Stochastic theory of ion movement in channels with single-ion occupancy. Application to sodium permeation of gramicidin channels.

The electrodiffusion equations were solved for the one-ion channel both by the analytical method due to Levitt and also by Brownian dynamic simulations. For both types of calculations equilibration of ion distribution between the bath and the ends of the channel was assumed. Potential profiles were found that give good fits to published data on Na+ permeation of gramicidin channels. The data were best fit by profiles that have no relative energy maximum at the mouth of the channel. This finding suggests that alignment of waters or channel charged groups inside the channel in response to an ion's approach may provide an energetically favorable situation for entry sufficient to overcome the energy required for removing bulk waters of hydration. An alternative possibility is that the barrier to ion entry is situated outside the region restricted to single-ion occupancy. Replacement of valine with more polar amino acids at the No. 1 location was found to correspond to a deepening of the potential minima near the channel mouths, an increase in height of the central barrier to ion translocation across the channel, and possibly a reduction in the mobility of the ion-water complex in the channel. The Levitt theory was extended to calculate passage times for ions to cross the channel and the blocking effects of ions that entered the channel but didn't cross. These quantities were also calculated by the Brownian dynamics method.     

Effect of pore structure on energy barriers and applied voltage profiles. I. Symmetrical channels.

This paper presents calculations of the image potential for an ion in an aqueous pore spanning a lipid membrane and for the electric field produced in such a pore when a transmembrane potential is applied. The pore diameter may be variable. As long as the length-to-radius ratio in the narrow portion of a channel is large enough, the image potential for an ion in or near the mouth of a channel is determined by the geometry of the mouth. Within the constriction, the image potential of the ion-pore system may be reasonably approximated by constructing an "equivalent pore" of uniform diameter spanning a somewhat thinner membrane. When a transmembrane potential is applied the electric field within a constricted, constant radius, section of the model pore is constant. If the length-to-radius ratio of the narrow part of the channel is not too large or the channel ensemble has wide mouths, the field extends a significant distance into the aqueous region. The method is used to model features of the gramicidin A channel. The energy barrier for hydration (for exiting the channel) is identified with the activation energy for gramicidin conductance (Bamberg and Läuger, 1974, Biochim. Biophys. Acta. 367:127).     

Simulation of voltage-driven hydrated cation transport through narrow transmembrane channels.

Molecular dynamics studies for the voltage-driven transport of the alkali metal ions lithium, sodium, and potassium through gramicidin A-type channels filled with water molecules are presented. The number of water molecules in the channel is obtained from a previous study (Skerra, A., and J. Brickmann, 1987, Biophys. J., 51:969-976). It is shown that the selectivity of the intrachannel ion diffusion through our model pore conforms to the experimentally observed selectivity of the gramicidin A channel. It is demonstrated that the number of water molecules in the channel plays a key role for the selectivity.     

Open channel noise. V. Fluctuating barriers to ion entry in gramicidin A channels.

We have measured the fluctuations in the current through gramicidin A (GA) channels in symmetrical solutions of monovalent cations of various concentrations, and compared the spectral density values with those computed using E. Frehland's theory for noise in discrete transport systems (Frehland, E. 1978. Biophys. Chem. 8:255-265). The noise for the transport of NH4+ and Na+ ions in glycerol-monooleate/squalene membranes could be accounted for entirely by "shot noise" in the process of transport through a single-filing pore with two ion binding sites. However, in confirmation of results in a previous paper (Sigworth, F. J., D. W. Urry, and K. U. Prasad. 1987. Biophys. J. 52:1055-1064) currents of Cs+ showed a substantial excess noise at low ion concentrations, as did currents of K+ and Rb+. The excess noise was increased in thicker membranes. The observations are accounted for by a theory that postulates fluctuations of the entry rates of ions into the channel on a time scale of approximately 1 microsecond. These fluctuations occur preferentially when the channel is empty; the presence of bound ions stabilizes the "high conductance" conformation of the channel. The fluctuations are sensed to different degrees by the various ion species, and their kinetics depend on membrane thickness.     

Electrostatic calculations for an ion channel. II. Kinetic behavior of the gramicidin A channel.

