Adaptive multilevel finite element solution of the Poisson-Boltzmann equation II. Refinement at solvent-accessible surfaces in biomolecular systems

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.

Impact of Ion-Mixing Entropy on Orientational Preferences of DNA Helices: FRET Measurements and Computer Simulations

Biological processes require DNA and RNA helices to pack together in specific interhelical orientations. While electrostatic repulsion between backbone charges is expected to be maximized when helices are in parallel alignment, such orientations are commonplace in nature. To better understand how the repulsion is overcome, we used experimental and computational approaches to investigate how the orientational preferences of DNA helices depend on the concentration and valence of mobile cations. We used Förster resonance energy transfer (FRET) to probe the relative orientations of two 24-bp helices held together via a freely rotating PEG linker. At low cation concentrations, the helices preferred more "cross"-like orientations over those closer to parallel, and this preference was reduced with increasing salt concentrations. The results were in good quantitative agreement with Poisson-Boltzmann (PB) calculations for monovalent salt (Na+). However, PB underestimated the ability of mixtures of monovalent and divalent ions (Mg2+) to reduce the conformational preference. As a complementary approach, we performed all-atom molecular dynamics (MD) simulations and found better agreement with the experimental results. While MD and PB predict similar electrostatic forces, MD predicts a greater accumulation of Mg2+ in the ion atmosphere surrounding the DNA. Mg2+ occupancy is predicted to be greater in conformations close to the parallel orientation than in conformations close to the crossed orientation, enabling a greater release of Na+ ions and providing an entropic gain (one bound ion for two released). MD predicts an entropy gain larger than that of PB because of the increased Mg2+ occupancy. The entropy changes have a negligible effect at low Mg2+ concentrations because the free energies are dominated by electrostatic repulsion. However, as the Mg2+ concentration increases, charge screening is more effective and the mixing entropy produces readily detectable changes in packing preferences. Our results underline the importance of mixing entropy of counterions in nucleic acid interactions and provide a new understanding on the impact of a mixed ion atmosphere on the packing of DNA helices.

Collagen Type II-Chitosan Interactions as Dependent on Hydroxylation and Acetylation Inferred from Molecular Dynamics Simulations

Chitosan-collagen blends have been widely applied in tissue engineering, joints diseases treatment, and many other biomedical fields. Understanding the affinity between chitosan and collagen type II is particularly relevant in the context of mechanical properties modulation, which is closely associated with designing biomaterials suitable for cartilage and synovial fluid regeneration. However, many structural features influence chitosan's affinity for collagen. One of the most important ones is the deacetylation degree (DD) in chitosan and the hydroxylation degree (HD) of proline (PRO) moieties in collagen. In this paper, combinations of both factors were analyzed using a very efficient molecular dynamics approach. It was found that DD and HD modifications significantly affect the structural features of the complex related to considered types of interactions, namely hydrogen bonds, hydrophobic, and ionic contacts. In the case of hydrogen bonds both direct and indirect (water bridges) contacts were examined. In case of the most collagen analogues, a very good correlation between binding free energy and DD was observed.

Collagen type II-hyaluronan interactions - the effect of proline hydroxylation: a molecular dynamics study

Hyaluronan-collagen composites have been employed in numerous biomedical applications. Understanding the interactions between hyaluronan and collagen is particularly important in the context of joint cartilage function and the treatment of joint diseases. Many factors affect the affinity of collagen for hyaluronan. One of the important factors is the ratio of 3- or 4-hydroxy proline to proline residues. This article presents the results from molecular dynamics calculations of HA-collagen type II interactions with hyaluronan. The applied protocol employed docking and geometry optimization of complexes built using collagen structures with different numbers of hydroxyl groups attached to proline moieties. It was established that the hydroxyproline/proline ratio affects both structural and energetic features of the collagen-hyaluronan complex. Proline hydroxylation was found to significantly influence the number of all identified types of molecular forces, hydrophobic interactions, water bridges and hydrogen bonds, which can be formed between collagen and hyaluronan. Importantly, an increase in the hydroxyproline/proline ratio in the collagen chain increases the binding affinity for hyaluronan. This is illustrated by the linear correlation between the binding free energy and the hydroxylation degree. A comparison of the results obtained for 3 and 4 hydroxylation of proline indicates that the hydroxyl group attachment position plays a minor role in complex stabilization. However, a slightly stronger affinity was observed for 4 hydroxylation. In order to evaluate the effect of the aqueous environment on the collagen-hyaluronan complex stability, the enthalpic and entropic contributions to the free energy of solvation were analyzed.

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.

