Simulations of ion current in realistic models of ion channels: the KcsA potassium channel

Validating a Coarse-Grained Voltage Activation Model by Comparing Its Performance to the Results of Monte Carlo Simulations

Simulating the nature of voltage-activated systems is a problem of major current interest, ranging from the action of voltage-gated ion channels to energy storage batteries. However, fully microscopic converging molecular simulations of external voltage effects present a major challenge, and macroscopic models are associated with major uncertainties about the dielectric treatment and the underlying physical basis. Recently we developed a coarse-grained (CG) model that represents explicitly the electrodes, the electrolytes, and the membrane/protein system. The CG model provides a semimacroscopic way of capturing the microscopic physics of voltage-activated systems. Our method was originally validated by reproducing macroscopic and analytical results for key test cases and then used in modeling voltage-activated ion channels and related problems. In this work, we further establish the reliability of the CG voltage model by comparing it to the results of Monte Carlo (MC) simulations with a microscopic electrolyte model. The comparison explores different aspects of membrane, electrolyte, and electrode systems ranging from the Gouy-Chapman model to the determination of the electrolyte charge distribution in the solution between two electrodes (without and with a separating membrane), as well as the evaluation of gating charges. Overall the agreement is very impressive. This provides confidence in the CG model and also shows that the MC model can be used in realistic simulation of voltage activation of membrane proteins with sufficient computer time.

Refining the treatment of membrane proteins by coarse-grained models

Obtaining a quantitative description of the membrane proteins stability is crucial for understanding many biological processes. However the advance in this direction has remained a major challenge for both experimental studies and molecular modeling. One of the possible directions is the use of coarse-grained models but such models must be carefully calibrated and validated. Here we use a recent progress in benchmark studies on the energetics of amino acid residue and peptide membrane insertion and membrane protein stability in refining our previously developed coarse-grained model (Vicatos et al., Proteins 2014;82:1168). Our refined model parameters were fitted and/or tested to reproduce water/membrane partitioning energetics of amino acid side chains and a couple of model peptides. This new model provides a reasonable agreement with experiment for absolute folding free energies of several β-barrel membrane proteins as well as effects of point mutations on a relative stability for one of those proteins, OmpLA. The consideration and ranking of different rotameric states for a mutated residue was found to be essential to achieve satisfactory agreement with the reference data.

Equilibrium fluctuation relations for voltage coupling in membrane proteins

A general theoretical framework is developed to account for the effects of an external potential on the energetics of membrane proteins. The framework is based on the free energy relation between two (forward/backward) probability densities, which was recently generalized to non-equilibrium processes, culminating in the work-fluctuation theorem. Starting from the probability densities of the conformational states along the "voltage coupling" reaction coordinate, we investigate several interconnected free energy relations between these two conformational states, considering voltage activation of ion channels. The free energy difference between the two conformational states at zero (depolarization) membrane potential (i.e., known as the chemical component of free energy change in ion channels) is shown to be equivalent to the free energy difference between the two "equilibrium" (resting and activated) conformational states along the one-dimensional voltage couplin reaction coordinate. Furthermore, the requirement that the application of linear response approximation to the free energy functionals of voltage coupling should satisfy the general free energy relations, yields a novel closed-form expression for the gating charge in terms of other basic properties of ion channels. This connection is familiar in statistical mechanics, known as the equilibrium fluctuation-response relation. The theory is illustrated by considering the coupling of a unit charge to the external voltage in the two sites near the surface of membrane, representing the activated and resting states. This is done using a coarse-graining (CG) model of membrane proteins, which includes the membrane, the electrolytes and the electrodes. The CG model yields Marcus-type voltage dependent free energy parabolas for the response of the electrostatic environment (electrolytes etc.) to the transition from the initial to the final configuratinal states, leading to equilibrium free energy difference and free energy barrier that follow the trend of the equilibrium fluctuation relation and the Marcus theory of electron transfer. These energetics also allow for a direct estimation of the voltage dependence of channel activation (Q-V curve), offering a quantitative rationale for a correlation between the voltage dependence parabolas and the Q-V curve, upon site-directed mutagenesis or drug binding. Taken together, by introducing the voltage coupling as the energy gap reaction coordinate, our framework brings new perspectives to the thermodynamic models of voltage activation in voltage-sensitive membrane proteins, offering an a framework for a better understating of the structure-function correlations of voltage gating in ion channels as well as electrogenic phenomena in ion pumps and transporters. Significantly, this formulation also provides a powerful bridge between the CG model of voltage coupling and the conventional macroscopic treatments.

Assembly and stability of Salmonella enterica ser. Typhi TolC protein in POPE and DMPE

In this work we assessed the suitability of two different lipid membranes for the simulation of a TolC protein from Salmonella enterica serovar Typhi. The TolC protein family is found in many pathogenic Gram-negative bacteria including Vibrio cholera and Pseudomonas aeruginosa and acts as an outer membrane channel for expulsion of drug and toxin from the cell. In S. typhi, the causative agent for typhoid fever, the TolC outer membrane protein is an antigen for the pathogen. The lipid environment is an important modulator of membrane protein structure and function. We evaluated the conformation of the TolC protein in the presence of DMPE and POPE bilayers using molecular dynamics simulation. The S. typhi TolC protein exhibited similar conformational dynamics to TolC and its homologues. Conformational flexibility of the protein is seen in the C-terminal, extracellular loops, and α-helical region. Despite differences in the two lipids, significant similarities in the motion of the protein in POPE and DMPE were observed, including the rotational motion of the C-terminal residues and the partially open extracellular loops. However, analysis of the trajectories demonstrated effects of hydrophobic matching of the TolC protein in the membrane, particularly in the lengthening of the lipids and subtle movements of the protein's β-barrel towards the lower leaflet in DMPE. The study exhibited the use of molecular dynamics simulation in revealing the differential effect of membrane proteins and lipids on each other. In this study, POPE is potentially a more suitable model for future simulation of the S. typhi TolC protein.

