Reducing grid-dependence in finite-difference Poisson-Boltzmann calculations

Inhibitory Activity of Flavonoid Scaffolds on SARS-CoV-2 3CLpro: Insights from the Computational and Experimental Investigations

The emergence of the COVID-19 situation has become a global issue due to the lack of effective antiviral drugs for treatment. Flavonoids are a class of plant secondary metabolites that have antiviral activity against SARS-CoV-2 through inhibition of the main protease (3CLpro). In this study, 22 flavonoids obtained from natural sources and semisynthetic approaches were investigated for their inhibitory activity against SARS-CoV-2 3CLpro, along with cytotoxicity on Vero cells. The protein-ligand interactions were examined using molecular dynamics simulation. Moreover, QSAR analysis was conducted to clarify the structural effects on bioactivity. Accordingly, the in vitro investigation demonstrated that four flavonoids, namely, tectochrysin (7), 6″,6″-dimethylchromeno-[2″,3″:7,8]-flavone (9), panduratin A (19), and genistein (20), showed higher protease inhibitory activity compared to the standard flavonoid baicalein. Finally, our finding suggests that genistein (20), an isoflavone discovered in Millettia brandisiana, has potential for further development as a SARS-CoV-2 3CLpro inhibitor.

AmberTools

AmberTools is a free and open-source collection of programs used to set up, run, and analyze molecular simulations. The newer features contained within AmberTools23 are briefly described in this Application note.

Grid-Robust Efficient Neural Interface Model for Universal Molecule Surface Construction from Point Clouds

Molecular surfaces play a pivotal role in elucidating the properties and functions of biological complexes. While various surfaces have been proposed for specific scenarios, their widespread adoption faces challenges due to limited efficiency stemming from hand-crafted modeling designs. In this work, we proposed a general framework that incorporates both the point cloud concept and neural networks. The use of matrix multiplication in this framework enables efficient implementation across diverse platforms and libraries. We applied this framework to develop the GENIUSES (Grid-robust Efficient Neural Interface for Universal Solvent-Excluded Surface) model for constructing SES. GENIUSES demonstrates high accuracy and efficiency across data sets with varying conformations and complexities. Compared to the classical implementation of SES in the AMBER software package, our framework achieved a 26-fold speedup while retaining ∼95% accuracy when ported to the GPU platform using CUDA. Greater speedups can be obtained in large-scale systems. Importantly, our model exhibits robustness against variations in the grid spacing. We have integrated this infrastructure into AMBER to enhance accessibility for research in drug screening and related fields, where efficiency is of paramount importance.

Cytotoxic Activity of Phytoconstituents Isolated from Monotheca buxifolia against Hepatocellular Carcinoma Cell Line HepG2: In Vitro and Molecular Docking Studies

Natural products and conventional chemotherapeutic drugs are believed to enhance anticancer treatment efficacy while lowering toxicity. The current study investigates the cytotoxic and apoptogenic effects of Monotheca buxifolia bioactive compounds on HepG2 cell lines. MTT assay was used to assess the effect on the viability of HepG2 cells. Morphological changes were investigated. Annexin-V-FITC/PI was used to demonstrate apoptotic activity. A molecular dynamics simulation study was carried out to investigate the compound binding pattern in the active site of the PPRAδ protein. MTT and annexin V-FITC/PI assays revealed that the isolated compounds lauric acid, oleanolic acid, and bis(2-ethylhexyl) phthalate inhibited the growth of hepatocellular cancer cells. The IC50 value for lauric acid was 56.46 ± 1.20 μg/mL, 31.94 ± 1.03 μg/mL for oleanolic acid, and 83.80 ± 2.18 μg/mL for bis(2-ethylhexyl) phthalate. Apoptosis was observed in 29.5, 52.1 and 22.4% of HepG2 cells treated with lauric acid, oleanolic acid, and bis(2-ethylhexyl) phthalate, respectively, after 24 h of treatment. Morphological assays and Hoechst staining microscopy revealed that the treatment caused morphological changes in the cell membrane and nuclear condensation. The high fluctuation indicates that various interactions were highly potent and widely adopted, and vice versa. Oleanolic acid displayed high residue fluctuation, remaining stable in the active site of the PPRAδ protein and involved in various interactions while remaining locally fluctuating in the binding sites of the other two compounds. These findings concluded that lauric acid, oleanolic acid, and bis(2-ethylhexyl) phthalate have a significant apoptogenic effect against HepG2 cells in inducing apoptosis. Our findings suggest that these bioactive compounds could be used as adjuvant therapies.

In silico study of HASDI (high-affinity selective DNA intercalator) as a new agent capable of highly selective recognition of the DNA sequence

Cancer as an acquired genetic disease is based on changes both in the genome itself and in transcription processes. Accordingly, it is at the DNA level that it makes sense to search for and design agents capable of effective and selective anticancer action. In this study, we used an iterative approach based on a molecular dynamics simulation to design a highly selective DNA-intercalating agent called HASDI. To confirm its selective affinity to DNA, we conducted two simulation experiments: HASDI in a complex with a DNA fragment of the EBNA1 gene (it targets 16 nucleotide pairs of this gene) and HASDI in a complex with a random DNA fragment of the KCNH2 gene. The molecular dynamics simulation was carried out in the GROMACS 2019 package. The binding energy was calculated by gmx_MMPBSA 1.5.2. The further analysis was performed using the built-in utilities of GROMACS, gmx_MMPBSA and also XMGRACE and Pymol 1.8. As a result, we determined that the EBNA1-50nt/HASDI complex was stable throughout the whole simulation trajectory. HASDI, due to the presence of a linker modified depending on a specific pair of nitrogenous bases, formed an average of 32 hydrogen bonds with a sequence of 16 nucleotide pairs. Phenazine rings were stably intercalated every 2 base pairs. The root-mean-square deviation of HASDI in such a complex fluctuated around the value of 6.5 Å and had no tendency to increase. The calculated value of the binding free energy was - 235.3 ± 7.77 kcal/mol. The KCNH2-50nt/HASDI complex, as an example of the intercalation of the designed structure into a random part of the human genome, maintained the stability of its position at a level comparable to the EBNA1-50nt/HASDI complex. The phenazine rings were constantly intercalated in their original positions, and the root-mean-square deviation fluctuated around one value, although it had a tendency to chaotic changes. At the same time, this complex was characterized by 17-19 hydrogen bonds, on average, and the binding free energy was - 193.47 ± 14.09 kcal/mol. Moreover, the DNA duplex had local single-nucleotide melting in the region of the 4th linker. According to a significant decrease in the number of hydrogen bonds, a decrease in energy gain, as well as a decrease in the stability of the DNA duplex characteristic of the KCNH2-50nt/HASDI complex compared to the target EBNA1-50nt/HASDI complex, the molecule we designed can be considered a potentially selective DNA polyintercalating agent capable of relatively accurate recognition of 16 base pairs.

