Study of ionic currents across a model membrane channel using Brownian dynamics

Experimental and Theoretical Brownian Dynamics Analysis of Ion Transport During Cellular Electroporation of E. coli Bacteria

Escherichia coli bacterium is a rod-shaped organism composed of a complex double membrane structure. Knowledge of electric field driven ion transport through both membranes and the evolution of their induced permeabilization has important applications in biomedical engineering, delivery of genes and antibacterial agents. However, few studies have been conducted on Gram-negative bacteria in this regard considering the contribution of all ion types. To address this gap in knowledge, we have developed a deterministic and stochastic Brownian dynamics model to simulate in 3D space the motion of ions through pores formed in the plasma membranes of E. coli cells during electroporation. The diffusion coefficient, mobility, and translation time of Ca2+, Mg2+, Na+, K+, and Cl- ions within the pore region are estimated from the numerical model. Calculations of pore's conductance have been validated with experiments conducted at Gustave Roussy. From the simulations, it was found that the main driving force of ionic uptake during the pulse is the one due to the externally applied electric field. The results from this work provide a better understanding of ion transport during electroporation, aiding in the design of electrical pulses for maximizing ion throughput, primarily for application in cancer treatment.

Resonance-driven ion transport and selectivity in prokaryotic ion channels

Ion channels exhibit a remarkably high accuracy in selecting uniquely its associated type of ion. The mechanisms behind ion selectivity are not well understood. Current explanations build mainly on molecular biology and bioinformatics. Here we propose a simple physical model for ion selectivity based on the driven damped harmonic oscillator (DDHO). The driving force for this oscillator is provided by self-organizing harmonic turbulent structures in the dehydrating ionic flow through the ion channel, namely, oscillating pressure waves in one dimension, and toroidal vortices in two and three dimensions. Density fluctuations caused by these turbulences efficiently transmit their energy to aqua ions that resonate with the driving frequency. Consequently, these release their hydration shell and exit the ion channel as free ions. Existing modeling frameworks do not express the required complex spatiotemporal dynamics, hence we introduce a macroscopic continuum model for ionic dehydration and transport, based on the hydrodynamics of a dissipative ionic flow through an ion channel, subject to electrostatic and amphiphilic interactions. This model combines three classical physical fields: Navier-Stokes equations from hydrodynamics, Gauss's law from Maxwell theory, and the convection-diffusion equation from continuum physics. Numerical experiments with mixtures of chemical species of ions in various degrees of hydration indeed reveal the emergence of strong oscillations in the ionic flow that are instrumental in the efficient dehydration and cause a strong ionic jet into the cell. As such, they provide a powerful engine for the DDHO mechanism. Theoretical predictions of the modeling framework match significantly with empirical patch-clamp data. The DDHO standard response curve defines a unique resonance frequency that depends on the mass and charge of the ion. In this way, the driving oscillations act as a selection mechanism that filters out one specific ion. Application of the DDHO model to real ion data shows that this mechanism indeed clearly distinguishes between chemical species and between aqua and bare ions with a large Mahalanobis distance and high oscillator quality. The DDHO framework helps to understand how SNP mutations can cause severe genetic pathologies as they destroy the geometry of the channel and so alter the resonance peaks of the required ion type.

Simulating Ion Transport with the NP+LEMC Method. Applications to Ion Channels and Nanopores

We describe a hybrid simulation technique that uses the Nernst-Planck (NP) transport equation to compute steady-state ionic flux in a non-equilibrium system and uses the Local Equilibrium Monte Carlo (LEMC) simulation technique to establish the statistical mechanical relation between the two crucial functions present in the NP equation: the concentration and the electrochemical potential profiles (Boda, D., Gillespie, D., J. Chem. Theor. Comput., 2012 8(3), 824–829). The LEMC method is an adaptation of the Grand Canonical Monte Carlo method to a non-equilibrium situation. We apply the resulting NP+LEMC method to ionic systems, where two reservoirs of electrolytes are separated by a membrane that allows the diffusion of ions through a nanopore. The nanopore can be natural (as the calcium selective Ryanodine Receptor ion channel) or synthetic (as a rectifying bipolar nanopore). We show results for these two systems and demonstrate the effectiveness of the NP+LEMC technique.

Simulating Current-Voltage Relationships for a Narrow Ion Channel Using the Weighted Ensemble Method

Ion channels are responsible for a myriad of fundamental biological processes via their role in controlling the flow of ions through water-filled membrane-spanning pores in response to environmental cues. Molecular simulation has played an important role in elucidating the mechanism of ion conduction, but connecting atomistically detailed structural models of the protein to electrophysiological measurements remains a broad challenge due to the computational cost of reaching the necessary time scales. Here, we introduce an enhanced sampling method for simulating the conduction properties of narrow ion channels using the Weighted ensemble (WE) sampling approach. We demonstrate the application of this method to calculate the current–voltage relationship as well as the nonequilibrium ion distribution at steady-state of a simple model ion channel. By direct comparisons with long brute force simulations, we show that the WE simulations rigorously reproduce the correct long-time scale kinetics of the system and are capable of determining these quantities using significantly less aggregate simulation time under conditions where permeation events are rare.

