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

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

Electrostatic tuning of permeation and selectivity in aquaporin water channels

Water permeation and electrostatic interactions between water and channel are investigated in the Escherichia coli glycerol uptake facilitator GlpF, a member of the aquaporin water channel family, by molecular dynamics simulations. A tetrameric model of the channel embedded in a 16:0/18:1c9-palmitoyloleylphosphatidylethanolamine membrane was used for the simulations. During the simulations, water molecules pass through the channel in single file. The movement of the single file water molecules through the channel is concerted, and we show that it can be described by a continuous-time random-walk model. The integrity of the single file remains intact during the permeation, indicating that a disrupted water chain is unlikely to be the mechanism of proton exclusion in aquaporins. Specific hydrogen bonds between permeating water and protein at the channel center (at two conserved Asp-Pro-Ala "NPA" motifs), together with the protein electrostatic fields enforce a bipolar water configuration inside the channel with dipole inversion at the NPA motifs. At the NPA motifs water-protein electrostatic interactions facilitate this inversion. Furthermore, water-water electrostatic interactions are in all regions inside the channel stronger than water-protein interactions, except near a conserved, positively charged Arg residue. We find that variations of the protein electrostatic field through the channel, owing to preserved structural features, completely explain the bipolar orientation of water. This orientation persists despite water translocation in single file and blocks proton transport. Furthermore, we find that for permeation of a cation, ion-protein electrostatic interactions are more unfavorable at the conserved NPA motifs than at the conserved Arg, suggesting that the major barrier against proton transport in aquaporins is faced at the NPA motifs.     

Transportation behavior of alkali ions through a cell membrane ion channel. A quantum chemical description of a simplified isolated model

Quantum chemical model calculations were carried out for modeling the ion transport through an isolated ion channel of a cell membrane. An isolated part of a natural ion channel was modeled. The model channel was a calixarene derivative, hydrated sodium and potassium ions were the models of the transported ion. The electrostatic potential of the channel and the energy of the channel-ion system were calculated as a function of the alkali ion position. Both attractive and repulsive ion-channel interactions were found. The calculations - namely the dependence of the system energy and the atomic charges of the water molecules with respect to the position of the alkali ion in the channel - revealed the molecular-structural background of the potassium selectivity of this artificial ion channel. It was concluded that the studied ion channel mimics real biological ion channel quite well.     

Effects of Multiple Occupancy and Interparticle Interactions on Selective Transport through Narrow Channels: Theory versus Experiment

Many biological and artificial transport channels function without direct input of metabolic energy during a transport event and without structural rearrangements involving transitions from a closed to an open state. Nevertheless, such channels are able to maintain efficient and selective transport. It has been proposed that attractive interactions between the transported molecules and the channel can increase the transport efficiency and that the selectivity of such channels can be based on the strength of the interaction of the specifically transported molecules with the channel. Herein, we study the transport through narrow channels in a framework of a general kinetic theory, which naturally incorporates multiparticle occupancy of the channel and non-single-file transport. We study how the transport efficiency and the probability of translocation through the channel are affected by interparticle interactions in the confined space inside the channel, and establish conditions for selective transport. We compare the predictions of the model with the available experimental data and find good semiquantitative agreement. Finally, we discuss applications of the theory to the design of artificial nanomolecular sieves.     

The gramicidin A channel: energetics and structural characteristics of the progression of a sodium ion in the presence of water

The distribution of water molecules in the Gramicidin A (GA) channel is determined by theoretical computations, and the role of this water on the energetics of the system upon progression of a sodium cation through the channel is investigated. In the absence of the ion, water molecules form a chain along the channel, hydrogen bonded to one another and to the L carbonyl oxygens, while others stay at the entrances of the channel, hydrogen-bonded to the free carbonyl oxygens of the L-Tryptophan residues. According to the definition adopted for the "inside" and the "outside" of the channel, it is found to contain at most 7 or 9 water molecules. When a hydrated sodium cation approaches and enters the channel, the structural properties corresponding to the minimized total energy of the system GA-water-Na+ indicate a reorganization, but not a destruction, of the chain of water molecules. The "energy profile" for the system GA-Na+-(22 waters) is analyzed in terms of its components and in comparison to the corresponding intrinsic profile computed earlier in vacuo. It appears that the presence of water does not unduely modify the pathway or the qualitative features of the energetics of the cation passage, except at the entrance, where the partial and progressive dehydration of the cation plays an important role. The presence and characteristics of the minimum found earlier at 10.5 A from the center are conserved.     

A microscopic electrostatic model for the amphotericin B channel.

