Statistical properties of ion channel records. Part II: estimation from the macroscopic current

Macroscopic ion channel current is the summation of the stochastic records of individual channel currents and therefore relates to their statistical properties. As a consequence of this relationship, it may be possible to derive certain statistical properties of single channel records or even generate some estimates of the records themselves from the macroscopic current when the direct measurement of single channel currents is not applicable. We present a procedure for generating the single channel records of an ion channel from its macroscopic current when the stochastic process of channel gating has the following two properties: (I) the open duration is independent of the time of opening event and has a single exponential probability density function (pdf), (II) all the channels have the same probability to open at time t. The application of this procedure is considered for cases where direct measurement of single channel records is difficult or impossible. First, the probability density function (pdf) of opening events, a statistical property of single channel records, is derived from the normalized macroscopic current and mean channel open duration. Second, it is shown that under the conditions (I) and (II), a non-stationary Markov model can represent the stochastic process of channel gating. Third, the non-stationary Markov model is calibrated using the results of the first step. The non-stationary formulation increases the model ability to generate a variety of different single channel records compared to common stationary Markov models. The model is then used to generate single channel records and to obtain other statistical properties of the records. Experimental single channel records of inactivating BK potassium channels are used to evaluate how accurately this procedure reconstructs measured single channel sweeps.     

Molecular dynamics simulation of cation motion in water-filled gramicidinlike pores.

A model calculation is carried out to study the potential energy profile of a sodium ion with several water molecules inside a simplified model of the gramicidin ion channel. The sodium ion is treated as a Lennard-Jones sphere with a point charge at its center. The Barnes polarizable water model is used to mimic the water molecules. A polarizable and deformable gramicidinlike channel is constructed based on the model obtained by Koeppe and Kimura. Potential minima and saddle points are located and the static energy barriers are computed. The potential minima at the two mouths of the channel exhibit an aqueous solvation structure very different from that at any of the interior minima. These sites are approximately 23.6 and 24.4 A apart for binding of a sodium ion and a cesium ion, respectively. Ionic motion from these exterior sites to the first interior minimum requires substantial rearrangement of the waters of solvation; this rearrangement may be the hydration/dehydration step in ionic permeation through the channel. Based on these results, a mechanism by which the sodium ion moves from the exterior binding site to the interior of the channel is proposed. Our model channel accommodates about eight water molecules and the transport of the ion and water within the channel is found to be single file. Results of less extensive calculations for Cs+ and Li+ ions in a channel with or without water are also reported.     

The role of the dielectric barrier in narrow biological channels: a novel composite approach to modeling single-channel currents

A composite continuum theory for calculating ion current through a protein channel of known structure is proposed, which incorporates information about the channel dynamics. The approach is utilized to predict current through the Gramicidin A ion channel, a narrow pore in which the applicability of conventional continuum theories is questionable. The proposed approach utilizes a modified version of Poisson-Nernst-Planck (PNP) theory, termed Potential-of-Mean-Force-Poisson-Nernst-Planck theory (PMFPNP), to compute ion currents. As in standard PNP, ion permeation is modeled as a continuum drift-diffusion process in a self-consistent electrostatic potential. In PMFPNP, however, information about the dynamic relaxation of the protein and the surrounding medium is incorporated into the model of ion permeation by including the free energy of inserting a single ion into the channel, i.e., the potential of mean force along the permeation pathway. In this way the dynamic flexibility of the channel environment is approximately accounted for. The PMF profile of the ion along the Gramicidin A channel is obtained by combining an equilibrium molecular dynamics (MD) simulation that samples dynamic protein configurations when an ion resides at a particular location in the channel with a continuum electrostatics calculation of the free energy. The diffusion coefficient of a potassium ion within the channel is also calculated using the MD trajectory. Therefore, except for a reasonable choice of dielectric constants, no direct fitting parameters enter into this model. The results of our study reveal that the channel response to the permeating ion produces significant electrostatic stabilization of the ion inside the channel. The dielectric self-energy of the ion remains essentially unchanged in the course of the MD simulation, indicating that no substantial changes in the protein geometry occur as the ion passes through it. Also, the model accounts for the experimentally observed saturation of ion current with increase of the electrolyte concentration, in contrast to the predictions of standard PNP theory.     

Exact reductions of Markovian dynamics for ion channels with a single permissive state

We show that the cooperative model for the kinetics of a tetrameric potassium ion channel derived in Nekouzadeh et al. (Biophys J 95(7):3510–3520, 2008) is an invariant manifold reduction of the full master equation for the channel kinetics. We further establish the validity of this reduction for ion channel models consisting of multiple independent subunits with cooperative transitions from a single permissive state to a conducting state. Finally, we conclude that solutions of the reduced model are globally asymptotically stable solutions of the full master equation system.     

