Evaluating changes of writhe in computer simulations of supercoiled DNA

We compute changes in the writhe of a polygonal space curve when one of the vertices is displaced. The resulting expressions can be used in simulations of supercoiled DNA. For Brownian dynamics simulations, the expressions can be used to eliminate the explicit twisting degree of freedom. For Monte Carlo simulations, they can be used in fast local moves. Preliminary Monte Carlo simulations using only such fast local moves show that these can be used to efficiently simulate small DNA supercoils.     

Thermodynamic properties of lattice hard-sphere models

Thermodynamic properties of several lattice hard-sphere models were obtained from grand canonical histogram- reweighting Monte Carlo simulations. Sphere centers occupy positions on a simple cubic lattice of unit spacing and exclude neighboring sites up to a distance sigma. The nearestneighbor exclusion model, sigma = radical2, was previously found to have a second-order transition. Models with integer values of sigma = 1 or 2 do not have any transitions. Models with sigma = radical3 and sigma = 3 have weak first-order fluid-solid transitions while those with sigma = 2 radical2, 2 radical3, and 3 radical2 have strong fluid-solid transitions. Pressure, chemical potential, and density are reported for all models and compared to the results for the continuum, theoretical predictions, and prior simulations when available.     

Brownian dynamics simulation of the diffusion of rods and wormlike chains in a gel modeled as a cubic lattice: application to DNA

The translational diffusion constant of a particle, D, in a congested medium or a gel can be written as the product of two terms that account for long-range hydrodynamic interaction between the gel or congested medium and the particle, DEM, and a short-range "steric" term, S. For particles of arbitrary shape, DEM has been examined previously within the framework of the effective medium, EM, model (S. Allison et al., J. Phys. Chem. B 2008, 112, 5858-5866). In the present work, we examine S for rod- and wormlike chain models of duplex DNA in the size range of 100 to over 2000 base pairs. The gel is modeled explicitly as a cubic lattice, and Brownian dynamics simulation is used to examine S for a wide range of rod/wormlike chain and gel parameters. For wormlike chains with P = 50 nm, an empirical formula is derived for S that should be valid over a wide range of wormlike chain/gel parameters. For duplex DNA in the size of several hundred to several thousand base pairs in an agarose gel of 2% or less, fair agreement between modeling and experiment is obtained. However, modeling overestimates the length dependence of D observed experimentally. Finally, the reduction of D of DNA (100 to over 1000 base pairs in length) in cytoplasm relative to water can be accounted for quite well using the effective medium plus steric correction approach.     

DNA electrophoresis in confined, periodic geometries: a new lakes-straits model

We present a method to study the dynamics of long DNA molecules inside a cubic array of confining spheres, connected through narrow openings. Our method is based on the coarse-grained, lakes-straits model of Zimm and is therefore much faster than Brownian dynamics simulations. In contrast to Zimm's approach, our method uses a standard stochastic kinetic simulation to account for the mass transfer through the narrow straits and the formation of new lakes. The different rates, or propensities, of the reactions are obtained using first-passage time statistics and a Monte Carlo sampling to compute the total free energy of the chain. The total free energy takes into account the self-avoiding nature of the chain as well as confinement effects from the impenetrable spheres. The mobilities of various chains agree with biased reptation theory at low and high fields. At moderate fields, confinement effects lead to a new regime of reptation where the mobility is a linear function of molecular weight and the dispersion is minimal.     

