Poisson-Boltzmann Solver Research Articles

An Unfitted Finite Element Poisson-Boltzmann Solver with Automatic Resolving of Curved Molecular Surface.

So far, the existing Poisson-Boltzmann (PB) solvers that accurately take into account the interface jump conditions need a pregenerated body-fitted mesh (molecular surface mesh). However, qualified biomolecular surface meshing and its implementation into numerical methods remains a challenging and laborious issue, which practically hinders the progress of further developments and applications of a bunch of numerical methods in this field. In addition, even with a molecular surface mesh, it is only a low-order approximation of the original curved surface. In this article, an interface-penalty finite element method (IPFEM), which is a typical unfitted finite element method, is proposed to solve the Poisson-Boltzmann equation (PBE) without requiring the user to generate a molecular surface mesh. The Gaussian molecular surface is used to represent the molecular surface and can be automatically resolved with a high-order approximation within our method. Theoretical convergence rates of the IPFEM for the linear PB equation have been provided and are well validated on a benchmark problem with an analytical solution (we also noticed from numerical examples that the IPFEM has similar convergence rates for the nonlinear PBE). Numerical results on a set of different-sized biomolecules demonstrate that the IPFEM is numerically stable and accurate in the calculation of biomolecular electrostatic solvation energy.

SARS-CoV-2: Prediction of critical ionic amino acid mutations

Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), that caused coronavirus disease 2019 (COVID-19), has been studied thoroughly, and several variants are revealed across the world with their corresponding mutations. Studies and vaccines development focus on the genetic mutations of the S protein due to its vital role in allowing the virus attach and fuse with the membrane of a host cell. In this perspective, we study the effects of all ionic amino acid mutations of the SARS-CoV-2 viral spike protein S1 when bound to Antibody CC12.1 within the SARS-CoV-2:CC12.1 complex model. Binding free energy calculations between SARS-CoV-2 and antibody CC12.1 are based on the Analysis of Electrostatic Similarities of Proteins (AESOP) framework, where the electrostatic potentials are calculated using Adaptive Poisson-Boltzmann Solver (APBS). The atomic radii and charges that feed into the APBS calculations are calculated using the PDB2PQR software. Our results are the first to propose in silico potential life-threatening mutations of SARS-CoV-2 beyond the present mutations found in the five common variants worldwide. We find each of the following mutations: K378A, R408A, K424A, R454A, R457A, K458A, and K462A, to play significant roles in the binding to Antibody CC12.1, since they are turned into strong inhibitors on both chains of the S1 protein, whereas the mutations D405A, D420A, and D427A, show to play important roles in this binding, as they are turned into mild inhibitors on both chains of the S1 protein.

Poisson-Boltzmann-based machine learning model for electrostatic analysis

Electrostatics is of paramount importance to chemistry, physics, biology, and medicine. The Poisson-Boltzmann (PB) theory is a primary model for electrostatic analysis. However, it is highly challenging to compute accurate PB electrostatic solvation free energies for macromolecules due to the nonlinearity, dielectric jumps, charge singularity, and geometric complexity associated with the PB equation. The present work introduces a PB-based machine learning (PBML) model for biomolecular electrostatic analysis. Trained with the second-order accurate MIBPB solver, the proposed PBML model is found to be more accurate and faster than several eminent PB solvers in electrostatic analysis. The proposed PBML model can provide highly accurate PB electrostatic solvation free energy of new biomolecules or new conformations generated by molecular dynamics with much reduced computational cost.

Optimized parallelization of boundary integral Poisson-Boltzmann solvers

The Poisson-Boltzmann (PB) model governs the electrostatics of solvated biomolecules, i.e., potential, field, energy, and force. These quantities can provide useful information about protein properties, functions, and dynamics. By considering the advantages of current algorithms and computer hardware, we focus on the parallelization of the treecode-accelerated boundary integral (TABI) PB solver using the Message Passing Interface (MPI) on CPUs and the direct-sum boundary integral (DSBI) PB solver using KOKKOS on GPUs. We provide optimization guidance for users when the DSBI solver on GPU or the TABI solver with MPI on CPU should be used depending on the size of the problem. Specifically, when the number of unknowns is smaller than a predetermined threshold, the GPU-accelerated DSBI solver converges rapidly thus has the potential to perform PB model-based molecular dynamics or Monte Carlo simulation. As practical applications, our parallelized boundary integral PB solvers are used to solve electrostatics on selected proteins that play significant roles in the spread, treatment, and prevention of COVID-19 virus diseases. For each selected protein, the simulation produces the electrostatic solvation energy as a global measurement and electrostatic surface potential for local details.

