MolecularDynamics Study of the Electric andDielectric Properties ofModel
DPPC and Dicaprin Insoluble Monolayers: Size Effect
Stanislav Tzvetanov,† Philip Shushkov,†,‡ Maria Velinova,† Anela Ivanova,† and Alia Tadjer*,†
†Laboratory of Quantum and Computational Chemistry, Department of Physical Chemistry, Faculty ofChemistry, University of Sofia, 1 James Bourchier Avenue, 1164 Sofia, Bulgaria, and ‡Department of
Chemistry, Yale University, 225 Prospect Street, New Haven, Connecticut 06520-8107
Received December 16, 2009. Revised Manuscript Received February 2, 2010
Atomisticmodeling of insolublemonolayers is currently used to inspect their organization and electric characteristics,providing a link between theory and experiment. Extensive molecular dynamics simulations at 300 K were carried outfor model films of the lipids dipalmitoylphosphatidylcholine (DPPC) and dicaprin (DC) at the air/water interface.Surface concentrations corresponding to a set of points along the surface pressure/area isotherms of the surfactants wereconsidered. The models contained 25 or 81 lipid molecules in hexagonal arrangement and explicit aqueous media(TIP3P) treated in periodic boundary conditions. Molecular dynamics simulations based on a classical force field(CHARMM27) were carried out and key characteristics of the studied films were estimated. The dielectric properties ofthe films in normal and tangential direction were quantified by means of dipole moment magnitude and orientationanalysis and bymonolayer dielectric permittivity. The contributions of lipids and interfacial water to each component ofthe considered characteristics were assessed and their variations upon film compression were discussed and comparedfor the two monolayers and to earlier results. The dielectric permittivity tensors were analyzed. Electrostatic potentialprofiles across the layers and surface pressure values were used for more detailed clarification of experimentalmeasurements. The results show dissimilar behavior of the two lipids at the air-water interface. While the averageelectric and dielectric properties of DPPCmonolayers result from opposite surfactant and water contributions, the twosubsystems are synergetic in the DC films. The anisotropy of the monolayer dipole moment and dielectric permittivity isexplained by domination of a different subsystem in the various components. Tangential characteristics turn out to bemore sensitive to the size of the model and to the degree of film compression.
Introduction
Electric and dielectric properties of insoluble mono-layers attract the interest of both experimentalists and theo-reticians because they can serve as indirect markers ofsurfactant organization at the air/water interface.1-3 More-over, some dielectric characteristics are used as parametersin the description of domain formation within Langmuirlipid monolayers.4
Probably the most frequently studied films of nonionogeniclipids are those of dipalmitoylphosphatidylcholine (DPPC). How-ever, the main focus of the experimental2a,5 and theoretical6-11
investigations of DPPC monolayers are thermodynamic and mor-phological characteristics of the lipids at the interface. Moreover,almost all molecular simulations are limited within a narrow rangeof surface concentrations (close to that present in living cells).6-11
Less attention is paid to the (di)electric features of these mono-layers, for instance tomeasurements of dipolemoments or dielectricpermittivities.
The most closely related recent theoretical publication, addres-sing electric properties of DPPC,12 has reported QM/MM calcula-tionson the surfactanthead in vacuumand in thepresenceof severalwatermolecules. The importance of the solvent was outlined, whichinfluences substantially the dipole moment by change of conforma-tion due to hydrogen bonding or by intermolecular charge transfer.
Probably the most frequently experimentally measured charac-teristic of DPPC monolayers is the surface pressure. It has beenused for structural and rheological studies almost from the begin-ning of Langmuir monolayer investigations. Lately, its evolutionuponmonolayer compression has beenused to extract the elasticityof monolayers (from the first derivatives with respect to area)13,14
*Corresponding author. E-mail: [email protected]. Telephone:þ35928161374. Fax: þ35929625438.(1) Gaines, G. L., Jr. Insoluble Monolayers at Liquid-Gas Interfaces; Wiley-
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or the free energy of mixing with other surfactants (throughintegration).14 Dipole-dipole interactions have been suggestedas the driving force for the mixing.14
There have also been attempts to estimate the normal dipolemoment of DPPC monolayers from experimental data. Twomethods for such evaluation dominate in the literature. The firstone is based on the classical Helmholtz equation,1 while the moremodern approach relies on Maxwell displacement current mea-surements.14,15 Using the latter, the profile of the normal dipolemoment along the entire surface/pressure area isotherm has beendetermined.14,16 In ref 14, the authors have suggested dipolemoments compensation as a possible reason for favorable mixingof DPPC with a second surfactant. Then, this contribution hasbeen ruled out as the dominating factor due to its negligiblemagnitude. On the other hand, the respective part of the tangen-tial dipoles has not been discussed. It is noteworthy that the dipolemoment curves in refs 14 and 16 are characterized by a minimumin the region of LE/LC phase coexistence. However, its origin hasnot been commented by the authors. Another indirect relation ofthe dipolemoment to the thickness ofDPPCmonolayers has beenthe hypothesis13 that the pHdependence of the latter is defined bythe orientation of the P-N vector of the choline group. Frommechanistic point of view, control of the monolayer dipolemoment has been deemed as a factor facilitating Langmuir-Blodgett film deposition.14
Dicaprin (DC) is another lipid forming stable insoluble mono-layers at the air/water interface.17 Unlike DPPC, the surfacecharacteristics of pure DC monolayers have been rarely studied.Most often,DC is employed as substrate for testing the hydrolyticactivity of various enzymes.18 The rheological properties ofdicaprin monolayers have been described solely by Ivanovaet al.19a and by Nannelli et al.19b They have recorded the surfacepressure/area and surface potential/area isotherms of DC films.Thereof, it has been seen that DC does not show any specificfeatures typical for lipid reorganization upon compression.This isin contrast to the behavior of DPPC, the isotherms of which arecharacterized byawell expressed plateau;a signof coexistence ofseveral phases.
The dissimilar conduct of the two lipids prompted us to usethem as targets of the present study, which is an attempt fordetailed theoretical interpretation of the electric and dielectric
properties of the respective monolayers formed at the gas/waterinterface at various surface concentrations.