A theoretical model of the gramicidin A channel is presented and the kinetic behavior of the model is derived and compared with previous experimental results. The major assumption of the model is that the only interaction between ions in a multiply-occupied channel is electrostatic. The electrostatic calculations indicate in a multiply-occupied channel is electrostatic. The electrostatic calculations indicate that there will be potential wells at each end of the channel and, at high concentrations, that both wells can be occupied. The kinetics are based on two reaction steps: movement of the ion from the bulk solution to the well and movement between the two wells. The kinetics for this reaction rate approach are identical to those based on the Nernst-Planck equation in the limit where the movement between the two wells is rate limiting. The experimental results for sodium and potassium are consistent with a maximum of two ions per channel. To explain the thallium results it is necessary to allow three ions per channel. It is shown that this case is compatible with the electrostatic calculations if the presence of an anion is included. The theoretical kinetics are in reasonable quantitative agreement with the following experimental measurements: single channel conductance of sodium, potassium, and thallium; bi-ionic potential and permeability ratio between sodium-potassium and potassium-thallium; the limiting conductance of potassium and thallium at high applied voltages; current-voltage curves for sodium and potassium at low (but not high) concentrations; and the inhibition of sodium conductance by thallium. The results suggest that the potential well is located close to the channel mouth and that the conductance is partially limited by the rate going from the bulk solution to the well. For thallium, this entrance rate is probably diffusion limited.     

A hypothesis for the mechanism of sodium channel opening by batrachotoxin and related toxins

The mechanism of action of one class of sodium channel opening agents (batrachotoxinin, veratridine, aconitine and grayanotoxin) is proposed to involve complexation of a triad of agent oxygen atoms with the ε-ammonium ion of a channel lysing side chain, holding open the mouth or exit of the ion channel. This idea complements the oxygen triad model derived by structural considerations (Masutani, T., Seyama, I., Narahashi, T. and Iwasa, J. (1981) J. Pharm. Exp. Therap. 217,812) and extended by crystal structure comparisons (Codding, P.W. (1983) J. Am. Chem. Soc. 105, 3172). The mechanism is based on results for acetylcholine receptor ion channel gating, structure and function, using single group rotation (SGR) theory (cf. Kosower, E.M. (1983) Biochem. Biophys. Res. Commun. 111, 1022 and in press (1983);FEBS Lett. (1983) 155, 245; ibid. 157, 144; Biophys. J. (1983) 45, in press).     

Theoretical study of the antiparallel double-stranded helical dimer of gramicidin as an ion channel.

Recent experimental studies by Durkin, J. T., O. S. Andersen, F. Heitz, Y. Trudelle, and R. E. Koeppe II (1987. Biophys. J. 51:451a) have suggested that the antiparallel double-stranded helical (APDS) dimer of gramicidin can form a transmembrane cation channel. This article reports a theoretical study that successfully rationalizes the channel properties of the APDS dimer. As in the case of the head-to-head (HH) dimer, the APDS exhibits a high potential energy barrier as anions approach the channel mouth, according for the observation of valence selectivity. The calculated potential energies of cations show two binding sites near the channel mouths, a typical feature of the HH channel. The potential energies of hydrated cations in the APDS are generally higher than those in the HH channel and show a larger pseudoperiodicity and higher barriers, an observation which suggests that the APDS should exhibit lower single 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.     

Molecular dynamics simulation of cation motion in water-filled gramicidinlike pores.

A model calculation is carried out to study the potential energy profile of a sodium ion with several water molecules inside a simplified model of the gramicidin ion channel. The sodium ion is treated as a Lennard-Jones sphere with a point charge at its center. The Barnes polarizable water model is used to mimic the water molecules. A polarizable and deformable gramicidinlike channel is constructed based on the model obtained by Koeppe and Kimura. Potential minima and saddle points are located and the static energy barriers are computed. The potential minima at the two mouths of the channel exhibit an aqueous solvation structure very different from that at any of the interior minima. These sites are approximately 23.6 and 24.4 A apart for binding of a sodium ion and a cesium ion, respectively. Ionic motion from these exterior sites to the first interior minimum requires substantial rearrangement of the waters of solvation; this rearrangement may be the hydration/dehydration step in ionic permeation through the channel. Based on these results, a mechanism by which the sodium ion moves from the exterior binding site to the interior of the channel is proposed. Our model channel accommodates about eight water molecules and the transport of the ion and water within the channel is found to be single file. Results of less extensive calculations for Cs+ and Li+ ions in a channel with or without water are also reported.     

Modulating dipoles for structure-function correlations in the gramicidin A channel.

Dipoles of the tryptophan indole side chains have a direct impact on ion conductance in the gramicidin channel. Here, fluorination of the indoles (both 5- and 6-fluoro) is used to manipulate both the orientations and the magnitudes of the dipoles. The orientations and positions with respect to the channel axis were determined using (2)H solid state NMR of uniformly aligned lipid bilayer preparations. By exchange of the remaining four protons in the indole ring for deuterium, comparison could be made to d(5)-indole spectra that have previously been recorded for each of the four indoles of gramicidin A. After making the assignments which were aided by the observation of (19)F-(2)H dipolar interactions, we found that fluorination caused only minor changes in side chain conformation. With the high-resolution structural characterization of the fluorinated indoles in position 11, 13, and 15, the electrostatic interactions with a cation at the channel and bilayer center can be predicted and the influence of the modified dipoles on ion conductance estimated. The importance of the long-range electrostatic interaction was recently documented with the observation of alpha-helical dipoles oriented toward the bilayer center on the ion conductance pathway for the Streptomyces K(+) channel. We present direct measurements of the orientation of gramicidin channel F-Trp positions for use in analysis of dipole effects on channel permeation.     