Discovery of All-d-Peptide Inhibitors of SARS-CoV-2 3C-like Protease

During the replication process of SARS-CoV-2, the main protease of the virus [3-chymotrypsin-like protease (3CL^pro)] plays a pivotal role and is essential for the life cycle of the pathogen. Numerous studies have been conducted so far, which have confirmed 3CL^pro as an attractive drug target to combat COVID-19. We describe a novel and efficient next-generation sequencing (NGS) supported phage display selection strategy for the identification of a set of SARS-CoV-2 3CL^pro targeting peptide ligands that inhibit the 3CL protease, in a competitive or noncompetitive mode, in the low μM range. From the most efficient l-peptides obtained from the phage display, we designed all-d-peptides based on the retro-inverso (ri) principle. They had IC₅₀ values also in the low μM range and in combination, even in the sub-micromolar range. Additionally, the combination with Rutinprivir decreases 10-fold the IC₅₀ value of the competitive inhibitor. The inhibition modes of these d-ri peptides were the same as their respective l-peptide versions. Our results demonstrate that retro-inverso obtained all-d-peptides interact with high affinity and inhibit the SARS-CoV-2 3CL protease, thus reinforcing their potential for further development toward therapeutic agents. The here described d-ri peptides address limitations associated with current l-peptide inhibitors and are promising lead compounds. Further optimization regarding pharmacokinetic properties will allow the development of even more potent d-peptides to be used for the prevention and treatment of COVID-19.

Targeting human thymidylate synthase: Ensemble-based virtual screening for drug repositioning and the role of water

A drug repositioning computational approach was carried to search inhibitors for human thymidylate synthase. An ensemble-based virtual screening of FDA-approved drugs showed the drugs Imatinib, Lumacaftor and Naldemedine to be likely candidates for repurposing. The role of water in the drug-receptor interactions was revealed by the application of an extended AutoDock scoring function that included the water forcefield. The binding affinity scores when hydrated ligands were docked were improved in the drugs considered. Further binding free energy calculations based on the Molecular Mechanics Poisson-Boltzmann Surface Area method revealed that Imatinib, Lumacaftor and Naldemedine scored -130.7 ± 28.1, -210.6 ± 29.9 and -238.0 ± 25.4 kJ/mol, respectively, showing good binding affinity for the candidates considered. Overall, the analysis of the molecular dynamics trajectory of the receptor-drug complexes revealed stable structures for Imatinib, Lumacaftor and Naldemedine, for the entire simulation time.

Electrostatic Contributions to the Binding Free Energy of Nicotine to the Acetylcholine Binding Protein

Biomolecular binding relies on specific attractive interactions between two partner molecules, including electrostatics, dispersion, hydrophobicity, and solvation. Assessing the contributions of electrostatic interactions to binding is key to the understanding of ligand binding mechanisms and the design of improved biomolecular binders. For example, nicotine is a well-known agonist of nicotinic acetylcholine receptors (nAChRs), but the molecular mechanisms for the differential action of nicotine on brain and muscle nAChRs remain elusive. In this work, we have chosen the acetylcholine binding protein (AChBP) in complex with nicotine as a model system to interrogate the electrostatic contributions to nicotine binding. Our absolute binding free energy simulations confirm that nicotine binds AChBP predominantly in its protonated (charged) form. By comparing energetic contributions from decomposed interactions for either neutral or charged nicotine, our calculations shed light on the nature of the binding of nicotine to the AChBP. The preferred binding of charged nicotine over neutral nicotine originates from its stronger electrostatic interactions with AChBP, a cation-π interaction to a tryptophan residue and a hydrogen bond between nicotine and the backbone carbonyl of the tryptophan, whereas the major force driving the binding process appears to be van der Waals interactions. The various nonelectrostatic terms can also indirectly modulate the electrostatic interactions through fine-tuning the binding pose of the ligand in the binding site, providing an explanation of why the binding specificity of nicotine to the brain versus muscle nAChRs is driven by electrostatic interaction, given that the immediate binding site residues, including the key tryptophan residue, are identical in the two receptors.

Finding alternatives to 5-fluorouracil: application of ensemble-based virtual screening for drug repositioning against human thymidylate synthase

5-fluorouracil and analogs are used in the treatment of many solid tumours. However, there are many cases of resistance and high toxicity associated with 5-fluorouracil chemotherapy. Repurposing FDA drugs against human thymidylate synthase revealed a number of FDA drugs that have a potential to be further developed for the treatment of various cancers for which 5-fluorouracil and analogs have been used for chemotherapy. Four FDA drugs prioritized for further validation included Erismodegib, Irinotecan, Conivaptan and Ergotamine. The role of water in mediating drug interactions and its contribution to the total binding energy was also shown. MM-PBSA calculations revealed that the binding affinity was the lowest for the hTS-Ergotamine complex (-66.702 ± 1.807 kJ/mol) suggesting moderate inhibition despite a large energetic contribution from van der Waal interactions (-190.889 ± 1.027 kJ/mol).Communicated by Ramaswamy H. Sarma.

gmx_MMPBSA: A New Tool to Perform End-State Free Energy Calculations with GROMACS

Molecular mechanics/Poisson-Boltzmann (Generalized-Born) surface area is one of the most popular methods to estimate binding free energies. This method has been proven to balance accuracy and computational efficiency, especially when dealing with large systems. As a result of its popularity, several programs have been developed for performing MM/PB(GB)SA calculations within the GROMACS community. These programs, however, present several limitations. Here we present gmx_MMPBSA, a new tool to perform end-state free energy calculations from GROMACS molecular dynamics trajectories. gmx_MMPBSA provides the user with several options, including binding free energy calculations with different solvation models (PB, GB, or 3D-RISM), stability calculations, computational alanine scanning, entropy corrections, and binding free energy decomposition. Noteworthy, several promising methodologies to calculate relative binding free energies such as alanine scanning with variable dielectric constant and interaction entropy have also been implemented in gmx_MMPBSA. Two additional tools-gmx_MMPBSA_test and gmx_MMPBSA_ana-have been integrated within gmx_MMPBSA to improve its usability. Multiple illustrating examples can be accessed through gmx_MMPBSA_test, while gmx_MMPBSA_ana provides fast, easy, and efficient access to different graphics plotted from gmx_MMPBSA output files. The latest version (v1.4.3, 26/05/2021) is available free of charge (documentation, test files, and tutorials included) at https://github.com/Valdes-Tresanco-MS/gmx_MMPBSA.

Machine-Learned Molecular Surface and Its Application to Implicit Solvent Simulations

Implicit solvent models, such as Poisson-Boltzmann models, play important roles in computational studies of biomolecules. A vital step in almost all implicit solvent models is to determine the solvent-solute interface, and the solvent excluded surface (SES) is the most widely used interface definition in these models. However, classical algorithms used for computing SES are geometry-based, so that they are neither suitable for parallel implementations nor convenient for obtaining surface derivatives. To address the limitations, we explored a machine learning strategy to obtain a level set formulation for the SES. The training process was conducted in three steps, eventually leading to a model with over 95% agreement with the classical SES. Visualization of tested molecular surfaces shows that the machine-learned SES overlaps with the classical SES in almost all situations. Further analyses show that the machine-learned SES is incredibly stable in terms of rotational variation of tested molecules. Our timing analysis shows that the machine-learned SES is roughly 2.5 times as efficient as the classical SES routine implemented in Amber/PBSA on a tested central processing unit (CPU) platform. We expect further performance gain on massively parallel platforms such as graphics processing units (GPUs) given the ease in converting the machine-learned SES to a parallel procedure. We also implemented the machine-learned SES into the Amber/PBSA program to study its performance on reaction field energy calculation. The analysis shows that the two sets of reaction field energies are highly consistent with a 1% deviation on average. Given its level set formulation, we expect the machine-learned SES to be applied in molecular simulations that require either surface derivatives or high efficiency on parallel computing platforms.

