A new program for optimizing periodic boundary models of solvated biomolecules (PBCAID)

Simulations of solvated macromolecules often use periodic lattices to account for long-range electrostatics and to approximate the surface effects of bulk solvent. The large percentage of solvent molecules in such models (compared to macromolecular atoms) makes these procedures computationally expensive. The cost can be reduced by using periodic cells containing an optimized number of solvent molecules (subject to a minimal distance between the solute and the periodic images). We introduce an easy-to-use program "PBCAID" to initialize and optimize a periodic lattice specified as one of several known space-filling polyhedra. PBCAID reduces the volume of the periodic cell by finding the solute rotation that yields the smallest periodic cell dimensions. The algorithm examines rotations by using only a subset of surface atoms to measure solute/image distances, and by optimizing the distance between the solute and the periodic cell surface. Once the cell dimension is optimized, PBCAID incorporates a procedure for solvating the domain with water by filling the cell with a water lattice derived from an ice structure scaled to the bulk density of water. Results show that PBCAID can optimize system volumes by 20 to 70% and lead to computational savings in the nonbonded computations from reduced solvent sizes. Copyright 2001 John Wiley & Sons, Inc. J Comput Chem 22: 1843-1850, 2001     

Efficient and accurate solvation energy calculation from polarizable continuum models

A new approach is proposed to enhance the efficiency and accuracy for calculation of the long-range electrostatic interaction from implicit solvation models, i.e., the polarizable continuum model (PCM) and its variants, conductorlike PCM/conductorlike screening model and integral equation formalism PCM. In these methods the solvent electrostatics effects are represented by a set of discrete apparent charges distributed on tesserae of the molecular cavity surface embedding the solute. In principle, the accuracy of these methods is improved if the cavity surface is tessellated to finer tesserae; however, the computational time is increased rapidly. We show that such undesired dependency between accuracy and efficiency is a result of the inaccurate treatment of the apparent charge self-contribution to the potential and/or electric field. By taking into account the full effects due to the size and curvature of the segment occupied by each apparent charge, the error in calculated electrostatic solvation free energy is essentially zero for ions (point charge at the center of a sphere) regardless of the degree of tessellation used. For molecules where gradient of apparent charge density is nonzero at the cavity surface, we propose a multiple-sampling technique which significantly lowers the calculated error compared to the original PCM methods, especially when very few numbers of tesserae are used.     

Effect of a low-dielectric interior on DNA electrostatic response to twisting and bending

We study the effects of a low-dielectric core of rod-like macromolecules on their electrostatic persistence lengths. We use the exact solution of the linear Poisson-Boltzmann equation for the potential of a charge on the surface of a low-dielectric cylinder. We apply the results to the B-DNA molecule, modeled as a double helical array of discrete charges wound on the surface of a low-dielectric rod. For this charge geometry, we calculate the change in the electrostatic twist persistence as compared to DNA with a water-permeable core. We also discuss possible effects of the low-dielectric molecular core on DNA bending persistence.     

Electrostatic screening and energy barriers of ions in low-dielectric membranes

We present exact solutions of the linear Poisson-Boltzmann equation for several problems relevant for ion translocation across low-dielectric membranes. Our results are obtained for a finite Debye screening length, and they generalize the classical results for pure Coulombic electrostatics (Parsegian, A. Nature (London) 1969, 221, 844). We calculate the electrostatic self-energy of an ion in the middle of a low-dielectric slab, its energy inside a cylindrical high-dielectric pore, and its energy inside a high-dielectric spherical jacket. We consider also the influence of negative charges distributed on the walls of the cylindrical pore. We show that ion self-energy barriers are considerably reduced due to screening of electrolyte. We compare our results with some numerical results for screened electrostatics of ion channels and wide pores.     

On the transferability of hydration-parametrized continuum electrostatics models to solvated binding calculations

Using molecular mechanics force field partial atomic charges, we show the nonuniqueness of the parametrization of continuum electrostatics models with respect to solute atomic radii and interior dielectric constant based on hydration (vacuum-to-water transfer) free energy data available for small molecules. Moreover, parameter sets that are optimal and equivalent for hydration free energy calculations lead to large variations of calculated absolute and relative electrostatic binding free energies. Hence, parametrization of solvation effects based on hydration data, although a necessary condition, is not sufficient to guarantee its transferability to the calculation of binding free energies in solution.     

