Molecular renormalization group coarse-graining of polymer chains: application to double-stranded DNA

CGeNArate: a sequence-dependent coarse-grained model of DNA for accurate atomistic MD simulations of kb-long duplexes

We present CGeNArate, a new model for molecular dynamics simulations of very long segments of B-DNA in the context of biotechnological or chromatin studies. The developed method uses a coarse-grained Hamiltonian with trajectories that are back-mapped to the atomistic resolution level with extreme accuracy by means of Machine Learning Approaches. The method is sequence-dependent and reproduces very well not only local, but also global physical properties of DNA. The efficiency of the method allows us to recover with a reduced computational effort high-quality atomic-resolution ensembles of segments containing many kilobases of DNA, entering into the gene range or even the entire DNA of certain cellular organelles.

Simulating the Lyotropic Phase Behavior of a Partially Self-Complementary DNA Tetramer

DNA oligomers in solution have been found to develop liquid crystal phases via a hierarchical process that involves Watson-Crick base pairing, supramolecular assembly into columns of duplexes, and long-range ordering. The multiscale nature of this phenomenon makes it difficult to quantitatively describe and assess the importance of the various contributions, particularly for very short strands. We performed molecular dynamics simulations based on the coarse-grained oxDNA model, aiming to depict all of the assembly processes involved and the phase behavior of solutions of the DNA GCCG tetramers. We find good quantitative matching to experimental data at both levels of molecular association (thermal melting) and collective ordering (phase diagram). We characterize the isotropic state and the low-density nematic and high-density columnar liquid crystal phases in terms of molecular order, size of aggregates, and structure, together with their effects on diffusivity processes. We observe a cooperative aggregation mechanism in which the formation of dimers is less thermodynamically favored than the formation of longer aggregates.

ANNaMo: Coarse-grained modeling for folding and assembly of RNA and DNA systems

The folding of RNA and DNA strands plays crucial roles in biological systems and bionanotechnology. However, studying these processes with high-resolution numerical models is beyond current computational capabilities due to the timescales and system sizes involved. In this article, we present a new coarse-grained model for investigating the folding dynamics of nucleic acids. Our model represents three nucleotides with a patchy particle and is parameterized using well-established nearest-neighbor models. Thanks to the reduction of degrees of freedom and to a bond-swapping mechanism, our model allows for simulations at timescales and length scales that are currently inaccessible to more detailed models. To validate the performance of our model, we conducted extensive simulations of various systems: We examined the thermodynamics of DNA hairpins, capturing their stability and structural transitions, the folding of an MMTV pseudoknot, which is a complex RNA structure involved in viral replication, and also explored the folding of an RNA tile containing a k-type pseudoknot. Finally, we evaluated the performance of the new model in reproducing the melting temperatures of oligomers and the dependence on the toehold length of the displacement rate in toehold-mediated displacement processes, a key reaction used in molecular computing. All in all, the successful reproduction of experimental data and favorable comparisons with existing coarse-grained models validate the effectiveness of the new model.

Anomalous Diffusion of Polyelectrolyte Segments on Supported Charged Lipid Bilayers

This work provides mesoscale models for the anomalous diffusion of a polymer chain on a heterogeneous surface with rearranging randomly distributed adsorption sites. Both the "bead-spring" model and oxDNA model were simulated on supported lipid bilayer membranes with various molar fractions of charged lipids, using Brownian dynamics method. Our simulation results demonstrate that "bead-spring" chains exhibit sub-diffusion on charged lipid bilayers which agrees with previous experimental observations for short-time dynamics of DNA segments on membranes. In addition, the non-Gaussian diffusive behaviors of DNA segments have not been observed in our simulations. However, a simulated 17 base pairs double stranded DNA, using oxDNA model, performs normal diffusion on supported cationic lipid bilayers. Due to the number of positively charged lipids attracted by short DNA is small, the energy landscape that the short DNA experiences during diffusion is not as heterogeneous as that experienced by long DNA chains, which results in normal diffusion rather than sub-diffusion for short DNA.