Computational Methods to Target Protein-Protein Interactions.

The chapter emphasizes the importance of understanding protein-protein interactions in cellular mechanisms and highlights the role of computational modeling in predicting these interactions. It discusses sequence-based approaches such as evolutionary trace (ET), correlated mutation analysis (CMA), and subtractive correlated mutation (SCM) for identifying crucial amino acid residues, considering interface conservation or evolutionary changes. The chapter also explores methods like differential ET, hidden-site class model, and spatial cluster detection (SCD) for interface specificity and spatial clustering. Furthermore, it examines approaches combining structural and sequential methodologies and evaluates modeled predictions through initiatives like critical assessment of prediction of interactions (CAPRI). Additionally, the chapter provides an overview of various software programs used for molecular docking, detailing their search, sampling, refinement and scoring stages, along with innovative techniques and tools like normal mode analysis (NMA) and adaptive Poisson-Boltzmann solver (APBS) for electrostatic calculations. These computational and experimental approaches are crucial for unraveling protein-protein interactions and aid in developing potential therapeutics for various diseases.

Bridging Eulerian and Lagrangian Poisson-Boltzmann solvers by ESES.

The Poisson-Boltzmann (PB) model is a widely used electrostatic model for biomolecular solvation analysis. Formulated as an elliptic interface problem, the PB model can be numerically solved on either Eulerian meshes using finite difference/finite element methods or Lagrangian meshes using boundary element methods. Molecular surface generators, which produce the discretized dielectric interfaces between solutes and solvents, are critical factors in determining the accuracy and efficiency of the PB solvers. In this work, we investigate the utility of the Eulerian Solvent Excluded Surface (ESES) software for rendering conjugated Eulerian and Lagrangian surface representations, which enables us to numerically validate and compare the quality of Eulerian PB solvers, such as the MIBPB solver, and the Lagrangian PB solvers, such as the TABI-PB solver. Furthermore, with the ESES software and its associated PB solvers, we are able to numerically validate an interesting and useful but often neglected source-target symmetric property associated with the linearized PB model.

SurfPB: A GPU-Accelerated Electrostatic Calculation and Visualization Tool for Biomolecules.

In this work, we present SurfPB as a useful tool for the study of biomolecules. It can do many typical calculations, including the molecular surface, electrostatic potential, solvation free energy, entropy, and binding free energy. Among all of the calculations, the entropy calculation is the most time-consuming one. In SurfPB, the calculation can be performed in a vacuum or implicit solvent and accelerated on GPU. The Poisson-Boltzmann equation solver is accelerated on GPU as well. Moreover, we developed a graphical user interface for SurfPB. It allows users to input the parameters and complete the whole calculation in a visual way. The calculated electrostatic potentials are shown on the molecular surface in a three-dimensional scene.

Fast iterative method for local steric Poisson–Boltzmann theories in biomolecular solvation

