INTRODUCTION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX REFERENCES

Most phospholipid acyl chains in animal cell membranes are saturated (only C-C bonds) or monounsaturated (one C==C bond) hydrocarbon polymers. However, it is surprising that membranes rich in polyunsaturated (multiple methylene-interrupted C==C bonds) lipids are found in certain animal tissues (like the brain, for instance), and the lengths of unsaturated lipid chains vary significantly. An obvious question to be asked is, how do hydrocarbon chain unsaturation and length affect membrane material properties important to the function and survival of cells? To address this question, we have used micropipette aspiration methods to test mechanical properties of giant-single bilayer vesicles made from fluid diacyl phosphatidylcholine (PC) bilayers with chain lengths of 13-22 carbons and a wide range of unsaturation (one, two, four, or six double bonds per lipid). Here we report results for equilibrium propertieselastic area and bending moduli. In a companion article (Olbrich et al., 2000), we present results for dynamic propertiesrupture strength (lysis tension) and water permeabilityfor the diC18 lipids with one to six cis double bonds. In both studies, major effects of unsaturation were found to occur when two or more cis double bonds punctuated by saturated bonds (C==C-C==C) appear in one or both chains. In the case of elasticity, the surprising outcome is that this type of poly-cis unsaturation leads to a precipitous reduction in the bending stiffness of the bilayer, which is accompanied by a prominent reduction in thickness, whereas neither chain length nor unsaturation significantly affects lateral area compressibility.

Several techniques have been used to quantitate mechanical stretch properties of bilayers. Most prominent are the micropipette approach we have pioneered for giant vesicles (Kwok and Evans, 1981; Evans and Needham, 1987), photon correlation spectroscopy and dynamic light scattering of small vesicles under osmotic stress (Rutkowski et al., 1991; Hallett et al., 1993), cryoelectron microscopy of vesicles subjected to osmotic stress (Mui et al., 1993), and NMR and x-ray diffraction of strongly dehydrated multibilayer arrays (Koenig et al., 1997). Of these approaches, only the micropipette method can be used to test the expansion of a single bilayer with a resolution of better than 0.1% relative change in area and verify elastic reversibility. In contrast to area stretch properties, the small bending stiffness of a bilayer has been derived traditionally from a detailed analysis of thermal shape fluctuations of flaccid vesicles (Schneider et al., 1984; Duwe et al., 1987; Faucon et al., 1989; and others) and more recently by measurement of the forces needed to pull nanoscale bilayer tubes from vesicles under tension (Bo and Waugh, 1989; Evans and Yeung, 1994; Evans et al., 1996). Much less complicated, however, is the micropipette pressurization of a single vesicle, which can also be used to quantitate the bilayer bending modulus. The approach is to measure the entropy-driven tension that arises as thermal bending undulations are smoothed under pipette pressurization of a giant vesicle (Evans and Rawicz, 1990, 1997). In the work reported here, both bending and elastic area stretch properties of the PC bilayers have been obtained by the micropipette pressurization technique.

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX REFERENCES

Lipids

Twelve synthetic species of diacyl-PC lipids were obtained from Avanti Polar Lipids (Alabaster, AL) in chloroform and used without further purification. Nine were cis unsaturated: 1-stearoyl-2-oleoyl-sn-glycero-3-phosphocholine (C18:0/1_c9); 1-oleoyl-2-stearoyl-sn-glycero-3-phosphocholine (C18:1_c9/0); 1,2-dipetroselinoleoyl-sn-glycero-3-phosphocholine (diC18:1_c6); 1,2-dioleoyl-sn-glycero-3-phosphocholine (diC18:1_c9); 1-stearoyl-2-linoleoyl-sn-glycero-3-phosphocholine (C18:0/2_c9,12); 1,2-dilinoleoyl-sn-glycero-3-phosphocholine (diC18:2_c9,12); 1,2-dilinolenoyl- sn-glycero-3-phosphatidylcholine (diC18:3_c9,12,15); 1,2-diarachidonoyl-sn-glycero-3-phosphocholine (diC20:4_c5,8,11,14); and 1,2-dierucoyl-sn- glycero-3-phosphocholine (diC22:1_c13). One was trans-unsaturated: 1,2-elaidoyl-sn-glycero-3-phosphocholine (diC18:1_t9). Two were fully saturated: 1,2-ditridecanoyl-sn-glycero-3-phosphocholine (diC13:0) and 1,2-dimyristoyl-sn-glycero-3-phosphocholine (diC14:0). The solutions were placed in amber glass screw-cap vials with Teflon-lined silicone septa. The vials were wrapped in aluminum foil and stored at 20°C under argon, which was especially important for the preservation of lipids containing oxidation-prone fatty acid chains (18:0/2, 18:2, 18:3, 20:4).