Computational methods of studying the binding of toxins from venomous animals to biological ion channels: theory and applications

The discovery of new drugs that selectively block or modulate ion channels has great potential to provide new treatments for a host of conditions. One promising avenue revolves around modifying or mimicking certain naturally occurring ion channel modulator toxins. This strategy appears to offer the prospect of designing drugs that are both potent and specific. The use of computational modeling is crucial to this endeavor, as it has the potential to provide lower cost alternatives for exploring the effects of new compounds on ion channels. In addition, computational modeling can provide structural information and theoretical understanding that is not easily derivable from experimental results. In this review, we look at the theory and computational methods that are applicable to the study of ion channel modulators. The first section provides an introduction to various theoretical concepts, including force-fields and the statistical mechanics of binding. We then look at various computational techniques available to the researcher, including molecular dynamics, brownian dynamics, and molecular docking systems. The latter section of the review explores applications of these techniques, concentrating on pore blocker and gating modifier toxins of potassium and sodium channels. After first discussing the structural features of these channels, and their modes of block, we provide an in-depth review of past computational work that has been carried out. Finally, we discuss prospects for future developments in the field.

Nonselective conduction in a mutated NaK channel with three cation-binding sites

The NaK channel is a cation-selective protein with similar permeability for K(+) and Na(+) ions. Crystallographic structures are available for the wild-type and mutated NaK channels with different numbers of cation-binding sites. We have performed a comparison between the potentials of mean force governing the translocation of K(+) ions and mixtures of one Na(+) and three K(+) ions in a mutated NaK channel with only three cation-binding sites (NaK-CNG). Since NaK-CNG is not selective for K(+) over Na(+), analysis of its multi-ion potential energy surfaces can provide clues about how selectivity originates. Comparison of the potentials of mean force of NaK-CNG and K(+)-selective channels yields observations that strongly suggest that the number of contiguous ion binding sites in a single-file mechanism is the key determinant of the channel's selectivity properties, as already proposed by experimental studies. We conclude that the presence of four binding sites in K(+)-selective channels is essential for highly selective and efficient permeation of K(+) ions, and that a key difference between K(+)-selective and nonselective channels is the absence/presence of a binding site for Na(+) ions at the boundary between S2 and S3 in the context of multi-ion permeation events.

Conductance properties of the inwardly rectifying channel, Kir3.2: molecular and Brownian dynamics study

Using the recently unveiled crystal structure, and molecular and Brownian dynamics simulations, we elucidate several conductance properties of the inwardly rectifying potassium channel, Kir3.2, which is implicated in cardiac and neurological disorders. We show that the pore is closed by a hydrophobic gating mechanism similar to that observed in Kv1.2. Once open, potassium ions move into, but not out of, the cell. The asymmetrical current-voltage relationship arises from the lack of negatively charged residues at the narrow intracellular mouth of the channel. When four phenylalanine residues guarding the intracellular gate are mutated to glutamate residues, the channel no longer shows inward rectification. Inward rectification is restored in the mutant Kir3.2 when it becomes blocked by intracellular Mg(2+). Tertiapin, a polypeptide toxin isolated from the honey bee, is known to block several subtypes of the inwardly rectifying channels with differing affinities. We identify critical residues in the toxin and Kir3.2 for the formation of the stable complex. A lysine residue of tertiapin protrudes into the selectivity filter of Kir3.2, while two other basic residues of the toxin form hydrogen bonds with acidic residues located just outside the channel entrance. The depth of the potential of mean force encountered by tertiapin is -16.1kT, thus indicating that the channel will be half-blocked by 0.4μM of the toxin.

On conduction in a bacterial sodium channel

Voltage-gated Na⁺-channels are transmembrane proteins that are responsible for the fast depolarizing phase of the action potential in nerve and muscular cells. Selective permeability of Na⁺ over Ca²⁺ or K⁺ ions is essential for the biological function of Na⁺-channels. After the emergence of the first high-resolution structure of a Na⁺-channel, an anionic coordination site was proposed to confer Na⁺ selectivity through partial dehydration of Na⁺ via its direct interaction with conserved glutamate side chains. By combining molecular dynamics simulations and free-energy calculations, a low-energy permeation pathway for Na⁺ ion translocation through the selectivity filter of the recently determined crystal structure of a prokaryotic sodium channel from Arcobacter butzleri is characterised. The picture that emerges is that of a pore preferentially occupied by two ions, which can switch between different configurations by crossing low free-energy barriers. In contrast to K⁺-channels, the movements of the ions appear to be weakly coupled in Na⁺-channels. When the free-energy maps for Na⁺ and K⁺ ions are compared, a selective site is characterised in the narrowest region of the filter, where a hydrated Na⁺ ion, and not a hydrated K⁺ ion, is energetically stable.

Ion channels are integral membrane proteins that control the passive diffusion of ions down their electrochemical gradient. According to the most permeating ion species, ion channels are classified in three categories: K⁺-channels, Na⁺-channels, and Ca²⁺-channels. The atomic structure of a K⁺-channel was the first to be solved experimentally more than 10 years ago. This structure inspired numerous computational studies, which revealed the mechanisms of conduction and selectivity in K⁺-channels. Recently, the first atomic structure of a Na⁺ selective channel has been solved. Here, molecular dynamics simulations and free-energy calculations are described and a possible mechanism for Na⁺ conduction is identified. In contrast to what it is observed in K⁺-channels, ion movements through Na⁺-channels appeared highly uncorrelated.

