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.     

Hybrid MD-Nernst Planck model of α-hemolysin conductance properties

Motivated by experiments in which an applied electric field translocates polynucleotides through an α-hemolysin protein channel causing ionic current transient blockade, a hybrid simulation model is proposed to predict the conductance properties of the open channel. Time scales corresponding to ion permeation processes are reached using the Poisson–Nernst–Planck (PNP) electro-diffusion model in which both solvent and local ion concentrations are represented as a continuum. The diffusion coefficients of the ions (K⁺ and Cl^?) input in the PNP model are, however, calculated from all-atom molecular dynamics (MD). In the MD simulations, a reduced representation of the channel is used. The channel is solvated in a 1?M KCl solution, and an external electric field is applied. The pore specific diffusion coefficients for both ionic species are reduced 5–7 times in comparison to bulk values. Significant statistical variations (17–45%) of the pore-ions diffusivities are observed. Within the statistics, the ionic diffusivities remain invariable for a range of external applied voltages between 30 and 240?mV. In the 2D-PNP calculations, the pore stem is approximated by a smooth cylinder of radius ～9?Å with two constriction blocks where the radius is reduced to ～6?Å. The electrostatic potential includes the contribution from the atomistic charges. The MD-PNP model shows that the atomic charges are responsible for the rectifying behaviour and for the slight anion selectivity of the α-hemolysin pore. Independent of the hierarchy between the anion and cation diffusivities, the anionic contribution to the total ionic current will dominate. The predictions of the MD-PNP model are in good agreement with experimental data and give confidence in the present approach of bridging time scales by combining a microscopic and macroscopic model.     

Investigating the dielectric effects of channel pore water on the electrostatic barriers of the permeation ion by the finite difference Poisson-Boltzmann method

In this paper, the finite difference Poisson-Boltzmann (FDPB) method with four dielectric constants is developed to study the effect of dielectric saturation on the electrostatic barriers of the permeation ion. In this method, the inner shape of the channel pore is explicitly represented, and the fact that the dielectric constant inside the channel pore is different from that of bulk water is taken into account. A model channel system which is a right-handed twist bundle with four α-helical segments is provided for this study. From the FDPB calculations, it is found that the difference of the ionic electrostatic solvation energy for wider domains depends strongly on the pore radius in the vicinity of the ion when the pore dielectric constant is changed from 78 to 5. However, the electrostatic solvation energy of the permeation ion can not be significantly affected by the dielectric constant in regions with small pore radii. Our results indicate that the local electrostatic interactions inside the ion channel are of major importance for ion electrostatic solvation energies, and the effect of dielectric saturation on the electrostatic barriers is coupled to the interior channel dimensions. Received: 28 January 1997 / Accepted: 24 September 1997     

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.     

Ion permeation and glutamate residues linked by Poisson-Nernst-Planck theory in L-type calcium channels.

L-type Ca channels contain a cluster of four charged glutamate residues (EEEE locus), which seem essential for high Ca specificity. To understand how this highly charged structure might produce the currents and selectivity observed in this channel, a theory is needed that relates charge to current. We use an extended Poisson-Nernst-Planck (PNP2) theory to compute (mean) Coulombic interactions and thus to examine the role of the mean field electrostatic interactions in producing current and selectivity. The pore was modeled as a central cylinder with tapered atria; the cylinder (i.e., "pore proper") contained a uniform volume density of fixed charge equivalent to that of one to four carboxyl groups. The pore proper was assigned ion-specific, but spatially uniform, diffusion coefficients and excess chemical potentials. Thus electrostatic selection by valency was computed self-consistently, and selection by other features was also allowed. The five external parameters needed for a system of four ionic species (Na, Ca, Cl, and H) were determined analytically from published measurements of thre limiting conductances and two critical ion concentrations, while treating the pore as a macroscopic ion-exchange system in equilibrium with a uniform bath solution. The extended PNP equations were solved with these parameters, and the predictions were compared to currents measured in a variety of solutions over a range of transmembrane voltages. The extended PNP theory accurately predicted current-voltage relations, anomalous mole fraction effects in the observed current, saturation effects of varied Ca and Na concentrations, and block by protons. Pore geometry, dielectric permittivity, and the number of carboxyl groups had only weak effects. The successful prediction of Ca fluxes in this paper demonstrates that ad hoc electrostatic parameters, multiple discrete binding sites, and logistic assumptions of single-file movement are all unnecessary for the prediction of permeation in Ca channels over a wide range of conditions. Further work is needed, however, to understand the atomic origin of the fixed charge, excess chemical potentials, and diffusion coefficients of the channel. The Appendix uses PNP2 theory to predict ionic currents for published "barrier-and-well" energy profiles of this channel.     

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.     

Concentration dependent ion selectivity in VDAC: a molecular dynamics simulation study

The voltage-dependent anion channel (VDAC) forms the major pore in the outer mitochondrial membrane. Its high conducting open state features a moderate anion selectivity. There is some evidence indicating that the electrophysiological properties of VDAC vary with the salt concentration. Using a theoretical approach the molecular basis for this concentration dependence was investigated. Molecular dynamics simulations and continuum electrostatic calculations performed on the mouse VDAC1 isoform clearly demonstrate that the distribution of fixed charges in the channel creates an electric field, which determines the anion preference of VDAC at low salt concentration. Increasing the salt concentration in the bulk results in a higher concentration of ions in the VDAC wide pore. This event induces a large electrostatic screening of the charged residues promoting a less anion selective channel. Residues that are responsible for the electrostatic pattern of the channel were identified using the molecular dynamics trajectories. Some of these residues are found to be conserved suggesting that ion permeation between different VDAC species occurs through a common mechanism. This inference is buttressed by electrophysiological experiments performed on bean VDAC32 protein akin to mouse VDAC.     

