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Department of Biophysical Chemistry, University of Groningen, Nijenborgh, Groningen, The Netherlands
Correspondence: Address reprint requests to Siewert J. Marrink, E-mail: s.j.marrink{at}chem.rug.nl.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONCONCLUSIONREFERENCES |
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38 mN/m the pores are stabilized. The minimum radius at which a pore can be stabilized is 0.7 nm. Based on the critical threshold tension the line tension of the bilayer was estimated to be 3 x 10-11 N, in good agreement with experimental measurements. The flux of water molecules through these stabilized pores was analyzed, and the structure and size of the pores characterized. When the lateral pressure exceeds the threshold tension, the pores become unstable and start to expand causing the rupture of the membrane. In the simulations the mechanical threshold tension necessary to cause rupture of the membrane on a nanosecond timescale is much higher in the case of the equilibrated bilayers, as compared with membranes containing preexisting pores. |
INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONCONCLUSIONREFERENCES |
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; Deamer and Bramhall, 1986; Lawaczeck, 1988). The presence of such pores could facilitate the passive transport of polar molecules across membranes as well as being initiation sites for structural defects associated with phase transitions, cell fusion, and lysis.
Electroporation and pipette aspiration experiments have been used extensively to study the formation and expansion of water pores in protein free lipid bilayers. Model membranes, vesicles and cells of different size and chemical composition have been used to study the phenomenon of membrane permeabilization under high electric fields and/or mechanical stress (Melikov et al., 2001; Tekle et al., 2001; Akinlaja and Sachs, 1998). The exact mechanism of pore formation due to electrocompressive stress is not completely understood. It is consistently found, however, that rupture of the membrane occurs at a critical transmembrane potential. Irreversible membrane breakdown (rupture) is believed to occur when the pore expands beyond a critical radius. In cases where the radius is below a critical value spontaneous resealing will occur. This is known as reversible electrical breakdown. The threshold for electroporation of a lipid bilayer depends strongly on the type of the model membrane and the duration and strength of the applied electrical pulse. Furthermore, experiments have shown that it is possible to control the evolution of these defect structures by applying surface tension to the membrane (Zhelev and Needham, 1993). The critical membrane tension for mechanical breakdown of electroporated SOPC liposomes was found to be 1–2.8 mN/m. At this tension large pores with a radius of 2–12 µm were stable for several seconds. It was observed that all porated membranes break down at lower tensions than the respective nonporated membranes. Recently it has been shown that the process of pore formation and expansion is primarily a kinetic process (Evans and Heinrich, 2003). Thus the critical tension necessary for rupturing a membrane depends strongly on the loading rate of the applied tension. At high loading rates (25 mN/m per second) the critical tension for a DOPC membrane ranges between 10 and 20 mN/m, whereas at low loading rates (0.07 mN/m per second) it is 4 mN/m.
A number of theoretical models have been proposed to describe the formation and evolution of these meta-stable pores (Barnett and Weaver, 1991; Glaser et al., 1988; Freeman et al., 1994). All are based on the idea that mechanical or electrocompressive stress generates defects in the lipid area. The initial pores that form are believed to be small and hydrophobic with the lipid tails exposed to water. The lipids then reorient, lining the water channel, to form a hydrophilic pore. In such simplified models the free energy of formation, E, of a cylindrical pore with a radius, r, is approximated by
 | (1) |
is the line tension that opposes the creation of the pore and 
is the surface tension that lowers the energetic barrier for pore creation and expansion. Therefore, changes in the surface tension of the membrane can either stabilize or destroy the meta-stable hydrophilic pores. From experimental data and theoretical studies it has been suggested that the critical radius for the formation of a hydrophilic pore is 0.3–0.5 nm. Simulation studies using simplified models have shown that it is possible to observe pore formation and closure (Groot and Rabone, 2001; Shillcock and Seifert, 1998). Shillcock and Seifert have used a two-dimensional fluid membrane model to study the formation of thermally induced pores. In their Monte Carlo simulations, lipids are represented by hard disks connected to each other, creating a fluid-like network. A more realistic model has been used recently by Groot and Rabone to study the effect of surfactants on the stability of lipid bilayers. Their model consists of beads that represent three carbon atoms or one water molecule. The dissipative particle dynamics method was used to relate the parameters of this simplified model to specific chemical details of the system. In general it is possible to obtain qualitative results by using coarse-grained and simplified models. However, the absence of the atomistic details of the system results in the loss of any quantitative information.
