Classical electrostatics for biomolecular simulations

Analytic Gradients for the Electrostatic Embedding QM/MM Model in Periodic Boundary Conditions Using Particle-Mesh Ewald Sums and Electrostatic Potential Fitted Charge Operators

Long-range electrostatic effects are fundamental for describing chemical reactivity in the condensed phase. Here, we present the methodology of an efficient quantum mechanical/molecular mechanical (QM/MM) model in periodic boundary conditions (PBC) compatible with QM/MM boundaries at chemical bonds. The method combines electrostatic potential fitted charge operators and electrostatic potentials derived from the smooth particle-mesh Ewald (PME) sum approach. The total energy and its analytic first derivatives with respect to QM, MM, and lattice vectors allow QM/MM molecular dynamics (MD) in the most common thermodynamic ensembles. We demonstrate the robustness of the method by performing a QM/MM MD equilibration of methanol in water. We simulate the cis/trans isomerization free-energy profiles in water of proline amino acid and a proline-containing oligopeptide, showing a correct description of the reaction barrier. Our PBC-compatible QM/MM model can efficiently be used to study the chemical reactivity in the condensed phase and enzymatic catalysis.

Perspective on Integrative Simulations of Bioenergetic Domains

Bioenergetic processes in cells, such as photosynthesis or respiration, integrate many time and length scales, which makes the simulation of energy conversion with a mere single level of theory impossible. Just like the myriad of experimental techniques required to examine each level of organization, an array of overlapping computational techniques is necessary to model energy conversion. Here, a perspective is presented on recent efforts for modeling bioenergetic phenomena with a focus on molecular dynamics simulations and its variants as a primary method. An overview of the various classical, quantum mechanical, enhanced sampling, coarse-grained, Brownian dynamics, and Monte Carlo methods is presented. Example applications discussed include multiscale simulations of membrane-wide electron transport, rate kinetics of ATP turnover from electrochemical gradients, and finally, integrative modeling of the chromatophore, a photosynthetic pseudo-organelle.

Computational insights into dynamics and conformational stability of N-acetylmannosamine kinase mutations

The activity of UDP-GlcNAc 2-epimerase/ManNAc kinase (GNE) is essential for the biosynthesis of sialic acid, which is involved in cellular processes in health and diseases. GNE contains an N-terminal epimerase domain and a C-terminal kinase domain (N-acetylmannosamine kinase, MNK). Mutations of the GNE protein led to hypoactivity of the enzyme and cause sialurea or autosomal recessive inclusion body myopathy/Nonaka myopathy. Here, we used all-atom molecular dynamics (MD) simulations to comprehend the folding, dynamics and conformational stability of MNK variants, including the wild type (WT) and three mutants (H677R, V696M and H677R/V696M). The deleterious and destabilizing nature of MNK mutants were predicted using different prediction tools. Results predicted that mutations modulate the stability, flexibility and function of MNK. The effect of mutations on the conformational stability and dynamics of MNK was next studied through the free-energy landscape (FEL), hydrogen-bonds and secondary structure changes. The FEL results show that the mutations interfere with various conformational transitions in both WT and mutants, exposing the structural underpinnings of protein destabilization and unfolding brought on by mutation. We discover that, when compared to the other two mutations, V696M and H677R/V696M, H677R has the most harmful effects. These findings have a strong correlation with published experimental studies that demonstrate how these mutations disrupt MNK activity. Hence, this computational study describes the structural details to unravel the mutant effects at the atomistic resolution and has implications for understanding the GNE's physiological and pathological role.Communicated by Ramaswamy H. Sarma.

Unsolved problem of long-range interactions: dipolar spin-ice study

Long-range interactions derive various strange phenomena. As illustrated by cutoff simulations of water, increasing cutoff length does not improve the simulation result necessarily; on the contrary, it makes the result worse. In the extreme situation, the structure of water transforms into a layer structure. In this study, to explore the underlying mechanism of this phenomenon, we performed Monte Carlo simulations on dipolar spins arranged on a pyrochlore spin-ice lattice. Like the water case, the present dipolar spin system also showed cutoff-induced dipole ordering and layer formation. The width of the layers depended on the cutoff length; and longer cutoff length led to a broader layer. These features are certainly consistent with the previous water case. This indicates that layer formation is the general behavior of dipolar systems whose interactions are truncated within a finite distance. The result is important for future exploration of the relationship between long-range interactions and resulting structures. In addition, it emphasizes the necessity of rigorous treatment of long-range interactions because increasing the cutoff length prevents convergence and provides an entirely different result from the rigorous Ewald calculation.

Polarizable AMOEBA Model for Simulating Mg2+·Protein·Nucleotide Complexes

Molecular mechanics (MM) simulations have the potential to provide detailed insights into the mechanisms of enzymes that utilize nucleotides as cofactors. In most cases, the activities of these enzymes also require the binding of divalent cations to catalytic sites. However, modeling divalent cations in MM simulations has been challenging. The inclusion of explicit polarization was considered promising, but despite improvements over nonpolarizable force fields and despite the inclusion of "Nonbonded-fix (NB-fix)" corrections, errors in interaction energies of divalent cations with proteins remain large. Importantly, the application of these models fails to reproduce the experimental structural data on Mg2+·Protein·ATP complexes. Focusing on these complexes, here we provide a systematic assessment of the polarizable AMOEBA model and recommend critical changes that substantially improve its predictive performance. Our key results are as follows. We first show that our recent revision of the AMOEBA protein model (AMOEBABIO18-HFC), which contains high field corrections (HFCs) to induced dipoles, dramatically improves Mg2+-protein interaction energies, reducing the mean absolute error (MAE) from 17 to 10 kcal/mol. This further supports the general applicability of AMOEBABIO18-HFC. The inclusion of many-body NB-fix corrections further reduces MAE to 6 kcal/mol, which amounts to less than 2% error. The errors are estimated with respect to vdW-inclusive density functional theory that we benchmark against CCSD(T) calculations and experiments. We also present a new model of ATP with revised polarization parameters to better capture its high field response, as well as new vdW and dihedral parameters. The ATP model accurately predicts experimental Mg2+-ATP binding free energy in the aqueous phase and provides new insights into how Mg2+ associates with ATP. Finally, we show that molecular dynamics (MD) simulations of Mg2+·Kinase·ATP complexes carried out with these improvements lead to a better agreement in global and local catalytic site structures between MD and X-ray crystallography.

A cutoff-based method with charge-distribution-data driven pair potentials for efficiently estimating electrostatic interactions in molecular systems

We introduce a simple cutoff-based method for precise electrostatic energy calculations in the molecular dynamics (MD) simulations of point-particle systems. Our method employs a theoretically derived smooth pair potential function to define electrostatic energy, offering stability and computational efficiency in MD simulations. Instead of imposing specific physical conditions, such as dielectric environments or charge neutrality, we focus on the relationship represented by a single summation formula of charge-weighted pair potentials. This approach allows an accurate energy approximation for each particle, enabling a straightforward error analysis. The resulting particle-dependent pair potential captures the charge distribution information, making it suitable for heterogeneous systems and ensuring an enhanced accuracy through distant information inclusion. Numerical investigations of the Madelung constants of crystalline systems validate the method's accuracy.