A microscopic model of an amphotericin B channel is proposed. The structure of the pores is generated using the atomic coordinates of the molecule in the structure determined experimentally by X-ray diffraction. The net charges of the atoms are determined by Mulliken analysis. With these charges the electrostatic energy profiles are calculated for a monovalent ion passing through the channels formed by different number of antibiotic molecules having different radii. The water inside the channel was considered through a continuum medium using the dielectric constant of the bulk, and the membrane contribution was included using the virtual images of the pore in a dielectric slab of epsilon = 3. The model satisfactorily explains the permeability and selectivity characteristics as well as other observations yet unexplained. The electrostatic profiles obtained reinforce the hypothesis of the existence of channels formed by a variable number of units.     

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

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

Computer simulation of water in cytochrome c oxidase

Statistical mechanics and molecular dynamics simulations have been carried out to study the distribution and dynamics of internal water molecules in bovine heart cytochrome c oxidase (CcO). CcO is found to be capable of holding plenty of water, which in subunit I alone amounts to about 165 molecules. The dynamic characterization of these water molecules is carried out. The nascent water molecules produced in the redox reaction at the heme a(3)-CuB binuclear site form an intriguing chain structure. The chain begins at the position of Glu242 at the end of the D channel, and has a fork structure, one branch of which leads to the binuclear center, and the other to the propionate d of heme a(3). The branch that leads to the binuclear center has dynamic access both to the site where the formation of water occurs, and to delta-nitrogen of His291. From the binuclear center, the chain continues to run into the K channel. The stability of this hydrogen bond network is examined dynamically. The catalytic site is located at the hydrophobic region, and the nascent water molecules are produced at the top of the energy hill. The energy gradient is utilized as the mechanism of water removal from the protein. The water exit channels are explored using high-temperature dynamics simulations. Two putative channels for water exit from the catalytic site have been identified. One is leading directly toward Mg(2+) site. However, this channel is only open when His291 is dissociated from CuB. If His291 is bound to CuB, the only channel for water exit is the one that originates at E242 and leads toward the middle of the membrane. This is the same channel that is presumably used for oxygen supply.     

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

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

Exploring the gating mechanism in the ClC chloride channel via metadynamics

Computer simulations have been used to probe the gating mechanism in the Salmonella serovar typhimurium chloride channel (st-ClC). Specifically, the recently developed metadynamics methodology has been exploited to construct free energy surfaces as a function of the positions of either one or two chloride ions inside the pore, the position and protonation state of the key E148 residue, and the number of water molecules coordinating the translocating ions. The present calculations confirm the multi-ion mechanism in which an ion-push-ion effect lowers the main barriers to chloride ion translocation. When a second anion is taken into account, the barrier for chloride passage through the E148 narrow region is computed to be 6 kcal/mol in the wild-type channel, irrespective of the protonation state of the E148 residue, which is shown to only affect the entrance barrier. In the E148A mutant, this barrier is much lower, amounting to 3 kcal/mol. The metadynamics calculations reported herein also demonstrate that before reaching the periplasmic solution, chloride ions have to overcome an additional barrier arising from two different effects, namely the rearrangement of their solvation shell and a flip in the backbone angles of the residues E148 and G149, which reside at the end of the alphaF helix.     

Model of a complex between the tetrahemic cytochrome c3 and the ferredoxin I from Desulfovibrio desulfuricans (Norway strain)

A three-dimensional model of an electron-transfer complex between the tetrahemic cytochrome c3 and the ferredoxin I from the sulfate-reducing bacterium Desulfovibrio desulfuricans (Norway strain) has been generated through computer graphics methods. The model is based on the known X-ray structure of the cytochrome and on a model of the ferredoxin that has been derived through computer graphics modeling and energy minimization methods, from the X-ray structure of the homologous ferredoxin from Peptococcus aerogenes. Four possible models of interaction between the two molecules were examined by bringing in close proximity each of the four hemes and the redox center (4Fe-4S) of the ferredoxin and by optimizing the ion pairs interactions. One of these models shows by far the "best" structure in terms of charges, interactions, and complementarity of the topology of the contact surfaces. In this complex, the distance between the iron atoms of the ferredoxin redox center and the hemic iron atom is 11.8 A, which compares well with those found between redox centers in other complexes. The contact surface area between the two molecules is 170 A2.     

Effective pore radius of the gramicidin channel. Electrostatic energies of ions calculated by a three-dielectric model.

