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Na+ and K+ ion transport through a solvated gramicidin A transmembrane channel: Molecular dynamics studies using parallel processors
Abstract
The energetics, structure, and dynamics of two different cations (Na+ and K+), water, and gramicidin A (GA) systems are studied by molecular dynamics methods with parallel computational techniques. The parallel computation techniques employed in this study are described. In our simulations, we study the dynamical behavior of 81 water molecules and a single ion of each kind, placed at one of two different locations along a channel. The ion and water move under the dynamic force fields from each other as well as the static force field of the channel, which is compared of the GA framework constrained to remain the Urry dimer conformation. All force fields are of ab initio origin. The trajectories illustrate the characteristic time scales of typical events at the chosen locations. The velocity and force autocorrelation functions from the simulations (the first computed for any GA model) are discussed.
... Many research themes have emerged from the study of gramicidin A. Because of the volume of publications reporting on the properties and structure of gramicidin A only a representative list is given here. These include the energetics of ion transport across membranes (Hladky and Haydon, 1972; Monoi, 1983; Eisenman and Sandblom, 1984; Jordan, 1984; Mackay et al., 1984; Urry et al., 1984; Etchebest et al., 1985; Kim et al., 1985; Prasad et al., 1986); the effect of gramicidin A on the order, dynamics, and phase stability of the supporting lipid (Chapman et al., 1977; Rice and Oldfield, 1979; Haigh et Address correspondence to B. A. Cornell. ...
Three analogues of the helical ionophore gramicidin A have been synthesized with (13)C-labeled carbonyls ((13)C=O) incorporated at either Gly(2), Ala(3), or Val(7). A fourth compound incorporated (13)C at both the carbonyl and alpha-carbon of Gly(2) within the same molecule. These labels were studied using solid-state, proton-enhanced, (13)C nuclear magnetic resonance (NMR) in hydrated dispersions of dimyristoylphosphatidylcholine (DMPC)-gramicidin A. The dispersions were aligned on glass coverslips whose orientation to the magnetic field could be varied through 180 degrees . The orientation dependence of the NMR spectrum was used to obtain an accurate measurement of the (13)C chemical shift anisotropy (CSA), and in the case of the fourth compound, the (13)C-(13)C dipolar coupling constant. From the measured CSA and estimates of the orientation of the (13)C shielding tensor, we are able to determine the direction of the (13)C=O bonds and to compare these with the predictions of the various reported models for the configuration of gramicidin A in phospholipid bilayers. Our results are consistent with the left-handed pipi(6.3) (LD) single-stranded helix (Urry, D. W., J. T. Walker, and T. L. Trapane. 1982. J. Membr. Biol. 69:225-231). The right-handed pipi(6.3) (LD) single-stranded helix observed for gramicidin A in sodium dodecyl sulfate micelles (Arseniev, A. S., I. L. Barsukov, V. F. Bystrov, A. L. Loize, and Yu A. Ovchinnikov. 1985. FEBS (Fed. Eur. Biochem. Soc.) Lett. 186:168-174) yields a poorer fit to the data. However, the width of the carbonyl resonances suggests a distribution of molecular geometries possibly resulting from a spread in the helix pitch and handedness. Double-stranded helices and beta sheet structures are excluded. In dispersions in which the lipid is in the L(alpha) phase, the gramicidin A undergoes rapid reorientation about an axis which is centered on the normal to the plane of the coverslips. When the supporting lipid is in the L(beta') phase the helices are rigid on the timescale of (13)C-NMR. The configuration of gramicidin A is unaltered by L(alpha)-L(beta') phase transition of the bilayer lipid.
... Our understanding of permeation might be judged complete to the extent that we can find a consistency between the results of molecular dynamics calculations and the results of computations of the sort presented in this paper. Several molecular dynamics studies including water-channel-ion interactions have been published that give explicit numbers for potentials inside the channel that govern Na+ permeation of gramicidin ( Lee and Jordan, 1984;Kim et al., 1985;Etchebest and Pullman, 1986). All of these studies concur that the energy of the system is reduced as the Na ion enters the channel, so that the Na ion tends to partition into the channel. ...
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.
... Several groups have done molecular dynamics studies on the movement of ions in gramicidin A channels including water. Two studies (Mackay et al., 1984; Kim et al., 1985) did relatively short and localized runs including the complete atomic structure of the gramicidin molecule. Another study (Lee and Jordan, 1984) reduced the degrees of freedom and explored more ion positions. ...
This paper shows how Brownian motion theory can be used to analyze features of individual ion trajectories in channels as calculated by molecular dynamics, and that its use permits more precise determinations of diffusion coefficients than would otherwise be possible. We also show how a consideration of trajectories of single particles can distinguish between effects due to the magnitude of the diffusion coefficient and effects due to barriers and wells in the potential profile, effects which can not be distinguished by consideration of average fluxes.
... The temperature of the transition is however much lower than would be predicted for water because the magnetic dipole is so much weaker than the electric dipole. Lattice sums (Edmonds, 1979, 1984) and molecular dynamic simulations (Mackay et al., 1984; Kim et al., 1985) both predict ordered water arrays in narrow water-filled pores. Because the hydrogen bonding in water is strong, any major disruption of the normal hydrogen-bonded water structures will lead to a high energy and thus to an improbable structure. ...
Today, the equilibrium behavior of ions in solution may be predicted with some confidence, essentially because rapid ionic diffusion over small distances ensures homogeneity throughout the solution. Equilibrium concepts such as ionic strength and pH apply. However, when attempting to understand the behavior of ions passing rapidly through narrow pores such as ion channels, no such equilibrium state may be assumed. The passing solution may have been in equilibrium with conditions at the mouth of the pore but will not be in equilibrium with charged molecules on the pore wall. In addition, the water in narrow pores will be partially ordered by contact with the pore walls and will not behave like bulk water.
To illustrate this difference, a simple equilibrium calculation of the ion concentrations near a plastic sheet penetrated by narrow pores and containing in its surface partially ionized carboxyl groups is shown to be in good agreement with experiment. However, to predict the non-equilibrium behavior within the narrow pores is much more difficult. To illustrate the difficulty, a Monte Carlo computer model is described which attempts to predict the rapid switching of ion current observed experimentally with these narrow pores.
