Synthetic nanopores as a test case for ion channel theories: the anomalous mole fraction effect without single filing

The Complex Proteolipidic Behavior of the SARS-CoV-2 Envelope Protein Channel: Weak Selectivity and Heterogeneous Oligomerization

The envelope (E) protein is a small polypeptide that can form ion channels in coronaviruses. In SARS coronavirus 2 (SARS-CoV-2), the agent that caused the recent COVID-19 pandemic, and its predecessor SARS-CoV-1, E protein is found in the endoplasmic reticulum-Golgi intermediate compartment (ERGIC), where virion budding takes place. Several reports claim that E protein promotes the formation of "cation-selective channels". However, whether this term represents specificity to certain ions (e.g., potassium or calcium) or the partial or total exclusion of anions is debatable. Herein, we discuss this claim based on the available data for SARS-CoV-1 and -2 E and on new experiments performed using the untagged full-length E protein from SARS-CoV-2 in planar lipid membranes of different types, including those that closely mimic the ERGIC membrane composition. We provide evidence that the selectivity of the E-induced channels is very mild and depends strongly on lipid environment. Thus, despite past and recent claims, we found no indication that the E protein forms cation-selective channels that prevent anion transport, and even less that E protein forms bona fide specific calcium channels. In fact, the E channel maintains its multi-ionic non-specific neutral character even in concentrated solutions of Ca2+ ions. Also, in contrast to previous studies, we found no evidence that SARS-CoV-2 E channel activation requires a particular voltage, high calcium concentrations or low pH, in agreement with available data from SARS-CoV-1 E. In addition, sedimentation velocity experiments suggest that the E channel population is mostly pentameric, but very dynamic and probably heterogeneous, consistent with the broad distribution of conductance values typically found in electrophysiological experiments. The latter has been explained by the presence of proteolipidic channel structures.

Heterogeneous Electrospinning Nanofiber Membranes with pH-regulated Ion Gating for Tunable Osmotic Power Harvesting

Biological ion channels existing in organisms are critical for many biological processes. Inspired by biological ion channels, the heterogeneous electrospinning nanofiber membranes (HENM) with functional ion channels are constructed by electrospinning technology. The HENM successfully realizes ion-gating effects, which can be used for tunable energy conversions. Introduction of pyridine and carboxylic acid groups into the HENM plays an important role in generating unique and stable ion transport behaviors, in which gates become alternative states of open and close, responding to symmetric/asymmetric pH stimulations. Then we used the HENM to convert osmotic energy into electric energy which reach a maximum value up to 12.34 W m^-2 and the output power density of HENM-based system could be regulated by ion-gating effects. The properties of the HENM provide widespread potentials in application of smart nanofluidic devices, energy conversion, and water treatment.

Physics of the Nuclear Pore Complex: Theory, Modeling and Experiment

The hallmark of eukaryotic cells is the nucleus that contains the genome, enclosed by a physical barrier known as the nuclear envelope (NE). On the one hand, this compartmentalization endows the eukaryotic cells with high regulatory complexity and flexibility. On the other hand, it poses a tremendous logistic and energetic problem of transporting millions of molecules per second across the nuclear envelope, to facilitate their biological function in all compartments of the cell. Therefore, eukaryotes have evolved a molecular "nanomachine" known as the Nuclear Pore Complex (NPC). Embedded in the nuclear envelope, NPCs control and regulate all the bi-directional transport between the cell nucleus and the cytoplasm. NPCs combine high molecular specificity of transport with high throughput and speed, and are highly robust with respect to molecular noise and structural perturbations. Remarkably, the functional mechanisms of NPC transport are highly conserved among eukaryotes, from yeast to humans, despite significant differences in the molecular components among various species. The NPC is the largest macromolecular complex in the cell. Yet, despite its significant complexity, it has become clear that its principles of operation can be largely understood based on fundamental physical concepts, as have emerged from a combination of experimental methods of molecular cell biology, biophysics, nanoscience and theoretical and computational modeling. Indeed, many aspects of NPC function can be recapitulated in artificial mimics with a drastically reduced complexity compared to biological pores. We review the current physical understanding of the NPC architecture and function, with the focus on the critical analysis of experimental studies in cells and artificial NPC mimics through the lens of theoretical and computational models. We also discuss the connections between the emerging concepts of NPC operation and other areas of biophysics and bionanotechnology.