Vesicle preparation and lipid oxidation assay

In our laboratories, the generic procedure for preparation of giant vesicles (15-30-µm diameter) is to rehydrate lipid films dried first from chloroform:methanol (2:1) onto the surface of a roughened Teflon disk (Needham et al., 1988). After deposition of the lipid film and evaporation of the organic solvent in vacuo, the Teflon disk is covered with a warm (37°C) sucrose solution (200 mOsm) and allowed to hydrate. To create a refractive index contrast between the inside and outside of vesicles and to sediment vesicles in the microscope chamber, an aliquot of vesicles is diluted manyfold in an equiosmolar solution of glucose or electrolyte buffer. The refractive index gradient is used to enhance optical detection of the projection length, as shown by the example in Fig. 1. Described below, accurate video tracking of the edge enables discrimination of <0.1% relative change in vesicle area or volume, even though optical measurements of total area and volume remain limited to an accuracy of a few percent by diffraction. For the formation of vesicles from polyunsaturated lipids, slight modifications were made in the procedure. First, argon-purged, deionized water was used to make the hydration and suspension solutions. Second, the container with the polyunsaturated lipid film was wrapped in aluminum foil (to minimize exposure to light). The polyunsaturated lipids were allowed to hydrate for only 3 h under argon and were used immediately (normally, lipids are left to hydrate overnight and then used the next day). In the initial phase of the study, samples of the polyunsaturated lipids were tested for possible oxidative damage over the time scales associated with preparation and experiment by spectrophotometric assay (Kim and LaBella, 1987; New, 1990). The absorbance of a solution of 1 mM lipid in absolute ethanol was measured at ~230 nm with a Beckman DU-7500 diode array spectrophotometer (Beckman, Fullerton, CA). Absorption at this wavelength indicates the presence of conjugated dienes in the hydrocarbon chain, which result from oxidation (Kim and LaBella, 1987; New, 1990). Lipid samples used soon after arrival from Avanti showed no detectable oxidative damage. Moreover, measurements of properties repeated with preparations from the same polyunsaturated lipid samples and those from new samples purchased at later times gave identical results.

View larger version (125K): [in this window] [in a new window]   FIGURE 1   Video micrograph of a vesicle area expansion test. (a) The vesicle at low tension. (b) The vesicle at high tension. The change in projection length Lp is proportional to the change in apparent surface area A.

Mechanical expansion of apparent vesicle area

Micropipette suction was used to pressurize vesicles and test elastic area expansion. Well established from mechanics (Kwok and Evans, 1981), the suction pressure P applied to a fluid bilayer vesicle produces a uniform membrane tension _m, which is described by a geometric relation based on the pipet caliber (diameter) D_p and diameter D_v of the vesicle spherical segment exterior to the pipette, i.e.,
The pipette suction (~10³ Pa) needed to expand the surface of a ~20-µm vesicle is small compared to osmotic driving forces (~10⁵ Pa) required to kinetically displace water on the time scale of the experiment and small compared to the osmotic activity of the trapped sucrose (~5 × 10⁵ Pa). Thus vesicle volume remains effectively fixed in an area expansion test. Movement of the projection length L_p inside the pipette provides a direct measure of the area increase. Using measurements of the vesicle spherical segment and cylindrical projection dimensions, we computed precise changes in apparent area A numerically for each displacement L_p of the projection length with simple geometric formulae and the constraint of a fixed vesicle volume (see the Appendix in Olbrich et al., 2000). The proportionality between apparent area and length is easily seen in the following first-order approximation:
Displacement of the projection length of an aspirated vesicle under change in suction pressure is shown in Fig. 1. Even though apparent area can be measured over a tension range from 10³ mN/m to vesicle rupture (~ 3-10 mN/m) as shown in Fig. 2, vesicle area expansion was examined in two separate regimes of low and high tension to minimize the duration of experiments and thereby avoid any reduction in vesicle volume caused by chamber dehydration at long times.

