Permeation of ions across the potassium channel: Brownian dynamics studies

Dynamics accelerate the kinetics of ion diffusion through channels: Continuous-time random walk models beyond the mean field approximation

Ion channels are proteins that play a significant role in physiological processes, including neuronal excitability and signal transduction. However, the precise mechanisms by which these proteins facilitate ion diffusion through cell membranes are not well understood. This is because experimental techniques to characterize ion channel activity operate on a time scale too large to understand the role of the various protein conformations on diffusion. Meanwhile, computational approaches operate on a time scale too short to rationalize the observed behavior at the microscopic scale. In this paper, we present a continuous-time random walk model that aims to bridge the scales between the atomistic models of ion channels and the experimental measurement of their conductance. We show how diffusion slows down in complex systems by using 3D lattices that map out the pore geometry of two channels: Nav1.7 and gramicidin. We also introduce spatial and dynamic site disorder to account for system heterogeneity beyond the mean field approximation. Computed diffusion coefficients show that an increase in spatial disorder slows down diffusion kinetics, while dynamic disorder has the opposite effect. Our results imply that microscopic or phenomenological models based on the potential of mean force data overlook the functional importance of protein dynamics on ion diffusion through channels.

Non-trivial dynamics in a model of glial membrane voltage driven by open potassium pores

Despite the molecular evidence that a nearly linear steady-state current-voltage relationship in mammalian astrocytes reflects a total current resulting from more than one differentially regulated K+ conductance, detailed ordinary differential equation (ODE) models of membrane voltage Vm are still lacking. Various experimental results reporting altered rectification of the major Kir currents in glia, dominated by Kir4.1, have motivated us to develop a detailed model of Vm dynamics incorporating the weaker potassium K2P-TREK1 current in addition to Kir4.1, and study the stability of the resting state Vr. The main question is whether, with the loss of monotonicity in glial I-V curve resulting from altered Kir rectification, the nominal resting state Vr remains stable, and the cell retains the trivial, potassium electrode behavior with Vm after EK. The minimal two-dimensional model of Vm near Vr showed that an N-shape deformed Kir I-V curve induces multistability of Vm in a model that incorporates K2P activation kinetics, and nonspecific K+ leak currents. More specifically, an asymmetrical, nonlinear decrease of outward Kir4.1 conductance, turning the channels into inward rectifiers, introduces instability of Vr. That happens through a robust bifurcation giving birth to a second, more depolarized stable resting state Vdr > -10 mV. Realistic recordings from electrographic seizures were used to perturb the model. Simulations of the model perturbed by constant current through gap junctions and seizure-like discharges as local field potentials led to depolarization and switching of Vm between the two stable states, in a downstate-upstate manner. In the event of prolonged depolarizations near Vdr, such catastrophic instability would affect all aspects of the glial function, from metabolic support to membrane transport, and practically all neuromodulatory roles assigned to glia.

Theoretical investigation on the effect of terahertz wave on Ca2+ transport in the calcium channel

The question of whether terahertz (THz) waves can interact with ions in channels of nerve cells and cause a further reaction has attracted much attention. To answer this question, we investigate the spontaneous radiation generated by Ca²⁺ moving in calcium channels and the effect of THz radiation on the transport of Ca²⁺ by solving the mathematical physical model through Brownian dynamics (BD) simulations. It is obtained that the moving Ca²⁺ in a calcium channel can generate electromagnetic radiation, the corresponding spectrum of which is concentrated in the THz range. Meanwhile, both the ion number in the channel and the background temperature are proved to have significant effects on the spontaneous emission spectra. The studies also show that external THz radiation can accelerate Ca²⁺ transport through the ion channel. These results are expected to provide a theoretical basis for the future treatment of THz waves in the neurological field.

The Persistent Question of Potassium Channel Permeation Mechanisms

Potassium channels play critical roles in many physiological processes, providing a selective permeation route for K⁺ ions in and out of a cell, by employing a carefully designed selectivity filter, evolutionarily conserved from viruses to mammals. The structure of the selectivity filter was determined at atomic resolution by x-ray crystallography, showing a tight coordination of desolvated K⁺ ions by the channel. However, the molecular mechanism of K⁺ ions permeation through potassium channels remains unclear, with structural, functional and computational studies often providing conflicting data and interpretations. In this review, we will present the proposed mechanisms, discuss their origins, and will critically assess them against all available data. General properties shared by all potassium channels are introduced first, followed by the introduction of two main mechanisms of ion permeation: soft and direct knock-on. Then, we will discuss critical computational and experimental studies that shaped the field. We will especially focus on molecular dynamics (MD) simulations, that provided mechanistic and energetic aspects of K⁺ permeation, but at the same time created long-standing controversies. Further challenges and possible solutions are presented as well.

