Water Conduction through a Peptide Nanotube - ACS Publications

Article pubs.acs.org/JPCC

Water Conduction through a Peptide Nanotube Jeffrey Comer,† François Dehez,† Wensheng Cai,‡ and Christophe Chipot*,†,§ †

Laboratoire International Associé Centre National de la Recherche Scientifique et University of Illinois at Urbana−Champaign, Unité Mixte de Recherche No. 7565, Université de Lorraine, B.P. 70239, 54506 Vandoeuvre-lès-Nancy cedex, France ‡ College of Chemistry, Nankai University, Tianjin 300071, People’s Republic of China § Beckman Institute for Advanced Science and Engineering, University of Illinois at UrbanaChampaign, 405 North Mathews, Urbana, Illinois 61801, United States S Supporting Information *

ABSTRACT: When inserted into lipid bilayers, synthetic channels formed by cyclic peptides of alternated D- and L-α-amino acids have been shown to modulate the permeability of the cell wall, thereby endowing them with potential bactericidal capability. Details of the underlying energetics of the permeation events remain, however, only fragmentary. Water conduction in a peptide nanotube formed by eight cyclo-(LW)4 subunits embedded in a fully hydrated palmitoyloleylphosphatidylcholine bilayer has been investigated using molecular-dynamics simulations with a time-dependent bias. The topology of the reconstructed free-energy landscape delineating the transport of water mirrors the arrangement of the cyclic peptides in the open-ended tubular structure. Within the nanotube, the small, periodic free-energy barriers, on the order of kBT, arising between adjacent peptide subunits, are suggestive of unhampered translocation. It still remains that translational diffusion of water in the hollow cylindrical cavity is necessarily affected by its interaction with the accessible polar moieties of the constituent D- and L-α-amino acids. By combining diffusivity measurements with the free-energy landscape, we put forth a reaction-rate theory to describe the conduction kinetics of water inside the peptide nanotube.

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INTRODUCTION In the realm of organic and inorganic tubular architectures, peptide nanotubes, which result from the self-assembly of cyclic subunits made of alternated D- and L-α-amino acids1 by dint of a network of intermolecular hydrogen bonds,2 constitute promising nanostructures eminently relevant for biological applications.3 At the structural level, successive D- and L-αamino acids cause all side chains of the cyclic peptides to point outward, resulting in a planar, flat conformation, germane for stacked arrangements into an antiparallel, β-sheet-like, openended tubular cavity.4,5 Upon insertion into a lipid environment, such hollow architectures are expected to modulate the permeability of the otherwise impervious cell membrane, thus endowing them with potential bactericidal activity.6−8 As a function of their amino-acid sequence, these peptide nanotubes self-assemble, adopting drastically distinct permeation modes. Amphipathic cyclic peptides self-assemble into tubular structures that subsequently permeate and diffuse in the lipid bilayer and aggregate into bundles, thereby forming macropores.9 Similarly amphipathic sequences, albeit involving titratable residues, organize into nanotubes that bind to the surface of the bilayer, discriminating between eukaryotic and prokaryotic, Gram-positive, and Gram-negative membranes, prior to partitioning into the lipid environment to disrupt the latter.10 In sharp contrast, predominantly hydrophobic cyclic peptides form independently stable synthetic channels, which span the biological membrane as individual entities,5 the hollow © 2013 American Chemical Society

tubular cavity offering a hospitable environment for the transport of water11−13 and ions.14−18 Simple individual tubular structures resulting from the selfassembly of hydrophobic cyclic peptides provide a convenient framework for the study of permeation events across membranes. In the present contribution, diffusion of water in a peptide nanotube is investigated, combining one- and twodimensional free-energy and Bayesian-inference calculations to reconcile the thermodynamics and kinetics of the process at hand. Although the particular sequence of amino acids chosen herein has been the object of prior attention,11−13 painting a qualitative structural picture of the transport properties of the peptide nanotube, detail of the underlying thermodynamic and dynamic features of water permeation remains is still incomplete. In particular, previous work13 has painted a fragmental picture of the free-energy landscape for water diffusion along the longitudinal direction of the synthetic channel and partial characterization of the radial distribution of water therein. Here, we obtain a free-energy landscape along both the longitudinal and radial directions for the complete nanotube. A full description of the transport process, however, requires kinetic Received: September 3, 2013 Revised: October 31, 2013 Published: November 1, 2013 26797

