Structure and interaction potentials in solid-supported lipid membranes studied by X-ray reflectivity at varied osmotic pressure
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Structure and interaction potentials in solid-supported lipidmembranes studied by X-ray reflectivity at varied osmoticpressure
Ulrike Mennicke a , Doru Constantin b , and Tim Salditt c Institut f¨ur R¨ontgenphysik, Friedrich-Hund-Platz 1, 37077 G¨ottingen, GermanyAugust 15, 2018
Abstract.
Highly oriented solid-supported lipid membranes in stacks of controlled number N ≃
16 (oligo-membranes) have been prepared by spin-coating using the uncharged lipid model system 1,2-dimyristoyl-sn-glycero-3-phosphocholine (DMPC). The samples have been immersed in aqueous polymer solutionsfor control of osmotic pressure and have been studied by X-ray reflectivity. The bilayer structure andfluctuations have been determined by modelling the data over the full q -range. Thermal fluctuations aredescribed using the continuous smectic hamiltonian with the appropriate boundary conditions at thesubstrate and at the free surface of the stack. The resulting fluctuation amplitudes and the pressure-distance relation are discussed in view of the inter-bilayer potential. PACS.
A quantitative understanding of the structure, fluctua-tions, interaction potential and elasticity properties of lipidmembranes, which represent model systems for biologicalmembranes, has been the goal of many theoretical andexperimental studies. Theoretically, they have been stud-ied as paradigmatic examples of quasi two-dimensionalmacromolecular structures governed by bending rigidity[1]. In aqueous solution, lipid bilayers assemble into stacksgoverned by distinct inter-bilayer interactions. A numberof seminal studies using high resolution synchrotron X-rays have been published on these systems [2,3,4,5]. Inthese studies, isotropic aqueous dispersions of multilamel-lar vesicles have been studied as a function of temperature T and/or osmotic pressure Π . Detailed quantitative in-formation on the interaction potentials and the elasticityproperties has thus been derived, see e.g. [2,3,4]. How-ever, information is lost in these small-angle scatteringstudies due to crystallographic powder averaging. In theanalysis, assumptions must therefore be made on the na-ture of the correlation functions in the framework of linearsmectic elasticity theory, leading to the Caill´e model [6]and related theories, see for instance [7]. In order to over-come the limitations of powder averaging, it is desirable a e-mail: [email protected] b e-mail: [email protected] Permanent address:Laboratoire de Physique des Solides, Universit´e Paris-Sud,Bˆat. 510, 91405 Orsay Cedex, France. c e-mail: [email protected] to work with aligned systems of lipid bilayers [8,9,10,11,12]. Under the same conditions of temperature and hy-dration, thermal fluctuations are not as strong for alignedsystems as in bulk studies due to the boundary conditionat the flat substrate, enabling a higher resolution in ρ ( z ).It is also advantageous to fit the data continuously over alarge range of momentum transfer q , as has been shownfor isotropic solutions [5] and for oriented stacks [12], andnot only in the vicinity of the Bragg peaks arising fromthe multilamellar structure. However, for aligned systemsof multilamellar membranes, satisfactory fits of the entirereflectivity curves and the formulation of a proper statis-tical model as well as of a scattering theory are still quitedifficult. Note that the best published fits are for systemsconsisting of monolayers at the air-water interface, for sin-gle bilayers or for a free-floating bilayer system developedby Fragneto and coworkers [13]. In multilamellar systemson the other hand, the reflectivity signal is typically muchmore complex and structured. As we show here, structuralparameters of the bilayer, interaction and fluctuation pa-rameters can be deduced from these curves, and are com-pared with the literature.Most previous studies on aligned multilamellar mem-branes suffered from a lack of control concerning the num-ber of bilayers N , the sample homogeneity and also the bi-layer hydration. Building on recent progress in the prepa-ration of bilayers on solid support by spin-coating [14],we use so-called oligo-membranes with a reduced num-ber of bilayers N ≃ −
20 resulting in very structuredand well resolved reflectivity signals. We have developed
Ulrike Mennicke et al.: X-ray reflectivity of lipid lamellar stacks under osmotic pressure a model for the thermal fluctuations calculated for theproper boundary conditions (rigid substrate and free up-per surface), which gives the fluctuation amplitudes (cid:10) u n (cid:11) of the bilayers as a function of their position in the stack, n = 1 , N . The values for (cid:10) u n (cid:11) are then inserted in the mul-tilamellar structure factor, along with a decreasing cover-age function (see below). The density profile ρ ( z ) is pa-rameterized in terms of its Fourier coefficients [15]. Thisapproach gives for the first time an agreement with themeasured reflectivity curve of multilamellar membranesover the full range of q z up to typically q z ≃ . − andover about seven orders of magnitude in the measured sig-nal.The main experimental parameter in this study is theosmotic pressure. The classical osmotic stress (OS) tech-nique as developed by Parsegian and coworkers [16] iswidely used for the measurement of force-distance curvesin colloidal systems. The osmotic pressure imposed to alamellar phase controls the interaction force experiencedby the membranes across the water layer by setting thechemical potential of the water molecules in the inter-bilayer