Specific ion effects on the electrophoretic mobility of small,highly charged peptides: A modeling study

In this work, we use coarse‐grained modeling to study the free solution electrophoretic mobility of small highly charged peptides (lysine, arginine, and short oligos thereof (up to nonapeptides)) in NaCl and Na₂SO₄ aqueous solutions at neutral pH and room temperature. The experimental data are taken from the literature. A bead modeling methodology that treats the electrostatics at the level of the nonlinear Poisson Boltzmann equation developed previously in our laboratory is able to account for the mobility of all peptides in NaCl, but not Na₂SO₄. The peptide mobilities in Na₂SO₄ can be accounted for by including sulfate binding in the model and this is proposed as one possible explanation for the discrepancy. Oligo arginine peptides bind more sulfate than oligo lysines and sulfate binding increases with the oligo length.     

Modeling the electrophoresis of oligolysines

CE is used to measure the electrophoretic mobility of low molecular mass oligo-L-lysines (n=1-8) in aqueous LiH?PO? buffer, BGE, at pH 2.5 over a range of temperatures (25-50 °C) and ionic strengths (10-100 mM). Mobilities are corrected for Joule heating and under the conditions of the experiment, interaction of the peptides with the capillary walls can be ignored. A "coarse grained" bead modeling methodology (BMM) (H. Pei et al., J. Chromatogr. A 2009, 1216, 1908-1916) is used to model the mobilities. This model partially accounts for peptide conformation as well as the assumed form of its secondary structure. For highly charged oligolysines, it is necessary to properly account for the relaxation effect. In the present study, the BMM approach tends to overestimate oligolysine mobility and that effect tends to increase with increasing ionic strength and peptide length. It is proposed that association between the oligolysines and buffer components (H?PO?? in this case) that go beyond classical electrostatic interactions are responsible for this discrepancy. A simple binding model is introduced that illustrates how this association can reconcile model and experiment.     

Recent advances in the electric double layer in colloid science

The interaction between charged colloidal particles is mediated by their electric double layers. Given that pairs of like-charged particles experience a repulsion, why do some dilute colloidal dispersions become unstable and condense at low ionic strengths? This puzzling paradox appears to have been largely resolved over the past year by a careful analysis of all the contributions to the thermodynamic potential of the dispersion. Condensation can be predicted using the traditional pair repulsion of the Poisson–Boltzmann theory without invoking any long-range attractions in the pair potential. However, it has emerged that one has to go beyond the Poisson–Boltzmann theory to account for the instability that occurs in confined colloidal dispersions. Other recent advances in the ubiquitous Poisson–Boltzmann theory have included effective surface charge approaches in calculating the electrokinetic zeta potential, and the modelling of charge regulation in colloidal systems.     

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.     

The Calculation of Surface Potential for Spherical Ionic Micelles

In this article, an approximate expression for the calculation of surface potential for spherical ionic micelles has been presented. This simple analytic form overcomes the complexity of original theory. The calculated values of surface potential of spherical ionic micelles are in quite good agreement with the exact numerical values of nonlinear Poisson‐Boltzmann equation.     

Multi-multigrid solution of modified Poisson–Boltzmann equation for arbitrarily shaped molecules

A new multi-multigrid method is presented for solving the modified Poisson–Boltzmann equation based on the Kirkwood Hierarchy of equations, with Loeb's closure, on a three-dimensional grid. The results are compared with standard Poisson–Boltzmann calculations, which are known to underestimate the local concentration of counterions near charged parts of molecules, mainly due to neglect of fluctuations in the ionic concentrations. In the present study, the Kirkwood hierarchy of equations is discretized with the finite volume method and solved using multigrid techniques. The new possibility for solution of the three-dimensional modified Poisson–Boltzmann equation, for the first time within a model including a dielectric discontinuity, and within reasonable computational time, enables the calculation of higher valence ion distributions around arbitrarily shaped biological macromolecules. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 893–901, 1998     