dx.doi.org/10.1021/jp4088223 | J. Phys. Chem. C 2013, 117, 26797−26803

The Journal of Physical Chemistry C

Article

distance between the tagged water molecule from û in the plane orthogonal to it. The definitions of the coordinates are shown schematically in Figure 1A. The reaction pathway spanned by the permeant extended over 40 Å in the ζ-direction and 3 Å in the ρ-direction. The free-energy change along ζ and ρ was measured using the adaptive biasing force method,27,28 which relies upon the integration of the average force acting on the model reaction coordinate, either one- or two-dimensional. The gradient of the free energy is estimated locally during the simulation, in bins 0.1 Å wide, thereby providing a continuous update of the biasing force. When applied to the relevant subset of atoms, this time-dependent bias converges to a Hamiltonian devoid of a net average force on the model reaction coordinate. It follows that all the values embraced by the latter are in principle sampled with an equal probability, hence improving markedly the accuracy of the computed free-energy differences. For numerical efficiency, the reaction pathway was stratified in four windows with a 4 Å overlap region, in each of which independent trajectories as long as 120 ns were generated, amounting to a total simulation time of 480 ns for both the one- and two-dimensional free-energy calculations. The mean-force profile along the longitudinal axis, ⟨fζ(ζ)⟩, was calculated from the longitudinal component of the twodimensional mean force, ⟨fζ(ζ,ρ)⟩ and the two-dimensional free-energy surface, G(ζ,ρ), by

information. Here, we show that variation of water transport kinetics within the open-ended tubular structure is appreciable. Water permeation through the peptide nanotube provides a convenient test bed of biased sampling methods for computing free energies and kinetic properties. Because of the relatively large number of water molecules within the channel (typically around 30), we are able to make direct comparisons between brute-force equilibrium simulations and biased simulations, providing validation of the biased sampling methods. These methods could then be applied to systems where the bruteforce route is less feasible. For instance, without the assistance of biased sampling techniques, multimicrosecond simulations would be required to obtain sufficient sampling for determining the free energy and the diffusivity of an ion permeating the channel.

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METHODS AND COMPUTATIONAL DETAILS The peptide nanotube utilized herein consisted of eight stacked, where the underlined letters denote D-amino acids.19 The synthetic channel was embedded in a model membrane formed by 78 palmitoyloleylphosphatidylcholine (POPC) lipids hydrated by 3215 water molecules, which altogether corresponds to a simulation cell of dimensions equal to about 55 × 55 × 68 Å3 at thermodynamic equilibrium. A snapshot of the simulation system is shown in Figure 1. Molecular-Dynamics Simulations. The molecular-dynamics simulations were performed in the isothermal−isobaric ensemble with the program NAMD20 and the all-atom CHARMM27 force field.21 To reduce computational cost, the tails of the POPC lipids were represented by a united-atom model22 adapted from the CHARMM36 force field for lipids.23 The united-atom lipid parameters employed in this work were specifically developed to be compatible with the all-atom CHARMM force field,2 including its protein component utilized to model the peptide nanotube. The temperature and the pressure were maintained, respectively, at 300 K and 1 atm, using Langevin dynamics and the Langevin piston method.24 Long-range electrostatic forces were taken into account by means of the particle mesh Ewald approach.25 The equations of motion were integrated with the r-RESPA multiple time-step propagator26 with an effective time step of 2 and 4 fs for shortand long-range interactions. All covalent bonds between heavy and hydrogen atoms were constrained to their equilibrium distance. To preserve the network of hydrogen bonds between adjacent cyclic peptides, the distance separating the nitrogen atoms of the amino moieties and the carbon atoms of the carbonyl groups was restrained to its equilibrium value using soft harmonic potentials. Figure S1 (Supporting Information) shows that the hollow tubular structure remains intact in the absence of restraints during a 40-ns equilibrium simulation. Additional similar potentials were applied to prevent the lipid head groups from entering the mouths of the nanotube, as in preliminary simulations, such a behavior was occasionally witnessed, slowing down convergence of the free-energy calculations. Free-Energy Calculations. Two model reaction coordinates were utilized to investigate water permeation in the synthetic channel. A one-dimensional transition coordinate, ζ, was defined as the distance separating the center of mass of a tagged water molecule from the center of mass of the hollow cylindrical structure, projected onto the longitudinal axis, û, of the latter. Moreover, a two-dimensional transition coordinate was devised by adding to ζ a radial component, ρ, i.e., the

