A New Multiscale Algorithm and Its Application to Coarse-Grained Peptide Models for Self-Assembly

Coarse-Grained Ions for Nucleic Acid Modeling

We present a general coarse-grained model of sodium, magnesium, spermidine, and chlorine in implicit solvent. The effective potentials between ions are systematically parametrized using a relative entropy coarse-graining approach [Carmichael, S. P. and M. S. Shell, J. Phys. Chem. B, 116, 8383-93 (2012)] that maximizes the information retained in a coarse-grained model. We describe the local distribution of ions in the vicinity of a recently published coarse-grained DNA model and demonstrate a dependence of persistence length on ionic strength that differs from that predicted by Odijk-Skolnick-Fixman theory. Consistent with experimental observations, we show that spermidine induces DNA condensation whereas magnesium and sodium do not. This model can be used alongside any coarse-grained DNA model that has explicit charges and an accurate reproduction of the excluded volume of dsDNA.

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.

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.

Simulations of Protein Aggregation: Insights from Atomistic and Coarse-Grained Models

This Perspective highlights recent computational approaches to protein aggregation, from coarse-grained models to atomistic simulations, using the islet amyloid polypeptide (IAPP) as a case study. We review salient open questions where simulations can make an impact, discuss the successes and challenges met by simulations, and explore new directions.

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.

Combining experiments and simulations using the maximum entropy principle

A key component of computational biology is to compare the results of computer modelling with experimental measurements. Despite substantial progress in the models and algorithms used in many areas of computational biology, such comparisons sometimes reveal that the computations are not in quantitative agreement with experimental data. The principle of maximum entropy is a general procedure for constructing probability distributions in the light of new data, making it a natural tool in cases when an initial model provides results that are at odds with experiments. The number of maximum entropy applications in our field has grown steadily in recent years, in areas as diverse as sequence analysis, structural modelling, and neurobiology. In this Perspectives article, we give a broad introduction to the method, in an attempt to encourage its further adoption. The general procedure is explained in the context of a simple example, after which we proceed with a real-world application in the field of molecular simulations, where the maximum entropy procedure has recently provided new insight. Given the limited accuracy of force fields, macromolecular simulations sometimes produce results that are at not in complete and quantitative accordance with experiments. A common solution to this problem is to explicitly ensure agreement between the two by perturbing the potential energy function towards the experimental data. So far, a general consensus for how such perturbations should be implemented has been lacking. Three very recent papers have explored this problem using the maximum entropy approach, providing both new theoretical and practical insights to the problem. We highlight each of these contributions in turn and conclude with a discussion on remaining challenges.

Efficient Parameter Estimation of Generalizable Coarse-Grained Protein Force Fields Using Contrastive Divergence: A Maximum Likelihood Approach

Maximum Likelihood (ML) optimization schemes are widely used for parameter inference. They maximize the likelihood of some experimentally observed data, with respect to the model parameters iteratively, following the gradient of the logarithm of the likelihood. Here, we employ a ML inference scheme to infer a generalizable, physics-based coarse-grained protein model (which includes Go̅-like biasing terms to stabilize secondary structure elements in room-temperature simulations), using native conformations of a training set of proteins as the observed data. Contrastive divergence, a novel statistical machine learning technique, is used to efficiently approximate the direction of the gradient ascent, which enables the use of a large training set of proteins. Unlike previous work, the generalizability of the protein model allows the folding of peptides and a protein (protein G) which are not part of the training set. We compare the same force field with different van der Waals (vdW) potential forms: a hard cutoff model, and a Lennard-Jones (LJ) potential with vdW parameters inferred or adopted from the CHARMM or AMBER force fields. Simulations of peptides and protein G show that the LJ model with inferred parameters outperforms the hard cutoff potential, which is consistent with previous observations. Simulations using the LJ potential with inferred vdW parameters also outperforms the protein models with adopted vdW parameter values, demonstrating that model parameters generally cannot be used with force fields with different energy functions. The software is available at https://sites.google.com/site/crankite/.

