Ionic exclusion phase transition in neutral and weakly charged cylindrical nanopores

A field theoretic variational approach is introduced to study ion penetration into water-filled cylindrical nanopores in equilibrium with a bulk reservoir [S. Buyukdagli, M. Manghi, and J. Palmeri, Phys. Rev. Lett. 105, 158103 (2010)]. It is shown that an ion located in a neutral pore undergoes two opposing mechanisms: (i) a deformation of its surrounding ionic cloud of opposite charge, with respect to the reservoir, which increases the surface tension and tends to exclude ions from the pore, and (ii) an attractive contribution to the ion self-energy due to the increased screening with ion penetration of the repulsive image forces associated with the dielectric jump between the solvent and the pore wall. For pore radii around 1 nm and bulk concentrations lower than 0.2 mol/l, this mechanism leads to a first-order phase transition, similar to capillary "evaporation," from an ionic-penetration state to an ionic-exclusion state. The discontinuous phase transition exists within the biological concentration range (～0.15 mol/l) for small enough membrane dielectric constants (ε(m) < 5). In the case of a weakly charged pore, counterion penetration exhibits a nonmonotonic behavior and is characterized by two regimes: at low reservoir concentrations or small pore radii, coions are excluded and counterions enter the pore to enforce electroneutrality; dielectric repulsion (image forces) remain strong and the counterion partition coefficient decreases with increasing reservoir concentration up to a characteristic value. For larger reservoir concentrations, image forces are screened and the partition coefficient of counterions increases with the reservoir concentration, as in the neutral pore case. Large surface charge densities (>2 × 10(-3) e/nm(2)) suppress the discontinuous transition by reducing the energy barrier for ion penetration and shifting the critical point toward very small pore sizes and reservoir concentrations. Our variational method is also compared to a previous self-consistent approach and yields important quantitative corrections. The role of the curvature of dielectric interfaces is highlighted by comparing ionic penetration into slit and cylindrical pores. Finally, a charge regulation model is introduced in order to explain the key effect of pH on ionic exclusion and explain the origin of observed time-dependent nanopore electric conductivity fluctuations and their correlation with those of the pore surface charge.     

Monte Carlo Calculations of Polymer Adsorption at the Entrance of Cylindrical Pores in Flat Adsorbing Surfaces

The influence of surface interactions on the conformation of flexible polymers partially confined inside narrow cylindrical pores in a flat surface is studied above the critical adsorption energy in a good solvent. We use a static configurational bias computational sampling method to calculate the adsorption free energy and the radius of gyration components parallel and perpendicular to the pore axis as a function of the polymer center of mass position at different degrees of confinement. We find strong free‐energy minima just in front of the pore entry for all degrees of confinement studied. At the location of the free‐energy minimum, polymers are partially adsorbed inside the pore and on the outer solid surface and adopt “drawing pin”‐like conformations. A distinct maximum in the average loop length at the pore entry indicates that the polymer bridges the pore entry of small pores.     

Dielectric exclusion of ions from membranes

Dielectric exclusion is caused by the interactions of ions with the bound electric charges induced by ions at interfaces between media of different dielectric constants. It is considered as one of mechanisms of nanofiltration. The transport properties of capillary model are expressed through ion distribution and diffusion coefficients. Due to local equilibrium the distribution coefficient is directly related to the excess solvation energy of ion. First, this energy is considered for single ions in single neutral pores in terms of pore size, ion charge, dielectric constants of solvent and membrane matrix and pore geometry. The dielectric exclusion from pores with closed geometry like circular cylinders is shown to be essentially stronger than that from pores with relatively open geometry like slits. Furthermore, the role of finite membrane porosity is analysed for the model of infinite slabs with alternating dielectric constants. The presence of other ions is accounted for within the scope of a mean-field approach, and the screening of dielectric exclusion is thus introduced and considered in some detail. A fixed electric charge is shown to cause additional screening. At the same time the dielectric exclusion makes the Donnan exclusion of ions stronger. Therefore the interaction between those two rejection mechanisms turns out to be non-trivial. Finally, the effect of solvent molecular structure is considered within the scope of non-local electrostatics. It is shown that the solvent non-locality typically results in somewhat stronger dielectric exclusion, however, its most important effect is slowing down the decline of dielectric exclusion with increasing bulk electrolyte concentration.     

