Collective membrane motions of high and low amplitude studied by dynamic light scattering and micro-interferometry Rainer Hirn,a Thomas M. Bayerl,*a Joachim O. Raé dlerb and Erich Sackmannb a Universitaé t W ué rzburg Physikalisches Institut EP-5 97074 Wué rzburg Germany b T echnische Universitaé t Mué nchen Physik Department E22 85747 Garching Germany Receiøed 9th October 1998 Undulations of lipid bilayers were experimentally studied for the two limiting cases of high and weak lateral tension using two well established model systems freely suspended planar lipid bilayers so-called black lipid membranes (BLM) for high-tension studies and large unilamellar vesicles (LUV) for measurements at weak tension. This variation in tension results in changes of undulation amplitudes from several hundred nm (LUV) down to 1 nm (BLM) thus requiring diÜerent physical methods for their detection.We have employed microinterferometric techniques (RICM) for studying the regime of weak tension and dynamic light scattering (DLS) for that of high tension. The dedicated DLS set-up allowed the measurements of undulations over a wide wave vector range of 250\q/cm~1\35 000 cm~1. This enabled the observation of collective membrane modes in two regimes the oscillating one at low q and the overdamped regime at high q. The transition between both regimes at the bifurcation point is rather abrupt and depends on the lateral tension of the bilayer as is demonstrated by comparing the dispersion curves of pure lipid and of lipid»cholestrol BLMs over the same q-range.The DLS measurements allowed a critical test of a hydrodynamic theory of the dispersion behaviour of membrane collective modes under tension. The DLS measurements are compared with RICM results of undulatory excitations of giant vesicles weakly adhering to substrates in the 10~6»2.5]10~7 m wavelength regime and at low frequencies (0.1»25 Hz). Experimental evidence for the strong decrease in the relaxation rate by the hydrodynamic coupling of the membrane with the wall is established. Introduction Collective motions (undulations) of —uid membranes are crucial for their mutual interaction at the nm lengthscale and thus for the swelling of lipids.1 The measurement of these motions provides not only a deeper understanding of membrane micromechanic properties down to the molecular scale but also sheds some light on the processes which are essential for the formation of contacts between cells and between cells and solid surfaces.In the low-frequency regime (up to 100 Hz) time-resolved interferometric microscopy is the method of choice to study collective motions of membranes of whole cells and of some microorganisms in great detail. On the other hand incoherent neutron scattering has been successfully employed for the measurement of collective membrane modes in the THz range.2 However there is quite a lack of experimental approaches for their study in length and frequency regimes which are commonly denoted as the mesoscopic range (kHz»MHz). So far only solid-state NMR techniques have been successful in detecting collective modes in this range,3 but their application is 17 Faraday Discuss.1998 111 17»30 limited by sensitivity sample geometry narrow accessible frequency range and NMR data analysis is hampered by the constraint that one-dimensional solid-state NMR measures temporal correlations only and not spatial correlations. The availability of a technique which can bridge the wide frequency gap of ca. six orders of magnitude between the well established low-frequency regime and the highest accessible frequency range would be extremely helpful in the testing of a number of theories based on membrane microelasticity as well as for sorting out the essential collective modes for each time domain. DLS has been used previously to measure collective motions of BLM and it demonstrated successfully the viscoelastic behaviour of such membranes4 but their usefulness was limited by the narrow range of accessible undulation modes from 600 to 1 800 cm~1.Here we report the –rst results of DLS measurements on BLMs obtained over a much wider wavevector range from 250 to 35 000 cm~1 (1.8\j/lm\250) and a timescale of four orders of magnitude (from 6]10~3 to 6]10~7 s) which is sufficient for a critical testing of theoretical predictions. A transition from single to two-exponential relaxation is encountered at high tension which exhibits much higher frequencies in the vicinity of the bifurcation point. The experiments in this new frequency regime enable one to study the eÜects of proteins peptides and polymers on collective membrane dynamics.Since BLMs exhibit generally a high lateral tension their undulations are dominated by lateral forces resulting in very low undulation amplitudes. On the other