Results Of Brownian Dynamics Simulations Research Articles

Characteristic features of self-avoiding active Brownian polymers under linear shear flow.

We present Brownian dynamics simulation results of a flexible linear polymer with excluded-volume interactions under shear flow in the presence of active noise. The active noise strongly affects the polymer's conformational and dynamical properties, such as the stretching in the flow direction and compression in the gradient direction, shear-induced alignment, and shear viscosity. In the asymptotic limit of large activities and shear rates, the power-law scaling exponents of these quantities differ significantly from those of passive polymers. The chain's shear-induced stretching at a given shear rate is reduced by active noise, and it displays a non-monotonic behavior, where an initial polymer compression is followed by its stretching with increasing active force. The compression of the polymer in the gradient direction follows the relation ∼WiPe-3/4 as a function of the activity-dependent Weissenberg number WiPe, which differs from the scaling observed in passive systems ∼WiPe-1/2. The flow-induced alignment at large Péclet numbers Pe ≫ 1, where Pe is the Péclet number, and large shear rates WiPe ≫ 1 displays the scaling behavior WiPe-1/2, with an exponent differing from the passive value -1/3. Furthermore, the polymer's zero-shear viscosity displays a non-monotonic behavior, decreasing in an intermediate activity regime due to excluded-volume interactions and increasing again for large Pe. Shear thinning appears with increasing Weissenberg number with the power-laws WiPe-1/2 and WiPe-3/4 for passive and active polymers, respectively. In addition, our simulation results are compared with the results of an analytical approach, which predicts quantitatively similar behaviors for the various aforementioned physical quantities.

Simulating wet active polymers by multiparticle collision dynamics.

The conformational and dynamical properties of active Brownian polymers embedded in a fluid depend on the nature of the driving mechanism, e.g., self-propulsion or external actuation of the monomers. Implementations of self-propelled and actuated active Brownian polymers in a multiparticle collision (MPC) dynamics fluid are presented, which capture the distinct differences between the two driving mechanisms. The active force-free nature of self-propelled monomers requires adaptations of the MPC simulation scheme, with its streaming and collision steps, where the monomer self-propulsion velocity has to be omitted in the collision step. Comparison of MPC simulation results for active polymers in dilute solution with results of Brownian dynamics simulations accounting for hydrodynamics via the Rotne-Prager-Yamakawa tensor confirm the suitability of the implementation. The polymer conformational and dynamical properties are analyzed by the static and dynamic structure factor, and the scaling behavior of the latter with respect to the wave number and time dependence are discussed. The dynamic structure factor displays various activity-induced temporal regimes, depending on the considered wave number, which reflect the persistent diffusive motion of the whole polymer at small wave numbers, and the activity-enhanced internal dynamics at large wave numbers. The obtained simulation results are compared with theoretical predictions.

Universality in Driven and Equilibrium Hard Sphere Liquid Dynamics.

We demonstrate that the time evolution of the van Hove dynamical pair correlation function is governed by adiabatic forces that arise from the free energy and by superadiabatic forces that are induced by the flow of the van Hove function. The superadiabatic forces consist of drag, viscous, and structural contributions, as occur in active Brownian particles, in liquids under shear and in lane forming mixtures. For hard sphere liquids, we present a power functional theory that predicts these universal force fields in quantitative agreement with our Brownian dynamics simulation results.

Interplay of reactive interference and crowding effects in the diffusion-influenced reaction kinetics.

We investigate the interplay of reactive interference and crowding effects in the irreversible diffusion-influenced bimolecular reactions of the type A+B→P+B by using the Brownian dynamics simulation method. It is known that the presence of nonreactive crowding agents retards the reaction rate when the volume fraction of the crowding agents is large enough. On the other hand, a high concentration of B is known to increase the reaction rate more than expected from the mass action law, although the B's may also act as crowders. Therefore, it would be interesting to see which effect dominates when the number density of B as well as the number density of the crowders increases. We will present an approximate theory that provides a reasonable account for the Brownian dynamics simulation results.

