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Theory and Simulation of Water Permeation in Aquaporin-1

Posted on: Sunday, 1 February 2004, 06:00 CST

ABSTRACT We discuss the difference between osmotic permeability p^sub f^ and diffusion permeability p^sub d^ of single-file water channels and demonstrate that the p^sub f^/p^sub d^ ratio corresponds to the number of effective steps a water molecule needs to take to permeate a channel. While p^sub d^ can be directly obtained from equilibrium molecular dynamics simulations, p^sub f^ can be best determined from simulations in which a chemical potential difference of water has been established on the two sides of the channel. In light of this, we suggest a method to induce in molecular dynamics simulations a hydrostatic pressure difference across the membrane, from which p^sub f^ can be measured. Simulations using this method are performed on aquaporin-1 channels in a lipid bilayer, resulting in a calculated p^sub f^ of 7.1 10^sup -14^ cm^sup 3^/s, which is in close agreement with observation. Using a previously determined p^sub d^ value, we conclude that p^sub f^/p^sub d^ for aquaporin-1 measures ~12. This number is explained in terms of channel architecture and conduction mechanism.

INTRODUCTION

Aquaporin-1 (AQP1), a membrane channel protein, is the first characterized member of the aquaporin (AQP) family (Hohmann et al., 2001). The protein is abundantly present in multiple human tissues, such as the kidneys. AQP1 forms homotetramers in cell membranes, each monomer forming a functionally independent water pore, which does not conduct protons, ions, or other charged solutes. A fifth pore is formed in the center of the tetramer. Recent experiments have indicated that the central pore of AQP1 tetramers may conduct ions (Saparov et al., 2001; Yool and Weinstein, 2002). However, the passive transport of water across cell membranes remains to be the major physiological function established for AQP1.

The first atomic structures of AQP1 were obtained by electron microscopy (Murata et al., 2000; Ren et al., 2001). A high- resolution structure of the Escherichia coli glycerol uptake facilitator (GlpF), a bacterial member of the AQP family, was solved by x-ray crystallography at about the same time (Fu et al., 2000). In light of the successful exploration of various membrane ion channels (Berneche and Roux, 2001; Biggin and Sansom, 2002; Randa et al., 1999; Roux, 2002), molecular dynamics (MD) simulations have also been performed on AQP water channels (de Groot et al., 2001; de Groot and Grubmuller, 2001; Jensen et al., 2001, 2002, 2003; Tajkhorshid et al., 2002; Zhu et al., 2001, 2002) soon after their structures became available. Recently, the structure of AQP1 was also solved by x-ray crystallography at high resolution (Sui et al., 2001), offering a better chance to study the dynamics and function of this water channel in atomic detail.

The key characteristics accounting for transport through water channels such as AQP1 are the osmotic permeability (p^sub f^) and the diffusion permeability (p^sub d^) (Finkelstein, 1987), which both can be measured experimentally. p^sub f^ is measured through application of osmotic pressure differences, whereas p^sub d^ is measured through isotopic labeling, e.g., use of heavy water. In this study, using a continuous-time random-walk model (Berezhkovskii and Hummer, 2002), we will demonstrate that p^sub f^ and p^sub d^ of a single-file water channel are related, but differ in value. We will further show that equilibrium MD simulations yield the p^sub d^ value, and propose a method to induce hydrostatic pressure differences across the membrane in MD simulations, allowing p^sub f^ of membrane channels to be determined from simulations. We will also describe MD simulations of AQP1 based on the structure reported in (Sui et al., 2001), in which pressure-induced water permeation was used to determine the channel's p^sub f^ value, which was found to agree well with experimental measurements.

THEORY AND METHODS

In this section, we will define p^sub f^ and p^sub d^ for water channels, and, in particular, investigate the relationship between the two for single-file water channels. We will also describe our simulations in which hydrostatic pressure differences across the membrane are established through application of external forces.

