Serdar Kuyucak

Tests of Continuum Theories as Models of Ion Channels. II. Poisson–Nernst–Planck Theory versus Brownian Dynamics

We test the validity of the mean-field approximation in Poisson-Nernst-Planck theory by contrasting its predictions with those of Brownian dynamics simulations in schematic cylindrical channels and in a realistic potassium channel. Equivalence of the two theories in bulk situations is demonstrated in a control study. In simple cylindrical channels, considerable differences are found between the two theories with regard to the concentration profiles in the channel and its conductance properties. These differences are at a maximum in narrow channels with a radius smaller than the Debye length and diminish with increasing radius. Convergence occurs when the channel radius is over 2 Debye lengths. These tests unequivocally demonstrate that the mean-field approximation in the Poisson-Nernst-Planck theory breaks down in narrow ion channels that have radii smaller than the Debye length.

Permeation of Ions Across the Potassium Channel: Brownian Dynamics Studies

The physical mechanisms underlying the transport of ions across a model potassium channel are described. The shape of the model channel corresponds closely to that deduced from crystallography. From electrostatic calculations, we show that an ion permeating the channel, in the absence of any residual charges, encounters an insurmountable energy barrier arising from induced surface charges. Carbonyl groups along the selectivity filter, helix dipoles near the oval chamber, and mouth dipoles near the channel entrances together transform the energy barrier into a deep energy well. Two ions are attracted to this well, and their presence in the channel permits ions to diffuse across it under the influence of an electric field. Using Brownian dynamics simulations, we determine the magnitude of currents flowing across the channel under various conditions. The conductance increases with increasing dipole strength and reaches its maximum rapidly; a further increase in dipole strength causes a steady decrease in the channel conductance. The current also decreases systematically when the effective dielectric constant of the channel is lowered. The conductance with the optimal choice of dipoles reproduces the experimental value when the dielectric constant of the channel is assumed to be 60. The current-voltage relationship obtained with symmetrical solutions is linear when the applied potential is less than approximately 100 mV but deviates from Ohms law at a higher applied potential. The reversal potentials obtained with asymmetrical solutions are in agreement with those predicted by the Nernst equation. The conductance exhibits the saturation property observed experimentally. We discuss the implications of these findings for the transport of ions across the potassium channels and membrane channels in general.

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.

Models of permeation in ion channels

Ion channels are formed by specific proteins embedded in the cell membrane and provide pathways for fast and controlled flow of selected ions down their electrochemical gradient. This activity generates action potentials in nerves, muscles and other excitable cells, and forms the basis of all movement, sensation and thought processes in living beings. While the functional properties of ion channels are well known from physiological studies, lack of structural knowledge has hindered development of realistic theoretical models necessary for understanding and interpretation of these properties. Recent determination of the molecular structures of potassium and mechanosensitive channels from x-ray crystallography has finally broken this impasse, heralding a new age in ion channel studies where study of structure– function relationships takes a central stage. In this paper, we present a critical review of various approaches to modelling of ion transport in membrane channels, including continuum theories, Brownian dynamics, and classical and ab initio molecular dynamics. Strengths and weaknesses of each approach are discussed and illustrated with applications to some specific ion channels.

Molecular Dynamics Study of the KcsA Potassium Channel

The structural, dynamical, and thermodynamic properties of a model potassium channel are studied using molecular dynamics simulations. We use the recently unveiled protein structure for the KcsA potassium channel from Streptomyces lividans. Total and free energy profiles of potassium and sodium ions reveal a considerable preference for the larger potassium ions. The selectivity of the channel arises from its ability to completely solvate the potassium ions, but not the smaller sodium ions. Self-diffusion of water within the narrow selectivity filter is found to be reduced by an order of magnitude from bulk levels, whereas the wider hydrophobic section of the pore maintains near-bulk self-diffusion. Simulations examining multiple ion configurations suggest a two-ion channel. Ion diffusion is found to be reduced to approximately 1/3 of bulk diffusion within the selectivity filter. The reduced ion mobility does not hinder the passage of ions, as permeation appears to be driven by Coulomb repulsion within this multiple ion channel.

Conducting-State Properties of the KcsA Potassium Channel from Molecular and Brownian Dynamics Simulations

The mechanisms underlying transport of ions across the potassium channel are examined using electrostatic calculations and three-dimensional Brownian dynamics simulations. We first build open-state configurations of the channel with molecular dynamics simulations, by pulling the transmembrane helices outward until the channel attains the desired interior radius. To gain insights into ion permeation, we construct potential energy profiles experienced by an ion traversing the channel in the presence of other resident ions. These profiles reveal that in the absence of an applied field the channel accommodates three potassium ions in a stable equilibrium, two in the selectivity filter and one in the central cavity. In the presence of a driving potential, this three-ion state becomes unstable, and ion permeation across the channel is observed. These qualitative explanations are confirmed by the results of three-dimensional Brownian dynamics simulations. We find that the channel conducts when the ionizable residues near the extracellular entrance are fully charged and those near the intracellular side are partially charged. The conductance increases steeply as the radius of the intracellular mouth of the channel is increased from 2 A to 5 A. Our simulation results reproduce several experimental observations, including the current-voltage curves, conductance-concentration relationships, and outward rectification of currents.