solution. Pressure-distance relations can be easilydetermined, e.g. if the lamellar periodicity d is measuredby X-ray scattering at different pressures Π .In this study we use a variant of the OS techniquewhere the oriented bilayers are put in direct contact withthe osmotic stress solution [18]. We have verified thatthe high molecular weight polymer with a radius of gyra-tion larger than d does not penetrate the lamellar phase.We emphasize that the osmotic pressure is one of themost important parameters in biomolecular systems, sincebiomolecular assemblies in the cell are mostly exposed tovarying Π while T is often constant. Therefore, it is ofgreat importance to study bilayer structure and elasticproperties such as bilayer bending rigidity κ or bilayer-bilayer interaction parameters as a function of osmoticpressure Π .From the relation between the osmotic pressure Π andthe lamellar spacing d the inter-bilayer interaction poten-tials can be determined. It is generally accepted that incharge neutral systems two main molecular interactionforces are dominant, in addition to the effective attrac-tive interaction by osmotic pressure : a repulsive hydra-tion potential f hyd ( d w ) and the attractive van der Waalspotential f vdW ( d w ) so that the total interaction in J / m is given by f ( d w ) = f hyd + f vdW + Πd w , defining the equi-librium distance (water layer thickness) d w as illustratedin Fig. 1 B). As discussed below, it is important to takethe correct form for f vdW as derived in [17], without theconventional half-space approximation. It can be arguedthat steric (Helfrich) repulsion forces have to be addedto the molecular forces in a mean field approach. Here,however, we will assume that thermal fluctuations in thinfilms of relatively stiff phospholipids do not have a signif-icant effect on the inter-bilayer interactions, in particularsince the flat boundary suppresses long range fluctuationsin the film. The paper is organized as follows : After thisintroduction, section 2 presents some experimental detailswhile the statistical model and data analysis are presented H O+polymer substrate k i k f (A)(B)
20 40 60-6.0x10 -6 -4.0x10 -6 -2.0x10 -6 -6 E ne r g y pe r A r ea [ J / m ] d w [ Å ]total Interaction potentialwith exact VdWwith approximated VdWharmonic approximation Fig. 1.
A): Sketch of the experimental setup where a solid-supported membrane stack is hydrated in an aqueous polymersolution. Arrows indicate the incoming and scattered X-raybeams. B): Interaction potential with van der Waals contri-bution according to Fenzl [17] (solid line) and with generalapproximation (dashed). The (dotted) parabola illustrates theharmonic approximation to the potential which enters in thesmectic elasticity theory. The arrow shows a typical fluctuationamplitude of a membrane inside a 16 bilayer stack. in section 3. Section 4 presents the results, followed by asection on the interaction potentials and the conclusionsin section 6.
Highly oriented oligo-membranes were prepared using thespin-coating method [14]. The uncharged lipid 1,2-dimy-ristoyl-sn-glycero-3-phosphocholine (DMPC) was boughtfrom Avanti (Alabaster, AL, USA) and used without fur-ther purification. The lipid was dissolved in chloroformat a concentration of 10 mg/ml. An amount of 100 µ lof the solution was pipetted onto carefully cleaned sili-con substrates of a size of 15 ×
25 mm cut from stan-dard commercial silicon wafers. The substrate was thenimmediately accelerated to rotation (3000 rpm), using aspin-coater. After 30 seconds the samples were dry and lrike Mennicke et al.: X-ray reflectivity of lipid lamellar stacks under osmotic pressure 3 subsequently exposed to high vacuum to remove any re-maining traces of solvent. The samples were then stored at4 ◦ C until the measurement. For the X-ray measurementsthe samples were hydrated in a stainless steel chamber[10] with kapton windows, which can be filled with wa-ter or with polymer solution to control the level of hy-dration by osmotic pressure, see the sketch in Fig. 1A.Temperature was controlled by an additional outer cham-ber at T = 40 ◦ C. The polymer polyethyleneglycol (PEG)of molar weight 20000 Da was bought from Fluka andused without further purification. PEG was dissolved inultrapure water (Millipore, Billerica, Mass.) at the con-centrations 1.5 %, 2.9 %, 3.6 %, 5.8 %, 9 %, 12.1 %,14.2 % and 25 % (wt. percent). The corresponding os-motic pressure values were taken from the literature. Thedata was obtained from the web site of the MembraneBiophysics Laboratory at the Brock University in Canada(http://aqueous.labs.brocku.ca/osfile.html). The value forthe osmotic pressure of PEG 20000 solutions is only avail-able at 20 ◦ C. At 40 ◦ C, temperature at which the experi-ments were performed, the pressure can be expected to besomewhat lower. However, the temperature coefficient issmall [16,19], and the corresponding discrepancy smallerthan the error bars in Fig 6 below, which is used in theanalysis of the interaction forces.An important issue in our method of direct contact ofthe lamellar phase with the polymer solution is that ofpossible interpenetration of the multilamellar phase. Ex-periments on polymer containing lyotropic lamellar phases[20,21] have shown that polymers can enter the water layerin between charged bilayers even if the radius of gyrationis larger than the water layer. However, in the present caseof neutral polymer in neutral bilayers, the amount of poly-mer in between the lamellae is negligible. The experimen-tal proof is given by (a) the density profile, which showsno deviation from the water density in between the bilay-ers, and (b) the force distance curve itself which shows noindication of such an effect.
