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# Phospholipid bilayers are viscoelastic

Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved September 27, 2010 (received for review July 21, 2010)

### This article has been retracted. Please see:

## Abstract

Lipid bilayers provide the structural framework for cellular membranes, and their character as two-dimensional fluids enables the mobility of membrane macromolecules. Though the existence of membrane fluidity is well established, the nature of this fluidity remains poorly characterized. Three-dimensional fluids as diverse as chocolates and cytoskeletal networks show a rich variety of Newtonian and non-Newtonian dynamics that have been illuminated by contemporary rheological techniques. Applying particle-tracking microrheology to freestanding phospholipid bilayers, we find that the membranes are not simply viscous but rather exhibit viscoelasticity, with an elastic modulus that dominates the response above a characteristic frequency that diverges at the fluid–gel (*L*_{α} - *L*_{β}) phase-transition temperature. These findings fundamentally alter our picture of the nature of lipid bilayers and the mechanics of membrane environments.

Lipid bilayers are nature’s most important two-dimensional fluid, forming the underlying architecture of cellular membranes. The fluidity of membranes is crucial to functions such as the assembly of proteins into signaling complexes and the controlled presentation of macromolecules at cell surfaces (1, 2), and the fundamental role of the lipid bilayer as the biophysical determinant of fluidity has been established for nearly four decades (3). It is generally assumed, for lack of evidence to the contrary, that homogeneous lipid bilayers are simple Newtonian fluids—that is, purely viscous two-dimensional liquids incapable of an in-plane elastic response. This assumption tacitly underlies prevailing treatments of many important phenomena. Attempts to relate molecular diffusion coefficients to bilayer viscosity (4–7), for example, model the membrane as a two-dimensional Newtonian fluid embedded in a three-dimensional Newtonian fluid (water) and calculate the resulting dissipative dynamics. The tracking of single membrane molecules, as another example, has often revealed diffusive behaviors that differ from those expected of particles in a homogeneous viscous fluid, interpreted as indicative of “lipid rafts” or other compositionally heterogeneous domains (1, 8). In recent years, microrheological techniques that examine the statistical mechanics of small tracer particles have yielded a wealth of insights into the properties of three-dimensional complex fluids and molecular monolayers at air–water interfaces, providing a means to explore materials that are not amenable to conventional macroscopic rheological analysis (9–16). Applying particle-tracking microrheology to lipid membranes, we find, strikingly, that phospholipid bilayers are not Newtonian fluids, but rather are viscoelastic, being dominated by viscous response at low frequencies and elastic response at high frequencies. Moreover, the cross-over frequency characterizing this viscoelasticity is sensitive to membrane phase transitions, exhibiting a strong peak at the transition temperature separating the fluid (*L*_{α}) and gel (*L*_{β}) phases. This discovery impacts our understanding of phenomena as diverse as macromolecular mobility, mechanical signal transduction, and the energetics of conformation changes of transmembrane proteins, in addition to fundamentally altering our conceptual picture of what lipid membranes are.

We constructed freestanding lipid membranes with lipid-anchored particles serving as tracers for microrheological analysis. Bilayers were formed from lipid monolayers self-assembled at air–water interfaces transferred by Langmuir–Schaefer deposition (17) onto hydrophobically treated substrates with arrays of 125-μm apertures (Fig. 1). In the aperture regions, apposing monolayers formed single bilayers (“black lipid membranes”) with aqueous buffer on both sides. Fluorescent neutroavidin-conjugated nanospheres (100 nm in radius unless otherwise noted) were bound to the membranes via a small fraction of lipids with biotin-conjugated headgroups. Experimental details are provided in *Methods*. A characteristic length scale relevant to two-dimensional fluids embedded in three-dimensional fluids is given by the ratio of their viscosities (λ), in our case the ratio of the viscosities of the lipid bilayer and bulk water. As we show below, membranes are not simply viscous, but a rough viscosity value of 10^{-9} to 10^{-8} Ns/m extracted from diffusion measurements (7, 18, 19) gives *λ* ∼ 1–10 μm, small compared to the 125-μm membrane extent, implying that edge-dependent effects on dynamics are minimal.

