Water Diffusion Behaviors and Transportation Properties in

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J. Phys. Chem. B 2010, 114, 12183–12192

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Water Diffusion Behaviors and Transportation Properties in Transmembrane Cyclic Hexa-, Octa- and Decapeptide Nanotubes Jian Liu,† Jianfen Fan,*,† Min Tang,† Min Cen,† Jianfeng Yan,‡ Zhao Liu,‡ and Weiqun Zhou† College of Chemistry, Chemical Engineering, and Materials Science, Soochow UniVersity, Suzhou 215123, China, and School of Computer Science & Technology, Soochow UniVersity, Suzhou 215006, China ReceiVed: April 30, 2010; ReVised Manuscript ReceiVed: August 1, 2010

Molecular dynamics simulations have been performed on three transmembrane cyclic peptide nanotubes, i.e., 8 × (WL)n)3,4,5/POPE (with uniform lengths but various radii) to investigate the radial dependences of the water-chain structures, diffusions, and transportation properties. The diffusions of individual water molecules and collective coordinates of all the channel-water in the three systems are certified as unbiased Brownian motions. From the very good linear relationships between MSDs and time intervals, the diffusion coefficients and transportation permeabilities have been deduced efficiently. Under the hydrostatic pressure differences across the membrane, a net unidirectional water flow rose up, and the osmotic permeabilities were determined. The ratios of the osmotic and diffusion permeabilities (pf/pd) were examined for all the three channels. 1. Introduction The efficient transportation of water through a cell membrane requires special pathways to cross the lipid bilayer. These pathways are provided by the nanopores embedded in the lipid membrane.1,2 For some biological channels, such as aquaporins (AQPs)3-10 and gramicidin A (GA),11,12 permeations of water play primary functions. Water permeations through channels of molecular dimensions are significant not only for the fundamental biological processes but also in the medical and industrial applications and have been the topics of theoretical and experimental studies over years. However, to explore the active transportation of water through a nanochannel is technically difficulty or even impossible at atomistic resolution. Molecular dynamics (MD) simulations allow us to track the motions of particles in such microscopic pores and offer a detailed view of the relevant contributions to the channels’ permeability and selectivity characteristics. Recently, more realistic computational strategies modeled the structural and dynamical properties of water in nanoscopic model pores, such as single-walled carbon nanotubes (SWCNT),13-21 helicalpeptide ones like GA,22,23 etc.,24,25 revealing concerted water motions and density distribution patterns. Cyclic peptide nanotubes (cyclo-PNTs, PNTs) are a class of synthetic nanomaterials formed by the self-assembly of closed peptide rings with an even number of alternating D- and L-amino acid residues.26 These peptide rings can stack on the top of one another through an H-bond network, producing an open-ended, hollow tubular structure that can provide a wide range of structural and functional capabilities of biological relevancy. The outer surface properties of PNTs can be controlled simply by adjusting the structures of amino acids employed as the peptide subunits.27 Appropriate hydrophobic side chains of PNTs are essential for partitioning the channels into lipid bilayers.28-31 The assembling PNTs in a membrane can act as artificial channels for transporting water,30,32-36 ions,37-40 and small molecules or mediating * To whom correspondence should be addressed. E-mail: jffan1305@ 163.com. † College of Chemistry, Chemical Engineering, and Materials Science. ‡ School of Computer Science & Technology.

the transportations of biologically relevant molecules41 across the lipid bilayer. The osmotic permeability (pf) and diffusion permeability (pd)42 are the two key characteristics accounting for the transportation through water channels. They both can be measured experimentally. pf is measured through the application of an osmotic pressure difference, whereas pd is measured through isotopic labeling, e.g., by isotopic replacement or by monitoring nuclear spin states. Alternatively, MD simulation provides another way to acquire these permeability properties. Usually, an equilibrium MD simulation can yield the pd value of a transmembrane channel. Zhu et al. proposed a nonequilibrium MD (NEMD) method by inducing a hydrostatic pressure difference across a membrane,5 allowing the pf value to be determined. Furthermore, by analyzing the hopping rates7 or collective coordinates16 of channel-water, Zhu et al. got the pf value from an equilibrium MD running directly. The single-file theory15,43 was believed to be fully understood and properly characterized. However, the transportation properties of non-single-files are acquainted scarcely. In this paper, three PNTs with different radii were used to investigate the dependences of water diffusion behaviors and transportation properties on the channel radii. For the first time, the osmotic and diffusion permeabilities of PNTs are reported. The ratios of the osmotic and diffusion permeabilities (pf/pd) were examined for these channels, and some relevances on the water-chain structures have been found. 2. Materials, Methods and Theory Modeling Systems. PNTs are a class of novel nanomaterials whose diameters and lengths can be controlled simply by adjusting the number of residues employed.27 A peptide nanotube composed of eight subunits would be long enough to span the average thickness (∼38 Å) of a biological membrane.20,36,44 Composed of the amino acid residues Trp (W) and Leu (L), the PNTs possess the desired hydrophobic surface characteristics for embedding themselves into a lipid bilayer. To understand the distinguished characteristics of the water-chain structures and permeability properties of the nanotubes with varying radius, three PNTs, i.e., 8 × (WL)n)3,4,5 were designed in this work, wherein the underlined letters correspond to the D-amino acids.

