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In order to permeate a nanopore, an ion must overcome a dehydration energy barrier caused by the redistribution of surrounding water molecules. The redistribution is inhomogeneous, anisotropic and strongly position-dependent, resulting in complex patterns that are routinely observed in molecular dynamics simulations. Here, we study the physical origin of these patterns and of how they can be predicted and controlled. We introduce an analytic model able to predict the patterns in a graphene nanopore in terms of experimentally accessible radial distribution functions, giving results that agree well with molecular dynamics simulations. The patterns are attributable to a complex interplay of ionic hydration shells with water layers adjacent to the graphene membrane and with the hydration cloud of the nanopore rim atoms, and we discuss ways of controlling them. Our findings pave the way to designing required transport properties into nanoionic devices by optimising the structure of the hydration patterns.

Biological ion channels are fundamental to maintaining life. In this manuscript we apply our recently developed statistical and linear response theory to investigate Na+ conduction through the prokaryotic Na+ channel NaChBac. This work is extended theoretically by the derivation of ionic conductivity and current in an electrochemical gradient, thus enabling us to compare to a range of whole-cell data sets performed on this channel. Furthermore, we also compare the magnitudes of the currents and populations at each binding site to previously published single-channel recordings and molecular dynamics simulations respectively. In doing so, we find excellent agreement between theory and data, with predicted energy barriers at each of the four binding sites of ∼4,2.9,3.6, and 4kT.

Mutation-induced transformations of conductivity and selectivity in NaChBac bacterial channels are studied experimentally and interpreted within the framework of ionic Coulomb blockade (ICB), while also taking account of resonant quantised dehydration (QD) and site protonation. Site-directed mutagenesis and whole-cell patch-clamp experiments are used to investigate how the fixed charge Qf at the selectivity filter (SF) affects both valence selectivity and same-charge selectivity. The new ICB/QD model predicts that increasing |Qf | should lead to a shift in selectivity sequences towards larger ion sizes, in agreement with the present experiments and with earlier work. Comparison of the model with experimental data leads to the introduction of an effective charge Qf∗ at the SF, which was found to differ between Aspartate and Glutamate charged rings, and also to depend on position within the SF. It is suggested that protonation of the residues within the restricted space of the SF is important in significantly reducing the effective charge of the EEEE ring. Values of Qf∗ derived from experiments on divalent blockade agree well with expectations based on the ICB/QD model and have led to the first demonstration of ICB oscillations in Ca2+ conduction as a function of the fixed charge. Preliminary studies of the dependence of Ca2+ conduction on pH are qualitatively consistent with the predictions of the model.

Selective permeation through graphene nanopores is attracting increasing interest as an efficient and cost-effective technique for water desalination and purification. In this work, using umbrella sampling and molecular dynamics simulations with constant electric field, we analyze the influence of pore charge on potassium and chloride ion permeation. As pore charge is increased, the barrier of the potential of mean force (PMF) gradually decreases until it turns into a well split in two subminima. While in the case of K+ this pattern can be explained as an increasing electrostatic compensation of the desolvation cost, in the case of Cl􀀀 the pattern can be attributed to the accumulation of a concentration polarization layer of potassium ions screening pore charge. Th analysis of potassium PMFs in terms of forces revealed a conflicting influence on permeation of van der Waals and electrostatic forces that both undergo an inversion of their direction as pore charge is increased. Even if the most important transition involves the interplay between the electrostatic forces exerted by graphene and water, the simulations also revealed an important role of the changing distribution of potassium and chloride ions. The influence of pore charge on the orientation of water molecules was also found to affect the van der Waals forces they exert on potassium.

We analyze selective ionic conduction through an artificial nanopore in a single graphene sheet, using molecular dynamics simulations and the prehistory probability distribution. We assess position-dependent changes in the number and orientation of water molecules in the first and second hydration shells of the ion as it crosses the nanopore. We reveal coupling between an ionic double layer near the sheet to the statistical properties of the hydration shells.

We consider the selective permeation of ions through narrow water-filled channels in the presence of strong interaction between the ions. These interactions lead to highly correlated ionic motion, which can conveniently be described via the concept of a quasiparticle. Here, we connect the quasiparticle’s effective potential and the multi-ion potential of the mean force, found through molecular dynamics simulations, and we validate the method on an analytical toy model of the KcsA channel. Possible future applications of the method to the connection between molecular dynamical calculations and the experimentally measured current-voltage and current-concentration characteristics of the channel are discussed.

We study the highly-concerted motion of ions in anarrow biological ion channel (KcsA) by considering the notionof a quasiparticle, with specific focus on the transition process.Namely, we show that the ion entering or exiting the channelis correlated with the position of the quasiparticle. This resultis of importance in the rate theories of ion conduction throughnarrow channels and artificial nanopores.

