Interactions Determining the Growth of Chiral Domains in

Publication Date (Web): September 28, 2000 ... Also unaccounted for is the noncompact growth of LC domains either at electrolyte ... The standard theo...
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Interactions Determining the Growth of Chiral Domains in Phospholipid Monolayers: Experimental Results and Comparison with Theory Suzanne Amador Kane,* Madison Compton, and Nicholas Wilder Physics Department, Haverford College, Haverford, Pennsylvania 19041 Received June 27, 2000. In Final Form: August 16, 2000 Results are presented from a quantitative analysis of liquid condensed domain shapes in monolayers of the phospholipid D-R-phosphatidylcholine, dipalmitoyl (dipalmitoylphosphatidylcholine or DPPC). We study the effect on liquid-condensed (LC) domain shape of varying in-plane dipole density by examining monolayers composed of different mixtures of the zwitterionic DPPC and anionic phospholipids at high and low electrolyte concentration. Comparison of our data and calculations with existing theories show that the dominant dependence of the shape factor (a measure of the noncompact geometry of LC domains) on in-plane density is described well by electric dipole-dipole interactions; however, the detailed chiral shapes of the liquid-condensed domains remains unexplained. Also unaccounted for is the noncompact growth of LC domains either at electrolyte concentrations high enough to screen out electric dipole interactions or in mixtures with anionic phospholipids. We have also tested the prediction of linear stability analyses that circular domain shapes should give way to a well-defined bilobe transition by computing the Fourier spectrum of our LC domains as a function of domain size and in-plane molecular density. We have performed Monte Carlo simulations of domain growth using an Eden model with energetic contributions from line tension and a novel short-range chiral interaction term. We show that these chiral interactions influence the geometry of the growing domains, yielding noncompact domain growth even in the absence of dipole-dipole interactions.

Introduction A surprising variety of physical and biological systems exhibit two-dimensional pattern growth with labyrinthine fingering. This phenomenon has been previously observed for many systems, including ferrofluids and the growth of condensed domains of the liquid condensed (LC) phase from the liquid expanded (LE) phase in phospholipid Langmuir monolayers, with and without the inclusion of cholesterol.1-5 A smaller subset of these systems evolves with an additional feature: the arms grow in a pattern with a handedness set by the chirality of the constituent molecules. The standard theoretical model for phospholipid domain growth has assumed a free energy competition between a line tension term (analogous to a twodimensional surface tension), which favors compact, approximately circular domains, and an electric dipoledipole free energy interaction, which favors less compact domains.6-11 The competition between these two interac* To whom correspondence should be addressed. (1) Weis, R. M.; McConnell, H. M. J. Phys. Chem. 1985, 89, 44534459. (2) Heckl, W. M.; Losche, M.; Cadenhead, D. A.; Mohwald, H. Eur. Biophys. J. 1986, 14, 11-17. (3) Dickstein, A. J.; Erramilli, S.; Goldstein, R. E.; Jackson, D. P.; Langer, S. A. Science 1993, 261, 1012. (4) Seul, M.; Sammon, M. J. Phys Rev Lett. 1990, 64, 1903-1906. (5) Seul, M.; Monar, L. R.; O’Gorman, L.; Wolfe, R. Science 1991, 254, 1616-1618. (6) Keller, D. J.; Korb, J. P.; McConnell, H. M. J. Phys. Chem. 1987, 91, 6417-6422. (7) Vanderlick, T. K.; Mohwald, H. J. Chem. Phys. 1990, 94, 886890. (8) Lubensky, D. K.; Goldstein, R. E. Phys. Fluids 1996, 8, 843-854. (9) Mayer, M. A.; Vanderlick, T. K. J. Chem. Phys. 1994, 100, 83998407. (10) Mayer, M. A.; Vanderlick, T. K. J. Chem. Phys. 1995, 10, 97889794. (11) Mayer, M. A.; Vanderlick, T. K. Phys. Rev. E 1997, 55, 11061119.

