Multiscale modeling of emergent materials: biological and soft matter

Uncertainty in integrative structural modeling

Integrative structural modeling uses multiple types of input information and proceeds in four stages: (i) gathering information, (ii) designing model representation and converting information into a scoring function, (iii) sampling good-scoring models, and (iv) analyzing models and information. In the first stage, uncertainty originates from data that are sparse, noisy, ambiguous, or derived from heterogeneous samples. In the second stage, uncertainty can originate from a representation that is too coarse for the available information or a scoring function that does not accurately capture the information. In the third stage, the major source of uncertainty is insufficient sampling. In the fourth stage, clustering, cross-validation, and other methods are used to estimate the precision and accuracy of the models and information.

Recent advances in transferable coarse-grained modeling of proteins

Computer simulations are indispensable tools for studying the structure and dynamics of biological macromolecules. Biochemical processes occur on different scales of length and time. Atomistic simulations cannot cover the relevant spatiotemporal scales at which the cellular processes occur. To address this challenge, coarse-grained (CG) modeling of the biological systems is employed. Over the last few years, many CG models for proteins continue to be developed. However, many of them are not transferable with respect to different systems and different environments. In this review, we discuss those CG protein models that are transferable and that retain chemical specificity. We restrict ourselves to CG models of soluble proteins only. We also briefly review recent progress made in the multiscale hybrid all-atom/CG simulations of proteins.

Coarse-Graining of TIP4P/2005, TIP4P-Ew, SPC/E, and TIP3P to Monatomic Anisotropic Water Models Using Relative Entropy Minimization

Coarse-grained models are becoming a competitive alternative for modeling processes that occur over time and length scales beyond the reach of fully atomistic molecular simulations. Ideally, coarse-grained models should not only achieve high computational efficiency but also provide accurate predictions and fundamental insight into the role of molecular interactions, the characteristic behavior, and properties of the system they model. In this work we derive a series of monatomic coarse-grained water models mX(REM) from the most popular atomistic water models X = TIP3P, SPC/E, TIP4P-Ew, and TIP4P/2005, using the relative entropy minimization (REM) method. Each coarse-grained water molecule is represented by a single particle that interacts through short-ranged anisotropic interactions that encourage the formation of "hydrogen-bonded" structures. We systematically investigate the features of the coarse-grained models in reproducing over 20 structural, dynamic, and thermodynamic properties of the reference atomistic water models-including the existence and locus of the characteristic density anomaly. The mX(REM) coarse-grained models reproduce quite faithfully the radial and angular distribution function of water, produce a temperature of maximum density (TMD), and stabilize the ice I crystal. Moreover, the ratio between the TMD and the melting temperature of the crystal in the mX(REM) models and liquid-ice equilibrium properties show reasonable agreement with the results of the corresponding atomistic models. The mX(REM) models, however, severely underestimate the cohesive energy of the condensed water phases. We investigate which specific limitations of the coarse-grained models arise from the REM methodology, from the monatomic nature of the models, and from the Stillinger-Weber interaction potential form. Our analysis indicates that a small compromise in the accuracy of structural properties can result in a significant increase of the overall accuracy and representability of the coarse-grained water models. We evaluate the accuracy of the atomistic and the monatomic anisotropic coarse-grained water models, including the mW water model, in reproducing experimental water properties. We find that mW and mTIP4P/2005(REM) score closer to experiment than widely used atomistic water models. We conclude that monatomic models of water with short-range, anisotropic "hydrogen-bonding" three-body interactions can be competitive in accuracy with fully atomistic models for the study of a wide range of properties and phenomena at less than 1/100th of the computational cost.

