Electrostatic gating of a nanometer water channel

Abstract

Water permeation across a single-walled carbon nanotube (SWNT) under the influence of a mobile external charge has been studied with molecular dynamics simulations. This designed nanopore shows an excellent on–off gating behavior by a single external charge (of value +1.0e): it is both sensitive to the available charge signal when it is close (less than a critical distance of 0.85 Å or about half the size of a water molecule) and effectively resistant to charge noise, i.e., the effect on the flow and net flux across the channel is found to be negligible when the charge is >0.85 Å away from the wall of the nanopore. This critical distance can be estimated from the interaction balance for the water molecule in the SWNT closest to the imposed charge with its neighboring water molecules and with the charge. The flow and net flux decay exponentially with respect to the difference between these two interaction energies when the charge gets closer to the wall of the SWNT and reaches a very small value once the charge crosses the wall, suggesting a dominating effect on the permeation properties from local water molecules near the external charge. These findings might have biological implications because membrane water channels share a similar single-file water chain inside these nanoscale channels.

The transportation of water molecules across nanometer water channels in membranes plays a key role in biological activities (1–8). It has been recognized that the existence of the charged residues in these water channels greatly reduces the permeation of protons across the channel but maintains quite stable water flows (4, 5, 9, 10). Moreover, because charges are indispensable in both membrane proteins and physiological solutions inside and outside the cells, it is also important to understand how external charges influence the water permeation.

By using molecular dynamics simulations, the importance of the charged residues in channel proteins such as aquaporin (AQP) and Glpf on the behavior of water molecules inside the channel has been studied recently (9–11). It has been found that the charged groups in the conserved NPA and ar/R regions that dominate the energetics of water permeation in these regions can interrupt the hydrogen bond along the water chain and generate the electrostatic field to exclude proton transfer (9–13). Furthermore, very recently, the electrostatic environment of the water channel has been found to be able to regulate water permeability, using mutational analysis (14).

However, the complex structure of biological channels and protein–water interactions often make further investigations of the mechanism of biological water channels very complicated. It has been well recognized that simple nanochannels can be used as model systems to exploit some of the primary behavior of the biological water channels. In 2001, Hummer et al. proposed that single-walled carbon nanotubes (SWNTs) can be designed as molecular channels for water (15–18). They showed that the free energy of filling the single-file water chain in the SWNT is sensitive to the small changes in the nanotube–water interaction, and the flow of the hydrogen bonded water chain through nanotubes occurs in bursts. The authors also pointed out that the water–nanotube interaction plays a significant role along with the hydrogen bond interaction in the water chain inside the SWNT in determining whether the tube is filled or empty (18–20). The single-file arrangement and the concerted movement of water molecules through the channel, together with the liquid–vapor transition in the channel due to the modification of the water–channel interaction, were thought to be the major characters shared by the biological channels (15, 21–23). Recently, Zhu and Schulten (24) assigned charges to the atoms of SWNTs (ntNPN) to mimic the biological channel and found that similar dipole orientation of water molecules as in AQP could be generated, along with a greatly reduced mobility for the water chain inside the SWNT. The effects of homogeneous electric field on the water chain polarization and the filling equilibrium inside the carbon nanotube were studied by Vaitheeswaran et al. (25). Dzubiella and Hansen (26) also found that a strong electric field generated by an ion concentration imbalance can decrease the critical radius R_c of hydrophobic nanopore for the water permeation. Zimmerli et al. (27) found that an increased filling and a geometrical rearrangement of the single-file water chain inside the tube could result from the nonuniform electric field generated by static dipole moment induced from the curvature of the carbon nanotubes.