Lattice model of oligonucleotide hybridization in solution. I. Model and thermodynamics

A coarse-grained lattice model of DNA oligonucleotides is proposed to investigate the general mechanisms by which single-stranded oligonucleotides hybridize to their complementary strands in solution. The model, based on a high-coordination cubic lattice, is simple enough to allow the direct simulation of DNA solutions, yet capturing how the fundamental thermodynamic processes are microscopically encoded in the nucleobase sequences. Physically relevant interactions are considered explicitly, such as interchain excluded volume, anisotropic base-pairing and base-stacking, and single-stranded bending rigidity. The model is studied in detail by a specially adapted Monte Carlo simulation method, based on parallel tempering and biased trials, which is designed to overcome the entropic and enthalpic barriers associated with the sampling of hybridization events of multiple single-stranded chains in solution. This methodology addresses both the configurational complexity of bringing together two complementary strands in a favorable orientation (entropic barrier) and the energetic penalty of breaking apart multiple associated bases in a double-stranded state (enthalpic barrier). For strands with sequences restricted to nonstaggering association and homogeneous pairing and stacking energies, base-pairing is found to dominate the hybridization over the translational and conformational entropy. For strands with sequence-dependent pairing corresponding to that of DNA, the complex dependence of the model's thermal stability on concentration, sequence, and degree of complementarity is shown to be qualitatively and quantitatively consistent both with experiment and with the predictions of statistical mechanical models.     

Monte Carlo simulations of polymer dynamics: Recent advances

A brief review is given of applications of Monte Carlo simulations to study the dynamical properties of coarse-grained models of polymer melts, emphasizing the crossover from the Rouse model toward reptation, and the glass transition. The extent to which Monte Carlo algorithms can mimic the actual chain dynamics is critically examined, and the need for the use of coarse-grained rather than fully atomistic models for such simulations is explained. It is shown that various lattice and continuum models yield qualitatively similar results, and the behavior agrees with the findings of corresponding molecular dynamics simulations and experiments, where available. It is argued that these simulations significantly enhance our understanding of the theoretical concepts on the dynamics of dense macromolecular systems. © 1997 John Wiley & Sons, Inc.     

Comparison of lattice and continuum treatments of n-alkanes with experimental data

Using a homologous series of n-alkanes as a model system, we compare the predictions of a lattice Born-Green-Yvon (BGY) theory and a continuum BGY theory with experimental results. We find that both theories are capable of describing the fluid properties and critical points of alkanes ranging from heptamers (n-C7) to nonadecamers (n-C19). We probe the connection between the lattice and continuum BGY models and extend our discussion to include a sampling of other lattice and continuum treatments.     

Contacts between segments in the random-flight model of polymer chains

The probabilities of single and double contacts, and correlations among contacts were calculated for the random-flight model of a polymer chain. The problem was studied earlier by various authors using the Gaussian statistics of the random-flight chain, which is applicable only for long chains, and for contacts which are formed by pair of residues (i,j) which are far from each other (i.e. |i−j|≫1). We used the exact (non-Gaussian) solution of the random flight model, so that our results are applicable also for residues which are relatively close to each other. We applied the results to close contacts of the β-sheet type and α-helix type in proteins, and compared the exact solutions of the problem with Gaussian approximations.     

Stretching semiflexible polymer chains: evidence for the importance of excluded volume effects from Monte Carlo simulation

Semiflexible macromolecules in dilute solution under very good solvent conditions are modeled by self-avoiding walks on the simple cubic lattice (d = 3 dimensions) and square lattice (d = 2 dimensions), varying chain stiffness by an energy penalty ε(b) for chain bending. In the absence of excluded volume interactions, the persistence length l(p) of the polymers would then simply be l(p) = l(b)(2d - 2)(-1)q(b) (-1) with q(b) = exp(-ε(b)/k(B)T), the bond length l(b) being the lattice spacing, and k(B)T is the thermal energy. Using Monte Carlo simulations applying the pruned-enriched Rosenbluth method (PERM), both q(b) and the chain length N are varied over a wide range (0.005 ≤ q(b) ≤ 1, N ≤ 50,000), and also a stretching force f is applied to one chain end (fixing the other end at the origin). In the absence of this force, in d = 2 a single crossover from rod-like behavior (for contour lengths less than l(p)) to swollen coils occurs, invalidating the Kratky-Porod model, while in d = 3 a double crossover occurs, from rods to Gaussian coils (as implied by the Kratky-Porod model) and then to coils that are swollen due to the excluded volume interaction. If the stretching force is applied, excluded volume interactions matter for the force versus extension relation irrespective of chain stiffness in d = 2, while theories based on the Kratky-Porod model are found to work in d = 3 for stiff chains in an intermediate regime of chain extensions. While for q(b) ? 1 in this model a persistence length can be estimated from the initial decay of bond-orientational correlations, it is argued that this is not possible for more complex wormlike chains (e.g., bottle-brush polymers). Consequences for the proper interpretation of experiments are briefly discussed.     

