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

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

Semi-Markov models for brownian dynamics permeation in biological ion channels

Constructing accurate computational models that explain how ions permeate through a biological ion channel is an important problem in biophysics and drug design. Brownian dynamics simulations are large-scale interacting particle computer simulations for modeling ion channel permeation but can be computationally prohibitive. In this paper, we show the somewhat surprising result that a small-dimensional semi-Markov model can generate events (such as conduction events and dwell times at binding sites in the protein) that are statistically indistinguishable from brownian dynamics computer simulation. This approach enables the use of extrapolation techniques to predict channel conduction when performing the actual brownian dynamics simulation that is computationally intractable. Numerical studies on the simulation of gramicidin A ion channels are presented.     

Not only enthalpy: large entropy contribution to ion permeation barriers in single-file channels

The effect of channel length on the barrier for potassium ion permeation through single-file channels has been studied by means of all-atom molecular dynamics simulations. Using series of peptidic gramicidin-like and simplified ring-structured channels, both embedded in model membranes, we obtained two distinct types of behavior: saturation of the central free energy barriers for peptidic channels and a linear increase in simplified ring-structured channels with increasing channel length. The saturation of the central free energy barrier for the peptidic channels occurs at relatively short lengths, and it is correlated with the desolvation from the bulk water. Remarkably, decomposition of free energy barriers into enthalpic and entropic terms reveals an entropic cost for ion permeation. Furthermore, this entropic cost dominates the ion permeation free energy barrier, since the corresponding free energy contribution is higher than the enthalpic barrier. We conclude that the length dependence of the free energy is enthalpy-dominated, but the entropy is the major contribution to the permeation barrier. The decrease in rotational water motion and the reduction of channel mobility are putative origins for the overall entropic penalty.     

Sodium in gramicidin: an example of a permion.

The reaction path and free energy profile of Na+ were computed in the interior of the channel protein gramicidin, with the program MOIL. Gramicidin was represented in atomic detail, but surrounding water and lipid molecules were not included. Thus, only short range interactions were investigated. The permeation path of the ion was an irregular spiral, far from a straight line. Permeation cannot be described by motions of a single Na+ ion. The minimal energy path includes significant motion of water and channel atoms as well as motion of the permeating ion. We think of permeation as motion of a permion, a quasi-particle that includes the many body character of the permeation process, comparable with quasi-particles like holes, phonons, and electrons of solid-state physics. Na+ is accompanied by a plug of water molecules, and motions of water, Na+, and the atoms of gramicidin are highly correlated. The permion moves like a linear polymer made of waters and ion linked and moving coherently along a zigzag line, following the reptation mechanism of polymer transport. The effective mass, free energy, and memory kernel (of the integral describing time-dependent friction) of short range interactions were calculated. The effective mass of the permion (properly normalized) is much less than Na+. Friction varies substantially along the path. The free energy profile has two deep minima and several maxima. In certain regions, the dominant motions along the reaction path are those of the channel protein, not the permeating ion: there, ion waits while the other atoms move. At these waiting sites, the permion's motion along the reaction path is a displacement of the atoms of gramicidin that prepare the way for the Na+ ion.     

Energetics of ion permeation, rejection, binding, and block in gramicidin A from free energy simulations

The rigid force fields currently used in molecular dynamics (MD) simulations of biomolecules are optimized for globular proteins. Whether they can also be used in MD simulations of membrane proteins is an important issue that needs to be resolved. Here we address this issue using the gramicidin A channel, which provides an ideal test case because of the simplicity of its structure and the availability of a wealth of functional data. Permeation properties of gramicidin A can be summarized as "it conducts monovalent cations, rejects anions, and binds divalent cations." Hence, a comprehensive test should consider the energetics of permeation for all three types of ions. To that end, we construct the potential of mean force for K(+), Cl(-), and Ca(2+) ions along the channel axis. For an independent check of the potential-of-mean-force results, we also calculate the free energy differences for these ions at the channel center and binding sites relative to bulk. We find that "rejection of anions" is satisfied but there are difficulties in accommodating the other two properties using the current MD force fields.     

