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Copyright © 2008 The Biophysical Society. All rights reserved.
Biophysical Journal, Volume 94, Issue 8, 2965-2986, 15 April 2008

doi:10.1529/biophysj.107.114215

Biophysical Theory and Modeling

Comparing Experimental and Simulated Pressure-Area Isotherms for DPPC

Susan L. Duncan and Ronald G. Larson^,  

Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan

Address reprint requests to Ronald G. Larson, Tel.: 734-936-0772.

Lung surfactant is the surface-active lining of the alveoli, and consists of ∼90% lipids and 5–10% proteins. Of the surfactant lipids, 80–90% are phospholipids, 70–80% of which are phosphatidylcholines, approximately half of which is dipalmitoylphosphatidylcholine (phosphatidylcholine with two palmatic acid tails, also known as dipalmitoyl lecithin) 1. Not only is dipalmitoylphosphatidylcholine (DPPC) the primary component of lung surfactant, but it is also thought to be primarily responsible for the reduction of surface tension in the lungs to near-zero. The surface film is thought to become enriched in DPPC at higher surface pressures due to selective adsorption of DPPC or the squeeze-out of non-DPPC components 2,3,4,5. Thus, understanding the response of DPPC to changes in surface area is fundamental to determining the functionality of lung surfactant and how to better design lung surfactant replacements for respiratory distress syndrome, both neonatal and adult.

Despite intensive research, the complex action of natural lung surfactant is poorly understood 6. Measurements of the surface behavior of surfactant films under dynamic compression have been among the most prevalent methods of study of pulmonary surfactant 7. The lipid phase transitions of lung surfactant are believed to be partially responsible for the regulation of surface tension in the lungs 5. A common feature of almost all lung surfactants and model mixtures is the coexistence of a semicrystalline solid phase known as the liquid-condensed (LC) phase and a disordered fluid phase called the liquid-expanded (LE) phase 8. In the LC/LE phase coexistence region, the surface film becomes a mesh of finely divided LC/LE domains, which may impart strength and flexibility 9. Lipid membrane phase transitions are also associated with density fluctuations, which are thought to play a very active role in membrane function 10. DPPC and other phospholipids are known to exhibit very rich phase behavior, which despite much research is not well understood. The current view is that the phase behavior of lipid monolayers displays subtle continuous changes between phases. The richness of phase behavior is indicative of the fact that monolayers are frustrated systems where local and global equilibria compete 11. This frustration is caused in part by the difference in the cross-sectional area of lipid headgroups and lipid tails, which induces a strain on the monolayer.

The defining features of a typical pressure-area isotherm for DPPC, in the proximity of the main phase transition temperature, are shown in Fig. 1 (left). The surface pressure π is calculated as: π=γ₀−γ, where γ₀ is the surface tension of pure water and γ is the surface tension of the monolayer-coated air-water interface 12. The monolayer area is typically given in terms of area/lipid. With increasing area and decreasing surface pressure, the phase transitions of the DPPC monolayer proceed in the following order: liquid-condensed (LC); coexistence between the liquid-condensed and liquid-expanded phases (LC-LE); liquid-expanded (LE); and coexistence between the liquid-expanded and gaseous phases (LE-G). The LC-LE phase transition is a first-order transition and is thus ideally represented by a perfectly horizontal plateau; however, experimental coexistence plateaus are only roughly horizontal. Once the monolayer has been compressed into a condensed phase, it becomes relatively incompressible and very low surface tensions (high surface pressures) are achieved with little change in area; thus, the LC portion of the isotherm has a steep slope. When the monolayer is compressed past its limiting area, monolayer collapse occurs. Collapse is signified by a decrease in area at constant surface pressure (a collapse plateau), resulting from the loss of lipids from the monolayer. In general, as the temperature is increased, DPPC isotherms shift to higher surface areas or equivalently higher surface pressures at a fixed area, and the coexistence region becomes less horizontal and is shifted to higher surface pressures 13. As shown in Fig. 1 (right), this behavior is seen in the isotherms of Crane et al. 14, which were obtained at 298.15K, 303.15K, and 310.15K using the captive bubble apparatus. This behavior is attributed to an increase in the thermal motion of the chains at higher temperature, which leads to an increase in surface pressure 15. Phillips and Chapman 16 found the static DPPC pressure-area isotherms obtained at various temperatures differed in the coexistence region, but converged at high (near-zero surface tension) and low (near-zero surface pressure) surface pressures. Similar observations can be seen in the isotherms obtained at various temperatures by Crane et al. 14 using the captive bubble apparatus (Fig. 1, right), and in the film balance experiments of Baldyga and Dluhy 17.

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Figure 1

(Left) The defining features of a typical pressure-area isotherm for DPPC near the main transition temperature. The phase regions include the liquid-condensed (LC), liquid-expanded (LE), and the LC-LE and LE-G transition regions. The LC-LE horizontal coexistence region and the horizontal collapse plateau are identified. (Right) Experimental results showing the effect of temperature on the shape of compression and expansion pressure-area isotherms of DPPC. These isotherms are reproduced from those published by Crane et al. 14, at 298.15K (dotted line), 303.15K (dashed line), and 310.15K (solid line). The experimental results presented in this figure (right) and in subsequent figures were obtained using Data Thief III, Ver.1 191.

Computer simulations of phospholipid systems are of great interest because they can yield molecular-level insight into the structure and dynamics of these systems on a resolution and timescale that may not be feasible experimentally. Coarse-grained simulations have the further advantage of realizing increased simulation times and larger system sizes. Like their experimental counterparts, pressure-area isotherms obtained from simulations of lipid monolayers also vary from study to study. For comparison, simulations of DPPC monolayers using both coarse-grained (CG) and atomistic models are included here, both from the work of other authors and from our own new simulations. To the best of our knowledge, there has not yet been a comprehensive review of the factors that could affect the shape of the pressure-area isotherm, nor a critical comparison of experimental and simulated pressure-area isotherms obtained from varying methods and experimental conditions. Therefore, here, in addition to presenting our new simulation work, we review a broad and diverse sample of the huge number of published isotherms for DPPC monolayers.

