Supported lipid bilayers (SLBs) have been studied extensively as simple but powerful models for cellular membranes. Yet, potential differences in the dynamics of the two leaflets of a SLB remain poorly understood. Here, using single particle tracking, we obtain a detailed picture of bilayer dynamics. We observe two clearly separate diffusing populations, fast and slow, that we associate with motion in the distal and proximal leaflets of the SLB, respectively, based on fluorescence quenching experiments. We estimate diffusion coefficients using standard techniques as well as a new method based on the blur of images due to motion. Fitting the observed diffusion coefficients to a two-leaflet membrane hydrodynamic model allows for the simultaneous determination of the intermonolayer friction coefficient and the substrate-membrane friction coefficient, without any prior assumptions on the strengths of the relevant interactions. Remarkably, our calculations suggest that the viscosity of the interfacial water confined between the membrane and the substrate is elevated by ∼10^{4} as compared to bulk water. Using hidden Markov model analysis, we then obtain insight into the transbilayer movement of lipids. We find that lipid flip-flop dynamics are very fast, with half times in the range of seconds. Importantly, we find little evidence for membrane defect mediated lipid flip-flop for SLBs at temperatures well above the solid-to-liquid transition, though defects seem to be involved when the SLBs are cooled down. Our work thus shows that the combination of single particle tracking and advanced hydrodynamic modeling provides a powerful means to obtain insight into membrane dynamics.

## INTRODUCTION

Supported lipid bilayers (SLBs) are commonly used as simplified mimics of cellular membranes in order to gain insight into basic membrane dynamics and function.^{1–7} For instance, the thermodynamics of phase behavior and lipid domain formation is largely preserved in SLBs and can be investigated by various surface-specific techniques.^{8–15} While lipids are considered free to diffuse in the fluid two-dimensional space formed by the SLB, it is well known that substrate proximity leads to a reduction in lipid mobility relative to free standing membranes.^{4,16,17} Depending on the preparation technique, lipid probe, and substrate type, either uniform^{11,16,18–21} or bimodal^{11,21–26} lipid diffusivity has been reported in the fluid phase of SLBs. Though it has been reasoned that the bimodal diffusion arises from different dynamics of the proximal and distal leaflets of the membrane,^{11,21,22,24} this proposal has only been clearly validated for membranes on silica beads.^{22} Alternatively, bilayer heterogeneity, defect structures, probe clustering, probe immobilization, and probe orientation have been suggested to cause the disparities in diffusivity.^{4,11,15,17,25–27}

In the popular description for membranes supported by a planar substrate, due to the work of Evans and Sackmann,^{18,28} it is assumed that momentum transfer in the membrane is dominated by the presence of the substrate. The amount of frictional coupling between the membrane and the substrate depends on the thickness and viscosity of the water layer separating the two.^{18,28} However, the hydrodynamic model of Evans and Sackmann treats the membrane as a single sheet, ignoring its dual leaflet structure. In the case of strong substrate-membrane coupling, partial decoupling may take place between the two leaflets of the membrane, which may be behind the appearance of two diffusion coefficients, as mentioned above.^{21,24,29} Such a decoupling was observed in studies of the solid-to-liquid phase transition of the proximal and distal leaflets of SLBs.^{8–10} Preparation of an asymmetric SLB can be used to shed light on this phenomenon. However, this approach is only meaningful if the bilayer does not equalize its composition within the experimental preparation times.^{14,24} Indeed, lipid flip-flop, i.e., the transbilayer movement of lipids, is considered to be faster for SLBs as compared to free standing membranes.^{30–34} Potential explanations for the high flip-flop rates in SLBs include membrane defects remaining from the preparation process,^{35} as well as surface induced acyl chain disorder.^{33} Experiments on SLBs in the fluid phase^{31,32} suggested flip-flop half times of the order of seconds.

Single-particle tracking (SPT) of membrane bound probes is extensively used to investigate the dynamics of lipid molecules or membrane-embedded proteins by direct observation at high temporal resolution and nanometer-scale precision.^{36} In contrast to ensemble-averaged measurements, SPT allows detecting subpopulations and rare dynamic events.^{37,38} This method is capable of revealing the nanoscale dynamics of either protein or lipid probes in the natural context of the cell plasma membrane,^{38–42} as well as in SLBs, where it has been used to reveal virus-membrane binding,^{6} switching between diffusive modes of membrane inserting peptides,^{3} or the influence of membrane curvature on lipid dynamics.^{5}

Here, we apply dual-color SPT to simultaneously track the motion of two labeled lipids within 1,2-dioleoyl-*sn*-glycero-3-phosphocholine (DOPC) and egg L-α-phosphatidylcholine (EPC) membranes in the fluid phase. We also introduce a novel method to estimate diffusion coefficients from individual frame images, based on the blur of the shapes of particles due to their motion. A Hidden Markov Model (HMM) analysis^{3,43} allows us to distinguish between different diffusive modes and to estimate their transitional dynamics. Iodine quenching experiments are used in order to associate slow and fast diffusing populations with the proximal and distal leaflets of the SLB, respectively. The transitions of probes between the two diffusive modes are therefore attributed to the flip-flop of lipids. Theoretical analysis based on the “Regularized Stokeslets” method, which takes into account the dual-leaflet nature of the bilayer,^{44} further allows us to obtain estimates for the membrane-substrate and inter-leaflet friction coefficients.

