Boundary conditions for- single-ion diffusion

Molecular dynamics investigation of Cl(-) transport through the closed and open states of the 2α12β2γ2 GABA(A) receptor

The α1β2γ2 gamma-aminobutyric type A receptor (GABA(A)R) is one of the most widely expressed GABA(A)R subtypes in the mammalian brain. GABA(A)Rsbelonging to the Cys-loop superfamily of ligand-gated ion channels have been identified as key targets for many clinical drugs, and the motions that govern the gating mechanism are still not well understood. In this study, an open-state GABA(A)R was constructed using the structure of the glutamate-gated chloride channel (GluCl), which has a high sequence identity to GABA(A)R. A closed-state model was constructed using the structure of the nicotinic acetylcholine receptor (nAChR). Molecular dynamics simulations of the open-state and closed-state GABA(A)R were performed. We calculated the electrostatic potential of the two conformations, the pore radius of the two ion channels and the root-mean-square fluctuation. We observed the presence of two positively charged girdles around the ion channel and found flexible regions in the GABA(A)R. Then, the free-energy of chloride ion permeations through the closed-state and open-state G GABA(A)R has been estimated using adaptive biasing force (ABF) simulation. For the closed-state G GABA(A)R, we observed two major energy barriers for chloride ion translocation in the transmembrane domain (TMD). For the open-state GABA(A)R, there was only one energy barrier formed by two Thr261 (α1), two Thr255 (β2) and one Thr271 (γ2). By using ABF simulation, the overall free-energy profile is obtained for Cl(-) transporting through GABA(A)R, which gives a complete map of the ion channel of Cl(-) permeation.

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.

Renormalizing SMD: the renormalization approach and its use in long time simulations and accelerated PMF calculations of macromolecules

Simulations of long time process in condensed phases, in general, and in biomolecules, in particular, present a major challenge that cannot be overcome at present by brute force molecular dynamics (MD) approaches. This work takes the renormalization method, intruded by us sometime ago, and establishes its reliability and potential in extending the time scale of molecular simulations. The validation involves a truncated gramicidin system in the gas phase. This system is small enough to allow for very long explicit simulations and sufficiently complex to present the physics of realistic ion channels. The renormalization approach is found to be reliable and arguably presents the first approach that allows one to exploit the otherwise problematic steered molecular dynamics (SMD) treatments in quantitative and meaningful studies. It is established that we can reproduce the long time behavior of large systems by using Langevin dynamics (LD) simulations of a renormalized implicit model. This is done without spending the enormous time needed to obtain such trajectories in the explicit system. The present study also provides a promising advance in accelerated evaluation of free energy barriers. This is done by adjusting the effective potential in the implicit model to reproduce the same passage time as that obtained in the explicit model under the influence of an external force. Here having a reasonable effective friction provides a way to extract the potential of mean force (PMF) without investing the time needed for regular PMF calculations. The renormalization approach, which is illustrated here in realistic calculations, is expected to provide a major help in studies of complex landscapes and in exploring long time dynamics of biomolecules.

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.

Single-occupancy binding in simple bounded and unbounded systems

The number of substrate molecules that can bind to the active site of an enzyme at one time is constrained. This paper develops boundary conditions that correspond to the constraint of single-occupancy binding. Two simple models of substrate molecules diffusing to a single-occupancy site are considered. In the interval model, a fixed number of substrate molecules diffuse in a bounded domain. In the spherical model, a varying number of molecules diffuse in a domain with boundary conditions that model contact with a reservoir containing a large number of substrate molecules. When the diffusive time scale is much shorter than the time scale for entering the single-occupancy site, the dynamics of binding are accurately described by simple approximations.

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.

