The nature of ion and water barrier crossings in a simulated ion channel

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.

Insights into the sliding movement of the lac repressor nonspecifically bound to DNA

The Lac repressor finds its DNA binding sequences with an association rate 2 orders of magnitude higher than what is expected for a random diffusive process. This experimental data stimulated numerous theoretical and experimental studies, which led to the facilitated diffusion model. In facilitated diffusion, the Lac repressor binds nonspecifically to DNA. This nonspecific binding is followed by an exploration of the DNA molecule in a reduced space. Single-molecule imaging confirmed that the Lac repressor may move along the DNA molecule; however, it is still under debate whether the LacI movement proceeds through sliding, with a continuous close contact between the protein and DNA, or through hopping between adjacent binding sites. We have investigated the one-dimensional sliding movement of the Lac repressor along nonspecific DNA by full-atomistic molecular dynamics simulations and free-energy calculations based on the umbrella sampling technique. The computed free-energy profile along a helical trajectory was periodic, with periodicity equal to the distance between successive nucleotides and an energy barrier between successive minima of 8.7 +/- 0.7 kcal/mol. The results from the molecular simulations were subsequently used in a Langevin dynamics framework to estimate the diffusion coefficient of the Lac repressor sliding along nonspecific DNA. The computed diffusion coefficient is close to the lower limit of the experimental range.

Extrusion of transmitter, water and ions generates forces to close fusion pore

During exocytosis the fusion pore opens rapidly, then dilates gradually, and may subsequently close completely, but what controls its dynamics is not well understood. In this study we focus our attention on forces acting on the pore wall, and which are generated solely by the passage of transmitter, ions and water through the open fusion pore. The transport through the charged cylindrical nano-size pore is simulated using a coupled system of Poisson-Nernst-Planck and Navier-Stokes equations and the forces that act radially on the wall of the fusion pore are then estimated. Four forces are considered: a) inertial force, b) pressure, c) viscotic force, and d) electrostatic force. The inertial and viscotic forces are small, but the electrostatic force and the pressure are typically significant. High vesicular pressure tends to open the fusion pore, but the pressure induced by the transport of charged particles (glutamate, ions), which is predominant when the pore wall charge density is high tends to close the pore. The electrostatic force, which also depends on the charge density on the pore wall, is weakly repulsive before the pore dilates, but becomes attractive and pronounced as the pore dilates. Given that the vesicular concentration of free transmitter can change rapidly due to the release, or owing to the dissociation from the gel matrix, we evaluated how much and how rapidly a change of the vesicular K(+)-glutamate(-) concentration affects the concentration of glutamate(-) and ions in the pore and how such changes alter the radial force on the wall of the fusion pore. A step-like rise of the vesicular K(+)-glutamate(-) concentration leads to a chain of events. Pore concentration (and efflux) of both K(+) and glutamate(-) rise reaching their new steady-state values in less than 100 ns. Interestingly within a similar time interval the pore concentration of Na(+) also rises, whereas that of Cl(-) diminishes, although their extra-cellular concentration does not change. Finally such changes affect also the water movement. Water efflux changes bi-phasically, first increasing before decreasing to a new, but lower steady-state value. Nevertheless, even under such conditions an overall approximate neutrality of the pore is maintained remarkably well, and the electrostatic, but also inertial, viscotic and pressure forces acting on the pore wall remain constant. In conclusion the extrusion of the vesicular content generates forces, primarily the force due to the electro-kinetically induced pressure and electrostatic force (both influenced by the pore radius and even more by the charge density on the pore wall), which tend to close the fusion pore.

Glutamate, water and ion transport through a charged nanosize pore

The transport of transmitter, ions and water through a positively-charged nanopore was investigated through computer simulations. The physics of the problem is described by a coupled set of Poisson-Nernst-Planck and Navier-Stokes equations in a computational domain consisting a cylindrical pore, whose radius ranged from 1 to 8 nm and which was flanked by two compartments representing the vesicular interior and extra-cellular space. The concentration of co-ions is suppressed and of counter-ions enhanced, especially near the pore wall owing to electrostatic interactions. Glutamate (i.e. the transmitter considered) is negatively charged and is simulated as a counter-ion. The electro-kinetically induced pressure due to the movement of ions is negative and very pronounced near the pore wall where the concentration and flux of counter-ions is very high. The water velocity peaks in the pore center, diminishes to zero at the pore wall, but is constant along the pore axis. The mean velocity of the water/fluid is proportional to the vesicular pressure and pore cross-sectional area. Interestingly it is inversely related to the vesicular glutamate concentration. The factors determining the glutamate flux are complex. The diffusive flux generally predominates for narrow pore, and convective flux may dominate for wide pore if the vesicular pressure is high. Surprisingly at low vesicular pressure the mean total glutamate flux per unit cross-sectional pore area is higher for narrow pores. Higher flux is probably due to the rise of glutamate concentration in the nanopore, which is much more pronounced for narrow nanopores, due to the maintenance of approximate neutrality of charges in the pore and on the pore wall. In conclusion intra-vesicular pressure helps 'flushing-out' the transmitter, but the induced pressure 'drags-out' the water into the extra-cellular space.