## MATERIALS AND METHODS

### SLB preparation and imaging

Flow cells were prepared using two microscope cover glasses (24 × 50 mm and 18 × 18 mm, No1, Marienfeld, Germany) that were first etched in 10% HF (hydrofluoric acid 40%, Merck, Germany) for 40 s while sonicating. The slides were then rinsed with Milli-Q H_{2}0, dried in a N_{2} stream, and glued together using two thin Parafilm (Bemis Company, USA) stripes followed by curing in an oven for 15 min at 115 °C. For the preparation of SLBs, 10 mg/ml Egg L-α-phosphatidylcholine (EPC, Avanti Polar Lipids Inc., USA) or 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC, Avanti Polar Lipids, Inc., USA) were dissolved in tert-butanol (Sigma, USA). Atto550 headgroup-labeled 1,2-dimyristoyl-sn-glycero-3-phosphoethanolamine (Atto550-DMPE, ATTO-TEC, Germany) as well as Atto647N-DMPE (ATTO-TEC, Germany) were dissolved in chloroform (Bio-Lab ltd., Israel). The labeled lipids were then separately added to the lipid solution to obtain a ∼5 × 10^{−5} mol % solution of labeled lipids. The lipid solutions were aliquoted, lyophilized, and stored at −20 °C. One aliquot of each lipid solution was then re-suspended to 10 mg/ml in the experimental buffer (50 mM TRIS, 100 mM NaCl, pH 7.5), extruded together through a 0.1 *μ*m filter (Whatman Anotop, GE Healthcare, USA) to create unilamellar vesicles^{45} and stored at 4 °C. The flow cells were rinsed four times with 200 *μ*l of the experimental buffer and then loaded and incubated with 50 *μ*l of the vesicle solution for 5 min. The flow cells were rinsed again four times with 200 *μ*l of the experimental buffer solution and directly used. In quenching experiments, the flow cells were rinsed two times with the experimental buffer solution and then four times with an iodide solution (50 mM TRIS, 100 mM KI, 0.1 mM Na_{2}S_{2}O_{3}, pH 7.5). SLBs were imaged using a dual color total internal reflection fluorescence (TIRF) microscope equipped with an electron-multiplying charge-coupled device (EMCCD) camera (Andor iXon 897), at a frame rate of 128.87 Hz. For a complete description of the TIRF setup, as well as ensemble fluorescence spectroscopy, please see Secs. S1 and S2 of the supplementary material.

### SPT analysis algorithm

The detection and tracking of particles was implemented in Matlab (MathWorks, USA). In brief (see Secs. S3–S6 of the supplementary material for more details), the extraction of spots was performed using a B3-spline version of the à trous wavelet transform.^{46} The detected spots were then fitted using a symmetric Gaussian model point spread function (PSF), with the expected photon count *E*_{n} at pixel *n* given by

where *b*^{2} is the background level, *I*_{ph} is the amplitude, *μ*_{x} and *μ*_{y} are the subpixel positions of a particle, *σ*_{a} is the Gaussian standard deviation, and *x*_{n}, *y*_{n} are the pixel positions. For iterative fitting of Eq. (1), we used a maximum likelihood estimation (MLE) method for Poisson distributed photon counting events,^{47} based on a modified Levenberg-Marquardt (LM) algorithm.^{48}

For particle tracking and linking between camera frames, we used an approach based on the linear assignment problem (LAP),^{49} with modifications to the cost matrix for particle assignment. After linking the particles, HMM analysis^{3,43} was applied to the full set of trajectories with a minimal length of five steps. The fast and slow diffusion states in each trajectory were then assigned by determining the single best state sequence, using the Viterbi algorithm.^{3,43} Following state assignment, we then used covariance-based estimation (CVE) analysis^{50} to extract diffusion coefficients from individual trajectories. For each SLB type, a total of 60 000–110 000 trajectories were analyzed for Atto550-DMPE and 30 000–45 000 for Atto647N-DMPE.

The SPT algorithm was carefully evaluated by using simulated EMCCD recordings that mimic the experimentally obtained data, including a full statistical treatment of the electron multiplication gain, noise contributions, and sub-frame molecular motion. For comparing flip-flop positions of the two differently labeled lipids, image registration was applied using a linear system of equations^{39} solved for the positions of microbeads, which were recorded before and after each SPT experiment, and the results were used to transform the positions of the lipid probes.

### Motion blur analysis

A motion blur analysis method was developed to obtain diffusion coefficients of the fast moving particles from the shapes of particle images in individual frames. To this end, we superimposed images of single particles (after assigning their trajectories to either the fast or slow state) by co-aligning them with respect to their subpixel position $(\mu ^x,\mu ^y)$ obtained using Eq. (1). Co-alignment was achieved by translating images in frequency space using the fast Fourier transform algorithm *fft2* from Matlab. As we co-aligned the particle images with respect to their estimated subpixel positions $(\mu ^x,\mu ^y)$, this procedure introduced the dynamic localization uncertainty *σ*_{i}(*D*_{i}) as an error in co-alignment, where the index *i* specifies the state, either “fast” or “slow.” Thus, for example, $\mu ^x=\mu \u0303x+\chi i$ where $\chi i\u223cN(0,\sigma i)$ is normally distributed with a zero mean and a standard deviation *σ*_{i}(*D*_{i}), and $\mu \u0303x$ denotes the true particle position. An analogous expression is relevant for *y*. In our analysis, we defined the dynamic localization uncertainty for a moving probe $\sigma i2Di$ in state *i*, including motion blur, as follows:

where the factor $Di\Delta tex/3$ is the variance in trajectory positions due to diffusion^{50,51} (i.e., causing motion blur), $\Delta tex=7.49\u2009ms$ is the exposure time of the camera, *s*_{0,th} = 0.21*λ*/*NA* is the theoretical diffraction limited PSF width^{52} as a function of wavelength *λ* and numerical aperture *NA*, and *N*_{corr,i} is correcting for the number of emitted photons. Eq. (2) is based on the approximation^{52}^{,}$\sigma i2\u2248si2/N$ in combination with^{50,51}$si2\u2248s0,th2+Di\Delta tex/3$. Here, the static localization uncertainty *σ*_{0}(*D* = 0) was approximated by the theoretical localization precision *σ*_{lp}, where *σ*_{lp} was calculated as explained in Ref. 47 from MLE of the slow diffusing probes, i.e., $\sigma 02\u2248\sigma lp,slow2$. Further, $\sigma i2(Di)$ was corrected for the average photon emission ⟨*N*_{i}⟩ of an individual state using *N*_{corr,i} = ⟨*N*_{slow}⟩/⟨*N*_{i}⟩, where ⟨*N*_{slow}⟩ is the average photon emission of the slow particles.

The superimages resulting from the superposition of co-aligned particle images were then least-squares fitted with the Gaussian PSF of Eq. (1) and related to the diffusion coefficient *D*_{i} using^{50}

where $sa2=s02+a2/12$ is the width of the diffraction limited Gaussian PSF *s*_{0}, corrected for the pixel size^{47} (*a* = 107 nm), and $\Delta tex$ is the camera exposure time per frame.

Here it is important to have an accurate measure of the diffraction limited PSF width $s02$. To obtain $s02$ experimentally for our setup, we used $\sigma a,slow2$ (i.e., for state *i* = “*slow*”) from the superimage of the slow moving probes to solve Eq. (3) for $s02$ using *D*_{slow} from CVE analysis and $\sigma slow2Dslow\u2248\sigma lp,slow2$ from the same experiment. The latter is a valid approximation for slow moving probes, i.e., if $D\Delta tex<s02$, as can be seen from Eq. (2). To finally obtain $Dfast\sigma a2$ (i.e., for state *i* = “*fast*”) from the superimage of the fast particles, we solved Eq. (3) using the experimental $s02$ as well as $\sigma fast2Dfast$ obtained from Eq. (2).

## EXPERIMENTAL RESULTS

### Slow and fast diffusion of lipids in SLBs

We recorded movies of fluorescent lipids diffusing in lipid bilayers supported on a microscope glass slide. Diffusive trajectories were created by detecting spots in individual frames and linking them in consecutive frames of the movies, as discussed in the section titled Materials and methods. They were then analyzed using a two-state HMM model in combination with the Viterbi algorithm.^{3,43} This enabled us to assign the sequence of states for individual trajectories, and by selecting the trajectories associated with each state, we obtained two well-separated populations of slow and fast diffusing particles. In Figs. 1(a) and 1(b), we show sample trajectories of slow and fast diffusing Atto647N-DMPE lipids in DOPC SLBs. Distributions of diffusion coefficients *D* of single trajectories of the two types of lipids (i.e., for Atto550-DMPE and Atto647N-DMPE) obtained using CVE analysis^{50} are shown in Figs. 1(c) and 1(d), respectively. *D* values associated with the fast state are shown in green whereas those associated with the slow state are shown in blue. In the histograms, each count represents a single trajectory. We note however that due to the way the experiment and analysis were conducted, the populations of the slow and fast diffusing molecules are not representative of their true state populations under equilibrium. As we measured with relatively high laser intensities, we reached a steady state where molecules within the field of view are photobleached and mainly fast particles are entering the image before they also photobleach. Further, the fast trajectories have fewer steps on average than the slow trajectories (i.e., for Atto647N-DMPE 14 fast steps vs. 37 slow steps). The values of the diffusion coefficients *D*_{fast} and *D*_{slow} for a single sample were obtained from fitting a Gaussian distribution to each population in the distributions. Table I summarizes the diffusion coefficients obtained for EPC and DOPC SLBs as averages over several samples.

. | EPC (RT) (μm^{2}/s)
. | DOPC (RT) (μm^{2}/s)
. | DOPC (10 °C) (μm^{2}/s)
. |
---|---|---|---|

Atto550-DMPE | D_{fast} = 4.22 ± 0.02 | D_{fast} = 4.89 ± 0.02 | D_{fast} = 2.47 ± 0.06 |

D_{slow} = 0.079 ± 0.002 | D_{slow} = 0.091 ± 0.002 | D_{slow} = 0.045 ± 0.001 | |

Atto647N-DMPE | D_{fast} = 3.94 ± 0.02 | D_{fast} = 4.30 ± 0.02 | D_{fast} = 2.22 ± 0.06 |

D_{slow} = 0.054 ± 0.002 | D_{slow} = 0.052 ± 0.002 | D_{slow} = 0.03 ± 0.001 |

. | EPC (RT) (μm^{2}/s)
. | DOPC (RT) (μm^{2}/s)
. | DOPC (10 °C) (μm^{2}/s)
. |
---|---|---|---|

Atto550-DMPE | D_{fast} = 4.22 ± 0.02 | D_{fast} = 4.89 ± 0.02 | D_{fast} = 2.47 ± 0.06 |

