Controlling Anomalous Diffusion in Lipid Membranes

Controlling Anomalous Diffusion in Lipid Membranes


Diffusion in cell membranes is not just simple two-dimensional Brownian motion, but typically depends on the timescale of the observation. The physical origins of this anomalous sub-diffusion are unresolved, and model systems capable of quantitative and reproducible control of membrane diffusion have been recognized as a key experimental bottleneck. Here we control anomalous diffusion using supported lipids bilayers containing lipids derivatized with polyethylene glycol (PEG) headgroups. Bilayers with specific excluded area fractions are formed by control of PEG-lipid mole fraction. These bilayers exhibit a switch in diffusive behavior, becoming anomalous as bilayer continuity is disrupted. Diffusion in these bilayers is well-described by a power-law dependence of the mean square displacement with observation time. The parameters describing this diffusion can be tailored by simply controlling the mole fraction of PEG-lipid, producing bilayers that exhibit anomalous behavior similar to biological membranes.

I Introduction

Diffusion is an essential transport mechanism in membrane biology, vital for a wide range of biological function including protein organization Sheets et al. (1997), signalling Choquet and Triller (2003); Kholodenko (2006) and cell survival Cheema et al. (2012). Interestingly, such living systems do not in general display the Brownian motion predicted by a simple random walk model, and instead exhibit ‘anomalous’ diffusion Saxton (1994) where the diffusivity is dependent on the timescale of observation. This phenomenon has been reported both for three-dimensional diffusion in the cell cytosol Regner et al. (2013) and two-dimensional diffusion in the plasma membrane Höfling and Franosch (2013); Fujiwara et al. (2016); Golan and Sherman (2017). Here we focus on membrane diffusion.

Why and how anomalous diffusion exists in the plasma membrane has been the subject of considerable investigation (reviewed in Saxton et al. (2012)). The common underlying mechanism is thought to be the crowded environment found in the cell membrane Kusumi et al. (2005), and the presence of slower-moving obstacles Saxton (1987); Berry and Chaté (2014), pinning sites, and compartmentalization Fujiwara et al. (2002); Murase et al. (2004); Kusumi et al. (2005) have all been suggested as potential contributors to anomaleity in membrane diffusion. Confinement in cellular membranes is observed on the order of tens to hundreds of nanometers, with anomalous diffusion reported in a large number of cell types Fujiwara et al. (2002); Murase et al. (2004). Overall this work has led to the adoption of a compartmentalized ‘picket fence’ model of the cell membrane as a proposed improvement to the ‘fluid-mosaic’ model Kusumi et al. (2005).

Artificial lipid bilayers have played a key role in improving our understanding of anomalous diffusion Schütz et al. (1997); Ratto and Longo (2003); Horton et al. (2010); Spillane et al. (2014); Wu et al. (2016); Rose et al. (2015), where both phase separation Ratto and Longo (2003) and protein binding Horton et al. (2010) in supported lipid bilayers (SLBs) have been used to generate anomalous diffusion. Simulations have also been vital in advancing our understanding, with much pioneering Saxton (1989, 1994, 2001) and recent Stachura and Kneller (2014); Mardoukhi et al. (2015); Koldsø et al. (2016); Jeon et al. (2016); Bakalis et al. (2015); Javanainen et al. (2013) work in this area. In particular, simulations have helped elucidate the role of mobile and immobile obstacles in causing anomalous behavior Saxton (1987); Berry and Chaté (2014). Relevant to our work, simulations have also been used to better interpret single particle tracking data Kepten et al. (2015) and provide methods to discriminate between classes of anomalous diffusion Metzler et al. (2014).