10.1021/jp1039207  2010 American Chemical Society Published on Web 09/01/2010

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TABLE 1: Main Parameters of Water Diffusion in the Octa-PNT Obtained from Four NEMD Simulations Denoted as sim1-sim4a sim1 -1

-1

F (kcal mol Å ) ∆P (MPa) j (ns-1) Vdrift (Å ns-1) F′ (kcal mol-1 Å-1) Dζ (kcal mol-1) jv (Å3 ns-1) pf (Å3 ns-1)

sim2

sim3

sim4

0.01 0.02 0.03 0.04 116 232 348 464 1.71 ( 0.15 3.86 ( 0.12 6.13 ( 0.07 8.24 ( 0.13 2.6 5.9 9.3 12.6 0.0152 0.0304 0.0456 0.0607 0.36 0.32 0.30 0.30 52.1 117.8 186.9 251.3 63.3 71.6 75.8 76.6

a Scaling of units: force (F), 1 kcal mol-1 Å-1 ) 69.5 pN; pressure (∆P), 1 kcal mol-1 Å-3 ) 6950 MPa; diffusion coefficient (D), 1 Å2 ns-1 ) 10-7 cm2 s-1; permeation coefficient (pd, pf), 1 Å3 ns-1 ) 10-15 cm3 s-1.

The internal diameters of the hexa-, octa- and decapeptide nanotubes are about 6, 8, and 10 Å, respectively. POPE (1palmitoyl-2-oleoylglycerophosphoethanolamine) was selected to model the lipid bilayer of a membrane. The simulation system of 8 × (WL)4/POPE was taken from our recent computational results.36 Those of 8 × (WL)n)3,5/POPE systems were generated by replacing the 8 × (WL)4 with 8 × (WL)n)3,5 ones, respectively. For the system of 8 × (WL)5/POPE, four lipid units were removed to avoid the bad connections between the lipid units and PNT. The two systems were equilibrated over 10 ns with the same simulation parameters as ref 36. The z-dimensions of the periodic unit cells for the systems with n ) 3-5 converged to 94, 97, and 97 Å after the 45, 50, and 45 ns equilibrium MD runs, respectively. The CHARMM27 force field45 and TIP3P water model46 were employed for all the MD simulations. Periodic boundary conditions were applied in three dimensions. The full electrostatic interactions were treated by the particle mesh Ewald (PME) approach.47 The Nose´-Hoover Langevin piston48 method was used to maintain the pressure of the MD simulation box at 1 bar during the simulation runs, and Langevin dynamics was used to control the temperature at 310 K. All the simulations were performed with the program NAMD 2.6.49 Analysis and visualization were made using the molecular graphics program VMD 1.8.7.50 Force Protocol. In this work, nonequilibrium MD simulations were also carried out for water permeation in the systems of 8 × (WL)n)3,4,5/POPE. A constant force (F) along the +z direction was applied on the oxygen atom of each water molecule within |z| > 18 Å in bulk, to induce a pressure difference across the membrane. This pressure difference is something like the osmotic pressure difference. If the pressure gradient is sufficiently small, the resultant water flux will be linear to the driving force. However, to obtain a significant transmembrane net flux of water to minimize the statistical uncertainty in a simulation, one would have to impose a quite large driving force. In fact, the resultant pressure difference is many times greater than those under physiological conditions. The four constant external forces, i.e., 0.01, 0.02, 0.03, and 0.04 kcal mol-1 Å-1, were used in this work, generating four transmembrane pressure differences, denoted as sim1 to sim4 in Table 1, respectively. To keep the membrane from being swept away, the lipid N and P atoms were constrained along the z-direction with spring constants of 1 kcal mol-1 Å-2. All atoms are free to move in the x- and y-directions. The CR atoms of the PNTs were also constrained, with spring constants of 1 kcal mol-1 Å-2. The simulations were performed under constant volume conditions to avoid any problems of too rapid dilation of the system volume by the exerted force. The other configurations are the same as

those in the equilibrium simulations. Starting from the last frames of the equilibrations, a total of 45 ns NEMD simulations for 8 × (WL)n)3,4,5/POPE were initiated and 30 000 frames were stored, respectively. Permeability Coefficients. The mobility of water molecules inside a channel was quantified by the osmotic (pf) and diffusion permeabilities (pd). pd quantifies the exchange of individual water molecules between the two reservoirs at equilibrium, while pf relates the net water flux through a channel due to the difference of the osmolyte concentrations (or equivalently the pressure difference) between the two compartments connected by the channel. pd and pf are the intrinsic properties of a water channel and are independent of tracer concentration and osmotic pressure. In particular, pd is easily measured in an equilibrium MD simulation. It quantifies the equilibrium flux through a channel and was extracted from the simulation as6