Modeling electrokinetics in ionic liquids

Using direct numerical simulations, we provide a thorough study regarding the electrokinetics of ionic liquids. In particular, modified Poisson–Nernst–Planck equations are solved to capture the crowding and overscreening effects characteristic of an ionic liquid. For modeling electrokinetic flows in an ionic liquid, the modified Poisson‐Nernst‐Planck equations are coupled with Navier–Stokes equations to study the coupling of ion transport, hydrodynamics, and electrostatic forces. Specifically, we consider the ion transport between two parallel charged surfaces, charging dynamics in a nanopore, capacitance of electric double‐layer capacitors, electroosmotic flow in a nanochannel, electroconvective instability on a plane ion‐selective surface, and electroconvective flow on a curved ion‐selective surface. We also discuss how crowding and overscreening and their interplay affect the electrokinetic behaviors of ionic liquids in these application problems.     

Ion binding to polyelectrolytes: Monte Carlo simulations versus classical mean field theories

The influence of ion size and surface charge model in titrations of ionizable polyelectrolytes is studied by means of the Semi Grand Canonical Monte Carlo simulation method in the context of the primitive model. Three models describing a discrete distribution of charged functional groups on the polyelectrolyte and different values for the radius of the background electrolyte spanning from ionic to hydrated radii values were analyzed. The polyelectrolyte titrations were simulated by calculating the degree of ionization versus pH curves at two ionic strengths. The results allow us to quantify the impact of the sizes of the background salt ions and surface functional groups of the polyelectrolyte on the dissociation degree. This influence is explained in terms of the effectiveness of the screening of the charged surface sites. Finally, by comparison with the Non-Linear Poisson–Boltzmann model, the influence of ionic correlations and finite size of the solution ions is assessed.     

Overcharging in Colloids: Beyond the Poisson–Boltzmann Approach

A broad range of manufactured products and biological fluids are colloids. The ability to understand and control the processes (of scientific, technological and industrial interest) in which such colloids are involved relies upon a precise knowledge of the electrical double layer. The traditional approach to describing this ion cloud around colloidal particles has been the Gouy–Chapman model, developed on the basis of the Poisson–Boltzmann equation. Since the early 1980s, however, more sophisticated theoretical treatments have revealed both quantitative and qualitative deficiencies in the Poisson–Boltzmann theory, particularly at high ionic strengths and/or high surface charge densities. This review deals with these novel approaches, which are mostly computer simulations and approximate integral equation theories based on the so‐called primitive model. Special attention is paid to phenomena that cannot be accounted for by the classic theory as a result of neglecting ion size correlations, such as overcharging, namely, the counterion concentration in the immediate neighborhood of the surface is so large that the particle surface is overcompensated. Other illustrative examples are the nonmonotonic behavior of the electrostatic potential and attractive interactions between equally charged surfaces. These predictions are certainly remarkable and, on paper, they can have an effect on experimentally measurable quantities (for instance, electrophoretic mobility). Even so, these new approaches have scarcely been applied in practice. Thus a critical survey on the relevance of ion size correlation in real systems is also included. Overcharging of macroions can also be brought about by adsorption of oppositely charged polyelectrolytes. Noteworthy examples and theoretical approaches for them are also briefly reviewed.     

Streptococcus mutans and Streptococcus intermedius adhesion to fibronectin films are oppositely influenced by ionic strength