The remainder of this article is outlined as follows: First, we provide details of our simulations, then present the simulation results, and finally compare them with experimental results with a brief discussion of factors that might contribute to the observed large variation among experimental results. Although our discussion will focus on DPPC, many of the factors discussed here affect the isotherms of other phospholipids similarly.

Our simulations are divided into five categories: coarse-grained (CG) pressure-area isotherm simulations using 1), surface tension coupling; 2), anisotropic pressure coupling; 3), semiisotropic pressure coupling; and 4), the NVT ensemble, as well as 5), atomistic pressure-area isotherm simulations using surface-tension coupling. Simulation parameters are given for each type of simulation below. For all simulations, temperature was maintained by coupling to a Berendsen thermostat with a 1-ps time constant 18. All simulations were run with periodic boundary conditions. All simulations and analysis were performed using GROMACS simulations software 19,20. The GROMACS analysis tool g_energy was used to extract the surface tensions and box dimensions at each time step 21. To obtain surface pressure from our surface tensions, pure water surface tensions of 72.8, 72.5, 72.0, 71.2, 69.6, and 67.9mN/m were used at temperatures of 293.15K, 295.15K, 298.15K, 303.15K, 313.15K, and 323.15K, which are roughly the surface-tension values given in the CRC Handbook of Chemistry and Physics 22. It should be noted that the simulated surface tensions at the air-water interface actually differ considerably from the experimental values, due to the peculiar nature of water 23,24. Vega and Miguel 25 calculated a surface tension of 54.7mN/m from their SPC water simulations at 300K, which underestimates the experimental value by ∼17mN/m. This could conceivably lead to an overestimation of surface pressures in our isotherms, which are calculated from the experimental surface tension. If this were the case, the low-surface-pressure expansion observed in our simulations at surface pressures near 30mN/m would actually be occurring at significantly lower surface pressures. However, errors in simulated water/vapor surface tension are thought to have little effect on the measurement of monolayer surface tension, which is dominated by headgroup/water and chain/vapor interactions 26. Thus, it is unlikely that our surface pressures are overestimated significantly. Because sources of error in simulation of water surface tension are likely to be particular to water and not expected to similarly affect the simulation of monolayer surface tensions, we believe that it is more accurate to use the experimental values of water surface tension instead of the simulated ones, in our calculation of monolayer surface pressure.

Experimental results are typically performed under atmospheric pressure, corresponding to a normal pressure of ∼1 bar. An applied normal pressure of 1 bar is commonly used in bilayer studies 27,28,29,30. However, the simulation of monolayers requires the use of empty space placed above the monolayer to prevent the monolayer from interacting with the periodic image of the simulation box. Despite the presence of the lipid/vacuum interface, implying a normal pressure of 0 bar, some monolayer studies have used an applied normal pressure of 1 bar 27,31,32. We have simulated several points along the CG isotherm at 298.15K using both a normal pressure of 0 bar and of 1 bar. Allowing the height of the box to fluctuate with an applied normal pressure of 1 bar leads to shrinkage in the z-dimension, upon lateral expansion, requiring the box size to be manually adjusted by periodic addition of more vacuum space. However, the use of 1 bar vs. 0 bar led to no detectable difference in the isotherm. Therefore, all results presented here will be for simulations performed at 1 bar. It has also been noted that due to large fluctuations in instantaneous pressure on the order of hundreds of atmospheres, in a simulation, 1 bar is essentially equivalent to 0 bar 27,33.

For all of our coarse-grained simulations, we utilize the peptide force-field parameters developed by Marrink et al. 34. The area/headgroup for DPPC bilayers using the coarse-grained model of Marrink et al. was found to match the experimental value, and many other properties have been found to match experiment at a quantitative or semiquantitative level 34. The CG model for DPPC has one bead representing the phosphate moiety, one bead representing the choline moiety, two tail beads representing the glycerol linkage, and four beads for each of the tails (each tail bead corresponds to four tail carbons). This model is used in conjunction with the coarse-grained model of Marrink et al. for water, which merges four water molecules into a single coarse-grain bead. The structure files for the CG DPPC monolayers were adapted from the CG structure files given on Marrink's website for DPPC bilayers in the fluid phase 35 and energy-minimized. The resulting fluid phase monolayer files contained two monolayers (composed of 256 lipid each) placed so that their headgroups were initially separated by ∼7nm of CG water molecules (10,654 CG molecules) and their tail groups were separated by ∼10nm of empty space. The resulting disordered monolayers were contained in a box of size 12.6847 nm×12.8295nm×23.2nm. However, in some of our CG simulations, spontaneous box shrinking became an issue, and intermittent addition of vacuum was necessary to prevent the two monolayers from merging into a single bilayer. For all simulations, the following parameters were taken from Marrink's website 35 and have been optimized for the coarse-grained model: short-range electrostatic and van der Waals cutoffs of 1.2nm, with van der Waals interaction shifting smoothly to Lennard-Jones interaction at 0.9nm, and with the Lennard-Jones cutoff set to 1.2nm. The neighbor list was updated every 10 steps using a grid with a 1.2nm cutoff distance. In all coarse-grained simulations, the energy parameters were saved every 0.4ps and used for analysis with the GROMACS analysis tool g_energy 21.

Most of our coarse-grained simulations were 20ns in duration. Marrink and Mark 36 suggested that only a few nanoseconds of simulation time are needed to measure area/lipid for CG simulation. However, our results have shown that ∼10ns of equilibration time was necessary before areas settled down to steady values. Thus, only the last 10ns of our 20-ns simulations were used for the calculation of average surface tension and area. The radial distribution functions and angle distributions were also averaged over the last 10ns of the 20-ns CG simulations. In some cases, near a phase transition, from mostly LE to mostly LC phase and vice versa, simulations appeared to be metastable, and longer simulation times up to 100ns were necessary. In each case, the last 10ns of simulation time were used for calculations. At large values of surface tension, the box size diverged and eventually exploded, making movement further down the isotherm to low-surface pressures impossible. The divergence of box size is attributed to the onset of hole formation, followed by expansion and ultimately the rupture of the monolayer. A plot of lateral area versus simulation time is given in the Supplementary Material for a CG simulation displaying uncontrollable box expansion.

