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Levy et al. [1] reported the breakdown of electroneutrality in confined nanopores embedded in a dielectric medium. A Robin boundary condition was derived which eliminates the need to include the dielectric medium explicitly when solving for the electric field within the nanopore. In this comment, we point out issues related to the approximations made during the derivation of the boundary condition. The errors caused by the use of this boundary condition can be significant even for nanochannels of large aspect (length to radius) ratio, a condition on which the approximations in Levy et al. [1] are based.

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Almost all biological reactions are pH dependent and understanding the origin of pH dependence requires knowledge of the pKa's of ionizable groups. Here we report a new edition of PKAD, the PKAD-2, which is a database of experimentally measured pKa's of proteins, both wild type and mutant proteins. The new additions include 117 wild type and 54 mutant pKa values, resulting in total 1742 experimentally measured pKa's. The new edition of PKAD-2 includes 8 new wild type and 12 new mutant proteins, resulting in total of 220 proteins. This new edition incorporates a visual 3D image of the highlighted residue of interest within the corresponding protein or protein complex. Hydrogen bonds were identified, counted, and implemented as a search feature. Other new search features include the number of neighboring residues <4A from the heaviest atom of the side chain of a given amino acid. Here, we present PKAD-2 with the intention to continuously incorporate novel features and current data with the goal to be used as benchmark for computational methods.

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In applications of bio-inspired nanoparticles (NPs), their composition is often optimised by including ionizable lipids. I use a generic statistical model to describe the charge and potential distributions in lipid nanoparticles (LNPs) containing such lipids. The LNP structure is considered to contain the biophase regions separated by narrow interphase boundaries with water. Ionizable lipids are uniformly distributed at the biophase-water boundaries. The potential is there described at the mean-filed level combining the Langmuir-Stern equation for ionizable lipids and the Poisson-Boltzmann equation for other charges in water. The latter equation is used outside a LNP as well. With physiologically reasonable parameters, the model predicts the scale of the potential in a LNP to be rather low, smaller or about [Formula: see text], and to change primarily near the LNP-solution interface or, more precisely, inside an NP near this interface because the charge of ionizable lipids becomes rapidly neutralized along the coordinate towards the center of a LNP. The extent of dissociation-mediated neutralization of ionizable lipids along this coordinate increases but only slightly. Thus, the neutralization is primarily due to the negative and positive ions related to the ionic strength in solution and located inside a LNP.

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The existence of an exclusion zone in which particles of a colloidal suspension in water are repelled from hydrophilic surfaces has been experimentally demonstrated in numerous studies, especially in the case of Nafion surfaces. Various explanations have been proposed for the origin of this phenomenon, which is not completely understood yet. In particular, the existence of a fourth phase of water has been proposed by G. Pollack and if this theory is proven correct, its implications on our understanding of the properties of water, especially in biological systems, would be profound and could give rise to new medical therapies. Here, a simple approach based on the linearized Poisson-Boltzmann equation is developed in order to study the repulsive forces mediated by ordered water and involving the following interacting biomolecules: 1) microtubule and a tubulin dimer, 2) two tubulin dimers and 3) a tubulin sheet and a tubulin dimer. The choice of microtubules in this study is motivated because they could be a good candidate for the generation of an exclusion zone in the cell and these models could be a starting point for detailed experimental investigations of this phenomenon.

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HYPOTHESIS: The origin of the negative charge at water/air interface was proved to be not only specific adsorption of OH- ions but that of HCO3- and/or CO32- ions in the previous study [1]. To determine which anionic species is primarily responsible for the surface charge, the surface density of ions in the Stern layer is numerically evaluated from foam-film thickness of aqueous solutions of NaHCO3 and Na2CO3. EXPERIMENTS: Equilibrium thickness (equivalent thickness at equilibrium) of the foam films formed from aqueous solutions of NaHCO3 and Na2CO3 was measured as a function of electrolyte concentration at 298.15 K. FINDINGS: Applying a modified Poisson-Boltzmann (PB) equation developed for various kinds of electrolytes to the equilibrium thickness gave the surface density of ions in the Stern layer for NaHCO3 and Na2CO3 systems. From the concentration dependence of the surface density together with that for NaCl and NaOH in the previous study [1], the negative surface charges for water and very dilute solutions were found to be due to specific adsorption of HCO3- ions. The surface charge at high electrolyte concentration is determined by the specific adsorption of electrolyte anions. The specific-adsorption ability of anion increases in the order CO32-≫ HCO3-> OH-≫ Cl-.

