Efficient treatment of induced dipoles

A Fast, Convenient, Polarizable Electrostatic Model for Molecular Dynamics

We present an efficient polarizable electrostatic model, utilizing typed, atom-centered polarizabilities and the fast direct approximation, designed for efficient use in molecular dynamics (MD) simulations. The model provides two convenient approaches for assigning partial charges in the context of atomic polarizabilities. One is a generalization of RESP, called RESP-dPol, and the other, AM1-BCC-dPol, is an adaptation of the widely used AM1-BCC method. Both are designed to accurately replicate gas-phase quantum mechanical electrostatic potentials. Benchmarks of this polarizable electrostatic model against gas-phase dipole moments, molecular polarizabilities, bulk liquid densities, and static dielectric constants of organic liquids show good agreement with the reference values. Of note, the model yields markedly more accurate dielectric constants of organic liquids, relative to a matched nonpolarizable force field. MD simulations with this method, which is currently parametrized for molecules containing elements C, N, O, and H, run only about 3.6-fold slower than fixed charge force fields, while simulations with the self-consistent mutual polarization average 4.5-fold slower. Our results suggest that RESP-dPol and AM1-BCC-dPol afford improved accuracy relative to fixed charge force fields and are good starting points for developing general, affordable, and transferable polarizable force fields. The software implementing these approaches has been designed to utilize the force field fitting frameworks developed and maintained by the Open Force Field Initiative, setting the stage for further exploration of this approach to polarizable force field development.

The Importance of Charge Transfer and Solvent Screening in the Interactions of Backbones and Functional Groups in Amino Acid Residues and Nucleotides

Quantum mechanical (QM) calculations at the level of density-functional tight-binding are applied to a protein–DNA complex (PDB: 2o8b) consisting of 3763 atoms, averaging 100 snapshots from molecular dynamics simulations. A detailed comparison of QM and force field (Amber) results is presented. It is shown that, when solvent screening is taken into account, the contributions of the backbones are small, and the binding of nucleotides in the double helix is governed by the base–base interactions. On the other hand, the backbones can make a substantial contribution to the binding of amino acid residues to nucleotides and other residues. The effect of charge transfer on the interactions is also analyzed, revealing that the actual charge of nucleotides and amino acid residues can differ by as much as 6 and 8% from the formal integer charge, respectively. The effect of interactions on topological models (protein -residue networks) is elucidated.

Evaluation of molecular photophysical and photochemical properties using linear response time-dependent density functional theory with classical embedding: Successes and challenges

Time-dependent density functional theory (TDDFT) based approaches have been developed in recent years to model the excited-state properties and transition processes of the molecules in the gas-phase and in a condensed medium, such as in a solution and protein microenvironment or near semiconductor and metal surfaces. In the latter case, usually, classical embedding models have been adopted to account for the molecular environmental effects, leading to the multi-scale approaches of TDDFT/polarizable continuum model (PCM) and TDDFT/molecular mechanics (MM), where a molecular system of interest is designated as the quantum mechanical region and treated with TDDFT, while the environment is usually described using either a PCM or (non-polarizable or polarizable) MM force fields. In this Perspective, we briefly review these TDDFT-related multi-scale models with a specific emphasis on the implementation of analytical energy derivatives, such as the energy gradient and Hessian, the nonadiabatic coupling, the spin–orbit coupling, and the transition dipole moment as well as their nuclear derivatives for various radiative and radiativeless transition processes among electronic states. Three variations of the TDDFT method, the Tamm–Dancoff approximation to TDDFT, spin–flip DFT, and spin-adiabatic TDDFT, are discussed. Moreover, using a model system (pyridine–Ag20 complex), we emphasize that caution is needed to properly account for system–environment interactions within the TDDFT/MM models. Specifically, one should appropriately damp the electrostatic embedding potential from MM atoms and carefully tune the van der Waals interaction potential between the system and the environment. We also highlight the lack of proper treatment of charge transfer between the quantum mechanics and MM regions as well as the need for accelerated TDDFT modelings and interpretability, which calls for new method developments.

