Effective diffusion coefficients of K+ and Cl- ions in ion channel models

Fast prediction of antibiotic permeability through membrane channels using Brownian dynamics

The efficient permeation across the Gram-negative bacterial membrane is an important step in the overall process of antibacterial action of a molecule and the one that has posed a significant hurdle on the way toward approved antibiotics. Predicting the permeability for a large library of molecules and assessing the effect of different molecular transformations on permeation rates of a given molecule is critical to the development of effective antibiotics. We present a computational approach for obtaining estimates of molecular permeability through a porin channel in a matter of hours using a Brownian dynamics approach. The fast sampling using a temperature acceleration scheme enables the approximate estimation of permeability using the inhomogeneous solubility diffusion model. Although the method is a significant approximation to similar all-atom approaches tested previously, we show that the present approach predicts permeabilities that correlate fairly well with the respective experimental permeation rates from liposome swelling experiments and accumulation rates from antibiotic accumulation assays, and is significantly, i.e., about 14 times, faster compared with a previously reported approach. The possible applications of the scheme in high-throughput screening for fast permeators are discussed.

Multi-oligomeric states of alamethicin ion channel: Assemblies and conductance

Transmembrane assemblies of the peptaibol alamethicin (ALM) are among the most extensively studied ion channels not only because of their antimicrobial activity but also as models for channel structure and aggregation. In this study, several oligomeric states of ALM are investigated with molecular dynamics simulations to establish properties of the channel and obtain free energy profiles for ion transport and the corresponding values of conductance. The hexamer, heptamer, and octamer of ALM in phospholipid membrane are found to be stable but highly dynamic in barrel-stave structures, with calculated conductance equal to 18, 195, and 1270 pS, respectively, in 1 M KCl ion solution. The corresponding free energy profiles, reported for the first time, are reconstructed from simulations at applied voltage of 200 mV with the aid of the electrodiffusion model both with and without the knowledge of diffusivity. The calculated free energy barriers are equal to 2.5, 1.5, and 0.5 kcal/mol for K+ and 4.0, 2.2, and 1.5 kcal/mol for Cl-, for hexamer, heptamer, and octamer, respectively. The calculated conductance and the ratio between conductance in consecutive states are in good agreement with those measured experimentally. This suggests that the hexamer is the lowest conducting state, with measured conductance equal to 19 pS. The selectivity of K+ over Cl- is calculated as 1.5 and 2.3 for the octameric and heptameric channels, close to the selectivity measured for high-conductance states. Selectivity increases to 13 in the hexameric channel in which the narrowest Gln7 site has a pore radius of only ∼1.6 Å, again in accord with experiment. A good agreement found between calculated and measured conductance through a hexamer templated on cyclodextrin lands additional support for the results of our simulations, and the comparison with ALM reveals the dependence of conductance on the nature of phospholipid membrane.

The Whole Is Bigger than the Sum of Its Parts: Drug Transport in the Context of Two Membranes with Active Efflux

Cell envelope plays a dual role in the life of bacteria by simultaneously protecting it from a hostile environment and facilitating access to beneficial molecules. At the heart of this ability lie the restrictive properties of the cellular membrane augmented by efflux transporters, which preclude intracellular penetration of most molecules except with the help of specialized uptake mediators. Recently, kinetic properties of the cell envelope came into focus driven on one hand by the urgent need in new antibiotics and, on the other hand, by experimental and theoretical advances in studies of transmembrane transport. A notable result from these studies is the development of a kinetic formalism that integrates the Michaelis-Menten behavior of individual transporters with transmembrane diffusion and offers a quantitative basis for the analysis of intracellular penetration of bioactive compounds. This review surveys key experimental and computational approaches to the investigation of transport by individual translocators and in whole cells, summarizes key findings from these studies and outlines implications for antibiotic discovery. Special emphasis is placed on Gram-negative bacteria, whose envelope contains two separate membranes. This feature sets these organisms apart from Gram-positive bacteria and eukaryotic cells by providing them with full benefits of the synergy between slow transmembrane diffusion and active efflux.

Millisecond-Long Simulations of Antibiotics Transport through Outer Membrane Channels

To reach their target site inside Gram-negative bacteria, almost all antibiotics need to cross the outer membrane. Computational modeling of such processes can be numerically demanding due to the size of the systems and especially due to the timescales involved. Recently, a hybrid Brownian and molecular dynamics approach, i.e., Brownian dynamics including explicit atoms (BRODEA), has been developed and evaluated for studying the transport of monoatomic ions through membrane channels. Later on, this numerically efficient scheme has been applied to determine the free energy surfaces of the ciprofloxacin and enrofloxacin translocation through the porin OmpC using temperature-accelerated simulations. To improve the usability and accuracy of the approach, schemes to approximate the position-dependent diffusion constant of the molecule while traversing the pore had to be established. To this end, we have studied the translocation of the charged phosphonic acid antibiotic fosfomycin through the porin OmpF from Escherichia coli devising and benchmarking several diffusion models. To test the efficiency and sensitivity of these models, the effect of OmpF mutations on the permeation of fosfomycin was analyzed. Permeation events have been recorded over millisecond-long biased and unbiased simulations, from which thermodynamics and kinetics quantities of the translocation processes were determined. As a result, the use of the BRODEA approach, together with the appropriate diffusion model, was seen to accurately reproduce the findings observed in electrophysiology experiments and all-atom molecular dynamics simulations. These results suggest that the BRODEA approach can become a valuable tool for screening numerous compounds to evaluate their outer membrane permeability, a property important in the development of new antibiotics.

