Filling and emptying kinetics of carbon nanotubes in water

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457

581

The molecular mechanism of a reaction in solution is reflected in its transition-state ensemble and transition paths. We use a Bayesian formula relating the equilibrium and transition-path ensembles to identify transition states, rank reaction coordinates, and estimate rate coefficients. We also introduce a variational procedure to optimize reaction coordinates. The theory is illustrated with applications to protein folding and the dipole reorientation of an ordered water chain inside a carbon nanotube. To describe the folding of a simple model of a three-helix bundle protein, we variationally optimize the weights of a projection onto the matrix of native and nonnative amino acid contacts. The resulting onedimensional reaction coordinate captures the folding transition state, with formation and packing of helix 2 and 3 constituting the bottleneck for folding.carbon nanotubes ͉ chemical kinetics ͉ protein folding ͉ transition-state theory ͉ Grotthuss mechanism I dentifying the molecular mechanism of a reaction in solution, such as protein folding or enzyme-catalyzed chemistry, poses serious challenges because of the large number of coupled degrees of freedom (1-5). The identification and characterization of populated intermediate states along reaction paths is only a first step. Understanding the mechanism at a molecular level requires in addition the characterization of the transitions between those populated intermediates. The goal then is (i) to identify what is common to the transitions in a rare molecular reaction (or significant subsets thereof), and (ii) to find coordinates that not only measure the progress of the reaction but also are useful to characterize the reaction dynamics. The former leads to the concept of a transition state, the latter to that of a reaction coordinate.Transition states can be thought of as configurations ''intermediate'' between reactants and products. In one widely used definition (6-11), the ensemble of transition states comprises those configurations that have an equal probability of reaching reactant and product regions. The chance of proceeding to reactants or products first can be quantified by the splitting (or commitment) probability introduced by Onsager for ion-pair recombination (12). The splitting probability is defined as the fraction of trajectories reaching the reactant region first when initiated from a given configuration with random MaxwellBoltzmann velocities, and possibly averaged over noise for stochastic dynamics. We note that one of the difficulties arising from the above definition of transition states is that splitting probabilities cannot be measured experimentally, not even in a single-molecule measurement with atomic resolution. The reason is that multiple initializations with precise atomic positions, including those of solvent molecules, are required. Here, we will show how this difficulty can be circumvented by calculating average splitting probabilities (13) from transitionpath and equilibrium ensembles that can also be measured experimentally.Fro...

Section: Resultsmentioning

confidence: 61%

Reaction coordinates and rates from transition paths

Best

Hummer

2005

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457

581

“…MD simulations of single-file water chains in CNTs yield residence times of order 0.1-1 ns (33). However, the calyx is closed at one end and is not as straight and smooth as a CNT, and these factors would slow down water exchange from the calyx.…”

Section: Discussionmentioning

confidence: 99%

A dry ligand-binding cavity in a solvated protein

Qvist

Davidovic

Hamelberg

et al. 2008

129

Ligands usually bind to proteins by displacing water from the binding site. The affinity and kinetics of binding therefore depend on the hydration characteristics of the site. Here, we show that the extreme case of a completely dehydrated free binding site is realized for the large nonpolar binding cavity in bovine ␤-lactoglobulin. Because spatially delocalized water molecules may escape detection by x-ray diffraction, we use water 17 O and 2 H magnetic relaxation dispersion (MRD), 13 C NMR spectroscopy, molecular dynamics simulations, and free energy calculations to establish the absence of water from the binding cavity. Whereas carbon nanotubes of the same diameter are filled by a hydrogenbonded water chain, the MRD data show that the binding pore in the apo protein is either empty or contains water molecules with subnanosecond residence times. However, the latter possibility is ruled out by the computed hydration free energies, so we conclude that the 315 Å 3 binding pore is completely empty. The apo protein is thus poised for efficient binding of fatty acids and other nonpolar ligands. The qualitatively different hydration of the ␤-lactoglobulin pore and carbon nanotubes is caused by subtle differences in water-wall interactions and water entropy.␤-lactoglobulin ͉ free energy simulation ͉ hydrophobic hydration ͉ magnetic relaxation dispersion G lobular proteins are stabilized by dense atomic packing and by water exclusion from the nonpolar core region. However, the packing density is not uniform and a typical protein contains approximately four cavities per 100 residues of sufficient size to accommodate at least one water molecule (1). Small nonpolar cavities are usually empty and can be regarded as packing defects, whereas small polar cavities tend to be occupied by structural water molecules that stabilize the folded protein by H-bonding to otherwise unsatisfied peptide partners (2). Large cavities are frequently linked to protein functions, such as ligand binding and transport, membrane translocation, and enzyme catalysis. Some of these large cavities are lined exclusively or predominantly by nonpolar sidechains. The question whether such large nonpolar cavities are empty or contain water molecules is of fundamental biological importance (3-8).The principal tool of structural biology, x-ray crystallography, cannot easily resolve this issue. There are three potential problems. (i) Because water molecules interact only weakly with the nonpolar cavity walls, they tend to be positionally disordered. As a result of such delocalization, the electron density of the water molecules may fall below the detection limit. (ii) At a typical resolution limit of Ϸ2 Å, a contaminating nonpolar ligand may be misinterpreted as a water cluster (9, 10). (iii) For structures determined at cryogenic temperatures, the hydration status of the cavity may be altered by re-equilibration during the flash cooling process (11).In favorable cases, water molecules in nonpolar protein cavities can be inferred from intermolecular nuclear O...