Fractional Poisson–Nernst–Planck Model for Ion Channels I: Basic Formulations and Algorithms

In this work, we propose a fractional Poisson–Nernst–Planck model to describe ion permeation in gated ion channels. Due to the intrinsic conformational changes, crowdedness in narrow channel pores, binding and trapping introduced by functioning units of channel proteins, ionic transport in the channel exhibits a power-law-like anomalous diffusion dynamics. We start from continuous-time random walk model for a single ion and use a long-tailed density distribution function for the particle jump waiting time, to derive the fractional Fokker–Planck equation. Then, it is generalized to the macroscopic fractional Poisson–Nernst–Planck model for ionic concentrations. Necessary computational algorithms are designed to implement numerical simulations for the proposed model, and the dynamics of gating current is investigated. Numerical simulations show that the fractional PNP model provides a more qualitatively reasonable match to the profile of gating currents from experimental observations. Meanwhile, the proposed model motivates new challenges in terms of mathematical modeling and computations.     

Microscopic model for selective permeation in ion channels.

Ionic permeation in the selectivity filter of ion channels is analyzed by a microscopic model based on molecular kinetic theory. The energy and flux equations are derived by assuming that: (a) the selectivity filter is formed by a symmetrical array of carbonyl groups; (b) ion movement is near the axis of the channel; (c) a fraction of water molecules is separated from the ion while it moves across the selectivity filter; (d) the applied voltage drops linearly across the selectivity filter; (e) ions move independently. Energy profiles, single channel conductances, and the degree of hydration of K+ in a hypothetical K+ channel are examined by varying the following microscopic parameters: ion radius and mass, channel radius, number of effective water dipoles, and number of carbonyl groups. The i-V curve is linear up to +/- 170 mV. If the positions of energy maxima and minima are fixed, this linear range is reduced to +/- 50 mV. Channel radius and ion-water interactions are found to be two major channel structural determinants for selectivity sequences. Both radius and mass of an ion are important in selectivity mediated by these interactions. The theory predicts a total of 15 possible kinetic selectivity sequences for alkali cations in ion channels with a single selectivity filter.     

Ion conductance and selectivity of single calcium-activated potassium channels in cultured rat muscle

The conductance and selectivity of the Ca-activated K channel in cultured rat muscle was studied. Shifts in the reversal potential of single channel currents when various cations were substituted for Ki+ were used with the Goldman-Hodgkin-Katz equation to calculate relative permeabilities. The selectivity was Tl+ greater than K+ greater than Rb+ greater than NH4+, with permeability ratios of 1.2, 1.0, 0.67, and 0.11. Na+, Li+, and Cs+ were not measurably permeant, with permeabilities less than 0.05 that of K+. Currents with the various ions were typically less than expected on the basis of the permeability ratios, which suggests that the movement of an ion through the channel was not independent of the other ions present. For a fixed activity of Ko+ (77 mM), plots of single channel conductance vs. activity of Ki+ were described by a two-barrier model with a single saturable site. This observation, plus the finding that the permeability ratios of Rb+ and NH+4 to K+ did not change with ion concentration, is consistent with a channel that can contain a maximum of one ion at any time. The empirically determined dissociation constant for the single saturable site was 100 mM, and the maximum calculated conductance for symmetrical solutions of K+ was 640 pS. TEAi+ (tetraethylammonium ion) reduced single channel current amplitude in a voltage-dependent manner. This effect was accounted for by assuming voltage-dependent block by TEA+ (apparent dissociation constant of 60 mM at 0 mV) at a site located 26% of the distance across the membrane potential, starting at the inner side. TEAo+ was much more effective in reducing single channel currents, with an apparent dissociation constant of approximately 0.3 mM.     

Modeling of ion permeation in calcium and sodium channel selectivity filters

Structure-function studies have shown that it is possible to convert a sodium channel to a calcium-selective channel by a single amino acid substitution in the selectivity filter locus. Ion permeation through the "model selectivity filter" was modeled with a reduced set of functional groups representative of the constituent amino acid side chains. Force-field minimizations were conducted to obtain the energy profile of the cations as they get desolvated and bind to the "model selectivity filter." The calculations suggest that the ion selectivity in the calcium channel is due to preferential binding, whereas in the sodium channel it is due to exclusion. Energetics of displacement of a bound cation from the calcium "model selectivity filter" by another cation suggest that "multi-ion mechanism" reduces the activation barrier for ion permeation. Thus, the simple model captures qualitatively most of the conduction characteristics of sodium and calcium channels. However, the computed barriers for permeation are fairly large, suggesting that ion interaction with additional residues along the transport path may be essential to effect desolvation.     