D_{slow} = 0.079 ± 0.002 | D_{slow} = 0.091 ± 0.002 | D_{slow} = 0.045 ± 0.001 | |

Atto647N-DMPE | D_{fast} = 3.94 ± 0.02 | D_{fast} = 4.30 ± 0.02 | D_{fast} = 2.22 ± 0.06 |

D_{slow} = 0.054 ± 0.002 | D_{slow} = 0.052 ± 0.002 | D_{slow} = 0.03 ± 0.001 |

All samples consistently revealed slow and fast diffusing populations of lipid probes. The values reported in Table I are in good agreement with diffusion coefficients reported before for similar bilayer systems based on either SPT^{23,25} or fluorescence correlation spectroscopy (FCS),^{20} though no slow population was identified by the latter method. We emphasize that the localization uncertainty is embedded into CVE analysis, and thus the values obtained are free of this contribution. When measuring at 10 °C, the diffusion slows down to about half the values measured at RT for the DOPC membranes. As the movies were recorded in dual color, this allowed us to directly compare the diffusion of the differently labeled lipids. We noticed that Atto550-DMPE lipids are diffusing slightly faster than Atto647N-DMPE lipids as consistently revealed under each experimental condition.

### Motion blur analysis

In addition to the trajectory-based analysis discussed above, we also considered extracting diffusion coefficients from individual frame images. The average width $\sigma a2$ of a particle image, according to Eq. (3), depends on the diffraction limited PSF width $s02$ and the diffusion coefficient *D*, which causes a variance $D\Delta tex/3$ in the particle position.^{51} For a finite exposure time $\Delta tex$ of the camera, this leads to larger particle images, a phenomenon known as motion blur. However, as shown in Fig. 2(a), the scattering in the estimates $\sigma ^a$ from Eq. (1) is too large to infer diffusional properties from single probe images. Therefore, in order to obtain average properties, we superimposed images of single particles by co-aligning them with respect to their subpixel localization position $(\mu ^x,\mu ^y)$. Contour plots of summed particle images are shown in Figs. 2(b) and 2(c) for slow and fast moving populations of Atto647N-DMPE in a DOPC SLB. The images were taken from the same trajectories as for the distributions shown in Fig. 1(d). As can be seen, the resulting superimages are highly symmetric, whereas the individual particle images are asymmetric due to stochastic Brownian motion and photon statistics [see insets in Fig. 2(a)]. We extracted the width $\sigma a,i2(Di)$ of the superimages of state *i* = “*fast*” or “*slow*,” by least squares fitting the Gaussian model PSF Eq. (1). As we did not know the exact width $s02$ of the diffraction limited PSF, we first used the slow superimage to estimate $s02$, given *D*_{slow,CVE} from CVE analysis and $\sigma lp,slow2$ from the same experiment, and subsequently used the fast superimage to estimate $Dfast(\sigma a2)$ using Eq. (3). For more details, see the section titled Materials and methods. We obtained $Dfast\sigma a2=4.31\u2009\mu m2/s$ from the superimage of the fast diffusing particles. This value is in good agreement with the diffusion coefficient *D*_{fast,CVE} = 4.22 ± 0.02 *μ*m^{2}/s obtained from CVE analysis. While the influence of localization precision is minor here, most critical to this method is to know $s02$ of the imaging system, and if available (we have deduced it from the slow particles here) it can be used to obtain good estimates for the diffusion of particles simply from the shape of the average blurred particle images.

While in Figs. 2(b) and 2(c) we superimposed thousands of particle images, we also tested whether motion blur analysis can be applied using a smaller number of images. In Fig. 2(d), we show the performance of the method as a function of the number of images superimposed. The figure shows that using only single images to obtain $\sigma a2$ and thus $D\sigma a2$ leads to a serious overestimation of the diffusion coefficient $Dfast\sigma a2$ when compared to *D*_{fast,CVE}. Further, the variance of the estimates is huge in this case. However, the more particles are superimposed, the better the estimate $Dfast\sigma a2$ gets. With a signal to noise ratio of $\u223c$3.5, good estimates are obtained from superimposing at least 100 probe images. We additionally show a value of $Dfast\sigma ^a2$ that was obtained by averaging the MLE particle widths $\sigma ^a2$ without superposition. As can be seen, this measure overestimates the diffusion constant.

The novel method presented here, based on motion blur, could in principle be used to extract diffusion coefficients without the need for particle tracking. This approach is similar to spot-size analysis,^{53} but uses a symmetric Gaussian function and is robust towards fitting bias and the effect of outliers due to the superposition of multiple particle images. The method can be strictly applied only to two dimensional systems, as out of focus detection leads to broadening of the image PSF.^{51} It could be used to extract local diffusion coefficients relative to a reference point. Possible applications include transient interactions of lipid probes with proteins^{54,55} or the transient partitioning of lipid probes into lipid rafts.^{56,57}

### Lipid diffusion in the proximal and distal leaflets

We hypothesized that the slow and fast populations of lipids arise from diffusion in the proximal and distal leaflets of the SLB, respectively. To verify this hypothesis, we performed fluorescence quenching experiments using molecular iodide.^{58–60} First, we assessed the bulk quenching of the fluorescence of Atto647N-DMPE and Atto550-DMPE in EPC vesicles, by performing iodide concentration dependence studies using a spectrofluorometer. We compared the results to similar results obtained with the same dyes conjugated to a small protein, protein L. In Fig. 3(a), we show the relative quenching of fluorescence calculated as 1 − *F*/*F*_{0}, where *F*_{0} and *F* are the fluorescence emission in the absence and presence of iodide, respectively. We observed much lower quenching for dyes conjugated to DMPE in EPC vesicles (squares with solid lines), as compared to the same dyes conjugated to protein L (circles with solid lines). In the case of vesicles, to a large extent only the outer leaflet is exposed to iodide, and therefore we can assume that half of the dyes remain unquenched. However, even after taking this into account by doubling the quenching (squares with dotted lines), the quenching in the vesicles is still lower than the quenching in the protein molecules. It thus follows that both dyes are not directly exposed to the solvent and partition into the bilayer at least partially, causing a reduction in fluorescence quenching for EPC vesicles.

