Langmuir 1991, 7, 1031-1034
Evaluation of Models for Surface Pressure-Area Behavior of Liquid-Expanded Monolayers J. M. Smaby and H. L. Brockman' The Hormel Institute, University of Minnesota, Austin, Minnesota 55912 Received September 4, 1990. In Final Form: October 25,1990
Introduction Monolayers of insoluble lipids at gas/liquid interfaces serve as models for biological lipid-water interfaces and are the basis of Langmuir-Blodgett film technology. The primary characterization of such films usually involves the measurement of surface pressure ( A ) and/or surface potential (AV) as a function of lipid molecular area ( A ) and film composition. These data provide useful information concerning the packing of lipid molecules, their orientations, their miscibilities, and specific interactions among species. Most acyl lipids found in living systems exist in a chain-melted, i.e., liquid-crystalline, state at physiological temperature. This state is most closely modeled in monolayers by the liquid-expanded state.'V2 The usefulness of A-A-AVdata obtained with such lipids for understanding the properties of biological interfaces and the regulation of biological processes depends, ultimately, on the degree to which the properties of monolayers of single lipid species can be defined. This arises because models to predict the behavior of complex mixtures are often phenomenological; i.e., they rely on measured properties of monolayers of the pure substances rather than first principles. Various thermodynamic models have been proposed to explain the *-A behavior of monolayers of pure lipids in the liquid-expanded state (e.g., ref 3). The equations of state fall into two major classes depending on whether water is considered a component of the surface phase, i.e., on the location of the dividing surface between the monolayer and bulk phases. Generally, these models enable a *-A isotherm to be reduced to two or three parameters. The models are imperfect in that they are admittedly approximations or rely on simplifying assumptions to limit the number of parameters. Nevertheless, if they are able to accurately regenerate the isotherm from which they were obtained, these models are of utility for predicting a from A and can serve as a basis for predicting the A-A properties of mixed films. In addition, the fitting parameters obtained may have physical significance in the context of the model itself, e.g., cross-sectional areas of lipid molecules. The existence of multiple, mathematically nonidentical equations of state to describe liquid-expanded monolayer behavior raises the questions (a) which best describes the data and (b) to what range of a-A data should they be applied? The latter question addresses not only what constitutes liquid-expanded behavior but over what range is the data reasonably unaffected by lipid-dependent experimental problems such as dissolution or metastability near the onset of a phase transition. In the past, investigators often chose a model and used it to analyze data over an arbitrarily defined or limited range of a or A values (e.g., refs 4 and 5). A more quantitative approach to the question was suggested by a recent study of the (1) MacDonald, R. C.; Simon, S. A. R o c . Natl. Acad. Sci. U.S.A. 1987, 84, 4089. (2) Ipeen, J. H.; Mouritaen, 0. G. J. Chem. Phys. 1989,91, 1855. (3) Gaines, G. L., Jr. Insoluble Monolayers at Liquid-Gas Interfaces; John Wiley and Sons: New York, 1966; pp 166-188. (4) Menger, F. M.; Wood, M. G., Jr.; Richardson, S.; Zhou, Q.; Elrington, A. R.; Sherrod, M. J. J. Am. Chem. Soc. 1988,110,6797.
