I. INTRODUCTION

TOP ABSTRACT I. INTRODUCTION II. MATERIALS AND METHODS III. EXPERIMENTAL PROCEDURES... IV. RESULTS V. DISCUSSION VI. CONCLUSIONS AND PROSPECTS APPENDIX REFERENCES

Lipid bilayers are conventionally accepted to be the simplest model that approximates some properties of biological membranes. Besides their structural resemblance, they are characterized by physical properties similar to those of biomembranes, including thickness, water permeability, bending rigidity, surface tension, and viscosity. Furthermore, artificial lipid membranes are well-defined systems and are readily prepared. Thus they provide a unique opportunity to investigate certain physiological functions and processes in biological membranes. In addition, bilayers are convenient systems for investigating two-dimensional (2-D) molecular motion (Saffman, 1976) and ordering (Nelson and Halperin, 1979; Nelson and Peliti, 1987; Seung and Nelson, 1988).

It has been established that the biological membranes are not rigid bodies but flexible and fluid materials. However, the biomembrane lipids exhibit a range of phase transitions (from fluid liquid crystalline to gel-like structures). It is known that the so-called growth temperature of some microorganisms is closely related to the membrane phase transition. Phospholipid phase transitions could also be important in regulating the activities of membrane proteins and their interaction with the lipid matrix. For instance, the lipid bilayer should be "softer" and not very viscous, to permit easy structural reconfiguration of the protein molecule. The lipid bilayer, being in the fluid state, would allow an inclusion to move without restoring force. On the other hand, when frozen or when subjected to a large deformation, the biological membrane exhibits an elastic response (Hochmuth et al., 1980; Waugh and Evans, 1979) (the red blood cell membrane has often been modeled as a thin rubber sheet; see, for instance, Skalak et al., 1973).

In terms of molecular structure, membrane fluidity in the L phase implies "melted" hydrocarbon chains of the lipid and positional disorder of the molecules in the bilayer plane. Conversely, in gel phase the lipid bilayer becomes "stiff," the acyl chains freeze in a nearly all-trans configuration, and the molecules (heads and/or chains) are apparently arranged in a 2-D hexagonal lattice. Thus the phase state of the lipid bilayer largely influences the mechanical properties of the membrane itself. A large variety of techniques have been employed to study the phase transition behavior of bilayer systems on both molecular and on macroscopic scales: differential scanning calorimetry (Janiak et al., 1976, 1979; Koynova and Caffrey, 1998; Heimburg, 1998), x-ray diffraction (Janiak et al., 1976, 1979; Brady and Fein, 1977, Smith et al., 1988), Raman spectroscopy (see refs. in Pink et al., 1980), NMR (Davis, 1979; MacKay, 1981; Wittebort et al., 1981), electron spin resonance (Tsuchida and Hatta, 1988), spectroscopic techniques describing molecular diffusion (see refs. in Tocanne et al., 1994), ultrasonic studies (Mitaku et al., 1978), and micropipette techniques (Evans and Kwok, 1982; Needham and Evans, 1988; Needham and Zhelev, 1996).

Dimyristoylphosphatidylcholine (DMPC) is a frequently studied artificial lipid because it undergoes a phase transition at a convenient temperature. Upon cooling below ~23.6°C (Koynova and Caffrey, 1998) it undergoes a transition from the liquid crystalline L phase to the P' solid rippled phase, characterized by periodic corrugations of the bilayer. Studies on the microscopic level (electron spin resonance, ¹³C NMR) showed that a significant fraction (~20%) of chain disorder still exists in the P' phase (Davis, 1979; MacKay, 1981; Wittebort et al., 1981; Tsuchida and Hatta, 1988). Lateral diffusion measurements (Derzko and Jacobson, 1980) detected heterogeneity in the self-diffusion coefficient and interpreted the results by assuming the existence of fast and slow components differing by several orders of magnitude. It was suggested (Schneider et al., 1983) that the P' phase comprises bands of ordered lipid separated by bands of disordered ones, the latter coinciding with the regions of high curvature in the rippled structure (as also proposed by Tsuchida and Hatta, 1988). A recent study (Jutila and Kinnunen, 1997) on the DMPC phase transition in large unilamellar vesicles reported evidence of pretransitional phenomena that were correlated to structure fluctuations and gel-like domain formation. The complexity of the melting process in giant vesicles was visualized by two-photon fluorescence microscopy (Bagatolli and Gratton, 1999). Although numerous studies have been performed, information on the physical characteristics of the lipid bilayer in the phase transition region is still needed. The work reported here is aimed at better understanding the mechanical properties of the lipid membrane on a macroscopic level (note that among the techniques cited above, few work on this scale).