Transport mechanisms of SARS-CoV-E viroporin in calcium solutions: Lipid-dependent Anomalous Mole Fraction Effect and regulation of pore conductance

The envelope protein E of the SARS-CoV coronavirus is an archetype of viroporin. It is a small hydrophobic protein displaying ion channel activity that has proven highly relevant in virus-host interaction and virulence. Ion transport through E channel was shown to alter Ca²⁺ homeostasis in the cell and trigger inflammation processes. Here, we study transport properties of the E viroporin in mixed solutions of potassium and calcium chloride that contain a fixed total concentration (mole fraction experiments). The channel is reconstituted in planar membranes of different lipid compositions, including a lipid mixture that mimics the endoplasmic reticulum-Golgi intermediate compartment (ERGIC) membrane where the virus localizes within the cell. We find that the E ion conductance changes non-monotonically with the total ionic concentration displaying an Anomalous Mole Fraction Effect (AMFE) only when charged lipids are present in the membrane. We also observe that E channel insertion in ERGIC-mimic membranes – including lipid with intrinsic negative curvature – enhances ion permeation at physiological concentrations of pure CaCl₂ or KCl solutions, with a preferential transport of Ca²⁺ in mixed KCl-CaCl₂ solutions. Altogether, our findings demonstrate that the presence of calcium modulates the transport properties of the E channel by interacting preferentially with charged lipids through different mechanisms including direct Coulombic interactions and possibly inducing changes in membrane morphology.

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.

The mixture effect on ionic selectivity and permeability of nanotubes

Ion-selective nanotubes have great potential in applications such as ion separation, desalination, and power generation. However, their performance is often limited by the deteriorated selectivity in mixed salt solutions. To reveal the underlying mechanism of the mixture effect on ion transport through nanotubes, we perform molecular dynamics (MD) simulations on ion transport through carbon nanotubes (CNTs) and polymer nanopores with a pore diameter of ∼1 nm and a charge density of -1 e nm^-2. Based on the simulation results, when a single salt solution is replaced by a mixed salt solution, the ionic selectivity drops as the permeability of higher permeable ions decreases much greater than that of lower permeable ions. This is because the adsorption of lower permeable ions on the inner surface of nanotubes blocks the ion flux and increases the entrance barrier to the nanotube, and the adsorption is significantly reduced in the mixed salt solution. Such a reduction results from the occupancy of higher permeable ions on the adsorption sites as they have a higher adsorption tendency albeit weaker adsorption compared with lower permeable ions. These studies will help design the next generation of nanostructures to circumvent the mixture effect and show high permeability and selectivity in real applications.

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.

Biomolecule-Functionalized Solid-State Ion Nanochannels/Nanopores: Features and Techniques

Solid-state ion nanochannels/nanopores, the biomimetic products of biological ion channels, are promising materials in real-world applications due to their robust mechanical and controllable chemical properties. Functionalizations of solid-state ion nanochannels/nanopores by biomolecules pave a wide way for the introduction of varied properties from biomolecules to solid-state ion nanochannels/nanopores, making them smart in response to analytes or external stimuli and regulating the transport of ions/molecules. In this review, two features for nanochannels/nanopores functionalized by biomolecules are abstracted, i.e., specificity and signal amplification. Both of the two features are demonstrated from three kinds of nanochannels/nanopores: nucleic acid-functionalized nanochannels/nanopores, protein-functionalized nanochannels/nanopores, and small biomolecule-functionalized nanochannels/nanopores, respectively. Meanwhile, the fundamental mechanisms of these combinations between biomolecules and nanochannels/nanopores are explored, providing reasonable constructs for applications in sensing, transport, and energy conversion. And then, the techniques of functionalizations and the basic principle about biomolecules onto the solid-state ion nanochannels/nanopores are summarized. Finally, some views about the future developments of the biomolecule-functionalized nanochannels/nanopores are proposed.

Multiscale analysis of the effect of surface charge pattern on a nanopore's rectification and selectivity properties: From all-atom model to Poisson-Nernst-Planck

We report a multiscale modeling study for charged cylindrical nanopores using three modeling levels that include (1) an all-atom explicit-water model studied with molecular dynamics, and reduced models with implicit water containing (2) hard-sphere ions studied with the Local Equilibrium Monte Carlo simulation method (computing ionic correlations accurately), and (3) point ions studied with Poisson-Nernst-Planck theory (mean-field approximation). We show that reduced models are able to reproduce device functions (rectification and selectivity) for a wide variety of charge patterns, that is, reduced models are useful in understanding the mesoscale physics of the device (i.e., how the current is produced). We also analyze the relationship of the reduced implicit-water models with the explicit-water model and show that diffusion coefficients in the reduced models can be used as adjustable parameters with which the results of the explicit- and implicit-water models can be related. We find that the values of the diffusion coefficients are sensitive to the net charge of the pore but are relatively transferable to different voltages and charge patterns with the same total charge.