For each freestanding membrane, the trajectories of 20–50 membrane-anchored particles were captured with fluorescence microscopy at 200 frames per second and analyzed using particle-tracking algorithms with ≈10 nm localization precision (20). Via the fluctuation–dissipation theorem, the thermally driven particle trajectories contain information about the linear viscoelastic response function of the material. Analysis of the trajectories, therefore, can yield the frequency-dependent complex shear modulus *G*^{∗}(*ω*), typically decomposed into the elastic storage modulus (*G*^{′}(*ω*)) and the viscous loss modulus (*G*^{′′}(*ω*)) (9–14) (see *Methods*). These techniques of passive microrheology are sensitive to several sorts of artifacts arising from the transformation of discretely sampled particle displacement data into shear moduli; we discuss these as well as several control measurements after presenting our results. We report the magnitudes of *G*^{′} and *G*^{′′} as effective three-dimensional moduli and comment later on the nontrivial mapping onto two-dimensional response functions.

## Results

We find that phospholipid bilayer membranes are viscoelastic, with an elastic modulus that can dominate the response at high frequency. We plot in Fig. 2 *G*^{′}(*ω*) and *G*^{′′}(*ω*) for the common zwitterionic phospholipid 1,2-dimyristoyl-sn-glycero-3-phosphocholine (DMPC). At temperatures (*T*) above and below *T*_{m}, which demarcates the high-temperature, disordered *L*_{α} phase and the low-temperature chain-ordered *L*_{β} phase, the membrane’s elastic modulus increases with frequency and becomes larger than the viscous modulus at a cross-over frequency denoted *ω*_{c}. The forms of *G*^{′}(*ω*) and *G*^{′′}(*ω*) are well-fit by a Maxwell model of viscoelasticity, namely, where *k* and *τ*_{c} are parameters. A simple realization of this model would be a spring with elastic constant *k* and a dashpot with viscosity η connected in series, for which *τ*_{c} = *η*/*k* and *ω*_{c} = 1/*τ*_{c}. We find that at *T* = *T*_{m}, *ω*_{c} shows a strong peak indicative of possible divergence and the membrane shows purely viscous character at all accessible frequencies (Fig. 2*B*). The mean-square displacements from which *G*^{′}(*ω*) and *G*^{′′}(*ω*) are calculated (see *Methods*) are plotted in Fig. S1.

Every phospholipid composition examined shows regimes of viscous-dominated and elastic-dominated response and a divergence of the cross-over frequency at the particular *T*_{m} of that membrane. In Fig. 3, we plot *ω*_{c} as a function of *T* - *T*_{m} for DMPC (Fig. 3*A*); DMPC with 10% 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC), a phospholipid with the same phosphatidylcholine headgroup but with unsaturated acyl chains (Fig. 3*B*); 1,2-dimyristoyl-sn-glycero-3-phospho-L-serine (DMPS), a phospholipid with a different, anionic, headgroup than zwitterionic DMPC but the same 14-carbon acyl chains, with 7.5% DOPC (Fig. 3*C*); 1,2-dilauroyl-sn-glycero-3-phosphate (DLPA), a phospholipid with a different anionic headgroup and shorter acyl chains (Fig. 3*D*); and 1,2-dinervonoyl-sn-glycero-3-phosphocholine (DNPC), featuring a phosphatidylcholine headgroup and long, 24-carbon, unsaturated acyl chains (Fig. 3*E*). Details of the compositions are given in *Methods*. The differences in lipid structure lead to a range of *T*_{m} spanning 13 °C, but a shared form for the viscoelasticity relative to the phase-transition temperature.