The finding that Atto550 and Atto647N partition into the lipid bilayer is in agreement with reported interactions of Atto647N and Atto550 with lipid bilayer membranes in bulk vesicle dialysis studies.^{61} In a molecular dynamics simulation of Texas Red DHPE (TR-DHPE) in DPPC bilayers, the dye was found to reside in the upper acyl chain region of the bilayer.^{62}

We then measured the amount of light emitted by single lipid probes, i.e., Atto647N-DMPE and Atto550-DMPE, in SPT experiments on SLBs in the presence and absence of iodide in the fluid above the membrane. Figure 3(b) shows the results, again presented as 1 − *F*/*F*_{0}, where *F*_{0} and *F* are now the average number of emitted photons per molecule in the presence of 100 mM NaCl and 100 mM KI buffer solution, respectively. When iodide was added to the SLB, the diffusing molecules emitted a lower number of photons. However, whereas the reduction was ∼30% for the fast diffusing population of lipids, it was only ∼5% for the slow diffusing population of lipids. Since iodide quenching affects the fast diffusing population much more than the slow diffusing population, we associate the fast diffusing molecules with the distal leaflet that is facing the bulk fluid and the slow diffusing molecules with the proximal leaflet facing the substrate. The quenching level of the fast lipids may in fact be somewhat underestimated, as fast lipids that are strongly quenched may not be detected anymore. The slight quenching of the slow lipids on the other hand may be due to residual leakage of iodide into the interfacial subfluid or into the lipid bilayer.^{58}

### Lipid flip-flop

Given that slow and fast diffusing probes are related to the proximal and distal leaflets, respectively, the state-to-state transitions we obtain from Viterbi analysis can be related to the transbilayer movement of lipids, also known as lipid flip-flop.^{63} It is therefore possible to obtain the flip-flop half-times, which for Atto550-DMPE in DOPC at RT are *t*_{1/2,slow} ≈ 6.3 ± 2.3 s and *t*_{1/2,fast} ≈ 3.3 ± 0.1 s. The full set of half-times is given in Table S1 of the supplementary material. The difference between slow and fast probes suggests an asymmetric probe distribution with some preference for the proximal leaflet (see Sec. S10 of the supplementary material for a detailed discussion of this issue).

It was speculated that membrane defects in SLBs may cause shorter flip-flop half times than in vesicles.^{35} To check whether this is indeed the case, we mapped the locations of transitions for both labels. To obtain a sufficient number of transitions for spatial point pattern analysis, we used rather high particle densities of ∼0.04–0.1 *μ*m^{−2}. Such high densities result in many instances where trajectories of two molecules cross each other, which may cause detection and particle assignment errors, and hence the assigned state-to-state transitions might be dominated by tracking errors. To diminish the influence of tracking errors, we therefore cropped trajectories as they got into close proximity (see Sec. S7 of the supplementary material). In simulations we found this was sufficient to remove the unwanted contribution from tracking errors and restore the correct transition probabilities using HMM (see Sec. S9 of the supplementary material). Additionally, we set the constraint that any transition in a diffusion trajectory has to be symmetrically flanked by a sequence of at least five frames, and we only considered trajectories that revealed either a single transition or none at all. This measure was taken in order to discard trajectories in which the analysis identified the highly unlikely appearance of multiple transitions and to include only well-defined transitions. Based on those constraints, we identified transition locations, as illustrated in Fig. 4(a) for a single trajectory. The sequence of particle track displacements $\Delta r$ used for state assignment is shown in Fig. 4(b) for the same trajectory. The resulting transition location maps are shown in Figs. 4(c) and 4(e) for the RT and 10 °C measurements, respectively.

The analysis of the resulting spatial distribution of points was then performed using Ripley’s *K* function,^{64} which computes the normalized number of events within a circle of radius *r* centered on a random point of the process. The functions *K*_{1} for Atto550-DMPE and *K*_{2} for Atto647N-DMPE are shown in Figs. 4(d) and 4(f), as well as the cross-function $K12*$, describing interdependencies of the two univariate processes. If the transition points are related to specific positions in the membrane, e.g., defects, we would expect the functions *K*_{1} and *K*_{2} to deviate from *K*_{CSR}, which is Ripley’s *K* function calculated for the case of complete spatial randomness (CSR). Similar results should hold for the function $K12*$ if the processes are dependent. We can see from Fig. 4(d) that at RT, there is considerable agreement between the three functions and *K*_{CSR}, indicating spatial randomness and independence of Atto550-DMPE and Atto647N-DMPE transition locations. However, at 10 °C there seems to be weak clustering of the transition points, as seen from some deviation of the *K* functions from *K*_{CSR} in Fig. 4(f). The results suggest that while at room temperature defects are unlikely to contribute to flip-flop reactions, there is some involvement of defects at 10 °C. We performed simulations to obtain more information on potential defect-based clustering, and the results suggest that at 10 °C around 40% of the flip-flop transitions belong to defects. This analysis is discussed in Sec. S11 and Fig. S5 of the supplementary material.