Application of molecular dynamic simulations in modeling the excited state behavior of confined molecules

Relative to isotropic organic solvent medium, the structure and conformation of a reactant molecule in an organized and confining medium are often different. In addition, because of the rigidity of the immediate environment, the reacting molecule have a little freedom to undergo large changes even upon gaining energy or modifications in the electronic structure. These alterations give rise to differences in the photochemistry of a molecular and supramolecular species. In this study, one such example is presented. α-Alkyl dibenzylketones upon excitation in isotropic solvents give products via Norrish type I and type II reactions that are independent of the chain length of the alkyl substituent. On the other hand, when these molecules are enclosed within an organic capsule of volume ~ 550 Å3, they give products that are strikingly dependent on the length of the α-alkyl substitution. These previously reported experimental observations are rationalized based on the structures generated by molecular modeling (docking and molecular dynamics (MD) simulations). It is shown that MD simulations that are utilized extensively in biologically important macromolecules can also be useful to understand the excited state behavior of reactive molecules that are part of supramolecular assemblies. These simulations can provide structural information of the reactant molecule and the surroundings complementing that with the one obtained from 1 and 2D NMR experiments. MD simulated structures of seven α-alkyl dibenzylketones encapsulated within the octa acid capsule provide a clear understanding of their unique behavior in this restricted medium. Because of the rigidity of the medium, these structures although generated in the ground state can rationalize the photochemical behavior of the molecules in the excited state.

Accurate Host-Guest Binding Free Energies Using the AMOEBA Polarizable Force Field

A grand challenge of computational biophysics is accurate prediction of interactions between molecules. Molecular dynamics (MD) simulations have recently gained much interest as a tool to directly compute rigorous intermolecular binding affinities. The choice of a fixed point-charge or polarizable multipole force field used in MD is a topic of ongoing discussion. To compare alternative methods, we participated in the SAMPL7 and SAMPL8 Gibb octaacid host-guest challenges to assess the Atomic Multipole Optimized Energetics for Biomolecular Applications (AMOEBA) polarizable multipole force field. Advantages of AMOEBA over fixed charge models include improved representation of molecular electrostatic potentials and better description of water occupying the unligated host cavity. Prospective predictions for 26 host-guest systems exhibit a mean unsigned error vs experiment of 0.848 kcal/mol across all absolute binding free energies, demonstrating excellent agreement between computational and experimental results. In addition, we explore two topics related to the inclusion of ions in MD simulations: use of a neutral co-alchemical protocol and the effect of salt concentration on binding affinity. Use of the co-alchemical method minimally affects computed energies, but salt concentration significantly perturbs our binding results. Higher salt concentration strengthens binding through classical charge screening. In particular, added Na+ ions screen negatively charged carboxylate groups near the binding cavity, thereby diminishing repulsive coulomb interactions with negatively charged guests. Overall, the AMOEBA results demonstrate the accuracy available through a force field providing a detailed energetic description of the four octaacid hosts and 13 charged organic guests. Use of the AMOEBA polarizable atomic multipole force field in conjunction with an alchemical free energy protocol can achieve chemical accuracy in application to realistic molecular systems.

Divide-and-Conquer Linear-Scaling Quantum Chemical Computations

Fragmentation and embedding schemes are of great importance when applying quantum-chemical calculations to more complex and attractive targets. The divide-and-conquer (DC)-based quantum-chemical model is a fragmentation scheme that can be connected to embedding schemes. This feature article explains several DC-based schemes developed by the authors over the last two decades, which was inspired by the pioneering study of DC self-consistent field (SCF) method by Yang and Lee (J. Chem. Phys. 1995, 103, 5674-5678). First, the theoretical aspects of the DC-based SCF, electron correlation, excited-state, and nuclear orbital methods are described, followed by the two-component relativistic theory, quantum-mechanical molecular dynamics simulation, and the introduction of three programs, including DC-based schemes. Illustrative applications confirmed the accuracy and feasibility of the DC-based schemes.

Treating Polarization Effects in Charged and Polar Bio-Molecules Through Variable Electrostatic Parameters

Polarization plays important roles in charged and hydrogen bonding containing systems. Much effort ranging from the construction of physics-based models to quantum mechanism (QM)-based and machine learning (ML)-assisted models have been devoted to incorporating the polarization effect into the conventional force fields at different levels, such as atomic and coarse grained (CG). The application of polarizable force fields or polarization models was limited by two aspects, namely, computational cost and transferability. Different from physics-based models, no predetermining parameters were required in the QM-based approaches. Taking advantage of both the accuracy of QM calculations and efficiency of molecular mechanism (MM) and ML, polarization effects could be treated more efficiently while maintaining the QM accuracy. The computational cost could be reduced with variable electrostatic parameters, such as the charge, dipole, and electronic dielectric constant with the help of linear scaling fragmentation-based QM calculations and ML models. Polarization and entropy effects on the prediction of partition coefficient of druglike molecules are demonstrated by using both explicit or implicit all-atom molecular dynamics simulations and machine learning-assisted models. Directions and challenges for future development are also envisioned.

Discovering potential inhibitors against SARS-CoV-2 by targeting Nsp13 Helicase

The rise in the incidence of COVID-19 as a result of SARS-CoV-2 infection has threatened public health globally. Till now, there have been no proper prophylactics available to fight COVID-19, necessitating the advancement and evolution of effective curative against SARS-CoV-2. This study aimed at the nonstructural protein 13 (nsp13) helicase as a promising target for drug development against COVID-19. A unique collection of nucleoside analogs was screened against the SARS-CoV-2 helicase protein, for which a molecular docking experiment was executed to depict the selected ligand's binding affinity with the SARS-CoV-2 helicase proteins. Simultaneously, molecular dynamic simulations were performed to examine the protein's binding site's conformational stability, flexibility, and interaction with the ligands. Key nucleoside ligands were selected for pharmacokinetic analysis based on their docking scores. Selected ligands (cordycepin and pritelivir) showed excellent pharmacokinetics and were well stabilized at the proteins' binding site throughout the MD simulation. We have also performed binding free energy analysis or the binding characteristics of ligands with Nsp13 by using MM-PBSA and MM-GBSA. Free energy calculation by MM-PBSA and MM-GBSA analysis suggests that pritelivir may work as viable therapeutics for efficient drug advancement against SARS-CoV-2 Nsp13 helicase, potentially arresting the SARS-CoV-2 replication.Communicated by Ramaswamy H. Sarma.