The Lorentz force on ions in membrane channels of neurons as a mechanism for transcranial static magnetic stimulation

Transcranial static magnetic stimulation is a novel noninvasive method of reduction of the cortical excitability in certain neurological diseases that makes use of static magnetic fields generated by permanent magnets. By contrast, ordinary transcranial magnetic stimulation makes use of pulsed magnetic fields generated by strong currents. Whereas the physical principle underlying ordinary transcranial magnetic stimulation is well known, that is, the Faraday´s law, the physical mechanism that explains the interaction between neurons and static magnetic fields in transcranial static magnetic stimulation remains unclear. In the present work, it is discussed the possibility that this mechanism might be the Lorentz force exerted on the ions flowing along the membrane channels of neurons. The overall effect of the static magnetic field would be to introduce an additional friction between the ions and the walls of the membrane channels, thus reducing its conductance. Calculations performed by using a Hodgkin-Huxley model demonstrate that even a slight reduction of the conductance of the membrane channels can lead to the suppression of the action potential, thus inhibiting neuronal activity.

Multiscale analysis of the effect of surface charge pattern on a nanopore's rectification and selectivity properties: From all-atom model to Poisson-Nernst-Planck

We report a multiscale modeling study for charged cylindrical nanopores using three modeling levels that include (1) an all-atom explicit-water model studied with molecular dynamics, and reduced models with implicit water containing (2) hard-sphere ions studied with the Local Equilibrium Monte Carlo simulation method (computing ionic correlations accurately), and (3) point ions studied with Poisson-Nernst-Planck theory (mean-field approximation). We show that reduced models are able to reproduce device functions (rectification and selectivity) for a wide variety of charge patterns, that is, reduced models are useful in understanding the mesoscale physics of the device (i.e., how the current is produced). We also analyze the relationship of the reduced implicit-water models with the explicit-water model and show that diffusion coefficients in the reduced models can be used as adjustable parameters with which the results of the explicit- and implicit-water models can be related. We find that the values of the diffusion coefficients are sensitive to the net charge of the pore but are relatively transferable to different voltages and charge patterns with the same total charge.

Biochemical and physiological properties of K+ channel-associated AKR6A (Kvβ) proteins

Voltage-gated potassium (Kv) channels play an essential role in the regulation of membrane excitability and thereby control physiological processes such as cardiac excitability, neural communication, muscle contraction, and hormone secretion. Members of the Kv1 and Kv4 families are known to associate with auxiliary intracellular Kvβ subunits, which belong to the aldo-keto reductase superfamily. Electrophysiological studies have shown that these proteins regulate the gating properties of Kv channels. Although the three gene products encoding Kvβ proteins are functional enzymes in that they catalyze the nicotinamide adenine dinucleotide phosphate (NAD[P]H)-dependent reduction of a wide range of aldehyde and ketone substrates, the physiological role for these proteins and how each subtype may perform unique roles in coupling membrane excitability with cellular metabolic processes remains unclear. Here, we discuss current knowledge of the enzymatic properties of Kvβ proteins from biochemical studies with their described and purported physiological and pathophysiological influences.

Controlling ion transport through nanopores: modeling transistor behavior

We present a modeling study of a nanopore-based transistor computed by a mean-field continuum theory (Poisson–Nernst–Planck, PNP) and a hybrid method including particle simulation (Local Equilibrium Monte Carlo, LEMC) that is able to take ionic correlations into account including the finite size of ions.

