Electrostatic calculations for an ion channel. I. Energy and potential profiles and interactions between ions

The electrostatic energy profile of one, two, or three ions in an aqueous channel through a lipid membrane is calculated. It is shown that the previous solution to this problem (based on the assumption that the channel is infinitely long) significantly overestimates the electrostatic energy barrier. For example, for a 3-A radius pore, the energy is 16 kT for the infinite channel and 6.7 kT for an ion in the center of a channel 25 A long. The energy as a function of the position of the ion is also determined. With this energy profile, the rate of crossing the membrane (using the Nernst-Planck equation) was estimated and found to be compatible with the maximum conductance observed for the gramicidin A channel. The total electrostatic energy (as a function of position) required to place two or three ions in the channel is also calculated. The electrostatic interaction is small for two ions at opposite ends of the channel and large for any positioning of the three ions. Finally, the gradient through the channel of an applied potential is calculated. The solution to these problems is based on solving an equivalent problem in which an appropriate surface charge is placed on the boundary between the lipid and aqueous regions. The magnitude of the surface charge is obtained from the numerical solution for a system of coupled integral equations.     

ON THE CAUSE OF THE INFLUENCE OF IONS ON THE RATE OF DIFFUSION OF WATER THROUGH COLLODION MEMBRANES. I

1. In three previous publications it had been shown that electrolytes influence the rate of diffusion of pure water through a collodion membrane into a solution in three different ways, which can be understood on the assumption of an electrification of the water or the watery phase at the boundary of the membrane; namely, (a) While the watery phase in contact with collodion is generally positively electrified, it happens that, when the membrane has received a treatment with a protein, the presence of hydrogen ions and of simple cations with a valency of three or above (beyond a certain concentration) causes the watery phase of the double layer at the boundary of membrane and solution to be negatively charged. (b) When pure water is separated from a solution by a collodion membrane, the initial rate of diffusion of water into a solution is accelerated by the ion with the opposite sign of charge and retarded by the ion with the same sign of charge as that of the water, both effects increasing with the valency of the ion and a second constitutional quantity of the ion which is still to be defined. (c) The relative influence of the oppositely charged ions, mentioned in (b), is not the same for all concentrations of electrolytes. For lower concentrations the influence of that ion usually prevails which has the opposite sign of charge from that of the watery phase of the double layer; while in higher concentrations the influence of that ion begins to prevail which has the same sign of charge as that of the watery phase of the double layer. For a number of solutions the turning point lies at a molecular concentration of about M/256 or M/512. In concentrations of M/8 or above the influence of the electrical charges of ions mentioned in (b) or (c) seems to become less noticeable or to disappear entirely. 2. It is shown in this paper that in electrical endosmose through a collodion membrane the influence of electrolytes on the rate of transport of liquids is the same as in free osmosis. Since the influence of electrolytes on the rate of transport in electrical endosmose must be ascribed to their influence on the quantity of electrical charge on the unit area of the membrane, we must conclude that the same explanation holds for the influence of electrolytes on the rate of transport of water into a solution through a collodion membrane in the case of free osmosis. 3. We may, therefore, conclude, that when pure water is separated from a solution of an electrolyte by a collodion membrane, the rate of diffusion of water into the solution by free osmosis is accelerated by the ion with the opposite sign of charge as that of the watery phase of the double layer, because this ion increases the quantity of charge on the unit area on the solution side of the membrane; and that the rate of diffusion of water is retarded by the ion with the same sign of charge as that of the watery phase for the reason that this ion diminishes the charge on the solution side of the membrane. When, therefore, the ions of an electrolyte raise the charge on the unit area of the membrane on the solution side above that on the side of pure water, a flow of the oppositely charged liquid must occur through the interstices of the membrane from the side of the water to the side of the solution (positive osmosis). When, however, the ions of an electrolyte lower the charge on the unit area of the solution side of the membrane below that on the pure water side of the membrane, liquid will diffuse from the solution into the pure water (negative osmosis). 4. We must, furthermore, conclude that in lower concentrations of many electrolytes the density of electrification of the double layer increases with an increase in concentration, while in higher concentrations of the same electrolytes it decreases with an increase in concentration. The turning point lies for a number of electrolytes at a molecular concentration of about M/512 or M/256. This explains why in lower concentrations of electrolytes the rate of diffusion of water through a collodion membrane from pure water into solution rises at first rapidly with an increase in concentration while beyond a certain concentration (which in a number of electrolytes is M/512 or M/256) the rate of diffusion of water diminishes with a further increase in concentration.     