pd ) q0Vw

(1)

where q0 is the rate of the unidirectional complete traversing of water in a channel and Vw is the volume of a water molecule, with a constant of 30.5 Å3. The pf value of a water channel can be obtained from either a nonequilibrium MD simulation by inducing an osmotic or hydrostatic pressure between the two sides of a membrane, or an equilibrium MD simulation using the analysis of collective coordinates. For the former, pf can be computed by the following equations.5

lp ) jVw /∆P pf ) (RT/Vw)lp

(2)

where j (ns-1) is the number of water molecules transferring through a channel every nanosecond and lp is referred to the hydraulic permeability. For every frame, a water molecule entering the channel at the lower boundary or exiting at the upper boundary contributes to the total flux number by +1/2, vice versa would be -1/2. Alternatively, pf of a non-single-file channel can be determined from an equilibrium MD simulation using the concept of collective coordinates,16 defined as the time dependent cumulative displacements of water molecules in a channel (S(t)), normalized to the channel length.

dn )

∑ dzi/L

(3a)

i∈S(t)

where L is the time-averaged channel length. In this work, L has a constant of 33.5 Å. The collective coordinate n was obtained by accumulating in every time step. n quantifies the net amount of water permeation, and its trajectory n(t) describes the time-evolution of the permeation. In our practice, the integral-differential equation is taken in the following form: ∆tft+∆tn )

∑

i∈S(t) i∈S(t+∆t)

∆zi /L +

1 ×( 2

∑

+

∑

i∈S(t)

i∉S(t)

i∉S(t+∆t)

i∈S(t+∆t)

)∆zi /L

(3b) where ∆t is the time interval between each frame of a MD trajectory, 1.5 ps of ∆t was used here.

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J. Phys. Chem. B, Vol. 114, No. 38, 2010 12185 occasionally. As the channel radius increases, the structure and diffusion properties are close to those in bulk. Water molecules in the deca-PNT with the largest radius are much like the bulk water (Figure 1c). The radius of a PNT is an important parameter for a water channel. A number of methods could get it. For example, radius can be determined from the distance between the opposite CR atoms in a peptide ring. In this paper, the effective radius for water movement in a channel was computed from cylindrical distribution function (CDF),32,36 which is a two-dimension radial distribution as a function of the distance of a partical from the channel axis. It is defined as

gcy )

Figure 1. Water-chains in the hexa- (a), octa- (b) and deca-PNTs (c), respectively. Only the backbones of PNTs are shown for clarity.

At equilibrium, n(t) can be described as a one-dimensional unbiased random walk. 〈n(t)〉 ) 0 denotes no net amount of water permeation through a channel. When t is much longer than the velocity correlation time of n, the MSD of n, 〈n(t)2〉, obeys the Einstein relation: 〈n(t)2〉 ) 2Dnt, where Dn is the diffusion coefficient of the collective coordinate. We therefore computed the osmotic permeability (pf) from

pf ) DnVw

(4)

3. Results and Discussions To understand the distinguished characteristics and permeability properties of the nanotubes with varying radius, three PNTs, i.e., 8 × (WL)n)3,4,5 by changing the number of the residues of each peptide ring, were designed, while the ring numbers of these PNTs were the same, which allows us to study the transmembrane channels with the different radii and same lengths. Water Chains in the Three Transmembrane PNTs. The knowledge of the structure-permeability relationship is an essential prerequisite for understanding the narrow channels. Water chains display different characteristics in the nanotubes with different radii and are the foundations of water diffusion properties. Water molecules in the hexa-PNT form a pure single-file chain, shown in Figure 1a. The interior-narrow channel prevents any exchanging or crossing events between individual channelwater molecules. Distinguished from the other single-file chains,6,8,11-13,15,18,22,23 these is no H-bonded-chain in the hexaPNT nanotube, as most water molecules locate at gaps, with a distance of about 4.8 Å between adjacent water molecules, showing no direct interactions between them. The water chain in the octa-PNT forms a novel 1-2-1-2 file structure32,36 shown in Figure 1b. Exchanging or crossing events occurred