An associative memory Hamiltonian model for DNA and nucleosomes

A model for DNA and nucleosomes is introduced with the goal of studying chromosomes from a single base level all the way to higher-order chromatin structures. This model, dubbed the Widely Editable Chromatin Model (WEChroM), reproduces the complex mechanics of the double helix including its bending persistence length and twisting persistence length, and their respective temperature dependence. The WEChroM Hamiltonian is composed of chain connectivity, steric interactions, and associative memory terms representing all remaining interactions leading to the structure, dynamics, and mechanical characteristics of the B-DNA. Several applications of this model are discussed to demonstrate its applicability. WEChroM is used to investigate the behavior of circular DNA in the presence of positive and negative supercoiling. We show that it recapitulates the formation of plectonemes and of structural defects that relax mechanical stress. The model spontaneously manifests an asymmetric behavior with respect to positive or negative supercoiling, similar to what was previously observed in experiments. Additionally, we show that the associative memory Hamiltonian is also capable of reproducing the free energy of partial DNA unwrapping from nucleosomes. WEChroM is designed to emulate the continuously variable mechanical properties of the 10nm fiber and, by virtue of its simplicity, is ready to be scaled up to molecular systems large enough to investigate the structural ensembles of genes. WEChroM is implemented in the OpenMM simulation toolkits and is freely available for public use.

Contrastive Learning of Coarse-Grained Force Fields

Coarse-grained models have proven helpful for simulating complex systems over long time scales to provide molecular insights into various processes. Methodologies for systematic parametrization of the underlying energy function or force field that describes the interactions among different components of the system are of great interest for ensuring simulation accuracy. We present a new method, potential contrasting, to enable efficient learning of force fields that can accurately reproduce the conformational distribution produced with all-atom simulations. Potential contrasting generalizes the noise contrastive estimation method with umbrella sampling to better learn the complex energy landscape of molecular systems. When applied to the Trp-cage protein, we found that the technique produces force fields that thoroughly capture the thermodynamics of the folding process despite the use of only α-carbons in the coarse-grained model. We further showed that potential contrasting could be applied over large data sets that combine the conformational ensembles of many proteins to improve force field transferability. We anticipate potential contrasting as a powerful tool for building general-purpose coarse-grained force fields.

Multiscale simulations of polyzwitterions in aqueous bulk solutions and brush array configurations

Zwitterionic polymers are very promising candidates for antifouling materials that exhibit high chemical stability as compared to polyethylene glycol-based systems. A number of simulation and experimental studies have emerged over recent years for the investigation of sulfobetaine-based zwitterionic polymers. Investigating the structural and thermodynamic properties of such polymers requires access to broad time and length regimes, thus necessitating the development of multiscale simulation strategies. The present article advocates a mesoscopic dissipative particle dynamics (DPD) model capable of addressing a wide range of time and length scales. The mesoscopic force field was developed hand-in-hand with atomistic simulations based on the OPLS force field through a bottom-up parameterization procedure that matches the atomistically calculated strand-length, strand-angle and pair distribution functions. The DPD model is validated against atomistic simulations conducted in this work, and against relevant atomistic simulation studies, theoretical predictions and experimental correlations from the literature. Properties examined include the conformations of SPE polymers in dilute bulk aqueous solution, the density profile and thickness of brush arrays as functions of the grafting density and chain length. In addition, we compute the potential of mean force of an approaching hydrophilic or hydrophobic foulant via umbrella sampling as a function of its position relative to the poly-zwitterion-covered surface. The aforementioned observables lead to important insights regarding the conformational tendencies of grafted polyzwitterions and their antifouling properties.

Potential of Mean Force for DNA Wrapping Around a Cationic Nanoparticle

Sharp bending and wrapping of DNA around proteins and nanoparticles (NPs) has been of extensive research interest. Here, we present the potential of mean force (PMF) for wrapping a DNA double helix around a cationic NP using coarse-grained models of a double-stranded DNA and a cationic NP. Starting from a NP wrapped around by DNA, the PMF was calculated along the distance between the center of the NP and one end of the DNA molecule. A relationship between the distance and the extent of DNA wrapping is used to calculate the PMF as a function of DNA wrapping around a NP. In particular, the PMF was compared for two DNA sequences of (AT)₂₅/(AT)₂₅ and (AC)₂₅/(GT)₂₅, for which the persistence lengths are different by ∼10 nm. The simulation results provide solid evidence of the thermodynamic preference for complex formation of a cationic NP with more flexible DNA over the less flexible DNA. Furthermore, we estimated the elastic energy of DNA bending, which was in good order-of-magnitude agreement with the theoretical prediction of elastic rods. This work suggests that the variation of sequence-dependent DNA flexibility can be utilized in DNA nanotechnologies, in which the position and dynamics of NPs are regulated on large-scale DNA structures, or the structural transformation of DNA is triggered by the sequence-dependent binding of NPs.

