Formulation of a new and simple nonuniform size-modified Poisson-Boltzmann description

Effect of Bjerrum pairs on the electrostatic properties of an electrolyte solution near charged surfaces: a mean-field approach

In this paper, we investigate the consequences of ion association, coupled with the considerations of finite size effects and orientational ordering of Bjerrum pairs as well as ions and water molecules, on the electric double layer near charged surfaces. Based on the lattice statistical mechanics accounting for finite sizes and dipole moments of ions, Bjerrum pairs and solvent molecules, we consider the formation of Bjerrum pairs and derive the mathematical expressions for Bjerrum pair number density as well as cation/anion number density and water molecule number density. We reveal several significant phenomena. Firstly, it is shown that our approach naturally yields the equilibrium constant for dissociation-association equilibrium between Bjerrum pairs and ions. Secondly, at low surface charge densities, an increase in the bulk concentration of Bjerrum pairs enhances the permittivity and decreases the differential capacitance. Next, for the cases where Bjerrum pairs in an alcohol electrolyte solution have a high value of dipole moment, the Bjerrum pair number density increases with decreasing distance from the charged surface, and the differential capacitance and permittivity are high compared to those for the cases with lower values of Bjerrum-pair dipole moments. Finally, we show that the difference in the concentration and dipole moment of Bjerrum pairs can lead to some variation in osmotic pressure between two similarly charged surfaces.

Diffuse double-layer structure in mixed electrolytes considering ions as dielectric spheres

The structure of the diffuse part of the electric double layer at solid-electrolyte solution interfaces is examined using a theoretical model that takes into account the finite ion size by modeling the solution as a suspension of polarizable insulating spheres in water. This formalism is applied to mixed electrolyte solutions using the "Boublik-Mansoori-Carnahan-Starling-Leland" (BMCSL) theory for the steric interactions among ions. It is shown that the ionic size differences have a strong bearing on the diffuse part of the electric double-layer structure of these systems. Moreover, for strong potential values, the different size-related effects become important even for binary electrolyte solutions due to the presence of H⁺ and OH^- ions that are substantially smaller than hydrated ions originated from salt dissociation. The obtained results display some of the qualitative features observed in experiments on aqueous systems that are generally interpreted in terms of totally different mechanisms.

Ion size effect on electrostatic and electroosmotic properties in soft nanochannels with pH-dependent charge density

We report a theoretical study of the ion size effect on various properties in a soft nanochannel with pH-dependent charge density.

Function-Space-Based Solution Scheme for the Size-Modified Poisson-Boltzmann Equation in Full-Potential DFT

The size-modified Poisson-Boltzmann (MPB) equation is an efficient implicit solvation model which also captures electrolytic solvent effects. It combines an account of the dielectric solvent response with a mean-field description of solvated finite-sized ions. We present a general solution scheme for the MPB equation based on a fast function-space-oriented Newton method and a Green's function preconditioned iterative linear solver. In contrast to popular multigrid solvers, this approach allows us to fully exploit specialized integration grids and optimized integration schemes. We describe a corresponding numerically efficient implementation for the full-potential density-functional theory (DFT) code FHI-aims. We show that together with an additional Stern layer correction the DFT+MPB approach can describe the mean activity coefficient of a KCl aqueous solution over a wide range of concentrations. The high sensitivity of the calculated activity coefficient on the employed ionic parameters thereby suggests to use extensively tabulated experimental activity coefficients of salt solutions for a systematic parametrization protocol.

Quantifying the Influence of the Crowded Cytoplasm on Small Molecule Diffusion

Cytosolic crowding can influence the thermodynamics and kinetics of in vivo chemical reactions. Most significantly, proteins and nucleic acid crowders reduce the accessible volume fraction, ϕ, available to a diffusing substrate, thereby reducing its effective diffusion rate, Deff, relative to its rate in bulk solution. However, Deff can be further hindered or even enhanced, when long-range crowder/diffuser interactions are significant. To probe these effects, we numerically estimated Deff values for small, charged molecules in representative, cytosolic protein lattices up to 0.1 × 0.1 × 0.1 μm(3) in volume via the homogenized Smoluchowski electro-diffusion equation. We further validated our predictions against Deff estimates from ϕ-dependent analytical relationships, such as the Maxwell-Garnett (MG) bound, as well as explicit solutions of the time-dependent electro-diffusion equation. We find that in typical, moderately crowded cell cytoplasm (ϕ ≈ 0.8), Deff is primarily determined by ϕ; in other words, diverse protein shapes and heterogeneous distributions only modestly impact Deff. However, electrostatic interactions between diffusers and crowders, particularly at low electrolyte ionic strengths, can substantially modulate Deff. These findings help delineate the extent that cytoplasmic crowders influence small molecule diffusion, which ultimately may shape the efficiency and timing of intracellular signaling pathways. More generally, the quantitative agreement between computationally expensive solutions of the time-dependent electro-diffusion equation and its comparatively cheaper homogenized form suggest that the latter is a broadly effective model for diffusion in wide-ranging, crowded biological media.

