Steered molecular dynamics simulations of Na+ permeation across the gramicidin A channel

The potential of mean forces (PMF) governing Na+ permeation through gramicidin A (gA) channels with explicit water and membrane was characterized using steered molecular dynamics (SMD) simulations. Constant-force SMD with a steering force parallel to the channel axis revealed at least seven energy wells in each monomer of the channel dimer. Except at the channel dimer interface, each energy well is associated with at least three and at most four backbone carbonyl oxygens and two water oxygens in a pseudo-hexahedral or pseudo-octahedral coordination with the Na+ ion. Repeated constant-velocity SMD by dragging a Na+ ion from each energy well in opposite directions parallel to the channel axis allowed the computation of the PMF across the gA channel, revealing a global minimum corresponding to Na+ binding sites near the entrance of gA at +/-9.3 A from the geometric center of the channel. The effect of volatile anesthetics on the PMF was also analyzed in the presence of halothane molecules. Although the accuracy of the current PMF calculation from SMD simulations is not yet sufficient to quantify the PMF difference with and without anesthetics, the comparison of the overall PMF profiles nevertheless confirms that the anesthetics cause insignificant changes to the structural makeup of the free energy wells along the channel and the overall permeation barrier. On average, the PMF appears less rugged in the outer part of the channel in the presence of anesthetics, consistent with our earlier finding that halothane interaction with anchoring residues makes the gA channel more dynamic. A causal relationship was observed between the reorientation of the coordinating backbone carbonyl oxygen and Na+ transit from one energy well to another, suggesting the possibility that even minute changes in the conformation of pore-lining residues due to dynamic motion could be sufficient to trigger the ion permeation. Because some of the carbonyl oxygens contribute to Na+ coordination in two adjacent energy wells, our SMD results reveal that the atomic picture of ion "hopping" through a gA channel actually involves a Na+ ion being carried in a relay by the coordinating oxygens from one energy well to the next. Steered molecular dynamics complements other computational approaches as an attractive means for the atomistic interpretation of experimental permeation studies.     

Energetics of ion transport in a peptide nanotube

Ion channels constitute an important family of integral membrane proteins responsible for the regulation of ion transport across the cell membrane. Yet, the underlying energetics of the permeation events and how the latter are modulated by the environment, specifically near the mouth of the pore, remain only partially characterized. Here, a synthetic membrane channel formed by cyclic peptides of alternated d- and l-hydrophobic alpha-amino acids was considered. The free energy delineating the translocation of a sodium ion was measured along the conduction pathway by means of molecular dynamics simulations. The free-energy profiles that underly the permeation of the open-ended tubular structure are shown to not only depend on the characteristics of the latter but also inherently on the location of the mouth of the synthetic channel with respect to the membrane surface.     

Calculating potentials of mean force from steered molecular dynamics simulations

Steered molecular dynamics (SMD) permits efficient investigations of molecular processes by focusing on selected degrees of freedom. We explain how one can, in the framework of SMD, employ Jarzynski's equality (also known as the nonequilibrium work relation) to calculate potentials of mean force (PMF). We outline the theory that serves this purpose and connects nonequilibrium processes (such as SMD simulations) with equilibrium properties (such as the PMF). We review the derivation of Jarzynski's equality, generalize it to isobaric--isothermal processes, and discuss its implications in relation to the second law of thermodynamics and computer simulations. In the relevant regime of steering by means of stiff springs, we demonstrate that the work on the system is Gaussian-distributed regardless of the speed of the process simulated. In this case, the cumulant expansion of Jarzynski's equality can be safely terminated at second order. We illustrate the PMF calculation method for an exemplary simulation and demonstrate the Gaussian nature of the resulting work distribution.     

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.     

Ranking protein–protein docking results using steered molecular dynamics and potential of mean force calculations

Crystallization of protein–protein complexes can often be problematic and therefore computational structural models are often relied on. Such models are often generated using protein–protein docking algorithms, where one of the main challenges is selecting which of several thousand potential predictions represents the most near‐native complex. We have developed a novel technique that involves the use of steered molecular dynamics (sMD) and umbrella sampling to identify near‐native complexes among protein–protein docking predictions. Using this technique, we have found a strong correlation between our predictions and the interface RMSD (iRMSD) in ten diverse test systems. On two of the systems, we investigated if the prediction results could be further improved using potential of mean force calculations. We demonstrated that a near‐native (<2.0 Å iRMSD) structure could be identified in the top‐1 ranked position for both systems. © 2016 Wiley Periodicals, Inc.     

