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PubMed2004

Dynamics of chain closure: approximate treatment of nonlocal interactions.

The Wilemski-Fixman model of diffusion controlled-reactions [J. Chem. Phys. 58, 4009 (1973)] is combined with a generalized random walk description of chain conformations to predict the dependence of the closure time tau on the chain length N of polymers with reactive end groups and nonlocal interactions. The nonlocal interactions are modeled by a modification to the connectivity term in the Edwards continuum representation of the polymer. The modification involves a parameter h lying between 0 and 1 that is a measure of the extent of correlation between adjacent monomers on the chain backbone. Different choices of h correspond to chain conformations of different average radial dimensions. In particular, the values 1/3, 1/2 and 3/5 provide approximations to the statistics of polymers in poor, theta and good solvents, respectively. The closure time tau of such chains is calculated analytically for different N. In all cases, tau is found to vary as a power law in N, Nb, with b a function of h. For the special case h = 1/3, which models collapsed polymers and globular proteins, b is about 1.6-1.7.

Biopolymers↗
PubMed1996

Deterministic diffusion in almost integrable systems.

Diffusion processes are usually associated with randomness in the system. In this paper we show that deterministic diffusion processes can also occur in systems with zero entropy that mimic chaos with any precision without being mathematically chaotic. A random walk model is used to predict the behavior of the diffusion coefficient. (c) 1996 American Institute of Physics.

Journal Article↗
PubMed1998

Motion in a stochastic layer described by symbolic dynamics.

The motion in the stochastic layer surrounding an island can be studied by using the standard map: This problem is of direct relevance to the diffusion of magnetic field lines in a tokamak. In a previous work it was shown that this process can be adequately modelled by a continuous time random walk (CTRW) describing transitions of the running point between three basins representing, respectively, trapped motion around the island, and passing motion above or below the island. The sticking property of the island deeply modifies the nature of the transport process, leading to subdiffusive behavior. In the present work it is shown that the motion can be analyzed in terms of a symbolic dynamics which leads to the possibility of an automatic measurement of the data necessary for the construction of the CTRW. The logical features of the procedure are described, and the method is applied to an analysis of long time series, thus completing the results of the previous work. (c) 1998 American Institute of Physics.

Journal Article↗
PubMed1999

Motion of grains down a bumpy surface.

We summarize in this article an extensive experimental and theoretical effort carried out to understand the behavior of a single ball when rolling down a bumpy surface. This may appear to be a simple problem but in fact is one that displays a rich variety of different behaviors which allow us to understand better dissipative systems such as granular media. Studies performed previously have shown that the motion of the single ball on the rough surface can be characterized by three different dynamic regimes according to the different values of the two control parameters, the inclination angle theta and the ratio Phi=R/r, where R is the radius of the rolling ball and r the radius of the glass beads which make up the rough surface. The three regimes are a decelerated regime A, a stationary regime B, characterized by a constant average velocity and a jumping regime C. This result was found to be independent of the composition of the rolling ball and the rough surface. It has been demonstrated that regime B is characterized by a viscous-like friction force that appears for specific parameter values. This friction force can be explained by a model whose central ingredient is the geometry of the surface. The trajectory of the ball in regime B can be pictured as a driven random walk motion where the fluctuations of the local velocities are due to collisions of the moving sphere and the surface grains. A detailed analysis of diffusive properties of the motion is discussed. (c) 1999 American Institute of Physics.

Journal Article↗
PubMed2004

Pores in bilayer membranes of amphiphilic molecules: coarse-grained molecular dynamics simulations compared with simple mesoscopic models.

We investigate pores in fluid membranes by molecular dynamics simulations of an amphiphile-solvent mixture, using a molecular coarse-grained model. The amphiphilic membranes self-assemble into a lamellar stack of amphiphilic bilayers separated by solvent layers. We focus on the particular case of tensionless membranes, in which pores spontaneously appear because of thermal fluctuations. Their spatial distribution is similar to that of a random set of repulsive hard disks. The size and shape distribution of individual pores can be described satisfactorily by a simple mesoscopic model, which accounts only for a pore independent core energy and a line tension penalty at the pore edges. In particular, the pores are not circular: their shapes are fractal and have the same characteristics as those of two-dimensional ring polymers. Finally, we study the size-fluctuation dynamics of the pores, and compare the time evolution of their contour length to a random walk in a linear potential.

Journal Article↗
PubMed2004

2H nuclear magnetic resonance study of the molecular motion in cyanoadamantane. I. Supercooled plastically crystalline phase.