Numerical Studies on Electrostatic Interaction Forces and the Free Energy between Parallel Colloidal Rods of Finite Size in Skewed Configurations

Formulas for interaction forces F(s) and the free energy G(s) between two parallel charged prismatic rods of various scaled values of d, ψ_s, and L in skewed configurations are obtained, where s is the lengthwise positional difference between the front-end faces of the respective rods, and d is the minimal distance between the opposing faces of the rods, ψ_s is the electric surface potential, L is the length of the rods. To obtain the free-energy function G(s), (i) 3D spatial distributions of the electric potential ψ around two rods were determined by numerically solving the nonlinear Poisson-Boltzmann equation with a finite element method, (ii) with the ψ distributions so determined, the lengthwise interaction electrostatic Maxwell stress tangential to the midplane between the rods was calculated to obtain the (discrete) s dependence of the stress, and (iii) by introducing two different fitting functions, the discrete s dependence was transformed into a continuous force function, F(s), which was then lengthwise integrated to derive G(s). It was found that the curves of G(s) linearly decreased with increasing s between 1 and L + 1 due to a localization of the stress. Although natural, it is of interest that the values of G(0) calculated for rods of various values of d, ψ_s, and L were in good agreement with those of the interaction free energy obtained in our preceding work by the widthwise integration of repulsive electrostatic forces normal to the midplane between the parallel rods in nonskewed configurations.

Comparison of the MSMS and NanoShaper molecular surface triangulation codes in the TABI Poisson-Boltzmann solver

The Poisson-Boltzmann (PB) implicit solvent model is a popular framework for studying the electrostatics of solvated biomolecules. In this model the dielectric interface between the biomolecule and solvent is often taken to be the molecular surface or solvent-excluded surface (SES), and the quality of the SES triangulation is critical in boundary element simulations of the model. This work compares the performance of the MSMS and NanoShaper surface triangulation codes for a set of 38 biomolecules. While MSMS produces triangles of exceedingly small area and large aspect ratio, the two codes yield comparable values for the SES surface area and electrostatic solvation energy, where the latter calculations were performed using the treecode-accelerated boundary integral (TABI) PB solver. However we found that NanoShaper is computationally more efficient and reliable than MSMS, especially when parameters are set to produce highly resolved triangulations.

Efficient mesh refinement for the Poisson-Boltzmann equation with boundary elements

The Poisson-Boltzmann equation is a widely used model to study electrostatics in molecular solvation. Its numerical solution using a boundary integral formulation requires a mesh on the molecular surface only, yielding accurate representations of the solute, which is usually a complicated geometry. Here, we utilize adjoint-based analyses to form two goal-oriented error estimates that allow us to determine the contribution of each discretization element (panel) to the numerical error in the solvation free energy. This information is useful to identify high-error panels to then refine them adaptively to find optimal surface meshes. We present results for spheres and real molecular geometries, and see that elements with large error tend to be in regions where there is a high electrostatic potential. We also find that even though both estimates predict different total errors, they have similar performance as part of an adaptive mesh refinement scheme. Our test cases suggest that the adaptive mesh refinement scheme is very effective, as we are able to reduce the error one order of magnitude by increasing the mesh size less than 20% and come out to be more efficient than uniform refinement when computing error estimations. This result sets the basis toward efficient automatic mesh refinement schemes that produce optimal meshes for solvation energy calculations.

Antimicrobial Peptides and Copper(II) Ions: Novel Therapeutic Opportunities

The emergence of new pathogens and multidrug resistant bacteria is an important public health issue that requires the development of novel classes of antibiotics. Antimicrobial peptides (AMPs) are a promising platform with great potential for the identification of new lead compounds that can combat the aforementioned pathogens due to their broad-spectrum antimicrobial activity and relatively low rate of resistance emergence. AMPs of multicellular organisms made their debut four decades ago thanks to ingenious researchers who asked simple questions about the resistance to bacterial infections of insects. Questions such as "Do fruit flies ever get sick?", combined with pioneering studies, have led to an understanding of AMPs as universal weapons of the immune system. This review focuses on a subclass of AMPs that feature a metal binding motif known as the amino terminal copper and nickel (ATCUN) motif. One of the metal-based strategies of hosts facing a pathogen, it includes wielding the inherent toxicity of copper and deliberately trafficking this metal ion into sites of infection. The sudden increase in the concentration of copper ions in the presence of ATCUN-containing AMPs (ATCUN-AMPs) likely results in a synergistic interaction. Herein, we examine common structural features in ATCUN-AMPs that exist across species, and we highlight unique features that deserve additional attention. We also present the current state of knowledge about the molecular mechanisms behind their antimicrobial activity and the methods available to study this promising class of AMPs.