hand LUV show only negligible or very weak tension and thus a diÜerent undulatory behaviour with high undulation amplitudes.5 The latter is constrained when the LUV comes into close contact with a wall. As a result a repulsive pressure is established which is gradually reduced with increasing lateral tension. A comparison between tension dominated BLMs and rather tensionless LUV thus allows a critical test of theoretical predictions on membrane micromechanics in the limit of two extremes. We have additionally performed time-resolved re—ection interference contrast microscopy (RICM) of LUV close to a solid surface.The results show that the relaxation modes are dominated by the hydrodynamic interaction between the LUV and the wall. (1) Theory of DLS of membranes under high tension One of the –rst comprehensive theoretical treatments of the dispersion behaviour resulting from collective motions of thin elastic membranes and the consequences for its dynamic light scattering properties was published by Kramer.6 The basic assumptions of this theory were The membrane elastic properties can be described by compression- and shear moduli and a membrane tension. All hydrodynamic equations (i.e. Navier»Stokes) are linear. The —uid symmetrically surrounding the membrane is incompressible. At the membrane/—uid interface the velocities of the two media are the same.Fluid velocity becomes zero at large distances from the membrane. The wavelength of the collective modes is large compared to the membrane thickness and small compared to its diameter. For a membrane of isotropic molecules separating two compartments of the same liquid transverse shear was identi–ed as the only DLS sensitive mode. Here the molecules perform out-ofplane motions only leading to an eÜective —uctuation of the membrane area. The dispersion relation of this mode is 2mou2]cq3(q[m)\0 where m\(q2[iou/g)1@2 q\2n/j is the scattering vector u\u0[iC is the complex frequency consisting of the eigenfrequency u and the damping constant C. o and g are the density and 0[iuc@ is a complex tension with the real part being the membrane c\c viscosity of the —uid and 0 tension and c@ being the surface viscosity.A plot of f and C vs. q of eqn. (1) using tensions typical for a free planar bilayer (BLM) shows and the faster by C 0 the following features (Fig. 1) which were discussed in greater detail by Kramer6 and later in the work of Earnshaw:4 For transverse shear there exists an oscillating and an overdamped regime in q the mesoscopic range and the transition between them at 0B5000 cm~1 is abrupt (bifurcation point). Above this q value the single damping constant C of the oscillating regime is replaced by 0 two exponential decays describing two overdamped modes with the slower one characterized by C 2 . No experimental veri–cation of this transition has been provided yet.For 1 Faraday Discuss. 1998 111 17»30 18 c0\1 mN m~1 Fig. 1 Theoretical curve according to Kramerœs theory [eqn. (1)] with membrane tension —uid density o\1 mg ml~1 and —uid viscosity g\1 mPs at diÜerent surface viscosities c@. increasing q-values above the bifurcation point the slower mode C approaches the value m\c0/c@ 1 asymptotically while the faster mode C merges at the same value of m with the bulk mode having 2 the dispersion relation (2) iu[gq2/o\0 Moreover this bulk mode is insensitive to DLS and therefore the faster mode disappears at q-values beyond m. By considering the anisotropy of lipids in a BLM an additional splay mode coupled to the transverse shear mode was found.7 The coupling strength between the two modes scales with the total free energy change a single lipid undergoes when its molecular director gets tilted by a certain angle away from that of its neighbour.This leads to a modi–cation of the membrane tension in eqn. (1) now becoming an eÜective tension (3) c0eff which includes the curvature energy i c0eff\c0]iq2 0 Hence at higher q the bending modulus i will dominate undulation behaviour over the membrane tension c (Fig. 2). In terms of the eÜective undulation amplitude ueff of a membrane of area A and driven by a thermal energy kT this corresponds to the Helfrich1 equation (4) Sueff 2 (q)T\ A(cq2]iq4) kT For standard BLM parameters (c0\1 mN m~1 i\10~19 J BLMdiameter\3.5 mm) eqn. (4) u gives effB0.09 ” at q\1000 cm~1 and a maximum amplitude of ueffB10 ” is obtained by summing the contributions over a q range limited by the cut-oÜs 1/membrane thickness and 1/membrane diameter.The reason for these tiny amplitudes compared to those reported below for LUV is the dominance of tension for BLMs while the —ickering systems are tension free and thus i dominated. For BLMs the eÜect of i is negligible up to very high q values of say q\300 000 cm~1 (j\200 nm) where the contribution from i amounts to 1/10 of that of the tension term. Nevertheless it has been shown experimentally4 that DLS is sensitive even to these very small amplitudes exhibited by BLMs in the q-range 600 to 1800 cm~1. 