Using atomic charges to model molecular polarization

We review different models for introducing electric polarization in force fields, with special focus on methods where polarization is modelled at the atomic charge level. While electric polarization has been included in several force fields, the common approach has been to focus on atomic dipole polarizability. Several approaches allow modelling electric polarization by using charge-flow between charge sites instead, but this has been less exploited, despite that atomic charges and charge-flow is expected to be more important than atomic dipoles and dipole polarizability. A number of challenges are required to be solved for charge-flow models to be incorporated into polarizable force fields, for example how to parameterize the models and how to make them computational efficient.

Characterization of a TP53 Somatic Variant of Unknown Function From an Ovarian Cancer Patient Using Organoid Culture and Computational Modeling

In our proof-of-concept study of 1 patient with stage IIIC carcinosarcoma of the ovary, we discovered a rare mutation in the tumor suppressor, TP53, that results in the deletion of N131. Immunofluorescence imaging of the organoid culture revealed hyperstaining of p53 protein. Computational modeling suggests this residue is important for maintaining protein conformation. Drug screening identified the combination of a proteasome inhibitor with a histone deacetylase inhibitor as the most effective treatment. These data provide evidence for the successful culture of a patient tumor and analysis of drug response ex vivo.

Statistical mechanics of polarizable force fields based on classical Drude oscillators with dynamical propagation by the dual-thermostat extended Lagrangian

Polarizable force fields based on classical Drude oscillators offer a practical and computationally efficient avenue to carry out molecular dynamics (MD) simulations of large biomolecular systems. To treat the polarizable electronic degrees of freedom, the Drude model introduces a virtual charged particle that is attached to its parent nucleus via a harmonic spring. Traditionally, the need to relax the electronic degrees of freedom for each fixed set of nuclear coordinates is achieved by performing an iterative self-consistent field (SCF) calculation to satisfy a selected tolerance. This is a computationally demanding procedure that can increase the computational cost of MD simulations by nearly one order of magnitude. To avoid the costly SCF procedure, a small mass is assigned to the Drude particles, which are then propagated as dynamic variables during the simulations via a dual-thermostat extended Lagrangian algorithm. To help clarify the significance of the dual-thermostat extended Lagrangian propagation in the context of the polarizable force field based on classical Drude oscillators, the statistical mechanics of a dual-temperature canonical ensemble is formulated. The conditions for dynamically maintaining the dual-temperature properties in the case of the classical Drude oscillator are analyzed using the generalized Langevin equation.

Induced Polarization in Molecular Dynamics Simulations of the 5-HT3 Receptor Channel

Ion channel proteins form water-filled nanoscale pores within lipid bilayers, and their properties are dependent on the complex behavior of water in a nanoconfined environment. Using a simplified model of the pore of the 5-HT3 receptor (5HT3R) which restrains the backbone structure to that of the parent channel protein from which it is derived, we compare additive with polarizable models in describing the behavior of water in nanopores. Molecular dynamics simulations were performed with four conformations of the channel: two closed state structures, an intermediate state, and an open state, each embedded in a phosphatidylcholine bilayer. Water density profiles revealed that for all water models, the closed and intermediate states exhibited strong dewetting within the central hydrophobic gate region of the pore. However, the open state conformation exhibited varying degrees of hydration, ranging from partial wetting for the TIP4P/2005 water model to complete wetting for the polarizable AMOEBA14 model. Water dipole moments calculated using polarizable force fields also revealed that water molecules remaining within dewetted sections of the pore resemble gas phase water. Free energy profiles for Na+ and for Cl- ions within the open state pore revealed more rugged energy landscapes using polarizable force fields, and the hydration number profiles of these ions were also sensitive to induced polarization resulting in a substantive reduction of the number of waters within the first hydration shell of Cl- while it permeates the pore. These results demonstrate that induced polarization can influence the complex behavior of water and ions within nanoscale pores and provides important new insights into their chemical properties.