〈∆N〉 2πs∆slF

(5)

where 〈∆N〉 is the average number of centers of mass (COMs) of water molecules in a cylindrical-shell of thickness ∆s and radius s from the axis of the nanotube, l is the tube length for analysis, and F is the numerical density of water molecules in bulk. Figure 2 illustrates the CDFs of water COMs in the three PNTs. The effective radius for water COM movement (Rwt) in a nanotube was defined as the position of the intersection point between the horizontal axis and inflectional tangent of the distal part of a CDF curve. The values obtained for the transmembrane hexa-, octa-, and decapeptide nanotubes are 0.7, 2.3, and 3.4 Å, respectively. Further consideration of the vdW radius of a water molecule (1.4 Å) deduces that the effective radii for water molecule movement in the three PNTs are 2.1, 3.7, and 4.8 Å, respectively. Information about the mobility of water inside the three PNTs can be superficially observed by tracking the time-dependent evolutions of the z-coordinates of individual channel-water molecules. The result for water in the octa-PNT is shown in Figure 3. It is evident that the diffusion mode of water molecules in the octa-PNT channel is not restrained to the single-file motion but includes exchanging and crossing events. It is of interest to distinguish the correlated from intrinsic movement of water molecules along the axis. To understand the concertedness of the water molecule movement through a channel, the nearest-neighbor cross correlation coefficient c(z)6,8 was introduced. It was defined as an average over all pairs of adjacent water molecules, i and j visiting z (meaning that the midpoint of the two correlated water oxygen atoms is z, in the octa- and decapeptides, two water molecules within a distance of 5.5 Å are counted):

c(z) )

〈δziδzj〉 (〈δziδzi〉〈δzjδzj〉)1/2

(6)

where δzi ) zoi(t + δt) - zoi(t), |zoi(t + δt/2) - zoj(t + δt/2)| < dww, z ≡ [zoi(t + δt/2) + zoj(t + δt/2)]/2, and δt ) 9 ps. Water-water correlations in the three transmembrane cyclic hexa-, octa- and deca-PNTs resolved along the channel axis z are presented in Figure 4. The correlation curves show that water translocations inside the channels are correlated and that the correlations are highest in the midplane regions of the hexaPNT channel. The average correlation coefficient 〈c〉 can be defined in the following:

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Figure 2. (a) Top views of the hexa-, octa- and deca-PNTs, respectively. (b) Cylindrical distribution functions of water molecules as functions of distances for water molecules from the nanotube axes in the three PNTs.

Figure 4. Water-water correlation coefficients between the nearestneighbor water molecules in the hexa-, octa- and deca-PNTs. Figure 3. Time evolutions of water molecule locations in the octaPNT during the equilibrium MD run. A running of 300 ps is given to improve clarity. Individual water molecules are assigned with different colors, the upper and lower boundaries of the PNT are (16-17 Å.

〈c〉 ) 〈L〉-1

∫PNT c(z) dz

(7)

where L is the average length of a PNT. The average correlation coefficients for water movement in the three PNTs are computed as 0.4, 0.2, and 0.03, respectively. Comparing with the corresponding data (0.5) in the single-file-channels of AqpZ and

GlpF8 indicates that the 〈c〉 value is not certified to be 0.5 for a single-file channel. As the channel radius increases, the water chain changes from single-file to non-single-file, the value of 〈c〉 decreases. The hexa-PNT was found to exhibit more pronounced singlefile characteristics. Overall, this channel features more concerted water motion than others do. The peaks of the correlation curve are in the gaps of the hexapeptide nanotubes while the valleys locate in R-plane regions. The fluctuating period is exactly the gap spacing of the channel. The correlation curve suggests that

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if two adjacent water molecules locate at the two adjacent R-plane regions, they must move in the same direction to avoid collision at a gap; but if the two are in the adjacent gaps, they can vibrate in much less concertedness. Water molecules in the octa- and deca-PNTs are less confined along the z-direction and more mobile laterally than in the hexaPNT. The average correlation in the octa-PNT is computed to be 0.2, indicating that there is still some concertedness of water motion in this non-single-file PNT. A slight fluctuation also can be found in the octa-PNT, with the fluctuating period as a half of the gap spacing. In the deca-PNT, water movement nearly loses all its correlation. Figure 4 depicts the reduced correlation with the increase of channel radius. Diffusion Behaviors of Water from Equilibrium MD. The translational mobility of a water molecule can be given by calculating the mean square displacement (MSD) of its COM, which is a measure of the average distance it travels and can be expressed as follows,

MSD(t) ) 〈r2(t)〉 )

〈

N

∑

1 |r (t) - ri(0)| 2 N i)0 i

〉

(8)