Sensing Single Molecule Penetration into Nanopores: Pushing the Time Resolution to the Diffusion Limit

To quantify small molecule penetration into and eventually permeation through nanopores, we applied an improved excess-noise analysis of the ion current fluctuation caused by entering molecules. The kinetic parameters of substrate entry and exit are derived from a two-state Markov model, analyzing the substrate concentration dependence of the average ion current and its variance. Including filter corrections allows one to detect the transition rates beyond the cutoff frequency, f_c, of the instrumental ion-current filter. As an application of the method, we performed an analysis of the single-channel ion current of Meropenem, an antibiotic of the carbapenem family, interacting with OmpF, the major general outer membrane channel of Escherichia coli bacteria. At 40 °C we detected the residence time of Meropenem inside OmpF of about 500 ns-more than 2 orders of magnitude smaller than f_c^-1 and close to the diffusion limit of few hundred nanoseconds. We also have established theoretical limit conditions under which the substrate-induced channel blockages can be detected and suggest that submicrosecond-scale gating kinetic parameters are accessible with existing experimental equipment.

Transmembrane dynamics of the Thr-5 phosphorylated sarcolipin pentameric channel

Sarcolipin (SLN), an important membrane protein expressed in the sarcoplasmic reticulum (SR), regulates muscle contractions in cardiac and skeletal muscle. The phosphorylation at amino acid Thr5 of the SLN protein modulates the amount of Ca(2+) that passes through the SR. Using molecular dynamics simulation, we evaluated the phosphorylation at Thr5 of pentameric SLN (phospho-SLN) channel's energy barrier and pore characteristics by calculating the potential of mean force (PMF) along the channel pore and determining the diffusion coefficient. The results indicate that pentameric phospho-SLN promotes penetration of monovalent and divalent ions through the channel. The analysis of PMF, pore radius and diffusion coefficient indicates that Leu21 is the hydrophobic gate of the pentameric SLN channel. In the channel, water molecules near the Leu21 pore demonstrated a clear hydrated-dehydrated transition; however, the mutation of Leu21 to an Alanine (L21A) destroyed the hydrated-dehydrated transitions. These water-dynamic behaviors and PMF confirm that Leu21 is the key residue that regulates the ion permeability of the pentameric SLN channel. These results provide the structural-basis insights and molecular-dynamic information that are needed to understand the regulatory mechanisms of ion permeability in the pentameric SLN channel.

Transport behavior of a single Ca(2+), K(+), and Na(+) in a water-filled transmembrane cyclic peptide nanotube

Molecular dynamics simulations have been performed to investigate the transport properties of a single Ca(2+), K(+), and Na(+) in a water-filled transmembrane cyclic peptide nanotube (CPNT). Two transmembrane CPNTs, i.e., 8×(WL)n=4,5/POPE (with uniform lengths but various radii), were applied to clarify the dependence of ionic transport properties on the channel radius. A huge energy barrier keeps Ca(2+) out of the octa-CPNT, while Na(+) and K(+) can be trapped in two CPNTs. The dominant electrostatic interaction of a cation with water molecules leads to a high distribution of channel water around the cation and D-defects in the first and last gaps, and significantly reduces the axial diffusion of channel water. Water-bridged interactions were mostly found between the artificially introduced Ca(2+) and the framework of the octa-CPNT, and direct coordinations with the tube wall mostly occur for K(+) in the octa-CPNT. A cation may drift rapidly or behave lazily in a CPNT. K(+) behaves most actively and can visit the whole deca-CPNT quickly. The first solvation shells of Ca(2+) and Na(+) are basically saturated in two CPNTs, while the hydration of K(+) is incomplete in the octa-CPNT. The solvation structure of Ca(2+) in the octa-CPNT is most stable, while that of K(+) in the deca-CPNT is most labile. Increasing the channel radius induces numerous interchange attempts between the first-shell water molecules of a cation and the ones in the outer region, especially for the K(+) system.