Realistic simulation of the activation of voltage-gated ion channels

Understanding the detailed mechanism of the activation of voltage-gated ion channels has been a problem of great current interest. Reliable molecular simulations of voltage effects present a major challenge because meaningful converging microscopic simulations are not yet available and macroscopic treatments involve major uncertainties regarding the dielectric constant used and other key features. The current work has overcome some of the above challenges by using our recently developed coarse-grained (CG) model in simulating the activation of the Kv1.2 channel. The CG model has allowed us to explore problems that cannot be addressed at present by fully microscopic simulations, while providing insights on some features that are not usually considered in continuum models, including the distribution of the electrolytes between the membrane and the electrodes during the activation process and thus the nature of the gating current. Furthermore, the clear connection to microscopic descriptions combined with the power of CG modeling offers a powerful tool for exploring the energy balance between the protein conformational energy and the interaction with the external potential in voltage-activated channels. Our simulations have reproduced the observed experimental trend of the gating charge and, most significantly, the correct trend in the free energies, where the closed channel is more stable at negative potential and the open channel is more stable at positive potential. Moreover, we provide a unique view of the activation landscape and the time dependence of the activation process.

Simulating electrostatic energies in proteins: perspectives and some recent studies of pKas, redox, and other crucial functional properties

Electrostatic energies provide what is arguably the most effective tool for structure-function correlation of biological molecules. Here, we provide an overview of the current state-of-the-art simulations of electrostatic energies in macromolecules, emphasizing the microscopic perspective but also relating it to macroscopic approaches. We comment on the convergence issue and other problems of the microscopic models and the ways of keeping the microscopic physics while moving to semi-macroscopic directions. We discuss the nature of the protein dielectric "constants" reiterating our long-standing point that the dielectric "constants" in semi-macroscopic models depend on the definition and the specific treatment. The advances and the challenges in the field are illustrated considering different functional properties including pK(a)'s, redox potentials, ion and proton channels, enzyme catalysis, ligand binding, and protein stability. We emphasize the microscopic overcharging approach for studying pK(a) 's of internal groups in proteins and give a demonstration of power of this approach. We also emphasize recent advances in coarse grained models with a physically based electrostatic treatment and provide some examples including further directions in treating voltage activated ion channels.

Coarse grained model for exploring voltage dependent ion channels

The relationship between the membrane voltage and the gating of voltage activated ion channels and other systems have been a problem of great current interest. Unfortunately, reliable molecular simulations of external voltage effects present a major challenge, since meaningful converging microscopic simulations are not yet available and macroscopic treatments involve major uncertainties in terms of the dielectric used and other key features. This work extends our coarse grained (CG) model to simulations of membrane/protein systems under external potential. Special attention is devoted to a consistent modeling of the effect of external potential due to the electrodes, emphasizing semimacroscopic description of the electrolytes in the solution regions between the membranes and the electrodes, as well as the coupling between the combined potential from the electrodes plus the electrolytes and the protein ionized groups. We also provide a clear connection to microscopic treatment of the electrolytes and thus can explore possible conceptual problems that are hard to resolve by other current approaches. For example, we obtain a clear description of the charge distribution in the entire electrolyte system, including near the electrodes in membrane/electrodes systems (where continuum models do not seem to provide the relevant results). Furthermore, the present treatment provides an insight on the distribution of the electrolyte charges before and after equilibration across the membrane, and thus on the nature of the gating charge. The different aspects of the model have been carefully validated by considering problems ranging for the simple Debye-Huckel, and the Gouy-Chapman models to the evaluation of the electrolyte distribution between two electrodes, as well as the effect of extending the simulation system by periodic replicas. Overall the clear connection to microscopic descriptions combined with the power of the CG modeling seems to offer a powerful tool for exploring the balance between the protein conformational energy and the interaction with the external potential in voltage activated channels. To illustrate these features we present a preliminary study of the gating charge in the voltage activated Kv1.2 channel, using the actual change in the electrolyte charge distribution rather than the conventional macroscopic estimate. We also discuss other special features of the model, which include the ability to capture the effect of changes in the protonation states of the protein residues during the close to open voltage induced transition. This article is part of a Special Issue entitled: Membrane protein structure and function.

Renormalizing SMD: the renormalization approach and its use in long time simulations and accelerated PMF calculations of macromolecules

Simulations of long time process in condensed phases, in general, and in biomolecules, in particular, present a major challenge that cannot be overcome at present by brute force molecular dynamics (MD) approaches. This work takes the renormalization method, intruded by us sometime ago, and establishes its reliability and potential in extending the time scale of molecular simulations. The validation involves a truncated gramicidin system in the gas phase. This system is small enough to allow for very long explicit simulations and sufficiently complex to present the physics of realistic ion channels. The renormalization approach is found to be reliable and arguably presents the first approach that allows one to exploit the otherwise problematic steered molecular dynamics (SMD) treatments in quantitative and meaningful studies. It is established that we can reproduce the long time behavior of large systems by using Langevin dynamics (LD) simulations of a renormalized implicit model. This is done without spending the enormous time needed to obtain such trajectories in the explicit system. The present study also provides a promising advance in accelerated evaluation of free energy barriers. This is done by adjusting the effective potential in the implicit model to reproduce the same passage time as that obtained in the explicit model under the influence of an external force. Here having a reasonable effective friction provides a way to extract the potential of mean force (PMF) without investing the time needed for regular PMF calculations. The renormalization approach, which is illustrated here in realistic calculations, is expected to provide a major help in studies of complex landscapes and in exploring long time dynamics of biomolecules.

Simulating the electrostatic guidance of the vectorial translocations in hexameric helicases and translocases

The molecular origin of the action of helicases is explored, starting with a model built based on the different X-ray structures of the large tumor antigen (LTag) hexameric helicase and a simplified model containing the ionized phosphate backbones of a single-strand DNA. The coupling between the protein structural changes and the translocation process is quantified using an effective electrostatic free-energy surface for the protein/DNA complex. This surface is then used in Langevin dynamics simulations of the time dependence of the translocation process. Remarkably, the simulated motion along the free-energy surface results in a vectorial translocation of the DNA, consistent with the biological process. The electrostatic energy of the system appears to reproduce the directionality of this process. Thus, we are able to provide a consistent structure-based molecular description of the energetic and dynamics of the translocation process. This analysis may have general implications for relating structural models to translocation directionality in helicases and other DNA translocases.