Multiscale modeling of a rectifying bipolar nanopore: explicit-water versus implicit-water simulations

In a multiscale modeling approach, we present computer simulation results for a rectifying bipolar nanopore at two modeling levels. In an all-atom model, we use explicit water to simulate ion transport directly with the molecular dynamics technique. In a reduced model, we use implicit water and apply the Local Equilibrium Monte Carlo method together with the Nernst-Planck transport equation. This hybrid method makes the fast calculation of ion transport possible at the price of lost details. We show that the implicit-water model is an appropriate representation of the explicit-water model when we look at the system at the device (i.e., input vs. output) level. The two models produce qualitatively similar behavior of the electrical current for different voltages and model parameters. Looking at the details of concentration and potential profiles, we find profound differences between the two models. These differences, however, do not influence the basic behavior of the model as a device because they do not influence the z-dependence of the concentration profiles which are the main determinants of current. These results then address an old paradox: how do reduced models, whose assumptions should break down in a nanoscale device, predict experimental data? Our simulations show that reduced models can still capture the overall device physics correctly, even though they get some important aspects of the molecular-scale physics quite wrong; reduced models work because they include the physics that is necessary from the point of view of device function. Therefore, reduced models can suffice for general device understanding and device design, but more detailed models might be needed for molecular level understanding.

Ion selectivity of graphene nanopores

As population growth continues to outpace development of water infrastructure in many countries, desalination (the removal of salts from seawater) at high energy efficiency will likely become a vital source of fresh water. Due to its atomic thinness combined with its mechanical strength, porous graphene may be particularly well-suited for electrodialysis desalination, in which ions are removed under an electric field via ion-selective pores. Here, we show that single graphene nanopores preferentially permit the passage of K(+) cations over Cl(-) anions with selectivity ratios of over 100 and conduct monovalent cations up to 5 times more rapidly than divalent cations. Surprisingly, the observed K(+)/Cl(-) selectivity persists in pores even as large as about 20 nm in diameter, suggesting that high throughput, highly selective graphene electrodialysis membranes can be fabricated without the need for subnanometer control over pore size.

Fabrication of Nanochannels

Nature has inspired the fabrication of intelligent devices to meet the needs of the advanced community and better understand the imitation of biology. As a biomimetic nanodevice, nanochannels/nanopores aroused increasing interest because of their potential applications in nanofluidic fields. In this review, we have summarized some recent results mainly focused on the design and fabrication of one-dimensional nanochannels, which can be made of many materials, including polymers, inorganics, biotic materials, and composite materials. These nanochannels have some properties similar to biological channels, such as selectivity, voltage-dependent current fluctuations, ionic rectification current and ionic gating, etc. Therefore, they show great potential for the fields of biosensing, filtration, and energy conversions. These advances can not only help people to understand the living processes in nature, but also inspire scientists to develop novel nanodevices with better performance for mankind.

Voltage-gated calcium channels: Determinants of channel function and modulation by inorganic cations

Voltage-gated calcium channels (VGCCs) represent a key link between electrical signals and non-electrical processes, such as contraction, secretion and transcription. Evolved to achieve high rates of Ca(2+)-selective flux, they possess an elaborate mechanism for selection of Ca(2+) over foreign ions. It has been convincingly linked to competitive binding in the pore, but the fundamental question of how this is reconcilable with high rates of Ca(2+) transfer remains unanswered. By virtue of their similarity to Ca(2+), polyvalent cations can interfere with the function of VGCCs and have proven instrumental in probing the mechanisms underlying selective permeation. Recent emergence of crystallographic data on a set of Ca(2+)-selective model channels provides a structural framework for permeation in VGCCs, and warrants a reconsideration of their diverse modulation by polyvalent cations, which can be roughly separated into three general mechanisms: (I) long-range interactions with charged regions on the surface, affecting the local potential sensed by the channel or influencing voltage-sensor movement by repulsive forces (electrostatic effects), (II) short-range interactions with sites in the ion-conducting pathway, leading to physical obstruction of the channel (pore block), and in some cases (III) short-range interactions with extracellular binding sites, leading to non-electrostatic modifications of channel gating (allosteric effects). These effects, together with the underlying molecular modifications, provide valuable insights into the function of VGCCs, and have important physiological and pathophysiological implications. Allosteric suppression of some of the pore-forming Cavα1-subunits (Cav2.3, Cav3.2) by Zn(2+) and Cu(2+) may play a major role for the regulation of excitability by endogenous transition metal ions. The fact that these ions can often traverse VGCCs can contribute to the detrimental intracellular accumulation of metal ions following excessive release of endogenous Cu(2+) and Zn(2+) or exposure to non-physiological toxic metal ions.