19 Faraday Discuss. 1998 111 17»30 1 Fig. 2 c Comparison of Kramerœs [eqn. (1)](2)] and Fanœs dispersion theory [eqn. (1)] for two pairs of and 0 i values.For the sake of clarity only the slow damping modes C are shown. The curves with the increasing slopes at higher q-values (curves 1 and 2) correspond to Fanœs theory. The upper two lines were calculated for c0\1 mN m~1 and i\0 (curve 3) or i\10~19 J (curve 2). The lower two lines are for tension c0\0.1 mN m~1 and i\0 (curve 4) or i\50]10~19 J (curve 1). However to obtain information about the surface viscosity c@ requires DLS measurements over a q-range that extends well into the overdamped regime since the dependence of C and of the 0\u0/2n on c@ is almost negligible in the oscillating regime (Fig. 1). Similarly a f eigenfrequency 0 representative test of eqn. (1) and (2) is impossible without measurements covering q in both the oscillating and overdamped regimes.(5) (6) (7) Theoretical predictions for —accid vesicles near a wall One of the most striking aspects of bending excitations of —accid membranes close to a wall is the strong repulsive interaction pressure pD(kB T )/ih0 3 arising from the entropy loss of a membrane con–ned to an average membrane»wall distance h0 . In the presence of membrane tension however the steric repulsion is strongly suppressed and decays exponentially pDexp([ch0 2/kB T ). For a weakly adhering liposome the entropic pressure is balanced by an attractive wall interaction. In this case the external forces experienced by the membrane can be described by an eÜective harmonic interaction at the equilibrium spacing h from the wall 0 V (h)\1/2V A(h[h0)2 where V A is the second derivative of V (h) at the equilibrium distance h0 .Since we are mainly interested in small-wavelength excitations (q~1) the undulation-induced dynamic surface roughness can be analysed in terms of plane waves h(r t)\; hq(t)expMiqrN q where q is the undulation wave vector. The amplitudes are determined by the equipartition theorem hq2\ kB T L2Miq4]cq2]V AN Faraday Discuss. 1998 111 17»30 20 where E(q)\iq4]cq2]V A is the energy of the Fourier mode q and i and c are again the membrane bending modulus and tension respectively. L2 is the system size over which the Fourier transformation was carried out. The dynamics of the randomly excited bending modes is characterized in terms of the time correlation function of the Fourier components which are given by Gq(t)\Shq(t)hq(0)T\kq2 expM[C(q)tN (8) with a relaxation rate C(q) to be discussed.In most LUV experiments we deal with membranes at weak tension (cB10~6 mN m~1). This situation has been treated theoretically in great detail by Seifert.8 One has to consider three situations (1) At low wave vectors (and thus low frequencies) the local density —uctuations within each monolayer are rapidly equilibrated by lateral diÜusion and the bending excitations are damped solely by coupling to the bending-induced hydrodynamic —ow in the surrounding —uid. (2) At increasing q the equilibration of the local density —uctuations is impeded by the friction between the monolayers which is characterized by a friction coefficient b\gm/dm where g is the m 2D membrane viscosity and d is the monolayer thickness.The bending modulus is renormalized m by the coupling of the bending excitations and the lateral density —uctuations to (9) i*\i]2dmK where K is the membrane modulus of isotropic compression. (3) At very high frequencies and q-vectors the damping is –nally determined by the in-plane shear deformation which is not considered here. Due to the interplay of bending and density —uctuations the membrane dynamics (of free and adherent vesicles) is determined by a low-frequency mode and a high-frequency mode. The relaxation of the latter is determined by monolayer friction. The cross-over wave vector between the two regimes is (10) q12\2gK/bi*B107 m~1 The number on the right-hand side holds for the data summarized in Fig.3. In the presence of a wall the situation is more complex. A third cross-over wave vector has to be considered since long wavelength excitations are impeded by the interaction of the membrane k F m m and are de–ned in eqn. (14) and (15). Note that at q\m both decay rates scale as C P q2. 0 h0\1 nm c\10~6 J m~2 Fig. 3 Theoretical dispersion relation of decay rates of bending excitation of adherent membrane as calculated by Kraus and Seifert.9 For weak adhesion corresponding to weak tension the two lowest frequency modes (out of three) are shown. C is the decay rate of the slow bending mode with rapid equilibration of local W density —uctuations within monolayers. C is the decay rate of the mode controlled by mutual friction between monolayers.The curves have been calculated for the following values of the parameters 0 V A\4]106 J m~4 b\5]108 J s m~4 g\10~3 J s m~3 K\0.1 J m~2 i\0.5]10~19 J. 21 Faraday Discuss. 