Estimation of Interaction Potentials through the Configurational Temperature Formalism

Molecular interaction potentials are difficult to measure experimentally and hard to compute from first principles, especially for large systems such as proteins. It is therefore desirable to estimate the potential energy that underlies a thermodynamic ensemble from simulated or experimentally determined configurations. This inverse problem of statistical mechanics is challenging because the various potential energy terms can exhibit subtle indirect and correlated effects on the resulting ensemble. A direct approach would try to adapt the force field parameters such that the given configurations are highly probable in the resulting ensemble. But this would require a full simulation of the system whenever a parameter changes. We introduce an extension of the configurational temperature formalism that allows us to circumvent these difficulties and efficiently estimate interaction potentials from molecular configurations. We illustrate the approach for various systems including fluids and a coarse-grained protein model.

Representability and Transferability of Kirkwood-Buff Iterative Boltzmann Inversion Models for Multicomponent Aqueous Systems

We discuss the application of the Kirkwood-Buff iterative Boltzmann inversion (KB-IBI) method for molecular coarse-graining (Ganguly et al. J. Chem. Theory Comput. 2012, 8, 1802) to multicomponent aqueous mixtures. Using a fixed set of effective single-site solvent-solvent potentials previously derived for binary urea-water systems, solute-solvent and solute-solute KB-IBI coarse-grained (CG) potentials have been derived for benzene in urea-water mixtures. Preferential solvation and salting-in coefficients of benzene are reproduced in quantitative agreement with the atomistic force field model. The transferability of the CG models is discussed, and it is shown that free energies of formation of hydrophobic benzene clusters obtained from simulations with the CG model are in good agreement with results obtained from all-atom simulations. The state-point representability of the CG models is discussed with respect to reproducing thermodynamic quantities such as pressure, isothermal compressibility, and preferential solvation. Combined use of KB-IBI and pressure corrections in deriving single-site CG models for pure-water, binary mixtures of urea and water, and ternary mixtures of benzene in urea-water at infinite benzene dilution provides an improved scheme to representing the atomistic pressure and the preferential solvation between the solution components. It is also found that the application of KB-IBI leads to a faster and improved convergence of the pressure and potential energy compared to the IBI method.

Variational Optimization of an All-Atom Implicit Solvent Force Field to Match Explicit Solvent Simulation Data

The development of accurate implicit solvation models with low computational cost is essential for addressing many large-scale biophysical problems. Here, we present an efficient solvation term based on a Gaussian solvent-exclusion model (EEF1) for simulations of proteins in aqueous environment, with the primary aim of having a good overlap with explicit solvent simulations, particularly for unfolded and disordered states - as would be needed for multiscale applications. In order to achieve this, we have used a recently proposed coarse-graining procedure based on minimization of an entropy-related objective function to train the model to reproduce the equilibrium distribution obtained from explicit water simulations. Via this methodology, we have optimized both a charge screening parameter and a backbone torsion term against explicit solvent simulations of an α-helical and a β-stranded peptide. The performance of the resulting effective energy function, termed EEF1-SB, is tested with respect to the properties of folded proteins, the folding of small peptides or fast-folding proteins, and NMR data for intrinsically disordered proteins. The results show that EEF1-SB provides a reasonable description of a wide range of systems, but its key advantage over other methods tested is that it captures very well the structure and dimension of disordered or weakly structured peptides. EEF1-SB is thus a computationally inexpensive (~ 10 times faster than Generalized-Born methods) and transferable approximation for treating solvent effects.

Electrostatic-Consistent Coarse-Grained Potentials for Molecular Simulations of Proteins

We present a new generation of coarse-grained (CG) potentials that account for a simplified electrostatic description of soluble proteins. The treatment of permanent electrostatic dipoles of the backbone and polar side-chains allows to simulate proteins, preserving an excellent structural and dynamic agreement with respective reference structures and all-atom molecular dynamics simulations. Moreover, multiprotein complexes can be well described maintaining their molecular interfaces thanks to the ability of this scheme to better describe the actual electrostatics at a CG level of resolution. An efficient and robust heuristic algorithm based on particle swarm optimization is used for the derivation of CG parameters via a force-matching procedure. The ability of this protocol to deal with high dimensional search spaces suggests that the extension of this optimization procedure to larger data sets may lead to the generation of a fully transferable CG force field. At the present stage, these electrostatic-consistent CG potentials are easily and efficiently parametrized, show a good degree of transferability, and can be used to simulate soluble proteins or, more interestingly, large macromolecular assemblies for which long all-atom simulations may not be easily affordable.