Calcein-modified multinanochannels on PET films for calcium-responsive nanogating

Calcein-modified multiporous films with conical channels are introduced in a nanofluid device to enhance the calcium-responsive intensity and stability of ionic currents. Calcein with more carboxyls enhances the response of channels to calcium ions, and the capability of immobilized calcein for Ca(2+)-binding could be regulated by the deprotonation of these carboxyls.

Accurate characterization of single track-etched, conical nanopores

Deviation from cone geometry significantly influences the ion current rectification through track-etched nanopores with tip radii smaller than 10 nm.

Interacting ions in biophysics: real is not ideal

Ions in water are important throughout biology, from molecules to organs. Classically, ions in water were treated as ideal noninteracting particles in a perfect gas. Excess free energy of each ion was zero. Mathematics was not available to deal consistently with flows, or interactions with other ions or boundaries. Nonclassical approaches are needed because ions in biological conditions flow and interact. The concentration gradient of one ion can drive the flow of another, even in a bulk solution. A variational multiscale approach is needed to deal with interactions and flow. The recently developed energetic variational approach to dissipative systems allows mathematically consistent treatment of the bio-ions Na(+), K(+), Ca(2+), and Cl(-) as they interact and flow. Interactions produce large excess free energy that dominate the properties of the high concentration of ions in and near protein active sites, ion channels, and nucleic acids: the number density of ions is often >10 M. Ions in such crowded quarters interact strongly with each other as well as with the surrounding protein. Nonideal behavior found in many experiments has classically been ascribed to allosteric interactions mediated by the protein and its conformation changes. The ion-ion interactions present in crowded solutions-independent of conformation changes of the protein-are likely to change the interpretation of many allosteric phenomena. Computation of all atoms is a popular alternative to the multiscale approach. Such computations involve formidable challenges. Biological systems exist on very different scales from atomic motion. Biological systems exist in ionic mixtures (like extracellular and intracellular solutions), and usually involve flow and trace concentrations of messenger ions (e.g., 10(-7) M Ca(2+)). Energetic variational methods can deal with these characteristic properties of biological systems as we await the maturation and calibration of all-atom simulations of ionic mixtures and divalents.

Ion correlations in nanofluidic channels: effects of ion size, valence, and concentration on voltage- and pressure-driven currents

The effects of ion-ion and ion-wall correlations in nanochannels are explored, specifically how they influence voltage- and pressure-driven currents and pressure-to-voltage energy conversion. Cations of different diameters (0.15, 0.3, and 0.9 nm) and different valences (+1, +2, and +3) at concentrations ranging from 10(-6) M to 1 M are considered in 50-nm- and 100-nm-wide nanoslits with wall surface charges ranging from 0 C/m(2) to -0.3 C/m(2). These parameters are typical of nanofluidic devices. Ion correlations have significant effects on device properties over large parts of this parameter space. These effects are the result of ion layering (oscillatory concentration profiles) for large monovalent cations and charge inversion (more cations in the first layer near the wall than necessary to neutralize the surface charge) for the multivalent cations. The ions were modeled as charged, hard spheres using density functional theory of fluids, and current was computed with the Navier-Stokes equations with two different no-slip conditions.

Slowing down DNA translocation through a nanopore in lithium chloride

The charge of a DNA molecule is a crucial parameter in many DNA detection and manipulation schemes such as gel electrophoresis and lab-on-a-chip applications. Here, we study the partial reduction of the DNA charge due to counterion binding by means of nanopore translocation experiments and all-atom molecular dynamics (MD) simulations. Surprisingly, we find that the translocation time of a DNA molecule through a solid-state nanopore strongly increases as the counterions decrease in size from K(+) to Na(+) to Li(+), both for double-stranded DNA (dsDNA) and single-stranded DNA (ssDNA). MD simulations elucidate the microscopic origin of this effect: Li(+) and Na(+) bind DNA stronger than K(+). These fundamental insights into the counterion binding to DNA also provide a practical method for achieving at least 10-fold enhanced resolution in nanopore applications.