Electrostatics of DNA complexes with cationic lipid membranes

We present the exact solutions of the linear Poisson-Boltzmann equation for several problems relevant to electrostatics of DNA complexes with cationic lipids. We calculate the electrostatic potential and electrostatic energy for lamellar and inverted hexagonal phases, concentrating on the effects of dielectric boundaries. We compare our results for the complex energy with the known results of numerical solution of the nonlinear Poisson-Boltzmann equation. Using the solution for the lamellar phase, we calculate the compressibility modulus and compare our findings with the experimental data available. Also, we treat charge-charge interactions across, along, and between two low-dielectric membranes. We obtain an estimate for the strength of electrostatic interactions of one-dimensional DNA smectic layers across the lipid membrane. We discuss in the end some aspects of two-dimensional DNA condensation and DNA-DNA attraction in the DNA-lipid lamellar phase in the presence of di- and trivalent cations. We analyze the equilibrium DNA-DNA separations in lamellar complexes using the recently developed theory of electrostatic interactions of DNA helical charge motifs.     

Charge state of the fast gate in chloride channels: insights from electrostatic calculations in a schematic model

Fast gating is a unique property of chloride channels, where a permeating Cl(-) ion acts as its own ligand in opening the channel. The glutamate residue implicated in fast gating normally carries a unit negative charge. Whether this charge needs to be protonated to enable permeation of a Cl(-) ion is an important question that will affect how models of chloride channels are constructed. We investigate the energetic consequences of the charge state of this glutamate residue from continuum electrostatics using a schematic cylindrical channel model. Both analytical solutions of the Poisson equation for an infinite cylinder and numerical ones for a finite cylinder are employed in the calculations.     

Molecular dynamics and continuum electrostatics studies of inactivation in the HERG potassium channel

Fast inactivation of the HERG potassium channel plays a critical role in normal cardiac function. Malfunction of these channels due to either genetic mutations or blockade by drugs leads to cardiac arrhythmias. An unusually long S5-P linker in the outer mouth of HERG is implicated in the fast inactivation mechanism. To examine the role of the S5-P linker in this inactivation mechanism, we study the permeation properties of the open and inactive states of a recent homology model of HERG. This model was constructed using the KcsA potassium channel as a template and contains specific conformations of the S5-P linker in the open and inactive states. We perform molecular dynamics simulations on the HERG model, followed by free energy, structural, and continuum electrostatics calculations. Our free energy calculations lead to selectivity results of the model channel (K+ over Na+) that are different in some respects from those of other potassium channels but consistent with experimental observations. Our structural results show that, in the inactive state, the S5-P linkers move closer to the channel axis, possibly causing a steric hindrance to permeating K+ ions. Our electrostatics calculations reveal, in the inactive state, an electrostatic potential energy barrier of approximately 14 kT at the extracellular pore entrance, again sufficient to stop K+ ion permeation through the pore. These results suggest that a steric and/or electrostatic plug mechanism contributes to inactivation in the HERG homology model.     

Converting the nanodomains of a diblock-copolymer thin film from spheres to cylinders with an external electric field

We investigate the ability of an applied electric field to convert the morphology of a diblock-copolymer thin film from a monolayer of spherical domains embedded in the matrix to cylindrical domains that penetrate through the matrix. As expected, the applied field increases the relative stability of cylindrical domains, while simultaneously reducing the energy barrier that impedes the transition to cylinders. The effectiveness of the field is enhanced by a large dielectric contrast between the two block-copolymer components, particularly when the low-dielectric contrast component forms the matrix. Furthermore, the energy barrier is minimized by selecting sphere-forming diblock copolymers that are as compositionally symmetric as possible. Our calculations, which are the most quantitatively reliable to date, are performed using a numerically precise spectral algorithm based on self-consistent-field theory supplemented with an exact treatment for linear dielectric materials.     