View larger version (17K): [in this window] [in a new window]   FIGURE 2   Examples of apparent area expansion measured over tensions from 0.001 to 8 mN/m for two vesicles made from C18:0/1 and diC18:3 PC. (A) Linear plot of tension versus apparent area expansion. The initial soft-exponential rise of tension with area expansion reveals smoothing of thermal shape fluctuations, which is followed by the onset of linear increase in tension as the bilayer begins to stretch. (B) Semilog plot of tension versus apparent area expansion. Slopes of the linear fits (dashed lines) applied to the range of very low tensions yield elastic bending moduli kc (× 8/kBT) for each bilayer (kc = 0.9 × 1019 J for C18:0/1 and kc = 0.4 × 1019 J for diC18:3). The solid curves in A and B are the fit of the elastic compressibility relation (Eq. 1) over the entire four-order-of-magnitude range of tension, using the values of bending elasticity and a common value of the direct expansion modulus (KA = 230 mN/m) for both lipid vesicles.

In the low-tension regime (0.001-0.5 mN/m), the vesicle surface increases by smoothing of subvisible thermal shape fluctuations (bending undulations). Even though a vesicle looks perfectly spherical in optical images under suction pressures of ~0.1 Pa or more (cf. Fig. 1), thermal shape fluctuations persist on the suboptical scale and act as a small reservoir for an increase in apparent surface area of the vesicle. To examine the low-tension regime, each vesicle was first prestressed under a tension of ~0.5 mN/m to ensure that small hidden projections of bilayer surface were pulled into the surface. After the vesicle was prestressed, suction pressures were dropped to ~0.1 Pa, which lowered the bilayer tension to 10³ mN/m. Then suction was increased in stepseach held steady for several secondsup to 2-3 × 10² Pa, where tensions approached 1 mN/m. Finally, a stepwise course of pressure reduction was used to verify reversibility. As shown by the data in Fig. 2 A, bilayer tension below ~0.5 mN/m increases as a very weak exponential of apparent area, which is verified by the linear increase in apparent area with log(tension) in Fig. 2 B. Tests of bending elasticity in the low-tension regime for the diC18 PCs were performed at 18°C (close to the dewpoint) to minimize chamber dehydration and to allow long working periods without having to change chamber solutions. But bending tests for diC13:0, diC14:0, and diC22:1 PC were performed at 22°C, 29°C, and 21°C, respectively, to be well separated from the main liquid crystal-crystalline phase transition. Measurements at high temperatures (lower relative humidities) are usually no problem but require more frequent sample changes to avoid chamber dehydration.

In the high-tension regime (>0.5 mN/m), direct surface stretch or dilation in area per lipid molecule becomes significant with a small, diminishing contribution from smoothing of thermal undulations. To examine the high-tension regime, each vesicle was first prestressed under a tension of ~0.5 mN/m to incorporate hidden excess area, and then suction was increased in stepseach held steady for several secondsup to a sublytic threshold of 1-3 × 10³ Pa, where tensions reached 3-8 mN/m. As before, a stepwise course of pressure reduction was used to verify reversibility. As shown by the data in Fig. 2 A, bilayer tension seems to increase in direct proportion to apparent area above ~1 mN/m but with different slopes for two different types of lipid bilayers. Much shorter in duration than the low-tension experiments, all area compressibility tests in the high-tension regime were performed at 21°C.

Bilayer thickness

X-ray diffraction was performed on oriented lipid multibilayers of six lipids by methods described previously (see McIntosh, 1987; McIntosh et al., 1989, 1992, for details). Partially hydrated, oriented bilayers of diC13:0, C18:0/1, diC18:1, diC18:2, diC18:3, and diC22:1 PC were formed by first placing a drop of lipid/chloroform solution on a curved glass support, slowly evaporating the chloroform, and then incubating the multilayers in a constant-humidity atmosphere. Afterward, the specimen was placed in a controlled humidity chamber on a single-mirror (line-focused) x-ray camera so that the x-ray beam was oriented at a grazing angle with respect to the multilayers. The relative humidity in the chamber was maintained at 98%, 93%, or 79% relative humidity with a cup of the appropriate saturated salt solution (McIntosh, 1987; McIntosh et al., 1989). To speed equilibration, a gentle stream of nitrogen gas was passed through a flask of the saturated salt and then through the chamber.