Influence of asymmetric depletion of solvents on the electric double layer of charged objects in binary polar solvent mixtures

The ratio of the dipole moment to the volume of solvent is the key factor for asymmetric depletion of solvents.

An electric double layer of colloidal particles in salt-free concentrated suspensions including non-uniform size effects and orientational ordering of water dipoles

Non-uniform size effects and orientational ordering of water dipoles influence the relative permittivity and electric potential in suspension.

Effect of thermodiffusion on pH-regulated surface charge properties of nanoparticle

Surface properties of nanoparticle are of high importance in the field of biotechnology, drug delivery and micro/nanofabrication. In this article, we developed a comprehensive theoretical model and subsequently solved that numerically to study the effect of thermodiffusion of ions on surface charge properties of nanoparticle. The theoretical study has been done considering silica nanoparticle for two aqueous solutions NaCl and KCl. The effect of solution pH in conjunction with nanoparticle temperature on surface charge density has been obtained for different salt concentrations (1, 10 and 100 mM) and nanoparticle size (diameter of 2 and 100 nm). It is observed from the results that with increasing temperature of the nanoparticle, the negative surface charge density gets higher due to increasing thermodiffusion effect. It is also found out that the magnitude of surface charge density is higher for KCl solution than NaCl solution under same condition which is attributed mostly due to less thermodiffusion of counterions for KCl than NaCl. Present study also shows that magnitude of surface charge density decreases with increasing nanoparticle size until it reaches a limiting value (called critical size) above which the effect of nanoparticle size on surface charge density is insignificant.

Competitive interaction of monovalent cations with DNA from 3D-RISM

The composition of the ion atmosphere surrounding nucleic acids affects their folding, condensation and binding to other molecules. It is thus of fundamental importance to gain predictive insight into the formation of the ion atmosphere and thermodynamic consequences when varying ionic conditions. An early step toward this goal is to benchmark computational models against quantitative experimental measurements. Herein, we test the ability of the three dimensional reference interaction site model (3D-RISM) to reproduce preferential interaction parameters determined from ion counting (IC) experiments for mixed alkali chlorides and dsDNA. Calculations agree well with experiment with slight deviations for salt concentrations >200 mM and capture the observed trend where the extent of cation accumulation around the DNA varies inversely with its ionic size. Ion distributions indicate that the smaller, more competitive cations accumulate to a greater extent near the phosphoryl groups, penetrating deeper into the grooves. In accord with experiment, calculated IC profiles do not vary with sequence, although the predicted ion distributions in the grooves are sequence and ion size dependent. Calculations on other nucleic acid conformations predict that the variation in linear charge density has a minor effect on the extent of cation competition.

Generalized Manning Condensation Model Captures the RNA Ion Atmosphere

RNA is highly sensitive to the ionic environment and typically requires Mg(2+) to form compact structures. There is a need for models capable of describing the ion atmosphere surrounding RNA with quantitative accuracy. We present a model of RNA electrostatics and apply it within coarse-grained molecular dynamics simulation. The model treats Mg(2+) ions explicitly to account for ion-ion correlations neglected by mean-field theories. Since mean-field theories capture KCl well, it is treated implicitly by a generalized Manning counterion condensation model. The model extends Manning condensation to deal with arbitrary RNA conformations, nonlimiting KCl concentrations, and the ion inaccessible volume of RNA. The model is tested against experimental measurements of the excess Mg(2+) associated with the RNA, Γ(2+), because Γ(2+) is directly related to the Mg(2+)-RNA interaction free energy. The excellent agreement with experiment demonstrates that the model captures the ionic dependence of the RNA free energy landscape.

Features of CPB: a Poisson-Boltzmann solver that uses an adaptive Cartesian grid

The capabilities of an adaptive Cartesian grid (ACG)-based Poisson-Boltzmann (PB) solver (CPB) are demonstrated. CPB solves various PB equations with an ACG, built from a hierarchical octree decomposition of the computational domain. This procedure decreases the number of points required, thereby reducing computational demands. Inside the molecule, CPB solves for the reaction-field component (ϕrf ) of the electrostatic potential (ϕ), eliminating the charge-induced singularities in ϕ. CPB can also use a least-squares reconstruction method to improve estimates of ϕ at the molecular surface. All surfaces, which include solvent excluded, Gaussians, and others, are created analytically, eliminating errors associated with triangulated surfaces. These features allow CPB to produce detailed surface maps of ϕ and compute polar solvation and binding free energies for large biomolecular assemblies, such as ribosomes and viruses, with reduced computational demands compared to other Poisson-Boltzmann equation solvers. The reader is referred to http://www.continuum-dynamics.com/solution-mm.html for how to obtain the CPB software.