The X-ray reflectivity measurements presented here werecarried out at the bending magnet beamline D4 of HA-SYLAB/DESY in Hamburg, Germany. At D4, a single-reflection Si(111) monochromator was used to select aphoton energy of 19.92 keV, after passing a Rh mirror toreduce higher harmonics. The chamber was mounted onthe z -axis diffractometer, and the reflected beam was mea-sured by a fast scintillation counter (Cyberstar, Oxford),using computer-controlled aluminum absorbers which at-tenuate the beam at small q z to prevent detector satura-tion. Incident and exit beams were defined by a system ofseveral motorized slits. The data is corrected for decreas-ing electron ring current and the diffuse contribution bysubtraction of an offset scan. Finally, an illumination cor-rection is performed. A typical measurement (reflectivityand offset scans) is shown in Figure 2, along with the cor-responding Fresnel reflectivity. The inset shows a rocking I [ c t s / s ] -0.4 -0.2 0.0 0.2 0.4q x [ 10 -3 Å -1 ] offset scan I n t e n s it y [ c oun t s / s ] z [Å -1 ] Fig. 2.
Specular reflectivity scan (open symbols) and offsetscan (red solid curve) for a sample immersed in pure water.The ”true specular” contribution (see Appendix) is given bythe difference of the two curves. In dotted line we also show theFresnel reflectivity profile corresponding to the same criticalangle q c . Inset : Rocking scan at the position of the secondBragg peak. The arrow shows the q x position of the offset scan. scan on the second Bragg peak, illustrating the separationbetween ”true specular” and diffuse components. In the semi-kinematic approximation the reflectivity ofa structured interface can be expressed by the so-calledmaster-formula of reflectivity [22] as : R ( q z ) = R F ( q z ) · (cid:12)(cid:12)(cid:12)(cid:12) ρ Z ∞−∞ d ρ ( z )d z e iq z z d z (cid:12)(cid:12)(cid:12)(cid:12) , (1)where R F denotes the Fresnel reflectivity of the sharp in-terface and ρ ( z ) is the intrinsic electron density profile,whereas ρ is the total step in electron density betweenthe two adjoining media. The electron density profile ofa solid-supported oligo-membrane stack, consisting of N membranes in water, can then be written as : ρ ( z ) = ρ (cid:18) z + d σ (cid:19) + N − X n =0 ρ ( z − nd + u n ) , (2)with erfc( z ) being the complementary error function and σ the rms substrate roughness. ρ = ρ Si − ρ H O is thedifference in electron density between the substrate andwater, d is the distance between the substrate and themidpoint of the first bilayer and ρ ( z ) is the electron den-sity profile of one bilayer in the stack. Thermal membrane Ulrike Mennicke et al.: X-ray reflectivity of lipid lamellar stacks under osmotic pressure fluctuations are considered in terms of the displacementfunction u n = u ( r || , z = nd ) of the position of the n -thmembrane from its average position z = nd along z . Re-placing the electron density profile (2) into (1) and takingthe ensemble average yields : R ( q z ) = R F ( q z ) · " e − q z σ −− i · e − q zσ N − X n =0 f ( n ) (cid:18) F f ( q z ) · sin( q z ( d + nd )) e − q z h u n i (cid:19) + | F f ( q z ) | · N − X m,n =0 f ( n ) f ( m ) e − iq z d ( m − n ) e − q z ( h u m i + h u n i ) . The first summand represents the reflectivity of the sub-strate. The second is a cross-term and represents inter-ference effects between the substrate and the membranestack. The third summand is the product of the form fac-tor | F f ( q z ) | containing the structural information aboutone bilayer in the stack, and the structure factor, repre-senting the periodic structure in the stack. The fluctua-tions are described by the correlation function h u n u m i . Inspecular reflectivity only the self-correlation function h u n i is important (see Appendix). The self-correlation function of the membrane fluctuationscan be calculated from linear smectic elasticity theorybased on a continuous model [23]. The complete theoryand calculations are described in [24]. Here only the es-sentials, which are important for specular reflectivity shallbe given. The linearized free energy can be written as afunction of the displacement u ( r || , z ) as : F = 12 Z V d r " B (cid:18) ∂u ( r || , z ) ∂z (cid:19) + K ( ∆ || u ( r || , z )) , (3)with K = κ/d the bending modulus and B the compres-sion modulus in the stack. We neglect the surface tensionbetween the lipid stack and the solvent. The discrete struc-ture of the stack consisting of N bilayers is taken into ac-count by expanding u over N independent modes. Also, weare only interested in the fluctuation amplitude at the po-sition of the bilayer midpoints, denoted by u n = u ( z = nd )for the n -th bilayer. From the equipartition theorem onecan calculate the correlation function of the membranefluctuations h u n i , which reads : h u n i = η (cid:18) dπ (cid:19) N X j =1 j − (cid:18) j − π nN (cid:19) (4)with the conventional Caill´e factor η = π d k B T √ BK . Figure5A shows the function for a 16 membrane stack with atypical η value for DMPC membranes at partial hydra-tion. In contrast to oligo-membranes which are partially hydrated from water vapor, oligo-membranes immersed inexcess water or polymer solution exhibit defects which re-sult in decreasing layer coverage with increasing distancefrom the substrate. This effect is evidenced experimen-tally by the suppression of thickness oscillations (Kiessigfringes) in the reflectivity curves. Thickness oscillationsare typically observed in vapor-hydrated samples but aresignificantly reduced or suppressed in samples immersed inaqueous solution [14,25]. In the model we take this effectinto account by multiplying the contribution of each mem-brane in the structure-factor with an empirical coverage-factor f ( n ) = [1 − ( n/N ) α ] , where α parameterizes thedecaying density due to decreasing coverage. A typicalexperimental value is α ≈ .