## THEORETICAL MODELING

### Numerical methods

To model the experiments, we consider a two-leaflet membrane suspended over solid support and use the Regularized Stockslets (RS) hydrodynamic numerical scheme^{65,66} to predict diffusion coefficients for the labeled lipids. For a complete description of the model, we refer the reader to Ref. 44, while this section offers a brief overview of the numerical method.

We consider a modified Seifert–Langer (SL)^{67} model of two coupled monolayers with a phenomenological drag between the upper (+) and lower (−) leaflets and a no-slip boundary condition for fluid flow a distance *H* below the bilayer [see Fig. 5(a)].^{44} The Green’s function *T* (or “membrane Oseen tensor”) relates the lipid velocity field to a given applied force density

where the indices i,j are summed over the (x,y) Cartesian directions and the indices α,β are summed over the leaflets (+,−). The Einstein summation convention is assumed. An explicit expression for the Green’s function of the two leaflet membrane system considered here is derived in Appendix A of Ref. 44 and is given by

The above expressions are given in Fourier space with the convention

In Eq. (6), −α represents the leaflet opposite to α, *b*_{leaflet} is the phenomenological interleaflet-friction coefficient, and *η* is the surface viscosity in each leaflet. The *K*^{±}(*q*) terms account for the traction of the bounding fluid below or above each leaflet.^{68} For the upper leaflet, which is adjacent to an infinite slab of fluid with viscosity $\eta f+$, $K+(q)=\u2212\eta f+q$. For the bottom leaflet, which is above the solid support, $K\u2212(q)=\u2212\eta f\u2212q\u2061coth(qH)$. The experiments are performed under conditions for which *H* is small and the viscosity of the fluid between the membrane and the support is large. (This assumption is validated by the fits to the experiment discussed below.) In this limit, $K\u2212q=\u2212\eta f\u2212/H=\u2212bsubstrate$, which means the lower leaflet is subjected to a simple drag by the underlying support.

With the hydrodynamic model defined, we use the RS scheme^{65,66} to calculate the diffusion coefficients of the labeled lipids in each leaflet. We define the regularized Green’s function

where $\varphi \u03f5r=12\pi \u03f52e\u2212r22\u03f52$ is a Gaussian blob function. A force density distributed over the blob envelope $f(r)=g\varphi \epsilon (r)$ will result in a velocity response $vi\alpha (r)=Tij\alpha \beta r;\epsilon gj\beta $ [Eq. (4)].

We represent the labeled lipids by a disc centered at the origin and composed of N blobs centered at positions {*R*_{n}} [see Fig. 5(b)]. The force on blob n is written $g\alpha [Rn]$ and the fluid response to all such forces [Eq. (4)] predicts the velocity at each *R*_{m} which yields the 2*N* × 2*N* matrix equation

We chose all blob velocities {*v*[*R*_{n}]} to be identical, corresponding to a rigid body translation of the disc in the $x^$ direction with velocity $V=Vx^$. Equation (10) is numerically solved to obtain the forces {*g*[*R*_{n}]} (for a detailed numerical method we refer the reader to Ref. 44) and the total force on the disc follows as $F=\u2211n=1Ng[Rn]$. The diffusion coefficient is then obtained from the Einstein relation

### Fitting procedure

We repeat the RS calculations for various inter-blob spacings *s*, and define *ϵ* to be half this spacing. As *s* decreases, the spatial resolution of the calculation increases. The mobility values were found to change approximately linearly with *s*^{44} thus allowing us to extrapolate to the infinite resolution limit *s* → 0; these extrapolated values are used to calculate the diffusion coefficients. While using thousands of blobs to construct each diffusing molecule is expected to produce diffusion coefficients with a maximal error of 0.4%,^{44} the calculation is time-consuming and cumbersome when used in a numerical fitting procedure (as discussed below), where the mobilities need to be evaluated thousands of times. Limiting the number of blobs to several hundred at the highest resolution and then performing the linear extrapolation produces results with maximal error of 2%, which is sufficient for our purposes and keeps the fitting procedure practical. (In practice, the results were obtained at four different resolutions: 106, 252, 440, and 534 blobs.) During the fitting procedure, we set the membrane viscosity *η* such that *l*_{sd} = *η*/*η*_{f} = 160 nm.^{69} The viscosity of the water layer above the membrane was set to bulk water viscosity: $\eta f+=1\u2009cP$^{70} and we set the labeled lipid radius *r* = 0.5 nm which is reasonable when taking an area per lipid molecule of *A* ≈ 0.43 nm^{2} (*r* ≈ 0.37 nm) for DMPE^{71} and allowing for a hydrodynamic radius slightly larger than the thermodynamic radius due to interactions between lipids.^{62} A global fitting algorithm was used to determine values for *b*_{leaflet} and *b*_{substrate} that minimize the error between the experimental diffusion coefficients and the coefficients obtained from the RS method, $(Dfastexp\u2212DfastRS)2+(Dslowexp\u2212DslowRS)2$. As Atto647N-DMPE shows significantly different flip-flop rates between the upper and lower leaflets depending upon the flip direction, potentially indicating some manner of dye-substrate interaction, while Atto550-DMPE reveals better balanced rates (see Table S1 of the supplementary material) and faster diffusion, we have fit only to the diffusion coefficients of the latter. The best fit parameters are presented in Table II for both membrane types.

