Molecular mechanism of H+ conduction in the single-file water chain of the gramicidin channel

The conduction of protons in the hydrogen-bonded chain of water molecules (or "proton wire") embedded in the lumen of gramicidin A is studied with molecular dynamics free energy simulations. The process may be described as a "hop-and-turn" or Grotthuss mechanism involving the chemical exchange (hop) of hydrogen nuclei between hydrogen-bonded water molecules arranged in single file in the lumen of the pore, and the subsequent reorganization (turn) of the hydrogen-bonded network. Accordingly, the conduction cycle is modeled by two complementary steps corresponding respectively to the translocation 1) of an ionic defect (H+) and 2) of a bonding defect along the hydrogen-bonded chain of water molecules in the pore interior. The molecular mechanism and the potential of mean force are analyzed for each of these two translocation steps. It is found that the mobility of protons in gramicidin A is essentially determined by the fine structure and the dynamic fluctuations of the hydrogen-bonded network. The translocation of H+ is mediated by spontaneous (thermal) fluctuations in the relative positions of oxygen atoms in the wire. In this diffusive mechanism, a shallow free-energy well slightly favors the presence of the excess proton near the middle of the channel. In the absence of H+, the water chain adopts either one of two polarized configurations, each of which corresponds to an oriented donor-acceptor hydrogen-bond pattern along the channel axis. Interconversion between these two conformations is an activated process that occurs through the sequential and directional reorientation of water molecules of the wire. The effect of hydrogen-bonding interactions between channel and water on proton translocation is analyzed from a comparison to the results obtained previously in a study of model nonpolar channels, in which such interactions were missing. Hydrogen-bond donation from water to the backbone carbonyl oxygen atoms lining the pore interior has a dual effect: it provides a coordination of water molecules well suited both to proton hydration and to high proton mobility, and it facilitates the slower reorientation or turn step of the Grotthuss mechanism by stabilizing intermediate configurations of the hydrogen-bonded network in which water molecules are in the process of flipping between their two preferred, polarized states. This mechanism offers a detailed molecular model for the rapid transport of protons in channels, in energy-transducing membrane proteins, and in enzymes.     

Hydroxide and proton migration in aquaporins

Hypothetical hydroxide and proton migration along the linear water chain in Aquaporin GlpF from Escherichia coli are studied by ab initio Car-Parrinello molecular dynamics simulations. It is found that the protein stabilizes a bipolar single file of water. The single file features a contiguous set of water-water hydrogen bonds in which polarization of the water molecules vary with position along the channel axis. Deprotonation of the water chain promotes the reorientation of water molecules while the hydroxide ion rapidly migrates by sequentially accepting protons from the neighboring water molecules. The hydroxide ion is not attracted by a conserved, channel-lining arginine residue, but is immobilized at two centrally located, conserved Asparagine-Proline-Alanine motifs where fourfold coordination stabilizes the ion. Hydroxide transition from the channel vestibules into the channel lumen is strongly influenced by electrostatic coupling to two conserved oppositely aligned macrodipoles. This suggests that the macrodipole's negative poles play a role in preventing hydroxide ions from entering into the channel's inner vestibules. Water protonation within the lumen facilitates water reorientation and subsequent proton expelling occurs. In the periplasmic half-channel, expelling occurs via the Grotthuss mechanism. Protonation within the cytoplasmic half-channel implies wire-breakage at the Asn-Pro-Ala motifs. The proton is here diffusively rejected as (H(5)O(2))(+).     

Density functional theory studies of model complexes for molybdenum-dependent nitrate reductase active sites