Because we are using the original CG model of Marrink et al. 37, all liquid-condensed phases simulated will be untilted. Marrink and co-workers have shown that tilted phases can be simulated using the CG model, if the model is altered to increase the size difference between the head- and tail-group beads. By decreasing the tail-group bead size by 10%, Marrink et al. 37 succeeded in simulating the tilted phase in a DPPC bilayer. It should also be noted that, due to the use of smoother potential functions for CG simulations, the dynamics of CG simulations are significantly faster (of course in computer time, but also even in physical time, as reported by the simulation) than for atomistic simulations. As a result, the effective time, which has been determined from water and lipid lateral diffusion rates, is roughly four times longer than the physical time 34. All times reported in this article will be physical time, as reported by the simulation not the effective times.

Three different pressure-coupling methods were employed: anisotropic, semiisotropic, and surface-tension pressure coupling. Anisotropic pressure coupling allows the box to flex independently in six directions (xx, yy, zz, xy/yx, xz/zx, and yz/zy) in response to a change in the pressure tensor. Semiisotropic pressure coupling only allows the box to change dimension laterally (x/y) and vertically (z). Surface tension coupling is similar to semiisotropic pressure coupling, but it uses normal pressure coupling for the z-direction, whereas the surface tension is coupled to the x/y dimensions of the box. The average surface tension γ(t) is calculated from the difference between the normal and the lateral pressure and the box is allowed to change dimension laterally (x/y) to adjust the surface tension back toward the set value. For more details on each coupling mechanism, the reader is referred to the GROMACS User Manual 21 and relevant simulation articles 27,29,30,33,38.

Simulations with surface tension coupling were run at 293.15K, 295.15K, 298.15K, 303.15K, and 323.15K. These simulations were run at several surface tensions varying between −50 and 62.5mN/m. For all simulations, the z pressure component was set to 1bar. Berendsen pressure coupling was used with a 1-ps time constant and with all compressibilities set to 5 e–6 bar⁻¹. A timestep of 0.04ps was used for most simulations. However, simulations undergoing a large change in box size (near the phase transition plateaus) required a smaller time-step of 0.02ps and longer simulation times. Two types of initial configurations were used:

Anisotropic and semiisotropic pressure simulations were run at 298.15K and at lateral pressures of 0, −10, −20, −30, and −40bar. For these simulations, the z-pressure component was set to 1bar and the off-diagonal pressure components of the anisotropic pressure tensor were all set to 0bar. Berendsen pressure coupling was used with a 1-ps time constant and with all compressibilities set to 5 e–6bar⁻¹. For all simulations, a timestep of 0.04ps was used. These simulations were run independently starting from the disorder configuration, containing 256 lipids/monolayer, described above.

Two NVT simulations were run at 298.15K. Both simulations were started form the disordered monolayer configuration, containing 256 lipids/monolayer, described above. The first simulation was run with the initial box size unchanged. The other simulation was run with the box size widened to 14nm×14nm×23.2nm and then energy-minimized. For both simulations, a timestep of 0.04ps was used.

Atomistic simulations were performed using the GROMACS force field 19,20. An atomistic structure file containing a 128-lipid DPPC bilayer was taken from the Tieleman group website 39 and modified to create a system containing two DPPC monolayers composed of 64 lipids each. The monolayers were placed with their headgroups facing each other and initially separated by ∼7nm of SPC water molecules (9662 molecules) and their tail-groups separated across a periodic boundary by ∼10nm of empty space. The resulting system was then energy-minimized and used as the starting configuration for each simulation. A 2-fs time step was used and each simulation was run for 10ns. The bond lengths were constrained using the LINCS algorithm 40. A particle-mesh Ewald summation 41 was used to calculate the electrostatic interactions with a Fourier spacing of 0.12nm and a fourth-order interpolation. The Coulomb cutoff was set to 0.9nm and the van der Waals cutoff was set to 1.2nm. The neighbor list was updated every 10 steps using a grid with a 0.9-nm cutoff distance. Temperature was maintained at 323.15K with a Berendsen thermostat 18. Surface-tension coupling was used with a Berendsen barostat and a time constant of 1.0ps with all compressibilities set to 4.5 e−5 bar⁻¹. The z-pressure component was set to 1bar. The simulations were run at several surface tensions varying between 0 and 60mN/m. Energies were output every 0.4ps for the calculation of pressure-area isotherms. Calculations were made over only the last 5ns of each simulation using the GROMACS analysis tool g_energy 21. The radial distribution functions and angle distributions were also averaged over the last 5ns of the 10-ns atomistic simulations.

We performed 20-ns cycling coarse-grained simulations of DPPC monolayers, using surface tension coupling, as described in Simulation Method, at 293.15K, 295.15K, 298.15K, 303.15K, and 323.15K. The resulting compression and expansion isotherms, for each temperature, are shown in Fig. 2. An increase in temperature results in an upward shift to larger surface pressures, a shortening of the LC-LE coexistence region of both the compression and expansion isotherms, and an increasing slope in the coexistence region of the compression isotherms. With the exception of the isotherm at 323.15K, which is shifted slightly to the right, all of the isotherms overlap except in the coexistence region. Although some experimental isotherms exhibit large hysteresis loops, the hysteresis seen in our isotherms is much larger than usually seen experimentally (Fig. 1, right), our LC-LE coexistence regions occur at much larger pressures, and our isotherms are also shifted to larger areas/lipid than those seen experimentally. Despite these differences, there are also some similarities. Experimental isotherms show, as seen in the simulations, that as the temperature is increased the coexistence region becomes less horizontal and is shifted to higher surface pressures, although the limiting high-pressure area of the isotherm remains invariant with temperature (Fig. 1, right). At 323.15K hysteresis can be seen between compression and expansion isotherms at near zero surface tension, suggesting metastability of the LC phase in the expansion isotherm at high surface pressure (Fig. 2).