Continuous development of schemes for parallel computing of the electrostatics in biological systems: Implementation in DelPhi

Due to the enormous importance of electrostatics in molecular biology, calculating the electrostatic potential and corresponding energies has become a standard computational approach for the study of biomolecules and nano‐objects immersed in water and salt phase or other media. However, the electrostatics of large macromolecules and macromolecular complexes, including nano‐objects, may not be obtainable via explicit methods and even the standard continuum electrostatics methods may not be applicable due to high computational time and memory requirements. Here, we report further development of the parallelization scheme reported in our previous work (Li, et al., J. Comput. Chem. 2012, 33, 1960) to include parallelization of the molecular surface and energy calculations components of the algorithm. The parallelization scheme utilizes different approaches such as space domain parallelization, algorithmic parallelization, multithreading, and task scheduling, depending on the quantity being calculated. This allows for efficient use of the computing resources of the corresponding computer cluster. The parallelization scheme is implemented in the popular software DelPhi and results in speedup of several folds. As a demonstration of the efficiency and capability of this methodology, the electrostatic potential, and electric field distributions are calculated for the bovine mitochondrial supercomplex illustrating their complex topology, which cannot be obtained by modeling the supercomplex components alone. © 2013 Wiley Periodicals, Inc.     

Modeling the acid-base surface chemistry of montmorillonite

Proton uptake on montmorillonite edge surfaces can control pore water pH, solute adsorption, dissolution kinetics and clay colloid behavior in engineered clay barriers and natural weathering environments. Knowledge of proton uptake reactions, however, is currently limited by strong discrepancies between reported montmorillonite titration data sets and by conflicting estimates of edge structure, reactivity and electrostatics. In the present study, we show that the apparent discrepancy between titration data sets results in large part from the widespread use of an erroneous assumption of zero specific net proton surface charge at the onset of titration. Using a novel simulation scheme involving a surface chemistry model to simulate both pretreatment and titration, we find that montmorillonite edge surface chemistry models that account for the "spillover" of electrostatic potential from basal onto edge surfaces and for the stabilization of deprotonated Al-Si bridging sites through bond-length relaxation at the edge surface can reproduce key features of the best available experimental titration data (the influence of pretreatment conditions on experimental results, the absence of a point of zero salt effect, buffer capacity in the acidic pH range). However, no combination of current models of edge surface structure, reactivity and electrostatics can quantitatively predict, without fitted parameters, the experimental titration data over the entire range of pH (4.5 to 9) and ionic strength (0.001 to 0.5 mol dm(-3)) covered by available data.     

Highly accurate biomolecular electrostatics in continuum dielectric environments

Implicit solvent models based on the Poisson-Boltzmann (PB) equation are frequently used to describe the interactions of a biomolecule with its dielectric continuum environment. A novel, highly accurate Poisson-Boltzmann solver is developed based on the matched interface and boundary (MIB) method, which rigorously enforces the continuity conditions of both the electrostatic potential and its flux at the molecular surface. The MIB based PB solver attains much better convergence rates as a function of mesh size compared to conventional finite difference and finite element based PB solvers. Consequently, highly accurate electrostatic potentials and solvation energies are obtained at coarse mesh sizes. In the context of biomolecular electrostatic calculations it is demonstrated that the MIB method generates substantially more accurate solutions of the PB equation than other established methods, thus providing a new level of reference values for such models. Initial results also indicate that the MIB method can significantly improve the quality of electrostatic surface potentials of biomolecules that are frequently used in the study of biomolecular interactions based on experimental structures.     

Atomic multipoles and perpendicular electrostatic forces in diatomic and planar molecules

Current methods for assigning atomic multipoles focus on reproduction of the molecular electrostatic potential. Another aspect of electrostatic interaction, which is usually overlooked, is the forces that an external electric field exerts on the nuclei of a molecule. In a self-consistent theory, both the electrostatic potential and force should be accounted for. However, in general it is not easy to meet this requirement for the force. For planar molecules, though, a formal solution is available in terms of atomic multipoles that are extracted from the molecular multipolar tensors. These Force-Related (FR) atomic multipoles are discussed in detail for some typical diatomics and planar polyatomics, and are shown to provide a solid uniform framework for treating both aspects of the electrostatics. In contrast, the commonly used potential-derived charges (i.e., the atomic charges obtained by fitting the electrostatic potential) can yield large deviations with respect to electrostatic forces on the nuclei, even when the electrostatic potential is very well reproduced.     