Testing the applicability of Nernst-Planck theory in ion channels: comparisons with Brownian dynamics simulations

The macroscopic Nernst-Planck (NP) theory has often been used for predicting ion channel currents in recent years, but the validity of this theory at the microscopic scale has not been tested. In this study we systematically tested the ability of the NP theory to accurately predict channel currents by combining and comparing the results with those of Brownian dynamics (BD) simulations. To thoroughly test the theory in a range of situations, calculations were made in a series of simplified cylindrical channels with radii ranging from 3 to 15 Å, in a more complex 'catenary' channel, and in a realistic model of the mechanosensitive channel MscS. The extensive tests indicate that the NP equation is applicable in narrow ion channels provided that accurate concentrations and potentials can be input as the currents obtained from the combination of BD and NP match well with those obtained directly from BD simulations, although some discrepancies are seen when the ion concentrations are not radially uniform. This finding opens a door to utilising the results of microscopic simulations in continuum theory, something that is likely to be useful in the investigation of a range of biophysical and nano-scale applications and should stimulate further studies in this direction.

Anion currents in yeast K+ transporters (TRK) characterize a structural homologue of ligand-gated ion channels

Patch clamp studies of the potassium-transport proteins TRK1,2 in Saccharomyces cerevisiae have revealed large chloride efflux currents: at clamp voltages negative to -100 mV, and intracellular chloride concentrations >10 mM (J. Membr. Biol. 198:177, 2004). Stationary-state current-voltage analysis led to an in-series two-barrier model for chloride activation: the lower barrier (α) being 10-13 kcal/mol located ~30% into the membrane from the cytoplasmic surface; and the higher one (β) being 12-16 kcal/mol located at the outer surface. Measurements carried out with lyotrophic anions and osmoprotective solutes have now demonstrated the following new properties: (1) selectivity for highly permeant anions changes with extracellular pH; at pH(o)= 5.5: I(-)≈ Br(-) >Cl(-) >SCN(-) >NO (3)(-) , and at pH(o) 7.5: I(-)≈ Br(-) > SCN(-) > NO(3)(-) >Cl(-). (2) NO(2)(-) acts like "superchoride", possibly enhancing the channel's intrinsic permeability to Cl(-). (3) SCN(-) and NO(3)(-) block chloride permeability. (4) The order of selectivity for several slightly permeant anions (at pH(o)= 5.5 only) is formate>gluconate>acetate>>phosphate(-1). (5) All anion conductances are modulated (choked) by osmoprotective solutes. (6) The data and descriptive two-barrier model evoke a hypothetical structure (Biophys. J. 77:789, 1999) consisting of an intramembrane homotetramer of fungal TRK molecules, arrayed radially around a central cluster of four single helices (TM7) from each monomer. (7) That tetrameric cluster would resemble the hydrophobic core of (pentameric) ligand-gated ion channels, and would suggest voltage-modulated hydrophobic gating to underlie anion permeation.

Selectivity sequences in a model calcium channel: role of electrostatic field strength

The energetics that give rise to selectivity sequences of ionic binding selectivity of Li(+), Na(+), K(+), Rb(+), and Cs(+) in a model of a calcium channel are considered. This work generalizes Eisenman's classic treatment (Biophys J 2(Suppl. 2):259, 1962) by including multiple, mobile binding site oxygens that coordinate many permeating ions (all modeled as charged, hard spheres). The selectivity filter of the model calcium channel allows the carboxyl terminal groups of glutamate and aspartate side chains to directly interact with and coordinate the permeating ions. Ion dehydration effects are represented with a Born energy between the dielectric coefficients of the selectivity filter and the bath. High oxygen concentration creates a high field strength site that prefers small ions, as in Eisenman's model. On the other hand, a low filter dielectric constant also creates a high field strength site, but this site prefers large ions, contrary to Eisenman's model. These results indicate that field strength does not have a unique effect on ionic binding selectivity sequences once entropic, electrostatic, and dehydration forces are included in the model. Thus, Eisenman's classic relationship between field strength and selectivity sequences must be supplemented with additional information about selectivity filters such as the calcium channel that has amino acid side chains mixing with ions to make a crowded permeation pathway.

DNA strands attached inside single conical nanopores: ionic pore characteristics and insight into DNA biophysics

Single nanopores attract a great deal of scientific interest as a basis for biosensors and as a system to study the interactions and behavior of molecules in a confined volume. Tuning the geometry and surface chemistry of nanopores helps create devices that control transport of ions and molecules in solution. Here, we present single conically shaped nanopores whose narrow opening of 8 or 12 nm is modified with single-stranded DNA molecules. We find that the DNA occludes the narrow opening of nanopores and that the blockade extent decreases with the ionic strength of the background electrolyte. The results are explained by the ionic strength dependence of the persistence length of DNA. At low KCl concentrations (10 mM) the molecules assume an extended and rigid conformation, thereby blocking the pore lumen and reducing the flow of ionic current to a greater extent than compacted DNA at high salt concentrations. Attaching flexible polymers to the pore walls hence creates a system with tunable opening diameters in order to regulate transport of both neutral and charged species.

