is explicitly computed. Results on the energy cumulants and free energy profiles provide direct evidences for the first‐order nature of the unfolding phase transition.     

Topological coarse graining of polymer chains using wavelet-accelerated Monte Carlo. II. Self-avoiding chains

In the preceding paper [A. E. Ismail, G. C. Rutledge, and G. Stephanopoulos J. Chem. Phys. (in press)] we introduced wavelet-accelerated Monte Carlo (WAMC), a coarse-graining methodology based on the wavelet transform, as a method for sampling polymer chains. In the present paper, we extend our analysis to consider excluded-volume effects by studying self-avoiding chains. We provide evidence that the coarse-grained potentials developed using the WAMC method obey phenomenological scaling laws, and use simple physical arguments for freely jointed chains to motivate these laws. We show that coarse-grained self-avoiding random walks can reproduce results obtained from simulations of the original, more-detailed chains to a high degree of accuracy, in orders of magnitude less time.     

Monte Carlo simulation algorithm for B‐DNA

Understanding the structure–function relationship of biomolecules containing DNA has motivated experiments aimed at determining molecular structure using methods such as small‐angle X‐ray and neutron scattering (SAXS and SANS). SAXS and SANS are useful for determining macromolecular shape in solution, a process which benefits by using atomistic models that reproduce the scattering data. The variety of algorithms available for creating and modifying model DNA structures lack the ability to rapidly modify all‐atom models to generate structure ensembles. This article describes a Monte Carlo algorithm for simulating DNA, not with the goal of predicting an equilibrium structure, but rather to generate an ensemble of plausible structures which can be filtered using experimental results to identify a sub‐ensemble of conformations that reproduce the solution scattering of DNA macromolecules. The algorithm generates an ensemble of atomic structures through an iterative cycle in which B‐DNA is represented using a wormlike bead–rod model, new configurations are generated by sampling bend and twist moves, then atomic detail is recovered by back mapping from the final coarse‐grained configuration. Using this algorithm on commodity computing hardware, one can rapidly generate an ensemble of atomic level models, each model representing a physically realistic configuration that could be further studied using molecular dynamics. © 2016 Wiley Periodicals, Inc.     

Topological coarse graining of polymer chains using wavelet-accelerated Monte Carlo. I. Freely jointed chains

We introduce a new, topologically-based method for coarse-graining polymer chains. Based on the wavelet transform, a multiresolution data analysis technique, this method assigns a cluster of particles to a coarse-grained bead located at the center of mass of the cluster, thereby reducing the complexity of the problem by dividing the simulation into several stages, each with a fraction of the number of beads as the overall chain. At each stage, we compute the distributions of coarse-grained internal coordinates as well as potential functions required for subsequent simulation stages. In this paper, we present the basic algorithm, and apply it to freely jointed chains; the companion paper describes its applications to self-avoiding chains.     

Mixing rules for binary Lennard–Jones chains: theory and Monte Carlo simulation

Theoretically-based van der Waals one-fluid (vdW1) mixing rules are derived for Lennard–Jones (LJ) chain mixtures. The rules provide equivalent one-fluid segment parameters for LJ size (σ) and energy () parameter as well as chain length (m) based on the parameters of the individual mixture components and the component mole fractions. The mixing rules are tested by performing Monte Carlo simulations of eight different binary mixtures and the equivalent vdW1 pure fluid, each at three densities. The simulations test the effects of changing LJ size parameter, LJ energy parameter and chain length individually and together. The effects of mole fraction and density are also examined. The mixing rules are tested for accuracy in predicting compressibility factors and radial distribution functions. It is found that the vdW1 rules provide excellent agreement when size and energy parameter are varied. Good agreement is found for mixtures with different chain lengths. The discrepancy is worst at very high densities when all component parameters are varied simultaneously.     