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Figure 2

Our pressure-area isotherms, obtained using cycling of coarse-grained simulations at 293.15K (squares), 295.15K (asterisks), 298.15K (circles), 303.15K (diamonds), and 323.15K (triangles). The arrows indicate the direction of cycling. In this and subsequent figures, the error bars (standard error) on our simulated isotherms are roughly the same size as the symbols.

Fig. 3 shows the coarse-grained cycling isotherm at 293.15K and the corresponding changes in the packing of the C2 tail beads with movement along the isotherm. Hexagonal packing, which is characteristic of the LC phase, is clearly visible at low areas/lipid. Whereas at larger areas/lipid the tail beads display disordered packing typical of the LE phase. As expected, the phase transition region, or plateau region, is accompanied by a visible change in the degree of order of the chain packing.

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Figure 3

Coarse-grained pressure-area isotherm obtained by cycling at 293.15K and corresponding images of the packing of C2 tail beads (from both monolayers) at various points along the isotherm.

We therefore compare our coarse-grained simulations to atomistic simulations, both our own and those obtained by others, as well as to the coarse-grained results of Adhangale et al. 32 all at 323.15K. In Fig. 4, our coarse-grained results, both from independent quenching and cycling, are compared to our atomistic results from independent quenching, as well as to the atomistic results of Kaznessis et al. 24, Skibinsky et al. 42, and Klauda et al. 26 and to the coarse-grained results of Adhangale et al. 32, and to the experimental results of Crane et al. 14. Kaznessis et al., Skibinsky et al., and Klauda et al. obtained their atomistic pressure-area isotherms using the NVT ensemble in CHARMM. Adhangale et al. used the coarse-grained model developed by Marrink et al. 34, with the NP_NγT ensemble in the simulation package NAMD. The experimental pressure-area isotherm of Crane et al. 14 was obtained using a captive bubble apparatus. Our coarse-grained results are very close to those obtained from our atomistic simulations. This indicates that the shift of the pressure-area isotherms to larger areas/lipid (relative to most experimental isotherms) is not an artifact of the coarse-grained model, but occurs for coarse-grained and atomistic simulations alike. Our simulations also resemble the atomistic results of Skibinsky et al. 42 and Klauda et al. 26 and the experimental results of Crane et al. 14, differing slightly in magnitude and slope, whereas the results of Adhangale et al. are shifted to considerably lower area/lipid, and the results of Kaznessis et al. are shifted to considerably lower surface pressures.

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Figure 4

Comparison of simulated and experimental pressure-area isotherms at 323.15K: our independent coarse-grained simulations (□), our cycling coarse-grained simulations (▵), our atomistic simulations (○), the atomistic simulations of Kaznessis et al. 24 (■), Klauda et al. 26 (▾), Skibinsky et al. 42 (▴), the coarse-grained simulations of Adhangale et al. 32 (●), and the experimental results obtained by Crane et al. 14 using the captive bubble apparatus (+). For simplicity, our simulations are denoted by open symbols and solid lines, experiments are denoted by characters and dashed lines, and solid symbols and dotted lines denote simulations by other groups.

Skibinsky et al. 42 obtained starting configurations for their NVT monolayer simulations at each area, from NP_nγT bilayer simulations. This provided a well-equilibrated starting point for the monolayer simulations, which is necessary to obtain an accurate surface pressure in constant volume simulation, which does not allow area to adjust to bring the system to equilibrium. The simulations of Klauda et al. 26 were started from the final coordinates obtained by Skibinsky et al., and run under the same conditions as used by Skibinsky et al. 42 but with the addition of the isotropic-periodic sum method to treat long-range Lennard-Jones interactions. This isotherm agrees very well with the Skibinsky isotherm, only shifted slightly, suggesting that the treatment of long-range LJ interactions has only a small effect on the isotherm. On the other hand, our results were obtained using the NP_nγT ensemble with two different starting conditions: independent quenching from a disordered state and cycling (stepping down and back up the isotherm point by point from an initially ordered state). The results of Adhangale were obtained using the same coarse-grained model used in our simulations (the CG model of Marrink et al. 34), but with long-range electrostatics added in the form of a smooth particle mesh Ewald summation. The large difference between the results of the simulations of Adhangale et al. 32 and our simulations may result from a problem with their periodic boundary conditions, which leads the monolayer to curve substantially at the edges, seemingly suggesting buckling, while maintaining disorder in the acyl chains even at increased surface pressure, where our simulations and experiments show highly ordered tails. The low surface pressures shown by the isotherm of Kaznessis et al. may result from the short simulation time of 1.3ns, which is not adequate for pressure convergence. Simulation of a DPPC monolayer has also been performed by Mauk et al. 43, using a united-atom model and the CHARMM22.0 force field at 21°C; however, in this very early article, only two points of the isotherm were simulated, and the timescale simulated was only 120ps, too short to provide reliable results.

For comparison, we ran two NVT simulations at 323.15K (Fig. 5, diamonds). The first simulation was run without making adjustments to the box size (63.6Ų/molecule), and the second simulation with the box size increased (76.6Ų/molecule). When the box size is increased, an unphysical increase in pressure is observed, suggesting that the NVT ensemble does not allow for sufficient pressure relaxation. Other authors have noted the inability of constant-area and constant-volume simulations to equilibrate to appropriate pressures. Simulations of DPPC bilayers performed by Feller et al. 27,38 also show that constant-area simulations tend to predict larger surface pressures at a given surface area than those predicted by constant-surface-tension simulations. Mauk et al. 43 found that the NπT ensemble was more favorable than the NAT ensemble, the latter of which yielded inaccurate equilibrium pressures and chain order. Furthermore, Mauk et al. 43 have suggested that the inaccuracy of NAT simulations of phospholipids monolayers is due to the lack of fluctuations in the periodic cell, which restricts the phospholipids from assuming energetically favorable conformations.

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Figure 5

Coarse-grained pressure-area isotherms obtained at 298.15K using the NVT ensemble (diamonds) and the NPT ensemble with three pressure coupling mechanisms: surface tension (squares), anisotropic (triangles), and semiisotropic (circles).