DNA stretching and optimization of nucleobase recognition in enzymatic nanopore sequencing

In nanopore sequencing, where single DNA strands are electrophoretically translocated through a nanopore and the resulting ionic signal is used to identify the four DNA bases, an enzyme has been used to ratchet the nucleic acid stepwise through the pore at a controlled speed. In this work, we investigated the ability of alpha-hemolysin nanopores to distinguish the four DNA bases under conditions that are compatible with the activity of DNA-handling enzymes. Our findings suggest that in immobilized strands, the applied potential exerts a force on DNA (∼10 pN at +160 mV) that increases the distance between nucleobases by about 2.2 Å V(-1). The four nucleobases can be resolved over wide ranges of applied potentials (from +60 to +220 mV in 1 m KCl) and ionic strengths (from 200 mM KCl to 1 M KCl at +160 mV) and nucleobase recognition can be improved when the ionic strength on the side of the DNA-handling enzyme is low, while the ionic strength on the opposite side is high.

Interacting ions in biophysics: real is not ideal

Ions in water are important throughout biology, from molecules to organs. Classically, ions in water were treated as ideal noninteracting particles in a perfect gas. Excess free energy of each ion was zero. Mathematics was not available to deal consistently with flows, or interactions with other ions or boundaries. Nonclassical approaches are needed because ions in biological conditions flow and interact. The concentration gradient of one ion can drive the flow of another, even in a bulk solution. A variational multiscale approach is needed to deal with interactions and flow. The recently developed energetic variational approach to dissipative systems allows mathematically consistent treatment of the bio-ions Na(+), K(+), Ca(2+), and Cl(-) as they interact and flow. Interactions produce large excess free energy that dominate the properties of the high concentration of ions in and near protein active sites, ion channels, and nucleic acids: the number density of ions is often >10 M. Ions in such crowded quarters interact strongly with each other as well as with the surrounding protein. Nonideal behavior found in many experiments has classically been ascribed to allosteric interactions mediated by the protein and its conformation changes. The ion-ion interactions present in crowded solutions-independent of conformation changes of the protein-are likely to change the interpretation of many allosteric phenomena. Computation of all atoms is a popular alternative to the multiscale approach. Such computations involve formidable challenges. Biological systems exist on very different scales from atomic motion. Biological systems exist in ionic mixtures (like extracellular and intracellular solutions), and usually involve flow and trace concentrations of messenger ions (e.g., 10(-7) M Ca(2+)). Energetic variational methods can deal with these characteristic properties of biological systems as we await the maturation and calibration of all-atom simulations of ionic mixtures and divalents.

Comparison of three-dimensional poisson solution methods for particle-based simulation and inhomogeneous dielectrics

Particle-based simulation represents a powerful approach to modeling physical systems in electronics, molecular biology, and chemical physics. Accounting for the interactions occurring among charged particles requires an accurate and efficient solution of Poisson's equation. For a system of discrete charges with inhomogeneous dielectrics, i.e., a system with discontinuities in the permittivity, the boundary element method (BEM) is frequently adopted. It provides the solution of Poisson's equation, accounting for polarization effects due to the discontinuity in the permittivity by computing the induced charges at the dielectric boundaries. In this framework, the total electrostatic potential is then found by superimposing the elemental contributions from both source and induced charges. In this paper, we present a comparison between two BEMs to solve a boundary-integral formulation of Poisson's equation, with emphasis on the BEMs' suitability for particle-based simulations in terms of solution accuracy and computation speed. The two approaches are the collocation and qualocation methods. Collocation is implemented following the induced-charge computation method of D. Boda et al. [J. Chem. Phys. 125, 034901 (2006)]. The qualocation method is described by J. Tausch et al. [IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 20, 1398 (2001)]. These approaches are studied using both flat and curved surface elements to discretize the dielectric boundary, using two challenging test cases: a dielectric sphere embedded in a different dielectric medium and a toy model of an ion channel. Earlier comparisons of the two BEM approaches did not address curved surface elements or semiatomistic models of ion channels. Our results support the earlier findings that for flat-element calculations, qualocation is always significantly more accurate than collocation. On the other hand, when the dielectric boundary is discretized with curved surface elements, the two methods are essentially equivalent; i.e., they have comparable accuracies for the same number of elements. We find that ions in water--charges embedded in a high-dielectric medium--are harder to compute accurately than charges in a low-dielectric medium.

