Traditional Poisson-Boltzmann Research Articles

Computational Study of the pH-Dependent Ionic Environment around β-Lactoglobulin.

Ions are involved in multiple biological processes and may exist bound to biomolecules or may be associated with their surface. Although the presence of ions in nucleic acids has traditionally gained more interest, ion-protein interactions, often with a marked dependency on pH, are beginning to gather attention. Here we present a detailed analysis on the binding and distribution of ions around β-lactoglobulin using a constant-pH MD (CpHMD) method, at a pH range 3-8, and compare it with the more traditional Poisson-Boltzmann (PB) model and the existing experimental data. Most analyses used ion concentration maps built around the protein, obtained from either the CpHMD simulations or PB calculations. The requirements of approximate charge neutrality and ionic strength equal to bulk, imposed on the MD box, imply that the absolute value of the ion excess should be half the protein charge, which is in agreement with experimental observation on other proteins ( Proc. Natl. Acad. Sci. U.S.A. 2021, 118, e2015879118) and lends support to this protocol. In addition, the protein total charge (including territorially bound ions) estimated with MD is in excellent agreement with electrophoretic measurements. Overall, the CpHMD simulations show good agreement with the nonlinear form of the PB (NLPB) model but not with its linear form, which involves a theoretical inconsistency in the calculation of the concentration maps. In several analyses, the observed pH-dependent trends for the counterions and co-ions are those generally expected, and the ion concentration maps correctly converge to the bulk ionic strength as one moves away from the protein. Despite the overall similarity, the CpHMD and NLPB approaches show some discrepancies when analyzed in more detail, which may be related to an apparent overestimation of counterion excess and underestimation of co-ion exclusion by the NLPB model, particularly at short distances from the protein.

Computation of magnetohydrodynamic electro-osmotic modulated rotating squeezing flow with zeta potential effects

Motivated by exploring novel intelligent functional materials deployed in electromagnetic squeezing flows such systems, a comprehensive mathematical model is developed to investigate the squeezing flow of a smart viscous ionic magneto-tribological fluid with zeta potential effects, intercalated between two parallel plates rotating in unison, under the simultaneous application of electric and magnetic fields. The lower disk permits lateral mass flux (suction or injection). Continuity and momentum equations are represented in the proposed three-dimensional mathematical model by a set of partial differential equations. The electro-viscous effects resulting from distorted electric double-capacity flow fields are comprehensively examined for various intensities of applied plate motion. The formulation features a more robust approach to the traditional Poisson-Boltzmann equation model. A similarity transformation is used to translate the governing equations into ordinary differential equations, which are then numerically solved with appropriate boundary conditions at the disks using MATLAB software. Via graphical visualization of velocity profiles, pressure gradients and the upper wall coefficient of skin friction, several characteristics of squeezing flow are analysed. The computations show that there is a rise in pressure near the plate walls and a fall in pressure in the centre with increment in rotational, electroosmosis, electric field, and magnetic parameters. However, by selecting the appropriate squeezing velocity, the viscous drag on the lower plate can be effectively reduced. It is also observed that as the disk (wall) suction parameter increases, both radial and transverse velocities are damped. The current study generalizes previous investigations with the novelty of rotation and also pressure gradient computations. Furthermore, it provides a useful benchmark for alternative numerical simulations.

A Focus on Two Electrokinetics Issues.

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.

Enhancing active electro-kinetic micro-mixer efficiency by introducing vertical electrodes and modifying chamber aspect ratio

Micro-mixers are considered as vital components of Micro Total Analysis systems (μTAS). Major objective in the design of micro-mixers is achieving high mixing quality in short mixing times. In this paper, numerical simulation of some micro-mixer designs has been carried out to understand the detailed flow pattern and thereby to propose modifications for improving mixing efficiency. It is well known that inducing convection will provide turbulent like behavior with corresponding mixing enhancement. In micro systems to drive the flow, electro-osmotic force is usually used by introducing electrodes. In this work, mixing electrodes have been implemented to induce convection and eddies. This numerical study addresses the problem by solving the system of inter-coupled Poisson-Nernst-Planck-Navier-Stokes equations using Rayan (in-house code). These equations compared to their simplified forms, provide much more realistic physical description by including diffusion, electro-migration and convection mechanisms. In cases of active AC electrodes, all of these mechanisms play important roles. The traditional Poisson-Boltzmann model loses its accuracy in cases of complex geometry or AC electrodes especially with high frequencies. As a result of modifications proposed herein by strategic placement of vertical and horizontal electrodes and extending mixing chamber length a mixing quality of 96.2% was achieved just in 1.8 s.

