pubs.acs.org/Langmuir © 2010 American Chemical Society
Molecular Dynamics Study of the Electric and Dielectric Properties of Model 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 of Chemistry, 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
Atomistic modeling of insoluble monolayers 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 out for 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 were considered. 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 of the films in normal and tangential direction were quantified by means of dipole moment magnitude and orientation analysis and by monolayer dielectric permittivity. The contributions of lipids and interfacial water to each component of the considered characteristics were assessed and their variations upon film compression were discussed and compared for the two monolayers and to earlier results. The dielectric permittivity tensors were analyzed. Electrostatic potential profiles across the layers and surface pressure values were used for more detailed clarification of experimental measurements. The results show dissimilar behavior of the two lipids at the air-water interface. While the average electric and dielectric properties of DPPC monolayers result from opposite surfactant and water contributions, the two subsystems are synergetic in the DC films. The anisotropy of the monolayer dipole moment and dielectric permittivity is explained by domination of a different subsystem in the various components. Tangential characteristics turn out to be more sensitive to the size of the model and to the degree of film compression.
Introduction Electric and dielectric properties of insoluble monolayers attract the interest of both experimentalists and theoreticians because they can serve as indirect markers of surfactant organization at the air/water interface.1-3 Moreover, some dielectric characteristics are used as parameters in the description of domain formation within Langmuir lipid monolayers.4 *Corresponding author. E-mail:
[email protected]. Telephone: þ35928161374. Fax: þ35929625438.
(1) Gaines, G. L., Jr. Insoluble Monolayers at Liquid-Gas Interfaces; WileyInterscience: New York, 1969. (2) (a) Moehwald, H. Annu. Rev. Phys. Chem. 1990, 41, 441. (b) Moehwald, H. In Handbook of Biological Physics; Lipowsky, R., Sackmann, E., Eds.; Elsevier: Amsterdam, 1995; Vol. 1, Chapter 4. (3) Balashev, K.; Panchev, N.; Petkov, I.; Panaiotov., I. Colloid Polym. Sci. 2000, 278, 301–311. (4) (a) Ivanov, T.; Radoev, B. Colloids Surf. A 2004, 224, 19–23. (b) Radoev, B.; Boev, T.; Avramov, M. Adv. Colloid Interface Sci. 2005, 114, 93–101. (c) Slavchov, R.; Ivanov, T.; Radoev, B. J. Phys.;Cond. Matt. 2006, 18, 5873–5879. (5) (a) Hoenig, D.; Moebius, D. J. Phys. Chem. 1991, 95, 4590–4592. (b) Lee, L.; Mann, E.; Langevin, D.; Farnoux, B. Langmuir 1991, 7, 3076–3080. (6) (a) Ivanova, Tz.; Grozev, N.; Panaiotov, I.; Proust, J. Colloid Polym. Sci. 1999, 277, 709–718. (b) Brasseur, R. In Molecular Description of Biological Membrane Components by Computer-Aided Conformational Analysis; Brasseur, R., Ed.; CRC Press: Boca Raton, FL, 1990; Vol. 1, p 203. (7) (a) de Vries, A. H.; Mark, A. E.; Marrink, S. J. J. Am. Chem. Soc. 2004, 126, 4488–4489. (b) Feller, S. E.; Venable, R. M.; Pastor, R. W. Langmuir 1997, 13, 6555– 6561. (c) Mauk, A. W.; Chaikof, E. L.; Ludovice, P. J. Langmuir 1998, 14, 5255–5266. (d) Perera, L.; Essmann, U.; Berkowitz, M. L. Langmuir 1996, 12, 2625–2629. (e) Snyder, R. G.; Tu, K.; Klein, M. L.; Mendelssohn, R.; Strauss, H. L.; Sun, W. J. Phys. Chem. B 2002, 106, 6273–6288. (8) (a) Shinoda, W.; Fukada, T.; Okazaki, S.; Okada, I. Chem. Phys. Lett. 1995, 232, 308–312. (b) Stern, H. A.; Feller, S. E. J. Chem. Phys. 2003, 118, 3401–3412. (c) Tieleman, D. P.; Berendsen, H. J. C. J. Chem. Phys. 1996, 105, 4871–4880. (d) Nandi, N.; Vollhardt, D. J. Phys. Chem. B 2002, 106, 10144–10149.
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Probably the most frequently studied films of nonionogenic lipids are those of dipalmitoylphosphatidylcholine (DPPC). However, the main focus of the experimental2a,5 and theoretical6-11 investigations of DPPC monolayers are thermodynamic and morphological characteristics of the lipids at the interface. Moreover, almost all molecular simulations are limited within a narrow range of surface concentrations (close to that present in living cells).6-11 Less attention is paid to the (di)electric features of these monolayers, for instance to measurements of dipole moments or dielectric permittivities. The most closely related recent theoretical publication, addressing electric properties of DPPC,12 has reported QM/MM calculations on the surfactant head in vacuum and in the presence of several water molecules. The importance of the solvent was outlined, which influences substantially the dipole moment by change of conformation due to hydrogen bonding or by intermolecular charge transfer. Probably the most frequently experimentally measured characteristic of DPPC monolayers is the surface pressure. It has been used for structural and rheological studies almost from the beginning of Langmuir monolayer investigations. Lately, its evolution upon monolayer compression has been used to extract the elasticity of monolayers (from the first derivatives with respect to area)13,14 (9) Kaznessis, Y.; Kim, S.; Larson, R. Biophys. J. 2002, 82, 1731–1742. (10) Knecht, V.; Muller, M.; Bonn, M.; Marrink, S.-J.; Mark, A. J. Chem. Phys. 2005, 122, 024704. (11) Wohlert, J.; Edholm, O. Biophys. J. 2004, 87, 2433–2445. (12) Yin, J.; Zhao, Y.-P. J. Colloid Interface Sci. 2009, 329, 410–415. (13) Lucero, A.; Rodrı´ guez Nino, M. R.; Gunning, A. P.; Morris, V. J.; Wilde, P. J.; Rodrı´ guez Patino, J. M. J. Phys. Chem. B 2008, 112, 7651–7661. (14) Beno, J.; Weis, M.; Dobrocka, E.; Hasko, D. Appl. Surf. Sci. 2008, 254, 6370–6375.