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A statistical theory is presented of the magnesium ion interacting with lysozyme under conditions where the latter is positively charged. Temporarily assuming magnesium is not noncovalently bound to the protein, I solve the nonlinear Poisson-Boltzmann equation accurately and uniformly in a perturbative fashion. The resulting expression for the effective charge, which is larger than nominal owing to overshooting, is subtle and cannot be asymptotically expanded at high ionic strengths that are practical. An adhesive potential taken from earlier work together with the assumption of possibly bound magnesium is then fitted to be in accord with measurements of the second virial coefficient by Tessier et al. The resulting numbers of bound magnesium ions as a function of MgBr[Formula: see text] concentration are entirely reasonable compared with densitometry measurements.

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An adaptive finite element solver for the numerical calculation of the electrostatic coupling between molecules in a solvent environment is developed and tested. At the heart of the solver is a goal-oriented a posteriori error estimate for the electrostatic coupling, derived and implemented in the present work, that gives rise to an orders of magnitude improved precision and a shorter computational time as compared to standard finite difference solvers. The accuracy of the new solver ARGOS is evaluated by numerical experiments on a series of problems with analytically known solutions. In addition, the solver is used to calculate electrostatic couplings between two chromophores, linked to polyproline helices of different lengths and between the spike protein of SARS-CoV-2 and the ACE2 receptor. All the calculations are repeated by using the well-known finite difference solvers MEAD and APBS, revealing the advantages of the present finite element solver.

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HYPOTHESIS: Some ions can prevent bubbles from coalescing in water. The Gibbs-Marangoni pressure has been proposed as an explanation of this phenomenon. This repulsive pressure occurs during thin film drainage whenever surface enhanced or surface depleted solutes are present. However, bubble coalescence inhibition is known to depend on which particular combination of ions are present in a peculiar and unexplained way. This dependence can be explained by the electrostatic surface potential created by the distribution of ions at the interface, which will alter the natural surface propensity of the ions and hence the Gibbs-Marangoni pressure. CALCULATIONS: A generalised form of the Gibbs-Marangoni pressure is derived for a mixture of solutes and the modified Poisson-Boltzmann equation is used to calculate this pressure for five different electrolyte solutions made up of four different ions. FINDINGS: Combining ions with differing surface propensities, i.e., one enhanced and one depleted, creates a significant electrostatic surface potential which dampens the natural surface propensity of these ions, resulting in a reduced Gibbs-Marangoni pressure, which allows bubble coalescence. This mechanism explains why the ability of electrolytes to inhibit bubble coalescence is correlated with surface tension for pure electrolytes but not for mixed electrolytes.

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Numerical treatment of singular charges is a grand challenge in solving the Poisson-Boltzmann (PB) equation for analyzing electrostatic interactions between the solute biomolecules and the surrounding solvent with ions. For diffuse interface PB models in which solute and solvent are separated by a smooth boundary, no effective algorithm for singular charges has been developed, because the fundamental solution with a space dependent dielectric function is intractable. In this work, a novel regularization formulation is proposed to capture the singularity analytically, which is the first of its kind for diffuse interface PB models. The success lies in a dual decomposition - besides decomposing the potential into Coulomb and reaction field components, the dielectric function is also split into a constant base plus space changing part. Using the constant dielectric base, the Coulomb potential is represented analytically via Green's functions. After removing the singularity, the reaction field potential satisfies a regularized PB equation with a smooth source. To validate the proposed regularization, a Gaussian convolution surface (GCS) is also introduced, which efficiently generates a diffuse interface for three-dimensional realistic biomolecules. The performance of the proposed regularization is examined by considering both analytical and GCS diffuse interfaces, and compared with the trilinear method. Moreover, the proposed GCS-regularization algorithm is validated by calculating electrostatic free energies for a set of proteins and by estimating salt affinities for seven protein complexes. The results are consistent with experimental data and estimates of sharp interface PB models.

