Article

Na+ and K+ ion transport through a solvated gramicidin A transmembrane channel: Molecular dynamics studies using parallel processors

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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.

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... 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. ...
Article
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.
... 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. ...
Article
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.
... 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. ...
Article
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. ...
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Chapter
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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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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.