Enforcing a constant surface area imposes a stronger restriction on the phase space available to the system then does enforcing a constant average pressure 44. Area is an extensive property that does not fluctuate when constrained. On the other hand, pressure is an intensive property, which is constrained as a time-averaged constant with fluctuations allowed. Also a change in pressure can be provoked by small intermolecular displacements, whereas a change in area requires large concerted motions of the lipids. Thus, the system is slow to equilibrate in response to imposed changes in area 44. However, it should be noted that constant-area simulations give reasonable results if the starting conditions are well equilibrated. In their simulations of DPPC bilayers, Feller and Pastor 38 found that order parameters, lateral diffusivities, magnitudes of area fluctuations, area fluctuation decay rates, and bilayer area compressibility moduli did not depend significantly on choice of ensemble (NP_NAT versus NP_NγT). In more recent studies, DPPC bilayer simulations showed that the pressure-area isotherms obtained using both ensembles were consistent with each other, suggesting the equivalence of the ensembles 42,45.

In addition to surface-tension coupling and NVT simulations, we also performed coarse-grained simulations using anisotropic and semiisotropic pressure coupling methods, to test the accuracy of each method. The isotherms obtained with each coupling method at 298.15K are shown in Fig. 5. At 298.15K, each coupling method gives nearly the same isotherm, differing only in LC-LE coexistence region, where they give different slopes. Although the choice of coupling method does not seem to have a big impact, the surface-tension coupling method yields the flattest plateau. Furthermore, in their simulations Feller et al. 27 set surface tension and allowed area to vary, regarding this as the most natural ensemble for simulating lipid/water interfaces. For these reasons surface-tension coupling was chosen as the preferred method and used for the majority of our simulations. Feller and Pastor 38 have suggested that simulation results depend much more on area than on ensemble used, which is consistent with our findings at 298.15K.

We calculated the distribution of P-N tilt with respect to the membrane normal from our atomistic simulations at 323.15K (Fig. 6). For comparison, the P-N tilt in our CG simulations is taken as the tilt of the bond connecting the PO4 and NC3 CG beads with respect to the membrane normal, which is calculated at 298.15K from simulations on the larger system size (1024 lipids/monolayer), and at 323.15K for the smaller size of 256 lipids/monolayer. The tilt angle was compared at areas/lipid corresponding to the two endpoints of each isotherm. No change is observed in the coarse-grained P-N tilt angle distribution as the area is changed, at either 298.15K or 323.15K. However, the atomistic simulations show a noticeable difference in the P-N tilt distribution as the monolayer is expanded from 56 to 73Ų/molecule. At 73Ų/molecule, the distribution is narrower than for the distribution at 56Ų/molecule and shifted so that although the probability of an angle below 60° is unchanged, the probability of an angle between 60° and 105° is increased, and the probability of an angle between 105° and 160° is decreased. The coarse-grained distributions are similar to the atomistic distribution at 56Ų/molecule. However, the CG PO4-NC3 tilt distribution does not exhibit the dependence on surface area seen in the atomistic simulations. The coarse-grained distributions show a shift to lower angles as the temperature is increased, and the distribution narrows slightly, excluding angles above 160°. Our atomistic simulations at 323.15K give a single peak centered at ∼90° at 56Ų/molecule and at ∼85° at 73Ų/molecule. Our coarse-grained simulations peak at 90° at 298.15K and 78° at 323.15K.

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Figure 6

P-N tilt angle distribution for atomistic simulations at 323.15K with areas 56Ų/molecule and 73Ų/molecule, for coarse-grained (CG) simulations with 1028lipids/monolayer at 298.15K with areas 48Ų/molecule and 68Ų/molecule, and for coarse-grained simulations with 256 lipids/monolayer at 323.15K with areas 56Ų/molecule and 71Ų/molecule. The solid, dark-shaded, and light-shaded lines represent the atomistic simulations, and CG simulations at 298.15K and 323.15K, respectively. For each shade, the solid and dotted lines represent the smaller and larger area per lipid, respectively. For clarity, the data shown here has been smoothed using time-averaged values.

Numerous experimental studies, including surface-potential measurements, on phospholipid bilayer systems suggest that the P-N orientation is parallel to the bilayer surface 46,47. A recent sum frequency generation spectroscopy study performed by Ma and Allen 48 suggests that the choline methyl groups are tilted from the surface normal and lie roughly parallel to the air-water interface. The sum frequency generation spectra obtained by Ma and Allen 48 at 12mN/m (LE phase) and 42mN/m (LC phase) are similar. These results suggest that the choline headgroup orientation is not significantly different in the LE and LC phases, in accordance with the previously held hypothesis that the overall conformation of the headgroup is not as sensitive to the aggregation state and the nature of the environment as the tails 48,49. The P-N tilt angle distributions obtained from our atomistic and coarse-grained simulations are also centered at or near 90°, in accord with experiments. Our results are also in agreement with previous atomistic simulations of a DPPC monolayer performed by Dominguez et al. 50, which showed that the average angle between the monolayer surface and the P-N vector was 5°. Although the shape of simulated P-N distributions vary, more recent atomistic 31 and coarse-grained 32 simulations have also shown average P-N tilt angles in the proximity of 90° with respect to the membrane normal.

As the DPPC monolayer undergoes a transition from the liquid-expanded to the liquid-condensed phase, the methylene groups of the DPPC tails transform from predominantly gauche conformations to all-trans conformations 48. The lipid tail dihedral distribution was calculated from the four CG tail beads for a system size of 1024 lipids/monolayer. At 298.15K we found that at 48Ų/molecule the trans tail configuration (180°) is highly preferred over the gauche configuration (±60°) and at 68Ų/molecule the trans configuration becomes less favorable and the distribution broadens such that all tail dihedrals are sampled almost equally, as is expected (data not shown).