Published on Web 03/25/2010
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Figure 1. Top view of monolayer elementary cells containing 25 molecules DPPC at 50 A˚2/molecule (left) and 25 molecules DC at 60 A˚2/ molecule (right). Unit translation vectors and orientation of the Cartesian coordinate system are shown, too.
or the free energy of mixing with other surfactants (through integration).14 Dipole-dipole interactions have been suggested as the driving force for the mixing.14 There have also been attempts to estimate the normal dipole moment of DPPC monolayers from experimental data. Two methods for such evaluation dominate in the literature. The first one is based on the classical Helmholtz equation,1 while the more modern approach relies on Maxwell displacement current measurements.14,15 Using the latter, the profile of the normal dipole moment along the entire surface/pressure area isotherm has been determined.14,16 In ref 14, the authors have suggested dipole moments compensation as a possible reason for favorable mixing of DPPC with a second surfactant. Then, this contribution has been ruled out as the dominating factor due to its negligible magnitude. On the other hand, the respective part of the tangential dipoles has not been discussed. It is noteworthy that the dipole moment curves in refs 14 and 16 are characterized by a minimum in the region of LE/LC phase coexistence. However, its origin has not been commented by the authors. Another indirect relation of the dipole moment to the thickness of DPPC monolayers has been the hypothesis13 that the pH dependence of the latter is defined by the orientation of the P-N vector of the choline group. From mechanistic point of view, control of the monolayer dipole moment has been deemed as a factor facilitating LangmuirBlodgett film deposition.14 Dicaprin (DC) is another lipid forming stable insoluble monolayers at the air/water interface.17 Unlike DPPC, the surface characteristics of pure DC monolayers have been rarely studied. Most often, DC is employed as substrate for testing the hydrolytic activity of various enzymes.18 The rheological properties of dicaprin monolayers have been described solely by Ivanova et al.19a and by Nannelli et al.19b They have recorded the surface pressure/area and surface potential/area isotherms of DC films. Thereof, it has been seen that DC does not show any specific features typical for lipid reorganization upon compression. This is in contrast to the behavior of DPPC, the isotherms of which are characterized by a well expressed plateau;a sign of coexistence of several phases. The dissimilar conduct of the two lipids prompted us to use them as targets of the present study, which is an attempt for detailed theoretical interpretation of the electric and dielectric (15) Iwamoto, M.; Majima, Y. J. Chem. Phys. 1991, 94, 5135. (16) Ou-yang, W.; Yamamoto, T.; Aida, T.; Manaka, T.; Iwamoto, M. Thin Solid Films 2008, 516, 2649–2651. (17) (a) Nannelli, F.; Puggelli, M.; Gabrielli, G. Mater. Sci. Eng., C 1999, 445– 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. Colloids Surf. B 2002, 24, 1–9.
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properties of the respective monolayers formed at the gas/water interface at various surface concentrations. An additional point of interest toward the (di)electric features of the monolayers stems from the fact that theoretical simulations allow correct estimation of the anisotropy of interfacial quantities and evaluation of their dependence on the degree of compression of the film. As evident from the above publications summary, theoretical calculations can also provide numerical values of some parameters, which are often needed by phenomenological models.4,20,21 For example, the monolayer dielectric constant and the profile of the electrostatic potential along the film have been included in models aimed at explaining ion binding to DPPC.20 Transition dipole moments of CH2 vibrations have also been of interest for description of monolayer phase states.21 Previously, we have reported22,23 some theoretical results for the dipole moment and the dielectric permittivity of the same two monolayers. The data were extracted from relatively small models comprising two to nine lipids in the elementary cell but permitted outlining the main tendencies of dipole moment and dielectric permittivity variation in directions normal and tangential to the interface upon film compression. Thus, another goal of the present study was to test the sensitivity of our results toward extension of the size of the models (see also ref 24). In addition, other thermodynamic parameters of the monolayers, such as surface potential and surface pressure, were assessed and discussed. The theoretical estimates were obtained by molecular dynamics simulations of DPPC and DC monolayers and are presented in terms of dipole moments and dielectric permittivities. Both quantities were decomposed into normal and tangential components. The shares of surfactants and water were determined, too.
Models and Computational Protocol This section outlines the construction of models and the computational scheme used. Models with 25 and 81 lipids were built in 3D periodic boundary conditions, as specified in our previous studies,22,23 in order to test the effect of the elementary cell (EC) size on the computational results. Hexagonal lattice and explicit water molecules were used throughout (Figure 1). The periodic box dimensions along the x and y axes were selected such that the corresponding areas per lipid molecule of the monolayer fell into the liquid condensed (LC) or the solid (20) (a) Leontidis, E.; Aroti, A.; Belloni, L. J. Phys. Chem. B 2009, 113, 1447– 1459. (b) Leontidis, E.; Aroti, A. J. Phys. Chem. B 2009, 113, 1460–1467. (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. Langmuir 2008, 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. Langmuir 2010, submitted as companion paper (DOI: 10.1021/la904734b).
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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), and Number of Water Molecules of the Studied Model Systems surfactants in EC
A, A˚2/ molecule
periodic box sizes, A˚
water molecules
25 DPPC
40 50 60 70 80 40 50 60 70 50 60 70 80 50 60 70 80
34.0 34.0 200 38.0 38.0 200 41.6 41.6 200 45.0 45.0 200 48.1 48.1 200 68.4 68.4 200 74.9 74.9 200 80.9 80.9 200 86.5 86.5 200 50.0 35.4 120 54.8 38.7 120 59.2 41.8 120 63.2 44.7 120 90.0 63.6 120 98.6 69.7 120 106.5 75.3 120 113.8 80.5 120
1048 1379 1718 2083 2463 3412 4922 5575 6853 829 1040 1235 1447 2700 3347 4068 4749
81 DPPC
25 DC
81 DC
condensed (SC) regions of the Π/A isotherms (Figure 2), namely, 40, 50, 60, 70, and 80 A˚2/molecule. Simulations of ECs built of 81 DPPC molecules at 80 A˚2/molecule were not carried out due to the limitations of the TIP3P water model, which gives lowered surface tension of the neat water surface and thus is not useful for modeling of monolayers at low surface concentration where pores are observed.25 A DC system at 40 A˚2/molecule was not considered because it falls beyond the collapse of the film,19 where multilayer structures start to form. The EC geometry was dictated by the shape of the surfactant molecules. A rhombic cell with angle of 60 and a parallelogramshaped cell with angle of 45 were chosen for DPPC and DC, respectively (Figure 1). Details of all models sizes are given in Table 1. For a given number of lipids in the EC the amount of water grew proportionally to the area/molecule so that the thickness of the water layer remained relatively constant for all surface concentrations considered. The size of the periodic box along the z-axis was set large enough to guarantee the lack of periodicity in this direction, thus ensuring the 2D dimensionality of the monolayer. The current models described in a more realistic way the studied systems than those with the smaller ECs reported before.22,23 When the periodic box comprised 25 surfactants, the nine central molecules (36%) interacted with eight real lipids each and those from the external shell (48%) of the EC interacted with five real surfactants and three periodic images. The four vertex molecules (25) (a) Vega, C.; de Miguel, E. J. Chem. Phys. 2007, 126, 154707. (b) Duncan, S. L.; Larson, R. G. Biophys. J. 2008, 94, 2965–2986.