Molecular dynamics - potential of mean force calculations as a tool for understanding ion permeation and selectivity in narrow channels

Ion channels catalyze the permeation of charged molecules across cell membranes and are essential for many vital physiological functions, including nerve and muscle activity. To understand better the mechanisms underlying ion conduction and valence selectivity of narrow ion channels, we have employed free energy techniques to calculate the potential of mean force (PMF) for ion movement through the prototypical gramicidin A channel. Employing modern all-atom molecular dynamics (MD) force fields with umbrella sampling methods that incorporate one hundred 1-2 ns trajectories, we find that it is possible to achieve semi-quantitative agreement with experimental binding and conductance measurements. We also examine the sensitivity of the MD-PMF results to the choice of MD force field and compare PMFs for potassium, calcium and chloride ions to explore the basis for the valence selectivity of this narrow and uncharged ion channel. A large central barrier is observed for both anions and divalent ions, consistent with lack of experimental conductance. Neither anion or divalent cation is seen to be stabilized inside the channel relative to the bulk electrolyte and each leads to large disruptions to the protein and membrane structure when held deep inside the channel. Weak binding of calcium ions outside the channel corresponds to a free energy well that is too shallow to demonstrate channel blocking. Our findings emphasize the success of the MD-PMF approach and the sensitivity of ion energetics to the choice of biomolecular force field.     

Influence of protein flexibility on the electrostatic energy landscape in gramicidin A

We describe an electrostatic model of the gramicidin A channel that allows protein atoms to move in response to the presence of a permeating ion. To do this, molecular dynamics simulations are carried out with a permeating ion at various positions within the channel. Then an ensemble of atomic coordinates taken from the simulations are used to construct energy profiles using macroscopic electrostatic calculations. The energy profiles constructed are compared to experimentally-determined conductance data by inserting them into Brownian dynamics simulations. We find that the energy landscape seen by a permeating ion changes significantly when we allow the protein atoms to move rather than using a rigid protein structure. However, the model developed cannot satisfactorily reproduce all of the experimental data. Thus, even when protein atoms are allowed to move, the dielectric model used in our electrostatic calculations breaks down when modeling the gramicidin channel.     

Conducting-state properties of the KcsA potassium channel from molecular and Brownian dynamics simulations.

The mechanisms underlying transport of ions across the potassium channel are examined using electrostatic calculations and three-dimensional Brownian dynamics simulations. We first build open-state configurations of the channel with molecular dynamics simulations, by pulling the transmembrane helices outward until the channel attains the desired interior radius. To gain insights into ion permeation, we construct potential energy profiles experienced by an ion traversing the channel in the presence of other resident ions. These profiles reveal that in the absence of an applied field the channel accommodates three potassium ions in a stable equilibrium, two in the selectivity filter and one in the central cavity. In the presence of a driving potential, this three-ion state becomes unstable, and ion permeation across the channel is observed. These qualitative explanations are confirmed by the results of three-dimensional Brownian dynamics simulations. We find that the channel conducts when the ionizable residues near the extracellular entrance are fully charged and those near the intracellular side are partially charged. The conductance increases steeply as the radius of the intracellular mouth of the channel is increased from 2 A to 5 A. Our simulation results reproduce several experimental observations, including the current-voltage curves, conductance-concentration relationships, and outward rectification of currents.     

Ion transport in a model gramicidin channel. Structure and thermodynamics.

The potential of mean force for Na+ and K+ ions as a function of position in the interior of a periodic poly(L,D)-alanine model for the gramicidin beta-helix is calculated with a detailed atomic model and realistic interactions. The calculated free energy barriers are 4.5 kcal/mol for Na+ and 1.0 kcal/mol for K+. A decomposition of the free energy demonstrates that the water molecules make a significant contribution to the free energy of activation. There is an increase in entropy at the transition state associated with greater fluctuations. Analysis reveals that the free energy profile of ions in the periodic channel is controlled not by the large interaction energy involving the ion but rather by the weaker water-water, water-peptide and peptide-peptide hydrogen bond interactions. The interior of the channel retains much of the solvation properties of a liquid in its interactions with the cations. Of particular importance is the flexibility of the helix, which permits it to respond to the presence of an ion in a fluidlike manner. The distortion of the helix is local (limited to a few carbonyls) because the structure is too flexible to transmit a perturbation to large distances. The plasticity of the structure (i.e., the property to deform without generating a large energy stress) appears to be an essential factor in the transport of ions, suggesting that a rigid helix model would be inappropriate.     

Time-correlation analysis of simulated water motion in flexible and rigid gramicidin channels.