tions can be characterized by the dimensionless parameter, Γ, defined as

Γ ) µ2/λ

(1)

where µ is the electric dipole moment in-plane density and λ is line tension. The line tension contribution adequately describes the growth of domains of the zwitterionic phospholipid D-R-phosphatidylcholine, dipalmitoyl (dipalmitoylphosphatidylcholine or DPPC), at low in-plane densities, resulting in circular or kidneyshaped domains. At higher in-plane densities, the effects of dipole-dipole interactions become important, leading to a transition to noncircular domain shapes and the evolution of sidearms and labyrinthine branching. One approach used to describe these more complex LC domain shapes involves a linear stability analysis of perturbations of circular domain shapes by n-fold Fourier terms. (That is, the transition of circular domains to more complex shapes is modeled by computing the stability of added bilobe, trilobe, etc. contributions when parameters such as Γ, domain area, etc. are systematically varied. In reality, actual phospholipid domain shapes are much more complex, exhibiting sharp singularities in the curvature along domain edges.) These results yielded the prediction that higher order Fourier modes would become increasingly stable as domain area increased for fixed Γ; the authors also found no evidence of coupling between Fourier modes.7 However, later Monte Carlo simulations to compute the stability of each Fourier mode found that only circular and bilobed solutions should be stable for small amplitude perturbations.11 These results also yielded detailed predictions for the dependence of domain geometry upon the value of Γ. While most theories have made the approximation that the competition between dipole-dipole interactions and

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line tension fully determines the actual domain shape, in fact the molecule’s chirality has a strong effect on domain structure.6,7,9-11 For example, the stereoisomers of DPPC form domains which are precise mirror images. As a result, while present theories do a good job of describing labyrinthine domains in many systems dominated by only dipole-dipole interactions and line tension, the actual shapes produced by such models do not exhibit the striking and specific chirality of actual DPPC domains. This effect could be directly due to molecular packing considerations or due to an induced chirality in the larger-scale ordering of the liquid-condensed domains.12 To model such effects, several authors have attempted to add chiral terms to a Landau free energy via interactions between the director and the dipole orientation.13,14 However, in these models the resulting pinwheel-shaped domains do not resemble actual observed DPPC domains. According to X-ray diffraction results, the phase traditionally called the LE or fluid phase corresponds to a two-dimensional fluid, with disordered hydrocarbon chains, while the LC, or gel phase, has longer distance positional correlations, and long-range bond orientational order possibly corresponding to a two-dimensional hexatic phase.15,16 The “solid” phase likely corresponds to a twodimensional crystal, with longer range positional order and ordered hydrocarbon chains. One might expect from this identification that chiral defects in the molecular orientational order parameters might be the underlying cause of the observed complex domain shapes. Indeed, fluorescence microscopy has shown that chiral defects can exist in Langmuir monolayers.17 However, earlier polarized light microscopy of DPPC monolayers has shown that DPPC LC domains do not exhibit similar internal defect structures.18,19 In this paper, we explore these issues by studying the effect on domain shape of varying in-plane dipole density by examining monolayers composed of different mixtures of the zwitterionic DPPC and the anionic phospholipid 1,2-dipalmitoyl-sn-glycero-3-[phospho-rac-(1-glycerol)] (dipalmitoylphosphatidylglycerol or DPPG) at high and low electrolyte concentration. We present a quantitative analysis of these data which shows that while simple theories for geometrical measures of domain complexity match our experimental results quite well, the detailed shapes are poorly predicted by existing theories. In addition, our work at high electrolyte concentrations shows that complex domain shapes persist even in the presence of strong electrostatic screening. We have also tested the prediction of the linear stability analysis that circular domain shapes should give way to a well-defined bilobe transition by computing the Fourier spectrum of our LC domains as a function of domain size and in-plane molecular density. To explain our results, we propose a novel chiral interaction term which mimics the effect of the actual stereochemistry of the DPPC enantiomer; we present results from an Eden growth model which includes this interaction. (12) Selinger, J. V.; Selinger, R. L. B. Phys. Rev. E 1995, 51, R860863. (13) Pikin, S. A. Physica A 1992, 191, 139-142. (14) Kam, R.; Levine, H. Phys. Rev. E 1996, 54, 2797-2801. (15) Kjaer, K.; Als-Nielsen, J.; Helm, C. A.; Laxhuber, L. A.; Mohwald, H. Phys. Rev. Lett. 1987, 58, 2224-2227. (16) Helm, C. A.; Mohwald, H.; Kjaer, K.; Als-Nielsen, J. Biophys. J. 1987, 52, 381-390. (17) Qiu, X.; Ruiz-Garcia, J.; Stine, K. J.; Knobler, C. M.; Selinger, J. V. Phys. Rev. Lett. 1991. 67, 703-706. (18) Moy, V. T.; Keller, D. J.; McConnell, H. M. J. Phys. Chem. 1986, 90, 3198-3202. (19) Moy, V. T.; Keller, D. J.; McConnell, H. M. J. Phys. Chem. 1988, 92, 5233-5238.