Drude Polarizable Force Field Parametrization of Carboxylate and N-Acetyl Amine Carbohydrate Derivatives

In this work, we report the development of Drude polarizable force field parameters for the carboxylate and N-acetyl amine derivatives, extending the functionality of the existing Drude polarizable carbohydrate force field. The force field parameters have been developed in a hierarchical manner, reproducing the quantum mechanical gas-phase properties of small model compounds representing the key functional group in the carbohydrate derivatives, including optimization of the electrostatic and bonded parameters. The optimized parameters were then used to generate the models for carboxylate and N-acetyl amine carbohydrate derivatives. The transferred parameters were further tested and optimized to reproduce crystal geometries and J-coupling data from nuclear magnetic resonance experiments. The parameter development resulted in the incorporation of d-glucuronate, l-iduronate, N-acetyl-d-glucosamine (GlcNAc), and N-acetyl-d-galactosamine (GalNAc) sugars into the Drude polarizable force field. The parameters developed in this study were then applied to study the conformational properties of glycosaminoglycan polymer hyaluronan, composed of d-glucuronate and N-acetyl-d-glucosamine, in aqueous solution. Upon comparing the results from the additive and polarizable simulations, it was found that the inclusion of polarization improved the description of the electrostatic interactions observed in hyaluronan, resulting in enhanced conformational flexibility. The developed Drude polarizable force field parameters in conjunction with the remainder of the Drude polarizable force field parameters can be used for future studies involving carbohydrates and their conjugates in complex, heterogeneous systems.

Molecular Dynamics Simulations of Ionic Liquids and Electrolytes Using Polarizable Force Fields

Many applications in chemistry, biology, and energy storage/conversion research rely on molecular simulations to provide fundamental insight into structural and transport properties of materials with high ionic concentrations. Whether the system is comprised entirely of ions, like ionic liquids, or is a mixture of a polar solvent with a salt, e.g., liquid electrolytes for battery applications, the presence of ions in these materials results in strong local electric fields polarizing solvent molecules and large ions. To predict properties of such systems from molecular simulations often requires either explicit or mean-field inclusion of the influence of polarization on electrostatic interactions. In this manuscript, we review the pros and cons of different treatments of polarization ranging from the mean-field approaches to the most popular explicit polarization models in molecular dynamics simulations of ionic materials. For each method, we discuss their advantages and disadvantages and emphasize key assumptions as well as their adjustable parameters. Strategies for the development of polarizable models are presented with a specific focus on extracting atomic polarizabilities. Finally, we compare simulations using polarizable and nonpolarizable models for several classes of ionic systems, discussing the underlying physics that each approach includes or ignores, implications for implementation and computational efficiency, and the accuracy of properties predicted by these methods compared to experiments.

Reduced computational cost of polarizable force fields by a modification of the always stable predictor-corrector

Classical polarizable force fields effectively incorporate the dynamic response of the electronic charge distributions into molecular dynamics simulations, but they do so at a significant increase in computational cost compared to simpler models. Here, we demonstrate how one can improve the stability of a polarizable force field molecular dynamics simulation or accelerate the evaluation of self-consistent polarization via a simple extension of the predictor in the always stable predictor-corrector method. Specifically, increasing the number of prior steps used in the predictor from 6 to 16 reduces the energy drift by an order of magnitude. Alternatively, for a given level of energy drift, the induced dipoles can be obtained ∼20% faster due to the reduced number of self-consistent field iterations required to maintain energetic stability. The extended-history predictor is straightforward to implement and involves minimal computational overhead.

Tinker 8: Software Tools for Molecular Design

The Tinker software, currently released as version 8, is a modular molecular mechanics and dynamics package written primarily in a standard, easily portable dialect of Fortran 95 with OpenMP extensions. It supports a wide variety of force fields, including polarizable models such as the Atomic Multipole Optimized Energetics for Biomolecular Applications (AMOEBA) force field. The package runs on Linux, macOS, and Windows systems. In addition to canonical Tinker, there are branches, Tinker-HP and Tinker-OpenMM, designed for use on message passing interface (MPI) parallel distributed memory supercomputers and state-of-the-art graphical processing units (GPUs), respectively. The Tinker suite also includes a tightly integrated Java-based graphical user interface called Force Field Explorer (FFE), which provides molecular visualization capabilities as well as the ability to launch and control Tinker calculations.