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in the EC (16%) were an exception because they “saw” just three real lipids and five periodic images. In the models with 81 lipids more than 60% of the surfactants (49 molecules) had only real partners and approximately 35% (28 molecules);five real lipids and three periodic images (Table 2). The force field CHARMM2726-29 as implemented in the program package GROMACS 3.3.330 was used in all calculations.29 TIP3P26 and TIP4P26 (only in the case of 81 DPPC at 70 A˚2/molecule due to artificial structuring of the monolayer yielded by the TIP3P model) models were employed for description of the water molecules. Bulk water below the monolayer was mimicked by imposing an additionally adjusted potential (see Appendix A) acting in direction normal to the interface on the oxygen atoms of the water molecules from the neat water surface. Throughout the simulations the O-H bonds of water were kept frozen with SETTLE31 and all other hydrogen-containing bonds were fixed with LINCS.32 Both algorithms were used as implemented in GROMACS 3.3.3.30 All molecular dynamics (MD) simulations were carried out with the following general protocol (for more details and for the specific lengths of the separate stages refer to Table S1 of the Supporting Information): (1) Three-step geometry optimization involving: (i) minimization of the water molecules with the lipid nonhydrogen atoms restrained by a harmonic potential (force constant k = 420 kJ mol-1 nm-2), (ii) short unrestrained optimization, and (iii) energy relaxation of the water molecules with softer restraints on the lipid non-hydrogen atoms (k = 210 kJ mol-1 nm-2). The three optimizations were performed by the L-BFGS method30 with convergence criterion of 50 kJ mol-1. (2) Heating of the system from 0 to 300 K in a series of steps with the lipid non-hydrogen atoms restrained (26) Jorgensen, W. L.; Chandrasekhar, J.; Madura, J. D.; Impey, R. W.; Klein, M. L. J. Chem. Phys. 1983, 79, 926–935. (27) MacKerell, A. D., Jr.; Bashford, D.; Bellott, M.; Dunbrack, R. L., Jr.; Evanseck, J. D.; Field, M. J.; Fischer, S.; Gao, J.; Guo, H.; Ha, S.; JosephMcCarthy, D.; Kuchnir, L.; Kuczera, K.; Lau, F. T. K.; Mattos, C.; Michnick, S.; Ngo, T.; Nguyen, D. T.; Prodhom, B.; Reiher, W.E., III.; Roux, B.; Schlenkrich, M.; Smith, J. C.; Stote, R.; Straub, J.; Watanabe, M.; Wiorkiewicz-Kuczera, J.; Yin, D.; Karplus, M. J. Phys. Chem. B 1998, 102, 3586–3616. (28) Schlenkrich, M.; Brickmann, J.; MacKerell, A. D., Jr., Karplus, M. In Biological Membranes: A Molecular Perspective from Computation and Experiment; Merz, K. M., Roux, B., Eds.; Birkhauser: Boston, MA, 1996; p 31. (29) Feller, S. E.; MacKerell, A. D., Jr. J. Phys. Chem. B 2000, 104, 7510–7515. (30) (a) Berendsen, H. J. C.; van der Spoel, D.; van Drunen, R. Comput. Phys. 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. (31) Miyamoto, Sh.; Kollman, P. J. Comput. Chem. 1992, 13, 952–962. (32) Hess, B.; Bekker, H.; Berendsen, H. J. C.; Fraaije, J. G. E. M. J. Comput. Chem. 1997, 18, 1463–1472.
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Tzvetanov et al. Table 2. Neighbor Distributions for EC with 25 and 81 Surfactants EC with 25 molecules lipids
real neighbors per lipid
images per lipid
lipids
real neighbors per lipid
images per lipid
9 12 4
8 5 3
0 3 5
49 28 4
8 5 3
0 3 5
central molecules molecules at the EC edge molecules at the EC vertex
(k = 210 kJ mol-1 nm-2) and subsequent equilibration at 300 K for several hundred picoseconds. (3) Two-stage restraint relaxation (k = 105 kJ mol-1 nm-2 during stage I followed by unrestrained stage II) and equilibration. The point of reaching equilibrium was verified by analysis of the energy, temperature, and pressure fluctuations along the trajectories; (4) Production simulation of 10 ns (15 ns for 25 DPPC at 40 A˚2/molecule and 5 ns with TIP4P for 81 DPPC at 70 A˚2/molecule) in NVT ensemble with time step 2 fs at 300 K. The constant temperature was maintained by the Berendsen thermostat33 implemented in GROMACS 3.3.3 with relaxation time of 0.1 ps. The short-range electrostatic interactions were calculated up to a cutoff distance of 14 A˚ with a switching function activated at 12 A˚ and the longrange electrostatics was taken into account with PME.34 The nonbonded interactions were evaluated with a switched cutoff of external radius 12 A˚ and internal radius 10 A˚. The stability of the generated MD trajectories was monitored through combined check of temperature and pressure fluctuations (Figures S1 and S2) and convergence of the potential energy to a constant average value (Figure S3). The structural relaxation of the system was verified by the fluctuations of the length of the P-N vector (for DPPC) and of the dihedral angle C-C-O-H involving the free OH-group of the DC head (Figures S4-S7). GROMACS 3.3.3, VMD 1.8.6,35 and original scripts were used for construction of the EC, batch calculations, vector decomposition, visualization and statistical analysis of the results. Snapshots were extracted from the trajectory for analysis at intervals of 1 ps, the entire analyzed ensembles consisting of 10 000 structures. All mean values shown below were averaged over the whole set and statistical accuracy was quantified by standard errors.
Estimation of Electric and Dielectric Monolayer Properties The electrostatic potential (φ) profile across the monolayers was obtained by integration of the charge distribution (Fc) along the z coordinate: φðzÞ -φð0Þ ¼ -
EC with 81 molecules
1 ε0
Z
z 0
dz0
Z
z0 0
dz00 FC ðz00 Þ
ð1Þ
The surface pressure of the monolayers was estimated from the virial relation: 0 1 2@ 1X Ekin þ rij Fij A P ¼ V 2 i, j
ð2Þ
where P is the pressure, V is the elementary cell volume, Ekin is the kinetic energy, rij is the interatomic distances, and Fij are the interatomic forces. Following the mechanical definition, the total surface tension (σ) of the system is: Pxx þ Pyy σ ¼ Lz Pzz 2
ð3Þ
where Lz is the periodic box length along z and the angle brackets denote averaging over all snapshots from the trajectory. Then the surface pressure (Π) is estimated by Π ¼ σ 0 -σm
ð4Þ
where σ0 denotes the experimental surface tension of water and σm is the calculated surface tension of the monolayer. In our case, σ0 is 72 mN/m36 while σm is obtained by subtracting the value reported for the TIP3P model (52.5 mN/m37) from the surface tension of the system (see the Appendix). In the case of 81 DPPC molecules in the EC at 70 A˚2/molecule, the simulation was conducted with the TIP4P/2005 water model and then a value of 65.5 mN/m was used.38 The latter was estimated from neat water slab simulation by using the same method. The (di)electric properties of the monolayers were characterized 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, was calculated in the monopole approximation. The separate contributions of surfactants and water were estimated and each of them was decomposed into normal and tangential components (with respect to the interface). The experimental estimates of the normal dipole moment (μ^) are related to the surface potential (ΔV) through the Helmholtz equation39 (eq 5), where ε^ is the normal monolayer dielectric permittivity and A is the mean area per surfactant. ΔV ¼
μ^ ε0 ε^ A
ð5Þ
In the present manuscript the theoretical normal dipole moment coincides with the z component of μ (μz μ^) and the
where ε0 is the dielectric permittivity of vacuum. (33) Berendsen, H. J. C.; Postma, J. P. M.; Van Gunsteren, W. F.; Dinola, A.; Haak, J. R. J. Chem. Phys. 1984, 81, 3684–3690. (34) (a) Darden, T.; York, D.; Pedersen, L. J. Chem. Phys. 1993, 98, 10089– 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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(36) White, H. J.; Sengers, J. V.; Neumann, D. B.; J. C. Bellows, D. B. IAPWS Release on the Surface Tension of Ordinary Water Substance; 1995; available from http://www.iapws.org. (37) Vega, C.; Abascal, J. L. F.; Conde, M. M.; Aragones, J. L. Faraday Discuss. 2009, 141, 251–276. (38) Jorgensen, W. L.; Tirado-Rives, J. Proc. Natl. Acad. Sci. U.S.A. 2005, 102, 6665–6670. (39) Davies, J. T.; Rideal, E. K. Interfacial Phenomena; Academic Press: New York, 1961.