Surface functionalization-dependent localization and affinity of SiO2 nanoparticles within the biofilm EPS matrix

The contribution of the biofilm extracellular polymeric substance (EPS) matrix to reduced antimicrobial susceptibility in biofilms is widely recognised. As such, the direct targeting of the EPS matrix is a promising biofilm control strategy that allows for the disruption of the matrix, thereby allowing a subsequent increase in susceptibility to antimicrobial agents. To this end, surface-functionalized nanoparticles (NPs) have received considerable attention. However, the fundamental understanding of the interactions occurring between engineered NPs and the biofilm EPS matrix has not yet been fully elucidated. An insight into the underlying mechanisms involved when a NP interacts with the EPS matrix will aid in the design of more efficient NPs for biofilm control. Here we demonstrate the use of highly specific fluorescent probes in confocal laser scanning microscopy (CLSM) to illustrate the distribution of EPS macromolecules within the biofilm. Thereafter, a three-dimensional (3D) colocalization analysis was used to assess the affinity of differently functionalized silica NPs (SiNPs) and EPS macromolecules from Pseudomonas fluorescens biofilms. Results show that both the charge and surface functional groups of SiNPs dramatically affected the extent to which SiNPs interacted and localized with EPS macromolecules, including proteins, polysaccharides and DNA. Hypotheses are also presented about the possible physicochemical interactions which may be dominant in EPS matrix-NP interactions. This research not only develops an innovative CLSM-based methodology for elucidating biofilm-nanoparticle interactions but also provides a platform on which to build more efficient NP systems for biofilm control.

Numerical Studies on Electrical Interaction Forces and Free Energy between Colloidal Plates of Finite Size

By solving the nonlinear Poisson-Boltzmann (PB) equation with a finite element method (FEM), three-dimensional (3D) spatial distributions of the electric potential (ψ, scaled) in electrolyte solutions having two charged parallel finite plates (including cubes and prismatic rods) are determined for various separations (d, scaled by the Debye length, κ^-1), surface potentials (ψ_s), and plate dimensions (length × width × thickness, each scaled by κ^-1). The total interaction force between two plates, F, is the sum of the electrostatic double-layer (EDL) repulsion (the osmotic pressure, F_osm) and the Maxwell electrostatic stress (F_es). The EDL repulsion is estimated using the distribution of ψ not only between the facing surfaces of two parallel plates but also around the other extremities of the plates. The Maxwell stress (F_es) is localized near the extremities to act as a repulsive force on the midplane between the two plates. The ratio F_es/F is 0.07-0.5, depending on d, ψ_s, and dimensions. It is found that, with increasing dimensions, the total F values per unit area calculated for finite plates, F̃, decreasingly approach the exact ones for parallel infinite plates, F̃^inf; for example, at d = 1 and ψ_s = 5, the ratio F̃/F̃^inf is 2.83 for plates with dimensions of 1 × 1 × 1 and 1.18 for plates of 10 × 10 × 1. The repulsions arising from the extremities cannot be neglected for plates with dimensions <10 × 10 × 1. Furthermore, the total interaction forces (F) are calculated at a series of discrete d values, respectively, for parallel plates. We introduce a force fitting function, F_f(d), with parameters that can be determined so that F_f(d) fits well to the calculated serial F values. By integrating the F_f(d), we obtain the interaction free energy, G(d), for finite parallel plates that consists of two Γ functions.

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.

Electrostatic Complementarity as a Fast and Effective Tool to Optimize Binding and Selectivity of Protein-Ligand Complexes

Electrostatic interactions between small molecules and their respective receptors are essential for molecular recognition and are also key contributors to the binding free energy. Assessing the electrostatic match of protein-ligand complexes therefore provides important insights into why ligands bind and what can be changed to improve binding. Ideally, the ligand and protein electrostatic potentials at the protein-ligand interaction interface should maximize their complementarity while minimizing desolvation penalties. In this work, we present a fast and efficient tool to calculate and visualize the electrostatic complementarity (EC) of protein-ligand complexes. We compiled benchmark sets demonstrating electrostatically driven structure-activity relationships (SAR) from literature data, including kinase, protein-protein interaction, and GPCR targets, and used these to demonstrate that the EC method can visualize, rationalize, and predict electrostatically driven ligand affinity changes and help to predict compound selectivity. The methodology presented here for the analysis of EC is a powerful and versatile tool for drug design.

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.

Implicit Solvation Using a Generalized Finite-Difference Approach in CRYSTAL: Implementation and Results for Molecules, Polymers, and Surfaces

We present the implementation of an implicit solvation model in the CRYSTAL code. The solvation energy is separated into two components: the electrostatic contribution arising from a self-consistent reaction field treatment obtained within a generalized finite-difference Poisson model, augmented by a nonelectrostatic contribution proportional to the solvent-accessible surface area of the solute. A discontinuous dielectric boundary is used, along with a solvent-excluded surface built from interlocking atom-centered spheres on which apparent surface point charges are mapped. The procedure is general and can be performed at both the Hartree-Fock and density functional theory levels, with pure or hybrid functionals, for systems periodic in 0, 1, and 2 directions, that is, for isolated molecules and extended polymers and surfaces. The Poisson equation resolution and apparent surface charge formalism is first validated on model analytical test cases. The good agreement obtained on solvation free energies is further confirmed by calculations performed on a large test set of 501 neutral molecules, for which a mean unsigned error of 1.3 kcal/mol is obtained when compared to the available experimental data. Importantly, the self-consistent reaction field procedure converges well for all molecules tested. This is further verified for all polymers and surfaces considered. In particular, for periodic systems, results obtained on an infinite glycine chain and on the wettability parameters of SiO₂ surfaces are in good agreement with previously published data. The size extensivity of the energetic terms involved in the electrostatic contribution to the solvation energy is also well verified. These encouraging results constitute a first step to take into account complex environments in the CRYSTAL code, potentially allowing for a more accurate modeling of complex processes for both periodic and nonperiodic systems.

