Coarse-grained Dynamics Research Articles

Simulations of dynamic crack propagation in brittle materials using nodal cohesive forces and continuum damage mechanics in the distinct element code LDEC

Experimental data indicates that the limiting crack speed in brittle materials is less than the Rayleigh wave speed. One reason for this is that dynamic instabilities produce surface roughness and microcracks that branch from the main crack. These processes increase dissipation near the crack tip over a range of crack speeds. When the scale of observation (or mesh resolution) becomes much larger than the typical sizes of these features, effective-medium theories are required to predict the coarse-grained fracture dynamics. Two approaches to modeling these phenomena are described and used in numerical simulations. The first approach is based on cohesive elements that utilize a rate-dependent weakening law for the nodal cohesive forces. The second approach uses a continuum damage model which has a weakening effect that lowers the effective Rayleigh wave speed in the material surrounding the crack tip. Simulations in this paper show that while both models are capable of increasing the energy dissipated during fracture when the mesh size is larger than the process zone size, only the continuum damage model is able to limit the crack speed over a range of applied loads. Numerical simulations of straight-running cracks demonstrate good agreement between the theoretical predictions of the combined models and experimental data on dynamic crack propagation in brittle materials. Simulations that model crack branching are also presented.

Coarse-grained computations of demixing in dense gas-fluidized beds

We use an "equation-free," coarse-grained computational approach to accelerate molecular dynamics-based computations of demixing (segregation) of dissimilar particles subject to an upward gas flow (gas-fluidized beds). We explore the coarse-grained dynamics of these phenomena in gently fluidized beds of solid mixtures of different densities, typically a slow process for which reasonable continuum models are currently unavailable.

Cytoskeletal dynamics of human erythrocyte.

The human erythrocyte (red blood cell, RBC) demonstrates extraordinary ability to undergo reversible large deformation and fluidity. Such mechanical response cannot be consistently rationalized on the basis of fixed connectivity of the cell cytoskeleton that comprises the spectrin molecular network tethered to phospholipid membrane. Active topological remodeling of spectrin network has been postulated, although detailed models of such dynamic reorganization are presently unavailable. Here we present a coarse-grained cytoskeletal dynamics simulation with breakable protein associations to elucidate the roles of shear stress, specific chemical agents, and thermal fluctuations in cytoskeleton remodeling. We demonstrate a clear solid-to-fluid transition depending on the metabolic energy influx. The solid network's plastic deformation also manifests creep and yield regimes depending on the strain rate. This cytoskeletal dynamics model offers a means to resolve long-standing questions regarding the reference state used in RBC elasticity theory for determining the equilibrium shape and deformation response. In addition, the simulations offer mechanistic insights into the onset of plasticity and void percolation in cytoskeleton. These phenomena may have implication for RBC membrane loss and shape change in the context of hereditary hemolytic disorders such as spherocytosis and elliptocytosis.

Metadynamics in Essential Coordinates: Free Energy Simulation of Conformational Changes

We propose an approach that combines an extraction of collective motions of a molecular system with a sampling of its free energy surface. A recently introduced method of metadynamics allows exploration of the free energy surface of a molecular system by means of coarse-grained dynamics with flooding of free energy minima. This free energy surface is defined as a function of a set of collective variables (e.g., interatomic distances, angles, torsions, and others). In this study, essential coordinates determined by essential dynamics (principle component analysis) were used as collective variables in metadynamics. First, dynamics of the model system (explicitly solvated alanine dipeptide, Ace-Ala-Nme) was simulated by a classical molecular dynamics simulation. The trajectory (1 ns) was then analyzed by essential dynamics to obtain essential coordinates. The free energy surface as a function of the first and second essential coordinates was then explored by metadynamics. The resulting free energy surface is in agreement with other studies of this system. We propose that a combination of these two methods (metadynamics and essential dynamics) has great potential in studies of conformational changes in peptides and proteins.