Recent Developments in Free Energy Calculations for Drug Discovery

The grand challenge in structure-based drug design is achieving accurate prediction of binding free energies. Molecular dynamics (MD) simulations enable modeling of conformational changes critical to the binding process, leading to calculation of thermodynamic quantities involved in estimation of binding affinities. With recent advancements in computing capability and predictive accuracy, MD based virtual screening has progressed from the domain of theoretical attempts to real application in drug development. Approaches including the Molecular Mechanics Poisson Boltzmann Surface Area (MM-PBSA), Linear Interaction Energy (LIE), and alchemical methods have been broadly applied to model molecular recognition for drug discovery and lead optimization. Here we review the varied methodology of these approaches, developments enhancing simulation efficiency and reliability, remaining challenges hindering predictive performance, and applications to problems in the fields of medicine and biochemistry.

Estimating the Roles of Protonation and Electronic Polarization in Absolute Binding Affinity Simulations

Accurate prediction of binding free energies is critical to streamlining the drug development and protein design process. With the advent of GPU acceleration, absolute alchemical methods, which simulate the removal of ligand electrostatics and van der Waals interactions with the protein, have become routinely accessible and provide a physically rigorous approach that enables full consideration of flexibility and solvent interaction. However, standard explicit solvent simulations are unable to model protonation or electronic polarization changes upon ligand transfer from water to the protein interior, leading to inaccurate prediction of binding affinities for charged molecules. Here, we perform extensive simulation totaling ∼540 μs to benchmark the impact of modeling conditions on predictive accuracy for absolute alchemical simulations. Binding to urokinase plasminogen activator (UPA), a protein frequently overexpressed in metastatic tumors, is evaluated for a set of 10 inhibitors with extended flexibility, highly charged character, and titratable properties. We demonstrate that the alchemical simulations can be adapted to utilize the MBAR/PBSA method to improve the accuracy upon incorporating electronic polarization, highlighting the importance of polarization in alchemical simulations of binding affinities. Comparison of binding energy prediction at various protonation states indicates that proper electrostatic setup is also crucial in binding affinity prediction of charged systems, prompting us to propose an alternative binding mode with protonated ligand phenol and Hid-46 at the binding site, a testable hypothesis for future experimental validation.

Molecular Basis for Polyketide Ketoreductase-Substrate Interactions

Polyketides are a large class of structurally and functionally diverse natural products with important bioactivities. Many polyketides are synthesized by reducing type II polyketide synthases (PKSs), containing transiently interacting standalone enzymes. During synthesis, ketoreductase (KR) catalyzes regiospecific carbonyl to hydroxyl reduction, determining the product outcome, yet little is known about what drives specific KR-substrate interactions. In this study, computational approaches were used to explore KR-substrate interactions based on previously solved apo and mimic cocrystal structures. We found five key factors guiding KR-substrate binding. First, two major substrate binding motifs were identified. Second, substrate length is the key determinant of substrate binding position. Third, two key residues in chain length specificity were confirmed. Fourth, phosphorylation of substrates is critical for binding. Finally, packing/hydrophobic effects primarily determine the binding stability. The molecular bases revealed here will help further engineering of type II PKSs and directed biosynthesis of new polyketides.

Elucidation of WW domain ligand binding specificities in the Hippo pathway reveals STXBP4 as YAP inhibitor

The Hippo pathway, which plays a critical role in organ size control and cancer, features numerous WW domain-based protein-protein interactions. However, ~100 WW domains and 2,000 PY motif-containing peptide ligands are found in the human proteome, raising a "WW-PY" binding specificity issue in the Hippo pathway. In this study, we have established the WW domain binding specificity for Hippo pathway components and uncovered a unique amino acid sequence required for it. By using this criterion, we have identified a WW domain-containing protein, STXBP4, as a negative regulator of YAP. Mechanistically, STXBP4 assembles a protein complex comprising α-catenin and a group of Hippo PY motif-containing components/regulators to inhibit YAP, a process that is regulated by actin cytoskeleton tension. Interestingly, STXBP4 is a potential tumor suppressor for human kidney cancer, whose downregulation is correlated with YAP activation in clear cell renal cell carcinoma. Taken together, our study not only elucidates the WW domain binding specificity for the Hippo pathway, but also reveals STXBP4 as a player in actin cytoskeleton tension-mediated Hippo pathway regulation.

Improved Poisson-Boltzmann Methods for High-Performance Computing

Implicit solvent models based on the Poisson-Boltzmann equation (PBE) have been widely used to study electrostatic interactions in biophysical processes. These models often treat the solvent and solute regions as high and low dielectric continua, leading to a large jump in dielectrics across the molecular surface which is difficult to handle. Higher order interface schemes are often needed to seek higher accuracy for PBE applications. However, these methods are usually very liberal in the use of grid points nearby the molecular surface, making them difficult to use on high-performance computing platforms. Alternatively, the harmonic average (HA) method has been used to approximate dielectric interface conditions near the molecular surface with surprisingly good convergence and is well suited for high-performance computing. By adopting a 7-point stencil, the HA method is advantageous in generating simple 7-banded coefficient matrices, which greatly facilitate linear system solution with dense data parallelism, on high-performance computing platforms such as a graphics processing unit (GPU). However, the HA method is limited due to its lower accuracy. Therefore, it would be of great interest for high-performance applications to develop more accurate methods while retaining the simplicity and effectiveness of the 7-point stencil discretization scheme. In this study, we have developed two new algorithms based on the spirit of the HA method by introducing more physical interface relations and imposing the discretized Poisson's equation to the second order, respectively. Our testing shows that, for typical biomolecules, the new methods significantly improve the numerical accuracy to that comparable to the second-order solvers and with ∼65% overall efficiency gain on widely available high-performance GPU platforms.

Heterogeneous Dielectric Implicit Membrane Model for the Calculation of MMPBSA Binding Free Energies

Membrane-bound protein receptors are a primary biological drug target, but the computational analysis of membrane proteins has been limited. In order to improve molecular mechanics Poisson-Boltzmann surface area (MMPBSA) binding free energy calculations for membrane protein-ligand systems, we have optimized a new heterogeneous dielectric implicit membrane model, with respect to free energy simulations in explicit membrane and explicit water, and implemented it into the Amber software suite. This new model supersedes our previous uniform, single dielectric implicit membrane model by allowing the dielectric constant to vary with depth within the membrane. We calculated MMPBSA binding free energies for the human purinergic platelet receptor (P2Y₁₂R) and two of the muscarinic acetylcholine receptors (M2R and M3R) bound to various antagonist ligands using both membrane models, and we found that the heterogeneous dielectric membrane model has a stronger correlation with experimental binding affinities compared to the older model under otherwise identical conditions. This improved membrane model increases the utility of MMPBSA calculations for the rational design and improvement of future drug candidates.

