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Physics of Ionic Conduction in Narrow Biological and Artificial Channels

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A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Statistical Physics".

Deadline for manuscript submissions: closed (30 October 2020) | Viewed by 38383

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Special Issue Editors

Prof. Peter V E McClintock

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Guest Editor

Department of Physics, Lancaster University, Lancaster LA1 4YB, UK
Interests: fluctuational phenomena; nonlinear dynamics; ion channels; biomedical physics; superfluidity; quantum turbulence

Dr. Dmitry G. Luchinsky

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Guest Editor

Department of Physics, Lancaster University, Lancaster LA1 4YB, UK
Interests: fluctuations and critical phenomena in nonlinear dynamical systems; classical mechanics; thermodynamics; Brownian motion in nonequilibrium systems; generalised nonequilibrium potential; experiments on time symmetry and detailed balance in stochastic dynamical systems; large fluctuations; ion channels

Special Issue Information

Dear Colleagues,

Biological ion channels are essential to life in all its forms. The key properties underlying their function are those of selectivity and conductivity—the ability to select between different kinds of ions, while allowing the favoured species to pass at nearly the rate of free diffusion. It is now appreciated that an understanding of selective conduction requires physics, and that the physics of biological ion channels has a great deal in common with that of artificial nanopores. In each case, there are intriguing analogies with the physics of quantum dots. Discovery of the atomic structures of many channels has brought significant progress, as has the building of subnanometer artificial channels and the experimental investigation of their selectivity and conduction; large-scale molecular dynamics simulations are yielding atomistic and statistical insights into many channel properties as a function of structure. However, the ability to predict the function of a channel from its structure, e.g., following a point mutation of a biological channel or the functionalization of a nanopore, remains elusive. Nonetheless, these recent advances have brought us tantalisingly close to a fundamental theory of ionic permeation, based on the statistical physics of ions within the channel. It promises to resolve the long-standing structure–function problem, thus enabling explicit current calculations for relatively complex structures.

The Special Issue aims to bring together original high-quality papers on ionic permeation through narrow water-filled channels, both biological and artificial. It will include papers on the statistical physics of the process, on molecular dynamics and Brownian dynamics simulations, and on relevant experiments. The time is ripe for bringing these mutually complementary approaches together, and we anticipate that they will facilitate major breakthroughs enabling the design of nanopores to meet particular technological requirements as well as improvements in drug design.

Prof. Peter V E McClintock
Dr. Dmitry G. Luchinsky
Guest Editors

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Keywords

biological ion channel
artificial nanopore
Statistical Physics
ionic coulomb blockade
fluctuations
excess chemical potential
potential of the mean force
ionic dehydration barrier
ionic binding energy
effective grand canonical ensemble
selectivity mechanism
selectivity sequence
linear response theory
molecular dynamics simulations
Brownian dynamics simulations drug design
desalination

Published Papers (14 papers)

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Editorial

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6 pages, 174 KiB

Open AccessEditorial

Introduction to the Physics of Ionic Conduction in Narrow Biological and Artificial Channels

by Dmitry G. Luchinsky and Peter V. E. McClintock

Entropy 2021, 23(6), 644; https://0-doi-org.brum.beds.ac.uk/10.3390/e23060644 - 21 May 2021

Cited by 1 | Viewed by 1411

Abstract

“There is plenty of room at the bottom” [...] Full article

(This article belongs to the Special Issue Physics of Ionic Conduction in Narrow Biological and Artificial Channels)

Research

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20 pages, 1174 KiB

Open AccessArticle

Electrophysiological Properties from Computations at a Single Voltage: Testing Theory with Stochastic Simulations

by Michael A. Wilson and Andrew Pohorille

Entropy 2021, 23(5), 571; https://0-doi-org.brum.beds.ac.uk/10.3390/e23050571 - 06 May 2021