Investigation of coarse-grained mappings via an iterative generalized Yvon-Born-Green method

Low resolution coarse-grained (CG) models enable highly efficient simulations of complex systems. The interactions in CG models are often iteratively refined over multiple simulations until they reproduce the one-dimensional (1-D) distribution functions, e.g., radial distribution functions (rdfs), of an all-atom (AA) model. In contrast, the multiscale coarse-graining (MS-CG) method employs a generalized Yvon-Born-Green (g-YBG) relation to determine CG potentials directly (i.e., without iteration) from the correlations observed for the AA model. However, MS-CG models do not necessarily reproduce the 1-D distribution functions of the AA model. Consequently, recent studies have incorporated the g-YBG equation into iterative methods for more accurately reproducing AA rdfs. In this work, we consider a theoretical framework for an iterative g-YBG method. We numerically demonstrate that the method robustly determines accurate models for both hexane and also a more complex molecule, 3-hexylthiophene. By examining the MS-CG and iterative g-YBG models for several distinct CG representations of both molecules, we investigate the approximations of the MS-CG method and their sensitivity to the CG mapping. More generally, we explicitly demonstrate that CG models often reproduce 1-D distribution functions of AA models at the expense of distorting the cross-correlations between the corresponding degrees of freedom. In particular, CG models that accurately reproduce intramolecular 1-D distribution functions may still provide a poor description of the molecular conformations sampled by the AA model. We demonstrate a simple and predictive analysis for determining CG mappings that promote an accurate description of these molecular conformations.

Modeling biology spanning different scales: an open challenge

It is coming nowadays more clear that in order to obtain a unified description of the different mechanisms governing the behavior and causality relations among the various parts of a living system, the development of comprehensive computational and mathematical models at different space and time scales is required. This is one of the most formidable challenges of modern biology characterized by the availability of huge amount of high throughput measurements. In this paper we draw attention to the importance of multiscale modeling in the framework of studies of biological systems in general and of the immune system in particular.

The power of coarse graining in biomolecular simulations

Computational modeling of biological systems is challenging because of the multitude of spatial and temporal scales involved. Replacing atomistic detail with lower resolution, coarse grained (CG), beads has opened the way to simulate large-scale biomolecular processes on time scales inaccessible to all-atom models. We provide an overview of some of the more popular CG models used in biomolecular applications to date, focusing on models that retain chemical specificity. A few state-of-the-art examples of protein folding, membrane protein gating and self-assembly, DNA hybridization, and modeling of carbohydrate fibers are used to illustrate the power and diversity of current CG modeling.

Probing the interplay between amyloidogenic proteins and membranes using lipid monolayers and bilayers

Many degenerative diseases such as Alzheimer's and Parkinson's involve proteins that have a tendency to misfold and aggregate eventually forming amyloid fibers. This review describes the use of monolayers, bilayers, supported membranes, and vesicles as model systems that have helped elucidate the mechanisms and consequences of the interactions between amyloidogenic proteins and membranes. These are twofold: membranes favor the formation of amyloid structures and these induce damage in those membranes. We describe studies that show how interfaces, especially charged ones, favor amyloidogenic protein aggregation by several means. First, surfaces increase the effective protein concentration reducing a three-dimensional system to a two-dimensional one. Second, charged surfaces allow electrostatic interactions with the protein. Anionic lipids as well as rafts, rich in cholesterol and gangliosides, prove to play an especially important role. Finally, these amphipathic systems also offer a hydrophobic environment favoring conformational changes, oligomerization, and eventual formation of mature fibers. In addition, we examine several models for membrane permeabilization: protein pores, leakage induced by extraction of lipids, chaotic pores, and membrane tension, presenting illustrative examples of experimental evidence in support of these models. The picture that emerges from recent work is one where more than one mechanism is in play. Which mechanism prevails depends on the protein, its aggregation state, and the lipid environment in which the interactions occur.