“…In particular, for fast water-mediated proton transport it is essential that water molecules form tightly connected chains in which all hydrogen bonds point into the same direction, up or down (11). At ambient conditions of 298 K and Ϸ1 bar pressure, both requirements are found to be fulfilled for short (6,6) carbon nanotubes immersed in water (9) and the filling/emptying of the hydrophobic tube interior has been studied in detail recently (3,(17)(18)(19)(20). We find here that subnanometer-wide tubes up to Ϸ0.1 mm length are densely filled by ordered water chains, thus providing excellent mediators of proton transfer over macroscopic dimensions.…”

mentioning

confidence: 99%

Macroscopically ordered water in nanopores

Köfinger

Hummer

Dellago

2008

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162

181

Water confined into the interior channels of narrow carbon nanotubes or transmembrane proteins forms collectively oriented molecular wires held together by tight hydrogen bonds. Here, we explore the thermodynamic stability and dipolar orientation of such 1D water chains from nanoscopic to macroscopic dimensions. We show that a dipole lattice model accurately recovers key properties of 1D confined water when compared to atomically detailed simulations. In a major reduction in computational complexity, we represent the dipole model in terms of effective Coulombic charges, which allows us to study pores of macroscopic lengths in equilibrium with a water bath (or vapor). We find that at ambient conditions, the water chains filling the tube are essentially continuous up to macroscopic dimensions. At reduced water vapor pressure, we observe a 1D Ising-like filling/emptying transition without a true phase transition in the thermodynamic limit. In the filled state, the chains of water molecules in the tube remain dipole-ordered up to macroscopic lengths of Ϸ0.1 mm, and the dipolar order is estimated to persist for times up to Ϸ0.1 s. The observed dipolar order in continuous water chains is a precondition for the use of nanoconfined 1D water as mediator of fast long-range proton transport, e.g., in fuel cells. For water-filled nanotube bundles and membranes, we expect anti-ferroelectric behavior, resulting in a rich phase diagram similar to that of a 2D Coulomb gas.1D confinement ͉ antiferro-electric ͉ carbon nanotubes ͉ proton transfer ͉ phase transition T he 1D wires formed by water in molecularly narrow pores are central to the function of many biomolecules, offer new possibilities for technological applications, and provide model systems to study the unique properties of dimensionally confined fluids (1). Proteins filled by water wires, such as aquaporins and gramicidin A, mediate the transport of water, protons, or ions across biological membranes (2, 3). Inspired in part by biology, narrow water-filled pores have been suggested as promising building blocks for high-selectivity/high-flux membranes in molecular separation devices and fuel cells (4-8). As an example of the rich properties of nanoscopically confined fluids, 1D water wires in carbon nanotubes have been found to exhibit first-order like drying transitions (9).Carbon nanotubes provide nearly ideal systems to study water in 1D confinement. Their smooth interior cavity surface interacts in a relatively nonspecific way with water molecules, confining them to a narrow, almost cylindrical volume. In pores with subnanometer diameters, the water molecules arrange themselves in a single-file structure, linked by hydrogen bonds. Such ordered chains of water molecules were found to permit rapid water flow (4, 9), and mediate proton transfer with mobilities exceeding those in bulk water (10-13).A key factor for the unique properties of 1D confined water is the nearly perfect molecular order, both translationally and orientationally, with uninterrupted chains of wate...