Accurate Atom-by-Atom Predictions of Solvation Electrostatics Using a Hydration-Shell Poisson-Boltzmann Model

Traditional Poisson-Boltzmann models of molecular electrostatics fail to model the behavior of the hydration-shell solvent, which deviates significantly from bulk. Here we present a hydration-shell Poisson-Boltzmann (HSPB) model that accounts for these deviations by perturbing the displacement-field boundary condition. The HSPB model correctly predicts charging free energies for Born ions, amino acids in multiple protonation states, and challenging test cases for charge-hydration asymmetry. Remarkably, the model achieves this accuracy using standard Lennard-Jones radii from molecular-dynamics force fields, and thus it has an order of magnitude fewer fitting parameters than traditional PB theories. Free-energy calculations in explicit solvent establish that HSPB also accurately predicts the charging free energies of individual atoms in biological molecules.In contrast, calculations using the traditional Poisson--Boltzmann model indicate substantial errors in intramolecular screening. The HSPB model's accuracy in predicting free-energy changes recommend it for drug design and protein engineering applications, where electrostatic contributions to binding are often assessed relative to hydrophobic isosteres.

An ionic concentration and size dependent dielectric permittivity Poisson-Boltzmann model for biomolecular solvation studies.

By considering the influence of volume exclusion on the solvent dielectric, a variable dielectric Poisson-Boltzmann (VDPB) model is explored for molecular solvation studies by using a dielectric as an explicit function of ionic sizes and concentrations. A finite element method is adopted and an iterative strategy is introduced to numerically solve the VDPB equation. According to our computations, the current dielectric model can result in considerable differences compared with the traditional Poisson-Boltzmann (PB) solutions, especially for those systems with highly charged biomolecule and/or under high salt concentration condition. The model to certain extent captures the fact of dielectric decrement of electrolyte solutions, which is especially remarkable in the vicinity of molecules. Counter-ion concentration very near the molecular surface in VDPB calculation is found higher than that in PB. The new dielectric model may also influence the charge compensation behavior near biomolecular surface. For a spherical cavity solvated in a concentrated ionic solution, charge inversion is observed in VDPB, which does not occur with the traditional PB model. Besides, the solvation energy predicted by VDPB will always be greater than that by PB. Moreover, differing from PB, the VDPB also allows non-monotonous dependencies of solvation energy on ionic strength.

Modeling the Influence of Salt on the Hydrophobic Effect and Protein Fold Stability

AbstractSalt influences protein stability through electrostatic mechanisms as well as through nonpolar Hofmeister effects. In the present work, a continuum solvation based model is developed to explore the impact of salt on protein stability. This model relies on a traditional Poisson-Boltzmann (PB) term to describe the polar or electrostatic effects of salt, and a surface area dependent term containing a salt concentration dependent microscopic surface tension function to capture the non-polar Hofmeister effects. The model is first validated against a series of cold-shock protein variants whose salt-dependent protein fold stability profiles have been previously determined experimentally. The approach is then applied to HIV-1 protease in order to explain an experimentally observed enhancement in stability and activity at high (1M) NaCl concentration. The inclusion of the salt-dependent non-polar term brings the model into quantitative agreement with experiment, and provides the basis for further studies into the impact of ionic strength on protein structure, function, and evolution.

Liquid flow retardation in nanospaces due to electroviscosity: Electrical double layer overlap, hydrodynamic slippage, and ambient atmospheric CO2 dissolution

A theoretical investigation is performed into the electroviscous-induced retardation of liquid flows through finitely long nanochannels or nanotubes with large wells at either end. Given the assumption of equilibrium conditions between the ionic solution in the wells and that within the nanochannel or nanotube, an exact solution is derived for the overlapped electrical double layer (EDL) for the case where the concentrations of the positive and negative ions in the wells may be unequal. The ion concentrations in the wells are determined by the conditions of global electroneutrality and mass conservation. It is shown that the overlapped EDL model proposed by Baldessari and Santiago [J. Colloid Interface Sci. 325, 526 (2008)10.1016/j.jcis.2008.06.007] is in fact the same as the “thick EDL model” (i.e., the traditional Poisson-Boltzmann model) when the positive and negative ion concentrations in the large enough wells are both equal to the bulk concentration of the salt solution. Utilizing the proposed overlapped EDL analytical model, an investigation is performed to evaluate the effects of hydrodynamic slippage on the flow retardation caused by electroviscosity in nanochannels or nanotubes. Furthermore, exact and approximate solutions are derived for the electroviscosity in ion-selective nanochannels and nanotubes. It is shown that in the absence of slip, the maximum electroviscosity in nanochannels and nanotubes containing a unipolar solution of simple monovalent counter-ions occurs at surface charge densities of h|σ| = 0.32 nm × C/m2 and a|σ| ≈ 0.4 nm × C/m2, respectively. In addition, it is shown that the electroviscosity in a nanotube is smaller than that in a nanochannel. For example, given a LiCl solution, the maximum electroviscosites in a non-slip nanochannel and non-slip nanotube are ηa/η ≈ 1.6 and 1.47, respectively. For both nanospaces, the electroviscosity is greatly increased when the liquid slip effect is taken into account. Significantly, under slip conditions, the electroviscosity in the nanotube is greater than that in the nanochannel. Finally, an investigation is performed into the effects of ambient atmospheric CO2 dissolution on the electroviscosities of salt/buffer solution and deionized (DI) water in silica nanochannels. The results show that the electroviscosity of CO2-saturated DI water (pH = 5.6) can be reasonably neglected in silica nanochannels with a height of less than 100 nm.