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Materials and Methods All phospholipid samples were purchased from Avanti Polar Lipids (Alabaster, Alabama) and used without further purification, except for D-R-phosphatidylcholine, dipalmitoyl (D-DPPC), which was purchased in powder form from Sigma (St. Louis, MO). The fluorescent probe used was the acyl-chain labeled 2-(12(7-nitrobenz-2-oxa-1,3-diazol-4-yl)amino)dodecanoyl-1-hexadecanoyl-sn-glycero-3-phosphocholine (NBD-PC). (Avanti Polar Lipids, Alabaster, AL). The NBD-PC probe is labeled on the acyl chain and partitions unequally between the gel (LC) and fluid (LE) phases, accumulating preferentially in the latter phase. The phase behaviors and domain shapes described here were independent of the specific probe used. Probe concentrations of 0.5 mol % were used to label the lipid samples. Studies of isotherms for various probe concentrations were performed to establish that this probe concentration has negligible effect on the monolayer behavior. All solvents used in the experiment were HPLC grade, purchased from either Aldrich or Fisher Scientific. Ultrapure Milli-Q water (Millipore Corp., Bedford, MA) was used to prepared the subphase. Subphases used consisted of either ultrapure water at pH 6.0 or 0.15 M NaCl titrated to 7.0 pH with HCl and NaOH. All chemicals used to prepare the subphase were highest grade ACS reagents. All samples were spread from a 2:1 v/v chloroform/methanol solution at lipid concentrations of 1 mg/mL. All phospholipid and lipid compositions are given as w/w ratios. The Langmuir trough used was a KSV Instruments (Riverside, CT) Minitrough, modified to mount on a Nikon Labophot 2A epifluorescence microscope. The trough itself was Teflon, and area compression was achieved using two hydrophilic Delrin barriers. The trough area was 7.5 cm by 25.5 cm, giving a surface area of 190 cm2. The subphase volume was 0.10 L. Surface pressure measurements were made using a KSV Wilhelmy electrobalance with a platinum plate. To eliminate air convection and dust, the entire microscope was enclosed in a sealed Plexiglas box. The trough itself was enclosed in another box covered with thermofoil heaters, which enabled the regulation of the temperature of the air above the monolayers. The temperature of the trough and its subphase was controlled using a water bath which circulated water through a heating block on the trough’s base. To avoid condensation, the microscope objective was also heated to just below the ambient temperature of the trough; this had the added advantage of minimizing domain drift. Calibrated Teflon-coated thermistors were used to measure the temperature of the subphase and air. Temperatures were controlled to (0.1 °C over the course of an isotherm, and subphase and air temperatures were carefully equilibrated before each data-taking run. Clean water isotherms were performed before the monolayer studies began, yielding reproducibly low surface pressure increases of 0.1 dyn/cm or less on a 10:1 area compression. Both the trough and Wilhelmy plate were cleaned and extensively rinsed with ultrapure water between sample runs with differing compositions. After spreading, films were allowed to equilibrate for 10 min before compression began. Compression times per isotherm were variable, ranging from 30 min of continuous compressions to 2 h of stepwise compressions for microscopy runs; either approach used compression rates which corresponded to 0.13 Å2/(molecule/s) for our trough area. A longer 6-h scan was performed for the L-DPPC 25 °C data to ensure that no changes in domain evolution resulted from equilibration at longer compression times; as a further check, upon reversing the direction of the barriers to expand the monolayers, we observed the LC domain shape evolution to reverse itself for L-DPPC at 25 °C. Isotherms corresponding to stepwise compression during fluorescence microscopy runs were in good agreement with those taken in one shorter, continuous compression. Imaging was performed both during compression and while the barriers were paused at fixed area; little evolution of domain shapes was seen during the pauses. Surface pressure measurements could be made simultaneously with the imaging, and the surface pressures changed by less than 10% during a typical pause for microscopy between compression steps. For microscopy, approximate maximum and minimum molecular areas were roughly 115 and 50

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Figure 1. Typical surface pressure/molecular area isotherm for Langmuir monolayers of DPPC with 0.5 mol % NBD-PC probe at 25 °C on ultrapure water, including edge tracings of characteristic liquid condensed domain shapes at the molecular areas and surface pressures indicated.