短杆菌肽A-DMPC通道内离子输运的分子动力学模拟

用最近提出的构建膜体系初始构象的有效方法 ,构建了在DMPC脂膜环境下短杆菌肽A通道模型 (GA -DMPC)。通过对Na 、Ca2 、Cl-三种不同离子在GA -DMPC通道内不同位置的分子动力学模拟 ,研究离子在通道内输运过程中与通道及通道内水分子的相互作用 ,从分子动力学的角度阐明离子在通道内的输运机制。主要计算结果表明 :(1)离子在通道内的输运使GA的构象发生变化 ,GA的柔性是离子在通道内通透的重要因素 ;(2)Cl- 离子可扩大通道半径 ,Na 离子和Ca2 离子则减小通道半径。Cl-离子不能在GA通道内通透 ;(3)离子的出现使通道内水分子的偶极方向发生变化。上述结果均与实验相符。     

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.     

Anomalous mole fraction effect, electrostatics, and binding in ionic channels.

Ionic channels bathed in mixed solutions of two permeant electrolytes often conduct less current than channels bathed in pure solutions of either. For many years, this anomalous mole fraction effect (AMFE) has been thought to occur only in single-file pores containing two or more ions at a time. Most thinking about channels incorporates this view. We show here that the AMFE arises naturally, as an electrostatic consequence of localized ion specific binding, if the average current through a channel is described by a theory (Poisson-Nernst-Planck, PNP) that computes the average electric field from the average concentration of charges in and near the channel. The theory contains only those ion-ion interactions mediated by the mean field, and it does not enforce single filing. The AMFE is predicted by PNP over a wide range of mean concentrations of ions in the channel; for example, it is predicted when (on the average) less, or much less, than one ion is found in the channel's pore. In this treatment, the AMFE arises, in large measure, from a depletion layer produced near a region of ion-specific binding. The small excess concentration of ions in the binding region repels all nearby ions of like charge, thereby creating a depletion layer. The overall conductance of the channel arises in effect from resistors in series, one from the binding region, one from the depletion zone, and one from the unbinding region. The highest value resistor (which occurs in the depletion zone) limits the overall series conductance. Here the AMFE is not the result of single filing or multiple occupancy, and so previous views of permeation need to be revised: the presence of an AMFE does not imply that ions permeate single file through a multiply occupied pore.     

Selectivity and permeation in calcium release channel of cardiac muscle: alkali metal ions

Current was measured from single open channels of the calcium release channel (CRC) of cardiac sarcoplasmic reticulum (over the range +/-180 mV) in pure and mixed solutions (e.g., biionic conditions) of the alkali metal ions Li+, K+, Na+, Rb+, Cs+, ranging in concentration from 25 mM to 2 M. The current-voltage (I-V) relations were analyzed by an extension of the Poisson-Nernst-Planck (PNP) formulation of electrodiffusion, which includes local chemical interaction described by an offset in chemical potential, which likely reflects the difference in dehydration/solvation/rehydration energies in the entry/exit steps of permeation. The theory fits all of the data with few adjustable parameters: the diffusion coefficient of each ion species, the average effective charge distribution on the wall of the pore, and an offset in chemical potential for lithium and sodium ions. In particular, the theory explains the discrepancy between "selectivities" defined by conductance sequence and "selectivities" determined by the permeability ratios (i.e., reversal potentials) in biionic conditions. The extended PNP formulation seems to offer a successful combined treatment of selectivity and permeation. Conductance selectivity in this channel arises mostly from friction: different species of ions have different diffusion coefficients in the channel. Permeability selectivity of an ion is determined by its electrochemical potential gradient and local chemical interaction with the channel. Neither selectivity (in CRC) seems to involve different electrostatic interaction of different ions with the channel protein, even though the ions have widely varying diameters.     

Energy barriers for passage of ions through channels. Exact solution of two electrostatic problems

Exact solutions are given to two electrostatic problems relevant to ion permeation through pores in membranes. The first assesses the importance of the pore forming molecule as a dielectric shield. It is shown on the basis of structural and dielectric considerations alone (neglecting effects attributable to possible charge distribution at the interior surface of the pre-former) that the minimum electrostatic barrier for monovalent ion passage through a gramicidin-like channel is 11 kT. It is further shown that given favorable circumstances, dielectric shielding might dramatically reduce the barrier to ion passage through potassium channels. The second problem considers the error introduced by treating ions as point charges. It is shown that for structureless pores the point charge approximation introduces no meaningful error, even if the ratio of ion radius to pore radius is as great as 0.95.     

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.     

Energetics of K+ permeability through Gramicidin A by forward-reverse steered molecular dynamics

The estimation of ion channel permeability poses a considerable challenge for computer simulations because of the significant free energy barriers involved, but also offers valuable molecular information on the ion permeation process not directly available from experiments. In this article we determine the equilibrium free energy barrier for potassium ion permeability in Gramicidin A in an efficient way by atomistic forward-reverse non-equilibrium steered molecular dynamics simulations, opening the way for its use in more complex biochemical systems. Our results indicate that the tent-shaped energetics of translocation of K+ ions in Gramicidin A is dictated by the different polarization responses to the ion of the external bulk water and the less polar environment of the membrane.