Spatially distributed stochastic systems: Equation-free and equation-assisted preconditioned computations

Spatially distributed problems are often approximately modeled in terms of partial differential equations (PDEs) for appropriate coarse-grained quantities (e.g., concentrations). The derivation of accurate such PDEs starting from finer scale, atomistic models, and using suitable averaging is often a challenging task; approximate PDEs are typically obtained through mathematical closure procedures (e.g., mean field approximations). In this paper, we show how such approximate macroscopic PDEs can be exploited in constructing preconditioners to accelerate stochastic computations for spatially distributed particle-based process models. We illustrate how such preconditioning can improve the convergence of equation-free coarse-grained methods based on coarse timesteppers. Our model problem is a stochastic reaction-diffusion model capable of exhibiting Turing instabilities.     

Eliminating fast reactions in stochastic simulations of biochemical networks: a bistable genetic switch

In many stochastic simulations of biochemical reaction networks, it is desirable to "coarse grain" the reaction set, removing fast reactions while retaining the correct system dynamics. Various coarse-graining methods have been proposed, but it remains unclear which methods are reliable and which reactions can safely be eliminated. We address these issues for a model gene regulatory network that is particularly sensitive to dynamical fluctuations: a bistable genetic switch. We remove protein-DNA and/or protein-protein association-dissociation reactions from the reaction set using various coarse-graining strategies. We determine the effects on the steady-state probability distribution function and on the rate of fluctuation-driven switch flipping transitions. We find that protein-protein interactions may be safely eliminated from the reaction set, but protein-DNA interactions may not. We also find that it is important to use the chemical master equation rather than macroscopic rate equations to compute effective propensity functions for the coarse-grained reactions.     

Structure-based coarse-graining in liquid slabs

Structure-based coarse-graining relies on matching the pair correlation functions of a reference (atomistic) and a coarse-grained system. As such, it is designed for systems with uniform density distributions. Here, we demonstrate how it can be generalized for inhomogeneous systems by coarse-graining slabs of liquid water and methanol in vacuum, as well as a single benzene molecule at the water-vacuum interface. Our conclusion is that coarse-graining performed in inhomogeneous systems improves thermodynamic properties and the structure of interfaces without significant alterations to the local structure of the bulk liquid.     

Coarse-grained lattice kinetic Monte Carlo simulation of systems of strongly interacting particles

A general approach is presented for spatially coarse-graining lattice kinetic Monte Carlo (LKMC) simulations of systems containing strongly interacting particles. While previous work has relied on approximations that are valid in the limit of weak interactions, here we show that it is possible to compute coarse-grained transition rates for strongly interacting systems without a large computational burden. A two-dimensional square lattice is employed on which a collection of (supersaturated) strongly interacting particles is allowed to reversibly evolve into clusters. A detailed analysis is presented of the various approximations applied in LKMC coarse graining, and a number of numerical closure rules are contrasted and compared. In each case, the overall cluster size distribution and individual cluster structures are used to assess the accuracy of the coarse-graining approach. The resulting closure approach is shown to provide an excellent coarse-grained representation of the systems considered in this study.     

Scaling behavior of quantum nanosystems: emergence of quasi-particles, collective modes, and mixed exchange symmetry states

Examples of quantum nanosystems are graphene nanoribbons, molecular wires, and superconducting nanoparticles. The objective of the multiscale theory presented here is to provide a new perspective on the coupling of processes across scales in space and time underlying the dynamics of these systems. The long range objective for this multiscale approach is to serve as an efficient computational algorithm. Long space-time dynamics is derived using a perturbation expansion in the ratio ? of the nearest-neighbor distance to a nanometer-scale characteristic length and a theorem on the equivalence of long time-averages and expectation values. This dynamics is shown to satisfy a coarse-grained wave equation (CGWE) which takes a Schro?dinger-like form with modified masses and interactions. The scaling of space and time is determined by the orders of magnitude of various contributions to the N-body potential. If the spatial scale of the coarse-graining is too large, the CGWE would imply an unbounded growth of gradients; if it is too short, the system's size would display uncontrolled growth inappropriate for the bound states of interest, i.e., collective motion or migration within a stable nanoassembly. The balance of these two extremes removes arbitrariness in the choice of the scaling of space-time. Since the long-scale dynamics of each Fermion involves its interaction with many others, we hypothesize that the solutions of the CGWE have mean-field character to good approximation, i.e., can be factorized into single-particle functions. This leads to a coarse-grained mean-field approximation that is distinct in character from traditional Hartree-Fock theory. A variational principle is used to derive equations for the single-particle functions. This theme is developed and used to derive an equation for low-lying disturbances from the ground state corresponding to long wavelength density disturbances or long-scale migration. An algorithm for the efficient simulation of quantum nanosystems is suggested.     