Hydrophilic pores have also been observed in simulations at atomic detail. During the spontaneous aggregation of lipids into bilayers, transient pores are observed as meta-stable intermediate structures (Marrink et al., 2001). Their lifetime ranged from 5 ns to 80 ns. In addition we recently reported atomistic simulations of DPPC bilayers in which pore formation was induced spontaneously on a nanosecond timescale (Tieleman et al., 2003). This was achieved either by applying a large mechanical tension or by the application of a strong electrical field. The pores induced by these methods led to irreversible breakdown of the membrane.
In this study we present extended, atomistic molecular dynamics (MD) simulations that have been conducted to enable us to understand, in detail, the structure and dynamics of transient membrane pores. The mechanism of pore stabilization and expansion under stress is investigated. Furthermore the passive transport of water molecules through the pores is calculated and the size and shape of these structures is characterized.
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONCONCLUSIONREFERENCES |
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. The GROMOS force field (van Gunsteren, et al., 1996; 1998) was used for the description of all bonds, angles, and improper dihedrals. The van der Waals interaction parameters of the united CHn atoms of the lipid tails were adjusted to pentadecane (Berger et al., 1997). For the electrostatic interactions the fractional charges suggested by Chiu et al. (1995) were used. The water was described by the simple point charge model (Berendsen et al., 1981). The MD simulations were performed using the GROMACS package version 3.05 (Lindahl et al., 2001). Periodic boundary conditions were applied and the temperature was coupled to 323 K. This is above the phase transition of DPPC, which is at 315 K. In all simulations the pressure in the direction normal to that of the membrane was kept constant at 1 bar (Berendsen, et al., 1984). To impose a surface tension onto the system the lateral pressure was varied systematically from -10 to -1000 bar. To evaluate the nonbonded interactions a group-based, twin-range cutoff scheme was used. Lennard-Jones and electrostatic interactions within a short-range cutoff of 1.0 nm were updated every step. Long-range electrostatic interactions within a cutoff of 1.5 nm were updated every 10 steps together with the pair list. For comparison, simulations were also performed including a reaction-field correction to account for the truncation of the long-range electrostatic interactions (Tironi et al., 1995). A 5-fs time step was used (Feenstra et al., 1999). Bond lengths were constrained using the LINCS algorithm (Hess et al., 1997). The system consisted of 128 DPPC lipids and 6029 water molecules.
An overview of the simulations performed is given in Table 1. The initial structure of the system was taken from a previous simulation of the spontaneous formation of a DPPC bilayer (Marrink et al., 2001). Two starting conformations were used in this study. One was an equilibrated bilayer and the other a bilayer that contained a hydrophilic pore formed during the process of aggregation. Both of the systems were simulated under a range of different surface tensions as well as under stress-free conditions (zero surface tension). A series of simulations was also performed at constant surface area. The starting structures for these simulations were taken from the simulations of the bilayer under tension.
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, in the simulations was calculated from the external pressure applied to the system as = Lz (Pz - 
PLat
) where Lz is the length of the simulation box in the z-direction, Pz is the pressure along the z axis, and PLat the average of the lateral pressure (in the x,y plane). The water pores studied in this work are very dynamic structures. To accurately determine their size and shape, all properties were averaged over long periods of simulation time (50–100 ns). To estimate the absolute size of the stabilized pores we calculated the difference in the average number of water molecules in the membrane between the bilayer with a pore and that without, under the same conditions. By integrating across the bilayer, the total number of water molecules Ntotal inside the pore could be estimated. The region between the peaks in the lipid density distribution was integrated over, which corresponds roughly to the positions of the phosphate groups at the interface (see Fig. 3). As the shape of the pores is nonuniform across the membrane, the number of water Nwater_I (and lipid Nlipids_I) molecules only in the central region of the pore was also calculated. The central region was defined as ±0.85 nm either side of the center of the bilayer. By averaging this property every nanosecond it is possible to estimate the fluctuations in the relative size of the central pore region. To estimate the area of a pore (Acylinder) it is necessary to make some assumptions in regard to its shape and the density of water molecules inside the pore. The central part of a pore was assumed to be perfectly cylindrical in shape, containing Nwater_I water molecules at the same density as bulk water. The stated pore area therefore reflects the area of the pore in the membrane interior. The regions used in this analysis are indicated in Fig. 1.