Application of the Poisson-Nernst-Planck theory with space-dependent diffusion coefficients to KcsA

The Poisson-Nernst-Planck electrodiffusion theory serves to compute charge fluxes and is here applied to the ion current through a protein channel. KcsA was selected as an example because of the abundance of experimental and theoretical data. The potassium channels MthK and KvAP were used as templates to define two open channel models for KcsA. Channel boundary surfaces and protein charge distributions were defined according to atomic radii and partial atomic charges. To establish the sensitivity of the results to these parameters, two different sets were used. Assigning the potassium diffusion coefficients equal to the value for free-diffusion in water (1.96 x 10(-9) m(2)/s), the computed currents overestimated the experimental data. Ion distributions inside the channel suggest that the overestimate is not due to an excess of charge shielding. A good agreement with the experimental data was achieved by reducing the potassium diffusion coefficient inside the channel to 1.96 x 10(-10) m(2)/s, a value of substantial motility but nonetheless in accord with the intuitive notion that the channel has a high affinity for the ions and therefore slows them down. These results are independent of the open channel model and the parameterization adopted for atomic radii and partial atomic charges. The method offers a reliable estimate of the channel current with low computational effort.     

Perturbation approximation of solutions of a nonlinear inverse problem arising in olfaction experimentation

In this paper, a mathematical model of the diffusion of cAMP into olfactory cilia and the resulting electrical activity is presented. The model, which consists of two nonlinear differential equations, is studied using perturbation techniques. The unknowns in the problem are the concentration of cAMP, the membrane potential, and the quantity of most interest in this work: the distribution of CNG channels along the length of a cilium. Experimental measurements of the total current during this diffusion process provide an extra boundary condition which helps determine the unknown distribution function. A simple perturbation approximation is derived and used to solve this inverse problem and thus obtain estimates of the spatial distribution of CNG ion channels along the length of a cilium. A one-dimensional computer minimization and a special delay iteration are used with the perturbation formulas to obtain approximate channel distributions in the cases of simulated and experimental data.      

BioMOCA—a Boltzmann transport Monte Carlo model for ion channel simulation

With the recent availability of high-resolution structural information for several key ion channel proteins and large-scale computational resources, Molecular Dynamics has become an increasingly popular tool for ion channel simulation. However, the CPU requirements for simulating ion transport on time scales relevant to conduction still exceed the resources presently available. To address this problem, we have developed Biology Monte Carlo (BioMOCA), a three-dimensional (3D) coarse-grained particle ion channel simulator based on the Boltzmann Transport Monte Carlo (BTMC) methodology. Although this approach is widely employed in the engineering community to study charge transport in electron devices, its application to molecular biology and electrolytes in general is new and hence must be validated. The pair correlation function, which is a measure of the microscopic structure of matter, provides a suitable benchmark to compare the BTMC method against the well-established Equilibrium Monte Carlo (EMC) approach. For validation purposes BioMOCA is used to simulate several simple homogeneous equilibrium electrolytes at concentrations of physiological interest. The ion–ion pair correlation functions computed from these simulations compare very well with those obtained from EMC simulations. We also demonstrate several performance-improving techniques that result in a several-fold speed-up without compromising the pair correlation function. BioMOCA is then used to perform full 3D simulations of ion transport in the gramicidin A channel in situ in a membrane environment, as well as to study the link between the electrostatic and dielectric properties of the protein and the channel's selectivity.     