⟨fζ (ζ )⟩ =

(∫ dρ e

−1

)

−βG(ζ , ρ)

∫ dρ e−βG(ζ ,ρ)⟨fζ (ζ ,ρ)⟩ (1)

where β = (kBT)−1. ⟨fζ(ζ)⟩ was then integrated to yield the onedimensional free-energy profile. The above formula seemed to produce smoother results than integrating G(ζ,ρ) directly, which might be explained by the fact that G(ζ,ρ) is not given directly by the adaptive biasing force method but must be inferred by inexact methods.17 All one-dimensional free-energy profiles have been shifted so that the average value of G(ζ) on the interval −10 Å ≤ ζ ≤ 10 Å is equal to zero. Diffusivity Calculations. The one-dimensional diffusivity as a function of the position along the pore long axis, D1(ζ), was estimated by the Bayesian inference scheme described in Comer et al.29 A slight variation of the scheme was employed, wherein the mean system force, fsys(ζ), was assumed to be conservative, so that it could be replaced with the gradient of the potential of mean force, −∇G1(ζ). Out of convenience, the Monte Carlo procedure was applied to G1(ζ) rather than fsys(ζ). This variation can be shown to have no perceptible effect on the final results. The functions G1(ζ) were interpolated by piecewise cubic polynomials from the discrete grid {ζi}ni=1, with a grid spacing of Δζ = 0.15 Å on the interval −19 Å ≤ ζ ≤ 19 Å. To ensure that D1(ζ) is always positive, it was interpolated by piecewise linear functions on the grid and its domain was constrained to D1(ζ) > 0.1 Å2/ns, which is orders of magnitude smaller than the expected values. The scale-invariant prior distribution Pscale = ∏i1/D(ζi) was applied to the diffusivity, whereas the smoothness of both G1(ζ) and D1(ζ) was enforced by including the prior, 26798



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Figure 1. (A) Synthetic channel formed by eight stacked cyclo-(LW)4 units embedded in a hydrated palmitoyloleylphosphatidylcholine lipid bilayer. L-Tryptophan and D-leucine residues of the nanotube are shown in yellow and magenta, respectively. Lipids are represented by translucent gray tubes, and the phosphate and choline in the headgroup are highlighed by gray translucent spheres. For clarity, water molecules and lipids appearing in front of the nanotube are not shown. The ζ-axis, with its origin at the center of mass of the nanotube, is aligned along the long axis of the nanotube. Its orientation fluctuates with respect to that of the bilayer normal, z. The ρ-axis is the radial coordinate orthogonal to ζ. (B) Onedimensional free-energy profiles delineating the translocation of a water molecule through the channel. The black curve was derived from an equilibrium simulation by G(ζ) = −kBT ln λ(ζ) where λ is the linear density of water along ζ. The red curve is the result of applying an adaptive biasing force along the ζ axis to a single tagged water molecule. An image of the peptide rings below the graph shows the correspondence of the free energy to the nanotube structure.