Framework models of ion permeation through membrane channels and the generalized King-Altman method

A modern approach to studying the detailed dynamics of biomolecules is to simulate them on computers. Framework models have been developed to incorporate information from these simulations in order to calculate properties of the biomolecules on much longer time scales than can be achieved by the simulations. They also provide a simple way to think about the simulated dynamics. This article develops a method for the solution of framework models, which generalizes the King-Altman method of enzyme kinetics. The generalized method is used to construct solutions of two framework models which have been introduced previously, the single-particle and Grotthuss (proton conduction) models. The solution of the Grotthuss model is greatly simplified in comparison with direct integration. In addition, a new framework model is introduced, generalizing the shaking stack model of ion conduction through the potassium channel.

Ionic permeation free energy in gramicidin: a semimicroscopic perspective

Ion permeation through the gramicidin channel is studied using a model that circumvents two major difficulties inherent to standard simulational methods. It exploits the timescale separation between electronic and structural contributions to dielectric stabilization, accounting for the influence of electronic polarization by embedding the channel in a dielectric milieu that describes this polarization in a mean sense. The explicit mobile moieties are the ion, multipolar waters, and the carbonyls and amides of the peptide backbone. The model treats the influence of aromatic residues and the membrane dipole potential. A new electrical geometry is introduced that treats long-range electrostatics exactly and avoids problems related to periodic boundary conditions. It permits the translocating ion to make a seamless transition from nearby electrolyte to the channel interior. Other degrees of freedom (more distant bulk electrolyte and nonpolar lipid) are treated as dielectric continua. Reasonable permeation free energy profiles are obtained for potassium, rubidium, and cesium; binding wells are shallow and the central barrier is small. Estimated cationic single-channel conductances are smaller than experiment, but only by factors between 2 (rubidium) and 50 (potassium). When applied to chloride the internal barrier is large, with a corresponding miniscule single-channel conductance. The estimated relative single-channel conductances of gramicidin A, B, and C agree well with experiment.

A framework model based on the Smoluchowski equation in two reaction coordinates

The general form of the Smoluchowski equation in two reaction coordinates is obtained as the diffusion limit of a random walk on an infinite square grid using transition probabilities that satisfy detailed balance at thermodynamic equilibrium. The diffusion limit is then used to construct a generalization of the single-particle model to two reaction coordinates. The state space includes a square on which diffusion takes place and an isolated empty state. Boundary conditions on opposite sides of the square correspond to transitions between the empty state and the square. The two-dimensional (2D) model can be reduced to a 1D single-particle model by adiabatic elimination. A finite element solution of the 2D boundary value problem is described. The method used to construct the 2D model can be adapted to state spaces that have been constructed by other authors to model K+ conduction through gramicidin, proton conduction through dioxolane-linked gramicidin, and chloride conduction through the bacterial H(+)-Cl- antiporter.

Ion Permeation and Selectivity of OmpF Porin: A Theoretical Study Based on Molecular Dynamics, Brownian Dynamics, and Continuum Electrodiffusion Theory

Three different theoretical approaches are used and compared to refine our understanding of ion permeation through the channel formed by OmpF porin from Escherichia coli. Those approaches are all-atom molecular dynamics (MD) in which ions, solvent, and lipids are represented explicitly, Brownian dynamics (BD) in which ions are represented explicitly, while solvent and lipids are represented as featureless dielectrics, and Poisson-Nernst-Planck (PNP) electrodiffusion theory in which both solvent and local ion concentrations are represented as a continuum. First, the ability of the different theoretical approaches in reproducing the equilibrium average ion density distribution in OmpF porin bathed by a 1M KCl symmetric salt solution is examined. Under those conditions the PNP theory is equivalent to the non-linear Poisson-Boltzmann (PB) theory. Analysis shows that all the three approaches are able to capture the important electrostatic interactions between ions and the charge distribution of the channel that govern ion permeation and selectivity in OmpF. The K(+) and Cl(-) density distributions obtained from the three approaches are very consistent with one another, which suggests that a treatment on the basis of a rigid protein and continuum dielectric solvent is valid in the case of OmpF. Interestingly, both BD and continuum electrostatics reproduce the distinct left-handed twisted ion pathways for K(+) and Cl(-) extending over the length of the pore which were observed previously in MD. Equilibrium BD simulations in the grand canonical ensemble indicate that the channel is very attractive for cations, particularly at low salt concentration. On an average there is 1.55 K(+) inside the pore in 10mM KCl. Remarkably, there is still 0.17 K(+) on average inside the pore even at a concentration as low as 1microM KCl. Secondly, non-equilibrium ion flow through OmpF is calculated using BD and PNP and compared with experimental data. The channel conductance in 0.2M and 1M KCl calculated using BD is in excellent accord with the experimental data. The calculations reproduce the experimentally well-known conductance-concentration relation and also reveal an asymmetry in the channel conductance (a larger conductance is observed under a positive transmembrane potential). Calculations of the channel conductance for three mutants (R168A, R132A, and K16A) in 1M KCl suggest that the asymmetry in the channel conductance arises mostly from the permanent charge distribution of the channel rather than the shape of the pore itself. Lastly, the calculated reversal potential in a tenfold salt gradient (0.1:1M KCl) is 27.4(+/-1.3)mV (BD) and 22.1(+/-0.6)mV (PNP), in excellent accord with the experimental value of 24.3mV. Although most of the results from PNP are qualitatively reasonable, the calculated channel conductance is about 50% higher than that calculated from BD probably because of a lack of some dynamical ion-ion correlations.