Perspective: Coarse-grained models for biomolecular systems

By focusing on essential features, while averaging over less important details, coarse-grained (CG) models provide significant computational and conceptual advantages with respect to more detailed models. Consequently, despite dramatic advances in computational methodologies and resources, CG models enjoy surging popularity and are becoming increasingly equal partners to atomically detailed models. This perspective surveys the rapidly developing landscape of CG models for biomolecular systems. In particular, this review seeks to provide a balanced, coherent, and unified presentation of several distinct approaches for developing CG models, including top-down, network-based, native-centric, knowledge-based, and bottom-up modeling strategies. The review summarizes their basic philosophies, theoretical foundations, typical applications, and recent developments. Additionally, the review identifies fundamental inter-relationships among the diverse approaches and discusses outstanding challenges in the field. When carefully applied and assessed, current CG models provide highly efficient means for investigating the biological consequences of basic physicochemical principles. Moreover, rigorous bottom-up approaches hold great promise for further improving the accuracy and scope of CG models for biomolecular systems.

More than the sum of its parts: coarse-grained peptide-lipid interactions from a simple cross-parametrization

Interfacial systems are at the core of fascinating phenomena in many disciplines, such as biochemistry, soft-matter physics, and food science. However, the parametrization of accurate, reliable, and consistent coarse-grained (CG) models for systems at interfaces remains a challenging endeavor. In the present work, we explore to what extent two independently developed solvent-free CG models of peptides and lipids--of different mapping schemes, parametrization methods, target functions, and validation criteria--can be combined by only tuning the cross-interactions. Our results show that the cross-parametrization can reproduce a number of structural properties of membrane peptides (for example, tilt and hydrophobic mismatch), in agreement with existing peptide-lipid CG force fields. We find encouraging results for two challenging biophysical problems: (i) membrane pore formation mediated by the cooperative action of several antimicrobial peptides, and (ii) the insertion and folding of the helix-forming peptide WALP23 in the membrane.

Automated optimization of water-water interaction parameters for a coarse-grained model

We have developed an automated parameter optimization software framework (ParOpt) that implements the Nelder-Mead simplex algorithm and applied it to a coarse-grained polarizable water model. The model employs a tabulated, modified Morse potential with decoupled short- and long-range interactions incorporating four water molecules per interaction site. Polarizability is introduced by the addition of a harmonic angle term defined among three charged points within each bead. The target function for parameter optimization was based on the experimental density, surface tension, electric field permittivity, and diffusion coefficient. The model was validated by comparison of statistical quantities with experimental observation. We found very good performance of the optimization procedure and good agreement of the model with experiment.

Nanotribology of biopolymer brushes in aqueous solution using dissipative particle dynamics simulations: an application to PEG covered liposomes in a theta solvent

We undertake the investigation of sheared polymer chains grafted onto flat surfaces to model liposomes covered with polyethylene glycol brushes as a case study for the mechanisms of efficient drug delivery in biologically relevant situations, for example, as carriers for topical treatments of illnesses in the human vasculature. For these applications, specific rheological properties are required, such as low viscosity at high shear rates, to improve the transport of the liposomes. Therefore, extensive non-equilibrium, coarse-grained dissipative particle dynamics simulations of polymer brushes of various lengths and shear rates are performed to obtain the average viscosity and the friction coefficient of the system as functions of the shear rate and polymerization degree under theta-solvent conditions, and we find that the brushes experience considerable shear thinning at large shear rates. The viscosity (η) and the friction coefficient (μ) are shown to obey the scaling laws η ∼ γ dot above (-0.31) and μ ∼ γ dot above (0.69) at high shear rates (γ dot above) in a theta solvent, irrespective of the degree of polymerization of brushes. These results confirm recent scaling predictions and reproduce very well trends in measurements of the viscosity at a high shear rate (γ dot above) of red blood cells in a liposome containing medium.

Morphological transition difference of linear and cyclic block copolymer with polymer blending in a selective solvent by combining dissipative particle dynamics and all-atom molecular dynamics simulations based on the ABEEM polarizable force field

AAMD based on ABEEM PFF were performed to obtain reliable DPD parameters for morphological transition.