The supercooled plastically crystalline phase (glassy crystal) of cyanoadamantane was investigated by multidimensional 2H NMR (T>Tg). Although the orientationally disordered crystalline phase always coexisted with the orientationally ordered crystalline phase, we were able to single out the signal from the glassy crystal by selective excitation and it was possible to carry out line shape measurements and two-dimensional (2D) experiments (in frequency and time domain). The latter directly reveal sixfold jumps with an reorientation of the molecular C3 axis via 90 degrees angles, thus reflecting the symmetry of the lattice. The motion around the C3 axis is found to be always fast. We can reproduce the line shape by random walk simulations properly taking into account these molecular motions. Both methods (line shape and 2D experiments) yield time constants which agree with those reported by other techniques. Refining the analysis a narrow distribution of correlation times is introduced to account for a weak stretching of the correlation function. We did not find any indication of a small angle process usually found in structural glasses. Thus, the motional process in the glassy crystal appears to be simple and quite different from that in structural glasses.

Journal Article↗
PubMed2004

A framework model based on the Smoluchowski equation in two reaction coordinates.

The general form of the Smoluchowski equation in two reaction coordinates is obtained as the diffusion limit of a random walk on an infinite square grid using transition probabilities that satisfy detailed balance at thermodynamic equilibrium. The diffusion limit is then used to construct a generalization of the single-particle model to two reaction coordinates. The state space includes a square on which diffusion takes place and an isolated empty state. Boundary conditions on opposite sides of the square correspond to transitions between the empty state and the square. The two-dimensional (2D) model can be reduced to a 1D single-particle model by adiabatic elimination. A finite element solution of the 2D boundary value problem is described. The method used to construct the 2D model can be adapted to state spaces that have been constructed by other authors to model K+ conduction through gramicidin, proton conduction through dioxolane-linked gramicidin, and chloride conduction through the bacterial H(+)-Cl- antiporter.

Journal Article↗
PubMed2004

Electron affinities with diffusion quantum Monte Carlo for C2 and BO molecules.

To verify the performance of the fixed-node diffusion quantum Monte Carlo method in electron affinities calculations, the adiabatic electron affinities of C(2) and BO molecules calculated by the fixed-node Ornstein-Uhlenbeck diffusion quantum Monte Carlo simulations guiding by trial wave functions constructed from the floating spherical Gaussian orbitals and spherical Gaussian geminals are presented in this work. The random walk based results, 3.264(43) and 2.507(32) eV for C(2) and BO, respectively, are compared with the available best experimental determined values, 3.269(6) and 2.508(8) eV for C(2) and BO, respectively, and the results of other theoretical calculations.

Journal Article↗
PubMed2005

On the shape of bottle-brush macromolecules: systematic variation of architectural parameters.

We measured the form factor of bottle-brush macromolecules under good solvent conditions with small-angle neutron scattering and static light scattering. The systems under investigation are brushes, synthesized via the grafting-from route, built from a poly(alkyl methacrylate) backbone to which poly(n-butyl acrylate) side chains are densely grafted. The aim of our work is to study how the systematic variation of structural parameters such as the side chain length and backbone length change the conformation of the polymer brushes in solution. All spectra can be consistently described by a model, considering the bottle-brush polymers as flexible rods with internal density fluctuations. Parameters discussed are (1) the contour length per main chain monomer l(b), (2) the fractal dimension of the side chains Ds, as well as (3) the fractal dimension D, and (4) the Kuhn length lambdak of the overall brush. l(b)=0.253+/-0.008 nm is found to be independent of the side chain length and equal to the value found for the bare main chain, indicating a strongly stretched conformation for the backbone due to the presence of the side chains. The fractal dimension of the side chains is determined to be Ds=1.75+/-0.07 which is very close to the value of 10.588 approximately 1.70 expected for a three-dimensional self-avoiding random walk (3D-SAW) under good solvent conditions. On larger length scales the overall brush appears to be a 3D-SAW itself (D=1.64+/-0.08) with a Kuhn-step length of lambdak=70+/-4 nm. The value is independent of the side chain length and 46 times larger than the Kuhn length of the bare backbone (lambdak=1.8+/-0.2 nm). The ratio of Kuhn length to brush diameter lambda(k)d>or=20 determines whether lyotropic behavior can be expected or not. Since longer side chains do not lead to more persistent structures, lambda(k)d decreases from 8 to 4 with increasing side chain length and lyotropic behavior becomes unlikely.

Journal Article↗
PubMed2005

Direct calculation of solid-liquid equilibria from density-of-states Monte Carlo simulations.