Molybdenum and tungsten complexes as models for the active sites of assimilatory or dissimilatory nitrate reductases (NR) were computed at the CPCM-B98/SDDp//B3LYP/Lanl2DZp* plus zero point energy level of density functional theory. The ligands were chosen on the basis of available experimental protein or small chemical model structures. A water molecule is found to bind to assimilatory NR models [(Me₂C₂S₂)MO(YMe)]⁻ (−11.5 kcal mol⁻¹ for M is Mo, Y is S) and may be replaced by nitrate (−4.5 kcal mol⁻¹) (but a hydroxy group may not). Nature’s choice of M is Mo and Y is S for NR has the largest activation energy for protein-free models (13.3 kcal mol⁻¹) and the least exothermic reaction energy for the nitrate reduction (−14.9 kcal mol⁻¹) compared with M is W and Y is O or Se alternatives. Water binding to dissimilatory NR model complexes [(Me₂C₂S₂)₂M(YR)]⁻ is considerably endothermic (10.3 kcal mol⁻¹); nitrate binding is only slightly so (1.5 kcal mol⁻¹ for RY⁻ is MeS⁻). The exchange of an oxo ligand (assimilatory NR) for a dithiolato ligand (dissimilatory NR model) reduces the exothermicity (−8.6 kcal mol⁻¹ relative to the fivefold-coordinate reduced complex) and raises the barrier for oxygen atom transfer (OAT) in the nitrate complex (19.2 kcal mol⁻¹). Not for the mono but only for the bisdithiolato complexes hydrogen bonding involving the coordinated substrate may significantly lower the OAT barrier as shown by explicitly adding water molecules. Substitution of tungsten for molybdenum generally lowers OAT activation energies and makes nitrate reduction reaction energies more negative. Bidentate carboxylato binding identified in Escherichia coli NarGHI is the preferred binding mode also for an acetato model. However, one dithiolato ligand folds when the Mo^VI center is bare of a good π-donor ligand, e.g., an oxo group. Computations on [(mnt)₂Mo^IV(YR)(PPh₃)]⁻ [mnt is (CN)₂C₂S₂ ²⁻] gave a smaller nitrate reduction activation energy for RY⁻ is Cl⁻, compared with RY⁻ is PhS⁻, although experimentally only the phenyl thiolato complex and not the chloro complex was found to be a functional NR model. Electronic supplementary material The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Noninnocent effect of axial ligand on the heme degradation process: a theoretical approach to hydrolysis pathway of verdoheme to biliverdin

Conversion of iron(II) verdoheme to iron(II) biliverdin in the presence of hydroxyl ion as a nucleophile and imidazole, pyridine, water, hydroxyl, cyanide, phenolate, chloride, thiolate and imidazolate as axial ligands was investigated using the B3LYP method and the 6-31G basis set. In the five-coordinated pathway the reactants and products are in the ground triplet state. In this path, hydroxyl ion directly attacks the macrocycle. The exothermic energy for addition of hydroxyl ion to iron(II) verdoheme with various ligands is 169.55, 166.34 and 164 kcal mol⁻¹ for water, pyridine and imidazole, energies which are around 30–60 kcal mol⁻¹ more exothermic than those for the other axial ligands used in this study. Therefore, imidazole, water and pyridine axial ligands can facilitate hydrolytic cleavage of iron(II) verdoheme to form open-chained helical iron(II) biliverdin complexes. The activation barrier for the conversion of iron(II) verdoheme hydroxyl species to the iron(II) biliverdin complex is estimated to be 5.2, 4.2, 4.35, 13.76 and 14.05 kcal mol⁻¹ for imidazole, water, cyanide, thiolate and imidazolate, respectively. Electronic supplementary material The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Ion Conduction through the hERG Potassium Channel

The inward rectifier voltage-gated potassium channel hERG is of primary importance for the regulation of the membrane potential of cardiomyocytes. Unlike most voltage-gated K⁺-channels, hERG shows a low elementary conductance at physiological voltage and potassium concentration. To investigate the molecular features underlying this unusual behavior, we simulated the ion conduction through the selectivity filter at a fully atomistic level by means of molecular dynamics-based methods, using a homology-derived model. According to our calculations, permeation of potassium ions can occur along two pathways, one involving site vacancies inside the filter (showing an energy barrier of about 6 kcal mol⁻¹), and the other characterized by the presence of a knock-on intermediate (about 8 kcal mol⁻¹). These barriers are indeed in accordance with a low conductance behavior, and can be explained in terms of a series of distinctive structural features displayed by the hERG ion permeation pathway.     

Possible Allosteric Significance of Water Structures in Proteins

Abstract

In recent theoretical molecular dynamics studies of ion solvation and transport through the model peptide ionophore, gramicidin A, it has been observed that the waters forming a linear single file within the channel have solvation and dynamic properties quite different from those found in bulk water. Strongly correlated motions among the interior single file column of waters persist over 20 Å. A speculation is entertained that related water structures could provide a mechanism for long range enzymatic allosteric effects as an alternative to chemical action at a distance propagated through the protein itself. Two possible specific mechanisms are discussed, hydraulic and “proton wire”. As a further control mechanism, the possibility is considered of modulating the allosteric effect though protein motion to open or close the channel thus producing a “valve” in the hydraulic line or a “switch” in the proton wire.     