Adaptive Brownian Dynamics Simulation for Estimating Potential Mean Force in Ion Channel Permeation

Ion channels are biological nanotubes formed by large protein molecules in the cell membrane. This paper presents a novel multiparticle simulation methodology, which we call adaptive controlled Brownian dynamics, for estimating the force experienced by a permeating ion at each discrete position along the ion-conducting pathway. The profile of this force, commonly known as the potential of mean force, results from the electrostatic interactions between the ions in the conduit and all the charges carried by atoms forming the channel the protein, as well as the induced charges on the protein wall. The current across the channel is solely determined by the potential of mean force encountered by the permeant ions. The simulation algorithm yields consistent estimates of this profile. The algorithm operates on an angstrom unit spatial scale and femtosecond time scale. Numerical simulations on the gramicidin ion channel show that the algorithm yields the potential of mean force profile that accurately reproduces experimental observations.

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.

Influence of protein flexibility on the electrostatic energy landscape in gramicidin A

We describe an electrostatic model of the gramicidin A channel that allows protein atoms to move in response to the presence of a permeating ion. To do this, molecular dynamics simulations are carried out with a permeating ion at various positions within the channel. Then an ensemble of atomic coordinates taken from the simulations are used to construct energy profiles using macroscopic electrostatic calculations. The energy profiles constructed are compared to experimentally-determined conductance data by inserting them into Brownian dynamics simulations. We find that the energy landscape seen by a permeating ion changes significantly when we allow the protein atoms to move rather than using a rigid protein structure. However, the model developed cannot satisfactorily reproduce all of the experimental data. Thus, even when protein atoms are allowed to move, the dielectric model used in our electrostatic calculations breaks down when modeling the gramicidin channel.

Saturation of conductance in single ion channels: the blocking effect of the near reaction field

The ionic current flowing through a protein channel in the membrane of a biological cell depends on the concentration of the permeant ion, as well as on many other variables. As the concentration increases, the rate of arrival of bath ions to the channel's entrance increases, and typically so does the net current. This concentration dependence is part of traditional diffusion and rate models that predict Michaelis-Menten current-concentration relations for a single ion channel. Such models, however, neglect other effects of bath concentrations on the net current. The net current depends not only on the entrance rate of ions into the channel, but also on forces acting on ions inside the channel. These forces, in turn, depend not only on the applied potential and charge distribution of the channel, but also on the long-range Coulombic interactions with the surrounding bath ions. In this paper, we study the effects of bath concentrations on the average force on an ion in a single ion channel. We show that the force of the reaction field on a discrete ion inside a channel embedded in an uncharged lipid membrane contains a blocking (shielding) term that is proportional to the square root of the ionic bath concentration. We then show that different blocking strengths yield different behavior of the current-concentration and conductance-concentration curves. Our theory shows that at low concentrations, when the blocking force is weak, conductance grows linearly with concentration, as in traditional models, e.g., Michaelis-Menten formulations. As the concentration increases to a range of moderate shielding, conductance grows as the square root of concentration, whereas at high concentrations, with high shielding, conductance may actually decrease with increasing concentrations: the conductance-concentration curve can invert. Therefore, electrostatic interactions between bath ions and the single ion inside the channel can explain the different regimes of conductance-concentration relations observed in experiments.

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.

A microscopic view of ion conduction through the K+ channel

Recent results from x-ray crystallography and molecular dynamics free-energy simulations have revealed the existence of a number of specific cation-binding sites disposed along the narrow pore of the K+ channel from Streptomyces lividans (KcsA), suggesting that K+ ions might literally "hop" in single file from one binding site to the next as permeation proceeds. In support of this view, it was found that the ion configurations correspond to energy wells of similar depth and that ion translocation is opposed only by small energy barriers. Although such features of the multiion potential energy surface are certainly essential for achieving a high throughput rate, diffusional and dissipative dynamical factors must also be taken into consideration to understand how rapid conduction of K+ is possible. To elucidate the mechanism of ion conduction, we established a framework theory enabling the direct simulation of nonequilibrium fluxes by extending the results of molecular dynamics over macroscopically long times. In good accord with experimental measurements, the simulated maximum conductance of the channel at saturating concentration is on the order of 550 and 360 pS for outward and inward ions flux, respectively, with a unidirectional flux-ratio exponent of 3. Analysis of the ion-conduction process reveals a lack of equivalence between the cation-binding sites in the selectivity filter.