Our experiments deal with micron-sized latex beads attached to giant vesicle membranes. The particles are manipulated by means of an optical trapping system (Velikov et al., 1997). In their motion the latex spheres directly "feel" the state of the membrane. We use them as macroscopic mechanical probes to characterize the viscous or/and elastic responses of the vesicle membrane. The general problem of the friction experienced by a single particle when it moves along a fluid vesicle surface has been studied (Dimova et al., 1999a), and this has allowed deduction of the membrane shear viscosity (_s) from the kinetics of particle motion. The general procedure (it is applicable to different particle and vesicle sizes and particle penetrations across the membrane) was tested with polystyrene latex beads and SOPC (L-stearoyl-oleoyl-phosphatidylcholine) membranes, which are fluid at room temperature.

In this work we study the DMPC membrane above and below the gel-fluid transition temperature. Basically, we investigate the temperature variation of the membrane viscosity in the fluid phase (T > T_m), using the above-mentioned single-bead method (Dimova et al., 1999a). In the gel phase (T < T_m), we probe the membrane elasticity by manipulating two beads simultaneously, and we measure the membrane elastic response by optical dynamometry up to T_m. As far as we know, these are the first experiments of that kind dedicated to lipid membranes and aimed at characterizing their pretransitional behavior on both sides of T_m. From the viewpoint of experimental techniques, ours has much in common with a number of recent experiments on biological membranes, using spherical particles as probes, either by optical (Bronkhorst et al., 1995; Hénon et al., 1999; see also Ashkin, 1997), or magnetic (Bausch et al., 1998, 1999; Boulbitch, 1999) dynamometry. All of these experiments are difficult to carry out and to interpret. An important point of this report is dedicated to interpreting the measured elastic response as a function of basic membrane elastic moduli. As we will explain, we do not read our data in terms of the membrane shear modulus (as one might believe a priori), but rather in terms of the membrane curvature modulus (k_C). In short, we report on the pretransitional behavior of k_C in the gel phase.

The paper is organized as follows. The next section (II) briefly introduces the materials and methods: sample preparation, experimental set-up, and procedure for the particle path analysis. In section III, we explain the principles of the different experimental methods and the kind of information that they provide. We start with the viscosimetry experiments: the procedures for measuring the particle friction coefficient and deducing the membrane viscosity are briefly reviewed in sections III.1 and III.2, respectively. The approach to the study of the gel phase elasticity, by optical dynamometry with two beads, is explained in section III.3. Our experimental results are reported in section IV: there we show the variation in _s above T_m and that in the membrane stiffness (k_M) below T_m. The pretransitional behaviors of _s and k_M are discussed and tentatively interpreted in section V. Our estimate of the amplitude of k_C in the gel phase is based not on a theory but on true analog simulations, which we carried out with macroscopic elastic sheets. These experiments are briefly described in the Appendix.

	II. MATERIALS AND METHODS

TOP ABSTRACT I. INTRODUCTION II. MATERIALS AND METHODS III. EXPERIMENTAL PROCEDURES... IV. RESULTS V. DISCUSSION VI. CONCLUSIONS AND PROSPECTS APPENDIX REFERENCES

II.1. Vesicle preparation

Giant vesicles were prepared from 1,2-dimyristoyl-sn-glycero-3-phosphocholine (DMPC) (Avanti Polar Lipids, Alabaster, AL; no additional purification of lipid was performed), using the method of electroformation (Angelova and Dimitrov, 1986; Angelova et al., 1992). During vesicle formation, the temperature (30°C) was kept well above the main phase transition of DMPC, and an electric field of a few V/mm was applied. The vesicles grow along platinum electrodes, on which the lipid was originally deposited. At the end of the preparation, the vesicles were usually interconnected and clustered. Target vesicles were selected at the outer rim of such clusters for experiments. There one easily finds vesicles that are unilamellar (as far as we can determine from phase contrast views) and without obvious internal structures. Most often, these outer vesicles were spherical and were connected just by a few contact points to the cluster. Sometimes they adhered by easily visible flat portions to neighbors or to the nearby platinum electrode.