The potassium channel: Structure, selectivity and diffusion

We employ the entire experimentally determined protein structure for the KcsA potassium channel from Streptomyces lividans in molecular dynamics calculations to observe hydrated channel protein structure, ion solvation, selectivity, multiple ion configurations, and diffusion. Free energy perturbation calculations display a significant ion discrimination of ∼9 kT in favor of the larger K+ ion. The protein forming the channel is very flexible yet is unable to fully solvate the Na+ ion because of its smaller size and large solvation energy. There is evidence that acidic and basic sidechains may dissociate in the presence of multiple K+ ions to explain experimental ion density maps. K+ diffusion is found to vary from approximately 10%–90% of bulk, supporting the high channel currents observed experimentally.

Study of Ionic Currents across a Model Membrane Channel Using Brownian Dynamics

Brownian dynamics simulations have been carried out to study ionic currents flowing across a model membrane channel under various conditions. The model channel we use has a cylindrical transmembrane segment that is joined to a catenary vestibule at each side. Two cylindrical reservoirs connected to the channel contain a fixed number of sodium and chloride ions. Under a driving force of 100 mV, the channel is virtually impermeable to sodium ions, owing to the repulsive dielectric force presented to ions by the vestibular wall. When two rings of dipoles, with their negative poles facing the pore lumen, are placed just above and below the constricted channel segment, sodium ions cross the channel. The conductance increases with increasing dipole strength and reaches its maximum rapidly; a further increase in dipole strength does not increase the channel conductance further. When only those ions that acquire a kinetic energy large enough to surmount a barrier are allowed to enter the narrow transmembrane segment, the channel conductance decreases monotonically with the barrier height. This barrier represents those interactions between an ion, water molecules, and the protein wall in the transmembrane segment that are not treated explicitly in the simulation. The conductance obtained from simulations closely matches that obtained from ACh channels when a step potential barrier of 2-3 kTr is placed at the channel neck. The current-voltage relationship obtained with symmetrical solutions is ohmic in the absence of a barrier. The current-voltage curve becomes nonlinear when the 3 kTr barrier is in place. With asymmetrical solutions, the relationship approximates the Goldman equation, with the reversal potential close to that predicted by the Nernst equation. The conductance first increases linearly with concentration and then begins to rise at a slower rate with higher ionic concentration. We discuss the implications of these findings for the transport of ions across the membrane and the structure of ion channels.

Continuum electrostatics fails to describe ion permeation in the gramicidin channel.

We investigate the validity of continuum electrostatics in the gramicidin A channel using a recently determined high-resolution structure. The potential and electric field acting on ions in and around the channel are computed by solving Poissons equation. These are then used in Brownian dynamics simulations to obtain concentration profiles and the current passing through the channel. We show that regardless of the effective dielectric constant used for water in the channel or the channel protein, it is not possible to reproduce all the experimental data on gramicidin A; thus, continuum electrostatics cannot provide a valid framework for the description of ion dynamics in gramicidin channels. Using experimental data and molecular dynamics simulations as guides, we have constructed potential energy profiles that can satisfactorily describe the available physiological data. These profiles provide useful benchmarks for future potential of mean force calculations of permeating ions from molecular dynamics simulations of gramicidin A. They also offer a convenient starting point for studying structure-function relationships in modified gramicidin channels.

Gramicidin A channel as a test ground for molecular dynamics force fields

We use the well-known structural and functional properties of the gramicidin A channel to test the appropriateness of force fields commonly used in molecular dynamics (MD) simulations of ion channels. For this purpose, the high-resolution structure of the gramicidin A dimer is embedded in a dimyristoylphosphatidylcholine bilayer, and the potential of mean force of a K(+) ion is calculated along the channel axis using the umbrella sampling method. Calculations are performed using two of the most common force fields in MD simulations: CHARMM and GROMACS. Both force fields lead to large central barriers for K(+) ion permeation, that are substantially higher than those deduced from the physiological data by inverse methods. In long MD simulations lasting over 60 ns, several ions are observed to enter the binding site but none of them crossed the channel despite the presence of a large driving field. The present results, taken together with many earlier studies, highlights the shortcomings of the standard force fields used in MD simulations of ion channels and calls for construction of more appropriate force fields for this purpose.