1998 111 17»30 with the hard wall. Due to the symmetry break induced by the wall there are two density modes (besides the bending mode) For negligible tension the undulations are determined by the wall at qPmk~1\(V A/i)1@4\2]108 m~1 while for membranes under tension the wall becomes eÜective at qPmc~1\(V A/c)1@2\106 m~1 The transition between the bending and the tension dominated regimes occurs at wave vectors (11) (12) (13) q\m\(c/i)1@2\3]106 m~1 The RICM experiments were performed at a tension cB10~6 N m~1 and the range of wave vectors studied was 0.6]106\q/m~1\4]106.We can thus assume that the tension can still be considered as weak. Summing up the above conditions let us conclude that there should be two modes accessible by microinterferometry A low-frequency long-wavelength mode CW which is aÜected by the presence of the wall (being at distance h0) but which is slow enough to allow for the equilibration of the monolayer density —uctuation and which is not aÜected by monolayer friction. Its damping constant is (14) CW(q)\(V Ah0 3 q2]ch0 3 q4)/12g q@(mi~1 h0~1)B107 m~1 and is thus observable by the This mode dominates the regime of microinterferometric technique. For bending modulus q nm h0~1AqAm~1 the damping constant is dominated by the C(q)\ih0 3 q6/12g which is however not observable by our technique since max\5]106 m~1.With the data summarized in Fig. 3 one –nds for qB106 m~1 and h0B40 CWB0.02 s~1 corresponding to a relaxation time qB50 s. The second interesting mode CF is controlled by the friction between the monolayers. It decays with a rate (15) CF\Kq2i/2bi* By inserting the data from Fig. 3 one can estimate a wave vector qB106 m~1 a decay constant CFB20»200 s~1 or a relaxation time q\0.05»0.005 s. A completely diÜerent type of behaviour is expected for permeable membranes as shown by Prost et al.10 The undulations can relax by permeation of water through the bilayer. This mechanism is expected to dominate over the hydrodynamic process for wave vectors q@q*\(jp g)1@2 ~3@2 where j is the water permeability of the bilayer.It is of the order jpB10~6 h0 m2 s kg~1 and p 0B40 nm one obtains for q@3]105 m~1. h The decay rate is (16) m~1) s C ~1 and in the case of bending dominated membranes C Cp\jp(cq2]iq4) For q\106 m~1 one expects in the case of tension dominated membranes (c[2]10~6 N pB0.1 pB1 s~1. Materials and methods Substances Cholesterol and n-decane 99]% were purchased from Sigma-Aldrich (Steinheim Germany) and the n-decane was further puri–ed by passing it through an alumina column until all coloured impurities were removed. The phospholipid 1,2-di-elaidoyl-sn-3-glycerophosphocholine (DEPC) was purchased from Avanti Polar Lipids Inc. (Alabaster AL USA) and was used without further puri–cation. All light scattering experiments were performed in 20 mM Hepes buÜer (Life Technologies Ltd.Daisley UK) containing 50 mM KCl (Fluka Chemie AG Buchs Switzerland) and the water used was from a Milli-Q puri–cation unit (Millipore Corp. Bedford USA). The buÜer was –ltered through a sterile –lter of 0.1 lm pore size (Millipore Corp. Bedford USA) and Faraday Discuss. 1998 111 17»30 22 degassed prior to its use in the scattering cell. Microinterferometric measurements used SOPC from Avanti Polar Lipids in Millipore water. Membrane preparation The scattering cell was a standard rectangular glass cuvette of 40]10]10 mm3 (Hellma GmbH & Co. Mué lheim Germany) separated into two compartments by a diagonally inserted Te—on wall (thickness 2 mm)11 with a circular aperture of 4.5 mm in the centre. Over this hole a 25 lm thick Te—on foil was spanned using Te—on glue (primer 770 and bonder 406 from Loctite GmbH Mué nchen Germany) which itself featured a hole of 3.5 mm diameter.The latter hole was punched with extreme care to ensure its rim was as smooth as possible (roughness was below the optical resolution limit of a light microscope). This smoothness together with the use of a very thin foil ensures the BLM long-time stability (up to 5 days) and prevents positional —uctuations of the BLM as a whole. The scattering cell was placed within a metal cell holder the temperature of which (22 °C for all measurements) was controlled by a water-bath thermostat. Before BLM preparation the scattering cell was –lled with buÜer solution to a level of ca. 3 mm above the upper edge of the Te—on wall so that both compartments were connected by the water.BLMs were prepared by sliding a Te—on loop containing the –lm-forming solution over the hole.12 Prior to this the hole was pretreated by spreading a methanol solution of 2 wt.