The role of solvation in the binding selectivity of the L-type calcium channel

Competition between Born solvation, dielectric exclusion, and Coulomb attraction in spherical nanopores

The recent measurement of a very low dielectric constant, ε, of water confined in nanometric slit pores leads us to reconsider the physical basis of ion partitioning into nanopores. For confined ions in chemical equilibrium with a bulk of dielectric constant ε_{b}>ε, three physical mechanisms, at the origin of ion exclusion in nanopores, are expected to be modified due to this dielectric mismatch: dielectric exclusion at the water-pore interface (with membrane dielectric constant, ε_{m}<ε), the solvation energy related to the difference in Debye-Hückel screening parameters in the pore, κ, and in the bulk κ_{b}, and the classical Born solvation self-energy proportional to ε^{-1}-ε_{b}^{-1}. Our goal is to clarify the interplay between these three mechanisms and investigate the role played by the Born contribution in ionic liquid-vapor (LV) phase separation in confined geometries. We first compute analytically the potential of mean force (PMF) of an ion of radius R_{i} located at the center of a nanometric spherical pore of radius R. Computing the variational grand potential for a solution of confined ions, we then deduce the partition coefficients of ions in the pore versus R and the bulk electrolyte concentration ρ_{b}. We show how the ionic LV transition, directly induced by the abrupt change of the dielectric contribution of the PMF with κ, is favored by the Born self-energy and explore the decrease of the concentration in the pore with ε both in the vapor and liquid states. Phase diagrams are established for various parameter values and we show that a signature of this phase transition can be detected by monitoring the total osmotic pressure as a function of R. For charged nanopores, these exclusion effects compete with the electrostatic attraction that imposes the entry of counterions into the pore to enforce electroneutrality. This study will therefore help in deciphering the respective roles of the Born self-energy and dielectric mismatch in experiments and simulations of ionic transport through nanopores.

Electrokinetic behavior of a pH-regulated dielectric cylindrical nanopore

A continuum model is adopted to describe the electrokinetic behavior of a pH-regulated cylindrical nanopore, the surface of which has charge-regulated carboxyl groups. We focus on the influences of the permittivity of the nanopore material, nanopore size, salt concentration, and solution pH on this behavior, and the underlying mechanisms. The influence of the nanopore permittivity becomes significant when a nanopore is shorter than ca. 50 nm. It is interesting to observe that if it is longer than ca. 100 nm, the nanopore conductance decreases with increasing permittivity. If it is sufficiently short, the conductance increases with increasing permittivity. If the nanopore length takes a medium level, the conductance is insensitive to the variation in the permittivity. For a short nanopore (~20 nm), the conductivity increases with increasing permittivity. However, if pH is sufficiently high, it becomes insensitive to permittivity. Although the larger the permittivity the greater the conductivity, in general, this effect becomes insignificant when the bulk salt concentration is sufficiently high, implying that the effect of membrane polarization is important only if the bulk salt concentration is sufficiently low.

Ion Pairing and Dielectric Decrement in Glycosaminoglycan Brushes

Cell-surface polysaccharides are essential to many aspects of physiology, serving as a highly conserved evolutionary feature of life and as an important part of the innate immune system in mammals. Here, as simplified biophysical models of these sugar coatings, we present results of molecular dynamics simulations of hyaluronic acid and heparin brushes that show important effects of ion pairing, water dielectric decrease, and coion exclusion. As in prior studies of macromolecular crowding under physiologically relevant salt concentrations, our results show equilibria with electroneutrality attained through screening and pairing of brush anionic charges by monovalent cations at the atomistic detail. Most surprising is the reversal of the Donnan potential obtained from both nonpolarizable and Drude polarizable force fields, in contrast to what would be expected based on electrostatic Boltzmann partitioning alone. Water dielectric decrement within the brush domain is also associated with Born hydration-driven cation exclusion from the brush. We observe that the primary partition energy attracting cations to attain brush electroneutrality is the ion pairing or salt-bridge energy. Potassium and sodium pairings to glycosaminoglycan carboxylates and sulfates show similar abundance of contact-pairing and solvent-separated pairing. We conclude that in these crowded macromolecular brushes, ion-pairing, Born-hydration, and electrostatic potential energies all contribute to attain electroneutrality and should therefore contribute in mean-field models to accurately represent brush electrostatics.