Non‐Newtonian viscosity in linear and star polymers

We theoretically investigate non‐Newtonian viscosity and coil deformation of linear and (regular) star polymers in dilute solution subject to large shear rates. A bead‐and‐spring model with preaveraged hydrodynamic interaction, accounting also approximately for good‐solvent expansion, is employed within the Rouse‐Zimm approach. We impose a constraint on the average spring lengths, so as to keep constant the average contour length of the arms under shear: this corresponds to assuming that the springs become increasingly stiffer. For any topology and a very large molecular mass, coil deformation modifies the hydrodynamic interaction, that goes to a maximum, and then decreases with a crossover from the Zimm to the Rouse regime with increasing shear rate. Correspondingly, the intrinsic viscosity decreases and then raises above its low‐shear value. This behavior is however much less pronounced under good‐solvent conditions. At very large shear rate, the constraint on the spring lengths becomes the dominant factor. This leads to a decrease of intrinsic viscosity with an asymptotic –2/3 power law for any draining condition. Simultaneously, the strongly elongated coil becomes fully aligned with flow.     

Monte Carlo simulation of tetrahedral chains, 7 the shape of linear and star-branched polymers near to theta-conditions

By use of the pivot algorithm, star-branched chains with F = 4, 8 and 12 arms of length n and linear chains (F = 2) are generated on a tetrahedral lattice (120 ≤ nF ≤ 3 840). By taking into account nearest neighbour interactions (each contact contributes an energy ϕ kT to the total energy of the configuration) a variation of the thermodynamic quality of the solvent is simulated by a variation of the energy parameter ϕ near the value of ϕ_θ = -0,475, characteristic of theta-conditions. For theta-conditions various quantities characteristic of the instantaneous shape of polymers exhibit similar values as found for nonreversal random walks; furthermore, while linear theta-chains are slightly less asymmetric than athermal ones, the opposite behaviour is found for star-branched polymers. Clearly, for all thermodynamic conditions the asymmetry of configurations decreases with increasing number of arms but remains appreciable even for F = 12.     

Monte Carlo and modified Tanford-Kirkwood results for macromolecular electrostatics calculations

The understanding of electrostatic interactions is an essential aspect of the complex correlation between structure and function of biological macromolecules. It is also important in protein engineering and design. Theoretical studies of such interactions are predominantly done within the framework of Debye-Hückel theory. A classical example is the Tanford-Kirkwood (TK) model. Besides other limitations, this model assumes an infinitesimally small macromolecule concentration. By comparison to Monte Carlo (MC) simulations, it is shown that TK predictions for the shifts in ion binding constants upon addition of salt become less reliable even at moderately macromolecular concentrations. A simple modification based on colloidal literature is suggested to the TK scheme. The modified TK models suggested here satisfactorily predict MC and experimental shifts in the calcium binding constant as a function of protein concentration for the calbindin D(9k) mutant and calmodulin.     

Fracture of flexible polymer chains in dilute solution under transient extensional flow

 Applying the technique of Brownian dynamics simulation, we have studied the fracture process of flexible polymer chains when they encounter an extensional flow field of transient character. For this purpose, a mathematical model was made of an experimental device used earlier by other authors to study fracture of polystyrene in dilute solution. The polymer/solvent system studied was a very dilute solution in theta conditions. The polymer molecule was modeled as a FENE bead-spring chain, including a modification to allow for chain fracture. The simulation results showed that the fracture yield depended strongly on flow rate and on molecular weight. We have characterized the molecular-weight dependence of the critical flow rate which is necessary for fracture to occur. The distribution of the result-ing fragments is interpreted in terms of the conformation of the chains prior to fracture. Received: 17 September 1996 Accepted: 29 May 1997     

Quality of random number generators significantly affects results of Monte Carlo simulations for organic and biological systems