7. Note that both α and thenumber of bilayers N are fit parameters.Since the form factor | F f ( q z ) | of the membranes con-sists of the squared Fourier transform of the z -derivativeof the electron density profile of the membrane, it is conve-nient to express the profile in terms of normalized Fouriercoefficients [15] ρ ( z ) = N X m =1 f m · cos (cid:16) πmzd (cid:17) · ρ + ¯ ρ − ρ w , with ¯ ρ being the average electron density of the membranestack and ρ w the electron density of water. For DMPC,¯ ρ = 0 . − / ˚A [4], very close to ρ w = 0 .
332 e − / ˚A , sothat | F f (0) | ≃ Figure 3 shows the reflectivity measurements (symbols)of 16 DMPC membranes on silicon substrates at four (outof nine measured) different osmotic pressures. The curveshave been stacked vertically for clarity, with increasingpressure from 4 kPa (bottom) to 195 kPa (top). The con-tinuous lines are simulations based on the model describedabove with the corresponding electron density profiles shownin Figure 4A. The reflectivity spectra presented in Figure3 were each scaled by the corresponding Fresnel reflectiv-ity. The error bars in the intensity at point n , ∆I n , areestimated considering Poissonian statistics (both for theraw reflectivity and for the offset scan). The error bar inthe q z direction is taken as ∆q n = ∆q = 5 × − ˚A − ,corresponding roughly to the symbol size, and is given bythe estimated precision in sample alignment.¿From the electron density profiles one can see thatthe increase of periodicity d with decreasing pressure ismainly due to changes in the water layer thickness, whilethe bilayer structure is essentially invariant over the rangein Π studied, with a headgroup spacing (distance be-tween the two maxima in the electron density profile) d HH = 34 ± . . et al. [4]. The simulations matchthe measured reflectivity curves not only at the positionof the Bragg peaks, but in the whole continuous q -range ofthe measurement. At lower osmotic pressure Π the higher lrike Mennicke et al.: X-ray reflectivity of lipid lamellar stacks under osmotic pressure 5 R / R F z [Å -1 ]0.11100.11100.1110 c=14% Π =195 kPac=9% Π =77 kPac=3.6% Π =14 kPac=1.5% Π =4 kPa -10010 (I n - F n ) / ∆ n z [Å -1 ]-10-50510-20-15-10-505-30-20-1001020 -4-20240.60.50.40.3 -4-2024-4-2024-4-2024 Fig. 3.
Left : Symbols : measured reflectivity curves of 16 DMPC membranes on silicon substrates under different osmoticpressures, normalized by the corresponding Fresnel reflectivity. Solid lines : show full q -range simulations using the describedmodel, with the density profiles given in Figure 4. The polymer concentration and the osmotic pressure are specified for eachcurve. Right : Residues of the reflectivity data, normalized by the estimated standard deviation (see text for details). The graphis horizontally split into two panels, with different y axes : Below 0 .
25 ˚A − , the large discrepancies correspond to systematicerrors, while for q z > .
25 ˚A − the residues are more randomly distributed and their scaled amplitude is below 5. -30 -15 0 1 5 3 00.20.30.4 z[Å]
195 77 14 4 ρ el electron density [e - /Å ] ρ el (H O) [kPa] Fig. 4.