A coupled reference interaction site model/molecular dynamics study of the potential of mean force curve of the SN2 Cl- + CH3Cl reaction in water

An application of the coupled reference interaction site model (RISM)/simulation methodology to the calculation of the potential of mean force (PMF) curve in aqueous solution for the identity nucleophilic substitution reaction Cl(-) + CH(3)Cl is performed. The free energy of activation is calculated to be 27.1 kcal/mol which compares very well with the experimentally determined barrier height of 26.6 kcal/mol. Furthermore, the calculated PMF is almost superimposed with that previously calculated using the computationally rigorous Monte Carlo with importance sampling method (Chandrasekhar, J.; Smith, S. F.; Jorgensen, W. L. J. Am. Chem. Soc. 1985, 107, 154). Using the calculated PMF, a crude estimate of the solvated kinetic transmission coefficient also compares well with that of previous more accurate simulations. These results indicate that the coupled RISM/simulation method provides a cost-effective methodology for studying reactions in solution.     

Reaction dynamics of Na n + in collision with molecular oxygen

Reaction dynamics of sodium cluster ions, Na _n ⁺ (n = 2–9), in collision with molecular oxygen, O₂ was investigated by measuring the absolute dissociation cross sections and the branching fractions by using a tandem mass spectrometer equipped with several octapole ion guides. The mass spectrum of the product ions show that the dominant reaction channels are production of oxide ions, Na_kO_i (i =1, 2), and intact ions, Na _p ⁺ (p < n). With increase in the collision energy, the cross section for the production of the oxide ions decreased, while that for the production of the intact ions increased. The collision-energy dependences of the cross section for the oxide formation reveals that electron harpooning from the molecule to Na _n ⁺ preludes the oxideion formation. On the other hand, the collision-energy dependences of the cross sections for the intact ion formation is explained by a hard-sphere-collision model similar to the collisional dissociation of Na _n ⁺ by rare-gas impact.     

Hydration of Y3+ ion: a Car-Parrinello molecular dynamics study

The solvation shell structure of Y3+ and the dynamics of the hydrated ion in an aqueous solution of 0.8 M YCl3 are studied in two conditions with and without an excess proton by using first principles molecular dynamics method. We find that the first solvation shell around Y3+ contains eight water molecules forming a square antiprism as expected from x-ray absorption near edge structure in both the conditions we examined. A detailed analysis relying upon localized orbitals reveals that the complexation of water molecules with yttrium cation leads to a substantial amount of charge redistribution particularly on the oxygen atoms, giving rise to the chemical shifts of approximately -20 ppm in 17O nuclear magnetic resonance relative to the computed nuclear shieldings of the bulk water.     

Binless estimation of the potential of mean force

For a system in thermal equilibrium, described by classical statistical mechanics, we derive an unbiased estimator for the marginal probability distribution of a coordinate of interest, rho( x). This result provides a "binless" method for estimating the potential of mean force, Phi = -beta (-1) ln rho, eliminating the need to construct histograms or perform numerical thermodynamic integration. In our method, the distribution that we seek to compute is expressed as the sum of a reference distribution, rho 0(x)essentially an initial guess or estimate of rho( x)and a correction term. While the method is valid for arbitrary rho 0, we speculate that an accurate choice of the reference distribution improves the convergence of the method. Using a model molecule, simulated both in vacuum and in solvent, we validate our proposed approach and compare its performance with the histogram and thermodynamic integration methods. We also discuss and validate an extension in which our approach is used in combination with a biasing force, meant to improve uniform sampling of the coordinate of interest.     

Poisson-Boltzmann calculations versus molecular dynamics simulations for calculating the electrostatic potential of a solvated peptide

We performed a very long molecular dynamics simulation of a peptide in explicit water molecules and ions and averaged the electrostatic potential caused by peptide, water and ions at eight points in the vicinity of the peptide. These electrostatic potential values were directly compared to the potential calculated by solving the non-linear Poisson-Boltzmann equation for the system, which describes the solvent using continuum electrostatics. We analyze the contribution of dielectric constant, conformational flexibility and solvation effects on the electrostatic potential at these eight points. Received: 24 April 1998 / Accepted: 4 August 1998 / Published online: 23 November 1998     