X-ray diffraction patterns were recorded at ambient temperatures on a stack of six sheets of Kodak DEF x-ray film loaded in a flat plate film cassette. After background subtraction, integrated intensities I(n) were obtained for each order n by measuring the area under each diffraction peak. The intensities of the oriented line-focused patterns were corrected by a single factor of n due to the cylindrical curvature of the multilayers (Blaurock and Worthington, 1966; Herbette et al., 1977), which yields a structure amplitude of F(n) = {nI(n)}^1/2. Then electron density profiles (x) were calculated on a relative scale from the Fourier transform,
where x is the distance from the center of the bilayer, d is the lamellar repeat period, (n) is the phase angle for order n, and the sum _n is over n. Phase angles were determined using the sampling theorem (Shannon, 1949) as described previously (McIntosh and Simon, 1986; McIntosh, 1987). Fig. 3 demonstrates typical structure factors for C18:0/1 and diC18:3, along with continuous Fourier transforms calculated using the sampling theorem. The resolution of the electron density profiles presented in this paper is d/2h_max  7 Å. However, as shown previously by McIntosh and Simon (1986), Nagle et al. (1996), and Petrache et al. (1998b), profiles with 7-Å resolution provide a quite accurate estimate ( Å) for the peak-to-peak headgroup separation across the bilayer. The critical requirement is that the profiles for all of the lipid measurements are obtained at the same resolution.

View larger version (22K): [in this window] [in a new window]   FIGURE 3   Examples of structure factors and continuous Fourier transforms obtained by x-ray diffraction from C18:0/1 () and diC18:3 () multibilayers.

	RESULTS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX REFERENCES

Measurements of bilayer elastic properties

As described above, measurements of aspiration length versus pipette pressurization were converted to apparent area expansion versus tension properties of bilayer vesicles in low- and high-tension regimes. Verified as reversible, the tension-apparent area results were correlated with a general constitutive relation for elastic dilation of fluid bilayers to obtain the direct lipid area expansion and bending moduli. Based on early concepts of Helfrich (Helfrich and Servuss, 1984), apparent area compressibility is derived from a superposition of smoothing of thermal bending undulations and reduction in lipid surface density (Evans and Rawicz, 1990, 1997),

(1)

 = A/A_o is the fractional increase in apparent or mean projected area A of the vesicle; K_A, k_c are the elastic moduli for direct stretch in area and for bilayer bending, respectively; thermal energy k_BT is ~4 × 10²¹ J; c is an unimportant constant of ~0.1 that depends on the type of modes (spherical harmonics or plane waves) used to describe surface undulations. Fig. 2 shows the correlation of Eq. 1 with measurements of apparent area for two different types of lipid vesicles over the full range of accessible tensions. In the soft-thermal regime at low tension, the apparent expansion relative to an initial state of tension _m(0) is dominated by smoothing of thermal undulations, and the bending stiffness is revealed by the logarithmic dependence of apparent area on tension, log_e[_m/_m(0)]  (8k_c/k_BT)A/A_o, as derived from the dashed line fits in Fig. 2 B. By comparison at high tension, an increase in apparent area approaches the direct stretch A'/A'_o, governed by the elastic area-stretch modulus, which increases linearly with tension, _m = K_A(A'/A'_o). Even so, residual thermal undulations introduce a small (but important) correction in the high-tension regime, as revealed by the slight curvature in the high-tension results of Fig. 2 A.

Bending moduli

Because of the logarithmic dependence on tension, the apparent expansion of vesicle area had to be measured over a large range of minuscule tensions to obtain the bending modulus. Limited by the 0.1 Pa pressure sensitivity, the regime of low tension accessible to pipette aspiration spanned nearly three orders of magnitude from 10³ to ~0.5 mN/m. Seen in Fig. 2 B, vesicle areas increase linearly with log(tension) over most of this range. Multiplied by k_BT/8  1.6 × 10²² J or 0.04k_BT, the slope of log_e(tension) versus fractional area expansion in the low-tension regime yielded the bending modulus of a bilayer as demonstrated by the dotted line fits to the data in Fig. 2 B. Tests of at least 10 vesicles were used to obtain bending moduli for each type of lipid as listed in Table 1 (± SD). The bending moduli of saturated/monounsaturated PCs increased progressively with chain length from k_c = 0.56 × 10¹⁹ J ( 14k_BT) for diC13:0 to k_c = 1.2 × 10¹⁹ J ( 30k_BT) for diC22:1. But unexpectedly for longer chains, the bending moduli of bilayers with two or more alternating cis double bonds along one or both chains (C18:0/2, diC18:2, diC18:3, and diC20:4) dropped precipitously to k_c  10-11k_BT, which shows that poly-cis unsaturated membranes are distinctly more flexible than saturated/monounsaturated PC bilayers. To illustrate the striking effect of polyunsaturation, Fig. 4 presents a histogram of the bending moduli (± SD) for diC18 bilayers arranged in order of increasing unsaturation.