ON THE CAUSE OF THE INFLUENCE OF IONS ON THE RATE OF DIFFUSION OF WATER THROUGH COLLODION MEMBRANES. II

1. It had been shown in previous publications that when pure water is separated from a solution of an electrolyte by a collodion membrane the ion with the same sign of charge as the membrane increases and the ion with the opposite sign of charge as the membrane diminishes the rate of diffusion of water into the solution; but that the relative influence of the oppositely charged ions upon the rate of diffusion of water through the membrane is not the same for different concentrations. Beginning with the lowest concentrations of electrolytes the attractive influence of that ion which has the same sign of charge as the collodion membrane upon the oppositely charged water increases more rapidly with increasing concentration of the electrolyte than the repelling effect of the ion possessing the opposite sign of charge as the membrane. When the concentration exceeds a certain critical value the repelling influence of the latter ion upon the water increases more rapidly with a further increase in the concentration of the electrolyte than the attractive influence of the ion having the same sign of charge as the membrane. 2. It is shown in this paper that the influence of the concentration of electrolytes on the rate of transport of water through collodion membranes in electrical endosmose is similar to that in the case of free osmosis. 3. On the basis of the Helmholtz theory of electrical double layers this seems to indicate that the influence of an electrolyte on the rate of diffusion of water through a collodion membrane in the case of free osmosis is due to the fact that the ion possessing the same sign of charge as the membrane increases the density of charge of the latter while the ion with the opposite sign diminishes the density of charge of the membrane. The relative influence of the oppositely charged ions on the density of charge of the membrane is not the same in all concentrations. The influence of the ion with the same sign of charge increases in the lowest concentrations more rapidly with increasing concentration than the influence of the ion with the opposite sign of charge, while for somewhat higher concentrations the reverse is true.     

Modulation of lateral transport of membrane components by spatial variations in diffusivity and solubility.

The effect of spatially varying diffusivity and solubility on the efficiency of intramembrane transport is investigated by obtaining solutions to the generalized lateral diffusion equation in which both the diffusion coefficient, D(r), and the partition coefficient, K(r), are functions of position. The mean-time-to-capture by a sink, tc, of particles diffusing in a plane is obtained analytically for the case of a sink surrounded by gradients in D(r) and K(r) with radially symmetrical geometry. It is shown that for particles originating at random locations, tc is shortened dramatically, if in an annular region around the sink, D and K are significantly greater than in the remainder of the plane. Similarly, a viscous boundary layer surrounding a sink is demonstrated to represent a significant barrier for diffusing particles. To investigate more complex geometries, a finite difference numerical integration method is used and is shown to provide comparable results for tc with modest computational power. The same method is used to calculate the tc for particles originating at a source that is joined to the sink by a channel. The increase in the rate with which particles travel from a source to a sink when they are joined by a high diffusivity and/or solubility channel is illustrated by several numerical examples and by graphical representations that show the equilibrium particle density (and hence the effective particle flow) in the presence of different sink, source, and channel combinations. These results are discussed in terms of fluidity domains and other membrane heterogeneities.     

Simultaneous optical and electrical recording of single gramicidin channels

We report here an approach for simultaneous fluorescence imaging and electrical recording of single ion channels in planar bilayer membranes. As a test case, fluorescently labeled (Cy3 and Cy5) gramicidin derivatives were imaged at the single-molecule level using far-field illumination and cooled CCD camera detection. Gramicidin monomers were observed to diffuse in the plane of the membrane with a diffusion coefficient of 3.3 x 10(-8) cm(2)s(-1). Simultaneous electrical recording detected gramicidin homodimer (Cy3/Cy3, Cy5/Cy5) and heterodimer (Cy3/Cy5) channels. Heterodimer formation was observed optically by the appearance of a fluorescence resonance energy transfer (FRET) signal (irradiation of Cy3, detection of Cy5). The number of FRET signals was significantly smaller than the number of Cy3 signals (Cy3 monomers plus Cy3 homodimers) as expected. The number of FRET signals increased with increasing channel activity. In numerous cases the appearance of a FRET signal was observed to correlate with a channel opening event detected electrically. The heterodimers also diffused in the plane of the membrane with a diffusion coefficient of 3.0 x 10(-8) cm(2)s(-1). These experiments demonstrate the feasibility of simultaneous optical and electrical detection of structural changes in single ion channels as well as suggesting strategies for improving the reliability of such measurements.     