The problem of predicting selective transport of ions through nano-pores from their structure in the biological and nano-technological systems is addressed. We use a molecular dynamics simulation to provide insight into the key physical parameters of nano-pores and develop a self-consistent analytic theory describing ionic conduction and selectivity through these devices. We analyse the ion's dehydration and excess chemical potential, derive an expression for the conductivity of the nano-pore, and emphasize the role of fluctuations in its performance. The theory is verified by comparison of the predicted current-voltage characteristics with the molecular dynamics results and experimental data obtained for a graphene nano-pore and the KcsA biological channel.

To establish the general statistical mechanical properties of highly conductive but selective nano-filters we develop an equilibrium statistical-mechanical theory of the KcsA filter, find the probabilities for the filter to bind ions from the mixed intra- and extra-cellular solutions, and evaluate the conductivity of the filter in its linear response regime. The results provide first principles analytical resolution of the long-standing paradox - how can narrow filter conduct potassium ions at nearly the rate of free diffusion while strongly selecting them over sodium ions - and are applicable to a wide range of biological and artificial channels.

Ionic Coulomb blockade (ICB) is an electrostatic phenomenon recently discovered in low-capacitance ion channels/nanopores. Depending on the fixed charge that is present, ICB strongly and selectively influences the ease with which a given type of ion can permeate the pore. The phenomenon arises from the discreteness of the charge-carriers, the dielectric self-energy, an electrostatic exclusion principle, and sequential pore neutralization, and it manifests itself strongly for divalent ions (e.g.\ Ca$^{2+}$). Ionic Coulomb blockade is closely analogous to electronic Coulomb blockade in quantum dots. In addition to the non-local 1D Coulomb interaction considered in the standard Coulomb blockade approach, we now propose a correction to take account of the singular part of the attraction to the binding site (i.e.\ local site binding). We show that this correction leads to a geometry-dependent shift of one of the barrierless resonant conduction points M$_0^{CB}$. We also show that local ion-ion repulsion accounts for a splitting of Ca$^{2+}$ profiles observed earlier in Brownian dynamics simulations., 4 pages, 4 figures, 25 references, ICNF2017 Replacement reasons: Some typo are fixed, notation is optimised)

Background. The selectivity of biological cation channels is defined by a short, narrow selectivity filter, having a negative net fixed charge Qf . Voltage gated bacterial channels (NaChBac and some others) are frequently used in biophysics as simplified models of mammalian calcium and sodium channels. We report an experimental, analytic and numerical study of the effects of Qf and bulk ionic concentrations of Ca2+ and Na+ on conduction and selectivity of NaChBac channels, wild type and Qf -varied mutants. Methods. Site-directed mutagenesis and voltage clamp recordings were used to investigate the Na+ /Ca2+ selectivity, divalent blockade and anomalous mole fraction effect (AMFE) for different NaChBac wild type/mutants channels and the properties dependence on Qf . Experimental results were compared with Brownian dynamics simulations and with analytic predictions of the ionic Coulomb blockade (ICB) model, which was extended to encompass bulk concentration effects. Results. It was shown that changing of Qf from –4e (for LESWAS wild type) to –8e (for LEDWAS mutant) leads to strong divalent blockade of the Na+ current by micromolar amounts of Ca2+ ions, similar to the effects seen in mammalian calcium channels. The BD simulations revealed a concentration-related logarithmic shift of the conduction bands. These results were shown to be consistent with ICB model predictions. Conclusions. The extended ICB model explains the experimental (divalent blockade and AMFE) and simulated (multi-ion bands and their concentration-related shifts) selectivity phenomena of NaChBac channel and its charge-varied mutants. These results extend the understanding of ion channel selectivity and may also be applicable to biomimetic nanopores with charged walls.

The highly selective permeation of ions through biological ion channels is an unsolved problem of noise and fluctuations. In this paper, we motivate and introduce a non-equilibrium and self-consistent multi-species kinetic model, with the express aims of comparing with experimental recordings of current versus voltage and concentration and extracting important permeation parameters. For self-consistency, the behavior of the model at the two-state, i.e., selective limit in linear response, must agree with recent results derived from an equilibrium statistical theory. The kinetic model provides a good fit to data, including the key result of an anomalous mole fraction effect.

The highly selective permeation of ions through biological ion channels can be described and explained in terms of fluctuational dynamics under the influence of powerful electrostatic forces. Hence valence selectivity, e.g. between Ca2+ and Na+ in calcium and sodium channels, can be described in terms of ionic Coulomb blockade, which gives rise to distinct conduction bands and stop-bands as the fixed negative charge Qf at the selectivity filter of the channel is varied. This picture accounts successfully for a wide range of conduction phenomena in a diversity of ion channels. A disturbing anomaly, however, is that what appears to be the same electrostatic charge and structure (the so-called EEEE motif) seems to select Na+ conduction in bacterial channels but Ca2+ conduction in mammalian channels. As a possible resolution of this paradox it is hypothesised that an additional charged protein residue on the permeation path of the mammalian channel increases |Qf | by e, thereby altering the selectivity from Na+ to Ca2+. Experiments are proposed that will enable the hypothesis to be tested.

