Molecular Dynamics Simulation Study of Correlated Motions in

Sep 15, 2009 - A 100 ns simulation of a fluid phase dioleoylphosphatidylcholine bilayer, consisting of 288 lipid molecules at full hydration, has been...
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J. Phys. Chem. B 2009, 113, 13229–13234

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Molecular Dynamics Simulation Study of Correlated Motions in Phospholipid Bilayer Membranes Matthew Roark and Scott E. Feller* Department of Chemistry, Wabash College, 301 West Wabash, CrawfordsVille, Indiana 47933 ReceiVed: March 11, 2009; ReVised Manuscript ReceiVed: August 12, 2009

A 100 ns simulation of a fluid phase dioleoylphosphatidylcholine bilayer, consisting of 288 lipid molecules at full hydration, has been studied to describe in detail the lateral translational motion of individual lipid molecules. Analysis of the simulation trajectories suggests that correlated motion between neighboring lipid molecules is an important aspect of lipid dynamics. The correlation among neighboring lipids within a monolayer is substantial and surprisingly long-ranged with a decay length of approximately 25 Å. This provides a mechanism for the previously published observation that lateral diffusion coefficients computed from molecular dynamics simulations exhibited a strong dependence on the size of the unit cell and for the recent suggestion that lipid flows on that nanoscale are an important component of translational diffusion within membranes. Additionally, we show that diffusive motion is only weakly correlated between lipids in opposing monolayers. Introduction The fluid nature of the cell membrane is central to our understanding of numerous biological processes, such as signaling and transport. A detailed characterization of this unique material is challenging, however, and different approaches to measure membrane fluidity can give conflicting descriptions. For example, nuclear magnetic resonance (NMR) relaxation experiments probing reorientational dynamics of lipid acyl chains1,2 suggest the viscosity of the bilayer interior is similar to that of an alkane. Experiments measuring lipid lateral diffusion coefficients, however, are consistent with a viscosity over 2 orders of magnitude greater than an alkane liquid.3,4 Pastor and co-workers, using molecular dynamics (MD) computer simulation methods, concurred with the idea that the bilayer interior very much resembled an alkane and suggested that the high viscosity of the membrane arises from molecular interactions at the lipid/water interface.5 This early simulation, however, could say relatively little about translational diffusion because of the limited time scale (190 ps) over which the molecules were observed. In the intervening years, the growth of computational power has allowed lipid bilayer simulations that are orders of magnitude longer in time6-8 and also has made possible simulations employing much larger unit cells containing many hundreds of molecules.9,10 In the past decade, several such simulation studies have been published examining lipid dynamics and lateral diffusion in particular.8,11-16 A recent report by Falck employed MD simulation to explore the role of collective motions within the bilayer as they affect lateral diffusion, including systems consisting of thousands of lipid molecules.16 A primary motivation for the present study is the recent advance in the experimental measurement of lipid diffusion coefficients. Several groups17-20 have applied pulsed field gradient (PFG) methods in combination with magic angle spinning (MAS) 1H NMR to determine lipid lateral diffusion coefficients by direct monitoring of lipid motion. This has several advantages over fluorescence techniques, including the Corresponding author. Phone: (765) 361-6175. Fax: (765) 361-6149. E-mail: [email protected].