Massively Parallel Implementation of Divide-and-Conquer Jacobi Iterations Using Particle-Mesh Ewald for Force Field Polarization

To accelerate the evaluation of the self-consistent polarization in large condensed-phase systems with polarizable force fields, the new divide-and-conquer Jacobi iterations (DC-JI) solver is adapted for periodic boundary conditions with particle-mesh Ewald and implemented in a massively parallel fashion within the Tinker-HP software package. DC-JI captures the mutual polarization of close-range interactions within subclusters of atoms using Cholesky decomposition and couples in the polarization effects between these clusters iteratively. Iterative convergence is accelerated with direct inversion of the iterative subspace (DIIS) extrapolation. Compared to widely used preconditioned conjugate gradient (PCG) or conventional Jacobi iterations (JI/DIIS) algorithms, DC-JI/DIIS solves the polarization equations ∼20-30% faster in protein systems ranging from ∼10,000-175,000 atoms run on hundreds of processor cores. This translates to ∼10-15% speed-ups in the number of nanoseconds of simulation time that can be achieved per day. Not only is DC-JI/DIIS faster than PCG, but it also gives more energetically robust solutions for a given convergence threshold. These improvements make numerically robust polarizable force field simulations more computationally tractable for chemical systems of interest.

Mapping the Drude polarizable force field onto a multipole and induced dipole model

The induced dipole and the classical Drude oscillator represent two major approaches for the explicit inclusion of electronic polarizability into force field-based molecular modeling and simulations. In this work, we explore the equivalency of these two models by comparing condensed phase properties computed using the Drude force field and a multipole and induced dipole (MPID) model. Presented is an approach to map the electrostatic model optimized in the context of the Drude force field onto the MPID model. Condensed phase simulations on water and 15 small model compounds show that without any reparametrization, the MPID model yields properties similar to the Drude force field with both models yielding satisfactory reproduction of a range of experimental values and quantum mechanical data. Our results illustrate that the Drude oscillator model and the point induced dipole model are different representations of essentially the same physical model. However, results indicate the presence of small differences between the use of atomic multipoles and off-center charge sites. Additionally, results on the use of dispersion particle mesh Ewald further support its utility for treating long-range Lennard Jones dispersion contributions in the context of polarizable force fields. The main motivation in demonstrating the transferability of parameters between the Drude and MPID models is that the more than 15 years of development of the Drude polarizable force field can now be used with MPID formalism without the need for dual-thermostat integrators nor self-consistent iterations. This opens up a wide range of new methodological opportunities for polarizable models.

The truncated conjugate gradient (TCG), a non-iterative/fixed-cost strategy for computing polarization in molecular dynamics: Fast evaluation of analytical forces

In a recent paper [F. Aviat et al., J. Chem. Theory Comput. 13, 180-190 (2017)], we proposed the Truncated Conjugate Gradient (TCG) approach to compute the polarization energy and forces in polarizable molecular simulations. The method consists in truncating the conjugate gradient algorithm at a fixed predetermined order leading to a fixed computational cost and can thus be considered "non-iterative." This gives the possibility to derive analytical forces avoiding the usual energy conservation (i.e., drifts) issues occurring with iterative approaches. A key point concerns the evaluation of the analytical gradients, which is more complex than that with a usual solver. In this paper, after reviewing the present state of the art of polarization solvers, we detail a viable strategy for the efficient implementation of the TCG calculation. The complete cost of the approach is then measured as it is tested using a multi-time step scheme and compared to timings using usual iterative approaches. We show that the TCG methods are more efficient than traditional techniques, making it a method of choice for future long molecular dynamics simulations using polarizable force fields where energy conservation matters. We detail the various steps required for the implementation of the complete method by software developers.

Assessing many-body contributions to intermolecular interactions of the AMOEBA force field using energy decomposition analysis of electronic structure calculations

In this work, we evaluate the accuracy of the classical AMOEBA model for representing many-body interactions, such as polarization, charge transfer, and Pauli repulsion and dispersion, through comparison against an energy decomposition method based on absolutely localized molecular orbitals (ALMO-EDA) for the water trimer and a variety of ion-water systems. When the 2- and 3-body contributions according to the many-body expansion are analyzed for the ion-water trimer systems examined here, the 3-body contributions to Pauli repulsion and dispersion are found to be negligible under ALMO-EDA, thereby supporting the validity of the pairwise-additive approximation in AMOEBA's 14-7 van der Waals term. However AMOEBA shows imperfect cancellation of errors for the missing effects of charge transfer and incorrectness in the distance dependence for polarization when compared with the corresponding ALMO-EDA terms. We trace the larger 2-body followed by 3-body polarization errors to the Thole damping scheme used in AMOEBA, and although the width parameter in Thole damping can be changed to improve agreement with the ALMO-EDA polarization for points about equilibrium, the correct profile of polarization as a function of intermolecular distance cannot be reproduced. The results suggest that there is a need for re-examining the damping and polarization model used in the AMOEBA force field and provide further insights into the formulations of polarizable force fields in general.