Arrayed lipid bilayer chambers allow single-molecule analysis of membrane transporter activity

Nano- to micron-size reaction chamber arrays (femtolitre chamber arrays) have facilitated the development of sensitive and quantitative biological assays, such as single-molecule enzymatic assays, digital PCR and digital ELISA. However, the versatility of femtolitre chamber arrays is limited to reactions that occur in aqueous solutions. Here we report an arrayed lipid bilayer chamber system (ALBiC) that contains sub-million femtolitre chambers, each sealed with a stable 4-μm-diameter lipid bilayer membrane. When reconstituted with a limiting amount of the membrane transporter proteins α-hemolysin or F0F1-ATP synthase, the chambers within the ALBiC exhibit stochastic and quantized transporting activities. This demonstrates that the single-molecule analysis of passive and active membrane transport is achievable with the ALBiC system. This new platform broadens the versatility of femtolitre chamber arrays and paves the way for novel applications aimed at furthering our mechanistic understanding of membrane proteins' function.

Physics of ion channels

We review the basic physics involved in transport of ions across membrane channels in cells. Electrochemical forces that control the diffusion of ions are discussed both from microscopic and macroscopic perspectives. A case is made for use of Brownian dynamics as the minimal phenomenological model that provides a bridge between experiments and more fundamental theoretical approaches. Application of Brownian and molecular dynamics methods to channels with known molecular structures is discussed.

Molecular dynamics simulation of the antiamoebin ion channel: linking structure and conductance

Molecular-dynamics simulations were carried out to ascertain which of the potential multimeric forms of the transmembrane peptaibol channel, antiamoebin, is consistent with its measured conductance. Estimates of the conductance obtained through counting ions that cross the channel and by solving the Nernst-Planck equation yield consistent results, indicating that the motion of ions inside the channel can be satisfactorily described as diffusive. The calculated conductance of octameric channels is markedly higher than the conductance measured in single channel recordings, whereas the tetramer appears to be nonconducting. The conductance of the hexamer was estimated to be 115 ± 34 pS and 74 ± 20 pS, at 150 mV and 75 mV, respectively, in satisfactory agreement with the value of 90 pS measured at 75 mV. On this basis, we propose that the antiamoebin channel consists of six monomers. Its pore is large enough to accommodate K⁺ and Cl⁻ with their first solvation shells intact. The free energy barrier encountered by K⁺ is only 2.2 kcal/mol whereas Cl⁻ encounters a substantially higher barrier of nearly 5 kcal/mol. This difference makes the channel selective for cations. Ion crossing events are shown to be uncorrelated and follow Poisson statistics.

Extrusion of transmitter, water and ions generates forces to close fusion pore

During exocytosis the fusion pore opens rapidly, then dilates gradually, and may subsequently close completely, but what controls its dynamics is not well understood. In this study we focus our attention on forces acting on the pore wall, and which are generated solely by the passage of transmitter, ions and water through the open fusion pore. The transport through the charged cylindrical nano-size pore is simulated using a coupled system of Poisson-Nernst-Planck and Navier-Stokes equations and the forces that act radially on the wall of the fusion pore are then estimated. Four forces are considered: a) inertial force, b) pressure, c) viscotic force, and d) electrostatic force. The inertial and viscotic forces are small, but the electrostatic force and the pressure are typically significant. High vesicular pressure tends to open the fusion pore, but the pressure induced by the transport of charged particles (glutamate, ions), which is predominant when the pore wall charge density is high tends to close the pore. The electrostatic force, which also depends on the charge density on the pore wall, is weakly repulsive before the pore dilates, but becomes attractive and pronounced as the pore dilates. Given that the vesicular concentration of free transmitter can change rapidly due to the release, or owing to the dissociation from the gel matrix, we evaluated how much and how rapidly a change of the vesicular K(+)-glutamate(-) concentration affects the concentration of glutamate(-) and ions in the pore and how such changes alter the radial force on the wall of the fusion pore. A step-like rise of the vesicular K(+)-glutamate(-) concentration leads to a chain of events. Pore concentration (and efflux) of both K(+) and glutamate(-) rise reaching their new steady-state values in less than 100 ns. Interestingly within a similar time interval the pore concentration of Na(+) also rises, whereas that of Cl(-) diminishes, although their extra-cellular concentration does not change. Finally such changes affect also the water movement. Water efflux changes bi-phasically, first increasing before decreasing to a new, but lower steady-state value. Nevertheless, even under such conditions an overall approximate neutrality of the pore is maintained remarkably well, and the electrostatic, but also inertial, viscotic and pressure forces acting on the pore wall remain constant. In conclusion the extrusion of the vesicular content generates forces, primarily the force due to the electro-kinetically induced pressure and electrostatic force (both influenced by the pore radius and even more by the charge density on the pore wall), which tend to close the fusion pore.