A density-of-states Monte Carlo method is proposed for simulations of solid-liquid phase equilibria. A modified Wang-Landau density-of-states sampling approach is used to perform a random walk in regions of potential energy and volume relevant to solid-liquid equilibrium. The method provides a direct estimate of the relative density of states [Omega(U,V)] and thus the relative free energy within these regions, which is subsequently used to determine portions of the melting curve over wide ranges of pressure and temperature. The validity and usefulness of the method are demonstrated by performing crystallization simulations for the Lennard-Jones fluid and for NaCl.

Journal Article↗
PubMed2005

Anomalous features of the kinetics of subdiffusion-assisted bimolecular reactions.

Some specific features of the kinetics of subdiffusion-assisted bimolecular reactions are analyzed in detail with the use of the non-Markovian stochastic Liouville equation (SLE) recently derived within the continuous time random walk approach. The SLE allows for describing important peculiarities of the reactions: Slow long time behavior of the kinetics, nonanalytical dependence of the reaction rate on the reactivity of particles, the onset of quasistatic regime independently of particle mobility in the case long-range reactivity, strong manifestation of fluctuation kinetics showing itself in very slow reaction kinetics at long times, etc.

Journal Article↗
PubMed2005

Soluble stochastic dynamics of quasi-one-dimensional single-file fluid self-diffusion.

We solve a model of random-walk stochastic dynamics for hard single-file fluids in the experimentally important quasi-one-dimensional regime. This is a nontrivial extension of exact solution beyond one dimension. We point out that quasi-one-dimensional single-file self-diffusion of one-component hard fluids of diameter a under stochastic forces is equivalent at long time to a one-dimensional hard-rod fluid with the same linear density but a different diameter, a(eff). This effective diameter is controlled by the details of the relative dynamics between the transverse and longitudinal directions. There are two regimes of limiting behavior. For very fast transverse motion, the system is likely (but we cannot prove rigorously) to be equivalent to the soluble-oriented hard-rectangle or cylinder systems, with a(eff)=a. With very slow transverse motion, the self-diffusion dynamics is described by an equivalent soluble one-dimensional mixture of fluids with a(eff)=a(ave), the average longitudinal separation between nearest-neighbor particles at contact. We have explored our theoretical predictions with Monte Carlo simulations.

Journal Article↗
PubMed2005

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.

Journal Article↗
PubMed2005

Polypeptide foldings obtained with effective pair potentials.

We present a model of protein folding which is based on a potential function that describes the effective interaction between two amino acids (alanines, in this case). Our model is consistent with the formation of two important secondary structures, namely, an alpha-helix and a beta-ladder. In each case, we estimate the density of states using a random walk in energy space. This function allows the direct calculation of certain thermodynamic properties. By means of the configurational temperature, we also verify that the obtained polypeptides are in their native state.

Alanine↗
PubMed2005

Adsorption of comb copolymers on weakly attractive solid surfaces.

In this work continuum and lattice Monte Carlo simulation methods are used to study the adsorption of linear and comb polymers on flat surfaces. Selected polymer segments, located at the tips of the side chains in comb polymers or equally spaced along the linear polymers, are attracted to each other and to the surface via square-well potentials. The rest of the polymer segments are modeled as tangent hard spheres in the continuum model and as self-avoiding random walks in the lattice model. Results are presented in terms of segment-density profiles, distribution functions, and radii of gyration of the adsorbed polymers. At infinite dilution the presence of short side chains promotes the adsorption of polymers favoring both a decrease in the depletion-layer thickness and a spreading of the polymer molecule on the surface. The presence of long side chains favors the adsorption of polymers on the surface, but does not permit the spreading of the polymers. At finite concentration linear polymers and comb polymers with long side chains readily adsorb on the solid surface, while comb polymers with short side chains are unlikely to adsorb. The simple models of comb copolymers with short side chains used here show properties similar to those of associating polymers and of globular proteins in aqueous solutions, and can be used as a first approximation to investigate the mechanism of adsorption of proteins onto hydrophobic surfaces.

Journal Article↗
PubMed2005

Hierarchical structure of the energy landscape of proteins revisited by time series analysis. II. Investigation of explicit solvent effects.