Understanding structural/functional properties of amidase from Rhodococcus erythropolis by computational approaches

The 3D structure of the amidase from Rhodococcus erythropolis (EC 3.5.1.4) built by homology-based modeling is presented. Propionamide and acetamide are docked to the amidase. The reaction models were used to characterize the explicit enzymatic reaction. The calculated free energy barrier at B3LYP/6-31G* level of Model A (Ser194 + propionamide) is 19.72 kcal mol⁻¹ in gas (6.47 kcal mol⁻¹ in solution), and of Model B (Ser194 + Gly193 + propionamide) is 18.71 kcal mol⁻¹ in gas (4.57 kcal mol⁻¹ in solution). The docking results reveal that propionamide binds more strongly than acetamide due to the ethyl moiety of propionamide, which makes the carboxyl oxygen center of the substrate slightly more negative, making formation of the positively charged tetrahedral intermediate slightly easier. The quantum mechanics results demonstrate that Ser194 is essential for the acyl-intermediate, and Gly193 plays a secondary role in stabilizing acyl-intermediate formation as the NH groups of Ser194 and Gly193 form hydrogen bonds with the carbonyl oxygen of propionamide. The new structural and mechanistic insights gained from this computational study should be useful in elucidating the detailed structures and mechanisms of amidase and other homologous members of the amidase signature family.     

The gramicidin a channel: A review of its permeability characteristics with special reference to the single-file aspect of transport

Summary Gramicidin A forms univalent cation-selective channels of 4 Å diameter in phospholipid bilayer membranes. The transport of ions and water throughout most of the channel length is by a singlefile process; that is, cations and water molecules cannot pass each other within the channel. The implications of this single-file mode of transport for ion movement are considered. In particular, we show that there is no significant electrostatic barrier to ion movement between the energy wells at the two ends of the channel. The rate of ion translocation (e.g., Na⁺ or Cs⁺) through the channel between these wells is limited by the necessity for an ion to move six water molecules in single file along with it; this also limits the maximum possible value for channel conductance. At all attainable concentrations of NaCl, the gramicidin A channel never contains more than one sodium ion, whereas even at 0.1M CsCl, some channels contain two cesium ions. There is no necessity to postulate more than two ion-binding sites in the channel or occupancy of the channel by more than two ions at any time.     

Car-Parrinello and path integral molecular dynamics study of the hydrogen bond in the acetic acid dimer in the gas phase

In the paper are described studies of the double proton transfer (DPT) processes in the cyclic dimer of acetic acid in the gas phase using Car-Parrinello (CPMD) and path integral molecular dynamics (PIMD). Structures, energies and proton trajectories have been determined. The results show the double proton transfer in 450 K. In the classical dynamics (CPMD) a clear process mechanism can be identified, where asynchronized DPT arises due to coupling between the O-H stretching oscillator and several low energy intermolecular vibrational modes. The DPT mechanism is also asynchronic when quantum tunneling has been allowed in the simulation. It has been found that the calculated values of barrier height for the proton transfer depends very strongly on the used approaches. Barrier received from the free-energy profile at the CPMD level is around 4.5 kcal mol^-1 whereas at the PIMD level is reduced to 1 kcal mol^-1. The nature of bonding in acetic acid dimer and rearrangement of electron density due to the proton movement has been also studied by the topological analysis of Electron Localization Function and Electron Localizability Indicator function.     