absence of perturbing probe molecules and the shorter time and length scale over which the measurement is taken. The PFG-MAS experiment provides sufficient sensitivity to quantify effects such as chain length, degree of unsaturation, and headgroup composition, as well as the presence of incorporated species, such as cholesterol. Importantly, these experiments provide a rigorous test of MD simulation methodologies by allowing close comparison between simulation and experiment. Such comparison, however, requires a careful analysis of both the measurement and the calculation. For example, the NMR experiment is measuring translation over a time scale of milliseconds, whereas the simulation is covering only nanoseconds. Similarly, the NMR is observing the motion of lipids on the micrometer length scale, as compared with the nanometer distances seen in the simulation. An additional consideration is the extent to which lipids are diffusing independently throughout the membrane. If some part of the experimentally measured diffusion arises from the motion of small patches of lipids, this component may not be observed in the simulation due to the finite number of lipid molecules and the influence of periodic boundary conditions (PBC). A critical issue in quantitative comparison of diffusion coefficients from experiment and simulation was recently identified by Klauda, Brooks, and Pastor.21 They showed that lateral diffusion coefficients computed from molecular dynamics (MD) simulations depended sensitively on system size, with artificial enhancement of lipid diffusion seen for small systems. This was attributed to enhanced interactions between neighboring lipids through the periodic boundary conditions for small systems where the correlation length is comparable to the unit cell dimension. Interestingly, the issue of PBC in molecular simulations has also been described by Yeh and Hummer,22,23 who demonstrated a sensitive dependence of the diffusion coefficient on simulation cell size for a variety of homogeneous liquids and aqueous solutions. However, in the case of diffusion in three dimensions examined by Yeh and Hummer, the small system size artifact had the opposite effect; that is, translational diffusion slowed in small systems as compared to large.

10.1021/jp902186f CCC: $40.75  2009 American Chemical Society Published on Web 09/15/2009

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In addition to the issue identified by molecular simulation of correlated translational diffusion involving neighbors in the same monolayer, the extent of correlation between molecules in opposing monolayers has been examined by experimental studies on trans-bilayer coupling. Motivated by questions regarding the correlation between inner and outer leaflets in biological membranes with differing compositions,24 domain registration experiments have demonstrated interactions between lipids in opposing leaflets.25 The measurement of lipid diffusion in an asymmetric bilayer, while possible in certain instances,26 is a challenging problem, one for which molecular simulation may be able to make a valuable contribution. To address a range of issues considering lateral diffusion in a lipid bilayer, we carried out a 100 ns MD simulation of a dioleoylphosphatidylcholine (DOPC) bilayer consisting of 288 lipids. In addition, smaller test systems containing 18 and 72 lipids were simulated to quantify the influence of artifacts induced by the periodic boundary conditions. The accuracy of the structure of the simulated bilayer was confirmed by comparison to X-ray scattering data. Details of the simulations are provided in the next section, followed by the results for diffusion coefficients and for measures of the degree of correlated motion within and across monolayers. The final section considers the implications of these results for the design of molecular simulation and for the understanding of lipid diffusion within a membrane. Methods Three MD simulations were carried out using the program CHARMM,27 differing only in the number of molecules in the simulation cell. DOPC was chosen for this study, in part because detailed structural comparisons with the X-ray scattering data of Nagle and co-workers28 showed excellent agreement between simulation and experiment, but also because precise data on lateral diffusion is available for this lipid from recent PFG-MAS NMR measurements.20,29 The simulations correspond to full hydration (32.5 H2O/lipid) at a temperature of 30 °C. All simulations were carried out under a constant normal pressure of 1 atm30 with fixed surface area corresponding to 72.2 Å2/ lipid.31 The simulation cells contained 18, 72, and 288 DOPC molecules with lateral cell dimensions of 25.49, 50.98, and 101.96 Å, respectively. van der Waals interactions and the real space portion of the Ewald summation were cut off at 10 Å. The smooth PME method32 was employed for evaluation of the reciprocal space contribution to the Coulombic forces using FFT grids with 24, 48, and 96 points, respectively, allowing force evaluations to be carried out with identical grid densities in each simulation. Equilibration involved 1 ns simulations of Langevin dynamics, followed by 5 ns of molecular dynamics. Equilibration was judged on the basis of the relaxation of the system internal energy (see the following section). Production runs were carried out using a Nose thermostat for temperature control with conformations saved every 1 ps. Production run lengths of 160, 40, and 100 ns were obtained for the 18, 72, and 288 lipid systems, respectively. Results The equilibration of each system was assessed by following the internal energy, U, as a function of time. Figure 1 plots the internal energy, on a per molecule basis, during the equilibration and early production run phases. The similarity of the data for all three system sizes is striking, with a decrease during the first 6 ns, followed by fluctuations around a very constant value

Roark and Feller

Figure 1. Internal energy per lipid molecule as a function of simulation time.