Electronic density profile of the bilayer, as a functionof the applied osmotic pressure Π . order peaks are suppressed due to increased thermal fluc-tuations, as quantified by the above model. As an illustra-tion, Fig. 5A shows the increase in the fluctuation ampli-tude as a function of the membrane index for a 16 bilayersample.As discussed above, the relevant parameters for thereflectivity curves are the mean squared fluctuation am-plitudes (cid:10) u n (cid:11) , which give access to the Caill´e parameter η . In order to compare our data to the bulk results [4]we then compute the interbilayer spacing fluctuation am- plitude σ = (cid:10) ( u n − u n − ) (cid:11) = η d π . Soft confinementtheories predict an exponential dependence [4] of param-eter σ with the interbilayer distance, which can be takenas the thickness of the water layer, given by d w = d − d B ,where d B = 44 ˚A is the thickness of the DMPC bilayer [4].Fig. 5B shows the reciprocal of the fluctuation ampli-tudes as a function of the water spacing σ − ( d w ), alongwith a fit to an exponential decay (solid line). The datapoints can be fitted to an exponential function σ − ∝ exp( − d w /λ fl ), with a decay length λ fl = 4 . ± . λ fl = 5 . d was measured for all valuesof the osmotic pressure Π , up to 870 kPa. We show Π ( d )in Figure 6 (diamonds). For comparison, we also plottedthe fit by Petrache et al. of the bulk data, Figure 7, up-per panel in their paper [4] (dashed line). They performedthe measurements at 30 ◦ C and obtain d = d ( Π = 0) =62 . ◦ C, yield d = 61 . d ( T ) for DMPC [26]. A quick test can be per-formed by shifting their curve by 0 . d values(solid line). The agreement is good, but a more detailedanalysis is obviously needed for a meaningful comparison. We saw above that our data can be brought to agreementwith the bulk equation of state Π ( d ) by the 0 . Ulrike Mennicke et al.: X-ray reflectivity of lipid lamellar stacks under osmotic pressure w [Å] σ -2 = π /( η d ) exponential fit bulk values(Petrache et al, [3]) N = 16 η = 0.065d = 59.5 Å < u n2 > [ Å ] σ - [ Å - ] A B
Fig. 5.
A) Amplitude of the bilayer position fluctuations (cid:10) u n (cid:11) in a solid-supported membrane stack consisting of 16 DMPCmembranes at partial hydration, controlled by an osmotic pressure of 4 kPa (used for the model in Fig. 3, bottom). B) Reciprocalof the squared fluctuation amplitudes σ − ( d ) = π / ( ηd ), as determined from the reflectivity fits, and exponential fit (solidline). Also shown is the exponential fit to the bulk data of Petrache et al. [4]. For clarity, their experimental data points are notshown. Π [ P a ] Fig. 6.
Osmotic pressure Π as a function of the lamellar d spacing. Diamonds : experimental data points. Dashed line :Fit to the bulk data of Petrache et al. [4]. For clarity, theirexperimental data points are not shown. Solid line: same curveshifted to lower d values to account for the effect of temperature(see text for details). bulk and the present data will also be identical or at leastsimilar. However, the choice of the functional forms forthe potentials, the geometric partitioning of the bilayerto calculate the van der Waals part and the choice of thebending rigidity κ can all be debated. We therefore firstgive a brief discussion of the different interaction poten-tials used for neutral lipid membranes, and then presentresults based on modelling the equation of state Π ( d ).The hydration potential is usually empirically describedby an exponential function of the water layer thickness d w [27] : f hyd ( d w ) = H exp( − d w /λ ) , (5) with a prefactor of the order of H = k B T / ˚A and a decaylength on the order of a few Angstroms λ = 1 − V vdW ( d w ) = H vdW π (cid:20) d w − d w + d B ) + 1( d w + 2 d B ) (cid:21) , (6)where d w is the water layer and d B = d − d w the bilayerthickness. The expression should be regarded as an ap-proximation to the result of a more detailed treatment, asdiscussed by Fenzl [17]. Accordingly, the potential shouldbe calculated from F vdW ( d h , T ) = 0 . k B T πd h ∞′ X n =0 Z ∞ r n d x x (7)ln " − (cid:18) ∆ n (1 − exp( − ax/d h ))1 − ∆ n exp( − ax/d h ) (cid:19) exp( − x ) , where d h is again the thickness of the hydrophilic lay-ers consisting of the water layer and the headgroups and ∆ n = ( ǫ H O ( ω n ) − ǫ CH ( ω n )) / ( ǫ H O ( ω n ) + ǫ CH ( ω n )) is afunction of the frequency-dependent dielectric constantsof hydrocarbon and water. The prime symbol ′ indicatesthat the static term ( n = 0) has to be multiplied by1 /
2. The calculation is somewhat involved, however Fenzlhas shown that a frequently used approximation of Eqs.6 is valid for the dispersion term, but not for the staticterm which dominates under salt-free conditions. More-over, Podgornik and coworkers have shown that nonpair-wise additive contributions to the van der Waals interac-tion play a significant role in multilayers at large swelling[28]. However, for the present parameters, the above treat-ment should be sufficient.Apart from the molecular forces discussed above, stericforces resulting from membrane bending elasticity should lrike Mennicke et al.: X-ray reflectivity of lipid lamellar stacks under osmotic pressure 7 be included, as first introduced by Helfrich [29]. Accord-ingly, a repulsive undulation force arises f U = 0 .