⎛ [D (ζ ) − D (ζ )]2 1 i 1 i−1

n

psmooth =

∑ exp⎜− i=2

−

⎝

2εD

excluded displacements of the tagged water molecule having initial positions within 1 Å of the window boundary, which is tantamount to excluding all segments of the trajectories for which the chance of reaching the boundary within Δt is less than 0.3%.

2

[G1(ζi) − G1(ζi − 1)]2 ⎞ ⎟ 2εG 2 ⎠

■

(2)

RESULTS AND DISCUSSION Here, we describe the free-energy change for a water molecule permeating along the longitudinal axis of the synthetic channel, and connect this free-energy change to the topology of the open-ended tubular structure. Next, we present the electrostatic potential of the peptide nanotube, which affects and is affected by the orientation of the water molecules found in the pore. Determining the free energy as a function of both longitudinal and radial coordinates is subsequently shown to render an enhanced picture of the typical permeation events and the water−peptide interactions at play. Finally, we consider the diffusivity of a water molecule within the hollow tubular structure, which again can be related to the topology of the latter, and estimate mean-first-passage times for movements in the direction of the longitudinal axis. One-Dimensional Free-Energy Profile. The one-dimensional free-energy landscape characterizing water permeation in the open-ended tubular structure is depicted in Figure 1. The potential of mean force along ζ was calculated in two ways. First, by calculating the linear density of all water molecules along ζ, averaged over an equilibrium simulation, and, second, by integrating the mean biasing force acting on the tagged water molecule in the adaptive biasing force simulations. As can be seen in Figure 1, the two curves are in good agreement and demonstrate a a pronounced ruggedness, which reflects the geometry of the synthetic channel. In particular, the features of the free-energy profile correlate with positions along ζ conventionally referred to as the α-plane, i.e., the plane of the Cα carbons of the cyclic peptides, and the midplane, i.e., the plane passing through the middle of the gap between two cyclic peptides. As discussed below in the subsection “Error Analysis of the Free Energies”, the statistical error for free-energy differences across the entire domain in the adaptive biasing force calculation is roughly 0.4 kcal/mol. Thus, any discrepancy between the PMF derived from the water density in an

where εD = 3 Å2/ns and εG = 0.75 kcal/mol. In Figure S2 (Supporting Information), we compare −ΔG1(ζ) resulting from adaptive biasing force and that inferred from the Bayesian scheme. The results are consistent within the statistical uncertainty, which was computed as described in the subsection “Error Analysis of the Free Energies” of Results and Discussion. Furthermore, Figure S3 (Supporting Information) demonstrates that performing the Bayesian analysis with G1(ζ) constrained to the output of the adaptive biasing force method does not yield significantly different results. As per the Bayesian scheme presented in Comer et al.,29 the motion of the water molecules was assumed to satisfy the discretized Brownian dynamics propagator, ζt +Δt = ζt + βD1(ζt ) f (ζt ,t ) + (2D1(ζt )Δt )1/2 gt + ∇D1(ζt )Δt

(3)

where f(ζ,t) = −∇G1(ζ) + f bias(t) is the sum of the system force and the biasing force and gt is a random number distributed as N(μ=0,σ2=1). The choice of the time interval, Δt, is crucial. Accurate discretization requires that D1(ζ) and f1(ζt,t) change little during Δt, so that, in practice, too large values for Δt result in poor resolution of D1(ζ). On the other hand, at very short times, inertia becomes significant, and consecutive displacements, ζt+Δt − ζt, are no longer uncorrelated, as would be assumed for overdamped Brownian dynamics. We chose Δt = 120 fs, which has adequate spatial resolution while exhibiting small but not negligible correlation of consecutive displacements. The autocorrelation function of the displacements is shown in Figure S4 (Supporting Information). Because the data were obtained from stratified simulations with four separate windows, care was taken to discard segments of the trajectories near the boundaries of the windows where the potential energy restraining the tagged water molecule to the desired range of ζ might ultimately affect the results. We 26799