Predictive local field theory for interacting active Brownian spheres in two spatial dimensions

We present a predictive local field theory for the nonequilibrium dynamics of interacting active Brownian particles with a spherical shape in two spatial dimensions. The theory is derived by a rigorous coarse-graining starting from the Langevin equations that describe the trajectories of the individual particles. For high accuracy and generality of the theory, it includes configurational order parameters and derivatives up to infinite order. In addition, we discuss possible approximations of the theory and present reduced models that are easier to apply. We show that our theory contains popular models such as Active Model B+ as special cases and that it provides explicit expressions for the coefficients occurring in these and other, often phenomenological, models. As a further outcome, the theory yields an analytical expression for the density-dependent mean swimming speed of the particles. To demonstrate an application of the new theory, we analyze a simple reduced model of the lowest nontrivial order in derivatives, which is able to predict the onset of motility-induced phase separation of the particles. By a linear stability analysis, an analytical expression for the spinodal corresponding to motility-induced phase separation is obtained. This expression is evaluated for the case of particles interacting repulsively by a Weeks–Chandler–Andersen potential. The analytical predictions for the spinodal associated with these particles are found to be in very good agreement with the results of Brownian dynamics simulations that are based on the same Langevin equations as our theory. Furthermore, the critical point predicted by our analytical results agrees excellently with recent computational results from the literature.

Effects of the Size, the Number, and the Spatial Arrangement of Reactive Patches on a Sphere on Diffusion-limited Reaction Kinetics: A Comprehensive Study.

We investigate how the size, the number, and the spatial arrangement of identical nonoverlapping reactive patches on a sphere influence the overall reaction kinetics of bimolecular diffusion-limited (or diffusion-controlled) reactions that occur between the patches and the reactants diffusing around the sphere. First, in the arrangement of two patches, it is known that the overall rate constant increases as the two patches become more separated from each other but decreases when they become closer to each other. In this work, we further study the dependence of the patch arrangement on the kinetics with three and four patches using the finite element method (FEM). In addition to the patch arrangement, the kinetics is also dependent on the number and size of the patches. Therefore, we study such dependences by calculating the overall rate constants using the FEM for various cases, especially for large-sized patches, and this study is complementary to the kinetic studies that were performed by Brownian dynamics (BD) simulation methods for small-sized patches. The numerical FEM and BD simulation results are compared with the results from various kinetic theories to evaluate the accuracies of the theories. Remarkably, this comparison indicates that our theory, which was recently developed based on the curvature-dependent kinetic theory, shows good agreement with the FEM and BD numerical results. From this validation, we use our theory to further study the variation of the overall rate constant when the patches are arbitrarily arranged on a sphere. Our theory also confirms that to maximize the overall rate constant, we need to break large-sized patches into smaller-sized patches and arrange them to be maximally separated to reduce their competition.

Brownian dynamics investigation of the Boltzmann superposition principle for orthogonal superposition rheology

The most general linear equation describing the stress response at time t to a time-dependent shearing perturbation may be written as the integral over the past history t' of a time dependent relaxation modulus, depending on t - t', multiplied by the perturbing shear rate at time t'. This is in agreement with the Boltzmann superposition principle, which says that the stress response of a system to a time dependent shearing deformation may be written as the sum of responses to a sequence of step-strain perturbations in the past. In equilibrium rheology, the Boltzmann superposition principle gives rise to the equality of the shear relaxation modulus, obtained from oscillatory experiments, and the stress relaxation modulus measured after a step-strain perturbation. In this paper, we describe the results of Brownian dynamics simulations of a simple soft matter system showing that the same conclusion does not hold when the system is steadily sheared in a direction perpendicular to the probing flows, and with a gradient parallel to that of the probing deformations, as in orthogonal superposition rheology. In fact, we find that the oscillatory relaxation modulus differs from the step-strain modulus even for the smallest orthogonal shear flows that we could simulate. We do find, however, that the initial or plateau levels of both methods agree and provide an equation relating the plateau value to the perturbation of the pair-function.

Operation of theoretical Brownian motors based on morphological adaptations

The dynamics of autonomous Brownian motors that achieve rectified motion due to morphological changes is analyzed. These motors are assumed to be deformable, with restitution internal forces, and to be able to adapt their shapes as function of their total elongation. A generic protocol for morphology adaptation is presented which guarantees unidirectional motion at a constant average speed. This protocol is illustrated by diverse particular examples. Motors operating under non-linear internal forces are analyzed. It is shown that non-linear forces produce considerable variations of the average speed and could serve as a control mechanism for the biased dynamics. The effects of hydrodynamic interactions are quantified as function of the morphological asymmetry. Such interactions are found to promote a very slight reduction of the rectification phenomenon, and could be observable only for motors operating at extremely reduced velocities. The theoretical analysis is found to be in very good agreement with the results of Brownian Dynamics simulations.