On the convergence of multi-scale free energy simulations

In this work we employ simple model systems to evaluate the relative performance of two of the most important free energy methods: The Zwanzig equation (also known as "Free energy perturbation") and Bennett's acceptance ratio method (BAR). Although our examples should be transferable to other kinds of free energy simulations, we focus on applications of multi-scale free energy simulations. Such calculations are especially complex, since they connect two different levels of theory with very different requirements in terms of speed, accuracy, sampling and parallelizability. We try to reconcile all those different factors by developing some simple criteria to guide the early stages of the development of a free energy protocol. This is accomplished by quantifying how many λ intermediate steps and how many potential energy evaluations are necessary in order to reach a certain level of convergence.

Advanced Potential Energy Surfaces for Molecular Simulation

Advanced potential energy surfaces are defined as theoretical models that explicitly include many-body effects that transcend the standard fixed-charge, pairwise-additive paradigm typically used in molecular simulation. However, several factors relating to their software implementation have precluded their widespread use in condensed-phase simulations: the computational cost of the theoretical models, a paucity of approximate models and algorithmic improvements that can ameliorate their cost, underdeveloped interfaces and limited dissemination in computational code bases that are widely used in the computational chemistry community, and software implementations that have not kept pace with modern high-performance computing (HPC) architectures, such as multicore CPUs and modern graphics processing units (GPUs). In this Feature Article we review recent progress made in these areas, including well-defined polarization approximations and new multipole electrostatic formulations, novel methods for solving the mutual polarization equations and increasing the MD time step, combining linear-scaling electronic structure methods with new QM/MM methods that account for mutual polarization between the two regions, and the greatly improved software deployment of these models and methods onto GPU and CPU hardware platforms. We have now approached an era where multipole-based polarizable force fields can be routinely used to obtain computational results comparable to state-of-the-art density functional theory while reaching sampling statistics that are acceptable when compared to that obtained from simpler fixed partial charge force fields.

A New Method for Treating Drude Polarization in Classical Molecular Simulation

With polarization becoming an increasingly common feature in classical molecular simulation, it is important to develop methods that can efficiently and accurately evaluate the many-body polarization solution. In this work, we expand the theoretical framework of our inertial extended Langrangian, self-consistent field iteration-free method (iEL/0-SCF), introduced for point induced dipoles, to the polarization model of a Drude oscillator. When applied to the polarizable simple point charge model (PSPC) for water, our iEL/0-SCF method for Drude polarization is as stable as a well-converged SCF solution and more stable than traditional extended Lagrangian (EL) approaches or EL formulations based on two temperature ensembles where Drude particles are kept "colder" than the real degrees of freedom. We show that the iEL/0-SCF method eliminates the need for mass repartitioning from parent atoms onto Drude particles, obeys system conservation of linear and angular momentum, and permits the extension of the integration time step of a basic molecular dynamics simulation to 6.0 fs for PSPC water.

OpenMM 7: Rapid development of high performance algorithms for molecular dynamics

OpenMM is a molecular dynamics simulation toolkit with a unique focus on extensibility. It allows users to easily add new features, including forces with novel functional forms, new integration algorithms, and new simulation protocols. Those features automatically work on all supported hardware types (including both CPUs and GPUs) and perform well on all of them. In many cases they require minimal coding, just a mathematical description of the desired function. They also require no modification to OpenMM itself and can be distributed independently of OpenMM. This makes it an ideal tool for researchers developing new simulation methods, and also allows those new methods to be immediately available to the larger community.