Accurate solution of multi-region continuum biomolecule electrostatic problems using the linearized Poisson-Boltzmann equation with curved boundary elements

We present a boundary-element method (BEM) implementation for accurately solving problems in biomolecular electrostatics using the linearized Poisson-Boltzmann equation. Motivating this implementation is the desire to create a solver capable of precisely describing the geometries and topologies prevalent in continuum models of biological molecules. This implementation is enabled by the synthesis of four technologies developed or implemented specifically for this work. First, molecular and accessible surfaces used to describe dielectric and ion-exclusion boundaries were discretized with curved boundary elements that faithfully reproduce molecular geometries. Second, we avoided explicitly forming the dense BEM matrices and instead solved the linear systems with a preconditioned iterative method (GMRES), using a matrix compression algorithm (FFTSVD) to accelerate matrix-vector multiplication. Third, robust numerical integration methods were employed to accurately evaluate singular and near-singular integrals over the curved boundary elements. Fourth, we present a general boundary-integral approach capable of modeling an arbitrary number of embedded homogeneous dielectric regions with differing dielectric constants, possible salt treatment, and point charges. A comparison of the presented BEM implementation and standard finite-difference techniques demonstrates that for certain classes of electrostatic calculations, such as determining absolute electrostatic solvation and rigid-binding free energies, the improved convergence properties of the BEM approach can have a significant impact on computed energetics. We also demonstrate that the improved accuracy offered by the curved-element BEM is important when more sophisticated techniques, such as nonrigid-binding models, are used to compute the relative electrostatic effects of molecular modifications. In addition, we show that electrostatic calculations requiring multiple solves using the same molecular geometry, such as charge optimization or component analysis, can be computed to high accuracy using the presented BEM approach, in compute times comparable to traditional finite-difference methods.     

Microscopic and semimicroscopic calculations of electrostatic energies in proteins by the POLARIS and ENZYMIX programs

Different microscopic and semimicroscopic approaches for calculations of electrostatic energies in macromolecules are examined. This includes the Protein Dipoles Langevin Dipoles (PDLD) method, the semimicroscopic PDLD (PDLD/S) method, and a free energy perturbation (FEP) method. The incorporation of these approaches in the POLARIS and ENZYMIX modules of the MOLARIS package is described in detail. The PDLD electrostatic calculations are augmented by estimates of the relevant hydrophobic and steric contributions, as well as the effects of the ionic strength and external pH. Determination of the hydrophobic energy involves an approach that considers the modification of the effective surface area of the solute by local field effects. The steric contributions are analyzed in terms of the corresponding reorganization energies. Ionic strength effects are studied by modeling the ionic environment around the given system using a grid of residual charges and evaluating the relevant interaction using Coulomb's law with the dielectric constant of water. The performance of the FEP calculations is significantly enhanced by using special boundary conditions and evaluating the long-range electrostatic contributions using the Local Reaction Field (LRF) model. A diverse set of electrostatic effects are examined, including the solvation energies of charges in proteins and solutions, energetics of ion pairs in proteins and solutions, interaction between surface charges in proteins, and effect of ionic strength on such interactions, as well as electrostatic contributions to binding and catalysis in solvated proteins. Encouraging results are obtained by the microscopic and semimicroscopic approaches and the problems associated with some macroscopic models are illustrated. The PDLD and PDLD/S methods appear to be much faster than the FEP approach and still give reasonable results. In particular, the speed and simplicity of the PDLD/S method make it an effective strategy for calculations of electrostatic free energies in interactive docking studies. Nevertheless, comparing the results of the three approaches can provide a useful estimate of the accuracy of the calculated energies. © 1993 John Wiley & Sons, Inc.     

SMPBS: Web server for computing biomolecular electrostatics using finite element solvers of size modified Poisson‐Boltzmann equation

SMPBS (Size Modified Poisson‐Boltzmann Solvers) is a web server for computing biomolecular electrostatics using finite element solvers of the size modified Poisson‐Boltzmann equation (SMPBE). SMPBE not only reflects ionic size effects but also includes the classic Poisson‐Boltzmann equation (PBE) as a special case. Thus, its web server is expected to have a broader range of applications than a PBE web server. SMPBS is designed with a dynamic, mobile‐friendly user interface, and features easily accessible help text, asynchronous data submission, and an interactive, hardware‐accelerated molecular visualization viewer based on the 3Dmol.js library. In particular, the viewer allows computed electrostatics to be directly mapped onto an irregular triangular mesh of a molecular surface. Due to this functionality and the fast SMPBE finite element solvers, the web server is very efficient in the calculation and visualization of electrostatics. In addition, SMPBE is reconstructed using a new objective electrostatic free energy, clearly showing that the electrostatics and ionic concentrations predicted by SMPBE are optimal in the sense of minimizing the objective electrostatic free energy. SMPBS is available at the URL: smpbs.math.uwm.edu © 2017 Wiley Periodicals, Inc.     