Structure prediction of cyclic peptides by molecular dynamics + machine learning

Recent computational methods have made strides in discovering well-structured cyclic peptides that preferentially populate a single conformation. However, many successful cyclic-peptide therapeutics adopt multiple conformations in solution. In fact, the chameleonic properties of some cyclic peptides are likely responsible for their high cell membrane permeability. Thus, we require the ability to predict complete structural ensembles for cyclic peptides, including the majority of cyclic peptides that have broad structural ensembles, to significantly improve our ability to rationally design cyclic-peptide therapeutics. Here, we introduce the idea of using molecular dynamics simulation results to train machine learning models to enable efficient structure prediction for cyclic peptides. Using molecular dynamics simulation results for several hundred cyclic pentapeptides as the training datasets, we developed machine-learning models that can provide molecular dynamics simulation-quality predictions of structural ensembles for all the hundreds of thousands of sequences in the entire sequence space. The prediction for each individual cyclic peptide can be made using less than 1 second of computation time. Even for the most challenging classes of poorly structured cyclic peptides with broad conformational ensembles, our predictions were similar to those one would normally obtain only after running multiple days of explicit-solvent molecular dynamics simulations. The resulting method, termed StrEAMM (Structural Ensembles Achieved by Molecular Dynamics and Machine Learning), is the first technique capable of efficiently predicting complete structural ensembles of cyclic peptides without relying on additional molecular dynamics simulations, constituting a seven-order-of-magnitude improvement in speed while retaining the same accuracy as explicit-solvent simulations.

The StrEAMM method enables predicting the structural ensembles of cyclic peptides that adopt multiple conformations in solution.     

Diffusion dynamics of the Li+ ion on a model surface of amorphous carbon: a direct molecular orbital dynamics study

Diffusion processes of the Li+ ion on a model surface of amorphous carbon (Li+C96H24 system) have been investigated by means of the direct molecular orbital (MO) dynamics method at the semiempirical AM1 level. The total energy and energy gradient on the full-dimensional AM1 potential energy surface were calculated at each time step in the dynamics calculation. The optimized structure, where Li+ is located in the center of the cluster, was used as the initial structure at time zero. The dynamics calculation was carried out in the temperature range 100-1000 K. The calculations showed that the Li+ ion vibrates around the equilibrium point below 200 K, while the Li+ ion moves on the surface above 250 K. At intermediate temperatures (300 K < T < 400 K), the ion moves on the surface and falls in the edge regions of the cluster. At higher temperatures (600 K < T), the Li+ ion transfers freely on the surface and edge regions. The diffusion pathway of the Li+ ion was discussed on the basis of theoretical results.     

Understanding the nitrate coordination to Eu3+ ions in solution by potential of mean force calculations

Coordination of nitrate anions with lanthanoid cations (Ln(3+)) in water, methanol and octanol-1 has been studied by means of molecular dynamics simulations with explicit polarization. Potential of mean force (PMF) profiles have been calculated for a mono-complex of lanthanoid nitrate (Ln(NO(3))(2+)) in these solvents using umbrella-sampling molecular dynamics. In pure water, no difference in the nitrato coordination to lanthanoids (Nd(3+), Eu(3+) and Dy(3+)) is observed, i.e. the nitrate anion prefers the monodentate coordination, which promotes the salt dissociation. Then, the influence of the nature of the solvating molecules on the nitrato coordination to Eu(3+) has been investigated. PMF profiles point out that both monodenate and bidentate coordinations are stable in neat methanol, while in neat octanol, only the bidentate one is. MD simulations of Eu(NO(3))(3) in water-octanol mixtures with different concentrations of water have been then performed and confirm the importance of the water molecules' presence on the nitrate ion's coordination mode.     

Structural and dynamical properties of the Hg2+ aqua ion: a molecular dynamics study

Molecular dynamics simulations of the Hg2+ ion in aqueous solution have been carried out using an effective two-body potential derived from quantum mechanical calculations. A stable heptacoordinated structure of the Hg2+ first hydration shell has been observed and confirmed by extended X-ray absorption fine structure (EXAFS) experimental data. The structural properties of the Hg2+ hydration shells have been investigated using radial and angular distribution functions, while the dynamical behavior has been discussed in terms of reorientational correlation functions, mean residence times of water molecules in the first and second hydration shells, and self-diffusion coefficients. The effect of water-water interactions on the Hg2+ hydration properties has been evaluated using the SPC/E and TIP5P water models.     