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Figure 3. Electrostatic potential profiles across the DPPC (left) and the DC (right) monolayers calculated with EC of 81 lipids at 50 A˚2/ molecule; arrows indicate surface potential jumps; the left sides of the graphs correspond to lipid tails. Solely box z-coordinates occupied by the monolayers are represented.
tangential component is estimated by the formula μjj ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi μx 2 þ μy 2
the total dipole moment being μtotal ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi μx 2 þ μy 2 þ μz 2
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 0
1 εjj 0 0 ε ¼ @ 0 εjj 0 A 0 0 ε^
ð6Þ
εRR ¼ 1 þ HRR =V
ð7Þ
HRR ¼
1 ðÆμ 2 æ -ÆμRR æ2 Þ ε0 kB T RR
ð8Þ
)
)
where μRR is the respective dipole moment component (μ^ or μ ) and V is the monolayer volume (Tables 1, 2 of the companion paper). Equations 6-8 were used for quantitative assessment of the dielectric permittivity components in normal (^) and tangential ( ) 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 the origin of the potential jump measured experimentally. The estimates for the DPPC and DC monolayers at 50 A˚2/molecule are shown for illustration in Figure 3. The profiles of the remaining systems are provided as Supporting Information (Figures S8 and S9). It should be noted first that the size effect on the potential profiles (Figures 3, S8, S9) is negligible because the smaller and the larger models of both DPPC and DC have identical curves both in terms of potential variation as a function of z and in terms of potential jumps across the interface. The DPPC potential profiles are characterized by large opposite contributions of lipids and water along the entire isotherm (Figure S8 (Supporting Information)), which compensate each other to result in a small total potential jump. The lipids govern Langmuir 2010, 26(11), 8093–8105
the potential in the segment of the tails but the polarization of water defines φ in the hydrated monolayer part determining also the resultant sign of the monolayer surface potential. The overall potential difference (ΔV), which can be compared to the measured surface potential, is minimal for 25 DPPC at 80 A˚2/ molecule, slightly larger at 70 A˚2/molecule and changing less appreciably to 50 A˚2/molecule; then it grows at 40 A˚2/molecule reaching the maximum value. For 81 DPPC the steeper growth begins already at 50 A˚2/molecule. This general trend is completely in line with the experimental ΔV/A isotherm of a DPPC monolayer40 (Figure 2). The DC films feature different potential profiles across the interface. Lipids and water are similarly polarized but the variation of the total potential within the monolayer is defined exclusively by the lipids. Water and lipid contributions add up to give the total surface potential of the film. The potential jump changes gradually along the isotherm (Figure S9 (Supporting Information)) from the smallest value at 80 A˚2/molecule to the largest ones at 50 A˚2/molecule for both EC sizes, which is in good correspondence 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, the absolute values are appreciably overestimated due to lack of explicit polarizability in the force field used. However, the ratio of ΔV for the two monolayers at the same surface concentration is reproduced well, which renders the comparison between them reliable. (Figure S10 (Supporting Information)). Another measure for the reliability of the chosen computational methods is the comparison between the calculated and the experimental average surface pressures of the two monolayers. The respective data are plotted in Figure 4. The surface pressure for the DPPC monolayer with 81 molecules in the EC at 50 A˚2/molecule agrees well with the experimental value while at lower surface concentrations the discrepancy increases. Nevertheless, the overall trend in the surface pressure is reproduced in the simulations. It has to be noted that the value at 70 A˚2/molecule is obtained with TIP4P/2005 water model. Initially, the system was simulated with the TIP3P water model but the measured surface pressure substantially increased (results not shown in the graph) due to opening of large pores in the monolayer, which tended to be stable during the rest of the run. An explanation is that the TIP3P surface tension is rather small (52.5 mN/m37) compared to the experimental value of 72 mN/m36 and this disparity renders possible the formation of (40) Physical Chemistry of Biological Interfaces; Baszkin, A.; Norde, W., Eds.; Marcel Dekker: New York and Basel, Swizerland, 2000.
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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.
surfactant-free patches in the monolayer at considerable surface concentrations. The change in the water model to TIP4P/2005, which has surface tension of 65.5 mN/m, resulted in at least qualitative agreement with the experiment. The system at 40 A˚2/ molecule is in the region of the monolayer collapse, so its value is not comparable to the experimental results. The negative values of the surface pressure, most prominent at 60 and 70 A˚2/molecule, show the tendency of the model monolayer to decrease its mean area per molecule. The periodic boundary conditions, used in the simulations, impose artificial periodicity on the organization of the film so that unit cells smaller than the extent of correlations pertinent to a particular phase will reinforce ordering of the molecular structure. Thus, the 25 DPPC clusters are insufficient to accommodate the length-scale of the correlations at 60 and 70 A˚2/molecule and hence tend to shrink to smaller areas per molecule. Consistent with this interpretation is the behavior at 50 and 80 A˚2/molecule, where the space correlations are shorter (Figure 5 of the companion paper) and the 25 DPPC clusters display positive surface pressures. Hence, the results obtained from the smaller systems will be regarded only for comparison. Although the values are slightly shifted to higher surface pressures, the results for DC monolayers are in good agreement with the experimental data. This can be attributed to the lack of highly charged moieties in the DC molecule as well as the good optimization of interactions between the lipid tails in the CHARMM22/27 force field. Also, formation of persistent water pores was not observed throughout the simulations of DC monolayers. Another interesting feature is that the results obtained with the smaller EC practically coincide with those obtained with 81 lipid molecules in the EC, which is consistent with the shorter range of interactions in these monolayers. Dipole Moments. Average Values. The calculated average values of the total dipole moments and of their normal and 8098 DOI: 10.1021/la9047352