Linear chain surfactants at a planar interface: a comparative Monte Carlo study of several lattice models

Linear chain surfactants in a densely packed arrangement (such as alkane chains in lipid monolayers in the “uniform tilt” structures) are described by a crude coarse-grained model where the endgroups grafted on the interface form a regular lattice and the chains are described by the bond fluctuation model with chains containing N = 4 effective monomers only. Square-well interactions between the monomers are studied for both the attractive and repulsive case for three choices of the interaction range. None of these models exhibits a structure with uniform tilt. For attractive interactions the last bond has a strong tendency to fold back thus leading to a very high density close to the interface. Only when an intrachain-potential favoring stiff chain configurations also is included one can obtain configurations with uniform tilt order. Although related models (with much longer chain lengths and lower grafting densities) are very useful for the study of polymer brushes, the present case of very short chains in a high-density state clearly is plagued by various lattice artefacts and it is concluded that for modelling linear chain surfactants one should use an off-lattice model even on a coarse-grained level.     

Polymer collapse, protein folding, and the percolation threshold

We study the transition of polymers in the dilute regime from a swollen shape at high temperatures to their low-temperature structures. The polymers are modeled by a single self-avoiding walk (SAW) on a lattice for which l of the monomers (the H monomers) are self-attracting, i.e., if two nonbonded H monomers become nearest neighbors on the lattice they gain energy of interaction (epsilon = -/epsilon/); the second type of monomers, denoted P, are neutral. This HP model was suggested by Lau and Dill (Macromolecules 1989, 22, 3986-3997) to study protein folding, where H and P are the hydrophobic and polar amino acid residues, respectively. The model is simulated on the square and simple cubic (SC) lattices using the scanning method. We show that the ground state and the sharpness of the transition depend on the lattice, the fraction g of the H monomers, as well as on their arrangement along the chain. In particular, if the H monomers are distributed at random and g is larger than the site percolation threshold of the lattice, a collapsed transition is very likely to occur. This conclusion, drawn for the lattice models, is also applicable to proteins where an effective lattice with coordination number between that of the SC lattice and the body centered cubic lattice is defined. Thus, the average fraction of hydrophobic amino acid residues in globular proteins is found to be close to the percolation threshold of the effective lattice.     

Continuum formulation of the Scheutjens-Fleer lattice statistical theory for homopolymer adsorption from solution

Homopolymer adsorption from a dilute solution on an interacting (attractive) surface under static equilibrium conditions is studied in the framework of a Hamiltonian model. The model makes use of the density of chain ends n(1,e) and utilizes the concept of the propagator G describing conformational probabilities to locally define the polymer segment density or volume fraction phi; both n(1,e) and phi enter into the expression for the system free energy. The propagator G obeys the Edwards diffusion equation for walks in a self-consistent potential field. The equilibrium distribution of chain ends and, consequently, of chain conformational probabilities is found by minimizing the system free energy. This results in a set of model equations that constitute the exact continuum-space analog of the Scheutjens-Fleer (SF) lattice statistical theory for the adsorption of interacting chains. Since for distances too close to the surface the continuum formulation breaks down, the continuum model is here employed to describe the probability of chain configurations only for distances z greater than 2l, where l denotes the segment length, from the surface; instead, for distances z < or = 2l, the SF lattice model is utilized. Through this novel formulation, the lattice solution at z = 2l provides the boundary condition for the continuum model. The resulting hybrid (lattice for distances z < or = 2l, continuum for distances z > 2l) model is solved numerically through an efficient implementation of the pseudospectral collocation method. Representative results obtained with the new model and a direct application of the SF lattice model are extensively compared with each other and, in all cases studied, are found to be practically identical.     