. | D _{ fast }
. | D _{ slow }
. | b _{ leaflet }
. | b _{ substrate }
. |
---|---|---|---|---|

. | (μm^{2}/s)
. | (μm^{2}/s)
. | (Pa s/m) . | (Pa s/m) . |

EPC (RT) | 4.22 | 0.079 | 2.1 · 10^{7} | 1.66 · 10^{10} |

DOPC (RT) | 4.89 | 0.091 | 1.07 · 10^{7} | 1.44 · 10^{10} |

. | D _{ fast }
. | D _{ slow }
. | b _{ leaflet }
. | b _{ substrate }
. |
---|---|---|---|---|

. | (μm^{2}/s)
. | (μm^{2}/s)
. | (Pa s/m) . | (Pa s/m) . |

EPC (RT) | 4.22 | 0.079 | 2.1 · 10^{7} | 1.66 · 10^{10} |

DOPC (RT) | 4.89 | 0.091 | 1.07 · 10^{7} | 1.44 · 10^{10} |

## DISCUSSION

### Lipid diffusion in SLBs

In this work, we analyzed the motion of lipid molecules within supported bilayers in the fluid phase. We identified two populations, slow and fast, with well-separated diffusion coefficients, using two different lipid types and two different probes. As noted in the Introduction, bimodal diffusivity has been observed previously in SLBs, yet its source has remained poorly understood. Here we demonstrated that the slow and fast diffusing populations of lipids in SLBs are related to leaflet dynamics, using iodine quenching of fluorescence combined with SPT.

Various ensemble-based methods have been applied to measure differences in the dynamics of the two leaflets of a SLB. Some authors used fluorescence quenching, as in this work, as a means to differentiate between the two leaflets.^{19,21} Another approach relied on asymmetric bilayer preparation,^{14,19,24} i.e., the incorporation of probes into one of the leaflets. However, this approach can lead to meaningful results only if the bilayer does not equalize its leaflet compositions within the experimental preparation times. Further, it has been suggested that whether or not heterogeneous diffusion is observed does not solely depend on the substrate type and SLB preparation technique,^{12,24} but also on the lipid probe used. Indeed, for the same bilayer preparation, either two-component diffusion or one-component diffusion has been observed, depending on the lipid probe.^{11,21} Instructive in this context is the observation that certain probes distribute highly asymmetrically between the two leaflets.^{14,21,72} This may lead to the observation of an effective one-component diffusion. (We note that in previous work from our lab,^{3} we observed a single diffusion coefficient using lissamine rhodamine B as a label, which might be asymmetrically distributed.) Another problem of ensemble-based measurements such as fluorescence recovery after photobleaching is that high flip-flop rates might average the two diffusion modes into a single apparent mode, especially when the measurement is done over large areas.^{25} Indeed, given a diffusion coefficient of *D*_{fast} ≈ 4 *μ*m^{2}/s of a probe in the upper leaflet and a flip-flop half time *t*_{1/2,fast} ≈ 5 s, we calculate that such a probe diffuses only $\Delta r\u22489\u2009\mu m$ before flipping to the lower leaflet. We also note that FCS is not optimal for measuring diffusion coefficients as slow as revealed in our study.^{73,74} Importantly, the measurements reported in this paper *do not* depend on asymmetric bilayer preparation nor do they depend on probe distribution. Indeed, in our experiments, the two subpopulations of diffusing lipids are *separately observed and analyzed*. SPT is clearly superior in that respect to ensemble-based methods.

### RS hydrodynamic calculations

We have estimated the physical parameters, *b*_{substrate} and *b*_{leaflet}, governing bimodal diffusion due to different leaflet dynamics using a generalized RS method.^{44} Previously, in diffusional studies, the Evans and Sackmann model^{28} was used to either estimate properties of the lubricating interfacial layer,^{12,18} i.e., the interfacial drag coefficient *b*_{substrate}, or the inter-monolayer friction coefficient *b*_{leaflet} inherent to the bilayer structure.^{18,29} The former approach relies on the approximation that *b*_{leaflet} ≫ *b*_{substrate}, and thus the membrane acts as a single sheet, whereas the latter approach assumes that *b*_{substrate} ≫ *b*_{leaflet}, with the upper leaflet sliding over a nearly fixed lower leaflet. Nano-rheological approaches and approaches relying on external shear forces to quantify SLB friction suffer from similar considerations.^{75,76} Here we have modeled the bilayer nature of a SLB based on a RS scheme that includes both substrate drag via *b*_{substrate} as well as inter-leaflet drag via *b*_{leaflet}, without imposing assumptions on the relative strength of these parameters. The interleaflet drag *b*_{leaflet} ≈ 1 · 10^{7} Pa s/m we find is in agreement with other measurements obtained for SLBs (i.e., 1 · 10^{7}–1 · 10^{8} Pa s/m).^{18,29,75} However, our calculations suggest that the viscosity of the confined interfacial water in the subfluid is elevated by ∼10^{4} as compared to bulk water. Assuming the membrane distance from the support at a constant value of *H* = 0.7 nm (which is within the range of values reported for supported membranes *H* ≈ 0.5–3 nm),^{72,77–81} and using^{28} $bsubstrate\u2248\eta f\u2212/H$ we obtain $\eta f\u2212\u224810\u2009Pa\u2009s$ (bulk water has $\eta f+\u22480.001\u2009Pa\u2009s$). Other approaches to measure *b*_{substrate} are limited by assuming weak substrate coupling and thus not surprisingly reveal lower coupling.^{12,18,76} It is likely that the surface treatment before SLB preparation affects the properties of the water layer under the SLB, for example, by changing the apparent viscosity of the nanoconfined water, as was shown for solid surfaces with different wettabilities by measuring shear viscous forces with an AFM tip.^{82} Though the value of the effective viscosity of nanometer scale interfacial water remains controversial,^{83} values ∼10^{2}-10^{6} times greater than bulk water have previously been measured for hydrophilic substrates,^{82,84} indicating that the present measurement may be reasonable. Another possibility however is that *b*_{substrate} inferred through the comparison between experiment and modeling in this work may not originate from simple hydrodynamic considerations. The modeling (in the parameter regime relevant to the experiments) is only sensitive to a friction coefficient of some origin and cannot distinguish whether the underlying cause of this drag is a high fluid viscosity in the trapped water region or reflects some other mechanism of dissipation.