Ions and counterions in a biological channel: a molecular dynamics simulation of OmpF porin from Escherichia coli in an explicit membrane with 1 M KCl aqueous salt solution

A 5 ns all-atom molecular dynamics trajectory of Escherichia coli OmpF porin embedded in an explicit dimyristoyl-phosphatidylcholine (DMPC) bilayer bathed by a 1 M [KCl] aqueous salt solution is generated to explore the microscopic details of the mechanism of ion permeation. The atomic model includes the OmpF trimer, 124 DMPC, 13470 water molecules as well as 231 K+ and 201 Cl-, for a total of 70,693 atoms. The structural and dynamical results are in excellent agreement with the X-ray data. The global root-mean-square deviation of the backbone atoms relative to the X-ray structure is 1.4 A. A cluster of three fully charged arginine (Arg42, Arg82, and Arg132) facing two acidic residues (Asp113 and Glu117) on L3 in the narrowest part of the aqueous pore is observed to be very stable in the crystallographic conformation. In this region of the pore, the water molecules are markedly oriented perpendicular to the channel axis due to the strong transversal electrostatic field arising from those residues. On average the size of the pore is smaller during the simulation than in the X-ray structure, undergoing small fluctuations. No large movements of loop L3 leading to a gating of the pore are observed. Remarkably, it is observed that K+ and Cl- follow two well-separated average pathways spanning over nearly 40 A along the axis of the pore. In the center of the monomer, the two screw-like pathways have a left-handed twist, undergoing a counter-clockwise rotation of 180 degrees from the extracellular vestibule to the pore periplasmic side. In the pore, the dynamical diffusion constants of the ions are reduced by about 50% relative to their value in bulk solvent. Analysis of ion solvation across the channel reveals that the contributions from the water and the protein are complementary, keeping the total solvation number of both ions nearly constant. Unsurprisingly, K+ have a higher propensity to occupy the aqueous pore than Cl-, consistent with the cation selectivity of the channel. However, further analysis suggests that ion-ion pairs play an important role. In particular, it is observed that the passage of Cl- occurs only in the presence of K+ counterions, and isolated K+ can move through the channel and permeate on their own. The presence of K+ in the pore screens the negative electrostatic potential arising from OmpF to help the translocation of Cl- by formation of ion pairs.

Hierarchical approach to predicting permeation in ion channels

A hierarchical computational strategy combining molecular modeling, electrostatics calculations, molecular dynamics, and Brownian dynamics simulations is developed and implemented to compute electrophysiologically measurable properties of the KcsA potassium channel. Models for a series of channels with different pore sizes are developed from the known x-ray structure, using insights into the gating conformational changes as suggested by a variety of published experiments. Information on the pH dependence of the channel gating is incorporated into the calculation of potential profiles for K(+) ions inside the channel, which are then combined with K(+) ion mobilities inside the channel, as computed by molecular dynamics simulations, to provide inputs into Brownian dynamics simulations for computing ion fluxes. The open model structure has a conductance of approximately 110 pS under symmetric 250 mM K(+) conditions, in reasonable agreement with experiments for the largest conducting substate. The dimensions of this channel are consistent with electrophysiologically determined size dependence of quaternary ammonium ion blocking from the intracellular end of this channel as well as with direct structural evidence that tetrabutylammonium ions can enter into the interior cavity of the channel. Realistic values of Ussing flux ratio exponents, distribution of ions within the channel, and shapes of the current-voltage and current-concentration curves are obtained. The Brownian dynamics calculations suggest passage of ions through the selectivity filter proceeds by a "knock-off" mechanism involving three ions, as has been previously inferred from functional and structural studies of barium ion blocking. These results suggest that the present calculations capture the essential nature of K(+) ion permeation in the KcsA channel and provide a proof-of-concept for the integrated microscopic/mesoscopic multitiered approach for predicting ion channel function from structure, which can be applied to other channel structures.

A Grand Canonical Monte Carlo-Brownian dynamics algorithm for simulating ion channels

A computational algorithm based on Grand Canonical Monte Carlo (GCMC) and Brownian Dynamics (BD) is described to simulate the movement of ions in membrane channels. The proposed algorithm, GCMC/BD, allows the simulation of ion channels with a realistic implementation of boundary conditions of concentration and transmembrane potential. The method is consistent with a statistical mechanical formulation of the equilibrium properties of ion channels (; Biophys. J. 77:139-153). The GCMC/BD algorithm is illustrated with simulations of simple test systems and of the OmpF porin of Escherichia coli. The approach provides a framework for simulating ion permeation in the context of detailed microscopic models.