The experimental cell is equipped with a circulating water jacket, allowing for a homogeneous temperature distribution in the chamber (see a detailed sketch in Fig. 1). The whole experimental unit is mounted on a motorized x-y stage. Basically, the optically trapped particles are immobile. To bring them in contact with vesicles, we moved the cell with the x-y stage. The temperature of the circulating water was kept constant to within ±0.1°C by means of a cryothermostat (Lauda RM6) and measured by a thermocouple located inside the cell (see Fig. 1 A) (some of the experibments reported were performed before construction of the chamber in Fig. 1, and the temperature was controlled only to within about ±0.4°C). The device can be operated from, say, 50°C down to ~15°C.

View larger version (41K): [in this window] [in a new window]   FIGURE 1   A schematic sketch of the experimental chamber. (A) Horizontal section: the cooling water cycle includes a cryothermostat. (B) Cross section: the optical trap is realized by two contrapropagating laser beams focused on the working cell by the two objectives.

II.2. Optical manipulation of latex beads

The optical trap, specially designed to ensure a long working distance (on the z axis) between manipulated particles and optical components, has been described in detail elsewhere (Angelova and Pouligny, 1993; see also Dietrich et al., 1997, for additional characteristics of the set-up). Basically, the trap consists of two contrapropagating laser beams focused inside the experimental chamber (standard optical tweezers are created from a single sharply focused beam; see Ashkin et al., 1986).

For the experiments, we used latex spheres (Polyscience, Warrington, PA) with diameters ranging from 2 to 12 µm. To avoid contamination with lipid, the beads are injected at some distance ( 15 mm) from the vesicle clusters at the electrodes (see Fig. 1 A). We pick up a particle with the laser trap and transport it to a previously selected vesicle. The bead sticks with a quick jump toward the vesicle interior as it comes in contact with the lipid bilayer. In our procedure, we attach the particle to the membrane in the fluid state, i.e., at T > T_m, but adhesion is possible below T_m as well. The way in which the particle stabilizes itself across the vesicle membrane is sometimes complex (see Dietrich et al., 1997, for details). A final equilibrium position is established within several seconds. This position is stable on the time scale of a single experiment (~1 h). By "stable" we mean that it does not change spontaneously and cannot be modified by the laser radiation pressure forces. When the lipid bilayer is in the fluid state, beads are allowed to move along the membrane. Driven by gravity, large and heavy beads sediment toward the bottom of the vesicle (see Fig. 2 A). Small and light particles exhibit Brownian motion (see Fig. 2 B). Beads can also be directed by the radiation pressure force (see Fig. 2 C). When the temperature is decreased below T_m, a membrane-bound particle becomes "frozen." It is no longer possible to make it move everywhere on the vesicle surface. Only a small lateral displacement can be achieved by means of the optical trap. When the laser beam is switched off, the particle returns to a point near its original position.

View larger version (48K): [in this window] [in a new window]   FIGURE 2   Schematic illustrations representing three different approaches to measurement of the mobility of a particle bound to a membrane. (A) Gravity-driven sedimentation. (B) Brownian motion at the bottom of the vesicle. (C) Optical trapping kinetics (the sketch is exaggerated in terms of distances).

Particle size calibration is performed before adhesion to the vesicle. The bead sedimentation velocity, v_sed, was measured in bulk water. Application of Stokes' law yields the bead radius a = (9_sed/2g)^1/2, where is the density difference between water and latex ( 0.05 g/cm³), g is the gravity acceleration, and  is the viscosity of water. For small particles, instead of measuring the sedimentation velocity, one can analyze their Brownian motion. Extracting the diffusion coefficient in the bulk aqueous suspension (D_free) provides the bead radius from the Stokes-Einstein equation [a = k_BT/(6D_free)], where k_BT is the Boltzmann energy.