The Folding Mechanics of a Knotted Protein

An increasing number of proteins are being discovered with a remarkable and somewhat surprising feature, a knot in their native structures. How the polypeptide chain is able to “knot” itself during the folding process to form these highly intricate protein topologies is not known. Here we perform a computational study on the 160-amino-acid homodimeric protein YibK, which, like other proteins in the SpoU family of MTases, contains a deep trefoil knot in its C-terminal region. In this study, we use a coarse-grained Cα-chain representation and Langevin dynamics to study folding kinetics. We find that specific, attractive nonnative interactions are critical for knot formation. In the absence of these interactions, i.e., in an energetics driven entirely by native interactions, knot formation is exceedingly unlikely. Further, we find, in concert with recent experimental data on YibK, two parallel folding pathways that we attribute to an early and a late formation of the trefoil knot, respectively. For both pathways, knot formation occurs before dimerization. A bioinformatics analysis of the SpoU family of proteins reveals further that the critical nonnative interactions may originate from evolutionary conserved hydrophobic segments around the knotted region.

Hydrodynamics of the Zero-Range Process in the Condensation Regime

We argue that the coarse-grained dynamics of the zero-range process in the condensation regime can be described by an extension of the standard hydrodynamic equation obtained from Eulerian scaling even though the system is not locally stationary. Our result is supported by Monte Carlo simulations.

Numerical Studies of a Coarse-grained Approximation for Dynamics of an Atomic Chain

In many applications materials are modeled by a large number of particles (or atoms) where each particle interacts with all others. Near or nearest-neighbor interaction is considered to be a good simplification of the full interaction in the engineering community. However, the resulting system is still too large to be solved using the existing computer power. In this paper we shall use the finite element and/or quasicontinuum idea to both position and velocity variables in order to reduce the number of degrees of freedom. The original and approximate particle systems are related to the discretization of the virtual internal bond model (continuum model). We focus more on the discrete system since the continuum description may not be physically complete as the stress-strain relation is not monotone and thus not necessarily well-posed. We provide numerical justification on how well the coarse-grained solution is close to the-fine grid solution in a temporal average sense. Descriptive title: Numerical studies of coarse-grained dynamics.

Heterogeneous animal group models and their group-level alignment dynamics: An equation-free approach

We study coarse-grained (group-level) alignment dynamics of individual-based animal group models for heterogeneous populations consisting of informed (on preferred directions) and uninformed individuals. The orientation of each individual is characterized by an angle, whose dynamics are nonlinearly coupled with those of all the other individuals, with an explicit dependence on the difference between the individual's orientation and the instantaneous average direction. Choosing convenient coarse-grained variables (suggested by uncertainty quantification methods) that account for rapidly developing correlations during initial transients, we perform efficient computations of coarse-grained steady states and their bifurcation analysis. We circumvent the derivation of coarse-grained governing equations, following an equation-free computational approach.

Coarse-Grained Brownian Dynamics Simulations of Electrophoresis of DNA Molecules from Generalized Reptation Models

We derive a course-grained Brownian dynamics simulation algorithm for the motion of DNA chains under an external electric field. We generalize a model for bulk rheological response of entangled polymers by Schieber et al. [J. Rheol. 2003, 47, 213], allowing both rheology and electrophoresis to be modeled under the same theoretical framework. The effect of the external electric field is included by modifying the strand free energy, which then influences both the monomer sliding dynamics between entanglements and the renewal of entanglements at the chain ends. We also modify the free energy to include the finite extensibility of the chain segments and account for changes in local chain mobility with chain configuration. The resulting model gives a detailed account of the chain dynamics on length scales above the tube diameter and is suitable to model a wide range of field conditions. Simulation results for a constant-field mobility are presented, which fully characterize the model under these conditions. We ensure that the model captures known scaling regimes from analytic theory and compare the predictions to a set of experimental data for constant-field mobility of double-stranded DNA in agarose gels at a range of gel concentrations. This data comparison provides a method to characterize DNA−matrix systems, providing possibilities for modeling of more general field conditions.

Dynamics of the SPRY domain–containing SOCS box protein 2: Flexibility of key functional loops