Theoretical study of the reduction of nitric oxide in an A-type flavoprotein

The mechanism for the reduction of nitric oxide to nitrous oxide and water in an A-type flavoprotein (FprA) in Moorella thermoacetica, which has been proposed to be a scavenging type of nitric oxide reductase, has been investigated using density functional theory (B3LYP). A dinitrosyl complex, [{FeNO}⁷]₂, has previously been proposed to be a key intermediate in the NO reduction catalyzed by FprA. The electrons and protons involved in the reduction were suggested to “super-reduce” the dinitrosyl intermediate to [{FeNO}⁸]₂ or the corresponding diprotonated form, [{FeNO(H)}⁸]₂. In this type of mechanism the electron and/or proton transfers will be a part of the rate-determining step. In the present study, on the other hand, a reaction mechanism is suggested in which N₂O can be formed before the protons and electrons enter the catalytic cycle. One of the irons in the diiron center is used to stabilize the formation of a hyponitrite dianion, instead of binding a second NO. Cleaving the N–O bond in the hyponitrite dianion intermediate is the rate-determining step in the proposed reaction mechanism. The barrier of 16.5 kcal mol⁻¹ is in good agreement with the barrier height of the experimental rate-determining step of 14.8 kcal mol⁻¹. The energetics of some intermediates in the “super-reduction” mechanism and the mechanism proceeding via a hyponitrite dianion are compared, favoring the latter. It is also discussed how to experimentally discriminate between the two mechanisms. Electronic supplementary material Supplementary material is available in the online version of this article at and is accessible for authorized users.     

Free energy profiles for H+ conduction along hydrogen-bonded chains of water molecules.

The molecular mechanism for proton conduction along hydrogen-bonded chains, or "proton wires," is studied with free energy simulations. The complete transport of a charge along a proton wire requires two complementary processes: 1) translocation of an excess proton (propagation of an ionic defect), and 2) reorientation of the hydrogen-bonded chain (propagation of a bonding defect). The potential of mean force profile for these two steps is computed in model systems comprising a single-file chain of nine dissociable and polarizable water molecules represented by the PM6 model of Stillinger and co-workers. Results of molecular dynamics simulations with umbrella sampling indicate that the unprotonated chain is preferably polarized, and that the inversion of its total dipole moment involves an activation free energy of 8 kcal/mol. In contrast, the rapid translocation of an excess H+ across a chain extending between two spherical solvent droplets is an activationless process. These results suggest that the propagation of a bonding defect constitutes a limiting step for the passage of several protons along single-file chains of water molecules, whereas the ionic translocation may be fast enough to occur within the lifetime of transient hydrogen-bonded water chains in biological membranes.     

Characterization of micellar-packaged gramicidin A channels.

The effects of heating, on an aqueous gramicidin A lysolecithin system, were examined by carbon-13 nuclear magnetic resonance (¹³C-NMR), circular dichroism (CD), and sodium-23 nuclear magnetic resonance (²³Na-NMR), and the results are collectively interpreted to indicate micellar-packaging of gramicidin channels and cation occupancy in the channel. ¹³C-NMR of the gramicidin-lysolecithin system demonstrates a decrease in mobility of the micellar lipid on heating which is indicative of incorporation of gramicidin into the hydrophobic core of the micelle. A unique and reproducible CD spectrum is obtained for the heat incorporated state. Sodium-23 spin-lattice relaxation times (T₁) demonstrated sodium interaction to be dependent on heat incorporation. The T₁ identified interaction is blocked by silver ion which is known to block sodium transport through the channel in lipid bilayer studies. The temperature dependence of the sodium-23 line width defines an exchange process with an activation energy of 6.8 kcal/mole which is essentially the same as the activation energy reported for transport through the channel in lecithin bilayer studies, and the sodium exchange process is blocked by thallium ion which is also known to block sodium transport through the channel.     

Possible allosteric significance of water structures in proteins

In recent theoretical molecular dynamics studies of ion solvation and transport through the model peptide ionophore, gramicidin A, it has been observed that the waters forming a linear single file within the channel have solvation and dynamic properties quite different from those found in bulk water. Strongly correlated motions among the interior single file column of waters persist over 20 A. A speculation is entertained that related water structures could provide a mechanism for long range enzymatic allosteric effects as an alternative to chemical action at a distance propagated through the protein itself. Two possible specific mechanisms are discussed, hydraulic and "proton wire". As a further control mechanism, the possibility is considered of modulating the allosteric effect though protein motion to open or close the channel thus producing a "valve" in the hydraulic line or a "switch" in the proton wire.     