tangential components (scaled per one surfactant molecule), as well as water and surfactant contributions therein, for all clusters studied are collected in Table 3. As seen from the data in Table 3, all standard errors are one to 2 orders of magnitude smaller than the respective average values, which renders the comparative analysis of the data reliable. The curves of the total dipole moment of DPPC calculated with EC of 25 and 81 lipids have similar profiles (Figures 5 and S11 (Supporting Information), left);a minimum is observed at 60 A˚2/molecule and 50 A˚2/molecule for 25 DPPC and 81 DPPC, respectively. The higher degrees of compression are characterized with steeper increase of μtotal. As could be expected, the variation of the dipole moment upon compression is governed by the lipid molecules, while μtotal of water remains virtually constant and decreases insignificantly at the highest surface concentration. The total dipole moment magnitude at low compression (60-80 A˚2/ molecule for 25 DPPC and 70 A˚2/molecule for 81 DPPC) originates primarily from water and the lipids begin to dominate only at the tightest packing. The total dipole moment of DC monolayers decreases smoothly upon compression (Figures 5 and S11 (Supporting Information), right). As we have shown previously,22 it is due mainly to the structuring of the water molecules around the surfactant heads at all surface concentrations studied. The lipid dipole moment varies immaterially along the entire isotherm. The calculated normal dipole moments of DPPC show that water and lipids have opposite shares, the former being the leading one (Figures 6 and S12 (Supporting Information), left). These results are fully in support of our previous findings based on EC of 9 lipids.22 μ^ almost does not depend on the lipid cluster size and varies in a very narrow range upon compression. It is noteworthy that shallow minima are observed for the DPPC Langmuir 2010, 26(11), 8093–8105
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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 DC Monolayers Modeled with Different Numbers of Lipids in the EC and at Various Degrees of Compression μlipid
μ^total
μ^water
μ^lipid
μ
otal
μ
ater
μ
ipid
)l
μwater
)w
μtotal
)t
area, A˚2/molecule
25DPPC 40 50 60 70 80
15.22 ( 0.03 11.42 ( 0.04 7.81 ( 0.04 8.15 ( 0.04 9.40 ( 0.05
6.92 ( 0.02 14.08 ( 0.01 7.63 ( 0.02 9.82 ( 0.01 7.64 ( 0.03 3.71 ( 0.01 8.36 ( 0.03 3.98 ( 0.01 8.45 ( 0.04 4.56 ( 0.02
1.78 ( 0.004 1.85 ( 0.01 2.13 ( 0.01 1.97 ( 0.01 1.67 ( 0.01
4.35 ( 0.01 4.55 ( 0.01 3.45 ( 0.01 4.07 ( 0.01 3.14 ( 0.01
-2.57 ( 0.004 15.01 ( 0.04 5.06 ( 0.03 13.62 ( 0.01 -2.70 ( 0.01 11.22 ( 0.05 5.79 ( 0.03 9.40 ( 0.01 -1.32 ( 0.01 7.38 ( 0.04 6.55 ( 0.03 3.32 ( 0.02 -2.10 ( 0.01 7.79 ( 0.04 6.98 ( 0.04 3.20 ( 0.01 -1.47 ( 0.01 9.15 ( 0.05 7.57 ( 0.04 4.09 ( 0.02
81DPPC 40 50 60 70
15.78 ( 0.01 6.49 ( 0.01 7.07 ( 0.01 4.64 ( 0.02
4.72 ( 0.02 15.07 ( 0.004 5.76 ( 0.02 5.58 ( 0.01 5.28 ( 0.02 5.45 ( 0.01 5.11 ( 0.02 2.95 ( 0.01
1.85 ( 0.003 2.24 ( 0.004 2.16 ( 0.004 2.18 ( 0.01
3.54 ( 0.004 4.49 ( 0.01 3.65 ( 0.01 3.93 ( 0.01
-1.69 ( 0.003 15.66 ( 0.02 2.90 ( 0.01 14.97 ( 0.004 -2.26 ( 0.01 5.99 ( 0.02 3.32 ( 0.02 5.09 ( 0.01 -1.53 ( 0.01 6.67 ( 0.02 3.58 ( 0.02 5.20 ( 0.01 -1.75 ( 0.01 3.95 ( 0.02 3.01 ( 0.02 2.28 ( 0.01
25DC 50 60 70 80
5.38 ( 0.02 5.54 ( 0.03 6.04 ( 0.03 6.65 ( 0.03
4.70 ( 0.02 5.05 ( 0.03 5.60 ( 0.03 6.24 ( 0.03
1.91 ( 0.003 1.78 ( 0.003 1.77 ( 0.003 1.87 ( 0.003
1.74 ( 0.01 1.69 ( 0.01 1.67 ( 0.01 1.70 ( 0.01
0.19 ( 0.01 0.09 ( 0.01 0.07 ( 0.01 -0.04 ( 0.01
1.55 ( 0.003 1.60 ( 0.003 1.61 ( 0.003 1.74 ( 0.003
4.98 ( 0.03 5.16 ( 0.03 5.68 ( 0.03 6.32 ( 0.03
4.66 ( 0.02 5.01 ( 0.03 5.55 ( 0.03 6.20 ( 0.03
1.03 ( 0.01 0.72 ( 0.004 0.64 ( 0.003 0.61 ( 0.003
1.55 ( 0.002 1.57 ( 0.002 1.62 ( 0.002 1.74 ( 0.002
2.65 ( 0.01 2.93 ( 0.02 3.31 ( 0.02 3.40 ( 0.02
2.56 ( 0.01 2.86 ( 0.01 3.21 ( 0.02 3.33 ( 0.02
0.44 ( 0.003 0.38 ( 0.002 0.40 ( 0.002 0.38 ( 0.002
81DC 50 60 70 80
3.27 ( 0.01 3.48 ( 0.01 3.80 ( 0.02 3.88 ( 0.02
2.59 ( 0.01 2.88 ( 0.02 3.24 ( 0.02 3.36 ( 0.02
1.63 ( 0.002 1.63 ( 0.002 1.68 ( 0.002 1.80 ( 0.002
1.74 ( 0.003 0.19 ( 0.003 1.67 ( 0.003 0.10 ( 0.003 1.66 ( 0.003 0.03 ( 0.004 1.64 ( 0.004 -0.10 ( 0.004
Figure 6. Normal dipole moment of DPPC (left) and DC (right) monolayers and its lipid and water shares.
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60 A˚2/molecule for 25 DPPC and at 50 A˚2/molecule for 81 DPPC and grows at the higher degrees of compression, as witnessed for the total dipole moment. After the minimum, the magnitude of μ is determined mainly by the lipid whereas water contributes less. The variation of this curve upon compression can be explained by a LE/LC transition and subsequent enhancement of lipid ordering, which is confirmed by the corresponding structural characteristics of the film.24 Even though the trendlines of the curves for the clusters with 25 and 81 DPPC are alike, the values of μ in the range 50 to 70 A˚2/molecule are very dissimilar. μ decreases with increase of EC, the only invariant value remaining the one at the tightest packing. No extrema are witnessed in the curve of μ describing the DC monolayers (Figure 7, right). The tangential dipole moment originates mainly from the water molecules and decreases almost monotonously upon compression. The lipid share remains practically constant with the exception of 25 DC at 50 A˚2/molecule. The size effect is similar to that of DPPC but here also the most compressed film is affected. Surprisingly, notwithstanding the substantially dissimilar structure and electrostatics of the two lipids, their normal dipole )
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normal components, similar to experimental estimates based on MDC measurements.13,15 Lipid and water shares of the normal dipole moment of DC have identical signs and the lipid magnitude of μ^ is larger (Figures 6 and S12 (Supporting Information), right). All values are smaller than those of DPPC, which is in agreement with the different electrostatics of DC and DPPC heads. The normal dipole moment of DC films grows slightly upon compression. It is also not sensitive to the EC size, which shows that the used models are large enough for accurate description of the normal dipole moment. The tangential dipole moment is especially sensitive to structural changes of the monolayer. This stems from the fact that its magnitude depends on the orientation of the lipid molecules in the monolayer plane (whereas the direction of the normal moment is fixed) at the various degrees of compression and thus provides information on the extent of lipid ordering. At low surface concentrations of the DPPC monolayers (Figure 7, left) the water share to μ prevails decreasing almost linearly upon compression, i.e. with decrease of the total number of water molecules in the film. The lipid value remains minimal at
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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.
moment is fairly alike both in qualitative behavior and as quantitative estimate (Figures 8 and S13 (Supporting Information)). The tangential component of both lipids determines the magnitude of the total monolayer dipole moment. Although most publications focus on discussion of the normal component due to its direct relation to experimental measurements, the 8100 DOI: 10.1021/la9047352
tangential dipole moment turns out to be much more lipid-specific and deserves detailed study. The components of the dipole moment scaled per unit molecular area have the meaning of polarization density in the respective direction. The density of the normal component is presented in Figures 9 and S14 (Supporting Information). Langmuir 2010, 26(11), 8093–8105
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Figure 10. Tangential polarization density scaled per one lipid molecule of DPPC (left) and DC (right) films with EC of 25 (top) and 81 (bottom) surfactants.