1031
AV-A behavior of a variety of lipid classes and species which give a-A behavior that is qualitatively of the liquidexpanded typeB6From A = 1mN/m to the surface pressure at which d2r/dA2fell below zero ( r d ) , most of these lipids showed good linearity of AV versus 1/A. This indicates a lack of significant dipole reorientation over this range of film compression. The range of l - A d encompasses 8090% of all data between A = 1mN/m and the onset of any phase transition or film collapse. Thus, it constitutes a reasonable definition of the range of liquid-expanded behavior and can be used to select a-A data for subsequent analysis. The availability of such criteria prompted us to compare the ability of representative equations to describe the liquid-expanded state for a variety of lipid species. The results show lipid-independent differences in the ability of the models to describe the data mathematically. Moreover, they reveal redundancies between pairs of models and between pairs of parameters for given models. Materials and Methods N-l1,14-Eicosadienoylgalactosyl sphingosinewas synthesized according to the method of Kishimoto' using psychosine instead of sphingosine. The P A isotherm for this lipid was collected by using an automated Langmuir film balance following the same procedure used earlier to obtain the other P A isothermsutilized in this study.6 The equations were fitted by using a Levenberg-Marquardt least-squaresalgorithm? Curve fitting calculationswere carried out to 15 significant figures. Convergence was defined as occurring when (a) two successivesummations of the squares of the residuals or (b) all parameters of two successive iterations agreed to seven significant figures. Results For comparison of models to describe A-A behavior in the liquid-expanded state, 18lipids were chosen. As shown in Table I there are 14 chemical classes with respect to their polar moieties. They include neutral, zwitterionic, and anionic species with one to three apolar substituents linked by alkyl or acyl groups. The latter are saturated, cis unsaturated, and methyl branched. Each of these lipids exhibits r-A behavior qualitatively of the liquidexpanded type and only three show discernible transitions to a more condensed surface phase at a values below collapse (data not shown). Qualitatively,liquid-expanded behavior is characterized by an abrupt, rather than gradual, increase in A from values near zero. As A is decreased, this 'lift off" occurs at an area 2-3 times the molecular cross-sectional area.3 For the lipids studied, this increase occurred from a values of 0.1-0.4 mN/m at areas per aliphatic chain of 43-60 A2. Moreover, for each, plots of AVversus 1/A show good linearity, Le., r 20.999 between A = 1 and A = A,+ The equations of state with which the a-A data were compared are given in Table 11. The origins and assumptions behind each can be found through the references in the table. With the exception of eq 6 (see below), these equations