An additional point of interest toward the (di)electric fea-tures of the monolayers stems from the fact that theoreticalsimulations allow correct estimation of the anisotropy ofinterfacial quantities and evaluation of their dependence onthe degree of compression of the film. As evident from theabove publications summary, theoretical calculations can alsoprovide numerical values of some parameters, which are oftenneeded by phenomenological models.4,20,21 For example, themonolayer dielectric constant and the profile of the electro-static potential along the film have been included in modelsaimed at explaining ion binding to DPPC.20 Transition dipolemoments of CH2 vibrations have also been of interest fordescription of monolayer phase states.21
Previously, we have reported22,23 some theoretical results forthe dipole moment and the dielectric permittivity of the same twomonolayers. The datawere extracted from relatively small modelscomprising two to nine lipids in the elementary cell but permittedoutlining the main tendencies of dipole moment and dielectricpermittivity variation in directions normal and tangential to theinterface upon film compression.
Thus, another goal of the present study was to test thesensitivity of our results toward extension of the size of themodels(see also ref 24). In addition, other thermodynamic parameters ofthe monolayers, such as surface potential and surface pressure,were assessed and discussed.
The theoretical estimateswere obtainedbymolecular dynamicssimulations of DPPC and DC monolayers and are presented interms of dipole moments and dielectric permittivities. Bothquantities were decomposed into normal and tangential compo-nents. The shares of surfactants and water were determined, too.
Models and Computational Protocol
This section outlines the construction of models and thecomputational scheme used. Models with 25 and 81 lipids werebuilt in 3D periodic boundary conditions, as specified in ourprevious studies,22,23 in order to test the effect of the elementarycell (EC) size on the computational results. Hexagonal lattice andexplicit water molecules were used throughout (Figure 1).
The periodic box dimensions along the x and y axes wereselected such that the corresponding areas per lipid molecule ofthe monolayer fell into the liquid condensed (LC) or the solid
Figure 1. Top view of monolayer elementary cells containing 25 molecules DPPC at 50 A2/molecule (left) and 25 molecules DC at 60 A2/molecule (right). Unit translation vectors and orientation of the Cartesian coordinate system are shown, too.
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450. (b) Ziomek, E.; Douchet, I.; Ivanova, M.; Verger, R.Chem. Phys. Lipids 1996, 81,1–9. (c) Rao, C. S.; Damodaran, S. Langmuir 2002, 18, 6294–6306.(18) (a) Cajal, Y.; Busquets, M. A.; Carvajal, H.; Girona, V.; Alsina, M. A.
J. Molec. Catal. B 2003, 22, 315–328. (b) Rogalska, E.; Ransac, S.; Verger, R. J. Biol.Chem. 1993, 268, 792–794. (c) Gargouri, Y.; Pitroni, G.; Rivicre, C.; Sardat, L.; Verger,R. Biochemistry 1986, 25, 1733–1738.(19) (a) Ivanova, M.; Svendsen, A.; Verger, R.; Panaiotov, I. Colloids Surf. B:
Biointerfaces 2000, 19, 137–146. (b) Nannelli, F.; Puggelli, M.; Gabrielli, G. ColloidsSurf. B 2002, 24, 1–9.
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(21) Mao, G.; Desai, J.; Flach, C. R.; Mendelsohn, R. Langmuir 2008, 24, 2025–2034.
(22) Shushkov, P. G.; Tzvetanov, S. A.; Ivanova, A. N.; Tadjer, A. V. Langmuir2008, 24, 4615–4624.
(23) Tadjer, A.; Ivanova, A.; Velkov, Y.; Tzvetanov, S.; Gotsev, M.; Radoev, B.Int. J. Quantum Chem. 2007, 107, 1719–1735.
(24) Shushkov, P. G.; Tzvetanov, S. A.; Ivanova, A. N.; Tadjer, A. V. Langmuir2010, submitted as companion paper (DOI: 10.1021/la904734b).
condensed (SC) regions of theΠ/A isotherms (Figure 2), namely,40, 50, 60, 70, and 80 A2/molecule. Simulations of ECs built of 81DPPC molecules at 80 A2/molecule were not carried out due tothe limitations of the TIP3P water model, which gives loweredsurface tension of the neat water surface and thus is not useful formodeling ofmonolayers at low surface concentrationwhere poresare observed.25 A DC system at 40 A2/molecule was not con-sidered because it falls beyond the collapse of the film,19 wheremultilayer structures start to form.
The EC geometry was dictated by the shape of the surfactantmolecules. A rhombic cell with angle of 60� and a parallelogram-shaped cell with angle of 45� were chosen for DPPC and DC,respectively (Figure 1). Details of all models sizes are given inTable 1.
For a given number of lipids in the EC the amount of watergrew proportionally to the area/molecule so that the thickness ofthe water layer remained relatively constant for all surfaceconcentrations considered. The size of the periodic box alongthe z-axis was set large enough to guarantee the lack of periodicityin this direction, thus ensuring the 2D dimensionality of themonolayer.
The currentmodels described in amore realisticway the studiedsystems than those with the smaller ECs reported before.22,23
When the periodic box comprised 25 surfactants, the nine centralmolecules (36%) interacted with eight real lipids each and thosefrom the external shell (48%) of the EC interacted with five realsurfactants and three periodic images. The four vertex molecules
in the EC (16%) were an exception because they “saw” just threereal lipids and five periodic images. In the models with 81 lipidsmore than 60% of the surfactants (49 molecules) had only realpartners and approximately 35% (28 molecules);five real lipidsand three periodic images (Table 2).
The force field CHARMM2726-29 as implemented in theprogram package GROMACS 3.3.330 was used in all calcula-tions.29 TIP3P26 and TIP4P26 (only in the case of 81 DPPC at 70A2/molecule due to artificial structuring of the monolayer yieldedby the TIP3Pmodel) models were employed for description of thewater molecules. Bulk water below the monolayer was mimickedby imposing an additionally adjusted potential (see Appendix A)acting in direction normal to the interface on the oxygen atoms ofthe water molecules from the neat water surface. Throughout thesimulations the O-H bonds of water were kept frozen withSETTLE31 and all other hydrogen-containing bonds were fixedwith LINCS.32 Both algorithms were used as implemented inGROMACS 3.3.3.30
All molecular dynamics (MD) simulations were carried outwith the following general protocol (for more details and for thespecific lengths of the separate stages refer to Table S1 of theSupporting Information):
(1) Three-step geometry optimization involving: (i) mini-mization of the water molecules with the lipid non-hydrogen atoms restrained by a harmonic potential(force constant k = 420 kJ mol-1 nm-2), (ii) shortunrestrained optimization, and (iii) energy relaxationof the water molecules with softer restraints on thelipid non-hydrogen atoms (k=210 kJ mol-1 nm-2).The three optimizations were performed by theL-BFGS method30 with convergence criterion of50 kJ mol-1.