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total and/or the tangential dipole moments reflect surface concentrations where there are well-defined states (maxima) or where structural rearrangements are taking place (minima)23 can be substantiated by an additional analysis of the tangential dipole moments. A more detailed interpretation of the molecular organization at the interface can be made by statistical analysis of the orientation of the vector of lipid μ with respect to the x-axis (quantified by the angle φ). The choice of this angle is dictated by the fact that μ can change its orientation in the monolayer plane upon compression and thus gives insight into the extent and type of lipid ordering. The distribution of μ orientation is based on the values of the angle φ closed by the tangential dipole moment with an 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 dissimilar way upon compression. For DPPC only the model with 25 lipids at the lowest surface concentration shows essentially uniform population of all angles. At the higher levels of compression the monolayers modeled by 25 and 81 lipids behave alike. Namely, at 70 A˚2/molecule still all directions of μ|| are present in the film but more definite peaks start to form almost equidistantly along the entire range. This may be assigned to the existence of a small share of ordered lipid domains surrounded by predominantly disordered lipids. At 60 A˚2/molecule multiple peaks varying irregularly in population and breadth and most of them overlapping indicate that the film is in the process of achieving long-range organization. At the next stage of compression already several well-shaped peaks are registered. The population of the remaining angles is substantially reduced but is still nonzero, especially for the larger models. Such angular distribution illustrates the presence of a main well-ordered lipid phase coexisting with a smaller portion of disordered molecules, which is in conjunction with the bimodal area distribution at 50 A˚2/molecule obtained from Voronoi analysis and with the multipeak phosphorus and nitrogen density distributions.24 Upon further compression the lipid structuring is )
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For both DPPC cluster sizes smooth increase of the normal polarization density upon compression is registered. The shares of both lipids and water are considerable the latter one prevailing. For the EC with 25 lipids the water share becomes more positive and that of lipids;more negative with increase of the surface concentration. A possible explanation is the enhanced mutual polarization of the two subsystems upon compression. Both shares feature local extrema at 60 A˚2/molecule marking the zone of phase transition. The films with EC of 81 DPPC preserves the same trend upon compression but the polarization density reaches saturation at the tighter packing (40-50 A˚2/molecule). This is an additional confirmation of the statement that DPPC normal dipole moment is described adequately by the larger models. The total normal polarization density of DC and its shares grow almost linearly upon compression. No extrema are detected, which means that the lipid molecules come closer without substantial reordering with the increase of surface concentration. There is practically no EC size effect, because the values vary in the 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 A˚2/ molecule for the smaller and at 50 A˚2/molecule for the larger clusters. Unlike the normal polarization density, there is no saturation of μ /A at the smaller areas, which indicates that the tangential dipole moment responds in a more pronounced way to changes of the surfactant surface concentration. For DC the monolayer polarization is governed mainly by water and grows linearly upon compression. The tangential polarization density of the lipids is close to zero and increases in a very narrow range with the surface concentration. There is strong EC size effect expressed in halving the values of the larger systems. Angle and Magnitude Distribution of the Tangential Dipole Moment. The hypothesis that the extrema in the curves of the
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Figure 11. Distribution of the tangential dipole moment (scaled per one lipid molecule) by direction (top panel) and magnitude (bottom panel) of clusters with 25 (top of each panel) and 81 (bottom of each panel) surfactants modeling DPPC (left) and DC (right) monolayers.
enhanced, which is illustrated by the very sharp peaks of φ in the most condensed DPPC films. The difference between the smaller and the larger models is that in the former one the peaks remain well separated from each other while in the latter the tangential dipole moments can adopt all possible orientations within the restricted range specified above. Anyway, the solid-type lipid arrangement at 40 A˚2/molecule24 seems to correspond to a preferred direction of the tangential dipole moments of the DPPC 8102 DOI: 10.1021/la9047352
molecules. Since the film is not monocrystalline, the dipoles in the separate domains are not strictly collinear, which causes the dispersion in the values of φ even at the tightest packing. Another reason for the relatively broad distribution of φ at 40 A˚2/molecule is the presence of irregularities in the perfect hexagonal packing both with respect to number and equidistance of nearest neighbors evidenced by the results from the Voronoi analysis and from the radial distribution functions.24 Overall, upon compression Langmuir 2010, 26(11), 8093–8105
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Article Table 4. Normal Dielectric Permittivity of DPPC and DC Monolayers and the Respective Standard Errors area, A˚2/molecule
ε^total
ε ^water
ε ^lipid
2.52 ( 1.20 3.49 ( 1.25 3.33 ( 1.17 3.80 ( 1.06 4.25 ( 1.16
2.10 ( 1.12 3.47 ( 1.18 3.31 ( 1.09 4.55 ( 1.16 5.40 ( 1.13
2.52 ( 1.29 2.57 ( 1.26 3.26 ( 1.28 2.76 ( 1.37
2.21 ( 1.14 3.60 ( 1.27 3.67 ( 1.17 2.69 ( 1.21
2.00 ( 1.02 1.99 ( 1.01 2.01 ( 1.01 2.03 ( 1.01
1.38 ( 1.04 1.39 ( 1.04 1.45 ( 1.05 1.39 ( 1.04
1.92 ( 1.02 1.96 ( 1.01 1.98 ( 1.01 1.94 ( 1.01
1.41 ( 1.08 1.47 ( 1.08 1.44 ( 1.08 1.51 ( 1.09
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25DPPC 1.67 ( 1.05 1.81 ( 1.05 1.84 ( 1.07 1.88 ( 1.17 1.93 ( 1.04
40 50 60 70 80
81DPPC 40 50 60 70
1.72 ( 1.09 1.83 ( 1.10 1.88 ( 1.10 1.83 ( 1.13
50 60 70 80
1.80 ( 1.06 1.84 ( 1.05 1.84 ( 1.05 1.83 ( 1.05
25DC
81DC 1.73 ( 1.09 1.79 ( 1.08 1.78 ( 1.08 1.79 ( 1.07
50 60 70 80
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Information)). Unlike the small average values of ε^, the tangential component has magnitudes close to that of water and varies in wide ranges depending on the lipid and on the surface concentration. The interpretation of the extrema in the profiles of ε is exactly the opposite to that of the dipole moments: maxima correspond 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 the lipids affect it negligibly. The total DPPC tangential dielectric permittivity has a maximum at 70 A˚2/molecule for 25 DPPC and at 60 A˚2/molecule for 81 DPPC, followed by decrease of ε upon further compression. This supports the hypothesis of phase transition at these areas and subsequent enhanced monolayer organization. The reorganization of the lipids begins earlier for the smaller models (at 70 A˚2/ molecule) followed by that of water at 60 A˚2/molecule. Within the models with larger EC the rearrangement of the two monolayer subsystems takes place at the same surface concentration (60 A˚2/ molecule). The area per molecule at which the reordering occurs can be related to the freedom of movement limitations imposed by the periodic box lateral dimensions. At the largest DPPC surface concentrations the dielectric permittivity of the aqueous layer is smaller than that of bulk water (82 for the TIP3P water model41); it increases upon film decompression and reaches saturation in the LE region, its magnitude there being close to that of bulk water. In general, the larger models are characterized by smaller values of ε , which may be rationalized in terms of decreased polarization effect of the larger number of real lipid neighbors. In addition, the value for 81 DPPC at 70 A˚2/molecule is too low, which is partly due to the different water model. The dielectric permittivity of DPPC bilayers has been reported by Stern and Feller.8b As their model contains 36 lipids with orthorhombic initial alignment in a leaflet and the simulation is at 62.9 A˚2/molecule and 323 K in NPnAT ensemble, the closest )
)
)
)