Interfacial interactions of glutamate, water and ions with carbon nanopore evaluated by molecular dynamics simulations

Molecular dynamics simulations were used to assess the transport of glutamate, water and ions (Na(+) and Cl(-)) in a single wall carbon nanopore. The spatial profiles of Na(+) and Cl(-) ions are largely determined by the pore wall charges. Co-ions are repelled whereas the counter-ions are attracted by the pore charges, but this 'rule' breaks down when the water concentration is set to a level significantly below that in the physiological bulk solution. In such cases water is less able to counteract the ion-wall interactions (electrostatic or non-electrostatic), co-ions are layered near the counter-ions attracted by the wall charges and are thus layered as counter-ions. Glutamate is concentrated near the pore wall even at physiological water concentration, and irrespective of whether the pore wall is neutral or charged (positively or negatively), and its peak levels are up to 40 times above mean values. The glutamate is thus always layered as a counter-ion. Layering of water near the wall is independent of charges on the pore wall, but its peak levels near the wall are 'only' 6-8 times above the pore mean values. However, if the mean concentration of water is significantly below the level in the physiological bulk solution, its layering is enhanced, whereas its concentration in the pore center diminishes to very low levels. Reasons for such a 'paradoxical' behavior of molecules (glutamate and water) are that the non-electrostatic interactions are (except at very short distances) attractive, and electrostatic interactions (between the charged atoms of the glutamate or water and the pore wall) are also attractive overall. Repulsive interactions (between equally charged atoms) exist, and they order the molecules near the wall, whereas in the pore center the glutamate (and water) angles are largely randomly distributed, except in the presence of an external electric field. Diffusion of molecules and ions is complex. The translational diffusion is in general both inhomogeneous and anisotropic. Non-electrostatic interactions (ion-wall, glutamate-wall or water-wall) powerfully influence diffusion. In the neutral nanopore the effective axial diffusion constants of glutamate, water and Na(+) and Cl(-) ions are all <10% of their values in the bulk, and the electrostatic interactions can reduce them further. Diffusion of molecules and ions is further reduced if the water concentration in the pore is low. Glutamate(-) is slowed more than water, and ions are reduced the most especially co-ions. In conclusion the interfacial interactions influence the spatial distribution of glutamate, water and ions, and regulate powerfully, in a complex manner and over a very wide range their transport through nanosize pores.

Water diffusion through a membrane protein channel: A first passage time approach

Water diffusion through OmpF, a porin in the outer membrane of Escherichia coli, is studied by molecular dynamics simulation. A first passage time approach allows characterizing the diffusive properties of a well-defined region of this channel. A carbon nanotube, which is considerably more homogeneous, serves as a model to validate the methodology. Here we find, in addition to the expected regular behavior, a gradient of the diffusion coefficient at the channel ends, witness of the transition from confinement in the channel to bulk behavior in the connected reservoirs. Moreover, we observe the effect of a kinetic boundary layer, which is the counterpart of the initial ballistic regime in a mean square displacement analysis. The overall diffusive behavior of water in OmpF shows remarkable similarity with that in a homogeneous channel. However, a small fraction of the water molecules appears to be trapped by the protein wall for considerable lengths of time. The distribution of trapping times exhibits a broad power law distribution psi(tau) approximately tau (-2.4), up to tau=10 ns, a bound set by the length of the simulation run. We discuss the effect of this distribution on the dynamic properties of water in OmpF in terms of incomplete sampling of phase space.

A hierarchical approach for predicting the transport properties of the gramicidin A channel

A hierarchical computational approach combining results of molecular dynamics (MD) simulations with continuous Poisson-Nernst-Planck (PNP) theory was used to investigate ion transport in a gramicidin A (gA) channel embedded within a 1,2-dimyristoylphosphatidylcholine (DMPC) bilayer. Molecular dynamics (MD) employing the CHARMM force field was used to investigate the diffusion of Na and K at different locations along the gA channel in both singly- and doubly-occupied states. Self-diffusion coefficients for single Na and K cations were determined to be 4.7 × 10 cm s and 6.2 × 10 cm s, respectively. Using these values, maximum ionic conductivities calculated from the Nernst-Einstein equation were 37 pS and 49 pS for Na and K, respectively, in the singly-occupied gA channel. These values agree with experimental data within an order of magnitude. Conductance of the gA channel was calculated from simulation results using the three-dimensional Poisson-Nernst-Planck (3D-PNP) model. Partial charge distributions for gA and for DMPC were assigned using the Poisson-Boltzmann module available in CHARMM. Diffusion coefficients were those obtained from the MD simulation. Results confirm that DMPC electrostatics have significant influence on channel conductivity. At low electrolyte concentrations, the channel cannot be occupied by more than one monovalent cation. Using ion diffusion coefficients obtained at different locations along the channel, current-voltage values obtained using 3D-PNP predictions for a channel immersed in an aqueous NaCl solution show good agreement with experimental values.

