Saturation of conductance in single ion channels: the blocking effect of the near reaction field

Biphasic concentration patterns in ionic transport under nanoconfinement revealed in steady-state and time-dependent properties

Ion permeation across nanoscopic structures differs considerably from microfluidics because of strong steric constraints, transformed solvent properties, and charge-regulation effects revealed mostly in diluted solutions. However, little is known about nanofluidics in moderately concentrated solutions, which are critically important for industrial applications and living systems. Here, we show that nanoconfinement triggers general biphasic concentration patterns in a myriad of ion transport properties by using two contrasting systems: a biological ion channel and a much larger synthetic nanopore. Our findings show a low-concentration regime ruled by classical Debye screening and another one where ion-ion correlations and enhanced ion-surface interactions contribute differently to each electrophysiological property. Thus, different quantities (e.g., conductance vs noise) measured under the same conditions may appear contradictory because they belong to different concentration regimes. In addition, non-linear effects that are barely visible in bulk conductivity only in extremely concentrated solutions become apparent in nanochannels around physiological conditions.

Ion transport in nanopores with highly overlapping electric double layers

Investigation of ion transport through nanopores with highly overlapping electric double layers is extremely challenging. This can be attributed to the non-linear Poisson-Boltzmann equation that governs the behavior of the electrical potential distribution as well as other characteristics of ion transport. In this work, we leverage the approach of Schnitzer and Yariv [Phys. Rev. E 87, 054301 (2013)] to reduce the complexity of the governing equation. An asymptotic solution is derived, which shows remarkable correspondence to simulations of the non-approximated equations. This new solution is leveraged to address a number of highly debated issues. We derive the equivalent of the Gouy-Chapman equation for systems with highly overlapping electric double layers. This new relationship between the surface charge density and the surface potential is then utilized to determine the power-law scaling of nanopore conductances as a function of the bulk concentrations. We derive the coefficients of transport for the case of overlapping electric double layers and compare it to the renowned uniform potential model. We show that the uniform potential model is only an approximation for the exact solution for small surface charges. The findings of this work can be leveraged to uncover additional hidden attributes of ion transport through nanopores.

Multiscale modeling shows that dielectric differences make NaV channels faster than KV channels

The generation of action potentials in excitable cells requires different activation kinetics of voltage-gated Na (Na_V) and K (K_V) channels. Na_V channels activate much faster and allow the initial Na⁺ influx that generates the depolarizing phase and propagates the signal. Recent experimental results suggest that the molecular basis for this kinetic difference is an amino acid side chain located in the gating pore of the voltage sensor domain, which is a highly conserved isoleucine in K_V channels but an equally highly conserved threonine in Na_V channels. Mutagenesis suggests that the hydrophobicity of this side chain in Shaker K_V channels regulates the energetic barrier that gating charges cross as they move through the gating pore and control the rate of channel opening. We use a multiscale modeling approach to test this hypothesis. We use high-resolution molecular dynamics to study the effect of the mutation on polarization charge within the gating pore. We then incorporate these results in a lower-resolution model of voltage gating to predict the effect of the mutation on the movement of gating charges. The predictions of our hierarchical model are fully consistent with the tested hypothesis, thus suggesting that the faster activation kinetics of Na_V channels comes from a stronger dielectric polarization by threonine (Na_V channel) produced as the first gating charge enters the gating pore compared with isoleucine (K_V channel).

Approximate time-dependent current-voltage relations for currents exceeding the diffusion limit

The time-dependent behavior of one-dimensional ion transport into a permselective medium is theoretically modeled in this work for currents exceeding the diffusion limit. Leveraging the findings of Yariv [E. Yariv, Phys. Rev. E 80, 051201 (2009)10.1103/PhysRevE.80.051201], we derive three separate expressions for the potential drop for short, intermediate, and long times. We show that the potential drop correlates to the time evolution of the space-charge layer adjacent to the permselective interface. Our approximate models show remarkable correspondence to numerical simulations.

Molecular determination of claudin-15 organization and channel selectivity

Members of the claudin family form tight junctions between adjacent epithelial and endothelial cells. Samanta et al. build an atomic model of claudin-15 using molecular dynamics simulations and conclude that four claudin-15 molecules each contribute an aspartic acid residue to form a selectivity filter.