(2) Heating of the system from 0 to 300 K in a series ofsteps with the lipid non-hydrogen atoms restrained
Figure 2. Surface pressure/area (Π/A) and surface potential/area (ΔV/A) isotherms of DPPC (left) and DC (right) monolayers at the air/water interface measured on ultrapure water [courtesy of Dr. Tz. Ivanova].
Table 1. Elementary Cell Sizes, Mean Molecular Area (A), andNumber of Water Molecules of the Studied Model Systems
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Commun. 1995, 91, 43–56. (b) Lindahl, E.; Hess, B.; van der Spoel, D. J. Mol. Mod.2001, 7, 306–317. (c) van der Spoel, D.; Lindahl, E.; Hess, B.; Groenhof, G.; Mark,A. E.; Berendsen, H. J. C. J. Comput. Chem. 2005, 26, 1701–1718.
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nm-2 during stage I followed by unrestrainedstage II) and equilibration. The point of reachingequilibrium was verified by analysis of the energy,temperature, and pressure fluctuations along thetrajectories;
(4) Production simulation of 10 ns (15 ns for 25 DPPCat 40 A2/molecule and 5 ns with TIP4P for 81 DPPCat 70 A2/molecule) in NVT ensemble with time step2 fs at 300 K. The constant temperature was main-tained by the Berendsen thermostat33 implementedin GROMACS 3.3.3 with relaxation time of 0.1 ps.The short-range electrostatic interactions werecalculated up to a cutoff distance of 14 A with aswitching function activated at 12 A and the long-range electrostatics was taken into account withPME.34 The nonbonded interactions were evaluatedwith a switched cutoff of external radius 12 A andinternal radius 10 A. The stability of the generatedMD trajectories was monitored through combinedcheck of temperature and pressure fluctuations(Figures S1 and S2) and convergence of the potentialenergy to a constant average value (Figure S3). Thestructural relaxation of the system was verified bythe fluctuations of the length of the P-N vector(for DPPC) and of the dihedral angle C-C-O-Hinvolving the free OH-group of the DC head(Figures S4-S7).
GROMACS3.3.3,VMD1.8.6,35 and original scripts were usedfor construction of the EC, batch calculations, vector decomposi-tion, visualization and statistical analysis of the results. Snapshotswere extracted from the trajectory for analysis at intervals of 1 ps,the entire analyzed ensembles consisting of 10 000 structures. Allmean values shown below were averaged over the whole set andstatistical accuracy was quantified by standard errors.
Estimation of Electric and Dielectric MonolayerProperties
The electrostatic potential (φ) profile across the monolayerswas obtained by integration of the charge distribution (Fc) alongthe z coordinate:
φðzÞ-φð0Þ ¼ -1
ε0
Z z
0
dz0Z z0
0
dz00FCðz00Þ ð1Þ
where ε0 is the dielectric permittivity of vacuum.
The surface pressure of the monolayers was estimated from thevirial relation:
P ¼ 2
VEkin þ 1
2
Xi, j
rij � Fij
0@
1A ð2Þ
whereP is the pressure,V is the elementary cell volume,Ekin is thekinetic energy, rij is the interatomic distances, and Fij are theinteratomic forces.
Following the mechanical definition, the total surface tension(σ) of the system is:
σ ¼ Lz Pzz -Pxx þPyy
2
� �� �ð3Þ
where Lz is the periodic box length along z and the angle bracketsdenote averaging over all snapshots from the trajectory.
Then the surface pressure (Π) is estimated by
Π ¼ σ0 -σm ð4Þwhere σ0 denotes the experimental surface tension of water andσm is the calculated surface tension of the monolayer. In our case,σ0 is 72 mN/m36 while σm is obtained by subtracting the valuereported for the TIP3P model (52.5 mN/m37) from the surfacetension of the system (see the Appendix). In the case of 81 DPPCmolecules in the EC at 70 A2/molecule, the simulation wasconducted with the TIP4P/2005 water model and then a valueof 65.5 mN/m was used.38 The latter was estimated from neatwater slab simulation by using the same method.
The (di)electric properties of the monolayers were character-ized by dipole moments and relative dielectric permittivities.Dipole moment magnitudes are given in debye (D).
The dipole moment, one of the main quantities of interest, wascalculated in themonopole approximation. The separate contribu-tions of surfactants andwaterwere estimated and each of themwasdecomposed into normal and tangential components (with respectto the interface). The experimental estimates of the normal dipolemoment (μ^) are related to the surface potential (ΔV) through theHelmholtz equation39 (eq 5), where ε^ is the normal monolayerdielectric permittivity and A is the mean area per surfactant.
ΔV ¼ μ^ε0ε^A
ð5Þ
In the present manuscript the theoretical normal dipole mo-ment coincides with the z component of μ (μz � μ^) and the
Table 2. Neighbor Distributions for EC with 25 and 81 Surfactants
EC with 25 molecules EC with 81 molecules
lipids real neighbors per lipid images per lipid lipids real neighbors per lipid images per lipid
central molecules 9 8 0 49 8 0molecules at the EC edge 12 5 3 28 5 3molecules at the EC vertex 4 3 5 4 3 5
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10092. (b) Essmann, U.; Perera, L.; Berkowitz, M. L.; Darden, T.; Lee, H.; Pedersen,L. G. J. Chem. Phys. 1995, 103, 8577–8593. (c) Toukmaji, A.; Sagui, C.; Board, J.;Darden, T. J. Chem. Phys. 2000, 113, 10913–10927.(35) Humphrey, W.; Dalke, A.; Schulten, K. J. Mol. Graphics 1996, 14, 33–38.
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The directions of x, y, and z are shown in Figure 1.The elements of the monolayer dielectric permittivity tensor
(εRR) were evaluated by the Kirkwood-Froehlich relations:8b
ε ¼εjj 0 00 εjj 00 0 ε^
0@
1A ð6Þ
εRR ¼ 1þHRR=V ð7Þ
HRR ¼ 1
ε0kBTðÆμRR2æ-ÆμRRæ
2Þ ð8Þ
where μRR is the respective dipole moment component (μ^ or μ ))and V is the monolayer volume (Tables 1, 2 of the companionpaper). Equations 6-8 were used for quantitative assessment ofthe dielectric permittivity components in normal (^) and tangen-tial ( )) direction, decomposed into lipid and water contributions.