Modeling the Device Behavior of Biological and Synthetic Nanopores with Reduced Models

Biological ion channels and synthetic nanopores are responsible for passive transport of ions through a membrane between two compartments. Modeling these ionic currents is especially amenable to reduced models because the device functions of these pores, the relation of input parameters (e.g., applied voltage, bath concentrations) and output parameters (e.g., current, rectification, selectivity), are well defined. Reduced models focus on the physics that produces the device functions (i.e., the physics of how inputs become outputs) rather than the atomic/molecular-scale physics inside the pore. Here, we propose four rules of thumb for constructing good reduced models of ion channels and nanopores. They are about (1) the importance of the axial concentration profiles, (2) the importance of the pore charges, (3) choosing the right explicit degrees of freedom, and (4) creating the proper response functions. We provide examples for how each rule of thumb helps in creating a reduced model of device behavior.

Rectification of bipolar nanopores in multivalent electrolytes: effect of charge inversion and strong ionic correlations

Bipolar nanopores have powerful rectification properties due to the asymmetry in the charge pattern on the wall of the nanopore. In particular, bipolar nanopores have positive and negative surface charges along the pore axis. Rectification is strong if the radius of the nanopore is small compared to the screening length of the electrolyte so that both cations and anions have depletion zones in the respective regions. The depths of these depletion zones is sensitive to sign of the external voltage. In this work, we are interested in the effect of the presence of strong ionic correlations (both between ions and between ions and surface charge) due to the presence of multivalent ions and large surface charges. We show that strong ionic correlations cause leakage of the coions, a phenomenon that is absent in mean field theories. In this modeling study, we use both the mean-field Poisson-Nernst-Planck (PNP) theory and a particle simulation method, Local Equilibrium Monte Carlo (LEMC), to show that phenomena such as overcharging and charge inversion cannot be reproduced with PNP, while LEMC is able to produce nonmonotonic dependence of currents and rectification as a function of surface charge strength.

Molecular Mean-Field Theory of Ionic Solutions: A Poisson-Nernst-Planck-Bikerman Model

We have developed a molecular mean-field theory—fourth-order Poisson–Nernst–Planck–Bikerman theory—for modeling ionic and water flows in biological ion channels by treating ions and water molecules of any volume and shape with interstitial voids, polarization of water, and ion-ion and ion-water correlations. The theory can also be used to study thermodynamic and electrokinetic properties of electrolyte solutions in batteries, fuel cells, nanopores, porous media including cement, geothermal brines, the oceanic system, etc. The theory can compute electric and steric energies from all atoms in a protein and all ions and water molecules in a channel pore while keeping electrolyte solutions in the extra- and intracellular baths as a continuum dielectric medium with complex properties that mimic experimental data. The theory has been verified with experiments and molecular dynamics data from the gramicidin A channel, L-type calcium channel, potassium channel, and sodium/calcium exchanger with real structures from the Protein Data Bank. It was also verified with the experimental or Monte Carlo data of electric double-layer differential capacitance and ion activities in aqueous electrolyte solutions. We give an in-depth review of the literature about the most novel properties of the theory, namely Fermi distributions of water and ions as classical particles with excluded volumes and dynamic correlations that depend on salt concentration, composition, temperature, pressure, far-field boundary conditions etc. in a complex and complicated way as reported in a wide range of experiments. The dynamic correlations are self-consistent output functions from a fourth-order differential operator that describes ion-ion and ion-water correlations, the dielectric response (permittivity) of ionic solutions, and the polarization of water molecules with a single correlation length parameter.

Energetics of counterion adsorption in the electrical double layer

The energetics of the electrical double layer (EDL) is studied in a systematic way to define how different components of the chemical potential help or hinder cation adsorption at a negatively charged wall. Specifically, the steric (i.e., excluded-volume interactions), mean electrostatic, and screening (i.e., electrostatic correlations beyond the mean field) components were computed using classical density functional theory of the primitive model of ions (i.e., ions as charged, hard spheres in a background dielectric). The reduced physics of the primitive model allows for an extensive analysis over a large parameter space: cation valences +1, +2, and +3, cation diameters 0.15, 0.30, 0.60, and 0.90 nm, bulk concentrations ranging from 1 µM to 1M, and surface charges ranging from 0 to -0.50 C/m². Our results show that all components are necessary to understand the physics of the EDL. The screening component is always significant; for small monovalent cations such as K⁺, it is generally much larger than the steric component, and for multivalent ions, charge inversion cannot occur without it. At moderate surface charges, the screening component makes the electrostatic potential less negative than in classical Poisson-Boltzmann theory, sometimes even positive (charge inversion). At high surface charges, this is overcome by the repulsive potential of the steric component as the first ion layer becomes extremely crowded. Large negative electrostatic potentials counteract this to draw even more cations into the first layer. Although we used an approximate model of the EDL, the physics inherent in these trends appears to be quite general.

