Comparison of Dynamic Lattice Monte Carlo Simulations and the Dielectric Self-Energy Poisson−Nernst−Planck Continuum Theory for Model Ion Channels

Self-energy-limited ion transport in subnanometer channels

The current-voltage characteristics of the alpha-Hemolysin protein pore during the passage of single-stranded DNA under varying ionic strength C are studied experimentally. We observe strong blockage of the current, weak superlinear growth of the current as a function of voltage, and a minimum of the current as a function of C. These observations are interpreted as the result of the ion electrostatic self-energy barrier originating from the large difference in the dielectric constants of water and the lipid bilayer. The dependence of DNA capture rate on C also agrees with our model.

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

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

Ion Current Calculations Based on Three Dimensional Poisson−Nernst−Planck Theory for a Cyclic Peptide Nanotube

Ion current calculations based on Poisson-Nernst-Planck (PNP) theory are performed for a synthetic cyclic peptide nanotube that consists of eight or ten cyclo[(-L-Trp-D-Leu-)4] embedded in a lipid bilayer membrane to investigate the ion transport properties of the nanotube. To explore systems with arbitrary geometries, three-dimensional PNP theory is implemented using a finite difference method. The influence of dipolar lipid molecules on the ion currents is also examined by turning on or off the charges of the lipid dipoles in dipalmitoylphosphatidylcholine (DPPC). Comparisons between the calculated and experimentally measured ion currents show that the PNP approach agrees well with the measurements at low ion concentrations but overestimates the currents at higher concentrations. Concentration profiles reveal the selectivity of the peptide nanotube to cations, which is attributed to the negatively charged carbonyl oxygens inside the nanotube. The dominant cation and the minimum anion concentrations inside the cyclic peptide nanotube suggest that these cyclic peptide nanotubes can be employed as ion sensors. In the case of the polar DPPC bilayer, smaller currents are obtained in the calculation. The variation of current with polarity of the lipids implies that both polar and nonpolar lipid bilayer membranes can be utilized to regulate ion currents in the peptide nanotube and other ion channels. Strengths and limitations of the PNP theory are also discussed.

Theoretical studies of the M2 transmembrane segment of the glycine receptor: models of the open pore structure and current-voltage characteristics

The pentameric glycine receptor (GlyR), a member of the nicotinicoid superfamily of ligand-gated ion channels, is an inhibitory Cl(-) channel that is gated by glycine. Using recently published NMR data of the second transmembrane segment (M2) of the human alpha1 GlyR, structural models of pentameric assemblies embedded in a lipid bilayer were constructed using a combination of experimentally determined constraints coupled with all-atom energy minimization. Based on this structure of the pentameric M2 "pore", Brownian dynamics simulations of ion permeation through this putative conducting open state of the channel were carried out. Simulated I-V curves were in good agreement with published experimental current-voltage curves and the anion/cation permeability ratio, suggesting that our open-state model may be representative of the conducting channel of the full-length receptor. These studies also predicted regions of chloride occupancy and suggested residues critical to anion permeation. Calculations of the conductance of the cation-selective mutant A251E channel are also consistent with experimental data. In addition, both rotation and untilting of the pore helices of our model were found to be broadly consistent with closing of the channel, albeit at distinct regions that may reflect alternate gates of the receptor.

Poisson–Nernst–Planck Theory Approach to the Calculation of Current Through Biological Ion Channels

The Poisson-Nernst-Planck (PNP) theory of electro-diffusion is reviewed. Techniques for numerical solution of the three-dimensional PNP equations are summarized, and several illustrative applications to ion transport through protein channels are presented. Strengths and weaknesses of the theory are discussed, as well as attempts to improve it via increasingly realistic evaluation of the force acting on each ion due to the protein/membrane environment.

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.

Kinetic models of ion transport through a nanopore

Kinetic equations for the stationary state distribution function of ions moving through narrow pores are solved for a number of 1D models of single ion transport. Ions move through pores of length L, under the action of a constant external field and of a concentration gradient. The interaction of single ions with the confining pore surface and with water molecules inside the pore are modeled by a Fokker-Planck term in the kinetic equation, or by uncorrelated collisions with thermalizing centers distributed along the pore. The temporary binding of ions to polar residues lining the pore is modeled by stopping traps or energy barriers. Analytic expressions for the stationary ion current through the pore are derived for several versions of the model, as functions of key physical parameters. In all cases, saturation of the current at high fields is predicted. Such simple models, for which results are analytic, may prove useful in the study of the current/voltage relations of ion channels through membranes.