Results and Discussion
Electrostatic Potential.The calculated electrostatic potential(φ) profiles across the monolayers can be used to identify theorigin of the potential jump measured experimentally. Theestimates for the DPPC and DC monolayers at 50 A2/moleculeare shown for illustration in Figure 3. The profiles of theremaining systems are provided as Supporting Information(Figures S8 and S9).
It should be noted first that the size effect on the potentialprofiles (Figures 3, S8, S9) is negligible because the smaller and thelargermodels of bothDPPCandDChave identical curves both interms of potential variation as a function of z and in terms ofpotential jumps across the interface.
The DPPC potential profiles are characterized by large oppo-site contributions of lipids and water along the entire isotherm(Figure S8 (Supporting Information)), which compensate eachother to result in a small total potential jump. The lipids govern
the potential in the segment of the tails but the polarization ofwater defines φ in the hydrated monolayer part determining alsothe resultant sign of the monolayer surface potential. The overallpotential difference (ΔV), which can be compared to the mea-sured surface potential, is minimal for 25 DPPC at 80 A2/molecule, slightly larger at 70 A2/molecule and changing lessappreciably to 50 A2/molecule; then it grows at 40 A2/moleculereaching the maximum value. For 81 DPPC the steeper growthbegins already at 50 A2/molecule. This general trend is completelyin line with the experimental ΔV/A isotherm of a DPPC mono-layer40 (Figure 2).
The DC films feature different potential profiles across theinterface. Lipids and water are similarly polarized but the varia-tion of the total potential within the monolayer is definedexclusively by the lipids. Water and lipid contributions add upto give the total surface potential of the film. The potential jumpchanges gradually along the isotherm (Figure S9 (SupportingInformation)) from the smallest value at 80 A2/molecule to thelargest ones at 50 A2/molecule for both EC sizes, which is in goodcorrespondence with the experimentally measured trend19
(Figure 2).It is noteworthy that even though the change in the potential
jump with surface concentration is reproduced correctly, theabsolute values are appreciably overestimated due to lack ofexplicit polarizability in the force field used. However, the ratio ofΔV for the two monolayers at the same surface concentration isreproduced well, which renders the comparison between themreliable. (Figure S10 (Supporting Information)).
Another measure for the reliability of the chosen computa-tional methods is the comparison between the calculated and theexperimental average surface pressures of the two monolayers.The respective data are plotted in Figure 4.
The surface pressure for the DPPC monolayer with 81 mole-cules in the EC at 50 A2/molecule agrees well with the experi-mental value while at lower surface concentrations thediscrepancy increases. Nevertheless, the overall trend in the sur-face pressure is reproduced in the simulations. It has to be notedthat the value at 70 A2/molecule is obtained with TIP4P/2005water model. Initially, the system was simulated with the TIP3Pwater model but the measured surface pressure substantiallyincreased (results not shown in the graph) due to opening of largepores in the monolayer, which tended to be stable during the restof the run. An explanation is that the TIP3P surface tension israther small (52.5 mN/m37) compared to the experimental valueof 72mN/m36 and this disparity renders possible the formation of
Figure 3. Electrostatic potential profiles across the DPPC (left) and the DC (right) monolayers calculated with EC of 81 lipids at 50 A2/molecule; arrows indicate surface potential jumps; the left sides of the graphs correspond to lipid tails. Solely box z-coordinates occupied bythe monolayers are represented.
(40) Physical Chemistry of Biological Interfaces; Baszkin, A.; Norde, W., Eds.;Marcel Dekker: New York and Basel, Swizerland, 2000.
surfactant-free patches in the monolayer at considerable surfaceconcentrations. The change in the water model to TIP4P/2005,which has surface tension of 65.5 mN/m, resulted in at leastqualitative agreement with the experiment. The system at 40 A2/molecule is in the region of the monolayer collapse, so its value isnot comparable to the experimental results. The negative values ofthe surface pressure, most prominent at 60 and 70 A2/molecule,show the tendency of the model monolayer to decrease its meanarea per molecule. The periodic boundary conditions, used in thesimulations, impose artificial periodicity on the organization ofthe film so that unit cells smaller than the extent of correlationspertinent to a particular phase will reinforce ordering of themolecular structure. Thus, the 25 DPPC clusters are insufficientto accommodate the length-scale of the correlations at 60 and70 A2/molecule and hence tend to shrink to smaller areas permolecule. Consistent with this interpretation is the behavior at50 and 80 A2/molecule, where the space correlations are shorter(Figure 5 of the companion paper) and the 25 DPPC clustersdisplay positive surface pressures. Hence, the results obtainedfrom the smaller systems will be regarded only for comparison.
Although the values are slightly shifted to higher surfacepressures, the results for DC monolayers are in good agreementwith the experimental data. This can be attributed to the lack ofhighly charged moieties in the DC molecule as well as the goodoptimization of interactions between the lipid tails in theCHARMM22/27 force field. Also, formation of persistent waterpores was not observed throughout the simulations of DCmonolayers. Another interesting feature is that the results ob-tained with the smaller EC practically coincide with thoseobtained with 81 lipid molecules in the EC, which is consistentwith the shorter range of interactions in these monolayers.DipoleMoments. AverageValues.The calculated average
values of the total dipole moments and of their normal and
tangential components (scaled per one surfactant molecule), aswell as water and surfactant contributions therein, for all clustersstudied are collected in Table 3.
As seen from the data inTable 3, all standard errors are one to 2orders of magnitude smaller than the respective average values,which renders the comparative analysis of the data reliable.
The curves of the total dipolemoment ofDPPC calculatedwithEC of 25 and 81 lipids have similar profiles (Figures 5 and S11(Supporting Information), left);a minimum is observed at60 A2/molecule and 50 A2/molecule for 25 DPPC and 81 DPPC,respectively. The higher degrees of compression are characterizedwith steeper increase of μtotal. As could be expected, the variationof the dipole moment upon compression is governed by the lipidmolecules, while μtotal of water remains virtually constant anddecreases insignificantly at the highest surface concentration. Thetotal dipole moment magnitude at low compression (60-80 A2/molecule for 25 DPPC and 70 A2/molecule for 81 DPPC)originates primarily from water and the lipids begin to dominateonly at the tightest packing.