Despite the great effect, the dependence of the dynamics of water molecules inside a channel on charges is still far from being understood. In this study, a SWNT with an external positive charge is used as a prototype to study the response to the indispensable charges as well as the possible charge noise near water channels. Molecular dynamics, which has been used widely for the studies of water dynamics in SWNT, in proteins, and in-between proteins (9–13, 15, 16, 28–35), is adopted here for this work. Unlike the previous studies, where the charge is fixed at the carbon atoms of the SWNT (24), the charge we introduced here has multiple positions and does not have to coincide with exact atom positions (5). Interestingly, in a very recent paper, Zimmerli et al. (27) have placed charges close to carbon atoms but not on carbon atoms to generate a static dipole moment. In addition to the imposed charge, hydrostatic pressure is applied in this study between two ends of the SWNT to investigate the net flux change due to the imposed charge. Remarkably, we find that the nanopore is an excellent on–off gate that is both effectively resistant to charge noises and sensitive to available signals. This exceptional property largely comes from the stable and strong hydrogen-bond chain inside the channel, and the critical charge distance for gating can be uniquely determined by comparing the interaction energy of the water molecule in the channel closest to the imposed charge with its neighboring water molecules and that with the imposed charge. Interestingly, in-between the fully opened and fully closed states, the flow and net flux decay exponentially according to the difference between those two interactions, indicating that the local properties near the external charge dominate the permeation of the water molecules. Meanwhile, the pattern of the single-file water chain inside the channel changes continuously from a completely concerted orientation, to two alternating orientations, and finally to a completely bipolar orientation, as the external charge moves closer to the wall of the SWNT. These findings from simple model systems might have implications in novel molecular switch design, as well as biological water channels on the on–off gating mechanism, because they share a similar single-file water molecule chain and bipolar orientation (9–11, 13). We believe that this excellent property is important for biological systems to achieve accurate information transfer in an environment full of thermal fluctuations.

Results and Discussion

The simulation framework is shown in Fig. 1. To mimic the biological channels in a membrane, an uncapped, single-walled carbon nanotube (36) 13.4 Å in length and 8.1 Å in diameter was embedded along the z direction in a graphite sheet that subdivides the SWNT and space into two equal parts. To study the behavior of the water permeation across the channel due to external charges on or outside the channel, a positive charge with a quantity 1.0e was set to move on the same plane of the graphite. The radial distance of the imposed charge from the carbon atoms of the nanotube is denoted by δ. The system without any external charges, namely the control system, is also prepared for the comparison.

Fig. 1.

Simulation framework. (A) Snapshot of the simulation system (side view). The dark gray cycles are the carbon atoms of the nanotube and the graphite. The light gray point is the imposed charge, and the water molecules are represented by lines. (B) Top view of the model channel system. There is no physical hole in between the SWNT and the graphite plate. The gap between the graphite plane and the SWNT is too small for a water molecule to penetrate.

When two sides of a membrane have a same hydrostatic pressure but different concentrations of an impermeable solute, an osmotic pressure difference is established, and water flows from the side with high solute concentration to the other side. Following the method proposed by Zhu et al. (37), and our previous paper (38), an external force is applied to each water molecule along +z direction (z axis is defined to be along the axis of the SWNT with origin at the graphite plane) to obtain a pressure difference between two ends of the SWNT. This pressure difference is something like the osmotic pressure difference (37) and is 181 MPa between two ends of the SWNT for an additional acceleration of 0.1 nm·ps⁻² at each atom (38).

As shown in Fig. 1, we set the imposed charge at different distances from the wall of the SWNT in our molecular dynamics simulation to study the influence of external charge on water flow and net flux. For each system with a different distance, a 120-ns molecular dynamics simulation is performed. The last 110 ns of the simulation were collected for analysis. Initially, there is no water molecule in the nanotube. The nanotube is rapidly filled by water molecules from the surrounding reservoir (15) in each simulation.

To our surprise, the water screening effect on the charge is so strong that the average flow and net flux across the channel are approximately equal to those in the control system when the distance δ is >0.85 Å (about half the size of a water molecule), as shown in Fig. 2. Here, the flow is defined as the total number of water molecules leaving the SWNT (entered the nanotube from the opposite side) per nanosecond, while the net flux is the difference of the numbers of water molecules leaving from the upper and bottom end (again entered from the opposite end) per nanosecond. Remarkably, the critical value δ_c ≈ 0.85 Å is only a little larger than half the radius of a water molecule (≈1.4 Å), and charges outside this distance will have minimal effect on the water dynamics inside the SWNT. Note that the interaction energy of the water molecules in the SWNT with the imposed effective charge is quite large, which reaches a value of about −20 kJ/mol. Thus, the permeation of the water molecules confined in the nanochannel can effectively resist to external charge noises when they are not very close. When δ decreases to within the critical distance, as shown in Fig. 2, the average flow and net flux decrease monotonically and sharply to about δ = 0.25 Å. At δ = 0.25 Å, both the average flow and net flux are very small with values of 1.1 ns⁻¹ and 0.8 ns⁻¹. The average flow and net flux further decrease slowly down to 0.7 ns⁻¹ and 0.6 ns⁻¹, respectively, at δ = 0, showing that the channel is almost closed, consistent with the previous findings (24). We note that the displacement of the imposed charge for the transition from the almost fully opened state (δ = 0.85 Å) to a nearly closed state (δ = 0) is only ≈0.85 Å. Thus, the single-file water chain in the nanochannel is also sensitive to the effective charge signal.

