Coarse-grained computations for a micellar system

Adaptive stochastic continuation with a modified lifting procedure applied to complex systems

Many complex systems occurring in the natural or social sciences or economics are frequently described on a microscopic level, e.g., by lattice- or agent-based models. To analyze the states of such systems and their bifurcation structure on the level of macroscopic observables, one has to rely on equation-free methods like stochastic continuation. Here we investigate how to improve stochastic continuation techniques by adaptively choosing the parameters of the algorithm. This allows one to obtain bifurcation diagrams quite accurately, especially near bifurcation points. We introduce lifting techniques which generate microscopic states with a naturally grown structure, which can be crucial for a reliable evaluation of macroscopic quantities. We show how to calculate fixed points of fluctuating functions by employing suitable linear fits. This procedure offers a simple measure of the statistical error. We demonstrate these improvements by applying the approach in analyses of (i) the Ising model in two dimensions, (ii) an active Ising model, and (iii) a stochastic Swift-Hohenberg model. We conclude by discussing the abilities and remaining problems of the technique.

Kinetic network study of the diversity and temperature dependence of Trp-Cage folding pathways: combining transition path theory with stochastic simulations

We present a new approach to study a multitude of folding pathways and different folding mechanisms for the 20-residue mini-protein Trp-Cage using the combined power of replica exchange molecular dynamics (REMD) simulations for conformational sampling, transition path theory (TPT) for constructing folding pathways, and stochastic simulations for sampling the pathways in a high dimensional structure space. REMD simulations of Trp-Cage with 16 replicas at temperatures between 270 and 566 K are carried out with an all-atom force field (OPLSAA) and an implicit solvent model (AGBNP). The conformations sampled from all temperatures are collected. They form a discretized state space that can be used to model the folding process. The equilibrium population for each state at a target temperature can be calculated using the weighted-histogram-analysis method (WHAM). By connecting states with similar structures and creating edges satisfying detailed balance conditions, we construct a kinetic network that preserves the equilibrium population distribution of the state space. After defining the folded and unfolded macrostates, committor probabilities (P(fold)) are calculated by solving a set of linear equations for each node in the network and pathways are extracted together with their fluxes using the TPT algorithm. By clustering the pathways into folding "tubes", a more physically meaningful picture of the diversity of folding routes emerges. Stochastic simulations are carried out on the network, and a procedure is developed to project sampled trajectories onto the folding tubes. The fluxes through the folding tubes calculated from the stochastic trajectories are in good agreement with the corresponding values obtained from the TPT analysis. The temperature dependence of the ensemble of Trp-Cage folding pathways is investigated. Above the folding temperature, a large number of diverse folding pathways with comparable fluxes flood the energy landscape. At low temperature, however, the folding transition is dominated by only a few localized pathways.

Recovering kinetics from a simplified protein folding model using replica exchange simulations: a kinetic network and effective stochastic dynamics

We present an approach to recover kinetics from a simplified protein folding model at different temperatures using the combined power of replica exchange (RE), a kinetic network, and effective stochastic dynamics. While RE simulations generate a large set of discrete states with the correct thermodynamics, kinetic information is lost due to the random exchange of temperatures. We show how we can recover the kinetics of a 2D continuous potential with an entropic barrier by using RE-generated discrete states as nodes of a kinetic network. By choosing the neighbors and the microscopic rates between the neighbors appropriately, the correct kinetics of the system can be recovered by running a kinetic simulation on the network. We fine-tune the parameters of the network by comparison with the effective drift velocities and diffusion coefficients of the system determined from short-time stochastic trajectories. One of the advantages of the kinetic network model is that the network can be built on a high-dimensional discretized state space, which can consist of multiple paths not consistent with a single reaction coordinate.

Equation-free multiscale computation: algorithms and applications

In traditional physicochemical modeling, one derives evolution equations at the (macroscopic, coarse) scale of interest; these are used to perform a variety of tasks (simulation, bifurcation analysis, optimization) using an arsenal of analytical and numerical techniques. For many complex systems, however, although one observes evolution at a macroscopic scale of interest, accurate models are only given at a more detailed (fine-scale, microscopic) level of description (e.g., lattice Boltzmann, kinetic Monte Carlo, molecular dynamics). Here, we review a framework for computer-aided multiscale analysis, which enables macroscopic computational tasks (over extended spatiotemporal scales) using only appropriately initialized microscopic simulation on short time and length scales. The methodology bypasses the derivation of macroscopic evolution equations when these equations conceptually exist but are not available in closed form-hence the term equation-free. We selectively discuss basic algorithms and underlying principles and illustrate the approach through representative applications. We also discuss potential difficulties and outline areas for future research.