The total dipole moment of DC monolayers decreasessmoothly upon compression (Figures 5 and S11 (SupportingInformation), right). As we have shown previously,22 it is duemainly to the structuring of the water molecules around thesurfactant heads at all surface concentrations studied. Thelipid dipole moment varies immaterially along the entireisotherm.
The calculated normal dipole moments of DPPC show thatwater and lipids have opposite shares, the former being theleading one (Figures 6 and S12 (Supporting Information), left).These results are fully in support of our previous findings basedon EC of 9 lipids.22 μ^ almost does not depend on the lipid clustersize and varies in a very narrow range upon compression. It isnoteworthy that shallow minima are observed for the DPPC
Figure 4. Calculated surface pressure values of the studied DPPC (left) and DC (right) monolayers compared to experimental values(Figure 2).
Figure 5. Total dipole moment of DPPC (left) and DC (right) monolayers and its surfactant and water shares.
normal components, similar to experimental estimates based onMDC measurements.13,15
Lipid and water shares of the normal dipole moment of DChave identical signs and the lipid magnitude of μ^ is larger(Figures 6 and S12 (Supporting Information), right). All valuesare smaller than those of DPPC, which is in agreement with thedifferent electrostatics of DC and DPPC heads. The normaldipole moment of DC films grows slightly upon compression. Itis also not sensitive to the EC size, which shows that the usedmodels are large enough for accurate description of the normaldipole moment.
The tangential dipole moment is especially sensitive to struc-tural changes of the monolayer. This stems from the fact that itsmagnitude depends on the orientation of the lipidmolecules in themonolayer plane (whereas the direction of the normal moment isfixed) at the various degrees of compression and thus providesinformation on the extent of lipid ordering.
At low surface concentrations of the DPPC monolayers(Figure 7, left) the water share to μ ) prevails decreasing almostlinearly upon compression, i.e. with decrease of the total numberof water molecules in the film. The lipid value remains minimal at
60 A2/molecule for 25DPPC and at 50 A2/molecule for 81DPPCand grows at the higher degrees of compression, as witnessed forthe total dipole moment. After the minimum, the magnitude ofμ ) is determined mainly by the lipid whereas water contributesless. The variation of this curve upon compression can beexplained by a LE/LC transition and subsequent enhancementof lipid ordering, which is confirmed by the correspondingstructural characteristics of the film.24 Even though the trendlinesof the curves for the clusters with 25 and 81 DPPC are alike, thevalues of μ ) in the range 50 to 70 A2/molecule are very dissimilar.μ ) decreases with increase of EC, the only invariant valueremaining the one at the tightest packing.
No extrema are witnessed in the curve of μ ) describing the DCmonolayers (Figure 7, right). The tangential dipole momentoriginates mainly from the water molecules and decreases almostmonotonously upon compression. The lipid share remains prac-tically constant with the exception of 25 DC at 50 A2/molecule.The size effect is similar to that of DPPC but here also the mostcompressed film is affected.
Surprisingly, notwithstanding the substantially dissimilarstructure and electrostatics of the two lipids, their normal dipole
Table 3. Average Values (in D) with Standard Errors of the Total Dipole Moment, Its Normal and Tangential Components and the Surfactant and
Water Contributions to These (Scaled per One Surfactant Molecule) of DPPC and DCMonolayers Modeled with Different Numbers of Lipids in
moment is fairly alike both in qualitative behavior and asquantitative estimate (Figures 8 and S13 (Supporting Infor-mation)). The tangential component of both lipids determinesthe magnitude of the total monolayer dipole moment. Althoughmost publications focus on discussion of the normal componentdue to its direct relation to experimental measurements, the
tangential dipolemoment turns out to bemuchmore lipid-specificand deserves detailed study.
The components of the dipole moment scaled per unit mole-cular area have the meaning of polarization density in therespective direction. The density of the normal component ispresented in Figures 9 and S14 (Supporting Information).
Figure 7. Tangential dipole moment of DPPC (left) and DC (right) monolayers and its lipid and water shares.
Figure 8. Total dipole moment of DPPC (left) and DC (right) monolayers decomposed into normal and tangential components.
Figure 9. Normal polarization density scaled per one lipid molecule of DPPC (left) and DC (right) monolayers with EC of 81 lipids.
For both DPPC cluster sizes smooth increase of the normalpolarization density upon compression is registered. The shares ofboth lipids and water are considerable the latter one prevailing.For the EC with 25 lipids the water share becomes more positiveand that of lipids;more negative with increase of the surfaceconcentration. A possible explanation is the enhanced mutualpolarization of the two subsystems upon compression. Bothshares feature local extrema at 60 A2/molecule marking the zoneof phase transition. The films with EC of 81 DPPC preserves thesame trendupon compressionbut the polarization density reachessaturation at the tighter packing (40-50 A2/molecule). This is anadditional confirmation of the statement that DPPC normaldipole moment is described adequately by the larger models.
The total normal polarization density of DC and its sharesgrowalmost linearly upon compression.No extrema are detected,which means that the lipid molecules come closer without sub-stantial reordering with the increase of surface concentration.There is practically no EC size effect, because the values vary inthe same range and to the same extent at every compression step.
The respective tangential components are shown in Figure 10.Only the water tangential polarization density of DPPC
monolayers increases linearly upon compression. The lipid share,which determines the variation of μ )/A, has a jump at 60 A2/molecule for the smaller and at 50 A2/molecule for the largerclusters. Unlike the normal polarization density, there is nosaturation of μ )/A at the smaller areas, which indicates that thetangential dipole moment responds in a more pronounced way tochanges of the surfactant surface concentration.
ForDC themonolayer polarization is governedmainly bywaterand grows linearly upon compression. The tangential polarizationdensity of the lipids is close to zero and increases in a very narrowrange with the surface concentration. There is strong EC size effectexpressed in halving the values of the larger systems.
Angle andMagnitude Distribution of the Tangential DipoleMoment. The hypothesis that the extrema in the curves of the
total and/or the tangential dipole moments reflect surface con-centrations where there are well-defined states (maxima) or wherestructural rearrangements are taking place (minima)23 can besubstantiated by an additional analysis of the tangential dipolemoments.Amore detailed interpretation of themolecular organi-zation at the interface can be made by statistical analysis of theorientation of the vector of lipid μ ) with respect to the x-axis(quantified by the angle φ). The choice of this angle is dictated bythe fact that μ ) can change its orientation in the monolayer planeupon compression and thus gives insight into the extent and typeof lipid ordering. The distribution ofμ ) orientation is based on thevalues of the angle φ closed by the tangential dipole moment withan arbitrary (in our case x) axis in the monolayer plane.