Brownian Dynamics simulations of model colloids in channel geometries and external fields

We review the results of Brownian Dynamics simulations of colloidal particles in external fields confined in channels. Super-paramagnetic Brownian particles are well suited two- dimensional model systems for a variety of problems on different length scales, ranging from pedestrian walking through a bottleneck to ions passing ion-channels in living cells. In such systems confinement into channels can have a great influence on the diffusion and transport properties. Especially we will discuss the crossover from single file diffusion in a narrow channel to the diffusion in the extended two-dimensional system. Therefore a new algorithm for computing the mean square displacement (MSD) on logarithmic time scales is presented. In a different study interacting colloidal particles were dragged over a washboard potential and are additionally confined in a two-dimensional micro-channel. In this system kink and anti-kink solitons determine the depinning process of the particles from the periodic potential.

Fully Anisotropic Rotational Diffusion Tensor from Molecular Dynamics Simulations.

We present a method to calculate the fully anisotropic rotational diffusion tensor from molecular dynamics simulations. Our approach is based on fitting the time-dependent covariance matrix of the quaternions that describe the rigid-body rotational dynamics. Explicit analytical expressions have been derived for the covariances by Favro, which are valid irrespective of the degree of anisotropy. We use these expressions to determine an optimal rotational diffusion tensor from trajectory data. The molecular structures are aligned against a reference by optimal rigid-body superposition. The quaternion covariances can then be obtained directly from the rotation matrices used in the alignment. The rotational diffusion tensor is determined by a fit to the time-dependent quaternion covariances, or directly by Laplace transformation and matrix diagonalization. To quantify uncertainties in the fit, we derive analytical expressions and compare them with the results of Brownian dynamics simulations of anisotropic rotational diffusion. We apply the method to microsecond long trajectories of the Dickerson-Drew B-DNA dodecamer and of horse heart myoglobin. The anisotropic rotational diffusion tensors calculated from simulations agree well with predictions from hydrodynamics.

Extensional Response and Equilibrium Kinetics of Torsionally Relaxed dsDNA Under Tension: A Brownian Dynamics Study

Deoxyribonucleic acid (DNA) is a long flexible polyelectrolyte that is housed in the aqueous environment within the cell of an organism. When a length of torsionally relaxed (untwisted) DNA is held in tension, such as is the case in many single molecule experiments, the thermal fluctuations arising from the constant bombardment of the DNA by the surrounding fluid molecules induce bending in it, while the applied tension tends to keep it extended. The combined effect of these influences is that DNA is never at its full extension but eventually attains an equilibrium value of end-to-end extension under these influences. An analytical model was developed to estimate the tension-dependent value of this extension. This model, however, does not provide any insight into the dynamics of the extensional response of DNA to applied tension nor the kinetics of DNA at equilibrium under said tension. This paper reports the results of Brownian dynamics simulations using a discrete wormlike-chain model of DNA that provide some insight into these dynamics and kinetics.

Unbiased diffusion in two-dimensional channels with corrugated walls.

This paper deals with diffusion of point particles in linearly corrugated two-dimensional channels. Such geometry allows one to obtain an approximate analytical expression that gives the particle effective diffusivity as a function of the geometric parameters of the channel. To establish its accuracy and the range of applicability, the expression is tested against Brownian dynamics simulation results. The test shows that the expression works very well for long channel periods, but fails when the period is not long enough compared to the minimum width of the channel. To fix this deficiency, we propose a simple empirical correction to the analytical expression. The resulting corrected expression for the effective diffusivity is in excellent agreement with the simulation results for all values of the channel period.

Anisotropic pair correlations in binary and multicomponent hard-sphere mixtures in the vicinity of a hard wall: A combined density functional theory and simulation study

The fundamental measure approach to classical density functional theory has been shown to be a powerful tool to predict various thermodynamic properties of hard-sphere systems. We employ this approach to determine not only one-particle densities but also two-particle correlations in binary and six-component mixtures of hard spheres in the vicinity of a hard wall. The broken isotropy enables us to carefully test a large variety of theoretically predicted two-particle features by quantitatively comparing them to the results of Brownian dynamics simulations. Specifically, we determine and compare the one-particle density, the total correlation functions, their contact values, and the force distributions acting on a particle. For this purpose, we follow the compressibility route and theoretically calculate the direct correlation functions by taking functional derivatives. We usually observe an excellent agreement between theory and simulations, except for small deviations in cases where local crystal-like order sets in. Our results set the course for further investigations on the consistency of functionals as well as for structural analysis on, e.g., the primitive model. In addition, we demonstrate that due to the suppression of local crystallization, the predictions of six-component mixtures are better than those in bidisperse or monodisperse systems. Finally, we are confident that our results of the structural modulations induced by the wall lead to a deeper understanding of ordering in anisotropic systems in general, the onset of heterogeneous crystallization, caging effects, and glassy dynamics close to a wall, as well as structural properties in systems with confinement.