The implementation of slab geometry for membrane-channel molecular dynamics simulations

Slab geometric boundary conditions are applied in the molecular dynamics simulation of a simple membrane-channel system. The results of the simulation were compared to those of an analogous system using normal three-dimensional periodic boundary conditions. Analysis of the dynamics and electrostatics of the system show that slab geometric periodicity eliminates the artificial bulk water orientational polarization that is present while using normal three-dimensional periodicity. Furthermore, even though the water occupancy and volume of our simple channel is the same when using either method, the electrostatic properties are considerably different when using slab geometry. In particular, the orientational polarization of water is seen to be different in the interior of the channel. This gives rise to a markedly different electric field within the channel. We discuss the implications of slab geometry for the future simulation of this type of system and for the study of channel transport properties.

Continuum electrostatics fails to describe ion permeation in the gramicidin channel

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

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.

Computational studies of the gramicidin channel

Ion channels are highly specific membrane-spanning protein structures which serve to facilitate the passage of selected ions across the lipid barrier. In the past decade, molecular dynamics simulations based on atomic models and realistic microscopic interactions with explicit solvent and membrane lipids have been used to gain insight into the function of these complex systems. These calculations have considerably expanded our view of ion permeation at the microscopic level. This Account will mainly focus on computational studies of the gramicidin A channel, one of the simplest and best characterized molecular pore.

Framework model for single proton conduction through gramicidin

This paper describes a framework model for proton conduction through gramicidin; a model designed to incorporate information from molecular dynamics and use this to predict conductance properties. The state diagram describes both motion of an excess proton within the pore as well as the reorientation of waters within the pore in the absence of an excess proton. The model is constructed as the diffusion limit of a random walk, allowing control over the boundary behavior of trajectories. Simple assumptions about the boundary behavior are made, which allow an analytical solution for the proton current and conductance. This is compared with corresponding expressions from statistical mechanics. The random walk construction allows diffusing trajectories underlying the model to be simulated in a simple way. Details of the numerical algorithm are described.

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.

Can we use rate constants and state models to describe ion transport through gramicidin channels?

Can we use rate constants and state models to describe ion transport through gramicidin channels? Maybe, but only if rate constants are just proportionality constants between rates and probabilities of observing states of the channel. This approach is natural if the system of channel plus ions (plus water) is almost always in one or another of a small number of identifiable states. Many features of ion transport through gramicidin, including the conductance-concentration relationship, concentration-dependent permeability ratios, anomalous mole fraction effect and to some extent flux ratio exponents, are consistent with a description in which there are four occupation 'states' of the pore: only water; an ion at one end; an ion at the other; and ions at both ends. Current-voltage relationships can (and must) also be fitted, but until there is a theory to predict the potential dependence of the rate constants this success will remain hollow. Other features have resisted interpretation. These include the failures to determine 'binding constants' consistent with all the data; the variation of flux ratio exponents with ion type; and, probably, the variation of the currents with asymmetrical ion concentrations. Nevertheless, state models still have one attractive feature, they allow consideration of the effects that one ion within the pore has on the movements of another.

Statistical mechanical equilibrium theory of selective ion channels

A rigorous statistical mechanical formulation of the equilibrium properties of selective ion channels is developed, incorporating the influence of the membrane potential, multiple occupancy, and saturation effects. The theory provides a framework for discussing familiar quantities and concepts in the context of detailed microscopic models. Statistical mechanical expressions for the free energy profile along the channel axis, the cross-sectional area of the pore, and probability of occupancy are given and discussed. In particular, the influence of the membrane voltage, the significance of the electric distance, and traditional assumptions concerning the linearity of the membrane electric field along the channel axis are examined. Important findings are: 1) the equilibrium probabilities of occupancy of multiply occupied channels have the familiar algebraic form of saturation properties which is obtained from kinetic models with discrete states of denumerable ion occupancy (although this does not prove the existence of specific binding sites; 2) the total free energy profile of an ion along the channel axis can be separated into an intrinsic ion-pore free energy potential of mean force, independent of the transmembrane potential, and other contributions that arise from the interfacial polarization; 3) the transmembrane potential calculated numerically for a detailed atomic configuration of the gramicidin A channel embedded in a bilayer membrane with explicit lipid molecules is shown to be closely linear over a distance of 25 A along the channel axis. Therefore, the present analysis provides some support for the constant membrane potential field approximation, a concept that has played a central role in the interpretation of flux data based on traditional models of ion permeation. It is hoped that this formulation will provide a sound physical basis for developing nonequilibrium theories of ion transport in selective biological channels.