Glutamate, water and ion transport through a charged nanosize pore

The transport of transmitter, ions and water through a positively-charged nanopore was investigated through computer simulations. The physics of the problem is described by a coupled set of Poisson-Nernst-Planck and Navier-Stokes equations in a computational domain consisting a cylindrical pore, whose radius ranged from 1 to 8 nm and which was flanked by two compartments representing the vesicular interior and extra-cellular space. The concentration of co-ions is suppressed and of counter-ions enhanced, especially near the pore wall owing to electrostatic interactions. Glutamate (i.e. the transmitter considered) is negatively charged and is simulated as a counter-ion. The electro-kinetically induced pressure due to the movement of ions is negative and very pronounced near the pore wall where the concentration and flux of counter-ions is very high. The water velocity peaks in the pore center, diminishes to zero at the pore wall, but is constant along the pore axis. The mean velocity of the water/fluid is proportional to the vesicular pressure and pore cross-sectional area. Interestingly it is inversely related to the vesicular glutamate concentration. The factors determining the glutamate flux are complex. The diffusive flux generally predominates for narrow pore, and convective flux may dominate for wide pore if the vesicular pressure is high. Surprisingly at low vesicular pressure the mean total glutamate flux per unit cross-sectional pore area is higher for narrow pores. Higher flux is probably due to the rise of glutamate concentration in the nanopore, which is much more pronounced for narrow nanopores, due to the maintenance of approximate neutrality of charges in the pore and on the pore wall. In conclusion intra-vesicular pressure helps 'flushing-out' the transmitter, but the induced pressure 'drags-out' the water into the extra-cellular space.

Diffusion constant of K+ inside Gramicidin A: a comparative study of four computational methods

The local diffusion constant of K(+) inside the Gramicidin A (GA) channel has been calculated using four computational methods based on molecular dynamics (MD) simulations, specifically: Mean Square Displacement (MSD), Velocity Autocorrelation Function (VACF), Second Fluctuation Dissipation Theorem (SFDT) and analysis of the Generalized Langevin Equation for a Harmonic Oscillator (GLE-HO). All methods were first tested and compared for K(+) in bulk water--all predicted the correct diffusion constant. Inside GA, MSD and VACF methods were found to be unreliable because they are biased by the systematic force exerted by the membrane-channel system on the ion. SFDT and GLE-HO techniques properly unbias the influence of the systematic force on the diffusion properties and predicted a similar diffusion constant of K(+) inside GA, namely, ca. 10 times smaller than in the bulk. It was found that both SFDT and GLE-HO methods require extensive MD sampling on the order of tens of nanoseconds to predict a reliable diffusion constant of K(+) inside GA.

Electrostatic properties of the mechanosensitive channel of small conductance MscS

The mechanosensitive channel of small conductance (MscS) belongs to a family of membrane proteins that are gated in response to changes in membrane tension, thereby protecting the cell from hypo-osmotic shock. Here we report on passive ion transport simulations of MscS in a POPC bilayer using a coarse-grained particle-based description based on the Boltzmann transport Monte Carlo method. Single channel current-voltage curves are computed over hundreds of nanoseconds for channel conformations derived from all-atom molecular dynamics simulations reaching an overall simulation time of over 5 micros. Channel conformations similar to that of the crystal structure exhibit low conductance, whereas conformations reached after opening the channel by means of steered molecular dynamics simulations match experimentally determined conductances. However, while experiments indicate a slight preference for anionic currents, the simulated channel strongly selects anions over cations and the direction of rectification at high voltages is opposite to what is observed in experiments. Three-dimensional maps of time-averaged ion distribution and equilibrium occupancy profiles constructed from trajectory data indicate separation of anions and cations inside and in the immediate vicinity of the large cytoplasmic domain of MscS, in accordance with earlier molecular dynamics simulations. This separation arises from the distribution of ionizable residues of MscS and suggests a specific, yet unknown, functional purpose.

Ionic permeation free energy in gramicidin: a semimicroscopic perspective

Ion permeation through the gramicidin channel is studied using a model that circumvents two major difficulties inherent to standard simulational methods. It exploits the timescale separation between electronic and structural contributions to dielectric stabilization, accounting for the influence of electronic polarization by embedding the channel in a dielectric milieu that describes this polarization in a mean sense. The explicit mobile moieties are the ion, multipolar waters, and the carbonyls and amides of the peptide backbone. The model treats the influence of aromatic residues and the membrane dipole potential. A new electrical geometry is introduced that treats long-range electrostatics exactly and avoids problems related to periodic boundary conditions. It permits the translocating ion to make a seamless transition from nearby electrolyte to the channel interior. Other degrees of freedom (more distant bulk electrolyte and nonpolar lipid) are treated as dielectric continua. Reasonable permeation free energy profiles are obtained for potassium, rubidium, and cesium; binding wells are shallow and the central barrier is small. Estimated cationic single-channel conductances are smaller than experiment, but only by factors between 2 (rubidium) and 50 (potassium). When applied to chloride the internal barrier is large, with a corresponding miniscule single-channel conductance. The estimated relative single-channel conductances of gramicidin A, B, and C agree well with experiment.