In an experiment based on electroosmotic ion transport, 3.6 nm high graphene nanochannels with a clean, smooth and hydrophobic surface and large slip length have 115 times greater ionic conductivity than SiO2 nanochannels.
Metal ions play significant roles in numerous fields including chemistry, geochemistry, biochemistry, and materials science. With computational tools increasingly becoming important in chemical research, methods have emerged to effectively face the challenge of modeling metal ions in the gas, aqueous, and solid phases. Herein, we review both quantum and classical modeling strategies for metal ion-containing systems that have been developed over the past few decades. This Review focuses on classical metal ion modeling based on unpolarized models (including the nonbonded, bonded, cationic dummy atom, and combined models), polarizable models (e.g., the fluctuating charge, Drude oscillator, and the induced dipole models), the angular overlap model, and valence bond-based models. Quantum mechanical studies of metal ion-containing systems at the semiempirical, ab initio, and density functional levels of theory are reviewed as well with a particular focus on how these methods inform classical modeling efforts. Finally, conclusions and future prospects and directions are offered that will further enhance the classical modeling of metal ion-containing systems.
Extensive molecular dynamics studies for “one dimensional” aqueous solutions of the alkali metal cations Li+, Na+ and K+ in gramicidin A-like channels, as well as for the water at the membrane surfaces are presented. It is demonstrated that the water structure significantly differs from the bulk water structure in both cases, and that the one dimensional solvation of the ions in the channel plays a key role for ionic transport rates.
Molecular dynamics studies for the voltage driven transport of the alkali metal ions Li+, Na+ and K+ through gramicidin A-type channels in the presence of water molecules are presented. It is demonstrated that the number of water molecules in the channel plays a key role for the cation selectivity.
Klassisch können die hier angesprochenen Monte Carlo- (MC) und Molekular-dynamische- (MD) Simulationsmethoden aus zwei Gründen genannt werden: Zum einen sind sie die ältesten Simulationsmethoden in der chemischen Forschung. Sie gehen zurück auf die Arbeiten von Metropolis und Mitarbeitern [1] sowie von Alder und Wainwright [2] in den fünfziger Jahren. Ihre Weiterentwicklung erfolgte parallel zur Entwicklung moderner Großrechner und waren teilweise auch Ansporn und Motivation zu deren Weiterentwicklung., insbesondere zur Steigerung der Rechengeschwindigkeit.
The general principles of Molecular Dynamics (MD) computer simulations are briefly sketched. Models used in simulations of aqueous ionic solutions are reviewed and static and dynamic results obtained from such simulations are discussed and compared with experimental ones.
The Ewald Summation technique is applied to compute the Coulombic expression of an empirical water-ferrierite pair potential utilized for Molecular Dynamics simulations including long-range electrostatic interactions. The trajectories of 33 water molecules inside 4 unit cells (11% of free capacity) created in the phase space allow the calculation at room temperature of energetic and dynamical properties as time-averages and of frequency spectrum characterizing the translational and reorientational molecular motions inside the framework.
There is now a rich history to those experiments studying the stimulation of biological systems using extremely weak electrical and magnetic fields. In the following, we review this work, with particular emphasis on the experimental evidence in support of ion cyclotron resonance as the interaction mechanism underlying the observed effects. The number of compartmentalized groups studying electromagnetic effects in tissue is growing rapidly, and it is wise to limit the area we shall cover. Thus, we restrict ourselves solely to those observations involving time-varying fields, primarily in the ELF range. We will begin with a short review of ion cyclotron resonance.
Following our previous studies on understanding the structure and dynamics of biomolecules in water we have performed molecular dynamics (MD) simulations for the following systems: i) the ion transport through Gramicidin A (GA) transmembrane channel, and ii) the hydration structure and dynamics of B and Z-DNA in the presence of counterions (K+). The results of these simulations are presented both for the structural and dynamical properties. In the case of gramicidin the results are most preliminary and seem to indicate that the waters close to the ion inside the channel point their dipoles towards the ion, while the waters away from the ion are influenced by the bulk waters. In the simulations of B and Z-DNA we find that the waters close to the helix retain their memory of their initial orientations for longer times than those waters away from the helix, in agreement with the relaxation studies of Lindsay et al. on Z-DNA. Further, the velocity auto correlation functions for the counterions (K+) exhibit a caged behavior unlike that in an ionic solution.
Computational chemistry is a very vast field dealing with atomic and molecular systems, considered at different complexity levels either as discretized quantum mechanical systems, or as statistical ensembles, amenable to Monte Carlo and Molecular Dynamic treatments, or as continuous matter fluid-dynamical distributions, modeled with Navier– Stokes equations. The mainstream computational chemistry was bent to fully solve the correlation problem with a single “technology.” Computational chemistry became a must for more and more chemists, even if the computer users had less and less awareness of the computational details of computer programs, and hardly understood that the computed answer could be incorrect, because of limitations of the selected method. In this computer generation and even more in the following years, internet, communications, commercial computer programs, computer servers, personal computers, desktop, graphics, Window, and Linux were common words, memory and disk space seemed unlimited, price/performance improved yearly, but faith in the computer replaced knowledge of the instrument and its software. Computational chemistry was becoming a part of the global economy.
This chapter discusses how the insights obtained from theoretical investigations of various cluster systems have enabled a researcher to predict structures and properties of novel functional molecular systems. Clusters are self-assembled structures comprised of a number of monomers under the given condition. Apart from aiding the development of novel materials, clusters are very useful for understanding the intrinsic and fundamental nature of molecular recognition and self-assembling phenomena. This is amply illustrated in a number of publications on a wide variety of atomic and molecular clusters, ranging from H-bonded clusters, p-system-containing clusters, and metal clusters. These investigations not only provide pertinent information useful for nanomaterial design but also highlight some of the important similarities and differences in their physical characteristics. These characteristics include structures, magnitudes of both attractive and repulsive interaction energies, vibrational frequencies, and charge redistributions. Additionally, one also obtains an insight into the contributions of cooperative and competitive forces, both of which govern self-assembly and molecular recognition.