Ion transport in mesoporous silica SBA-16 thin films with 3D cubic structures

Mesoporous silica SBA-16 thin films with highly ordered 3D cubic structures were synthesized on a Si substrate via the dip-coating method. After these films were filled with KCl aqueous solutions, the ionic current passing through the mesopores was measured by applying dc electric fields. At low ion concentrations, the measured I-V curves were nonlinear and the current increased exponentially with respect to voltage. As the ion concentration increased, the I-V curve approached linear behavior. The nonlinear behavior of I-V curves can be reasonably attributed to the electric potential barrier created in nanopores.

Particle-based simulation of charge transport in discrete-charge nano-scale systems: the electrostatic problem

The fast and accurate computation of the electric forces that drive the motion of charged particles at the nanometer scale represents a computational challenge. For this kind of system, where the discrete nature of the charges cannot be neglected, boundary element methods (BEM) represent a better approach than finite differences/finite elements methods. In this article, we compare two different BEM approaches to a canonical electrostatic problem in a three-dimensional space with inhomogeneous dielectrics, emphasizing their suitability for particle-based simulations: the iterative method proposed by Hoyles et al. and the Induced Charge Computation introduced by Boda et al.

SIMULATION OF ION CONDUCTION IN α-HEMOLYSIN NANOPORES WITH COVALENTLY ATTACHED β-CYCLODEXTRIN BASED ON BOLTZMANN TRANSPORT MONTE CARLO MODEL

Ion channels, as natures' solution to regulating biological environments, are particularly interesting to device engineers seeking to understand how natural molecular systems realize device-like functions, such as stochastic sensing of organic analytes. What's more, attaching molecular adaptors in desired orientations inside genetically engineered ion channels, enhances the system functionality as a biosensor. In general, a hierarchy of simulation methodologies is needed to study different aspects of a biological system like ion channels. Biology Monte Carlo (BioMOCA), a three-dimensional coarse-grained particle ion channel simulator, offers a powerful and general approach to study ion channel permeation. BioMOCA is based on the Boltzmann Transport Monte Carlo (BTMC) and Particle-Particle-Particle-Mesh (P(3)M) methodologies developed at the University of Illinois at Urbana-Champaign. In this paper, we have employed BioMOCA to study two engineered mutations of α-HL, namely (M113F)(6)(M113C-D8RL2)(1)-β-CD and (M113N)(6)(T117C-D8RL3)(1)-β-CD. The channel conductance calculated by BioMOCA is slightly higher than experimental values. Permanent charge distributions and the geometrical shape of the channels gives rise to selectivity towards anions and also an asymmetry in I-V curves, promoting a rectification largely for cations.

Algorithm for rigid-body Brownian dynamics

We present an algorithm for performing rigid-body Brownian dynamics that can take into account the hydrodynamic properties (translational and rotational friction tensors and the coupling between them) of each rigid body. In the zero temperature limit, the error term scales as Delta;{4} for time step Delta , while at nonzero temperatures the error scaling is Delta;{5/2} . We test the algorithm by applying it to a molecule of four-aminopyridine in water. We intend to use the algorithm to model the interaction between biological ion channels and other channel blocker molecules, but it may also have applicability to modeling other small particles such as colloids or nanoparticles.

Reconstructing potentials of mean force from short steered molecular dynamics simulations of Vpu from HIV-1

Vpu from human immunodeficiency virus type-1 (HIV-1) is an 81 amino acid type I integral membrane protein. Vpu forms ion conducting homooligomeric assemblies. To assess the energy landscape of an ion traversing the channel or pore single ion potentials of mean force (PMF) are reconstructed from short (1.2 ns) steered molecular dynamics (SMD) simulations using the Langevin equation of motion. For the simulations a section of the first 32 amino acids including the transmembrane domain of the Vpu protein is used. The values for the friction coefficient are estimated as a function of time using the velocity autocorrelation method. The PMFs of K(+), Na(+), and C(-) adopt a wave like pattern with a maximum around the hydrophobic stretch of the pore and a minimum at the hydrophilic site (C terminus). Independent of the pore size the amplitude of the PMF of at least one cation is always the lowest.

Single-channel current through nicotinic receptor produced by closure of binding site C-loop

We investigated the initial coupling of agonist binding to channel gating of the nicotinic acetylcholine receptor using targeted molecular-dynamics (TMD) simulation. After TMD simulation to accelerate closure of the C-loops at the agonist binding sites, the region of the pore that passes through the cell membrane expands. To determine whether the structural changes in the pore result in ion conduction, we used a coarse-grained ion conduction simulator, Biology Boltzmann transport Monte Carlo, and applied it to two structural frames taken before and after TMD simulation. The structural model before TMD simulation represents the channel in the proposed "resting" state, whereas the model after TMD simulation represents the channel in the proposed "active" state. Under external voltage biases, the channel in the "active" state was permeable to cations. Our simulated ion conductance approaches that obtained experimentally and recapitulates several functional properties characteristic of the nicotinic acetylcholine receptor. Thus, closure of the C-loop triggers a structural change in the channel sufficient to account for the open channel current. This approach of applying Biology Boltzmann transport Monte Carlo simulation can be used to further investigate the binding to gating transduction mechanism and the structural bases for ion selection and translocation.