Computer simulation of the epitaxy of surfactant-templated inorganic nanomaterials on patterned surfaces

For a surfactant-inorganic-water system, we study systemically the epitaxy process of the surfactant-templated inorganic nanomaterial on a patterned surface with a lattice Monte Carlo method. It is found that by using ring-patterned substrate as a template, vertically oriented inorganic material may be formed through the nanometer-scale epitaxy. For all the cases studied in this work, a hemicylinder initially formed on the ring pattern behaves as nucleation sites for the following growth process. Different parameters, such as surfactant architecture, bulk surfactant concentration, fraction of inorganic component, and pattern size, are found to affect the epitaxial growth process of the inorganic nanomaterial. The change of surfactant architecture alters the structure of formed aggregates significantly, thus affecting the epitaxial growth. For the effects of surfactant concentration, it is found that there exists a critical value. If and only if the bulk surfactant concentration is higher than the critical value is the epitaxial growth of inorganic material nucleated from the patterned substrate possible. For the effects of the fraction of the inorganic component, simulation results indicate that there also exists a volume fraction above which the nanomaterial growth is dominated by macrophase separation but not templated by the substrate pattern. The geometry of the chemical modification of the surface also shows significant effects on the nanometer-scale epitaxy, depending on whether its sizes are commensurate with the morphology of the formed inorganic nanomaterials or not.

Macrophase and microphase separations for surfactants adsorbed on solid surfaces: a gauge cell monte carlo study in the lattice model

By combining the gauge cell method and lattice model, we study the surface phase transition and adsorption behaviors of surfactants on a solid surface. Two different cases are considered in this work: macrophase transition and adsorption in a single-phase region. For the case of macrophase transition, where two phases coexist, we investigate the shape and size of the critical nuclei and determine the height of the nucleation barrier. It is found that the nucleation depends on the bulk surfactant concentration. Our simulations show that there exist a critical temperature and critical adsorption energy, below which the transition from low-affinity adsorption to the bilayer structure shows the characteristic of a typical first-order phase transition. Such a surface phase transition in the adsorption isotherm is featured by a hysteresis loop. The hysteresis loop becomes narrower at higher temperature and weaker adsorption energy and finally disappears at the critical value. For the case where no macrophase transition occurs, we study the adsorption isotherm and microphase separation in a single-phase region. The simulation results indicate that the adsorption isotherm in adsorption processes is divided into four regions in a log-log plot, being in agreement with experimental observations. In this work, the four regions are called the low-affinity adsorption region, the hemimicelle region, the morphological transition region, and the plateau region. Simulation results reveal that in the second region the adsorbed monomers aggregate and nucleate hemimicelles, while adsorption in the third region is accompanied by morphological transitions.

Computer simulation of adsorption kinetics of surfactants on solid surfaces

Adsorption kinetics of surfactants on solid surfaces has been studied by using computer simulation. Both bulk surfactant concentration and diffusion region are explicitly integrated in our model. Depending on the head-surface interaction, our simulation results indicate that there exist two different kinetic modes in adsorption process of surfactants on solid surfaces. One is the four-regime mode and the other is step-wise mode. For the strongly attractive head-surface interaction, four distinct regimes of surfactant adsorption are found: a diffusion-controlled regime, a self-assembly controlled regime, an intermediate coverage regime and a saturated regime. In particular, the adsorption in second regime displays a power-law time dependence with an exponent unrelated to bulk concentrations and diffusion coefficients. While for the weaker adsorption surfaces, the step-wise mode is found. The mode includes a low-coverage nucleation regime and the saturated regime after a sudden aggregation of surfactants on the substrates which occurs stochastically. Besides the head-surface interaction, in this work, the effects of surfactant diffusivity, bulk concentration, the length of diffusion zone and surfactant architecture on the adsorption kinetics are also considered. The simulated adsorption kinetics is compared qualitatively with experimental results.

Confinement induced critical micelle concentration shift

In this work, extensive lattice Monte Carlo simulations were performed to investigate the influence of confinement on critical micelle concentration (CMC). It is found that the CMC of surfactants in a confined space is shifted from its bulk value, and the shift is affected by the presence of the confining boundaries, which induces both the finite size effect and the wall-surfactant interaction. In general, for strongly confined system (the system with narrow pore size), the finite size effect dominates the CMC shift because the confined space cannot accommodate fully developed micelles, and the rapid increase of the entropic loss due to the decrease of the pore size results in the rapid increase of CMC. In contrast, for a weakly confined space, the CMC shift depends on the interaction between the walls and surfactants. For the systems with two weakly hydrophilic surfaces, the local density depletion of the surfactants near the walls results in lower CMCs than the bulk value, and the CMC shifts to a higher value as the pore size increases. For the systems with moderately hydrophilic surfaces, the shifts of CMCs show a similar behavior as those for weakly hydrophilic surfaces, but the CMCs are near their bulk values in the range of weak confinement. For the systems with strongly attractive wall-surfactant interactions, the strong adsorption also results in lower CMCs than their bulk value, but the CMCs decrease with the increase of pore size.