Å2/molecule. For 7:3 mixtures of DPPC:DPPG, reproducible isotherms were observed for three successive cycles of compression and expansion covering the entire LE-LC coexistence region. For the epifluorescence measurements, samples were illuminated with a 100-W mercury lamp source filtered through a dichroic mirror/filter combination (Omega Optical). No photobleaching occurred because of a slight drift in domain position during the course of a typical measurement. A long working distance 40× objective and a 5× projection lens combination was used to give a field-of-view of 105 µm by 135 µm. Images were collected using a Quantex QC-100 image-intensified camera, and stored on VHS videotape. For quantitative analysis, individual images were digitized using a PCVisionPlus framegrabber from Imaging Technologies in a Pentium personal computer. The resulting 8 bit 480 by 640 pixel images were analyzed using the MOCHA image processing package (Jandel Scientific, California). The simpler calculations and edge tracking were all performed within MOCHA. More complex calculations were performed on spreadsheets into which the edge tracking data had been saved. All Fourier transforms were performed with FFT subroutines.20

Experimental Results A representative isotherm and sample LC domain images for DPPC monolayers on ultrapure water at pH 6.0 with no additional buffer or salt are shown in Figure 1. The fluorescence images pictured in Figure 1 as edge tracings were collected across the LC-LE coexistence region at fixed temperature. For each data set, a uniformly bright background, corresponding to uniformly distributed fluorescent probe, was observed before the beginning of the LC/LE coexistence region. The exclusion of probes from the LC phases made their growing domain shapes visible throughout the coexistence region. For each temperature, small, circular nuclei were observed soon after the kink in compressibility that indicates the onset of the coexistence region. These initially circular domains evolve to heart shapes and then develop additional protrusions. These finger into chiral arms, which further develop chiral branching. For each monolayer, images of multiple domains were collected at a variety of points along the LC/LE coexistence region. These images were digitized and analyzed to compute the domain perimeter (P) and domain area (A). (20) Press: W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T. Numerical Recipes in C: The Art of Scientific Computing; Cambridge University Press: Cambridge, U.K., 1992.

Figure 2. Surface pressure/molecular area isotherm for Langmuir monolayers of DPPC with 0.5 mol % NBD-PC probe at 25 °C in 0.15 M NaCl, pH 7.0, including sample epifluorescence images of characteristic liquid condensed domain shapes at various values of molecular area: (a) 71 Å2/molecule, 8.88 dyn/cm; (b) 66 Å2/molecule, 9.98 dyn/cm; (c) 61 Å2/molecule, 11.1 dyn/cm; (d) 58 Å2/molecule, 11.8 dyn/cm; (e) 54 Å2/molecule, 14.0 dyn/cm; (f) 51 Å2/molecule, 16.6 dyn/cm; (g) isotherm.

These quantities were then used to compute the shape factor, S, defined as

S ) P2/4πA

(2)

The values of this geometrical measure of compactness range from a low of 1 for a perfectly circular shape to higher values for less compact shapes. Thus the shape factor provides a way to characterize the evolution of domains from more to less compact geometries.21

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Figure 3. Dependence of shape factor on molecular area for (a) D-DPPC (O) and L-DPPC (0) at 25 °C on ultrapure water subphase; (b) L-DPPC at 21 °C on ultrapure water subphase; and (c) L-DPPC (O) and mixtures of 7:3 L-DPPC/DPPG (0) at 25 °C on 0.10 M NaCl, pH 7.0 subphases. (The relatively compact domain formation for (c) compared to (a) and (b) is evident in both plots.) (d) Shape factor as a function of LC domain area for D-DPPC (O) and L-DPPC (0) at 25 °C on ultrapure water subphase.

Similar results were seen for DPPC monolayers at temperatures ranging from approximately 18 to 28 °C. We will discuss here results from detailed analyses of data taken at 21 and 25 °C. We performed measurements for two different subphases: ultrapure water at pH 6.0 with no additional buffer or salt, and 0.15 M NaCl, unbuffered subphase, titrated to pH 7.0 with NaOH and HCl. Very different results were found for these two cases. For the ultrapure water subphase (Figure 1), very complex domain shapes resulted upon compression, as domains grew to their largest sizes. The average shape factor is constant throughout the beginning of the coexistence region (which has its onset at approximately 75 Å2 at 25 °C) but begins to increase with decreasing molecular area, Amol, once the surface pressure increases at the end of this region (Figure 3a). Similar results were seen for 21 °C samples on ultrapure water, although there were occasional variations in the usual monotonic increase of shape factor due to twinning of domains (Figure 3b). This increase is reproducible, and agrees for L- and D-DPPC (21) Stine, K. J.; Stratmann, D. T. Langmuir 1992, 8, 2509-2514.

monolayers on ultrapure water (Figure 3a). For the 0.15 M NaCl subphase, the DPPC monolayers grew relatively compactly, as shown by Figure 2. These general impressions can be made more quantitative in a plot of shape factor as a function of molecular area. While the shape factor of LC domains grown on ultrapure water achieves very high values (Figure 3a,b), values for the 0.15 M NaCl subphase remain consistent with approximately circular domains (Figure 3c). This agrees with calculations on screening at high electrolyte concentration using the Gouy-Chapman model to predict complete charge screening by monovalent counterions for >0.1 M electrolyte solutions;22 this would lead to a prediction of much more compact domain growth throughout the coexistence region for the 0.15 M NaCl subphase samples compared to the samples grown on ultrapure water subphase, as observed. The ultrapure water subphase yielded a minimum shape factor of 1.03 ( 0.05, while the monolayers on 0.15 M NaCl had slightly more oblong shapes with a shape factor (22) Helm, C. A.; Laxhuber, L.; Losche, M.; Mohwald, H. Colloid Polym. Sci. 1986, 264, 46-55.