This chapter discusses theoretical approaches to the design of functional nanomaterials. A theoretical description of nanomaterials that belongs to the mesoscopic phase is fraught with a number of problems because neither quantum–chemical methods used to investigate the microscopic phase nor solid-state physics methods used to investigate the macroscopic phase can adequately describe all the desired properties. A judicious combination of both these methods apart from providing a detailed insight of the properties of these nanomaterials also aids the de novo design of novel molecular systems and functional materials. Another area of interest is molecular recognition, wherein quantum chemical calculations aid the design of microscopic chemical/biochemical sensors/monitors such as DNA chips. This chapter examines the ways in which an effective combination of diverse theoretical methods ranging from traditional ab initio, density functional, tight binding, Monte Carlo to molecular dynamics simulations, have helped elucidate the nature of intermolecular interactions in a large number of widely disparate molecular complexes (aqueous and metal clusters, biomolecules), and have also aided the design of novel nanomaterials (ionophores, receptors, endo/exohedral fullerenes, fullerides, nanotori, nanotubes, nanowires, and molecular devices).
This chapter reviews the biological membranes that have complex effects on passive transport by diffusion because they are structured as a mosaic of the regions with distinct hydrophobic and hydrophilic properties. The ability of membrane proteins, specifically to admit some polar molecules and exclude others dramatically, affects the responses of biological membranes to the concentration gradients. The chapter examines that nuclear magnetic resonance (NMR) spectroscopy has proven to be a very powerful technique for the study of the transport process and the molecular systems responsible for the transport. NMR techniques can be used to determine the three-dimensional structure of the channel, to determine the thermodynamic parameters for the incorporation of the transport system into the membrane environment, to obtain the thermodynamic parameters for the binding of the cations to the channel, to determine the kinetic activation enthalpy for the transport process, and to study the internal motion of the peptide or protein that forms the channel.
We present here an example of the global simulation approach to complex systems, selecting, as example, the study of a liquid, specifically water. We start by building the molecules of the liquid from nuclei and electrons using quantum mechanics. Next we obtain the interaction potentials (two, three and four body), again by quantum mechanics. Then we use Monte Carlo and molecular dynamics to study the motions of a water molecule within its Onsager sphere, and the collective properties; subsequently we overlap fluid dynamics by considering a flow along a channel with or without obstacles; finally we extend further and report a preliminary simulation of a Benard problem, using Newton's equations. These simulations are performed on a parallel supercomputer which we have recently assembled; the system is briefly described in the second part of this work. A number of applications in science and engineering are analysed with attention to the degree of parallelization achievable with and without special hardwares such as busses and bulk shared memory. To conclude the implication of the global simulation methodology and supercomputers evolution is discussed in terms of productivity of information.
We investigated five lowest energy structures of the water hexamer ring, book, bag, cage, and prism using extensive ab initio calculations. High levels of theory using various basis sets were employed. On the basis of Mo "ller–Plesset second order perturbation MP2 calculations using a large basis set 9s6 p4d2 f 1g/6s4 p2ddiffuse(2sp/s), the lowest energy structure with zero point energy ZPE correction is the cage conformer, followed by the book within 0.1 kcal/mol and the prism within 0.2 kcal/mol. The spectra of the five conformers have been investigated. The predicted rotational constants and dipole moments of the cage conformer are in good agreement with the experiment Liu et al., Nature 381, 501 1996 as compared to other structures. This proves that the experiment surely found the cage structure, which was first reported by one of the authors Kim et al., Chem. Phys. Lett. 131, 451 1986. However, the five structures would still be nearly isoenergetic within 0.7 kcal/mol at 0 K. Above 40 K, the free energy of the book is slightly lower than the cage, which might imply that the book structure would be detected. Upon deuteration, the cage structure is the lowest energy conformer, followed by two competing structures of the book and prism whose energies are only 0.2 kcal/mol higher at 0 K; above 55 K the book would be more populated than the cage. © 1998 American Institute of Physics. S0021-96069830738-2
A number of conformers of aqua-K complexes, K H 2 O n (n1 – 10) have been investigated using high level ab initio calculations, to elucidate the structures and thermodynamic energies of the hydrated potassium ions. Since the coordination number of K is around six in the bulk water, the focus of the present study has been the n5 and 6 clusters. In contrast to previous studies which have used only the enthalpies to compare against the experimental numbers, the present study also employs free energies. As a result, the predictions of a number of hitherto unknown conformers are in excellent agreement with the experimental results. The maximum coordination number for K in ligands containing O atoms is evaluated to be around eight from the energetics of structures possessing only the first hydration shell of water molecules around the K ion. It is of interest to note that the hydration of the K ion is less structured than that of the Na ion, since the water– water interaction becomes more important in the aqua–K clusters. The predicted vibrational frequencies of the aqua–K clusters reflect the H-bonding signature, and hence, could be utilized in the identification of the hydration structures of K in experiments. © 1999 American Institute of Physics. S0021-96069930533-X
Our previous findings provide strong evidence that transport of a given ionic species through a cell membrane can be precisely controlled by tuning externally applied magnetic fields to the ion cyclotron resonance (CR) gyrofrequency for the ion in question. Experiment and theory have shown that certain odd harmonics of the fundamental resonance frequency are also effective. In the present experiment we again used the model of Ca-dependent motility of benthic diatoms, extending our harmonic studies through N = 17, for both Ca2+ and K+ fundamentals. Eight separate Ca2+ fundamental frequencies: 8, 12, 16, 23, 31, 32, 46, 64 Hz were attempted and each was found to obey the CR condition. We also report two observations: (1) tuning to K+ results in inhibition of motility, directly opposite to the enhancement that occurs when tuning for Ca2+; (2) both the K+ inhibition and Ca2+ enhancement are independently observed at exactly the same harmonics: N = 1, 3, 5, 15. This strongly suggests that despite differences in ionic selectivity, K+ and Ca2+ channels may share fundamentally similar protein configurations.