The SPRY domain was identified originally as a sequence repeat in the dual-specificity kinase splA and ryanodine receptors and subsequently found in many other distinct proteins, including more than 70 encoded in the human genome. It is a subdomain of the B30.2/SPRY domain and is believed to function as a protein-protein interaction module. Three-dimensional structures of several B30.2/SPRY domain-containing proteins have been reported recently: murine SSB-2 in solution by NMR spectroscopy, a Drosophila SSB (GUSTAVUS), and human PRYSPRY protein by X-ray crystallography. The three structures share a core of two antiparallel beta-sheets for the B30.2/SPRY domain but show differences located mainly at one end of the beta-sandwich. Analysis of SSB-2 residues required for interactions with its intracellular ligands has provided insights into B30.2/SPRY binding specificity and identified loop residues critical for the function of this domain. We have investigated the backbone dynamics of SSB-2 by means of Modelfree analysis of its backbone (15)N relaxation parameters and carried out coarse-grained dynamics simulation of B30.2/SPRY domain-containing proteins using normal mode analysis. Translational self-diffusion coefficients of SSB-2 measured using pulsed field gradient NMR were used to confirm the monomeric state of SSB-2 in solution. These results, together with previously reported amide exchange data, highlight the underlying flexibility of the loop regions of B30.2/SPRY domain-containing proteins that have been shown to be important for protein-protein interactions. The underlying flexibility of certain regions of the B30.2/SPRY domain-containing proteins may also contribute to some apparent structural differences observed between GUSTAVUS or PRYSPRY and SSB-2.

Revaluation of the first-order upwind difference scheme to solve coarse-grained master equations

This study addresses the initial-boundary value problem of coarse-grained probability measure on the state space in which a differentiable vector field v is given and, as a consequence, the differenced continuity equation using the first-order upwind difference scheme (UDS) based on the finite volume method appears as the physical substance on the coarse-grained dynamics. At first, the UDS is theoretically shown to be equivalent to a class of coarse-grained master equations (CGME), brought by a principle that we cannot distinguish state points in the same partition with each other. The principle is based on the formulation of non-equilibrium statistical mechanics to resolve the macroscopic irreversibility. Moreover the entropy production evaluated by the UDS is also shown to be in accord with the average volume contraction rate in the steady state. This is essential for the non-equilibrium statistical dynamics and was numerically confirmed. Under the principle of coarse graining the UDS is very superior to the conventional Monte-Carlo method in computer time and storage and is very useful to solve the CGME.

BLOCK ENTROPY ANALYSIS OF HEART RATE VARIABILITY SIGNALS

In this paper we present a novel approach to the analysis of Heat Rate Variability (HRV) data, by coarse-graining analysis using the estimation of Block Entropies with the technique of lumping. HRV time series are generated from long recordings of Electrocardiograms (ECGs) and are then filtered in order to produce a coarse-grained symbolic dynamics. Block Entropy analysis is applied to these dynamics in order to examine its coarse-grained statistics. Our data set is comprised of two subsets, one of healthy subjects and another of Coronary Artery Disease (CAD) patients. It is found that Entropy analysis provides a quick and efficient tool for the differentiation of these series according to subject category. Healthy subjects provided more complex statistics compared to patients; specifically, the healthy data files provided higher values of block Entropies compared to patient ones. We also compare these results with the Correlation Dimension Estimation in order to establish coherency. We believe that this analysis may provide a useful statistical method towards the better understanding of the human cardiac system.

Dynamics of random packings in granular flow

We present a multiscale simulation algorithm for amorphous materials, which we illustrate and validate in a canonical case of dense granular flow. Our algorithm is based on the recently proposed spot model, where particles in a dense random packing undergo chainlike collective displacements in response to diffusing "spots" of influence, carrying a slight excess of interstitial free volume. We reconstruct the microscopic dynamics of particles from the "coarse grained" dynamics of spots by introducing a localized particle relaxation step after each spot-induced block displacement, simply to enforce packing constraints with a (fairly arbitrary) soft-core repulsion. To test the model, we study to what extent it can describe the dynamics of up to 135,000 frictional, viscoelastic spheres in granular drainage simulated by the discrete-element method (DEM). With only five fitting parameters (the radius, volume, diffusivity, drift velocity, and injection rate of spots), we find that the spot simulations are able to largely reproduce not only the mean flow and diffusion, but also some subtle statistics of the flowing packings, such as spatial velocity correlations and many-body structural correlations. The spot simulations run over 100 times faster than the DEM and demonstrate the possibility of multiscale modeling for amorphous materials, whenever a suitable model can be devised for the coarse-grained spot dynamics.

Coarse Graining the Dynamics of Coupled Oscillators

We present an equation-free computational approach to the study of the coarse-grained dynamics of finite assemblies of nonidentical coupled oscillators at and near full synchronization. We use coarse-grained observables which account for the (rapidly developing) correlations between phase angles and natural frequencies. Exploiting short bursts of appropriately initialized detailed simulations, we circumvent the derivation of closures for the long-term dynamics of the assembly statistics.