Coarse-grained analysis of stochasticity-induced switching between collective motion states

A single animal group can display different types of collective motion at different times. For a one-dimensional individual-based model of self-organizing group formation, we show that repeated switching between distinct ordered collective states can occur entirely because of stochastic effects. We introduce a framework for the coarse-grained, computer-assisted analysis of such stochasticity-induced switching in animal groups. This involves the characterization of the behavior of the system with a single dynamically meaningful "coarse observable" whose dynamics are described by an effective Fokker-Planck equation. A "lifting" procedure is presented, which enables efficient estimation of the necessary macroscopic quantities for this description through short bursts of appropriately initialized computations. This leads to the construction of an effective potential, which is used to locate metastable collective states, and their parametric dependence, as well as estimate mean switching times.

Gyration-radius dynamics in structural transitions of atomic clusters

This paper is concerned with the structural transition dynamics of the six-atom Morse cluster with zero total angular momentum, which serves as an illustrative example of the general reaction dynamics of isolated polyatomic molecules. It develops a methodology that highlights the interplay between the effects of the potential energy topography and those of the intrinsic geometry of the molecular internal space. The method focuses on the dynamics of three coarse variables, the molecular gyration radii. By using the framework of geometric mechanics and hyperspherical coordinates, the internal motions of a molecule are described in terms of these three gyration radii and hyperangular modes. The gyration radii serve as slow collective variables, while the remaining hyperangular modes serve as rapidly oscillating "bath" modes. Internal equations of motion reveal that the gyration radii are subject to two different kinds of forces: One is the ordinary force that originates from the potential energy function of the system, while the other is an internal centrifugal force. The latter originates from the dynamical coupling of the gyration radii with the hyperangular modes. The effects of these two forces often counteract each other: The potential force generally works to keep the internal mass distribution of the system compact and symmetric, while the internal centrifugal force works to inflate and elongate it. Averaged fields of these two forces are calculated numerically along a reaction path for the structural transition of the molecule in the three-dimensional space of gyration radii. By integrating the sum of these two force fields along the reaction path, an effective energy curve is deduced, which quantifies the gross work necessary for the system to change its mass distribution along the reaction path. This effective energy curve elucidates the energy-dependent switching of the structural preference between symmetric and asymmetric conformations. The present methodology should be of wide use for the systematic reduction of dimensionality as well as for the identification of kinematic barriers associated with the rearrangement of mass distribution in a variety of molecular reaction dynamics in vacuum.

Adsorption and morphology transition of surfactants on hydrophobic surfaces: a lattice Monte Carlo study

In this work, we first show that there are only five independent interchange parameters in the surfactant-solvent-interface system in Larson's model, and then adsorption and morphology transition of surfactants on hydrophobic surfaces are studied by extensive lattice Monte Carlo simulations. In our simulations, we found that there exist six adsorbed morphologies: (1) premature admicelle, (2) hemisphere, (3) hemisphere-hemicylinder mixture, (4) wormlike hemicylinder, (5) perforated monolayer, and (6) monolayer. The surface morphologies and the amount of adsorption on hydrophobic surfaces are found to be affected obviously by two interchange parameters. One is the attractive interaction between tail groups and surface (chiTS), and the other is the solubility of head groups in bulk (chiHW). Phase diagrams in chiHW versus chiTS planes for surfactants with different hydrophobicities (chiTW) and for surfactants with different molecular structures are determined in this work, from which the transitions of surface morphologies and adsorption behaviors are discussed.

Multiscale simulation of transmembrane proteins

Multiscale simulation is employed to examine changes in atomistic-level protein structure due to long wavelength membrane undulations and plane stress fields. An ensemble of atomistic-level simulations of a model of a transmembrane influenza A virus M2 proton channel in a dimyristoylphosphatidylcholine (DMPC) bilayer is coupled to a corresponding mesoscopic model of a DMPC bilayer in an explicit mesoscopic solvent. Structural variations in the key proton gating His37 residues of the M2 channel are examined. Small, but distinct variations in the structure of the His37 residues are observed in both the open and closed states of the channel as a result of the coupling to mesoscopic-level membrane motions.

Coarse-grained kinetic computations for rare events: application to micelle formation

We discuss a coarse-grained approach to the computation of rare events in the context of grand canonical Monte Carlo (GCMC) simulations of self-assembly of surfactant molecules into micelles. The basic assumption is that the computational system dynamics can be decomposed into two parts-fast (noise) and slow (reaction coordinates) dynamics, so that the system can be described by an effective, coarse-grained Fokker-Planck (FP) equation. While such an assumption may be valid in many circumstances, an explicit form of FP equation is not always available. In our computations we bypass the analytic derivation of such an effective FP equation. The effective free energy gradient and the state-dependent magnitude of the random noise, which are necessary to formulate the effective Fokker-Planck equation, are obtained from ensembles of short bursts of microscopic simulations with judiciously chosen initial conditions. The reaction coordinate in our micelle formation problem is taken to be the size of a cluster of surfactant molecules. We test the validity of the effective FP description in this system and reconstruct a coarse-grained free energy surface in good agreement with full-scale GCMC simulations. We also show that, for very small clusters, the cluster size ceases to be a good reaction coordinate for a one-dimensional effective description. We discuss possible ways to improve the current model and to take higher-dimensional coarse-grained dynamics into account.