Simulations of ion channels--watching ions and water move

Ion channels mediate electrical excitability in neurons and muscle. Three-dimensional structures for model peptide channels and for a potassium (K+) channel have been combined with computer simulations to permit rigorous exploration of structure-function relations of channels. Water molecules and ions within transbilayer pores tend to diffuse more slowly than in bulk solutions. In the narrow selectivity filter of the bacterial K+ channel (i.e. the region of the channel that discriminates between different species of ions) a column of water molecules and K+ ions moves in a concerted fashion. By combining atomistic simulations (in which all atoms of the channel molecule, water and ions are treated explicitly) with continuum methods (in which the description of the channel system is considerably simplified) it is possible to simulate some of the physiological properties of channels.

Accelerated diffusion of Na+ in a hydrophobic region revealed by molecular dynamics simulations of a synthetic ion channel

To get insight into the significance of the hydrophobic lining on the ion permeation, we performed molecular dynamics simulations on a Na(+) permeation through a de novo synthetic hydrophobic channel. Electrophysiological study has suggested that the channel is formed from a tail-to-tail associated dimer of a cyclic octa-peptide coupled with hydrophobic acyl chains. The acyl chains line the channel pore while the cyclic peptide forms the channel entrance [Z. Qi, M. Sokabe, K. Donowaki, H. Ishida, Biophys. J. 76 (1999) 631]. Molecular dynamics simulation of water in the channel indicated that the inferred structure is physically reasonable [Z. Qi, M. Sokabe, Biophys. Chem. 71 (1998) 35]. In the present study, the potential energy profile of the Na(+) and the energy contributions from each component of the system at different positions along the channel axis were calculated. An energy well instead of a peak is located at the central hydrophobic cavity of the channel, due to its ability of accommodating at least five water molecules to hydrate the ion. Interestingly, the ion diffuses much faster in the hydrophobic acyl chain region, particularly in the central hydrophobic cavity, than it does in the peptide ring region and even surprisingly faster than that in the bulk phase. These results provide a physical basis for an idea that the hydrophobic lining of the K(+) channel [D.A. Doyle, J.M. Cabral, R.A. Pfuetzner, A. Kuo, J.M. Gulbis, S.L. Cohen, B.T. Chait, R. MacKinnon, Science 280 (1998) 69] plays an active role to facilitate the ion permeation through the channel pore.

Permeation of ions across the potassium channel: Brownian dynamics studies

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

The binding site of sodium in the gramicidin A channel

The available information concerning the structure and location of the main binding site for sodium in the gramicidin A channel is reviewed and discussed. Results from molecular dynamics simulations using an atomic model of the channel embedded in a lipid bilayer are compared with experimental observations. The combined information from experiment and simulation suggests that the main binding sites for sodium are near the channel's mouth, approximately 9.2 A from the centre of the dimer channel, although the motion along the axis could be as large as 1 to 2 A. In the binding site, the sodium ion is lying off axis, making contact with two carbonyl oxygens and two single-file water molecules. The main channel ligand is provided by the carbonyl group of the Leu10-Trp11 peptide linkage, which exhibits the largest deflection from the ion-free channel structure.

Effective diffusion coefficients of K+ and Cl- ions in ion channel models

We have used molecular dynamics simulations, corresponding to a total simulation time of 11 ns, to investigate the effective short-time local diffusion coefficient of potassium and chloride ions in a series of model ion channels. These models, which include channels formed by the fungal peptide alamethicin, by a synthetic leucine-serine peptide, and by the pore-lining M2 helix bundle of the nicotinic acetylcholine receptor, have a range of different secondary structures, diameters and hydrophobicities. We find that the diffusion coefficients of both ions are appreciably reduced in the narrower channels, the extent of the reduction being similar for both the anionic and cationic species. This suggests that a difference in mobility cannot be the source of the ion selectivity exhibited by some of the channels (for example, the leucine-serine peptide). We find no evidence for a reduction in mobility of either ion in the nAChR model. These results are broadly in line with a previous similar study of Na+ ions, and may be useful in Poisson-Nernst-Planck, Eyring rate theory or Brownian dynamics calculations of channel conductance.

Simulation study of a gramicidin/lipid bilayer system in excess water and lipid. II. Rates and mechanisms of water transport

A gramicidin channel in a fluid phase DMPC bilayer with excess lipid and water has been simulated. By use of the formal correspondence between diffusion and random walk, a permeability for water through the channel was calculated, and was found to agree closely with the experimental results of Rosenberg and Finkelstein (Rosenberg, P.A., and A. Finkelstein. 1978. J. Gen. Physiol. 72:327-340; 341-350) for permeation of water through gramicidin in a phospholipid membrane. By using fluctuation analysis, components of resistance to permeation were computed for movement through the channel interior, for the transition step at the channel mouth where the water molecule solvation environment changes, and for the process of diffusion up to the channel mouth. The majority of the resistance to permeation appears to occur in the transition step at the channel mouth. A significant amount is also due to structurally based free energy barriers within the channel. Only small amounts are due to local friction within the channel or to diffusive resistance for approaching the channel mouth.