Statistical properties of ion channel records. Part I: relationship to the macroscopic current

Macroscopic ion channel current can be derived by summation of the stochastic records of individual channel currents. In this paper, we present two probability density functions of single channel records that can uniquely determine the macroscopic current regardless of other statistical properties of records or the stochastic model of channel gating (presented often with stationary Markov models). We show that H(t), probability density function of channel opening events (introduced explicitly in this paper), and D(t), probability density function of the open duration (sometimes has named dwell time distribution as well), determine the normalized macroscopic current, G(t), through G(t) = P(t) - H(t) * Q(t) where P(t) is the cumulative density function of H(t), Q(t) is the cumulative density function of D(t), * is the symbol of convolution integral and G(t) is the macroscopic current divided by the amplitude of single channel current and the number of single channel sweeps. Compared to other equations for the macroscopic current, here the macroscopic current is expressed only in terms of the statistical properties of single channel current and not the stochastic model of ion channel gating or a conditioned form of macroscopic current. Single channel currents of an inactivating BK channel were used to validate this relationship experimentally too. In this paper, we used median filters as they can remove the unwanted noise without smoothing the transitions between open and closed states (compare to low pass filters). This filtering leads to more accurate measurement of transition times and less amount of missed events.     

Interaction of amiloride and one of its derivatives with Vpu from HIV-1: a molecular dynamics simulation

Vpu is an 81-residue membrane protein, with a single transmembrane segment that is encoded by HIV-1 and is involved in the enhancement of virion release via formation of an ion channel. Cyclohexamethylene amiloride (Hma) has been shown to inhibit ion channel activity. In the present 12-ns simulation study a putative binding site of Hma blockers in a pentameric model bundle built of parallel aligned helices of the first 32 residues of Vpu was found near Ser-23. Hma orientates along the channel axis with its alkyl ring pointing inside the pore, which leads to a blockage of the pore.     

Detection of DNA via an ion channel switch biosensor

Detection of DNA by an ion channel switch biosensor has been demonstrated in a model system, using single-stranded oligonucleotide sequences of 52-84 bases in length. Two different biotinylated probes are bound, via streptavidin, either to the outer region of a gramicidin ion channel dimer or to an immobilized membrane component. The ion channels are switched off upon detection of DNA containing complementary epitopes to these probes, separated by a nonbinding region, at nanomolar levels. The DNA cross-links the ion channel to the immobilized species, preventing ions passing through the channel. Addition of DNase I after the target DNA has been added switches the ion channels on. The DNA response is dependent on the rate of hybridization of the individual probes to their complementary epitopes, as shown by using a single probe against DNA containing a repeat of the complementary epitope. These results were correlated with hybridization rates determined using surface plasmon resonance (BIAcore 2000), and with free energies of dimer formation for the probes.     

Stochastic Markovian modeling of electrophysiology of ion channels: reconstruction of standard deviations in macroscopic currents

Markovian models of ion channels have proven useful in the reconstruction of experimental data and prediction of cellular electrophysiology. We present the stochastic Galerkin method as an alternative to Monte Carlo and other stochastic methods for assessing the impact of uncertain rate coefficients on the predictions of Markovian ion channel models. We extend and study two different ion channel models: a simple model with only a single open and a closed state and a detailed model of the cardiac rapidly activating delayed rectifier potassium current. We demonstrate the efficacy of stochastic Galerkin methods for computing solutions to systems with random model parameters. Our studies illustrate the characteristic changes in distributions of state transitions and electrical currents through ion channels due to random rate coefficients. Furthermore, the studies indicate the applicability of the stochastic Galerkin technique for uncertainty and sensitivity analysis of bio-mathematical models.     

Free energy of a potassium ion in a model of the channel formed by an amphipathic leucine-serine peptide

We use molecular dynamics simulations to investigate the position-dependent free energy of a potassium ion in a model of an ion channel formed by the synthetic amphipathic leucine-serine peptide, LS3. The channel model is a parallel bundle of six LS3 helices around which are packed 146 methane-like spheres in order to mimic a membrane. At either end of and within the channel are 1051 water molecules, plus four ions (two potassium and two chloride). The free energy of a potassium ion in the channel was estimated using the weighted histogram analysis (WHAM) method. This is the first time to our knowledge that such a calculation has been carried out as a function of the position of an ion in three dimensions within a channel. The results indicate that for this channel, which is lined by hydrophilic serine sidechains, there is a relatively weak dependence of the free energy on the axial/off-axial position of the ion. There are some off-axis local minima, especially in the C-terminal half of the channel. Using the free energy results, a single channel current-voltage curve was estimated using a one-dimensional Nernst-Planck equation. Although reasonable agreement with experiment is achieved for K(+) ions flowing from the N-terminal to the C-terminal mouth, in the opposite direction the current is underestimated. This underestimation may be a consequence of under-sampling of the conformational dynamics of the channel. We suggest that our simulations may have captured, for example, a sub-conductance level (i.e. an incompletely open state) of the LS3 channel.     