This work proposes a fast iterative method for local steric Poisson–Boltzmann (PB) theories, in which the electrostatic potential is governed by the Poisson's equation and ionic concentrations satisfy equilibrium conditions. To present the method, we focus on a local steric PB theory derived from a lattice-gas model, as an example. The advantages of the proposed method in efficiency are achieved by treating ionic concentrations as scalar implicit functions of the electrostatic potential, though such functions are only numerically achievable. The existence, uniqueness, boundness, and smoothness of such functions are rigorously established. A Newton iteration method with truncation is proposed to solve a nonlinear system discretized from the generalized PB equations. The existence and uniqueness of the solution to the discretized nonlinear system are established by showing that it is a unique minimizer of a constructed convex energy. Thanks to the boundness of ionic concentrations, truncation bounds for the potential are obtained by using the extremum principle. The truncation step in iterations is shown to be energy and error decreasing. To further speed-up computations, we propose a novel precomputing-interpolation strategy, which is applicable to other local steric PB theories and makes the proposed methods for solving steric PB theories as efficient as for solving the classical PB theory. Analysis on the Newton iteration method with truncation shows local quadratic convergence for the proposed numerical methods. Applications to realistic biomolecular solvation systems reveal that counterions with steric hindrance stratify in an order prescribed by the parameter of ionic valence-to-volume ratio. Finally, we remark that the proposed iterative methods for local steric PB theories can be readily incorporated in well-known classical PB solvers.

A Putative New Role of Tv-PSP1 Recognizes IRE and ERE Hairpin Structures from Trichomonas vaginalis.

To understand whether protein Tv-PSP1 from Trichomonas vaginalis recognizes mRNA parasite stem-loop structures, we conducted REMSA and intrinsic fluorescence assays. We found the recombinant Tv-PSP1 structure, determined with X-ray crystallography, showed unusual thermal stability of the quaternary structure, associated with a disulfide bridge CYS76-CYS104. To gain deeper insight into the Tv-PSP1 interaction with mRNA stem-loops (mRNAsl) and its relationship with thermal stability, we also used an integrated computational protocol that combined molecular dynamics simulations, docking assays, and binding energy calculations. Docking models allowed us to determine a putative contact surface interaction region between Tv-PSP1 and mRNAsl. We determined the contributions of these complexes to the binding free energy (ΔGb) in the electrostatic (ΔGelec) and nonelectrostatic (ΔGnon-elec) components using the Adaptive Poisson-Boltzmann Solver (APBS) program. We are the first, to the best of our knowledge, to show the interaction between Tv-PSP1 and the stem-loop structures of mRNA.

TABI-PB 2.0: An Improved Version of the Treecode-Accelerated Boundary Integral Poisson-Boltzmann Solver.

This work describes TABI-PB 2.0, an improved version of the treecode-accelerated boundary integral Poisson-Boltzmann solver. The code computes the electrostatic potential on the molecular surface of a solvated biomolecule, and further processing yields the electrostatic solvation energy. The new implementation utilizes the NanoShaper surface triangulation code, node-patch boundary integral discretization, a block preconditioner, and a fast multipole method based on barycentric Lagrange interpolation and dual tree traversal. Performance-critical portions of the code were implemented on a GPU. Numerical results for protein 1A63 and two viral capsids (Zika, H1N1) demonstrate the code's accuracy and efficiency.

Computing electrostatic binding energy with the TABI Poisson–Boltzmann solver

Computing electrostatic binding energy with the TABI Poisson–Boltzmann solver

Functional annotation and evaluation of hypothetical proteins in cyanobacterium Synechocystis sp. PCC 6803

The cyanobacterium Synechocystis sp. PCC 6803 is well reported as a potential species for producing valuable bioactive compounds as well as model microorganism for investigating photosynthesis. However, there are a great number of proteins in the genome of this microorganism with unclear functions. To this end, the present study employed numerous bioinformatics techniques in conjunction with experimental validation to describe the function of four hypothetical proteins in cyanobacterium Synechocystis sp. PCC 6803. Four potential proteins, namely sll1388, sll1512 (cytosolic), sll1164, and slr0964 (membrane), were selected for functional annotation. The sll1388 and sll1512 proteins containing adenosine triphosphate (ATP) and deoxyribonucleic acid (DNA) are expected to be induced by toxic concentrations of NaCl and hydrogen peroxide (H2O2). Sll1164 and slr0964 are predicted as membrane proteins involved in cysteine (Cys) and ferrous (Fe2+) transport. The Adaptive Poisson-Boltzmann Solver (APBS) findings suggested negative and positive charges for ligand-binding sites and negative and uncharged residues for predicted pores. Transcript levels of sll1388 and sll1512 increased at high concentrations of NaCl and H2O2. An increase in the expression level of sll1164 and slr0964 was observed under Sulfur (S) and Fe2+ depletion. Transcript analysis along with experimental gene expression research gave us vital indications concerning the function of the above-mentioned hypothetical proteins. The present study used a combination of in silico approaches in conjugation with experimental methods, which finally increased our understanding of the stress tolerance of Synechocystis sp. PCC 6803 as a potential model microorganism with industrial applications.