Despite these advances, the specific molecular mechanisms that give rise to anomalous diffusion in vivo remain elusive. This is most clearly highlighted by Saxton et al., who published a call for ‘a positive control for anomalous diffusion’ as a solution to this problem Saxton et al. (2012). This positive control would be a simple and reproducible experimental model exhibiting ‘readily tuneable’ anomalous diffusion spanning several orders of magnitude in timescale. Here we seek to address this call by engineering a simple experimental model in which it is possible to select the anomalous behavior. We take advantage of previous work on the disruption of SLB formation by PEG-DPPE Kaufmann et al. (2009) to control nanoscale obstacle formation in a bilayer (Fig. 1A). By varying the PEG-DPPE composition in a bilayer, we expect that an increased fraction of polymer in the brush regime will result in the formation of specific defects in the bilayer, similar to interfacial or grain boundary defects caused by phases separating mixtures Keller et al. (2005). Similar defects have also been reported using Atomic Force Microscopy (AFM) of incomplete SLB formation from small unilamellar vesicles (SUVs) Richter and Brisson (2005) as well as in SLBs formed in the presence of membrane active peptides Oliynyk et al. (2007).

The complex nanometer-scale confinement reported in cell membranes gives rise to anomalous behavior that spans from microseconds to seconds. Thus to properly characterize anomalous diffusion, it is important to apply techniques capable of studying these timescales. Here we exploit a combination of single-molecule Total Internal Reflection Fluorescence (smTIRF) Axelrod et al. (1984) and Interferometric Scattering (iSCAT) microscopies Lindfors et al. (2004); Ortega-Arroyo and Kukura (2012) (Figs. 1B and C, Supplementary Methods) to characterize diffusion using single-particle tracking that spans over four orders of magnitude in time.

Figure 1: PEG bilayer model. (A) Schematic of supported lipid bilayers. As the mole fraction of PEGylated lipids increases (left to right), defects form in the bilayer that that act as obstacles, generating anomalous diffusion. Representative single-particle tracking of smTIRF (B) and iSCAT (C) images (scale 10 m and 1 m, respectively).

Ii Theory

Anomalous diffusion describes random molecular motion that does not display a linear scaling of the second moment with time. The most common model for anomalous diffusion is to allow the second moment to scale as a power of time Havlin and Ben-avraham (1987); Saxton (1994),


where is the anomalous exponent and is replaced by , the anomalous transport coefficient. Anomalous sub- and super-diffusion are defined by and . Given the form of equation 1, can be determined from the gradient of a logarithmic plot of vs. .

The transport coefficient is somewhat more difficult to interpret as it has dimension of [L]/[T], thus its dimensions are changing for different degrees of anomalous behavior. This apparent problem can be overcome by de-dimensionalizing the observation time Saxton (1994) using a ‘jump time’, :


can be interpreted in terms of a length scale () associated with the anomalous behavior in 2D ().

Figure 2: Anomalous diffusion in PEG bilayers. (A) AFM shows an increase in defect area fraction with increasing mol% PEG-DPPE (scale 500 nm). (B) Anomalous sub-diffusion increases as amount of PEG-DPPE increases from 0 (black) to 10 (yellow) mol%, here for PEG(2K)-DPPE. (C) Equivalent iSCAT data for 0 to 2.6 mol% PEG(2K)-DPPE. Variation of and with mol% PEG-DPPE (D&F), and excluded area fractions (E&G). PEG(1K)- (blue triangles), PEG(2K)- (red circles) and PEG(5K)-DPPE (green squares). Error bars (grey) throughout represent standard errors.

Iii Results

iii.1 Supported Lipid Bilayers

We produced SUVs from DOPC doped with PEG-DPPE (0 - 10 mol% PEG-DPPE; 1,2, & 5 kDa PEG-). Fusion of these SUVs onto a glass coverslip created a SLB. We confirmed the physical nature of the bilayers using AFM (Fig. 2A): As PEG-DPPE content increases, small defects appear in the bilayer. Further increase in the concentration of PEG-DPPE results in the extension of the interfacial defects, until the system crosses the percolation threshold. This leads to confined bilayer patches. Image binarization and autocorrelation were used to calculate the excluded area fraction and the length scale associated with the defects.