that continues without drift to the end of the simulations (166 ns in the case of the smallest system). Important structural features of the lipid bilayer can be described by the electron density profile along the bilayer normal, given in Figure 2A. The peak-to-peak distance provides information on the location of the electron-rich phosphate groups, and the width of the trough region gives insight into the distribution of the methyl groups due to their relatively low electron density. Figure 2 compares the simulation results to the experimental data of Nagle and co-workers28 both in real space and in reciprocal space (where a model-free, direct comparison with experiment can be made). The agreement is exceptional, indicating that the bilayer dimensions (thickness and molecular area) in the simulation correspond to those of the real system. These measures of bilayer structure are largely insensitive to the number of lipids in the simulation cell with differences between the simulation curves comparable to the small changes observed with varying bin sizes in analyzing the simulation results. Lateral diffusion is most often characterized from the simulation by the displacement correlation function,

c(τ) ) 〈(x(t0 + τ) - x(t0))2 + (y(t0 + τ) - y(t0))2〉

(1) where the brackets denote an averaging over all lipids and over all starting points, t0. Figure 3 shows as an example the c(τ) computed from the largest simulation. The connection to experimental measurement is typically made through the lateral diffusion coefficient, D, using the Einstein relation,

D ) limτf∞

r (t0 + τ) - b r (t0) | Z) 2 (|b 2d τ

(2)

where the dimension, d, is equal to 2. The slope of c(τ) vs τ at long times is, thus, equated with the diffusion coefficient. A significant challenge in analyzing the simulations is to determine which region of the correlation function corresponds to “long time”. In the present work, we have used an approach described previously33 in which the mean and standard deviation at each time point of the displacement correlation function is computed from the ensemble average of the N lipids in the system (with N ) 18, 72, or 288). The diffusion coefficient is then determined

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Figure 2. (A) Electron density profiles computed from the three simulations (lines) and from the model based on experimental measurements in ref 28 (symbols). (B) X-ray structure factors computed from the three simulations (lines) and determined experimentally in ref 28 (symbols).

Figure 3. Displacement correlation functions as a function of lag time for centers of mass of individual lipids (symbols) along with best linear fit (line) for the 288 lipid system.

by a linear fit that begins at tlong and continues until the end of the correlation function, with each data point in the fit weighted by the inverse of the standard deviation at that time point. This has the advantage of weighting appropriately those points computed at long times where the statistical errors are large. The only choice that must be made in computing the diffusion coefficient is the value of tlong, that is, where the short time “cage rattling” motions give way to true translational diffusion. The data in Figure 3 show that the slope of the correlation function may not obtain a constant value until the ∼25 ns point (longer than many atomistic MD simulations, especially for systems containing hundreds of lipid molecules). Analysis of the largest data set (i.e. the 100 ns simulation of 288 lipids (Figure 3)) showed that over a range of tlong values covering 3 orders of magnitude (50 ps to 50 ns) the value of the diffusion coefficient varied by ∼30% with values ranging from 0.75 to 0.99 × 10-7 cm2/s. These values compare favorably with experimental measurements in the range 0.99-1.05 × 10-7 cm2/s. The choice of tlong appears to be the limiting factor in precisely extracting a diffusion coefficient from the simulation data, although statistical uncertainty arising from a finite length simulation with a fixed number of particles is also a consideration.