The calibration of the optical trap forces is described in the next subsection. For the latex spheres of interest, no heating effects (absorption) were detected (Angelova and Pouligny, 1993). The applied laser powers were weak (less than 6 mW in the sample cell) compared with experiments with optical tweezers, in which the highly focused laser spot (a few hundred mW) could induce heating of the lipid membrane (Liu et al., 1995) or even mechanical effects (Granek et al., 1995; Bar-Ziv et al., 1995).

For two-bead experiments a double trap configuration of the optical system is used. The laser beam is split into two pairs forming two traps (Angelova and Pouligny, 1993; Martinot-Lagarde et al., 1995). One trap is fixed, while the other can be moved by means of a mirror mounted on a one-direction motorized stage. The distance between the two traps can be adjusted between 0 and ~35 µm.

II.3. Optical dynamometry

The force (several pN) exerted on a trapped bead depends on its size and on the refractive indices of the particle and of the surrounding media; it is proportional to the applied laser power. Radiation pressure forces are directed through the center of the manipulated (supposedly spherical) particle (Martinot-Lagarde et al., 1995; Polaert et al., 1998). The radiation pressure for the beam geometry used was computed with the Generalized Lorenz-Mie Theory (GLMT) (Gouesbet et al., 1988; Ren et al., 1994; Martinot-Lagarde et al., 1995). In bulk water, the trap force in the x-y plane (transverse force component) is roughly proportional to the distance () between the bead center and the beam axis, when it is less than ~0.6 times the particle radius, a (the deviation is within ±10%, which is a reasonable accuracy for the data interpretation, keeping in mind the experimental error). Fig. 3 presents the theoretically computed transverse optical trapping force, F_RP, versus (the force is calculated for an incident laser power of 5 mW, which is a typical value). Different curves correspond to different particle radii.

View larger version (13K): [in this window] [in a new window]   FIGURE 3   Numerical calculations of optical trap force, FRP (transverse force component), as a function of particle off-centering, , for beads of different sizes. The computations are performed for the trap configuration used in experiments and an applied laser power of 5 mW. The wavelength of the incident beam is 514 nm in air; the beam waist is 4.15 µm.

Another method for deducing the magnitude of F_RP is to submit the trapped sphere to a constant counterflow of known velocity and determine the "escape" velocity, v_esc, at which the particle leaves the trap. The corresponding trapping force, which is the maximum of F_RP(), exactly balances the viscous drag force (Stokes' law):

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Both methods were used to estimate the trapping force.

Knowing the radiation pressure force applied through the particle center, we probed the vesicle membrane for forces in the piconewton range. At temperatures at and above T_m, we measured the bilayer shear viscosity. Below T_m, by means of the two-particle manipulation, the elastic restoring force was studied.

II.4. Image processing

A classical microscope with elements integrated in the optical trap set-up allows us to observe bead and vesicle position from above (top view). While the vesicle contour and a small sphere (<4 µm in diameter) are best represented in phase-contrast mode, larger beads are preferably imaged in simple transmission (amplitude contrast) mode. Applying digital image processing allows the bead motion (horizontal projection) to be followed with a rate of ~6 Hz. Essentially, the algorithm is based on subtraction of a previously recorded background frame (without particle) and discrimination of the resulting image in 0 (no particle) and 1 (particle) levels. The accuracy is set by the pixel resolution (0.156 × 0.159 µm) of the CCD camera (Hamamatsu). Image sequences are recorded with standard video equipment (U-matic; SONY).

	III. EXPERIMENTAL PROCEDURES AND DATA ANALYSIS

TOP ABSTRACT I. INTRODUCTION II. MATERIALS AND METHODS III. EXPERIMENTAL PROCEDURES... IV. RESULTS V. DISCUSSION VI. CONCLUSIONS AND PROSPECTS APPENDIX REFERENCES

III.1. Viscosimetry of fluid membranes

In this subsection we discuss three different one-bead-one-vesicle scenarios followed in our experiments and their interpretations. By these single-particle manipulations, we determine the viscosity of the lipid bilayer in the fluid state (T  T_m). The parameter measured in all three experiments is the friction coefficient, . It relates the bead velocity (v) and the drag force (F_fr) experienced by the particle:

(2)

The particle motion can be driven by gravity (sedimentation experiment, Fig. 2 A), by thermal fluctuations (Brownian motion, Fig. 2 B), or by radiation pressure force (Fig. 2 C). We suppose that the friction coefficient is constant when the membrane is in the fluid state. This amounts to hypothesizing that the bilayer shear viscosity is constant, i.e., independent of frequency. As we will see, the experimental results are in line with this assumption.