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Figure 2. (A) Instantaneous snapshot illustrating the biorientated water organization in the nanotube. Cyclic peptides are depicted in a stick representation using the following color code: cyan for carbon, red for oxygen, blue for nitrogen, and white for the hydrogen forming hydrogen bonds with adjacent rings (all other hydrogens being omitted for clarity). Purple spheres represent the phospholipid headgroups. (B) Mean angle between the dipole of water molecules and the longitudinal axis of the tube, averaged for each midplane region (between the peptide rings). The analysis is provided for each individual window employed for calculating the one-dimensional free-energy profile and further averaged over all the windows. (C) Cross-sectional view of the electrostatic potential mapped in three dimensions and averaged over a 1 ns trajectory. The cross-section is made along a plane containing the longitudinal axis of the nanotube and the bilayer normal.

While engaged in a network of hydrogen bonds that forms the antiparallel β-sheet scaffold of the synthetic channel, these chemical bonds also participate in the solvation of the permeant. As a water molecule moves toward the next cyclic peptide, the space available to accommodate other hydrating water molecules diminishes, resulting in an increase of the free energy. The free energy subsequently decreases as the permeant further translocates to the α-plane of the cyclic peptides, where it is coordinated by water molecules lying in the two contiguous midplanes. Water Orientation and the Electrostatic Field. When trapped in the nanotube, water molecules adopt a preferential orientation depending on their position along the pore axis. In the upper and the lower region of the self-assembled structure, water dipoles are distributed asymmetrically, pointing toward the interior of the membrane, as can be observed in Figure 2A. These observations are essentially in line with the data reported by Liu et al.13 Figure 2 seems to suggest that there is a statistically significant difference between the average orientation of water molecules within the simulation where the tagged water molecule lies within window 4 and those simulations where the tagged water molecule is found in another window. Due to the large fluctuations in the instantaneous orientation and in the absence of substantially longer simulations, it is difficult to determine without ambiguity whether the collective orientation is, indeed, evolving slowly. The apparent difference in the water orientation profile between the simulation where the tagged water molecule lies in window 4 and that where it is found in other windows (Figure 2B) does not appear to lead to markedly discrepant free-energy profiles, as the main characteristics in window 1, i.e., −20 < z < −8 Å, and in window 4, i.e., 8 < z < 20 Å, are similar (Figure 1B). Yet, if there are changes in the collective orientation of the water molecules enclosed in the nanotube, which is one conclusion that could be drawn from the difference in the orientation profiles, and if these changes correspond to radically different free energies, our importancesampling calculations may reflect quasi nonergodicity. A dually oriented water-file motif was also reported for aquaporins.30−32 In such channels, water dipoles are oriented from the center of the protein toward the bulk. Water organization inside the nanotube can be better understood by

equilibrium simulation and that derived from the adaptive biasing force algorithm in Figure 1B cannot be considered significant. The deepest free-energy minima appear at the midplanes, i.e., at each of the seven gaps between the eight cyclic peptides. Each minimum is surrounded by barriers of about 1 kBT. The barriers within the synthetic channel are sufficiently low, i.e., on the order of kBT, to guarantee the unhindered transport of water. Each barrier may be seen as two peaks that have partially coalesced, with an appreciably small minimum, i.e., typically less than kBT, separating them. The latter minimum coincides exactly with the position of the α-plane of the cyclic peptides, i.e., near the center of mass of the peptide cycles. Such secondary minima are present at the center of the inner cyclic peptides, whereas the two rings forming the mouths of the synthetic channel clearly do not show this feature. Noteworthily, the distance separating adjacent minima or maxima is consistent with the geometry of the open-ended tubular structure, notably the average distance between consecutive cyclic peptides, found in the case of cyclo-(LW)4 to be equal to 4.7 Å.5 It is worth noting that the present free-energy profile is consistent with that calculated by Liu et al.13 for the central cyclic peptides in the midst of the tubular structure. As is particularly visible in the potential of mean force derived from the equilibrium density, translocation of water in the cylindrical cavity is prefaced at both ends of the latter by an energetic penalty of about 1 kBT, i.e., the cost for partially dehydrating the permeant as it moves from the bulk liquid to the particular confined environment of the nanotube examined in this work. The barrier that a water molecule needs to overcome to enter the peptide nanotube is similar to that for a sodium ion,16 albeit substantially lower than that for the bulkier chloride ion18typically over 20 kBT. As illuminated by Figure 1, water permeation proceeds through hopping between adjacent free-energy barriers, which may be interpreted as successive energetic penalties for transiently dehydrating the permeant. At the midplanes, coordination of the latter is optimal, in particular not only by other water molecules but also by the carbonyl and the amino functional groups of the cyclic peptides. Interestingly enough, both the −CO and −N−H chemical bonds are somewhat tilted with respect to the longitudinal axis of the tubular cavity. 26800



