Two-Point Microrheology of Phase-Separated Domains in Lipid Bilayers

Abstract

Though the importance of membrane fluidity for cellular function has been well established for decades, methods for measuring lipid bilayer viscosity remain challenging to devise and implement. Recently, approaches based on characterizing the Brownian dynamics of individual tracers such as colloidal particles or lipid domains have provided insights into bilayer viscosity. For fluids in general, however, methods based on single-particle trajectories provide a limited view of hydrodynamic response. The technique of two-point microrheology, in which correlations between the Brownian dynamics of pairs of tracers report on the properties of the intervening medium, characterizes viscosity at length-scales that are larger than that of individual tracers and has less sensitivity to tracer-induced distortions, but has never been applied to lipid membranes. We present, to our knowledge, the first two-point microrheological study of lipid bilayers, examining the correlated motion of domains in phase-separated lipid vesicles and comparing one- and two-point results. We measure two-point correlation functions in excellent agreement with the forms predicted by two-dimensional hydrodynamic models, analysis of which reveals a viscosity intermediate between those of the two lipid phases, indicative of global fluid properties rather than the viscosity of the local neighborhood of the tracer.

Introduction

The lipid bilayer is one of the most important structures in biology. In cells, these bilayers perform a myriad of functions, enabled by several structural features. Membrane fluidity has long been recognized as paramount among these (1). In particular, fluidity allows molecules at the surfaces of cells and organelles to spatially reorganize in response to various stimuli (2,3). The timescales for any such fluidity-mediated processes will be set in part by the viscosity of the lipid membrane itself, and so quantifying membrane viscosity is an important prerequisite for the development of predictive models of cellular dynamics. However, measurements of this key material property remain sparse, even if we extend the literature to include studies of related systems such as monolayers or multilayers. Furthermore, such studies have often relied on complicated protocols or methods that are suitable only for specific model systems (4–8). Many of the most precise measurements of lipid bilayer viscosity have used passive microrheology, in which the Brownian trajectories of membrane-anchored particles or of lipid domains in the bilayer are recorded and analyzed to reveal insight into the fluid properties of the sample (9–11). Although powerful, all such studies to date have made use of single-point methods, in which the statistics of individual tracer motions are analyzed, which report on the local environment of the tracer and hence may not be representative of global characteristics, perhaps because of the influence of the tracer itself.

The methodology of two-point microrheology compliments single-point techniques by considering the correlated displacements of pairs of particles. This extends the length-scale examined from the tracer radius to the separation distance between tracer pairs, and is therefore sensitive to the separating medium in addition to the individual tracer neighborhoods (12–15). Disparities between two-point and single-point microrheology, then, demonstrate length-scale dependent effective viscosities. For a cellular membrane, such length-scale separation could imply that the viscosity relevant for protein diffusion, for example, would be different than the viscosity relevant for large-scale membrane deformations.

Two-point microrheology has been applied to a wide variety of three-dimensional materials (12,16,17) and has been extended both theoretically and experimentally to two-dimensional fluids. Levine and MacKintosh have derived the response functions that characterize a membrane embedded in a three-dimensional fluid, providing explicit forms for interparticle correlation functions that can be compared with measured correlations (18). Weeks and co-workers have used two-point microrheology with colloidal microspheres as tracers to examine thin soap films as well as proteins at an air-water interface, quantifying the two-dimensional viscosity and establishing the hydrodynamic response functions of these systems (19–21). To date, these pioneering studies have been the only published reports, to our knowledge, of two-point microrheology of two-dimensional fluids, leaving open the question of what two-point analysis will reveal for lipid membranes. In particular it is unknown, before the studies reported in this article, whether simple viscous fluid models are adequate to describe correlated diffusion of membrane inclusions, and if so, whether the reported viscosity will be the same as that shown by single-point methods.

We examined the rheology of giant unilamellar vesicles (GUVs) exhibiting cholesterol-dependent phase separation into coexisting liquid phases. Such systems can be broadly controlled to tune the degree and scale of compositional heterogeneity. Furthermore, phase-separated bilayers are well-characterized model systems (22) and, importantly, have been studied using single-point microrheology because phase-separated domains themselves can be used as tracers that report the viscosity of the majority phase (9,23).