% lipid onto it followed by methanol removal by drying in air.8 The term lipid refers to pure DEPC for BLM sample 1 and to a mixture of DEPC with 30 mol% cholesterol for BLM sample 2. The –lm-forming solution was n-decane containing 1 wt.% lipid. BLM formation was con–rmed by the formation of a sharp spot of specular re—ected incident laser light. A rectangular Te—on pressure cap was then carefully lowered from the top into the scattering cell down to the water level thereby eliminating any air bubbles and eÜectively damping out all surface capillary waves of the water which might couple with the BLM below.Dynamic light scattering (DLS) Experimental set-up. The scattering geometry of our set-up is shown in Fig. 4. Note that undulations like the transverse shear mode represent plane waves u(r t) with (17) u(r t)\u0 e~i(qr`ut) and can spread in all directions perpendicular to the membrane normal thus giving rise to both in-plane and out-of-plane scattering of the incident light vector ki . Thus in a plane perpendicular to the membrane normal at the in—ection point all possible q-vectors belonging to the same undulation mode are con–ned within a circle. However owing to the inclination of the detector i k (scattering angle 45°) the inplane wavevector q the in-plane scattered vector ks (with quasi the ki i.e.quasi elastic light scattering) the out-of-plane wave vector q@ and the out-of-plane Fig. 4 Schematic depiction of the scattering geometry with the incoming vector k and specular re—ected vector same length as ir scattered vector ks @ . 23 Faraday Discuss. 1998 111 17»30 ir plane by 45° with respect to the membrane normal (i.e. the plane perpendicular to the specular re—ected vector k k ir) the region of scattered vectors at the detector site can be approximated to s the –rst order by an elliptical aperture with an aspect ratio of 1.41 in front of the detector and with k going through its centre. The DLS set-up was mounted on a 3]2]0.2 m3 (200 kg weight) laser table (Melles Griot) supported by four air damping modules for mechanical shock protection.A 4 W argon ion laser (Coherent Inc. Santa Clara CA USA) was used operating at 457.9 nm in TEM mode with a 00 maximum power of 150 mW. The beam was focused onto the BLM at an incident angle of 0.11° by a lens of f\50 cm to obtain high q-resolution giving an illuminated BLM spot of 160 lm. Its polarisation was adjusted parallel to the membrane plane to give the highest amount of scattering intensity. At a re—ection angle of 45° and at a distance of 30 cm from the BLM the photomultiplier (PM) was arranged at a goniometer arm. Two pinholes between BLM and PM one right after the BLM (£\700 lm) and one in front of the PM (£\80 lm) were used for q-range selection. For measurements at q[3400 cm~1 the PM pinhole was replaced by a vertical slit aperture (2000]200 lm2) to allow additionally the detection of out-of-plane scattering giving an increase in PM signal by a factor of 80 without any reduction in signal quality compared to the use of two pinholes.The slit aperture represents an approximation of the above-mentioned elliptical shape of the out-of-plane scattering at detector site by simply rotating the PM with respect to the BLM centre thus replacing the ellipse by its tangent in each point. The error made by this approach compared to the use of diÜerent elliptical pinholes for each q-value is 3% at q\3400 cm~1. The PM signal was preampli–ed and the heterodyne autocorrelation function calculated in 388 channels using an ALV 3000 correlator (ALV GmbH Langen Germany).The time increments used were in the range 0.1 to 15 ls. Here the diÜuse scattering arising from the molecular roughness of the membrane was used as local oscillator for the heterodyne mode. Data analysis. For the oscillating regime the theoretically expected autocorrelation function2 (18) G0 @ (t)\a]b cos(u0 @ t]U) e~\0t]ct was –tted to the experimental data. Here ct is a linear baseline correction and U is a phase factor. Corrections of the above autocorrelation function at small q for instrumental eÜects as suggested in ref. 4 were not performed since deviations from eqn. (5) were completely negligible for q[400 cm~1. At the transition point (bifurcation point) between the oscillating and the overdamped regime and within a narrow q-range above it (250»400 cm~1 depending on the membrane used) the data were –tted according to (19) Gt(t)\a](b]ct) e~Ct]dt Hence in this regime both overdamped modes C and C were generally detectable.In the over- 1 2 damped regime where two damping modes are expected according to eqn. (1) a function of the form (20) Gd(t)\a]b e~\1t]c e~\2t]dt was –tted to the data again including a linear baseline correction dt. However beyond C2\c0/c the fast overdamped mode becomes undetectable and the remaining slow mode C can be –tted by 1 a single exponential. Examples of representative data sets for q-values in the oscillating and the overdamped regime together with the –tting results according to eqn. (18) and (2) are shown in Fig. 5. Microinterferometric technique The microinterferometric technique developed for the analysis of undulatory excitations of weakly adherent vesicles (i.e.weak lateral tension) has been described previously by Raé dler et al.13 The vesicles are observed by RICM enabling local measurements of the distance between substrate Faraday Discuss. 