The angular distribution of the tangential dipole moments(Figure 11, top) of the two monolayers varies in a dissimilarway upon compression. For DPPC only the model with 25 lipidsat the lowest surface concentration shows essentially uniformpopulation of all angles. At the higher levels of compression themonolayers modeled by 25 and 81 lipids behave alike. Namely, at70 A2/molecule still all directions of μ|| are present in the film butmore definite peaks start to form almost equidistantly along theentire range. Thismay be assigned to the existence of a small shareof ordered lipid domains surrounded by predominantly disor-dered lipids. At 60 A2/moleculemultiple peaks varying irregularlyin population and breadth andmost of them overlapping indicatethat the film is in the process of achieving long-range organiza-tion. At the next stage of compression already several well-shapedpeaks are registered. The population of the remaining angles issubstantially reduced but is still nonzero, especially for the largermodels. Such angular distribution illustrates the presence of amain well-ordered lipid phase coexisting with a smaller portion ofdisordered molecules, which is in conjunction with the bimodalarea distribution at 50 A2/molecule obtained from Voronoianalysis and with the multipeak phosphorus and nitrogen densitydistributions.24 Upon further compression the lipid structuring is
Figure 10. Tangential polarization density scaled per one lipid molecule of DPPC (left) and DC (right) films with EC of 25 (top) and81 (bottom) surfactants.
enhanced, which is illustrated by the very sharp peaks of φ in themost condensed DPPC films. The difference between the smallerand the larger models is that in the former one the peaks remainwell separated from each other while in the latter the tangentialdipole moments can adopt all possible orientations within therestricted range specified above. Anyway, the solid-type lipidarrangement at 40 A2/molecule24 seems to correspond to apreferred direction of the tangential dipolemoments of theDPPC
molecules. Since the film is not monocrystalline, the dipoles in theseparate domains are not strictly collinear, which causes thedispersion in the values of φ even at the tightest packing. Anotherreason for the relatively broad distribution of φ at 40 A2/moleculeis the presence of irregularities in the perfect hexagonal packingboth with respect to number and equidistance of nearest neigh-bors evidenced by the results from the Voronoi analysis and fromthe radial distribution functions.24 Overall, upon compression
Figure 11. Distribution of the tangential dipole moment (scaled per one lipid molecule) by direction (top panel) and magnitude (bottompanel) of clusters with 25 (top of each panel) and 81 (bottom of each panel) surfactants modeling DPPC (left) and DC (right) monolayers.
both DPPC models feature convergence of the peaks accompa-nied by increase of the population indicating enhancement of themolecular organization.
The angular distributions of μ ) of the DC films (Figure 11, top)have rather different behavior than their DPPC analogues. Onlythe points of 25DCand81DCat the lowest surface concentrationhave similar profiles to the 25 DPPC point at 80 A2/molecule;uniform population of all angles without any visible extrema. Themodels at 70 and 60 A2/molecule are characterized by single verybroad maxima, which are even broader for the larger model,especially the one at 60 A2/molecule. The DC peaks at 50 A2/molecule extend over much more angles than the correspondingDPPC maxima at 40 A2/molecule. This leads to the most sub-stantial dissimilarity between the twomonolayers;there is no zeroprobability for any angle φ in the DC films, irrespective of thedegree of compression. It means that the DC dipoles retainappreciable freedom for reorientation even at the highest externalstress. This is in line with the significant share of relativelydisordered lipids in these films, as yielded by theVoronoi analysis.24
With respect to distribution of the tangential dipole momentmagnitudes (Figure 11, bottom) the two monolayers differ, too.The variation is less pronounced in the DCmodels. The values ofμ ) for 25 DPPC at 60-80 A2/molecule are described by muchdistorted Gaussian peaks spanning a range of about 6 D while at50 A2/molecule μ ) is centered around much higher values;ca.9 D. The point with the highest surface concentration is char-acterized with a very narrow sharp maximum at ca. 14 D withoutany visible tails. The latter is in line with the solid-type lipidorganization at this degree of compression and in good agreementwith quantummechanical estimates.23When the elementary cell isextended to 81 lipids the peaks become narrower in general andwithout definite tails and the systems are grouped in a differentway. This illustrates the enhanced ordering in the larger modelsdue to increased number of real neighbors.
Unlike DPPC, the DC films are characterized by Boltzmann-type distributions of the tangential dipole moment magnitudesshowing that solid-state ordering is not achieved at any degree ofcompression. The size effect here is expressed in the fact that forthe smaller systems the maxima shift to higher values and thecurves become broader upon compression, while for the largermodels the maxima remain invariant. Overall, whatever reorga-nization takes place in theDCmonolayers upon compression, it isnot reflected in the magnitude distribution of μ ).Dielectric Permittivity. The above estimates of the dipole
moment show that the properties of the studied insoluble mono-layers are characterized by pronounced anisotropy. Anotherrelated parameter is the dielectric permittivity, which is essentialfor description of 2D-nucleation. Its normal and tangentialcomponents were calculated by eq 7. It has to be noted that thetangential dielectric permittivity can not be measured directlyexperimentally. The volume of each monolayer is obtained fromits density profile.24 The calculated values of ε of the monolayersshow that it varies significantly depending on the degree ofcompression and on the surfactant type.
Table 4 contains the normal component of the dielectric per-mittivity tensor for DPPC and DC monolayers and the respectivestandard errors. Because of the small values of ε^, the differencesbetween the average estimates at different surface concentrationsfall within the standard errors. Therefore, we will refrain fromfurther comments on the variation of ε^ upon compression. Theonly noteworthy comment is that the normal dielectric permittivityof DPPC varies in a range twice as large as that of DC.
The tangential dielectric permittivity of the monolayerswas estimated by eq 7, too (Figures 12 and S15 (Supporting
Information)). Unlike the small average values of ε^, the tangen-tial component hasmagnitudes close to that ofwater and varies inwide ranges depending on the lipid and on the surface concentra-tion. The interpretation of the extrema in the profiles of ε ) isexactly the opposite to that of the dipole moments: maximacorrespond to disorder and minima;to organized structures.