Lipid bilayers are not amenable to conventional bulk rheological techniques, motivating our adoption of passive microrheological methods in which the spectral decomposition of thermally driven motion reveals the membrane shear modulus. Several potential complications can affect these sorts of microrheological analyses, requiring careful assessment (14). First, the tracers should be small enough that their Brownian dynamics are not dominated by the viscosity of the surrounding water but rather by the properties of the membrane to which they are linked. This condition is met if the particle sizes are not large compared to the characteristic length scale λ defined above; in our experiments, tracer radii are 100 nm and *λ* ∼ 1–10 μm. The number of protein-lipid linkages between the particles and the bilayer is not known, depends on particle size, and likely has a high variance even between particles of the same size. Although the number of linkages will affect the particles’ diffusion coefficients and the overall magnitudes of *G*^{′}(*ω*) and *G*^{′′}(*ω*), it should not alter the character of the membrane as being viscous or viscoelastic. More directly, we performed experiments using tracers of different radii to verify that our measurements are independent of particle size. We plot in Fig. S2 the temperature dependence of *ω*_{c} for bilayers composed of DMPC with 10% DOPC probed using tracers with radii of 100 and 260 nm; both datasets show viscoelastic shear moduli with a peak in *ω*_{c} at *T*_{m}.

Second, the finite frame rate of the data collection set limits on the range of frequencies for which *G*^{∗}(*ω*) can reliably be extracted. One expects artifacts in *G*^{∗}(*ω*) near the extremes of ω, a consequence of transforming necessarily discretely sampled temporal data on particle positions into a frequency-space representation (21). We assess the range of reliable *G*^{∗} values using two independent methods. First, using the same tracer particle sizes and imaging parameters as applied to our membrane measurements, we measure the passive microrheology of two materials with known viscoelastic properties: 40% glycerol in water, a viscous liquid, and a 0.1% gellan gum solution, a viscoelastic liquid. The glycerol solution shows a dominant viscous response (Fig. S3*A*), but with an artifactual elastic component that grows to exceed the viscous modulus for frequencies over 100 Hz. This cross-over could be taken to be a measure of the upper bound of the reliable frequency window. We adopt instead a more stringent measure, considering only frequencies for which the glycerol *G*^{′′}(*ω*) > *G*^{′}(*ω*) *and* the derivative of *G*^{′′}(*ω*) with respect to ω is also greater than that of *G*^{′}(*ω*); this cutoff frequency is 33 Hz. Although all cutoffs are to some degree arbitrary, this value characterizes the range in which trends in *G*^{′} are unlikely to be due to sampling artifacts. The gellan gum shows a viscoelastic cross-over of *G*^{′} and *G*^{′′} within our reliable frequency window as expected (22) (Fig. S3*B*). As our second, independent assessment of sampling rate dependence, we assess our data analysis methods by simulating the motion of a spring-and-dashpot Maxwell material whose parameters match those typical of our lipid membranes, and sampling the calculated trajectories at various effective frame rates. At the frame rates of our experiments, the sampling-induced error in the determination of *ω*_{c} is ≈3% (Fig. S4). The errors in determining *G*^{′} and *G*^{′′} are similarly small.

Third, the local environment sampled by the tracer may be altered by the presence of the probe, and hence may not reliably report the unperturbed material. The insensitivity of our measurements on particle size in itself provides an indication that the particles do not dominate the observed response. Moreover, in addition to characterizing individual particle displacements relative to a fixed laboratory frame, we can examine the displacement of each particle relative to any neighbor and determine an effective two-particle *G*^{∗}(*ω*). In other words, rather than using the displacements of single particles, Δ*r*_{i}, to determine *G*^{∗}, where the Δ indicates differences in position at different times and *i* labels a single particle, we can examine the behavior of Δ(*r*_{i} - *r*_{j}), where the subscripts label particle pairs. These relative displacements should exhibit the Brownian dynamics of an inclusion of size equal to the interparticle separation and are therefore sensitive to the properties of the medium separating the pair of particles. One could imagine scenarios in which the relative displacements showed different dynamics than that of single tracers, for example, if the particles were locally “solidifying” a membrane that was purely viscous in the interparticle space. Examining relative displacements of all particle pairs in a membrane, we find that the resulting two-particle *G*^{∗}(*ω*), like the one-particle complex modulus, shows viscoelasticity with the same apparent divergence of *ω*_{c} at *T*_{m} (Fig. 4). We stress that though it shares the philosophy of using information beyond that carried in single-particle trajectories, this approach is not the same as the technique of two-point microrheology. In the latter, cross-correlations of the displacements of pairs of particles reveal microrheological properties (14, 23). A full theoretical framework for pair correlations in a two-dimensional fluid has yet to be developed, discussed further below.