The diffusive properties of these bilayers were assessed using single-particle tracking: smTIRF microscopy was used to follow Texas Red-labelled lipids ( mol% TR-DHPE) at 200 Hz; iSCAT tracked 40 nm antibiotin-conjugated gold nanoparticles (AuNPs) tethered to biotinylated lipids at 5 kHz.

As the concentration of PEG(2K)-DPPE was increased from 0 to 6 mol%, the gradient of the vs. plot deviates from zero (Fig. 2B). Figure 2C shows the equivalent PEG(2K)-DPPE dataset arising from iSCAT. Similar plots were produced for PEG(1K)- and PEG(5K)-DPPE (Fig. S1). Gradients extracted from these plots () allow calculation of the anomalous exponent, while the y-axis intercept reports the transport coefficient. The values of for all three PEG molecular weights are collated in Figure 2D; transitions from 1 (free diffusion) to 0 (confined diffusion). These data were fit empirically by a simple sigmoid. The midpoint of each sigmoid is a measure of the transition between continuous and discontinuous diffusion. Both smTIRF and iSCAT measurements give rise to the same trend (Table S1). Figure 2F shows the equivalent variation of with mol% PEG-DPPE. Again, in agreement with Figure 2D, the apparent diffusion coefficient slows as the particles become confined. It is worth emphasizing that as scales with , only points with the same values can be compared directly. Limiting values for differ as expected between smTIRF and iSCAT experiments due to the size difference of fluorescently labelled lipids and AuNPs Mascalchi et al. (2012).

Using our AFM calibration (Fig. S2), we are able to convert mol% to excluded area fraction (Fig. 2E). For both and , the sigmoids for the three different PEG molecular weights now overlap, showing the same trend with excluded area fraction. For vs. excluded area fraction a single sigmoid fit yields a midpoint at .

iii.2 Monte Carlo Simulations

Figure 3: Monte Carlo simulations of anomalous diffusion. (A) Schematic of the unit cell. (B) Diffusion analysis of the resultant tracks showed similar behavior to experiment ( nm, m s). (C&D) A similar trend to experiment was also present for and for 150 nm, 500 nm and 1 m (grey triangles, black circles, teal squares respectively). (E) Plots of vs. show the expected linear relation, dependent on obstacle size. (F) Similar plots for our experimental data show an essentially static linear relationship for different PEG lengths. PEG(1K)- (blue triangles), PEG(2K)- (red circles) and PEG(5K)- (green squares).

To help improve our understanding of these experimental results we constructed a simple Monte-Carlo simulation of anomalous diffusion: A periodic square lattice of circular, immobile obstacles of radius was simulated using a unit cell with side-length (Fig. 4A). A discrete-time random walk was subject to the constraint that the walk cannot enter the circular obstacle. As expected, anomalous behavior arises in the simulation as the excluded area fraction was increased (Fig. 4B), and again a plateau of normal diffusion at short times was observed. When values are collated a sigmoidal trend was present with being reached near to the percolation threshold for circular obstacles on a square lattice (0.785). was best fit by a double-sigmoid (Fig. 4D, Table S2).

Combining equations 1 and 2, a linear variation of with is expected; and reproduced by our data. The characteristic length scales () calculated from both simulation (Fig. 4E) and experiment (Fig. 4F) are summarized in Table S3.

Iv Discussion

The presence of PEG-DPPE disrupts SLB formation leaving a network of defects whose area fraction is dependent on the concentration. We have exploited this defect formation to create predictable and tuneable anomalous behavior. It is bilayer continuity (not the presence of PEG as an obstacle) that causes the anomalous behavior. Our controls (Fig. S4) confirm that normal diffusive behavior can be rescued by filling in bilayer defects.

The variation of and with excluded area fraction that we observe shows a sigmoidal transition between free and confined diffusion. This relationship can be used to tune anomalous behavior. It has been shown using simulations Stachura and Kneller (2014) and in cell membranes Murase et al. (2004); Schwille et al. (1999); Feder et al. (1996); Smith et al. (1999); Schütz et al. (1997) that values of 0.5 to 0.7 are most biologically relevant. Using our model, we can make specific and controlled changes to obstacle extent that match this range; providing an opportunity to use this simple model to help predict and study biological systems that exhibit complex membrane diffusion in vivo.