An efficient second-order poisson-boltzmann method

Immersed interface method (IIM) is a promising high-accuracy numerical scheme for the Poisson-Boltzmann model that has been widely used to study electrostatic interactions in biomolecules. However, the IIM suffers from instability and slow convergence for typical applications. In this study, we introduced both analytical interface and surface regulation into IIM to address these issues. The analytical interface setup leads to better accuracy and its convergence closely follows a quadratic manner as predicted by theory. The surface regulation further speeds up the convergence for nontrivial biomolecules. In addition, uncertainties of the numerical energies for tested systems are also reduced by about half. More interestingly, the analytical setup significantly improves the linear solver efficiency and stability by generating more precise and better-conditioned linear systems. Finally, we implemented the bottleneck linear system solver on GPUs to further improve the efficiency of the method, so it can be widely used for practical biomolecular applications. © 2019 Wiley Periodicals, Inc.

Robustness and Efficiency of Poisson-Boltzmann Modeling on Graphics Processing Units

Poisson-Boltzmann equation (PBE) based continuum electrostatics models have been widely used in modeling electrostatic interactions in biochemical processes, particularly in estimating protein-ligand binding affinities. Fast convergence of PBE solvers is crucial in binding affinity computations as numerous snapshots need to be processed. Efforts have been reported to develop PBE solvers on graphics processing units (GPUs) for efficient modeling of biomolecules, though only relatively simple successive over-relaxation and conjugate gradient methods were implemented. However, neither convergence nor scaling properties of the two methods are optimal for large biomolecules. On the other hand, geometric multigrid (MG) has been shown to be an optimal solver on CPUs, though no MG have been reported for biomolecular applications on GPUs. This is not a surprise as it is a more complex method and depends on simpler but limited iterative methods such as Gauss-Seidel in its core relaxation procedure. The robustness and efficiency of MG on GPUs are also unclear. Here we present an implementation and a thorough analysis of MG on GPUs. Our analysis shows that robustness is a more pronounced issue than efficiency for both MG and other tested solvers when the single precision is used for complex biomolecules. We further show how to balance robustness and efficiency utilizing MG's overall efficiency and conjugate gradient's robustness, pointing to a hybrid GPU solver with a good balance of efficiency and accuracy. The new PBE solver will significantly improve the computational throughput for a range of biomolecular applications on the GPU platforms.

Exploring a multi-scale method for molecular simulation in continuum solvent model: Explicit simulation of continuum solvent as an incompressible fluid

We explored a multi-scale algorithm for the Poisson-Boltzmann continuum solvent model for more robust simulations of biomolecules. In this method, the continuum solvent/solute interface is explicitly simulated with a numerical fluid dynamics procedure, which is tightly coupled to the solute molecular dynamics simulation. There are multiple benefits to adopt such a strategy as presented below. At this stage of the development, only nonelectrostatic interactions, i.e., van der Waals and hydrophobic interactions, are included in the algorithm to assess the quality of the solvent-solute interface generated by the new method. Nevertheless, numerical challenges exist in accurately interpolating the highly nonlinear van der Waals term when solving the finite-difference fluid dynamics equations. We were able to bypass the challenge rigorously by merging the van der Waals potential and pressure together when solving the fluid dynamics equations and by considering its contribution in the free-boundary condition analytically. The multi-scale simulation method was first validated by reproducing the solute-solvent interface of a single atom with analytical solution. Next, we performed the relaxation simulation of a restrained symmetrical monomer and observed a symmetrical solvent interface at equilibrium with detailed surface features resembling those found on the solvent excluded surface. Four typical small molecular complexes were then tested, both volume and force balancing analyses showing that these simple complexes can reach equilibrium within the simulation time window. Finally, we studied the quality of the multi-scale solute-solvent interfaces for the four tested dimer complexes and found that they agree well with the boundaries as sampled in the explicit water simulations.

Computational Analysis for the Rational Design of Anti-Amyloid Beta (Aβ) Antibodies

Alzheimer's disease (AD) is a neurodegenerative disorder that lacks effective treatment options. Anti-amyloid beta (Aβ) antibodies are the leading drug candidates to treat AD, but the results of clinical trials have been disappointing. Introducing rational mutations into anti-Aβ antibodies to increase their effectiveness is a way forward, but the path to take is unclear. In this study, we demonstrate the use of computational fragment-based docking and MMPBSA binding free energy calculations in the analysis of anti-Aβ antibodies for rational drug design efforts. Our fragment-based docking method successfully predicts the emergence of the common EFRH epitope. MD simulations coupled with MMPBSA binding free energy calculations are used to analyze scenarios described in prior studies, and we computationally introduce rational mutations into PFA1 to predict mutations that can improve its binding affinity toward the pE3-Aβ_3-8 form of Aβ. Two out of our four proposed mutations are predicted to stabilize binding. Our study demonstrates that a computational approach may lead to an improved drug candidate for AD in the future.

Synergistic regulation mechanism of iperoxo and LY2119620 for muscarinic acetylcholine M2 receptor

Muscarinic acetylcholine receptors are GPCRs that regulate the activity of a diverse array of central and peripheral functions in the human body, including the parasympathetic actions of acetylcholine. The M2 muscarinic receptor subtype plays a key role in modulating cardiac function and many important central processes. The orthosteric agonist and allosteric modulator can bind the pocket of M2. However, the detailed relationship between orthosteric agonist and allosteric modulator of M2 is still unclear. In this study, we intend to elucidate the residue-level regulation mechanism and pathway via a combined approach of dynamical correlation network and molecular dynamics simulation. Specifically computational residue-level fluctuation correlation data was analyzed to reveal detailed dynamics signatures in the regulation process. A hypothesis of "synergistic regulation" is proposed to reveal the cooperation affection between the orthosteric agonist and allosteric modulator, which is subsequently validated by perturbation and mutation analyses. Two possible synergistic regulation pathways of 2CU-I178-Y403-W400-F396-L114-Y440-Nb9 and IXO-V111-F396-L114-Y440-Nb9 were identified by the shortest path algorithm and were confirmed by the mutation of junction node. Furthermore, the efficiency of information transfer of bound M2 is significant higher than any single binding system. Our study shows that targeting the synergistic regulation pathways may better regulate the calcium channel of M2. The knowledge gained in this study may help develop drugs for diseases of the central nervous system and metabolic disorders.