Coarse-grained description of a passive scalar

The issue of the parameterization of small-scale dynamics is addressed in the context of passive-scalar turbulence. The basic idea of our strategy is to identify dynamical equations for the coarse-grained scalar dynamics starting from closed equations for two-point statistical indicators. With the aim of performing a fully-analytical study, the Kraichnan advection model is considered. The white-in-time character of the latter model indeed leads to closed equations for the equal-time scalar correlation functions. The classical closure problem however still arises if a standard filtering procedure is applied to those equations in the spirit of the large-eddy-simulation strategy. We show both how to perform exact closures and how to identify the corresponding coarse-grained scalar evolution.

Short- and Long-Term Joint Symbolic Dynamics of Heart Rate and Blood Pressure in Dilated Cardiomyopathy

Autonomic cardiovascular control involves complex interactions of heart rate and blood pressure. In patients with dilated cardiomyopathy (DCM), this control is impaired and parameters for its quantification might be of prognostic importance. In this paper, we introduce methods based on joint symbolic dynamics (JSD) for the enhanced analysis of heart rate and blood pressure interactions. To assess the coarse-grained dynamics beat-to-beat changes of heart rate and blood pressure are encoded in symbol strings. Subsequently, the distribution properties of short symbol sequences (words) as well as the scaling properties of the whole symbol string are assessed. The comparison of joint symbolic heart rate and blood pressure dynamics in DCM (n = 75) with those in healthy controls (n = 75) showed significant changes. Both, the distribution of words and the scaling properties indicate a loss in heart rate dynamics associated with blood pressure regulation in DCM. In conclusion, the analyses of short- and long-term JSDs provide insights into complex physiological heart rate and blood pressure interactions and furthermore reveal patho-physiological cardiovascular control in DCM.

Nanoparticle Dynamics: A Multiscale Analysis of the Liouville Equation

A theory of nanoparticle dynamics based on scaling arguments and the Liouville equation is presented. We start with a delineation of the scales characterizing the behavior of the nanoparticle/host fluid system. Asymptotic expansions, multiple time and space scale techniques, the resulting coarse-grained dynamics of the probability densities of the Fokker-Planck-Chandrasekhar (FPC) type for the nanoparticle(s), and the hydrodynamic models of the host medium are obtained. Collections of nanoparticles are considered so that problems such as viral self-assembly and the transition from a particle suspension to a solid porous matrix can be addressed via a deductive approach that starts with the Liouville equation and a calibrated atomic force field, and yields a generalized FPC equation. Extensions allowing for the investigation of the rotation and deformation of the nanoparticles are considered in the context of the space-warping formalism. Thermodynamic forces and dissipative effects are accounted for. The notion of configuration-dependent drag coefficients and their implications for coagulation and consolidation are shown to be natural consequences of the analysis. All results are obtained via formal asymptotic expansions in mass, size, and other physical and kinetic parameter ratios.

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.

Mapping potential energy surfaces.

A recently proposed dynamical method [A. Laio and M. Parrinello, Proc. Natl. Acad. Sci. U.S.A. 99, 12562 (2002)] allows us to globally sample the free energy surface. This approach uses a coarse-grained non-Markovian dynamics to bias microscopic atomic trajectories. After a sufficiently long simulation time, the global free energy surface can be reconstructed from the non-Markovian dynamics. Here we apply this scheme to study the T=0 free energy surface, i.e., the potential energy surface in coarse-grained space. We show that the accuracy of the reconstructed potential energy surface can be dramatically improved by a simple postprocessing procedure with only minor computational overhead. We illustrate this approach by conducting conformational analysis on a small organic molecule, demonstrating its superiority over traditional unbiased approaches in sampling potential energy surfaces in coarse-grained space.

A new perspective on the coarse-grained dynamics of fluids.

A computational methodology is presented that is designed to model, at a coarse-grained level, the mesoscale dynamics of fluids and potentially other forms of soft matter. Within a molecular dynamics simulation, "ghost" particles of a specific size, corresponding to the fundamental length-scale of coarse-graining, are used as micro-probes designed to respond to local mesoscale fluid flows and stress gradients. A subsequent coarse-grained model is then developed that incorporates both the coarse-grained mesoscale dynamics and isothermal compressibility of the original microscopic system. The method is applied to water and methanol. A contrast with dissipative particle dynamics (DPD) is also presented.