The conduction and selectivity of calcium/sodium ion channels are described in terms of ionic Coulomb blockade, a phenomenon based on charge discreteness, an electrostatic exclusion principle, and stochastic ion motion through the channel. This novel approach provides a unified explanation of numerous observed and modelled conductance and selectivity phenomena, including the anomalous mole fraction effect and discrete conduction bands. Ionic Coulomb blockade and resonant conduction are similar to electronic Coulomb blockade and resonant tunnelling in quantum dots. The model is equally applicable to other nanopores.

The ability of the potassium channel to conduct K+ at almost the rate of free diffusion, while discriminating strongly against the (smaller) Na+ ion, is of enormous biological importance [1]. Yet its function remains at the center of a “many-voiced debate” [2,3]. In this presentation, a first-principles explanation is provided for the seemingly paradoxical coexistence of high conductivity with high selectivity between monovalent ions within the channel. It is shown that the conductivity of the selectivity filter is described by the generalized Einstein relation. A novel analytic approach to the analysis of the conductivity is proposed, based on the derivation of an effective grand canonical ensemble for ions within the filter. The conditions for barrier-less diffusion-limited conduction through the KcsA filter are introduced, and the relationships between system parameters required to satisfy these conditions are derived. It is shown that the Eisenman selectivity equation is one of these, and that it follows directly from the condition for barrier-less conduction. The proposed theory provides analytical insight into the “knock-on” [1] and Coulomb blockade [4] mechanisms of K+ conduction through the KcsA filter. It confirms and illuminates an earlier argument [3] that the “snug-fit" model cannot describe the fast diffusion-limited conduction seen in experiments. Numerical examples are provided illustrating agreement of the theory with experimentally-measured I-V curves. The results are not restricted to biological systems, but also carry implications for the design of artificial nanopores.

The flow of ions through a biological ion channel can be considered as transitions between occupied energy levels in the channel and either of the connecting bulk reservoirs [2]. Discreteness of ions and an electrostatic exclusion principle ensure that the number of channel energy levels equals the number of occupying ions. Using these fundamental physical principles we have recently introduced [1] an ionic Coulomb blockade (ICB) theory developed by analogy with the similar phenomenon of electron tunnelling in quantum dots [2,3]. In this picture channel selectivity is governed by energy level changes [1]. We present details of the ICB theory for ion transitions through the channel. It incorporates physiological solutions and channel properties: physical dimension, voltage drop and fixed charge, and hence allows for comparison with physiological data. The set of kinetic equations obtained using ICB is analysed. The channel probability of occupancy as a function of transition rates (and hence fixed charge and number of ions) is obtained in the steady-state approximation. It is shown that this probability displays the staircase structure familiar from analysis of occupancy in quantum dots. It is also shown that current through the channel displays sharp peaks as a function of fixed charge, hence relating channel selectivity to the structure and position of energy levels. The contribution of hydration energy is also discussed. We anticipate that inclusion of this energy into ICB theory will provide an important insight into the selectivity and conductivity of ion channels.

The effect of fluctuations on the conductivity of ion channels is investigated. It is shown that modulation of the potential barrier at the selectivity site due to electrostatic amplification of charge fluctuations at the channel mouth exerts a leading-order effect on the channel conductivity. A Brownian dynamical model of ion motion in a channel is derived that takes into account both fluctuations at the channel mouth and vibrational modes of the wall. The charge fluctuations are modeled as a short noise flipping the height of the potential barrier. The wall fluctuations are introduced as a slow vibrational mode of protein motion that modulates ion conductance both stochastically and periodically. The model is used to estimate the contribution of the electrostatic amplification of charge fluctuations to the conductivity of ion channels., Conference, UPON2008, 14 pages

Electrostatic interactions between ions in an ionic channel and the charge fluctuations in the channel mouth are considered. It is shown that the charge fluctuations can be enhanced in channels of low dielectric constant, resulting in strong modulation of the potential barrier at the selectivity site. It is conjectured that similar effects can alter transition probabilities in other molecular dynamical systems.

The effect of charge fluctuations at the mouth of an open ion channel on its conduction mechanism is analyzed within the framework of self‐consistent Brownian dynamics simulations. It is shown that volume charge fluctuations at the channel mouth can be modelled as a generalized shot noise and result in strong modulation of the potential barrier for an ion at the selectivity site, on a sub‐nanosecond time scale.