Time series analysis tools are employed on the principal modes obtained from the C(alpha) trajectories from two independent molecular-dynamics simulations of alpha-amylase inhibitor (tendamistat). Fluctuations inside an energy minimum (intraminimum motions), transitions between minima (interminimum motions), and relaxations in different hierarchical energy levels are investigated and compared with those encountered in vacuum by using different sampling window sizes and intervals. The low-frequency low-indexed mode relationship, established in vacuum, is also encountered in water, which shows the reliability of the important dynamics information offered by principal components analysis in water. It has been shown that examining a short data collection period (100 ps) may result in a high population of overdamped modes, while some of the low-frequency oscillations (<10 cm(-1)) can be captured in water by using a longer data collection period (1200 ps). Simultaneous analysis of short and long sampling window sizes gives the following picture of the effect of water on protein dynamics. Water makes the protein lose its memory: future conformations are less dependent on previous conformations due to the lowering of energy barriers in hierarchical levels of the energy landscape. In short-time dynamics (<10 ps), damping factors extracted from time series model parameters are lowered. For tendamistat, the friction coefficient in the Langevin equation is found to be around 40-60 cm(-1) for the low-indexed modes, compatible with literature. The fact that water has increased the friction and that on the other hand has lubrication effect at first sight contradicts. However, this comes about because water enhances the transitions between minima and forces the protein to reduce its already inherent inability to maintain oscillations observed in vacuum. Some of the frequencies lower than 10 cm(-1) are found to be overdamped, while those higher than 20 cm(-1) are slightly increased. As for the long-time dynamics in water, it is found that random-walk motion is maintained for approximately 200 ps (about five times of that in vacuum) in the low-indexed modes, showing the lowering of energy barriers between the higher-level minima.

Computer Simulation↗
PubMed2005

Convergence of replica exchange molecular dynamics.

Replica exchange molecular dynamics (REMD) method is one of the generalized-ensemble algorithms which performs random walk in energy space and helps a system to escape from local energy traps. In this work, we studied the accuracy and efficiency of REMD by examining its ability to reproduce the results of multiple extended conventional molecular dynamics (MD) simulations and to enhance conformational sampling. Two sets of REMD simulations with different initial configurations, one from the fully extended and the other from fully helical conformations, were conducted on a fast-folding 21-amino-acid peptide with a continuum solvent model. Remarkably, the two REMD simulation sets started to converge even within 1.0 ns, despite their dramatically different starting conformations. In contrast, the conventional MD within the same time and with identical starting conformations did not show obvious signs of convergence. Excellent convergence between the REMD sets for T>300 K was observed after 14.0 ns REMD simulations as measured by the average helicity and free-energy profiles. We also conducted a set of 45 MD simulations at nine different temperatures with each trajectory simulated to 100.0 and 200.0 ns. An excellent agreement between the REMD and the extended MD simulation results was observed for T>300 K, showing that REMD can accurately reproduce long-time MD results with high efficiency. The autocorrelation times of the calculated helicity demonstrate that REMD can significantly enhance the sampling efficiency by 14.3+/-6.4, 35.1+/-0.2, and 71.5+/-20.4 times at, respectively, approximately 360, approximately 300, and approximately 275 K in comparison to the regular MD. Convergence was less satisfactory at low temperatures (T<300 K) and a slow oscillatory behavior suggests that longer simulation time was needed to reach equilibrium. Other technical issues, including choice of exchange frequency, were also examined.

Algorithms↗
PubMed2006

Brownian dynamics simulations with stiff finitely extensible nonlinear elastic-Fraenkel springs as approximations to rods in bead-rod models.

A very stiff finitely extensible nonlinear elastic (FENE)-Fraenkel spring is proposed to replace the rigid rod in the bead-rod model. This allows the adoption of a fast predictor-corrector method so that large time steps can be taken in Brownian dynamics (BD) simulations without over- or understretching the stiff springs. In contrast to the simple bead-rod model, BD simulations with beads and FENE-Fraenkel (FF) springs yield a random-walk configuration at equilibrium. We compare the simulation results of the free-draining bead-FF-spring model with those for the bead-rod model in relaxation, start-up of uniaxial extensional, and simple shear flows, and find that both methods generate nearly identical results. The computational cost per time step for a free-draining BD simulation with the proposed bead-FF-spring model is about twice as high as the traditional bead-rod model with the midpoint algorithm of Liu [J. Chem. Phys. 90, 5826 (1989)]. Nevertheless, computations with the bead-FF-spring model are as efficient as those with the bead-rod model in extensional flow because the former allows larger time steps. Moreover, the Brownian contribution to the stress for the bead-FF-spring model is isotropic and therefore simplifies the calculation of the polymer stresses. In addition, hydrodynamic interaction can more easily be incorporated into the bead-FF-spring model than into the bead-rod model since the metric force arising from the non-Cartesian coordinates used in bead-rod simulations is absent from bead-spring simulations. Finally, with our newly developed bead-FF-spring model, existing computer codes for the bead-spring models can trivially be converted to ones for effective bead-rod simulations merely by replacing the usual FENE or Cohen spring law with a FENE-Fraenkel law, and this convertibility provides a very convenient way to perform multiscale BD simulations.

Algorithms↗