Cited by 4 | Viewed by 1950

Abstract

We use stochastic simulations to investigate the performance of two recently developed methods for calculating the free energy profiles of ion channels and their electrophysiological properties, such as current–voltage dependence and reversal potential, from molecular dynamics simulations at a single applied voltage. These methods require neither knowledge of the diffusivity nor simulations at multiple voltages, which greatly reduces the computational effort required to probe the electrophysiological properties of ion channels. They can be used to determine the free energy profiles from either forward or backward one-sided properties of ions in the channel, such as ion fluxes, density profiles, committor probabilities, or from their two-sided combination. By generating large sets of stochastic trajectories, which are individually designed to mimic the molecular dynamics crossing statistics of models of channels of trichotoxin, p7 from hepatitis C and a bacterial homolog of the pentameric ligand-gated ion channel, GLIC, we find that the free energy profiles obtained from stochastic simulations corresponding to molecular dynamics simulations of even a modest length are burdened with statistical errors of only 0.3 kcal/mol. Even with many crossing events, applying two-sided formulas substantially reduces statistical errors compared to one-sided formulas. With a properly chosen reference voltage, the current–voltage curves can be reproduced with good accuracy from simulations at a single voltage in a range extending for over 200 mV. If possible, the reference voltages should be chosen not simply to drive a large current in one direction, but to observe crossing events in both directions. Full article

(This article belongs to the Special Issue Physics of Ionic Conduction in Narrow Biological and Artificial Channels)

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15 pages, 2213 KiB

Open AccessArticle

Application of a Statistical and Linear Response Theory to Multi-Ion Na⁺ Conduction in NaChBac

by William A. T. Gibby, Olena A. Fedorenko, Carlo Guardiani, Miraslau L. Barabash, Thomas Mumby, Stephen K. Roberts, Dmitry G. Luchinsky and Peter V. E. McClintock

Entropy 2021, 23(2), 249; https://0-doi-org.brum.beds.ac.uk/10.3390/e23020249 - 21 Feb 2021

Cited by 3 | Viewed by 2257

Abstract

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 [...] Read more.

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

\sim 4, 2.9, 3.6

, and

4 k T

. Full article

(This article belongs to the Special Issue Physics of Ionic Conduction in Narrow Biological and Artificial Channels)

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22 pages, 1151 KiB

Open AccessArticle

Maxwell Equations without a Polarization Field, Using a Paradigm from Biophysics

by Robert S. Eisenberg

Entropy 2021, 23(2), 172; https://0-doi-org.brum.beds.ac.uk/10.3390/e23020172 - 30 Jan 2021

Cited by 7 | Viewed by 2853

Abstract

When forces are applied to matter, the distribution of mass changes. Similarly, when an electric field is applied to matter with charge, the distribution of charge changes. The change in the distribution of charge (when a local electric field is applied) might in general be called the induced charge. When the change in charge is simply related to the applied local electric field, the polarization field P is widely used to describe the induced charge. This approach does not allow electrical measurements (in themselves) to determine the structure of the polarization fields. Many polarization fields will produce the same electrical forces because only the divergence of polarization enters Maxwell’s first equation, relating charge and electric forces and field. The curl of any function can be added to a polarization field P without changing the electric field at all. The divergence of the curl is always zero. Additional information is needed to specify the curl and thus the structure of the P field. When the structure of charge changes substantially with the local electric field, the induced charge is a nonlinear and time dependent function of the field and P is not a useful framework to describe either the electrical or structural basis-induced charge. In the nonlinear, time dependent case, models must describe the charge distribution and how it varies as the field changes. One class of models has been used widely in biophysics to describe field dependent charge, i.e., the phenomenon of nonlinear time dependent induced charge, called ‘gating current’ in the biophysical literature. The operational definition of gating current has worked well in biophysics for fifty years, where it has been found to makes neurons respond sensitively to voltage. Theoretical estimates of polarization computed with this definition fit experimental data. I propose that the operational definition of gating current be used to define voltage and time dependent induced charge, although other definitions may be needed as well, for example if the induced charge is fundamentally current dependent. Gating currents involve substantial changes in structure and so need to be computed from a combination of electrodynamics and mechanics because everything charged interacts with everything charged as well as most things mechanical. It may be useful to separate the classical polarization field as a component of the total induced charge, as it is in biophysics. When nothing is known about polarization, it is necessary to use an approximate representation of polarization with a dielectric constant that is a single real positive number. This approximation allows important results in some cases, e.g., design of integrated circuits in silicon semiconductors, but can be seriously misleading in other cases, e.g., ionic solutions. Full article

(This article belongs to the Special Issue Physics of Ionic Conduction in Narrow Biological and Artificial Channels)

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23 pages, 10934 KiB

Open AccessArticle

Unraveling of a Strongly Correlated Dynamical Network of Residues Controlling the Permeation of Potassium in KcsA Ion Channel

by Salvatore M. Cosseddu, Eunju Julia Choe and Igor A. Khovanov

Entropy 2021, 23(1), 72; https://0-doi-org.brum.beds.ac.uk/10.3390/e23010072 - 06 Jan 2021