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Monte Carlo (MC) methods are important computational tools for molecular structure optimizations and predictions. When solvent effects are explicitly considered, MC methods become very expensive due to the large degree of freedom associated with the water molecules and mobile ions. Alternatively implicit-solvent MC can largely reduce the computational cost by applying a mean field approximation to solvent effects and meanwhile maintains the atomic detail of the target molecule. The two most popular implicit-solvent models are the Poisson-Boltzmann (PB) model and the Generalized Born (GB) model in a way such that the GB model is an approximation to the PB model but is much faster in simulation time. In this work, we develop a machine learning-based implicit-solvent Monte Carlo (MLIMC) method by combining the advantages of both implicit solvent models in accuracy and efficiency. Specifically, the MLIMC method uses a fast and accurate PB-based machine learning (PBML) scheme to compute the electrostatic solvation free energy at each step. We validate our MLIMC method by using a benzene-water system and a protein-water system. We show that the proposed MLIMC method has great advantages in speed and accuracy for molecular structure optimization and prediction.

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Ultrafiltration/diafiltration (UF/DF) operations are employed for achieving the desired therapeutic monoclonal antibody (mAb) formulations. Due to electrostatic interactions between the charged proteins, solute ions, and uncharged excipients, the final pH and concentration values are not always equal to those in the DF buffer. At high protein concentrations, typical for industrial formulations, this effect becomes predominant. To account for challenges occurring in industrial environments, a robust mathematical framework enabling the prediction of pH and concentration profiles throughout the UF/DF process is provided. The proposed mechanistic model combines a macroscopic mass balance approach with a molecular approach based on a Poisson-Boltzmann equation dealing with electrostatic interactions and accounting for protein exclusion volume effect. The mathematical model was validated with experimental data of two commercially relevant mAbs obtained from an industrial UF/DF process using scalable laboratory equipment. The robustness and flexibility of the model were tested by using proteins with different isoelectric points and net charges. The latter was determined via a titration curve, enabling realistic protein charge-pH evaluation. In addition, the model was tested for different DF buffer types containing both monovalent and polyvalent ions, with various types of uncharged excipients. The model generality enables its implementation for the UF/DF processes of other protein varieties.

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DelPhi is a popular scientific program which numerically solves the Poisson-Boltzmann equation (PBE) for electrostatic potentials and energies of biomolecules immersed in water via finite difference method. It is well known for its accuracy, reliability, flexibility, and efficiency. In this work, a new edition of DelPhi that uses a novel Newton-like method to solve the nonlinear PBE, in addition to the already implemented Successive Over Relaxation (SOR) algorithm, is introduced. Our tests on various examples have shown that this new method is superior to the SOR method in terms of stability when solving the nonlinear PBE, being able to converge even for problems involving very strong nonlinearity.