Implicit Solvents for the Polarizable Atomic Multipole AMOEBA Force Field.

Computational protein design, ab initio protein/RNA folding, and protein-ligand screening can be too computationally demanding for explicit treatment of solvent. For these applications, implicit solvent offers a compelling alternative, which we describe here for the polarizable atomic multipole AMOEBA force field based on three treatments of continuum electrostatics: numerical solutions to the nonlinear and linearized versions of the Poisson-Boltzmann equation (PBE), the domain-decomposition conductor-like screening model (ddCOSMO) approximation to the PBE, and the analytic generalized Kirkwood (GK) approximation. The continuum electrostatics models are combined with a nonpolar estimator based on novel cavitation and dispersion terms. Electrostatic model parameters are numerically optimized using a least-squares style target function based on a library of 103 small-molecule solvation free energy differences. Mean signed errors for the adaptive Poisson-Boltzmann solver (APBS), ddCOSMO, and GK models are 0.05, 0.00, and 0.00 kcal/mol, respectively, while the mean unsigned errors are 0.70, 0.63, and 0.58 kcal/mol, respectively. Validation of the electrostatic response of the resulting implicit solvents, which are available in the Tinker (or Tinker-HP), OpenMM, and Force Field X software packages, is based on comparisons to explicit solvent simulations for a series of proteins and nucleic acids. Overall, the emergence of performative implicit solvent models for polarizable force fields opens the door to their use for folding and design applications.

Advanced Spectroscopy and APBS Modeling for Determination of the Role of His190 and Trp103 in Mouse Thymidylate Synthase Interaction with Selected dUMP Analogues.

A homo-dimeric enzyme, thymidylate synthase (TS), has been a long-standing molecular target in chemotherapy. To further elucidate properties and interactions with ligands of wild-type mouse thymidylate synthase (mTS) and its two single mutants, H190A and W103G, spectroscopic and theoretical investigations have been employed. In these mutants, histidine at position 190 and tryptophan at position 103 are substituted with alanine and glycine, respectively. Several emission-based spectroscopy methods used in the paper demonstrate an especially important role for Trp 103 in TS ligands binding. In addition, the Advanced Poisson–Boltzmann Solver (APBS) results show considerable differences in the distribution of electrostatic potential around Trp 103, as compared to distributions observed for all remaining Trp residues in the mTS family of structures. Together, spectroscopic and APBS results reveal a possible interplay between Trp 103 and His190, which contributes to a reduction in enzymatic activity in the case of H190A mutation. Comparison of electrostatic potential for mTS complexes, and their mutants, with the substrate, dUMP, and inhibitors, FdUMP and N-OH-dCMP, suggests its weaker influence on the enzyme–ligand interactions in NOH-dCMP-mTS compared to dUMP-mTS and FdUMP-mTS complexes. This difference may be crucial for the explanation of the ”abortive reaction” inhibitory mechanism of NOH-dCMP towards TS. In addition, based on structural analyses and the H190A mutant capacity to form a denaturation-resistant complex with N-OH-dCMP in the mTHF-dependent reaction, His190 is apparently responsible for a strong preference of the enzyme active center for the anti rotamer of the imino inhibitor form.