Cited by 2 | Viewed by 2933

Abstract

The complicated patterns of the single-channel currents in potassium ion channel KcsA are governed by the structural variability of the selectivity filter. A comparative analysis of the dynamics of the wild type KcsA channel and several of its mutants showing different conducting patterns was performed. A strongly correlated dynamical network of interacting residues is found to play a key role in regulating the state of the wild type channel. The network is centered on the aspartate D80 which plays the role of a hub by strong interacting via hydrogen bonds with residues E71, R64, R89, and W67. Residue D80 also affects the selectivity filter via its backbones. This network further compromises ions and water molecules located inside the channel that results in the mutual influence: the permeation depends on the configuration of residues in the network, and the dynamics of network’s residues depends on locations of ions and water molecules inside the selectivity filter. Some features of the network provide a further understanding of experimental results describing the KcsA activity. In particular, the necessity of anionic lipids to be present for functioning the channel is explained by the interaction between the lipids and the arginine residues R64 and R89 that prevents destabilizing the structure of the selectivity filter. Full article

(This article belongs to the Special Issue Physics of Ionic Conduction in Narrow Biological and Artificial Channels)

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18 pages, 1845 KiB

Open AccessArticle

Changes in Ion Selectivity Following the Asymmetrical Addition of Charge to the Selectivity Filter of Bacterial Sodium Channels

by Olena A. Fedorenko, Igor A. Khovanov, Stephen K. Roberts and Carlo Guardiani

Entropy 2020, 22(12), 1390; https://0-doi-org.brum.beds.ac.uk/10.3390/e22121390 - 09 Dec 2020

Cited by 1 | Viewed by 1826

Abstract

Voltage-gated sodium channels (NaVs) play fundamental roles in eukaryotes, but their exceptional size hinders their structural resolution. Bacterial NaVs are simplified homologues of their eukaryotic counterparts, but their use as models of eukaryotic Na⁺ channels is limited by their homotetrameric structure at odds with the asymmetric Selectivity Filter (SF) of eukaryotic NaVs. This work aims at mimicking the SF of eukaryotic NaVs by engineering radial asymmetry into the SF of bacterial channels. This goal was pursued with two approaches: the co-expression of different monomers of the NaChBac bacterial channel to induce the random assembly of heterotetramers, and the concatenation of four bacterial monomers to form a concatemer that can be targeted by site-specific mutagenesis. Patch-clamp measurements and Molecular Dynamics simulations showed that an additional gating charge in the SF leads to a significant increase in Na⁺ and a modest increase in the Ca²⁺ conductance in the NavMs concatemer in agreement with the behavior of the population of random heterotetramers with the highest proportion of channels with charge −5e. We thus showed that charge, despite being important, is not the only determinant of conduction and selectivity, and we created new tools extending the use of bacterial channels as models of eukaryotic counterparts. Full article

(This article belongs to the Special Issue Physics of Ionic Conduction in Narrow Biological and Artificial Channels)

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25 pages, 1361 KiB

Open AccessEditor’s ChoiceArticle

Diffusion Limitations and Translocation Barriers in Atomically Thin Biomimetic Pores

by Subin Sahu and Michael Zwolak

Entropy 2020, 22(11), 1326; https://0-doi-org.brum.beds.ac.uk/10.3390/e22111326 - 20 Nov 2020

Cited by 3 | Viewed by 2747

Abstract

Ionic transport in nano- to sub-nano-scale pores is highly dependent on translocation barriers and potential wells. These features in the free-energy landscape are primarily the result of ion dehydration and electrostatic interactions. For pores in atomically thin membranes, such as graphene, other factors come into play. Ion dynamics both inside and outside the geometric volume of the pore can be critical in determining the transport properties of the channel due to several commensurate length scales, such as the effective membrane thickness, radii of the first and the second hydration layers, pore radius, and Debye length. In particular, for biomimetic pores, such as the graphene crown ether we examine here, there are regimes where transport is highly sensitive to the pore size due to the interplay of dehydration and interaction with pore charge. Picometer changes in the size, e.g., due to a minute strain, can lead to a large change in conductance. Outside of these regimes, the small pore size itself gives a large resistance, even when electrostatic factors and dehydration compensate each other to give a relatively flat—e.g., near barrierless—free energy landscape. The permeability, though, can still be large and ions will translocate rapidly after they arrive within the capture radius of the pore. This, in turn, leads to diffusion and drift effects dominating the conductance. The current thus plateaus and becomes effectively independent of pore-free energy characteristics. Measurement of this effect will give an estimate of the magnitude of kinetically limiting features, and experimentally constrain the local electromechanical conditions. Full article