Design of a high-sensitivity negative ion source time-of-flight mass analyzer assembly created by cylindrical electrodes with a common axis

The new design incorporates the negative ion source and the mass analyzer, both constructed from cylindrical electrodes. The ion source is formed by three gridded cylindrical electrodes: a pulsed grid, the intermediate grid and the final accelerating grid. During a first time lapse, the electrons penetrate through the pulsed grid into the retarding field between this grid and the intermediate grid. The electrons are turning at some depth inside this intergrid space, where the attachment to neutral molecules most probably occurs. Next, the pulsed grid becoming strongly negative and ions are extracted towards the final acceleration grid. The ions from the cylindrical surface where they were created concentrate on the common axis of the electrodes (lateral focusing). The source lateral and time focus are coincident. A cylindrical electrostatic mirror is fitted to the source. The design, with a single stage, ensures also lateral focusing of the ions diverging from the common axis of the electrodes. The mirror electric and geometric parameters were selected to ensure both lateral and time focusing on the final detector with subsequent high luminosity. The basic parameters of the specific negative ion source time-of-flight mass analyzer design proposed here, are ion source final acceleration, intermediate, pulsed cylindrical grid radii 10, 20 and 30 mm, respectively, electrostatic mirror earthed grid and ion turning points surface radii 0.6 and 0.8 m, respectively. Ion packet smearing by the ion energy spread (resulting from the initial electron energy spread as electrons are turning at different depths inside the ionization region, from the moment when ions were created, being accelerated towards the pulsed grid during ionization) and by the turnaround time inside the cylindrical field was accounted for. Maintaining very high sensitivity, a resolution of the order of 100 is expected.     

Ion current rectification at nanopores in glass membranes

The origin of ion current rectification observed at conical-shaped nanopores in glass membranes immersed in KCl solutions has been investigated using finite-element simulations. The ion concentrations and fluxes (due to diffusion, migration, and electroosmotic convection) were determined by the simultaneous solution of the Nernst-Planck, Poisson, and Navier-Stokes equations for the two-ion (K+ and Cl-) system. Fixed surface charge on both the internal and external glass surfaces that define the pore structure was included to account for electric fields and nonuniform ion conductivity within the nanopores and electric fields in the external solution near the pore mouth. We demonstrate that previous observations of ion current rectification in conical-shaped glass nanopores are a consequence of the voltage-dependent solution conductivity in the vicinity of the pore mouth, both inside and outside of the pore. The simulations also demonstrate that current rectification is maximized at intermediate bulk ion concentrations, a combination of (i) the electrical screening of surface charge at high concentrations and (ii) a fixed number of charge-carrying ions in the pore at lower concentration, which are physical conditions where the voltage dependence of the conductivity disappears. In addition, we have quantitatively shown that electroosmotic flow gives rise to a significant but small contribution to current rectification.     

Dispersion self-free energies and interaction free energies of finite-sized ions in salt solutions

The role for many-body dipolar (dispersion) potentials in ion-solvent and ion-solvent-interface interactions is explored. Such many-body potentials, accessible in principle from measured dielectric data, are necessary in accounting for Hofmeister specific ion effects. Dispersion self-energy is the quantum electrodynamic analogue of the Born electrostatic self-energy of an ion. We here describe calculations of dispersion self-free energies of four different anions (OH-, Cl-, Br-, and I-) that take finite ion size into account. Three different examples of self-free energy calculations are presented. These are the self-free energy of transfer of an ion to bulk solution, which influences solubility; the dispersion potential acting between one ion and an air-water interface (important for surface tension calculations); and the dispersion potential acting between two ions (relevant to activity coefficient calculations). To illustrate the importance of dispersion self-free energies, we compare the Born and dispersion contributions to the free energy of ion transfer from water to air (oil). We have also calculated the change in interfacial tension with added salt for air (oil)-water interfaces. A new model is used that includes dispersion potentials acting on the ions near the interface, image potentials, and ions of finite size that are allowed to spill over the solution-air interface. It is shown that interfacial free energies require a knowledge of solvent profiles at the interface.     

