Short-chain behaviour of star-branched nonreversal random walks embedded in a tetrahedral lattice

Analytical results are presented for the mean-square dimensions of star-branched nonreversal random walk polymers on a tetrahedral lattice considering short-chain effects and the influence of the core (centre) of the star. They are checked against numerical results obtained from highly precise Monte Carlo calculations. Short-chain effects are studied in detail. In addition, the deviation from proportionality between mean-square dimensions and the total chain-length is evaluated and compared to that for self-avoiding chains and stars near theta-conditions, the total chain-lengths ranging from 125 to 3845 segments.     

Shape asymmetry of star-branched random walks and nonreversal random walks

Star-branched random walks with 3, 4, 6, 8 and 12 arms (the total chain-length ranging from N = 49 to 1925) have been produced and analysed with respect to their instantaneous shape. The short-chain behaviour of nonreversal random walk stars (NRRWs) embedded in various lattices is compared to that of star-branched freely jointed (off-lattice) chains (RWs). While for all types of NRRW-stars examined as well as for RW-stars with bonds of constant length shape-asymmetry increases with increasing chain-lengths, the opposite behaviour is found for RW-stars with Gaussian-distributed bond-lengths. The amount of short-chain effects is strongly dependent on the number of arms and on the lattice type used. In the limit of infinitely large molecules, however, quantities characteristic of the shape converge to common values for all types of RWs and NRRWs examined.     

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 simulation studies of the size and shape of linear and star‐branched polymers embedded in the tetrahedral lattice

In recent years the author investigated the size and shape of linear and star‐branched model chains as a function of functionality (number of arms) F and chain length (total number of beads) N for several lattice and off‐lattice random walk models as well as for self‐avoiding random walks (embedded in the tetrahedral lattice) subjected to various thermodynamic conditions. Not only mean values – characteristic quantities averaged over a large number of configurations of given length, functionality and thermodynamic conditions – have been computed, but also distributions of shape parameters, and the correlation operative between size and shape has been explored. The present feature article in principle summarizes these results, however, the data given in part are recomputed for still larger sample sizes and chain‐lengths as in the underlying papers. In addition to stars with F = 4, 8 and 12 arms, so far unpublished investigations on stars with F = 3, F = 6 and F = 10 arms are presented and discussed.     

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.     

Monte Carlo study of star-branched polymers on the tetrahedral lattice. I. Conformation of the macromolecule

The star-branched polymers on the tetrahedral lattice are studied by means of the Monte Carlo method. The influence of solvent quality on the dimensions of the coil is described for both linear and branched polymer systems of different functionality. It has been observed that the ratios of gyration radii 〈S²〉_b/〈S²〉_l are greater than those predicted theoretically for the random-flight model. The fourth reduced moment of S² distribution and the mean-square separation of the branch ends from the center of gravity have been also computed. The changes in segment arrangement in the coil with increasing number of branches have been observed.     

Monte carlo study of star-branched polymers on the tetrahedral lattice. II. Statistical thermodynamics of single macromolecules

The thermodynamical properties of the star-branched polymers on the tetrahedral lattice are studied taking into account nearest-neighbor interactions. The excess free energy and energy and heat capacities are computed for wide ranges of chain lengths, reduced potential ?/kT, and number of branches. A significant influence of the degree of branching on long-range interactions in the polymer random coil is observed. The possibilities of phase transitions in both linear and branched systems are discussed on the basis of the Monte Carlo data.     

Resistance distance in graphs and random walks

We study the resistance distance on connected undirected graphs, linking this concept to the fruitful area of random walks on graphs. We provide two short proofs of a general lower bound for the resistance, or Kirchhoff index, of graphs on N vertices, as well as an upper bound and a general formula to compute it exactly, whose complexity is that of inverting an N×N matrix. We argue that the formulas for the resistance in the case of the Platonic solids can be generalized to all distance‐transitive graphs. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 81: 29–33, 2001     

Electrophoretic mobility of linear and star-branched DNA in semidilute polymer solutions

Electrophoresis of large linear T2 (162 kbp) and 3-arm star-branched (N(Arm) = 48.5 kbp) DNA in linear polyacrylamide (LPA) solutions above the overlap concentration c* has been investigated using a fluorescence visualization technique that allows both the conformation and mobility mu of the DNA to be determined. LPA solutions of moderate polydispersity index (PI approximately 1.7-2.1) and variable polymer molecular weight Mw (0.59-2.05 MDa) are used as the sieving media. In unentangled semidilute solutions (c* < c < c(e)), we find that the conformational dynamics of linear and star-branched DNA in electric fields are strikingly different; the former migrating in predominantly U- or I-shaped conformations, depending on electric field strength E, and the latter migrating in a squid-like profile with the star-arms outstretched in the direction opposite to E and dragging the branch point through the sieving medium. Despite these visual differences, mu for linear and star-branched DNA of comparable size are found to be nearly identical in semidilute, unentangled LPA solutions. For LPA concentrations above the entanglement threshold (c > c(e)), the conformation of migrating linear and star-shaped DNA manifest only subtle changes from their unentangled solution features, but mu for the stars decreases strongly with increasing LPA concentration and molecular weight, while mu for linear DNA becomes nearly independent of c and Mw. These findings are discussed in the context of current theories for electrophoresis of large polyelectrolytes.     

Properties of star-branched and linear chains in confined space: a computer simulation study

We studied the properties of simple models of linear and star-branched polymer chains confined in a slit. The polymer chains were built of united atoms and were restricted to a simple cubic lattice. Two macromolecular architectures of the chain linear and star-branched with three branches (of equal length) were studied. The excluded volume was the only potential introduced into the model and thus, the system was athermal. The chains were put between two parallel and impenetrable surfaces. Monte Carlo simulations with a sampling algorithm based on chain’s local changes of conformation were carried out. The differences and similarities in the global size and the structure and of linear and star-branched chains were shown and discussed.     