Effect of DNA Flexibility on Complex Formation of a Cationic Nanoparticle with Double-Stranded DNA

We present extensive molecular dynamics simulations of a cationic nanoparticle and a double-stranded DNA molecule to discuss the effect of DNA flexibility on the complex formation of a cationic nanoparticle with double-stranded DNA. Martini coarse-grained models were employed to describe double-stranded DNA molecules with two different flexibilities and cationic nanoparticles with three different electric charges. As the electric charge of a cationic nanoparticle increases, the degree of DNA bending increases, eventually leading to the wrapping of DNA around the nanoparticle at high electric charges. However, a small increase in the persistence length of DNA by 10 nm requires a cationic nanoparticle with a markedly increased electric charge to bend and wrap DNA around. Thus, a more flexible DNA molecule bends and wraps around a cationic nanoparticle with an intermediate electric charge, whereas a less flexible DNA molecule binds to a nanoparticle with the same electric charge without notable bending. This work provides solid evidence that a small difference in DNA flexibility (as small as 10 nm in persistence length) has a substantial influence on the complex formation of DNA with proteins from a biological perspective and suggests that the variation of sequence-dependent DNA flexibility can be utilized in DNA nanotechnology as a new tool to manipulate the structure of DNA molecules mediated by nanoparticle binding.

A Primer on the oxDNA Model of DNA: When to Use it, How to Simulate it and How to Interpret the Results

The oxDNA model of Deoxyribonucleic acid has been applied widely to systems in biology, biophysics and nanotechnology. It is currently available via two independent open source packages. Here we present a set of clearly documented exemplar simulations that simultaneously provide both an introduction to simulating the model, and a review of the model's fundamental properties. We outline how simulation results can be interpreted in terms of-and feed into our understanding of-less detailed models that operate at larger length scales, and provide guidance on whether simulating a system with oxDNA is worthwhile.

Bottom-Up Coarse-Grained Modeling of DNA

Recent advances in methodology enable effective coarse-grained modeling of deoxyribonucleic acid (DNA) based on underlying atomistic force field simulations. The so-called bottom-up coarse-graining practice separates fast and slow dynamic processes in molecular systems by averaging out fast degrees of freedom represented by the underlying fine-grained model. The resulting effective potential of interaction includes the contribution from fast degrees of freedom effectively in the form of potential of mean force. The pair-wise additive potential is usually adopted to construct the coarse-grained Hamiltonian for its efficiency in a computer simulation. In this review, we present a few well-developed bottom-up coarse-graining methods, discussing their application in modeling DNA properties such as DNA flexibility (persistence length), conformation, "melting," and DNA condensation.

Bottom-up coarse-grained models for external fields and interfaces

Bottom-up coarse-grained (CG) models accurately describe the structure of homogeneous systems but sometimes provide limited transferability and a poor description of thermodynamic properties. Consequently, inhomogeneous systems present a severe challenge for bottom-up models. In this work, we examine bottom-up CG models for interfaces and inhomogeneous systems. We first analyze the effect of external fields upon the many-body potential of mean force. We also demonstrate that the multiscale CG (MS-CG) variational principle for modeling the external field corresponds to a generalization of the first Yvon-Born-Green equation. This provides an important connection with liquid state theory, as well as physical insight into the structure of interfaces and the resulting MS-CG models. We then develop and assess MS-CG models for a film of liquid methanol that is adsorbed on an attractive wall and in coexistence with its vapor phase. While pair-additive potentials provide unsatisfactory accuracy and transferability, the inclusion of local-density (LD) potentials dramatically improves the accuracy and transferability of the MS-CG model. The MS-CG model with LD potentials quite accurately describes the wall-liquid interface, the bulk liquid density, and the liquid-vapor interface while simultaneously providing a much improved description of the vapor phase. This model also provides an excellent description of the pair structure and pressure-density equation of state for the bulk liquid. Thus, LD potentials hold considerable promise for transferable bottom-up models that accurately describe the structure and thermodynamic properties of both bulk and interfacial systems.