DNA hybridization kinetics: zippering, internal displacement and sequence dependence

Although the thermodynamics of DNA hybridization is generally well established, the kinetics of this classic transition is less well understood. Providing such understanding has new urgency because DNA nanotechnology often depends critically on binding rates. Here, we explore DNA oligomer hybridization kinetics using a coarse-grained model. Strand association proceeds through a complex set of intermediate states, with successful binding events initiated by a few metastable base-pairing interactions, followed by zippering of the remaining bonds. But despite reasonably strong interstrand interactions, initial contacts frequently dissociate because typical configurations in which they form differ from typical states of similar enthalpy in the double-stranded equilibrium ensemble. Initial contacts must be stabilized by two or three base pairs before full zippering is likely, resulting in negative effective activation enthalpies. Non-Arrhenius behavior arises because the number of base pairs required for nucleation increases with temperature. In addition, we observe two alternative pathways—pseudoknot and inchworm internal displacement—through which misaligned duplexes can rearrange to form duplexes. These pathways accelerate hybridization. Our results explain why experimentally observed association rates of GC-rich oligomers are higher than rates of AT- rich equivalents, and more generally demonstrate how association rates can be modulated by sequence choice.

Hierarchical Multiscale Modeling of Macromolecules and their Assemblies

Soft materials (e.g., enveloped viruses, liposomes, membranes and supercooled liquids) simultaneously deform or display collective behaviors, while undergoing atomic scale vibrations and collisions. While the multiple space-time character of such systems often makes traditional molecular dynamics simulation impractical, a multiscale approach has been presented that allows for long-time simulation with atomic detail based on the co-evolution of slowly-varying order parameters (OPs) with the quasi-equilibrium probability density of atomic configurations. However, this approach breaks down when the structural change is extreme, or when nearest-neighbor connectivity of atoms is not maintained. In the current study, a self-consistent approach is presented wherein OPs and a reference structure co-evolve slowly to yield long-time simulation for dynamical soft-matter phenomena such as structural transitions and self-assembly. The development begins with the Liouville equation for N classical atoms and an ansatz on the form of the associated N-atom probability density. Multiscale techniques are used to derive Langevin equations for the coupled OP-configurational dynamics. The net result is a set of equations for the coupled stochastic dynamics of the OPs and centers of mass of the subsystems that constitute a soft material body. The theory is based on an all-atom methodology and an interatomic force field, and therefore enables calibration-free simulations of soft matter, such as macromolecular assemblies.

The Theory of Ultra-Coarse-Graining. 1. General Principles

Coarse-grained (CG) models provide a computationally efficient means to study biomolecular and other soft matter processes involving large numbers of atoms correlated over distance scales of many covalent bond lengths and long time scales. Variational methods based on information from simulations of finer-grained (e.g., all-atom) models, for example the multiscale coarse-graining (MS-CG) and relative entropy minimization methods, provide attractive tools for the systematic development of CG models. However, these methods have important drawbacks when used in the "ultra-coarse-grained" (UCG) regime, e.g., at a resolution level coarser or much coarser than one amino acid residue per effective CG particle in proteins. This is due to the possible existence of multiple metastable states "within" the CG sites for a given UCG model configuration. In this work, systematic variational UCG methods are presented that are specifically designed to CG entire protein domains and subdomains into single effective CG particles. This is accomplished by augmenting existing effective particle CG schemes to allow for discrete state transitions and configuration-dependent resolution. Additionally, certain conclusions of this work connect back to single-state force matching and open up new avenues for method development in that area. These results provide a formal statistical mechanical basis for UCG methods related to force matching and relative entropy CG methods and suggest practical algorithms for constructing optimal approximate UCG models from fine-grained simulation data.