In this equation, t denotes the correlation time and r stands for the coordinate vector of the COM of a molecule. ri(t) - ri(0) is the (vector) distance traveled by molecule i over time interval t. The squared magnitude of this vector is averaged (as indicated by angle brackets) over many such time intervals. Usually, this quantity is also averaged over all the molecules in a channel at both the start and end moments, summing i from 1 to N, and divided by N. The motion of channel water molecules can be well described as a random walk, for which the MSD increases only linearly with time. The growth rate of the MSD depends on the interaction between the channel-water molecules and the internal surface of peptide nanotubes, rather than the running length or time interval over which the straight line (MSD ∼ t) fitted. The limiting slope of MSD(t), considered for time intervals sufficiently long for it to be in the linear regime, is related to the self-diffusion constant (D), according to the Einstein equation: D ) MSD(t)/2dt, where d is the dimension of the space. Figure 5 shows the MSD curves along x-, y-, and z-dimensions and the total one. In the hexa-PNT, it was observed that after an initial subdiffusion regime, the evolution was linear with the time for t g 50 ps. The root of MSD at this point is about 1.5 Å, which is much less than the distance between the channel-water molecules, indicating that it still follows the Fickian diffusion mechanism rather than the single-file diffusion one17 (MSD ∝ t). The Fickian diffusion coeffient can also be used to describe the dynamics of water molecules in the hexaPNT. All curves tend to satisfy the Einstein equation in the longtime run. From the resultant slope of MSD(t), the diffusion coefficients (D) of water molecules in the three filled PNTs can be determined. Linear fits in the time range of 50-750 ps yield the tracer diffusion coefficients of the hexa-, octa- and decaPNTs as 5.0 ( 0.5, 62 ( 8, and 130 ( 15 Å2 ns-1 along the z-dimension, respectively, and nearly zero along the lateral directions of x and y. A simple relationship seems to exist between the diffusion coefficients (D) and effective radii (Rwt), viz. D ∝ Rwt2 (see Figure 6). This can be coincidentally explained by combining the Hagen-Poiseuille equation (j ∝ ∆P × R-4)24,51 with the Einstein formula (D ∝ ζ-1)52. However, the continuum theory is not expected to be applicable for such systems within

Figure 5. MSD curves of water molecules constrained in the nanotubes of the hexa- (a), octa- (b), and deca-PNTs (c), respectively. The MSDs were calculated in the longitudinal (z-axis) and lateral (x- and y-axes) directions, respectively. The total MSDs are nearly in superposition with MSDs_z, while MSDs_x and MSDs_y are approximately zero. The MSDs of water in the PNTs demonstrate that the movements of water molecules along the z-axis are one-dimensional diffusions.

Figure 6. Linear relationship between channel-water self-diffusion coefficients (D) and the square of the effective radius of water COM movement (Rwt2) in the three PNTs. Error bars computed from the standard deviation of five sample bins are shown in vertical lines.

molecular dimensions,14 and the value of viscosity coefficient of water is unreasonable to extended from macroscopy to microscopy. Furthermore, this proportionality would not be kept in the channels with relative large radii, as all the properties of channel-water are converged to those in the bulk (∼800 Å2 ns-1, the diffusion coefficient of TIP3P water under 310 K53). Therefore, we decline that this relationship is nothing but a

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numerical coincidence. Systematic studies should employ nanotubes with various degrees of radii, such as SWCNT, heli-PNTs but rather cyclo-PNTs. To quantify the exchange of individual water molecules between the two reservoirs at equilibrium, a “permeation event” was defined as a complete transportation of a water molecule through a channel from one reservoir to the other. Let q0 be the total number of such permeation events in unidirection per unit time. Our 50 ns equilibrium MD run shows that 36 water molecules enter the octa-PNT channel and exit from the opposite entrance. If we assume that the permeation is a Poissan process, the variance of permeation event would be equal to its expectation. Thus we can estimate the standard deviation of the permeation event to be 6 (361/2), giving q0 ) 0.72 ( 0.12 ns-1. We also found such events over the last two 15 ns trajectories in the +z- and -z-directions to be 11, 11, 13, and 9, respectively. Normal distribution analysis shows that the value of the permeation event is 11 ( (8/3)1/2, giving q0 ) 0.73 ( 0.13 ns-1, which is nearly the same as above. Similarly, about 145 water molecules travel through the deca-PNT unidirectionally, acquiring the q0 value of 3.2 ( 0.3 ns-1. With the q0 value, it is easy to compute the diffusion permeability (pd) of a channel according to eq 1. The pd value obtained in this way for the octa- and deca-PNTs are 22.0 ( 3.6 and 97.6 ( 9.1 Å3 ns-1, respectively. According to eq 1, to get the value of pd, it is necessary to know the total events of diffusion permeation in a simulation. However, for water in the hexa-PNT, this permeability is too small (actually only 0.7 events) to get a significant net flux of water across the nanotube in the simulation. In our practice, only none or a few permeation events can be observed in the 45 ns equilibrium MD trajectories of water molecules in the hexa-PNT. Fortunately, there is a relationship between the self-diffusion coefficient (D) and q0. Dimensional analysis shows that the rate of a water molecule traveling from one end to the other one would be proportional to D/L2. If 〈N〉 water molecules travel parallel, such a rate would be 〈N〉 times one single molecule’s:

q0 ) R

〈N〉D L2

(9)