Definition of p^sub f^ and p^sub d^

An interesting method, which we refer to as the "two-chamber setup," has been used to study osmotically driven water flow in MD simulations (Kalra et al., 2003), where the unit cell consists of two membranes and two water layers containing different concentrations of solutes. In the present study, we chose our proposed method rather than the two-chamber setup for two reasons. Firstly, to observe on the ns timescale a statistically significant water flux through an AQP1 channel, one has to induce in the two- chamber setup a large chemical potential difference ([Delta][mu]) of water. However, it is noteworthy that Eq. 10 is valid only for dilute solutions; when the solute concentration is high, [Delta][mu] is no longer linearly proportional to the concentration difference. In contrast, in our method, [Delta][mu] can be linearly controlled (see Eq. 18). Secondly, the osmotic water flux in the two-chamber setup will decrease with time and eventually stop (Kalra et al., 2003), whereas our method generates a stationary flux, which permits sampling for as long as one can afford.

Simulation setup

The AQP1 (Sui et al., 2001) tetramer was embedded in a palmitoyl- oleoylphosphatidyl-ethanolamine (POPE) lipid bilayer and solvated by adding layers of water molecules on both sides of the membrane. The whole system (shown in Fig.2) contains 81,065 atoms. The system was first equilibrated for 500 ps with the protein fixed, under constant temperature (310 K) and constant pressure (1 atm) conditions. Then the protein was released and another 450 ps equilibration performed.

Starting from the last frame of the equilibration, four simulations were initiated. In these simulations (to which we refer as sim1, sim2, sim3, and sim4), a constant force (f) was applied on the oxygen atoms of the water molecules in region III, defined as a 7.7-[Angstrom] thick layer (shown in Fig. 2) in our system, to induce a pressure difference across the membrane. In principle, the position and thickness of region III can be arbitrarily defined and should not affect the results, as long as the induced pressure difference is set to the same value (by choosing a proper f); in practice, one would partition the bulk water in such a way that each of the three regions (I, II, III) has a sufficiently large thickness (relative to the diameter of a water molecule). The constant forces used in the four simulations differ in their direction or magnitude, generating four pressure differences, as summarized in Table 1. The simulations were performed under constant temperature (310 K) and constant volume conditions.

FIGURE 2 Side view of the unit cell including the AQP1 tetramer, POPE lipid molecules, and water molecules. The protein is shown in tube representation; lipids in line representation (hydrogen atoms not shown); phosphorus atoms of lipids are drawn as vdW spheres; water molecules are shown in line representation, with those in region III (see Fig. 1) colored blue.

TABLE 1 Summary of the four simulations reported in this study

As mentioned earlier, the membrane needs to be constrained to prevent the overall movement of the system under the external forces. This is done by applying harmonic constraints to the C[alpha] atoms of the protein and the phosphorus atoms of the lipid molecules, with spring constants of 0.12 kcal/ mol/[Angstrom]^sup 2^ and 0.8 kcal/mol/[Angstrom]^sup 2^, respectively. These spring constants are chosen to fully balance the external forces when the whole membrane is displaced by ~1 [Angstrom] along z from its reference position under a pressure difference of 200 MPa (as in sim1 and sim4). The constraints are applied only in the z- direction, and all atoms are free to move in the x- and y- directions. Note that the constraints on the protein are fairly weak and act only on the backbone C[alpha] atoms; therefore, significant flexibility of the protein (especially its side chains) is still realized during the simulations.