Multiscale modeling of a rectifying bipolar nanopore: explicit-water versus implicit-water simulations

In a multiscale modeling approach, we present computer simulation results for a rectifying bipolar nanopore at two modeling levels. In an all-atom model, we use explicit water to simulate ion transport directly with the molecular dynamics technique. In a reduced model, we use implicit water and apply the Local Equilibrium Monte Carlo method together with the Nernst-Planck transport equation. This hybrid method makes the fast calculation of ion transport possible at the price of lost details. We show that the implicit-water model is an appropriate representation of the explicit-water model when we look at the system at the device (i.e., input vs. output) level. The two models produce qualitatively similar behavior of the electrical current for different voltages and model parameters. Looking at the details of concentration and potential profiles, we find profound differences between the two models. These differences, however, do not influence the basic behavior of the model as a device because they do not influence the z-dependence of the concentration profiles which are the main determinants of current. These results then address an old paradox: how do reduced models, whose assumptions should break down in a nanoscale device, predict experimental data? Our simulations show that reduced models can still capture the overall device physics correctly, even though they get some important aspects of the molecular-scale physics quite wrong; reduced models work because they include the physics that is necessary from the point of view of device function. Therefore, reduced models can suffice for general device understanding and device design, but more detailed models might be needed for molecular level understanding.

Multi-Dielectric Brownian Dynamics and Design-Space-Exploration Studies of Permeation in Ion Channels

This paper proposes a multi-dielectric Brownian dynamics simulation framework for design-space-exploration (DSE) studies of ion-channel permeation. The goal of such DSE studies is to estimate the channel modeling-parameters that minimize the mean-squared error between the simulated and expected "permeation characteristics." To address this computational challenge, we use a methodology based on statistical inference that utilizes the knowledge of channel structure to prune the design space. We demonstrate the proposed framework and DSE methodology using a case study based on the KcsA ion channel, in which the design space is successfully reduced from a 6-D space to a 2-D space. Our results show that the channel dielectric map computed using the framework matches with that computed directly using molecular dynamics with an error of 7%. Finally, the scalability and resolution of the model used are explored, and it is shown that the memory requirements needed for DSE remain constant as the number of parameters (degree of heterogeneity) increases.

Multiscale modeling of a rectifying bipolar nanopore: Comparing Poisson-Nernst-Planck to Monte Carlo

In the framework of a multiscale modeling approach, we present a systematic study of a bipolar rectifying nanopore using a continuum and a particle simulation method. The common ground in the two methods is the application of the Nernst-Planck (NP) equation to compute ion transport in the framework of the implicit-water electrolytemodel. The difference is that the Poisson-Boltzmann theory is used in the Poisson-Nernst-Planck (PNP) approach, while the Local Equilibrium Monte Carlo (LEMC) method is used in the particle simulation approach (NP+LEMC) to relate the concentration profile to the electrochemical potential profile. Since we consider a bipolar pore which is short and narrow, we perform simulations using two-dimensional PNP. In addition, results of a non-linear version of PNP that takes crowding of ions into account are shown. We observe that the mean field approximation applied in PNP is appropriate to reproduce the basic behavior of the bipolar nanopore (e.g., rectification) for varying parameters of the system (voltage, surface charge,electrolyte concentration, and pore radius). We present current data that characterize the nanopore's behavior as a device, as well as concentration, electrical potential, and electrochemical potential profiles.

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.

Effects of electromagnetic field exposure on conduction and concentration of voltage gated calcium channels: A Brownian dynamics study

A three-dimensional Brownian Dynamics (BD) in combination with electrostatic calculations is employed to specifically study the effects of radiation of high frequency electromagnetic fields on the conduction and concentration profile of calcium ions inside the voltage-gated calcium channels. The electrostatic calculations are performed using COMSOL Multiphysics by considering dielectric interfaces effectively. The simulations are performed for different frequencies and intensities. The simulation results show the variations of conductance, average number of ions and the concentration profiles of ions inside the channels in response to high frequency radiation. The ionic current inside the channel increases in response to high frequency electromagnetic field radiation, and the concentration profiles show that the residency of ions in the channel decreases accordingly.

Fast assessment of structural models of ion channels based on their predicted current-voltage characteristics

Computational prediction of protein structures is a difficult task, which involves fast and accurate evaluation of candidate model structures. We propose to enhance single-model quality assessment with a functionality evaluation phase for proteins whose quantitative functional characteristics are known. In particular, this idea can be applied to evaluation of structural models of ion channels, whose main function - conducting ions - can be quantitatively measured with the patch-clamp technique providing the current-voltage characteristics. The study was performed on a set of KcsA channel models obtained from complete and incomplete contact maps. A fast continuous electrodiffusion model was used for calculating the current-voltage characteristics of structural models. We found that the computed charge selectivity and total current were sensitive to structural and electrostatic quality of models. In practical terms, we show that evaluating predicted conductance values is an appropriate method to eliminate models with an occluded pore or with multiple erroneously created pores. Moreover, filtering models on the basis of their predicted charge selectivity results in a substantial enrichment of the candidate set in highly accurate models. Tests on three other ion channels indicate that, in addition to being a proof of the concept, our function-oriented single-model quality assessment method can be directly applied to evaluation of structural models of some classes of protein channels. Finally, our work raises an important question whether a computational validation of functionality should be included in the evaluation process of structural models, whenever possible.