III.1.1. Sedimentation

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g

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gt

g

g

View larger version (15K): [in this window] [in a new window]   FIGURE 4   (A) The sketch illustrates the particle (latex bead) movement along the spherical surface of a vesicle; see text for notations. (B) A recorded particle trajectory, top view (the bead contour corresponds to the moment when the particle passes through the equatorial plane of the vesicle). The recorded trace is the horizontal projection of the particle path.

III.1.2. Brownian motion

g

D

III.1.3. Optical trapping dynamics

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3

III.2. From particle friction to membrane viscosity

Deducing the value of the membrane shear viscosity, _s, from that of  necessitates a theory for the particle motion. We used the theory of Danov et al., either in the simple version for flat Langmuir films (Danov et al., 1995) or in the recent general version for vesicles (Danov et al., manuscript submitted for publication). As discussed by Danov et al. (manuscript submitted for publication) and Dimova et al. (1999a), two important assumptions of the theory are that the membrane behaves like a single film and that the membrane-particle contact line is locked on the particle surface ("contact line pinning"). These two assumptions greatly simplify the theory, and it was shown by Dimova et al. (1999a) and Dietrich et al. (1997) that they are satisfied by the latex bead-lipid vesicle system, indicating that the particle does not roll on the membrane and that lipids do not slip along the particle surface.

In the case of a small particle on a large vesicle the bead "sees" the membrane as a flat surface. For such systems, _s can be simply deduced from the friction coefficient, , following the procedure of Velikov et al. (1997). There the theory of Danov et al. (1995) was employed for the -to-_s inversion: the model for the motion of a particle along a flat infinite film at the air/water surface is adapted to a particle moving along a membrane (i.e., water/bilayer/water interface). The adaptation requires 1) that the bead be much smaller than the vesicle size, a R (to satisfy the condition for a flat surface), and 2) that the membrane intercept the particle through its equator, i.e., the contact angle has to be ~90° (to allow for a superposition of two equivalent air/water systems).

For a large particle and an arbitrary radial penetration (or an arbitrary contact angle) of the particle, one needs to account for possible finite size effects (e.g., increased friction due to recirculation of the water enclosed in the vesicle bulk); a generalized theory accounting for these factors is available (Danov et al., manuscript submitted for publication). For membranes of moderate surface viscosity (e.g., on the order of 5 × 10⁶ surface poise) the recirculation effect may have a considerable impact on the value of . Indeed, the theory shows that  definitely increases beyond , the value corresponding to the flat membrane limit (R/a  ), when R/a < 10 and when the particle penetrates more toward the vesicle interior. The theory was successfully applied to the interpretation of data from sedimentation and diffusion measurements (Dimova et al., 1999a) and allowed for a robust determination of the shear surface viscosity. For SOPC lipid membranes at room temperature, _s was found to be ~3 × 10⁶ surface poise (dyn·s/cm or sp; note that the commonly used "surface shear viscosity" has units of [bulk membrane viscosity × membrane thickness]).

The geometry of some of the experimental systems reported here permits application of the mathematically simpler model (Velikov et al., 1997). For others (e.g., when the bead is predominantly situated on one side of the vesicle surface), it was necessary to introduce a finite-size correction factor deduced from the theoretical predictions (Danov et al., manuscript submitted for publication). However, for measurements on highly viscous membranes (at temperatures close to T_m), we did not correct the raw data because the finite size correction was within experimental error.

III.3. Gel phase elastic response

III.3.1. Static elasticity

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View larger version (28K): [in this window] [in a new window]   FIGURE 5   Experimental steps in the static elasticity experiment. d = R   indicates the particle penetration depth (d is positive when the particle center is external to the membrane surface). (A) Initial position of particles. (B) Mobile trap displacement. (C) "Three-spring" model.