A Finite Element Solution of Lateral Periodic Poisson-Boltzmann Model for Membrane Channel Proteins

Membrane channel proteins control the diffusion of ions across biological membranes. They are closely related to the processes of various organizational mechanisms, such as: cardiac impulse, muscle contraction and hormone secretion. Introducing a membrane region into implicit solvation models extends the ability of the Poisson–Boltzmann (PB) equation to handle membrane proteins. The use of lateral periodic boundary conditions can properly simulate the discrete distribution of membrane proteins on the membrane plane and avoid boundary effects, which are caused by the finite box size in the traditional PB calculations. In this work, we: (1) develop a first finite element solver (FEPB) to solve the PB equation with a two-dimensional periodicity for membrane channel proteins, with different numerical treatments of the singular charges distributions in the channel protein; (2) add the membrane as a dielectric slab in the PB model, and use an improved mesh construction method to automatically identify the membrane channel/pore region even with a tilt angle relative to the z-axis; and (3) add a non-polar solvation energy term to complete the estimation of the total solvation energy of a membrane protein. A mesh resolution of about 0.25 Å (cubic grid space)/0.36 Å (tetrahedron edge length) is found to be most accurate in linear finite element calculation of the PB solvation energy. Computational studies are performed on a few exemplary molecules. The results indicate that all factors, the membrane thickness, the length of periodic box, membrane dielectric constant, pore region dielectric constant, and ionic strength, have individually considerable influence on the solvation energy of a channel protein. This demonstrates the necessity to treat all of those effects in the PB model for membrane protein simulations.

Improvements to the APBS biomolecular solvation software suite

The Adaptive Poisson-Boltzmann Solver (APBS) software was developed to solve the equations of continuum electrostatics for large biomolecular assemblages that have provided impact in the study of a broad range of chemical, biological, and biomedical applications. APBS addresses the three key technology challenges for understanding solvation and electrostatics in biomedical applications: accurate and efficient models for biomolecular solvation and electrostatics, robust and scalable software for applying those theories to biomolecular systems, and mechanisms for sharing and analyzing biomolecular electrostatics data in the scientific community. To address new research applications and advancing computational capabilities, we have continually updated APBS and its suite of accompanying software since its release in 2001. In this article, we discuss the models and capabilities that have recently been implemented within the APBS software package including a Poisson-Boltzmann analytical and a semi-analytical solver, an optimized boundary element solver, a geometry-based geometric flow solvation model, a graph theory-based algorithm for determining pK_a values, and an improved web-based visualization tool for viewing electrostatics.

Treating ion distribution with Gaussian-based smooth dielectric function in DelPhi

The standard treatment of ions in the framework of the Poisson-Boltzmann equation relies on molecular surfaces, which are commonly constructed along with the Stern layer. The molecular surface determines where ions can be present. In the Gaussian-based smooth dielectric function in DelPhi, smooth boundaries between the solute and solvent take the place of molecular surface. Therefore, this invokes the question of how to model mobile ions in the water phase without a definite solute-solvent boundary. This article reports a natural extension of the Gaussian-based smooth dielectric function approach that treats mobile ions via Boltzmann distribution with an added desolvation penalty. Thus, ion concentration near macromolecules is governed by the local electrostatic potential and the desolvation penalty (from being partially desolvated). The approach is tested against the experimental salt dependence of binding free energy on 7 protein-protein complexes and 12 DNA-protein complexes, resulting in Pearson correlations of 0.95 and 0.88, respectively. © 2017 Wiley Periodicals, Inc.

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.

Acceleration of Linear Finite-Difference Poisson-Boltzmann Methods on Graphics Processing Units

Electrostatic interactions play crucial roles in biophysical processes such as protein folding and molecular recognition. Poisson-Boltzmann equation (PBE)-based models have emerged as widely used in modeling these important processes. Though great efforts have been put into developing efficient PBE numerical models, challenges still remain due to the high dimensionality of typical biomolecular systems. In this study, we implemented and analyzed commonly used linear PBE solvers for the ever-improving graphics processing units (GPU) for biomolecular simulations, including both standard and preconditioned conjugate gradient (CG) solvers with several alternative preconditioners. Our implementation utilizes the standard Nvidia CUDA libraries cuSPARSE, cuBLAS, and CUSP. Extensive tests show that good numerical accuracy can be achieved given that the single precision is often used for numerical applications on GPU platforms. The optimal GPU performance was observed with the Jacobi-preconditioned CG solver, with a significant speedup over standard CG solver on CPU in our diversified test cases. Our analysis further shows that different matrix storage formats also considerably affect the efficiency of different linear PBE solvers on GPU, with the diagonal format best suited for our standard finite-difference linear systems. Further efficiency may be possible with matrix-free operations and integrated grid stencil setup specifically tailored for the banded matrices in PBE-specific linear systems.