Mathematical modeling and simulation of nanopore blocking by precipitation

High surface charges of polymer pore walls and applied electric fields can lead to the formation and subsequent dissolution of precipitates in nanopores. These precipitates block the pore, leading to current fluctuations. We present an extended Poisson-Nernst-Planck system which includes chemical reactions of precipitation and dissolution. We discuss the mathematical modeling and present 2D numerical simulations.

Crowding effects in non-equilibrium transport through nano-channels

Transport through nano-channels plays an important role in many biological processes and industrial applications. Gaining insights into the functioning of biological transport processes and the design of man-made nano-devices requires an understanding of the basic physics of such transport. A simple exclusion process has proven to be very useful in explaining the properties of several artificial and biological nano-channels. It is particularly useful for modeling the influence of inter-particle interactions on transport characteristics. In this paper, we explore several models of the exclusion process using a mean field approach and computer simulations. We examine the effects of crowding inside the channel and in its immediate vicinity on the mean flux and the transport times of single molecules. Finally, we discuss the robustness of the theory's predictions with respect to the crucial characteristics of the hindered diffusion in nano-channels that need to be included in the model.

Energy variational analysis of ions in water and channels: Field theory for primitive models of complex ionic fluids

Ionic solutions are mixtures of interacting anions and cations. They hardly resemble dilute gases of uncharged noninteracting point particles described in elementary textbooks. Biological and electrochemical solutions have many components that interact strongly as they flow in concentrated environments near electrodes, ion channels, or active sites of enzymes. Interactions in concentrated environments help determine the characteristic properties of electrodes, enzymes, and ion channels. Flows are driven by a combination of electrical and chemical potentials that depend on the charges, concentrations, and sizes of all ions, not just the same type of ion. We use a variational method EnVarA (energy variational analysis) that combines Hamilton's least action and Rayleigh's dissipation principles to create a variational field theory that includes flow, friction, and complex structure with physical boundary conditions. EnVarA optimizes both the action integral functional of classical mechanics and the dissipation functional. These functionals can include entropy and dissipation as well as potential energy. The stationary point of the action is determined with respect to the trajectory of particles. The stationary point of the dissipation is determined with respect to rate functions (such as velocity). Both variations are written in one Eulerian (laboratory) framework. In variational analysis, an "extra layer" of mathematics is used to derive partial differential equations. Energies and dissipations of different components are combined in EnVarA and Euler-Lagrange equations are then derived. These partial differential equations are the unique consequence of the contributions of individual components. The form and parameters of the partial differential equations are determined by algebra without additional physical content or assumptions. The partial differential equations of mixtures automatically combine physical properties of individual (unmixed) components. If a new component is added to the energy or dissipation, the Euler-Lagrange equations change form and interaction terms appear without additional adjustable parameters. EnVarA has previously been used to compute properties of liquid crystals, polymer fluids, and electrorheological fluids containing solid balls and charged oil droplets that fission and fuse. Here we apply EnVarA to the primitive model of electrolytes in which ions are spheres in a frictional dielectric. The resulting Euler-Lagrange equations include electrostatics and diffusion and friction. They are a time dependent generalization of the Poisson-Nernst-Planck equations of semiconductors, electrochemistry, and molecular biophysics. They include the finite diameter of ions. The EnVarA treatment is applied to ions next to a charged wall, where layering is observed. Applied to an ion channel, EnVarA calculates a quick transient pile-up of electric charge, transient and steady flow through the channel, stationary "binding" in the channel, and the eventual accumulation of salts in "unstirred layers" near channels. EnVarA treats electrolytes in a unified way as complex rather than simple fluids. Ad hoc descriptions of interactions and flow have been used in many areas of science to deal with the nonideal properties of electrolytes. It seems likely that the variational treatment can simplify, unify, and perhaps derive and improve those descriptions.

Simulations of calcium channel block by trivalent cations: Gd(3+) competes with permeant ions for the selectivity filter

Current through L-type calcium channels (Ca(V)1.2 or dihydropyridine receptor) can be blocked by micromolar concentrations of trivalent cations like the lanthanide gadolinium (Gd(3+)). It has been proposed that trivalent block is due to ions competing for a binding site in both the open and closed configuration, but possibly with different trivalent affinities. Here, we corroborate this general view of trivalent block by computing conductance of a model L-type calcium channel. The model qualitatively reproduces the Gd(3+) concentration dependence and the effect that substantially more Gd(3+) is required to produce similar block in the presence of Sr(2+) (compared to Ba(2+)) and even more in the presence of Ca(2+). Trivalent block is explained in this model by cations binding in the selectivity filter with the charge/space competition mechanism. This is the same mechanism that in the model channel governs other selectivity properties. Specifically, selectivity is determined by the combination of ions that most effectively screen the negative glutamates of the protein while finding space in the midst of the closely packed carboxylate groups of the glutamate residues.