Dynamic properties of Na+ ions in models of ion channels: a molecular dynamics study

We present simulation results for the effective diffusion coefficients of a sodium ion in a series of model ion channels of different diameters and hydrophobicities, including models of alamethicin, a leucine-serine peptide, and the M2 helix bundle of the nicotinic acetylcholine receptor. The diffusion coefficient, which in the simulations has a value of 0.15(2) A2ps-1 in bulk water, is found to be reduced to as little as 0.02(1) A2ps-1 in the narrower channels, and to about 0.10(5) A2ps-1 in wider channels such as the nicotinic acetylcholine receptor. It is anticipated that this work will be useful in connection with calculations of channel conductivity using such techniques as the Poisson-Nernst-Planck equation, Eyring rate theory, or Brownian dynamics.

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

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

Modelling and simulation of ion channels: applications to the nicotinic acetylcholine receptor

Molecular dynamics simulations with experimentally derived restraints have been used to develop atomic models of M2 helix bundles forming the pore-lining domains of the nicotinic acetylcholine receptor and related ligand-gated ion channels. M2 helix bundles have been used in microscopic simulations of the dynamics and energetics of water and ions within an ion channel. Translational and rotational motion of water are restricted within the pore, and water dipoles are aligned relative to the pore axis by the surrounding helix dipoles. Potential energy profiles for translation of a Na+ ion along the pore suggest that the protein and water components of the interaction energy exert an opposing effect on the ion, resulting in a relatively flat profile which favors cation permeation. Empirical conductance calculations based on a pore radius profile suggest that the M2 helix model is consistent with a single channel conductance of ca. 50 pS. Continuum electrostatics calculations indicate that a ring of glutamate residues at the cytoplasmic mouth of the alpha 7 nicotinic receptor M2 helix bundle may not be fully ionized. A simplified model of the remainder of the channel protein when added to the M2 helix bundle plays a significant role in enhancing the ion selectivity of the channel.

Using theory and simulation to understand permeation and selectivity in ion channels

It is clear that the function of ion channels must flow from their structure. With recent advances in computational power and methodology, it appears feasible to correlate structure to ion channel permeation at an atomistically detailed level of description. The overall strategy is to structure the calculations in a hierarchy, ranging from coarse-grained thermodynamic and kinetic descriptions to fine-grained molecular dynamics descriptions with atomic detail. Each level of description is connected to the others by appropriate statistical mechanical theory. The coarse-grained descriptions can be correlated directly with electrophysiological experiment. The fine-grained descriptions are used to parameterize the coarse-grained descriptions and to describe the permeation process at the most detailed level. This strategy has so far had varying degrees of success. It has successfully described water permeation through lipid bilayers and gramicidin channels. It has revealed the essential events of ion permeation through gramicidin channels at an atomistically detailed level. The role of channel protein motions in permeation has been elucidated. However, it appears that force fields used to describe molecular dynamics must be refined further to achieve completely accurate predictions of the permeation of such small ions as sodium. Channels with more complex structure and more multiion occupancy than gramicidin pose major computational challenges with respect to sampling protein conformations and ion distributions involved in the permeation process. Possible approaches to meeting these challenges are discussed.

Ion-water and water-water interactions in a gramicidinlike channel: effects due to group polarizability and backbone flexibility

In the gramicidin channel, ionic transport and water transport occur simultaneously. Gramicidin's transport properties are influenced by ionic interactions with both the polypeptide and the channel waters. We present results of molecular dynamics studies on a series of alkali metal ions interacting with a water-filled gramicidinlike channel (a configurationally constrained polyglycine analog) at the dimer junction, in mid-monomer, and near the channel entrance. We investigate details of both short and long range ion-water and water-water correlation; these are notably dependent on the explicit consideration of polarizability and the degree of backbone flexibility. The nature of ion-water and water-water correlations changes as ionic size decreases and these changes may be augmented or attenuated by manipulation of the two parameters under study. Incorporating polarizability generally shortens ion-water distances and enhances ion-induced electrostriction (decreased water-water separations), while simultaneously reducing the long range orientational correlation of the single filing waters within the channel. Increasing flexibility predictably results in a broadening of the distribution of water-water and ion-water separations and contributes to the loss of long range orientational correlations. Both effects are ion specific; Cs+ and Na+ interact with the channel in distinctly different ways, while K+ represents an intermediate case more closely resembling Cs+. Our results demonstrate that incorporation of polarizability in the potential function has significant effects on the properties of channel water and, consequently, on the ionic transport process. While ion-water and water-water distances are decreased due to this feature, thereby fostering longer ranged correlations within the channel, enhanced interactions between water molecules and peptide groups tend to mitigate this effect. Possible implications for the multiple occupancy states of gramicidin and long range information transfer via a single file water chain are considered.

Solvation, water permeation, and ionic selectivity of a putative model for the pore region of the voltage-gated sodium channel

This paper describes a molecular dynamics and molecular mechanics study of the solvation and selectivity of the narrow pore and vestibule region of a model-built structure for the voltage-gated sodium channel. The particular structure used was one proposed by Guy and Durell. However, many of the features we saw would likely be shared with other possible models for this channel, such as the one proposed by Lipkind and Fozzard. It was found that the water mobility was reduced in the channel and the water orientations were significantly ordered by the channel environment. Water mobility depended on protein mobility; in a computer experiment in which the protein was artificially frozen, channel water at 300 degrees K was immobilized. Water motions were defined in significant part by a series of discrete moves from one pattern of hydrogen bonding with particular amino acids to another. However, there are so many different hydrogen bonding patterns that a description of the motion in terms of transitions among a small number of discrete states is not appropriate. In the model whose solvation we explored, several charged residues seem to play a particularly significant role in determining solvation and water motions. Based on energy minimization studies, the structure clearly shows selectivity for univalent cations over anions.