Quantitatively, ε ) of the DPPC layers is influenced by the lipid,modulating the water contribution, while in the DC films thelipids affect it negligibly.
The total DPPC tangential dielectric permittivity has a maxi-mum at 70 A2/molecule for 25 DPPC and at 60 A2/molecule for81 DPPC, followed by decrease of ε ) upon further compression.This supports the hypothesis of phase transition at these areas andsubsequent enhanced monolayer organization. The reorganiza-tion of the lipids begins earlier for the smaller models (at 70 A2/molecule) followed by that of water at 60 A2/molecule.Within themodels with larger EC the rearrangement of the two monolayersubsystems takes place at the same surface concentration (60 A2/molecule). The area per molecule at which the reordering occurscan be related to the freedomofmovement limitations imposed bythe periodic box lateral dimensions.
At the largest DPPC surface concentrations the dielectricpermittivity of the aqueous layer is smaller than that of bulkwater (82 for the TIP3P water model41); it increases upon filmdecompression and reaches saturation in the LE region, itsmagnitude there being close to that of bulk water. In general,the larger models are characterized by smaller values of ε ), whichmay be rationalized in terms of decreased polarization effect ofthe larger number of real lipid neighbors. In addition, the valuefor 81 DPPC at 70 A2/molecule is too low, which is partly due tothe different water model.
The dielectric permittivity of DPPC bilayers has been reportedby Stern and Feller.8b As their model contains 36 lipids withorthorhombic initial alignment in a leaflet and the simulation is at62.9 A2/molecule and 323 K in NPnAT ensemble, the closest
Table 4. Normal Dielectric Permittivity of DPPC and DC
analogue is our 25DPPC system at 60 A2/molecule. The differ-ences are too many to arrive at identical values but trendlinescould be compared. Numerically, the results of Stern and Fellerfor the normal dielectric permittivity are of the same order ofmagnitude (4.6 ( 0.4 for DPPC, 5.5 ( 0.5 for water, and 3 ( 1total), the total value of ε^ being times smaller than the sumof thecomponents, in line with the opposite correlation of normaldipole moment contributions of DPPC and water. The valuesfor the tangential components are significantly different: on theaverage the share ofwater in this study is twice as large and that ofDPPC twice as small, the total tangential permittivity being lowerthan the one reported by Stern andFeller, probably indicating theimpact of the second leaflet on the dielectric properties ofinsoluble lipid monolayers or due to the differences in thesimulation conditions.
The situation with the DC films is different: the surfactant andwater shares of ε ) are relatively constant along the entire isotherm.Water has the dominating role both qualitatively and quantita-tively. Irrespective of that, the total tangential dielectric permit-tivity is always lower than that of bulk water. This is mostprobably due to the depolarizing effect of the surfactant heads.Analysis of the curve for 81 DC also shows a maximum at 70 A2/molecule and increasing organization upon further compression.This reorganization, however, involves only the water molecules,unlike in the DPPC monolayers.
In spite of the very dissimilar polarity of the two lipids, theoverall low level of organization within the monolayers24 leads tohigh values of ε ) (about 45 for the two lipids) even at the tightestcompression.
In general, the results from the simulations reflect properly thetendency toward decrease of the total tangential dielectric per-mittivity with enhanced degree of ordering in the monolayers.More accurate numerical estimates especially for the normalpermittivity require longer simulations.
Conclusions
Molecular dynamics simulations on model DPPC and dicaprin(DC) insoluble monolayers at the air/water interface were carriedout. The size effect on the estimated properties was tested byemploying elementary cells of 25 and 81 lipids and imposing 2Dperiodic boundary conditions. The calculations were performed forseveral surface concentrations located in the LE and LC regions ofthe surface pressure/area isotherm. A refinement of the model wasthe introduction of a specially adapted potential mimicking thepresence of bulk water and thus mitigating the influence of water/vacuum interface below the film. The (di)electric properties of themonolayers, dipolemoment, dielectric permittivity, and electrostaticpotential as well as surface pressure, were estimated by analysis of
the MD trajectories. The data were discussed both for the entiresystems and in terms of normal and tangential to the interfacecomponents and of surfactant and water contributions.
Marked EC size effect was observed only for the tangentialcomponents. The parameters in normal direction were practicallynot affected by the increase of the elementary cell dimensions.This signifies that the current models are sufficient to describeaccurately the normal characteristics of the monolayers, whiletangential properties are given correctly only qualitatively. Forquantitative values comparable to experiment the models have tobe extended further.
The two studied monolayers have very different behavior withrespect to their (di)electric features. It was shown that the normalpolarizationof theDPPCmonolayer results fromabalance betweenthe lipid and the water contribution while that of DC is governedsolely by the surfactant. The DPPC tangential dipole moment isdictatedmainly by the lipid while the DC one stems from organiza-tion of the water molecules around the surfactant heads. Theextrema in the dipole moment and dielectric permittivity curveswere interpreted in terms of monolayer rearrangement or orderedstates. It was shown that the tangential dielectric permittivitydominates over the normal one and that it is much more sensitiveto the structural rearrangements and to the model size within thelipid monolayers. All these conclusions support our previoushypotheses and findings based on smaller models of the two filmsand recent structural analyses of the current models. The size-effectis expressed in the fact that the concentration indicating essentialmonolayer reorganization tends to be higher in larger systems.
Distribution of the lipid tangential dipolemoments by in-planeorientation and by magnitude reveal distinct structuring of theDPPC monolayers and more amorphous arrangement of thelipids in the DC films. It can be generalized that the EC size hasmostly quantitative effect.
Inspection of the electrostatic potential and surface pressureprofiles shows qualitative correspondence with the experimentalcurves although the former values are somewhat overestimated forboth lipids, whereas the latter are underestimated for DPPC andoverestimated for DC. The nice qualitative agreement betweentheory and experiment of the trends of φ andΠ upon compressionrenders the used computational procedure reliable for the descrip-tion of the properties of various insoluble lipid monolayers at themolecular level. The results definitely show that whereas localorganization can be discussed based on comparatively smallmodels, thermodynamic parameters require a sufficiently largeunit cell as exemplified by the surface pressure results in this study.