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Figure 3. Two dimesional free-energy landscapes. (A) Surface plot showing the free energy as a function of the ζ, the coordinate along the nanotube axis, and ρ, the radial coordinate perpendicular to the axis. This free energy was calculated from simulations with the adaptive biasing force applied both the ζ and ρ dimensions. (B) Free-energy profile along ζ comparing the results of the one- (red, dashed curve) and two-dimensional (blue, solid curve) adaptive biasing force calculations. For the two-dimensional case, the profile along ζ is calculated by integration over ρ (see Methods and Computational Details).

deviations from ideal symmetry are smaller than the overall error associated with the calculation (see subsection “Error Analysis of the Free Energies”), it is not clear whether these small deviations from ideal symmetry are due to statistical error, are deficits in ergodic sampling, or are real features due to the intrinsically asymmetric arrangement of the nanotube in the bilayer. Error Analysis of the Free Energies. The statistical uncertainty in the free-energy landscapes was estimated as follows. For each bin, the force of the system on the transition coordinate was collected at 40 ps intervals. Because force values display significant correlation on this time scale, we applied the renormalization group method of Flyvbjerg and Petersen33 to obtain uncertainty estimates for each bin. The free-energy profiles are obtained by integrating numerically the negative of the mean system force over each bin. The uncertainty in the free-energy difference between two different values of the reaction coordinate, ΔG = G1(ζk) − G1(ζj), therefore, depends on the uncertainty values of all intervening bins. If we assume independence of the fluctuations in each bin, the uncertainty in such free-energy differences can be calculated by

considering the electrostatic signature of the cylindrical cavity shown in Figure 2C. The cross-sectional view of the electrostatic potential reveals that the latter quantity increases when moving from the bulk to the interior of the tubular structure. The resulting electric field, pointing toward the interior of the channel, is distributed antisymmetrically with respect to the region encompassing the second and the third gaps, the latter orientation being responsible for the dually oriented water-file organization along the longitudinal axis. Two-Dimensional Free-Energy Landscape. Although water does not diffuse in the synthetic channel as a truly single file, the eight stacked cyclic peptides altogether form an appreciably confined space to the extent that the free-energy landscape effectively mirrors the topology of the tubular cavity. Compared to its one-dimensional counterpart, the twodimensional free-energy landscape of Figure 3A sheds additional light on the permeation event, revealing that the minimum-action path followed by water is not the longitudinal axis, û, of the peptide nanotube, but instead, a narrow valley snaking 1.0−1.5 Å away from û. When ρ < 1.0 Å, the permeant dwells in the neighborhood of the α-plane of the cyclic peptides, whereas it preferentially remains in the midplane region when ρ > 1.5 Å. These two repetitive alternate metastable states correspond to periodic oblate free-energy minima that extend in the radial direction and intersect roughly when 1.0 < ρ < 1.5 Å. Liu et al.13 observed a similar behavior by comparing the water density as a function of ρ for two values of ζ, corresponding to the mid- and α-planes. Notwithstanding the simplicity of the molecular assembly at hand and the significant length of the present importancesampling simulations, the free-energy landscapes of Figures 1 and 3 are not perfectly symmetric. Some deviation from ideal symmetry is to be expected, given that the nanotube is not symmetrically arranged within the bilayer, as can be clearly seen in Figure 2. Not too surprisingly, the free-energy profile with respect to ζ = 0 Å is most nearly symmetric over the range of the model reaction coordinate corresponding to the core of the tubular cavity, i.e., the four central cyclic peptides. Asymmetry is more conspicuous toward the mouths of the synthetic channel, remembering that while the inner cyclic peptides are engaged on both of their faces in eight hydrogen bonds, termini units are more loosely noncovalently bonded to the rest of the cylindrical structure. Still, symmetry defects never exceed onethird of a kBT across the reaction pathway. Because the