Molecular dynamics simulations have been done on a system consisting of the polypeptide membrane channel former gramicidin, plus water molecules in the channel and caps of waters at the two ends of the channel. In the absence of explicit simulation of the surrounding membrane, the helical form of the channel was maintained by artificial restraints on the peptide motion. The characteristic time constant of the artificial restraint was varied to assess the effect of the restraints on the channel structure and water motions. Time-correlation analysis was done on the motions of individual channel waters and on the motions of the center of mass of the channel waters. It is found that individual water molecules confined in the channel execute higher frequency motions than bulk water, for all degrees of channel peptide restraint. The center-of-mass motion of the chain of channel waters (which is the motion that is critical for transmembrane transport, due to the mandatory single filing of water in the channel) does not exhibit these higher frequency motions. The mobility of the water chain is dramatically reduced by holding the channel rigid. Thus permeation through the channel is not like flow through a rigid pipe; rather permeation is facilitated by peptide motion. For the looser restraints we used, the mobility of the water chain was not very much affected by the degree of restraint. Depending on which set of experiments is considered, the computed mobility of our water chain in the flexible channel is four to twenty times too high to account for the experimentally measured resistance of the gramicidin channel to water flow. From this result it appears likely that the peptide motions of an actual gramicidin channel embedded in a lipid membrane may be more restrained than in our flexible channel model, and that these restraints may be a significant modulator of channel permeability. For the completely rigid channel model the "trapping" of the water molecules in preferred positions throughout the molecular dynamics run precludes a reasonable assessment of mobility, but it seems to be quite low.     

Ion transport in the gramicidin channel: molecular dynamics study of single and double occupancy.

The structural and thermodynamic factors responsible for the singly and doubly occupied saturation states of the gramicidin channel are investigated with molecular dynamics simulations and free energy perturbation methods. The relative free energy of binding of all of the five common cations Li+, Na+, K+, Rb+, and Cs+ is calculated in the singly and doubly occupied channel and in bulk water. The atomic system, which includes the gramicidin channel, a model membrane made of neutral Lennard-Jones particles and 190 explicit water molecules to form the bulk region, is similar to the one used in previous work to calculate the free energy profile of a Na+ ion along the axis of the channel. In all of the calculations, the ions are positioned in the main binding sites located near the entrances of the channel. The calculations reveal that the doubly occupied state is relatively more favorable for the larger ions. Thermodynamic decomposition is used to show that the origin of the trend observed in the calculations is due to the loss of favorable interactions between the ion and the single file water molecules inside the channel. Small ions are better solvated by the internal water molecules in the singly occupied state than in the doubly occupied state; bigger ions are solvated almost as well in both occupation states. Water-channel interactions play a role in the channel response. The observed trends are related to general thermodynamical properties of electrolyte solutions.     

Simulation study of a gramicidin/lipid bilayer system in excess water and lipid. II. Rates and mechanisms of water transport

A gramicidin channel in a fluid phase DMPC bilayer with excess lipid and water has been simulated. By use of the formal correspondence between diffusion and random walk, a permeability for water through the channel was calculated, and was found to agree closely with the experimental results of Rosenberg and Finkelstein (Rosenberg, P.A., and A. Finkelstein. 1978. J. Gen. Physiol. 72:327-340; 341-350) for permeation of water through gramicidin in a phospholipid membrane. By using fluctuation analysis, components of resistance to permeation were computed for movement through the channel interior, for the transition step at the channel mouth where the water molecule solvation environment changes, and for the process of diffusion up to the channel mouth. The majority of the resistance to permeation appears to occur in the transition step at the channel mouth. A significant amount is also due to structurally based free energy barriers within the channel. Only small amounts are due to local friction within the channel or to diffusive resistance for approaching the channel mouth.     

"Adaptive" behavior of ligand-gated ion channels: constraints by thermodynamics.

The calcium-induced calcium release channel of the cardiac sarcoplasmic reticulum has been reported to inactivate in a novel manner (termed "adaptation"), which permits reactivation by exposure to successively higher concentrations of calcium. I examined the limitations placed by thermodynamics on the possible kinetic mechanisms for such behavior. The mechanism suggested by Gyorke and Fill, in which the affinity of a calcium-binding site decreases during adaptation, is not thermodynamically feasible for a passive system, but requires an external input of free energy. Possible sources of such energy are 1) metabolic energy, which is excluded by the fact that adaptation was observed in isolated channels in the absence of ATP, or 2) coupling of ion permeation to gating, for which there is currently no evidence. I derived a general limit on the thermodynamic feasibility of a sequence of channel activations and adaptations, irrespective of channel kinetics, from the requirement that the free energy must decrease during the spontaneous evolution of the system from the state existing immediately after a step increase in [Ca2+] to the state of maximum open probability that follows. The opening of the channel must involve an increase in free energy, which must be compensated by the free energy released by the incremental binding of calcium. This requirement leads to a complicated system of inequalities, which was simplified and manipulated algebraically into the form of a linear programming problem. Numerical solution of this problem showed that the sequence of adaptations of the SR channel observed by Gyorke and Fill requires the presence of at least 10 calcium-binding sites on the channel if it is to occur in the absence of exogenous sources of free energy. This indicates either that a large number of calcium-binding sites participate in the regulation of the SR calcium release channel, or that the existing data are significantly flawed with respect to the low open probability in the resting state, the importance of "calcium spike" artifacts from flash photolysis, or both.     