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.

Kinetics of nucleotide entry into RNA polymerase active site provides mechanism for efficiency and fidelity

During transcription, RNA polymerase II elongates RNA by adding nucleotide triphosphates (NTPs) complementary to a DNA template. Structural studies have suggested that NTPs enter and exit the active site via the narrow secondary pore but details have remained unclear. A kinetic model is presented that integrates molecular dynamics simulations with experimental data. Previous simulations of trigger loop dynamics and the dynamics of matched and mismatched NTPs in and near the active site were combined with new simulations describing NTP exit from the active site via the secondary pore. Markov state analysis was applied to identify major states and estimate kinetic rates for transitions between those states. The kinetic model predicts elongation and misincorporation rates in close agreement with experiment and provides mechanistic hypotheses for how NTP entry and exit via the secondary pore is feasible and a key feature for achieving high elongation and low misincorporation rates during RNA elongation.

Accurate estimation of electrostatic binding energy with Poisson-Boltzmann equation solver DelPhi program

Poisson–Boltzmann (PB) model is a widely used implicit solvent approximation in biophysical modeling because of its ability to provide accurate and reliable PB electrostatic salvation free energies ([Formula: see text] as well as electrostatic binding free energy ([Formula: see text] estimations. However, a recent study has warned that the 0.5[Formula: see text]Å grid spacing which is normally adopted can produce unacceptable errors in [Formula: see text] estimation with the solvent excluded surface (SES) (Harris RC, Boschitsch AH and Fenley MO, Influence of grid spacing in Poisson–Boltzmann equation binding energy estimation, J Chem Theory Comput 19: 3677–3685, 2013). In this work, we investigate the grid dependence of the widely used PB solver DelPhi v6.2 with molecular surface (MS) for estimating both electrostatic solvation free energies and electrostatic binding free energies. Our results indicate that, for the molecular complex and components the absolute errors of [Formula: see text] are smaller than that of [Formula: see text], and grid spacing of 0.8[Formula: see text]Å with DelPhi program ensures the accuracy and reliability of [Formula: see text]; however, the accuracy of [Formula: see text] largely relies on the order of magnitude of [Formula: see text] itself rather than that of [Formula: see text] or [Formula: see text]. Our findings suggest that grid spacing of 0.5[Formula: see text]Å is enough to produce accurate [Formula: see text] for molecules whose [Formula: see text] are large, but finer grids are needed when [Formula: see text] is very small.

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.

Blinded predictions of distribution coefficients in the SAMPL5 challenge

In the context of the SAMPL5 challenge water-cyclohexane distribution coefficients for 53 drug-like molecules were predicted. Four different models based on molecular dynamics free energy calculations were tested. All models initially assumed only one chemical state present in aqueous or organic phases. Model A is based on results from an alchemical annihilation scheme; model B adds a long range correction for the Lennard Jones potentials to model A; model C adds charging free energy corrections; model D applies the charging correction from model C to ionizable species only. Model A and B perform better in terms of mean-unsigned error ([Formula: see text] D units - 95 % confidence interval) and determination coefficient [Formula: see text], while charging corrections lead to poorer results with model D ([Formula: see text] and [Formula: see text]). Because overall errors were large, a retrospective analysis that allowed co-existence of ionisable and neutral species of a molecule in aqueous phase was investigated. This considerably reduced systematic errors ([Formula: see text] and [Formula: see text]). Overall accurate [Formula: see text] predictions for drug-like molecules that may adopt multiple tautomers and charge states proved difficult, indicating a need for methodological advances to enable satisfactory treatment by explicit-solvent molecular simulations.

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.

KECSA-Movable Type Implicit Solvation Model (KMTISM)

Computation of the solvation free energy for chemical and biological processes has long been of significant interest. The key challenges to effective solvation modeling center on the choice of potential function and configurational sampling. Herein, an energy sampling approach termed the “Movable Type” (MT) method, and a statistical energy function for solvation modeling, “Knowledge-based and Empirical Combined Scoring Algorithm” (KECSA) are developed and utilized to create an implicit solvation model: KECSA-Movable Type Implicit Solvation Model (KMTISM) suitable for the study of chemical and biological systems. KMTISM is an implicit solvation model, but the MT method performs energy sampling at the atom pairwise level. For a specific molecular system, the MT method collects energies from prebuilt databases for the requisite atom pairs at all relevant distance ranges, which by its very construction encodes all possible molecular configurations simultaneously. Unlike traditional statistical energy functions, KECSA converts structural statistical information into categorized atom pairwise interaction energies as a function of the radial distance instead of a mean force energy function. Within the implicit solvent model approximation, aqueous solvation free energies are then obtained from the NVT ensemble partition function generated by the MT method. Validation is performed against several subsets selected from the Minnesota Solvation Database v2012. Results are compared with several solvation free energy calculation methods, including a one-to-one comparison against two commonly used classical implicit solvation models: MM-GBSA and MM-PBSA. Comparison against a quantum mechanics based polarizable continuum model is also discussed (Cramer and Truhlar’s Solvation Model 12).