Progress in ab initio QM/MM free-energy simulations of electrostatic energies in proteins: accelerated QM/MM studies of pKa, redox reactions and solvation free energies

Hybrid quantum mechanical/molecular mechanical (QM/MM) approaches have been used to provide a general scheme for chemical reactions in proteins. However, such approaches still present a major challenge to computational chemists, not only because of the need for very large computer time in order to evaluate the QM energy but also because of the need for proper computational sampling. This review focuses on the sampling issue in QM/MM evaluations of electrostatic energies in proteins. We chose this example since electrostatic energies play a major role in controlling the function of proteins and are key to the structure-function correlation of biological molecules. Thus, the correct treatment of electrostatics is essential for the accurate simulation of biological systems. Although we will be presenting different types of QM/MM calculations of electrostatic energies (and related properties) here, our focus will be on pKa calculations. This reflects the fact that pKa's of ionizable groups in proteins provide one of the most direct benchmarks for the accuracy of electrostatic models of macromolecules. While pKa calculations by semimacroscopic models have given reasonable results in many cases, existing attempts to perform pKa calculations using QM/MM-FEP have led to discrepancies between calculated and experimental values. In this work, we accelerate our QM/MM calculations using an updated mean charge distribution and a classical reference potential. We examine both a surface residue (Asp3) of the bovine pancreatic trypsin inhibitor and a residue buried in a hydrophobic pocket (Lys102) of the T4-lysozyme mutant. We demonstrate that, by using this approach, we are able to reproduce the relevant side chain pKa's with an accuracy of 3 kcal/mol. This is well within the 7 kcal/mol energy difference observed in studies of enzymatic catalysis, and is thus sufficient accuracy to determine the main contributions to the catalytic energies of enzymes. We also provide an overall perspective of the potential of QM/MM calculations in general evaluations of electrostatic free energies, pointing out that our approach should provide a very powerful and accurate tool to predict the electrostatics of not only solution but also enzymatic reactions, as well as the solvation free energies of even larger systems, such as nucleic acid bases incorporated into DNA.

Non-Markovian noise mediated through anomalous diffusion within ion channels

It is evident from a wide range of experimental findings that ion channel gating is inherently stochastic. The issue of "memory effects" (diffusional retardation due to local changes in water viscosity) in ionic flow has been recently addressed using Brownian dynamics simulations. The results presented indicate such memory effects are negligible, unless the diffusional barrier is much higher than that of free solute. In this paper using differential stochastic methods we conclude that the Markovian property of exponential dwell times gives rise to a high barrier, resulting in diffusional memory effects that cannot be ignored in determining ionic flow through channels. We have addressed this question using a generalized Langevin equation that contains a combination of Markovian and non-Markovian processes with different time scales. This approach afforded the development of an algorithm that describes an oscillatory ionic diffusional sequence. The resulting oscillatory function behavior, with exponential decay, was obtained at the weak non-Markovian limit with two distinct time scales corresponding to the processes of ionic diffusion and drift. This will be analyzed further in future studies using molecular dynamics simulations. We propose that the rise of time scales and memory effects is related to differences of shear viscosity in the cytoplasm and extracellular matrix.

Conduction of Na+ and K+ through the NaK channel: molecular and Brownian dynamics studies

Conduction of ions through the NaK channel, with M0 helix removed, was studied using both Brownian dynamics and molecular dynamics. Brownian dynamics simulations predict that the truncated NaK has approximately a third of the conductance of the related KcsA K+ channel, is outwardly rectifying, and has a Michaelis-Menten current-concentration relationship. Current magnitude increases when the glutamine residue located near the intracellular gate is replaced with a glutamate residue. The channel is blocked by extracellular Ca2+. Molecular dynamics simulations show that, under the influence of a strong applied potential, both Na+ and K+ move across the selectivity filter, although conduction rates for Na+ ions are somewhat lower. The mechanism of conduction of Na+ differs significantly from that of K+ in that Na+ is preferentially coordinated by single planes of pore-lining carbonyl oxygens, instead of two planes as in the usual K+ binding sites. The water-containing filter pocket resulting from a single change in the selectivity filter sequence (compared to potassium channels) disrupts several of the planes of carbonyl oxygens, and thus reduces the filter's ability to discriminate against sodium.

Calculation of protein-ligand binding free energy by using a polarizable potential

The binding of charged ligands benzamidine and diazamidine to trypsin was investigated by using a polarizable potential energy function and explicit-water molecular dynamics simulations. The binding free energies were computed from the difference between the free energies of decoupling the ligand from water and protein environments. Both the absolute and the relative free energies from the perturbation simulations agree with experimental measurements to within 0.5 kcal.mol(-1). Comparison of free-energy components sampled from different thermodynamic paths indicates that electrostatics is the main driving force behind benzamidine recognition of trypsin. The contribution of electronic polarization to binding appears to be crucial. By computing the free-energy contribution caused by the polarization between the ligand and its surroundings, we found that polarization has the opposite effect in dissimilar environments. Although polarization favors ligand solvation in water, it weakens the protein-ligand attraction by screening the electrostatic interaction between trypsin and benzamidine. We also examined the relative binding free energies of a benzamidine analog diazamidine to trypsin. The changes in free energy on benzamidine-diazamidine substitution were tens of kilocalories in both water and trypsin environments; however, the change in the total binding free energy is <2 kcal.mol(-1) because of cancellation, consistent with the experimental results. Overall, our results suggest that the use of a polarizable force field, given adequate sampling, is capable of achieving chemical accuracy in molecular simulations of protein-ligand recognition.