Simulation of charge transport in ion channels and nanopores with anisotropic permittivity

Ion channels are part of nature's solution for regulating biological environments. Every ion channel consists of a chain of amino acids carrying a strong and sharply varying permanent charge, folded in such a way that it creates a nanoscopic aqueous pore spanning the otherwise mostly impermeable membranes of biological cells. These naturally occurring proteins are particularly interesting to device engineers seeking to understand how such nanoscale systems realize device-like functions. Availability of high-resolution structural information from X-ray crystallography, as well as large-scale computational resources, makes it possible to conduct realistic ion channel simulations. In general, a hierarchy of simulation methodologies is needed to study different aspects of a biological system like ion channels. Biology Monte Carlo (BioMOCA), a three-dimensional coarse-grained particle ion channel simulator, offers a powerful and general approach to study ion channel permeation. BioMOCA is based on the Boltzmann Transport Monte Carlo (BTMC) and Particle-Particle-Particle-Mesh (P(3)M) methodologies developed at the University of Illinois at Urbana-Champaign. In this paper we briefly discuss the various approaches to simulating ion flow in channel systems that are currently being pursued by the biophysics and engineering communities, and present the effect of having anisotropic dielectric constants on ion flow through a number of nanopores with different effective diameters.

Conduction of Na+ and K+ through the NaK channel: molecular and Brownian dynamics studies

Conduction of ions through the NaK channel, with M0 helix removed, was studied using both Brownian dynamics and molecular dynamics. Brownian dynamics simulations predict that the truncated NaK has approximately a third of the conductance of the related KcsA K+ channel, is outwardly rectifying, and has a Michaelis-Menten current-concentration relationship. Current magnitude increases when the glutamine residue located near the intracellular gate is replaced with a glutamate residue. The channel is blocked by extracellular Ca2+. Molecular dynamics simulations show that, under the influence of a strong applied potential, both Na+ and K+ move across the selectivity filter, although conduction rates for Na+ ions are somewhat lower. The mechanism of conduction of Na+ differs significantly from that of K+ in that Na+ is preferentially coordinated by single planes of pore-lining carbonyl oxygens, instead of two planes as in the usual K+ binding sites. The water-containing filter pocket resulting from a single change in the selectivity filter sequence (compared to potassium channels) disrupts several of the planes of carbonyl oxygens, and thus reduces the filter's ability to discriminate against sodium.

Estimating the dielectric constant of the channel protein and pore

When modelling biological ion channels using Brownian dynamics (BD) or Poisson-Nernst-Planck theory, the force encountered by permeant ions is calculated by solving Poisson's equation. Two free parameters needed to solve this equation are the dielectric constant of water in the pore and the dielectric constant of the protein forming the channel. Although these values can in theory be deduced by various methods, they do not give a reliable answer when applied to channel-like geometries that contain charged particles. To determine the appropriate values of the dielectric constants, here we solve the inverse problem. Given the structure of the MthK channel, we attempt to determine the values of the protein and pore dielectric constants that minimize the discrepancies between the experimentally-determined current-voltage curve and the curve obtained from BD simulations. Two different methods have been applied to determine these values. First, we use all possible pairs of the pore dielectric constant of water, ranging from 20 to 80 in steps of 10, and the protein dielectric constant of 2-10 in steps of 2, and compare the simulated results with the experimental values. We find that the best agreement is obtained with experiment when a protein dielectric constant of 2 and a pore water dielectric constant of 60 is used. Second, we employ a learning-based stochastic optimization algorithm to pick out the optimum combination of the two dielectric constants. From the algorithm we obtain an optimum value of 2 for the protein dielectric constant and 64 for the pore dielectric constant.

Solution of the Poisson-Nernst-Planck equations in the cell-substrate interface

Electrogenic cells are able to generate electrical signals which can be measured by various invasive electrophysiological methods such as patch-clamp or sharp microelectrode recordings. Growing cells on the surfaces of e.g. metal microelectrodes or field-effect transistors allows the recording of an extracellular component of these signals. For an understanding of such extracellular signals it is mandatory to get detailed topographical as well as electrical information about the cell-sensor interface. In a first approximation, this interface can be described by a flat disk between cell membrane and sensor surface. For a correct description of the signals, the electrodiffusion of ions in this interface is modeled by using the stationary Poisson-Nernst-Planck equations. We solve the equations analytically, and derive expressions for the potential, the ionic charge densities, and the seal resistance. The results provide a method for determining the distance h between sensor surface and cell membrane. For human embryonic kidney cells, we receive h approximately 70 nm. Comparison with literature shows good agreement.