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C, where J is the flux and C the concentration gradient. The flux was estimated as described above and the concentration gradient was taken as equal to 55.5 mol/L. |
RESULTS |
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38 mN/m) and -10 bar (10 mN/m) the pores appear to be stable (on the timescale of our simulations 160 ns) and their structural characteristics can be analyzed. In Fig. 3 the distribution of water and lipid molecules along the normal of the membrane is illustrated. Distributions are shown both for a bilayer containing a pore and for a bilayer without a pore simulated under the same tension. The difference between the water density profiles in these two systems (shaded region) reveals the distribution of water molecules inside the pore. This distribution is clearly not constant. A higher water density is found near both interfaces, at the pore openings. This is consistent with the pore having an hourglass shape.
The absolute number of water molecules Ntotal inside the pore was estimated by integrating the water density distribution inside the pore across the membrane. The boundaries of the membrane were taken as the peaks in the electron density profiles as shown in Fig. 3. The results are summarized in Table 2. Under stress-free conditions the water pore contains 92 water molecules. More water molecules enter the pore when the membrane is under tension. The size of the pore increases to 163–180 molecules for tensions between 9 and 32 mN/m. However, when the tension reaches a value of 38 mN/m the number of water molecules in the interior of the membrane increases to 224.
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85 water molecules and 7 lipid headgroups. The amount of interior water increases only slightly as the tension is increased (98 water molecules at 32 mN/m). The number of lipid headgroups does not change. However, when the tension reaches 38 mN/m both the number of water molecules (124) and lipid headgroups (9) increase significantly. The values presented in Table 2 are 50-ns averages as the pore is highly dynamic. Fig. 4 shows the dependence of the relative number of water molecules in the central region of the pore on time, for the simulation under a lateral pressure of -30 bar (25 mN/m). Notice that although the size of the pore fluctuates rapidly around the mean value, the overall structure remains stable. This holds for all simulations with a surface tension below the critical value of 38 mN/m.
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0.7 nm. The radius increases slowly to 0.9 nm at the critical surface tension. For the porated bilayers stabilized under lateral pressures of -30, -40, and -50 bar, constant area simulations have also been conducted. Starting from the preformed pores that were equilibrated under constant surface tension the area was fixed and the system allowed to evolve. The details of the simulations are given in Table 1. Note that after equilibration the lateral pressure is almost the same in all systems. The relative number of water molecules in the middle of the membrane was estimated and the averaged values are presented in Table 2. In all cases the pore remains more or less constant during the 50–100-ns simulation. The pores show no tendency to close. A constant area simulation was also performed for the bilayer under a pressure of -70 bar. The starting configuration for this simulation was a frame taken 4 ns after tension had been applied to the bilayer, at a time when the pore was already enlarged. The pore was allowed to equilibrate for 45 ns at this area. Note that the average lateral pressure drops from -70 bar to -55 bar, although the relative size of the pore does not change significantly. The flux of water molecules that pass through the equilibrated pores is also given in Table 2. For the simulations under constant surface tension it is observed that there is a small systematic increase in the flux as the tension rises. The flux increases markedly when the tension is high (38 mN/m). The simulations at constant area give similar results. The flux is slightly enhanced, however, for the case of Caxy30 and Caxy40 as compared to Stxy30 and Stxy40, respectively. Note that the flux is greatly increased when the pore is large and expanded as in the case of Caxy70. The permeability coefficient of a single pore was calculated to be in the range of 7 x 10-13 cm3/s for the Stxy50 simulation and 2 x 10-13 cm3/s for the simulation Stxy10.
Membrane rupture
When the applied pressure is more than a critical value of -50 bar (38 mN/m), the bilayer that contains a preformed pore becomes highly unstable. The time evolution of the relative size of the pore under different tensions is illustrated in Fig. 5. When the lateral pressure is high enough the pore expands. More water molecules penetrate the membrane whereas the bilayer thickness does not change significantly. The process continues until the lipid bilayer is completely disrupted. The mechanism by which pore expansion leads to the rupture of the membrane is similar for all the simulations Stxy60–100. Nevertheless, the time lag before the pore starts to grow, becomes less as the lateral pressure is increased. In the Stxy60 simulation the pore starts expanding after 15 ns whereas for Stxy100 only 4 ns are needed. The process of pore expansion is illustrated in Fig. 6 for a membrane under a lateral pressure of -60 bar (34 mN/m). Initially the minimum distance between the lipid headgroups that are in the middle of the pore is 0.3–0.5 nm and it can be seen that the lipids within the pore are quite disordered. The size of the pore remains stable for the first 15 ns, and then starts to grow. More water molecules concentrate in the membrane whereas the lipid headgroups move in and line the pore. The radius of the pore continues to increase until the rupture of the membrane after 30 ns. Notice that the thickness of the bilayer remains the same. Although the lipid headgroups within the pore rearrange during the expansion, their relative population is constant.
