## 1. Introduction

Early models of the plasma membrane, notably the fluid mosaic model [

1], postulated that transmembrane proteins were freely diffusing in a sea of lipids. During these two last decades, it has become apparent that cell surface membranes are far from being a homogeneous mixture of their lipid and protein components. They are compartmented into domains whose composition, physical properties and function are different. Numerous studies on transmembrane proteins and plasma membrane lipids by means of single particle tracking (SPT), fluorescence correlation spectroscopy (FCS) or fluorescence recovery after photobleaching (FRAP) have shown the existence of micro and nanometer size domains on both model membrane [

2,

3] and living cells [

4,

5,

6]. In the plasma membrane of living cells, these domains can come from different origins but are generally classified into two main groups:

“Rafts” model where lipid/lipid phase separation drives the lateral partitioning of transmembrane proteins [

7].

“Cytoskeleton fence” model in which transmembrane proteins are coralled by a fence of cytoskeleton just beneath the plasma membrane [

8,

9].

First, variable radii FRAP [

5,

6], and then spot variation FCS [

10,

11,

12] helped in discriminating amongst these two models the nature of the deviation to pure Brownian diffusion of membrane components in living cells.

Fluorescence recovery after photobleachingexperiments have been used for determination of long-range molecular diffusion of proteins and lipids on both the model system and cells for more than 30 years [

13,

14]. Briefly, fluorescently labelled molecules localized within a predefined area are irreversibly photodestructed by a short and intense laser pulse. The recovery of the fluorescence in this area is then measured against time. Since no reversible photoreaction occurs, recovery of the fluorescence in the photobleached area is due to diffusion. Fluorescence Recovery After Photobleaching data are generally interpreted by assuming classical Brownian diffusion. Two parameters can then be obtained: D, the lateral diffusion coefficient and M, the mobile fraction of the diffusing molecule. When the radius of the photobleached area is small compared to the diffusion area, M must be equal to 1 for freely diffusing species. In fact, most of the data reported so far in biological membranes for transmembrane proteins shows a value of M

$<1$. This lack in total fluorescence recovery can be interpreted as a restriction to free-diffusion behaviour. Parameters obtained then have to be re-evaluated to recognize the effect of time-dependent interactions in a field of random energy barriers.

An experimental approach to that question has been proposed by Feder et al. [

15] by introducing anomalous subdiffusion in the motion of transmembrane proteins. Many sources of motion restriction can lead to anomalous diffusion (for a review, see [

16,

17]). Saxton has performed extensive numerical simulations to help with identifying the sources of anomalous diffusion (obstacles, binding, etc.) using SPT measurements [

18,

19] and he declined this more recently to FRAP experiments [

20] using fractional Brownian motion (fBm) or continuous time random walk (CTRW) models as sources of anomalous diffusion.

Membrane bound proteins should also be submitted to several interactions with their surrounding environment that should account for an anomalous subdiffusion behavior. Sources of deviation from Brownian motion in their lateral diffusion may include lipid domain trapping, binding to immobile proteins and/or obstruction by cytoskeletal elements. These different possible interactions can exhibit different characteristic times or different distributions of characteristic times. Here, diffusion of an intracellular membrane-bound protein domain (pleckstrin homology (PH) domain of EFA6, the ARF6 exchange factor) has been analyzed inside living cells by FRAP experiments. Previous studies have shown that these proteins are linked to the polar head of PI(4,5)P

${}_{2}$ lipids by means of electrostatic interactions [

21]. Furthermore, the protein used in this study appears to have a functional requirement to be associated with the plasma membrane within cells [

22,

23]. In this paper, numerical simulation of the CTRW model of anomalous subdiffusion was first performed for a single spot size. Based on the quality of the fit using different analytical expression, we tested the ability to retrieve this anomalous diffusion in the simulated recovery curves first and in the experimental one afterwards. We showed that performing FRAP experiments for a single spot size did not allow for discriminating between the CTRW-induced anomalous diffusion case and the empirical classical approach using mobile and immobile fraction.

We then computed and performed experimental FRAP at variable radii. By plotting changes in the anomaly of the diffusion or in the mobile fraction as a function of the inverse of the bleached radius, as in Salomé et al. [

5], we showed that it was possible to discriminate between the two models. Interestingly, we observed that the restriction to the mobility of the PH-EFA6 domain is not due to CTRW anomalous subdiffusion, but more certainly to the subcortical actin fences.

## 3. Discussion

This work has been initiated to characterize the nature of the diffusion of molecules binding the inner leaflet of the cell plasma membranes by means of FRAP experiments. In a first attempt, we decided to compare experimental data obtained with the PH domain of EFA6 expressed in BHK cells to FRAP curves generated from anomalous subdiffusive particles numerically simulated. Then, we analyzed the recoveries with three different diffusion models, namely the free Brownian motion (Bm), the restricted Brownian motion (rBm) and the CTRW anomalous subdiffusion (aDm). Four parameters can be extracted from these diffusion models. The Brownian diffusion coefficient

D and the mobile fraction M (M = 1 in the case of Bm) on one side, and the anomalous subdiffusion exponent

$\alpha $ and its related anomalous diffusion coefficient

${D}_{\alpha}$ on the other side. The aDm model has been extensively studied by numerical simulations. Direct analysis of numerically simulated curves lead to an underestimation of