On the classical vibrational coherence of carbonyl groups in the selectivity filter backbone of the KcsA ion channel

It has been suggested that quantum coherence in the selectivity filter of ion channel may play a key role in fast conduction and selectivity of ions. However, it has not been clearly elucidated yet why classical coherence is not sufficient for this purpose. In this paper, we investigate the classical vibrational coherence between carbonyl groups oscillations in the selectivity filter of KcsA ion channels based on the data obtained from molecular dynamics simulations. Our results show that classical coherence plays no effective role in fast ionic conduction.

Channel function reconstitution and re-animation: a single-channel strategy in the postcrystal age

The most essential properties of ion channels for their physiologically relevant functions are ion-selective permeation and gating. Among the channel species, the potassium channel is primordial and the most ubiquitous in the biological world, and knowledge of this channel underlies the understanding of features of other ion channels. The strategy applied to studying channels changed dramatically after the crystal structure of the potassium channel was resolved. Given the abundant structural information available, we exploited the bacterial KcsA potassium channel as a simple model channel. In the postcrystal age, there are two effective frameworks with which to decipher the functional codes present in the channel structure, namely reconstitution and re-animation. Complex channel proteins are decomposed into essential functional components, and well-examined parts are rebuilt for integrating channel function in the membrane (reconstitution). Permeation and gating are dynamic operations, and one imagines the active channel by breathing life into the 'frozen' crystal (re-animation). Capturing the motion of channels at the single-molecule level is necessary to characterize the behaviour of functioning channels. Advanced techniques, including diffracted X-ray tracking, lipid bilayer methods and high-speed atomic force microscopy, have been used. Here, I present dynamic pictures of the KcsA potassium channel from the submolecular conformational changes to the supramolecular collective behaviour of channels in the membrane. These results form an integrated picture of the active channel and offer insights into the processes underlying the physiological function of the channel in the cell membrane.

A study of quantum mechanical probabilities in the classical Hodgkin-Huxley model

The Hodgkin-Huxley (HH) model is a powerful model to explain different aspects of spike generation in excitable cells. However, the HH model was proposed in 1952 when the real structure of the ion channel was unknown. It is now common knowledge that in many ion-channel proteins the flow of ions through the pore is governed by a gate, comprising a so-called "selectivity filter" inside the ion channel, which can be controlled by electrical interactions. The selectivity filter (SF) is believed to be responsible for the selection and fast conduction of particular ions across the membrane of an excitable cell. Other (generally larger) parts of the molecule such as the pore-domain gate control the access of ions to the channel protein. In fact, two types of gates are considered here for ion channels: the "external gate", which is the voltage sensitive gate, and the "internal gate" which is the selectivity filter gate (SFG). Some quantum effects are expected in the SFG due to its small dimensions, which may play an important role in the operation of an ion channel. Here, we examine parameters in a generalized model of HH to see whether any parameter affects the spike generation. Our results indicate that the previously suggested semi-quantum-classical equation proposed by Bernroider and Summhammer (BS) agrees strongly with the HH equation under different conditions and may even provide a better explanation in some cases. We conclude that the BS model can refine the classical HH model substantially.

Multi-ion versus single-ion conduction mechanisms can yield current rectification in biological ion channels

There is clear evidence that the net magnitude of negative charge at the intracellular end of inwardly rectifying potassium channels helps to generate an asymmetry in the magnitude of the current that will pass in each direction. However, a complete understanding of the physical mechanism that links these charges to current rectification has yet to be obtained. Using Brownian dynamics, we compare the conduction mechanism and binding sites in rectifying and non-rectifying channel models. We find that in our models, rectification is a consequence of asymmetry in the hydrophobicity and charge of the pore lining. As a consequence, inward conduction can occur by a multi-ion conduction mechanism. However, outward conduction is restricted, since there are fewer ions at the intracellular entrance and outwardly moving ions must cross the pore on their own. We pose the question as to whether the same mechanism could be at play in inwardly rectifying potassium channels.