Tight junctions are macromolecular structures that traverse the space between adjacent cells in epithelia and endothelia. Members of the claudin family are known to determine tight junction permeability in a charge- and size-selective manner. Here, we use molecular dynamics simulations to build and refine an atomic model of claudin-15 channels and study its transport properties. Our simulations indicate that claudin-15 forms well-defined channels for ions and molecules and otherwise “seals” the paracellular space through hydrophobic interactions. Ionic currents, calculated from simulation trajectories of wild-type as well as mutant channels, reflect in vitro measurements. The simulations suggest that the selectivity filter is formed by a cage of four aspartic acid residues (D55), contributed by four claudin-15 molecules, which creates a negative electrostatic potential to favor cation flux over anion flux. Charge reversal or charge ablation mutations of D55 significantly reduce cation permeability in silico and in vitro, whereas mutations of other negatively charged pore amino acid residues have a significantly smaller impact on channel permeability and selectivity. The simulations also indicate that water and small ions can pass through the channel, but larger cations, such as tetramethylammonium, do not traverse the pore. Thus, our model provides an atomic view of claudin channels, their transport function, and a potential three-dimensional organization of its selectivity filter.

Interplay between Nanochannel and Microchannel Resistances

Current nanochannel system paradigm commonly neglects the role of the interfacing microchannels and assumes that the ohmic electrical response of a microchannel-nanochannel system is solely determined by the geometric properties of the nanochannel. In this work, we demonstrate that the overall response is determined by the interplay between the nanochannel resistance and various microchannel attributed resistances. Our experiments confirm a recent theoretical prediction that in contrast to what was previously assumed at very low concentrations the role of the interfacing microchannels on the overall resistance becomes increasingly important. We argue that the current nanochannel-dominated conductance paradigm can be replaced with a more correct and intuitive microchannel-nanochannel-resistance-model-based paradigm.

Energetics of discrete selectivity bands and mutation-induced transitions in the calcium-sodium ion channels family

We use Brownian dynamics (BD) simulations to study the ionic conduction and valence selectivity of a generic electrostatic model of a biological ion channel as functions of the fixed charge Q(f) at its selectivity filter. We are thus able to reconcile the discrete calcium conduction bands recently revealed in our BD simulations, M0 (Q(f)=1e), M1 (3e), M2 (5e), with a set of sodium conduction bands L0 (0.5e), L1 (1.5e), thereby obtaining a completed pattern of conduction and selectivity bands vs Q(f) for the sodium-calcium channels family. An increase of Q(f) leads to an increase of calcium selectivity: L0 (sodium-selective, nonblocking channel) → M0 (nonselective channel) → L1 (sodium-selective channel with divalent block) → M1 (calcium-selective channel exhibiting the anomalous mole fraction effect). We create a consistent identification scheme where the L0 band is putatively identified with the eukaryotic sodium channel The scheme created is able to account for the experimentally observed mutation-induced transformations between nonselective channels, sodium-selective channels, and calcium-selective channels, which we interpret as transitions between different rows of the identification table. By considering the potential energy changes during permeation, we show explicitly that the multi-ion conduction bands of calcium and sodium channels arise as the result of resonant barrierless conduction. The pattern of periodic conduction bands is explained on the basis of sequential neutralization taking account of self-energy, as Q(f)(z,i)=ze(1/2+i), where i is the order of the band and z is the valence of the ion. Our results confirm the crucial influence of electrostatic interactions on conduction and on the Ca(2+)/Na(+) valence selectivity of calcium and sodium ion channels. The model and results could be also applicable to biomimetic nanopores with charged walls.

Analyzing the components of the free-energy landscape in a calcium selective ion channel by Widom's particle insertion method

The selectivity filter of the L-type calcium channel works as a Ca(2+) binding site with a very large affinity for Ca(2+) versus Na(+). Ca(2+) replaces half of the Na(+) ions in the filter even when these ions are present in 1 μM and 30 mM concentrations in the bath, respectively. The energetics of this strong selectivity is analyzed in this paper. We use Widom's particle insertion method to compute the space-dependent profiles of excess chemical potential in our grand canonical Monte Carlo simulations. These profiles define the free-energy landscape for the various ions. Following Gillespie [Biophys. J. 94, 1169 (2008)], the difference of the excess chemical potentials for the two competing ions defines the advantage that one of the ions has over the other in the competition for space in the crowded selectivity filter. These advantages depend on ionic bath concentrations: the ion that is present in the bath in larger quantity (Na(+)) has the "number" advantage which is balanced by the free-energy advantage of the other ion (Ca(2+)). The excess chemical potentials are decomposed into hard sphere exclusion and electrostatic components. The electrostatic terms correspond to interactions with the mean electric field produced by ions and induced charges as well to ionic correlations beyond the mean field description. Dielectrics are needed to produce micromolar Ca(2+) versus Na(+) selectivity in the L-type channel. We study the behavior of these terms with changes in bath concentrations of ions, charges, and diameters of ions, as well as geometrical parameters such as radius of the pore and the dielectric constant of the protein. Ion selectivity in calcium binding proteins probably has a similar mechanism.