Although we have chosen to analyze the uncertainty from the variance among lipids, it is also possible to break up the simulation trajectory into samples defined by time intervals. For example, the simulation by Bockmann et al. used such an approach and found a relative uncertainty in the computed diffusion coefficient of 7.7% from a 100 ns simulation of 128 lipids.34 Accounting for the greater number of lipids in our largest simulation, this would correspond to ∼5% statistical error, significantly less than the range of diffusion coefficients we report here. Data shown in Figure 4 (upper panel) show that the computed displacement correlation functions exhibit a pronounced dependence on the number of lipids. The smaller systems show significantly greater displacements, with the diffusion coefficient for the 72 and 18 lipid systems increased by 21% and 73%, respectively, over the large (288 lipid) system. Thus, for the larger system studied, excellent agreement between simulation and experiment was obtained; however, for the smallest system, the rate of diffusion observed in the simulations was substantially greater than expected. This system size dependence for translation is especially striking when compared to the negligible difference in thermodynamics (e.g. the internal energy plotted in Figure 1) and structural measures (e.g. the transverse density profile plotted in Figure 2). It has previously been suggested8,34,35 that relative motion of the two monolayers is responsible for an artificial enhancement of the lipid displacement function in bilayer simulations. These authors removed the center of mass (COM) motion of the individual monolayers before calculating the displacement of the individual lipids. Following this method, we obtain the data displayed in Figure 4 (lower panel). Notice the reVersal of the ordering of the curves from Figure 4 (upper panel). The largest system now has the greatest rate of translational diffusion, whereas the small system has the lowest diffusion coefficient. Importantly, the largest (288 lipid) system shows no effect from differences in the treatment of the data, with diffusion coefficients having a statistically insignificant difference. The 72 and 18 lipid systems, however, have their diffusion coefficients reduced by factors of 2 and 6, respectively. The effect of system size has been noted previously,35 where the calculated diffusion coefficient in a dipalmitoylphosphatidylcholine simulation in-

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Figure 4. Displacement correlation functions as a function of lag time for centers of mass of individual lipids: (upper panel) raw simulation data, (lower panel) simulation data after removal of motion of monolayer COM drift. The symbol size in the upper panel represents the standard error in the least precisely determined points; that is, those at the longest times.

creased by more than a factor of 3 when 1024 lipids were simulated, as compared with 64 lipids; however, the authors attributed this difference to issues of time scale and convergence. The system size dependence seen in Figure 4 (lower panel) can be interpreted on the basis of the work of Yeh and Hummer,22,23 who found that a system-size-dependent correction, proportional to 1/L, describes a variety of homogeneous systems undergoing three-dimensional diffusion. Their correction factor, derived from hydrodynamic theory and tested on systems as diverse as liquid argon and aqueous RNA segments, is based on the idea that the constraint of zero net momentum on the system restricts the diffusive motion of the fluid sample. The severity of this restraint decreases with larger system size. The case of a lipid bilayer is somewhat more complex due to the separate phases (lipid and water) present in the inhomogenous simulation cell. In addition, the simulation setup imposes the momentum restraint only on the system COM, whereas the lipid and water phases are free to move individually, subject only to the restraint on their combined COM. Even if one imposed COM invariance on the lipid bilayer, for example, by altering the equations of motion or by postprocessing the coordinates to shift the bilayer COM at the origin for each time step, each of the two lipid monolayers would still be able to move. The postprocessing of the data used to produce Figure 4 (lower panel) is essentially an imposition of zero net momentum on

Roark and Feller

Figure 5. Displacement correlation functions as a function of lag time for centers of mass of entire monolayers: (upper panel) monolayer motion as a function of system size, (lower panel) motion observed for the monolayer of the 72 lipid system (dashed) and the motion expected for a random sample of 36 lipids taken from the 288 lipid system (solid).

each lipid monolayer individually, and as predicted by the theory of Yeh and Hummer, this constraint reduces the diffusion coefficient for the small systems. The displacement correlation functions for the monolayer COMs (Figure 5, upper panel) were calculated from the simulation trajectory to show the magnitude of the motion of these lipid patches. The displacement of these monolayers is inversely proportional to system size, consistent with the enhanced diffusion observed for small systems in Figure 3 (upper panel). For the smallest system, the magnitude of the COM motion is actually larger than the mean displacement of individual lipids in the largest system. It is important to note that even in the large system, there is motion of the individual monolayers. Each monolayer is expected to show some motion, consistent with the random fluctuations of any group of 9, 36, or 144 lipids. The problem is that this motion is artificially enhanced for the small systems. This is shown quantitatively in Figure 5 (lower panel), where the displacement correlation function for the monolayer COMs (72 lipid system) is plotted along with the function for the COM of random samples of 36 lipids (sampled from the 288 lipid system). This figure shows clearly both why the diffusion coefficient is overestimated in small systems (Figure 4, upper panel) and why the monolayer COM subtracted data presented in Figure 4 (lower panel) underestimates the diffusion coefficient for small systems. Motion of small patches of lipid are an essential component of