Accurate Classical Polarization Solution with No Self-Consistent Field Iterations

We present a new solution for classical polarization that does not require any self-consistent field iterations, the aspect of classical polarization that makes it computationally expensive. The new method builds upon our iEL/SCF Lagrangian scheme that defines a set of auxiliary induced dipoles whose original purpose was to serve as a time-reversible initial guess to the SCF solution of the set of real induced dipoles. In the new iEL/0-SCF approach the auxiliary dipoles now drive the time evolution of the real induced dipoles such that they stay close to the Born-Oppenheimer surface in order to achieve a truly SCF-less method. We show that the iEL/0-SCF exhibits no loss of simulation accuracy when analyzed across bulk water, low to high concentration salt solutions, and small solutes to large proteins in water. In addition, iEL/0-SCF offers significant computational savings over more expensive SCF calculations based on traditional 1 fs time step integration using symplectic integrators and is as fast as reversible reference system propagator algorithms with an outer 2 fs time step.

Isotropic periodic sum for multipole interactions and a vector relation for calculation of the Cartesian multipole tensor

Isotropic periodic sum (IPS) is a method to calculate long-range interactions based on the homogeneity of simulation systems. By using the isotropic periodic images of a local region to represent remote structures, long-range interactions become a function of the local conformation. This function is called the IPS potential; it folds long-ranged interactions into a short-ranged potential and can be calculated as efficiently as a cutoff method. It has been demonstrated that the IPS method produces consistent simulation results, including free energies, as the particle mesh Ewald (PME) method. By introducing the multipole homogeneous background approximation, this work derives multipole IPS potentials, abbreviated as IPSMm, with m being the maximum order of multipole interactions. To efficiently calculate the multipole interactions in Cartesian space, we propose a vector relation that calculates a multipole tensor as a dot product of a radial potential vector and a directional vector. Using model systems with charges, dipoles, and/or quadrupoles, with and without polarizability, we demonstrate that multipole interactions of order m can be described accurately with the multipole IPS potential of order 2 or m - 1, whichever is higher. Through simulations with the multipole IPS potentials, we examined energetic, structural, and dynamic properties of the model systems and demonstrated that the multipole IPS potentials produce very similar results as PME with a local region radius (cutoff distance) as small as 6 Å.

An empirical extrapolation scheme for efficient treatment of induced dipoles

Many cutting edge force fields include polarization, to enhance their accuracy and range of applicability. In this work, we develop efficient strategies for the induced dipole polarization method. By fitting various orders of perturbation theory (PT) dipoles to a diverse training set, we arrive at a family of fully analytic methods - whose nth order is referred to OPTn - that span the full spectrum of polarization methods from the fast zeroth-order approach that neglects mutual dipole coupling, approaching the fully variational approach at high order. Our training set contains many difficult cases where the PT series diverges, and we demonstrate that our OPTn methods still deliver excellent results in these cases. Our tests show that the OPTn methods exhibit rapid convergence towards the exact answer with each increasing PT order. The fourth order OPT4 method, whose costs are commensurate with three iterations of the leading conjugate gradient method, is a particularly promising candidate to be used as a drop-in replacement for existing solvers without further parameterization.

An Estimation of Hybrid Quantum Mechanical Molecular Mechanical Polarization Energies for Small Molecules Using Polarizable Force-Field Approaches

In this work, we report two polarizable molecular mechanics (polMM) force field models for estimating the polarization energy in hybrid quantum mechanical molecular mechanical (QM/MM) calculations. These two models, named the potential of atomic charges (PAC) and potential of atomic dipoles (PAD), are formulated from the ab initio quantum mechanical (QM) response kernels for the prediction of the QM density response to an external molecular mechanical (MM) environment (as described by external point charges). The PAC model is similar to fluctuating charge (FQ) models because the energy depends on external electrostatic potential values at QM atomic sites; the PAD energy depends on external electrostatic field values at QM atomic sites, resembling induced dipole (ID) models. To demonstrate their uses, we apply the PAC and PAD models to 12 small molecules, which are solvated by TIP3P water. The PAC model reproduces the QM/MM polarization energy with a R² value of 0.71 for aniline (in 10,000 TIP3P water configurations) and 0.87 or higher for other 11 solute molecules, while the PAD model has a much better performance with R² values of 0.98 or higher. The PAC model reproduces reference QM/MM hydration free energies for 12 solute molecules with a RMSD of 0.59 kcal/mol. The PAD model is even more accurate, with a much smaller RMSD of 0.12 kcal/mol, with respect to the reference. This suggests that polarization effects, including both local charge distortion and intramolecular charge transfer, can be well captured by induced dipole type models with proper parametrization.