Matching of additive and polarizable force fields for multiscale condensed phase simulations

Inclusion of electronic polarization effects is one of the key aspects in which the accuracy of current biomolecular force fields may be improved. The principal drawback of such approaches is the computational cost, which typically ranges from 3 - 10 times that of the equivalent additive model, and may be greater for more sophisticated treatments of polarization or other many-body effects. Here, we present a multiscale approach which may be used to enhance the sampling in simulations with polarizable models, by using the additive model as a tool to explore configuration space. We use a method based on information theory to determine the charges for an additive model that has optimal overlap with the polarizable one, and we demonstrate the feasibility of enhancing sampling via a hybrid replica exchange scheme for several model systems. An additional advantage is that, in the process, we obtain a systematic method for deriving charges for an additive model that will be the natural complement to its polarizable parent. The additive charges are found by an effective coarse-graining of the polarizable force field, rather than by ad hoc procedures.

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.

Ligand Binding Pathway Elucidation for Cryptophane Host-Guest Complexes

Modeling binding pathways can provide insight into molecular recognition, including kinetic mechanisms, barriers to binding, and gating effects. This work represents a novel computational approach, Hopping Minima, for the determination of conformational transitions of single molecules as well as binding pathways for molecular complexes. The method begins by thoroughly sampling a set of conformational minima for a molecular system. The natural motions of the system are modeled using the normal modes of the sampled minima. The natural motions are utilized to connect conformational minima and are finally combined to form association/binding pathways in the case of molecular complexes. We provide an implementation and example application of the method using alanine dipeptide and a set of chemical host-guest systems: two cryptophane hosts with two guest cations, trimethylammonium and tetramethylammonium. Our results demonstrate that conformational transitions can be modeled and extended to find binding pathways as well as energetic information relevant to the minimum conformations involved. This approach has advantages over simulation-based methods for studying systems with slow binding processes and can help design molecules with preferred binding kinetics.

A general method for spatially coarse-graining Metropolis Monte Carlo simulations onto a lattice

A recently introduced method for coarse-graining standard continuous Metropolis Monte Carlo simulations of atomic or molecular fluids onto a rigid lattice of variable scale [X. Liu, W. D. Seider, and T. Sinno, Phys. Rev. E 86, 026708 (2012)] is further analyzed and extended. The coarse-grained Metropolis Monte Carlo technique is demonstrated to be highly consistent with the underlying full-resolution problem using a series of detailed comparisons, including vapor-liquid equilibrium phase envelopes and spatial density distributions for the Lennard-Jones argon and simple point charge water models. In addition, the principal computational bottleneck associated with computing a coarse-grained interaction function for evolving particle positions on the discretized domain is addressed by the introduction of new closure approximations. In particular, it is shown that the coarse-grained potential, which is generally a function of temperature and coarse-graining level, can be computed at multiple temperatures and scales using a single set of free energy calculations. The computational performance of the method relative to standard Monte Carlo simulation is also discussed.

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 coarse-graining of potential energy landscapes and dynamics in liquids

Recent efforts have shown that the dynamic properties of a wide class of liquids can be mapped onto semi-universal scaling laws and constitutive relations that are motivated by thermodynamic analyses of much simpler models. In particular, it has been found that many systems exhibit dynamics whose behavior in state space closely follows that of soft-sphere particles interacting through an inverse power repulsion. In the present work, we show that a recently developed coarse-graining theory provides a natural way to understand how arbitrary liquids can be mapped onto effective soft-sphere models and hence how one might potentially be able to extract underlying dynamical scaling laws. The theory is based on the relative entropy, an information metric that quantifies how well a soft-sphere approximation to a liquid's multidimensional potential energy landscape performs. We show that optimization of the relative entropy not only enables one to extract effective soft-sphere potentials that suggest an inherent scaling of thermodynamic and dynamic properties in temperature-density space, but that also has rather interesting connections to excess entropy based theories of liquid dynamics. We apply the approach to a binary mixture of Lennard-Jones particles, and show that it gives effective soft-sphere scaling laws that well-describe the behavior of the diffusion constants. Our results suggest that the relative entropy formalism may be useful for "perturbative" type theories of dynamics, offering a general strategy for systematically connecting complex energy landscapes to simpler reference ones with better understood dynamic behavior.