Binding characteristics of staphylococcal protein A and streptococcal protein G for fragment crystallizable portion of human immunoglobulin G

In the wide array of physiological processes, protein–protein interactions and their binding are the most basal activities for achieving adequate biological metabolism. Among the studies on binding proteins, the examination of interactions between immunoglobulin G (IgG) and natural immunoglobulin-binding ligands, such as staphylococcal protein A (spA) and streptococcal protein G (spG), is essential in the development of pharmaceutical science, biotechnology, and affinity chromatography. The widespread utilization of IgG-spA/spG binding characteristics has allowed researchers to investigate these molecular interactions. However, the detailed binding strength of each ligand and the corresponding binding mechanisms have yet to be fully investigated. In this study, the authors analyzed the binding strengths of IgG–spA and IgG–spG complexes and identified the mechanisms enabling these bindings using molecular dynamics simulation, steered molecular dynamics, and advanced Poisson–Boltzmann Solver simulations. Based on the presented data, the binding strength of the spA ligand was found to significantly exceed that of the spG ligand. To find out which non-covalent interactions or amino acid sites have a dominant role in the tight binding of these ligands, further detailed analyses of electrostatic interactions, hydrophobic bonding, and binding free energies have been performed. In investigating their binding affinity, a relatively independent and different unbinding mechanism was found in each ligand. These distinctly different mechanisms were observed to be highly correlated to the protein secondary and tertiary structures of spA and spG ligands, as explicated from the perspective of hydrogen bonding.

Computational Hot-Spot Analysis of the SARS-CoV-2 Receptor Binding Domain/ACE2 Complex*.

Infection and replication of SARS CoV-2 (the virus that causes COVID-19) requires entry to the interior of host cells. In humans, a protein-protein interaction (PPI) between the SARS CoV-2 receptor-binding domain (RBD) and the extracellular peptidase domain of ACE2 on the surface of cells in the lower respiratory tract is an initial step in the entry pathway. Inhibition of the SARS CoV-2 RBD/ACE2 PPI is currently being evaluated as a target for therapeutic and/or prophylactic intervention. However, relatively little is known about the molecular underpinnings of this complex. Employing multiple computational platforms, we predicted "hot-spot" residues in a positive-control PPI (PMI/MDM2) and the CoV-2 RBD/ACE2 complex. Computational alanine scanning mutagenesis was performed to predict changes in Gibbs' free energy that are associated with mutating residues at the positive control (PMI/MDM2) or SARS RBD/ACE2 binding interface to alanine. Additionally, we used the Adaptive Poisson-Boltzmann Solver to calculate macromolecular electrostatic surfaces at the interface of the positive-control PPI and SARS CoV-2/ACE2 PPI. Finally, a comparative analysis of hot-spot residues for SARS-CoV and SARS-CoV-2, in complex with ACE2, is provided. Collectively, this study illuminates predicted hot-spot residues, and clusters, at the SARS CoV-2 RBD/ACE2 binding interface, potentially guiding the development of reagents capable of disrupting this complex and halting COVID-19.

Electronic structure calculations in electrolyte solutions: Methods for neutralization of extended charged interfaces.

Density functional theory (DFT) is often used for simulating extended materials such as infinite crystals or surfaces, under periodic boundary conditions (PBCs). In such calculations, when the simulation cell has non-zero charge, electrical neutrality has to be imposed, and this is often done via a uniform background charge of opposite sign ("jellium"). This artificial neutralization does not occur in reality, where a different mechanism is followed as in the example of a charged electrode in electrolyte solution, where the surrounding electrolyte screens the local charge at the interface. The neutralizing effect of the surrounding electrolyte can be incorporated within a hybrid quantum-continuum model based on a modified Poisson-Boltzmann equation, where the concentrations of electrolyte ions are modified to achieve electroneutrality. Among the infinite possible ways of modifying the electrolyte charge, we propose here a physically optimal solution, which minimizes the deviation of concentrations of electrolyte ions from those in open boundary conditions (OBCs). This principle of correspondence of PBCs with OBCs leads to the correct concentration profiles of electrolyte ions, and electroneutrality within the simulation cell and in the bulk electrolyte is maintained simultaneously, as observed in experiments. This approach, which we call the Neutralization by Electrolyte Concentration Shift (NECS), is implemented in our electrolyte model in the Order-N Electronic Total Energy Package (ONETEP) linear-scaling DFT code, which makes use of a bespoke highly parallel Poisson-Boltzmann solver, DL_MG. We further propose another neutralization scheme ("accessible jellium"), which is a simplification of NECS. We demonstrate and compare the different neutralization schemes on several examples.