1998 111 17»30 24 Fig. 5 Examples of typical correlation functions G(t) measured in the damped and in the overdamped regime for a free planar bilayer (BLM). The full lines represent –ts to the data according to eqn. (5) and (7). To distinguish the –t from the data more clearly not all data points are represented by a circle. Note that the overdamped dataset is –tted only with a mono-exponential function. and vesicle with high precision (^1 nm). The method is illustrated in Fig.6(a). The adhering liposome shows interference fringes at the contact rim where the contour bends away from the surface with a –nite contact angle. The —at centre part of the liposome appears dark and exhibits dynamical intensity —uctuations due to height —uctuations. The heights h(x y) are obtained from the digitized interference intensities by inverse cosine transform using a Pixelpipeline frame grabber (Perceptics) and NIH image software. A one-dimensional contour line hx\Sh(x y)Ty was evaluated by averaging over the width of a –nite stripe as shown in Fig. 6(b). The height pro–le hx along the stripe with length L was numerically Fourier transformed and the resulting one- G dimensional modes correlated in time qx(t)\Sh 8 qx(t)/h 8 qx(0)T.In this case the correlation function Gqx(t) is given by (21) Gqx(t)\PdqyShqx qy 2 TexpM[C(qx qy)tN Note that the correlation function Gqx(t) exhibits a superposition of relaxation modes qy which is a result of the data processing due to the integration over the –nite width of the evaluated stripe.9 25 Faraday Discuss. 1998 111 17»30 Fig. 6 (a) Analysis of undulations of a —accid vesicle near a transparent substrate by re—ection interference contrast microscopy. The image is generated by interference of light re—ected from the substrate and the adhering body respectively. For contrast enhancement the glass substrate is covered by an MgF –lm. (b) Snapshot of weakly adherent vesicle. The leopard-like pattern is due to —uctuations of the distance h(x y) 2 between substrate and vesicle.Bright areas correspond to large and dark areas to small values of h(x y). The white frame shows the stripe over which a one-dimensional time correlation analysis was carried out. f0(q) and the damping C(q) as shown in Fig. 7A. For the overdamped data set shown in Results BLMS measured by DLS Typical autocorrelation functions measured by DLS at selected q values corresponding to the damped and to the overdamped regime of the transverse-splay mode of BLM (3.5 mm diameter) of DEPC are shown in Fig. 5. The data were –tted according to eqn. (6) and (8) giving the mode frequency Fig. 5 a single exponential corresponds to the slow mode C1. However in the vicinity of the bifurcation point only two-exponential –ts [eqn. (7)] gave satisfactory results.This clearly indicates that around the bifurcation point the fast mode C is indeed detectable while it decreases 2 rapidly and merges with the non-detectable bulk mode [eqn. (2)] at higher q. This behaviour strongly suggests a non-negligible eÜect of the surface viscosity c being not less than 10~7 mN s m~1. The data in Fig 7A are compared with best –ts according to hydrodynamic theory [eqn. (1) full lines in Fig. 7A] to obtain the average lateral tension c and the shear interfacial viscosity c@ 0 acting in the normal direction of the membrane. It is obvious that the agreement between experiment and theory is excellent over almost 3 orders of magnitude of q. The predicted transition from the damped to the overdamped case (cf. Fig. 1) is clearly observed experimentally above q\3300 cm~1.Minor deviations of f0(q) from the theory at lowest measurable q in Fig. 5A can be ascribed to a slight average overall equilibrium deformation of the BLM giving rise to some diÜuse re—ection of the incident laser light at the lowest values of q. Moreover at q\400 cm~1 there could be a non-negligible contribution arising from the (Gaussian) instrumental function of the set-up. 0\0.42 mN m~1 From the –ts to the data in Fig. 7A an average value of the lateral tension c with a maximum deviation of ^0.03 mN m~1 is obtained while the viscosity c@ was in the range 2]10~7 mN s. The rather weak in—uence of c@ seems justi–ed considering that the BLM is not expected to exhibit any signi–cant frictional drag between its two constituent monolayers owing to the presence of retained solvent (decane) between them.The decane retained in the BLM increases the internal volume thereby eÜectively reducing the van der Waals interactions between adjacent lipid tails and between the two monolayers. The maximum thickness of this decane layer is ca. 2 nm.11 26 Faraday Discuss. 