where 〈N〉 is the average water number in a channel. Results in the octa- and deca-PNTs with L of 33.5 Å, q0 of 0.72 and 3.2 ns-1, D of 62 and 130 Å2 ns-1, and 〈N〉 of 22 and 51 well match this equation, respectively, when we set R to be 1/2. This equation provides an alternative way to calculate the “permeation rate” (q0) from the self-diffusion coefficient (D), further giving the pd of a channel. The pd values of the hexa-, octa-, and deca-PNTs obtained in this way are 0.54, 18.6, and 94.6 Å3 ns-1, respectively. The osmotic permeability (pf) relating to the net water flux through a channel due to a difference in osmolyte concentration (or equivalently a pressure difference) between the two compartments connected by the channel, can be obtained from an equilibrium simulation by applying analysis of the collective coordinates (n) of channel-water molecules. Figure 7a illustrates the time evolution of n for water in the octa-PNT channel, showing the characteristics of an one-dimensional unbiased Brownian motion. Running through the full simulation trajectories gives the MSDn ∼ t curves in the hexa-, octa-, and decaPNTs, depicted in Figure 7b-d, all exhibiting the MSD of n, 〈n(t)2〉, very well linear to time. From the slopes of the three curves, the diffusion coefficients of the collective coordinates, Dn, can be computed from the Einstein relation 〈n(t)2〉 ) 2Dnt.

Figure 7. Time-evolution of the collective coordinates (n) of channelwater molecules in the octa-PNT obtained from an equilibrium MD simulation (a). n is accumulated by ∆n between every frame. MSD curves of n in the hexa- (b), octa- (c), and deca-PNTs (d) show very good linear relationships with time. The three curves were computed through the full trajectories.

Thus, the pf values of the hexa-, octa-, and deca-PNTs were computed to be 3.97, 66.2, and 496.9 Å3 ns-1 with eq 4, respectively. Transportation Properties of Water under Hydrostatic Pressures. To gauge the extent to which the applied forces used in our simulations mimic the hydrostatic pressure differences across an actual lipid bilayer, the normal pressure produced by the applied force in the 8 × (WL)4/POPE system as a function of depth in the bulk was calculated and shown in Figure 8. Consequently, a net water flux j through the transmembrane channel can be induced. Four NEMD simulations were carried out for water diffusion in 8 × (WL)4/POPE by applying the forces of 0.01, 0.02, 0.03, and 0.04 kcal mol-1 Å-1, respectively. During the simulations, the applied forces produce water density gradients in bulk water, increasing with the z-direction, as illustrated in Figure 9. The induced transmembrane hydrostatic pressure differences and the

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Figure 8. (Right) our scheme to induce a hydrostatic pressure difference in a nonequilibrium MD simulation. In a periodic system, the unit cell is replicated in three dimensions. Thus, water layers and membranes alternate along the z-direction, defined as the membrane normal. A constant force F along the z-direction is exerted on all water molecules with |z| > 18 Å, generating a pressure difference at the two entrances of a nanotube: ∆P ) nF/A, where n is the number of water molecules exerted by a force and A is the area of the membrane. (Left) pressure profile produced by the applied force of 0.02 kcal mol-1 Å-1 along the z-axis of the modeled system 8 × (WL)4/POPE. Data were collected from a 3 ns pressure profile simulation44 and stored in 60 bins of thickness 1.5 Å. The pressure increasing in the bulk region and the difference (232 MPa) between the two entrances can be read from the figure.

Figure 9. Water density distributions along the z-direction in the bulk water region between two membranes in the adjacent periodic unit cells. Data points are taken from four NEMD simulations, where external forces of 0.01, 0.02, 0.03, and 0.04 kcal mol-1 Å-2 along +z were applied on all bulk water molecules, respectively. The density is measured by averaging the number of water molecules within a slab of 0.1 Å thickness over the full trajectory. A line with best-fit slope for the data points is also shown. Departure from the linearity at the low end of curve 4 means that the force acting on a water molecule is probably big somehow. Fortunately, the osmotic properties concerning different forces were almost unaffected in spite of this nonlinearity. The natural values of a water molecule’s volume (Vw) and numerical density at 310 K are respectively 30.5 Å3 and 0.0328 Å-3, which agree well with our simulations.

corresponding net water fluxes are summarized in Table 1. Two sets of j, obtained from counting the number of water molecules

Figure 10. Linear dependence of the water flux rates (j) on the applied pressure differences (∆P). The data are taken from Table 1. Error bars are estimated from the standard deviations of 10 noncorrelated results analyzed from the last 15 ns of each trajectory.

entering to/exiting from the PNTs and tracking the timeevolution of Dn, show almost no distinctive difference. According to eq 2, the net water flux (j) should be linear to ∆P, which is summarized in Table 1 and illustrated in Figure 10. Then, the osmotic permeability (pf) can be determined by the ratio of j and ∆P. The values of pf in the octa-PNT nanotube are 63.3, 71.6, 75.8, and 76.6 Å3 ns-1 from the four NEMD simulations, respectively. The osmotic permeability was determined to be pf ) 71.8 ( 6 Å3 ns-1, averaging the four simulations. Comparison with the result of pf (66.5 Å3 ns-1) obtained from the above equilibrium MD simulation shows that there is no significant difference between them. The pf values