The heme-sensitive regulator SbnI has a bifunctional role in staphyloferrin B production by Staphylococcus aureus

Staphylococcus aureus infection relies on iron acquisition from its host. S. aureus takes up iron through heme uptake by the iron-responsive surface determinant (Isd) system and by the production of iron-scavenging siderophores. Staphyloferrin B (SB) is a siderophore produced by the 9-gene sbn gene cluster for SB biosynthesis and efflux. Recently, the ninth gene product, SbnI, was determined to be a free l-serine kinase that produces O-phospho-l-serine (OPS), a substrate for SB biosynthesis. Previous studies have also characterized SbnI as a DNA-binding regulatory protein that senses heme to control sbn gene expression for SB synthesis. Here, we present crystal structures at 1.9-2.1 Å resolution of a SbnI homolog from Staphylococcus pseudintermedius (SpSbnI) in both apo form and in complex with ADP, a product of the kinase reaction; the latter confirmed the active-site location. The structures revealed that SpSbnI forms a dimer through C-terminal domain swapping and a dimer of dimers through intermolecular disulfide formation. Heme binding had only a modest effect on SbnI enzymatic activity, suggesting that its two functions are independent and structurally distinct. We identified a heme-binding site and observed catalytic heme transfer between a heme-degrading protein of the Isd system, IsdI, and SbnI. These findings support the notion that SbnI has a bifunctional role contributing precursor OPS to SB synthesis and directly sensing heme to control expression of the sbn locus. We propose that heme transfer from IsdI to SbnI enables S. aureus to control iron source preference according to the sources available in the environment.

Polycation-globular protein complex: Ionic strength and chain length effects on the structure and properties

AbstractThe effect of high salt concentration and pH on the binding of globular protein to polycation at different molar masses has been investigated by dynamic light scattering (DLS), turbidimetry and electrostatic modeling for the protein. In dilute concentration regime, DLS and pH titrations showed three remarkable pH transitions: pHc, the pH where soluble complexes of bovine serum albumin (BSA) and linear synthetic Polyethylenemine (PEI) are formed, pHc’ presents the end of primary soluble complex and pHφ presents the first appearance of microcoacervate droplets. All pH transitions increase with increasing NaCl concentration. The Adaptive Poisson-Boltzmann Solver (APBS) identify with precision the functional sites at the surface of BSA and shows that the strength of electrostatic interactions depends hugely on the variation of pH and ionic strength. At a very high concentration of salt, no remarkable effect on a mixture formed of a long chain of polycation and globular protein.

A Boundary‐Integral Approach for the Poisson–Boltzmann Equation with Polarizable Force Fields

Implicit-solvent models are widely used to study the electrostatics in dissolved biomolecules, which are parameterized using force fields. Standard force fields treat the charge distribution with point charges; however, other force fields have emerged which offer a more realistic description by considering polarizability. In this work, we present the implementation of the polarizable and multipolar force field atomic multipole optimized energetics for biomolecular applications (AMOEBA), in the boundary integral Poisson-Boltzmann solver PyGBe. Previous work from other researchers coupled AMOEBA with the finite-difference solver APBS, and found difficulties to effectively transfer the multipolar charge description to the mesh. A boundary integral formulation treats the charge distribution analytically, overlooking such limitations. This becomes particularly important in simulations that need high accuracy, for example, when the quantity of interest is the difference between solvation energies obtained from separate calculations, like happens for binding energy. We present verification and validation results of our software, compare it with the implementation on APBS, and assess the efficiency of AMOEBA and classical point-charge force fields in a Poisson-Boltzmann solver. We found that a boundary integral approach performs similarly to a volumetric method on CPU. Also, we present a GPU implementation of our solver. Moreover, with a boundary element method, the mesh density to correctly resolve the electrostatic potential is the same for standard point-charge and multipolar force fields. Finally, we saw that for binding energy calculations, a boundary integral approach presents more consistent results than a finite difference approximation for multipolar force fields. © 2019 Wiley Periodicals, Inc.