Truncated Conjugate Gradient: An Optimal Strategy for the Analytical Evaluation of the Many-Body Polarization Energy and Forces in Molecular Simulations

We introduce a new class of methods, denoted as Truncated Conjugate Gradient(TCG), to solve the many-body polarization energy and its associated forces in molecular simulations (i.e. molecular dynamics (MD) and Monte Carlo). The method consists in a fixed number of Conjugate Gradient (CG) iterations. TCG approaches provide a scalable solution to the polarization problem at a user-chosen cost and a corresponding optimal accuracy. The optimality of the CG-method guarantees that the number of the required matrix-vector products are reduced to a minimum compared to other iterative methods. This family of methods is non-empirical, fully adaptive, and provides analytical gradients, avoiding therefore any energy drift in MD as compared to popular iterative solvers. Besides speed, one great advantage of this class of approximate methods is that their accuracy is systematically improvable. Indeed, as the CG-method is a Krylov subspace method, the associated error is monotonically reduced at each iteration. On top of that, two improvements can be proposed at virtually no cost: (i) the use of preconditioners can be employed, which leads to the Truncated Preconditioned Conjugate Gradient (TPCG); (ii) since the residual of the final step of the CG-method is available, one additional Picard fixed point iteration (“peek”), equivalent to one step of Jacobi Over Relaxation (JOR) with relaxation parameter ω, can be made at almost no cost. This method is denoted by TCG-n(ω). Black-box adaptive methods to find good choices of ω are provided and discussed. Results show that TPCG-3(ω) is converged to high accuracy (a few kcal/mol) for various types of systems including proteins and highly charged systems at the fixed cost of four matrix-vector products: three CG iterations plus the initial CG descent direction. Alternatively, T(P)CG-2(ω) provides robust results at a reduced cost (three matrix-vector products) and offers new perspectives for long polarizable MD as a production algorithm. The T(P)CG-1(ω) level provides less accurate solutions for inhomogeneous systems, but its applicability to well-conditioned problems such as water is remarkable, with only two matrix-vector product evaluations.

Calculating distribution coefficients based on multi-scale free energy simulations: an evaluation of MM and QM/MM explicit solvent simulations of water-cyclohexane transfer in the SAMPL5 challenge

One of the central aspects of biomolecular recognition is the hydrophobic effect, which is experimentally evaluated by measuring the distribution coefficients of compounds between polar and apolar phases. We use our predictions of the distribution coefficients between water and cyclohexane from the SAMPL5 challenge to estimate the hydrophobicity of different explicit solvent simulation techniques. Based on molecular dynamics trajectories with the CHARMM General Force Field, we compare pure molecular mechanics (MM) with quantum-mechanical (QM) calculations based on QM/MM schemes that treat the solvent at the MM level. We perform QM/MM with both density functional theory (BLYP) and semi-empirical methods (OM1, OM2, OM3, PM3). The calculations also serve to test the sensitivity of partition coefficients to solute polarizability as well as the interplay of the quantum-mechanical region with the fixed-charge molecular mechanics environment. Our results indicate that QM/MM with both BLYP and OM2 outperforms pure MM. However, this observation is limited to a subset of cases where convergence of the free energy can be achieved.