View larger version (21K): [in this window] [in a new window]   FIGURE 4   Elastic bending moduli for diC18 PC bilayers arranged in order of increasing unsaturation.

Lipid area dilation moduli

Ideally, both bending and direct stretch moduli can be derived from a nonlinear fit of Eq. 1 to measurements of apparent area over the full range of high and low tensions. However, for reasons given above and just as accurately within experimental resolution, the bending modulus can be derived first from the low-tension regime then used with the constitutive Eq. 1 to carry out a one-parameter fit to the apparent area expansion in the high-tension regime (>0.5 mN/m). The result is indistinguishable from the correlation over the full range of tension as demonstrated in Fig. 2. When the value of the bending modulus is known, the contribution to apparent area expansion from smoothing thermal undulations can be determined uniquely in the high-tension regime. For each ith data pair of apparent area expansion and tension, the direct area dilation '(i) = (i) + (i) is found relative to an initial tension state _m(1), using the correction

(2)

The direct area expansion ' found in this way is plotted versus tension in Fig. 5 for the vesicle tests in Fig. 2. Although the slope K_app of tension versus apparent area expansion above 1 mN/m differs significantly between the two lipid bilayers in Fig. 2 A, a common value of K_A = 230 mN/m was found for the direct stretch moduluseither by fit of the constitutive equation over the full range of data in Fig. 2 or linear regression to the tension versus direct area expansion in Fig. 5.

View larger version (12K): [in this window] [in a new window]   FIGURE 5   The direct area expansion versus increase in tension over the high-tension regime (>0.5 mN/m) for the C18:0/1 and diC18:3 vesicles in Fig. 2. The direct area expansion was obtained from the increase in apparent area, using the correction for smoothing of thermal shape fluctuations (Eq. 2) and mean values of elastic bending moduli measured for each type of lipid bilayer. A common linear regression fits both sets of data and yields the same direct expansion modulus (KA = 230 mN/m), which matches the result from the fit of the elastic compressibility relation (Eq. 1) over the full four-order-of-magnitude range in tension.

To obtain direct stretch properties for a population of vesicles, we used the average bending modulus to correct the values of apparent area expansion measured in the high-tension regime (>0.5 mN/m) with Eq. 2. The statistical uncertainty introduced into the determination of direct-stretch moduli due to the ±10% variation in bending modulus ranges from only ±3% for the stiffest bilayers to ±8-10% for the most flexible bilayers, which lies within the experimental standard deviation. Tests were performed on at least 10 vesicles for each type of lipid to obtain values of K_A (± SD), which are listed in Table 1, along with apparent expansion moduli K_app. Comparison of K_app and K_A demonstrates that the smoothing of thermal undulations is quite significant for highly flexible bilayers. Unlike bending elasticity, the direct-stretch moduli varied little with chain unsaturation or length, as shown by the histogram in Fig. 6.

View larger version (32K): [in this window] [in a new window]   FIGURE 6   Direct area stretch moduli for fluid-phase diacyl PC bilayers with chain lengths from 13 to 22 carbons and range of unsaturation from 0 to 6 double bonds per lipid.

Measurements of bilayer thicknesses

View larger version (17K): [in this window] [in a new window]   FIGURE 7   Electron density profiles for (A) C18:0/1 and (B) diC18:3 at 98%, 93%, and 79% relative humidities. (C) Electron density profiles of C18:0/1 (solid line) and diC18:3 (dotted line) compared at 98% relative humidity, which shows the major reduction in peak-to-peak headgroup distance from ~4.1 nm for C18:0/1 to ~3.4 nm for diC18:3.

View larger version (14K): [in this window] [in a new window]   FIGURE 8   Correlation of the elastic ratio (kc/KA)1/2 for bending:direct stretch with the peak-to-peak headgroup separations hpp taken from Table 1. The solid line is the polymer brush model (kc/KA)1/2 = (hpp  ho)/(24)1/2 for bilayer elasticity derived in the Appendix, with ho = 1 nm.