42 ( k B T ) κd w , (8)which cannot, however, be simply added to the molecularforces. Instead, steric forces must be treated by field theo-retical approaches which go beyond the mean field approx-imation [1], or by self-consistent models [30], but whichto date have not been combined with realistic molecu-lar potentials in multilamellar stacks. Facing these com-plications, Petrache and coworkers [3] have pointed outthat the measured rms-fluctuation of the next neighbordistance σ = √ ηd/π can be used to experimentally de-termine the fluctuation pressure P fluct , which they thenadded to the pressure calculated from the molecular po-tentials to fit their data Π ( d ) = P mol + P fluct . Obviously,this approach avoids the problematic identification of thethermodynamic compression modulus to the bulk mod-ulus B as defined in the Caill´e model, but still assumesthat the total pressure can be written as a sum, whichmay strictly only be true in mean field approximation.The advantage of the approach is that it makes clever useof the experimental information from either of the inter-connected functions B ( d ), η ( d ), or σ ( d ) to compute thepressure. According to [3] P fluct = − (4 k B T ) (8 π ) κ dσ − d ( d w ) . (9)Fig. 5 shows the measured parameter the inverse of thefluctuation amplitudes σ − ( d ), along with a fit to an ex-ponential decay (solid line). The data points can be fit-ted to an exponential function 2 .
14 exp( − d w / .
17) ˚A − .Subtracting the corresponding fluctuation pressure P fluct obtained by differentiation according to Eq. 9 for a givenparameter κ from Π ( d ), the molecular interactions (hy-dration and van der Waals interactions) can then be mod-elled and compared to the data, as shown below in Fig.7. Below we give results for two different approaches inthe data analysis. The first approach is described in detailin [31]. It is based on the assumption that the periodicity d is dominated by the molecular forces, and that steric forcesare comparatively small for relatively stiff phospholipidmembranes.First approach: The calculation of the van der Waalsinteraction was based on equation (7) for the static con-tribution. The static part for n = 0 was numerically inte-grated between 0 and 100. For water ǫ H O (0) = 80 andfor hydrocarbon (tetradecane) ǫ CH (0) = 2 was taken.For the dispersion term a hydrophobic bilayer thickness26 ˚A and a Hamaker constant H dis = 0 .
297 was used. Thelatter value has been chosen to approximate (7), evalu-ated for dispersion relations ǫ H O ( ω ) and ǫ CH ( ω ) whichhave been parameterized by oscillator models as in [17].Note that in this approach there is no free parameter forthe vdW interaction. Figure 1 B shows the total interac-tion potential (solid line) with separately calculated static and dispersion terms as described above. A calculationbased only on Eq. 6 (with adjusted Hamaker constant) isshown for comparison (dashed curve). The parameters ofthe hydration interaction which were then freely adjustedwere H = 4 . k B T / ˚A and λ = 1 .
88 ˚A. The 10% reduc-tion from the fixed values in [17] in the van der Waals termfor equation (7) may be attributed to the fact that Helfrichrepulsive forces have not been included in the force bal-ance. At the same time, it is interesting to note the valueobtained for κ in this approach. To estimate κ we note thatthe Caill´e parameter η = π d k B T √ Bκ/d has been determinedfrom the full q -range fits to the reflectivity curves at differ-ent pressures. As the compression modulus B ( d ) = ∂Π/∂d has been independently determined by numerical deriva-tion of the measured Π ( d ) curve, one can now estimate thebending modulus κ from the experimental values of η ( d ).The best agreement was obtained for κ = 23 . ± . k B T .Within these uncertainties κ ≃ k B T compares quitewell with the value of κ = 19 k B T at 30 ◦ C determinedfrom bulk suspensions of DMPC by Petrache et al. [3].Note however that another study employing full q -rangefits and osmotic pressure variation reports κ = 11 . k B T [32], again at full hydration and comparable T . Finally,thermal diffuse scattering analysis points to significantlysmaller values κ = 7 k B T [33]. Note that the determina-tion of κ from the osmotic pressure series is based on aproblematic assumption, i.e. that the identification of thebulk modulus B as defined in the Caill´e model and thethermodynamic compression modulus is correct. More de-tails on the data analysis following this approach based onmolecular interactions only are given in [31].In the second approach we followed exactly the pro-cedure given by [3]. First, the fluctuation pressure is sub-tracted from P ( d ), and then the resulting bare pressureis modelled in terms of the molecular interactions. How-ever, the calculation of the fluctuation pressure accordingto equation 9 needs the bending rigidity κ as an additionalparameter. To illustrate the range of parameter variabil-ity, we present a comparison of two different choices ofparameters: (a) all parameters and functions are kept asclose as possible to those used in [3], in particular keep-ing κ = 18 . k B T fixed. The corresponding values for H = 0 .