We must also address the limitations of this model system. Confinement in cell membranes is not created directly by membrane defects, but is likely due to the excluded area created by membrane proteins and their interactions with lipids. Despite these fundamental differences, both result in a similar restriction to free diffusion in the bilayer, and a parallel can be drawn between the excluded area controlled in this simple model and that inaccessible to diffusing species in cell membranes.

Theory predicts that in a system with finite hierarchy, the diffusion will return to normal behavior (with a reduced diffusion coefficient) at sufficiently long observation times Saxton (2007). Figure 2B shows that over the time scales observed in these experiments, the diffusion here remains anomalous. As normal behavior returns at around 100 ms for similarly sized compartments in cells Murase et al. (2004), our model must not possess the restricted range of compartment size that are presumably present in cell membranes. However, we predict that additional control of bilayer defect formation would enable a return to normal diffusion at these timescales, for example by nano-patterning of the substrate before SLB formation Tsai et al. (2008).

By using two single-particle microscopy techniques we have sampled the anomalous behavior of this model over four orders of magnitude of time. Alone, fluorescence microscopy cannot access the divergence of the diffusivity at short times and the onset of anomalous behavior. However iSCAT is not without its own limitations: iSCAT image analysis requires efficient background subtraction Kukura et al. (2009); Ortega-Arroyo and Kukura (2012) which fails if particles do not move sufficiently e.g. within the confined regime. The high frame rate of iSCAT also presents its own challenges in data management, preventing us from probing all the relevant timescales with a single technique.

Our simulations helped us to understand the relationship between and excluded area fraction. In the square lattice model, an excluded area fraction of 0.785 represents the point at which the obstacle diameter is equal to the size of the unit cell, . This is therefore the point at which confinement occurs. The mid point of the sigmoidal fit occurs at an excluded area fraction of 0.773 and the fit effectively reaches = 0 at 0.808. If we consider to sample the probability of being confined, we can say that the mid-point is indicative of the percolation threshold. Using this to interpret the experimental data we find that the percolation threshold of our model falls around an excluded area fraction of .

Perhaps of greatest interest is the difference between plots of vs. between our simulations and experiment (Table S3): Only our simulation shows a variation in length scale with obstacle size. In contrast, our experimental data shows a similar gradient for all three molecular weights of PEG-DPPE. This suggests that in the experiment the size (but not number or extent) of defects produced are of a similar scale ( 150 nm) and are independent of the PEG molecular weight. The values extracted from AFM FWHMs are smaller by around a factor of 3. There is also a modest negative correlation between FWHM and excluded area fraction (see Fig. S4). The differences between experiment and simulation are most likely due to the different topologies for defects in the two cases: To preserve simplicity, defects in the simulation were simple circles. This is in contrast to our experimental AFM images (Fig. 2A) that show not circular, but rather interfacial defects for intermediate PEG-lipid concentrations. We therefore interpret this characteristic length scale as a descriptor of the barrier to diffusion. In our model, that is the scale associated with bilayer defects, and in cells it is likely to be associated with the cytoskeleton.

V Conclusion

By controlling SLB formation using PEGylated lipids we are able to produce bilayers with defined anomalous diffusive properties, dependent on the excluded area fraction. Thus, we hope our work in part answers the call for a simple and reproducible experimental model, readily tuneable in anomaleity over the length scales observed in vivo. This study also opens the way to further experiments that exclude membrane area using more complex methods than the simple inclusion of PEG-lipids presented here. Future work must be directed to expand our understanding of cell membranes - to recreate biological pathways controlled by diffusion, and to enable the rational design of devices with tailored bilayer properties.

Vi Acknowledgements

We thank the European Research Council for providing funding for this work (ERC-2012-StG-106913 CoSMiC).

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