Computer simulation of bottle-brush polymers with flexible backbone: good solvent versus theta solvent conditions

By molecular dynamics simulation of a coarse-grained bead-spring-type model for a cylindrical molecular brush with a backbone chain of N(b) effective monomers to which with grafting density σ side chains with N effective monomers are tethered, several characteristic length scales are studied for variable solvent quality. Side chain lengths are in the range 5 ≤ N ≤ 40, backbone chain lengths are in the range 50 ≤ N(b) ≤ 200, and we perform a comparison to results for the bond fluctuation model on the simple cubic lattice (for which much longer chains are accessible, N(b) ≤ 1027, and which corresponds to an athermal, very good, solvent). We obtain linear dimensions of the side chains and the backbone chain and discuss their N-dependence in terms of power laws and the associated effective exponents. We show that even at the theta point the side chains are considerably stretched, their linear dimension depending on the solvent quality only weakly. Effective persistence lengths are extracted both from the orientational correlations and from the backbone end-to-end distance; it is shown that different measures of the persistence length (which would all agree for Gaussian chains) are not mutually consistent with each other and depend distinctly both on N(b) and the solvent quality. A brief discussion of pertinent experiments is given.     

Modeling ssDNA electrophoretic migration with band broadening in an entangled or cross-linked network

We use a coarse-grained model proposed by Graham and Larson based on the temporary network model by Schieber et al.. [1] to simulate the electrophoretic motion of ssDNA and corresponding band broadening due to dispersion. With dimensionless numbers reflecting the experimental physical properties, we are able to simulate ssDNA behavior under weak to moderate electric field strengths for chains with 8-50 entanglements per chain ( approximately 1000-8500 base pairs), and model smoothly the transition from reptation to oriented reptation. These results are fitted with an interpolation equation, which allows the user to calculate dimensionless mobilities easily from input parameters characterizing the gel matrix, DNA molecules, and field strengths. Dimensionless peak widths are predicted from mobility fluctuations using the central limit theorem and the assumption that the mobility fluctuations are Gaussian. Using results from previous studies of ssDNA physical properties (effective charge xiq and Kuhn step length b(K)) and sieving matrix properties (pore size or tube diameter a), we give scaling factors to convert the dimensionless values to "real" experimental values, including the mobility, migration distance, and time. We find that the interpolation equation fits well the experimental data of ssDNA mobilities and peak widths, supporting the validity of the coarse-grained model. The model does not account for constraint release and hernia formation, and assumes that the sieving network is a homogeneous microstructure with no temperature gradients and no peak width due to injection. These assumptions can be relaxed in future work for more accurate prediction.     

A coarse-grained model for the formation of alpha helix with a noninteger period on simple cubic lattices

Periodicity is an important parameter in the characterization of a helix in proteins. In this work, a coarse-grained model for a homopolypeptide in simple cubic lattices has been extended to build an alpha helix with a noninteger period. The lattice model is based on the bond fluctuation algorithm in which bond lengths and orientations are altered in a quasicontinuous way, and the simulation is performed via dynamic Monte Carlo simulation. Hydrogen bonds are assumed to be formed between a virtual-carbonyl group in a residue and a downstream virtual-imino group in another residue. Consequently, the period of the formed alpha helix is a noninteger. An improved spatial correlation function has been suggested to quantitatively describe the periodicity of the helical conformation, by which helical period and helical persistent length can be calculated by statistics. The resultant periods are very close to 3.6 residues; the persistent length based upon the improved definition can be larger or smaller than the chain length and reflect the inherent regularity of a chain including one or multiple helical blocks. The simulation outputs agree with the calculation of the Zimm-Bragg theory based upon the associated nucleation parameter and propagation parameter as well.