Electrokinetic flow-induced currents in silica nanofluidic channels

Electrokinetic flow-induced currents inside slit-shaped silica nanochannels are investigated. The unusual features observed experimentally in silica nanochannels are described successfully using a new theoretical framework. First, a simple and reliable physicochemical boundary condition at the interface between the channel surface and the solution is suggested. It accounts for the surface conduction effect through the Stern layer and the dependence of the surface charge on the salt concentration and pH, which were commonly neglected in previous studies. Second, the proposed boundary condition is then incorporated into the traditional Poisson–Boltzmann and Nernst–Planck models to complete the self-consistent model. Model predictions are validated by comparison with experimental data. It is found that the direct numerical predictions of the concentration polarization and the induced potential or pressure field are possible, and these allow us to describe the dependence of currents on the solution properties in the nanofluidic channel more accurately than the models proposed in previous studies.

Multiple Protonation Equilibria in Electrostatics of Protein−Protein Binding

All proteins contain groups capable of exchanging protons with their environment. We present here an approach, based on a rigorous thermodynamic cycle and the partition functions for energy levels characterizing protonation states of the associating proteins and their complex, to compute the electrostatic pH-dependent contribution to the free energy of protein-protein binding. The computed electrostatic binding free energies include the pH of the solution as the variable of state, mutual "polarization" of associating proteins reflected as changes in the distribution of their protonation states upon binding and fluctuations between available protonation states. The only fixed property of both proteins is the conformation; the structure of the monomers is kept in the same conformation as they have in the complex structure. As a reference, we use the electrostatic binding free energies obtained from the traditional Poisson-Boltzmann model, computed for a single macromolecular conformation fixed in a given protonation state, appropriate for given solution conditions. The new approach was tested for 12 protein-protein complexes. It is shown that explicit inclusion of protonation degrees of freedom might lead to a substantially different estimation of the electrostatic contribution to the binding free energy than that based on the traditional Poisson-Boltzmann model. This has important implications for the balancing of different contributions to the energetics of protein-protein binding and other related problems, for example, the choice of protein models for Brownian dynamics simulations of their association. Our procedure can be generalized to include conformational degrees of freedom by combining it with molecular dynamics simulations at constant pH. Unfortunately, in practice, a prohibitive factor is an enormous requirement for computer time and power. However, there may be some hope for solving this problem by combining existing constant pH molecular dynamics algorithms with so-called accelerated molecular dynamics algorithms.

Numerical analysis on electroosmotic flows in a microchannel with rectangle-waved surface roughness using the Poisson–Nernst–Planck model

The present study has numerically investigated two-dimensional electroosmotic flows in a microchannel with dielectric walls of rectangle-waved surface roughness to understand the roughness effect. For the study, numerical simulations are performed by employing the Nernst–Planck equation for the ionic species and the Poisson equation for the electric potential, together with the traditional Navier–Stokes equation. Results show that the steady electroosmotic flow and ionic-species transport in a microscale channel are well predicted by the Poisson–Nernst–Planck model and depend significantly on the shape of surface roughness such as the amplitude and periodic length of wall wave. It is found that the fluid flows along the surface of waved wall without involving any flow separation because of the very strong normal component of EDL (electric double layer) electric field. The flow rate decreases exponentially with the amplitude of wall wave, whereas it increases linearly with the periodic length. It is mainly due to the fact that the external electric-potential distribution plays a crucial role in driving the electroosmotic flow through a microscale channel with surface roughness. Finally, the present results using the Poisson–Nernst–Planck model are compared with those using the traditional Poisson–Boltzmann model which may be valid in these scales.

On the Restabilization of Protein-Covered Latex Colloids at High Ionic Strengths

Recent experiments on restabilization of protein-covered latex colloids at high ionic strengths reported by Lopez-Leon et al.(1) revealed strong specific anion effects. The same authors also emphasized that a recent polarization model, which involves both hydration and double layer forces, can account only for some of their experimental results but are in disagreement with other experimental results. The aim of the present paper is to show that most experimental results of ref 1 can be described, more than qualitatively, when the association equilibria for all the ions (with both the acidic and basic sites of the protein) are taken into account. As the traditional Poisson-Boltzmann approach, the polarization model neglects additional interactions between ions, and ions and surfaces, not included in the "mean field" electrical potential; therefore, a complete quantitative agreement should not be expected. While many of the discrepancies between calculations and experiment occur at low ionic strengths (10(-)(4)-10(-)(2) M), in the range of validity of the traditional DLVO theory, the latter can neither explain them. It is suggested that the structural modifications of the protein configuration induced by the electrolyte are responsible for some of the disagreements between experiment and calculations.