The structure and dynamics of confined water in cylindrical pores have been investigated by molecular dynamics simulations. Both rigid (TIP4P) and flexible (BJH) models have been used. Pore radii between 4.2 and 20 Å have been studied; the pore walls are modeled either as a smooth (10−4) Lennard-Jones wall or as a structured wall consisting of (12−6) Lennard-Jones particles. Polar functional groups on the pore surface are modeled by arrays of point charges. We present results on density and orientational distribution functions and on the water mobility. We observe that water transport through nonpolar pores is fast and dominated by the surface layer, whereas transport in polar pores is slowed down relative to bulk liquid water and occurs preferentially through the center of the pore.
To find a novel amphi-ionophore which strongly binds a cation or an anion alternatively in aqueous solution, we have investigated the host−guest complexation of a cyclohexapeptide composed of six alanine molecules with either alkali metal ions or halide anions, using ab initio calculations, molecular mechanics, and molecular dynamics simulations. The cyclohexapeptide is indeed found to be a novel amphi-ionophore in the aqueous phase as well as in the gas phase. In the presence of cations (Li+, Na+), the CO group tends to fold inward to capture a cation, whereas in the presence of anion, the N−H group tends to fold inward to capture an anion (F-, Cl-). In the aqueous phase, the ions tend to preferentially bind both cyclohexaalanyl and water molecules at the surface of cyclohexaalanyl.
Using the computer-aided molecular design approach, we recently reported the synthesis of calix[4]hydroquinone (CHQ) nanotube arrays self-assembled with infinitely long one-dimensional (1-D) short hydrogen bonds (H-bonds) and aromatic−aromatic interactions. Here, we assess various calculation methods employed for both the design of the CHQ nanotubes and the study of their assembly process. Our calculations include ab initio and density functional theories and first principles calculations using ultrasoft pseudopotential plane wave methods. The assembly phenomena predicted prior to the synthesis of the nanotubes and details of the refined structure and electronic properties obtained after the experimental characterization of the nanotube crystal are reported. For better characterization of intriguing 1-D short H-bonds and exemplary displaced π−π stacks, the X-ray structures have been further refined with samples grown in different solvent conditions. Since X-ray structures do not contain the positions of H atoms, it is necessary to analyze the system using quantum theoretical calculations. The competition between H-bonding and displaced π−π stacking in the assembling process has been clarified. The IR spectroscopic features and NMR chemical shifts of 1-D short H-bonds have been investigated both experimentally and theoretically. The dissection of the two most important interaction components leading to self-assembly processes would help design new functional materials and nanomaterials.
Advances in computer simulation techniques have made it possible to model complex macromolecular assemblies. It is now reasonable to believe that these methods will become useful tools for understanding the properties and behavior of transmembrane ion channels, the physiological functional units facilitating ion transport across cell membranes. This article is a progress report on modeling the behavior of the simplest such system - gramicidin A and its analogues. Theory can account for many of the qualitative features of alkali cation-channel interaction. Quantitative predictions still leave much to be desired, suggesting a need for more accurate treatment of the long-range interactions between ions and water and ions and membranes.
Using previously reported ab initio potentials of the intermolecular interaction energies of phospholipid (PL), Lysophosphatidyl Ethanolamine, with one Na+ ion and one water molecule, we performed Monte Carlo simulations for PL-water and PL-Na+-water systems. Water-water and PL-water interaction energetics of PL hydration sites are analyzed to understand, in a qualitative way, why the PL head part shows hydrophilicity and the tail part shows hydrophobicity. The interaction of Na+ with PL, as well as the interaction of water with PL, is visualized from the analysis of the hydration structures near PL, and the radial distribution functions are analyzed for selected hydration sites. The PL molecule shows much stronger interaction with Na+ than with water. The Na+ ion is likely to be strongly bound to PO, even to the extent of being trapped, whereas, for water, there exist two strong binding regions near NH and PO. Three water molecules near NH are much more strongly bound than four water molecules near the double-bonded oxygens of PO. The hydrogens of CH2 adjacent to NH show somewhat strong hydrophilicity, while the hydrogens of CH2 adjacent to PO does not show such characteristics. The CH2 groups at the PL tail part give repulsive interactions with water molecules, showing hydrophobicity. Water molecules near the PL tail are stabilized only by water-water interactions.
In this paper, we present Monte Carlo and molecular dynamics simulations of water molecules inside a ferrierite-type framework. Detailed analyses of the energetic, structural, and dynamical properties are carried out and compared with liquid water results in order to study the influence of the framework on the physisorbed water molecules.
The interface between a membrane, modelled by an ensemble of COO− functional groups with rotational and translational degrees of freedom, and liquid water, represented by the TIP4P model, is studied by molecular dynamics (MD) computer simulations. The structure of the bare membrane and of the membrane in the presence of water are studied by means of pair correlation functions. The influence of the membrane on the structure of the water is similarly discussed in terms of density profiles and distribution functions. Three ordered layers of water are found above the membrane, followed by a transition region to the bulk structure. Finally, the single molecule dynamics is examined. The translational self-diffusion of the water is lowered by one order of magnitude in the vicinity of the membrane and remains anisotropic even above the layered water.
Transport of ions through membrane by different chemical compounds is analyzed on the basis of conformational analysis and Monte Carlo calculations performed in the author's works. Ionophores, siderophores, and channels are considered. The complex formation of ionophores is accompanied by the formation of intermediate exclusive and solvent-shared states. The siderophores' hydration determines their activity. The changes in the channels' permeability are related to the changes in the pore diameter. © 1992 John Wiley & Sons, Inc.
Since theoretical investigations of gas phase clusters enable the evaluation of intrinsic molecular properties and intermolecular interactions, one can predict the macroscopic properties of bulk matter, from a microscopic determination of the properties of individual atoms, molecules, or clusters. Based on the insights obtained from theoretical investigations of the properties of a large number of cluster systems (ranging from simple water clusters to large π-systems), we have investigated the properties of various novel molecular systems including endo/exohedral fullerenes, nanotori, nonlinear optical materials, ionophores/receptors, polypeptides, enzymes, organic nanotubes, nanowires, and electronic and nano-mechanical molecular devices. The present mini-review highlights some of the interesting results obtained in the course of our extensive theoretical investigations of clusters and nanomaterials.