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both DPPC models feature convergence of the peaks accompanied by increase of the population indicating enhancement of the molecular organization. The angular distributions of μ of the DC films (Figure 11, top) have rather different behavior than their DPPC analogues. Only the points of 25 DC and 81 DC at the lowest surface concentration have similar profiles to the 25 DPPC point at 80 A˚2/molecule; uniform population of all angles without any visible extrema. The models at 70 and 60 A˚2/molecule are characterized by single very broad maxima, which are even broader for the larger model, especially the one at 60 A˚2/molecule. The DC peaks at 50 A˚2/ molecule extend over much more angles than the corresponding DPPC maxima at 40 A˚2/molecule. This leads to the most substantial dissimilarity between the two monolayers;there is no zero probability for any angle φ in the DC films, irrespective of the degree of compression. It means that the DC dipoles retain appreciable freedom for reorientation even at the highest external stress. This is in line with the significant share of relatively disordered lipids in these films, as yielded by the Voronoi analysis.24 With respect to distribution of the tangential dipole moment magnitudes (Figure 11, bottom) the two monolayers differ, too. The variation is less pronounced in the DC models. The values of μ for 25 DPPC at 60-80 A˚2/molecule are described by much distorted Gaussian peaks spanning a range of about 6 D while at 50 A˚2/molecule μ is centered around much higher values;ca. 9 D. The point with the highest surface concentration is characterized with a very narrow sharp maximum at ca. 14 D without any visible tails. The latter is in line with the solid-type lipid organization at this degree of compression and in good agreement with quantum mechanical estimates.23 When the elementary cell is extended to 81 lipids the peaks become narrower in general and without definite tails and the systems are grouped in a different way. This illustrates the enhanced ordering in the larger models due to increased number of real neighbors. Unlike DPPC, the DC films are characterized by Boltzmanntype distributions of the tangential dipole moment magnitudes showing that solid-state ordering is not achieved at any degree of compression. The size effect here is expressed in the fact that for the smaller systems the maxima shift to higher values and the curves become broader upon compression, while for the larger models the maxima remain invariant. Overall, whatever reorganization takes place in the DC monolayers upon compression, it is not reflected in the magnitude distribution of μ . Dielectric Permittivity. The above estimates of the dipole moment show that the properties of the studied insoluble monolayers are characterized by pronounced anisotropy. Another related parameter is the dielectric permittivity, which is essential for description of 2D-nucleation. Its normal and tangential components were calculated by eq 7. It has to be noted that the tangential dielectric permittivity can not be measured directly experimentally. The volume of each monolayer is obtained from its density profile.24 The calculated values of ε of the monolayers show that it varies significantly depending on the degree of compression and on the surfactant type. Table 4 contains the normal component of the dielectric permittivity tensor for DPPC and DC monolayers and the respective standard errors. Because of the small values of ε^, the differences between the average estimates at different surface concentrations fall within the standard errors. Therefore, we will refrain from further comments on the variation of ε^ upon compression. The only noteworthy comment is that the normal dielectric permittivity of DPPC varies in a range twice as large as that of DC. The tangential dielectric permittivity of the monolayers was estimated by eq 7, too (Figures 12 and S15 (Supporting
(41) Tieleman, D. P.; Marrink, S. J.; Berendsen, H. J. C. Biochim. Biophys. Acta 1997, 1331, 235–270.
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Figure 12. Variation of the tangential component of the relative tangential dielectric permittivity and its surfactant and water shares for DPPC (left) and DC (right) monolayers.
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analogue is our 25DPPC system at 60 A˚2/molecule. The differences are too many to arrive at identical values but trendlines could be compared. Numerically, the results of Stern and Feller for the normal dielectric permittivity are of the same order of magnitude (4.6 ( 0.4 for DPPC, 5.5 ( 0.5 for water, and 3 ( 1 total), the total value of ε^ being times smaller than the sum of the components, in line with the opposite correlation of normal dipole moment contributions of DPPC and water. The values for the tangential components are significantly different: on the average the share of water in this study is twice as large and that of DPPC twice as small, the total tangential permittivity being lower than the one reported by Stern and Feller, probably indicating the impact of the second leaflet on the dielectric properties of insoluble lipid monolayers or due to the differences in the simulation conditions. The situation with the DC films is different: the surfactant and water shares of ε are relatively constant along the entire isotherm. Water has the dominating role both qualitatively and quantitatively. Irrespective of that, the total tangential dielectric permittivity is always lower than that of bulk water. This is most probably due to the depolarizing effect of the surfactant heads. Analysis of the curve for 81 DC also shows a maximum at 70 A˚2/ 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, the overall low level of organization within the monolayers24 leads to high values of ε (about 45 for the two lipids) even at the tightest compression. In general, the results from the simulations reflect properly the tendency toward decrease of the total tangential dielectric permittivity with enhanced degree of ordering in the monolayers. More accurate numerical estimates especially for the normal permittivity require longer simulations.
Conclusions Molecular dynamics simulations on model DPPC and dicaprin (DC) insoluble monolayers at the air/water interface were carried out. The size effect on the estimated properties was tested by employing elementary cells of 25 and 81 lipids and imposing 2D periodic boundary conditions. The calculations were performed for several surface concentrations located in the LE and LC regions of the surface pressure/area isotherm. A refinement of the model was the introduction of a specially adapted potential mimicking the presence of bulk water and thus mitigating the influence of water/ vacuum interface below the film. The (di)electric properties of the monolayers, dipole moment, dielectric permittivity, and electrostatic potential as well as surface pressure, were estimated by analysis of 8104 DOI: 10.1021/la9047352
the MD trajectories. The data were discussed both for the entire systems and in terms of normal and tangential to the interface components and of surfactant and water contributions. Marked EC size effect was observed only for the tangential components. The parameters in normal direction were practically not affected by the increase of the elementary cell dimensions. This signifies that the current models are sufficient to describe accurately the normal characteristics of the monolayers, while tangential properties are given correctly only qualitatively. For quantitative values comparable to experiment the models have to be extended further. The two studied monolayers have very different behavior with respect to their (di)electric features. It was shown that the normal polarization of the DPPC monolayer results from a balance between the lipid and the water contribution while that of DC is governed solely by the surfactant. The DPPC tangential dipole moment is dictated mainly by the lipid while the DC one stems from organization of the water molecules around the surfactant heads. The extrema in the dipole moment and dielectric permittivity curves were interpreted in terms of monolayer rearrangement or ordered states. It was shown that the tangential dielectric permittivity dominates over the normal one and that it is much more sensitive to the structural rearrangements and to the model size within the lipid monolayers. All these conclusions support our previous hypotheses and findings based on smaller models of the two films and recent structural analyses of the current models. The size-effect is expressed in the fact that the concentration indicating essential monolayer reorganization tends to be higher in larger systems. Distribution of the lipid tangential dipole moments by in-plane orientation and by magnitude reveal distinct structuring of the DPPC monolayers and more amorphous arrangement of the lipids in the DC films. It can be generalized that the EC size has mostly quantitative effect. Inspection of the electrostatic potential and surface pressure profiles shows qualitative correspondence with the experimental curves although the former values are somewhat overestimated for both lipids, whereas the latter are underestimated for DPPC and overestimated for DC. The nice qualitative agreement between theory and experiment of the trends of φ and Π upon compression renders the used computational procedure reliable for the description of the properties of various insoluble lipid monolayers at the molecular level. The results definitely show that whereas local organization can be discussed based on comparatively small models, thermodynamic parameters require a sufficiently large unit cell as exemplified by the surface pressure results in this study. An improvement of the computational scheme aimed at simulation of systems with lower surface concentration (from the LE region of the isotherms) will demand, however, a change of Langmuir 2010, 26(11), 8093–8105
Tzvetanov et al.