020 J / m and H vdW = 4 . · − J are practi-cally identical to the values in [3], showing that the sameapproach can explain both bulk and thin film data. Thesame treatment has then been carried out for a differentchoice of κ = 8 k B T . Again the simulations can be broughtinto agreement with the data, but only for a different setof parameters H = 0 .
028 J / m and H vdW = 1 . · − J.Thus values at the lower and upper range of the κ valuesreported in the literature both lead to reasonable agree-ment, indicating that extra information from other ex-periments is needed to unambiguously determine the po-tentials. The potentials are shown for the two cases, andcan be compared also to the potential in Fig. 1(b), de-rived from the data analysis under the assumption thatthe fluctuation repulsion is negligible. Ulrike Mennicke et al.: X-ray reflectivity of lipid lamellar stacks under osmotic pressure P r e ss u r e [ J / m ] periodicity d [Å] -6 P o t en t i a l E ne r g y [ J / m ] oligo DMPC datamolecular potential troughpressure of molecular potentialfluctuation pressuretotal pressure H hyd = 0.028 J/m vdW =1.11e-20 J k = 8 k B T P r e ss u r e [ J / m ] periodicity d [Å] -6 P o t en t i a l E ne r g y [ J / m ] oligo DMPC datamolecular potential troughpressure of molecular potentialfluctuation pressuretotal pressure H hyd = 0.020 J/m vdW =4.91e-21 J k = 18.5 k B T Fig. 7.
Osmotic pressure Π as a function of lamellar peri-odicity d (same data as in Fig. 5(A)). The simulations havebeen carried out following the approach of [3] for (a) fixed κ = 18 . k B T and (b) κ = 8 k B T . The parameters of thehydration interaction H (Eq. 5) and the Hamaker constant H vdW (Eq. 6) are varied to match the data. The simulationsin (a) and (b) show the total pressure (solid line), the fluctua-tion pressure P fluct (dotted line), the pressure corresponding tothe molecular potential (dashed line), and the potential troughcorresponding to the molecular forces (dash-dotted line). Thefluctuation pressure as determined from Eq. 9 and the data inFig. 5 was added to the pressure stemming from the molecularforces. In conclusion, we have presented an osmotic pressure ex-periment on thin solid-supported lipid multilayers (oligo-membranes). The x-ray reflectivity has been measured andmodelled over the full q z -range up to 0 . − . From thisanalysis fluctuation and structural parameters can be ob-tained, similar to the lineshape analysis of bulk suspen-sions [3,5]. Solid-supported oligo-membranes offer someadvantages both over thick multilamellar films and thebulk counterpart. Long range thermal fluctuations are notas strong as for bulk samples, and the scattering can beprobed up to higher momentum transfer. Owing to the smaller number of bilayers, destructive interference in-between the Bragg peaks is not quite as strong as inthick stacks of several hundred bilayers. To achieve sat-isfactory fits, two important effects were taken into ac-count: (i) the static defects leading to a decreasing cov-erage of the bilayers from the substrate to the top of thefilm, and (ii) the thermal fluctuations of the bilayers sub-ject to the boundary condition of a flat substrate [24].While (i) most likely reflects non-equilibrium aspects ofsample deposition and/or equilibrium wetting properties(not further analyzed here), (ii) is exploited to deduce in-teraction parameters in the framework of linear smectictheory. The curve B ( d ) derived from the osmotic pressureseries Π ( d ) is subsequently modelled based on differentinteraction potentials. However, this modelling cannot becarried out without assumptions or additional theoreticarguments.In the data analysis, we have presented two entirelydifferent approaches to illustrate how the determinationof interaction forces depends on the specific assumptions,theoretical arguments, or extra information taken fromother experiments. The first approach builds upon therather strong assumption that steric forces are negligibleand that the derivative of the equation of state ∂Π ( d ) /∂d can be identified with the modulus B ( d ) which controlsthe thermal fluctuations. It then yields the parameters ofthe hydration force necessary to balance the van der Waalsattraction at each given osmotic pressure. This approachalso gives a value for the bending constant from simulta-neous inspection of B ( d ) and η ( d ). However, the resulting κ ≃ k B T , is probably an overestimation. The ratherlarge value may point to the fact that B is underestimatedby the contribution of only the bare potentials. Adding afluctuation pressure would tend to increase B and thusdecrease κ = K d . Note that this determination of κ isconceptually very different from a more direct assessmentof κ , e.g. from the measurement of diffuse scattering.The second approach includes the steric Helfrich un-dulation forces. This contribution is a subtle issue for thefollowing reasons: (a) it has been shown by Lipowsky andcoworkers that the Helfrich term cannot be simply addedto the molecular forces. If one nevertheless uses a meanfield approach, (b) the functional form to be used as well asthe numerical prefactor are still under debate [34]. There-fore, we have followed an idea of Petrache et al. [3], whohave carried out an osmotic pressure study on DMPC,which is the bulk analogue of the