The mechanism of ion transport by carrier ionophores is investigated. The electrostatic potential is used as index of the
binding energy of a cation with valinomycin and enniatin B. The ion binding capacities of these ionophores are studied as
functions of conformation and of distance of an approaching ion-complex. The energetics of dirnerisation and the binding energy
profile of an ion in dimers of valinomycin and enniatin B are examined. The binding energy profiles and the electrostatic
potential surfaces of valinomycin and enniatin B are compared in relation to their biological activities.
A statistical method based on Monte Carlo sampling of paths associated with Markovianstochastic processes is proposed for the location of transition states in molecularly complex reactions. It is discussed in what respects the proposed method would identify transition state regions, and an algorithm for Metropolis sampling of Metropolis paths (based on the Metropolis random walk process) is devised. Physical aspects of this new algorithm are examined, and the relation of this method to simulated annealing algorithms for physical design of circuits is noted.
We discuss two experimental parallel computer systems 1CAP-1 and 1CAP-2 which can be applied to the entire spectrum of scientific and engineering applications. These systems achieve ‘supercomputer’ levels of performance by spreading large scale computations across multiple cooperating processors—several with vector capabilities. We outline system hardware and software, and discuss our programming strategy for migrating codes from a conventional sequential system to a parallel one. The performance of a variety of applications programs is analyzed to demonstrate the merits of this approach. Finally, we discuss 1CAP-3, an extension to this computing system, which has been recently assembled.
The hydration of gramicidin incorporated in lipid bilayers has been studied by FT-IR-ATR spectroscopy.The dependence of the stability of the system on the water environment of the polypeptide moiety and of the structural water of the lipid bilayer is discussed. The results indicate that partial drying of the gramicidin-lipid system first removes the water molecules bound to the peptide groups of the polypeptide moiety stabilizing its helical structure. Removal of water molecules bound tightly to the phosphate and choline head groups of the lipid produces a destabilization of the lipid bilayer, perturbing all the system. Our observations are compatible with a conformational transformation of gramicidin upon drying.
We discuss the vectorization of a classical mechanical trajectory code which simulated the interaction of a rotationally excited rigid rotator with a colliding atom. This code was optimized in FORTRAN 77 and can be run on either a VAX 11/780 minicomputer or a CRAY-1 vector pipeline supercomputer. The article presents a global strategy for vectorizing a classical mechanical trajectory code, a set of performance criteria for characterizing the vectorization of a computer code, and an analysis of the four major subroutines of the vectorized trajectory code in terms of this strategy and these criteria.
The study of southern bean mosaic virus protein coat high resolution model revealed a structure with properties of a natural protein-ion channel. Coat protein pentamers form a 30-A long channel and the amino acid composition of its wall bears some homology with the pentameric structure proposed for the nicotinic acetylcholine receptor channel. Ion transport properties were analyzed by computing ion-protein interaction energies on the basis of quantum chemistry methods. Energy maps show a channel attractive for cations, fully permeable to Li(+) and a narrow barrier for other cations and water. The energy profiles found are similar to the profiles determined for the K(+) channel of the sarcoplasmic reticulum. Comparisons with other icosahedral virus structures, including picornaviruses, suggest that ion channels would be a common feature of viral capsids. Biological roles for these channels are proposed.
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.
The structure and dynamics of solvated alkali metal cations in transmembrane channels are treated using the molecular dynamics simulation technique. The simulations are based on a modified Fischer-Brickmann model (Fischer, W., and J. Brickmann, 1983, Biophys. Chem., 18:323-337) for gramicidin A-type channels. The trajectories of all particles in the channel as well as two-dimensional pair correlation functions are analyzed. It is found from the analysis of the stationary simulation state that one-dimensional solvation complexes are formed and that the number of water molecules in the channel varies for different alkali metal cations.
Gramicidin A forms ion-conducting channels which can traverse the hydrocarbon core of lipid bilayer membranes. The structures formed by gramicidin A are among the best characterized of all membrane-bound polypeptides or proteins. In this review a brief summary is given of the occurrence, conformation, and synthesis of gramicidin A, and of its use as a model for ion transport and the interaction of proteins and lipids in biological membranes.
The helical polypeptide, gramicidin A has been widely studied as a model for the interactions of hydrophobic proteins with lipid bilayer membranes. Many reports are now available of the physical effects of mixing gramicidin A with phospholipid membranes, however, the interpretation of these data remains unclear. The purpose of this communication is to examine the controversial claim that high concentrations of gramicidin A′ cause disorder within the L
α phase of phosphatidylcholine-water dispersions. Solid-state nuclear magnetic resonance (NMR), density gradient and X-ray diffraction techniques are used to confirm the existence of such an effect and mechanisms are discussed which account for the known effects of gramicidin A on lipid bilayers.
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.
Extending previous work (Sung & Jordan, 1987 a, Biophys. J. 51, 661-672; 1988, Biophys. J.54, 519-526), we describe channel properties of five possible gramicidin dimers by studying dimerization energies and axial electrical potentials. Unlike the head-to-head dimer (the predominant channel former), both tail-to-tail and head-to-tail dimers with the same beta-helical monomer structure as the head-to-head dimer only form four intermonomer hydrogen bonds and are much less stable. Were channels formed from these dimers to be observed, their electrical potential profiles suggest that they should be cation selective, probably conduct less than the head-to-head dimer, have a central cation binding site, bind cations preferentially if crystallizable, and in the case of the head-to-tail dimer, rectify. Like the antiparallel double stranded helical dimer (a possible minor conducting pathway) the parallel double stranded helical dimer has 28 interstrand hydrogen bonds, but its hydrogen bond network is quite distorted and it is much less stable. If it formed, its electrical potential profile suggests that it would be cation selective, bind anions preferentially if crystallizable, rectify, and at high enough voltages, might exhibit a conductance greater than that of the antiparallel form.
Numerical solutions are presented to the equation of motion for an ion confined to a region of space by a restoring force and subject to DC and AC magnetic fields. We have expanded on the theoretical work of Durney et al. [1988] by including a potential well as a confining factor. This additional term in the equation of motion, being nondissipative, could allow for the buildup of stored energy within the system to a level necessary for a macroscopic resonant phenomenon. Resonant behaviour has been studied, including calculation of the trajectory and energy (kinetic and potential) of a confined ion, with emphases on the appearance of both amplitude and frequency windows. The results are discussed in relation to ion transport through transmembrane channels exposed to magnetic fields. When realistic values of the frictional and restoring-force coefficients are considered, all predicted resonant behaviour disappears, except at very high field strengths.