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0.6 nm2) as the lateral pressure increases, rupture does not occur even at pressures as high as -150 bar. The area expansion of the bilayer under low tension (9 mN/m) is 1–3%, in agreement with experimental measurements (Needham et al., 1988). Only when the pressure reaches the value of -200 bar, do the lipids start to rearrange significantly. The fluctuations induced in the lipid matrix result in the formation of a water channel in the membrane. In Fig. 7 the formation and expansion of this channel is illustrated. The starting conformation is the equilibrated DPPC bilayer. During the first few nanoseconds the membrane thins and the area almost doubles (1.2 nm2). After 12 ns a pore is formed. The distance between the lipid headgroups in the middle of the pore is initially 0.5 nm but it grows rapidly. Within 2 ns after the formation of the pore, the membrane is severely disrupted. Pore formation starts with the penetration of water molecules from both sides of the membrane. When the water molecules meet in the middle of the bilayer a complete water channel is created. Once the pore has formed, the mechanism of further expansion and membrane rupture is the same as that described previously.
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DISCUSSION |
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0.7 nm. Above a critical tension (38 mN/m) the pores become unstable and membrane rupture occurs. The same type of pores can be formed starting from an equilibrated bilayer if the tension is sufficiently high. Because the time required for pore formation at tensions close to the critical one is very slow compared with the simulation timescale, a high tension (90 mN/m) must be applied to accelerate the process. In this way spontaneous pore formation can be observed. However, this results in significant thinning of the bilayer before pore formation. The pore itself is very unstable and rupture occurs within only 2 ns. In our simulations the critical tension at which rupture occurs is higher than the reported experimental values. In general lipid bilayers are known to rupture under tensions that are 1–25 mN/m. However, rupture is a dynamical process and depends strongly on the rate at which loading is applied (Evans and Heinrich, 2003). Increasing the loading rate shifts the tension at which rupture occurs to higher values. For DOPC and SOPC vesicles it is found that the rupture tension increases by a factor of 4 when the loading rate is increased by 3 orders of magnitude. Using a kinetic model, Evans and Heinrich have shown that two regimes can be distinguished for membrane rupture. At low loading rates, the kinetics of rupture are determined by the rate at which the pore can expand, whereas at high loading rates the formation of the pore itself becomes rate-limiting. Thus, at high loading rates the tension required for pore formation is relatively high and inevitably leads to membrane rupture. This is what is observed in our simulations of pore formation in equilibrated bilayers, where the effective loading rate is very high. The timescale at which pores form and membranes rupture in biological systems under low tensions is still beyond that achievable using fully atomistic molecular dynamic simulation techniques. At low loading rates, the kinetic model predicts that the height of the barrier to pore expansion is tension-dependent. At a critical threshold tension the kinetic impedance disappears. Fitting their experimental measurements on DOPC and SOPC vesicles to the kinetic theory, Evans and Heinrich (2003) estimate this critical tension to be between 30 and 100 mN/m. The value of 38 mN/m that we find for unimpeded pore expansion of a DPPC membrane is within this range.
From the critical tension we can estimate the line tension of the membrane. At the critical tension * the free energy change upon pore expansion dE/dr is zero. Using the simple free energy expression given in Eq. 1 the line tension is then given by
 | (2) |
* = 38 mN/m the radius of the pore was estimated to be r* = 0.9 nm, resulting in a line tension for our DPPC membrane of 3 x 10-11 N. This value agrees well with the value of 1 x 10-11 N reported by Zhelev and Needham (1993), based on experimental measurements. It is interesting to point out that the line tension appears to be smaller closer to the opening of the pore. Table 2 shows that, for tensions lower than the critical tension, the opening of the pore expands whereas the pore interior remains essentially unchanged. This effect is probably a consequence of the curvature along the pore not being constant. Although the curvature perpendicular to the membrane does not change, the lateral curvature of the pore is much larger in the middle of membrane than at the rim. Therefore, the line tension at both pore openings is less, allowing it to expand at lower tensions.