Modeling negative ion defect migration through the gramicidin A channel

The results of potential of mean force (PMF) calculations for the distinct stages of proton conduction through the gramicidin A channel, including proton migration, reorientation of the water file and negative ion defect migration, are presented. The negative ion defect migration mechanism was hypothesized in experimental studies but was not considered previously in molecular dynamics simulations. The model system consisted of the peptide chains constructed on the base of the structure PDBID:1JNO, the inner file of nine water molecules and external clusters of water molecules placed at both ends of the channel. Potential energy functions were computed with the CHARMM/PM6/TIP3P parameters. The results obtained for proton migration and water file reorientation are basically consistent with those reported previously by Pómès and Roux (Biophys J 82:2304, 2002) within the similar approach. For the newly considered mechanism of negative ion defect migration from the channel center to the end of the water file we obtain the energy 3.8 kcal mol⁻¹ which is not considerably different from the activation energy of water reorientation, 5.4 kcal mol⁻¹. Therefore this mechanism may principally compete for the rate-limiting step in proton conduction in gramicidin.     

Membrane surface-charge titration probed by gramicidin A channel conductance.

We manipulate lipid bilayer surface charge and gauge its influence on gramicidin A channel conductance by two strategies: titration of the lipid charge through bulk solution pH and dilution of a charged lipid by neutral. Using diphytanoyl phosphatidylserine (PS) bilayers with CsCl aqueous solutions, we show that the effects of lipid charge titration on channel conductance are masked 1) by conductance saturation with Cs+ ions in the neutral pH range and 2) by increased proton concentration when the bathing solution pH is less than 3. A smeared charge model permits us to separate different contributions to the channel conductance and to introduce a new method for "bilayer pKa" determination. We use the Gouy-Chapman expression for the charged surface potential to obtain equilibria of protons and cations with lipid charges. To calculate cation concentration at the channel mouth, we compare different models for the ion distribution, exact and linearized forms of the planar Poisson-Boltzmann equation, as well as the construction of a "Gibbs dividing surface" between salt bath and charged membrane. All approximations yield the intrinsic pKain of PS lipid in 0.1 M CsCl to be in the range 2.5-3.0. By diluting PS surface charge at a fixed pH with admixed neutral diphytanoyl phosphatidylcholine (PC), we obtain a conductance decrease in magnitude greater than expected from the electrostatic model. This observation is in accord with the different conductance saturation values for PS and PC lipids reported earlier (, Biochim. Biophys. Acta. 552:369-378) and verified in the present work for solvent-free membranes. In addition to electrostatic effects of surface charge, gramicidin A channel conductance is also influenced by lipid-dependent structural factors.     

How shortening a channel may lower its conductance. The case of des-Val7-DVal8-gramicidin A

A shortened analog of gramicidin A has been shown by Urry et al. (Biochim. Biophys. Acta 775, 115-119) to have lower conductance than native gramicidin A. They argue this suggests that the major current carrier is the doubly occupied channel. A different perspective is presented here. Channel formation does not alter bilayer width. In a shortened channel an ion approaching the binding site moves further toward the center of the lipid-pore system. The electrostatic contribution to the energy barrier near the constriction mouth is greater for the shorter channel. As long as entry to the channel is rate limiting singly occupied short channels should exhibit lower conductance. The data are not inconsistent with singly occupied channels being the major current carriers. Experiments on other gramicidin analogs are suggested to more clearly distinguish between singly and doubly occupied channels as the dominant conducting species.     

Ion currents through pores. The roles of diffusion and external access steps in determining the currents through narrow pores.

External access steps, which may include restricted aqueous diffusion, are introduced into a kinetic model for ion transport through narrow pores. The conductance-concentration relation and the concentration dependence of the biionic permeability are calculated using two alternative assumptions: (a) access to the mouth of the pore is allowed only when no ion is within the lumen or at either mouth; (b) ions remain at the mouth only very transiently. With either assumption the concentration dependence of the fluxes is the same as in previous treatments in which all steps in access were lumped into a single process. Also as before, the biionic permeability ratio is independent of concentration so long as the lumen is never doubly occupied. For narrow pores, such as those formed by gramicidin A, the slowest external portion of the access process must occur close to the pore's mouth, and thus the region an ion must occupy to gain access is small. As a consequence, the probability of finding an ion within this region is also small. On this basis, it is argued that the second assumption is appropriate for these pores. The kinetic equations that result are identical to those used by Urban, B., S.B. Hladky, and D.A. Haydon (1980, Biochim. Biophys. Acta. 602:331-354).     