Fig. 2.

The averaged water flow (black) and net flux (gray) through the channel with respect to δ, together with those for the control systems (dash lines). δ is the radial distance of the imposed charge to the centerline minus the radius of the SWNT. The flow is defined as the total number of water molecules leaving the SWNT and entering the nanotube from the other side per nanosecond, whereas the net flux is the difference of the numbers of water molecules leaving from the upper and bottom end, respectively, that entered the nanotube from the other end, per nanosecond.

To understand the mechanism behind this excellent on–off gating with charge signals, we calculate the interaction energies of the water molecule at position z inside the SWNT (total about five such water molecules) with its neighbor water molecules, denoted by P_WW, and the electrostatic potential with the imposed charge, P_WC. The results are shown in Fig. 3 A and B. As δ decreases, the absolute value of P_WC increases. It is interesting to find that, in the middle of the SWNT, P_WC forms a valley within the region, which faces the position of the imposed charge, with a range of about one-fifth of the length of the SWNT. Consider that the average number of water molecules inside the channel is 5.0 ± 0.1, which is quite stable for different positions of the imposed charge in this article, the z range of the valley corresponds to about one water molecule facing to the external charge. For easy discussion, we denote the region with z coordinate ranging from −1.34 Å to 1.34 Å, which is about one-fifth the length of the channel, by the central-region, and the water molecule in this region by the central-water-molecule hereafter. In the following we will find that the main properties of the flow and net flux, including the value of the critical value δ_c and the decreasing of the flow and net flux, can be dominatingly determined by the hydrogen bonds and the dipole orientation of the water molecule facing to the imposed charge. P_WW also weakens as δ decreases.

Fig. 3.

Water–charge and water–water interactions. (A and B) The interaction energies of a water molecule at z with the imposed charge, P_WC (A), and with the neighbored water molecules, P_WW (B), for different radial distance δ. (C) Values of P_WC and P_WW/2 at z = 0, denoted by P_WC^C and P_WW^C, with respect to δ. (D) Decreasing of the average value of flow and the net flux with respect of P_WW^C − P_WC^C. Exponential laws can fit the data very well in the interval of 0.25 Å ≤ δ ≤ 0.75 Å.

Fig. 3C plots P_WC^C and P_WW^C, the value of P_WC and P_WW/2, respectively, as a function of δ at z = 0, where the imposed charge is directly faced. P_WW^C equals to the interaction energy of a water molecule at z = 0 with either side of its neighboring water molecules due to the symmetry. It is clear that P_WW^C is stronger than P_WC^C when δ is large (larger than ≈0.8 Å) and weaker than P_WC^C when δ is small. Interestingly, P_WW^C ≈ P_WC^C at δ = δ_c. The transition point can be determined from the competition between P_WW^C and P_WC^C. It is clear that the stronger the hydrogen bond between the water molecules, the larger the value of P_WW^C, and correspondingly, the smaller the value of δ_c. The small value of δ_c means that water molecules in the nanochannel can effectively shield the external charge noise, thanks to the exceptional property of water with strong hydrogen bond between molecules.