Influence of protein flexibility on the electrostatic energy landscape in gramicidin A

We describe an electrostatic model of the gramicidin A channel that allows protein atoms to move in response to the presence of a permeating ion. To do this, molecular dynamics simulations are carried out with a permeating ion at various positions within the channel. Then an ensemble of atomic coordinates taken from the simulations are used to construct energy profiles using macroscopic electrostatic calculations. The energy profiles constructed are compared to experimentally-determined conductance data by inserting them into Brownian dynamics simulations. We find that the energy landscape seen by a permeating ion changes significantly when we allow the protein atoms to move rather than using a rigid protein structure. However, the model developed cannot satisfactorily reproduce all of the experimental data. Thus, even when protein atoms are allowed to move, the dielectric model used in our electrostatic calculations breaks down when modeling the gramicidin channel.

Ligand-gated ion channels: mechanisms underlying ion selectivity

Anion/cation selectivity is a critical property of ion channels and underpins their physiological function. Recently, there have been numerous mutagenesis studies, which have mapped sites within the ion channel-forming segments of ligand-gated ion channels that are determinants of the ion selectivity. Site-directed mutations to specific amino acids within or flanking the M2 transmembrane segments of the anion-selective glycine, GABA(A) and GABA(C) receptors and the cation-selective nicotinic acetylcholine and serotonin (type 3) receptors have revealed discrete, equivalent regions within the ion channel that form the principal selectivity filter, leading to plausible molecular mechanisms and mathematical models to describe how ions preferentially permeate these channels. In particular, the dominant factor determining anion/cation selectivity seems to be the sign and exposure of charged amino acids lining the selectivity filter region of the open channel. In addition, the minimum pore diameter, which can be influenced by the presence of a local proline residue, also makes a contribution to such ion selectivity in LGICs with smaller diameters increasing anion/cation selectivity and larger ones decreasing it.

Improved 3D continuum calculations of ion flux through membrane channels

A continuum model, based on the Poisson-Nernst-Planck (PNP) theory, is applied to simulate steady-state ion flux through protein channels. The PNP equations are modified to explicitly account (1) for the desolvation of mobile ions in the membrane pore and (2) for effects related to ion sizes. The proposed algorithm for a three-dimensional self-consistent solution of PNP equations, in which final results are refined by a focusing technique, is shown to be suitable for arbitrary channel geometry and arbitrary protein charge distribution. The role of the pore shape and protein charge distribution in formation of basic electrodiffusion properties, such as channel conductivity and selectivity, as well as concentration distributions of mobile ions in the pore region, are illustrated by simulations on model channels. The influence of the ionic strength in the bulk solution and of the externally applied electric field on channel properties are also discussed.

Exploring the origin of the ion selectivity of the KcsA potassium channel

The availability of structural information about biological ion channels provides an opportunity to gain a detailed understanding of the control of ion selectivity by biological systems. However, accomplishing this task by computer simulation approaches is very challenging. First, although the activation barriers for ion transport can be evaluated by microscopic simulations, it is hard to obtain accurate results by such approaches. Second, the selectivity is related to the actual ion current and not directly to the individual activation barriers. Thus, it is essential to simulate the ion currents and this cannot be accomplished at present by microscopic MD approaches. In order to address this challenge, we developed and refined an approach capable of evaluating ion current while still reflecting the realistic features of the given channel. Our method involves generation of semimacroscopic free energy surfaces for the channel/ions system and Brownian dynamics (BD) simulations of the corresponding ion current. In contrast to most alternative macroscopic models, our approach is able to reproduce the difference between the free energy surfaces of different ions and thus to address the selectivity problem. Our method is used in a study of the selectivity of the KcsA channel toward the K+ and Na+ ions. The BD simulations with the calculated free energy profiles produce an appreciable selectivity. To the best of our knowledge, this is the first time that the trend in the selectivity in the ion current is produced by a computer simulation approach. Nevertheless, the calculated selectivity is still smaller than its experimental estimate. Recognizing that the calculated profiles are not perfect, we examine how changes in these profiles can account for the observed selectivity. It is found that the origin of the selectivity is more complex than generally assumed. The observed selectivity can be reproduced by increasing the barrier at the exit and the entrance of the selectivity filter, but the necessary changes in the barrier approach the limit of the error in the PDLD/S-LRA calculations. Other options that can increase the selectivity are also considered, including the difference between the Na+...Na+ and K+...K+ interaction. However, this interesting effect does not appear to lead to a major difference in selectivity since the Na+ ions at the limit of strong interaction tend to move in a less concerted way than the K+ ions. Changes in the relative binding energies at the different binding sites are also not so effective in changing the selectivity. Finally, it is pointed out that using the calculated profiles as a starting point and forcing the model to satisfy different experimentally based constraints, should eventually provide more detailed understanding of the different complex factors involved in ion selectivity of biological channels.

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.