Materials and Methods

Giant unilamellar vesicle composition and preparation

We formed GUVs by electroformation (24) in 0.1 molar sucrose and used the same solution for the exterior environment in our experiments. The diameters of the vesicles examined were in the range 50 to 100 μm. We considered five different GUV compositions with differing fractions of DPPC (1,2-dipalmitoyl-snglycero-3-phospocholine), DOPC (1,2-dioleoyl-sn-glycero-3-phosphocholine), and cholesterol. All lipids were purchased from Avanti Polar Lipids (Alabaster, AL). Vesicles with such compositions readily partition in to liquid-ordered (${\text{L}}_{\text{O}}$) and liquid-disordered (${\text{L}}_{\text{D}}$) domains (22,23,25–27). This phase separation can be observed experimentally through inclusion of a small amount of Texas Red DHPE, a fluorescent lipid probe that preferentially partitions into the ${\text{L}}_{\text{D}}$ phase (Fig. 1). In our experiments, we formed GUVs with 1 mol% Texas Red DHPE. Control experiments using a lower probe concentration (0.2 mol%) showed similar domain diffusion coefficients, but poor signal-to-noise ratios prohibited precise assessments of viscosity (see Supporting Material). We provide in the Supporting Material(Fig. S6) a DOPC:DPPC:cholesterol phase diagram constructed from earlier studies (22,28), which also indicates the miscibility transition temperatures, on which we have indicated the compositions used in this work.

Figure 1.

Fluorescence images from GUVs with diffusing phase-separated domains: (top row) ${\text{L}}_{\text{D}}$ majority and (bottom row) ${\text{L}}_{\text{O}}$ majority. In each case the images shown were captured 4 sec apart.

Fluorescence microscopy and domain tracking

We recorded epifluorescence images at 10 to 40 frames per second with a Hamamatsu ORCA CCD camera (Hamamatsu City, Japan) on a Nikon TE2000 inverted fluorescence microscope with a 60× magnification objective. All measurements were made at room temperature (296 K).

We identified phase-separated domains in images by intensity-based thresholding and estimated domain centers by fitting two-dimensional Gaussian profiles using maximum likelihood estimation. From tracking simulated images with similar signal-to-noise characteristics, and from the statistical assessment of Vestergaard et al. (29), we estimate our localization error to be less than 0.07 μm. This localization uncertainty contributes to the uncertainty in measured diffusion coefficients and correlations. Domain boundaries were determined using a bilateral filter (30); the enclosed area was used to determine the domain radius. Uncertainties in the domain radius because of growth and bulging of domains out of the membrane plane have negligible effects on the assessed viscosities, as discussed in Supporting Material. We considered only domains that are located within at most one-third of the vesicle radius from the GUV pole, as these appear in-focus in images, and only small components of their motion are perpendicular to the focal plane. Furthermore, we selected only domains that were continuously imaged for at least 100 frames; this ensures sufficient statistics to characterize domain diffusion (see below). We linked domain positions into trajectories using a nearest neighbor linking algorithm. This process yields a time series of domain positions for several domains per GUV that can be analyzed to obtain diffusion coefficients and other statistics. The number of domains per GUV ranged from three to more than fifty.

Analysis of tracer motion

Given a set of single-particle positions $\mathbf{x}\left(t_n\right)$ measured at times $t_1,t_2,\dots ,t_N$, each separated by an interval $\Delta t$ (for instance from time series data such as we obtained using the process detailed above), estimating the particle’s diffusion coefficient is a well-defined task, for which there exists an explicit, unbiased, and nearly optimal estimator based on the covariance of the displacements (29). (A linear fit of the mean-squared displacements $\Delta {\mathbf{x}}^2\left(\tau \right)$ versus the time interval of the displacement, τ, though often employed, does not provide a good estimate because of the correlation of the values with one another; in fact, it can have the perverse property of becoming less accurate as the number of data points increases (29).) The covariance-based estimator is given by the following:

$$D=\langle \Delta {\mathbf{x}}_n^2\rangle /\left(2\Delta t\right)+\langle \Delta {\mathbf{x}}_n\Delta {\mathbf{x}}_{n+1}\rangle /\Delta t,$$ (1)

where $\Delta {\mathbf{x}}_n=\mathbf{x}\left(t_n\right)-\mathbf{x}\left(t_{n-1}\right)$ is the displacement over one time step and the angle bracket indicates an average over all time steps. Moreover, the approach described in Vestergaard et al. (29) provides a measure of the goodness of fit (${\chi }^2$) of an observed trajectory to a purely Brownian one via comparison of its periodogram (roughly, the power spectrum of $\Delta {\mathbf{x}}_n$) with the functional form for free diffusion (see the Supporting Material for a brief discussion as well as examples of mean-square-displacements of trajectories). For our lipid domains, we find the average reduced ${\chi }^2=1.25$, indicative of pure diffusion in a viscous liquid (Fig. 2).

Figure 2.

