Improved 3D continuum calculations of ion flux through membrane channels

Electrokinetic translocation of a deformable nanoparticle controlled by field effect in nanopores

Nanopores have become a popular single-molecule manipulation and detection technology. In this paper, we have constructed a continuum model of the nanopore; the arbitrary Lagrangian-Eulerian (ALE) method is used to describe the motion of particles and fluid. The mathematical model couples the stress-strain equation for the dynamics of a deformable particle, the Poisson equation for the electric field, the Navier-Stokes equations for the flow field, and the Nernst-Planck equations for ionic transport. Based on the model, the mechanism of field-effect regulation of particles passing through a nanopore is investigated. The results show that the transport of particles which is controlled by the field effect depends on the electroosmotic flow (EOF) generated by the gate electrode in the nanopore and the electrostatic interaction between the nanopore and particles. That also explains the asymmetry of particle transport velocity in the nanopore with a gate electrode. When the gate potential is negative, or the gate electrode length is small, the maximum deformation of the particles is increased. The field-effect regulation in the nanopore provides an active and compatible method for nanopore detection, and provides a convenient method for the active control of the particle deformation in the nanopore.

Electrodiffusion Phenomena in Neuroscience and the Nernst–Planck–Poisson Equations

This work is aimed to give an electrochemical insight into the ionic transport phenomena in the cellular environment of organized brain tissue. The Nernst–Planck–Poisson (NPP) model is presented, and its applications in the description of electrodiffusion phenomena relevant in nanoscale neurophysiology are reviewed. These phenomena include: the signal propagation in neurons, the liquid junction potential in extracellular space, electrochemical transport in ion channels, the electrical potential distortions invisible to patch-clamp technique, and calcium transport through mitochondrial membrane. The limitations, as well as the extensions of the NPP model that allow us to overcome these limitations, are also discussed.

Optimization of 3D Poisson-Nernst-Planck model for fast evaluation of diverse protein channels

We show the accuracy and applicability of our fast algorithmic implementation of a three-dimensional Poisson-Nernst-Planck (3D-PNP) flow model for characterizing different protein channels. Due to its high computational efficiency, our model can predict the full current-voltage characteristics of a channel within minutes, based on the experimental 3D structure of the channel or its computational model structure. Compared with other methods, such as Brownian dynamics, which currently needs a few weeks of the computational time, or even much more demanding molecular dynamics modeling, 3D-PNP is the only available method for a function-based evaluation of very numerous tentative structural channel models. Flow model tests of our algorithm and its optimal parametrization are provided for five native channels whose experimental structures are available in the protein data bank (PDB) in an open conductive state, and whose experimental current-voltage characteristics have been published. The channels represent very different geometric and structural properties, which makes it the widest test to date of the accuracy of 3D-PNP on real channels. We test whether the channel conductance, rectification, and charge selectivity obtained from the flow model, could be sufficiently sensitive to single-point mutations, related to unsignificant changes in the channel structure. Our results show that the classical 3D-PNP model, under proper parametrization, is able to achieve a qualitative agreement with experimental data for a majority of the tested characteristics and channels, including channels with narrow and irregular conductivity pores. We propose that although the standard PNP model cannot provide insight into complex physical phenomena due to its intrinsic limitations, its semiquantitative agreement is achievable for rectification and selectivity at a level sufficient for the bioinformatical purpose of selecting the best structural models with a great advantage of a very short computational time.

Computational electrophysiology: the molecular dynamics of ion channel permeation and selectivity in atomistic detail

Presently, most simulations of ion channel function rely upon nonatomistic Brownian dynamics calculations, indirect interpretation of energy maps, or application of external electric fields. We present a computational method to directly simulate ion flux through membrane channels based on biologically realistic electrochemical gradients. In close analogy to single-channel electrophysiology, physiologically and experimentally relevant timescales are achieved. We apply our method to the bacterial channel PorB from pathogenic Neisseria meningitidis, which, during Neisserial infection, inserts into the mitochondrial membrane of target cells and elicits apoptosis by dissipating the membrane potential. We show that our method accurately predicts ion conductance and selectivity and elucidates ion conduction mechanisms in great detail. Handles for overcoming channel-related antibiotic resistance are identified.