Ion permeation through the alpha-hemolysin channel: theoretical studies based on Brownian dynamics and Poisson-Nernst-Plank electrodiffusion theory

Identification of the molecular interaction governing ion conduction through biological pores is one of the most important goals of modern electrophysiology. Grand canonical Monte Carlo Brownian dynamics (GCMC/BD) and three-dimensional Poisson-Nernst-Plank (3d-PNP) electrodiffusion algorithms offer powerful and general approaches to study of ion permeation through wide molecular pores. A detailed analysis of ion flows through the staphylococcal alpha-hemolysin channel based on series of simulations at different concentrations and transmembrane potentials is presented. The position-dependent diffusion coefficient is approximated on the basis of a hydrodynamic model. The channel conductance calculated by GCMC/BD is approximately 10% higher than (electrophysiologically measured) experimental values, whereas results from 3d-PNP are always 30-50% larger. Both methods are able to capture all important electrostatic interactions in equilibrium conditions. The asymmetric conductance upon the polarity of the transmembrane potential observed experimentally is reproduced by GCMC/BD and 3d-PNP. The separation of geometrical and energetic influence of the channel on ion conduction reveals that such asymmetries arise from the permanent charge distribution inside the pore. The major determinant of the asymmetry is unbalanced charge in the triad of polar residues D127, D128, and K131. The GCMC/BD or 3d-PNP calculations reproduce also experimental reversal potentials and permeability rations in asymmetric ionic solutions. The weak anionic selectivity of the channel results from the presence of the salt bridge between E111 and K147 in the constriction zone. The calculations also reproduce the experimentally derived dependence of the reversible potential to the direction of the salt gradient. The origin of such effect arises from the asymmetrical distribution of energetic barriers along the channel axis, which modulates the preferential ion passage in different directions.     

Helix 4 of the Bacillus thuringiensis Cry1Aa toxin lines the lumen of the ion channel.

The mode of action of Bacillus thuringiensis insecticidal proteins is not well understood. Based on analogies with other bacterial toxins and ion channels, we hypothesized that charged amino acids in helix 4 of the Cry1Aa toxin are critical for toxicity and ion channel function. Using Plutella xylostella as a model target, we analyzed responses to Cry1Aa and eight proteins with altered helix 4 residues. Toxicity was abolished in five charged residue mutants (E129K, R131Q, R131D, D136N, D136C), however, two charged (R127E and R127N) and one polar (N138C) residue mutant retained wild-type toxicity. Compared with Cry1Aa and toxic mutants, nontoxic mutants did not show greatly reduced binding to brush border membrane vesicles, but their ion channel conductance was greatly reduced in planar lipid bilayers. Substituted cysteine accessibility tests showed that in situ restoration of the negative charge of D136C restored conductance to wild-type levels. The results imply that charged amino acids on the Asp-136 side of helix 4 are essential for toxicity and passage of ions through the channel. These results also support a refined version of the umbrella model of membrane integration in which the side of helix 4 containing Asp-136 faces the aqueous lumen of the ion channel.     

Model channel ion currents in NaCl-extended simple point charge water solution with applied-field molecular dynamics.

Using periodic boundary conditions and a constant applied field, we have simulated current flow through an 8.125-A internal diameter, rigid, atomistic channel with polar walls in a rigid membrane using explicit ions and extended simple point charge water. Channel and bath currents were computed from 10 10-ns trajectories for each of 10 different conditions of concentration and applied voltage. An electric field was applied uniformly throughout the system to all mobile atoms. On average, the resultant net electric field falls primarily across the membrane channel, as expected for two conductive baths separated by a membrane capacitance. The channel is rarely occupied by more than one ion. Current-voltage relations are concentration dependent and superlinear at high concentrations.     

Physical descriptions of experimental selectivity measurements in ion channels

Three experiments that quantify the amount of selectivity exhibited by a biological ion channel are examined with Poisson-Nernst-Planck (PNP) theory. Conductance ratios and the conductance mole fraction experiments are examined by considering a simple model ion channel for which an approximate solution to the PNP equations with Donnan boundary conditions is derived. A more general result is derived for the Goldman-Hodgkin-Katz permeability ratio. The results show that (1) the conductance ratio measures the ratio of the diffusion coefficients of the ions inside the channel, (2) the mole fraction experiment measures the difference of the excess chemical potentials of the ions inside the channel, and (3) the permeability ratio measures both diffusion coefficients and excess chemical potentials. The results are used to divide selectivity into two components: partitioning, an equilibrium measure of how well the ions enter the channel, and diffusion, a nonequilibrium measure of how well the ions move through the channel.     