Electrodiffusive model for astrocytic and neuronal ion concentration dynamics

The cable equation is a proper framework for modeling electrical neural signalling that takes place at a timescale at which the ionic concentrations vary little. However, in neural tissue there are also key dynamic processes that occur at longer timescales. For example, endured periods of intense neural signaling may cause the local extracellular K⁺-concentration to increase by several millimolars. The clearance of this excess K⁺ depends partly on diffusion in the extracellular space, partly on local uptake by astrocytes, and partly on intracellular transport (spatial buffering) within astrocytes. These processes, that take place at the time scale of seconds, demand a mathematical description able to account for the spatiotemporal variations in ion concentrations as well as the subsequent effects of these variations on the membrane potential. Here, we present a general electrodiffusive formalism for modeling of ion concentration dynamics in a one-dimensional geometry, including both the intra- and extracellular domains. Based on the Nernst-Planck equations, this formalism ensures that the membrane potential and ion concentrations are in consistency, it ensures global particle/charge conservation and it accounts for diffusion and concentration dependent variations in resistivity. We apply the formalism to a model of astrocytes exchanging ions with the extracellular space. The simulations show that K⁺-removal from high-concentration regions is driven by a local depolarization of the astrocyte membrane, which concertedly (i) increases the local astrocytic uptake of K⁺, (ii) suppresses extracellular transport of K⁺, (iii) increases axial transport of K⁺ within astrocytes, and (iv) facilitates astrocytic relase of K⁺ in regions where the extracellular concentration is low. Together, these mechanisms seem to provide a robust regulatory scheme for shielding the extracellular space from excess K⁺.

When neurons generate electrical signals they release potassium ions (K⁺) into the extracellular space. During periods of intense neural activity, the local extracellular K⁺ may increase drastically. If it becomes too high, it can lead to neural dysfunction. Astrocytes (a kind of glial cells) are involved in preventing this from happening. Astrocytes can take up excess K⁺, transport it intracellularly, and release it in regions where the concentration is lower. This process is called spatial buffering, and a full mechanistic understanding of it is currently lacking. The aim of this work is twofold: First, we develop a formalism for modeling ion concentration dynamics in the intra- and extracellular space. The formalism is general, and could be used to simulate many cellular processes. It accounts for ion transports due to diffusion (along concentration gradients) as well as electrical migration (along voltage gradients). It extends previous, related formalisms, which have focused only on intracellular dynamics. Secondly, we apply the formalism to model how astrocytes exchange ions with the extracellular space. We conclude that the membrane mechanisms possessed by astrocytes seem optimal for shielding the extracellular space from excess K⁺, and provide a full mechanistic description of the spatial (K⁺) buffering process.

Potassium ions in the cavity of a KcsA channel model

The high rate of ion flux and selectivity of potassium channels has been attributed to the conformation and dynamics of the ions in the filter which connects the channel cavity and the extracellular environment. The cavity serves as the reservoir for potassium ions diffusing from the intracellular medium. The cavity is believed to decrease the dielectric barrier for the ions to enter the filter. We study here the equilibrium and dynamic properties of potassium ions entering the water-filled cavity of a KcsA channel model. Atomistic molecular dynamics simulations that are supplemented by electrostatic calculations reveal the important role of water molecules and the partially charged protein helices at the bottom of the cavity in overcoming the energy barrier and stabilizing the potassium ion in the cavity. We further show that the average time for a potassium ion to enter the cavity is much shorter than the conduction rate of a potassium passing through the filter, and this time duration is insensitive over a wide range of the membrane potential. The conclusions drawn from the study of the channel model are applicable in generalized contexts, including the entry of ions in artificial ion channels and other confined geometries.

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.

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.

Physics of ion channels

We review the basic physics involved in transport of ions across membrane channels in cells. Electrochemical forces that control the diffusion of ions are discussed both from microscopic and macroscopic perspectives. A case is made for use of Brownian dynamics as the minimal phenomenological model that provides a bridge between experiments and more fundamental theoretical approaches. Application of Brownian and molecular dynamics methods to channels with known molecular structures is discussed.