Role of the dielectric constants of membrane proteins and channel water in ion permeation

Using both analytical solutions obtained from simplified systems and numerical results from more realistic cases, we investigate the role played by the dielectric constant of membrane proteins epsilon(p) and pore water epsilon(w) in permeation of ions across channels. We show that the boundary and its curvature are the crucial factors in determining how an ion's potential energy depends on the dielectric constants near an interface. The potential energy of an ion outside a globular protein has a dominant 1/epsilon(w) dependence, but this becomes 1/epsilon(p) for an ion inside a cavity. For channels, where the boundaries are in between these two extremes, the situation is more complex. In general, we find that variations in epsilon(w) have a much larger impact on the potential energy of an ion compared to those in epsilon(p). Therefore a better understanding of the effective epsilon(w) values employed in channel models is desirable. Although the precise value of epsilon(p) is not a crucial determinant of ion permeation properties, it still needs to be chosen carefully when quantitative comparisons with data are made.

Ion Permeation and Selectivity of OmpF Porin: A Theoretical Study Based on Molecular Dynamics, Brownian Dynamics, and Continuum Electrodiffusion Theory

Three different theoretical approaches are used and compared to refine our understanding of ion permeation through the channel formed by OmpF porin from Escherichia coli. Those approaches are all-atom molecular dynamics (MD) in which ions, solvent, and lipids are represented explicitly, Brownian dynamics (BD) in which ions are represented explicitly, while solvent and lipids are represented as featureless dielectrics, and Poisson-Nernst-Planck (PNP) electrodiffusion theory in which both solvent and local ion concentrations are represented as a continuum. First, the ability of the different theoretical approaches in reproducing the equilibrium average ion density distribution in OmpF porin bathed by a 1M KCl symmetric salt solution is examined. Under those conditions the PNP theory is equivalent to the non-linear Poisson-Boltzmann (PB) theory. Analysis shows that all the three approaches are able to capture the important electrostatic interactions between ions and the charge distribution of the channel that govern ion permeation and selectivity in OmpF. The K(+) and Cl(-) density distributions obtained from the three approaches are very consistent with one another, which suggests that a treatment on the basis of a rigid protein and continuum dielectric solvent is valid in the case of OmpF. Interestingly, both BD and continuum electrostatics reproduce the distinct left-handed twisted ion pathways for K(+) and Cl(-) extending over the length of the pore which were observed previously in MD. Equilibrium BD simulations in the grand canonical ensemble indicate that the channel is very attractive for cations, particularly at low salt concentration. On an average there is 1.55 K(+) inside the pore in 10mM KCl. Remarkably, there is still 0.17 K(+) on average inside the pore even at a concentration as low as 1microM KCl. Secondly, non-equilibrium ion flow through OmpF is calculated using BD and PNP and compared with experimental data. The channel conductance in 0.2M and 1M KCl calculated using BD is in excellent accord with the experimental data. The calculations reproduce the experimentally well-known conductance-concentration relation and also reveal an asymmetry in the channel conductance (a larger conductance is observed under a positive transmembrane potential). Calculations of the channel conductance for three mutants (R168A, R132A, and K16A) in 1M KCl suggest that the asymmetry in the channel conductance arises mostly from the permanent charge distribution of the channel rather than the shape of the pore itself. Lastly, the calculated reversal potential in a tenfold salt gradient (0.1:1M KCl) is 27.4(+/-1.3)mV (BD) and 22.1(+/-0.6)mV (PNP), in excellent accord with the experimental value of 24.3mV. Although most of the results from PNP are qualitatively reasonable, the calculated channel conductance is about 50% higher than that calculated from BD probably because of a lack of some dynamical ion-ion correlations.

OmpA: a pore or not a pore? Simulation and modeling studies

The bacterial outer membrane protein OmpA is composed of an N-terminal 171-residue beta-barrel domain (OmpA(171)) that spans the bilayer and a periplasmic, C-terminal domain of unknown structure. OmpA has been suggested to primarily serve a structural role, as no continuous pore through the center of the barrel can be discerned in the crystal structure of OmpA(171). However, several groups have recorded ionic conductances for bilayer-reconstituted OmpA(171). To resolve this apparent paradox we have used molecular dynamics (MD) simulations on OmpA(171) to explore the conformational dynamics of the protein, in particular the possibility of transient formation of a central pore. A total of 19 ns of MD simulations of OmpA(171) have been run, and the results were analyzed in terms of 1) comparative behavior of OmpA(171) in different bilayer and bilayer-mimetic environments, 2) solvation states of OmpA(171), and 3) pore characteristics in different MD simulations. Significant mobility was observed for residues and water molecules within the beta-barrel. A simulation in which putative gate region side chains of the barrel interior were held in a non-native conformation led to an open pore, with a predicted conductance similar to experimental measurements. The OmpA(171) pore has been shown to be somewhat more dynamic than suggested by the crystal structure. A gating mechanism is proposed to explain its documented channel properties, involving a flickering isomerization of Arg138, forming alternate salt bridges with Glu52 (closed state) and Glu128 (open state).