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View larger version (45K): [in this window] [in a new window]   FIGURE 6   Test for linearity. The hatched zone covers the area of applicability of the spring model (the lower boundary is approximate). The broken lines are linear fits (least squares) to data at different temperatures (see legend). The slope of the solid line is 1.

III.3.2. Dynamic elasticity

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	IV. RESULTS

TOP ABSTRACT I. INTRODUCTION II. MATERIALS AND METHODS III. EXPERIMENTAL PROCEDURES... IV. RESULTS V. DISCUSSION VI. CONCLUSIONS AND PROSPECTS APPENDIX REFERENCES

All five experimental techniques described in the previous section focus on studying the temperature dependence of membrane viscoelastic characteristics. The viscous resistance enters the theoretical descriptions of the experiments through the friction coefficient . The elastic response below T_m is interpreted by introducing the effective spring constant k_M. In the following section we present results on these two characteristic parameters and demonstrate the temperature dependence of the membrane behavior in the region of the phase transition.

IV.1. Fluid membrane viscosity

For a membrane in the fluid phase, the three different viscosimetry methods that we presented in section III.1 are equivalent. This can be tested with the same particle. The bead has to be big enough to show a clear sedimentation path and, on the other hand, small enough to show Brownian excursions significantly larger than the resolution of the digital image processing. The following results concern one vesicle/particle system (a = 3.2 µm, R = 21 µm, d = 0.2a; see Fig. 5 A for the penetration d) where the bead satisfies these conditions.

Fig. 7 A presents two examples of sedimentation paths at two different temperatures. For the region (t) > /2, the measured sedimentation trajectories are well described by Eq. 5 (Pe  340). The experimental data fit provides the maximum sedimentation velocity, v_max. The latter is inversely proportional to the particle friction coefficient. For the two examples presented in Fig. 7 A, the temperature decrease (T = 0.7°C) induces a ~20-fold increase in .

View larger version (12K): [in this window] [in a new window]   FIGURE 7   (A) Sedimentation particle trajectories, (t), recorded with one bead-vesicle system (a = 3.2 µm, R = 21 µm) at two different temperatures. The fits are performed according to Eq. 5. (B) Brownian diffusion: calculated mean squared displacements (MSD) of the trajectories of a latex bead (a = 3.2 µm) near the bottom of a vesicle (R = 21 µm) for two different temperatures. Note: As established by computer simulations, MSD plots are affected by detection noise causing an offset: MSD = 4D dt + Cnoise. For corrected analysis, slopes (= 4D) were obtained from line fits that cover the time regime from points 1 ( 1/6 s) to 4 ( 4/6 s), asillustrated with solid lines. (C) Optical trapping kinetics: displacements (logarithmic scale) of a latex bead (a = 3.2 µm, R = 21 µm), after switching on the optical trap at t = 0. Straight lines are fits according to Eq. 7 for the region  < 0.6a.

The Brownian motion of the same particle was studied at the lower pole of the vesicle. The corresponding averaged squared displacements are given in Fig. 7 B. The slopes of the line fits (solid lines) for t  1 s yield values for the diffusion coefficient. Similar to the case with sedimentation, decreasing the temperature considerably slows the Brownian motion.

Finally, Fig. 7 C shows the optical trapping kinetics of the same particle, in displacement, , versus t presentation. The time origin is defined as the instant when the trap is switched on. In the region x < 0.6a, the recorded trajectories are found to be exponential (Eq. 7). The characteristic time of the process, _c (fit parameter), shows a 25-fold increase when T is decreased from 23.3°C to 22.5°C.