An improvement of the computational scheme aimed atsimulation of systems with lower surface concentration (fromtheLE regionof the isotherms)will demand, however, a changeof
Figure 12. Variation of the tangential component of the relative tangential dielectric permittivity and its surfactant and water shares forDPPC (left) and DC (right) monolayers.
the water model from TIP3P to a more sophisticated one, e.g.,TIP4P/2005 or the polarizable POL3, which reproduce moresuccessfully the thermodynamic properties of surfacial water.
In our opinion themost essential result is the importance of thetangential characteristics of the lipid films for their overallbehavior, usually overlooked by theorists and experimentalistsbut surely deserving a more detailed study.
Acknowledgment. The research is funded by Projects DO-2-256/2008 and DO-2-82/2008 of the National Science Fund ofBulgaria. The Alexander von Humboldt Foundation is acknowl-edged for an Equipment Grant.
Appendix A
Lipid monolayers can be modeled by two geometries of thesimulation cell, which ensure the periodicity in the lateral x and ydirections and guarantee for the inhomogeneity of the system inthe normal z-direction. The symmetric geometry consists of amiddle layer of water molecules covered by two layers ofsurfactants in opposite orientation. The water slab is usually ofconsiderable thickness so as to prevent interaction between themonolayers at the opposing sides as well as the number of lipidmolecules is doubled than actually needed formodeling of the realsystem. Although this type of geometry is suitable for directcalculation of the surface tension, the nearly 2-fold increase in thenumber of particles renders it computationally very expensive.Onthe other hand, the asymmetric geometry of the periodic cell,which contains only one surfactant layer spread out over a waterfilm, allows the simulation of monolayers with the minimumamount of particles and hence substantial decrease in the compu-tational burden. However, it requires the application of wallpotential on the water/vacuum interface, which prevents theevaporation of water molecules and reduces the polarizationeffects arising from the latter phase boundary. In the currentwork, a wall potential that fulfilled both requirements wasdeveloped beforehand, and it was used in the simulations ofsurfactant films throughout.
The guiding idea in the parametrization of the wall potentialwas tomimic the presence of implicit bulk water whileminimizingthe disturbance of the mass density of the explicit water slab. Theevaporation of water molecules can be neatly eliminated byapplication of a simple repulsive potential and the polarizationof surface water can be mitigated by imposing an orientationalrestraint on the water molecules. The functional form of the twocontributions to the wall potential was derived from properties ofan explicit water film of about 4000 molecules simulated insymmetric geometry; i.e., the z-dimension of the elementary cellwas long enough to prevent interaction between the periodicimages. For consistency, the water model was CHARMM-modified TIP3P and the simulation conditions matched thoseof the monolayer simulations, i.e. the temperature was main-tained at 300 K in NVT ensemble. The position of a watermolecule was evaluated from the coordinates of the oxygen atomand its orientation to the interfacewas determined from the cosineof the angle between the unit surface normal and the unit vectoralong the bisector of the HOH valence angle.
The repulsive portion of the wall potential was chosen to be of22-12 Lennard-Jones type and it was fitted to the TIP3Poxygen-oxygen radial distribution function. The resulting formof the potential is:
ULJðzÞ ¼ 3:35� 10-11
ðz-z0Þ22-1:19� 10-5
ðz-z0Þ12
where ULJ is in kJ mol-1, z (nm) is the z-coordinate of the oxygenatom, and z0 (nm) is the position of the wall. Furthermore, theposition of the wall was varied slightly in the different simulatedsystems to ensure minimal influence of the potential on the waterdensity. In all cases, the repulsion was truncated and shifted tozero at 0.85 nm away from the wall in the water layer.
The orientational portion of the wall potential was assumed tobe of squared cosine form, as suggested by the orientationaldistribution of water molecules at the water/vacuum interface.Additionally, the polarization varied with the distance to thephase boundary and the strength of the interaction was weightedwith the z-position. The resulting form of the potential as well asthe values of the parameters after self-consistent optimization in aseries of restrained simulations is:
UðθÞ ¼ U0 cos2 θ
whereU is in kJmol-1,θ is the angle between the unit vector alongthe bisector of theHOHvalence angle and the unit surface normalvector, and the strength of the restrainedU0 isweighted accordingto the formula:
U0 ¼ -2:245ðz-zcutÞþ 3:118ðz-zcutÞ2
Here z (nm) is the z-coordinate of the oxygen atomand zcut (nm) isthe cutoff of the orientational potential where it is shifted to zero.The cutoff distance was set at 1.58 nm away from the position ofthe wall. In addition, the parabolic weighting was switched toconstant between the wall and 0.98 nm away from it in thedirection of the water film. It is well-known42 that such simplepotentials are not able to completely smooth out the orientationalpolarization of the water molecules at the interface and theproposed wall potential only succeeded to diminish these effects.
Another consequence of the asymmetric geometry of the simula-tion cell and the need for correcting potential is that the surfacetension cannot be estimatedwith high accuracy. The surface tensionof the system as a whole is the sum of the surface tensions of bothinterfaces, one of which, the water/vacuum boundary, is perturbedby the wall potential. Since the wall position (as defined above)slightly varied from system to system, the disturbance caused by thewall potential also changed. Nevertheless, the evaluation of thesurface tension of doubly restrainedwater slab showed that thewallpotential influenced inconsequentially the value for the correspond-ingmodel. Then, it was reasonable to use the surface tension of neatwater as obtained from unrestrained simulation for the calculationof the surface pressure of the lipid monolayers, as discussed above.
Supporting Information Available: Details of the separatecomputational stages (Table S1), temperature (Figure S1),pressure (Figure S2), and potential energy (Figure S3) fluctua-tions along the production part of the MD trajectories of thestudied systems at 60 A2/molecule, variation of the P-N vectorlength of DPPC monolayers (Figures S4 and S5) and of thedihedral angle C-C-O-H of DC films (Figures S6 and S7)along theproductionpartof the trajectory, electrostaticpotentialprofiles of the simulatedDPPC (Figure S8) andDC (Figure S9)monolayers, calculated and experimental values of the surfacepotential of all systems (Figure S10), average total and normaldipole moments (Figures S11, S12) and normal polarizationdensity (Figures S14), decompositionof the total dipolemomentinto normal and tangential contributions (Figure S13), andtangential dielectric permittivity (Figure S15) for the modelswith 25 lipids in the EC. This material is available free of chargevia the Internet at http://pubs.acs.org.
(42) Beglov, D.; Roux, B. J. Chem. Phys. 1994, 100, 9050–9063.