Hydropathic analysis and comparison of KcsA and Shaker potassium channels

The similarity in structure of potassium (K(+)) channels from different families has been revealed by only recently available crystallographic 3D structural data. The hydropathic analysis presented in this work illuminates whether homologous residues perform the same functions in channels that use different gating mechanisms. We calculated and compared the hydropathic profiles of two K(+) channels, KcsA and Kv1.2 (the latter a member of the Shaker family), at their pore-forming domain. Quantitative information describing important interactions stabilizing the protein beyond obvious secondary-structure elements was extracted from the analysis and applied as a template for subsequent molecular-dynamics (MD) analyses. For example, two key groups of interactions, defining the turns that connect the transmembrane helices and responsible for the orientation of the pore helix, were identified. Our results also indicate that Asp(80) and Asp(379) play a similar role in stabilizing the P-loop of KcsA and Kv1.2, respectively, but to significantly different extents.

Incorporation of inhomogeneous ion diffusion coefficients into kinetic lattice grand canonical monte carlo simulations and application to ion current calculations in a simple model ion channel

To deal with inhomogeneous diffusion coefficients of ions without altering the lattice spacing in the kinetic lattice grand canonical Monte Carlo (KLGCMC) simulation, an algorithm that incorporates diffusion coefficient variation into move probabilities is proposed and implemented into KLGCMC calculations. Using this algorithm, the KLGCMC simulation method is applied to the calculation of ion currents in a simple model ion channel system. Comparisons of ion currents and ion concentrations from these simulations with Poisson-Nernst-Planck (PNP) results show good agreement between the two methods for parameters where the latter method is expected to be accurate.

Cooperative transport in a potassium ion channel

Our current understanding of ion permeation through the selectivity filter of the KcsA potassium channel is based on the concept of a multi-ion transport mechanism. The details of this concerted movement, however, are not well understood. In the present paper we report on molecular dynamics simulations which provides new insights. It is shown that ion translocation is based on the collective hopping of ions and water molecules which is mediated by the flexible charged carbonyl groups lining the backbone of the pore. In particular, there is strong evidence for pairwise translocations where one ion and one water molecule form a bound state. We suggest a physical explanation of the observed phenomena employing a simple lattice model. It is argued that the water molecules can act as rectifiers during the hopping of ion-water pairs.

An accurate and efficient empirical approach for calculating the dielectric self-energy and ion-ion pair potential in continuum models of biological ion channels

This paper presents empirical formulas for calculating the dielectric self-energy and ion-ion pair interactions in cylindrical ion channels. The proposed approach can be extended to more complex channel structures, for example, (i) a "straight" channel with variable radius and (ii) a "curved" channel with constant radius. For calibration purposes, we compare results obtained based on the approximate effective potentials developed herein to exact electrostatic calculations obtained via the algorithm of Graf et al.: the agreement is satisfactory. A dynamic lattice Monte Carlo (DLMC) technique is used to further assess the accuracy and efficiency of the proposed empirical potentials. The concentration profiles and current-voltage curves produced with our simple empirical energy formulas are in excellent agreement with numerical results obtained using the algorithm of Graf et al., which calculates all relevant electrostatic forces exactly. The use of effective ion-ion potentials greatly reduces the computer memory required to perform DLMC ion permeation simulations in dielectrically inhomogeneous environments, thus enabling treatment of larger systems than can be handled by numerically exact techniques.

Quantum mechanical calculations of charge effects on gating the KcsA channel

A series of ab initio (density functional) calculations were carried out on side chains of a set of amino acids, plus water, from the (intracellular) gating region of the KcsA K(+) channel. Their atomic coordinates, except hydrogen, are known from X-ray structures [D.A. Doyle, J.M. Cabral, R.A. Pfuetzner, A. Kuo, J.M. Gulbis, S.L. Cohen, B.T. Chait, R. MacKinnon, The structure of the potassium channel: molecular basis of K(+) conduction and selectivity, Science 280 (1998) 69-77; R. MacKinnon, S.L. Cohen, A. Kuo, A. Lee, B.T. Chait, Structural conservation in prokaryotic and eukaryotic potassium channels, Science 280 (1998) 106-109; Y. Jiang, A. Lee, J. Chen, M. Cadene, B.T. Chait, R. MacKinnon, The open pore conformation of potassium channels. Nature 417 (2001) 523-526], as are the coordinates of some water oxygen atoms. The 1k4c structure is used for the starting coordinates. Quantum mechanical optimization, in spite of the starting configuration, places the atoms in positions much closer to the 1j95, more tightly closed, configuration. This state shows four water molecules forming a "basket" under the Q119 side chains, blocking the channel. When a hydrated K(+) approaches this "basket", the optimized system shows a strong set of hydrogen bonds with the K(+) at defined positions, preventing further approach of the K(+) to the basket. This optimized structure with hydrated K(+) added shows an ice-like 12 molecule nanocrystal of water. If the water molecules exchange, unless they do it as a group, the channel will remain blocked. The "basket" itself appears to be very stable, although it is possible that the K(+) with its hydrating water molecules may be more mobile, capable of withdrawing from the gate. It is also not surprising that water essentially freezes, or forms a kind of glue, in a nanometer space; this agrees with experimental results on a rather different, but similarly sized (nm dimensions) system [K.B. Jinesh, J.W.M. Frenken, Capillary condensation in atomic scale friction: how water acts like a glue, Phys. Rev. Lett. 96 (2006) 166103/1-4]. It also agrees qualitatively with simulations on channels [A. Anishkin, S. Sukharev, Water dynamics and dewetting transitions in the small mechanosensitive channel MscS, Biophys. J. 86 (2004) 2883-2895; O. Beckstein, M.S.P. Sansom, Liquid-vapor oscillations of water in hydrophobic nanopores, Proc. Natl Acad. Sci. U. S. A. 100 (2003) 7063-7068] and on featureless channel-like systems [J. Lu, M.E. Green, Simulation of water in a pore with charges: application to a gating mechanism for ion channels, Prog. Colloid Polym. Sci. 103 (1997) 121-129], in that it forms a boundary on water that is not obvious from the liquid state. The idea that a structure is stable, even if individual molecules exchange, is well known, for example from the hydration shell of ions. We show that when charges are added in the form of protons to the domains (one proton per domain), the optimized structure is open. No stable water hydrogen bonds hold it together; an opening of 11.0 A appears, measured diagonally between non-neighboring domains as glutamine 119 carbonyl O-O distance. This is comparable to the opening in the MthK potassium channel structure that is generally agreed to be open. The appearance of the opening is in rather good agreement with that found by Perozo and coworkers. In contrast, in the uncharged structure this diagonal distance is 6.5 A, and the water "basket" constricts the uncharged opening still further, with the ice-like structure that couples the K(+) ion to the gating region freezing the entrance to the channel. Comparison with our earlier model for voltage gated channels suggests that a similar mechanism may apply in those channels.