⎞1/2 ⎛ k 2 δ ΔG(ζj ,ζk) = Δζ ⎜⎜∑ δfsys (ζi)⎟⎟ ⎠ ⎝ i=j

(4)

where Δζ is the size of the bins and δfsys is the uncertainty of the mean system force for each bin, obtained by the method of Flyvberg and Petersen. For the calculations here, δfsys is in the range 0.09−0.4 kcal/(mol Å). Using this formula to calculate the uncertainty of the one-dimensional profile in Figure 1A, we calculate a free-energy difference of 0.6 ± 0.1 kcal/mol between the minimum near ζ = 0 and the top of the barrier to the right, i.e., ζ = 1.6 Å. If we instead calculate the free-energy difference between this minimum and the top of the barrier to the left, i.e., ζ = −1.6 Å, we obtain 0.5 ± 0.1 kcal/mol. Although there appears to be a small asymmetry in the height of these barriers, the difference cannot be considered significant. The upper limit for the estimated uncertainties of free-energy differences on the one-dimensional profile comes from considering the freeenergy difference across the entire sampled range, for which we obtain 0.4 kcal/mol. Diffusivity Profiles and Mean First Passage Times. Figure 4 shows the one-dimensional translational diffusivity profile as a function of ζ. To ensure that the effect of the biasing 26801



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permeant along the transition pathway, τa→b, from one region, a, of the synthetic channel to another, b:34 τa → b =

∫a ×

b

dζ exp[β ΔG(ζ )]D−1(ζ )

∫a

ζ

dζ′ exp[−β ΔG(ζ′)]

(5)

Applying eq 5 to the free-energy and diffusivity profiles given in Figure 1B and diffusion, respectively, we calculate a mean time of about 0.32−0.38 ns for the transfer between any two midplane free-energy minima. To cross essentially the entire nanotube, from ζ = −19 Å to ζ = 19 Å, we estimate a mean time of 2.2 ns. For comparison, we directly calculated first passage times from trajectories of water molecules in a 14 ns equilibrium simulation. Averaging over the 40 complete permeation events observed, we obtain a mean first passage time of 2.5 ± 0.2 ns, in nice agreement with the kinetic model.

Figure 4. One-dimensional diffusivity profile as a function of ζ calculated by a Bayesian inference scheme. The black curve is calculated from the trajectories of all water molecules in the nanotube during an equilibrium simulation, and the red curve is calculated from the trajectories and biasing force on the tagged water molecule in four adaptive biasing force runs covering different parts of the domain.