Boundary conditions for- single-ion diffusion

We have constructed a theory for diffusion through the pore of a single-ion channel by taking a limit of a random walk around a cycle of states. Similar to Levitt's theory of single-ion diffusion, one obtains boundary conditions for the Nernst-Planck equation that guarantee that the pore is occupied by at most one ion. Two of the terms in the boundary conditions are identical to those given by Levitt. However, the construction gives rise to a third term not found in Levitt's theory. With this term, the channel spends exponentially distributed intervals in the empty state. Ion sample paths have been simulated to help visualize trajectories near the channel entrances, with and without the new term. We use the modified Levitt theory to fit several potential profiles to the conductance data of Russell et al. In particular, we have analyzed the profile for Na+ in gramicidin calculated by Roux and Karplus. The peak-to-peak amplitude of their result must be reduced to at most 35% of its original value to fit the data. But with this reduction, excellent fits are obtained.

Expression of members of the proteolipid protein gene family in the developing murine central nervous system

Two homologous cDNAs were previously isolated by expression cloning with a monoclonal antibody that recognized a CNS neuronal membrane protein. Both cDNAs, M6a and M6b, bore significant homology with the major myelin proteolipid protein, PLP/DM20. Our initial studies of M6 gene expression in the adult mouse brain showed that M6a was present in neurons, PLP/DM20 in oligodendrocytes, and M6b in both neurons and glia. This led to the recognition of a novel gene family that included the oligodendrocyte-specific PLP/DM20 gene and the neuronal M6 genes. These observations supported the idea that PLP/DM20 may have functions other than myelination. In this report, we describe the spatial and temporal patterns of expression of M6a, M6b, and PLP/DM20 in the developing nervous system. PLP expression was limited to the white matter. M6a appeared in post-mitotic neurons of the brain and spinal cord as early as E10, and later in the hippocampus, cerebral cortex, and the granule cells of the cerebellum. In contrast, M6b was expressed at early embryonic stages in the ventricular zone of the spinal cord, and later during development in both neurons and glia. The early appearance of M6a and M6b mRNAs in the murine CNS suggested that these molecules might play an important role in the development of a variety of neural cell types.

Structure and dynamics of a proton wire: a theoretical study of H+ translocation along the single-file water chain in the gramicidin A channel

The rapid translocation of H+ along a chain of hydrogen-bonded water molecules, or proton wire, is thought to be an important mechanism for proton permeation through transmembrane channels. Computer simulations are used to study the properties of the proton wire formed by the single-file waters in the gramicidin A channel. The model includes the polypeptidic dimer, with 22 water molecules and one excess proton. The dissociation of the water molecules is taken into account by the "polarization model" of Stillinger and co-workers. The importance of quantum effects due to the light mass of the hydrogen nuclei is examined with the use of discretized Feynman path integral molecular dynamics simulations. Results show that the presence of an excess proton in the pore orients the single-file water molecules and affects the geometry of water-water hydrogen bonding interactions. Rather than a well-defined hydronium ion OH3+ in the single-file region, the protonated species is characterized by a strong hydrogen bond resembling that of O2H5+. The quantum dispersion of protons has a small but significant effect on the equilibrium structure of the hydrogen-bonded water chain. During classical trajectories, proton transfer between consecutive water molecules is a very fast spontaneous process that takes place in the subpicosecond time scale. The translocation along extended regions of the chain takes place neither via a totally concerted mechanism in which the donor-acceptor pattern would flip over the entire chain in a single step, nor via a succession of incoherent hops between well-defined intermediates. Rather, proton transfer in the wire is a semicollective process that results from the subtle interplay of rapid hydrogen-bond length fluctuations along the water chain. These rapid structural fluctuations of the protonated single file of waters around an average position and the slow movements of the average position of the excess proton along the channel axis occur on two very different time scales. Ultimately, it is the slow reorganization of hydrogen bonds between single-file water molecules and channel backbone carbonyl groups that, by affecting the connectivity and the dynamics of the single-file water chain, also limits the translocation of the proton across the pore.