Polarizable Molecular Block Model: Toward the Development of an Induced Dipole Force Field for DNA

For flexible and highly ionized macromolecules such as DNA, it is important to correctly evaluate the intramolecular polarization in an induced dipole force field. In a proposed polarizable molecular block (PMB) model, a large molecule is divided into several molecular blocks. The atomic charges of the blocks are optimized by using the respective electrostatic potentials (ESPs) on the molecular surface. By using the capped hydrogen removal operation, the total charge of the blocks is controlled exactly to have an integer charge. The atomic polarizabilities of the blocks are optimized by using the respective polarized one-electron potentials that are the differences between ESPs with and without an external test charge. Induced dipole-charge interactions between the blocks are all included, but those interactions within the blocks are strictly excluded. All dipole-dipole interactions are included, but the damping functions are applied to the close dipole-dipole pairs. Several types of damping (simple scaling, exponential, linear, and Gaussian) are evaluated. The validity of the PMB model was verified by using trinucleotide duplexes which have A-, B-, and Z-DNA forms. The reference energies of trinucleotide duplexes including counterions (GGT3Na-ACC3Na, GAC3Na-GTC3Na, and GCG3Na-CGC3Na) are calculated using ωB97XD/aug-cc-pVDZ. All damping types reproduced well the reference interaction energies, dipole moments, and ESPs. Among them, the simple scaling with strong attenuation to 1-2 atomic pairs showed the highest stability against the polarization catastrophe. This study shows that it is possible to develop a high-quality polarizable force field by treating the intramolecular polarization on a block-by-block basis.

Multiscale Modeling of Phosphate···π Contacts in RNA U-Turns Exposes Differences between Quantum-Chemical and AMBER Force Field Descriptions

Phosphate···π, also called anion···π, contacts occur between nucleobases and anionic phosphate oxygens (OP2) in r(GNRA) and r(UNNN) U-turn motifs (N = A,G,C,U; R = A,G). These contacts were investigated using state-of-the-art quantum-chemical methods (QM) to characterize their physicochemical properties and to serve as a reference to evaluate AMBER force field (AFF) performance. We found that phosphate···π interaction energies calculated with the AFF for dimethyl phosphate···nucleobase model systems are less stabilizing in comparison with double-hybrid DFT and that minimum contact distances are larger for all nucleobases. These distance stretches are also observed in large-scale AFF vs QM/MM computations and classical molecular dynamics (MD) simulations on several r(gcGNRAgc) tetraloop hairpins when compared to experimental data extracted from X-ray/cryo-EM structures (res. ≤ 2.5 Å) using the WebFR3D bioinformatic tool. MD simulations further revealed shifted OP2/nucleobase positions. We propose that discrepancies between the QM and AFF result from a combination of missing polarization in the AFF combined with too large AFF Lennard-Jones (LJ) radii of nucleobase carbon atoms in addition to an exaggerated short-range repulsion of the r^-12 LJ repulsive term. We compared these results with earlier data gathered on lone pair···π contacts in CpG Z-steps occurring in r(UNCG) tetraloops. In both instances, charge transfer calculations do not support any significant n → π* donation effects. We also investigated thiophosphate···π contacts that showed reduced stabilizing interaction energies when compared to phosphate···π contacts. Thus, we challenge suggestions that the experimentally observed enhanced thermodynamic stability of phosphorothioated r(GNRA) tetraloops can be explained by larger London dispersion.

Non-Ewald methods for evaluating the electrostatic interactions of charge systems: similarity and difference

In molecular simulations, it is essential to properly calculate the electrostatic interactions of particles in the physical system of interest. Here we consider a method called the non-Ewald method, which does not rely on the standard Ewald method with periodic boundary conditions, but instead relies on the cutoff-based techniques. We focus on the physicochemical and mathematical conceptual aspects of the method in order to gain a deeper understanding of the simulation methodology. In particular, we take into account the reaction field (RF) method, the isotropic periodic sum (IPS) method, and the zero-multipole summation method (ZMM). These cutoff-based methods are based on different physical ideas and are completely distinguishable in their underlying concepts. The RF and IPS methods are "additive" methods that incorporate information outside the cutoff region, via dielectric medium and isotropic boundary condition, respectively. In contrast, the ZMM is a "subtraction" method that tries to remove the artificial effects, generated near the boundary, from the cutoff sphere. Nonetheless, we find physical and/or mathematical similarities between these methods. In particular, the modified RF method can be derived by the principle of neutralization utilized in the ZMM, and we also found a direct relationship between IPS and ZMM.

Inclusion of High-Field Target Data in AMOEBA's Calibration Improves Predictions of Protein-Ion Interactions

The reliability of molecular mechanics simulations to predict effects of ion binding to proteins depends on their ability to simultaneously describe ion-protein, ion-water, and protein-water interactions. Force fields (FFs) to describe protein-water and ion-water interactions have been constructed carefully and have also been refined routinely to improve accuracy. Descriptions for ion-protein interactions have also been refined, although in an a posteriori manner through the use of "nonbonded-fix (NB-fix)" approaches in which parameters from default Lennard-Jones mixing rules are replaced with those optimized against some reference data. However, even after NB-fix corrections, there remains a significant need for improvement. This is also true for polarizable FFs that include self-consistent inducible moments. Our recent studies on the polarizable AMOEBA FF suggested that the problem associated with modeling ion-protein interactions could be alleviated by recalibrating polarization models of cation-coordinating functional groups so that they respond better to the high electric fields present near ions. Here, we present such a recalibration of carbonyls, carboxylates, and hydroxyls in the AMOEBA protein FF and report that it does improve predictions substantially─mean absolute errors in Na⁺-protein and K⁺-protein interaction energies decrease from 8.7 to 5.3 and 9.6 to 6.3 kcal/mol, respectively. Errors are computed with respect to estimates from van der Waals-inclusive density functional theory benchmarked against high-level quantum mechanical calculations and experiments. While recalibration does improve ion-protein interaction energies, they still remain underestimated, suggesting that further improvements can be made in a systematic manner through modifications in classical formalism. Nevertheless, we show that by applying our many-body NB-fix correction to Lennard-Jones components, these errors are further reduced to 2.7 and 2.6 kcal/mol, respectively, for Na⁺ and K⁺ ions. Finally, we show that the recalibrated AMOEBA protein FF retains its intrinsic reliability in predicting protein structure and dynamics in the condensed phase.