Sensitivities to parameterization in the size-modified Poisson-Boltzmann equation

Experimental results have demonstrated that the numbers of counterions surrounding nucleic acids differ from those predicted by the nonlinear Poisson-Boltzmann equation, NLPBE. Some studies have fit these data against the ion size in the size-modified Poisson-Boltzmann equation, SMPBE, but the present study demonstrates that other parameters, such as the Stern layer thickness and the molecular surface definition, can change the number of bound ions by amounts comparable to varying the ion size. These parameters will therefore have to be fit simultaneously against experimental data. In addition, the data presented here demonstrate that the derivative, SK, of the electrostatic binding free energy, ΔGel, with respect to the logarithm of the salt concentration is sensitive to these parameters, and experimental measurements of SK could be used to parameterize the model. However, although better values for the Stern layer thickness and ion size and better molecular surface definitions could improve the model's predictions of the numbers of ions around biomolecules and SK, ΔGel itself is more sensitive to parameters, such as the interior dielectric constant, which in turn do not significantly affect the distributions of ions around biomolecules. Therefore, improved estimates of the ion size and Stern layer thickness to use in the SMPBE will not necessarily improve the model's predictions of ΔGel.

Reduced model captures Mg(2+)-RNA interaction free energy of riboswitches

The stability of RNA tertiary structures depends heavily on Mg(2+). The Mg(2+)-RNA interaction free energy that stabilizes an RNA structure can be computed experimentally through fluorescence-based assays that measure Γ2+, the number of excess Mg(2+) associated with an RNA molecule. Previous explicit-solvent simulations predict that the majority of excess Mg(2+) ions interact closely and strongly with the RNA, unlike monovalent ions such as K(+), suggesting that an explicit treatment of Mg(2+) is important for capturing RNA dynamics. Here we present a reduced model that accurately reproduces the thermodynamics of Mg(2+)-RNA interactions. This model is able to characterize long-timescale RNA dynamics coupled to Mg(2+) through the explicit representation of Mg(2+) ions. KCl is described by Debye-Hückel screening and a Manning condensation parameter, which represents condensed K(+) and models its competition with condensed Mg(2+). The model contains one fitted parameter, the number of condensed K(+) ions in the absence of Mg(2+). Values of Γ2+ computed from molecular dynamics simulations using the model show excellent agreement with both experimental data on the adenine riboswitch and previous explicit-solvent simulations of the SAM-I riboswitch. This agreement confirms the thermodynamic accuracy of the model via the direct relation of Γ2+ to the Mg(2+)-RNA interaction free energy, and provides further support for the predictions from explicit-solvent calculations. This reduced model will be useful for future studies of the interplay between Mg(2+) and RNA dynamics.

Ionic Size Effects: Generalized Boltzmann Distributions, Counterion Stratification, and Modified Debye Length

Near a charged surface, counterions of different valences and sizes cluster; and their concentration profiles stratify. At a distance from such a surface larger than the Debye length, the electric field is screened by counterions. Recent studies by a variational mean-field approach that includes ionic size effects and by Monte Carlo simulations both suggest that the counterion stratification is determined by the ionic valence-to-volume ratios. Central in the mean-field approach is a free-energy functional of ionic concentrations in which the ionic size effects are included through the entropic effect of solvent molecules. The corresponding equilibrium conditions define the generalized Boltzmann distributions relating the ionic concentrations to the electrostatic potential. This paper presents a detailed analysis and numerical calculations of such a free-energy functional to understand the dependence of the ionic charge density on the electrostatic potential through the generalized Boltzmann distributions, the role of ionic valence-to-volume ratios in the counterion stratification, and the modification of Debye length due to the effect of ionic sizes.

Competitive adsorption and ordered packing of counterions near highly charged surfaces: From mean-field theory to Monte Carlo simulations

Competitive adsorption of counterions of multiple species to charged surfaces is studied by a size-effect-included mean-field theory and Monte Carlo (MC) simulations. The mean-field electrostatic free-energy functional of ionic concentrations, constrained by Poisson's equation, is numerically minimized by an augmented Lagrangian multiplier method. Unrestricted primitive models and canonical ensemble MC simulations with the Metropolis criterion are used to predict the ionic distributions around a charged surface. It is found that, for a low surface charge density, the adsorption of ions with a higher valence is preferable, agreeing with existing studies. For a highly charged surface, both the mean-field theory and the MC simulations demonstrate that the counterions bind tightly around the charged surface, resulting in a stratification of counterions of different species. The competition between mixed entropy and electrostatic energetics leads to a compromise that the ionic species with a higher valence-to-volume ratio has a larger probability to form the first layer of stratification. In particular, the MC simulations confirm the crucial role of ionic valence-to-volume ratios in the competitive adsorption to charged surfaces that had been previously predicted by the mean-field theory. The charge inversion for ionic systems with salt is predicted by the MC simulations but not by the mean-field theory. This work provides a better understanding of competitive adsorption of counterions to charged surfaces and calls for further studies on the ionic size effect with application to large-scale biomolecular modeling.