A physical model of potassium channel activation: from energy landscape to gating kinetics

We have developed a method for rapidly computing gating currents from a multiparticle ion channel model. Our approach is appropriate for energy landscapes that can be characterized by a network of well-defined activation pathways with barriers. To illustrate, we represented the gating apparatus of a channel subunit by an interacting pair of charged gating particles. Each particle underwent spatial diffusion along a bistable potential of mean force, with electrostatic forces coupling the two trajectories. After a step in membrane potential, relaxation of the smaller barrier charge led to a time-dependent reduction in the activation barrier of the principal gate charge. The resulting gating current exhibited a rising phase similar to that measured in voltage-dependent ion channels. Reduction of the two-dimensional diffusion landscape to a circular Markov model with four states accurately preserved the time course of gating currents on the slow timescale. A composite system containing four subunits leading to a concerted opening transition was used to fit a series of gating currents from the Shaker potassium channel. We end with a critique of the model with regard to current views on potassium channel structure.     

A molecular level prototype for mechanoelectrical transducer in mammalian hair cells

The mechanoelectrical transducer (MET) is a crucial component of mammalian auditory system. The gating mechanism of the MET channel remains a puzzling issue, though there are many speculations, due to the lack of essential molecular building blocks. To understand the working principle of mammalian MET, we propose a molecular level prototype which constitutes a charged blocker, a realistic ion channel and its surrounding membrane. To validate the proposed prototype, we make use of a well-established ion channel theory, the Poisson–Nernst–Planck equations, for three-dimensional (3D) numerical simulations. A wide variety of model parameters, including bulk ion concentration, applied external voltage, blocker charge and blocker displacement, are explored to understand the basic function of the proposed MET prototype. We show that our prototype prediction of channel open probability in response to blocker relative displacement is in remarkable accordance with experimental observation of rat cochlea outer hair cells. Our results appear to suggest that tip links which connect hair bundles gate MET channels.     

Laser‐Assisted Ultrafast Exfoliation of Black Phosphorus in Liquid with Tunable Thickness for Li‐Ion Batteries

Few‐layer black phosphorus (BP) is an emerging 2D material suitable for energy applications. However, its controllable preparation remains challenging. Herein, a highly efficient route is presented for the scalable production of few‐layered BP nanosheets using a pulsed laser in low‐boiling point solvents. Changing the laser irradiation time, energy, and solvent type leads to precise control over the layer number and lateral size of the nanosheets with a narrow distribution. The process is understood by a plasma quenching mechanism and interlayer interaction weakened by the in situ generated vapor bubbles. The excellent control of the BP nanosheets enables morphological effects on Li‐ion battery performance to be studied. Low layer numbers benefit both charge transfer and Li⁺ ion diffusion, while a high aspect ratio can not only improve the charge transfer but also increase the Li⁺ ion diffusion path. This study delivers insights on the tailored fabrication of thin 2D materials using lasers for morphology‐dependent electrochemical energy conversion and storage.     

Membranes with the same ion channel populations but different excitabilities

Electrical signaling allows communication within and between different tissues and is necessary for the survival of multicellular organisms. The ionic transport that underlies transmembrane currents in cells is mediated by transporters and channels. Fast ionic transport through channels is typically modeled with a conductance-based formulation that describes current in terms of electrical drift without diffusion. In contrast, currents written in terms of drift and diffusion are not as widely used in the literature in spite of being more realistic and capable of displaying experimentally observable phenomena that conductance-based models cannot reproduce (e.g. rectification). The two formulations are mathematically related: conductance-based currents are linear approximations of drift-diffusion currents. However, conductance-based models of membrane potential are not first-order approximations of drift-diffusion models. Bifurcation analysis and numerical simulations show that the two approaches predict qualitatively and quantitatively different behaviors in the dynamics of membrane potential. For instance, two neuronal membrane models with identical populations of ion channels, one written with conductance-based currents, the other with drift-diffusion currents, undergo transitions into and out of repetitive oscillations through different mechanisms and for different levels of stimulation. These differences in excitability are observed in response to excitatory synaptic input, and across different levels of ion channel expression. In general, the electrophysiological profiles of membranes modeled with drift-diffusion and conductance-based models having identical ion channel populations are different, potentially causing the input-output and computational properties of networks constructed with these models to be different as well. The drift-diffusion formulation is thus proposed as a theoretical improvement over conductance-based models that may lead to more accurate predictions and interpretations of experimental data at the single cell and network levels.     