Bacterial adhesion to protein-coated surfaces is mediated by an interplay of specific and nonspecific interactions. Although nonspecific interactions are ubiquitously present, little is known about the physicochemical mechanisms of specific interactions. The aim of this paper is to determine the influence of ionic strength on the adhesion of two streptococcal strains to fibronectin films. Streptococcus mutans LT11 and Streptococcus intermedius NCTC11324 both possess antigen I/II with the ability to bind fibronectin from solution, but S. intermedius binds approximately 20x less fibronectin than does the S. mutans strain under identical conditions. Both strains as well as fibronectin films are negatively charged in low ionic strength phosphate buffered saline (PBS, 10x diluted), but bacteria appear uncharged in high ionic strength PBS. Physicochemical modeling on the basis of overall cell surface properties (cell surface hydrophobicity and zeta potentials) demonstrates that both strains should favor adhesion to fibronectin films in a high ionic strength environment as compared to in a low ionic strength environment, where electrostatic repulsion between equally charged surfaces is dominant. Adhesion of S. intermedius to fibronectin films in a parallel plate flow chamber was completely in line with this modeling, while in addition atomic force microscopy (AFM) indicated stronger adhesion forces upon retraction between fibronectin-coated tips and the cell surfaces in high ionic strength PBS than in low ionic strength PBS. Thus, the dependence of the interaction on ionic strength is dominated by the overall negative charge on the interacting surfaces. Adhesion of S. mutans to fibronectin films, however, was completely at odds with theoretical modeling, and the strain adhered best in low ionic strength PBS. Moreover, AFM indicated weaker repulsive forces upon approach between fibronectin-coated tips and the cell surfaces in low ionic strength PBS than in high ionic strength PBS. This indicated that the dependence of the interaction on ionic strength is dominated by electrostatic attraction between oppositely charged, localized domains on the interacting surfaces, despite their overall negative charge. In summary, this study shows that physicochemical modeling of bacterial adhesion to protein-coated surfaces is only valid provided the number of specific interaction sites on the cell surfaces is low, such as on S. intermedius NCTC11324. Nonspecific interactions are dominated by specific interactions if the number of specific interaction sites is large, such as on S. mutans LT11. Its ionic strength dependence indicates that the specific interaction is electrostatic in nature and operative between oppositely charged domains on the interacting surfaces, despite the generally overall negatively charged character of the surfaces.     

The Calculation of Interfacial Tension and Electrostatic Free Energy of Spherical Ionic Micelles with High Surface Potentials

Based on extended Langmuir's method on the dressed micelles, approximate expressions for the calculation of interfacial tension and electrostatic free energy of spherical ionic micelles with high surface potentials have been presented. These expressions are derived from nonlinear Poisson‐Boltzmann equation. The present formulae for the calculation of interfacial tension and electrostatic free energy of spherical ionic micelles are in quite good agreement with Hayter's results.     

Reversed Hofmeister series—The rule rather than the exception

Over recent years, the supposedly universal Hofmeister series has been replaced by a diverse spectrum of direct, partially altered and reversed series. This review aims to provide a detailed understanding of the full spectrum by combining results from molecular dynamics simulations, Poisson–Boltzmann theory and AFM experiments. Primary insight into the origin of the Hofmeister series and its reversal is gained from simulation-derived ion–surface interaction potentials at surfaces containing non-polar, polar and charged functional groups for halide anions and alkali cations. In a second step, the detailed microscopic interactions of ions, water and functional surface groups are incorporated into Poisson–Boltzmann theory. This allows us to quantify ion-specific binding affinities to surface groups of varying polarity and charge, and to provide a connection to the experimentally measured long-ranged electrostatic forces that stabilize colloids, proteins and other particles against precipitation. Based on the stabilizing efficiency, the direct Hofmeister series is obtained for negatively charged hydrophobic surfaces. Hofmeister series reversal is induced by changing the sign of the surface charge from negative to positive, by changing the nature of the functional surface groups from hydrophobic to hydrophilic, by increasing the salt concentration, or by changing the pH. The resulting diverse spectrum reflects that alterations of Hofmeister series are the rule rather than the exception and originate from the variation of ion-surface interactions upon changing surface properties.     

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.     