Molecular insights into substrate binding mechanism of undecaprenyl pyrophosphate with membrane integrated phosphatidyl glycerophosphate phosphatase B (PgpB) using molecular dynamics simulation approach

Undecaprenyl phosphate (C55-P) acts as carrier lipid in the synthesis of peptidoglycan, which is de novo synthesized from dephosphorylation of undecaprenyl pyrophosphate (C55-PP). The phosphatidylglycerol phosphate phosphatase B (PgpB) catalyzes the dephosphorylation of C55-PP and forms C55-P. As no structural study has been made regarding the binding of C55-PP to PgpB, in the current study, in silico molecular docking, followed by 150 ns molecular dynamics simulation of the putative binding complex in membrane/solvent environment has been performed to understand conformational dynamics. Results are compared with simulated apo form and PE inhibitor-bound form. Analysis of correlated residual fluctuation network in apo form, C55-PP bound and PE inhibitor-bound form suggests that difference in dynamic coupling between TM domain and α2 and α3 helix of periplasmic domain provides ligand binding to facilitate catalysis or to show inhibitory activity. Distance distribution in catalytic residual pair, H207-R104; H207-R201 and H207-D211 which stabilizes phosphate-enzyme intermediate shows a narrow peak in 2.4-3.6 Å in substrate-bound compared to apo form. Binding interactions and binding free energy analyses complement the partial inhibition of PE where PE has less binding free energy compared to the C55-PP substrate as well as the difference in binding interaction with catalytic pocket. Thus, the present study provides how substrate binding couples the movement in TM domain and periplasmic domain which might help in the understanding of active site communication in PgpB. C55-PP phosphatase interactions with a catalytic pocket of PgpB provide new insight for designing drugs against bacterial infection.

Recent Developments and Applications of the MMPBSA Method

The Molecular Mechanics Poisson-Boltzmann Surface Area (MMPBSA) approach has been widely applied as an efficient and reliable free energy simulation method to model molecular recognition, such as for protein-ligand binding interactions. In this review, we focus on recent developments and applications of the MMPBSA method. The methodology review covers solvation terms, the entropy term, extensions to membrane proteins and high-speed screening, and new automation toolkits. Recent applications in various important biomedical and chemical fields are also reviewed. We conclude with a few future directions aimed at making MMPBSA a more robust and efficient method.

Ionic Solution: What Goes Right and Wrong with Continuum Solvation Modeling

Solvent-mediated electrostatic interactions were well recognized to be important in the structure and function of molecular systems. Ionic interaction is an important component in electrostatic interactions, especially in highly charged molecules, such as nucleic acids. Here, we focus on the quality of the widely used Poisson-Boltzmann surface area (PBSA) continuum models in modeling ionic interactions by comparing with both explicit solvent simulations and the experiment. In this work, the molality-dependent chemical potentials for sodium chloride (NaCl) electrolyte were first simulated in the SPC/E explicit solvent. Our high-quality simulation agrees well with both the previous study and the experiment. Given the free-energy simulations in SPC/E as the benchmark, we used the same sets of snapshots collected in the SPC/E solvent model for PBSA free-energy calculations in the hope to achieve the maximum consistency between the two solvent models. Our comparative analysis shows that the molality-dependent chemical potentials of NaCl were reproduced well with both linear PB and nonlinear PB methods, although nonlinear PB agrees better with SPC/E and the experiment. Our free-energy simulations also show that the presence of salt increases the hydrophobic effect in a nonlinear fashion, in qualitative agreement with previous theoretical studies of Onsager and Samaras. However, the lack of molality-dependency in the nonelectrostatics continuum models dramatically reduces the overall quality of PBSA methods in modeling salt-dependent energetics. These analyses point to further improvements needed for more robust modeling of solvent-mediated interactions by the continuum solvation frameworks.

Fractional Poisson-Nernst-Planck Model for Ion Channels I: Basic Formulations and Algorithms

In this work, we propose a fractional Poisson-Nernst-Planck model to describe ion permeation in gated ion channels. Due to the intrinsic conformational changes, crowdedness in narrow channel pores, binding and trapping introduced by functioning units of channel proteins, ionic transport in the channel exhibits a power-law-like anomalous diffusion dynamics. We start from continuous-time random walk model for a single ion and use a long-tailed density distribution function for the particle jump waiting time, to derive the fractional Fokker-Planck equation. Then, it is generalized to the macroscopic fractional Poisson-Nernst-Planck model for ionic concentrations. Necessary computational algorithms are designed to implement numerical simulations for the proposed model, and the dynamics of gating current is investigated. Numerical simulations show that the fractional PNP model provides a more qualitatively reasonable match to the profile of gating currents from experimental observations. Meanwhile, the proposed model motivates new challenges in terms of mathematical modeling and computations.

A Continuum Poisson-Boltzmann Model for Membrane Channel Proteins

Membrane proteins constitute a large portion of the human proteome and perform a variety of important functions as membrane receptors, transport proteins, enzymes, signaling proteins, and more. Computational studies of membrane proteins are usually much more complicated than those of globular proteins. Here, we propose a new continuum model for Poisson-Boltzmann calculations of membrane channel proteins. Major improvements over the existing continuum slab model are as follows: (1) The location and thickness of the slab model are fine-tuned based on explicit-solvent MD simulations. (2) The highly different accessibilities in the membrane and water regions are addressed with a two-step, two-probe grid-labeling procedure. (3) The water pores/channels are automatically identified. The new continuum membrane model is optimized (by adjusting the membrane probe, as well as the slab thickness and center) to best reproduce the distributions of buried water molecules in the membrane region as sampled in explicit water simulations. Our optimization also shows that the widely adopted water probe of 1.4 Å for globular proteins is a very reasonable default value for membrane protein simulations. It gives the best compromise in reproducing the explicit water distributions in membrane channel proteins, at least in the water accessible pore/channel regions. Finally, we validate the new membrane model by carrying out binding affinity calculations for a potassium channel, and we observe good agreement with the experimental results.

Numerical interpretation of molecular surface field in dielectric modeling of solvation

Continuum solvent models, particularly those based on the Poisson-Boltzmann equation (PBE), are widely used in the studies of biomolecular structures and functions. Existing PBE developments have been mainly focused on how to obtain more accurate and/or more efficient numerical potentials and energies. However to adopt the PBE models for molecular dynamics simulations, a difficulty is how to interpret dielectric boundary forces accurately and efficiently for robust dynamics simulations. This study documents the implementation and analysis of a range of standard fitting schemes, including both one-sided and two-sided methods with both first-order and second-order Taylor expansions, to calculate molecular surface electric fields to facilitate the numerical calculation of dielectric boundary forces. These efforts prompted us to develop an efficient approximated one-dimensional method, which is to fit the surface field one dimension at a time, for biomolecular applications without much compromise in accuracy. We also developed a surface-to-atom force partition scheme given a level set representation of analytical molecular surfaces to facilitate their applications to molecular simulations. Testing of these fitting methods in the dielectric boundary force calculations shows that the second-order methods, including the one-dimensional method, consistently perform among the best in the molecular test cases. Finally, the timing analysis shows the approximated one-dimensional method is far more efficient than standard second-order methods in the PBE force calculations. © 2017 Wiley Periodicals, Inc.