Enhancement of transport selectivity through nano-channels by non-specific competition

The functioning of living cells requires efficient and selective transport of materials into and out of the cell, and between different cellular compartments. Much of this transport occurs through nano-scale channels that do not require large scale molecular re-arrangements (such as transition from a 'closed' to an 'open' state) and do not require a direct input of metabolic energy during transport. Nevertheless, these 'always open' channels are highly selective and pass only their cognate molecules, while efficiently excluding all others; indeed, these channels can efficiently transport specific molecules even in the presence of a vast excess of non-specific molecules. Such biological transporters have inspired the creation of artificial nano-channels. These channels can be used as nano-molecular sorters, and can also serve as testbeds for examining modes of biological transport. In this paper, we propose a simple kinetic mechanism that explains how the selectivity of such 'always open' channels can be based on the exclusion of non-specific molecules by specific ones, due to the competition for limited space inside the channel. The predictions of the theory account for the behavior of the nuclear pore complex and of artificial nanopores that mimic its function. This theory provides the basis for future work aimed at understanding the selectivity of various biological transport phenomena.

Sieving experiments and pore diameter: it's not a simple relationship

The classic sieving experiment for estimating an ion channel's diameter with successively larger ions is re-examined. Using a very reduced model of a calcium channel, it is shown that sieving experiments measure a combination of three mechanisms: the cross-sectional area available to the sieving ions (the classic interpretation), the exclusion of the sieving ions from a pore crowded with amino acid side chains that protrude into the permeation pathway, and competitive selectivity of the sieving ions with other ions in the bath (even if those are present only at trace concentrations). The latter two can be called sieving-by-crowding because they stem from the excluded volume of the amino acids in the permeation pathway. The model shows that--to a first--order approximation-sieving experiments measure the available volume inside a selectivity filter, rather than the available cross-sectional area. The two are only the same if the narrow part of the pore does not have flexible amino acid side chains interacting directly with the permeant ions; this may be true of potassium channels, but not calcium, sodium, and other channels with "crowded" selectivity filters.

Molecular control of ionic conduction in polymer nanopores

Polymeric nanopores show unique transport properties and have attracted a great deal of scientific interest as a test system to study ionic and molecular transport at the nanoscale. By means of all-atom molecular dynamics, we simulated the ion dynamics inside polymeric polyethylene terephthalate nanopores. For this purpose, we established a protocol to assemble atomic models of polymeric material into which we sculpted a nanopore model with the key features of experimental devices, namely a conical geometry and a negative surface charge density. Molecular dynamics simulations of ion currents through the pore show that the protonation state of the carboxyl group of exposed residues have a considerable effect on ion selectivity, by affecting ionic densities and electrostatic potentials inside the nanopores. The role of high concentrations of Ca2+ ions was investigated in detail.

Reinterpreting the anomalous mole fraction effect: the ryanodine receptor case study

The origin of the anomalous mole fraction effect (AMFE) in calcium channels is explored with a model of the ryanodine receptor. This model predicted and experiments verified new AMFEs in the cardiac isoform. In mole fraction experiments, conductance is measured in mixtures of ion species X and Y as their relative amounts (mole fractions) vary. This curve can have a minimum (an AMFE). The traditional interpretation of the AMFE is that multiple interacting ions move through the pore in a single file. Mole fraction curves without minima (no AMFEs) are generally interpreted as X displacing Y from the pore in a proportion larger than its bath mole fraction (preferential selectivity). We find that the AMFE is also caused by preferential selectivity of X over Y, if X and Y have similar conductances. This is a prediction applicable to any channel and provides a fundamentally different explanation of the AMFE that does not require single filing or multiple occupancy: preferential selectivity causes the resistances to current flow in the baths, channel vestibules, and selectivity filter to change differently with mole fraction, and produce the AMFE.