Electric field-derived point charges to mimic the electrostatics in molecular crystals

Because of the way the electrostatic potential is defined in a crystal, it is not possible to determine potential-derived charges for atoms in a crystal. To overcome this limitation, we present a novel method for determining atomic charges for a molecule in a crystal based on a fit to the electric field at points on a surface around the molecule. Examples of fits to the electric field at points on a Hirshfeld surface, using crystal Hartree-Fock electron densities computed with a DZP basis set are presented for several organic molecular crystals. The field-derived charges for common functional groups are transferable, and reflect chemical functionality as well as the subtle effects of intermolecular interactions. The charges also yield an excellent approximation to the electric field surrounding a molecule in a crystal for use in cluster calculations on molecules in solids.     

Binding of aminoglycoside antibiotics to the small ribosomal subunit: a continuum electrostatics investigation

The binding of paromomycin and similar antibiotics to the small (30S) ribosomal subunit has been studied using continuum electrostatics methods. Crystallographic information from a complex of paromomycin with the 30S subunit was used as a framework to develop structures of similar antibiotics in the same ribosomal binding site. Total binding energies were calculated from electrostatic properties obtained by solution of the Poisson-Boltzmann equation combined with a surface area-dependent apolar term. These computed results showed good correlation with experimental data. Additionally, calculation of the ribosomal electrostatic potential in the paromomycin binding site provided insight into the electrostatic mechanisms for aminoglycoside binding and clues for the rational design of more effective antibiotics.     

Advances in electric field and atomic surface derived properties from experimental electron densities

The present study is devoted to a general use of the Gauss law. This is applied to the atomic surfaces derived from the topological analysis of the electron density. The method proposed here is entirely numerical, robust and does not necessitate any specific parametrization of the atomic surfaces. We focus on two fundamental properties: the atomic charges and the electrostatic forces acting on atoms in molecules. Application is made on experimental electron densities modelized by the Hansen-Coppens model from which the electric field is derived for a heterogenic set of compounds: water molecule, NO(3) anion, bis-triazine molecule and MgO cluster. Charges and electrostatic forces are estimated by the atomic surface flux of the electric field and the Maxwell stress tensor, respectively. The charges obtained from the present method are in good agreement with those issued from the conventional volume integration. Both Feynman and Ehrenfest forces as well as the electrostatic potential at the nuclei (EPN) are here estimated from the experimental electron densities. The values found for the molecular compounds are presented and discussed in the scope of the mechanics of atomic interactions.     

Treatment of geometric singularities in implicit solvent models

Geometric singularities, such as cusps and self-intersecting surfaces, are major obstacles to the accuracy, convergence, and stability of the numerical solution of the Poisson-Boltzmann (PB) equation. In earlier work, an interface technique based PB solver was developed using the matched interface and boundary (MIB) method, which explicitly enforces the flux jump condition at the solvent-solute interfaces and leads to highly accurate biomolecular electrostatics in continuum electric environments. However, such a PB solver, denoted as MIBPB-I, cannot maintain the designed second order convergence whenever there are geometric singularities, such as cusps and self-intersecting surfaces. Moreover, the matrix of the MIBPB-I is not optimally symmetrical, resulting in the convergence difficulty. The present work presents a new interface method based PB solver, denoted as MIBPB-II, to address the aforementioned problems. The present MIBPB-II solver is systematical and robust in treating geometric singularities and delivers second order convergence for arbitrarily complex molecular surfaces of proteins. A new procedure is introduced to make the MIBPB-II matrix optimally symmetrical and diagonally dominant. The MIBPB-II solver is extensively validated by the molecular surfaces of few-atom systems and a set of 24 proteins. Converged electrostatic potentials and solvation free energies are obtained at a coarse grid spacing of 0.5 A and are considerably more accurate than those obtained by the PBEQ and the APBS at finer grid spacings.