In Fig. 7, the PO4-PO4, PO4-NC3, NC3-NC3, and C2-C2 radial distribution functions (RDFs) are shown, where PO4 is the phosphate moiety, NC3 is the choline moiety, and C2 is the second CG tail bead from the glycerol linkage (which corresponds to the fifth through eighth carbon atoms from the glycerol linkage); each of these sites is represented by a single coarse-grained bead. Each radial distribution function is normalized so that the integral is equal to the total number of lipids (twice the number of lipids in the case of the C2-C2 distribution because there are two C2 sites/lipid). The atomistic results compared in Fig. 7 were obtained using the following atoms: P, N, and the sixth tail carbon from the glycerol linkage. The two endpoints of each isotherm are selected to observe the effect of surface area on the shape of the radial distributions. Each isotherm used was obtained from independent runs rather then cycling. At 298.15K (Fig. 7, left) the RDFs are compared at areas of 48 and 68Ų/molecule for the larger CG system size (1024 lipids/monolayer). At 323.15K (Fig. 7, center and right) the RDFs are compared at areas of 56 and 71Ų/molecule for a CG system of size 256 lipids/monolayer and at areas of 56 and 73Ų/molecule for an atomistic system size of 64 lipids/monolayer. The difference in the areas shown at 298.15K and 323.15K reflects the shift in the isotherms to larger areas/lipid as temperature is increased.

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Figure 7

Radial distribution functions. (Left) Independent coarse-grained (CG) simulations at 298.15K for the larger system size (1024 lipids/monolayer) at both 48Ų/molecule (black) and 68Ų/molecule (red). (A) PO4-PO4 distribution. (D) C2-C2 distribution. (Center and right) Atomistic (atom.) simulations at 323.15K with 64 lipids/monolayer at both 56Ų/molecule (black) and 73Ų/molecule (red) and independent CG simulations at 323.15K with 256 lipids/monolayer at both 56Ų/molecule (green) and 71Ų/molecule (blue). (B) PO4-PO4 distribution; (C) NC3-NC3 distribution; (E) C2-C2 distribution. (F) PO4-NC3 distribution.

At 298.15K, the CG PO4-PO4 (Figure 7A), PO4-NC3 (not shown), and NC3-NC3 (not shown) RDFs show little difference as area is changed from 48 to 68Ų/molecule; however, the C2-C2 (Figure 7D) RDF changes significantly. At 48Ų/molecule, the C2-C2 RDF reflects the highly ordered tails expected for a system in the LC phase, whereas at 68Ų/molecule it reflects the disordering of the system. These CG results are in contrast to the atomistic results of Knecht et al. 5 at 293K, which show that decreasing the area/lipid causes lipids to bind closer together, leading to an increase in the phosphate-phosphate correlation in addition to the increase in tail order observed here. Although our CG radial distribution functions show a clear increase in tail order as area is decreased, unlike the atomistic simulations of Knecht et al, we see only a small increase in the height of the first phosphate-phosphate correlation peak. These results suggest that the coarse-grained model is better at capturing the effect of changing surface area on lipid tails than on lipid headgroups.

At 323.15K the simulated isotherms are in the expanded phase. The CG C2-C2 (Figure 7E) distribution indicates that the tails are slightly more ordered at 56Ų/molecule than at 71Ų/molecule. However, both areas/lipid give an RDF that reflects considerably less order than does the LC RDF at 298.15K and 48Ų/molecule (Figure 7D), and is comparable to the less ordered distribution at 298.15K and 68Ų/molecule (Figure 7D). At 323.15K, the CG PO4-PO4 (Figure 7B), PO4-NC3 (Figure 7F), and NC3-NC3 (Figure 7C) RDFs show little difference between the two areas/lipid and are almost identical to those at 298.15K (Figure 7A, PO4-NC3 and NC3-NC3 distributions are not shown), suggesting that temperature has a larger effect on the RDF of lipid tails than that of lipid headgroups.

For the atomistic simulations at 323.15K (Fig. 7, center and right), a change in surface area from 56 to 73Ų/molecule does not strongly affect any of the RDFs; however, the distributions do appear to fluctuate more at 73Ų/molecule. Overall the CG and atomistic radial distribution functions match reasonably well at 323.15K. Despite some differences, the C2-C2 and C-C (Figure 7E), PO4-NC3 and P-N (Figure 7F), and PO4-PO4 and P-P (Figure 7B) RDFs correlate well. However, the NC3-NC3 and N-N (Figure 7C) RDFs differ from each other considerably, whereas the NC3-NC3 (Figure 7C) RDF is very similar to the PO4-PO4 RDF (Figure 7B), indicating that the coarse-grained model is unable to capture the difference in the N-N and P-P interactions present in the atomistic simulations, which ultimately leads to inaccuracy in the NC3-NC3 RDF. The (inaccurate) similarity between the NC3-NC3 and PO4-PO4 distributions in the CG simulations is a direct result of an oversimplification contained in the CG model. The CG model uses bead types Qd (charged hydrogen-bond donor) and Qa (charged hydrogen-bond acceptor) to represent NC3 and PO4 sites, respectively. Qa-Qa and Qd-Qd Lennard-Jones interactions are both considered intermediate and have the same LJ parameters 25.

The shape and location of the peaks of our atomistic P-N and P-P RDFs correlate well with the atomistic results of Kaznessis et al. 24 for a DPPC monolayer and Sun 31 for a 1,2-dilignoceroylphosphatidylcholine monolayer. Both our P04-NC3 (CG) and P-N (atomistic) RDFs show a strong attraction between choline and phosphate groups, in agreement with the atomistic results of Kaznessis et al. 24. It has been proposed that electrostatic interactions between neighboring choline and phosphate groups are responsible for attraction between neighboring phospholipids 51.