Relationship between membrane excitability and single channel open-close kinetics.

We have developed a novel technique for simulating the influence of the effects of single channel kinetics on the voltage changes associated with membrane excitability. The technique uses probability distribution functions for the durations of channel open- and closed-state lifetimes, which can be calculated for any model of the ion conductance process. To illustrate the technique, we have used the Hodgkin and Huxley model of nerve membrane ion conductances to simulate channel kinetics during predetermined voltage changes, such as a voltage jump and an action potential. We have also simulated the influence of channels on voltage changes in a free running, non-voltage-clamped patch of membrane 1 micron2 or less in area. The latter results provide a direct illustration of the relationship between fluctuations of membrane excitability and fluctuations in channel open- and closed-state lifetimes.     

Ion channels: does each subunit do something on its own?

The advent of the patch-clamp technique 25 years ago revolutionized the study of ion channels. This method also made it possible to measure the kinetic behavior of single protein molecules. The low-noise recordings of ionic currents through single channels, coupled with other cutting-edge technologies, have revealed a rich complexity of functional states that are not readily explained by simple allosteric protein models such as the popular concerted model and the sequential model. Although these models can each account for elements of ion channel function, we propose that variations or extensions of the lesser-known general allosteric model provide a more promising framework for explaining the intricate behaviors of ion channels.     

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 peptaibol antiamoebin as a model ion channel: similarities to bacterial potassium channels.

Antiamoebin (AAM) is a polypeptide antibiotic that is capable of forming ion channels in phospholipid membranes: planar bilayer studies have suggested the channels are octamers. The crystal structure of a monomeric form of AAM has provided the basis for molecular modelling of an octameric helical bundle channel. The channel model is funnel-shaped due to a substantial bend in the middle of the polypeptide chain caused by the presence of several imino acids. Inter-monomer hydrogen bonds orientate a ring of glutamine side chains to form a constriction in the pore lumen. The channel lumen is lined both by side chains of Gln11 and by polypeptide backbone carbonyl groups. Electrostatic calculations on the model are compatible with a channel that transports cations across membranes. The AAM channel model is compared with the crystal structures of two bacterial (KcsA andMthK) potassium channels. AAM and the potassium channels exhibit common functional features, such as cation-selectivity and similar single channel conductances. Common structural features include being multimers, each formed from a bundle of eight transmembrane helices, with lengths roughly comparable to the thickness of lipid bilayers. In addition, they all have aromatic amino acids that lie at the bilayer interfaces and which may aid in the stabilization of the transmembrane helices, as well as narrower constrictions that define the ion binding sites or selectivity filters in the pore lumen. The commonality of structural and functional features in these channels thus suggests that antiamoebin is a good, simple model for more complex bacterial and eukaryotic ion channels, capable of providing insight into details of the mechanisms of ion transport and multimeric channel stability.     

Simple model of ion transport through alamethicin channels in lipid membranes

The electrical characteristics of wide membrane channels such as those induced in lipid membranes by alamethicin have been analyzed using an electrodiffusion model. The channel is considered to be a water filled cylinder in which the potential energy barrier is a result of the difference in polarization energy of the ion environment when the ion is located inside as compared to outside of the channel. In addition, an electric field related to the channel structure is assumed. It is shown that without postulating any specific chemical ion-channel interaction one can reproduce experimental membrane potentials for NaCl, KCl, and CaCl₂ concentration gradients with a single set of channel parameters. The calculations also yield experimental J-V characteristics of discrete conduction states. In addition, a simple mechanism of interchannel coupling based on the above model is discussed. The model suggests a unifying approach to the problem of the origin of interionic selectivity of membrane channels induced by polyene antibiotics.     

The ion channel inverse problem: neuroinformatics meets biophysics

Ion channels are the building blocks of the information processing capability of neurons: any realistic computational model of a neuron must include reliable and effective ion channel components. Sophisticated statistical and computational tools have been developed to study the ion channel structure–function relationship, but this work is rarely incorporated into the models used for single neurons or small networks. The disjunction is partly a matter of convention. Structure–function studies typically use a single Markov model for the whole channel whereas until recently whole-cell modeling software has focused on serial, independent, two-state subunits that can be represented by the Hodgkin–Huxley equations. More fundamentally, there is a difference in purpose that prevents models being easily reused. Biophysical models are typically developed to study one particular aspect of channel gating in detail, whereas neural modelers require broad coverage of the entire range of channel behavior that is often best achieved with approximate representations that omit structural features that cannot be adequately constrained. To bridge the gap so that more recent channel data can be used in neural models requires new computational infrastructure for bringing together diverse sources of data to arrive at best-fit models for whole-cell modeling. We review the current state of channel modeling and explore the developments needed for its conclusions to be integrated into whole-cell modeling.