Mesoscale model of polymer melt structure: self-consistent mapping of molecular correlations to coarse-grained potentials

Development and application of coarse-graining methods to condensed phases of macromolecules is an active area of research. Multiscale modeling of polymeric systems using coarse-graining methods presents unique challenges. Here we apply a coarse-graining method that self-consistently maps structural correlations from detailed molecular dynamics (MD) simulations of alkane oligomers onto coarse-grained potentials using a combination of MD and inverse Monte Carlo methods. Once derived, the coarse-grained potentials allow computationally efficient sampling of ensemble of conformations of significantly longer polyethylene chains. Conformational properties derived from coarse-grained simulations are in excellent agreement with experiments. The level of coarse graining provides a control over the balance of computational efficiency and retention of chemical identity of the underlying polymeric system. Challenges to extension and application of this and similar structure-based coarse-graining methods to model dynamics and phase behavior in polymeric systems are briefly discussed.     

On developing coarse-grained models for biomolecular simulation: a review

So-called coarse-grained models are a popular type of model for accessing long time scales in simulations of biomolecular processes. Such models are coarse-grained with respect to atomic models. But any modelling of processes or substances involves coarse-graining, i.e. the elimination of non-essential degrees of freedom and interactions from a more fine-grained level of modelling. The basic ingredients of developing coarse-grained models based on the properties of fine-grained models are reviewed, together with the conditions that must be satisfied in order to preserve the correct physical mechanisms in the coarse-graining process. This overview should help the reader to determine how realistic a coarse-grained model of a biomolecular system is, i.e. whether it reflects the underlying physical mechanisms or merely provides a set of pretty pictures of the process or substances of interest.     

Binomial tau-leap spatial stochastic simulation algorithm for applications in chemical kinetics

In cell biology, cell signaling pathway problems are often tackled with deterministic temporal models, well mixed stochastic simulators, and/or hybrid methods. But, in fact, three dimensional stochastic spatial modeling of reactions happening inside the cell is needed in order to fully understand these cell signaling pathways. This is because noise effects, low molecular concentrations, and spatial heterogeneity can all affect the cellular dynamics. However, there are ways in which important effects can be accounted without going to the extent of using highly resolved spatial simulators (such as single-particle software), hence reducing the overall computation time significantly. We present a new coarse grained modified version of the next subvolume method that allows the user to consider both diffusion and reaction events in relatively long simulation time spans as compared with the original method and other commonly used fully stochastic computational methods. Benchmarking of the simulation algorithm was performed through comparison with the next subvolume method and well mixed models (MATLAB), as well as stochastic particle reaction and transport simulations (CHEMCELL, Sandia National Laboratories). Additionally, we construct a model based on a set of chemical reactions in the epidermal growth factor receptor pathway. For this particular application and a bistable chemical system example, we analyze and outline the advantages of our presented binomial tau-leap spatial stochastic simulation algorithm, in terms of efficiency and accuracy, in scenarios of both molecular homogeneity and heterogeneity.     

Dynamics of large proteins through hierarchical levels of coarse-grained structures

Elastic network models have been successful in elucidating the largest scale collective motions of proteins. These models are based on a set of highly coupled springs, where only the close neighboring amino acids interact, without any residue specificity. Our objective here is to determine whether the equivalent cooperative motions can be obtained upon further coarse-graining of the protein structure along the backbone. The influenza virus hemagglutinin A (HA), composed of N = 1509 residues, is utilized for this analysis. Elastic network model calculations are performed for coarse-grained HA structures containing only N/2, N/10, N/20, and N/40 residues along the backbone. High correlations (>0.95) between residue fluctuations are obtained for the first dominant (slowest) mode of motion between the original model and the coarse-grained models. In the case of coarse-graining by a factor of 1/40, the slowest mode shape for HA is reconstructed for all residues by successively selecting different subsets of residues, shifting one residue at a time. The correlation for this reconstructed first mode shape with the original all-residue case is 0.73, while the computational time is reduced by about three orders of magnitude. The reduction in computational time will be much more significant for larger targeted structures. Thus, the dominant motions of protein structures are robust enough to be captured at extremely high levels of coarse-graining. And more importantly, the dynamics of extremely large complexes are now accessible with this new methodology.     

Reaction Brownian dynamics and the effect of spatial fluctuations on the gain of a push-pull network

Brownian Dynamics algorithms have been widely used for simulating systems in soft-condensed matter physics. In recent times, their application has been extended to the simulation of coarse-grained models of biochemical networks. In these models, components move by diffusion and interact with one another upon contact. However, when reactions are incorporated into a Brownian dynamics algorithm, care must be taken to avoid violations of the detailed-balance rule, which would introduce systematic errors in the simulation. We present a Brownian dynamics algorithm for simulating reaction-diffusion systems that rigorously obeys detailed balance for equilibrium reactions. By comparing the simulation results to exact analytical results for a bimolecular reaction, we show that the algorithm correctly reproduces both equilibrium and dynamical quantities. We apply our scheme to a "push-pull" network in which two antagonistic enzymes covalently modify a substrate. Our results highlight that spatial fluctuations of the network components can strongly reduce the gain of the response of a biochemical network.     