Electrostatic calculation of the gramicidin channel is performed on the basis of a three-dielectric model in which the peptide backbone of the channel is added as a third dielectric region to the conventional two-dielectric channel model (whose pore radius is often referred to as the effective pore radius reff). A basic principle for calculating electrostatic fields in three-dielectric models is introduced. It is shown that the gramicidin channel has no unique value of reff. The reff with respect to the "self-image energy" (i.e., the image energy in the presence of a single ion) is 2.6-2.7 A, slightly depending upon the position of the ion (the least-square value over the whole length of the pore is 2.6 A). In contrast, the reff with respect to the electric potential due to an ion (and hence the reff with respect to the interaction energy between two ions) is dependent upon the distance s of separation; it ranges from 2.6 to greater than 5 A, increasing with an increase in s. However, for the purpose of rough estimation, the reff with respect to the self-image energy can also be used in calculating the electric potential and the interaction energy, because the error introduced by this approximation is an overestimation of the order of 30% at most. It is also shown that the apparent dielectric constant for the interaction between two charges depends markedly upon the positions of the charges. In the course of this study, the dielectric constant and polarizability of the peptide backbone in the beta-sheet structure is estimated to be 10 and 8.22 A3.     

Proton permeation of lipid bilayers

Proton permeation of the lipid bilayer barrier has two unique features. First, permeability coefficients measured at neutral pH ranges are six to seven orders of magnitude greater than expected from knowledge of other monovalent cations. Second, proton conductance across planar lipid bilayers varies at most by a factor of 10 when pH is varied from near 1 to near 11. Two mechanisms have been proposed to account for this anomalous behavior: proton conductance related to contaminants of lipid bilayers, and proton translocation along transient hydrogen-bonded chains (tHBC) of associated water molecules in the membrane. The weight of evidence suggests that trace contaminants may contribute to proton conductance across planar lipid membranes at certain pH ranges, but cannot account for the anomalous proton flux in liposome systems.Two new results will be reported here which were designed to test the tHBC model. These include measurements of relative proton/potassium permeability in the gramicidin channel, and plots of proton flux against the magnitude of pH gradients. (1) The relative permeabilities of protons and potassium through the gramicidin channel, which contains a single strand of hydrogenbonded water molecules, were found to differ by at least four orders of magnitude when measured at neutral pH ranges. This result demonstrates that a hydrogen-bonded chain of water molecules can provide substantial discrimination between protons and other cations. It was also possible to calculate that if approximately 7% of bilayer water was present in a transient configuration similar to that of the gramicidin channel, it could account for the measured proton flux. (2) The plot of proton conductance against pH gradient across liposome membranes was superlinear, a result that is consistent with one of three alternative tHBC models for proton conductance described by Nagle elsewhere in this volume.     

Ion permeation in normal and batrachotoxin-modified Na+ channels in the squid giant axon

Na+ permeation through normal and batrachotoxin (BTX)-modified squid axon Na+ channels was characterized. Unmodified and toxin-modified Na+ channels were studied simultaneously in outside-out membrane patches using the cut-open axon technique. Current-voltage relations for both normal and BTX-modified channels were measured over a wide range of Na+ concentrations and voltages. Channel conductance as a function of Na+ concentration curves showed that within the range 0.015-1 M Na+ the normal channel conductance is 1.7-2.5-fold larger than the BTX-modified conductance. These relations cannot be fitted by a simple Langmuir isotherm. Channel conductance at low concentrations was larger than expected from a Michaelis-Menten behavior. The deviations from the simple case were accounted for by fixed negative charges located in the vicinity of the channel entrances. Fixed negative charges near the pore mouths would have the effect of increasing the local Na+ concentration. The results are discussed in terms of energy profiles with three barriers and two sites, taking into consideration the effect of the fixed negative charges. Either single- or multi-ion pore models can account for all the permeation data obtained in both symmetric and asymmetric conditions. In a temperature range of 5-15 degrees C, the estimated Q10 for the conductance of the BTX-modified Na+ channel was 1.53. BTX appears not to change the Na+ channel ion selectively (for the conditions used) or the surface charge located near the channel entrances.     

Solvent effects in ionic transport through transmembrane protein channels

Ion transport through a gramicidin A like channel in the presence of solvent molecules with van der Waals parameters of water has been studied by means of the molecular dynamics simulation technique. It was found that the presence of solvent molecules in the channel has a tendency to equalize the effective masses of the ions through "association" thus giving the experimentally found ion selectivity to the gramicidin A channel.     

Permeation of ions across the potassium channel: brownian dynamics studies

The physical mechanisms underlying the transport of ions across a model potassium channel are described. The shape of the model channel corresponds closely to that deduced from crystallography. From electrostatic calculations, we show that an ion permeating the channel, in the absence of any residual charges, encounters an insurmountable energy barrier arising from induced surface charges. Carbonyl groups along the selectivity filter, helix dipoles near the oval chamber, and mouth dipoles near the channel entrances together transform the energy barrier into a deep energy well. Two ions are attracted to this well, and their presence in the channel permits ions to diffuse across it under the influence of an electric field. Using Brownian dynamics simulations, we determine the magnitude of currents flowing across the channel under various conditions. The conductance increases with increasing dipole strength and reaches its maximum rapidly; a further increase in dipole strength causes a steady decrease in the channel conductance. The current also decreases systematically when the effective dielectric constant of the channel is lowered. The conductance with the optimal choice of dipoles reproduces the experimental value when the dielectric constant of the channel is assumed to be 60. The current-voltage relationship obtained with symmetrical solutions is linear when the applied potential is less than approximately 100 mV but deviates from Ohm's law at a higher applied potential. The reversal potentials obtained with asymmetrical solutions are in agreement with those predicted by the Nernst equation. The conductance exhibits the saturation property observed experimentally. We discuss the implications of these findings for the transport of ions across the potassium channels and membrane channels in general.     