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force has been properly accounted for in the Bayesian scheme, we compare the results accrued by applying this scheme to both the trajectory of the tagged water molecule in adaptive biasing force simulations and the trajectories of all water molecules in the nanotube in an equilibrium simulation. The results are not identical, but similar enough to have a negligible effect on such quantities as the mean first-passage time.34 Overall, the diffusivity inside the nanotube is lower than that for bulk TIP3P water35 with the temperature and Langevin thermostat used in this work, namely 514 ± 5 Å2/ns. It is worth noting that the self-diffusion coefficient of TIP3P, the standard water model of the CHARMM force field,21 is substantially higher than in experiments. The computed diffusivities and passage times should thus be considered semiquantitative and are likely to be higher and lower, respectively, than in reality. The diffusivity within the channel in the present set of simulations varies from about 60% to 80% of the bulk value. This reduction of the diffusivity is somewhat less than in water channels, such as AQP1, namely from 40 to 80 Å2/ns,36 which might be expected due to the fact that the diameter of the nanotube is considerably larger and permeation is not strictly single-file. Indeed, in Figure 4, diffusivity minima appear where the water molecule is most tightly constricted by the nanotube, i.e., at the eight α-planes. On the other hand, the diffusivity maxima appear at the seven midplanes. For example, z = 0 corresponds to a midplanei.e., it lies in between two cyclic peptidesand is characterized by a free-energy minimum (Figure 1A), and a diffusivity maximum (Figure 3). Although there is a rule of thumb that free-energy minima often correspond to diffusivity minima, this rule appears to be violated for the nanotube, likely because a water molecule lying in the midplane experiences much less lateral confinement than in the α-plane. Diffusion is most rapid at the center of the gaps between the cyclic peptides. Another major feature of diffusion is that the diffusivity appears to be highest at the center of the nanotube, decreasing slightly toward the edges, and then falling precipitously to global minima at the α-planes of the first and last peptide units. The position-dependent diffusivity shown here is only moderately smaller than the values reported by Engels et al.37 for a homologous nanotube (410−480 Å2/ns), where the difference might be ascribed to the use of different thermostatting algorithms. A significantly lower value for the diffusivity within the nanotube was suggested by Tarek et al.11 Translational diffusivity may be combined with the freeenergy landscape to infer the mean first passage time of the

CONCLUSION In the present contribution, we have determined the thermodynamic and kinetic properties of water conduction through a peptide nanotube, highlighting the advantages of considering multiple degrees of freedomnamely longitudinal, radial, and orientationalas well as considering positiondependent diffusivity. The former demonstrated the tendency for an indirect, zigzag-like permeation path and changes in orientation in the course of the permeation process, whereas the latter showed that diffusion was the fastest in the midplanes, at equidistance between the cycles, where the water molecules are less constrained spatially, and the slowest within the αplanes, in particular at the two mouths of the peptide nanotube. Put together, the thermodynamics and kinetics suggest that entering the open-ended tubular structure constitutes the slowest part of the water permeation process. Such an insight could help in the design of peptide nanotubes endowed with specific properties, such as controlled permeabilities. The set of simulations reported here admittedly only involved a single synthetic channel with a given pore radius. Yet, hollow tubular structures formed by larger, 12 amino-acid cyclic peptides have been found to yield a similar electrostatic signature, either in water or in membrane environments. Thus, some of the results in this contribution, such as the description of the orientational order, are likely to extend to slightly larger structures as well. Furthermore, the methods presented and validated here could also be leveraged in understanding passive transport mechanisms in biological channels as well as in engineering particular behavior, such as ion specificity, in both synthetic and biological channels. The present simulations also are anticipated to help improve the accuracy of continuum approaches targeted at investigating conduction events across membranes.

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ASSOCIATED CONTENT

S Supporting Information *

Figures of comparison of the root-mean-square deviation of the Cα atomic positions, mean force profile, diffusivity profiles, and autocorrelation function of the displacements of the tagged water molecule. This information is available free of charge via the Internet at http://pubs.acs.org/. 26802



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AUTHOR INFORMATION

Corresponding Author

*C. Chipot: e-mail, [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge the Centre Informatique National de l’Enseignement Supérieur and the Grand Équipement National de Calcul Intensif for generous provision of computer time. They are also grateful to the Délégation Régionale à la Recherche et à la Technologie de Lorraine for funding. C.C. is indebted to the Embassy of France in China for travel support. The Cai Yuanpei program is also appreciatively acknowledged for its support of the international collaboration between the research groups of C.C. and W.C.

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