Optimizing DNA nanotechnology through coarse-grained modeling: a two-footed DNA walker

DNA has enormous potential as a programmable material for creating artificial nanoscale structures and devices. For more complex systems, however, rational design and optimization can become difficult. We have recently proposed a coarse-grained model of DNA that captures the basic thermodynamic, structural, and mechanical changes associated with the fundamental process in much of DNA nanotechnology, the formation of duplexes from single strands. In this article, we demonstrate that the model can provide powerful insight into the operation of complex nanotechnological systems through a detailed investigation of a two-footed DNA walker that is designed to step along a reusable track, thereby offering the possibility of optimizing the design of such systems. We find that applying moderate tension to the track can have a large influence on the operation of the walker, providing a bias for stepping forward and helping the walker to recover from undesirable overstepped states. Further, we show that the process by which spent fuel detaches from the walker can have a significant impact on the rebinding of the walker to the track, strongly influencing walker efficiency and speed. Finally, using the results of the simulations, we propose a number of modifications to the walker to improve its operation.

MagiC: Software Package for Multiscale Modeling

We present software package MagiC, which is designed to perform systematic structure-based coarse graining of molecular models. The effective pairwise potentials between coarse-grained sites of low-resolution molecular models are constructed to reproduce structural distribution functions obtained from the modeling of the system in a high resolution (atomistic) description. The software supports coarse-grained tabulated intramolecular bond and angle interactions, as well as tabulated nonbonded interactions between different site types in the coarse-grained system, with the treatment of long-range electrostatic forces by the Ewald summation. Two methods of effective potential refinement are implemented: iterative Boltzmann inversion and inverse Monte Carlo, the latter accounting for cross-correlations between pair interactions. MagiC uses its own Metropolis Monte Carlo sampling engine, allowing parallel simulation of many copies of the system with subsequent averaging of the properties, which provides fast convergence of the method with nearly linear scaling at parallel execution.

Systematic methods for structurally consistent coarse-grained models

This chapter provides a primer on theories for coarse-grained (CG) modeling and, in particular, reviews several systematic methods for determining effective potentials for CG models. The chapter first reviews a statistical mechanics framework for relating atomistic and CG models. This framework naturally leads to a quantitative criterion for CG models that are "consistent" with a particular atomistic model for the same system. This consistency criterion is equivalent to minimizing the relative entropy between the two models. This criterion implies that a many-body PMF is the appropriate potential for a CG model that is consistent with a particular atomistic model. This chapter then presents a unified exposition of the theory and numerical methods for several approaches for approximating this many-body PMF. Finally, this chapter closes with a brief discussion of a few of the outstanding challenges facing the field of systematic coarse-graining.

Vesicles and vesicle fusion: coarse-grained simulations

Biological cells are highly dynamic, and continually move material around their own volume and between their interior and exterior. Much of this transport encapsulates the material inside phospholipid vesicles that shuttle to and from, fusing with, and budding from, other membranes. A feature of vesicles that is crucial for this transport is their ability to fuse to target membranes and release their contents to the distal side. In industry, some personal care products contain vesicles to help transport reagents across the skin, and research on drug formulation shows that packaging active compounds inside vesicles delays their clearance from the blood stream. In this chapter, we survey the biological role and physicochemical properties of phospholipids, and describe progress in coarse-grained simulations of vesicles and vesicle fusion. Because coarse-grained simulations retain only those molecular details that are thought to influence the large-scale processes of interest, they act as a model embodying our current understanding. Comparing the predictions of these models with experiments reveals the importance of the retained microscopic details and also the deficiencies that can suggest missing details, thereby furthering our understanding of the complex dynamic world of vesicles.