Non-ideal diffusion effects, short-range ordering, and unsteady-state effects strongly influence Brownian aggregation rates in concentrated dispersions of interacting spheres.

Brownian aggregation rates are determined for concentrated dispersions of interacting particles with Brownian dynamics (BD) simulations and various theoretical models. Using simulation results as benchmarks, the predictions of the classical Fuchs-Smoluchowski (FS) model are shown to be quite inaccurate for concentrated dispersions. A new aggregation model is presented which provides significantly improved predictions. This model is developed on the basis of the fundamental measure theory (FMT) which is a rigorous "liquid-state" dynamic density-functional theory (DDFT) approach. It provides a major improvement of the FS model by considering short-range ordering, non-ideal diffusion, and unsteady-state effects. These were recently shown by the authors to play important roles in Brownian aggregation of hard spheres at high concentrations. Two types of interparticle interaction potentials are examined, the purely attractive van der Waals potential and the DLVO potential which includes van der Waals attraction and electrostatic double layer repulsion. For dispersions of particles with purely attractive interactions, the FS model underpredicts the aggregation rates by up to 1000 fold. In the presence of strong interparticle repulsive forces, its predictions are in fair agreement with the BD simulation results for dilute systems with particle volume fractions ϕ < < 0.1. In contrast, the predictions of the new FM-DDFT based model compare favorably with the BD simulation results, in both cases, up to ϕ = 0.3. A new quantitative measure for colloidal dispersion stability, different from the classical FS stability ratio, is proposed on the basis of aggregation half-times. Hence, a better mechanistic understanding of Brownian aggregation is obtained for concentrated dispersions of particles with either attractive or repulsive interactions, or both.

Introducing Molecular Flexibility in Efficient Simulations of Many-Protein Systems

A novel multiple conformations Monte Carlo (mcMC) computational method is presented that allows the modeling of protein-protein interaction and aggregation. Such processes are relevant in realistic biological environments, such as the cytoplasm and the extracellular matrix, which are characterized by high concentrations of biomolecular solutes, e.g. of 300-400 mg/mL for proteins and RNA in the cytoplasm of E. coli. Simulation of such environments necessitates the inclusion of a large number of protein molecules and therefore computationally inexpensive methods, such as rigid-body Brownian dynamics (BD) and Monte Carlo (MC) methods, must be used. However, the rigid-body representation typically employed in simulations of many-protein systems give rise to certain artifacts in protein-protein interactions. We present a methodology that allows us to incorporate molecular flexibility in MC simulations at low computational cost, and thereby eliminate ambiguities based on the structure selection in rigid molecule simulations. We benchmark and validate the methodology on solutions of hen egg white lysozyme (HEWL), an extraordinarily well-studied system for which extensive experimental data, including osmotic virial coefficients, solution structure factors, and multiple structures determined by x-ray and neutron crystallography and solution NMR, as well as previous BD simulation results, are available.

Electrostatic Channeling in P. falciparum DHFR-TS: Brownian Dynamics and Smoluchowski Modeling

We perform Brownian dynamics simulations and Smoluchowski continuum modeling of the bifunctional Plasmodium falciparum dihydrofolate reductase-thymidylate synthase (P. falciparum DHFR-TS) with the objective of understanding the electrostatic channeling of dihydrofolate generated at the TS active site to the DHFR active site. The results of Brownian dynamics simulations and Smoluchowski continuum modeling suggest that compared to Leishmania major DHFR-TS, P. falciparum DHFR-TS has a lower but significant electrostatic-mediated channeling efficiency (∼15–25%) at physiological pH (7.0) and ionic strength (150 mM). We also find that removing the electric charges from key basic residues located between the DHFR and TS active sites significantly reduces the channeling efficiency of P. falciparum DHFR-TS. Although several protozoan DHFR-TS enzymes are known to have similar tertiary and quaternary structure, subtle differences in structure, active-site geometry, and charge distribution appear to influence both electrostatic-mediated and proximity-based substrate channeling.

Construction of a closed polymer network for computer simulations.