Ionic selectivity in L-type calcium channels by electrostatics and hard-core repulsion

A physical model of selective "ion binding" in the L-type calcium channel is constructed, and consequences of the model are compared with experimental data. This reduced model treats only ions and the carboxylate oxygens of the EEEE locus explicitly and restricts interactions to hard-core repulsion and ion-ion and ion-dielectric electrostatic forces. The structural atoms provide a flexible environment for passing cations, thus resulting in a self-organized induced-fit model of the selectivity filter. Experimental conditions involving binary mixtures of alkali and/or alkaline earth metal ions are computed using equilibrium Monte Carlo simulations in the grand canonical ensemble. The model pore rejects alkali metal ions in the presence of biological concentrations of Ca(2+) and predicts the blockade of alkali metal ion currents by micromolar Ca(2+). Conductance patterns observed in varied mixtures containing Na(+) and Li(+), or Ba(2+) and Ca(2+), are predicted. Ca(2+) is substantially more potent in blocking Na(+) current than Ba(2+). In apparent contrast to experiments using buffered Ca(2+) solutions, the predicted potency of Ca(2+) in blocking alkali metal ion currents depends on the species and concentration of the alkali metal ion, as is expected if these ions compete with Ca(2+) for the pore. These experiments depend on the problematic estimation of Ca(2+) activity in solutions buffered for Ca(2+) and pH in a varying background of bulk salt. Simulations of Ca(2+) distribution with the model pore bathed in solutions containing a varied amount of Li(+) reveal a "barrier and well" pattern. The entry/exit barrier for Ca(2+) is strongly modulated by the Li(+) concentration of the bath, suggesting a physical explanation for observed kinetic phenomena. Our simulations show that the selectivity of L-type calcium channels can arise from an interplay of electrostatic and hard-core repulsion forces among ions and a few crucial channel atoms. The reduced system selects for the cation that delivers the largest charge in the smallest ion volume.

Modeling Transport Through Synthetic Nanopores

Nanopores in thin synthetic membranes have emerged as convenient tools for high-throughput single-molecule manipulation and analysis. Because of their small sizes and their ability to selectively transport solutes through otherwise impermeable membranes, nanopores have numerous potential applications in nanobiotechnology. For most applications, properties of the nanopore systems have to be characterize at the atomic level, which is currently beyond the limit of experimental methods. Molecular dynamics (MD) simulations can provide the desired information, however several technical challenges have to be met before this method can be applied to synthetic nanopore systems. Here, we highlight our recent work on modeling synthetic nanopores of the most common types. First, we describe a novel graphical tool for setting up all-atom systems incorporating inorganic materials and biomolecules. Next, we illustrate the application of the MD method for silica, silicon nitride, and polyethylene terephthalate nanopores. Following that, we describe a method for modeling synthetic surfaces using a bias potential. Future directions for tool development and nanopore modeling are briefly discussed at the end of this article.

The anomalous mole fraction effect in calcium channels: a measure of preferential selectivity

The cause of the anomalous mole fraction effect (AMFE) in calcium-selective ion channels is studied. An AMFE occurs when the conductance through a channel is lower in a mixture of salts than in the pure salts at the same concentration. The textbook interpretation of the AMFE is that multiple ions move through the pore in coordinated, single-file motion. Instead of this, we find that at its most basic level an AMFE reflects a channel's preferential binding selectivity for one ion species over another. The AMFE is explained by considering the charged and uncharged regions of the pore as electrical resistors in series: the AMFE is produced by these regions of high and low ion concentration changing differently with mole fraction due to the preferential ion selectivity. This is demonstrated with simulations of a model L-type calcium channel and a mathematical analysis of a simplistic point-charge model. The particle simulations reproduce the experimental data of two L-type channel AMFEs. Conditions under which an AMFE may be found experimentally are discussed. The resistors-in-series model provides a fundamentally different explanation of the AMFE than the traditional theory and does not require single filing, multiple occupancy, or momentum-correlated ion motion.

The cardiac ryanodine receptor: looking for anomalies in permeation properties

Anomalies in the permeation properties of the cardiac RyR channel reconstituted into bilayer lipid membranes were investigated systematically. We tested the presence of the anomalous mole fraction effect (AMFE) for the ion conductance and the reversal potential with varying mole fractions of two permeant ions, while the total ion concentration was lower, as in previous studies, to avoid the masking effect of the channel pore saturation with ions. Mixtures of Ba(2+) with other divalents (Ca(2+), Sr(2+)), of Ca(2+) with monovalents (Li(+), Cs(+)), and of Na(+) with other monovalents (Cs(+), Li(+)) were used. We revealed a clear anomaly only for the ion conductance measured in the Na(+)-Cs(+) and Ca(2+)-Li(+) mixtures as computed by a Poisson-Nernst-Planck/density functional theory (PNP/DFT) model. Furthermore, we found a significant minimum in the concentration dependence of the reversal potential determined under Li(+)/Ca(2+) bi-ionic conditions. Our study led to new observations that may have important implications for understanding the mechanisms involved in ion handling in the RyR channel pore; furthermore our results could be useful for further validation of ion permeation models developed for the RyR channel.