	DISCUSSION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX REFERENCES

The important outcomes of this work are first that diacyl PC bilayers with a wide range of chain length and unsaturation exhibited little difference in direct area expansion moduli once increases in apparent area were corrected for smoothing of subvisible thermal bending undulations. Second, by comparison, bending rigidity increased in a steady, progressive manner with the number of carbons for saturated/monounsaturated chain bilayers. However, most striking was a major reduction in bending rigidity, which occurred when two or more cis double bonds were present in one or both chains of the lipid. Because of the unusual bending flexibility of polyunsaturated chain bilayers, a much larger fraction of surface area is incorporated into thermal undulations, which significantly lowers the apparent area modulus, as seen by comparison of K_A to K_app in Table 1. This feature accounts for the ~20-40% variations in elastic area moduli of different PC bilayers published in the past by our groups and others. Recently, for example, Koenig et al. (1997) used NMR and x-ray methods to derive the elastic area moduli for C18:0/1 and diC14:0. Their values were ~220 mN/m and ~140 mN/m for C18:0/1 and diC14:0, respectively, which are consistent with the apparent moduli K_app in Table 1 and point to thermal undulations as the source of deviation from the direct-stretch moduli K_A. Finally, measurements of peak-to-peak headgroup separation by x-ray diffraction confirmed the hypothesis that the prominent variations in bending stiffness emanated principally from variations in bilayer thickness. In the case of saturated/monounsaturated chain bilayers, the measurements of the elastic ratio (k_c/K_A)^1/2 and structural thickness h_pp correlated well with the linear prediction obtained from the polymer brush model for bilayer elasticity (Appendix), which implied a fixed length of h_o = 1 nm for the combined separations of headgroup centers from the deformable hydrocarbon core. Based on purely geometric considerations and molecular structure, Hitchcock et al. (1974) and McIntosh and Simon (1986) also concluded that h_o  1 nm was needed to estimate the thickness of the compact hydrocarbon region and, thereby, obtain the area per lipid 2a, using the hydrocarbon packing relation 2a × (h_pp  h_o) = 4a_c × L. Similarly, Petrache et al. (1998b) arrived at a length of h_o = 0.82 nm for dimyristoylphosphatidylcholine. On the other hand, correlation of the elastic ratio to thickness for diC18:2, diC18:3, and diC20:4 PC bilayers showed that poly-cis unsaturated chain bilayers are thinner and more flexible than saturated/monounsaturated chain bilayers.

Although naïve in many respects, some useful insights into the origins of bilayer elasticity and thickness are found in the idealized (polymer brush) model of lipid monolayers as structureless polymer chains confined by a constant hydrophobic energy density. First of all, the polymer brush model predicts that the surface pressure  in each monolayer and the direct expansion modulus of a bilayer are related by a constant factor of 6, i.e., K_A = 2(3). The numerical factor reflects the inverse cube dependence of surface pressure on area per acyl chain governed by the Gaussian-harmonic regime of polymer extension. In the absence of mechanical tension, the surface pressure in each monolayer is balanced by the interfacial energy density  for hydrophobic interactions,  = . Consequently, the model predicts that elastic area compressibility is constant if the interfacial energy density  is constant (per the idealized concept of hydrophobic interactions). Consistent with this prediction, direct-stretch moduli of the 12 PC bilayers vary little in a range of 230-250 mN/m, except for diC18:1_c9 and diC22:1_c13 at 263-265 mN/m. As such, the mean value of 243 mN/m implies that monolayer surface pressure varies little from   40 mJ/m² (mN/m). The < ±10% variations from 243 mN/m obviously reflect deviations from the idealized model, which could arise from many sources: e.g., departure from Gaussian-harmonic behavior of chains, headgroup interactions, and chemical variations in hydrophobic and other interfacial interactions.