present work. Calculat-ing the partition function within the linear smectic elastic-ity model, they have derived derived an expression for thefluctuation pressure as a function of a measurable quan-tity, namely the derivative of the fluctuation amplitudes,see Eq. 5. In a mean field treatment, they add this pres-sure to the bare pressure calculated from the interactionpotentials and fit the sum to the measured curve Π ( d ). Inthis step, an assumption of κ has to be made, e.g. fromother experimental data. This approach has been carriedout for two choices of κ , see Fig. 7.We point out, however, that the questions related tothe interaction potentials arise only on a secondary level, lrike Mennicke et al.: X-ray reflectivity of lipid lamellar stacks under osmotic pressure 9 where structural results ( d and σ ) are interpreted andtransformed to elasticity and interaction parameters. Onthe primary level that the structural results presentedhere, i.e. Π ( d ), ρ ( z, Π ) and (cid:10) u n (cid:11) ( d ) are well supportedby the curves and fits shown here. The second level isnecessarily model-dependent. We have presented two al-ternative approaches to illustrate the relation and interde-pendencies of different assumptions and results. It may bejustified to conclude that the second approach as proposedand used by [3] is more appropriate, since steric repulsionis known to be important. However, the choice of the vander Waals expression may have to be improved to a moreaccurate form, and the choice of κ is also an importantissue. Unfortunately, the second approach also relies onthe validity of a mean field approximation. This simplifi-cation could be eliminated in the future by generalizationof a recent self-consistent calculation for bilayer fluctu-ations and interactions [30] to the case of several mem-branes or by use of the approach developed in [35]. Fur-thermore, non-linear effects due to the asymmetry of thepotential could also be included by more general models[36,35] and/or numeric simulations. To elucidate the valid-ity of the linearized model a posteriori , the rms-deviation p h ( u n − u n +1 ) i between neighboring membranes can becompared to the width of the inter-bilayer potential well,see Figure 1 (B). For N = 16 and η = 0 .
08 (full hydration)the bilayers in the center of the stack already exhibit con-siderable next-neighbor distance fluctuations in the rangeof 4 − Appendix
In this appendix, we give a sketch of the derivation ofour formula (3.1) for the reflectivity, insisting upon theseparation between the specular and diffuse components.This is a classical result and a more detailed derivationcan be found in references [37] (equation 2.28) and [22](subsection 3.8.3) for the case of single interfaces and in[38] (section 3) for multiple interfaces.It is well known that bulk lamellar phases exhibit theLandau-Peierls instability, leading to a characteristic power-law variation of the scattered signal [6]. In such a systemthe fluctuation amplitude h u n i diverges. It is then moreappropriate to use the correlation of the height difference,which remains finite for all finite values of r : g mn ( r k ) = D(cid:0) u m ( r k ) − u n ( ) (cid:1) E (10)= h u m i + h u n i − h u m ( r k ) u n ( ) i It is then easy to show [37,38] that the structure fac-tor of the lamellar stack (without taking into account the substrate contribution, so only the third term in Eq. (3.1)is described) reads : S ( q ) = X m,n e − iq z d ( m − n ) Z d r k e − i q k r k e − q z g mn ( r ) . (11)where r = | r k | , assuming that the fluctuations are isotropicin the membrane plane. For bulk systems, lim r →∞ g mn ( r ) = ∞ , so that lim r →∞ exp (cid:2) − q z g mn ( r ) (cid:3) = 0 and the Fouriertransform with respect to r k in formula (11) yields a ”smooth”function S ( q k ) at fixed q z . If, however, g mn ( r ) does not di-verge for r → ∞ , the function exp (cid:2) − q z g mn ( r ) (cid:3) now hasa constant background, at a value of exp (cid:2) − q z g mn ( ∞ ) (cid:3) =exp (cid:2) − q z (cid:0) h u m i + h u n i (cid:1)(cid:3) , quantifying the ”remanent or-der” in the system. Its Fourier transform is a Dirac deltafunction δ ( q k ) (in practice, its width is given by a com-bination of resolution effects, beam coherence and systemsize). This term is sometimes called the ”true specularcomponent”, because the smooth function discussed above(the ”diffuse” component) also contributes to the specu-lar signal S ( q k = , q z ). However, as the diffuse scatteringvaries over a much larger q k range, it can be accountedfor in the first approximation by an offset scan taken closeenough to the specular sharp peak (see figure 2). Finally,we can write : S spec ( q k = , q z ) = X m,n e − iq z d ( m − n ) e − q z ( h u m i + h u n i )(12)which is the form employed in equation (3.1). It is note-worthy that this ”true specular” contribution is distinctin nature from the signal measured in SAXS experimentson powder samples, where only the diffuse signal persists. Guillaume Brotons is acknowledged for helpful discussions onthe osmotic stress technique. D. C. has been supported bya Marie Curie Fellowship of the European Community pro-gramme
Improving the Human Research Potential under con-tract number HPMF-CT-2002-01903.
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