1998 111 17»30 Fig. 7 Mode frequency f0\u0/2n and damping C vs. mode wave vector q of a free planar bilayer (BLM) of DEPC (A) and of DEPC with 30 mol% cholesterol (B) calculated from the autocorrelation functions measured at the corresponding q-values. The full lines represents the theoretical prediction with the average lateral tension c and the viscosity c@ (see text) as parameters. 0 In a second experiment we have studied the eÜect of cholesterol on the collective modes of a BLM as measured by DLS.This steroid is well known from a number of bilayer studies to cause a stiÜening of the —uid bilayer and a drastic increase in its molecular order. Fig. 7B shows results for f0(q) and C(q) for the case of a DEPC»cholesterol (30 mol%) BLM for a comparable q range as for the pure DEPC BLM from Fig. 7A. The eÜect of cholesterol manifests itself by a signi–cant f increase in the transition of 0(q) from the damped to the overdamped case and a corresponding change in the damping C(q). Fits of the data to eqn. (1) now give an average lateral tension of c0\1.55 mN m~1 with a maximum deviation of ^0.07 mN m~1 while c@ is within the same order of magnitude as for pure DEPC.LUV measured by RICM h Fig. 8 shows the relaxation measurements of the bending oscillations for a vesicle weakly adhering to the substrate. In a previous analysis by Raé dler et al.13 the equilibrium contour and the static —uctuations of adhering vesicles were evaluated. Under the given conditions it was shown that the equilibrium distance 0B30 nm and the membrane tension cB10~6 J m~2. Here we evaluated 27 Faraday Discuss. 1998 111 17»30 (1)\0.67]106 m~1. Fig. 8 Time correlation of the Fourier modes hqx of the one-dimensional intensity pro–le h shown in Fig. 0\30 nm c\10~6 J m~2 h 6(b). All relaxation curves are –tted using eqn. (14) and eqn. (21) with –xed values x and V AB5]108 J m~4. The data are in agreement with hydrodynamic modes CW which are slowed down by the presence of the wall.The semi-logarithmic presentation of the same data (inset) shows the predicted multi-exponential decay. The –rst mode (=) corresponds to a wave vector qx the time correlation functions by numerically –tting eqn. (21) to a set of time correlation functions. We tested the diÜerent models for relaxation rate C. We found the hydrodynamically damped modes eqn. (14) –tted best. As shown in Fig. 8 the relaxation of all –ve modes can be simultah neously –tted with one set of constants 0\30 nm c\10~6 J m~2 g\10~3 J s m~3 and V AB5]108 J m~4. The parameter V A was left variable to improve the individual –ts but the resulting values did not scatter by more than 15%. The fact that the data show indeed a superposition of relaxation modes is demonstrated in the semi-logarithmic plot (inset in Fig.8) which shows clearly deviations from a straight line at low q. Hence the data are in good agreement with the slow mode C described by Kraus and Seifert.9 The fast frictional mode C W F on the other hand was not observed even though the theoretical cross-over should have allowed its detection. We have to assume that this mode is too fast to be detected by video microscopy. In fact even the fourth and –fth mode of the hydrodynamic damping in Fig. 8 are at the signal-to-noise detection limit. It must be mentioned that the data can be brought to reasonable agreement with the dispersion relation predicted by the permeation model. In this case the parameters c\10~6 J m~2 and jB10~6 m2 s kg~1 obtained best simultaneous –ts.However the –ts did not follow the multiexponential decay as seen in the semi-logarithmic plot shown in Fig. 8. q Discussion We have combined two physical techniques and two model systems to study undulations in two limiting cases high tension (BLM) and weak tension (LUV). The data shown in Fig. 7 provide for the –rst time the viscoelastic dispersion behaviour of a transverse shear mode of a BLM over an exceedingly wide (mesoscopic) q-range. For comparison previously reported measurements by this method were limited to maxB1800 cm~1 while for the present work we have qmaxB35 000 Faraday Discuss. 