An accurate and efficient empirical approach for calculating the dielectric self-energy and ion-ion pair potential in continuum models of biological ion channels

This paper presents empirical formulas for calculating the dielectric self-energy and ion-ion pair interactions in cylindrical ion channels. The proposed approach can be extended to more complex channel structures, for example, (i) a "straight" channel with variable radius and (ii) a "curved" channel with constant radius. For calibration purposes, we compare results obtained based on the approximate effective potentials developed herein to exact electrostatic calculations obtained via the algorithm of Graf et al.: the agreement is satisfactory. A dynamic lattice Monte Carlo (DLMC) technique is used to further assess the accuracy and efficiency of the proposed empirical potentials. The concentration profiles and current-voltage curves produced with our simple empirical energy formulas are in excellent agreement with numerical results obtained using the algorithm of Graf et al., which calculates all relevant electrostatic forces exactly. The use of effective ion-ion potentials greatly reduces the computer memory required to perform DLMC ion permeation simulations in dielectrically inhomogeneous environments, thus enabling treatment of larger systems than can be handled by numerically exact techniques.     

Comparing the Predictions of the Nonlinear Poisson-Boltzmann Equation and the Ion Size-Modified Poisson-Boltzmann Equation for a Low-Dielectric Charged Spherical Cavity in an Aqueous Salt Solution

The ion size-modified Poisson Boltzmann equation (SMPBE) is applied to the simple model problem of a low-dielectric spherical cavity containing a central charge, in an aqueous salt solution to investigate the finite ion size effect upon the electrostatic free energy and its sensitivity to changes in salt concentration. The SMPBE is shown to predict a very different electrostatic free energy than the nonlinear Poisson-Boltzmann equation (NLPBE) due to the additional entropic cost of placing ions in solution. Although the energy predictions of the SMPBE can be reproduced by fitting an appropriatelysized Stern layer, or ion-exclusion layer to the NLPBE calculations, the size of the Stern layer is difficult to estimate a priori. The SMPBE also produces a saturation layer when the central charge becomes sufficiently large. Ion-competition effects on various integrated quantities such the total number of ions predicted by the SMPBE are qualitatively similar to those given by the NLPBE and those found in available experimental results.     

Nonlocal continuum electrostatic theory predicts surprisingly small energetic penalties for charge burial in proteins

We study the energetics of burying charges, ion pairs, and ionizable groups in a simple protein model using nonlocal continuum electrostatics. Our primary finding is that the nonlocal response leads to markedly reduced solvent screening, comparable to the use of application-specific protein dielectric constants. Employing the same parameters as used in other nonlocal studies, we find that for a sphere of radius 13.4 ? containing a single +1e charge, the nonlocal solvation free energy varies less than 18 kcal/mol as the charge moves from the surface to the center, whereas the difference in the local Poisson model is ～35 kcal/mol. Because an ion pair (salt bridge) generates a comparatively more rapidly varying Coulomb potential, energetics for salt bridges are even more significantly reduced in the nonlocal model. By varying the central parameter in nonlocal theory, which is an effective length scale associated with correlations between solvent molecules, nonlocal-model energetics can be varied from the standard local results to essentially zero; however, the existence of the reduction in charge-burial penalties is quite robust to variations in the protein dielectric constant and the correlation length. Finally, as a simple exploratory test of the implications of nonlocal response, we calculate glutamate pK(a) shifts and find that using standard protein parameters (ε(protein) = 2-4), nonlocal results match local-model predictions with much higher dielectric constants. Nonlocality may, therefore, be one factor in resolving discrepancies between measured protein dielectric constants and the model parameters often used to match titration experiments. Nonlocal models may hold significant promise to deepen our understanding of macromolecular electrostatics without substantially increasing computational complexity.     

Electric-field-controlled water and ion permeation of a hydrophobic nanopore

The permeation of hydrophobic, cylindrical nanopores by water molecules and ions is investigated under equilibrium and out-of-equilibrium conditions by extensive molecular-dynamics simulations. Neglecting the chemical structure of the confining pore surface, we focus on the effects of pore radius and electric field on permeation. The simulations confirm the intermittent filling of the pore by water, reported earlier under equilibrium conditions for pore radii larger than a critical radius R(c). Below this radius, water can still permeate the pore under the action of a strong electric field generated by an ion concentration imbalance at both ends of the pore embedded in a structureless membrane. The water driven into the channel undergoes considerable electrostriction characterized by a mean density up to twice the bulk density and by a dramatic drop in dielectric permittivity which can be traced back to a considerable distortion of the hydrogen-bond network inside the pore. The free-energy barrier to ion permeation is estimated by a variant of umbrella sampling for Na(+), K(+), Ca(2+), and Cl(-) ions, and correlates well with known solvation free energies in bulk water. Starting from an initial imbalance in ion concentration, equilibrium is gradually restored by successive ion passages through the water-filled pore. At each passage the electric field across the pore drops, reducing the initial electrostriction, until the pore, of radius less than R(c), closes to water and hence to ion transport, thus providing a possible mechanism for voltage-dependent gating of hydrophobic pores.     