During preliminary simulations, the side chain of ARG^sub 197^ (PDB entry 1J4N) was found to deviate from its original position (and in some cases even blocked the channel) due to the breaking of an H-bond between its guanidinium group and its backbone oxygen. A similar behavior of this ARG residue was observed in simulations of G1pF with induced pressure differences (Zhu et al., 2002), but not in equilibrium simulations of AQP1 or G1pF (de Groot and Grubmuller, 2001; Tajkhorshid et al., 2002). Therefore, the inward motion of the ARG appears to arise from the application of large pressure differences in the present study. We constrained the above mentioned H-bond in our present simulations to avoid blockage of the channel, a measure also applied in our earlier simulations (Zhu et al., 2002). In the crystal structure of AQP1 (1J4N), a water molecule (HOH:383) is buried inside the protein, at a position close to an intrinsic H-bond of an [alpha]-helix. In our preliminary simulations, this water molecule was usually squeezed out of the protein; however, occasionally it attracted other water molecules from the outside into this reg\ion and made the protein unstable. To avoid the instability, the stated water molecule was deleted for the simulations reported in this article.

All simulations were performed using the CHARMM27 force field (MacKerell et al., 1998; Schlenkrich et al., 1996), the TIP3P (Jorgensen et al., 1983) water model, and the MD program NAMD2 (Kale et al., 1999). Full electrostatics was employed using the particle mesh Ewald (PME) method (Essmann et al., 1995). Simulations sim1, sim2, sim3, and sim4 were each run for 5 ns, with the first 1 ns discarded and the remaining 4 ns used for analysis. One nanosecond of simulation took 22.4 h on 128 1-GHz Alpha processors.

RESULTS

During the simulations, the water density distribution in regions I, II, and III exhibited different patterns, as shown in Fig. 3, where the dashed lines are the boundaries separating these regions. In region III, where the external forces are applied, a gradient of water density is observed; in regions I and II, the density of water is roughly constant, indicating that the hydrostatic pressure in these regions is uniform. The water density gradient in region III and, hence, the density difference between regions I and II, differ in the four simulations. From the observed water density difference and the calculated pressure difference (see Table 1) in these simulations, the compressibility of water is estimated to be 4.9 10^sup -5^ atm^sup -1^, which is in satisfactory agreement with its experimental value of 4.5 10^sup -5^ atm^sup -1^ (Sperelakis, 1998).

FIGURE 3 Water density distribution along the z-direction in region III (within the dashed lines) and part of regions I and II. Data points marked by circles, diamonds, stars, and squares represent sim1, sim2, sim3, and sim4, respectively. The density is measured by averaging the number of water molecules within a I- [Angstrom] thick slab over the last 4 ns of each trajectory.

Water molecules in the channels were usually found in the single- file configuration (as shown in Fig. 4 a) and moved concertedly during the simulations (as shown in Fig. 4 b), despite occasional exceptions when more water molecules were accommodated in the channel, or when the water file appeared broken in a part of the channel. Nevertheless, the continuous-time random-walk model can be used to provide a simplified quantitative description of water movement in AQP1 channels. According to the model, the parameters characterizing the water movement are the hopping rates k^sub r^ and k^sub 1^. In the following, we suggest a method to determine these rates from MD trajectories.

FIGURE 4 (a) An AQP1 monomer with channel water and nearby bulk water. Water molecules in the constriction (single-file) region, the vestibules of the channel, and in the bulk are rendered in vdW, CPK, and line representations, respectively. The two bars indicate the 15- [Angstrom] long region in which water movement is analyzed as described in the text, (b) Trajectories (from sim1) of seven water molecules in the constriction region during 500 ps.

We define a region that spans the constriction region of an AQP1 channel, with length L = 15 [Angstrom] (as indicated by the two bars in Fig. 4 a), and only look at water movement in this region. We also define a coordinate, X, by cumulating the sum of one- dimensional displacements of all water molecules in the mentioned region every picosecond. If a water molecule enters or exits the defined region within a picosecond, only the portion of its displacement within the region contributes to the sum. In this way, the actual many-body water movement is reduced to a single-particle trajectory (see Fig. 5), which represents the collective water movement inside the channel.