To test the dependence of the results on the force field used in the simulations, additional simulations were performed in which a reaction-field correction (Tironi et al., 1995) was included in the calculation of the long-range electrostatic interactions (see Methods). Including a reaction-field correction avoids the artificial correlations that are seen when using the cutoff methods (C. Anezo et al., 2003). Qualitatively the results found are the same. However, with the reaction-field correction the pores appear less stable. The critical tension at which expansion occurs falls to 18 mN/m, a difference of 20 mN/m compared to the straight cutoff simulations. At lower tensions the pores remain stable with the same properties as was observed when using just a cutoff. The effect of the reaction-field correction therefore is a reduction of the line tension by roughly a factor of 2 (bringing it closer to the experimental values). The different properties of the reaction-field bilayer are also reflected in the area per headgroup. For an equilibrated bilayer the area increases from 0.62 to 0.66 nm2 upon including a reaction-field correction. Interestingly, when simulating with a straight cutoff a similar area of 0.66 nm2 is obtained at tensions of 20 mN/m (see Table 2). Compared to the simulations using a straight cutoff, the simulations including a reaction-field correction can be seen as equivalent to a bilayer under an effective tension of 20 mN/m, with respect to the conditions under which the model was originally parameterized.
In Fig. 8 a cartoon is presented summarizing the effect of tension on the structure and stability of the pore as observed in the simulations. At zero tension, a bilayer without a pore is the most thermodynamically stable state (Stxy0). At small tensions, pores can be stabilized inside the bilayer (Stxy10–40). When the tension is increased, the pore expands slightly. This expansion takes place predominantly at the pore openings. Once the critical tension is reached (Stxy50), expansion of the pore occurs at no free energy cost, and pores of any size can be trapped (Caxy70). Above the critical tension, the pores are unstable and will grow unhindered, leading to the rupture of the membrane (Stxy60–100). A gradual thinning of the bilayer is concomitant with an increased tension.
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6–24 water molecules per nanosecond passing through the pores that are stabilized at tensions below the critical tension. As expected, the magnitude of the flux correlates with the area of the pore and a larger flux is obtained for pores that are expanded, as in the simulation Caxy70. The flux through the nonexpanded pores corresponds to a single-channel permeability coefficient of 2–7 x 10-13 cm3/s. This is relatively close to the value of 1.1 x 10-12 cm3/s reported for desformyl-gramicidin channel, a channel with an extremely high water permeation rate (Sarapov et al., 2000). The water permeability coefficient for a pure DPPC lipid vesicle is in the order of 10-4 cm/s. If the dominant mechanism for water permeation was via water pores, 108 pores/cm2 or 1 pore/106 lipids would be sufficient to account for the flux of water molecules through the vesicle. Assuming a surface density for water of the order of 10-9 mol/cm2, this corresponds to a free energy of the pore of 40 kJ/mol. The generally accepted mechanism by which small neutral molecules permeate the membrane is diffusion through the membrane and not through water pores (Weaver et al., 1984). Forty kJ/mol thus represents a lower bound for the free energy of the pore. Typical permeation rates for ions are 10-12 cm/s. If it is assumed that the diffusion rate of ions through the pore is of the same order as that of water molecules, only 1 pore/cm2 would be required to account for the permeation rate of ions. The corresponding free energy of the pore in that case is 90 kJ/mol. The assumption that the diffusion rate of ions through the pore is of the same order as the water might be an overestimation.
The above discussion of pore formation relates to pure lipid vesicles. The question that remains is whether spontaneous pore formation is significant in a biological context. In prebiotic systems it is probable that spontaneously formed pores played a role in the transport of ions and organic compounds. Modern cells, however, have evolved a variety of mechanisms to allow for and stabilize transient pores. For example, in hyposmotic conditions which lead to the swelling of cells and high membrane tension, the rupture of the cell is prevented by specialized mechanoselective proteins such as the MscL channels which open and prevent cell lysis. These proteins have evolved to gate just below the tension required for spontaneous pore formation (Moe et al., 2000; Colombo et al., 2003). A number of experimental studies indicate that peptides are capable of permeabilizing membranes by means of pore creation. The antimicrobial peptide magainin 2, for instance, imposes positive curvature strain to the membrane, resulting in the creation of a dynamic peptide-lipid complex pore (Matsuzaki, 1998). Membrane composition is also important for the stability of the pores in cell membranes. The presence of cholesterol or short-chain lipids influences the mechanical and thermodynamic properties of the membranes, altering in this way their barrier properties.