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reported in this paper cannot be certified because there are no experimental results reported up to the time. However, there are some experimental and simulation results for some similar systems. For example, the pf value of the AQP1 monomer, a transmembrane water-channel with similar size, was determined to be 71 Å3 ns-1 by MD simulation.7 Some experiments have reported its pf value ranges from 54.39 to 117 Å3 ns-1.10 Usually, the agreements of permeabilities between MD simulations and experiments are favorable, providing a sensitive evidence of a simulation. The osmotic permeability (pf) is an intrinsic property of a water channel and independent of the transmembrane hydrostatic pressure difference (∆P). For the sake of simplicity, only an external force of 0.019 kcal mol-1 Å-1 was applied on the bulk water molecules in the systems of the hexa- and deca-PNTs to mimic the transmembrane pressure differences of 232 and 229 MPa in the two systems, separately. Counting the numbers of permeation events, or monitoring the evolutions of collective coordinates (n) in the full NEMD simulation trajectories, the flux rates (j) were given as 0.12 and 25.2 ns-1 for water molecules in the hexa- and deca-PNTs, respectively. Using eq 2, the osmotic permeabilities (pf) of the hexa- and deca-PNTs were obtained as 2.3 and 473.3 Å3 ns-1, respectively. The results are consistent with the above-obtained pf values of 4.0 Å3 ns-1 for the hexa-PNT and 496 Å3 ns-1 for the deca-PNT by employing the diffusion of collective coordinates. The relatively large difference of pf for the hexa-PNT is probably due to not enough simulation time for this narrow nanotube. Only a dozen of the flux events during the 45 ns simulation probably bring relatively large statistic uncertainty. Fortunately, the MSD of the collective coordinate (n) of channel-water molecules in the hexa-PNT is well aligned with the statistics time. Thus, the pf of the hexa-PNT obtained from the collective coordinate analysis was thought to be more reliable than the data directly from NEMD simulation. When water molecules translate through a channel at a net flux, the drift vector of water molecule (Vdrift) can be obtained from jL/〈N〉, where L is the total length of the channel and 〈N〉 is the mean number of water molecules in the channel. For each water molecule crossing the channel from the reservoir with higher hydrostatic pressure to the other, a work of ∆PVw will be done by the external force to the system and then dissipated in the Langevin simulation. If we suppose there is a friction (F′) acting on each channel-water molecule to drive a steady net flux of water, the F′ would be excepted as ∆PVw/L. Then, the fluid friction coefficient (ζ) of water molecules in a channel would be

ζ)

∆PVw /L F′ ) Vdrift jL/〈N〉

(10)

The drift vectors (Vdrift), friction forces (F′), and friction coefficients (ζ) of water obtained in the octa-PNT are summarized in Table 1. With the self-diffusion coefficient D of 62 Å2 ns-1, we obtained the average value of Dζ as 0.32 kcal mol-1. Similarly, the Dζ values for water in the hexa- and deca-PNTs were computed to be 0.32 and 0.25 kcal mol-1, respectively. The Einstein formula52 written as Dζ ) kBT predicts the value of Dζ would be 0.62 kcal mol-1 (when T ) 310 K). Our results are about half of the predictions by the Einstein formula. It seems that this universal equation does not fully meet the conditions in this work. The reasons are probably that there is not a real external force driving on the channel-water molecules,

TABLE 2: Radii and Some Transportation Properties in the Three PNTs Rca (Å) Rwt (Å) 〈N〉 D (Å2 ns-1) Dζ (kcal mol-1) pf (Å3 ns-1) pd (Å3 ns-1) (〈N〉 + 1)/(pf/pd)

hexa-PNT

octa-PNT

deca-PNT

3.4 0.7 7.3 5.5 ( 0.5 0.35 3.97 0.54 1.13

4.6 2.3 22 62 ( 8 0.28 66.2 22.0 7.36

5.8 3.4 51 130 ( 15 0.25 496.9 97.6 9.85

a The Rca radius is half the diameter of a PNT, which is defined as the distance between the opposite R-carbon atoms, while Rwt is the effective radius for water molecule movement in a PNT. Further consideration of the vdW radius of a water molecule (1.35 Å) can deduce the effective radii for water molecule movement in the three PNTs. D is the self-diffusion coefficient of water molecules in a filled PNT. All data in this table are obtained from the equilibrium MD simulations.