电毛细管中非线性PB积分方程及其数值解

依据电毛细管非线性Ｐｏｉｓｓｏｎ Ｂｏｌｔｚｍａｎｎ微分方程的物理原理,导出其积分形式的ＰＢ方程．并采用数值迭代法给出相应方程的数值解．数值计算只用到电势Ψ的离散值,不需要Ψ的导数值,从根本上解决了因电势在管壁陡然变化引起数值解法的困难．文中给出的计算实例表明该算法是正确的、有效的和高精度的（相对误差小于０．０１％）,且在ＰＣ机上容易实现．     

New method for the computation of ionic distribution around rod-like polyelectrolytes with the helical distribution of charges. I. General approach and a nonlinearized Poisson–Boltzmann equation

This article is a first step in an attempt to reevaluate the relative role of different contributions to the energetics of DNA in salt solutions. To identify individual terms yielding such contributions a new derivation is given of the generalized Poisson–Boltzmann equation, which includes correlation effects, and explicitly shows terms ignored in the regular Poisson–Boltzmann approach. A general method based on the Boundary Element Technique is discussed, which can be used to evaluate these terms in the next steps of the reevaluation. An implementation of this method for the solution of the nonlinear Poisson–Boltzmann equation is described in detail, and is used to compute the ionic atmosphere around DNAs modeled as cylinders with helical distributions of charges. In the B-type DNA models, it is found that the ion densities in the minor and major grooves near the DNA surface differ by up to threefold. This difference is ca. 10-fold for Z-type DNA models. There are 20–25% differences in the magnitude of the maximum ionic charge density between DNA models of the same type. The addition of excess salt (up to 0.15 M) changes this maximum by only 10–15%. This change is not proportional to the concentration of excess salt. The contributions of different factors to the stabilization of alternative forms of DNA are evaluated. These factors are: (1) interactions between the phosphates, (2) interactions of phosphates with water, (3) interactions of phosphates with the ionic cloud, (4) interactions within the ionic cloud, (5) entropy of the ionic cloud. It is found that regardless of large variations in the counterion distributions around different DNAs, energetic contributions from these distributions are similar (?12.65 ± 0.6 kcal/mol · cell). The calculated change in stabilization per unit cell of models of B and Z-type DNAs due to 0.15 M excess NaCl is only ?0.56 ± 0.02 kcal/mol, indicating no tendency toward B-Z transition in this concentration range. Significantly larger variations of the order of 10 kcal/mol per unit cell can result from factors 1–2. Possible effects of the realistic DNA-solvent boundaries on the energetics of DNA solutions are discussed.     

Lattice Poisson-Boltzmann simulations of electro-osmotic flows in microchannels

This paper presents the numerical results of electro-osmotic flows in micro- and nanofluidics using a lattice Poisson-Boltzmann method (LPBM) which combines a potential evolution method on discrete lattices to solve the nonlinear Poisson equation (lattice Poisson method) with a density evolution method on discrete lattices to solve the Boltzmann-BGK equation (lattice Boltzmann method). In an electrically driven osmotic flow field, the flow velocity increases with both the external electrical field strength and the surface zeta potential for flows in a homogeneous channel. However, for a given electrical field strength and zeta potential, electrically driven flows have an optimal ionic concentration and an optimum width that maximize the flow velocity. For pressure-driven flows, the electro-viscosity effect increases with the surface zeta potential, but has an ionic concentration that yields the largest electro-viscosity effect. The zeta potential arrangement has little effect on the electro-viscosity for heterogeneous channels. For flows driven by both an electrical force and a pressure gradient, various zeta potential arrangements were considered for maximize the mixing enhancement with a less energy dissipation.     