Theoretical study of N<Subscript>4</Subscript>X (X = O,S, Se) systems

A series of N₄X (X = O, S, Se) compounds have been examined with ab initio and density functional theory (DFT) methods. To our knowledge, these compounds, except for the C_2v ring and the C_3v towerlike isomers of N₄O, are first reported here. The ring structures are the most energetically favored for N₄X (X = O and S) systems. For N₄Se, the cagelike structure is the most energetically favored. Several decomposition and isomerization pathways for the N₄X species have been investigated. The dissociation of C_2v ring N₄O and N₄S structures via ring breaking and the barrier height are only 1.1 and −0.2 kcal mol⁻¹ at the CCSD(T)/6-311+G*//MP2/6-311+G* level of theory. The dissociation of the cagelike N₄X species is at a cost of 12.1–16.2 kcal mol⁻¹. As for the towerlike and triangle bipyramidal isomers, their decomposition or isomerization barrier heights are all lower than 10.0 kcal mol⁻¹. Although the C_S cagelike N₄S isomer has a moderate isomerization barrier (18.3–29.1 kcal mol⁻¹), the low dissociation barrier (−1.0 kcal mol⁻¹) indicates that it will disappear when going to the higher CCSD(T) level. Electronic supplementary material The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

The state of water in the outer barrier of the isolated frog skin

Summary The flux of water across the outer barrier of the frog skin is generally regarded as the rate-limiting step in the movement of water across the whole membrane. This paper presents some evidence that, at room temperature, the flux of water across the outer barrier occurs through water in a non-liquid state. The organization of water in a non-liquid state lowers the diffusion coefficient of water through water by several orders of magnitude. The study employs a method recently developed in this laboratory which permits measurement of unidirectional fluxes at the outermost part of an epithelial membrane mounted as a flat sheet. Only above 25°C is the activation energy for the flow of tritiated water (4.3 kcal mole⁻¹) similar to the one observed in free water (4.6 kcal mole⁻¹). At temperatures around 15°C, the energy of activation is 8.5 kcal mole⁻¹. At temperatures near 0°C, at which the frog lives only part of the year, the energy of activation is 16.7 kcal mole⁻¹.     

Optimization of parameters for semiempirical methods V: Modification of NDDO approximations and application to 70 elements

Several modifications that have been made to the NDDO core-core interaction term and to the method of parameter optimization are described. These changes have resulted in a more complete parameter optimization, called PM6, which has, in turn, allowed 70 elements to be parameterized. The average unsigned error (AUE) between calculated and reference heats of formation for 4,492 species was 8.0 kcal mol⁻¹. For the subset of 1,373 compounds involving only the elements H, C, N, O, F, P, S, Cl, and Br, the PM6 AUE was 4.4 kcal mol⁻¹. The equivalent AUE for other methods were: RM1: 5.0, B3LYP 6–31G*: 5.2, PM5: 5.7, PM3: 6.3, HF 6–31G*: 7.4, and AM1: 10.0 kcal mol⁻¹. Several long-standing faults in AM1 and PM3 have been corrected and significant improvements have been made in the prediction of geometries. Figure Calculated structure of the complex ion [Ta₆Cl₁₂]²⁺ (footnote): Reference value in parenthesis Electronic supplementary material The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Ion selectivity mechanism in a bacterial pentameric ligand-gated ion channel

The proton-gated ion channel from Gloeobacter violaceus (GLIC) is a prokaryotic homolog of the eukaryotic nicotinic acetylcholine receptor that responds to the binding of neurotransmitter acetylcholine and mediates fast signal transmission. Recent emergence of a high-resolution crystal structure of GLIC captured in a potentially open state allowed detailed, atomic-level insight into ion conduction and selectivity mechanisms in these channels. Herein, we have examined the barriers to ion conduction and origins of ion selectivity in the GLIC channel by the construction of potential-of-mean-force profiles for sodium and chloride ions inside the transmembrane region. Our calculations reveal that the GLIC channel is open for a sodium ion to transport, but presents a ∼11 kcal/mol free energy barrier for a chloride ion. Our collective findings identify three distinct contributions to the observed preference for the permeant ions. First, there is a substantial contribution due to a ring of negatively charged glutamate residues (E-2′) at the narrow intracellular end of the channel. The negative electrostatics of this region and the ability of the glutamate side chains to directly bind cations would strongly favor the passage of sodium ions while hindering translocation of chloride ions. Second, our results imply a significant hydrophobic contribution to selectivity linked to differences in the desolvation penalty for the sodium versus chloride ions in the central hydrophobic region of the pore. This hydrophobic contribution is evidenced by the large free energy barriers experienced by Cl⁻ in the middle of the pore for both GLIC and the E-2′A mutant. Finally, there is a distinct contribution arising from the overall negative electrostatics of the channel.     