Ions and counterions in a biological channel: a molecular dynamics simulation of OmpF porin from Escherichia coli in an explicit membrane with 1 M KCl aqueous salt solution

A 5 ns all-atom molecular dynamics trajectory of Escherichia coli OmpF porin embedded in an explicit dimyristoyl-phosphatidylcholine (DMPC) bilayer bathed by a 1 M [KCl] aqueous salt solution is generated to explore the microscopic details of the mechanism of ion permeation. The atomic model includes the OmpF trimer, 124 DMPC, 13470 water molecules as well as 231 K+ and 201 Cl-, for a total of 70,693 atoms. The structural and dynamical results are in excellent agreement with the X-ray data. The global root-mean-square deviation of the backbone atoms relative to the X-ray structure is 1.4 A. A cluster of three fully charged arginine (Arg42, Arg82, and Arg132) facing two acidic residues (Asp113 and Glu117) on L3 in the narrowest part of the aqueous pore is observed to be very stable in the crystallographic conformation. In this region of the pore, the water molecules are markedly oriented perpendicular to the channel axis due to the strong transversal electrostatic field arising from those residues. On average the size of the pore is smaller during the simulation than in the X-ray structure, undergoing small fluctuations. No large movements of loop L3 leading to a gating of the pore are observed. Remarkably, it is observed that K+ and Cl- follow two well-separated average pathways spanning over nearly 40 A along the axis of the pore. In the center of the monomer, the two screw-like pathways have a left-handed twist, undergoing a counter-clockwise rotation of 180 degrees from the extracellular vestibule to the pore periplasmic side. In the pore, the dynamical diffusion constants of the ions are reduced by about 50% relative to their value in bulk solvent. Analysis of ion solvation across the channel reveals that the contributions from the water and the protein are complementary, keeping the total solvation number of both ions nearly constant. Unsurprisingly, K+ have a higher propensity to occupy the aqueous pore than Cl-, consistent with the cation selectivity of the channel. However, further analysis suggests that ion-ion pairs play an important role. In particular, it is observed that the passage of Cl- occurs only in the presence of K+ counterions, and isolated K+ can move through the channel and permeate on their own. The presence of K+ in the pore screens the negative electrostatic potential arising from OmpF to help the translocation of Cl- by formation of ion pairs.

Potassium and sodium ions in a potassium channel studied by molecular dynamics simulations

We have performed simulations of both a single potassium ion and a single sodium ion within the pore of the bacterial potassium channel KcsA. For both ions there is a dehydration energy barrier at the cytoplasmic mouth suggesting that the crystal structure is a closed conformation of the channel. There is a potential energy barrier for a sodium ion in the selectivity filter that is not seen for potassium. Radial distribution functions for both ions with the carbonyl oxygens of the selectivity filter indicate that sodium may interact more tightly with the filter than does potassium. This suggests that the key to the ion selectivity of KcsA is the greater dehydration energy of Na(+) ions, and helps to explain the block of KcsA by internal Na(+) ions.

Mechanisms of permeation and selectivity in calcium channels

The mechanisms underlying ion transport and selectivity in calcium channels are examined using electrostatic calculations and Brownian dynamics simulations. We model the channel as a rigid structure with fixed charges in the walls, representing glutamate residues thought to be responsible for ion selectivity. Potential energy profiles obtained from multi-ion electrostatic calculations provide insights into ion permeation and many other observed features of L-type calcium channels. These qualitative explanations are confirmed by the results of Brownian dynamics simulations, which closely reproduce several experimental observations. These include the current-voltage curves, current-concentration relationship, block of monovalent currents by divalent ions, the anomalous mole fraction effect between sodium and calcium ions, attenuation of calcium current by external sodium ions, and the effects of mutating glutamate residues in the amino acid sequence.

Simulations of ion channels--watching ions and water move

Ion channels mediate electrical excitability in neurons and muscle. Three-dimensional structures for model peptide channels and for a potassium (K+) channel have been combined with computer simulations to permit rigorous exploration of structure-function relations of channels. Water molecules and ions within transbilayer pores tend to diffuse more slowly than in bulk solutions. In the narrow selectivity filter of the bacterial K+ channel (i.e. the region of the channel that discriminates between different species of ions) a column of water molecules and K+ ions moves in a concerted fashion. By combining atomistic simulations (in which all atoms of the channel molecule, water and ions are treated explicitly) with continuum methods (in which the description of the channel system is considerably simplified) it is possible to simulate some of the physiological properties of channels.

Molecular dynamics estimates of ion diffusion in model hydrophobic and KcsA potassium channels

Molecular dynamics simulations are carried out to obtain estimates of diffusion coefficients of biologically important Na+, K+, Ca2+ and Cl- ions in hydrophobic cylindrical channels with varying radii and large reservoirs. Calculations for the cylindrical channels are compared to those for the KcsA potassium channel, for which the protein structure has recently been determined from X-ray diffraction experiments. Our results show that ion diffusion is maintained at reasonably high levels even within narrow channels, and does not support the very small diffusion coefficients used in some continuum models in order to fit experimental data. The present estimates of ion diffusion coefficients are useful in the calculation of channel conductance using the Poisson-Nernst-Planck theory, or Brownian dynamics.