Article
the water model from TIP3P to a more sophisticated one, e.g., TIP4P/2005 or the polarizable POL3, which reproduce more successfully the thermodynamic properties of surfacial water. In our opinion the most essential result is the importance of the tangential characteristics of the lipid films for their overall behavior, usually overlooked by theorists and experimentalists but surely deserving a more detailed study. Acknowledgment. The research is funded by Projects DO-2256/2008 and DO-2-82/2008 of the National Science Fund of Bulgaria. The Alexander von Humboldt Foundation is acknowledged for an Equipment Grant.
Appendix A Lipid monolayers can be modeled by two geometries of the simulation cell, which ensure the periodicity in the lateral x and y directions and guarantee for the inhomogeneity of the system in the normal z-direction. The symmetric geometry consists of a middle layer of water molecules covered by two layers of surfactants in opposite orientation. The water slab is usually of considerable thickness so as to prevent interaction between the monolayers at the opposing sides as well as the number of lipid molecules is doubled than actually needed for modeling of the real system. Although this type of geometry is suitable for direct calculation of the surface tension, the nearly 2-fold increase in the number of particles renders it computationally very expensive. On the other hand, the asymmetric geometry of the periodic cell, which contains only one surfactant layer spread out over a water film, allows the simulation of monolayers with the minimum amount of particles and hence substantial decrease in the computational burden. However, it requires the application of wall potential on the water/vacuum interface, which prevents the evaporation of water molecules and reduces the polarization effects arising from the latter phase boundary. In the current work, a wall potential that fulfilled both requirements was developed beforehand, and it was used in the simulations of surfactant films throughout. The guiding idea in the parametrization of the wall potential was to mimic the presence of implicit bulk water while minimizing the disturbance of the mass density of the explicit water slab. The evaporation of water molecules can be neatly eliminated by application of a simple repulsive potential and the polarization of surface water can be mitigated by imposing an orientational restraint on the water molecules. The functional form of the two contributions to the wall potential was derived from properties of an explicit water film of about 4000 molecules simulated in symmetric geometry; i.e., the z-dimension of the elementary cell was long enough to prevent interaction between the periodic images. For consistency, the water model was CHARMMmodified TIP3P and the simulation conditions matched those of the monolayer simulations, i.e. the temperature was maintained at 300 K in NVT ensemble. The position of a water molecule was evaluated from the coordinates of the oxygen atom and its orientation to the interface was determined from the cosine of the angle between the unit surface normal and the unit vector along the bisector of the HOH valence angle. The repulsive portion of the wall potential was chosen to be of 22-12 Lennard-Jones type and it was fitted to the TIP3P oxygen-oxygen radial distribution function. The resulting form of the potential is: ULJ ðzÞ ¼
3:35 10 -11 1:19 10 -5 ðz -z0 Þ22 ðz -z0 Þ12
Langmuir 2010, 26(11), 8093–8105
where ULJ is in kJ mol-1, z (nm) is the z-coordinate of the oxygen atom, and z0 (nm) is the position of the wall. Furthermore, the position of the wall was varied slightly in the different simulated systems to ensure minimal influence of the potential on the water density. In all cases, the repulsion was truncated and shifted to zero at 0.85 nm away from the wall in the water layer. The orientational portion of the wall potential was assumed to be of squared cosine form, as suggested by the orientational distribution of water molecules at the water/vacuum interface. Additionally, the polarization varied with the distance to the phase boundary and the strength of the interaction was weighted with the z-position. The resulting form of the potential as well as the values of the parameters after self-consistent optimization in a series of restrained simulations is: UðθÞ ¼ U0 cos2 θ where U is in kJ mol-1, θ is the angle between the unit vector along the bisector of the HOH valence angle and the unit surface normal vector, and the strength of the restrained U0 is weighted according to the formula: U0 ¼ -2:245ðz -zcut Þ þ 3:118ðz -zcut Þ2 Here z (nm) is the z-coordinate of the oxygen atom and zcut (nm) is the cutoff of the orientational potential where it is shifted to zero. The cutoff distance was set at 1.58 nm away from the position of the wall. In addition, the parabolic weighting was switched to constant between the wall and 0.98 nm away from it in the direction of the water film. It is well-known42 that such simple potentials are not able to completely smooth out the orientational polarization of the water molecules at the interface and the proposed wall potential only succeeded to diminish these effects. Another consequence of the asymmetric geometry of the simulation cell and the need for correcting potential is that the surface tension cannot be estimated with high accuracy. The surface tension of the system as a whole is the sum of the surface tensions of both interfaces, one of which, the water/vacuum boundary, is perturbed by the wall potential. Since the wall position (as defined above) slightly varied from system to system, the disturbance caused by the wall potential also changed. Nevertheless, the evaluation of the surface tension of doubly restrained water slab showed that the wall potential influenced inconsequentially the value for the corresponding model. Then, it was reasonable to use the surface tension of neat water as obtained from unrestrained simulation for the calculation of the surface pressure of the lipid monolayers, as discussed above. Supporting Information Available: Details of the separate computational stages (Table S1), temperature (Figure S1), pressure (Figure S2), and potential energy (Figure S3) fluctuations along the production part of the MD trajectories of the studied systems at 60 A˚2/molecule, variation of the P-N vector length of DPPC monolayers (Figures S4 and S5) and of the dihedral angle C-C-O-H of DC films (Figures S6 and S7) along the production part of the trajectory, electrostatic potential profiles of the simulated DPPC (Figure S8) and DC (Figure S9) monolayers, calculated and experimental values of the surface potential of all systems (Figure S10), average total and normal dipole moments (Figures S11, S12) and normal polarization density (Figures S14), decomposition of the total dipole moment into normal and tangential contributions (Figure S13), and tangential dielectric permittivity (Figure S15) for the models with 25 lipids in the EC. This material is available free of charge via the Internet at http://pubs.acs.org. (42) Beglov, D.; Roux, B. J. Chem. Phys. 1994, 100, 9050–9063.
DOI: 10.1021/la9047352
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