Finally, the sensitivity of our rheological measurements to membrane phase transitions in itself strongly implies that the tracer particles are faithful reporters of bilayer properties. All image acquisition and analysis parameters are the same for experiments at different temperatures, and so the observed temperature dependence of *G*^{∗} and the divergence of *ω*_{c} at *T*_{m} must reflect properties of the membrane and cannot arise due to details of the experimental procedure.

## Discussion

Our work demonstrates viscoelastic response in phospholipid bilayers. [We note that a prior study of the fatty acid glycerol monooleate found signatures of a sizeable elastic modulus at high frequencies, over 10^{4} Hz (24). The photon correlation spectroscopy technique employed was incapable of probing the lower frequency regime of viscoelastic cross-over, which is crucial to establishing the form of the rheological response.] Our data raise a host of questions of deep biophysical significance. It will be important to determine the microscopic basis of viscoelasticity. It is plausible that entanglement of hydrophobic chains generates an elastic response, but the appearance of significant viscoelasticity in membranes composed of both saturated and unsaturated lipids suggests that correlations between entanglement and elasticity are nontrivial.

The existence of viscoelasticity in lipid bilayers has important consequences for studies of lipid and protein mobility. At sufficiently short timescales, the Saffman–Delbrück or Hughes–Pailthorpe–White models of membranes as simple Newtonian fluids (4, 25) cannot be applicable, and attempts to fit molecular diffusion data into such frameworks may be flawed. Gambin et al., for example, measured diffusion coefficients for membrane inclusions that were found to scale inversely with the inclusion radius (6), interpreted as implying a viscous hydrodynamic drag behavior that is different than the logarithmic scaling predicted by Saffman–Delbrück theory. Alternatively, one might imagine that membrane elasticity could give rise to dynamical data that, if forced into purely viscous models, would appear to give “anomalous” scaling behavior of an effective diffusion coefficient. Put simply, hydrodynamic analyses involving Newtonian models of membranes rely on both the correctness of the models themselves and their applicability to lipid membranes. Viscoelasticity calls the latter assumption into question. At low frequencies, we find that the viscous response of phospholipid bilayers is dominant. One would therefore expect that measurements at timescales larger than roughly hundreds of milliseconds, as probed for example in experiments tracking large phase-separated lipid domains (19), could be interpreted in the context of purely viscous bilayers with negligible inaccuracy. The most important effects of membrane elasticity are likely to be found in the fast motions of individual molecules and in the rapid conformational changes of membrane proteins, noted below.

Membrane viscoelasticity must be integrated into our still-developing understanding of two-dimensional fluids. In three dimensions, it is straightforward to relate Brownian dynamics to fluid properties. For Newtonian liquids of viscosity η, the Stokes–Einstein–Sutherland relation connects η to the mean-squared displacement, 〈Δ*r*^{2}〉 traveled by a particle of radius *a* over time τ: , where *k*_{B} is Boltzmann’s constant and *T* is the absolute temperature. A generalization of the Stokes–Einstein relation to viscoelastic fluids underpins the microrheological analysis employed above (9, 14). We use this analysis without modification from investigations of three-dimensional complex fluids, and so the *G*^{∗}(*ω*) we report is an effective modulus with units being that of a three-dimensional material. More rigorously, the particle dynamics give a memory function ζ(ω) that, for a three-dimensional fluid, is related to the viscosity via , where *d* = 3 is the dimensionality. The complex modulus is then given by *G*^{∗}(*ω*) = *ωη*(*ω*). In two dimensions the simple relation above between η and ζ breaks down, a consequence of the well-known Stokes Paradox, motivating a body of work on how to treat two-dimensional viscosity that remains controversial (4–6, 19, 26). For simplicity, we report an effective *G*^{∗} as simply , i.e., using the three-dimensional relationships but with *d* = 2. This *G*^{∗} could be crudely mapped onto a two-dimensional modulus , which is dimensionally correct though physically unmotivated. We therefore caution against such a mapping, which is in any case unnecessary to our conclusions on the existence and temperature dependence of viscoelasticity in membranes. Recently, two-point correlation functions for Brownian particles in a two-dimensional fluid that provide an alternative route to determining viscosities have been developed theoretically (27) and tested experimentally on proteins at air–water interfaces (15). The corresponding analytic relations for viscoelastic fluids do not yet exist, though the framework for determining these is known (27) and tractable relations will likely be elaborated in the near future.