Intrinsic electrostatic potential in the BK channel pore: role in determining single channel conductance and block

The internal vestibule of large-conductance Ca²⁺ voltage-activated K⁺ (BK) channels contains a ring of eight negative charges not present in K⁺ channels of lower conductance (Glu386 and Glu389 in hSlo) that modulates channel conductance through an electrostatic mechanism (Brelidze, T.I., X. Niu, and K.L. Magleby. 2003. Proc. Natl. Acad. Sci. USA. 100:9017–9022). In BK channels there are also two acidic amino acid residues in an extracellular loop (Asp326 and Glu329 in hSlo). To determine the electrostatic influence of these charges on channel conductance, we expressed wild-type BK channels and mutants E386N/E389N, D326N, E329Q, and D326N/E329Q channels on Xenopus laevis oocytes, and measured the expressed currents under patch clamp. Contribution of E329 to the conductance is negligible and single channel conductance of D326N/E329Q channels measured at 0 mV in symmetrical 110 mM K⁺ was 18% lower than the control. Current–voltage curves displayed weak outward rectification for D326N and the double mutant. The conductance differences between the mutants and wild-type BK were caused by an electrostatic effect since they were enhanced at low K⁺ (30 mM) and vanished at high K⁺ (1 M K⁺). We determine the electrostatic potential change, Δφ, caused by the charge neutralization using TEA⁺ block for the extracellular charges and Ba²⁺ for intracellular charges. We measured 13 ± 2 mV for Δφ at the TEA⁺ site when turning off the extracellular charges, and 17 ± 2 mV for the Δφ at the Ba²⁺ site when the intracellular charges were turned off. To understand the electrostatic effect of charge neutralizations, we determined Δφ using a BK channel molecular model embedded in a lipid bilayer and solving the Poisson-Boltzmann equation. The model explains the experimental results adequately and, in particular, gives an economical explanation to the differential effect on the conductance of the neutralization of charges D326 and E329.

Stochastic study of the effect of ionic strength on noncovalent interactions in protein pores

Salt plays a critical role in the physiological activities of cells. We show that ionic strength significantly affects the kinetics of noncovalent interactions in protein channels, as observed in stochastic studies of the transfer of various analytes through pores of wild-type and mutant alpha-hemolysin proteins. As the ionic strength increased, the association rate constant of electrostatic interactions was accelerated, whereas those of both hydrophobic and aromatic interactions were retarded. Dramatic decreases in the dissociation rate constants, and thus increases in the overall reaction formation constants, were observed for all noncovalent interactions studied. The results suggest that with the increase of salt concentration, the streaming potentials for all the protein pores decrease, whereas the preferential selectivities of the pores for either cations or anions drop. Furthermore, results also show that the salt effect on the rate of association of analytes to a pore differs significantly depending on the nature of the noncovalent interactions occurring in the protein channel. In addition to providing new insights into the nature of analyte-protein pore interactions, the salt-dependence of noncovalent interactions in protein pores observed provides a useful means to greatly enhance the sensitivity of the nanopore, which may find useful application in stochastic sensing.

Steric selectivity in Na channels arising from protein polarization and mobile side chains

Monte Carlo simulations of equilibrium selectivity of Na channels with a DEKA locus are performed over a range of radius R and protein dielectric coefficient epsilon(p). Selectivity arises from the balance of electrostatic forces and steric repulsion by excluded volume of ions and side chains of the channel protein in the highly concentrated and charged (approximately 30 M) selectivity filter resembling an ionic liquid. Ions and structural side chains are described as mobile charged hard spheres that assume positions of minimal free energy. Water is a dielectric continuum. Size selectivity (ratio of Na+ occupancy to K+ occupancy) and charge selectivity (Na+ to Ca2+) are computed in concentrations as low as 10(-5) M Ca2+. In general, small R reduces ion occupancy and favors Na+ over K+ because of steric repulsion. Small epsilon(p) increases occupancy and favors Na+ over Ca2+ because protein polarization amplifies the pore's net charge. Size selectivity depends on R and is independent of epsilon(p); charge selectivity depends on both R and epsilon(p). Thus, small R and epsilon(p) make an efficient Na channel that excludes K+ and Ca2+ while maximizing Na+ occupancy. Selectivity properties depend on interactions that cannot be described by qualitative or verbal models or by quantitative models with a fixed free energy landscape.