Coarse-graining of polyisoprene melts using inverse Monte Carlo and local density potentials

Bottom-up coarse-graining of polymers is commonly performed by matching structural order parameters such as distribution of bond lengths, bending and dihedral angles, and pair distribution functions. In this study, we introduce the distribution of nearest-neighbors as an additional order parameter in the concept of local density potentials. We describe how the inverse-Monte Carlo method provides a framework for forcefield development that is capable of overcoming challenges associated with the parameterization of interaction terms in polymer systems. The technique is applied on polyisoprene melts as a prototype system. We demonstrate that while different forcefields can be developed that perform equally in terms of matching target distributions, the inclusion of nearest-neighbors provides a straightforward route to match both thermodynamic and conformational properties. We find that several temperature state points can also be addressed, provided that the forcefield is refined accordingly. Finally, we examine both the single-particle and the collective dynamics of the coarse-grain models, demonstrating that all forcefields present a similar acceleration relative to the atomistic systems.

Direct route to reproducing pair distribution functions with coarse-grained models via transformed atomistic cross correlations

Coarse-grained (CG) models are often parameterized to reproduce one-dimensional structural correlation functions of an atomically detailed model along the degrees of freedom governing each interaction potential. While cross correlations between these degrees of freedom inform the optimal set of interaction parameters, the correlations generated from the higher-resolution simulations are often too complex to act as an accurate proxy for the CG correlations. Instead, the most popular methods determine the interaction parameters iteratively while assuming that individual interactions are uncorrelated. While these iterative methods have been validated for a wide range of systems, they also have disadvantages when parameterizing models for multicomponent systems or when refining previously established models to better reproduce particular structural features. In this work, we propose two distinct approaches for the direct (i.e., noniterative) parameterization of a CG model by adjusting the high-resolution cross correlations of an atomistic model in order to more accurately reflect correlations that will be generated by the resulting CG model. The derived models more accurately describe the low-order structural features of the underlying AA model while necessarily generating inherently distinct cross correlations compared with the atomically detailed reference model. We demonstrate the proposed methods for a one-site-per-molecule representation of liquid water, where pairwise interactions are incapable of reproducing the true tetrahedral solvation structure. We then investigate the precise role that distinct cross-correlation features play in determining the correct pair correlation functions, evaluating the importance of the placement of correlation features as well as the balance between features appearing in different solvation shells.

Adversarial-residual-coarse-graining: Applying machine learning theory to systematic molecular coarse-graining

We utilize connections between molecular coarse-graining (CG) approaches and implicit generative models in machine learning to describe a new framework for systematic molecular CG. Focus is placed on the formalism encompassing generative adversarial networks. The resulting method enables a variety of model parameterization strategies, some of which show similarity to previous CG methods. We demonstrate that the resulting framework can rigorously parameterize CG models containing CG sites with no prescribed connection to the reference atomistic system (termed virtual sites); however, this advantage is offset by the lack of a closed-form expression for the CG Hamiltonian at the resolution obtained after integration over the virtual CG sites. Computational examples are provided for cases in which these methods ideally return identical parameters as relative entropy minimization CG but where traditional relative entropy minimization CG optimization equations are not applicable.

TacoxDNA: A user-friendly web server for simulations of complex DNA structures, from single strands to origami

Simulations of nucleic acids at different levels of structural details are increasingly used to complement and interpret experiments in different fields, from biophysics to medicine and materials science. However, the various structural models currently available for DNA and RNA and their accompanying suites of computational tools can be very rarely used in a synergistic fashion. The tacoxDNA webserver and standalone software package presented here are a step toward a long-sought interoperability of nucleic acids models. The webserver offers a simple interface for converting various common input formats of DNA structures and setting up molecular dynamics (MD) simulations. Users can, for instance, design DNA rings with different topologies, such as knots, with and without supercoiling, by simply providing an XYZ coordinate file of the DNA centre-line. More complex DNA geometries, as designable in the cadnano, CanDo and Tiamat tools, can also be converted to all-atom or oxDNA representations, which can then be used to run MD simulations. Though the latter are currently geared toward the native and LAMMPS oxDNA representations, the open-source package is designed to be further expandable. TacoxDNA is available at http://tacoxdna.sissa.it. © 2019 Wiley Periodicals, Inc.