and the friction force mainly comes from the two ends of the channel.22 The analysis here is only meaningful at the estimated magnitude. pf/pd Ratios in the Three PNTs. One of the most celebrated equalities characterizing the conduction properties of a water channel is the ratio of the osmotic (pf) and diffusive permeabilities (pd). The ratio pf/pd can be measured experimentally for a channel of constant length without knowing the density of channels in contrast to pf and pd individually. For a singlefile channel, a generally accepted result is

pf /pd ) 〈N〉 + 1

(11)

where 〈N〉 is the number of water molecules in a channel. This is raised by Berezhkovskii,43 verified by Zhu,15 and widely agreed at present for single-file water channels.7,8,22,23 That is, if the channel radius is narrow enough and channel-water forms a single-file in it, the value of (〈N〉 + 1)/(pf/pd) would be a constant of 1. For water diffusion in the hexa-PNT with a small radius, the number of water molecules (〈N〉) in it is counted to be 7.0. It is found that the value of (〈N〉 + 1)/(pf/pd) for this nanotube is 1.13 (Table 2). Water molecules move in a singlefile pattern, although there are no H-bonds inside the waterchain. They satisfy eq 11 entirely. As the radius of a channel increases, the water-chain deviates the single-file pattern, eq 11 is no longer applicable. Portella et al.23 carried out a systematic study on the helical peptide systems with arbitrarily adjusted diameters and gave a promotion of eq 11. They found that when the radius increases, and the water-chain in the channel is no longer a single-file form, the value of (〈N〉 + 1)/(pf/pd) increases linearly with the radius of a channel and approximately equals to dwwN/L, where dww is the typical water-water distance, established to be 2.75 Å. In this work, the numbers of water molecules (〈N〉) in the octa- and deca-PNTs are 22 and 51, respectively. The values of dwwN/L are 1.84 and 4.35, while those of (〈N〉 + 1)/(pf/pd) are 7.36 and 9.85, respectively. Obviously, there are large deviations between (〈N〉 + 1)/(pf/pd) and dwwN/L. Therefore, the phenomenological formula proposed by Portella et al. is not still applicable for our systems. The contradiction probably comes from the fact that the radii of the cyclo-PNTs in the R-plane and midplane regions are different, while the heli-PNTs are radium-uniform systems. It

Cyclic Hexa-, Octa-, and Decapeptide Nanotubes is difficult to find well-defined radius values to compare the radius-dependent properties for the cyclo-PNTs and heli-PNTs. The radii values of the cyclo-PNTs used in this work are mainly contributed from the midplane regions. It was found that 〈N〉, dwwN/L, and some other parameters are smaller than those in the heli-PNTs with the corresponding radii. However, the values of (〈N〉 + 1)/(pf/pd) are much larger than those in the corresponding heli-PNTs. Although the cyclo-PNTs and heliPNTs both consist of β-sheet-like internal surfaces, possessing similar affinities, their microstructures are quite different, which are important to the water-chain patterns. The periodic radius and hydrophilicity providing a series of binding sites in the cyclo-PNTs channels cause transportation mechanisms much different from those in the heli-PNTs. For example, it has been found that the pf values of the cyclo-PNTs are in inverse proportion to the channel length54 rather than independent.22 The formula raised by Portella et al. in the heli-PNTs is no longer applicable for the cyclo-PNTs. In this work, it was also found that the value of (〈N〉 + 1)/(pf/pd) increases with the radius of a channel but goes far out of the dwwN/L relationship. However, the relationship between the pf/pd ratio and radius of a cycloPNT channel needs further study. Deeper and more systematic studies will be held in the future. 4. Summary A systematic MD study has been performed for water in the three transmembrane cyclic peptide nanotubes with different inner diameters, expanding the discussion of water permeability to non-single-file nanochannels. From equilibrium MD simulations, the self-diffusion coefficients (D) and rates of complete translocations (q0) of water molecules in the hexa-, octa-, and decapeptide nanotubes were studied, and the diffusion permeabilities (pd) were gained further from q0 or D. By measuring the diffusion coefficient of the collective coordinate (n), we determined the value of osmotic permeability (pf). In the simulation under a hydrostatic pressure, the density of water increases linearly with the z-coordinate, and the rate of the increase is linear to the hydrostatic pressure difference. The permeation rate (j) of a water-channel is proved to be proportional to the hydrostatic pressure difference, even in the very high regions. The pf values from the slopes of the permeation rate-pressure curves are agreeable with those from equilibrium MD simulations with collective coordinate analysis. A new way to measure the effective inner radii (Rwt) for water COM movements in the hexa-, octa-, and deca-PNTs was proposed, and they were determined to be 0.7, 2.3, and 3.4 Å, standing for the transportation abilities of the three channels, respectively. A simple relationship seems to exist between the self-diffusion coefficients (D) and effective channel-radii (Rwt). Comparison of the correlations of channel-water movement in the three PNTs suggests that water molecules in the hexa-PNT form a typical single file, possess some concertedness in the octa-PNT, and are completely chaotic in the deca-PNT. In the cyclo-PNTs, radius affects the behavior of water transportation in a complex and subtle way. The results reported here are crucial for achieving the complete understanding of the properties of confined water, which is undoubtedly necessary for the design of novel nanomaterials based on cyclo-PNTs. References and Notes (1) Agre, P. Angew. Chem., Int. Ed. 2004, 43, 4278–4290. (2) Khalili-Araghi, F.; Gumbart, J.; Wen, P. C.; Sotomayor, M.; Tajkhorshid, E.; Schulten, K. Curr. Opin. Struct. Biol. 2009, 19, 128–137.

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