are frequently given in monographs or have been reported recently. In general, models 1-3 are variations on the two-dimensional, ideal gas equation, aA = kT,and do not consider water as a monolayer component, whereas models 4-6 are based on the osmotic model of surface pressure. Because a is usually determined as a function of A, the equations are written in the form a = f ( A ) .All contain the constants k and T,where k is the Boltzmann (5) Momsen, W. E.; Smaby, J. M.; Brockman, H. L. J . Biol. Chem. 1979.254. _ _ ., .-- -,8855.
(6) Smaby, J. M.; Brockman, H. L.Biophys. J. 1990, 58, 195. (7) Kishimoto, Y. Chem. Phys. Liprds 1975, 16, 33. (8) Brown, K. M.; Dennis, J. E. Numer. Math. 1972, 18, 289.
0743-7463/91/2407-lO31$02.50/0 0 1991 American Chemical Society
1032 Langmuir, Vol. 7,No. 5, 1991
Notes
Table I. Statistical ComDarison of Models for Surface Pressure-Area Isotherms subphase: standard deviation of r, mN/m largest absolute r residual, mN/m number of 1 2 3 lipida temp(OC) 1 2 3 4 5 6 1 2 3 4 5 6 0.18 0.12 0.02 0.18 0.02 0.03 0.42 0.35 0.06 0.43 0.06 0.10 4 3 47 A, 24 trioleoylglycerol 2.05 1.41 0.34 2.22 0.19 0.31 4.91 3.21 1.00 5.31 0.63 1.09 4 5 11 A, 24 l,2-diphytanoylGPC 1.82 1.64 0.21 2.00 0.08 0.09 4.13 3.95 0.63 4.59 0.31 0.37 2 3 15 1-palmitoyl-2-oleoylGPC A, 15 1.55 1.50 0.22 1.70 0.10 0.12 3.82 3.97 0.65 4.24 0.35 0.39 2 3 15 1-palmitoyl-2-oleoylGPC B, 24 1.80 1.58 0.26 1.96 0.13 0.13 4.07 3.76 0.74 4.49 0.48 0.47 2 3 5 1-palmitoyl-2-oleoylGPC A, 24 2.19 1.85 0.27 2.37 0.12 0.14 4.75 4.20 0.77 5.20 0.37 0.51 2 3 3 1-palmitoyl-2-oleoylGPC C, 24 2.02 1.68 0.23 2.19 0.08 0.12 4.42 3.97 0.66 4.88 0.28 0.46 2 3 11 1-palmitoyl-2-oleoylGPC D, 24 1.97 1.64 0.24 2.14 0.10 0.14 4.36 3.87 0.67 4.81 0.32 0.51 2 3 9 1-palmitoyl-2-oleoylGPC E, 24 1.89 1.54 0.23 2.04 0.10 0.12 4.24 3.62 0.66 4.64 0.35 0.48 2 3 11 1-palmitoyl-2-oleoylGPC A, 30 2.01 1.97 0.26 2.17 0.13 0.24 4.44 3.10 0.76 4.80 0.42 0.75 2 3 7 1-palmitoyl-2-oleoylGPC A, 37 1.39 1.19 0.23 1.50 0.14 0.17 3.41 3.10 0.67 3.70 0.42 0.67 4 3 9 A, 24 1,2-dimyristoylGPC 1.62 1.36 0.23 1.76 0.13 0.13 4.61 3.62 0.95 5.02 0.65 0.67 2 5 13 A, 24 rac-1-0-alkyl-2-acylGPC 0.92 1.34 0.16 1.06 0.10 0.18 2.34 3.53 0.50 2.72 0.31 0.62 2 3 13 1-palmitoyl-2-oleoylGPE A, 24 A, 24 0.57 0.90 0.09 0.65 0.06 0.23 1.64 2.75 0.35 1.85 0.24 0.54 4 3 29 1-0-alkyl-2-acylGPE F, 24 2.06 1.51 0.30 2.22 0.16 0.18 4.91 3.48 0.92 5.30 0.59 0.72 2 3 9 1,2-dioleoylGPS 1.81 1.49 0.16 1.94 0.06 0.09 4.05 3.76 0.52 4.41 0.22 0.40 2 3 15 F, 24 bovine liver diacylGP1 2.43 1.66 0.34 2.60 0.19 0.20 5.50 3.59 1.16 5.91 0.78 0.88 2 3 11 G, 24 l,%dioleoylGPG 0.78 1.19 0.15 0.88 0.09 0.38 1.97 3.03 0.50 2.24 0.35 1.07 4 3 11 A, 24 1-palmitoyl-2-oleoylGP A, 24 1.55 1.21 0.34 1.72 0.24 0.42 4.09 2.91 1.00 4.52 0.71 1.37 6 3 13 N-11.14-eicosadienoylgalactosylSPH A, 24 0.69 0.81 0.09 0.76 0.05 0.10 1.60 2.21 0.27 1.78 0.15 0.36 2 3 19 1,2-dioleolylglycero1 A, 24 0.79 0.67 0.06 0.84 0.03 0.03 1.81 1.80 0.18 1.94 0.09 0.10 2 3 23 1,3-dioleoylglycerol A, 24 0.48 1.21 0.27 0.64 0.26 0.26 1.49 2.58 0.77 1.85 0.76 0.75 2 3 9 oleic acid A, 24 0.21 0.99 0.06 0.13 0.09 0.09 0.74 2.57 0.22 0.54 0.33 0.34 16 3 33 . oleyl alcohol A, 24 oleyl nitrile 0.10 0.25 0.03 0.09 0.03 0.05 0.32 0.79 0.09 0.29 0.11 0.21 12 3 66 1.13 1.29 0.24 1.29 0.18 0.21 2.71 3.35 0.87 3.13 0.73 0.60 4 3 13 A, 24 N-oleylethanolamine ~~