A Hybrid Approach for Highly Coarse-grained Lipid Bilayer Models

We present a systematic methodology to develop highly coarse-grained (CG) lipid models for large scale bio-membrane simulations, in which we derive CG interactions using a powerful combination of the multiscale coarse-graining (MS-CG) method, and an analytical form of the CG potential to model interactions at short range. The resulting hybrid coarse-graining (HCG) methodology is used to develop a three-site solvent-free model for 1,2-dilauroyl-sn-glycero-3-phosphocholine (DLPC), 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC), and a 1:1 mixture of 1,2-dioleoyl-sn-glycero-3-phospho-L-serine (DOPS) and DOPC. In addition, we developed a four-site model of DOPC, demonstrating the capability of the HCG methodology in designing model lipid systems of a desired resolution. We carried out microsecond-scale molecular dynamics (MD) simulations of large vesicles, highlighting the ability of the model to study systems at mesoscopic length and time scales. The models of DLPC, DOPC and DOPC-DOPS have elastic properties consistent with experiment and structural properties such as the radial distribution functions (RDF), bond and angle distributions, and the z-density distributions that compare well with reference all-atom systems.

Understanding the solvent polarity effects on surfactant-capped nanoparticles

Understanding the molecular interactions between suspended nanoparticles (NPs) and the suspending solvent fluid may provide a useful avenue to create and to study exotic NP ensembles. This study focused on using a coarse-grained computational model to investigate the molecular interactions between oleate-capped NPs in various solvents, and to relate the results to experimental features of solvent-suspended, oleate-capped CdSe quantum dots (QDs). The QDs were modeled as a closed-shell fullerene molecule with an oleate-like ligand attached to each vertex. Solvent polarity was found to correlate to the simulation and experimental results more strongly than either dielectric constant or dipole moment. Computational results showed that the nonpolar solvents of hexane, toluene, and benzene (polarity index E(T)(N) < 0.120) kept NPs in suspension and solvated the oleate chains such that the oleate layer swelled to full extension. In contrast, as the most polar solvent tested (E(T)(N) = 1.000), water caused NPs to aggregate and precipitate. It partially solvated the oleate chains and compressed the layer to 86% of full extension. For solvents of intermediate polarity like ethanol, acetone, and chloroform, the oleate layer swelled with decreasing polarity index values, with rapid swelling occurring close to E(T)(N) = 0.307 (~50:50 vol % chloroform/acetone) below which QDs were colloidally stable. This study represents the first attempt to delineate the solvent effect on surfactant-coated NP hydrodynamic size, colloidal stability, and aggregation behavior.

Coarse-grain Protein Models

Coarse-graining is a powerful approach for modeling biomolecules that, over the last few decades, has been extensively applied to proteins. Coarse-grain models offer access to large systems and to slow processes without becoming computationally unmanageable. In addition, they are very versatile, enabling both the protein representation and the energy function to be adapted to the biological problem in hand. This review concentrates on modeling soluble proteins and their assemblies. It presents an overview of the coarse-grain representations, of the associated interaction potentials, and of the optimization procedures used to define them. It then shows how coarse-grain models have been used to understand processes involving proteins, from their initial folding to their functional properties, their binary interactions, and the assembly of large complexes.

Coarse-graining entropy, forces, and structures

Coarse-grained (CG) models enable highly efficient simulations of complex processes that cannot be effectively studied with more detailed models. CG models are often parameterized using either force- or structure-motivated approaches. The present work investigates parallels between these seemingly divergent approaches by examining the relative entropy and multiscale coarse-graining (MS-CG) methods. We demonstrate that both approaches can be expressed in terms of an information function that discriminates between the ensembles generated by atomistic and CG models. While it is well known that the relative entropy approach minimizes the average of this information function, the present work demonstrates that the MS-CG method minimizes the average of its gradient squared. We generalize previous results by establishing conditions for the uniqueness of structure-based potentials and identify similarities with corresponding conditions for the uniqueness of MS-CG potentials. We analyze the mapping entropy and extend the MS-CG and generalized-Yvon-Born-Green formalisms for more complex potentials. Finally, we present numerical calculations that highlight similarities and differences between structure- and force-based approaches. We demonstrate that both methods obtain identical results, not only for a complete basis set, but also for an incomplete harmonic basis set in Cartesian coordinates. However, the two methods differ when the incomplete basis set includes higher order polynomials of Cartesian coordinates or is expressed as functions of curvilinear coordinates.