Loop-erased self-avoiding random walks in four and five dimensions

Monte Carlo simulations of loop-erased self-avoiding random walks in four and five dimensions were performed, using two distinct algorithms. We find consistency between these methods in their estimates of critical exponents. The upper critical dimension for this phenomenon is four, and it has been shown that the mean square end-to-end distance grows as n(log n)^α. It has recently been established that the mean square end-to-end distance is asymptotically bounded by n(log n)^1/3 (see Ref. 21). Our results show that asymptotic convergence to n(log n)^1/3 in fact obtains and does so rather quickly. In five dimensions we examine the rate of asymptotic convergence to the mean-field model. © 1996 by John Wiley & Sons, Inc.     

Monte Carlo simulation of tetrahedral chains, 6. 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 non-bonded 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 in the range −0,425 to −0,525. The energy parameter ϕ_⊙ = −0,475, characteristic of theta conditions, was evaluated by use of an intramolecular criterion (proportionality between mean-square dimensions and total chain-length) and confirmed by a crossover scaling analysis. Theta dimensions are found to be larger than those of nonreversal random walks, the deviation growing with increasing number of arms.     

Comparison of the rheological properties of linear and star-branched polyisoprenes in shear and elongational flows

The rheological properties of narrow-molecular-weight-distribution linear and star-branched polyisoprenes have been determined using both shearing and stretching deformations. At all strain rates studied the tensile stress measured under transient and steady-state conditions did not increase above the linear viscoelastic value. The absence of an enhanced tensile stress for the branched polymer is in contrast to what is observed for branched low-density polyethylene. An explanation for the difference is proposed. Additional remarks are made about the broad distribution of relaxation times observed for star-branched polyisoprenes and about the approach to steady state in constant-strain-rate and constant-stress tests.     

Two-particle continuous-time random walks and binary reactions in disordered media

The relation between unary-(trappihg) and binary (mutual annihilation) reactions in disordered systems is studied in the framework of the continuous-time random walk. It is found that if the waiting-time distribution of the walk has infinite moments, a time-independent binary rate constant may exist even though a unary one does not.     

Dynamics of macromolecular chains. I. Theory of motions on a tetrahedral lattice

The bond correlation function for a macromolecular model chain on a tetrahedral lattice is derived by considering elementary three-bond and four-bond motions. The method of calculation allows the conformational structure of the chain to be involved in the final equations. Moreover, when the three-bond motions are considered alone, no linearization assumption is required; hence, the theory is valid at short times.     

Properties of amorphous and crystallizable hydrocarbon polymers. II. Rheology of linear and star-branched polybutadiene

Some results are reported on the linear viscoelastic properties of polybutadienes with narrow-molecular-weight distributions. The zero shear viscosity η₀ varies as M^3.4 in the linear samples, and viscosity enhancement is found in star-branched samples with long arms, in good agreement with results reported earlier by Kraus and Gruver. The temperature coefficient of viscosity appears to be slightly larger in stars when the arms become long. The steady state recoverable compliance J is 2.1 × 10 ^?7 cm²/dyn in linear samples of high molecular weight, but it increases to values as much as 10 times larger in the stars. The plateau modulus G, obtained from a composite curve for the linear samples, is 1.32 × 10⁷ dyn/cm². The terminal relaxation spectrum of the stars is too broad to allow an evaluation of plateau modulus.     

Phase behavior of linear heterogeneous trimers on a square lattice

Monte Carlo simulations in the grand canonical ensemble, the multiple-histogram analysis and finite-size scaling techniques have been used to study a phase behavior of trimer BAB on a square lattice. The systems with the same energies u(AA) = u(BB) and different strengths of interactions between unlike segments are considered. The AB-contacts are energetically unprofitable. There are two phase transitions: the first-order vapor-liquid transition and the second-order structural transition in the supercritical fluid. The phase diagram topology depends on the energy u(AB). The crossover between the tricritical point phase diagram topology and the critical end phase diagram topology is found. It is demonstrated that the transition to the ordered strip-like phase is non-universal.     

Importance of chirality and reduced flexibility of protein side chains: a study with square and tetrahedral lattice models

Side chains of amino acid residues are the determining factor that distinguishes proteins from other unstable chain polymers. In simple models they are often represented implicitly (e.g., by spin states) or simplified as one atom. Here we study side chain effects using two-dimensional square lattice and three-dimensional tetrahedral lattice models, with explicitly constructed side chains formed by two atoms of different chirality and flexibility. We distinguish effects due to chirality and effects due to side chain flexibilities, since residues in proteins are L residues, and their side chains adopt different rotameric states. For short chains, we enumerate exhaustively all possible conformations. For long chains, we sample effectively rare events such as compact conformations and obtain complete pictures of ensemble properties of conformations of these models at all compactness region. This is made possible by using sequential Monte Carlo techniques based on chain growth method. Our results show that both chirality and reduced side chain flexibility lower the folding entropy significantly for globally compact conformations, suggesting that they are important properties of residues to ensure fast folding and stable native structure. This corresponds well with our finding that natural amino acid residues have reduced effective flexibility, as evidenced by statistical analysis of rotamer libraries and side chain rotatable bonds. We further develop a method calculating the exact side chain entropy for a given backbone structure. We show that simple rotamer counting underestimates side chain entropy significantly for both extended and near maximally compact conformations. We find that side chain entropy does not always correlate well with main chain packing. With explicit side chains, extended backbones do not have the largest side chain entropy. Among compact backbones with maximum side chain entropy, helical structures emerge as the dominating configurations. Our results suggest that side chain entropy may be an important factor contributing to the formation of alpha helices for compact conformations.