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TABLE 1: Mean Value of the Cross Correlation Coefficient for the Three Lipid Simulations, with and without Removal of the Monolayer Center of Mass from the Trajectories 〈Cij〉

18 lipids

72 lipids

288 lipids

raw data mono COM removed

0.704 -0.125

0.267 -0.028

0.071 -0.007

the bilayer dynamics, and their removal, for the reasons described by Yeh and Hummer,23 decreases the calculated diffusion coefficient. However, it is also clear from the data that the lipids forming a monolayer are moving substantially more than would be expected from the random motion of 36 molecules. The results presented above provide clear evidence for excessive diffusion of individual monolayers for small systems. It has previously been shown10,36 that equilibrium properties, such as the surface tension or area per molecule, can be influenced by the size of the system. In general, one expects to see such small system effects when the correlation length between molecules becomes comparable to the unit cell dimensions such that a molecule simultaneously interacts with a second molecule and its periodic image. In the present case, we are interested in the possibility of correlated translational motion; thus, we have calculated the cross-correlation function,

Cij )

〈δxi δxj〉 〈δx2〉

Figure 6. Cross-correlation function for the lateral motion of individual lipids as a function of intermolecular separation: (upper panel) crosscorrelation as a function of system size with δt ) 10 ps, (lower panel) cross-correlation as a function of time lag for the 288 lipid system.

(3)

where δx is the x component of the displacement of the COM of the lipid during the time δt, and the brackets denote an averaging over all lipid pairs (i, j) residing in the same monolayer and over all starting times. The cross-correlation function was also calculated from the y components of the displacements, and the results presented here are the average of the x and y results. The results will depend somewhat on the time interval over which the displacement is measured; thus, the calculation has been carried out with δt ranging from 1 ps to 1 ns. Table 1 gives the mean value of the correlation function for the three simulations using δt ) 10 ps. The extent of correlated motion between pairs of lipids depends, not surprisingly, on the system size because in the smaller systems, the intermolecular distances are reduced, on average (there are few pairs of lipids separated by large distances in a small box). Table 1 also shows the results obtained after removing the monolayer COM from the trajectory. This effective restraint on the system produces an overall anticorrelation that can be understood by considering that the motion of any individual lipid must be exactly balanced by motion in the opposite direction by the other N-1 molecules. To better understand the nature of the correlated motion, we have calculated the correlation coefficient as a function of the distance between lipid molecules (Figure 6, upper panel). From the figure, the long-range nature of the correlated motion is apparent with a decay length on the order of 25 Å. Although this may seem surprisingly large, it is less than three molecular diameters for this lipid, and interactions on this length scale could easily arise from the highly disordered acyl chains or long-range dipolar interactions among headgroups. From analysis of very large MD simulations of lipid bilayers, Falck et al. observed evidence of concerted motions on even longer length scales that involved multiple lipid molecules.16 It appears from the data that even the largest system investigated may have a small correlation through the periodic cell; however, the displacement correlation function data (Figure 3) suggests this effect is small or at least

Figure 7. Cross-correlation function for the lateral motion of individual lipids as a function of intermolecular separation for lipids in opposing monolayers for the 288 lipid system.

within the statistical error of these finite duration simulations. The short time periods (10 ps) over which the displacements have been calculated in Table 1 and Figure 5 lead to the question of what type of motion is being studied; that is, is this “rattling in a cage” or true translational diffusion. The cross-correlation functions have been calculated for time periods ranging over 3 orders of magnitude (1 ps to 1 ns) for the largest system. The results (not shown) are similar to those in Figure 5 and even indicate that the magnitude of the correlation increases with longer lag time, suggesting that the correlation is measuring true translational diffusion. Having identified correlated motion among neighboring lipids in the same monolayer, acting over a substantial distance, we now examine possible correlations between molecules in opposing leaflets of the bilayer. Equation 3 was applied to compute the correlation coefficient as a function of distance between lipids in the x-y plane. The results from the 288 lipid system with a 1 ns time lag, presented in Figure 7, show that the motions of lipids in opposing monolayers are correlated, but much more weakly than the intraleaflet case. Discussion The simulation results suggest that the diffusive motions of individual phospholipids within a bilayer are not independent,