Asymmetric gramicidin channels: heterodimeric channels with a single F6Val1 residue.

Substitution of Val1 by 4,4,4,4',4',4'-F6Val in [Val1]gramicidin A ([Val1]gA) produces channels in which the effects of amino acid replacements on dimer stability and ion permeation are nonadditive. If only one Val1 (in a symmetric [Val1]gA channel) is substituted by F6Val, the resulting heterodimeric channels are destabilized relative to both homodimeric parent channels and the single-channel conductance of the heterodimeric channels is reduced relative to the parent channels (Russell, E. W. B., L. B. Weiss, F. I. Navetta, R. E. Koeppe II, and O. S. Andersen. 1986. Single-channel studies on linear gramicidins with altered amino acid side chains. Effects of altering the polarity of the side chain at position #1 in gramicidin A. Biophys. J. 49:673; Durkin, J. T., R. E. Koeppe II, and O. S. Andersen. 1990. Energetics of gramicidin hybrid channel formation as a test for structural equivalence. Side-chain substitutions in the native sequence. J. Mol. Biol. 211:221-234). To understand the basis for this destabilization, we have examined further the characteristics of [F6Val1]/[Xxx1]gA heterodimers, where Xxx = Gly, Val, and Ala. These heterodimeric channels show rapid current transitions between (at least) two current levels and display asymmetric i-V characteristics. The orientation of the heterodimers relative to the applied potential was determined by asymmetric addition of the gramicidin analogs, one to each side of a preformed bilayer. The current transitions are most clearly illustrated for [F6Val1]/[Gly1]gA heterodimers, which possess two finite and well defined current levels. Based on the existence of these two conductance states and the analysis of duration and interval distributions, we conclude that the transitions between the two current levels correspond to conformational transitions in "stable" heterodimers. In the case of [F6Val1]/[Val1]gA and [F6Val1]/[Ala1]gA heterodimers, the low-conductance state is indistinguishable from zero. The two (or more) conductance states presumably correspond to different orientations of the dipolar F6Val1 side chain. The distribution between the high- and the low-conductance states varies as a function of potential in [F6Val1]/[Gly1]gA channels. These characteristics cause the [F6Val1]/nonpolar (Val, Ala, Gly)gA hybrid channels to serve as a "simple" model for understanding gating transitions in membrane-spanning channels.     

On the mechanism of channel-length dependence of gramicidin single-channel conductance

Single-channel conductance data on four different gramicidin channel lengths demonstrate that conductance magnitude is neither inversely dependent on the square of the channel length nor on the image force arising from differences in the extent of lipid dimpling (Jordan and Vayl (1985) Biochim. Biophys. Acta 818, 416-420). Rather the conductance differences are consistent with the decreased off-rate constant for the singly occupied state as the ionic radius decreases from that of cesium ion to sodium ion coupled with the decreased probability of the doubly occupied channel due to increased ion-ion repulsion as the channel is shortened (Urry et al. (1984) Biochim. Biophys. Acta 774, 115-119).     

Modulation of gramicidin A open channel lifetime by ion occupancy.

The hypothesis that the gramicidin A channel stability depends on the level of ion occupancy of the channel was used to derive a mathematical model relating channel lifetime to channel occupancy. Eyring barrier permeation models were examined for their ability to fit the zero-voltage conductance, current-voltage, as well as lifetime data. The simplest permeation model required to explain the major features of the experimental data consists of three barriers and four sites (3B4S) with a maximum of two ions occupying the channel. The average lifetime of the channel was calculated from the barrier model by assuming the closing rate constant to be proportional to the probability of the internal channel sites being empty. The link between permeation and lifetime has as its single parameter the experimentally determined averaged lifetime of gramicidin A channels in the limit of infinitely dilute solutions and has therefore no adjustable parameters. This simple assumption that one or more ions inside the channel completely stabilize the dimer conformation is successful in explaining the experimental data considering the fact that this model for stabilization is independent of ion species and configurational occupancy. The model is used to examine, by comparison with experimental data, the asymmetrical voltage dependence of the lifetime in asymmetrical solutions, the effects of blockers, and the effects of elevated osmotic pressure.     

Engineering aspects of biological ion channels--from biosensors to computational models for permeation

This paper presents two important engineering aspects of biological ion channels-how to build sensors out of gramicidin channels and how to construct computational models for ion channel permeation. We describe our recent research in these areas, potential challenges and possible solutions.