The effect of protein dielectric coefficient on the ionic selectivity of a calcium channel

Calcium-selective ion channels are known to have carboxylate-rich selectivity filters, a common motif that is primarily responsible for their high Ca(2+) affinity. Different Ca(2+) affinities ranging from micromolar (the L-type Ca channel) to millimolar (the ryanodine receptor channel) are closely related to the different physiological functions of these channels. To understand the physical mechanism for this range of affinities given similar amino acids in their selectivity filters, we use grand canonical Monte Carlo simulations to assess the binding of monovalent and divalent ions in the selectivity filter of a model Ca channel. We use a reduced model where the electolyte is modeled by hard-sphere ions embedded in a continuum dielectric solvent, while the interior of protein surrounding the channel is allowed to have a dielectric coefficient different from that of the electrolyte. The induced charges that appear on the protein/lumen interface are calculated by the induced charge computation method [Boda et al., Phys. Rev. E 69, 046702 (2004)]. It is shown that decreasing the dielectric coefficient of the protein attracts more cations into the pore because the protein's carboxyl groups induce negative charges on the dielectric boundary. As the density of the hard-sphere ions increases in the filter, Ca(2+) is absorbed into the filter with higher probability than Na(+) because Ca(2+) provides twice the charge to neutralize the negative charge of the pore (both structural carboxylate oxygens and induced charges) than Na(+) while occupying about the same space (the charge/space competition mechanism). As a result, Ca(2+) affinity is improved an order of magnitude by decreasing the protein dielectric coefficient from 80 to 5. Our results indicate that adjusting the dielectric properties of the protein surrounding the permeation pathway is a possible way for evolution to regulate the Ca(2+) affinity of the common four-carboxylate motif.

Conduction properties of KcsA measured using brownian dynamics with flexible carbonyl groups in the selectivity filter

In the narrow segment of an ion conducting pathway, it is likely that a permeating ion influences the positions of the nearby atoms that carry partial or full electronic charges. Here we introduce a method of incorporating the motion of charged atoms lining the pore into Brownian dynamics simulations of ion conduction. The movements of the carbonyl groups in the selectivity filter of the KcsA channel are calculated explicitly, allowing their bond lengths, bond angles, and dihedral angels to change in response to the forces acting upon them. By systematically changing the coefficients of bond stretching and of angle bending, the carbon and oxygen atoms can be made to fluctuate from their fixed positions by varying mean distances. We show that incorporating carbonyl motion in this way does not alter the mechanism of ion conduction and only has a small influence on the computed current. The slope conductance of the channel increases by approximately 25% when the root mean-square fluctuations of the carbonyl groups are increased from 0.01 to 0.61 A. The energy profiles and the number of resident ions in the channel remain unchanged. The method we utilized here can be extended to allow the movement of glutamate or aspartate side chains lining the selectivity filters of other ionic channels.

Self-consistent molecular dynamics formulation for electric-field-mediated electrolyte transport through nanochannels

A self-consistent molecular dynamics (SCMD) formulation is presented for electric-field-mediated transport of water and ions through a nanochannel connected to reservoirs or baths. The SCMD formulation is compared with a uniform field MD approach, where the applied electric field is assumed to be uniform, for 2nm and 3.5nm wide nanochannels immersed in a 0.5M KCl solution. Reservoir ionic concentrations are maintained using the dual-control-volume grand canonical molecular dynamics technique. Simulation results with varying channel height indicate that the SCMD approach calculates the electrostatic potential in the simulation domain more accurately compared to the uniform field approach, with the deviation in results increasing with the channel height. The translocation times and ionic fluxes predicted by uniform field MD can be substantially different from those predicted by the SCMD approach. Our results also indicate that during a 2ns simulation time K+ ions can permeate through a 1nm channel when the applied electric field is computed self-consistently, while the permeation is not observed when the electric field is assumed to be uniform.

Modeling the fast gating mechanism in the ClC-0 chloride channel

A simplified three-dimensional model ClC-0 chloride channel is constructed to couple the permeation of Cl- ions to the motion of a glutamate side chain that acts as the putative fast gate in the ClC-0 channel. The gate is treated as a single spherical particle attached by a rod to a pivot point. This particle moves in a one-dimensional arc under the influence of a bistable potential, which mimics the isomerization process by which the glutamate side chain moves from an open state (not blocking the channel pore) to a closed state (blocking the channel pore, at a position which also acts as a binding site for Cl- ions moving through the channel). A dynamic Monte Carlo (DMC) technique is utilized to perform Brownian dynamics simulations to investigate the dependence of the gate closing rate on both internal and external chloride concentration and the gate charge as well. To accelerate the simulation of gate closing to a time scale that can be accommodated with current methodology and computer power, namely, microseconds, parameters that govern the motion of the bare gate (i.e., in the absence of coupling to the permeating ions) are chosen appropriately. Our simulation results are in qualitative agreement with experimental observations and consistent with the "foot-in-the-door" mechanism (Chen et al. J. Gen. Physiol. 2003, 122, 641; Chen and Miller J. Gen. Physiol. 1996, 108, 237), although the absolute time scale of gate closing in the real channel is much longer (millisecond time scale). A simple model based on the fractional occupation probability of the Cl- binding site that is ultimately blocked by the fast gate suggests straightforward scalability of simulation results for the model channel considered herein to experimentally realistic time scales.