Best Practices in Constant pH MD Simulations: Accuracy and Sampling

Various approaches have been proposed to include the effect of pH in molecular dynamics (MD) simulations. Among these, the λ-dynamics approach proposed by Brooks and co-workers [Kong, X.; Brooks III, C. L. J. Chem. Phys.1996, 105, 2414−2423] can be performed with little computational overhead and hfor each typeence be used to routinely perform MD simulations at microsecond time scales, as shown in the accompanying paper [Aho, N. et al. J. Chem. Theory Comput.2022, DOI: 10.1021/acs.jctc.2c00516]. At such time scales, however, the accuracy of the molecular mechanics force field and the parametrization becomes critical. Here, we address these issues and provide the community with guidelines on how to set up and perform long time scale constant pH MD simulations. We found that barriers associated with the torsions of side chains in the CHARMM36m force field are too high for reaching convergence in constant pH MD simulations on microsecond time scales. To avoid the high computational cost of extending the sampling, we propose small modifications to the force field to selectively reduce the torsional barriers. We demonstrate that with such modifications we obtain converged distributions of both protonation and torsional degrees of freedom and hence consistent pK_a estimates, while the sampling of the overall configurational space accessible to proteins is unaffected as compared to normal MD simulations. We also show that the results of constant pH MD depend on the accuracy of the correction potentials. While these potentials are typically obtained by fitting a low-order polynomial to calculated free energy profiles, we find that higher order fits are essential to provide accurate and consistent results. By resolving problems in accuracy and sampling, the work described in this and the accompanying paper paves the way to the widespread application of constant pH MD beyond pK_a prediction.

Galvani Offset Potential and Constant-pH Simulations of Membrane Proteins

A central problem in computational biophysics is the treatment of titratable residues in molecular dynamics simulations of large biological macromolecular systems. Conventional simulation methods ascribe a fixed ionization state to titratable residues in accordance with their pK_a and the pH of the system, assuming that an effective average model will be able to capture the predominant behavior of the system. While this assumption may be justifiable in many cases, it is certainly limited, and it is important to design alternative methodologies allowing a more realistic treatment. Constant-pH simulation methods provide powerful approaches to handle titratable residues more realistically by allowing the ionization state to vary statistically during the simulation. Extending the molecular mechanical (MM) potential energy function to a family of potential functions accounting for different ionization states, constant-pH simulations are designed to sample all accessible configurations and ionization states, properly weighted according to their Boltzmann factor. Because protonation and deprotonation events correspond to a change in the total charge, difficulties arise when the long-range Coulomb interaction is treated on the basis of an idealized infinite simulation model and periodic boundary conditions with particle-mesh Ewald lattice sums. Charging free-energy calculations performed under these conditions in aqueous solution depend on the Galvani potential of the bulk water phase. This has important implications for the equilibrium and nonequilibrium constant-pH simulation methods grounded in the relative free-energy difference corresponding to the protonated and unprotonated residues. Here, the effect of the Galvani potential is clarified, and a simple practical solution is introduced to address this issue in constant-pH simulations of the acid-sensing ion channel (ASIC).

Hydration of divalent lanthanides, Sm2+ and Eu2+ : A molecular dynamics study with polarizable AMOEBA force field

The chemistry of divalent lanthanides, Ln2+ , is a growing sub-field of heavy element chemistry owing to new synthetic approaches. However, some theoretical aspects of these unusual cations are currently underdeveloped, especially as they relate to their dynamic properties in solution. In this work, we address the hydration of two of the classical Ln2+ cations, Sm2+ and Eu2+ , using atomic multipole optimized energetic for biomolecular applications (AMOEBA) force fields. These cations have not been parameterized to date with AMOEBA, and few studies are available because of their instability with respect to oxidation in aqueous media. Coordination numbers (CN's) of 8.2 and 8.1 respectively for Sm2+ and Eu2+ , and 8.8 for both Sm3+ and Eu3+ have been obtained and are in good agreement with the few available AIMD and X-ray absorption fine structures studies. The decreased CN of Ln2+ compared with Ln3+ arises from progressive water exchange events that indicates the gradual stabilization of 8-coordinate structures with respect to 9-coordinate geometries. Moreover, the effects of the chloride counter anions on the coordination of Ln2+ cations have been studied at different chloride concentrations in this work. Lastly, water exchange times of Ln2+ cations have been calculated to provide a comprehensive understanding of the behavior of Eu2+ and Sm2+ in aqueous chloride media.

Better force fields start with better data: A data set of cation dipeptide interactions

We present a data set from a first-principles study of amino-methylated and acetylated (capped) dipeptides of the 20 proteinogenic amino acids - including alternative possible side chain protonation states and their interactions with selected divalent cations (Ca²⁺, Mg²⁺ and Ba²⁺). The data covers 21,909 stationary points on the respective potential-energy surfaces in a wide relative energy range of up to 4 eV (390 kJ/mol). Relevant properties of interest, like partial charges, were derived for the conformers. The motivation was to provide a solid data basis for force field parameterization and further applications like machine learning or benchmarking. In particular the process of creating all this data on the same first-principles footing, i.e. density-functional theory calculations employing the generalized gradient approximation with a van der Waals correction, makes this data suitable for first principles data-driven force field development. To make the data accessible across domain borders and to machines, we formalized the metadata in an ontology.

Modified Poisson–Boltzmann theory for polyelectrolytes in monovalent salt solutions with finite-size ions

We present a soft-potential-enhanced Poisson–Boltzmann (SPB) theory to efficiently capture ion distributions and electrostatic potential around rodlike charged macromolecules. The SPB model is calibrated with a coarse-grained particle-based model for polyelectrolytes (PEs) in monovalent salt solutions as well as compared to a full atomistic molecular dynamics simulation with the explicit solvent. We demonstrate that our modification enables the SPB theory to accurately predict monovalent ion distributions around a rodlike PE in a wide range of ion and charge distribution conditions in the weak-coupling regime. These include excess salt concentrations up to 1M and ion sizes ranging from small ions, such as Na+ or Cl−, to softer and larger ions with a size comparable to the PE diameter. The work provides a simple way to implement an enhancement that effectively captures the influence of ion size and species into the PB theory in the context of PEs in aqueous salt solutions.

Grotthuss Molecular Dynamics Simulations for Modeling Proton Hopping in Electrosprayed Water Droplets

Excess protons in water exhibit unique transport properties because they can rapidly hop along H-bonded water wires. Considerable progress has been made in unraveling this Grotthuss diffusion mechanism using quantum mechanical-based computational techniques. Unfortunately, high computational cost tends to restrict those techniques to small systems and short times. Molecular dynamics (MD) simulations can be applied to much larger systems and longer time windows. However, standard MD methods do not permit the dissociation/formation of covalent bonds, such that Grotthuss diffusion cannot be captured. Here, we bridge this gap by combining atomistic MD simulations (using Gromacs and TIP4P/2005 water) with proton hopping. Excess protons are modeled as hydronium ions that undergo H₃O⁺ + H₂O → H₂O + H₃O⁺ transitions. In accordance with ab initio MD data, these Grotthuss hopping events are executed in "bursts" with quasi-instantaneous hopping across one or more waters. The bursts are separated by regular MD periods during which H₃O⁺ ions undergo Brownian diffusion. The resulting proton diffusion coefficient agrees with the literature value. We apply this Grotthuss MD technique to highly charged water droplets that are in a size regime encountered during electrospray ionization (5 nm radius, ∼17,000 H₂O). The droplets undergo rapid solvent evaporation and occasional H₃O⁺ ejection, keeping them at ca. 81% of the Rayleigh limit. The simulated behavior is consistent with phase Doppler anemometry data. The Grotthuss MD technique developed here should be useful for modeling the behavior of various proton-containing systems that are too large for high-level computational approaches. In particular, we envision future applications related to electrospray processes, where earlier simulations used metal cations while in reality excess protons dominate.