Role of convection and diffusion in a single pore with adsorptive walls

The importance of intraparticle convection during and after the pressurization step of a pressure swing adsorption process is assessed by considering a single, cylindrical, closed-end pore with adsorptive walls exposed to a binary mixture of an adsorbable component and an inert gas. Gas-phase mass transfer is comprised of pore diffusion and convection, and surface diffusion occurs in the adsorbed phase. Concentration, velocity, and flux profiles are obtained inside the pore both during and after pressurization. Solutions are obtained analytically for the limiting cases of no adsorption, no diffusion, and no inert gas. Complete solutions of the material balance equations are obtained by orthogonal collocation. The pressurization rate, the adsorptive capacity of the pore wall, and the gas-phase mole fraction are varied over a wide range to study the relative importance of convection and diffusion under different conditions. Results show that convection makes a large contribution to transport in the pore except when the adsorbable component has a small mole fraction.     

Thermodynamic characteristics of the hydrate shell of a Na+ ion in a plane nanopore with hydrophobic walls

The chemical potential, free energy, and work of hydration of a single-charged sodium cation are calculated using the Monte Carlo method for a bicanonic statistic ensemble at the molecular level at 298 K in plane model nanopores 0.5 and 0.7 nm wide. It is shown that the nanopores have a stabilizing effect on the hydrate shell of an ion. It is concluded that the crisis of stability that occurs outside a pore is transformed into an abrupt acceleration of growth with the conservation of a stable equilibrium with vapor under the conditions of plane nanopores. It is established that the mechanism of the threshold acceleration of growth inside a pore is associated with an ion being displaced from its own hydrate shell.     

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.     

Energies of ions in water and nanopores within density functional theory

Accurate calculations of electrostatic potentials and treatment of substrate polarizability are critical for predicting the permeation of ions inside water-filled nanopores. The ab initio molecular dynamics method, based on density functional theory (DFT), accounts for the polarizability of materials, water, and solutes, and it should be the method of choice for predicting accurate electrostatic energies of ions. In practice, DFT coupled with the use of periodic boundary conditions in a charged system leads to large energy shifts. Results obtained using different DFT packages may vary because of the way pseudopotentials and long-range electrostatics are implemented. Using maximally localized Wannier functions, we apply robust corrections that yield relatively unambiguous ion energies in select molecular and aqueous systems and inside carbon nanotubes. Large binding energies are predicted for ions in metallic carbon nanotube arrays, while Na+ and Cl- energies are found to exhibit asymmetry in water that is smaller than but comparable with those computed using nonpolarizable water force fields.     

Adsorption of block copolymers in nanoporous alumina

We have studied the adsorption of end-attaching block copolymer chains inside the cylindrical pores of nanoporous alumina. Highly asymmetric PS-PEO block copolymers, with a small PEO anchoring block and a long PS dangling block, were allowed to adsorb onto porous alumina substrates with an average pore diameter of ∼200 nm from toluene solution. The adsorption process was monitored using FTIR spectroscopy, whereas depth profile analysis was performed by means of XPS and Ar⁺ ion sputtering. It is found that the PS-PEO adsorption kinetics in porous alumina are ∼4 orders of magnitude slower than the corresponding case of a flat alumina substrate. It appears that chains adsorbed near the pore entrance early on tend to form a barrier for chains entering the pore at later times, thereby slowing down the adsorption process significantly. This effect is much more pronounced for large chains whose dimensions are comparable with the pore diameter. The equilibrium adsorbance value is also affected by chain size due to the additional entropic penalty associated with chain confinement, the adsorbance falling substantially when the chain dimensions become comparable with the pore diameter. © 2010 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 48: 1676–1682, 2010