Simulation of biological ion channels with technology computer-aided design

Computer simulations of realistic ion channel structures have always been challenging and a subject of rigorous study. Simulations based on continuum electrostatics have proven to be computationally cheap and reasonably accurate in predicting a channel's behavior. In this paper we discuss the use of a device simulator, SILVACO, to build a solid-state model for KcsA channel and study its steady-state response. SILVACO is a well-established program, typically used by electrical engineers to simulate the process flow and electrical characteristics of solid-state devices. By employing this simulation program, we have presented an alternative computing platform for performing ion channel simulations, besides the known methods of writing codes in programming languages. With the ease of varying the different parameters in the channel's vestibule and the ability of incorporating surface charges, we have shown the wide-ranging possibilities of using a device simulator for ion channel simulations. Our simulated results closely agree with the experimental data, validating our model.

Brownian dynamic model of the glycine receptor chloride channel: effect of the position of charged amino acids on ion membrane currents

Glycine is the major inhibitory neurotransmitter in the brainstem and spinal cord, where it participates in a variety of motor and sensory functions. It activates a special type of ligand-gated membrane receptor, which provides for Cl- ion conductance of the neuronal membrane. Computer simulations of a single-channel current through this receptor have been carried out on the basis of Brownian (Langevin) dynamics. The dependence of the currents on pore diameter and the location of the charged amino acid residues have been obtained. It has been shown that the presence and the symmetry of the filter-forming residues determined not only the ion-selectivity of the channel but also increased transmembrane anion current.

A hydrophobic gate in an ion channel: the closed state of the nicotinic acetylcholine receptor

The nicotinic acetylcholine receptor (nAChR) is the prototypic member of the 'Cys-loop' superfamily of ligand-gated ion channels which mediate synaptic neurotransmission, and whose other members include receptors for glycine, gamma-aminobutyric acid and serotonin. Cryo-electron microscopy has yielded a three-dimensional structure of the nAChR in its closed state. However, the exact nature and location of the channel gate remains uncertain. Although the transmembrane pore is constricted close to its center, it is not completely occluded. Rather, the pore has a central hydrophobic zone of radius about 3 A. Model calculations suggest that such a constriction may form a hydrophobic gate, preventing movement of ions through a channel. We present a detailed and quantitative simulation study of the hydrophobic gating model of the nicotinic receptor, in order to fully evaluate this hypothesis. We demonstrate that the hydrophobic constriction of the nAChR pore indeed forms a closed gate. Potential of mean force (PMF) calculations reveal that the constriction presents a barrier of height about 10 kT to the permeation of sodium ions, placing an upper bound on the closed channel conductance of 0.3 pS. Thus, a 3 A radius hydrophobic pore can form a functional barrier to the permeation of a 1 A radius Na+ ion. Using a united-atom force field for the protein instead of an all-atom one retains the qualitative features but results in differing conductances, showing that the PMF is sensitive to the detailed molecular interactions.

Brownian dynamics investigation into the conductance state of the MscS channel crystal structure

We suggest that the crystal structure of the mechanosensitive channel of small conductance is in a minimally conductive state rather than being fully activated. Performing Brownian dynamics simulations on the crystal structure show that no ions pass through it. When simulations are conducted on just the transmembrane domain (excluding the cytoplasmic residues 128 to 280) ions are seen to pass through the channel, but the conductance of approximately 30 pS is well below experimentally measured values. The mutation L109S that replaces a pore lining hydrophobic residue with a polar one is found to have little effect on the conductance of the channel. Widening the hydrophobic region of the pore by 2.5 Angstrom however, increases the channel conductance to over 200 pS suggesting that only a minimal conformational change is required to gate the pore.

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.