Thermodynamic stability of water molecules in the bacteriorhodopsin proton channel: a molecular dynamics free energy perturbation study.

The proton transfer activity of the light-driven proton pump, bacteriorhodopsin (bR) in the photochemical cycle might imply internal water molecules. The free energy of inserting water molecules in specific sites along the bR transmembrane channel has been calculated using molecular dynamics simulations based on a microscopic model. The existence of internal hydration is related to the free energy change on transfer of a water molecule from bulk solvent into a specific binding site. Thermodynamic integration and perturbation methods were used to calculate free energies of hydration for each hydrated model from molecular dynamics simulations of the creation of water molecules into specific protein-binding sites. A rigorous statistical mechanical formulation allowing the calculation of the free energy of transfer of water molecules from the bulk to a protein cavity is used to estimate the probabilities of occupancy in the putative bR proton channel. The channel contains a region lined primarily by nonpolar side-chains. Nevertheless, the results indicate that the transfer of four water molecules from bulk water to this apparently hydrophobic region is thermodynamically permitted. The column forms a continuous hydrogen-bonded chain over 12 A between a proton donor, Asp 96, and the retinal Schiff base acceptor. The presence of two water molecules in direct hydrogen-bonding association with the Schiff base is found to be strongly favorable thermodynamically. The implications of these results for the mechanism of proton transfer in bR are discussed.     

Simulation of Chain-length Partitioning in a Microfabricated Channel via Entropic Trapping

A molecular simulation approach has been used to investigate the mechanism of entropic trapping for model linear DNA molecules as they go from a deep channel to a shallow channel driven by an electric field. In such a system, a molecule whose radius of gyration is larger than the gap of the shallow channel will tend to get temporarily trapped at its entrance. The free energy of the molecules as a function of chain position and the trapping times were obtained via Monte Carlo simulations. In weak to moderate electrical fields, the free energy barrier for escape ( F max increases with chain length approaching a plateau value; in a strong electrical field, ( F max exhibits a mild decreasing trend with chain length. At weak electric fields, shorter chains escape faster than longer chains because of their lower associated ( F max . At moderate and strong fields, longer chains escape faster than shorter ones because, in the absence of significant differences in ( F max , larger chains access a larger entrance area to the narrow channel; these results are in agreement with reported experimental observations. Preliminary results on the effect of chain branching on the escape rate are also presented.     

Structure of hydrated Na+ ions around a region of A- or B-DNA helix

A molecular-dynamics simulation was used to carry out an introductory study of the hydration of a section of a rigid single A- or B-DNA helix with one Na⁺ counterion per nucleotide. Four Na⁺ ions and four nucleotides and periodic boundary conditions were used to mimic an infinite helix. The atoms of the helix and the Na⁺ ions were assumed to be Lennard-Jones spheres that also carried charges. Stillinger four-point charge model water molecules were used. We carried out five calculations, for 26 and 46 water molecules in B-DNA and 20, 32, and 46 in A-DNA fragments. The arrangements of the Na⁺ ions are found to have some similarities to those obtained by Clementi and Corongiu. In the calculations with 46 water molecules, we found that two Na⁺ ions can be bridged by about two water molecules and form a hydrated bound pair, which in turn forms a bridge between the guanine N7 and a near phosphate group. These bound pairs may be important in stabilizing the helix structure of DNA molecules.     

Simple combined explicit/implicit water model

A simple combined water model (SCW model) for the calculation of the hydration free energy is presented. In the frame of the model a solute is placed in the centre of the spherical cavity with explicit water molecules, which are considered at the atomistic level. Rigid wall potential at the boundary of the cavity restricts the moving of the explicit water molecules. Water outside the sphere is considered as the conducting continuum (implicit part of the model). Simulation is performed in the frame of the NVT ensemble (constant number of particles, volume and temperature), density of water is fixed and equal to experimental value 1 g/cm³. The energy of electrostatic interaction of atomic point charges of the explicit water molecules with conducting continuum is calculated analytically by means of the image charges method. It provides high computational efficiency of the SCW model. For the averaging of the calculated thermodynamic and structural values over microstates of the system the thermodynamic integration method is used. The possible using of SCW for the docking problem is discussed.