Nucleosome Crowding in Chromatin Slows the Diffusion but Can Promote Target Search of Proteins

Dynamics of nuclear proteins in crowded chromatin has only been poorly understood. Here, we address the diffusion, target search, and structural dynamics of three proteins in a model chromatin using coarse-grained molecular simulations run on the K computer. We prepared two structures of chromatin made of 20 nucleosomes with different nucleosome densities and investigated dynamics of two transcription factors, HMGB1 and p53, and one signaling protein, ERK, embedded in the chromatin. We found fast and normal diffusion of the nuclear proteins in the low-density chromatins and slow and subdiffusional movements in the high-density chromatin. The diffusion of the largest transcription factor, p53, is slowed by high-density chromatin most markedly. The on rates and off rates for DNA binding are increased and decreased, respectively, in the high-density chromatin. To our surprise, the DNA sequence search was faster in chromatin with high nucleosome density, though the diffusion is slower. We also found that the three nuclear proteins preferred to bind on the linker DNA and the entry and exit regions of nucleosomal DNA. In addition to these regions, HMGB1 and p53 also bound to the dyad.

The "sugar" coarse-grained DNA model

More than 20 coarse-grained (CG) DNA models have been developed for simulating the behavior of this molecule under various conditions, including those required for nanotechnology. However, none of these models reproduces the DNA polymorphism associated with conformational changes in the ribose rings of the DNA backbone. These changes make an essential contribution to the DNA local deformability and provide the possibility of the transition of the DNA double helix from the B-form to the A-form during interactions with biological molecules. We propose a CG representation of the ribose conformational flexibility. We substantiate the choice of the CG sites (six per nucleotide) needed for the "sugar" GC DNA model, and obtain the potentials of the CG interactions between the sites by the "bottom-up" approach using the all-atom AMBER force field. We show that the representation of the ribose flexibility requires one non-harmonic and one three-particle potential, the forms of both the potentials being different from the ones generally used. The model also includes (i) explicit representation of ions (in an implicit solvent) and (ii) sequence dependence. With these features, the sugar CG DNA model reproduces (with the same parameters) both the B- and A- stable forms under corresponding conditions and demonstrates both the A to B and the B to A phase transitions. Graphical Abstract The proposed coarse-grained DNA model allows to reproduce both the B- and A- DNA forms and the transitions between them under corresponding conditions.

A Review of Multiscale Computational Methods in Polymeric Materials

Polymeric materials display distinguished characteristics which stem from the interplay of phenomena at various length and time scales. Further development of polymer systems critically relies on a comprehensive understanding of the fundamentals of their hierarchical structure and behaviors. As such, the inherent multiscale nature of polymer systems is only reflected by a multiscale analysis which accounts for all important mechanisms. Since multiscale modelling is a rapidly growing multidisciplinary field, the emerging possibilities and challenges can be of a truly diverse nature. The present review attempts to provide a rather comprehensive overview of the recent developments in the field of multiscale modelling and simulation of polymeric materials. In order to understand the characteristics of the building blocks of multiscale methods, first a brief review of some significant computational methods at individual length and time scales is provided. These methods cover quantum mechanical scale, atomistic domain (Monte Carlo and molecular dynamics), mesoscopic scale (Brownian dynamics, dissipative particle dynamics, and lattice Boltzmann method), and finally macroscopic realm (finite element and volume methods). Afterwards, different prescriptions to envelope these methods in a multiscale strategy are discussed in details. Sequential, concurrent, and adaptive resolution schemes are presented along with the latest updates and ongoing challenges in research. In sequential methods, various systematic coarse-graining and backmapping approaches are addressed. For the concurrent strategy, we aimed to introduce the fundamentals and significant methods including the handshaking concept, energy-based, and force-based coupling approaches. Although such methods are very popular in metals and carbon nanomaterials, their use in polymeric materials is still limited. We have illustrated their applications in polymer science by several examples hoping for raising attention towards the existing possibilities. The relatively new adaptive resolution schemes are then covered including their advantages and shortcomings. Finally, some novel ideas in order to extend the reaches of atomistic techniques are reviewed. We conclude the review by outlining the existing challenges and possibilities for future research.