Computational Insights of Unfolding of N-Terminal Domain of TDP-43 Reveal the Conformational Heterogeneity in the Unfolding Pathway

TDP-43 proteinopathies is a disease hallmark that characterizes amyotrophic lateral sclerosis (ALS) and frontotemporal lobar degeneration (FTLD). The N-terminal domain of TDP-43 (NTD) is important to both TDP-43 physiology and TDP-43 proteinopathy. However, its folding and dimerization process is still poorly characterized. In the present study, we have investigated the folding/unfolding of NTD employing all-atom molecular dynamics (MD) simulations in 8 M dimethylsulfoxide (DMSO) at high temperatures. The MD results showed that the unfolding of the NTD at high temperature evolves through the formation of a number of conformational states differing in their stability and free energy. The presence of structurally heterogeneous population of intermediate ensembles was further characterized by the different extents of solvent exposure of Trp80 during unfolding. We suggest that these non-natives unfolded intermediate ensembles may facilitate NTD oligomerization and subsequently TDP-43 oligomerization, which might lead to the formation of irreversible pathological aggregates, characteristics of disease pathogenesis.

DFT-based QM/MM with Particle-Mesh Ewald for Direct, Long-Range Electrostatic Embedding

We present a DFT-based, QM/MM implementation with long-range electrostatic embedding achieved by direct real-space integration of the particle mesh Ewald (PME) computed electrostatic potential. The key transformation is the interpolation of the electrostatic potential from the PME grid to the DFT quadrature grid, from which integrals are easily evaluated utilizing standard DFT machinery. We provide benchmarks of the numerical accuracy with choice of grid size and real-space corrections, and demonstrate that good convergence is achieved while introducing nominal computational overhead. Furthermore, the approach requires only small modification to existing software packages, as is demonstrated with our implementation in the OpenMM and Psi4 software. After presenting convergence benchmarks, we evaluate the importance of long-range electrostatic embedding in three solute/solvent systems modeled with QM/MM. Water and BMIM/BF4 ionic liquid were considered as ``simple' and ``complex' solvents respectively, with water and p-phenylenediamine (PPD) solute molecules treated at QM level of theory. While electrostatic embedding with standard real-space truncation may introduce negligible error for simple systems such as water solute in water solvent, errors become more significant when QM/MM is applied to complex solvents such as ionic liquids. An extreme example is the electrostatic embedding energy for oxidized PPD in BMIM/BF4 for which real-space truncation produces severe error even at 2-3 nm cutoff distances. This latter example illustrates that utilization of QM/MM to compute redox potentials within concentrated electrolytes/ionic media requires carefully chosen long-range electrostatic embedding algorithms, with our presented algorithm providing a general and robust approach.

Accelerating therapeutic protein design

Protein structures provide for defined microenvironments that can support complex pharmacological functions, otherwise unachievable by small molecules. The advent of therapeutic proteins has thus greatly broadened the range of manageable disorders. Leveraging the knowledge and recent advances in de novo protein design methods has the prospect of revolutionizing how protein drugs are discovered and developed. This review lays out the main challenges facing therapeutic proteins discovery and development, and how present and future advancements of protein design can accelerate the protein drug pipelines.

Electrostatic Discovery Atomic Force Microscopy

While offering high resolution atomic and electronic structure, scanning probe microscopy techniques have found greater challenges in providing reliable electrostatic characterization on the same scale. In this work, we offer electrostatic discovery atomic force microscopy, a machine learning based method which provides immediate maps of the electrostatic potential directly from atomic force microscopy images with functionalized tips. We apply this to characterize the electrostatic properties of a variety of molecular systems and compare directly to reference simulations, demonstrating good agreement. This approach offers reliable atomic scale electrostatic maps on any system with minimal computational overhead.

Changes in the Local Conformational States Caused by Simple Na+ and K+ Ions in Polyelectrolyte Simulations: Comparison of Seven Force Fields with and without NBFIX and ECC Corrections

Electrostatic interactions have a determining role in the conformational and dynamic behavior of polyelectrolyte molecules. In this study, anionic polyelectrolyte molecules, poly(glutamic acid) (PGA) and poly(aspartic acid) (PASA), in a water solution with the most commonly used K+ or Na+ counterions, were investigated using atomistic molecular dynamics (MD) simulations. We performed a comparison of seven popular force fields, namely AMBER99SB-ILDN, AMBER14SB, AMBER-FB15, CHARMM22*, CHARMM27, CHARMM36m and OPLS-AA/L, both with their native parameters and using two common corrections for overbinding of ions, the non-bonded fix (NBFIX), and electronic continuum corrections (ECC). These corrections were originally introduced to correct for the often-reported problem concerning the overbinding of ions to the charged groups of polyelectrolytes. In this work, a comparison of the simulation results with existing experimental data revealed several differences between the investigated force fields. The data from these simulations and comparisons with previous experimental data were then used to determine the limitations and strengths of these force fields in the context of the structural and dynamic properties of anionic polyamino acids. Physical properties, such as molecular sizes, local structure, and dynamics, were studied using two types of common counterions, namely potassium and sodium. The results show that, in some cases, both the macroion size and dynamics depend strongly on the models (parameters) for the counterions due to strong overbinding of the ions and charged side chain groups. The local structures and dynamics are more sensitive to dihedral angle parameterization, resulting in a preference for defined monomer conformations and the type of correction used. We also provide recommendations based on the results.