Three computational methods for studying permeation, selectivity and dynamics in biological ion channels

The cell membrane, confining some ions and molecules on one side and exchanging others with the other side, is the ultimate unit of the physiology of life. The delicate task of regulating the transport of ions across the membrane is carried out by biological nanotubes called 'ion channels'. Recently, there have been enormous strides in our understanding of the structure-function relationships of biological ion channels. The molecular structures of several ion channels have been determined from crystallographic analysis, including potassium channels, mechanosensitive channels, a chloride channel, as well as gramicidin channels and porins. It is expected that the X-ray structures of other ion channels will soon follow these discoveries, ushering in a new era of ion channel studies in which predicting the function of channels from their atomic structures will become the main quest. In parallel to these experimental findings, there have been important advances in computational biophysics. Here we summarize three theoretical approaches that have been utilized to understand the dynamics of ion permeation across bio-nanotubes, highlighting their advantages and shortcomings, and briefly describe some of the salient properties of ion channels uncovered through computational studies.

The influence of amino acid protonation states on molecular dynamics simulations of the bacterial porin OmpF

Several groups, including our own, have found molecular dynamics (MD) calculations to result in the size of the pore of an outer membrane bacterial porin, OmpF, to be reduced relative to its size in the x-ray crystal structure. At the narrowest portion of its pore, loop L3 was found to move toward the opposite face of the pore, resulting in decreasing the cross-section area by a factor of approximately 2. In an earlier work, we computed the protonation states of titratable residues for this system and obtained values different from those that had been used in previous MD simulations. Here, we show that MD simulations carried out with these recently computed protonation states accurately reproduce the cross-sectional area profile of the channel lumen in agreement with the x-ray structure. Our calculations include the investigation of the effect of assigning different protonation states to the one residue, D(127), whose protonation state could not be modeled in our earlier calculation. We found that both assumptions of charge states for D(127) reproduced the lumen size profile of the x-ray structure. We also found that the charged state of D(127) had a higher degree of hydration and it induced greater mobility of polar side chains in its vicinity, indicating that the apparent polarizability of the D(127) microenvironment is a function of the D(127) protonation state.

Ion transport through membrane-spanning nanopores studied by molecular dynamics simulations and continuum electrostatics calculations

Narrow hydrophobic regions are a common feature of biological channels, with possible roles in ion-channel gating. We study the principles that govern ion transport through narrow hydrophobic membrane pores by molecular dynamics simulation of model membranes formed of hexagonally packed carbon nanotubes. We focus on the factors that determine the energetics of ion translocation through such nonpolar nanopores and compare the resulting free-energy barriers for pores with different diameters corresponding to the gating regions in closed and open forms of potassium channels. Our model system also allows us to compare the results from molecular dynamics simulations directly to continuum electrostatics calculations. Both simulations and continuum calculations show that subnanometer wide pores pose a huge free-energy barrier for ions, but a small increase in the pore diameter to approximately 1 nm nearly eliminates that barrier. We also find that in those wider channels the ion mobility is comparable to that in the bulk phase. By calculating local electrostatic potentials, we show that the long range Coulomb interactions of ions are strongly screened in the wide water-filled channels. Whereas continuum calculations capture the overall energetics reasonably well, the local water structure, which is not accounted for in this model, leads to interesting effects such as the preference of hydrated ions to move along the pore wall rather than through the center of the pore.

Electrostatic basis of valence selectivity in cationic channels

We examine how a variety of cationic channels discriminate between ions of differing charge. We construct models of the KcsA potassium channel, voltage gated sodium channel and L-type calcium channel, and show that they all conduct monovalent cations, but that only the calcium channel conducts divalent cations. In the KcsA and sodium channels divalent ions block the channel and prevent any further conduction. We demonstrate that in each case, this discrimination and some of the more complex conductance properties of the channels is a consequence of the electrostatic interaction of the ions with the charges in the channel protein. The KcsA and sodium channels bind divalent ions strongly enough that they cannot be displaced by other ions and thereby block the channel. On the other hand, the calcium channel binds them less strongly such that they can be destabilized by the repulsion of another incoming divalent ion, but not by the lesser repulsion from monovalent ions.

A model of sodium channels

We have explored the permeation and blockage of ions in sodium channels, relating the channel structure to function using electrostatic profiles and Brownian dynamics simulations. The model used resembles the KcsA potassium channel with an added external vestibule and a shorter selectivity filter. The electrostatic energy landscape seen by permeating ions is determined by solving Poisson's equation. The two charged amino acid rings of Glu-Glu-Asp-Asp (EEDD) and Asp-Glu-Lys-Ala (DEKA) around the selectivity filter region are seen to play a crucial role in making the channel sodium selective, and strongly binding calcium ions such that they block the channel. Our model closely reproduces a range of experimental data including the current-voltage curves, current-concentration curves and blockage of monovalent ions by divalent ions.