Systematic hierarchical coarse-graining with the inverse Monte Carlo method

We outline our coarse-graining strategy for linking micro- and mesoscales of soft matter and biological systems. The method is based on effective pairwise interaction potentials obtained in detailed ab initio or classical atomistic Molecular Dynamics (MD) simulations, which can be used in simulations at less accurate level after scaling up the size. The effective potentials are obtained by applying the inverse Monte Carlo (IMC) method [A. P. Lyubartsev and A. Laaksonen, Phys. Rev. E 52(4), 3730-3737 (1995)] on a chosen subset of degrees of freedom described in terms of radial distribution functions. An in-house software package MagiC is developed to obtain the effective potentials for arbitrary molecular systems. In this work we compute effective potentials to model DNA-protein interactions (bacterial LiaR regulator bound to a 26 base pairs DNA fragment) at physiological salt concentration at a coarse-grained (CG) level. Normally the IMC CG pair-potentials are used directly as look-up tables but here we have fitted them to five Gaussians and a repulsive wall. Results show stable association between DNA and the model protein as well as similar position fluctuation profile.

Multiscale coarse-grained modelling of chromatin components: DNA and the nucleosome

To model large biomolecular systems, such as cell and organelles an atomistic description is not currently achievable and is not generally practical. Therefore, simplified coarse-grained (CG) modelling becomes a necessity. One of the most important cellular components is chromatin, a large DNA-protein complex where DNA is highly compacted. Recent progress in coarse graining modelling of the major chromatin components, double helical DNA and the nucleosome core particle (NCP) is presented. First, general principles and approaches allowing rigorous bottom-to-top generation of interaction potentials in the CG models are presented. Then, recent CG models of DNA are reviewed and their adequacy is benchmarked against experimental data on the salt dependence of DNA flexibility (persistence length). Furthermore, a few recent CG models of the NCP are described and their application for studying salt-dependent NCP-NCP interaction is discussed. An example of a multiscale approach to CG modelling of chromatin is presented where interactions and self-assembly of thousands of NCPs in solution are observed.

A Solvent-Mediated Coarse-Grained Model of DNA Derived with the Systematic Newton Inversion Method

We present a new class of coarse-grained (CG) force fields (FFs) for B-DNA with explicit ions suited for large-scale mesoscale simulations at microsecond-micrometer scale using a wide spectrum of particle simulation methods from molecular dynamics to dissipative particle dynamics. The effective solvent-mediated pairwise interactions making up the FFs are obtained by inverting radial distribution functions and other particle-particle distributions obtained from all-atom simulations of numbers of octadecamer DNA fragments from the Ascona B-DNA library. The inverse Monte Carlo (IMC) method, later known as Newton inversion (NI) (Lyubartsev, A. P.; Laaksonen, A. Phys. Rev. E, 1995, 52, 3730-3737), was used together with the iterative Boltzmann inversion (IBI) scheme to compute the effective CG potentials. We show that this systematic structure-based approach is capable of providing converged potentials that accurately reproduce the structural features of the underlying atomistic system within a few percents of relative difference. We also show that a simple one-site-per-nucleotide model with 10 intramolecular pair interaction potentials is able to reproduce key features of DNA, for example, the persistence length and its dependence on the ionic concentration, experimentally determined around 50 nm at physiological salt concentration.

RedMDStream: Parameterization and Simulation Toolbox for Coarse-Grained Molecular Dynamics Models

Coarse-grained (CG) models in molecular dynamics (MD) are powerful tools to simulate the dynamics of large biomolecular systems on micro- to millisecond timescales. However, the CG model, potential energy terms, and parameters are typically not transferable between different molecules and problems. So parameterizing CG force fields, which is both tedious and time-consuming, is often necessary. We present RedMDStream, a software for developing, testing, and simulating biomolecules with CG MD models. Development includes an automatic procedure for the optimization of potential energy parameters based on metaheuristic methods. As an example we describe the parameterization of a simple CG MD model of an RNA hairpin.