Crystal Structure of StnA for the Biosynthesis of Antitumor Drug Streptonigrin Reveals a Unique Substrate Binding Mode

Streptonigrin methylesterase A (StnA) is one of the tailoring enzymes that modify the aminoquinone skeleton in the biosynthesis pathway of Streptomyces species. Although StnA has no significant sequence homology with the reported α/β-fold hydrolases, it shows typical hydrolytic activity in vivo and in vitro. In order to reveal its functional characteristics, the crystal structures of the selenomethionine substituted StnA (SeMet-StnA) and the complex (S185A mutant) with its substrate were resolved to the resolution of 2.71 Å and 2.90 Å, respectively. The overall structure of StnA can be described as an α-helix cap domain on top of a common α/β hydrolase domain. The substrate methyl ester of 10'-demethoxystreptonigrin binds in a hydrophobic pocket that mainly consists of cap domain residues and is close to the catalytic triad Ser185-His349-Asp308. The transition state is stabilized by an oxyanion hole formed by the backbone amides of Ala102 and Leu186. The substrate binding appears to be dominated by interactions with several specific hydrophobic contacts and hydrogen bonds in the cap domain. The molecular dynamics simulation and site-directed mutagenesis confirmed the important roles of the key interacting residues in the cap domain. Structural alignment and phylogenetic tree analysis indicate that StnA represents a new subfamily of lipolytic enzymes with the specific binding pocket located at the cap domain instead of the interface between the two domains.

ESES: Software for Eulerian solvent excluded surface

Solvent excluded surface (SES) is one of the most popular surface definitions in biophysics and molecular biology. In addition to its usage in biomolecular visualization, it has been widely used in implicit solvent models, in which SES is usually immersed in a Cartesian mesh. Therefore, it is important to construct SESs in the Eulerian representation for biophysical modeling and computation. This work describes a software package called Eulerian solvent excluded surface (ESES) for the generation of accurate SESs in Cartesian grids. ESES offers the description of the solvent and solute domains by specifying all the intersection points between the SES and the Cartesian grid lines. Additionally, the interface normal at each intersection point is evaluated. Furthermore, for a given biomolecule, the ESES software not only provides the whole surface area, but also partitions the surface area according to atomic types. Homology theory is utilized to detect topological features, such as loops and cavities, on the complex formed by the SES. The sizes of loops and cavities are measured based on persistent homology with an evolutionary partial differential equation-based filtration. ESES is extensively validated by surface visualization, electrostatic solvation free energy computation, surface area and volume calculations, and loop and cavity detection and their size estimation. We used the Amber PBSA test set in our electrostatic solvation energy, area, and volume validations. Our results are either calibrated by analytical values or compared with those from the MSMS software. © 2017 Wiley Periodicals, Inc.

Modeling Membrane Protein-Ligand Binding Interactions: The Human Purinergic Platelet Receptor

Membrane proteins, due to their roles as cell receptors and signaling mediators, make prime candidates for drug targets. The computational analysis of protein-ligand binding affinities has been widely employed as a tool in rational drug design efforts. Although efficient implicit solvent-based methods for modeling globular protein-ligand binding have been around for many years, the extension of such methods to membrane protein-ligand binding is still in its infancy. In this study, we extended the widely used Amber/MMPBSA method to model membrane protein-ligand systems, and we used it to analyze protein-ligand binding for the human purinergic platelet receptor (P2Y12R), a prominent drug target in the inhibition of platelet aggregation for the prevention of myocardial infarction and stroke. The binding affinities, computed by the Amber/MMPBSA method using standard parameters, correlate well with experiment. A detailed investigation of these parameters was conducted to assess their impact on the accuracy of the method. These analyses show the importance of properly treating the nonpolar solvation interactions and the electrostatic polarization in the binding of nucleotide agonists and non-nucleotide antagonists to P2Y12R. On the basis of the crystal structures and the experimental conditions in the binding assay, we further hypothesized that the nucleotide agonists lose their bound magnesium ion upon binding to P2Y12R, and our computational study supports this hypothesis. Ultimately, this work illustrates the value of computational analysis in the interpretation of experimental binding reactions.

Coupling between ATP hydrolysis and protein conformational change in maltose transporter

As the intracellular part of maltose transporter, MalK dimer utilizes the energy of ATP hydrolysis to drive protein conformational change, which then facilitates substrate transport. Free energy evaluation of the complete conformational change before and after ATP hydrolysis is helpful to elucidate the mechanism of chemical-to-mechanical energy conversion in MalK dimer, but is lacking in previous studies. In this work, we used molecular dynamics simulations to investigate the structural transition of MalK dimer among closed, semi-open and open states. We observed spontaneous structural transition from closed to open state in the ADP-bound system and partial closure of MalK dimer from the semi-open state in the ATP-bound system. Subsequently, we calculated the reaction pathways connecting the closed and open states for the ATP- and ADP-bound systems and evaluated the free energy profiles along the paths. Our results suggested that the closed state is stable in the presence of ATP but is markedly destabilized when ATP is hydrolyzed to ADP, which thus explains the coupling between ATP hydrolysis and protein conformational change of MalK dimer in thermodynamics. Proteins 2017; 85:207-220. © 2016 Wiley Periodicals, Inc.

Calculating protein-ligand binding affinities with MMPBSA: Method and error analysis

Molecular Mechanics Poisson-Boltzmann Surface Area (MMPBSA) methods have become widely adopted in estimating protein-ligand binding affinities due to their efficiency and high correlation with experiment. Here different computational alternatives were investigated to assess their impact to the agreement of MMPBSA calculations with experiment. Seven receptor families with both high-quality crystal structures and binding affinities were selected. First the performance of nonpolar solvation models was studied and it was found that the modern approach that separately models hydrophobic and dispersion interactions dramatically reduces RMSD's of computed relative binding affinities. The numerical setup of the Poisson-Boltzmann methods was analyzed next. The data shows that the impact of grid spacing to the quality of MMPBSA calculations is small: the numerical error at the grid spacing of 0.5 Å is already small enough to be negligible. The impact of different atomic radius sets and different molecular surface definitions was further analyzed and weak influences were found on the agreement with experiment. The influence of solute dielectric constant was also analyzed: a higher dielectric constant generally improves the overall agreement with experiment, especially for highly charged binding pockets. The data also showed that the converged simulations caused slight reduction in the agreement with experiment. Finally the direction of estimating absolute binding free energies was briefly explored. Upon correction of the binding-induced rearrangement free energy and the binding entropy lost, the errors in absolute binding affinities were also reduced dramatically when the modern nonpolar solvent model was used, although further developments were apparently necessary to further improve the MMPBSA methods. © 2016 Wiley Periodicals, Inc.

Dynamics Correlation Network for Allosteric Switching of PreQ1 Riboswitch

Riboswitches are a class of metabolism control elements mostly found in bacteria. Due to their fundamental importance in bacteria gene regulation, riboswitches have been proposed as antibacterial drug targets. Prequeuosine (preQ₁) is the last free precursor in the biosynthetic pathway of queuosine that is crucial for translation efficiency and fidelity. However, the regulation mechanism for the preQ₁ riboswitch remains unclear. Here we constructed fluctuation correlation network based on all-atom molecular dynamics simulations to reveal the regulation mechanism. The results suggest that the correlation network in the bound riboswitch is distinctly different from that in the apo riboswitch. The community network indicates that the information freely transfers from the binding site of preQ₁ to the expression platform of the P3 helix in the bound riboswitch and the P3 helix is a bottleneck in the apo riboswitch. Thus, a hypothesis of “preQ₁-binding induced allosteric switching” is proposed to link riboswitch and translation regulation. The community networks of mutants support this hypothesis. Finally, a possible allosteric pathway of A50-A51-A52-U10-A11-G12-G56 was also identified based on the shortest path algorithm and confirmed by mutations and network perturbation. The novel fluctuation network analysis method can be used as a general strategy in studies of riboswitch structure-function relationship.