Evaluation of Representations and Response Models for Polarizable Force Fields

For classical simulations of condensed-phase systems, such as organic liquids and biomolecules, to achieve high accuracy, they will probably need to incorporate an accurate, efficient model of conformation-dependent electronic polarization. Thus, it is of interest to understand what determines the accuracy of a polarizable electrostatics model. This study approaches this problem by breaking polarization models down into two main components: the representation of electronic polarization and the response model used for mapping from an inducing field to the polarization within the chosen representation. Among the most common polarization representations are redistribution of atom-centered charges, such as those used in the fluctuating charge model, and atom-centered point dipoles, such as those used in a number of different polarization models. Each of these representations has been combined with one or more response models. The response model of fluctuating charge, for example, is based on the idea of electronegativity equalization in the context of changing electrostatic potentials (ESPs), whereas point-dipole representations typically use a response model based on point polarizabilities whose induced dipoles are computed based on interaction with other charges and dipoles. Here, we decouple polarization representations from their typical response models to analyze the strengths and weaknesses of various polarization approximations. First, we compare the maximal possible accuracies achievable by the charge redistribution and point-dipole model representations, by testing their ability to replicate quantum mechanical (QM) ESPs around small molecules polarized by external inducing charges. Perhaps not surprisingly, the atom-centered dipole model can yield higher accuracy. Next, we test two of the most commonly used response functions used for the point-dipole representations, self-consistent and direct (or first-order) inducible point polarizabilities, where the polarizabilities are optimized to best fit the full set of polarized QM potentials for each molecule studied. Strikingly, the induced-dipole response model markedly degrades accuracy relative to that obtainable with optimal point dipoles. In fact, the maximal accuracy achievable with this response model is even lower than that afforded by an optimal charge-redistribution representation. This means that, if coupled with a sufficiently accurate response function, the point-charge representation could outperform the standard induced-dipole model. Furthermore, although a key advantage of the point-dipole representation, relative to charge redistribution, is its ability to capture out-of-plane polarization, the inducible dipole response model causes it to be less accurate than optimal charge redistribution for out-of-plane induction of the planar nitrobenzene molecule. Thus, the widely used inducible dipole response function falls short of the full potential accuracy achievable with the point-dipole representation it employs. Additional results reported here bear on the relative accuracy of self-consistent inducible dipoles versus that of the first-order, or direct, approximation and on methods for assigning partial atomic charges for use in conjunction with inducible dipole models. In sum, these results point to the improvement of polarization response models as an important direction for future research aimed at improving the accuracy of molecular simulations.

Ambient-Potential Composite Ewald Method for ab Initio Quantum Mechanical/Molecular Mechanical Molecular Dynamics Simulation

A new approach for performing Particle Mesh Ewald in ab initio quantum mechanical/molecular mechanical (QM/MM) simulations with extended atomic orbital basis sets is presented. The new approach, the Ambient-Potential Composite Ewald (CEw) method, does not perform the QM/MM interaction with Mulliken charges nor electrostatically fit charges. Instead the nuclei and electron density interact directly with the MM environment, but in a manner that avoids the use of dense Fourier transform grids. By performing the electrostatics with the underlying QM density, the CEw method avoids self-consistent field instabilities that have been encountered with simple charge mapping procedures. Potential of mean force (PMF) profiles of the p-nitrophenyl phosphate dissociation reaction in explicit solvent are computed from PBE0/6-31G* QM/MM molecular dynamics simulations with various electrostatic protocols. The CEw profiles are shown to be stable with respect to real-space Ewald cutoff, whereas the PMFs computed from truncated and switched electrostatics produce artifacts. PBE0/6-311G**, AM1/d-PhoT, and DFTB2 QM/MM simulations are performed to generate two-dimensional PMF profiles of the phosphoryl transesterification reactions with ethoxide and phenoxide leaving groups. The semiempirical models incorrectly produce a concerted ethoxide mechanism, whereas PBE0 correctly produces a stepwise mechanism. The ab initio reaction barriers agree more closely to experiment than the semiempirical models. The failure of Mulliken-charge QM/MM-Ewald is analyzed.

An Empirical Polarizable Force Field Based on the Classical Drude Oscillator Model: Development History and Recent Applications

Molecular mechanics force fields that explicitly account for induced polarization represent the next generation of physical models for molecular dynamics simulations. Several methods exist for modeling induced polarization, and here we review the classical Drude oscillator model, in which electronic degrees of freedom are modeled by charged particles attached to the nuclei of their core atoms by harmonic springs. We describe the latest developments in Drude force field parametrization and application, primarily in the last 15 years. Emphasis is placed on the Drude-2013 polarizable force field for proteins, DNA, lipids, and carbohydrates. We discuss its parametrization protocol, development history, and recent simulations of biologically interesting systems, highlighting specific studies in which induced polarization plays a critical role in reproducing experimental observables and understanding physical behavior. As the Drude oscillator model is computationally tractable and available in a wide range of simulation packages, it is anticipated that use of these more complex physical models will lead to new and important discoveries of the physical forces driving a range of chemical and biological phenomena.