In the interval of 0.25 Å ≤ δ ≤ 0.75 Å, which is below the critical value δ_c, both the average flow and net flux can be fit to an exponential function with respect to (P_WW^C − P_WC^C) very well, as shown in Fig. 3D.

where flow₀ = 18.2 ns⁻¹ and flux₀ = 10.3 ns⁻¹, respectively, slightly larger than the values of 16.2 ns⁻¹ and 9.2 ns⁻¹ for the control system. ε_flow ≈ ε_flux = 14.7 kJ/mol, implying a similar rule governing the behavior of the flow and net flux. Importantly, both the critical value and the exponential laws clearly show that the flow and net flux in this δ interval are mainly determined locally by the competition of the interactions that the water molecule in the SWNT closest to the charge experiences with its neighboring water molecules and with the external charge.

It has been widely recognized that the positive residues lying in the central-region of the AQPs water channels results in bipolar water orientations (4, 9, 10). The hydrogen bond chain was usually interrupted by the charged residues which can exclude the proton permeation but maintain a stable water flow (9, 10, 13). The bipolar water orientation is also found in the SWNT for small δ (more discussions below for the differences between AQPs and the current nanopore). Fig. 4A displays the distribution of the averaged dipole orientation of water molecules inside the nanochannel for various δ, where 〈θ〉 is the average of the angle between a water dipole and the nanotube axis (z axis), and the average is taken over all of the water molecules inside the tube (38). For the control system and the systems with a large δ (e.g., δ ≥ 1 Å), the value of cos 〈θ〉 falls into two ranges, −1 < cos 〈θ〉 < −0.65 and 0.65 < cos 〈θ〉 < 1, resulting from the concerted dipole orientations of all of the water molecules inside, either upward or downward. On the other hand, there is only a single peak at cos 〈θ〉 = 0 for δ = 0.5 Å, indicating a majority of the bipolar orientation of water chain inside the nanochannel (see below for further description). When 0.5 Å < δ < 1 Å, there are three peaks in the distribution of averaged water dipole orientation, showing that the water molecules inside adopt either a uniform orientation or bipolar orientation, alternatively. As δ decreases, the peak at cos <θ> = 0 increases and those at cos 〈θ〉 = ±0.85 decrease.

Fig. 4.

Water dipole orientation inside the SWNT. (A) Distributions of the averaged dipole orientation of water molecules inside the SWNT, θ, for various distances δ. The distribution has two peaks at about cos 〈θ〉 = ±0.85 for δ = 1 Å and a single peak at cos 〈θ〉 = 0 for δ = 0.5 Å. (B) Profile of 〈cos ϕ〉 along the SWNT channel. (C) Examples of the average values of ϕ, denoted by 〈ϕ〉, in three different regions for δ = 1, 0.75, and 0.5 Å: central-region (black), other regions for positive and negative z values (green and red). (D) Probability of bipolar orientation of all water molecules inside the channel. The red line is a linear fit for probability = −1.91 × (δ − 1.0).

To further characterize the dipole orientation of the water chain, the average value of cos ϕ, 〈cos ϕ〉, along the SWNT channel is shown in Fig. 4B, where ϕ is the angle between a water dipole and the nanotube axis (z axis), and the average is taken over all of the data from simulations. All of the curves are symmetric with respect to the center (z = 0) of the SWNT, the position in the nanochannel closest to the external charge. Transitions can be found at the boundary of the central-region, and the smaller the value of δ, the sharper the transition. It is clear that 〈cos ϕ〉 = 0 everywhere inside the SWNT when there is no external charge, because the probabilities are equal for both directions, either along the z axis or the opposite, for the dipoles of all water molecules inside the SWNT. As the external charge gets closer, |〈cos ϕ〉| at the outer-region increases. Here the outer-region is defined as the other two parts besides the central-region in the nanochannel. At δ = 0.5 Å, 〈cos ϕ〉 ≈ ±0.85 in the outer-region, indicating that the water dipoles in the outer-region are almost parallel to the axis of the SWNT. This is consistent with the previous results on SWNTs (24) and the AQP channels (9, 10). To characterize how the water molecules go from the concerted orientation to the bipolar orientation, we have plotted in Fig. 4C some examples of the average values of ϕ, denoted by 〈ϕ〉, in three different regions, i.e., the central-region and other regions with positive or negative z values, respectively. As δ = 1 Å, 〈ϕ〉 falls in the same regions as for all three regions, consistent with the concerted orientation behavior. When δ = 0.5 Å, 〈ϕ〉 vibrates around 90° for the central-region and have values of ≈30° and 150° for the other regions, showing bipolar orientation. It is interesting to find that when δ = 0.75 Å, 〈ϕ〉 oscillates between the concerted orientation and the bipolar orientation. To obtain a quantitative description on the transition from a concerted orientation to a bipolar orientation as δ decreases from 1 Å to 0.5 Å, we have computed the probability of bipolar orientation. Here we assume that the system shows a bipolar orientation when all of the water dipoles in the outer-region point away from the center of the SWNT. The probability is calculated based on data from all simulations. The result is shown in Fig. 4D. It is interesting to find that the probability increases linearly from 8% to 98% when δ decreases from 1 Å to 0.5 Å.