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Figure 4. (a) Shape factor and (b) LC domain area vs molecular area for L-DPPC (O), mixtures of 7:3 L-DPPC/DPPG (0) and mixtures of 7:3 L-DPPC: DPPG (4). The increase in shape factor for the mixed phospholipid samples was primarily due to domain twinning.

of 1.5 ( 0.08. We also analyzed the dependence of shape factor on LC domain area, which was expected to be a well-defined function for constant values of Γ. However, our results yielded nonmonotonic and irreproducible curves for different samples (Figure 3d), with several distinct points at which a transition from circular to less compact shapes occurred for varying films. Mixtures of L-DPPC with various concentrations of DPPG were also studied on 0.15 M NaCl, pH 7.0 subphases at 25 °C. The basic domain shapes were unaltered, but the shape factors computed for these mixed phospholipid films were dominated by the increase in domain size caused by the more extensive twinning of domains observed for these mixed monolayers, as evident in the increased domain areas (Figure 4).19 In Figure 5, we see that domain shapes are unchanged when DPPG is introduced at mol/mol concentrations of 3:7 with respect to DPPC. This is in accordance with earlier studies of these systems, where no change in domain shapes over a wider range of DPPG concentrations was observed.23 Since DPPG is an anionic phospholipid which cannot

Figure 5. Surface pressure/molecular area isotherm for Langmuir monolayers of mixtures of 7:3 L-DPPC/DPPG with 0.5 mol % NBD-PC probe at 25 °C in 0.15 M NaCl, pH 7.0, including sample epifluorescence images of characteristic liquid condensed domain shapes at various values of molecular area: (a) 65 Å2/molecule, 10.5 dyn/cm; (b) 63 Å2/molecule, 11.0 dyn/ cm; (c) 60 Å2/molecule, 11.5 dyn/cm; (d) 55 Å2/molecule, 12.6 dyn/cm; (e) 53 Å2/molecule, 13.4 dyn/cm; (f) isotherm.

participate in the same dipole-dipole interactions as DPPC, this lends force to the argument that the dominant forces driving domain growth in this system in the high electrolyte concentration case must be stereochemical. (23) Nag, K.; Rich, N. H.; Keough, K. M. W. Thin Solid Films 1994, 244, 841-844.

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Fourier Analysis of Liquid Condensed Domain Shapes To enable a quantitative analysis of the evolution of LC domain shapes, edge tracings of the LC domains were obtained for images taken of 17 typical domains for L-DPPC monolayers at 25 °C on ultrapure water subphases at pH 6.0 (Figure 1). The edge-tracing files were used to compute the center-of-mass of each domain and the radius, R, at each point on the perimeter was defined as the distance from the edge to the center-of-mass. Each edge-tracing image was scaled to have unit area, and then converted into a file containing values of the edge’s radial distance from the center-of-mass, R, as a function of angle, θ. The resulting function, R(θ), was subjected to Fourier analysis to determine the spectra content from each n-fold angular term, as described in ref 7 (Figure 6 inset). On the basis of the theoretical work, our expectations were that the domains would be approximately circular for small LC domain areas and large molecular areas and that their power spectra would consequently be dominated by the 0-fold circular terms in the Fourier series.7 At larger values of domain area (smaller molecular area), the theory predicts well-defined transitions to domain shapes with a significant bilobe (2-fold) component.9 However, our results yielded Fourier power spectra with very different behavior from the theory (Figure 6). In fact, for every range of LC domain area and molecular area studied, we did not see appreciable trends in the Fourier spectral content of our domains. The reason for this is evident from a careful inspection of the actual domain shapes, as shown in Figure 1. There, it is clear that the cusps and other sharp contributions which give the domains their peculiar shapes must derive from higherorder Fourier modes, and these contributions come into play early in the coexistence region. These higher-order Fourier modes do not prevent our observing any significant trends toward circular to bilobe, trilobe, etc. transitions, since each mode’s contribution to the domain edge tracing is independently determined in the Fourier analysis. Thus, we not only did not see any clear-cut transition from purely circular to n-fold order but we also did not see evidence for any region in which the domains could usefully be described as small perturbations about an approximately circular form. Comparison of Liquid Condensed Domain Shapes with Theory We have used two simple models to compare our data with results from Monte Carlo simulations of domain shapes, assuming the dominant energetic interactions are due to line tension and electric dipole-dipole interactions.11 The results from Figures 3 and 4 of this reference were used to compute values of shape factor vs the dimensionless parameter Γ. To compare the simulations results with our shape factor vs molecular area curves, we need to assume a relationship between Γ and molecular area, Amol. At the beginning of the LC-LE coexistence region (which has its onset at approximately 75 Å2 and ends at approximately 48 Å2 for our data) the value of the in-plane dipole density should remain unchanged, and the shape factor remains unchanged in accordance with expectations. At smaller molecular area (higher in-plane densities), the transition is virtually complete, and the molecules undergo compression, leading to changes in the molecular density and hence Γ. We assumed two different trial relationships:

Figure 6. Results of Fourier analysis of edge tracings of the LC domain shapes for L-DPPC samples in ultrapure water at 25 °C. Contributions from the FFT moduli for 0-fold (O), 2-fold (0), 3-fold (4), and 4-fold (3) terms are plotted vs (a) LC domain area and (b) molecular area. No strong dependence on either molecular area or LC domain area is shown.

Model 1. No Molecular Tilt Dependence. This model assumes that the in-plane dipole moment density, µ, is simply proportional to in-plane molecular density

µ ) µ o/Amol

(3)

where µo ) dipole moment of a single DPPC and Amol is its molecular area. If we assume that line tension, λ, is independent of Amol, we have

Γ ) µ o2/(λAmol2) ∝ 1/Amol2

(4)

Model 2. Molecular Tilt Dependence. Alternatively, if the molecular tilt angle, θ, decreases during compression,

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assumed model 1, for which Amol ∝ 1/Γ1/2 (no change in tilt angle). The break in slope of the shape factor vs 1/Γ1/2 curve occurred for Γ between 0.20 and 0.22; the simulations included six data points in the range of our data, including three data points on the steeply sloping part of the curve. The interesting values of Γ thus ranged from 0.20 to 0.274. We needed to scale 1/Γ1/2 by dividing it by an empirically determined factor to plot the experimental shape factor data as a function of molecular area, Amol, and the simulations results for shape factor as a function of 1/Γ1/2 on one graph. Model 1 gives this factor as xλ/µo (see eq 4). The best agreement between the experimental data and simulations results was obtained for

xλ/µo ) 0.0344 ( 0.0015/Å2 (90% CI)

Figure 7. (a) Illustration of the dependence of in-plane molecular area (Amol) on molecular tilt angle, θ. (b) Comparison of our LC domain image analysis results for shape factor vs molecular areas (experimental data) or with simulations results from ref 11 plotted as shape factor vs 1/Γ1/2 scaled by an appropriate factor, as described in the text. The data shown are the same as those included in Figure 3a: D-DPPC (O) and L-DPPC (0) at 25 °C on ultrapure water subphase, while the results from simulations as shown as a straight line and (b).

Γ could also depend on Amol. One way this could happen would be to assume that the dipole-dipole interactions are due to µ||, the component of the molecular dipole moment normal to the layers (Figure 7a)

µ|| ) µo cos θ

(5a)

Amol ) Ao/cos θ

(5b)

where Ao is the molecule’s cross-sectional area when θ ) 0. Thus, if the tilt angle changes

Γ ) µ||2/(λA2) ) µo2 cos θ2/(λAmol2)

(6a)

Γ ∝ 1/Amol4

(6b)

We can test these predictions by plotting the experimental shape factor vs molecular area, Amol, results from Figure 3a and the simulations results for the shape factor as a function of 1/Γ1/2 or 1/Γ1/4. Agreement between the experimental data and simulation results was poor for the 1/Γ1/4 plots. The results of plotting the experimental data and the simulations results for shape factor vs 1/Γ1/2 are shown in Figure 7b. This means that a comparison between L- and D-DPPC data in ultrapure water at 25 °C and Monte Carlo simulations showed good agreement when we

(7)