Molecular dynamics simulations of water within models of ion channels

The transbilayer pores formed by ion channel proteins contain extended columns of water molecules. The dynamic properties of such waters have been suggested to differ from those of water in its bulk state. Molecular dynamics simulations of ion channel models solvated within and at the mouths of their pores are used to investigate the dynamics and structure of intra-pore water. Three classes of channel model are investigated: a) parallel bundles of hydrophobic (Ala20) alpha-helices; b) eight-stranded hydrophobic (Ala10) antiparallel beta-barrels; and c) parallel bundles of amphipathic alpha-helices (namely, delta-toxin, alamethicin, and nicotinic acetylcholine receptor M2 helix). The self-diffusion coefficients of water molecules within the pores are reduced significantly relative to bulk water in all of the models. Water rotational reorientation rates are also reduced within the pores, particularly in those pores formed by alpha-helix bundles. In the narrowest pore (that of the Ala20 pentameric helix bundle) self-diffusion coefficients and reorientation rates of intra-pore waters are reduced by approximately an order of magnitude relative to bulk solvent. In Ala20 helix bundles the water dipoles orient antiparallel to the helix dipoles. Such dipole/dipole interaction between water and pore may explain how water-filled ion channels may be formed by hydrophobic helices. In the bundles of amphipathic helices the orientation of water dipoles is modulated by the presence of charged side chains. No preferential orientation of water dipoles relative to the pore axis is observed in the hydrophobic beta-barrel models.

Water in channel-like cavities: structure and dynamics

Ion channels contain narrow columns of water molecules. It is of interest to compare the structure and dynamics of such intrapore water with those of the bulk solvent. Molecular dynamics simulations of modified TIP3P water molecules confined within channel-like cavities have been performed and the orientation and dynamics of the water molecules analyzed. Channels were modeled as cylindrical cavities with lengths ranging from 15 to 60 A and radii from 3 to 12 A. At the end of the molecular dynamics simulations water molecules were observed to be ordered into approximately concentric cylindrical shells. The waters of the outermost shell were oriented such that their dipoles were on average perpendicular to the normal of the wall of the cavity. Water dynamics were analyzed in terms of self-diffusion coefficients and rotational reorientation rates. For cavities of radii 3 and 6 A, water mobility was reduced relative to that of simulated bulk water. For 9- and 12-A radii confined water molecules exhibited mobilities comparable with that of the bulk solvent. If water molecules were confined within an hourglass-shaped cavity (with a central radius of 3 A increasing to 12 A at either end) a gradient of water mobility was observed along the cavity axis. Thus, water within simple models of transbilayer channels exhibits perturbations of structure and dynamics relative to bulk water. In particular the reduction of rotational reorientation rate is expected to alter the local dielectric constant within a transbilayer pore.

Water-mediated conformational transitions in nicotinic receptor M2 helix bundles: a molecular dynamics study

The ion channel of the nicotinic acetylcholine receptor is a water-filled pore formed by five M2 helix segments, one from each subunit. Molecular dynamics simulations on bundles of five M2 alpha 7 helices surrounding a central column of water and with caps of water molecules at either end of the pore have been used to explore the effects of intrapore water on helix packing. Interactions of water molecules with the N-terminal polar sidechains lead to a conformational transition from right- to left-handed supercoils during these stimulations. These studies reveal that the pore formed by the bundle of M2 helices is flexible. A structural role is proposed for water molecules in determining the geometry of bundles of isolated pore-forming helices.

Osmotic permeability in a molecular dynamics simulation of water transport through a single-occupancy pore

The aim of this work is to determine plausible values for the rate constants of kinetic models representing water transport through narrow pores. We present here the results of molecular dynamics simulations of the movement of water molecules through a single-site hydrophilic pore. The system consists of a rectangular box of water molecules, some of which are positionally restrained so as to act as a membrane. This membrane separates two compartments where water molecules move freely; one of the positions in the membrane is initially vacant (the 'single-site pore'), but can be occupied by mobile molecules. To analyze the results, we represented the pore by a two-state kinetic diagram in which the vacant and occupied states are linked by transitions corresponding to the binding and release of water molecules. The mean occupancy and vacancy times directly yield the rate constants of binding and release, which in turn yield the osmotic water permeability coefficient per pore pf. We also compute the apparent activation energies delta E* for the rate constants and for pf. The pf value was (1.56 +/- 0.04).10(-11) cm3/s (at 307 K), which is much larger than those determined for CHIP28 and for gramicidin A (of about 10(-13) and 10(-14) cm3/s, respectively). These values were compared with those arising from a model of a symmetric single-file pore through which one-vacancy-mediated water transport takes place. The model yields an expression for pf as a function of the rate constants and of the number of molecular positions (n) in the file. When n = 1, this expression becomes the one corresponding to the single-site pore studied in our current simulation. Using the rate constants of binding and release derived from our simulation, the pf values are consistent with an occupancy value of 5-6 found for gramicidin A, and with occupancies of 4-7 that can be estimated for the single-file pore of a recently proposed model for CHIP28. delta E* for pf is 3.0 kcal/mol, a value similar to that determined for CHIP28. Hence, the system simulated here appears plausible and can be used to mimic some physical properties of water transport through biological pores.