### Lipid flip-flop in SLBs

We observed transitions between slow and fast diffusion in some of our single-molecule trajectories and assigned them to flip-flop of lipids between the two membrane leaflets. Flip-flop half times, *t*_{1/2}, in the range of hours to days were reported for unilamellar vesicles.^{35,85} In contrast, the transbilayer motion is faster by orders of magnitudes in SLBs, with measured half-times in the range of minutes^{30} or even faster.^{31–34} In this work, we found that the flip-flop half-times are indeed in the range of seconds. We would like to emphasize that single molecule analysis with high temporal resolution is essential in determining these short flip-flop half times, whereas many ensemble-based approaches lack the temporal resolution necessary to measure sub-minute half times.^{30–33} Interestingly, our values are 2-3 orders of magnitude shorter than the half times in ensemble experiments on a similar system of Atto633-DOPE in DOPC/DOPS SLBs.^{30} It is possible that labeling of the lipids alters the free energy barrier for flip-flop, and indeed label-induced alterations in lipid flip-flop rates have been proposed in the literature.^{32,86,85,87} However, label-free measurements^{31–34} also suggested very fast flip-flop dynamics, which seem to agree, at least qualitatively, with our results. The similarity of these results to ours suggests that hydrophobic or membrane interacting dyes,^{61} as used in our study, do not perturb the flip-flop process.

Lipid flip-flop dynamics in free standing membranes are considered to be mediated by transient structural defects that lower the resistance for the polar lipid head groups to pass the hydrophobic membrane core.^{63} It has been speculated that enhanced flip-flop rates in SLBs are due to similar defects.^{35} By contrast, in a recent study, it was proposed that in defect free SLBs, fast flip-flop is driven by surface induced disorder of the bilayer’s acyl chain packing.^{33} Indeed, it has been shown that in the fluid phase, sufficiently above the main transition temperature, vesicle-deposited SLBs form continuous and largely defect-free bilayers.^{9,10,13,88} Our analysis allowed us to directly test the involvement of defects in the flip-flop reactions in a SLB. We find that at RT, transbilayer transitions are not clustered in specific points on the surface but are rather randomly distributed, consistent with a flip-flop mechanism that does not involve defects. At a lower temperature, it seems that membrane defects play some role in enhancing the flip-flop rates. The transition temperature (*T*_{m}) of DOPC^{89} is −16.5 °C and thus the SLB is supposed to remain sufficiently far from its main liquid-to-solid transition at 10 °C, although the transition broadens considerably due to the presence of a support.^{13} It is therefore likely that the cooling of the sample itself creates defects due to a reduction in the area per molecule^{90} of the DOPC lipids in the bilayer.

## CONCLUSION

In this work, we presented a detailed analysis of lipid diffusion in SLBs. We found two well-separated populations of slow and fast diffusing lipids that we can associate with the lower and upper leaflet, respectively. While labeling effects are of some concern in deducing self-contained lipid behavior, the direct observation of lipid subpopulations clearly demonstrates the benefits of SPT over ensemble-based approaches. Unlocking the immanent dynamics of each population, we could also characterize the flip-flop kinetics of the lipids, which were found to occur on a time scale of seconds. We have further supported our measurements with hydrodynamic calculations that revealed the physical parameters governing bimodal diffusion due to the bilayer structure of SLBs and showed that the viscosity of the water layer trapped under the SLB is ∼10^{4} larger than the viscosity of free water. The complex and rich picture emerging from these studies reveals the various influences of the support on lipid behavior and dynamics in the two distinct leaflets of a supported membrane.

## SUPPLEMENTARY MATERIAL

See supplementary material for detailed description of the experimental setup and methods used for particle tracking and data analysis.

## ACKNOWLEDGMENTS

We thank Dr. Haim Aviram for providing protein L and for his help with building the microscopy setup. We further thank Dr. Menahem Pirchi for contributing code for the HMM analysis, and Professor Haim Diamant for many discussions. This work was supported by Grant No. 2012084 from the U.S.-Israel Binational Science Foundation, and in part by Grant No. CHE-1465162 from the National Science Foundation of the United States. R.L.S. was supported by a post-doctoral fellowship from the Swiss National Science Foundation (SNSF).