A model of NaCl and water flow through paracellular pathways of renal proximal tubules.

To explain how hydrostatic pressure differences between tubule lumen and interstitium modulate isotonic reabsorption rates, we developed a model of NaCl and water flow through paracellular pathways of the proximal tubule. Structural elements of the model are a tight junction membrane, an intercellular channel whose walls transport NaCl actively at a constant rate, and a basement membrane. Equations of change were derived for the channel, boundary conditions were formulated from irreversible thermodynamics, and a pressure-area relationship typical of thin-walled tubing was assumed. The boundary value problem was solved numerically. The principal conclusions are: 1) channel NaCl concentration must remain within a few mOsm of isotonic values for reabsorption rates to be modulated by transtubular pressure differences known to affect this system: 2) basement membrane and channel wall parameters determine reabsorbate tonicity; tight junction parameters affect the sensitivity of reabsorption to transmural pressure; 3) channel NaCl concentration varies inversely with transmural pressure difference; this concentration variation controls NaCl diffusion through the tight junction; 4) modulation of NaCl diffusion through the tight junction controls the rate of isotonic reabsorption; modulation of water flow can increase sensitivity to transmural pressure; 5) no pressure-induced change in permeability of the tight junction or basement membrane is needed for pressure to modulate reabsorption; and 6) system performance is indifferent to the distribution of active transport sites, to the numerical value of the compliance function, and to the relationship between lumen and cell pressures.     

Computing transient gating charge movement of voltage-dependent ion channels

The opening of voltage-gated sodium, potassium, and calcium ion channels has a steep relationship with voltage. In response to changes in the transmembrane voltage, structural movements of an ion channel that precede channel opening generate a capacitative gating current. The net gating charge displacement due to membrane depolarization is an index of the voltage sensitivity of the ion channel activation process. Understanding the molecular basis of voltage-dependent gating of ion channels requires the measurement and computation of the gating charge, Q. We derive a simple and accurate semianalytic approach to computing the voltage dependence of transient gating charge movement (Q–V relationship) of discrete Markov state models of ion channels using matrix methods. This approach allows rapid computation of Q–V curves for finite and infinite length step depolarizations and is consistent with experimentally measured transient gating charge. This computational approach was applied to Shaker potassium channel gating, including the impact of inactivating particles on potassium channel gating currents.     

Design of an Efficient Inhibitor for the Influenza A Virus M2 Ion Channel

Influenza A virus is capable of rapidly infecting large human populations, warranting the development of novel drugs to efficiently inhibit virus replication. A transmembrane ion channel formed by the M2 protein plays an important role in influenza virus replication. A reasonable approach to designing an effective antivirus drug is constructing a molecule that binds in the M2 transmembrane proton channel, blocks H+ proton diffusion through the channel, and thus the influenza A virus cycle. The known anti-influenza drugs amantadine and rimantadine have a weak effect on influenza A virus replication. A new class of positively charged molecules, diazabicyclooctane derivatives with a constant charge of +2, was proposed to block proton diffusion through the M2 ion channel. Molecular dynamics simulations were performed to study the temperature fluctuations in the M2 structure, and ionization states of histidine residues were established at physiological pH values. Two types of diazabicyclooctane derivatives were analyzed for binding with the M2 ion channel. An optimal structure was determined for a blocker to most efficiently bind with the M2 ion channel and block proton diffusion. The new molecule is advantageous over amantadine and rimantadine in having a positive charge of +2, which creates a positive electrostatic potential barrier to proton transport through the M2 ion channel in addition to a steric barrier.