${D}_{\alpha}$ and

$\alpha $. This was already observed by Feder et al. who proposed, in order to circumvent this underestimation, to add a mobile fraction (M) as a new parameter [

15]. On a physical point of view, this is incorrect since the phenomenological parameter “mobile fraction” is indeed a part of

$\alpha $ as discussed by Nagle et al. [

24]. This underestimation of

${D}_{\alpha}$ and

$\alpha $ is mainly due to a finite size effect (space and time) that cannot be easily overcome either in simulations or in experiments. We directly tested for anomalous subdiffusion in the simulated and experimental curves by fitting the recovery curves with normal and anomalous equations and look for systematic deviations of the fit, both in linear plots to see the fit at large times and log-log plots to see the fit at short times. From this approach, we could see that the Bm can be immediately discarded. The difference between the aDm and the rBm could only be observed at very short times (log-log plots) and very long times. Unfortunately, these two extreme times are hardly easy to analyze in experiments. Indeed, at short times, the curve may be distorted by diffusion during the bleach pulse [

26] and limits in the frequency of data collection. At long times, motion of the membrane or photobleaching of the fluorescent probe might appear. This is illustrated here in our experimental data. Fits of single spot fluorescence recoveries did not allow for determination without uncertainties which of the aDm or the rBm model reflect the nature of PH-EFA6 diffusion in the plasma membrane of BHK cells. Although underestimated, the

$\alpha $ value we found here, when fitting with the aDm model, reflect a strong deviation from the Brownian motion and suggest that PH-EFA6 explores a strongly compartmentalized landscape while traveling in the inner leaflet of the plasma membrane. Nevertheless, this

$\alpha $ value, as well as the M value in the case of the rBm model, are higher than the one found for the IgE receptor transmembrane protein in RBL cells (

$\alpha =0.46\pm 0.22$) [

15]. Using single particle tracking experiments, other transmembrane proteins such as MHC class I in HeLa cells have also been shown to exhibit anomalous subdiffusion with an

$\alpha $ value close to 0.5 [

27]. On the contrary, other transmembrane proteins exhibit high

$\alpha $ values (

$\alpha $ = 0.8) (Kv2.1 potassium channel in HEK293T cells [

28]) or pure Brownian motion (MHC class II in CHO cells [

29] or aquaporin-1 in MDCK cells [

30].

The inability of FRAP to cover several decades of time as SPT or FCS techniques will do can be overcome by probing the environment at different space scale using variable radii FRAP [

5,

6]. Here, we have simulated recoveries in the case of CTRW anomalous subdiffusion at different space scales and fit them with the aDm model in order to extract the set of parameters (

$\alpha $,

D). By monitoring the change of fitted (

$\alpha $,

D) parameters as a function of space (1/R), we observed that the fitted values of

$\alpha $ decreased with an increasing radius of observation. This was an unexpected result, since, in our CTRW model, the anomalous sub-diffusion exponent

$\alpha $ is supposed to be spatially invariant. Nevertheless, this could be explained by finite size effects (finite space and time used in our numerical simulations). We showed that, in the case of the CTRW model, the correct values of

$\alpha $ and

D could be determined at 1/R = 0, i.e., when R

$\to \infty $. This is one way to overcome the finite size effect of the simulations.

Then, we applied this approach to the experimental recoveries obtained at different radii. Surprisingly, we observed the opposite tendency to the one observed in our simulation, suggesting that the CTRW anomalous subdiffusion is not the correct model to describe the motion of PH-EFA6 in BHK cells. On the contrary, when monitoring the change of the mobile fraction obtained by fitting the experimental recoveries with the rBm model, we observed the same tendency as the one described in [

5,

31], i.e., an increase of the mobile fraction with a decreasing radius. Using this approach, we could determine that 25% of PH-EFA6 molecules are confined in domains of a 90 nm radius.

As stated in the introduction, CTRW is not the only source of anomalous subdiffusion. The increase of the experimentally determined

$\alpha $ with a decreasing radius can also be an apparent consequence of a crossover regime with two different diffusion coefficients as it is described by the rBm model in this study. Using FCS experiments and simulations at different radii of a two-phase, two component lipid mixtures at different temperatures, Favard et al. showed that changes in an anomalous subdiffusion exponent

$\alpha $ could nicely predict the phase transitions temperatures but failed in determining the average size of domains coexisting in the two phases [

2]. On the contrary, by monitoring the change in diffusion regimes, they could nicely determine the mean size of the gel-phase domains. If we extend this approach to our

$\alpha $ plot as a function of the probe’s radius, we see that the transition from anomalous subdiffusion (

$\alpha <1$) to normal diffusion (

$\alpha =1$) occurs at a radius of 160 nm, i.e., not far from the values obtained with the rBm model.

The range of domain sizes observed here (90 to 160 nm radius), independently from the model used to describe the dynamics of PH-EFA6, is likely to be due to subcortical actin cytoskeleton. Equivalent sizes have been observed in NRK cells [

32] using electron microscopy, and recently in several cell lines, by monitoring membrane lipids dynamics using STED-FCS [

33]. Interestingly, Krapf et al. described that this meshwork has a fractal dimension and could therefore lead to anomalous subdiffusion [

34]. Therefore, further investigations and numerical simulations using a meshwork with a fractal dimension as the origin of the anomalous subdiffusion are likely to be conducted in order to understand the origin of our

$\alpha =f$(1/R) behavior in our vrFRAP experiments.