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This review article intends to communicate the new understanding and viewpoints on two fundamental electrokinetics topics that have only become available recently. The first is on the holistic approach to the Poisson-Boltzmann equation that can account for the effects arising from the interaction between the mobile ions in the Debye layer and the surface charge. The second is on the physical picture of the inner electro-hydrodynamic flow field of an electrophoretic particle and its drag coefficient. For the first issue, the traditional Poisson-Boltzmann equation focuses only on the mobile ions in the Debye layer; effects such as charge regulation and the isoelectronic point arising from the interaction between the mobile ions in the Debye layer and the surface charge are left to supplemental measures. However, a holistic treatment is entirely possible in which the whole electrical double layer-the Debye layer and the surface charge-is treated consistently from the beginning. While the derived form of the Poisson-Boltzmann equation remains unchanged, the zeta potential boundary condition becomes a calculated quantity that can reflect the various effects due to the interaction between the surface charges and the mobile ions in the liquid. The second issue, regarding the drag coefficient of a spherical electrophoretic particle, has existed ever since the breakthrough by Smoluchowski a century ago that linked the zeta potential of the particle to its mobility. Due to the highly nonlinear mathematics involved in the electro-hydrodynamics inside the Debye layer, there has been a lack of an exact solution for the electrophoretic flow field. Recent numerical simulation results show that the flow field comprises an inner region and an outer region, separated by a rather sharp interface. As the inner flow field is carried along by the particle, the measured drag is that at the inner/outer interface rather than at the solid/liquid interface. This identification and its associated physical picture of the inner flow field resolves a long-standing puzzle regarding the electrophoretic drag coefficient.

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Accurate calculation of electrostatic potential and gradient on the molecular surface is highly desirable for the continuum and hybrid modeling of large scale deformation of biomolecules in solvent. In this article a new numerical method is proposed to calculate these quantities on the dielectric interface from the numerical solutions of the Poisson-Boltzmann equation. Our method reconstructs a potential field locally in the least square sense on the polynomial basis enriched with Green's functions, the latter characterize the Coulomb potential induced by charges near the position of reconstruction. This enrichment resembles the decomposition of electrostatic potential into singular Coulomb component and the regular reaction field in the Generalized Born methods. Numerical experiments demonstrate that the enrichment recovery produces drastically more accurate and stable potential gradients on molecular surfaces compared to classical recovery techniques.

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This chapter describes the current status of development of the fragment molecular orbital (FMO) method for analyzing the electronic state and intermolecular interactions of biomolecular systems in solvent. The orbital energies and the inter-fragment interaction energies (IFIEs) for a specific molecular structure can be obtained directly by performing FMO calculations by exposing water molecules and counterions around biomolecular systems. Then, it is necessary to pay attention to the thickness of the water shell surrounding the biomolecules. The single-point calculation for snapshots from MD trajectory does not incorporate the effects of temperature and configurational fluctuation, but the SCIFIE (statistically corrected IFIE) method is proposed as a many-body correlated method that partially compensates for this deficiency. Furthermore, implicit continuous dielectric models have been developed as effective approaches to incorporating the screening effect of the solvent in thermal equilibrium, and we illustrate their usefulness for theoretical evaluation of IFIEs and ligand-binding free energy on the basis of the FMO-PBSA (Poisson-Boltzmann surface area) method and other computational methods.

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HYPOTHESIS: The origin of negative surface charge at water/air interface is still not clear. The most probable origin is specific adsorption of OH- ions. From diffuse layer potential, we can evaluate the surface density of ions in the Stern layer which can be a measure for the specific adsorption of ions and determines whether the surface charge is solely due to the specific adsorption of OH- ions. EXPERIMENTS: Equilibrium thickness of foam films of pure water and aqueous solutions of NaCl, HCl, and NaOH was measured as a function of disjoining pressure for water and as a function of concentration for the aqueous solutions at 298.15 K. Quartz-glass cells thoroughly cleaned and immersed in pure water before use were used for the measurement. FINDINGS: Application of a modified Poisson-Boltzmann equation to the equilibrium film thickness gave the diffuse layer potential and the surface density of ions in the Stern layer. From the concentration dependence of the surface density, it was concluded that not only OH- ions but also Cl- ions and HCO3- and/or CO32- ions adsorb specifically at the water/air interface.