Attenuation of proton currents by methanol in a dioxolane-linked gramicidin A channel in different lipid bilayers.

The mobility of protons in a dioxolane-linked gramicidin A channel (D1) is comparable to the mobility of protons in aqueous solutions (Cukierman, S., E. P. Quigley, and D. S. Crumrine. 1997. Biophys. J. 73:2489-2502). Aliphatic alcohols decrease the mobility of H+ in aqueous solutions. In this study, the effects of methanol on proton conduction through D1 channels were investigated in different lipid bilayers and at different HCl concentrations. Methanol attenuated H+ currents in a voltage-independent manner. Attenuation of proton currents was also independent of H+ concentrations in solution. In phospholipid bilayers, methanol decreased the single channel conductance to protons without affecting the binding affinity of protons to bilayers. In glycerylmonooleate membranes, the attenuation of single channel proton conductances qualitatively resembled the decrease of conductivities of HCl solutions by methanol. However, in both types of lipid bilayers, single channel proton conductances through D1 channels were considerably more attenuated than the conductivities of different HCl solutions. This suggests that methanol modulates single proton currents through D1 channels. It is proposed that, on average, one methanol molecule binds to a D1 channel, and attenuates H+ conductance. The Gibbs free energy of this process (DeltaG0) is approximately 1.2 kcal/mol, which is comparable to the free energy of decrease of HCl conductivity in methanol solutions (1.6 kcal/mol). Apolar substances like urea and glucose that do not transport protons in HCl solutions and do not permeate D1 channels decreased solution conductivity and single channel conductance by a considerably larger proportion than methanol. Cs+ currents through D1 channels were considerably less (fivefold) attenuated by methanol than proton currents. It is proposed that methanol partitions inside the pore of gramicidin channels and delays the transfer of protons between water and methanol molecules, causing a significant attenuation of the single channel proton conductance. Gramicidin channels offer an interesting experimental model to study proton hopping along a single chain of water molecules interrupted by a single methanol molecule.     

Diffusion and diffusion-osmosis models of the transfer of charged molecules across biological barriers

Mathematical models of the transfer of large enough charged molecules (macroions) have been constructed on the basis of the classical equations of electromigration diffusion (Helmholtz-Smoluchowski, Goldman, and Goldman-Hodgkin-Katz). It is shown that ion transfer in placental barriers (mimicking lipid-protein membrane barriers) and in muscle barriers proceeds by different mechanisms. In placental barriers, the electromigration diffusion takes place through lipid-protein channels formed by conformational alteration of phospholipid and protein molecules, with diffusion coefficients D = (2.6–3.6) × 10⁻⁸ cm²/s. The transfer in muscle barriers is due to migration via charged interfibrillar channels with negative diffusion activation energy (explained by changes in the structure of muscle fibers and expenditures of thermal energy for the displacement of Cl⁻ from channel walls), and D = (6.0–10.0) × 10⁻⁶ cm²/s.     

Dielectric relaxation studies of ionic processes in lysolecithin-packaged gramicidin channels

Summary Dielectric permittivities have been determined for suspensions of lysolecithin packaged malonyl gramicidin channels over the frequency range of 5kHz to 900 MHz and under conditions of approximately equimolar concentrations (10mM) of channels and salts. The salts were lithium chloride, sodium chloride and thallium acetate. A relaxation process unique to the thallium acetate-channel system was observed which on analysis gave rise to a relaxation time at 25⁰ of 120 nsec. The permittivity data, as well as a comparison of binding constants, indicate that the relaxation process results from Tl⁺ being bound within the channel and more specifically from an intrachannel ion translocation with a rate constant of approximately 4×10⁶ sec^–1 and with an energy of activation of less than 6.7 kcal/mole. These data compare favorably with data from conductance studies on planar bilayers and with ion and carbon-13 nuclear magnetic studies on the lysolecithin packaged malonyl gramicidin channels which combine to indicate that the relaxation process is due to the jump of the thallium ion across a central barrier.