Next, if we accept that the mechanical thickness h_pp  h_o is equal to the hydrocarbon thickness, we can use the polymer brush model to predict hydrocarbon thickness and area per lipid 2a, which can be compared to structural measurements. The simple result is that hydrocarbon thickness and area per molecule depend only on a fixed ratio c = (3k_BT/a_c) of thermal to interfacial energy and the number n_s = L/2b of statistical segments per chain, i.e., h_pp  h_o  2L/(cn_s)^1/3 and 2a  0.40 nm² × (cn_s)^1/3, where 2a_c  0.40 nm² and L = n_carbons × 0.125 nm in the all-trans configuration. (The polymer brush model also predicts that bending stiffnessscaled by the characteristic energy L²depends uniquely on the ratios c and n_s, i.e., k_c = (L²)/(cn_s)^2/3, which yields a range from k_c  0.57 × 10¹⁹ J for C13 to k_c  1.2 × 10¹⁹ J for C22 chains.) Taking the value of   40 mJ/m² deduced from the mean stretch modulus defines c  1.5, so all relations reduce to a simple dependence on the number n_s of statistical segments per chain, i.e., thickness h_pp  h_o  1.75 L/n_s^1/3 and area per lipid 2a  0.46 nm² × n_s^1/3. As shown in the Appendix, correlating the dependence of surface pressure on area per chain predicted by the model to the published monolayer isotherm for C18:0/1 yields the mean number L/2b  2.25 of statistical segments per chain and a persistence length of b  0.5 nm. Hence the structural thickness of C18:0/1 is predicted to be h_pp  3.0 nm + h_o, which agrees with the value in Table 1 if we take h_o = 1 nm. Furthermore, assuming common values of persistence length b  0.5 nm and h_o = 1 nm, thicknesses can be predicted over the full range of chain length from h_pp  3.4 nm for diC13:0 to h_pp  4.4 nm for diC22:1, which is again consistent with the span in Table 1. As an experimental comparison for area per molecule, Koenig et al. (1997) used deuterium NMR to derive hydrocarbon thickness and area per lipid by measuring the integrated-average order parameter for ²H-labeled chains. The molecular areas of 0.59 nm² for diC14:0 and 0.61 nm² for C18:0/1 from their experiments are close to the estimates of 0.55 nm² and 0.6 nm², respectively, predicted by the polymer brush model.

In contrast to the predictive success for saturated/monounsaturated chain bilayers, neither the thickness nor the bending modulus calculated with the polymer brush relations matches the measured values for bilayers with polyunsaturated chains. Most likely, this stems from increased chain repulsion and peaked stress distributions within the monolayers. For example, a shorter persistence length b could be an origin of larger repulsion between poly-cis unsaturated chains. Evidence for this comes from Monte Carlo simulations by Rabinovich and Ripatti (1991), which showed that the mean end-to-end length of a chain diminished as the number of methylene-interrupted cis double bonds were increased in a chain. For 18 carbon chains, the apparent increase in chain "flexibility" implied a reduction of ~33% in persistence length. In the context of the polymer brush model, the increase in chain flexibility would reduce the peak-to-peak headgroup thickness by ~0.4 nm to ~3.6 nm for bilayers with C18 chainsagain taking h_o = 1 nm. However, even though a 33% reduction in persistence length may account for a large portion of the reduction in thickness, changes in persistence length alone cannot lead to a departure from the linear correlation of elastic ratio to mechanical thickness seen in Fig. 8 for saturated/monounsaturated chain bilayers. This deviation implies a narrowing in the distribution of chain stresses within monolayers of the bilayer or an increase in h_o.

Distributions of chain stresses in PC monolayers have been nicely modeled in recent simulations by Cantor (1999). Invoking the same entropic physics and interfacial energy for hydrocarbon-water interaction as used in our simple theory, Cantor used a mean-field lattice calculation to obtain distributions of lateral stress across monolayers in bilayers. For a wide range of hydrocarbon chain length and unsaturation, the stress distributions resembled flattened half-period sine functions. Generally, monolayers of chains with cis double bonds were found to be thinner than layers with saturated chains of the same length. But important asymmetries arose when multiple cis bonds appeared at specific locations. In particular, noticeably peaked and asymmetrical distributions were obtained when unsaturated bonds were present near the interface, e.g., C18:1_c6, C18:3_c6,9,12, C20:4_c5,8,11,14, and C22:6_{c4,7,10,13,16,19}. It was puzzling, however, that there were essentially no differences in stress distributions or thicknesses between mono- and poly-cis unsaturated chains when double bonds started in the nine position, i.e., C18:1_c9 vis-a-vis C18:3_c9,12 and C18:3_c9,12,15, unlike the thickness measurements listed in Table 1. So perhaps a different algorithm is needed to describe the energetics of chain configurations on the lattice in the case of methylene-interrupted cis double bonds. As a possible clue, the molecular-model computations of Applegate and Glomset (1991) have shown that angle-iron-shaped conformations of polyene sequences shorten chains significantly, and the reduction in length is larger when polyene sequences are more distal from the water-hydrocarbon interface, as is the case for C18:3_c9,12,15.