Simulation of biological ion channels with technology computer-aided design

Computer simulations of realistic ion channel structures have always been challenging and a subject of rigorous study. Simulations based on continuum electrostatics have proven to be computationally cheap and reasonably accurate in predicting a channel's behavior. In this paper we discuss the use of a device simulator, SILVACO, to build a solid-state model for KcsA channel and study its steady-state response. SILVACO is a well-established program, typically used by electrical engineers to simulate the process flow and electrical characteristics of solid-state devices. By employing this simulation program, we have presented an alternative computing platform for performing ion channel simulations, besides the known methods of writing codes in programming languages. With the ease of varying the different parameters in the channel's vestibule and the ability of incorporating surface charges, we have shown the wide-ranging possibilities of using a device simulator for ion channel simulations. Our simulated results closely agree with the experimental data, validating our model.

Mechanism of Ion Permeation in a Model Channel: Free Energy Surface and Dynamics of K+Ion Transport in an Anion-Doped Carbon Nanotube

The mechanism of the ion permeation is investigated for an anion-doped carbon nanotube, as a model of the K+ channel, by analyzing the free energy surface and the dynamics of the ion permeation through the model channel. It is found that the main rate-determining step is how an ion enters the channel. The entrance of the ion is mostly blocked by a water molecule located at this entrance. Only about 10% of K+ ions which reach the mouth of the channel can really enter the channel. The rejection rate sensitively depends on the location of this water molecule, which is easily controlled by the charge of the carbon nanotube; for example, the maximum permeation is obtained when the anion charge is at a certain value, -5.4e in the present model. At this charge, the facile translocation of the ion inside the channel is also induced due to the number of fluctuations of the ions inside the channel. Therefore, the so-called "Newton's balls", a toy model, combined with a simple ion diffusion model for explaining the fast ion permeation should be modified. The present analysis thus suggests that there exists an optimum combination of the length and the charge of the carbon nanotube for the most efficient ion permeation.

A rigid-body Newtonian propagation scheme based on instantaneous decomposition into rotation and translation blocks

The rotation and translation block (RTB) method of Durand et al. [Biopolymers 34, 759 (1994)] and Tama et al. [Proteins 41, 1 (2000)] provides an appealing way to calculate low-frequency normal modes of large biomolecules by restricting the space of motions to exclude internal motions of preselected rigid fragments within the molecule. These fragments are modeled essentially as rigid bodies and the need to calculate high-frequency relative motions of the atoms that form them is obviated in a natural way. Here we extend the RTB approach into a method for computing the classical (Newtonian) dynamics of a biomolecule, or any large molecule, with effective rigid-body constraints applied to a prechosen set of internal molecular fragments. This method, to be termed RTB dynamics, is easy to implement, conserves the total energy of the system, does not require the construction of the matrix of second spatial derivatives of the potential-energy function (Hessian matrix), and can be used to compute the classical dynamics of a system moving in an arbitrary anharmonic force field. An elementary numerical application to signal propagation in the small membrane-bound polypeptide gramicidin-A is presented for illustration purposes.

Modeling electrostatic effects in proteins

Electrostatic energies provide what is perhaps the most effective tool for structure-function correlation of biological molecules. This review considers the current state of simulations of electrostatic energies in macromolecules as well as the early developments of this field. We focus on the relationship between microscopic and macroscopic models, considering the convergence problems of the microscopic models and the fact that the dielectric 'constants' in semimacroscopic models depend on the definition and the specific treatment. The advances and the challenges in the field are illustrated considering a wide range of functional properties including pK(a)'s, redox potentials, ion and proton channels, enzyme catalysis, ligand binding and protein stability. We conclude by pointing out that, despite the current problems and the significant misunderstandings in the field, there is an overall progress that should lead eventually to quantitative descriptions of electrostatic effects in proteins and thus to quantitative descriptions of the function of proteins.

Application of the Poisson-Nernst-Planck theory with space-dependent diffusion coefficients to KcsA

The Poisson-Nernst-Planck electrodiffusion theory serves to compute charge fluxes and is here applied to the ion current through a protein channel. KcsA was selected as an example because of the abundance of experimental and theoretical data. The potassium channels MthK and KvAP were used as templates to define two open channel models for KcsA. Channel boundary surfaces and protein charge distributions were defined according to atomic radii and partial atomic charges. To establish the sensitivity of the results to these parameters, two different sets were used. Assigning the potassium diffusion coefficients equal to the value for free-diffusion in water (1.96 x 10(-9) m(2)/s), the computed currents overestimated the experimental data. Ion distributions inside the channel suggest that the overestimate is not due to an excess of charge shielding. A good agreement with the experimental data was achieved by reducing the potassium diffusion coefficient inside the channel to 1.96 x 10(-10) m(2)/s, a value of substantial motility but nonetheless in accord with the intuitive notion that the channel has a high affinity for the ions and therefore slows them down. These results are independent of the open channel model and the parameterization adopted for atomic radii and partial atomic charges. The method offers a reliable estimate of the channel current with low computational effort.