Ca2+ selectivity of a chemically modified OmpF with reduced pore volume

We studied an E. coli OmpF mutant (LECE) containing both an EEEE-like locus, typical of Ca(2+) channels, and an accessible and reactive cysteine. After chemical modification with the cysteine-specific, negatively charged (-1e) reagents MTSES or glutathione, this LECE mutant was tested for Ca(2+) versus alkali metal selectivity. Selectivity was measured by conductance and zero-current potential. Conductance measurements showed that glutathione-modified LECE had reduced conductance at Ca(2+) mole fractions <10(-3). MTSES-modified LECE did not. Apparently, the LECE protein is (somehow) a better Ca(2+) chelator after modification with the larger glutathione. Zero-current potential measurements revealed a Ca(2+) versus monovalent cation selectivity that was highest in the presence of Li(+) and lowest in the presence of Cs(+). Our data clearly show that after the binding of Ca(2+) the LECE pore (even with the bulky glutathione present) is spacious enough to allow monovalent cations to pass. Theoretical computations based on density functional theory combined with Poisson-Nernst-Planck theory and a reduced pore model suggest a functional separation of ionic pathways in the pore, one that is specific for small and highly charged ions, and one that accepts preferentially large ions, such as Cs(+).

Ion exchange phase transitions in water-filled channels with charged walls

Ion transport through narrow water-filled channels is impeded by a high electrostatic barrier. The latter originates from the large ratio of the dielectric constants of the water and the surrounding media. We show that "doping," i.e., immobile charges attached to the walls of the channel, substantially reduces the barrier. This explains why most of the biological ion channels are "doped." We show that at rather generic conditions the channels may undergo ion exchange phase transitions (typically of the first order). Upon such a transition a finite latent concentration of ions may either enter or leave the channel, or be exchanged between the ions of different valences. We discuss possible implications of these transitions for the Ca-vs-Na selectivity of biological Ca channels. We also show that transport of divalent Ca ions is assisted by their fractionalization into two separate excitations.

Et tu, Grotthuss! and other unfinished stories

This review article is divided into three sections. In Section 1, a short biographical note on Freiherr von Grotthuss is followed by a detailed summary of the main findings and ideas present in his 1806 paper. Attempts to place Grotthuss contribution in the context of the science done at his time were also made. In Section 2, the modern version of the Grotthuss mechanism is reviewed. The classical Grotthuss model has been recently questioned and new mechanisms and ideas regarding proton transfer are briefly discussed. The last section discusses the significance of a classical Grotthuss mechanism for proton transfer in water chains inside protein cavities. This has been an interesting new twist in the ongoing history of the Grotthuss mechanism. A summary and discussion of what was learned from probably the simplest currently available experimental models of proton transfer in water wires in semi-synthetic ion channels are critically presented. This review ends discussing some of the questions that need to be addressed in the near future.

Computing numerically the access resistance of a pore

The access resistance (AR) of a channel is an important component of the conductance of ion channels, particularly in wide and short channels, where it accounts for a substantial fraction of the total resistance to the movement of ions. The AR is usually calculated by using a classical and simple expression derived by Hall from electrostatics (J.E. Hall 1975 J. Gen. Phys. 66:531-532), though other expressions, both analytical and numerical, have been proposed. Here we report some numerical results for the AR of a channel obtained by solving the Poisson-Nernst-Planck equations at the entrance of a circular pore. Agreement is found between numerical calculations and analytical results from Hall's equation for uncharged pores in neutral membranes. However, for channels embedded in charged membranes, Hall's expression overestimates the AR, which is much lower and can even be neglected in some cases. The weak dependence of AR on the pore radius for charged membranes at low salt concentration can be exploited to separate the channel and the access contributions to the measured conductance.