(This article belongs to the Special Issue Physics of Ionic Conduction in Narrow Biological and Artificial Channels)

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26 pages, 1291 KiB

Open AccessArticle

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

by Dezső Boda, Mónika Valiskó and Dirk Gillespie

Entropy 2020, 22(11), 1259; https://0-doi-org.brum.beds.ac.uk/10.3390/e22111259 - 05 Nov 2020

Cited by 7 | Viewed by 2005

Abstract

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. Full article

(This article belongs to the Special Issue Physics of Ionic Conduction in Narrow Biological and Artificial Channels)

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19 pages, 2684 KiB

Open AccessArticle

Electric Double Layer and Orientational Ordering of Water Dipoles in Narrow Channels within a Modified Langevin Poisson-Boltzmann Model

by Mitja Drab, Ekaterina Gongadze, Veronika Kralj-Iglič and Aleš Iglič

Entropy 2020, 22(9), 1054; https://0-doi-org.brum.beds.ac.uk/10.3390/e22091054 - 21 Sep 2020

Cited by 10 | Viewed by 2944

Abstract

The electric double layer (EDL) is an important phenomenon that arises in systems where a charged surface comes into contact with an electrolyte solution. In this work we describe the generalization of classic Poisson-Boltzmann (PB) theory for point-like ions by taking into account orientational ordering of water molecules. The modified Langevin Poisson-Boltzmann (LPB) model of EDL is derived by minimizing the corresponding Helmholtz free energy functional, which includes also orientational entropy contribution of water dipoles. The formation of EDL is important in many artificial and biological systems bound by a cylindrical geometry. We therefore numerically solve the modified LPB equation in cylindrical coordinates, determining the spatial dependencies of electric potential, relative permittivity and average orientations of water dipoles within charged tubes of different radii. Results show that for tubes of a large radius, macroscopic (net) volume charge density of coions and counterions is zero at the geometrical axis. This is attributed to effective electrolyte charge screening in the vicinity of the inner charged surface of the tube. For tubes of small radii, the screening region extends into the whole inner space of the tube, leading to non-zero net volume charge density and non-zero orientational ordering of water dipoles near the axis. Full article

(This article belongs to the Special Issue Physics of Ionic Conduction in Narrow Biological and Artificial Channels)

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15 pages, 1007 KiB

Open AccessArticle

Review and Modification of Entropy Modeling for Steric Effects in the Poisson-Boltzmann Equation

by Tzyy-Leng Horng

Entropy 2020, 22(6), 632; https://0-doi-org.brum.beds.ac.uk/10.3390/e22060632 - 08 Jun 2020

Cited by 8 | Viewed by 2560

Abstract

The classical Poisson-Boltzmann model can only work when ion concentrations are very dilute, which often does not match the experimental conditions. Researchers have been working on the modification of the model to include the steric effect of ions, which is non-negligible when the ion concentrations are not dilute. Generally the steric effect was modeled to correct the Helmholtz free energy either through its internal energy or entropy, and an overview is given here. The Bikerman model, based on adding solvent entropy to the free energy through the concept of volume exclusion, is a rather popular steric-effect model nowadays. However, ion sizes are treated as identical in the Bikerman model, making an extension of the Bikerman model to include specific ion sizes desirable. Directly replacing the ions of non-specific size by specific ones in the model seems natural and has been accepted by many researchers in this field. However, this straightforward modification does not have a free energy formula to support it. Here modifications of the Bikerman model to include specific ion sizes have been developed iteratively, and such a model is achieved with a guarantee that: (1) it can approach Boltzmann distribution at diluteness; (2) it can reach saturation limit as the reciprocal of specific ion size under extreme electrostatic conditions; (3) its entropy can be derived by mean-field lattice gas model. Full article

(This article belongs to the Special Issue Physics of Ionic Conduction in Narrow Biological and Artificial Channels)

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23 pages, 898 KiB

Open AccessFeature PaperArticle

Effects of Diffusion Coefficients and Permanent Charge on Reversal Potentials in Ionic Channels

by Hamid Mofidi, Bob Eisenberg and Weishi Liu

Entropy 2020, 22(3), 325; https://0-doi-org.brum.beds.ac.uk/10.3390/e22030325 - 12 Mar 2020