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but rather, highly correlated with neighboring molecules. From the longest time lags examined, a correlation length of approximately 25 Å is extracted, although the poor statistics inherent in the long lag time data preclude a more quantitative determination. This finding may explain the apparent discrepancy between the small diffusion coefficients for lipid molecules and the relatively low viscosity of the hydrocarbon interior.5 This may also be important in developing models of obstructed diffusion in cellular membranes37 and in understanding concerted translational motion within the bilayer membrane.16 Including correlation among neighboring molecules could be an avenue for refining various models for lipid diffusion. For example, models based on the concept of free area have been applied to the study of lipid diffusion, diffusion as a function of temperature and sterol content, and diffusion in the presence of membrane incorporated protein species.38,39 This approach is based on a hard sphere type interaction of molecules translating independently through the medium, whereas the correlation among neighbors demonstrated by the present work suggests a more complicated mechanism. At the very least, these observations are critically important for evaluating diffusion coefficients from molecular simulations. Periodic boundary conditions produce a situation in which a molecule can interact simultaneously with a second molecule and with the second molecule’s periodic image if the first and second molecules are separated by a distance L/2. Since the dynamics of the second molecule and its periodic image are perfectly correlated, an artificial environment is created for the first molecule. The usual step to mitigate this problem is to ensure that the unit cell dimension is more than twice the distance of the interaction potential (typically 10-12 Å). Even the smallest system studied here adheres to this guideline. This approach, however, fails to consider indirect interactions between two lipids that are mediated by neighboring molecules. The present results suggest that L/2 should be several decay lengths to safely study diffusion without artifacts from PBC. The largest system examined in this study had L/2 equal to just two decay lengths and, to within the statistical imprecision of the calculations, showed no deleterious effects. For example, motion of the monolayer COM had no effect on the calculated diffusion coefficient. Simulations with fewer molecules, however, showed pronounced system size effects. It is tempting to solve this problem by removing monolayer COM motion in analyzing diffusion coefficients; however, as described theoretically by Yeh and Hummer (2004) and shown explicitly in Figure 4 (lower panel), this approach also results in a system size dependence to the diffusion coefficient. Furthermore, the enhanced motion of the individual monolayers in small systems will disrupt the overall dynamics by introducing spurious frictional forces between monolayer and water and between the two opposing monolayers, and these errant forces cannot be remedied by any postprocessing of the data. Acknowledgment. We thank the National Science Foundation for support through award MCB-0543124.