Molecular dynamics simulations of membrane proteins

Membrane proteins control the traffic across cell membranes and thereby play an essential role in cell function from transport of various solutes to immune response via molecular recognition. Because it is very difficult to determine the structures of membrane proteins experimentally, computational methods have been increasingly used to study their structure and function. Here we focus on two classes of membrane proteins-ion channels and transporters-which are responsible for the generation of action potentials in nerves, muscles, and other excitable cells. We describe how computational methods have been used to construct models for these proteins and to study the transport mechanism. The main computational tool is the molecular dynamics (MD) simulation, which can be used for everything from refinement of protein structures to free energy calculations of transport processes. We illustrate with specific examples from gramicidin and potassium channels and aspartate transporters how the function of these membrane proteins can be investigated using MD simulations.

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.

On the selective ion binding hypothesis for potassium channels

The mechanism by which K(+) channels select for K(+) over Na(+) ions has been debated for the better part of a century. The prevailing view is that K(+) channels contain highly conserved sites that selectively bind K(+) over Na(+) ions through optimal coordination. We demonstrate that a series of alternating sites within the KcsA channel selectivity filter exists, which are thermodynamically selective for either K(+) (cage made from two planes of oxygen atoms) or Na(+) ions (a single plane of four oxygen atoms). By combining Bennett free energy perturbation calculations with umbrella sampling, we show that when K(+) and Na(+) are both permitted to move into their preferred positions, the thermodynamic preference for K(+) over Na(+) is significantly reduced throughout the entire selectivity filter. We offer a rationale for experimental measures of thermodynamic preference for K(+) over Na(+) from Ba(2+) blocking data, by demonstrating that the presence of Ba(2+) ions exaggerates K(+) over Na(+) thermodynamic stability due to the different binding locations of these ions. These studies reveal that K(+) channel selectivity may not be associated with the thermodynamics of ions in crystallographic K(+) binding sites, but requires consideration of the kinetic barriers associated with the different multi-ion permeation mechanisms.

A method for treating the passage of a charged hard sphere ion as it passes through a sharp dielectric boundary

In the implicit solvent models of electrolytes (such as the primitive model (PM)), the ions are modeled as point charges in the centers of spheres (hard spheres in the case of the PM). The surfaces of the spheres are not polarizable which makes these models appropriate to use in computer simulations of electrolyte systems where these ions do not leave their host dielectrics. The same assumption makes them inappropriate in simulations where these ions cross dielectric boundaries because the interaction energy of the point charge with the polarization charge induced on the dielectric boundary diverges. In this paper, we propose a procedure to treat the passage of such ions through dielectric interfaces with an interpolation method. Inspired by the "bubble ion" model (in which the ion's surface is polarizable), we define a space-dependent effective dielectric coefficient, ε(eff)(r), for the ion that overlaps with the dielectric boundary. Then, we replace the "bubble ion" with a point charge that has an effective charge q/ε(eff)(r) and remove the portion of the dielectric boundary where the ion overlaps with it. We implement the interpolation procedure using the induced charge computation method [D. Boda, D. Gillespie, W. Nonner, D. Henderson, and B. Eisenberg, Phys. Rev. E 69, 046702 (2004)]. We analyze the various energy terms using a spherical ion passing through an infinite flat dielectric boundary as an example.

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.

Comparative study of the energetics of ion permeation in Kv1.2 and KcsA potassium channels

Biological ion channels rely on a multi-ion transport mechanism for fast yet selective permeation of ions. The crystal structure of the KcsA potassium channel provided the first microscopic picture of this process. A similar mechanism is assumed to operate in all potassium channels, but the validity of this assumption has not been well investigated. Here, we examine the energetics of ion permeation in Shaker Kv1.2 and KcsA channels, which exemplify the six-transmembrane voltage-gated and two-transmembrane inward-rectifier channels. We study the feasibility of binding a third ion to the filter and the concerted motion of ions in the channel by constructing the potential of mean force for K(+) ions in various configurations. For both channels, we find that a pair of K(+) ions can move almost freely within the filter, but a relatively large free-energy barrier hinders the K(+) ion from stepping outside the filter. We discuss the effect of the CMAP dihedral energy correction that was recently incorporated into the CHARMM force field on ion permeation dynamics.