Cited by 11 | Viewed by 2627

Abstract

In this work, the dependence of reversal potentials and zero-current fluxes on diffusion coefficients are examined for ionic flows through membrane channels. The study is conducted for the setup of a simple structure defined by the profile of permanent charges with two mobile ion species, one positively charged (cation) and one negatively charged (anion). Numerical observations are obtained from analytical results established using geometric singular perturbation analysis of classical Poisson–Nernst–Planck models. For 1:1 ionic mixtures with arbitrary diffusion constants, Mofidi and Liu (arXiv:1909.01192) conducted a rigorous mathematical analysis and derived an equation for reversal potentials. We summarize and extend these results with numerical observations for biological relevant situations. The numerical investigations on profiles of the electrochemical potentials, ion concentrations, and electrical potential across ion channels are also presented for the zero-current case. Moreover, the dependence of current and fluxes on voltages and permanent charges is investigated. In the opinion of the authors, many results in the paper are not intuitive, and it is difficult, if not impossible, to reveal all cases without investigations of this type. Full article

(This article belongs to the Special Issue Physics of Ionic Conduction in Narrow Biological and Artificial Channels)

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Review

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25 pages, 958 KiB

Open AccessEditor’s ChoiceReview

Dynamics of Ion Channels via Non-Hermitian Quantum Mechanics

by Tobias Gulden and Alex Kamenev

Entropy 2021, 23(1), 125; https://0-doi-org.brum.beds.ac.uk/10.3390/e23010125 - 19 Jan 2021

Cited by 2 | Viewed by 2822

Abstract

We study dynamics and thermodynamics of ion transport in narrow, water-filled channels, considered as effective 1D Coulomb systems. The long range nature of the inter-ion interactions comes about due to the dielectric constants mismatch between the water and the surrounding medium, confining the electric filed to stay mostly within the water-filled channel. Statistical mechanics of such Coulomb systems is dominated by entropic effects which may be accurately accounted for by mapping onto an effective quantum mechanics. In presence of multivalent ions the corresponding quantum mechanics appears to be non-Hermitian. In this review we discuss a framework for semiclassical calculations for the effective non-Hermitian Hamiltonians. Non-Hermiticity elevates WKB action integrals from the real line to closed cycles on a complex Riemann surfaces where direct calculations are not attainable. We circumvent this issue by applying tools from algebraic topology, such as the Picard-Fuchs equation. We discuss how its solutions relate to the thermodynamics and correlation functions of multivalent solutions within narrow, water-filled channels. Full article

(This article belongs to the Special Issue Physics of Ionic Conduction in Narrow Biological and Artificial Channels)

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39 pages, 1231 KiB

Open AccessReview

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

by Jinn-Liang Liu and Bob Eisenberg

Entropy 2020, 22(5), 550; https://0-doi-org.brum.beds.ac.uk/10.3390/e22050550 - 14 May 2020

Cited by 41 | Viewed by 5490

Abstract

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. Full article

(This article belongs to the Special Issue Physics of Ionic Conduction in Narrow Biological and Artificial Channels)

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Other

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14 pages, 2023 KiB

Open AccessPerspective

Prospects of Observing Ionic Coulomb Blockade in Artificial Ion Confinements

by Andrey Chernev, Sanjin Marion and Aleksandra Radenovic

Entropy 2020, 22(12), 1430; https://0-doi-org.brum.beds.ac.uk/10.3390/e22121430 - 18 Dec 2020

Cited by 6 | Viewed by 2943

Abstract

Nanofluidics encompasses a wide range of advanced approaches to study charge and mass transport at the nanoscale. Modern technologies allow us to develop and improve artificial nanofluidic platforms that confine ions in a way similar to single-ion channels in living cells. Therefore, nanofluidic platforms show great potential to act as a test field for theoretical models. This review aims to highlight ionic Coulomb blockade (ICB)—an effect that is proposed to be the key player of ion channel selectivity, which is based upon electrostatic exclusion limiting ion transport. Thus, in this perspective, we focus on the most promising approaches that have been reported on the subject. We consider ion confinements of various dimensionalities and highlight the most recent advancements in the field. Furthermore, we concentrate on the most critical obstacles associated with these studies and suggest possible solutions to advance the field further. Full article

(This article belongs to the Special Issue Physics of Ionic Conduction in Narrow Biological and Artificial Channels)

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