Our simulations show hole formation at areas in the proximity of 100Ų/lipid, which could represent the onset of the liquid-gas phase transition. For the CG surface tension coupling simulations, at 323.15K, calculations were made for specified surface tensions between 0mN/m and 46.6mN/m, which yielded average surface pressures between 68.8mN/m and 21.5mN/m. When the specified surface tension was increased further to 47mN/m, a jump in area/lipid was observed from ∼71.4Ų to∼129Ų. As shown in Fig. 8, this jump in area/lipid is accompanied by hole formation, which is not an artifact of the coarse-grained method of simulation, because hole formation was also observed in our atomistic simulations (left). The holes are unstable and expanding, ultimately leading to the rupture of the monolayer. Knecht et al. 5 also saw hole formation in their united-atom simulations of DPPC monolayers. They observed the transient formation of holes at ∼98Ų/molecule and stable pore formation at ∼105Ų/molecule. According to Knecht et al. the appearance of holes suggests the onset of the LE-G phase transition. Fluorescence microscopy has revealed that in the LE-G coexistence region the gas phase is present as holes in an interconnected liquid phase 52. Due to limited spatial resolution of fluorescence images, the LE-G coexistence region cannot be directly determined using fluorescence microscopy 5. However, the LE-G phase transition is thought to occur at areas of hundreds of Ų/molecule 53. Knecht et al. propose that the hole formation in their MD simulations corresponds to the sharp transition in the order of lipid chains recently detected by vibrational sum frequency generation spectra at 110Ų/molecule, which they suggest could be associated with the onset of the gas-liquid coexistence region 5. Knecht et al. also observed LC domain formation away from pore boundaries 5. Whether LC domain formation can be seen in CG simulations at conditions beyond those needed to generate holes has not yet been tested. In contrast to our results and those of Knecht et al. 5, the results of Nielsen et al. 54 using a CG model (which is structurally similar to the model of Marrink et al. 34, but includes long-range electrostatics) showed that at large area/lipid, monolayer lipids become highly disordered and spread on the surface instead of forming holes. In the simulations of Nielsen et al. 54, the entropic benefit of spreading on the surface outweighs the van der Waals interaction energy, which suggests a possible problem with their energy parameterization, which they admit is exploratory and not yet validated. Hole formation has also been observed in atomistic simulations of DPPC bilayers. Leontiadou et al. 28 observed a critical surface tension (∼38mN/m) above which pores in the bilayer expand becoming unstable and ultimately leading to the rupture of the bilayer. Feller and Pastor 38 have also described large and sudden expansions at a surface tension of 50mN/m, which may suggest the disruption of the bilayer.

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Figure 8

Hole formation in atomistic (left) and coarse-grained (right) simulations at 323.15K, from the side (top) and corresponding top view (bottom). The lipid tails and glycerol groups are shown in green, the headgroups in red, and the waters in blue. The corresponding surface tensions and simulation times are given below the images.

It is generally agreed that the packing of DPPC molecules is determined by the size difference between the head- and tail-groups, with the area required by the headgroup being substantially larger than that required for the tails, leading to packing adjustments such as lipid-chain tilting and headgroup overlap 46,49. The coarse-grained model of Marrink et al. 34 utilizes a Lennard-Jones bead size of σ=0.47nm, for all bead types. Thus it does not capture the large difference in limiting area between the phosphatidylcholine headgroup and the acyl chains. To test the effect of the relative size difference between the headgroup and acyl chains on the packing of DPPC monolayers, we ran simulations (results not shown) with the bead size of the tails including the glycerols decreased, while the headgroup bead size remained at 0.47nm. Our simulations showed that decreasing the tail-bead size by the proper amount allows the monolayer to achieve smaller minimum areas closer to the experimentally determined limiting area, while maintaining the correct packing arrangement. On the other hand, decreasing tail-bead size too much impairs packing and the area is not minimized.

Many studies containing experimentally measured pressure-area isotherms for pure DPPC monolayers have been reported 3,13,14,15,16,17,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167. However, very few studies compare their isotherms with those of others, and those that do tend to compare with only one or two selected isotherms that resemble their own. A major reason for this lack of comparison is due to the diverse conditions under which isotherms are obtained, making reproducibility problematic. Thus, even though the pressure-area isotherm of a monolayer is a thermodynamic relationship that, like pressure-volume isotherms for bulk substances, ought to be a universal function if measured accurately and under equilibrium conditions, in practice isotherms vary considerably, due to variability in compression rate, type, and geometry of experimental apparatus, and experimental artifacts (leakage, impurities, etc.), as well as pH, ionic strength, and spreading solvent 168. The variation among selected experimental isotherms is illustrated in Fig. 9, at 293.15K (top left), 295.15K (top right), 298.15K (bottom left), and 303.15K (bottom right) with our simulated isotherms included.

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Figure 9

Comparison of simulated CG pressure-area isotherms with various experimental ones at 293.15K (top left), 295.15K (top right), 298.15K (bottom left), and 303.15K (bottom right).

The complexity of phospholipid phase behavior and the many experimental factors involved can lead to results that are ambiguous and apparently conflicting. The difficulty in finding isotherms obtained under similar conditions has been noted before 16,76. Experimental artifacts can also lead to results that can be easily misinterpreted. Different authors may come to remarkably different, and often contradictory, interpretations of monolayer behavior, involving factors such as collapse mechanism, relaxation times, and the effect of the experimental conditions (spreading method, compression rate, etc.). These differences are not inconsequential; the shape of the isotherm is physiologically relevant, making accurate determination of it very important. For example, the very low surface tension when the film is compressed toward collapse is thought to be a mechanism for preventing alveolar closure at end-expiration 169, and the steep slope of DPPC postcollapse expansion isotherms is thought to be important for alveolar recruitment and stabilization of lung units during inspiration 4. Furthermore, the shape of the isotherm is crucial to obtaining a proper understanding the behavior of the monolayer on the molecular level; for example, the compressibility is determined from the slope of the isotherm 158.