Molecular dynamics and brownian dynamics investigation of ion permeation and anesthetic halothane effects on a proton-gated ion channel

Bacterial Gloeobacter violaceus pentameric ligand-gated ion channel (GLIC) is activated to cation permeation upon lowering the solution pH. Its function can be modulated by anesthetic halothane. In the present work, we integrate molecular dynamics (MD) and Brownian dynamics (BD) simulations to elucidate the ion conduction, charge selectivity, and halothane modulation mechanisms in GLIC, based on recently resolved X-ray crystal structures of the open-channel GLIC. MD calculations of the potential of mean force (PMF) for a Na(+) revealed two energy barriers in the extracellular domain (R109 and K38) and at the hydrophobic gate of transmembrane domain (I233), respectively. An energy well for Na(+) was near the intracellular entrance: the depth of this energy well was modulated strongly by the protonation state of E222. The energy barrier for Cl(-) was found to be 3-4 times higher than that for Na(+). Ion permeation characteristics were determined through BD simulations using a hybrid MD/continuum electrostatics approach to evaluate the energy profiles governing the ion movement. The resultant channel conductance and a near-zero permeability ratio (P(Cl)/P(Na)) were comparable to experimental data. On the basis of these calculations, we suggest that a ring of five E222 residues may act as an electrostatic gate. In addition, the hydrophobic gate region may play a role in charge selectivity due to a higher dehydration energy barrier for Cl(-) ions. The effect of halothane on the Na(+) PMF was also evaluated. Halothane was found to perturb salt bridges in GLIC that may be crucial for channel gating and open-channel stability, but had no significant impact on the single ion PMF profiles.

Approximation of super-ions for single-file diffusion of multiple ions through narrow pores

The general theory of the single-file multiparticle diffusion in the narrow pores could be greatly simplified in the case of inverted bell-like shape of the single-particle energy profile, which is often observed in biological ion channels. There is a narrow and deep groove in the energy landscape of multiple interacting ions in such profiles, which corresponds to the pre-defined optimal conduction pathway in the configurational space. If such groove exists, the motion of multiple ions can be reduced to the motion of single quasiparticle, called the superion, which moves in one-dimensional effective potential. The concept of the superions dramatically reduces the computational complexity of the problem and provides very clear physical interpretation of conduction phenomena in the narrow pores.

An iterative, fast, linear-scaling method for computing induced charges on arbitrary dielectric boundaries

Simulating coarse-grained models of charged soft-condensed matter systems in presence of dielectric discontinuities between different media requires an efficient calculation of polarization effects. This is almost always the case if implicit solvent models are used near interfaces or large macromolecules. We present a fast and accurate method (ICC( small star, filled)) that allows to simulate the presence of an arbitrary number of interfaces of arbitrary shape, each characterized by a different dielectric permittivity in one-, two-, and three-dimensional periodic boundary conditions. The scaling behavior and accuracy of the underlying electrostatic algorithms allow to choose the most appropriate scheme for the system under investigation in terms of precision and computational speed. Due to these characteristics the method is particularly suited to include nonplanar dielectric boundaries in coarse-grained molecular dynamics simulations.

Reinterpreting the anomalous mole fraction effect: the ryanodine receptor case study

The origin of the anomalous mole fraction effect (AMFE) in calcium channels is explored with a model of the ryanodine receptor. This model predicted and experiments verified new AMFEs in the cardiac isoform. In mole fraction experiments, conductance is measured in mixtures of ion species X and Y as their relative amounts (mole fractions) vary. This curve can have a minimum (an AMFE). The traditional interpretation of the AMFE is that multiple interacting ions move through the pore in a single file. Mole fraction curves without minima (no AMFEs) are generally interpreted as X displacing Y from the pore in a proportion larger than its bath mole fraction (preferential selectivity). We find that the AMFE is also caused by preferential selectivity of X over Y, if X and Y have similar conductances. This is a prediction applicable to any channel and provides a fundamentally different explanation of the AMFE that does not require single filing or multiple occupancy: preferential selectivity causes the resistances to current flow in the baths, channel vestibules, and selectivity filter to change differently with mole fraction, and produce the AMFE.