This value was primarily determined by the location of the sharp break in slope of the shape factor vs 1/Γ1/2 curve for Γ between 0.20 and 0.22. We estimate that the line tension, λ, should be equal to approximately 1.4 × 10-12 J/cm, based on two arguments: (1) The average van der Waals interaction energy in vacuo between two methyl groups at a distance of 0.4 nm and infinity is equal to 0.6 kJ/mol, allowing an estimation of the interaction between DPPC molecules with their two 14-carbon alkane chains.24 An estimate for line tension results from taking the ratio between the estimated interaction energy between two DPPC molecules and dividing by effective radius, here 0.4 nm/2: λ = 2 × 14 × 0.6 (kJ/mol)/0.2 nm ) 1.4 × 10-12 J/cm). (2) Values derived from lattice simulations models for energy differences between two 14-chain zwitterionic phospholipids at Amol ) 50 and 75 Å2/mol.25 These values give an estimated coordination energy of 0.32 eV for every molecule added to the perimeter of a growing domain; since each molecule adds 2Rmol (the molecular radius, equal to approximately 3.6 Å for our system) to the perimeter and adds two coordination energies, this results in the stated value for the line tension (λ = 0.32 eV/ 3.6 Å ) 1.4 × 10-12 J/cm). This value for line tension and our result for xλ/µo gives µo = 10 D, in reasonable agreement with the upper limit of published values of 0.5-15 D.26,27 While the overall agreement between the experimental data and the theoretical model is quite good, inspection of the domain shapes predicted for higher shape factors reveals that the actual DPPC LC domain shapes disagree significantly with the simple labyrinthine of shapes predicted for this limit. To address this discrepancy, we evolved a new model to include the notable chiral features of these more complex domain shapes. Molecular Simulations Results To improve upon the success of existing models of phospholipid domain growth, we wished to include a local interaction which mimics the effect of the actual chiral stereochemistry of DPPC, while still including the effects of line tension, dipole interactions, and temperature. To this purpose, we performed simulations of domain growth adding a novel interaction which simulates chiral stereochemistry. We used the Eden growth rule to add walkers onto a growing cluster because of the high concentration of phospholipids at the perimeter of the actual experimental clusters; in such a situation, diffusion is not a limiting factor. (24) McCammon, J. A.; Harvey, S. C. Dynamics of proteins and nucleic acids; Cambridge University Press: Cambridge, U.K., 1987; p 16. (25) Tobochnik, J.; Gould, H. Comput. Phys. 1996, 10, 542-547. (26) Bowen, P. J.; Lewis, T. J. Thin Solid Films 1983, 99, 157-163. (27) Brockman, H. Chemistry Phys. Lipids 1994, 73, 57-79.

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Our lattice was a 100 × 100 triangular lattice with characteristic dimensions corresponding to a step size of 7.1 Å. Our maximum domain size of 2200 occupied sites (332 Å per side) was much smaller than those actually observed experimentally. Clusters evolved from a hexagonal seed approximately 200 walkers in size. A Metropolis algorithm was used to decide on the addition of a walker to the growing domain; a value of kBT ) 0.026 eV was used to accord with the experimental temperatures used. The algorithm used to add particles to the perimeter was as follows: (1) A site on the perimeter was chosen randomly. (2) The energy, E, of adding a particle at that site was computed as described below. (3) A logistical function

g(E) )

1 1 + eE/kBT

Figure 8. Geometry for describing the chiral interaction energy. The vectors are described in the text. Previously occupied sites are shown as open circles, while new sites are shown as light gray circles. (a) Additional walker added to a nonchiral site. (b) Two sites at which chiral terms to the energy would occur, with the energy contributions having the signs indicated.

was used to compute the probability, g(E), of a particle being added or not to the perimeter site chosen. After each growth step, we implemented a subroutine which permitted diffusion within the cluster at a rate of 0.1 diffusion per occupied site per growth step (tstep ) 5 × 10-7 s); this diffusion used the same Metropolis algorithm as did the growth rule, but with the constraint that coordination remains constant or increases. To reduce the perimeter roughness, a final diffusion of 1 diffusion step per occupied sites (1 ms) was done after the maximum domain size had been achieved. Our energy function was the following:

Etotal ) -EnnNi,nn +

∑ j)i

M2 |r bi - b r j|3

+C

rˆ i‚(rˆ nn × zˆ ) ∑ nn (8)