Arguably the most important set of questions motivated by our findings is on the role of lipid bilayer viscoelasticity in living cells. The mechanisms by which the physical properties of the membrane influence membrane-localized signal transduction remain poorly understood (2). At the level of single proteins, rapid conformational changes on the part of transmembrane proteins such as ion channels and pumps must couple to the local lipid environment; whether this environment is viscous or elastic must therefore influence any molecular model of protein function. At larger length scales, it is increasingly realized that cellular membranes are heterogeneous, home to domains of different compositions (1). Membrane composition is already known to mediate long-range mechanical interactions via differential bending rigidities (2, 28), i.e., spatial couplings to the third dimension. In principle, differential modulation of in-plane rheological properties provides a fundamentally distinct route to mechanical signal transduction and intermolecular communication at membranes.

## Methods

### Lipids and Other Materials.

The following lipids were purchased from Avanti Polar Lipids: DOPC, DMPC, DMPS, DNPC, DLPA, and 1,2-dipalmitoyl-sn-glycero-3-phosphoethanolamine-N-(cap Biotinyl) (biotin-PE). The fluorescent lipid Texas red 1,2-dihexadecanoyl-sn-glycero-3-phosphoethanolamine (TR-DHPE) was purchased from Invitrogen (Carlsbad, CA). The peak excitation and emission wavelengths for Texas red are 583 and 601 nm, respectively. Yellow-green fluorescent neutravidin-conjugated polystyrene nanospheres (radii 20 and 100 nm) were purchased from Invitrogen. FITC and biotin-conjugated scilica nanospheres (radius 260 nm) were purchased from Bangs Laboratories. Gellan gum (Kelcogel) was graciously provided by C.P. Kelco.

### Lipid Compositions.

The molar percentages of lipids in the membranes examined to provide the data shown are as follows. Figs. 2, 3*A*, and 4, and Fig. S1: 96.5% DMPC, 0.5% TR-DHPE, 3% biotin-PE. Figs. 3*B* and Fig. S2: 90% DMPC, 9% DOPC, 0.5% TR-DHPE, 0.5% biotin-PE. Fig. 3*C*: 89% DMPS, 7.5% DOPC, 0.5% TR-DHPE, 3% biotin-PE. Fig. 3*D*: 96.5% DLPA, 0.5% TR-DHPE, 3% biotin-PE. Fig. 3*E*: 96.5% DNPC, 0.5% TR-DHPE, 3% biotin-PE.

### Substrate Preparation.

Gilded hexagonal transmission electron microscope (TEM) grids with aperture diameters of 125 μm were purchased from Structure Probe, Inc. The grids were hydrophobically treated by incubation with a 50 mM solution of octadecylthiol (ODT) in ethanol for 2 h. Prior to dip coating with lipid, the grids were also coated with 2 μL of 40% squalene in hexane to ensure the hydrophobicity of the hexagonal aperture regions (29, 30).