The Theory of Ultra-Coarse-Graining. 2. Numerical Implementation

The increasing interest in the modeling of complex macromolecular systems in recent years has spurred the development of numerous coarse-graining (CG) techniques. However, many of the CG models are constructed assuming that all details beneath the resolution of CG degrees of freedom are fast and average out, which sets limits on the resolution of feasible coarse-grainings and on the range of applications of the CG models. Ultra-coarse-graining (UCG) makes it possible to construct models at any desired resolution while accounting for discrete conformational or chemical changes within the CG sites that can modulate the interactions between them. Here, we discuss the UCG methodology and its numerical implementation. We pay particular attention to the numerical mechanism for including state transitions between different conformations within CG sites because this has not been discussed previously. Using a simple example of 1,2-dichloroethane, we demonstrate the ability of the UCG model to reproduce the multiconfigurational behavior of this molecular liquid, even when each molecule is modeled with only one CG site. The methodology can also be applied to other molecular liquids and macromolecular systems with time scale separation between conformational transitions and other intramolecular motions and rotations.

Monte-Carlo simulations of PAMAM dendrimer-DNA interactions

We use Monte Carlo simulations to determine the influence of poly(amido amine) (PAMAM) dendrimer size and charge on its interactions with double-stranded DNA conformation and interaction strength. To achieve a compromise between simulation speed and molecular detail, we combine the coarse-grained DNA model of de Pablo et al. which resolves each DNA base using three beads - and thereby retains the double-helix structure - with a dendrimer model with resolution similar to that of the DNA. The resulting predictions of the effects of dendrimer generation, dendrimer surface charge density, and salt concentration on dendrimer-DNA complexes are in agreement with both experiments and all-atom MD simulations. The model predicts that DNA wraps a fully charged G5 or G6 dendrimer at low salt concentration (10 mM) similarly to a histone octamer, and for the G5 dendrimer, DNA super helices with both handednesses occur. At salt concentrations above 50 mM, or when a high fraction of dendrimer surface charges are neutralized by acetylation, DNA adheres but does not compactly wrap the dendrimer, in agreement with experimental findings. We are also able to simulate pairs of dendrimers binding to the same DNA strand. Thus, our mesoscale simulation not only elucidates dendrimer-DNA interactions, but also provides a methodology for efficiently simulating chromatin formation and other cationic macroion-DNA complexes.

Wrapping/unwrapping transition of double-stranded DNA in DNA-nanosphere complexes induced by multivalent anions

Wrapping and unwrapping behaviors of double-stranded DNA around a positively charged nanosphere in solution are studied by using the coarse-grained molecular dynamics (CGMD) simulation method. When monovalent, divalent and trivalent anions are added to the DNA-nanosphere complex solution, double-stranded DNA binds with a nanosphere owing to strong electrostatic attraction. However, when tetravalent anions are added to the DNA-nanosphere complex solution, local charge inversion is observed for a high anion concentration of tetravalent anions and the double-stranded DNA can be unwrapped from the nanosphere because of the local charge inversion near the nanosphere. Moreover, the helical structure of DNA is damaged when double-stranded DNA wraps around the nanosphere and the helical structure can be rebuilt when the double-stranded DNA unwraps from the nanosphere. This study can help us understand how to control the release of DNA in DNA-nanosphere complexes.

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.

Fitting coarse-grained distribution functions through an iterative force-matching method

An iterative coarse-graining method is developed for systematically converting an atomistic force field to a model at lower resolution that is able to accurately reproduce the distribution functions defined in the coarse-grained potential. The method starts from the multiscale coarse-graining (MS-CG) approach, and it iteratively refines the distribution functions using repeated applications of the MS-CG algorithm. It is justified on the basis of the force matching normal equation, which can be considered a discrete form of the Yvon-Born-Green equation in liquid state theory. Numerical results for molecular systems involving pairwise nonbonded and three-body bonded interactions are obtained, and comparison with other approaches in literature is provided.

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.

Evolutionary Algorithm in the Optimization of a Coarse-Grained Force Field

Simulations using residue-scale coarse-grained models of biomolecules are less computationally demanding than simulations employing full-atomistic force fields. However, the coarse-grained models are often difficult and tedious to parametrize for certain applications. Therefore, a systematic and objective method to help develop or adapt the coarse-grained models is needed. We present an automatic method that implements an evolutionary algorithm to find a set of optimal force field parameters for a one-bead coarse-grained model. In addition to an optimized force field, parameter correlations and significance of the potential energy terms can be determined. The method is applied to two classes of problems: the dynamics of an RNA helix and the RNA structure prediction.