The first term represents the effective line tension interaction term, the second the usual dipole-dipole interactions throughout the cluster, and the third the chiral interaction between nearest-neighbors. In computing the line tension contributions, we assigned each walker an energy proportional to the number of nearest neighbors, Ni,nn, for the ith site, and assumed an interaction energy equal to the van der Waals interactions between DPPC molecules, given the two 14-carbon alkane chains of two DPPC molecules, on the order of Enn ) 0.32 eV.25 The dipole-dipole interaction term has the usual form, with rˆ i ) a unit vector connecting the new site to a nearest neighbor, and parameters for the molecular dipole moments, M, chosen as 0.25 eV g M g 0.30 eV, and Γ ) M2/ Enn; 0.28 g Γ g 0.20. These parameters correspond to our best guesses for matches to our experimental situation, although we are limited by computational time in going to realistic domain areas and equilibration times. The third term is the chiral interaction energy; its effect can best be understood by examining Figure 8. There, two possible sites for adding a new walker are shown relative to an established interface. The chiral energy term has opposite signs depending upon the “handedness” of the added walker, established by computing the average of the dot-product between rˆ i with the cross-product between zˆ , the normal to the plane of the growing domain, and rˆ nn, the local normal to the perimeter. Figure 8a shows these vectors for one possible new walker site, labeled “+ C”, which would yield a contribution to the energy equal to + C; the site labeled “- C” (Figure 8b) would add - C to the energy because of its opposite “handedness”. Inspection of the resulting structures shows that the interface indeed would have a leftward (rightward) curve upon the addition

Figure 9. (a) Results from the Eden growth model with no dipole-dipole interactions, shown as a plot of shape factor vs chirality parameter, C. Sample domain shapes are shown corresponding to the chirality parameters indicated. (b) Dependence of shape factor on diffusion rate, for C ) 1.0, Enn ) 0.32, and T ) 0.025.

of a walker at one of these sites. By this means, the orientation of this cross-product (parallel or antiparallel

Growth of Chiral Domains

to the average orientation) establishes an effective measure of local “chirality” of the site relative to an existing interface. The sign of the chiral interaction parameter, C, determines which “handedness” results in a negative or positive contribution to the energy, and the magnitude of C determines the strength of the chiral interaction energy relative to the line tension and dipolar contributions. Figure 9a shows the dependence of shape factor and domain shape on C for the case of no dipole-dipole interactions (Γ ) 0). For high values of C, domains with very rough perimeters are seen. For intermediate values of C, deviations from compact domain growth are seen. As C goes to zero, the shape factor does not go to exactly 1 because of two factors: (1) there is a remaining roughness due to the statistical nature of the deposition process and subsequent diffusion, and (2) the underlying triangular lattice results in a predisposition toward hexagonal domains, for which the shape factor = 1.1. There is little overall obvious “handedness” of the resulting domains, although domain shapes do typically evolve larger scale finger-like protrusions as shown. Figure 9b shows the dependence of shape factor on diffusion rate for C ) 1.0, Γ ) 0. (A diffusion rate of 0.1 represents the value used during growth for the results shown in Figure 9a; the final diffusion step was done at the maximum diffusion rate shown here.) Inspection of domains grown at varying diffusion rates shows the main effect of more frequent diffusion is to smooth out small-scale roughness. Conclusions We have successfully compared our data to Monte Carlo simulation results which use a model of phospholipid monolayer domain shape; comparisons with our data yielded a value of xλ/µo2 ) 0.0344 ( 0.0015/Å2. This corresponds to a dipole moment for DPPC of ∼10 D, in good agreement with surface potential measurements. The data and theory agree if the molecular tilt angle does not alter the dipole moment. Contrary to predictions from linear stability analyses, Fourier transforms of ap-

Langmuir, Vol. 16, No. 22, 2000 8455

proximately circular domain shapes do not show clear transitions between circular and higher order Fourier modes. This is because of significant singularities in the curvature as a function of angle, which introduce higherorder Fourier modes early on the in the growth process. Our analysis shows that the Fourier amplitudes for higher order modes show no discernible trends when plotted as a function of LC domain area or molecular area. Our simulations results show that chiral interactions do influence the value of the shape factor for low values of C. Hence, we would expect even samples in which electric dipole-dipole interactions are screened out appreciably to exhibit complex geometries, consistent with our data. We have not been able to reproduce the complex domain shapes seen in actual phospholipid LC domains in detail, but we think the inclusion of a variable dipole orientation in-plane may result in closer agreement. We also are interested in developing a new theoretical measure of chirality, since there is no good current measure of how “handed” a particular cluster is. Existing measures cannot distinguish between merely labyrinthine shapes and visibly chiral ones.28 Acknowledgment. This work was supported in part by National Science Foundation Grant No. NSF-DMB9109460, by a William and Floral Hewlett Foundation Award from the Research Corporation, and through a grant from the Undergraduate Biological Sciences Education Initiative of the Howard Hughes Medical Institute. Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for partial support of this research. We wish to express our appreciation to Lyle Roelofs for his assistance in performing the simulations. We have also benefited greatly from conversations with Tom Lubensky, Randy Kamien, and T. Kyle Vanderlick. LA000902D (28) Gilat, G. J. Phys. A: Math. Gen. 1989, 22, L545-550.