1.36 1.28 0.21 1.48 0.11 0.16 3.23 3.08 0.62 3.54 0.40 0.58
averages
4 3 17
sign changes 4 5 6 6 51 26 2 19 14 2 31 33 4 41 31 2 13 13 2 15 15 2 47 23 2 17 27 2 13 21 2 17 14 2 19 17 2 33 39 2 33 28 6 41 20 2 13 15 2 47 73 2 19 15 2 19 4 4 17 6
2 25 10 4 129 125 2 5 5 21 17 17 16 55 78 2 17 13 4
30
27
Abbreviations: GPC, sn-glycero-3-phosphocholine; GPE, sn-glycero-3-phosphoethanolamine; GPS, sn-glycero-3-phosphoserine; GPI, snglycero-3-phosphoinositol;GPG, sn-glyero-3-phosphoglycerol; GP, sn-glycerol-3-phosphate;SPH, spingosine. Subphases: A, 0.01 M phosphate, 0.1 M NaCl, pH 6.6; B, H20; C, 0.01 M phosphate, 2 M NaCl, pH 6.6; D, 0.01 M PIPES, 0.2 M NaCL 0.1 mM EGTA, pH 7.4; E, 0.01 M PIPES, 0.1 M NaCl, 0.005 M MgC12, pH 7.4; F, 0.01 M phosphate, 0.1 NaCl, 0.1 mM EGTA, pH 6.6; G, 0.15 MM KC1, 0.2 MM EGTA, pH 7.0. a
*
Table 11. Models for Liquid-Expanded Monolayers
no. 1 2
equation K A
= KO + kT/(A - Ao) [A -A, & ((A - A d 2 +
adjustable parameters
Ao A,. n, K TO,
refs 3, 13 4
.
0.2
E
0.0
E
z
-e¶
9
8
-0.2
U
1
-0.4
constant and T is the absolute temperature. Models 4-6 also include 01, the cross-sectional area of an interfacial water molecule which is set at 9.65 A2.9 The remaining two or three terms are adjustable parameters which are listed in the table. It should be noted that one mathematical form may involve different interpretations of the fitting parameters, Le., see eq 1 references, and that a given adjustable parameter symbol, such as Ao, may have a unique interpretation in the context of each model. Using the T-A isotherm for each of the lipids given in Table I between the limits of ?r = 1 and T = ?rd, approximately 500 *-A data points were selected at equal increments of x. This data set was then fitted by using each of the six equations to obtain the best values of the adjustable parameters. For model 2,the fitting program was generally unable to converge prior to the occurrence of fatal errors involving the square root function. It was, therefore, used in the form A = A, + n k T / a - A,KT. To evaluate the degree to which the parameters describe the originaldata, statistical analysis of the residuals (*experimental - xcalcUlatedwas performed. Three criteria of evaluation were used; the standard deviation of T , the number of sign changes in the residuals as T increases monotonically, and (9)Fowkss, F. M.J . Phys. Chem. 1962, 66,385.
2.0
4.0
6.0
8.0
10.0
Experimental n , mN I m
Figure 1. Comparison of surface pressure residuals for triomodel 2 (A),model 3 leoylglycerol from fits using model l ,).( ( O ) , model 4 (o),and model 5 (m), as described in the text.