Simulation of a model nanopore sensor: Ion competition underlies device behavior

We study a model nanopore sensor with which a very low concentration of analyte molecules can be detected on the basis of the selective binding of the analyte molecules to the binding sites on the pore wall. The bound analyte ions partially replace the current-carrier cations in a thermodynamic competition. This competition depends both on the properties of the nanopore and the concentrations of the competing ions (through their chemical potentials). The output signal given by the device is the current reduction caused by the presence of the analyte ions. The concentration of the analyte ions can be determined through calibration curves. We model the binding site with the square-well potential and the electrolyte as charged hard spheres in an implicit background solvent. We study the system with a hybrid method in which we compute the ion flux with the Nernst-Planck (NP) equation coupled with the Local Equilibrium Monte Carlo (LEMC) simulation technique. The resulting NP+LEMC method is able to handle both strong ionic correlations inside the pore (including finite size of ions) and bulk concentrations as low as micromolar. We analyze the effect of bulk ion concentrations, pore parameters, binding site parameters, electrolyte properties, and voltage on the behavior of the device.

Incorporating Born solvation energy into the three-dimensional Poisson-Nernst-Planck model to study ion selectivity in KcsA K^{+} channels

Potassium channels are much more permeable to potassium than sodium ions, although potassium ions are larger and both carry the same positive charge. This puzzle cannot be solved based on the traditional Poisson-Nernst-Planck (PNP) theory of electrodiffusion because the PNP model treats all ions as point charges, does not incorporate ion size information, and therefore cannot discriminate potassium from sodium ions. The PNP model can qualitatively capture some macroscopic properties of certain channel systems such as current-voltage characteristics, conductance rectification, and inverse membrane potential. However, the traditional PNP model is a continuum mean-field model and has no or underestimates the discrete ion effects, in particular the ion solvation or self-energy (which can be described by Born model). It is known that the dehydration effect (closely related to ion size) is crucial to selective permeation in potassium channels. Therefore, we incorporated Born solvation energy into the PNP model to account for ion hydration and dehydration effects when passing through inhomogeneous dielectric channel environments. A variational approach was adopted to derive a Born-energy-modified PNP (BPNP) model. The model was applied to study a cylindrical nanopore and a realistic KcsA channel, and three-dimensional finite element simulations were performed. The BPNP model can distinguish different ion species by ion radius and predict selectivity for K^{+} over Na^{+} in KcsA channels. Furthermore, ion current rectification in the KcsA channel was observed by both the PNP and BPNP models. The I-V curve of the BPNP model for the KcsA channel indicated an inward rectifier effect for K^{+} (rectification ratio of ∼3/2) but indicated an outward rectifier effect for Na^{+} (rectification ratio of ∼1/6).

Effects of electromagnetic field exposure on conduction and concentration of voltage gated calcium channels: A Brownian dynamics study

A three-dimensional Brownian Dynamics (BD) in combination with electrostatic calculations is employed to specifically study the effects of radiation of high frequency electromagnetic fields on the conduction and concentration profile of calcium ions inside the voltage-gated calcium channels. The electrostatic calculations are performed using COMSOL Multiphysics by considering dielectric interfaces effectively. The simulations are performed for different frequencies and intensities. The simulation results show the variations of conductance, average number of ions and the concentration profiles of ions inside the channels in response to high frequency radiation. The ionic current inside the channel increases in response to high frequency electromagnetic field radiation, and the concentration profiles show that the residency of ions in the channel decreases accordingly.

Electrolytes between dielectric charged surfaces: Simulations and theory

We present a simulation method to study electrolyte solutions in a dielectric slab geometry using a modified 3D Ewald summation. The method is fast and easy to implement, allowing us to rapidly resum an infinite series of image charges. In the weak coupling limit, we also develop a mean-field theory which allows us to predict the ionic distribution between the dielectric charged plates. The agreement between both approaches, theoretical and simulational, is very good, validating both methods. Examples of ionic density profiles in the strong electrostatic coupling limit are also presented. Finally, we explore the confinement of charge asymmetric electrolytes between neutral surfaces.

Selecting ions by size in a calcium channel: the ryanodine receptor case study

Many calcium channels can distinguish between ions of the same charge but different size. For example, when cations are in direct competition with each other, the ryanodine receptor (RyR) calcium channel preferentially conducts smaller cations such as Li(+) and Na(+) over larger ones such as K(+) and Cs(+). Here, we analyze the physical basis for this preference using a previously established model of RyR permeation and selectivity. Like other calcium channels, RyR has four aspartate residues in its GGGIGDE selectivity filter. These aspartates have their terminal carboxyl group in the pore lumen, which take up much of the available space for permeating ions. We find that small ions are preferred by RyR because they can fit into this crowded environment more easily.