In AQPs the bipolar water order stops proton permeation but not water transport. In the current nanopore, water flow can also be fully stopped, if the charge signal is too close. To some extent, the current charge signal is stronger (because of its single point charge nature) than that in biological channels where the positive charge is spread over several atoms in lysine or arginine residues. Nevertheless, the current results show no apparent discrepancy on the bipolar water order and the water flow; in other words, it is possible to have both the bipolar water order and water transport at the same time. As shown in Fig. 4D, the probability of bipolar orientation of water molecules inside the SWNT approaches 1.0 when δ = 0.5 Å or less. Meanwhile, even at this small charge distance of δ = 0.5 Å, the average flow and net flux still have considerable values of 2.6 ns⁻¹ and 1.9 ns⁻¹, respectively (16% and 21% of those of the fully opened state), which are comparable to the measured 3.9 ± 0.6 ns⁻¹ for AQP1 (3). Thus, it is feasible for the nanopore to have the flow and net flux comparable to the biological channels while maintaining the bipolar water order, which is linked to the exclusion of proton transfer in biological channels. It should be noted that the charge distribution and the structure of AQPs are much more complicated (several charged residues can be involved with charges spread to many atoms), and these charges effectively exclude the proton permeation but maintain a stable water flow by inducing a bipolar water orientation (4, 9, 10). In the present study, we focus on the model nanopore with only one external charge (+1.0e, same as one basic residue) but probably stronger local electrostatic strength on the nearest water molecule inside the SWNT.

The potential of mean force (PMF) is often used to characterize the behavior of water molecules inside the channel (39, 40). Fig. 5A displays the PMF curves of water molecules along the axis of channel. The wave-like structure of PMF retains after the introduction of the imposed charge, but the peak-to-peak (min-to-max) value increases as δ decreases. To further study the relationship of the PMF profile with the flow and net flux quantitatively, the height of the barrier ΔG (41) is calculated. It is interesting to note that, in the region of 0 Å ≤ δ ≤ 0.75 Å, the relationship can be fit to a power law very well with factors of α_flow = −1.85 (flow = 0.9 × ΔG^−1.85) and α_flux = −1.61 (flux = 0.9 × ΔG^−1.61), respectively. Similar power law with a factor of −2 has been observed (41). We note that the exponential laws for the relations of the flow and net flux with (P_WW^C − P_WC^C) shown above, with a same factor found in the z interval of 0.25 Å ≤ δ ≤ 0.75 Å (see Fig. 3D), whereas the power law observed here is available in the region of 0 Å ≤ δ ≤ 0.75 Å with different factors for flow and net flux. The physical origin for those laws is still unclear.

Fig. 5.

PMF profiles for water inside SWNTs and their relationship with the flow and wet flux. (A) PMF profiles of water molecules along the axis of channel in several representative systems with δ = 1.5 Å (black), 1 Å (blue), 0.75 Å (red), 0.5 Å (magenta), and 0 Å (green). (B) Relationships of the flow and net flux with the height of the barrier ΔG in PMF profiles. The black and red lines are flow = 0.9 × ΔG^−1.85 and flux = 0.9 × ΔG^−1.61.