Rabbits with a fibular ostectomy were exposed for 28 days to magnetic fields that satisfied the ion resonance conditions for calcium or magnesium. The rabbits were exposed to whole body treatment for 1/2 hour, 3 hours, or 24 hours per day. The fibulae from the experimental and control animals were removed surgically and were subjected to force-deflection testing to establish the stiffness of the healed fracture. The fibulae from the rabbits exposed to the ion resonance magnetic fields were found to be 55-299% (p < 0.01) more robust than the fibulae from the control animals.
A comprehensive state-of-the-art ab initio study is performed on the wet electron-an electron interacting with a small cluster of water molecules-in the water hexamer system. Predictions include two previously unknown distinctive geometries which bind the excess electron as internal and external states, photoemission ionization energies in agreement with experiment, identification of generic electrophilic sites involving dangling hydrogen atoms, and the tendency of all hydrogen atoms to be saturated in hydrogen bonding or in interaction with the excess electron. An emerging insight is the capability of electrophilic sites to be actuators of electron transport pathways in biomolecular systems.
The capture of chloride from water by the tetraprotonated form of the spherical macrotricyclic molecule SC24 was studied using molecular dynamics simulation methods. This model ionophore represents a broad class of molecules which remove ions from water. Two binding sites for the chloride were found, one inside and one outside the ligand. These sites are separated by a potential energy barrier of approximately 20 kcal mol-1. The major contribution to this barrier comes from dehydration of the chloride. The large, unfavorable dehydration effect is compensated for by an increase in electrostatic attraction between the oppositely charged chloride and cryptand, and by energetically favorable rearrangements of water structure. Additional assistance in crossing the barrier and completing the dehydration of the ion is provided by the shift of three positively charged hydrogen atoms of the cryptand towards the chloride. This structural rigidity is partially responsible for its selectivity.
On the basis of recently synthesized calix[4]hydroquinone (CHQ) nanotubes which were self-assembled with infinitely long one-dimensional (1-D) short hydrogen bonds (SHB), we have investigated the nature of 1-D SHB using first-principles calculations for all the systems including the solvent water. The H-bonds relay (i.e., contiguous H-bonds) effect in CHQs shortens the H...O bond distances significantly (by more than 0.2 A) and increases the bond dissociation energy to a large extent (by more than approximately 4 kcal/mol) due to the highly enhanced polarization effect along the H-bond relay chain. The H-bonds relay effect shows a large increase in the chemical shift associated with the SHB. The average binding energies for the infinite 1-D H-bond arrays of dioles and dions increase by approximately 4 and approximately 9 kcal/mol per H-bond, respectively. The solvent effect (due to nonbridging water molecules) has been studied by explicitly adding water molecules in the CHQ tube crystals. This effect is found to be small with slight weakening of the SHB strength; the H...O bond distance increases only by 0.02 A, and the average binding energy decreases by approximately 1 kcal/mol per H-bond. All these results based on the first-principles calculations are the first detailed analysis of energy gain by SHB and energy loss by solvent effect, based on a partitioning scheme of the interaction energy components. These reliable results elucidate not only the self-assembly phenomena based on the H-bond relay but also the solvent effect on the SHB strength.
Upon excitation of Cl(-)(H(2)O)(3) and I(-)(H(2)O)(3) clusters, the electron transfers from the anionic precursor to the solvent, and then the excess electron is stabilized by polar solvent molecules. This process has been investigated using ab initio molecular dynamics (AIMD) simulations of excited states of Cl(-)(H(2)O)(3) and I(-)(H(2)O)(3) clusters. The AIMD simulation results of Cl(-)(H(2)O)(3) and I(-)(H(2)O)(3) are compared, and they are found to be similar. Because the role of the halogen atom in the photoexcitation mechanism is controversial, we also carried out AIMD simulations for the ground-state bare excess electron -- water trimer [e(-)(H(2)O)(3)] at 300 K, the results of which are similar to those for the excited state of X(-)(H(2)O)(3) with zero kinetic energy at the initial excitation. This indicates that the rearrangement of the complex is closely related to that of e(-)(H(2)O)(3), whereas the role of the halide anion is not as important.
In contrast to the extensive theoretical investigation of the solvation phenomena, the dissolution phenomena have hardly been investigated theoretically. Upon the excitation of hydrated halides, which are important substances in atmospheric chemistry, an excess electron transfers from the anionic precursor (halide anion) to the solvent and is stabilized by the water cluster. This results in the dissociation of hydrated halides into halide radicals and electron-water clusters. Here we demonstrate the charge-transfer-to-solvent (CTTS)-driven femtosecond-scale dissolution dynamics for I-(H2O)n=2-5 clusters using excited state (ES) ab initio molecular dynamics (AIMD) simulations employing the complete-active-space self-consistent-field (CASSCF) method. This study shows that after the iodine radical is released from I-(H2O)n=2-5, a simple population decay is observed for small clusters (2 </= n </= 4), while rearrangement to stabilize the excess electron to an entropy-driven structure is seen for n = 5. These results are in excellent agreement with the previous ultrafast pump-probe experiments. For the first approximately 30 fs of the simulations, the iodine plays an important role in rearranging the hydrogen orientations (although the water network hardly changes), which increases the kinetic energy of the cluster. However, approximately 50 fs after the excitation, the role of the iodine radical is no longer significant. After approximately 100 fs, the iodine radical is released, and the solvent molecules rearrange themselves to a lower free energy structure. The CTTS-driven dissolution dynamics could be useful in designing the receptors which are able to bind and release ions in host-guest chemistry.
Using a transformation of the rigid body equations of motion due to
Evans [4], a new algorithm is presented for the molecular dynamics
simulation of rigid polyatomic molecules. The algorithm consists of
solving the eulerian rigid body equations, using quaternions to
represent orientations, by a fifth-order predictor corrector method.
Compared to previous methods, it is shown that this algorithm leads to
an order of magnitude increase in computing speed.