The results of all experiments, performed with the same particle-vesicle system at different temperatures in the 22.5-28°C range are gathered in Fig. 8. Different symbols correspond to sedimentation (diamonds), Brownian motion (filled squares), or optical trapping dynamics (asterisks). While errors of sedimentation and diffusion experiments were estimated from the precision of the applied fits, trapping dynamics experiments were repeated at least six times, and the observed standard deviation was taken as a measure for the error. In all performed experiments, starting from higher temperatures and passing through the main phase transition, we observe a drastic decrease of particle mobility (i.e., increase in ). This drop is due to the well-known main phase transition of DMPC. In the fluid phase (>24°C) data from the three different methods are in fairly good agreement. For temperatures below ~22.5°C, no long-range particle movement was detected, which is a direct experimental indication that the lipid membrane became solid. Short-distance motion is still possible if the membrane is elastic enough (small elastic moduli) in the gel-like phase. Indeed, at 22°C we clearly detected an elastic response to optical bead displacements, and we observed short-range displacements caused by thermal agitation (hindered Brownian motion). Consequently, scattering of data near T_m for the three different methods might be the result of different sensitivities to partial elastic and/or restrained bead movement. Disappearance of the long-distance motion (when in sedimentation   ) is the best criterion for locating T_m.

View larger version (14K): [in this window] [in a new window]   FIGURE 8   Thermal behavior of the drag coefficient of a particle bound to a vesicle membrane in the fluid phase. The graph gathers the data obtained from sedimentation-, diffusion-, and optical trap-driven motion with the same particle-vesicle system (a = 3.2 µm, R = 21 µm). Data were obtained with a primitive experimental cell (a generation before the one in Fig. 1), where the temperature was controlled within ±0.4°C.

Instead of gathering the results of three different techniques with one and the same particle-vesicle system, as we did in Fig. 8, we now show the results obtained with three different particle-vesicle systems with the same technique, e.g., sedimentation. The surface viscosity is calculated using the procedure explained in section III.2. In two of the systems, the particles penetrated the vesicle, so that the flow confinement effect could not be neglected. In these cases it was necessary to use the full theory (Danov et al., manuscript submitted for publication) to correctly deduce _s from . It is evident from Fig. 9 that the three systems give coherent results. Far above the transition temperature, the surface viscosity is 5 ± 2 ×10⁶ sp. This value is only slightly larger than that found for SOPC bilayers at room temperature (Dimova et al., 1999a) and about the same as that found for egg phosphatidylcholine, using the filament-pulling technique (Waugh, 1982a,b). While macroscopic techniques give membrane viscosities on the order of a few 10⁶ sp, measurements based on diffusion of molecular probes and Saffman's theory (Saffman, 1976; Hughes et al., 1981) give smaller values of ~10⁷ sp (see, e.g., Vaz et al., 1984; Merkel et al., 1989). We will comment on this point in section V.1.

View larger version (10K): [in this window] [in a new window]   FIGURE 9   Temperature dependence of the shear surface viscosity, s, obtained from sedimentation experimental data for three systems: a = 2.64 µm, R = 30.4 µm, d = 0.8 (); a = 2.7 µm, R = 46.4 µm, d = 0.9 (); a = 2.72 µm, R = 30.5 µm, d = 0.1 (*). The solid curve shows the fit by: s = 25 × 106|T  23.4|1.4 sp.

Near the transition, _s increased by more than two decades. We tentatively fitted a power law to the data, _s  |T  T_m|^w, but because of the data scatter, different sets of the three adjustable parameters (T_m, w, and the proportionality factor) were found to be equally acceptable. We restrained the choice by putting T_m = 23.4°C, which is approximately the temperature at which the membrane elastic response (approaching from T < T_m) vanishes, within experimental error. We thus found _s  25 × 10⁶|T  23.4|^1.4 sp; this is the solid line in Fig. 9.

Not all of the particle-vesicle systems investigated behaved in the same way. In some cases, the divergence of  (or _s) was less gradual than that in Fig. 9, i.e., _s was greatly increased only very close to T_m. However, the vesicles in these cases were probably multilamellar. Indeed, these membranes were anomalously dark in the phase-contrast images. This is an indication that pretransitional phenomena show up far from T_m (a few degrees apart) only with unilamellar membranes (Jutila and Kinnunen, 1997; Bagatolli and Gratton, 1999).