Histogram of reduced ${\chi }^2$ values obtained by comparing tracer motion with the expected form for free diffusion. Binned values for the ${\text{L}}_{\text{O}}$ majority composition 2:1 DPPC:DOPC with 40% cholesterol are in blue; corresponding bins for the ${\text{L}}_{\text{D}}$ majority composition 1:2 DPPC:DOPC with 20% cholesterol are in orange. To see this figure in color, go online.

In two-point microrheology, one considers the correlations of displacements. The radial component of the correlation tensor, $D_{rr}$, is determined from the following measurements of the displacements of tracers i and j over time τ (18,19):

$$D_{rr}\left(R,\tau \right)={\langle \Delta r^i\left(\tau \right)\Delta r^j\left(\tau \right)\delta \left[R-R^{ij}\right]\rangle }_{i\ne j},$$ (2)

where R is the particle separation distance, $R^{ij}$ is the distance between tracers i and j, the $\Delta r$ are the components of the displacements calculated along the axis connecting the tracers’ positions, and the angle brackets indicate an average over all time points and particle pairs $i\ne j$. $D_{rr}$ increases linearly with τ for Brownian particles, much like a particle’s mean square displacements. As is the case for single-point diffusion, however, this linearity is misleading and does not yield an optimal estimator for the correlation. The aim of quantifying the correlation between Brownian processes arises often in finance (31,32), and the associated literature shows that the optimal estimator, just like in the single-particle case, is formed from the covariance of single-frame displacements (32).

Results

We first consider single-point viscosity measurements, determined from the dependence of the diffusion coefficients, D, of phase-separated domains on their radii, a. Our single-domain diffusion data for two of the compositions examined, a ${\text{L}}_{\text{D}}$ majority phase (2:1 DOPC:DPPC with 20% cholesterol) and a ${\text{L}}_{\text{O}}$ majority phase (1:2 DOPC:DPPC with 40% cholesterol), are shown in Fig. 3. Fitting to the classic two-dimensional hydrodynamic model of Hughes, Pailthorpe, and White (HPW) (33), as in earlier work (9,23), gives membrane viscosity ${\eta }_{1\text{pt}}=0.75\pm 0.15$ nPa$\cdot$ s$\cdot$ m for the ${\text{L}}_{\text{D}}$ majority phase, and ${\eta }_{1\text{pt}}=3.90\pm 0.42$ nPa$\cdot$ s$\cdot$ m for the ${\text{L}}_{\text{O}}$ majority phase, with uncertainties assessed by jackknife error estimation (34). As expected, the ${\text{L}}_{\text{O}}$ phase is more viscous than the ${\text{L}}_{\text{D}}$ phase (23). The HPW model applies to solid inclusions; however, recent calculations of the mobility of liquid domains in fluid membranes (35,36) indicate that for the conditions observed in our experiments, corrections to the HPW model will be approximately a few percent in magnitude, well within the uncertainty of our measurements.

Figure 3.

Single-point microrheology and radius dependence of domain diffusion coefficients. Data from two different GUV compositions are shown: a ${\text{L}}_{\text{D}}$ majority phase (2:1 DOPC:DPPC with 20% cholesterol) in orange, and a ${\text{L}}_{\text{O}}$ majority phase (1:2 DOPC:DPPC with 40% cholesterol) in blue. Points are averages from roughly 10 domains each, with error bars indicating the standard deviation. Curves represent fits to the HPW model of $D\left(a\right)$ with bilayer viscosity as the single free parameter. To see this figure in color, go online.

Similarly, we can calculate viscosity from the two-point correlations measured from the same trajectories. From Levine and MacKintosh (18), the expected form is the following:

$$\frac{D_{rr}\left(R,\tau \right)}{\tau }=\frac{k_{B}T}{2\pi \eta }\left[\frac{\pi }{\beta }{H}_1\left(\beta \right)-\frac{2}{{\beta }^2}-\frac{\pi }{2}\left[Y_0\left(\beta \right)+Y_2\left(\beta \right)\right]\right],$$ (3)

where $\beta =2R{\eta }_B/\eta$ is the reduced tracer separation, ${\eta }_B$ is the viscosity of the aqueous medium surrounding the bilayer, $H_{\nu }$ are Struve functions, and $Y_{\nu }$ are Bessel functions of the second kind.

We find that the dependence of the measured correlations on domain separation agrees with the expected theoretical form (Fig. 4). However, in contrast to the single-point measurements, we find that the viscosities of the two compositions as measured from two-point correlations are indistinguishable within uncertainties: ${\eta }_{2\text{pt}}=2.76\pm 1.45$ nPa$\cdot$ s$\cdot$ m for the ${\text{L}}_{\text{D}}$ majority phase, and ${\eta }_{2\text{pt}}=2.95\pm 0.61$ nPa$\cdot$ s$\cdot$ m for ${\text{L}}_{\text{O}}$ majority phase. Furthermore, we note that these values are similar to the average of the single-point results for the two majority phases: 2.32 nPa$\cdot$ s$\cdot$ m.