When comparing experimental pressure-area isotherms, there are a few key experimental trends to keep in mind. Varying the dynamic compression rate is not expected to have a large effect 72,158,170, and in many cases the presence of relatively small concentrations of ions leads to little or no change in the isotherm of zwitterionic monolayers 63,140,141,142,171. At moderate pH, the isotherm shows little sensitivity to pH. However, at low pH, decreased hydrogen-bonding leads to an increase in the maximum surface pressure and can cause a shift to smaller areas due to hindered solvation, and at high pH, solvation is increased and equilibrium is shifted toward the fluid phase 130,143,172. The type of experimental apparatus used is known to have an effect on the shape of pressure-area isotherms, and each type has a unique set of conditions and limitations to take into account 7,12,72,82,151,163,168,173,174,175,176,177,178,179,180,181. The geometry should be considered because of curvature effects, area available for creep and leakage, and disordering of lipids near walls that all effect the measurement of area/lipid. The potential for leakage is greatest at high temperatures and large dynamic pressures, and is greatest in a conventional Langmuir trough; however, the use of devices such as ribbon barriers help minimize or even eliminate leakage 72,82,174,182. The pulsating bubble surfactometer also suffers from leakage, whereas the captive bubble apparatus is free from the effects of leakage. Leakage leads to a shift in the isotherm to lower surface pressures and a decrease in its slope, which can be mistaken as premature collapse 82. Even in the absence of leakage, creep along the walls can be an issue and problems with contact angle can give erroneously low surface tensions 176,182. Impurities may also arise from many sources including the experimental apparatus itself, and lead to isotherms that do not have a well-defined phase-transition region, are shifted, or do not reach near-zero surface tensions upon end-compression 16,145,148. Careful consideration of the choice of spreading solvent is necessary, because it can have a large effect on the displacement of isotherms along the area/molecule axis and can impair film stability 76,150,179. Polar components are surface-active and may solubilize the lipids, causing a shift in the isotherm to very low areas/lipid due to the loss of lipid from the interface. The effects of compression rate, pH, ionic strength, experimental apparatus, spreading agent, and impurities are discussed in more detail in the Supplementary Material .

As noted by others 43, simulations of phospholipid monolayers are limited to the nanosecond timescale, which cannot account for long time adjustments that the monolayers undergo to reach equilibrium. Thus, the results of computer simulations of phospholipid monolayers must not be interpreted as equilibrium behavior, but rather as dynamic (i.e., metastable or quasiequilibrium). This is important to take into account when comparing simulation results to experimental data. It is important to compare simulation results with dynamic isotherms (isotherms compressed relatively rapidly and thus allowed to reach near-zero surface tensions), rather than static isotherms, which have relaxed to equilibrium and reach substantially lower surface pressures.

In Fig. 9, our simulated pressure-area isotherms are compared to experimental isotherms at 293.15K 15,55,56,57,58,59,60,61 (top left), 295.15K 62,63,64,65,66 (top right), 298.15K 14,67,68,69,70,71,72 (bottom left), and 303.15K 14,17,73 (bottom right). For each temperature, our simulations were run both independently from an initially disordered state (black triangles) and cycled beginning from an initially ordered state (blue squares). At 303.15K, the results from cycling simulations are compared for run durations of 20ns and 100ns at each point (Fig. 9, bottom right). The experimental conditions for each isotherm are given in Table 1, including compression rate, type of experimental apparatus, subphase composition, pH, and spreading solvent.

These experimental isotherms in Fig. 9 vary greatly from one to the next in shape and magnitude. All of the isotherms presented here were obtained at moderate pH, except those of Kanintronkul et al. 68 (pH 9) and Nakahara et al. 70 (pH 2), both at 298.15K. pH is not expected to be a major factor affecting the isotherms that were obtained at moderate pH values, for which the monolayer is thought to be insensitive to pH. The isotherm of Kanintronkul et al. 68 is shifted to a larger area/lipid relative to the other isotherms; it also displays elevated surface pressures at large areas/lipid, and does not display well-defined phase transitions. This can likely be attributed to increased solvation and a shift in equilibrium toward the fluid phase, resulting from the high pH. In contrast, the isotherm of Nakahara et al. 70 is shifted to lower areas/lipid reflecting hindered solvation attributed to the acidic medium.

No defining trends associated with the type of apparatus used are evident from the isotherms shown in Fig. 9. All of the isotherms obtained at 293.15K and 295.15K were obtained in a trough (see Table 1), yet much variation among them remains. At 298.15K and 303.15K, all pressure-area isotherms were obtained with a trough, except for the isotherms reported by Crane et al. 14, which utilized the captive bubble apparatus. Despite this, the isotherms presented by Crane et al. 14 do not have any defining features that distinguish them from the other isotherms presented here. Leakage could be an issue in any of the experiments except those of Crane et al. (because of the use of the captive bubble apparatus), the equilibrium isotherm of Tabak et al. 72 (because spreading inside a beaker was used), and the dynamic isotherms of Tabak et al. 72 and Taneva et al. 66 (because of the use of a ribbon barrier). Furthermore, experiments performed without the use of a Wilhelmy plate or with discontinuous compression may be especially susceptible to leakage. Thus, leakage is a likely factor attributing to the large variation between the experimental isotherms shown here.

Dynamic compression rate appears to play a role in the slope of the isotherms at high surface pressures (low areas/lipid). The slope tends to become steeper as compression rate is increased. Isotherms compressed the quickest, such as those obtained by Bordi et al. 56, Williams et al. 62, Rana et al. 64, Slotte and Mattjus 65, and Crane et al. 14 have the steepest slopes. This is made more evident by the magnitude of the area compressibility moduli calculated for these isotherms (discussed in detail in the next section). Note that although slower compression leads to better equilibration, it does not necessarily produce more accurate isotherms. Additionally, isotherms that compressed quickly better mimic the physiological conditions.

The spreading solvent is typically not thought to have a large effect when used in a trough, which is open to air circulation and takes up a relatively large surface area. Nevertheless, in comparing these isotherms, spreading solvent does appear to have played a major role. At 293.15K, the isotherms obtained by Borissevitch et al. 57, Dubreil et al. 58, Ahuja and Möbius 55, and Yun et al. 15 all reach relatively low surface pressures at end compressions of roughly 42, 48, 50, and 55mN/m, respectively. Collapse does not appear to have been reached before measurement was halted for the isotherms of Dubreil et al. 58 and Ahuja and Möbius 55, and it remains uncertain what the actual collapse pressure would have been. For all of these isotherms the spreading solvent was pure or almost pure chloroform (98% in the case of 55). At 295.15K, the isotherm obtained by Dynarowicz-Łątka et al. 62 used the highest concentration of chloroform in the spreading solvent (90% by volume), and also has the lowest maximum surface pressure (highest minimum surface tension). At 298.15K, slightly low dynamic maximum surface pressures are obtained by Nakahara et al. 70 (∼64mN/m), Kanintronkul et al. 68 (∼65mN/m), Shen et al.