Computer simulation study on the concentration distribution of spherical colloids within confined spaces of well‐defined pores

Aside from the virial expansion and density functional methods, theoretical results on the concentration partitioning behavior for charged colloids within cylindrical pores have not been presented so far. With the increase of relative solute size as well as solute concentration, however, the approximate analytic methods have proven to be unreliable. A suitable Monte Carlo simulation, which is proved as a rigorous technique for concentrated colloids, has been applied in the present study. The concentration profiles within the pore representing the effects of solute concentration as well as solution ionic strength are obtained via a stochastic process, from which the partition coefficient is estimated. Previously developed analyses on the linearized Poisson‐Boltzmann (P‐B) equation are employed for the estimation of long‐range electrostatic interaction. Both the singularity method and the analytical solution with series representation properly determine respective interaction energies between pairs of solute particles and between the solute particle and the pore wall. The effect of solute‐solute and solute‐wall interactions associated with repulsive energy is presented on the partitioning of colloids. Simulation results show that the partition coefficient is evidently enhanced when no particle‐wall interaction exists. Hindered diffusion can be predicted by the simplifying assumption of the centerline approximation analogy, where a dependence on the solute concentration becomes greater as the solution ionic strength decreases.     

Electrophoresis of soft particles with hydrophobic inner core grafted with pH-regulated and highly charged polyelectrolyte layer

Electrophoresis of core–shell composite soft particles possessing hydrophobic inner core grafted with highly charged polyelectrolyte layer (PEL) has been studied analytically. The PEL bears pH-dependent charge properties due to the presence of zwitterionic functional groups. The dielectric permittivity of the PEL and bulk aqueous medium were taken to be different, which resulted in the ion-partitioning effect. Objective of this study was to provide a simple expression for the mobility of such core–shell soft particles under Donnan limit where the thickness of the PEL well exceeds the electric double layer thickness. Going beyond the widely used Debye–Hückel linearization, the nonlinear Poisson–Boltzmann equation coupled with Stokes–Darcy–Brinkman equations was solved to determine the electrophoretic mobility. The derived expression further recovers all the existing results for the electrophoretic mobility under various simplified cases. The graphical presentation of the results illustrated the impact of pertinent parameters on the electrophoretic mobility of such a soft particle.     

Accuracy assessment of the linear Poisson–Boltzmann equation and reparametrization of the OBC generalized Born model for nucleic acids and nucleic acid–protein complexes

The generalized Born model in the Onufriev, Bashford, and Case (Onufriev et al., Proteins: Struct Funct Genet 2004, 55, 383) implementation has emerged as one of the best compromises between accuracy and speed of computation. For simulations of nucleic acids, however, a number of issues should be addressed: (1) the generalized Born model is based on a linear model and the linearization of the reference Poisson–Boltmann equation may be questioned for highly charged systems as nucleic acids; (2) although much attention has been given to potentials, solvation forces could be much less sensitive to linearization than the potentials; and (3) the accuracy of the Onufriev–Bashford–Case (OBC) model for nucleic acids depends on fine tuning of parameters. Here, we show that the linearization of the Poisson Boltzmann equation has mild effects on computed forces, and that with optimal choice of the OBC model parameters, solvation forces, essential for molecular dynamics simulations, agree well with those computed using the reference Poisson–Boltzmann model. © 2015 Wiley Periodicals, Inc.     

Fragment‐based quantum mechanical calculation of protein–protein binding affinities

The electrostatically embedded generalized molecular fractionation with conjugate caps (EE‐GMFCC) method has been successfully utilized for efficient linear‐scaling quantum mechanical (QM) calculation of protein energies. In this work, we applied the EE‐GMFCC method for calculation of binding affinity of Endonuclease colicin–immunity protein complex. The binding free energy changes between the wild‐type and mutants of the complex calculated by EE‐GMFCC are in good agreement with experimental results. The correlation coefficient (R) between the predicted binding energy changes and experimental values is 0.906 at the B3LYP/6‐31G*‐D level, based on the snapshot whose binding affinity is closest to the average result from the molecular mechanics/Poisson–Boltzmann surface area (MM/PBSA) calculation. The inclusion of the QM effects is important for accurate prediction of protein–protein binding affinities. Moreover, the self‐consistent calculation of PB solvation energy is required for accurate calculations of protein–protein binding free energies. This study demonstrates that the EE‐GMFCC method is capable of providing reliable prediction of relative binding affinities for protein–protein complexes. © 2018 Wiley Periodicals, Inc.