Figure 4.

Two-point correlations $D_{rr}$ and determination of viscosities. Data from the same two GUV compositions as in Fig. 3 are shown: ${\text{L}}_{\text{D}}$ majority data in orange and ${\text{L}}_{\text{O}}$ majority data in blue. Curves represent fits to Eq. 3. Each point is the average of more than 100 measurements of $D_{rr}$, with error bars indicating the standard deviation. To see this figure in color, go online.

It appears that two-point microrheology of minority-phase lipid domains returns a value for membrane viscosity that is intermediate between those of the majority and minority phases. To test whether this is merely a peculiarity of the compositions examined for the data in Figs. 2 and 3 or a more robust feature of phase-separated vesicles, we measured one- and two-point-derived viscosities over a range of GUV compositions for which the single-point-derived majority phase viscosity spans nearly two orders of magnitude (Fig. 5). For ${\text{L}}_{\text{O}}$ majority-phase lipids spanning 1:1 to 1:9 DOPC:DPPC, we find that the ratios of ${\eta }_{2\text{pt}}/{\eta }_{1\text{pt}}$ are less than 1, with ${\eta }_{2\text{pt}}=\left(0.77\pm 0.05\right)\times$ the single-point viscosity (Fig. 5, blue symbols). Analogously, for ${\text{L}}_{\text{D}}$ majority-phase lipids of 2:1 and 4:1 DOPC:DPPC, ${\eta }_{2\text{pt}}/{\eta }_{1\text{pt}}$ is greater than 1, with ${\eta }_{2\text{pt}}=\left(3.07\pm 0.86\right)\times$ the single-point viscosity (Fig. 5, orange symbols).

Figure 5.

Ratio of the viscosities derived from two- and one-point methods as a function of single-point viscosity for a range of different compositions. Blue icons indicate compositions that are majority ${\text{L}}_{\text{O}}$, whereas orange symbols indicate compositions that are majority ${\text{L}}_{\text{D}}$. Compositions are given as mol$\%$ DOPC:DPPC:cholesterol. To see this figure in color, go online.

Discussion

We report, to our knowledge, the first demonstration that two-point microrheology can be applied to lipid membranes, providing a sensitive test of the applicability of continuum two-dimensional hydrodynamic models to lipid systems. Despite their topographic distortions (37) and potential for long-range interactions (38), phase-separated membrane domains show a distance-dependent correlation in their Brownian dynamics with a functional form in remarkably good agreement with theories of two-dimensional fluid response.

More importantly, our results imply that two-point measurements report an effective global membrane viscosity, amalgamating the characteristics of compositionally different regions of the membrane, whereas single-point measurements probe the viscosity of the majority phase surrounding tracer domains. This is perhaps to be expected for a two-dimensional fluid, because hydrodynamic correlations in two dimensions are intrinsically long-ranged. Relatedly, recent theoretical work points to a strong sensitivity of in-plane correlations to static inclusions, even at low concentrations, again driven by long-range interactions (39). It would be interesting to develop methods to examine, both theoretically and experimentally, two-point viscosity as a function of the area fraction of the minority phase to determine the weighting of the properties of different regions toward the overall response. We also note that existing theories of two-point correlations are formulated for small, rigid inclusions. Though their forms fit our observations, we hope that our work will spur the development of models that explicitly consider the dynamics of finite-sized fluid domains, as has recently been done for single-domain diffusion (35,36).

We stress that, in contrast to various three-dimensional complex fluids for which two-point methods give measures of viscosity relatively uncontaminated by the distortions of local probes, our results imply that two-point methods applied to phase-separated membranes should not be considered better than single-point methods. Rather, the latter provide insights into the viscosity of particular phases, whereas the former provide insights into the larger-scale effective viscosity of a heterogeneous fluid. We note that cellular membranes exhibit a far greater degree of heterogeneity in structure and composition than the model bilayers examined here. It would be interesting to examine whether, similarly, two-point viscosity using various cellular membrane probes would show robust features that average over small-scale heterogeneity.

Author Contributions

T.T.H. and R.P. designed the research; T.T.H. and M.A.R. performed the research; T.T.H., M.A.R., and R.P. analyzed data; and T.T.H. and R.P. wrote the article.

Editor: Ka Yee Lee.