The conformation of species 3 of Val-gramicidin A in dioxane has been determined by two-dimensional NMR spectroscopy. It is presented by the left handed ⇅ππ5.6LD double helix, a suitable model of an ion permeable pore across the membrane matrix.
The potential energies for the water dimer in various geometrical configurations have been calculated with a configuration-interaction method. The computed dimerization binding energies corresponding to the potential minima for the linear, cyclic, and bifurcated configurations are -5.6, -4.9, and -4.2 kcal/mol, respectively; the correlation effects account for -1.1, -1.2, and -0.9 kcal/mol, respectively, of the total binding energy for these three dimeric forms. The correlation effects for the entire potential surface have been analyzed in terms of inter- and intramolecular effects; the substantial coupling found between these effects, particularly in the vicinity of equilibrium position, is discussed. The computational technique employed, in particular an analysis on the selection criteria for the configuration state functions, is discussed, and its reliability is assessed. Two analytical expressions for the water dimer potential surface obtained by fitting the calculated energies are presented. The potential surface given here is being used to determine the structure of liquid water (in the pairwise approximation and with Monte Carlo techniques); this latter work will be reported elsewhere.
The energy profile for Na+ in the channel formed by the gramicidin A β-helical dimer backbone was computed introducing all the terms in the theory of intermolecular interactions. The effect of allowing the ion to reach its successive optimal positions shows the presence of a series of energy minima associated with different carbonyls. The presence of a second ion lowers the central barrier for the first one and facilitates its progression and exit. The energy profile for double occupancy indicates the presence of two symmetrical minima at about 13 Å from the center.
The inclusion of the presence and flexibility of the CH2CH2OH end chain in the computation of the energy profile for single occupancy by Na+ of the gramicidin A channel modifies substantially the profile obtained without that chain. The binding site (deepest minimum) in the profile is situated at 10.5 Å from the center of the channel, in satisfactory agreement with the conclusions based on 13C-NMR studies. The existence of an external minimum at the mouth is confirmed.
In order to obtain the heat of formation ΔH, for the ion‐water complexes previously studied in the Hartree‐Fock approximation in the first three papers of this series, we have computed the normal frequencies of the complexes, the zero‐point energy correction to ΔH, and the molecular extra correlation energy. The main contribution to ΔH is due to the Hartree‐Fock binding; the least important contribution results from the correlation effects. The Hartree‐Fock binding varies from about 35 kcal∕mole (Li+☒H2O) to about 12 kcal∕mole (Cl−☒H2O); the zero‐point correction is between 1 and 2 kcal∕mole; and the molecular extra correlation correction is less than 1 kcal∕mole. The computation of ΔH is analyzed in order to estimate upper and lower bounds. We conclude that the calculated ΔH values are accurate to about 2.0 kcal∕mole. Experimental data support this conclusion. In the Appendix, the potentials for water‐ion complexes have been presented in the form of a simple analytical expansion. The expansion has been obtained by fitting the Hartree‐Fock computed energies for the water‐ion complexes.
The conductance induced by gramicidin A in lipid bilayer membranes has been shown to be made up of discrete, well-defined units. In 0.1 M NaCl, and for 100 mV applied, the integral conductance of the unit channel at 20 °C is 5.8·10−12 Ω−1.The channels are formed by transitions involving inactive gramicidin molecules already in the membrane. The precise nature of a transition is not certain, but circumstantial evidence suggests that the final conducting structure consists of at least two polypeptide molecules. From the temperature coefficient of the average duration of the channels it has been shown that the activation energy for channel closure must be ≳ 19 kcal mole−1. The frequency of occurrence and the average duration of the channels both become larger the thinner the membrane. The equilibrium between non-conducting and conducting species therefore shifts towards the conducting species as the membrane thickness decreases.With one exception, the same unit channel conductance was observed for a range of membranes having hydrocarbon thicknesses from 26 to 64 Å and, from this and other evidence, it has been concluded that the conducting channel is a pore, rather than a carrier. The length of the pore has been estimated to be less than 35 Å.The pore passes univalent cations but completely excludes polyvalent cations and anions. The selectivity between the univalent cations is not great, and the sequence of the various ion conductances is similar to that for the corresponding electrolytes in aqueous solution. The activation energies for the conduction of the ions are also similar to those in aqueous solution.The current-voltage relationships for the single channel tend to be curved towards the current axis at high electrolyte concentrations, linear at intermediate concentrations and curved towards the voltage axis at low concentrations. For each of the electrolytes studied the conductance of the single channel tends towards a limiting value at high concentrations.It is noted that one of the dimeric helical structures (that which contains the helix) proposed by Urry et al.22 could be consistent with some of the properties of the single channel.
The temperature dependence of the mean life-time and of the conductance Λ of single ion channels induced by gramicidin A in lipid bilayer membranes has been measured. In a second series of experiments, the rate constants of formation (kR) and dissociation (kD) of the conducting channel have been determined at different temperatures from electrical relaxation experiments. It has been found that the channel formation proceeds according to a second-order reaction in the whole temperature range (10–40 °C). Furthermore, the mean life-time of the channel approximately agreed with the reciprocal of the dissociation rate constant. Both findings are consistent with the view that the channel consists of a dimer of gramicidin A. From the temperature dependence of Λ, kR, and kD the activation energies of ion migration through the channel (EΛ) as well as the activation energy of formation (ER) and dissociation (ED) of the dimer may be calculated. The magnitude of ED (17 kcal/mole) is consistent with the assumption that dissociation involves the breakage of several hydrogen bonds. The value of ER (20 kcal/mole) is tentatively explained by the energy required for the structural rearrangement of the lipid matrix in the vicinity of the channel.