Local Charge Distributions, Electric Dipole Moments, and Local Electric Fields Influence Reactivity Patterns and Guide Regioselectivities in α-Ketoglutarate-Dependent Non-heme Iron Dioxygenases

Non-heme iron dioxygenases catalyze vital processes for human health related to the biosynthesis of essential products and the biodegradation of toxic metabolites. Often the natural product biosyntheses by these non-heme iron dioxygenases is highly regio- and chemoselective, which are commonly assigned to tight substrate-binding and positioning. However, recent high-level computational modeling has shown that substrate-binding and positioning is only part of the story and long-range electrostatic interactions can play a major additional role.In this Account, we review and summarize computational viewpoints on the high regio- and chemoselectivity of α-ketoglutarate-dependent non-heme iron dioxygenases and how external perturbations affect the catalysis. In particular, studies from our groups have shown that often a regioselectivity in enzymes can be accomplished by stabilization of the rate-determining transition state for the reaction through external charges, electric dipole moments, or local electric field effects. Furthermore, bond dissociation energies in molecules are shown to be influenced by an electric field effect, and through targeting a specific bond in an electric field, this can lead to an unusually specific reaction. For instance, in the carbon-induced starvation protein, we studied two substrate-bound conformations and showed that regardless of what C-H bond of the substrate is closest to the iron(IV)-oxo oxidant, the lowest hydrogen atom abstraction barrier is always for the pro-S C²-H abstraction due to an induced dipole moment of the protein that weakens this bond. In another example of the hygromycin biosynthesis enzyme, an oxidative ring-closure reaction in the substrate forms an ortho-δ-ester ring. Calculations on this enzyme show that the selectivity is guided by a protonated lysine residue in the active site that, through its positive charge, triggers a low energy hydrogen atom abstraction barrier. A final set of examples in this Account discuss the viomycin biosynthesis enzyme and the 2-(trimethylammonio)ethylphosphonate dioxygenase (TmpA) enzyme. Both of these enzymes are shown to possess a significant local dipole moment and local electric field effect due to charged residues surrounding the substrate and oxidant binding pockets. The protein dipole moment and local electric field strength changes the C-H bond strengths of the substrate as compared to the gas-phase triggers the regioselectivity of substrate activation. In particular, we show that in the gas phase and in a protein environment C-H bond strengths are different due to local electric dipole moments and electric field strengths. These examples show that enzymes have an intricately designed structure that enables a chemical reaction under ambient conditions through the positioning of positively and negatively charged residues that influence and enhance reaction mechanisms. These computational insights create huge possibilities in bioengineering to apply local electric field and dipole moments in proteins to achieve an unusual selectivity and specificity and trigger a fit-for-purpose biocatalyst for unique biotransformations.

First Hyperpolarizability of Water in Bulk Liquid Phase: Long-Range Electrostatic Effects Included via the Second Hyperpolarizability

The molecular first hyperpolarizability β contributes to second-order optical non-linear signals collected from molecular liquids. For the Second Harmonic Generation (SHG) response, the first hyperpolarizability β (2ω,ω,ω) often depends on...

Short-Range Imbalances in the AMBER Lennard-Jones Potential for (Deoxy)Ribose···Nucleobase Lone-Pair···π Contacts in Nucleic Acids

The lone-pair···π (lp···π) (deoxy)ribose···nucleobase stacking is a recurring interaction in Z-DNA and RNAs that is characterized by sub-van der Waals lp···π contacts (<3.0 Å). It is a part of the structural signature of CpG Z-step motifs in Z-DNA and r(UNCG) tetraloops that are known to behave poorly in molecular dynamics (MD) simulations. Although the exact origin of the MD simulation issues remains unclear, a significant part of the problem might be due to an imbalanced description of nonbonded interactions, including the characteristic lp···π stacking. To gain insights into the links between lp···π stacking and MD, we present an in-depth comparison between accurate large-basis-set double-hybrid Kohn-Sham density functional theory calculations DSD-BLYP-D3/ma-def2-QZVPP (DHDF-D3) and data obtained with the nonbonded potential of the AMBER force field (AFF) for NpN Z-steps (N = G, A, C, and U). Among other differences, we found that the AFF overestimates the DHDF-D3 lp···π distances by ∼0.1-0.2 Å, while the deviation between the DHDF-D3 and AFF descriptions sharply increases in the short-range region of the interaction. Based on atom-in-molecule polarizabilities and symmetry-adapted perturbation theory analysis, we inferred that the DHDF-D3 versus AFF differences partly originate in identical nucleobase carbon atom Lennard-Jones (LJ) parameters despite the presence/absence of connected electron-withdrawing groups that lead to different effective volumes or vdW radii. Thus, to precisely model the very short CpG lp···π contact distances, we recommend revision of the nucleobase atom LJ parameters. Additionally, we suggest that the large discrepancy between DHDF-D3 and AFF short-range repulsive part of the interaction energy potential may significantly contribute to the poor performances of MD simulations of nucleic acid systems containing Z-steps. Understanding where, and if possible why, the point-charge-type effective potentials reach their limits is vital for developing next-generation FFs and for addressing specific issues in contemporary MD simulations.

N,N-Dimethylacetamide-Diluted Nitrate Electrolyte for Aqueous Zn//LiMn2O4 Hybrid Ion Batteries

N,N-Dimethylacetamide (DMA) cooperated with LiNO₃ salt has previously shown to be a promising electrolyte for a Li//O₂ battery, showing good stability against both the O₂ electrode reaction and Li stripping/plating. In this work, DMA is hybridized with a concentrated nitrate electrolyte [2.5 m Zn(NO₃)₂ + 13 m LiNO₃ aqueous solution] for better electrochemical stability while using less dissolved salts. The widest electrochemical stability window for this DMA-diluted electrolyte is determined as 3.1 V, the negative critical stability potential of which is -1.6 V versus Ag/AgCl, indicating desirable stability against hydrogen evolution and Zn deposition. The findings can be attributed to the weakened Li⁺/Zn²⁺ solvation sheath caused by low permittivity of DMA, as revealed through Raman spectra characterization and molecular dynamics simulation. A Zn//Zn symmetrical cell and Zn//LiMn₂O₄ hybrid ion batteries are assembled in air directly, attributed to the stability of DMA toward O₂. Zn stripping/plating with a dendrite-free morphology is delivered for 110 h and 200 charge/discharge cycles under 1 C rate, achieving 99.0% Coulombic efficiency. The maximum capacity of the battery is 121.0 mA h·g^-1 under 0.2 C rate (based on the mass of LiMn₂O₄), delivering an energy density of 165.8 W h·kg^-1 together with 2.0 V working voltage. This work demonstrates the feasibility and validity of utilizing a relatively dilute electrolyte dissolved in oxygen for a highly stable aqueous rechargeable battery.