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Screened repulsion between uniformly charged plates with an intervening electrolyte is analyzed for strongly overlapped electrical double layers (EDL), accounting for the steric effect of ions and their expulsion from EDL edges into the surrounding solution. As a generalization of a study by Philipse et al. which does not account for these effects, an analytical expression is derived for the repulsion pressure in the limit of infinitely long plates with a zero-field assumption, which agrees closely with the corresponding numerical solution at low inter-plate separations. Our results show an augmented repulsive pressure for finite-sized ions at strong EDL overlaps. For plates with a finite lateral size, we demonstrate a further extended domain of low inter-plate gaps where the repulsion pressure increases with ion size due to a strong interplay between the steric interaction of ions and the EDL overspill phenomenon, considered earlier in a study by Ghosal & Sherwood limited to the linear Debye-Hückel regime (which cannot account for the steric effect of ions). This investigation on a simple model should enhance our understanding of the interaction between charged particles in electrophoresis, nanoscale self-assembly, active particles, and various other electrokinetic systems.

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We present a new size-modified Poisson-Boltzmann ion channel (SMPBIC) model and use it to calculate the electrostatic potential, ionic concentrations, and electrostatic solvation free energy for a voltage-dependent anion channel (VDAC) on a biological membrane in a solution mixture of multiple ionic species. In particular, the new SMPBIC model adopts a membrane surface charge density and a natural Neumann boundary condition to reflect the charge effect of the membrane on the electrostatics of VDAC. To avoid the singularity difficulties caused by the atomic charges of VDAC, the new SMPBIC model is split into three submodels such that the solution of one of the submodels is obtained analytically and contains all the singularity points of the SMPBIC model. The other two submodels are then solved numerically much more efficiently than the original SMPBIC model. As an application of this SMPBIC submodel partitioning scheme, we derive a new formula for computing the electrostatic solvation free energy. Numerical results for a human VDAC isoform 1 (hVDAC1) in three different salt solutions, each with up to five different ionic species, confirm the significant effects of membrane surface charges on both the electrostatics and ionic concentrations. The results also show that the new SMPBIC model can describe well the anion selectivity property of hVDAC1, and that the new electrostatic solvation free energy formula can significantly improve the accuracy of the currently used formula. © 2019 Wiley Periodicals, Inc.

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Within the framework of analytical theories for soft surface electrophoresis, soft particles are classically defined by a hard impermeable core of given surface charge density surrounded by a polyelectrolyte shell layer permeable to both electroosmotic flow and ions from background electrolyte. This definition excludes practical core-shell particles, e.g. dendrimers, viruses or multi-layered polymeric particles, defined by a polyelectrolytic core where structural charges are distributed and where counter-ions concentration and electroosmotic flow velocity can be significant. Whereas a number of important approximate expressions has been derived for the electrophoretic mobility of hard and soft particles, none of them is applicable to such generic composite core-shell particles with differentiated ions- and fluid flow-permeabilities of their core and shell components. In this work, we elaborate an original closed-form electrophoretic mobility expression for this generic composite particle type within the Debye-Hückel electrostatic framework and thin double layer approximation. The expression explicitly involves the screening Debye layer thickness and the Brinkman core and shell hydrodynamic length scales, which favors so-far missing analysis of the respective core and shell contributions to overall particle mobility. Limits of this expression successfully reproduce results from Ohshima's electrophoresis theory solely applicable to soft particles with or without hard core.

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Ions play significant roles in biological processes-they may specifically bind to a protein site or bind non-specifically on its surface. Although the role of specifically bound ions ranges from actively providing structural compactness via coordination of charge-charge interactions to numerous enzymatic activities, non-specifically surface-bound ions are also crucial to maintaining a protein's stability, responding to pH and ion concentration changes, and contributing to other biological processes. However, the experimental determination of the positions of non-specifically bound ions is not trivial, since they may have a low residential time and experience significant thermal fluctuation of their positions. Here, we report a new release of a computational method, the BION-2 method, that predicts the positions of non-specifically surface-bound ions. The BION-2 utilizes the Gaussian-based treatment of ions within the framework of the modified Poisson-Boltzmann equation, which does not require a sharp boundary between the protein and water phase. Thus, the predictions are done by the balance of the energy of interaction between the protein charges and the corresponding ions and the de-solvation penalty of the ions as they approach the protein. The BION-2 is tested against experimentally determined ion's positions and it is demonstrated that it outperforms the old BION and other available tools.

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