Structural determinants in bacterial 2-keto-3-deoxy-D-gluconate dehydrogenase KduD for dual-coenzyme specificity

Short-chain dehydrogenase/reductase (SDR) is distributed in many organisms, from bacteria to humans, and has significant roles in metabolism of carbohydrates, lipids, amino acids, and other biomolecules. An important intermediate in acidic polysaccharide metabolism is 2-keto-3-deoxy-d-gluconate (KDG). Recently, two short and long loops in Sphingomonas KDG-producing SDR enzymes (NADPH-dependent A1-R and NADH-dependent A1-R') involved in alginate metabolism were shown to be crucial for NADPH or NADH coenzyme specificity. Two SDR family enzymes-KduD from Pectobacterium carotovorum (PcaKduD) and DhuD from Streptococcus pyogenes (SpyDhuD)-prefer NADH as coenzyme, although only PcaKduD can utilize both NADPH and NADH. Both enzymes reduce 2,5-diketo-3-deoxy-d-gluconate to produce KDG. Tertiary and quaternary structures of SpyDhuD and PcaKduD and its complex with NADH were determined at high resolution (approximately 1.6 Å) by X-ray crystallography. Both PcaKduD and SpyDhuD consist of a three-layered structure, α/β/α, with a coenzyme-binding site in the Rossmann fold; similar to enzymes A1-R and A1-R', both arrange the two short and long loops close to the coenzyme-binding site. The primary structures of the two loops in PcaKduD and SpyDhuD were similar to those in A1-R' but not A1-R. Charge neutrality and moderate space at the binding site of the nucleoside ribose 2' coenzyme region were determined to be structurally crucial for dual-coenzyme specificity in PcaKduD by structural comparison of the NADH- and NADPH-specific SDR enzymes. The corresponding site in SpyDhuD was negatively charged and spatially shallow. This is the first reported study on structural determinants in SDR family KduD related to dual-coenzyme specificity. Proteins 2016; 84:934-947. © 2016 Wiley Periodicals, Inc.

SDPBS Web Server for Calculation of Electrostatics of Ionic Solvated Biomolecules

The Poisson-Boltzmann equation (PBE) is one important implicit solvent continuum model for calculating electrostatics of protein in ionic solvent. We recently developed a PBE solver library, called SDPBS, that incorporates the finite element, finite difference, solution decomposition, domain decomposition, and multigrid methods. To make SDPBS more accessible to the scientific community, we present an SDPBS web server in this paper that allows clients to visualize and manipulate the molecular structure of a biomolecule, and to calculate PBE solutions in a remote and user friendly fashion. The web server is available on the website https://lsextrnprod.uwm.edu/electrostatics/.

Hybrid peptide ATCUN-sh-Buforin: Influence of the ATCUN charge and stereochemistry on antimicrobial activity

The emergence of antibiotic resistant strains of bacteria has resulted in the need to develop more potent antimicrobials that target microorganisms in a novel manner. Antimicrobial Peptides (AMPs) show great potential for drug development because of their broad activity and unique mechanism of action. Several AMPs contain an Amino Terminal Copper and Nickel (ATCUN) binding motif; however, its function has not yet been determined. We have previously demonstrated that the activity of a truncated version of Buforin II (sh-Buforin, RAGLQFPVGRVHRLLRK-NH2) increases by the addition of an ATCUN motif. We now focus our current studies on understanding the effect of: 1) a positively charged ATCUN sequence, and 2) l-to-d amino acid substitution on the hybrid peptides. We identified that the addition of a positively charged ATCUN motif increases the affinity of the ATCUN-AMP for DNA but does not always result in an enhanced antimicrobial activity over a neutral ATCUN motif. The all-d peptides exhibited up to a 32-fold increase in antimicrobial activity compared to the all-l peptides. The larger activity of the all-d peptides is the result of a larger DNA cleavage activity and higher stability towards proteolysis. Cytotoxicity assays determined that, at their MIC, these peptides caused less than 8% hemolysis and, at 128 μM, no toxicity to HeLa and HEK293 cell lines. These results indicate that the ATCUN-AMP hybrids are an attractive alternative for treating infectious diseases and provide key insights into the role of the ATCUN motif in naturally-occurring AMPs.

TBI server: a web server for predicting ion effects in RNA folding

Background

Metal ions play a critical role in the stabilization of RNA structures. Therefore, accurate prediction of the ion effects in RNA folding can have a far-reaching impact on our understanding of RNA structure and function. Multivalent ions, especially Mg²⁺, are essential for RNA tertiary structure formation. These ions can possibly become strongly correlated in the close vicinity of RNA surface. Most of the currently available software packages, which have widespread success in predicting ion effects in biomolecular systems, however, do not explicitly account for the ion correlation effect. Therefore, it is important to develop a software package/web server for the prediction of ion electrostatics in RNA folding by including ion correlation effects.