The estimated hopping rates for each simulation are provided in Table 2. To obtain Var[[Delta]X(t)] and (k^sub r^ + k^sub l^), the 4- ns trajectory of X was divided into M = 20 subtrajectories, each l = 200 ps long, and Var[[Delta]X(t)] was calculated from the 20 displacements in different subtrajectories at t. Because the results may depend on how we split the trajectory, we checked two alternative dividing schemes, i.e., (M = 10, l = 400 ps) and (M = 40, l = 100 ps), respectively, but in both cases obtained (k^sub r^ + k^sub l^) values close to those listed in Table 2. Note that it is not feasible to choose very large / and very small M, because that would result in a large uncertainty in the variance.

FIGURE 5 Trajectories of the collective coordinate X for water molecules in the defined region in an AQP1 monomer, determined as described in the text. The four curves were obtained from simulations sim1, sim2, sim3, and sim4, respectively.

TABLE 3 Water flux observed in the four simulations

The obtained k^sub r^ and k^sub 1^ values are in the range of 1- 5/ns. In contrast, the equilibrium hopping rate (k^sub 0^) for a narrow carbon nanotube was determined to be ~14-38/ns (Berezhkovskii and Hummer, 2002; Zhu and Schulten, 2003), indicating much faster water movement than in AQP1. The slow kinetics in AQP1 and the relatively short sampling time introduce large statistical errors to the resulting k^sub r^ and k^sub l^ values. According to Eq. 6, the ratio k^sub r^/k^sub l^ can be predicted from the chemical potential difference ([Delta][mu]) of water. Indeed, one can see from Table 2 that the obtained ratio for each simulation is of the same order of magnitude as the prediction (compare the last two columns of the table).

TABLE 2 Hopping rates estimated from the simulations

FIGURE 6 The dependence of water flux on the applied pressure difference. Values of pressure differences and water fluxes are taken from Tables 1 and 3, respectively. A line with the best-fit slope for the four data points is also shown in the figure.

The net water fluxes, directly determined from the simulations, are given in Table 3. These values are plotted versus the applied pressure difference in Fig. 6. From their best-fit slope, and according to Eqs. 1 and 14, the osmotic permeability was determined to be p^sub f^ = (7.1 0.9) 10^sup -14^ cm^sup 3^/s. Different experiments have reported p^sub f^ values for AQP1 monomers in the range of 1-16 10^sup -14^ cm^sup 3^/s, the variation being probably due to uncertainties in the number of channels per unit membrane area (Heymann and Engel, 1999); typically referenced p^sub f^ values range from 5.43 10^sup -14^ cm^sup 3^/s (Walz et al., 1994) to 11.7 10^sup -14^ cm^sup 3^/s (Zeidel et al., 1992). In light of this, the p^sub f^ value calculated from our simulations agrees satisfactorily with experiments. According to Eq. 11, the equilibrium hopping rate in the continuous-time random-walk model for AQP1 is then k^sub 0^ = 2.4/ns. Assuming that the AQP1 channel is symmetric, and according to Eq. 7, this corresponds to a (k^sub r^ + k^sub l^ = 2k^sub 0^) value of 4.8/ns. This value is indeed consistent with the (k^sub r^ + k^sub l^) values (4-6/ns) in Table 2, determined from monitoring Var[[Delta]X(t)] in our simulations.

Equilibrium MD simulations of AQP1 were performed by other researchers, where a total of 16 permeation events (in four AQP1 monomers in either direction) were observed in 10 ns (de Groot and Grubmuller, 2001). Therefore the rate of unidirectional permeation events in a monomer is q^sub 0^ = 0.2 H2O/ns. According to Eq. 3, this 170 value translates into a diffusion permeability of p^sub d^ = 6.0 10^sup -15^ cm^sup 3^/s. Using this p^sub d^ value and the calculated p^sub f^ value of this study, one obtains a p^sub f^/ p^sub d^ ratio of 11.9, in good agreement with the experimentally measured ratio of 13.2 for AQP1 (Mathai et al., 1996). The ratio corresponds to the number of effective steps in which a water molecule needs to participate to cross AQP1.