(In)validity of the constant field and constant currents assumptions in theories of ion transport

Constant electric fields and constant ion currents are often considered in theories of ion transport. Therefore, it is important to understand the validity of these helpful concepts. The constant field assumption requires that the charge density of permeant ions and flexible polar groups is virtually voltage independent. We present analytic relations that indicate the conditions under which the constant field approximation applies. Barrier models are frequently fitted to experimental current-voltage curves to describe ion transport. These models are based on three fundamental characteristics: a constant electric field, negligible concerted motions of ions inside the channel (an ion can enter only an empty site), and concentration-independent energy profiles. An analysis of those fundamental assumptions of barrier models shows that those approximations require large barriers because the electrostatic interaction is strong and has a long range. In the constant currents assumption, the current of each permeating ion species is considered to be constant throughout the channel; thus ion pairing is explicitly ignored. In inhomogeneous steady-state systems, the association rate constant determines the strength of ion pairing. Among permeable ions, however, the ion association rate constants are not small, according to modern diffusion-limited reaction rate theories. A mathematical formulation of a constant currents condition indicates that ion pairing very likely has an effect but does not dominate ion transport.

Noncontact dipole effects on channel permeation. I. Experiments with (5F-indole)Trp13 gramicidin A channels

Gramicidin A (gA), with four Trp residues per monomer, has an increased conductance compared to its Phe replacement analogs. When the dipole moment of the Trp13 side chain is increased by fluorination at indole position 5 (FgA), the conductance is expected to increase further. gA and FgA conductances to Na+, K+, and H+ were measured in planar diphytanoylphosphatidylcholine (DPhPC) or glycerylmonoolein (GMO) bilayers. In DPhPC bilayers, Na+ and K+ conductances increased upon fluorination, whereas in GMO they decreased. The low ratio in the monoglyceride bilayer was not reversed in GMO-ether bilayers, solvent-inflated or -deflated bilayers, or variable fatty acid chain monoglyceride bilayers. In both GMO and DPhPC bilayers, fluorination decreased conductance to H+ but increased conductance in the mixed solution, 1 M KCl at pH 2.0, where K+ dominates conduction. Eadie-Hofstee plot slopes suggest similar destabilization of K+ binding in both lipids. Channel lifetimes were not affected by fluorination in either lipid. These observations indicate that fluorination does not change the rotameric conformation of the side chain. The expected difference in the rate-limiting step for transport through channels in the two bilayers qualitatively explains all of the above trends.

Far-field analysis of coupled bulk and boundary layer diffusion toward an ion channel entrance

We present a far-field analysis of ion diffusion toward a channel embedded in a membrane with a fixed charge density. The Smoluchowski equation, which represents the 3D problem, is approximated by a system of coupled three- and two-dimensional diffusions. The 2D diffusion models the quasi-two-dimensional diffusion of ions in a boundary layer in which the electrical potential interaction with the membrane surface charge is important. The 3D diffusion models ion transport in the bulk region outside the boundary layer. Analytical expressions for concentration and flux are developed that are accurate far from the channel entrance. These provide boundary conditions for a numerical solution of the problem. Our results are used to calculate far-field ion flows corresponding to experiments of Bell and Miller (Biophys. J. 45:279, 1984).

Electrostatic effects in macromolecules: fundamental concepts and practical modeling

The past few years have seen an exponential growth in the calculations of electrostatic energies of macromolecules and an increased recognition of the crucial role of electrostatic effects. This review considers the current state of the field. Focus is placed on calculations of pKas, redox potentials and binding energies in macromolecules and clarification of the fact that the value of the dielectric 'constant' of a protein depends on its definition and that small dielectric constants should not be used in describing charge-charge interactions by current continuum models.

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.