Results

The TBI web server http://rna.physics.missouri.edu/tbi_index.html provides predictions for the total electrostatic free energy, the different free energy components, and the mean number and the most probable distributions of the bound ions. A novel feature of the TBI server is its ability to account for ion correlation and ion distribution fluctuation effects.

Conclusions

By accounting for the ion correlation and fluctuation effects, the TBI server is a unique online tool for computing ion-mediated electrostatic properties for given RNA structures. The results can provide important data for in-depth analysis for ion effects in RNA folding including the ion-dependence of folding stability, ion uptake in the folding process, and the interplay between the different energetic components.

Performance comparison of Poisson–Boltzmann equation solvers DelPhi and PBSA in calculation of electrostatic solvation energies

Many Poisson–Boltzmann equation (PBE) solvers have been developed to calculate electrostatic energy of biomolecules, and each PBE solver has its advantages. In this study two PBE solvers — DelPhi and PBSA module in AMBER — are compared in terms of calculation results, convergence and program-running time by calculating the electrostatic solvation energy for a test set composed of 4 Kirkwood models and another test set of 25 protein structures. The protein structures were pretreated by AMBER and AMBER99SB force field parameters were used in all calculations. It is found that (i) At fine grids, both PBE solvers can produce accurate results on test set 1 and consistent results with each other on test set 2, with differences between each other varying from several to ~ 10 kcal/mol. (ii) Under convergence criterion "absolute value of relative error is less than or equal to 2% (|RE| ≤ 2%)", both PBE solvers need very fine grids to produce convergent results on small and complex Kirkwood models, while grid spacings of ≤ 0.5 Å–0.6 Å are well enough for them to achieve good convergent results on various molecular structures. We recommend users to adopt such grid spacing in using of these PBE solvers so as to get good enough convergent results. (iii) In terms of time consumption, DelPhi appears to be more time-saving than PBSA. In summary, according to our comparison, DelPhi and PBSA are paralleled good PBE solvers. The convergence of PBSA is a little better than DelPhi, while DelPhi exceeds PBSA in running speed.

Biological applications of classical electrostatics methods

Continuum electrostatics modeling of solvation based on the Poisson–Boltzmann (PB) equation has gained wide acceptance in biomolecular applications such as energetic analysis and structural visualization. Successful application of the PB solvent models requires careful calibration of the solvation parameters. Extensive testing and validation is also important to ensure accuracy in their applications. Limitation in the continuum modeling of solvation is also a known issue in certain biomolecular applications. Growing interest in membrane systems has further spurred developmental efforts to allow inclusion of membrane in the PB solvent models. Despite their past successes due to careful parameterization, algorithm development and parallel implementation, there is still much to be done to improve their transferability from the small molecular systems upon which they were developed and validated to complex macromolecular systems as advances in technology continue to push forward, providing ever greater computational resources to researchers to study more interesting biological systems of higher complexity.

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.

Improved Coarse-Grained Modeling of Cholesterol-Containing Lipid Bilayers

Cholesterol trafficking, which is an essential function in mammalian cells, is intimately connected to molecular-scale interactions through cholesterol modulation of membrane structure and dynamics and interaction with membrane receptors. Since these effects of cholesterol occur on micro- to millisecond timescales, it is essential to develop accurate coarse-grained simulation models that can reach these timescales. Cholesterol has been shown experimentally to thicken the membrane and increase phospholipid tail order between 0-40% cholesterol, above which these effects plateau or slightly decrease. Here, we showed that the published MARTINI coarse-grained force-field for phospholipid (POPC) and cholesterol fails to capture these effects. Using reference atomistic simulations, we systematically modified POPC and cholesterol bonded parameters in MARTINI to improve its performance. We showed that the corrections to pseudo-bond angles between glycerol and the lipid tails and around the oleoyl double bond particle (the "angle-corrected model") slightly improves the agreement of MARTINI with experimentally measured thermal, elastic, and dynamic properties of POPC membranes. The angle-corrected model improves prediction of the thickening and ordering effects up to 40% cholesterol but overestimates these effects at higher cholesterol concentration. In accordance with prior work that showed the cholesterol rough face methyl groups are important for limiting cholesterol self-association, we revised the coarse-grained representation of these methyl groups to better match cholesterol-cholesterol radial distribution functions from atomistic simulations. In addition, by using a finer-grained representation of the branched cholesterol tail than MARTINI, we improved predictions of lipid tail order and bilayer thickness across a wide range of concentrations. Finally, transferability testing shows that a model incorporating our revised parameters into DOPC outperforms other CG models in a DOPC/cholesterol simulation series, which further argues for its efficacy and generalizability. These results argue for the importance of systematic optimization for coarse-graining biologically important molecules like cholesterol with complicated molecular structure.

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.

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.