Role of protein flexibility in ion permeation: a case study in gramicidin A

Proteins have a flexible structure, and their atoms exhibit considerable fluctuations under normal operating conditions. However, apart from some enzyme reactions involving ligand binding, our understanding of the role of flexibility in protein function remains mostly incomplete. Here we investigate this question in the realm of membrane proteins that form ion channels. Specifically, we consider ion permeation in the gramicidin A channel, and study how the energetics of ion conduction changes as the channel structure is progressively changed from completely flexible to a fixed one. For each channel structure, the potential of mean force for a permeating potassium ion is determined from molecular dynamics (MD) simulations. Using the same molecular dynamics data for completely flexible gramicidin A, we also calculate the average densities and fluctuations of the peptide atoms and investigate the correlations between these fluctuations and the motion of a permeating ion. Our results show conclusively that peptide flexibility plays an important role in ion permeation in the gramicidin A channel, thus providing another reason--besides the well-known problem with the description of single file pore water--why this channel cannot be modeled using continuum electrostatics with a fixed structure. The new method developed here for studying the role of protein flexibility on its function clarifies the contributions of the fluctuations to energy and entropy, and places limits on the level of detail required in a coarse-grained model.

Through the channel and around the channel: Validating and comparing microscopic approaches for the evaluation of free energy profiles for ion penetration through ion channels

Microscopic calculations of free energy profiles for ion transport through biological ion channels present a very serious challenge to modern simulation approaches. The main problem is due to the major convergence problems associated with the heterogeneous landscape of the electrostatic environment in ion channels and with the need to evaluate the profile associated with the transfer of the ion from bulk water to the channel environment. This problem is compounded by the lack of reliable and relevant benchmarks that can discriminate between alternative approaches. The present study is aimed at reducing the above problems by defining benchmarks that are directly relevant to ion channels and can also give converging results. This is done by constructing a series of models of a truncated gramicidin channel with different numbers of water molecules and by comparing the profiles for going around the channel and through the channel. These discriminating models are then used to validate and compare the adiabatic charging free energy perturbation (FEP) approach combined with an umbrella sampling approach (Warshel, A. J. Phys. Chem. 1982, 86, 2218) and the potential of mean force (PMF) approach used frequently in studies of ion channels. It is found that both approaches work quite well until one moves to the case of the fully solvated channel. In this limit, the PMF approach may give different results for the overall work of going through the channel and around the channel, while the FEP approach gives physically consistent results. The present benchmark also indicates that the weighted histogram analysis method (WHAM) approach does not offer a significant advantage over earlier approaches at least as much as studies of ion channels are concerned. Finally, it is concluded that the FEP approach may be more useful in evaluating the overall barrier for moving ions from water to ion channels and that in some cases it might be beneficial to use the FEP approach for selective points along the channel and then to connect these points by PMF calculations.

Molecular dynamics of a protein surface: ion-residues interactions

Time-resolved measurements indicated that protons could propagate on the surface of a protein or a membrane by a special mechanism that enhanced the shuttle of the proton toward a specific site. It was proposed that a suitable location of residues on the surface contributes to the proton shuttling function. In this study, this notion was further investigated by the use of molecular dynamics simulations, where Na(+) and Cl(-) are the ions under study, thus avoiding the necessity for quantum mechanical calculations. Molecular dynamics simulations were carried out using as a model a few Na(+) and Cl(-) ions enclosed in a fully hydrated simulation box with a small globular protein (the S6 of the bacterial ribosome). Three independent 10-ns-long simulations indicated that the ions and the protein's surface were in equilibrium, with rapid passage of the ions between the protein's surface and the bulk. However, it was noted that close to some domains the ions extended their duration near the surface, thus suggesting that the local electrostatic potential hindered their diffusion to the bulk. During the time frame in which the ions were detained next to the surface, they could rapidly shuttle between various attractor sites located under the electrostatic umbrella. Statistical analysis of the molecular dynamics and electrostatic potential/entropy consideration indicated that the detainment state is an energetic compromise between attractive forces and entropy of dilution. The similarity between the motion of free ions next to a protein and the proton transfer on the protein's surface are discussed.

Ion conduction and selectivity in K(+) channels

Potassium (K(+)) channels are tetrameric membrane-spanning proteins that provide a selective pore for the conductance of K(+) across the cell membranes. These channels are most remarkable in their ability to discriminate K(+) from Na(+) by more than a thousandfold and conduct at a throughput rate near diffusion limit. The recent progress in the structural characterization of K(+) channel provides us with a unique opportunity to understand their function at the atomic level. With their ability to go beyond static structures, molecular dynamics simulations based on atomic models can play an important role in shaping our view of how ion channels carry out their function. The purpose of this review is to summarize the most important findings from experiments and computations and to highlight a number of fundamental mechanistic questions about ion conduction and selectivity that will require further work.

On the importance of atomic fluctuations, protein flexibility, and solvent in ion permeation

Proteins, including ion channels, often are described in terms of some average structure and pictured as rigid entities immersed in a featureless solvent continuum. This simplified view, which provides for a convenient representation of the protein's overall structure, incurs the risk of deemphasizing important features underlying protein function, such as thermal fluctuations in the atom positions and the discreteness of the solvent molecules. These factors become particularly important in the case of ion movement through narrow pores, where the magnitude of the thermal fluctuations may be comparable to the ion pore atom separations, such that the strength of the ion channel interactions may vary dramatically as a function of the instantaneous configuration of the ion and the surrounding protein and pore water. Descriptions of ion permeation through narrow pores, which employ static protein structures and a macroscopic continuum dielectric solvent, thus face fundamental difficulties. We illustrate this using simple model calculations based on the gramicidin A and KcsA potassium channels, which show that thermal atomic fluctuations lead to energy profiles that vary by tens of kcal/mol. Consequently, within the framework of a rigid pore model, ion-channel energetics is extremely sensitive to the choice of experimental structure and how the space-dependent dielectric constant is assigned. Given these observations, the significance of any description based on a rigid structure appears limited. Creating a conducting channel model from one single structure requires substantial and arbitrary engineering of the model parameters, making it difficult for such approaches to contribute to our understanding of ion permeation at a microscopic level.