Quantifying the Separation of Positive and Negative Areas in Electrostatic Potential for Predicting Feasibility of Ammonium Sulfate for Protein Crystallization

Ammonium sulfate (AS) and poly(ethylene glycol) (PEG) are the most popular precipitants in protein crystallization. Some proteins are preferably crystallized by AS, while some are by PEG. The electrostatic potential is related to the preference of the precipitant agents. The iso-surfaces of the electrostatic potentials for the AS-crystallized proteins display a common shape and a distinct separation between the positive and negative areas. In contrast, the PEG-crystallized proteins show unclear positive and negative separation. In this work, we propose schemes to quantitatively evaluate the separation for predicting which precipitant is favorable for crystal growth between AS or PEG. Three methods were attempted to quantify the amplitude of the separation, separation distance, dipole moment, and shape regularity. The positive and negative areas are approximated to the spherical potentials caused by point charges. The first method is a measurement of the distance between the positive and negative point charges. The second one is an assessment including the quantity of electric charge into the distance. The last one is an approach monitoring the clarity of the positive and negative separation. The average value for 25 kinds of AS-preferring proteins was higher than that for the PEG-preferring ones in all three methods. Therefore, every method can distinguish the proteins preferring AS for crystal growth from those preferring PEG. These methods require an iso-surface of the electrostatic potential depicted at a certain contouring value. The shape of the iso-surface depends on the contouring value. The dependency on contour was examined by depicting the iso-surfaces of electrostatic potential with three values at ±0.8, ±0.5, and ±0.2 kT/e. While reducing the contouring value leads to the increase in separation distance and the decrease in shape regularity, dipole moment is independent of the alteration of contouring value. While the AS-preferring proteins are distinguishable from the PEG-preferring ones in any contouring values, the iso-surface at ±0.5 kT/e seems adequate for regular use. The dipole moment assessment is feasible for the choice of potent precipitants for crystal growth in experiments.

Integration of theory, simulation, artificial intelligence and virtual reality: a four-pillar approach for reconciling accuracy and interpretability in computational spectroscopy

The established pillars of computational spectroscopy are theory and computer based simulations. Recently, artificial intelligence and virtual reality are becoming the third and fourth pillars of an integrated strategy for the investigation of complex phenomena. The main goal of the present contribution is the description of some new perspectives for computational spectroscopy, in the framework of a strategy in which computational methodologies at the state of the art, high-performance computing, artificial intelligence and virtual reality tools are integrated with the aim of improving research throughput and achieving goals otherwise not possible. Some of the key tools (e.g., continuous molecular perception model and virtual multifrequency spectrometer) and theoretical developments (e.g., non-periodic boundaries, joint variational-perturbative models) are shortly sketched and their application illustrated by means of representative case studies taken from recent work by the authors. Some of the results presented are already well beyond the state of the art in the field of computational spectroscopy, thereby also providing a proof of concept for other research fields.

A Quantum Chemical Topology Picture of Intermolecular Electrostatic Interactions and Charge Penetration Energy

Based on the Interacting Quantum Atoms approach, we present herein a conceptual and theoretical framework of short-range electrostatic interactions, whose accurate description is still a challenging problem in molecular modeling. For all the noncovalent complexes in the S66 database, the fragment-based and atomic decomposition of the electrostatic binding energies is performed using both the charge density of the dimers and the unrelaxed densities of the monomers. This energy decomposition together with dispersion corrections gives rise to a pairwise approximation to the total binding energy. It also provides energetic descriptors at varying distance that directly address the atomic and molecular electrostatic interactions as described by point-charge or multipole-based potentials. Additionally, we propose a consistent definition of the charge penetration energy within quantum chemical topology, which is mainly characterized in terms of the intramolecular electrostatic energy. Finally, we discuss some practical implications of our results for the design and validation of electrostatic potentials.

g_elpot: A Tool for Quantifying Biomolecular Electrostatics from Molecular Dynamics Trajectories

Electrostatic forces drive a wide variety of biomolecular processes by defining the energetics of the interaction between biomolecules and charged substances. Molecular dynamics (MD) simulations provide trajectories that contain ensembles of structural configurations sampled by biomolecules and their environment. Although this information can be used for high-resolution characterization of biomolecular electrostatics, it has not yet been possible to calculate electrostatic potentials from MD trajectories in a way allowing for quantitative connection to energetics. Here, we present g_elpot, a GROMACS-based tool that utilizes the smooth particle mesh Ewald method to quantify the electrostatics of biomolecules by calculating potential within water molecules that are explicitly present in biomolecular MD simulations. g_elpot can extract the global distribution of the electrostatic potential from MD trajectories and measure its time course in functionally important regions of a biomolecule. To demonstrate that g_elpot can be used to gain biophysical insights into various biomolecular processes, we applied the tool to MD trajectories of the P2X3 receptor, TMEM16 lipid scramblases, the secondary-active transporter Glt_Ph, and DNA complexed with cationic polymers. Our results indicate that g_elpot is well suited for quantifying electrostatics in biomolecular systems to provide a deeper understanding of its role in biomolecular processes.

Analytical Hessians for Ewald and particle mesh Ewald electrostatics

The particle mesh Ewald (PME) method has become ubiquitous in the molecular simulation community due to its ability to deliver long range electrostatics accurately with ON⁡log(N) complexity. Despite this widespread use, spanning more than two decades, second derivatives (Hessians) have not been available. In this work, we describe the theory and implementation of PME Hessians, which have applications in normal mode analysis, characterization of stationary points, phonon dispersion curve calculation, crystal structure prediction, and efficient geometry optimization. We outline an exact strategy that requires O(1) effort for each Hessian element; after discussing the excessive memory requirements of such an approach, we develop an accurate, efficient approximation that is far more tractable on commodity hardware.

A Benchmark of Electrostatic Method Performance in Relative Binding Free Energy Calculations

Relative free energy calculations are fast becoming a critical part of early stage pharmaceutical design, making it important to know how to obtain the best performance with these calculations in applications that could span hundreds of calculations and molecules. In this work, we compared two different treatments of long-range electrostatics, Particle Mesh Ewald (PME) and Reaction Field (RF), in relative binding free energy calculations using a nonequilibrium switching protocol. We found simulations using RF achieve comparable results to those using PME but gain more efficiency when using CPU and similar performance using GPU. The results from this work encourage more use of RF in molecular simulations.

A compression strategy for particle mesh Ewald theory

Particle Mesh Ewald (PME) has become a standard method for treating long-range electrostatics in molecular simulations. Although the method has inferior asymptotic computational complexity to its linear scaling competitors, it remains enormously popular due to its high efficiency, which stems from the use of fast Fourier transforms (FFTs). This use of FFTs provides great challenges for scaling the method up to massively parallel systems, in large part because of the need to transfer large amounts of data. In this work, we demonstrate that this data transfer volume can be greatly reduced as a natural consequence of the structure of the PME equations. We also suggest an alternative algorithm that supplants the FFT with a linear algebra approach, which further decreases communication costs at the expense of increased asymptotic computational complexity. This linear algebra based approach is demonstrated to have great potential for latency hiding by interleaving communication and computation steps of the short- and long-range electrostatic terms.