We have simulated pure liquid butane, methanol, and hydrated alanine polypeptide with the Monte Carlo technique using three kinds of random number generators (RNG's)—the standard Linear Congruential Generator (LCG), a modification of the LCG with additional randomization used in the BOSS software, and the “Mersenne Twister” generator by Matsumoto and Nishimura. While using the latter two RNG's leads to reasonably similar physical features, the LCG produces significant different results. For the pure fluids, a noticeable expansion occurs. Using the original LCG on butane yields, a molecular volume of 171.4 ų per molecule compared to about 163.6–163.9 ų for the other two generators, a deviation of about 5%. For methanol, the LCG produces an average volume of 86.3 ų per molecule, which is about 24% higher than the 68.8–70.2 ų obtained with the RNG's in BOSS and the generator by Matsumoto and Nishimura. In case of the hydrated tridecaalanine peptide, the volume and energy tend to be noticeably greater with the LCG than with the BOSS (modified LCG) RNG's. For the simulated hydrated extended conformation of tridecaalanine, the difference in volume reached about 87%. The uniformity and periodicity of the generators do not seem to play the crucial role in these phenomena. We conclude that, it is important to test a RNG's by modeling a system such as the pure liquid methanol with a well‐established force field before routinely employing it in Monte Carlo simulations. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2011     

A gradient‐directed Monte Carlo approach for protein design

We develop a new global optimization strategy, gradient‐directed Monte Carlo (GDMC) sampling, to optimize protein sequence for a target structure using RosettaDesign. GDMC significantly improves the sampling of sequence space, compared to the classical Monte Carlo search protocol, for a fixed backbone conformation as well as for the simultaneous optimization of sequence and structure. As such, GDMC sampling enhances the efficiency of protein design. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2010     

Lattice Monte Carlo simulations of three-dimensional charged polymer chains. II. Added salt

The configurational properties of strongly charged polyelectrolytes accompanied by counterions and salt ions in dilute solutions are simulated using the cooperative motion algorithm on the face-centered-cubic lattice. The calculations indicate that both positive and negative ions condense on the chains at sufficiently low temperatures and their amount depends on the concentration of added salt. As the temperature decreases from high to low, the chains undergo conformational changes from neutral-like, self-avoiding polymers by more outstretched forms to compact globules. The observed extension of the chains at intermediate temperatures is also affected by the amount of salt. Furthermore, configurations with the lowest energies recorded at the lowest temperatures are aggregates of single or more entangled chains and ions of both types.     

A Novel Method for Fixed‐node Quantum Monte Carlo

In this paper, a novel method for fixed‐node quantum Monte Carlo is given. By comparing this method with the traditional fixed‐node one, this novel method can be applied to calculate molecular energy more exactly. An expansion of the eigenvalue of the energy for a system has been derived. It is proved that the value of the energy calculated using the traditional fixed‐node method is only the zeroth order approximation of the eigenvalue of the energy. But when using this novel method, in the case of only increasing less computing amounts ( < 1%), the first order approximation, the second order approximation, and so on can be obtained conveniently with the detailed equations and steps in the practical calculation to calculate the values of the zeroth, first and second approximation of the energies of 1 ¹A, state of CH₂, ¹A₂(C_4h, acet) state of C₈ and the ground‐states of H₂, LiH, Li₂, and H₂O The results indicate that for these states it needs only the second order approximation to obtain over 97% of electronic correlation energy, which demonstrates that this novel method is very excellent in both the computing accuracy and the amount of calculation required.     

Spatial Electron Relaxation: Comparison of Monte Carlo and Boltzmann Equation Results

In former investigations on the spatial relaxation of the electron component of weakly ionized plasmas a considerable spectrum of distinct spatial structures has been found in the velocity distribution function as well as in various macroscopic quantities of the electrons. These results have been mainly obtained by solving the spatially inhomogeneous electron Boltzmann equation considering the action of uniform electric fields and the impact of elastic and inelastic collisions of the electrons with the gas atoms. To verify these partly unexpected results on the complex structure formation, analogous Monte Carlo simulations were performed now for a helium plasma at various reduced electric field strengths. Detailed comparisons were made between the results of the two independent kinetic approaches with respect to the spatial evolution of the velocity distribution function as well as of associated macroscopic quantities. Good agreement was generally found, thus confirming the earlier results on the complex spatial relaxation structures.