Malonyl gramicidin is incorporated into lysolecithin micelles in a manner which satisfies a number of previously demonstrated criteria for the formation of the transmembrane channel structure. By means of sodium-23 nuclear magnetic resonance, two binding sites are observed: a tight site and a weak site with binding constants of approximately 100 M-1 and 1 M-1 respectively. In addition, off-rate constants from the two sites were estimated from NMR analyses to be kofft congruent to 3 X 10(5)/sec and koffw congruent to 2 X 10(7)/sec giving, with the binding constants, the on-rate constants, kont congruent to 3 X 10(7)/Msec and konw congruent to 2 X 10(7)/Msec. Five different multiple occupancy models with NMR-restricted energy profiles were considered for the purpose of calculating single-channel currents as a function of voltage and concentration utilizing the four NMR-derived rate constants (and an NMR-limit placed on a fifth rate constant for intrachannel ion translocation) in combination with Eyring rate theory for the introduction of voltage dependence. Using the X-ray diffraction results of Koeppe et al. (1979) for limiting the positions of the tight sites, the two-site model and a three-site model in which the weak sites occur after the tight site is filled were found to satisfactorily calculate the experimental currents (also reported here) and to fit the experimental currents extraordinarily well when the experimentally derived values were allowed to vary to a least squares best fit. Surprisingly the "best fit" values differed by only about a factor of two from the NMR-derived values, a variation that is well within the estimated experimental error of the rate constants. These results demonstrate the utility of ion nuclear magnetic resonance to determine rate constants relevant to transport through the gramicidin channel and of the Eyring rate theory to introduce voltage dependence.
Ion-induced chemical shifts in the carbonyl carbon resonances of synthesized ad verified (1-13C)D-Val8 gramicidin A and (1-13C)D-Leu14 gramicidin A are utilized in combination with the previously determined location of the ion binding sites of the gramicidin A channel (using the carbonyls of L-residues) to determine that the helix sense of the gramicidin A channel) is left-handed. Having resolved the handedness issue, the location of the ion binding sites (which are fundamental to understanding the mechanism of ion transport) are further delineated with the results indicating two sites separated by just over 20 A. Furthermore, the demonstration that the divalent barium ion interacts at the binding site while not being transported through the channel is used to argue that the mechanism of monovalent vs. divalent cation selectivity is due to the positive image force contribution to the central barrier.
We treat the transport of univalent cations through pore-like protein channels in biological membranes analytically, using two models (A + B) for the channel and the ion-channel interaction. A Lennard-Jones-type repulsion between the ions and the pore wall is introduced. We also include Van der Waals- and coulomb-type interactions between polar ligands of the pore-forming protein (e.g., carbonyl groups directed towards the axis of the channel) and the migrating particles. In model A, the polar groups are assumed to occur in pairs of dipoles pointing in opposite directions (as in the gramicidin A channel), while in model B the channel is treated as a pore with a radially isotropic charge distribution. In both models the ion-channel interaction leads to the occurrence of periodic potentials, corresponding to quasi-equilibrium and transition state sites of the ion in the pore. The diffusion rate can be calculated employing rate-theoretical concepts on the basis of microscopic parameters. It is demonstrated that the anomaly (inversion of the normal mass effect) for the transport rates of different ions can be related to differences in the activation entropy. The latter quantity is estimated analytically for both models. As a test, we performed numerical calculations with parameters based on the gramicidin A model. The results are in good agreement with experimental data and data from computer simulations. This shows that simple analytic expressions are well suited for predicting trends in the ionic conductivity of protein channels on the basis of microscopic interactions.
The energy profiles for single occupancy by Cs+, K+ and Na+ in the gramicidin A channel assumed to be in a head-to-head beta 6.3 3.3 helical dimeric structure, were computed: (A) allowing complete conformational freedom to the ethanolamine end, (B) constraining it to stay in its intrinsically preferred conformation. Whatever the constraint, both the entrance barrier and the central barrier appear in the order Cs+ less than K+ less than Na+. Introducing the flexibility of the tail modifies appreciably the profiles and the location of the extrema along it.
In rate-theory analysis of ion transport in channels, the energy of binding sites and the height of activation barriers are usually considered to be time-independent and not influenced by the movement of the ion. The assumption of a fixed barrier structure seems questionable, however, in view of the fact that proteins may exist in a large number of conformational states and may rapidly move from one state to the other. In this study, some of the effects of multiple conformational states of a channel on ion transport are analyzed. In the first part of the paper, the ion permeability of a channel with n binding sites is treated on the assumption that interconversion of channel states is much faster than ion transfer between binding sites. Under this condition, the form of the flux equation remains the same as for a channel with fixed barriers, provided that the rate constants for ion jumps are replaced by weighted averages over the rate constants for the individual conformational states. In the second part, a channel with two (main) barriers and a single (main) binding site is considered, with the rates of conformational transitions being arbitrary. This case, in particular, includes the situation where a jump of the ion is followed by a slow transition to a more polarized state of the binding site. Under this condition, the conductance of the channel exhibits a nonlinear dependence on ion concentration which is different from a simple saturation behavior. Under non-stationary conditions damped oscillations may occur.
Ion transport through biological membranes often takes place via pore-like protein channels. The elementary process of this transport can be described as a motion of the ion in a quasi-periodic multi-well potential. In this study molecular dynamics simulations of ion transport in a model channel were performed in order to test the validity of reaction-rate theory for this process. The channel is modelled as a hexagonal helix of infinite length, and the ligand groups interacting with the ion are represented by dipoles lining the central hole of the channel. The dipoles interact electrostatically with each other and are allowed to oscillate around an equilibrium orientation. The coupled equations or motion for the ion and the dipoles were solved simultaneously with the aid of a numerical integration procedure. From the calculated ion trajectories it is seen that. particularly at low temperatures, the ion oscillates back and forth in the trapping site many times before it leaves the site and jumps over the barrier. The observed oscillation frequency was found to be virtually temperature-independent (vo ~2 × 1012 s−1) so that the strong increase of transport rate with temperature results almost exclusively from the Arrhenius-type exponential dependence of jump probability w on 1/T. At higher temperatures simultaneous jumps over several barriers occasionally occur. Although the exponential form of w(T) was in agreement with the predictions of rate theory, the activation energy Ea as determined from w(T) was different from the barrier height which was calculated from the static potential of the ion in the channel: the actual transport rate was 1 × 103 times higher than the rate predicted from the calculated barrier height. This observation was interpreted by the notion that ion transport in the channel is strongly influenced by thermal fluctuations in the conformation of the ligand system which in turn give rise to fluctuations of barrier height.