Coarse-graining DNA for simulations of DNA nanotechnology

To simulate long time and length scale processes involving DNA it is necessary to use a coarse-grained description. Here we provide an overview of different approaches to such coarse-graining, focussing on those at the nucleotide level that allow the self-assembly processes associated with DNA nanotechnology to be studied. OxDNA, our recently-developed coarse-grained DNA model, is particularly suited to this task, and has opened up this field to systematic study by simulations. We illustrate some of the range of DNA nanotechnology systems to which the model is being applied, as well as the insights it can provide into fundamental biophysical properties of DNA.

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.

An advanced coarse-grained nucleosome core particle model for computer simulations of nucleosome-nucleosome interactions under varying ionic conditions

In the eukaryotic cell nucleus, DNA exists as chromatin, a compact but dynamic complex with histone proteins. The first level of DNA organization is the linear array of nucleosome core particles (NCPs). The NCP is a well-defined complex of 147 bp DNA with an octamer of histones. Interactions between NCPs are of paramount importance for higher levels of chromatin compaction. The polyelectrolyte nature of the NCP implies that nucleosome-nucleosome interactions must exhibit a great influence from both the ionic environment as well as the positively charged and highly flexible N-terminal histone tails, protruding out from the NCP. The large size of the system precludes a modelling analysis of chromatin at an all-atom level and calls for coarse-grained approximations. Here, a model of the NCP that include the globular histone core and the flexible histone tails described by one particle per each amino acid and taking into account their net charge is proposed. DNA wrapped around the histone core was approximated at the level of two base pairs represented by one bead (bases and sugar) plus four beads of charged phosphate groups. Computer simulations, using a Langevin thermostat, in a dielectric continuum with explicit monovalent (K(+)), divalent (Mg(2+)) or trivalent (Co(NH(3))(6) (3+)) cations were performed for systems with one or ten NCPs. Increase of the counterion charge results in a switch from repulsive NCP-NCP interaction in the presence of K(+), to partial aggregation with Mg(2+) and to strong mutual attraction of all 10 NCPs in the presence of CoHex(3+). The new model reproduced experimental results and the structure of the NCP-NCP contacts is in agreement with available data. Cation screening, ion-ion correlations and tail bridging contribute to the NCP-NCP attraction and the new NCP model accounts for these interactions.

Ab initio determination of coarse-grained interactions in double-stranded DNA

We derive the coarse-grained interactions between DNA nucleotides from ab initio total-energy calculations based on density functional theory (DFT). The interactions take into account base and sequence specificity, and are decomposed into physically distinct contributions that include hydrogen bonding, stacking interactions, backbone, and backbone-base interactions. The interaction energies of each contribution are calculated from DFT for a wide range of configurations and are fitted by simple analytical expressions for use in the coarse-grained model, which reduces each nucleotide into two sites. This model is not derived from experimental data, yet it successfully reproduces the stable B-DNA structure and gives good predictions for the persistence length. It may be used to realistically probe dynamics of DNA strands in various environments at the μs time scale and the μm length scale.

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.

AWSEM-MD: protein structure prediction using coarse-grained physical potentials and bioinformatically based local structure biasing

The associative memory, water mediated, structure and energy model (AWSEM) is a coarse-grained protein force field. AWSEM contains physically motivated terms, such as hydrogen bonding, as well as a bioinformatically based local structure biasing term, which efficiently takes into account many-body effects that are modulated by the local sequence. When combined with appropriate local or global alignments to choose memories, AWSEM can be used to perform de novo protein structure prediction. Herein we present structure prediction results for a particular choice of local sequence alignment method based on short residue sequences called fragments. We demonstrate the model's structure prediction capabilities for three levels of global homology between the target sequence and those proteins used for local structure biasing, all of which assume that the structure of the target sequence is not known. When there are no homologues in the database of structures used for local structure biasing, AWSEM calculations produce structural predictions that are somewhat improved compared with prior works using related approaches. The inclusion of a small number of structures from homologous sequences improves structure prediction only marginally, but when the fragment search is restricted to only homologous sequences, AWSEM can perform high resolution structure prediction and can be used for kinetics and dynamics studies.