Understanding photosynthetic light-harvesting: a bottom up theoretical approach

We discuss a bottom up approach for modeling photosynthetic light-harvesting. Methods are reviewed for a full structure-based parameterization of the Hamiltonian of pigment-protein complexes (PPCs). These parameters comprise (i) the local transition energies of the pigments in their binding sites in the protein, the site energies; (ii) the couplings between optical transitions of the pigments, the excitonic couplings; and (iii) the spectral density characterizing the dynamic modulation of pigment transition energies and excitonic couplings by protein vibrations. Starting with quantum mechanics perturbation theory, we provide a microscopic foundation for the standard PPC Hamiltonian and relate the expressions obtained for its matrix elements to quantities that can be calculated with classical molecular mechanics/electrostatics approaches including the whole PPC in atomic detail and using charge and transition densities obtained with quantum chemical calculations on the isolated building blocks of the PPC. In the second part of this perspective, the Hamiltonian is utilized to describe the quantum dynamics of excitons. Situations are discussed that differ in the relative strength of excitonic and exciton-vibrational coupling. The predictive power of the approaches is demonstrated in application to different PPCs, and challenges for future work are outlined.

Numerical Poisson-Boltzmann Model for Continuum Membrane Systems

Membrane protein systems are important computational research topics due to their roles in rational drug design. In this study, we developed a continuum membrane model utilizing a level set formulation under the numerical Poisson-Boltzmann framework within the AMBER molecular mechanics suite for applications such as protein-ligand binding affinity and docking pose predictions. Two numerical solvers were adapted for periodic systems to alleviate possible edge effects. Validation on systems ranging from organic molecules to membrane proteins up to 200 residues, demonstrated good numerical properties. This lays foundations for sophisticated models with variable dielectric treatments and second-order accurate modeling of solvation interactions.

Exploring a charge-central strategy in the solution of Poisson's equation for biomolecular applications

Continuum solvent treatments based on the Poisson-Boltzmann equation have been widely accepted for energetic analysis of biomolecular systems. In these approaches, the molecular solute is treated as a low dielectric region and the solvent is treated as a high dielectric continuum. The existence of a sharp dielectric jump at the solute-solvent interface poses a challenge to model the solvation energetics accurately with such a simple mathematical model. In this study, we explored and evaluated a strategy based on the "induced surface charge" to eliminate the dielectric jump within the finite-difference discretization scheme. In addition to the use of the induced surface charges in solving the equation, the second-order accurate immersed interface method is also incorporated to discretize the equation. The resultant linear system is solved with the GMRES algorithm to explicitly impose the flux conservation condition across the solvent-solute interface. The new strategy was evaluated on both analytical and realistic biomolecular systems. The numerical tests demonstrate the feasibility of utilizing induced surface charge in the finite-difference solution of the Poisson-Boltzmann equation. The analysis data further show that the strategy is consistent with theory and the classical finite-difference method on the tested systems. Limitations of the current implementations and further improvements are also analyzed and discussed to fully bring out its potential of achieving higher numerical accuracy.

Biomolecular electrostatics and solvation: a computational perspective

An understanding of molecular interactions is essential for insight into biological systems at the molecular scale. Among the various components of molecular interactions, electrostatics are of special importance because of their long-range nature and their influence on polar or charged molecules, including water, aqueous ions, proteins, nucleic acids, carbohydrates, and membrane lipids. In particular, robust models of electrostatic interactions are essential for understanding the solvation properties of biomolecules and the effects of solvation upon biomolecular folding, binding, enzyme catalysis, and dynamics. Electrostatics, therefore, are of central importance to understanding biomolecular structure and modeling interactions within and among biological molecules. This review discusses the solvation of biomolecules with a computational biophysics view toward describing the phenomenon. While our main focus lies on the computational aspect of the models, we provide an overview of the basic elements of biomolecular solvation (e.g. solvent structure, polarization, ion binding, and non-polar behavior) in order to provide a background to understand the different types of solvation models.

Dielectric pressure in continuum electrostatic solvation of biomolecules

Continuum solvation representations based on the Poisson-Boltzmann equation have become widely accepted in biomolecular applications after years of basic research and development. Since analytical solution of the differential equation can be achieved only in a few specific cases with simple solute geometry, only numerical solution is possible for biomolecular applications. However, it is conceptually difficult to assign solvation forces in the numerical methods, limiting their applications into direct simulations of energy minimization and molecular dynamics. In this study a dielectric pressure formulation was derived from the general Maxwell stress tensor for continuum solvation of biomolecules modeled with the widely used abrupt-transitioned dielectrics. A charge-central strategy was then proposed to improve the numerical behavior of the computed pressure. An interesting observation is the highly similar charge-central formulations between the smooth-transition dielectric and the abrupt-transition dielectric models utilized in the biomolecular solvation treatments. The connections of the new formulation with both the Davis-McCammon and Gilson et al. approaches were further presented after applying the normal field approximation. The consistency was verified with the numerical tests on a realistic biomolecule. The numerical experiments on the tested biomolecule further indicate that the charge-central strategy combined with the normal field approximation not only improves the accuracy of the dielectric boundary force but also reduces its grid dependence for biomolecular applications.