Dielectric continuum model examination of real-space electrostatic treatments

Electrostatic interaction is long ranged; thus, the accurate calculation is not an easy task in molecular dynamics or Monte Carlo simulations. Though the rigorous Ewald method based on the reciprocal space has been established, real-space treatments have recently become an attractive alternative because of the efficient calculation. However, the construction is not yet completed and is now a challenging subject. In an earlier theoretical study, Neumann and Steinhauser employed the Onsager dielectric continuum model to explain how simple real-space cutoff produces artificial dipolar orientation. In the present study, we employ this continuum model to explore the fundamental properties of the recently developed real-space treatments of three shifting schemes. The result of the distance-dependent Kirkwood function G_K(R) showed that the simple bare cutoff produces a well-known hole-shaped artifact, whereas the shift treatments do not. Two-dimensional mapping of electric field well explained how these shift treatments remove the hole-shaped artifact. Still, the shift treatments are not sufficient because they do not produce a flat G_K(R) profile unlike ideal no-cutoff treatment. To test the continuum model results, we also performed Monte Carlo simulations of dipolar particles. The results found that the continuum model could predict the qualitative tendency as to whether each electrostatic treatment produces the hole-shaped artifact of G_K(R) or not. We expect that the present study using the continuum model offers a stringent criterion to judge whether the primitive electrostatic behavior is correctly described or not, which will be useful for future construction of electrostatic treatments.

A simplified charge projection scheme for long-range electrostatics in ab initio QM/MM calculations

In a previous work [Pan et al., Molecules 23, 2500 (2018)], a charge projection scheme was reported, where outer molecular mechanical (MM) charges [>10 Å from the quantum mechanical (QM) region] were projected onto the electrostatic potential (ESP) grid of the QM region to accurately and efficiently capture long-range electrostatics in ab initio QM/MM calculations. Here, a further simplification to the model is proposed, where the outer MM charges are projected onto inner MM atom positions (instead of ESP grid positions). This enables a representation of the long-range MM electrostatic potential via augmentary charges (AC) on inner MM atoms. Combined with the long-range electrostatic correction function from Cisneros et al. [J. Chem. Phys. 143, 044103 (2015)] to smoothly switch between inner and outer MM regions, this new QM/MM-AC electrostatic model yields accurate and continuous ab initio QM/MM electrostatic energies with a 10 Å cutoff between inner and outer MM regions. This model enables efficient QM/MM cluster calculations with a large number of MM atoms as well as QM/MM calculations with periodic boundary conditions.

Branched versus linear lactide chains for cellulose nanoparticle modification: an atomistic molecular dynamics study

We studied the structure of brushes consisting of branched oligolactide (OLA) chains grafted onto the surface of cellulose nanoparticles (CNPs) in polylactide (PLA) and compared the outcomes to the case of grafting linear OLA chains using atomistic molecular dynamics simulations. The systems were considered in a melt state. The branched model OLA chains comprised one branching point and three branches, while the linear OLA chains examined had a molecular weight similar to the branched chains. It was shown that free branches of the branched OLA chains tend to fold back toward the CNPs due to dipole-dipole interactions within the grafted layer, in contrast to the well-established behavior of the grafted uncharged branched chains. This result, however, is in qualitative agreement with the conformational behavior known for linear OLA chains. At the same time, no significant difference in the effectiveness of covering the filler surface with grafted branched or linear OLA chains was found. In terms of the expelling ability of the grafted chains and the interaction between PLA and CNP or OLA, the linear chains were broadly similar (sparse grafting) or better (intermediate or dense grafting) compared to the branched ones. Thus, the grafted lactide chains with a linear architecture, rather than their branched counterpart, may be preferable for the covalent modification of cellulose nanoparticles.

Electrostatics of charged dielectric spheres with application to biological systems. III. Rigorous ionic screening at the Debye-Hückel level

The unequivocal role of electrostatic forces in biological (and colloidal) systems underscores the importance of attaining accurate and rapid calculations of electrostatic forces if one wishes to faithfully simulate the electrostatic aspect of a biological system. This paper makes significant progress toward this aspect as it rigorously incorporates ionic screening at the Debye-Hückel level for an electrolyte system containing dielectric spheres of finite radii. We investigated earlier this system without mobile ions via a surface charge method. However, the need for computing a large number of Wigner rotation matrix elements per configuration can significantly slow down the numerical calculations. This difficulty was recently overcome by our Wigner-matrix-free formalism. Unfortunately, in that method ions can only be included individually, making it impractical to investigate, for example, ionic screening in a system modeled by charged dielectric spheres immersed in a solution of mobile ions. Here, we overcome this difficulty by extending the surface charge method to treat ions implicitly. Previous treatments of charged dielectric spheres in a solution of mobile ions did not emphasize the energy reciprocity of electrostatics and are largely limited to a few spheres and/or special symmetries. Our new formalism respects reciprocity and accommodates arbitrarily many dielectric spheres of different dielectric constants and sizes while being rigorous at the Debye-Hückel level. The differences, and the relationship, between our new implicit ion treatment and our previous ion-free (or explicit ion) approach are described. A closed form for the electrostatic energy with implicit ions is also provided. This new formalism speeds up the computation of the electrostatic energy in the presence of ions, and accommodates permanent and induced multipoles that are very important when the polarization effect needs to be correctly included. We also mention how the proposed method can be transformed to a numerical method for use with arbitrary nonspherical surfaces.

A Newton-like iterative method implemented in the DelPhi for solving the nonlinear Poisson-Boltzmann equation

DelPhi is a popular scientific program which numerically solves the Poisson-Boltzmann equation (PBE) for electrostatic potentials and energies of biomolecules immersed in water via finite difference method. It is well known for its accuracy, reliability, flexibility, and efficiency. In this work, a new edition of DelPhi that uses a novel Newton-like method to solve the nonlinear PBE, in addition to the already implemented Successive Over Relaxation (SOR) algorithm, is introduced. Our tests on various examples have shown that this new method is superior to the SOR method in terms of stability when solving the nonlinear PBE, being able to converge even for problems involving very strong nonlinearity.

Computational Spectroscopy in Solution by Integration of Variational and Perturbative Approaches on Top of Clusterized Molecular Dynamics

Multiscale QM/MM approaches have become the most suitable and effective methods for the investigation of spectroscopic properties of medium- or large-size chromophores in condensed phases. On these grounds, we are developing a novel workflow aimed at improving the generality, reliability, and ease of use of the available tools. In the present paper, we report the latest developments of such an approach with specific reference to a general workplan starting with the addition of acetonitrile to the panel of solvents already available in the General Liquid Optimized Boundary (GLOB) model enforcing nonperiodic boundary conditions (NPBC). Next, the solvatochromic shifts induced by acetonitrile on both rigid (uracil and thymine) and flexible (thyrosine) chromophores have been studied introducing in our software a number of new features ranging from rigid-geometry NPBC molecular dynamics based on the quaternion formalism to a full integration of variational (ONIOM) and perturbative (perturbed matrix method (PMM)) approaches for describing different solute–solvent topologies and local fluctuations, respectively. Finally, thymine and uracil have been studied also in methanol to point out the generality of the computational strategy. While further developments are surely needed, the strengths of our integrated approach even in its present version are demonstrated by the accuracy of the results obtained by an unsupervised approach and coupled to a computational cost strongly reduced with respect to that of conventional QM/MM models without any appreciable accuracy deterioration.