The number (~12) of effective steps in a complete permeation event should be interpreted as follows. In the bulk, water conduction is essentially uncorrelated, i.e., the bulk phase does not contribute to the p^sub f^ / p^sub d^ ratio. In the constriction region of the channel, however, (average) N = 7 water molecules move essentially in single file, i.e., in a correlated and concerted fashion, such that N +1 = 8 steps are needed to transport a water molecule through. Water molecules in the vestibules (also shown in Fig. 4 a) at the termini of the channel are not forming a single file, but nevertheless move in a somewhat concerted fashion, contributing the remainder of the p^sub f^ / p^sub d^ ratio.

DISCUSSION

We have discussed the difference between osmotic and diffusion permeabilities through theoretical analysis of single-file water transport and computational investigation of AQP1. Understanding this difference is important for the correct interpretation of simulations studying water transport. In particular, although it is a common practice to count permeation events in equilibrium simulations, one should note that it is not correct to use this count to calculate p^sub f^. Although the present study focuses on single-file water transport in channels, the issue is even more critical for larger water pores, in which the p^sub f^/p^sub d^ ratio can be even larger (Finkelstein, 1987) and improper comparison between simulations and experiments would lead to a more serious discrepancy.

It is noteworthy that the magnitudes of the chemical potential difference ([Delta][mu]) of water in our simulations are 1.32 k^sub B^T and 0.66 k^sub B^T. In this case, because [Delta][mu] is not much smaller than k^sub B^T, it may not be safe to keep only the linear terms of [Delta][mu]/k^sub B^T in Eq. 7. Consequently, the net water flux is not guaranteed to be linear in [Delta][mu]. However, the water flux was not found to significantly deviate from linearity in the present study. In earlier simulations where even higher pressure differences were induced (Zhu et al., 2002), nonlinear behavior was not observed either. For single-file water channel\s, breaking of the single-file configuration at high pressure differences poses an upper limit for the validity of the continuous-time random-walk model, and probably also for the linear range of the flux-pressure relationship. However, nonlinear effects may already arise before the pressure difference reaches this upper limit. It is desirable to know when and how the water flux deviates from linearity at high pressure differences, and to even develop a nonlinear model to describe the general behavior. For this purpose, one could first study the flux-pressure relationship for some simpler water channels, e.g., carbon nanotubes (Hummer et al., 2001; Zhu and Schulten, 2003) or other nanopores (Beckstein and Sansom, 2003), because due to their stability and small size, they can be more easily simulated for both high and low pressure differences.

In this study, we have demonstrated that the major experimental quantity for AQP1 channels, p^sub f^, can be reproduced in MD simulations with induced hydrostatic pressure difference. We expect that our method of inducing pressure differences can be used also to determine the permeability of other water channels. In light of ever increasing computing power, the accuracy of such computational measurements will be further improved when longer simulations, in which the induced pressure difference can be lower and closer to experimental conditions, become affordable. Consequently, the presented method may serve as a complementary technique to experiments for quantitative characterizations of water channels.

Molecular images in this paper were generated with the molecular graphics program VMD (Humphrey et al., 1996).

This work was supported by grants from the National Institutes of Health (NIH PHS 5 P41 RR05969 and R01 7M067887) and from the National Science Foundation (NSF CCR 02-10843). The authors also acknowledge computer time provided at the NSF centers by the grant NRAC MCA93S028. F.Z. acknowledges a graduate fellowship awarded by the UIUC Beckman Institute.

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Fangqiang Zhu, Emad Tajkhorshid, and Klaus Schulten

Theoretical and Computational Biophysics Group, Beckman Institute, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801

Submitted June 23, 2003, and accepted for publication September 10, 2003.

Address reprint requests to Klaus Schulten, 405 N. Mathews, Urbana, IL 61801.

Tel.: 217-244-1604; Fax: 217-244-6078; E-mail: kschulte@ks.uiuc.edu.

2004 by the Biophysical Society

0006-3495/04/01/50/08 $2.00

Copyright Biophysical Society Jan 2004

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