the largest absolute residual. The abilities of eqs 1-5 to describe the data are exemplified by the residual plots shown in Figures 1-3. Equation 6 is a variant of eq 5 and will be discussed separately below. The figures show results for trioleoylglycerol, a three-chain lipid that collapses at 11.9 mN/m and 101.3 A2/molecule, 1,2-dioleoyl-sn-glycero-3-phosphoserine, a two-chain phospholipid that collapses at 45.6 mN/m and 57.4A2/molecule, and oleyl alcohol, a one-chain lipid that collapses a t 32.1 mN/m and 28.5 A2/molecule.6 Thus, they represent a range of lipid types, collapse pressures, and molecular areas. For each lipid shown in Figures 1-3, the results differ greatly among models. Moreover, a pattern is apparent. Namely, pairs of models, i.e., 1 and 4 (filled circles and open boxes) and 3 and 5 (open circles and filled boxes), show nearly identical residuals as a function of the experimentally measured T . Like the figures, Table I shows differences in the ability of the equations to
Langmuir, Vol. 7, No.5, 1991 1033
Notes
t
Y 6.0
16.0
26.0
36.0
Experimental n , mN / m
Figure 2. Comparison of surface pressure residuals for 1,2-dioleoyl-sn-glycero-3-phosphoserine from fits using model 1 (01, model 2 (A), model 3 (01, model 4 (01, and model 5 (HI,as
n
described in the test. 2.4
. z
1.6
E
0.8
7 0
0.0
E
F
d
-0.8
4.0
10.0
16.0
22.0
Experimental n , mN / m
Figure 3. Comparison of surface pressure residuals for oleyl alcohol from fits using model 1 (a),model 2 (A),model 3 (O), model 4 (a), and model 5 (m), as described in the text.
accurately describe the data, as well as the patterns noted above. For each lipid the standard deviation of A differed more than an order of magnitude among the models. The lowest values, i.e., best fits, were obtained with model 5 (average = 0.11 mN/m) followed by model 3 (average = 0.21 mN/m). Model 4 gave the poorest fitswith an average standard deviation of 1.48 mN/m. The same trend is seen with the maximum absolute residuals, which averaged 0.40, 0.62, and 3.54 mN/m for models 5,3, and 4, respectively. The number of residual sign changes, a measure of the randomness of the residual values, generally shows the largest values with model 5 (average = 301, followed by model 3 (average = 17). Model 4 averaged only four sign changes. Comparison of all statistics shows that only with 3 of the 18 lipids was any fitting criterion better than that obtained with model 5. Thus, from the viewpoint of predicting H from A for liquid-expanded films, model 5 is best suited. It was expected that the three-parameter models should out-perform those withonly two. Although generally true, model 2 gave rather poor fits of the data which were comparable to the two-parameter forms, models 1and 4. This may be the reason it has only been applied to data obtained over rather narrow ranges of H and A.4 Equation 5 gives the best representation of the H-A data, and the values of the three adjustable parameters for the 18lipids are given in Table 111. They differ slightly from the those reported earlierefor most of the lipids because in the prior study eq 5 was used in the form A = f(d. Because two
of the adjustable parameter, fl and q (Table 11),reflect activity coefficients,1° we examined their possible correlation. A plot of q versus f1 for the 18lipids studied at 24O on comparable subphases, Le., A, F, and G (Table I), gave a curve suggestive of an exponential relation (not shown). Replotting the data as In ( q ) versus f~ (Figure 4, open circles) gave reasonable linearity with a slope of -5.179, an intercept of 7.061, and a correlation coefficientof 0.957. Thus, to a good approximation q = exp(7.061- 5.197f1) and this relationship can be used to reduce to two the number of adjustable parameters in eq 5. This modified eq 5 is denoted as model 6 (Table 11). Because models 3 and 5 showed similar residual patterns (Figures 1-3), a similar comparison of pairs of parameter values was made for eq 3. This showed a linear relationship between parameters C1 and C2 with a correlation coefficient of 0.938. Thus, the total number of empirical parameters in model 3could also be reduced to two. This was not pursued because, overall, model 5 gave better fits of the data and showed slightly better correlation of f l and q than did C1 and C2 of model 3. The data for the lipids were fitted by using eq 6, and the results are presented in Tables I and 111. Comparison of the statistical parameters (Table I) and the adjustable parameters (Table 111) with those obtained by using model 5 shows that model 6 gives a good description of the data in most cases. Most affected by the parameter substitution were the values of Ao, which were changed an absolute averageof 3.2 % with a maximum change of 10.4%. In contrast, f1 was changed