Synergistic Allosteric Mechanism of Fructose-1,6-bisphosphate and Serine for Pyruvate Kinase M2 via Dynamics Fluctuation Network Analysis

Pyruvate kinase M2 (PKM2) plays a key role in tumor metabolism and regulates the rate-limiting final step of glycolysis. In tumor cells, there are two allosteric effectors for PKM2: fructose-1,6-bisphosphate (FBP) and serine. However, the relationship between FBP and serine for allosteric regulation of PKM2 is unknown. Here we constructed residue/residue fluctuation correlation network based on all-atom molecular dynamics simulations to reveal the regulation mechanism. The results suggest that the correlation network in bound PKM2 is distinctly different from that in the free state, FBP/PKM2, or Ser/PKM2. The community network analysis indicates that the information can freely transfer from the allosteric sites of FBP and serine to the substrate site in bound PKM2, while there exists a bottleneck for information transfer in the network of the free state. Furthermore, the binding free energy between the substrate and PKM2 for bound PKM2 is significantly lower than either of FBP/PKM2 or Ser/PKM2. Thus, a hypothesis of "synergistic allosteric mechanism" is proposed for the allosteric regulation of FBP and serine. This hypothesis was further confirmed by the perturbational and mutational analyses of community networks and binding free energies. Finally, two possible synergistic allosteric pathways of FBP-K433-T459-R461-A109-V71-R73-MG2-OXL and Ser-I47-C49-R73-MG2-OXL were identified based on the shortest path algorithm and were confirmed by the network perturbation analysis. Interestingly, no similar pathways could be found in the free state. The process targeting on the allosteric pathways can better regulate the glycolysis of PKM2 and significantly inhibit the progression of tumor.

Charge Central Interpretation of the Full Nonlinear PB Equation: Implications for Accurate and Scalable Modeling of Solvation Interactions

Continuum solvation modeling based upon the Poisson-Boltzmann equation (PBE) is widely used in structural and functional analysis of biomolecules. In this work, we propose a charge-central interpretation of the full nonlinear PBE electrostatic interactions. The validity of the charge-central view or simply charge view, as formulated as a vacuum Poisson equation with effective charges, was first demonstrated by reproducing both electrostatic potentials and energies from the original solvated full nonlinear PBE. There are at least two benefits when the charge-central framework is applied. First the convergence analyses show that the use of polarization charges allows a much faster converging numerical procedure for electrostatic energy and forces calculation for the full nonlinear PBE. Second, the formulation of the solvated electrostatic interactions as effective charges in vacuum allows scalable algorithms to be deployed for large biomolecular systems. Here, we exploited the charge-view interpretation and developed a particle-particle particle-mesh (P3M) strategy for the full nonlinear PBE systems. We also studied the accuracy and convergence of solvation forces with the charge-view and the P3M methods. It is interesting to note that the convergence of both the charge-view and the P3M methods is more rapid than the original full nonlinear PBE method. Given the developments and validations documented here, we are working to adapt the P3M treatment of the full nonlinear PBE model to molecular dynamics simulations.

Synergistic Modification Induced Specific Recognition between Histone and TRIM24 via Fluctuation Correlation Network Analysis

Histone modification plays a key role in gene regulation and gene expression. TRIM24 as a histone reader can recognize histone modification. However the specific recognition mechanism between TRIM24 and histone modification is unsolved. Here, systems biology method of dynamics correlation network based on molecular dynamics simulation was used to answer the question. Our network analysis shows that the dynamics correlation network of H3K23ac is distinctly different from that of wild type and other modifications. A hypothesis of “synergistic modification induced recognition” is then proposed to link histone modification and TRIM24 binding. These observations were further confirmed from community analysis of networks with mutation and network perturbation. Finally, a possible recognition pathway is also identified based on the shortest path search for H3K23ac. Significant difference of recognition pathway was found among different systems due to methylation and acetylation modifications. The analysis presented here and other studies show that the dynamic network-based analysis might be a useful general strategy to study the biology of protein post-translational modification and associated recognition.

Applications of MMPBSA to Membrane Proteins I: Efficient Numerical Solutions of Periodic Poisson-Boltzmann Equation

Continuum solvent models have been widely used in biomolecular modeling applications. Recently much attention has been given to inclusion of implicit membranes into existing continuum Poisson-Boltzmann solvent models to extend their applications to membrane systems. Inclusion of an implicit membrane complicates numerical solutions of the underlining Poisson-Boltzmann equation due to the dielectric inhomogeneity on the boundary surfaces of a computation grid. This can be alleviated by the use of the periodic boundary condition, a common practice in electrostatic computations in particle simulations. The conjugate gradient and successive over-relaxation methods are relatively straightforward to be adapted to periodic calculations, but their convergence rates are quite low, limiting their applications to free energy simulations that require a large number of conformations to be processed. To accelerate convergence, the Incomplete Cholesky preconditioning and the geometric multigrid methods have been extended to incorporate periodicity for biomolecular applications. Impressive convergence behaviors were found as in the previous applications of these numerical methods to tested biomolecules and MMPBSA calculations.

On the energy components governing molecular recognition in the framework of continuum approaches

Molecular recognition is a process that brings together several biological macromolecules to form a complex and one of the most important characteristics of the process is the binding free energy. Various approaches exist to model the binding free energy, provided the knowledge of the 3D structures of bound and unbound molecules. Among them, continuum approaches are quite appealing due to their computational efficiency while at the same time providing predictions with reasonable accuracy. Here we review recent developments in the field emphasizing on the importance of adopting adequate description of physical processes taking place upon the binding. In particular, we focus on the efforts aiming at capturing some of the atomistic details of the binding phenomena into the continuum framework. When possible, the energy components are reviewed independently of each other. However, it is pointed out that rigorous approaches should consider all energy contributions on the same footage. The two major schemes for utilizing the individual energy components to predict binding affinity are outlined as well.

g_mmpbsa--a GROMACS tool for high-throughput MM-PBSA calculations

Molecular mechanics Poisson-Boltzmann surface area (MM-PBSA), a method to estimate interaction free energies, has been increasingly used in the study of biomolecular interactions. Recently, this method has also been applied as a scoring function in computational drug design. Here a new tool g_mmpbsa, which implements the MM-PBSA approach using subroutines written in-house or sourced from the GROMACS and APBS packages is described. g_mmpbsa was developed as part of the Open Source Drug Discovery (OSDD) consortium. Its aim is to integrate high-throughput molecular dynamics (MD) simulations with binding energy calculations. The tool provides options to select alternative atomic radii and different nonpolar solvation models including models based on the solvent accessible surface area (SASA), solvent accessible volume (SAV), and a model which contains both repulsive (SASA-SAV) and attractive components (described using a Weeks-Chandler-Andersen like integral method). We showcase the effectiveness of the tool by comparing the calculated interaction energy of 37 structurally diverse HIV-1 protease inhibitor complexes with their experimental binding free energies. The effect of varying several combinations of input parameters such as atomic radii, dielectric constant, grid resolution, solute-solvent dielectric boundary definition, and nonpolar models was investigated. g_mmpbsa can also be used to estimate the energy contribution per residue to the binding energy. It has been used to identify those residues in HIV-1 protease that are most critical for binding a range of inhibitors.