Transbilayer pores formed by beta-barrels: molecular modeling of pore structures and properties

Transmembrane beta-barrels, first observed in bacterial porins, are possible models for a number of membrane channels. Restrained molecular dynamics simulations based on idealized C alpha beta templates have been used to generate models of such beta-barrels. Model beta-barrels have been analyzed in terms of their conformational, energetic, and pore properties. Model beta-barrels formed by N = 4, 8, 12 and 16 anti-parallel Ala10 strands have been developed. For each N, beta-barrels with shear numbers S = N to 2N have been modeled. In all beta-barrel models the constituent beta-strands adopt a pronounced right-handed twist. Interstrand interactions are of approximately equal stability for all models with N > or = 8, whereas such interactions are weaker for the N = 4 beta-barrels. In N = 4 beta-barrels the pore is too narrow (minimum radius approximately 0.6 A) to allow ion permeation. For N > or = 8, the pore radius depends on both N and S; for a given value of N an increase in S from N to 2N is predicted to result in an approximately threefold increase in pore conductance. Calculated maximal conductances for the beta-barrel models are compared with experimental values for porins and for K+ channels.

Sodium in gramicidin: an example of a permion

The reaction path and free energy profile of Na+ were computed in the interior of the channel protein gramicidin, with the program MOIL. Gramicidin was represented in atomic detail, but surrounding water and lipid molecules were not included. Thus, only short range interactions were investigated. The permeation path of the ion was an irregular spiral, far from a straight line. Permeation cannot be described by motions of a single Na+ ion. The minimal energy path includes significant motion of water and channel atoms as well as motion of the permeating ion. We think of permeation as motion of a permion, a quasi-particle that includes the many body character of the permeation process, comparable with quasi-particles like holes, phonons, and electrons of solid-state physics. Na+ is accompanied by a plug of water molecules, and motions of water, Na+, and the atoms of gramicidin are highly correlated. The permion moves like a linear polymer made of waters and ion linked and moving coherently along a zigzag line, following the reptation mechanism of polymer transport. The effective mass, free energy, and memory kernel (of the integral describing time-dependent friction) of short range interactions were calculated. The effective mass of the permion (properly normalized) is much less than Na+. Friction varies substantially along the path. The free energy profile has two deep minima and several maxima. In certain regions, the dominant motions along the reaction path are those of the channel protein, not the permeating ion: there, ion waits while the other atoms move. At these waiting sites, the permion's motion along the reaction path is a displacement of the atoms of gramicidin that prepare the way for the Na+ ion.

Extended dipolar chain model for ion channels: electrostriction effects and the translocational energy barrier

We reinvestigate the dipolar chain model for an ion channel. Our goal is to account for the influence that ion-induced electrostriction of channel water has on the translocational energy barriers experienced by different ions in the channel. For this purpose, we refine our former model by relaxing the positional constraint on the ion and the water dipoles and by including Lennard-Jones contributions in addition to the electrostatic interactions. The positions of the ion and the waters are established by minimization of the free energy. As before, interaction with the external medium is described via the image forces. Application to alkali cations show that the short range interactions modulate the free energy profiles leading to a selectivity sequence for translocation. We study the influence of some structural parameters on this sequence and compare our theoretical predictions with observed results for gramicidin.

The "independence principle" in the processes of water transport

The processes of membrane transport exhibiting permeability coefficients depending on the species activities do not obey the "independence principle" and are assumed to take place by a mechanism of discrete nature, analyzable by a kinetic formalism. In this article, we study the dependence of the osmotic permeability coefficient on the water activities, from the steady-state analysis of a kinetic model of single-file water transport that simultaneously incorporates the vacancy-mediated and "knock-on" mechanisms into the state diagram. In particular, we study the relation between the near-equilibrium osmotic permeability (Pe) and the equilibrium water activity of the compartments (w). The analysis and numerical calculations performed for a simple case of the model show that, for values of the parameters consistent with experimental data, Pe exhibits only a small variation with w within the physiological range in the majority of the situations considered here. It is not possible to predict, from the study of these simple models, whether more complicated kinetic diagrams of water transport may be characterized by permeability coefficients with a more evident dependence on the water activities. Nevertheless, the results obtained here suggest that, for the case of physiological water pores, the analysis of the kinetic dependence of the permeability coefficients on the water activities may not yield evidence pointing to a discrete nature for the transport process.

Brownian dynamics study of a multiply-occupied cation channel: application to understanding permeation in potassium channels

The behavior of a multiply-occupied cation-selective channel has been computed by Brownian dynamics. The length, cross-section, ion-ion repulsion force, and ionic mobility within the channel are all estimated from data and physical reasoning. The only free parameter is a partition energy at the mouth of the channel, defining the free energy of an ion in the channel compared to the bath. It is presumed that this partition energy is associated with the energetics of exchanging a bulk hydration environment for a channel hydration environment. Varying the partition energy alone, keeping all other parameters fixed, gives approximately the full range of magnitudes of single channel conductances seen experimentally for K channels. Setting the partition energy at -11 kT makes the computed channel look similar to a squid axon K channel with respect to magnitude of conductance, shape of the I-V curve, non-unity of Ussing flux ratio exponents, decrease of current and increase of conductance with extracellular ion accumulation, and saturation at high ion concentration in the bathing solution. The model includes no preferred binding sites (local free energy minima) for ions in the channel. Therefore it follows that none of the above-mentioned properties of K channels are strong evidence for the existence of such sites. The model does not show supersaturation of current at very high bathing concentrations nor any pronounced voltage-dependence of the Ussing flux ratio exponent, suggesting that these features would require additional details not included in the model presented herein.