Investigation of structures and properties of cyclic peptide nanotubes by experiment and molecular dynamics

In order to investigate the structures and properties of cyclic peptide nanotubes of cyclo[(-D: -Phe-L: -Ala)( n = 3,4,5,6)-], cyclo[(-D: -Phe-L: -Ala)( n = 4)-] was synthesized and self-assembled to nanotubes, and its structure and morphology of the nanotube were characterized by mass spectrometry (MS), fourier transform infrared spectroscopy (FT-IR) and scanning electron microscopy (SEM). On the basis of these experimental results, the structures of cyclo[(-D: -Phe-L: -Ala)( n = 3,4,5,6)-] were characterized by molecular dynamics. In addition, the motion behaviors of H(2)O molecules in nanotubes were investigated by molecular dynamics using a COMPASS force field. Experimental results show that cyclo[(-D: -Phe-L: -Ala)( n = 4)-] peptides self-assemble into nanotube bundles. Molecular modeling results indicate that cyclic peptide nanotubes with n = 3, 4, 5 and 6 are very stable; these nanotubes have internal diameters of 5.9 A, 8.1 A, 10.8 A and 13.1 A and outer diameters of 18.2 A, 21.7 A, 23.4 A and 25.9 A respectively. Modeling results demonstrate that H(2)O molecules move in cooperation in single nanotube and they diffuse in one dimension, but they did not diffuse unilaterally due to the antiparallel ring stacking arrangement.     

Monte Carlo calculations of the mean squared force in molecular liquids

The mean squared intermolecular force (F²₁) on a molecule in a diatomic liquid has been evaluated by a Monte Carlo calculation at reduced density p^* = 0.8 and reduced temperature 7^* = 0.72. The isotropic intermolecular potential is of Lennard-Jones form; anisotropic multipolar and overlap potentials of various strengths are used. For the largest strengths studied, the anisotropic contribution to (F²₁) is about 50–60% of the isotropic one.     

Self-assembling cyclic peptides: molecular dynamics studies of dimers in polar and nonpolar solvents

The self-assembly of cyclic D,L-alpha-peptides into hollow nanotubes is a crucial mechanistic step in their application as antibacterial and drug-delivery agents. To understand this process, molecular dynamics (MD) simulations were performed on dimers of cyclic peptides formed from cyclo [(-L-Trp-D-N-MeLeu-)4-]2 and cyclo [(-L-Trp-D-Leu-)4-]2 subunits in nonpolar (nonane) and polar (water) solvent. The dimers were observed to be stable only in nonpolar solvent over the full 10 ns length of the MD trajectory. The behavior of the dimers in different solvents is rationalized in terms of the intersubunit hydrogen bonding, hydrogen bonding with the solvent, and planarity of the rings. It is shown that the phi and psi dihedral angles of a single uncapped ring in nonane lie in the beta-sheet region of the Ramachandran plot, and the ring stays in a flat conformation. Steered MD (SMD) simulations based on Jarzynski's equality were performed to obtain the potential of mean force as a function of the distance between the two rings of the capped dimer in nonane. It is also shown that a single peptide subunit prefers to reside close to the nonane/water interface rather than in bulk solvent because of the amphiphilic character of the peptide ring. The present MD results build the foundation for using MD simulations to study the mechanism of the formation of cyclic peptide nanotubes in lipid bilayers.     

Molecular dynamics simulations of the Ag+ or Na+ cation with an excess electron in bulk water

The properties of an excess electron interacting with a monovalent cation in bulk water are studied by molecular dynamics simulations. Sodium and silver cations are chosen as prototypical cases because of their very different redox properties. In both cases, mixed quantum classical molecular dynamics simulations reproduce the experimental UV-Vis spectra. In the case of silver, we observe a highly polarized neutral atom, corresponding to a dipolar excitonic state. For sodium a contact cation/electron pair is observed. Free energy curves along the cation electron coordinate are calculated using quantum Umbrella Sampling technique. The relative stability of the different chemical species is discussed.     

Structure‐Driven Selection of Adaptive Transmembrane Na+ Carriers or K+ Channels

Self‐assembled alkyl‐ureido‐benzo‐15‐crown‐5‐ethers are selective ionophores for K⁺ cations, which are preferred to Na⁺ cations. The transport mechanism is determined by the optimal coordination rather than classical dimensional compatibility between the crown ether hole and the cation diameter. Herein, we demonstrate that systematic changes of the structure lead to unexpected modifications in the cation‐transport activity and suffice to produce adaptive selection. We show that the main contribution to performance arises from optimal constraints on the conformational freedom, which are determined by the binding macrocycles, the nature of the hydrogen‐bonding groups, and the hydrophobic tails. Simple changes to the flexible 15‐crown‐5‐ether lead to selective carriers for Na⁺. Hydrophobic stabilization of the channels through mutual interactions between lipids and variable hydrophobic tails appears to be an important cause of increased activity. Oppositely, restricted translocation is achieved when constrained hydrogen‐bonded macrocyclic relays are less dynamic in a pore superstructure.