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

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

Ion flux through membrane channels--an enhanced algorithm for the Poisson-Nernst-Planck model

A novel algorithmic scheme for numerical solution of the 3D Poisson-Nernst-Planck model is proposed. The algorithmic improvements are universal and independent of the detailed physical model. They include three major steps: an adjustable gradient-based step value, an adjustable relaxation coefficient, and an optimized segmentation of the modeled space. The enhanced algorithm significantly accelerates the speed of computation and reduces the computational demands. The theoretical model was tested on a regular artificial channel and validated on a real protein channel-alpha-hemolysin, proving its efficiency.

Rectification properties and pH-dependent selectivity of meningococcal class 1 porin

We studied the current rectification properties and selectivity of class 1 porin (PorA) from Neisseria meningitidis (strain H44/76 Delta 3 Delta 4) reconstituted in planar lipid membranes varying salt concentrations and pH. PorA channel shows voltage gating with a characteristic time remarkably longer than other porins. Its current-voltage asymmetry, evaluated as the current rectification ratio, changes nonmonotonically with salt concentration. Interestingly, it reaches its maximum value at physiological concentration. Porin selectivity, quantified by reversal potential measurements, is also significantly asymmetric. Depending on the direction of the salt gradient, the channel becomes more or less selective (10:1 vs. 5:1 Na(+)/Cl(-)). Besides, the reversal potential measurements suggest that porin inserts directionally following the concentration gradient. Measurements over a wide range of pH show that although PorA is strongly cation selective at pH >6, its selectivity gradually changes to anionic in an acidic medium (pH < 4). We show that a continuum electrodiffusion model quantitatively accounts for conductance and reversal potential measurements at positive and negative applied voltages.

Electrostatic Screening and Energy Barriers of Ions in Low-dielectric Membranes

We present exact solutions of the linear Poisson-Boltzmann equation for several problems relevant for ion translocation across low-dielectric membranes. Our results are obtained for a finite Debye screening length, and they generalize the classical results for pure Coulombic electrostatics (Parsegian, A. Nature (London) 1969, 221, 844). We calculate the electrostatic self-energy of an ion in the middle of a low-dielectric slab, its energy inside a cylindrical high-dielectric pore, and its energy inside a high-dielectric spherical jacket. We consider also the influence of negative charges distributed on the walls of the cylindrical pore. We show that ion self-energy barriers are considerably reduced due to screening of electrolyte. We compare our results with some numerical results for screened electrostatics of ion channels and wide pores.

Finite element solution of the steady-state Smoluchowski equation for rate constant calculations

This article describes the development and implementation of algorithms to study diffusion in biomolecular systems using continuum mechanics equations. Specifically, finite element methods have been developed to solve the steady-state Smoluchowski equation to calculate ligand binding rate constants for large biomolecules. The resulting software has been validated and applied to mouse acetylcholinesterase. Rates for inhibitor binding to mAChE were calculated at various ionic strengths with several different reaction criteria. The calculated rates were compared with experimental data and show very good agreement when the correct reaction criterion is used. Additionally, these finite element methods require significantly less computational resources than existing particle-based Brownian dynamics methods.

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.

Saturation of conductance in single ion channels: the blocking effect of the near reaction field

The ionic current flowing through a protein channel in the membrane of a biological cell depends on the concentration of the permeant ion, as well as on many other variables. As the concentration increases, the rate of arrival of bath ions to the channel's entrance increases, and typically so does the net current. This concentration dependence is part of traditional diffusion and rate models that predict Michaelis-Menten current-concentration relations for a single ion channel. Such models, however, neglect other effects of bath concentrations on the net current. The net current depends not only on the entrance rate of ions into the channel, but also on forces acting on ions inside the channel. These forces, in turn, depend not only on the applied potential and charge distribution of the channel, but also on the long-range Coulombic interactions with the surrounding bath ions. In this paper, we study the effects of bath concentrations on the average force on an ion in a single ion channel. We show that the force of the reaction field on a discrete ion inside a channel embedded in an uncharged lipid membrane contains a blocking (shielding) term that is proportional to the square root of the ionic bath concentration. We then show that different blocking strengths yield different behavior of the current-concentration and conductance-concentration curves. Our theory shows that at low concentrations, when the blocking force is weak, conductance grows linearly with concentration, as in traditional models, e.g., Michaelis-Menten formulations. As the concentration increases to a range of moderate shielding, conductance grows as the square root of concentration, whereas at high concentrations, with high shielding, conductance may actually decrease with increasing concentrations: the conductance-concentration curve can invert. Therefore, electrostatic interactions between bath ions and the single ion inside the channel can explain the different regimes of conductance-concentration relations observed in experiments.