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CONCLUSION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONCONCLUSIONREFERENCES |
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0.7 nm in the interior of the membrane. Lipid headgroups line the pore, which typically contains 100 water molecules. The water permeability coefficient of these pores is in the order of 10-13 cm3/s per pore, similar to that of a gramicidin dimer channel. In accordance with experimental measurements and theoretical predictions, the pores become unstable when the tension is increased beyond a certain threshold. For the DPPC membrane simulated we find a critical tension of 38 mN/m. At this tension the line tension opposing pore expansion is overcome. The line tension is found to be 3 x 10-11 N. At higher tensions the pore expands, leading to the rupture of the membrane. Pores can also be created in the bilayer by applying an even larger stress (90 mN/m), in which case the creation of the pore is immediately followed by rupture. Submitted on June 24, 2003; accepted for publication October 13, 2003.
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H. Leontiadou, A. E. Mark, and S.-J. Marrink Ion Transport across Transmembrane Pores Biophys. J., June 15, 2007; 92(12): 4209 - 4215. [Abstract] [Full Text] [PDF]
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V. Knecht and S.-J. Marrink Molecular Dynamics Simulations of Lipid Vesicle Fusion in Atomic Detail Biophys. J., June 15, 2007; 92(12): 4254 - 4261. [Abstract] [Full Text] [PDF]
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P.-A. Boucher, B. Joos, M. J. Zuckermann, and L. Fournier Pore Formation in a Lipid Bilayer under a Tension Ramp: Modeling the Distribution of Rupture Tensions Biophys. J., June 15, 2007; 92(12): 4344 - 4355. [Abstract] [Full Text] [PDF]
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J. Y. Lee and M. Schick Field Theoretic Study of Bilayer Membrane Fusion III: Membranes with Leaves of Different Composition Biophys. J., June 1, 2007; 92(11): 3938 - 3948. [Abstract] [Full Text] [PDF]
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M. Kotulska Natural Fluctuations of an Electropore Show Fractional Levy Stable Motion Biophys. J., April 1, 2007; 92(7): 2412 - 2421. [Abstract] [Full Text] [PDF]
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A. A. Gurtovenko and I. Vattulainen Ion Leakage through Transient Water Pores in Protein-Free Lipid Membranes Driven by Transmembrane Ionic Charge Imbalance Biophys. J., March 15, 2007; 92(6): 1878 - 1890. [Abstract] [Full Text] [PDF]
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S. Esteban-Martin and J. Salgado Self-Assembling of Peptide/Membrane Complexes by Atomistic Molecular Dynamics Simulations Biophys. J., February 1, 2007; 92(3): 903 - 912. [Abstract] [Full Text] [PDF]
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K. Koshiyama, T. Kodama, T. Yano, and S. Fujikawa Structural Change in Lipid Bilayers and Water Penetration Induced by Shock Waves: Molecular Dynamics Simulations Biophys. J., September 15, 2006; 91(6): 2198 - 2205. [Abstract] [Full Text] [PDF]
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K. Katsov, M. Muller, and M. Schick Field Theoretic Study of Bilayer Membrane Fusion: II. Mechanism of a Stalk-Hole Complex Biophys. J., February 1, 2006; 90(3): 915 - 926. [Abstract] [Full Text] [PDF]
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M. Pavlin, M. Kanduser, M. Rebersek, G. Pucihar, F. X. Hart, R. Magjarevic, and D. Miklavcic Effect of Cell Electroporation on the Conductivity of a Cell Suspension Biophys. J., June 1, 2005; 88(6): 4378 - 4390. [Abstract] [Full Text] [PDF]
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M. Tarek Membrane Electroporation: A Molecular Dynamics Simulation Biophys. J., June 1, 2005; 88(6): 4045 - 4053. [Abstract] [Full Text] [PDF]
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H. L. Tepper and G. A. Voth Protons May Leak through Pure Lipid Bilayers via a Concerted Mechanism Biophys. J., May 1, 2005; 88(5): 3095 - 3108. [Abstract] [Full Text] [PDF]
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