### Lipid Membrane Deposition.

Freestanding lipid bilayers were formed by Langmuir–Schaefer deposition. A glass bottomed chamber was filled with 3 mL of PBS, pH 7.4, and heated to a temperature above *T*_{m} with a temperature controlled stage (QE-1; Warner Instruments). Lipids of the desire composition were mixed in chloroform and deposited on the surface of the heated PBS with a microsyringe forming a surfactant monolayer. The surface tension was constantly monitored by a tensiometer (Kibron Microtrough, Kibron Instruments) that measured the capillary force of a tungsten wire probe. Lipid addition to the surface continued until a saturated surface tension was reached (typically between 30 and 35 mN/m). Gold, ODT-coated TEM grids (described above) were then lowered vertically into the lipid monolayer at 1 mm/ min, remaining in the PBS buffer. The TEM grid was then transferred to a new chamber while underwater to rinse away lipids in solution and reduce background.

### Nanosphere Binding.

Lipid membranes were incubated with nanomolar concentrations of nanospheres for at least 15 min and then rinsed of unbound beads. Typical nanosphere densities were 50 bound particles per 125-μm-diameter freestanding lipid bilayer.

### Microscopy.

Sample imaging was performed using a Nikon TE2000-E inverted fluorescence microscope (Nikon). Nanosphere images were acquired at 200 frames per second with a complementary metal-oxide-semiconductor camera (PCO.1200s; Cooke Corporation) using CAMWARE software. Membrane fluorescence images to verify bilayer integrity were taken with a CCD camera (ORCA-ER; Hamamatsu) using Nikon Elements software. Temperature control was maintained throughout imaging with a stability within ± 0.3 °C.

### Particle Tracking.

Nanosphere positions were tracked using in-house–developed software written in MATLAB. The programs implement previously described and well-established tracking algorithms (20) with a precision of approximately 10 nm, as verified by analysis of nanospheres deliberately immobilized on surfaces.

### Microrheological Analysis.

The mean-squared displacement, 〈Δ*r*^{2}(*τ*)〉, was calculated for each particle trajectory and then averaged to yield one function per membrane at each temperature. We considered only displacements larger than the 10 nm precision noted above, thereby excluding steps for which there is ambiguity as to whether the step size indicates true particle motion or positional uncertainty. Calculation of *G*^{∗}(*ω*) from 〈Δ*r*^{2}(*τ*)〉 was performed by the method outlined by Mason and Weitz (9, 21). In brief, generalization of the Stokes–Einstein relation to viscoelastic media gives a relationship between *G*^{∗}(*ω*) and the unilateral Fourier transform of 〈Δ*r*^{2}(*τ*)〉. This frequency-space characterization of the mean-square displacements cannot be determined exactly but can be approximated using power law expansions around the frequencies of interest and relating *G*^{∗}(*ω*) to the Fourier transform of this local power law fit (14, 21).

### Determination of *T*_{m}.

*T*_{m} was determined by fitting to linear forms above and below the temperature at which was minimal and identifying the intersection of the two linear fits as the chain melting temperature. This procedure yielded values within about 1 °C of literature values of lipid transition temperatures; from ref. 31, *T*_{m,lit} = 23.5 °C for DMPC, 36 °C for DMPS, 33.5 °C for DLPA, and 26.7 °C for DNPC.

### Determination of *ω*_{c}.

Cross-over frequencies are determined from fits of *G*^{′} and *G*^{′′} to a Maxwell model and are in close agreement with the *ω*_{c} evident by eye for temperatures a few degrees away from *T*_{m} (Fig. 2 *A* and *C*). Near *T*_{m}, where *ω*_{c} is large, the fit-derived *ω*_{c} values provide a measure of the rheological cross-over in frequency regimes outside the directly accessible range.

## Acknowledgments

We thank Jennifer Curtis, John Toner, and Eric Weeks for useful discussions. This work was supported by the Alfred P. Sloan Foundation (R.P.), the Office of Naval Research through the Oregon Nanoscience and Microtechnologies Institute (R.P. and C.W.H.), and the National Science Foundation via Awards 1006171 (to R.P.), 0742540 (to C.W.H.), and 0622620 (to M.J.B.).

## Footnotes

^{1}To whom correspondence should be addressed. E-mail: raghu{at}uoregon.edu.Author contributions: C.W.H. and R.P. designed research; C.W.H. and M.J.B. performed research; C.W.H. analyzed data; and R.P. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1010700107/-/DCSupplemental.

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