Ion conduction in ligand-gated ion channels: Brownian dynamics studies of four recent crystal structures

Four x-ray crystal structures of prokaryotic homologs of ligand-gated ion channels have recently been determined: ELIC from Erwinia chrysanthemi, two structures of a proton-activated channel from Gloebacter violaceus (GLIC1 and GLIC2) and that of the E221A mutant (GLIC1M). The availability of numerous structures of channels in this family allows for aspects of channel gating and ion conduction to be examined. Here, we determine the likely conduction states of the four structures as well as IV curves, ion selectivity, and steps involved in ion permeation by performing extensive Brownian dynamics simulations. Our results show that the ELIC structure is indeed nonconductive, but that GLIC1 and GLIC1M are both conductive of ions with properties different from those seen in experimental studies of the channel. GLIC2 appears to reflect an open state of the channel with a predicted conductance of 10.8-12.4 pS in 140 mM NaCl solution, which is comparable to the experimental value 8 +/- 2 pS. The extracellular domain of the channel is shown to have an important influence on the channel current, but a less significant role in ion selectivity.

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.

Boron nitride nanotubes selectively permeable to cations or anions

Biological ion channels in membranes are selectively permeable to specific ionic species. They maintain the resting membrane potential, generate propagated action potentials, and control a wide variety of cell functions. Here it is demonstrated theoretically that boron nitride nanotubes have the ability to carry out some of the important functions of biological ion channels. Boron nitride nanotubes with radii of 4.83 and 5.52 A embedded in a silicon nitride membrane are selectively permeable to cations and anions, respectively. They broadly mimic some of the permeation characteristics of gramicidin and chloride channels. Using distributional molecular dynamics, which is a combination of molecular and stochastic dynamics simulations, the properties of these engineered nanotubes are characterized, such as the free energy encountered by charged particles, the water-ion structure within the pore, and the current-voltage and current-concentration profiles. These engineered nanotubes have potential applications as sensitive biosensors, antibiotics, or filtration devices.

Theory of single-file multiparticle diffusion in narrow pores

Single-file diffusion of multiple strongly interacting particles in a one-dimensional pore is described within a general analytical framework. The theory accounts for nonequilibrium conditions, explicit particle-particle interactions, external potential acting on the particles and the fluctuations of the number of particles due to their exchange with external equilibrium reservoirs. It is shown that the problem can be reduced to a closed hierarchical set of partial differential equations of increasing dimensionality, which can be solved numerically. Our framework allows computing any macroscopic characteristic of multiparticle diffusion in the pore. It is shown that the pore occupancy probabilities and the current are rational functions of external concentrations in the steady state. The theory is tested on a simplified model of the narrow rigid pore inspired by the selectivity filter of biological ion channel. Perspectives and limitations of the theory are discussed.

Discretization of the induced-charge boundary integral equation

Boundary-element methods (BEMs) for solving integral equations numerically have been used in many fields to compute the induced charges at dielectric boundaries. In this paper, we consider a more accurate implementation of BEM in the context of ions in aqueous solution near proteins, but our results are applicable more generally. The ions that modulate protein function are often within a few angstroms of the protein, which leads to the significant accumulation of polarization charge at the protein-solvent interface. Computing the induced charge accurately and quickly poses a numerical challenge in solving a popular integral equation using BEM. In particular, the accuracy of simulations can depend strongly on seemingly minor details of how the entries of the BEM matrix are calculated. We demonstrate that when the dielectric interface is discretized into flat tiles, the qualocation method of Tausch [IEEE Trans Comput.-Comput.-Aided Des. 20, 1398 (2001)] to compute the BEM matrix elements is always more accurate than the traditional centroid-collocation method. Qualocation is not more expensive to implement than collocation and can save significant computational time by reducing the number of boundary elements needed to discretize the dielectric interfaces.

Computational study of the transmembrane domain of the acetylcholine receptor

The nicotinic acetylcholine receptor (nAChR) is a ligand-gated ion channel protein whose transmembrane domain (TM-domain) is believed to be responsible for channel gating via a hydrophobic effect. In this work, we perform molecular dynamics and Brownian dynamics simulations to investigate the effect of transmembrane potential on the conformation and water occupancy of TM-domain, and the resulting ion permeation events. The results show that the behavior of the hydrophobic gate is voltage-dependent. Large hyperpolarized membrane potential can change the conformation of TM-domain and water occupancy in this region, which may enable ion conduction. An electrostatic gating mechanism is also proposed from our simulations, which seems to play a role in addition to the well-known hydrophobic effect.