1998 111 17»30 28 cm~1. This allows the experimental observation of the transition from the oscillatory or damped to the overdamped regime of the transverse shear mode and thus oÜers a critical test of the validity of the hydrodynamic theory suggested previously for such modes.The general agreement of our DLS data (Fig. 7) with the Kramer theory [eqn. (1)] is quite excellent and thus provides strong support for its validity within the mesoscopic q range covered by our experiments. In particular we can draw the following conclusions (1) The dispersion behaviour of the BLMs studied is clearly dominated by the lateral tension only at highest q values might there be some minor contributions arising from membrane bending rigidity. (2) Accordingly the transverse shear interfacial viscosity c@ is nearly negligible over the q-range considered most likely re—ecting the fact that qA1/d in the mesoscopic q range (d being the bilayer thickness).(3) In contrast to the theoretically predicted existence of two overdamped modes our DLS data show in the overdamped regime with the exception of the vicinity of the bifurcation point only one mode which is compatible with the slower of the two predicted ones. Considering the changes in the DEPC-BLM dispersion behaviour upon the addition of cholesterol (cf. Fig. 7B) at an amount (30 mol%) that causes in a —uid bilayer the creation of the so-called liquid-ordered (l0) phase we can conclude that this system shows an approximately three-fold higher lateral tension c while the transverse shear interfacial viscosity c@ remains nearly 0 negligible. The increase in c is probably caused by the homogeneous distribution of the hydro- 0 phobic cholesterol over the bilayer in the l -phase.The tails of DEPC molecules adjacent to the 0 stiÜ steroid body are forced into a state of higher molecular order thereby reducing their area per molecule projected in the normal direction and thus giving rise to an increase in c0 . Note that DEPC is a synthetic lipid which can pack rather densely because it exhibits just one trans-double bond in each acyl chain. A surprising result is that in spite of the excellent agreement between experiment and theory only one of the two theoretically predicted overdamped modes C and C 1 2 [eqn. (20)] seems to be detectable by DLS. While C describes the slow recovery of the system 1 driven by tension and viscosity C is supposedly driven by the inertia of the system arising within 2 the approximations of the theory from the —uid surrounding the BLM.Hence while C is readily 1 detectable in our experiment with the data following the predicted course up to the qmax limit relaxation via the C -process is only seen in the region of the bifurcation point. A rationale for this 2 behaviour is not given here but there is room for some speculation. One possible reason could be that the occurrence of surface viscosity c@ drives the fast mode C into a merger with the bulk 2 mode which in turn is not detectable by DLS. The interferometric measurements are limited to a smaller q-range but are unique in the analysis of membranes interacting with solid supports. We provided quantitative evidence for hydrodynamic coupling between the undulating membrane and the wall.The hydrodynamic mode was found to be in better agreement than the permeation mode. However discrimination between the modes is difficult since in both cases the dominating terms scale as CDq2. It will be necessary for future measurements to extend the q-range of the interferometric microscopy using ultra-fast cameras and two-dimensional image processing. In this case the fast frictional mode might become visible. On the other hand the fast modes are even more likely to be biased by water permeation. Conclusions The dynamic undulations of lipid bilayers reveal a hierachy of relaxation mechanisms if measured over a wide range of wave vectors. These relaxation modes are sensitive to the bilayer tension the elastic bending modulus as well as the local hydrodynamic environment in case of membranes close to a solid wall.The study of multi-component membranes where the dominating relaxation parameters can be controlled by incorporation of steroids transmembrane pores or short membrane binding peptides will further elucidate the dynamics of membranes. Acknowledgements The authors R.H. and T.M.B. are indebted to Professor Lorenz Kramer (Universitaé t Bayreuth) for many helpful discussions. 29 Faraday Discuss. 1998 111 17»30 Paper 8/07883A References 1 W. Helfrich and R-M. Servuss Il Nuovo Climento 1984 3D 137. 2 W. PfeiÜer S. Koé nig J. F. Legrand T. Bayerl and E. Sackmann Europhys. L ett. 1993 23 457. 3 C. Dolainsky A. Mops and T. Bayerl J. Chem. Phys. 1993 98 1712. 4 J. F. Crilly and J. C. Earnshaw Biophys. J. 1983 41 197. 5 W. Haé ckl U. Seifert and E. Sackmann J. Phys. France 1997 7 1141. 6 L. Kramer J. Chem. 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