Theoretical and computational models of biological ion channels

The goal of this review is to establish a broad and rigorous theoretical framework to describe ion permeation through biological channels. This framework is developed in the context of atomic models on the basis of the statistical mechanical projection-operator formalism of Mori and Zwanzig. The review is divided into two main parts. The first part introduces the fundamental concepts needed to construct a hierarchy of dynamical models at different level of approximation. In particular, the potential of mean force (PMF) as a configuration-dependent free energy is introduced, and its significance concerning equilibrium and non-equilibrium phenomena is discussed. In addition, fundamental aspects of membrane electrostatics, with a particular emphasis on the influence of the transmembrane potential, as well as important computational techniques for extracting essential information from all-atom molecular dynamics (MD) simulations are described and discussed. The first part of the review provides a theoretical formalism to 'translate' the information from the atomic structure into the familiar language of phenomenological models of ion permeation. The second part is aimed at reviewing and contrasting results obtained in recent computational studies of three very different channels: the gramicidin A (gA) channel, which is a narrow one-ion pore (at moderate concentration), the KcsA channel from Streptomyces lividans, which is a narrow multi-ion pore, and the outer membrane matrix porin F (OmpF) from Escherichia coli, which is a trimer of three beta-barrel subunits each forming wide aqueous multi-ion pores. Comparison with experiments demonstrates that current computational models are approaching semi-quantitative accuracy and are able to provide significant insight into the microscopic mechanisms of ion conduction and selectivity. We conclude that all-atom MD with explicit water molecules can represent important structural features of complex biological channels accurately, including such features as the location of ion-binding sites along the permeation pathway. We finally discuss the broader issue of the validity of ion permeation models and an outlook to the future.     

How electrolyte shielding influences the electrical potential in transmembrane ion channels.

The electrical potential due to fixed charge distributions is strongly altered in the vicinity of a membrane and notably dependent on aqueous electrolyte concentration. We present an efficient way to solve the nonlinear Poisson-Boltzmann equation applicable to general cylindrically symmetric dielectric geometries. It generalizes Gouy-Chapman theory to systems containing transmembrane channels. The method is applied to three channel systems: gramicidin, gap junction, and porin. We find that for a long, narrow channel such as gramicidin concentration variation has little influence on the electrical image barrier to ion permeation. However, electrolyte shielding reduces the image induced contribution to the energy required for multiple occupancy. In addition, the presence of electrolyte significantly affects the voltage profile due to an applied potential, substantially compressing the electric field to the immediate vicinity of the pore itself. In the large diameter channels, where bulk electrolyte may be assumed to enter the pore, the electrolyte greatly reduces the image barrier to ion permeation. At physiological ionic strengths this barrier is negligible and the channel may be readily multiply occupied. At all ionic strengths considered (l greater than 0.005 M) the image barrier saturates rapidly and is essentially constant more than one channel radius from the entrance to the pore. At lower ionic strengths (l less than 0.016 M) there are noticeable (greater than 20 mV) energy penalties associated with multiple occupancy.     

The gramicidin ion channel: a model membrane protein

The linear peptide gramicidin forms prototypical ion channels specific for monovalent cations and has been extensively used to study the organization, dynamics and function of membrane-spanning channels. In recent times, the availability of crystal structures of complex ion channels has challenged the role of gramicidin as a model membrane protein and ion channel. This review focuses on the suitability of gramicidin as a model membrane protein in general, and the information gained from gramicidin to understand lipid-protein interactions in particular. Special emphasis is given to the role and orientation of tryptophan residues in channel structure and function and recent spectroscopic approaches that have highlighted the organization and dynamics of the channel in membrane and membrane-mimetic media.     

Molecular dynamics computations and solid state nuclear magnetic resonance of the gramicidin cation channel.

This paper reports on a coupled approach to determining the structure of the gramicidin A ion channel, utilizing solid state nuclear magnetic resonance (NMR) of isotopically labeled gramicidin channels aligned parallel to the magnetic field direction, and molecular dynamics (MD). MD computations using an idealized right-handed beta-helix as a starting point produce a refined molecular structure that is in excellent agreement with atomic resolution solid state NMR data. The data provided by NMR and MD are complementary to each other. When applied in a coordinated manner they provide a powerful approach to structure determination in molecular systems not readily amenable to x-ray diffraction.