Computer simulations are an important tool for linking the behaviour of polymer materials to the properties of the constituent polymer chains. In simulations, one normally uses periodic boundary conditions to mimic a macroscopic system. For a cross-linked polymer network, this will impose restrictions on the motion of the polymer chains at the borders of the simulation cell. We present a new method for constructing a three-dimensional closed network without periodic boundaries by embedding the system onto the surface of a sphere in four dimensions. This method can also be used to construct finite-sized gel particles for simulating the swelling of particles in a surrounding solvent. The method is described in algorithmic detail to allow the incorporation of the method into different types of simulation programs. We also present the results of Brownian dynamics simulations, analyzing the end-to-end distribution, radial distribution function, and the pore size distribution for different volume fractions and for chains with varying stiffness.

Nonideal Diffusion Effects and Short-Range Ordering Lead to Higher Aggregation Rates in Concentrated Hard-Sphere Dispersions

Brownian aggregation in concentrated hard-sphere dispersions is studied using models and Brownian dynamics (BD) simulations. Two new theoretical models are presented and compared to several existing approaches and BD simulation results, which serve as benchmarks. The first new model is an improvement over an existing local density approximation (LDA)-based model. The other is based on the more rigorous Fundamental measure theory (FMT) applied to the "liquid-state" dynamic density-functional theory (DDFT). Both models provide significant improvements over the classical Smoluchowski model. The predictions of the new FM-DDFT-based model for aggregation kinetics are in excellent agreement with BD simulation results for dispersions with initial particle volume fractions, ϕ, up to 0.35 (close to the hard-sphere freezing transition at ϕ = 0.494). In contrast to previous approaches, the nonideal particle diffusion effects and the initial and time-dependent short-range ordering in concentrated dispersions due to entropic packing effects are explicitly considered here, in addition to the unsteady-state effects. The greater accuracy of the FM-DDFT-based model compared to that of the LDA-based models indicates that nonlocal contributions to particle diffusion (only accounted for in the former) play important roles in aggregation. At high concentrations, the FM-DDFT-based model predicts aggregation half-times and gelation times that are up to 2 orders of magnitude shorter than those of the Smoluchowski model. Moreover, the FM-DDFT-based model predicts asymmetric cluster-cluster aggregation rate constants, at least for short times. Overall, a rigorous mechanistic understanding of the enhancement of aggregation kinetics in concentrated dispersions is provided.

Numerical and Experimental Study of Suspensions Containing Carbon Blacks Used as Conductive Additives in Composite Electrodes for Lithium Batteries

Suspensions of carbon blacks and spherical carbon particles are studied experimentally and numerically to understand the role of the particle shape on the tendency to percolation. Two commercial carbon blacks and one lab-synthesized spherical carbon are used. The percolation thresholds in suspensions are experimentally determined by two complementary methods: impedance spectroscopy and rheology. Brownian dynamics simulations are performed to explain the experimental results taking into account the fractal shape of the aggregates in the carbon blacks. The results of Brownian dynamics simulations are in good agreement with the experimental results and allow one to explain the experimental behavior of suspensions.

A parallel finite element simulator for ion transport through three‐dimensional ion channel systems

A parallel finite element simulator, ichannel, is developed for ion transport through three-dimensional ion channel systems that consist of protein and membrane. The coordinates of heavy atoms of the protein are taken from the Protein Data Bank and the membrane is represented as a slab. The simulator contains two components: a parallel adaptive finite element solver for a set of Poisson-Nernst-Planck (PNP) equations that describe the electrodiffusion process of ion transport, and a mesh generation tool chain for ion channel systems, which is an essential component for the finite element computations. The finite element method has advantages in modeling irregular geometries and complex boundary conditions. We have built a tool chain to get the surface and volume mesh for ion channel systems, which consists of a set of mesh generation tools. The adaptive finite element solver in our simulator is implemented using the parallel adaptive finite element package Parallel Hierarchical Grid (PHG) developed by one of the authors, which provides the capability of doing large scale parallel computations with high parallel efficiency and the flexibility of choosing high order elements to achieve high order accuracy. The simulator is applied to a real transmembrane protein, the gramicidin A (gA) channel protein, to calculate the electrostatic potential, ion concentrations and I - V curve, with which both primitive and transformed PNP equations are studied and their numerical performances are compared. To further validate the method, we also apply the simulator to two other ion channel systems, the voltage dependent anion channel (VDAC) and α-Hemolysin (α-HL). The simulation results agree well with Brownian dynamics (BD) simulation results and experimental results. Moreover, because ionic finite size effects can be included in PNP model now, we also perform simulations using a size-modified PNP (SMPNP) model on VDAC and α-HL. It is shown that the size effects in SMPNP can effectively lead to reduced current in the channel, and the results are closer to BD simulation results.