IV.2. Gel-phase elastic response

The temperature dependence of the membrane stiffness in terms of the elastic spring constant, k_M, obtained from static elastic experiments is given in Fig. 10. Different sets of symbols correspond to different temperature scans. Filled and empty symbols refer to two individual two-particle-one-vesicle systems. The solid curve is a fit function (least-square minimization) to the whole set of data: k_M  2.4 × 10⁴ |T  23.4|^1.5 dyn/cm. For lower temperatures (i.e., stiffer membranes) the experimental error increases because the optical trap-induced displacements (x_f and x_m; see Fig. 5) approach the pixel resolution of the camera. When the lipid enters the fluid L phase, the membrane looses its elastic properties and k_M = 0 (within the experimental accuracy). The continuous decrease in the effective stiffness as T_m is approached is an indication of a continuous gel-fluid phase transition.

View larger version (11K): [in this window] [in a new window]   FIGURE 10   The measured membrane effective stiffness, kM, plotted as a function of the temperature. The different sets of symbols correspond to different temperature scans. Filled and empty symbols correspond to two individual particle-vesicle systems; for all particles a  5.3 µm, d  0.8a. The solid curve shows the fit by kM = 2.4 × 104|T  23.4|1.5 dyn/cm.

A very important experimental detail to note, as we will see further, is the penetration depth of the latex beads, d. In the experiments reported here, particles are protruding predominantly on one side of the vesicle wall, d  0.8a. For one system with d  0.8a (i.e., the particle centers were external to the vesicle surface; data not shown), k_M does not distinctly differ.

Finally, following the procedure described in section III.3.2, we studied the membrane relaxation response in terms of recorded interparticle distance (l) versus time (Fig. 11). First we induce a displacement of the bead held by the mobile trap. Then the particle is released, and it slowly relaxes toward its initial position. A simple exponential function (solid line) described by Eq. 7 adequately fits the relaxation branch of the curve. The characteristic time for membrane relaxation in this example is _r = 5 s at T = 19°C.

View larger version (13K): [in this window] [in a new window]   FIGURE 11   A dynamic elasticity experiment performed at T = 19°C. The points represent the measured interparticle distance, l, and the solid curve is an exponential fit following Eq. 11 (r = 5 s) to the relaxation branch of the trace.

Not all of the recorded particles relaxed according to a single exponential and returned exactly to their initial positions. In many other examples, we noticed that the particles' final positions differed from the original ones (mainly for the particle in the mobile trap), beyond experimental uncertainty, and that the relaxations, though still monotonic, were not single exponentials. In these situations, we estimated a half-time for relaxation as the point where the induced displacement dropped to half of its maximum value; the average half-time was ~7 s at all temperatures. Thus there was no acceleration or a reduction of the relaxation kinetics when the temperature was increased toward T_m. Why beads do not return to their initial positions is not obvious from the observation. As we explained in section II.1, the vesicles are connected to the cluster by contact points or contact zones. If the vesicle were free, e  x_m and x_f would be equal, from symmetry. The fact that x_f in general is definitely smaller than e  x_m is due to the above-mentioned contacts, because they hinder the overall rotation of the vesicle. It is possible that the forces exerted on the beads by the laser beams make the vesicle move slightly, precluding full reversibility.

In an individual case of large bead penetration (d  1) the particle could be repositioned to a relatively larger distance (> a). The average relaxation time for this particular system was about four times longer than the rest. The effect may be ascribed to eventual connection of the bead to the vesicle membrane in the form of a tether and not a definite contact line.

	V. DISCUSSION

TOP ABSTRACT I. INTRODUCTION II. MATERIALS AND METHODS III. EXPERIMENTAL PROCEDURES... IV. RESULTS V. DISCUSSION VI. CONCLUSIONS AND PROSPECTS APPENDIX REFERENCES

V.1. Fluid phase viscosity

A simple viscous fluid would have a constant viscosity. If the fluid is viscoelastic, _s is defined as a complex number depending on the frequency. In the time domain, the friction would not be defined as a constant but as a time-dependent response (see, for instance, Berne and Pecora, 1976). As we explained, we analyzed our data under the assumption that  was a constant. If this was not so, the recorded trajectories would show systematic deviations from the equations set out in section III.1. In fact, experiments are consistent with the assumption of  = constant, within experimental error. We may translate this statement in terms of frequency: in Fig. 7 C, the tr