Role of ions on structure and stability of a synthetic gramicidin ion channel in solution. A molecular dynamics study

We performed a molecular dynamics (MD) simulation to the investigate structure and stability of a synthetic gramicidin-like peptide in solution with and without ions. The starting structures of the MD simulations were taken from two recently solved NMR structures of this peptide in isotropic solution, which forms stable monomers or dimers in the presence or absence of ions, respectively. The monomeric structure is channel-like and is assumed to be stabilized by the presence of two Cs(+) ions bound in the channel, each one close to one channel entrance. In our MD simulations, we observed how the Cs(+) ions bind in the channel formed by the monomeric gramicidin-like peptide using implicit solvent and explicit ions with a concentration of 2 M. MD simulations were performed with and without explicit ions but with an implicit solvent model defined by the generalized Born approximation, which was used to mimic the dielectric properties of the solvent and to speed up the computations.

Homology model of the GABAA receptor examined using Brownian dynamics

We have developed a homology model of the GABA(A) receptor, using the subunit combination of alpha1beta2gamma2, the most prevalent type in the mammalian brain. The model is produced in two parts: the membrane-embedded channel domain and the extracellular N-terminal domain. The pentameric transmembrane domain model is built by modeling each subunit by homology with the equivalent subunit of the heteropentameric acetylcholine receptor transmembrane domain. This segment is then joined with the extracellular domain built by homology with the acetylcholine binding protein. The all-atom model forms a wide extracellular vestibule that is connected to an oval chamber near the external surface of the membrane. A narrow, cylindrical transmembrane channel links the outer segment of the pore to a shallow intracellular vestibule. The physiological properties of the model so constructed are examined using electrostatic calculations and Brownian dynamics simulations. A deep energy well of approximately 80 kT accommodates three Cl(-) ions in the narrow transmembrane channel and seven Cl(-) ions in the external vestibule. Inward permeation takes place when one of the ions queued in the external vestibule enters the narrow segment and ejects the innermost ion. The model, when incorporated into Brownian dynamics, reproduces key experimental features, such as the single-channel current-voltage-concentration profiles. Finally, we simulate the gamma2 K289M epilepsy inducing mutation and examine Cl(-) ion permeation through the mutant receptor.

Poisson–Nernst–Planck Theory Approach to the Calculation of Current Through Biological Ion Channels

The Poisson-Nernst-Planck (PNP) theory of electro-diffusion is reviewed. Techniques for numerical solution of the three-dimensional PNP equations are summarized, and several illustrative applications to ion transport through protein channels are presented. Strengths and weaknesses of the theory are discussed, as well as attempts to improve it via increasingly realistic evaluation of the force acting on each ion due to the protein/membrane environment.

Influence of the Water Molecule on Cation−π Interaction: Ab Initio Second Order Møller−Plesset Perturbation Theory (MP2) Calculations

The influence of introducing water molecules into a cation-pi complex on the interaction between the cation and the pi system was investigated using the MP2/6-311++G method to explore how a cation-pi complex changes in terms of both its geometry and its binding strength during the hydration. The calculation on the methylammonium-benzene complex showed that the cation-pi interaction is weakened by introducing H(2)O molecules into the system. For example, the optimized interaction distance between the cation and the benzene becomes longer and longer, the transferred charge between them becomes less and less, and the cation-pi binding strength becomes weaker and weaker as the water molecule is introduced one by one. Furthermore, the introduction of the third water molecule leads to a dramatic change in both the complex geometry and the binding energy, resulting in the destruction of the cation-pi interaction. The decomposition on the binding energy shows that the influence is mostly brought out through the electrostatic and induction interactions. This study also demonstrated that the basis set superposition error, thermal energy, and zero-point vibrational energy are significant and needed to be corrected for accurately predicting the binding strength in a hydrated cation-pi complex at the MP2/6-311++G level. Therefore, the results are helpful to better understand the role of water molecules in some biological processes involving cation-pi interactions.

Fifty Years of Progress in Ion Channel Research

Fifty years ago, ion channels were but a reasonable hypothesis. I outline some major steps in transforming this idea from a plausible description of the biological assemblies responsible for controlling passive ion transport across membranes to established fact. Important electrophysiological, biochemical, molecular biological, structural, and theoretical tools are discussed in the context of the transition from studying whole cell preparations, containing many channels, to investigating single channel behavior. Six channel families are exemplified: the model peptide, gramicidin, the acetylcholine receptor, and the sodium, potassium, calcium, and chloride channels. Some questions of current interest are posed.

Brownian Dynamics Simulation for Modeling Ion Permeation Across Bionanotubes

The principles underlying Brownian dynamics (BD), its statistical consistency, and algorithms for practical implementation are outlined here. The ability to compute current flow across ion channels confers a distinct advantage to BD simulations compared to other simulation techniques. Thus, two obvious applications of BD ion channels are in calculation of the current-voltage and current-concentration curves, which can be directly compared to the physiological measurements to assess the reliability of the model and predictive power of the method. We illustrate how BD simulations are used to unravel the permeation dynamics in two biological ion channels-the KcsA K+ channel and CIC Cl- channel.

Ions and blockers in potassium channels: insights from free energy simulations

Potassium ion channels enable efficient and selective permeation of K+ ions across nonpolar biological membranes. Here we review the results of recent free energy calculations related to the permeation of monovalent cations through K+ channels and to the channel inhibition by blocker compounds. In particular, the progress in computational studies of the bacterial KcsA channel is discussed.

Computer simulations of membrane proteins

Computer simulations are rapidly becoming a standard tool to study the structure and dynamics of lipids and membrane proteins. Increasing computer capacity allows unbiased simulations of lipid and membrane-active peptides. With the increasing number of high-resolution structures of membrane proteins, which also enables homology modelling of more structures, a wide range of membrane proteins can now be simulated over time spans that capture essential biological processes. Longer time scales are accessible by special computational methods. We review recent progress in simulations of membrane proteins.