On the Modeling of Polar Component of Solvation Energy using Smooth Gaussian-Based Dielectric Function

Traditional implicit methods for modeling electrostatics in biomolecules use a two-dielectric approach: a biomolecule is assigned low dielectric constant while the water phase is considered as a high dielectric constant medium. However, such an approach treats the biomolecule-water interface as a sharp dielectric border between two homogeneous dielectric media and does not account for inhomogeneous dielectric properties of the macromolecule as well. Recently we reported a new development, a smooth Gaussian-based dielectric function which treats the entire system, the solute and the water phase, as inhomogeneous dielectric medium (J Chem Theory Comput. 2013 Apr 9; 9(4): 2126-2136.). Here we examine various aspects of the modeling of polar solvation energy in such inhomogeneous systems in terms of the solute-water boundary and the inhomogeneity of the solute in the absence of water surrounding. The smooth Gaussian-based dielectric function is implemented in the DelPhi finite-difference program, and therefore the sensitivity of the results with respect to the grid parameters is investigated, and it is shown that the calculated polar solvation energy is almost grid independent. Furthermore, the results are compared with the standard two-media model and it is demonstrated that on average, the standard method overestimates the magnitude of the polar solvation energy by a factor 2.5. Lastly, the possibility of the solute to have local dielectric constant larger than of a bulk water is investigated in a benchmarking test against experimentally determined set of pKa's and it is speculated that side chain rearrangements could result in local dielectric constant larger than 80.

Recent progress in adapting Poisson–Boltzmann methods to molecular simulations

Electrostatic solvation modeling based upon the Poisson–Boltzmann equation is widely used in studies of biomolecular structures and functions. This manuscript provides a thorough review of published efforts to adapt the numerical Poisson–Boltzmann methods to molecular simulations so that these methods can be extended to biomolecular studies involving conformational fluctuation and/or dynamics. We first review the fundamental works on how to define the electrostatic free energy and the Maxwell stress tensor. These topics are followed by three different strategies in developing algorithms to compute electrostatic forces and how to improve their numerical performance. Finally procedures are also presented in detail on how to discretize these algorithms for numerical calculations. Given the pioneer works reviewed here, further developmental efforts will be on how to balance efficiency and accuracy in these theoretical sound approaches — two important issues in applying any numerical algorithms for routine biomolecular applications. Even if not reviewed here, more advanced numerical solvers are certainly necessary to achieve higher accuracy than the widely used classical methods to improve the overall performance of the numerical Poisson–Boltzmann methods.

Electrostatic forces in the Poisson-Boltzmann systems

Continuum modeling of electrostatic interactions based upon numerical solutions of the Poisson-Boltzmann equation has been widely used in structural and functional analyses of biomolecules. A limitation of the numerical strategies is that it is conceptually difficult to incorporate these types of models into molecular mechanics simulations, mainly because of the issue in assigning atomic forces. In this theoretical study, we first derived the Maxwell stress tensor for molecular systems obeying the full nonlinear Poisson-Boltzmann equation. We further derived formulations of analytical electrostatic forces given the Maxwell stress tensor and discussed the relations of the formulations with those published in the literature. We showed that the formulations derived from the Maxwell stress tensor require a weaker condition for its validity, applicable to nonlinear Poisson-Boltzmann systems with a finite number of singularities such as atomic point charges and the existence of discontinuous dielectric as in the widely used classical piece-wise constant dielectric models.

Calculating the binding free energies of charged species based on explicit-solvent simulations employing lattice-sum methods: an accurate correction scheme for electrostatic finite-size effects

The calculation of a protein-ligand binding free energy based on molecular dynamics (MD) simulations generally relies on a thermodynamic cycle in which the ligand is alchemically inserted into the system, both in the solvated protein and free in solution. The corresponding ligand-insertion free energies are typically calculated in nanoscale computational boxes simulated under periodic boundary conditions and considering electrostatic interactions defined by a periodic lattice-sum. This is distinct from the ideal bulk situation of a system of macroscopic size simulated under non-periodic boundary conditions with Coulombic electrostatic interactions. This discrepancy results in finite-size effects, which affect primarily the charging component of the insertion free energy, are dependent on the box size, and can be large when the ligand bears a net charge, especially if the protein is charged as well. This article investigates finite-size effects on calculated charging free energies using as a test case the binding of the ligand 2-amino-5-methylthiazole (net charge +1 e) to a mutant form of yeast cytochrome c peroxidase in water. Considering different charge isoforms of the protein (net charges -5, 0, +3, or +9 e), either in the absence or the presence of neutralizing counter-ions, and sizes of the cubic computational box (edges ranging from 7.42 to 11.02 nm), the potentially large magnitude of finite-size effects on the raw charging free energies (up to 17.1 kJ mol(-1)) is demonstrated. Two correction schemes are then proposed to eliminate these effects, a numerical and an analytical one. Both schemes are based on a continuum-electrostatics analysis and require performing Poisson-Boltzmann (PB) calculations on the protein-ligand system. While the numerical scheme requires PB calculations under both non-periodic and periodic boundary conditions, the latter at the box size considered in the MD simulations, the analytical scheme only requires three non-periodic PB calculations for a given system, its dependence on the box size being analytical. The latter scheme also provides insight into the physical origin of the finite-size effects. These two schemes also encompass a correction for discrete solvent effects that persists even in the limit of infinite box sizes. Application of either scheme essentially eliminates the size dependence of the corrected charging free energies (maximal deviation of 1.5 kJ mol(-1)). Because it is simple to apply, the analytical correction scheme offers a general solution to the problem of finite-size effects in free-energy calculations involving charged solutes, as encountered in calculations concerning, e.g., protein-ligand binding, biomolecular association, residue mutation, pKa and redox potential estimation, substrate transformation, solvation, and solvent-solvent partitioning.

Exploring accurate Poisson-Boltzmann methods for biomolecular simulations

Accurate and efficient treatment of electrostatics is a crucial step in computational analyses of biomolecular structures and dynamics. In this study, we have explored a second-order finite-difference numerical method to solve the widely used Poisson-Boltzmann equation for electrostatic analyses of realistic bio-molecules. The so-called immersed interface method was first validated and found to be consistent with the classical weighted harmonic averaging method for a diversified set of test biomolecules. The numerical accuracy and convergence behaviors of the new method were next analyzed in its computation of numerical reaction field grid potentials, energies, and atomic solvation forces. Overall similar convergence behaviors were observed as those by the classical method. Interestingly, the new method was found to deliver more accurate and better-converged grid potentials than the classical method on or nearby the molecular surface, though the numerical advantage of the new method is reduced when grid potentials are extrapolated to the molecular surface. Our exploratory study indicates the need for further improving interpolation/extrapolation schemes in addition to the developments of higher-order numerical methods that have attracted most attention in the field.