Homology model of the GABAA receptor examined using Brownian dynamics

We have developed a homology model of the GABA(A) receptor, using the subunit combination of alpha1beta2gamma2, the most prevalent type in the mammalian brain. The model is produced in two parts: the membrane-embedded channel domain and the extracellular N-terminal domain. The pentameric transmembrane domain model is built by modeling each subunit by homology with the equivalent subunit of the heteropentameric acetylcholine receptor transmembrane domain. This segment is then joined with the extracellular domain built by homology with the acetylcholine binding protein. The all-atom model forms a wide extracellular vestibule that is connected to an oval chamber near the external surface of the membrane. A narrow, cylindrical transmembrane channel links the outer segment of the pore to a shallow intracellular vestibule. The physiological properties of the model so constructed are examined using electrostatic calculations and Brownian dynamics simulations. A deep energy well of approximately 80 kT accommodates three Cl(-) ions in the narrow transmembrane channel and seven Cl(-) ions in the external vestibule. Inward permeation takes place when one of the ions queued in the external vestibule enters the narrow segment and ejects the innermost ion. The model, when incorporated into Brownian dynamics, reproduces key experimental features, such as the single-channel current-voltage-concentration profiles. Finally, we simulate the gamma2 K289M epilepsy inducing mutation and examine Cl(-) ion permeation through the mutant 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.

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.

Relating microscopic charge movement to macroscopic currents: the Ramo-Shockley theorem applied to ion channels

Since the discovery of gating current, electrophysiologists have studied the movement of charged groups within channel proteins by changing potential and measuring the resulting capacitive current. The relation of atomic-scale movements of charged groups to the gating current measured in an external circuit, however, is not obvious. We report here that a general solution to this problem exists in the form of the Ramo-Shockley theorem. For systems with different amounts of atomic detail, we use the theorem to calculate the gating charge produced by movements of protein charges. Even without calculation or simulation, the Ramo-Shockley theorem eliminates a class of interpretations of experimental results. The theorem may also be used at each time step of simulations to compute external current.

Conduction mechanisms of chloride ions in ClC-type channels

The conduction properties of ClC-0 and ClC-1 chloride channels are examined using electrostatic calculations and three-dimensional Brownian dynamics simulations. We create an open-state configuration of the prokaryotic ClC Cl(-) channel using its known crystallographic structure as a basis. Two residues that are occluding the channel are slowly pushed outward with molecular dynamics to create a continuous ion-conducting path with the minimum radius of 2.5 A. Then, retaining the same pore shape, the prokaryotic ClC channel is converted to either ClC-0 or ClC-1 by replacing all the nonconserved dipole-containing and charged amino acid residues. Employing open-state ClC-0 and ClC-1 channel models, current-voltage curves consistent with experimental measurements are obtained. We find that conduction in these pores involves three ions. We locate the binding sites, as well as pinpointing the rate-limiting steps in conduction, and make testable predictions about how the single channel current across ClC-0 and ClC-1 will vary as the ionic concentrations are increased. Finally, we demonstrate that a ClC-0 homology model created from an alternative sequence alignment fails to replicate any of the experimental observations.

Computing induced charges in inhomogeneous dielectric media: application in a Monte Carlo simulation of complex ionic systems

The efficient calculation of induced charges in an inhomogeneous dielectric is important in simulations and coarse-grained models in molecular biology, chemical physics, and electrochemistry. We present the induced charge computation (ICC) method for the calculation of the polarization charges based on the variational formulation of Allen et al. [Phys. Chem. Chem. Phys. 3, 4177 (2001)]. We give a different solution for their extremum condition that produces a matrix formulation. The induced charges are directly calculated by solving the linear matrix equation Ah=c, where h contains the discretized induced charge density, c depends only on the source charges-the ions moved in the simulation-and the matrix A depends on the geometry of dielectrics, which is assumed to be unchanged during the simulation. Thus, the matrix need be inverted only once at the beginning of the simulation. We verify the efficiency and accuracy of the method by means of Monte Carlo simulations for two special cases. In the simplest case, a single sharp planar dielectric boundary is present, which allows comparison with exact results calculated using the method of electrostatic images. The other special case is a particularly simple case where the matrix A is not diagonal: a slab with two parallel flat boundaries. Our results for electrolyte solutions in these special cases show that the ICC method is both accurate and efficient.

Proteins, channels and crowded ions

Ion channels are proteins with a hole down their middle that control a vast range of biological function in health and disease. Selectivity is an important biological function determined by the open channel, which does not change conformation on the biological time scale. The challenge is to predict the function-the current of ions of different types and concentrations through a variety of channels-from structure, given fundamental physical laws. Walls of ion channels, like active sites of enzymes, often contain several fixed charges. Those fixed charges demand counter ions nearby, and the density of those counter ions is very high, greater than 5 molar, because of the tiny volumes of the channel's pore. Physical chemists can now calculate the free energy per mole of salt solutions (e.g. the activity coefficient) from infinite dilution to saturation, even in ionic melts. Such calculations of a model of the L-type calcium channel show that the large energies needed to crowd charges into the channel can account for the substantial selectivity and complex properties found experimentally. The properties of such crowded charge are likely to be an important determinant of the properties of proteins in general because channels are nearly enzymes.