Roark and Feller References and Notes (1) Brown, M. F.; Ribeiro, A. A.; Williams, G. D. Proc. Natl. Acad. Sci. U.S.A. 1983, 80, 4325. (2) Eldho, N. V.; Feller, S. E.; Tristram-Nagle, S.; Polozov, I. V.; Gawrisch, K. J. Am. Chem. Soc. 2003, 125, 6409. (3) Saffman, P. G.; Delbruck, M. Proc. Natl. Acad. Sci. U.S.A. 1975, 72, 3111. (4) Hughes, B. D.; Pailthorpe, B. A.; White, L. R.; Sawyer, W. H. Biophys. J. 1982, 37, 673. (5) Venable, R. M.; Zhang, Y.; Hardy, B. J.; Pastor, R. W. Science 1993, 262, 223. (6) Knecht, V.; Marrink, S. J. Biophys. J. 2007, 92, 4254. (7) Niemela, P. S.; Hyvonen, M. T.; Vattulainen, I. Biochim. Biophys. Acta 2009, 1788, 122. (8) Lindahl, E.; Edholm, O. J. Chem. Phys. 2001, 115, 4938. (9) Niemela, P. S.; Ollila, S.; Hyvonen, M. T.; Karttunen, M.; Vattulainen, I. PLoS Comput. Biol. 2007, 3, 304. (10) Lindahl, E.; Edholm, O. Biophys. J. 2000, 79, 426. (11) Essmann, U.; Berkowitz, M. L. Biophys. J. 1999, 76, 2081. (12) Moore, P. B.; Lopez, C. F.; Klein, M. L. Biophys. J. 2001, 81, 2484. (13) Edholm, O. Comput. Model. Membr. Bilayers 2008, 60, 91. (14) Klauda, J. B.; Eldho, N. V.; Gawrisch, K.; Brooks, B. R.; Pastor, R. W. J. Phys. Chem. B 2008, 112, 5924. (15) Klauda, J. B.; Roberts, M. F.; Redfield, A. G.; Brooks, B. R.; Pastor, R. W. Biophys. J. 2008, 94, 3074. (16) Falck, E.; Rog, T.; Karttunen, M.; Vattulainen, I. J. Am. Chem. Soc. 2008, 130, 44. (17) Gaede, H. C.; Gawrisch, K. Magn. Reson. Chem. 2004, 42, 115. (18) Gaede, H. C.; Gawrisch, K. Biophys. J. 2003, 85, 1734. (19) Pampel, A.; Zick, K.; Glauner, H.; Engelke, F. J. Am. Chem. Soc. 2004, 126, 9534. (20) Filippov, A.; Oradd, G.; Lindblom, G. Biophys. J. 2003, 84, 3079. (21) Klauda, J. B.; Brooks, B. R.; Pastor, R. W. J. Chem. Phys. 2006, 125, 144710. (22) Yeh, I. C.; Hummer, G. Biophys. J. 2004, 86, 681. (23) Yeh, I. C.; Hummer, G. System-Size Dependence of Diffusion Coefficients and Viscosities from Molecular Dynamics Simulations with Periodic Boundary Conditions. J. Phys. Chem. B 2004, 108, 15873. (24) Wagner, A. J.; Loew, S.; May, S. Biophys. J. 2007, 93, 4268. (25) Collins, M. D. Biophys. J. 2008, 94, L32. (26) Hetzer, M.; Heinz, S.; Grage, S.; Bayerl, T. M. Langmuir 1998, 14, 982. (27) Brooks, B. R.; Bruccoler, R. E.; Olafson, B. D.; States, D. J.; Swaminathan, S.; Karplus, M. J. Comput. Chem. 1983, 4, 187. (28) Liu, Y.; Nagle, J. F. Phys. ReV. E: Stat., Nonlinear, Soft Matter Phys. 2004, 69, 040901. (29) Gaede, H. C.; Yau, W. M.; Gawrisch, K. J. Phys. Chem. B 2005, 109, 13014. (30) Feller, S. E.; Zhang, Y.; Pastor, R. W.; Brooks, B. R. J. Chem. Phys. 1995, 103, 4613. (31) Tristram-Nagle, S.; Petrache, H. I.; Nagle, J. F. Biophys. J. 1998, 75, 917. (32) Essmann, U.; Perera, L.; Berkowitz, M. L.; Darden, T.; Lee, H.; Pedersen, L. G. J. Chem. Phys. 1995, 103, 8577. (33) Zhang, Y.; Venable, R. M.; Pastor, R. W. J. Phys. Chem. 1996, 100, 2652. (34) Bockmann, R. A.; Hac, A.; Heimburg, T.; Grubmuller, H. Biophys. J. 2003, 85, 1647. (35) Hofsass, C.; Lindahl, E.; Edholm, O. Biophys. J. 2003, 84, 2192. (36) Feller, S. E.; Pastor, R. W. Biophys. J. 1996, 71, 1350. (37) Deverall, M. A.; Gindl, E.; Sinner, E. K.; Besir, H.; Ruehe, J.; Saxton, M. J.; Naumann, C. A. Biophys. J. 2005, 88, 1875. (38) Almeida, P. F. F.; Vaz, W. L. C.; Thompson, T. E. Biochemistry 1992, 31, 6739. (39) O’Leary, T. J. Proc. Natl. Acad. Sci. U.S.A. 1987, 84, 429.

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