Conclusion

The single-file water molecules inside a SWNT have a transition-like behavior with a critical radial distance δ_c ≈ 0.85 Å (the radial distance from the SWNT wall for the external charge) under the influence of an external charge. When the radial distance of the imposed charge to the carbon atom of the SWNT δ > δ_c, the average flow and net flux of the water across the channel are almost the same as those without any external charge. The flow and net flux decrease as δ decreases. When δ = 0, the flow and net flux are very small, showing that the nanochannel is effectively closed. This shows that the SWNT has distinguished properties as an excellent nano-controllable on–off gate for water molecules under an external charge. Explicitly, on one hand, the fully opened state can effectively resist the external charge noise even if the charge is quite close to the channel boundary (as long as it is ≈0.85 Å away). On the other hand, the channel is sensitive to effective charge signals once the distance of the charge is <0.85 Å, from an almost opened state to a closed state.

The strong hydrogen bonds between the water molecule in the SWNT closest to the charge, and its neighboring water molecules play the critical role on this important on–off gating behavior. The critical radial distance δ_c can be approximately determined by the balance of the interaction of this water molecule with its neighboring molecules and with the external charge. In the states between fully opened and closed, the flow and net flux decay exponential according to the difference between those two interactions. This suggests that the local properties near the imposed charge dominate the water permeation across the channel.

The water molecules in the SWNT show the bipolar orientation for a small radial distance δ. As δ decreases from 1 Å, which is slightly larger than the critical distance for the flow and net flux, the probability of bipolar orientation increases linearly with respect to the radial distance δ, and the water chain flips between two states, the concerted orientation state and bipolar orientation state. The probability of bipolar orientation reaches ≈100% before the channel is closed. In addition, the height of barrier in PMF profile (peak-to-peak value) increases monotonically as δ decreases. The relationship of the height of the PMF barrier with flow and net flux can be fit to a power law very well, with power factors of −1.85 and −1.61, respectively. This indicates the strong water orientational preferences induced by the charge lead to a substantial free energy barrier to water chain translocation of several kT, if the chain translocates in a concerted fashion.

The single-file water molecules chain and the strong hydrogen bonds between water molecules in the SWNT play crucial role on the excellent on–off gating behavior of the channel. We are reasonably confident that biological water channels share the properties of this simple nanochannel due to similar single-file water molecules chain and bipolar orientation with charges on the proteins (4, 9, 10). We believe that this excellent property is important for biological systems to achieve accurate information transfer in an environment full of thermal fluctuations.

System and Methods

The 144-carbon (6,6) nanotube was formed by folding a graphite sheet of 5 × 12 carbon rings to a cylinder and then relaxed with interactions between carbon atoms. The carbon–carbon interaction parameters were adopted from the previous work by Brenner (42) using the Tersoff formulism (43). Periodic boundary conditions were applied in all directions. The distance of the imposed charge from the central axis of the SWNT ranges from 5.55 Å to 4.05 Å. For the convenience of analysis, this distance is denoted by δ + r, where δ is the radial distance of the imposed charge from the carbon atoms of the nanotube; r = 4.05 Å is the radius of the channel. Thus, 0 < δ < 1.5 Å. δ = 0 implies that the charge has a same distance to the centerline with the atoms of the SWNT while its position may not coincide with any atom of the nanochannel. Numerically, we find that the axial angle of the imposed charge does not change the result much in this article. The SWNT and the graphite sheet usually will deform because of the external charges by altering the electronic structure of the channel atoms. Such polarization and deformation have not been accounted for in our present simulations (23).

The molecular dynamics simulations were carried out at a constant pressure (1 bar with initial box size L_x = 5.0 nm, L_y = 5.0 nm, L_z = 5.0 nm) and temperature (300 K) with GROMACS 3.2.1 (www.gromacs.org). Here the TIP3P (44) water model was applied. Periodic boundary conditions were applied in all directions. The particle-mesh Ewald method (45) was used to treat the long-range electrostatic interactions. A time step of 2 fs was used, and data were collected every 0.5 ps. In the simulations, the carbon atoms were modeled as uncharged Lennard–Jones particles with a cross-section of σ_CC = 0.34 nm, σ_CO = 0.3275 nm, and a depth of the potential well of ε_CC = 0.3612 kJ·mol⁻¹, ε_CO = 0.4802 kJ·mol⁻¹ (15). To prevent the SWNT from being swept away, the carbon atoms at the inlet and outlet were fixed in the simulations.

Abbreviations

AQP

aquaporin

PMF

potential of mean force

SWNT

single-walled carbon nanotube.