Consistent and transferable coarse-grained model for semidilute polymer solutions in good solvent

We present a coarse-grained model for linear polymers with a tunable number of effective atoms (blobs) per chain interacting by intra- and intermolecular potentials obtained at zero density. We show how this model is able to accurately reproduce the universal properties of the underlying solution of athermal linear chains at various levels of coarse-graining and in a range of chain densities which can be widened by increasing the spatial resolution of the multiblob representation, i.e., the number of blobs per chain. The present model is unique in its ability to quantitatively predict thermodynamic and large scale structural properties of polymer solutions deep in the semidilute regime with a very limited computational effort, overcoming most of the problems related to the simulations of semidilute polymer solutions in good solvent conditions.     

Coarse-grained nonequilibrium approach to the molecular modeling of permeation through microporous membranes

We present a modeling technique that combines a statistical-mechanical coarse-graining scheme with a nonequilibrium molecular simulation algorithm to provide an efficient simulation of steady-state permeation across a microporous material. The coarse-graining scheme is based on the mapping of an atomistic model to a lattice using multidimensional free-energy and transition-state calculations. The nonequilibrium simulation algorithm is a stochastic, lattice version of the recently developed atomistic dual-control-volume grand canonical molecular dynamics. We demonstrate the approach on a model of methane permeating through a bulk portion of siliceous zeolite ZK4 at 300 K under imposed fugacity differences. We predict the coarse-grained (cage-level) density profiles and observe the development of nonlinearities as the magnitude of the fugacity difference is increased. From the net flux of methane we also predict a mean permeability coefficient under the various conditions. The simulation results are obtained over time scales on the order of microseconds and length scales on the order of dozens of nanometers.     

Fluctuating hydrodynamics for multiscale simulation of inhomogeneous fluids: mapping all-atom molecular dynamics to capillary waves

We introduce a multiscale framework to simulate inhomogeneous fluids by coarse-graining an all-atom molecular dynamics (MD) trajectory onto sequential snapshots of hydrodynamic fields. We show that the field representation of an atomistic trajectory is quantitatively described by a dynamic field-theoretic model that couples hydrodynamic fluctuations with a Ginzburg-Landau free energy. For liquid-vapor interfaces of argon and water, the parameters of the field model can be adjusted to reproduce the bulk compressibility and surface tension calculated from the positions and forces of atoms in an MD simulation. These optimized parameters also enable the field model to reproduce the static and dynamic capillary wave spectra calculated from atomistic coordinates at the liquid-vapor interface. In addition, we show that a density-dependent gradient coefficient in the Ginzburg-Landau free energy enables bulk and interfacial fluctuations to be controlled separately. For water, this additional degree of freedom is necessary to capture both the bulk compressibility and surface tension emergent from the atomistic trajectory. The proposed multiscale framework illustrates that bottom-up coarse-graining and top-down phenomenology can be integrated with quantitative consistency to simulate the interfacial fluctuations in nanoscale transport processes.     

Coarse-graining errors and numerical optimization using a relative entropy framework

The ability to generate accurate coarse-grained models from reference fully atomic (or otherwise "first-principles") ones has become an important component in modeling the behavior of complex molecular systems with large length and time scales. We recently proposed a novel coarse-graining approach based upon variational minimization of a configuration-space functional called the relative entropy, S(rel), that measures the information lost upon coarse-graining. Here, we develop a broad theoretical framework for this methodology and numerical strategies for its use in practical coarse-graining settings. In particular, we show that the relative entropy offers tight control over the errors due to coarse-graining in arbitrary microscopic properties, and suggests a systematic approach to reducing them. We also describe fundamental connections between this optimization methodology and other coarse-graining strategies like inverse Monte Carlo, force matching, energy matching, and variational mean-field theory. We suggest several new numerical approaches to its minimization that provide new coarse-graining strategies. Finally, we demonstrate the application of these theoretical considerations and algorithms to a simple, instructive system and characterize convergence and errors within the relative entropy framework.     

Coarse-grained computations for a micellar system

We establish, through coarse-grained computation, a connection between traditional, continuum numerical algorithms (initial value problems as well as fixed point algorithms), and atomistic simulations of the Larson model of micelle formation. The procedure hinges on the (expected) evolution of a few slow, coarse-grained mesoscopic observables of the Monte Carlo simulation, and on (computational) time scale separation between these and the remaining "slaved," fast variables. Short bursts of appropriately initialized atomistic simulation are used to estimate the (coarse grained, deterministic) local dynamics of the evolution of the observables. These estimates are then in turn used to accelerate the evolution to computational stationarity through traditional continuum algorithms (forward Euler integration, Newton-Raphson fixed point computation). This "equation-free" framework, bypassing the derivation of explicit, closed equations for the observables (e.g., equations of state), may provide a computational bridge between direct atomistic/stochastic simulation and the analysis of its macroscopic, system-level consequences.