# Numerical Simulation and FRAP Experiments Show That the Plasma Membrane Binding Protein PH-EFA6 Does Not Exhibit Anomalous Subdiffusion in Cells

## Abstract

**:**

## 1. Introduction

- “Rafts” model where lipid/lipid phase separation drives the lateral partitioning of transmembrane proteins [7].

## 2. Results

#### 2.1. Anomalous Subdiffusion Modeling

#### 2.2. Validating Numerical Simulation and Analytical Models

#### 2.3. Challenging Analytical Models to Identify Numerically Simulated Anomalous Diffusion Fluorescence Recoveries

- Anomalous diffusion motion (aDm): see Equation (8) in Section 2.1,
- Free Brownian motion (Bm):$${I}_{R}\left(t\right)=\sum _{1}^{\infty}\frac{{(-K)}^{n}}{n!}\frac{1}{1+n+\frac{8nDt}{{R}^{2}}},$$
- Restricted Brownian motion (rBm):$${I}_{R}\left(t\right)=(1-M)\frac{1-{e}^{-K}}{K}+M\sum _{1}^{\infty}\frac{{(-K)}^{n}}{n!}\frac{1}{1+n+\frac{8nDt}{{R}^{2}}},$$

#### 2.4. Single Spot Fluorescence Recovery After Photobleaching Does Not Allow for Identifying the Nature of PH-EFA6 Diffusion in Cells

#### 2.5. Variable Radii Fluorescence Recovery After Photobleaching Allows Correct Estimation of the Anomalous Sub-Diffusion Exponent $\alpha $

#### 2.6. Continuous Time Random Walk Anomalous Subdiffusion Does Not Explain PH-EFA6 Motions in the Plasma Membrane of BHK Cells

## 3. Discussion

## 4. Materials and Methods

#### 4.1. Monte Carlo Simulation

#### 4.2. Cell Culture and Transfection

#### 4.3. Fluorescence Recovery After Photobleaching Experiments

## 5. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Examples of Numerically Generated Fluorescence Recovery Curves

**Figure A1.**Monte Carlo simulation of normalized fluorescence recoveries in the case of CTRW anomalous subdiffusion. Different values of D and $\alpha $ have been tested in the simulations. Here, values of $D=0.5;1;1.5$ are represented from bottom to top with different $\alpha $ in each case: 0.6 (dots); 0.7 (thin line); 0.8 (thick line). The Monte Carlo simulation has been constructed with ${10}^{7}$ individual trajectories.

## Appendix B. Comparison of Input and Fitted D and α at Variable Radii

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**Figure 1.**Values of the parameters($D,\alpha $) obtained from the fit of the simulated recoveries; (

**a**)—D values obtained after fit of the fluorescence recovery with Equation (8) (${D}_{output}$) as a function of D values used in the numerical simulation (${D}_{input}$) for different $\alpha $. Note that the slope is always less than 1; (

**b**)—values of $\alpha $ obtained after fit of the simulated curves with Equation (8) for different $\alpha $ used in the simulation and as a function of the D values used in the simulation. Note that the original $\alpha $ value used in the simulation is never reached by fitting of the simulated recoveries.

**Figure 2.**Best fits using the different models of normal and log–log plots of simulated continuous time random walk CTRW anomalous sub-diffusion recoveries. $\alpha =0.6$ is the value used for the simulation in the four plots. In blue, the free Brownian motion Bm model (Equation (9)), in red restricted Brownian motion rBm model (Equation (10)) and in green anomalous diffusion motion (aDm) (Equation (8)) fits. In (

**a**,

**b**), D = 2 and, in (

**c**,

**d**), D = 0.1. (

**a**,

**c**) Are the normal plots while (

**b**,

**d**) are the log–log plots. From these graphs, it can be seen that one can hardly distinguish between the rBm model (red) and the aDm model (green) fits of the simulated recovery.

**Figure 3.**Average The three models Bm, rBm and aDm) were used to fit the average recovery curve obtained from 45 different experiments. The normal (

**a**) plot shows the residual from the fit of the different models. Note that only the Bm model fit is inaccurate. The log–log plot (

**b**) illustrates again the difficulty to discriminate between the aDm and the rBm model in the goodness of the fit.

**Figure 4.**Values of $\alpha $ obtained from the fit of Continuous Time Random Walk model simulated recoveries as a function of 1/R. Values of $\alpha $ introduced in the fit were respectively 0.6 (in blue), 0.7 (in red) and 0.8 (in green). Dots represent the mean ± s.d. values of $\alpha $ obtained with the fit using Equation (8) of simulated recoveries for a set of D values (0.01, 0.05, 0.1, 0.5, 1, 2, 3, black dots in the graph). Extrapolation at 1/R = 0of the linear fit of the different $\alpha $ obtained from the fits of recoveries at different radii gives $\alpha $ values close to the one used for the simulations.

**Figure 5.**Comparison of the two models aDm and rBm using variable radii FRAP (vr-FRAP) in the case of PH-EFA6 diffusion in the plasma membrane of BHK cells. (

**a**) plot of $\alpha $ values obtained by fitting the experimental recoveries with Equation (8) as a function of 1/R. The plot exhibit a positive slot in opposite to the one observed in Figure 4, suggesting an absence of a CTRW anomalous subdiffusion in the motion of PH-EFA6; (

**b**) plot of M value obtained by fitting the experimental recoveries with Equation (10) as a function of 1/R. The plot exhibits a positive slope as observed in [5,25], suggesting a diffusion with trapping in spatial domains.

**Table 1.**Parameter values obtained by the fit of the average experimental recovery with the different analytical models.

Model | D ($\mathsf{\mu}\mathbf{m}\xb7{\mathbf{s}}^{-1}$) ${}^{1}$ | M | $\mathit{\alpha}$ | ${\mathit{D}}_{\mathit{\alpha}}$ ($\mathsf{\mu}\mathbf{m}\xb7{\mathbf{s}}^{-\mathit{\alpha}}$) ${}^{2}$ | ${\mathit{\chi}}^{2}$${}^{3}$ |
---|---|---|---|---|---|

Bm ${}^{4}$ | 0.12 ± 0.06 | 1 | - | - | 5.7 ± 0.5 |

rBm ${}^{5}$ | 0.22 ± 0.01 | 0.92 ± 0.01 | - | - | 3.8 ± 1.6 |

aDm ${}^{6}$ | - | - | 0.65 ± 0.02 | 1.48 ± 0.05 | 2.9 ± 1.6 |

Objective | R ($\mathsf{\mu}\mathbf{m}$) | ΔR ($\mathsf{\mu}\mathbf{m}$) | Laser Waist (nm) |
---|---|---|---|

16× (NA ${}^{1}$ = 1.0) | 0.74 | 0.04 | 370 |

40× (NA = 1.3) | 0.44 | 0.03 | 222 |

63× (NA = 1.4) | 0.37 | 0.01 | 185 |

100× (NA = 1.4) | 0.32 | 0.01 | 160 |

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**MDPI and ACS Style**

Favard, C.
Numerical Simulation and FRAP Experiments Show That the Plasma Membrane Binding Protein PH-EFA6 Does Not Exhibit Anomalous Subdiffusion in Cells. *Biomolecules* **2018**, *8*, 90.
https://doi.org/10.3390/biom8030090

**AMA Style**

Favard C.
Numerical Simulation and FRAP Experiments Show That the Plasma Membrane Binding Protein PH-EFA6 Does Not Exhibit Anomalous Subdiffusion in Cells. *Biomolecules*. 2018; 8(3):90.
https://doi.org/10.3390/biom8030090

**Chicago/Turabian Style**

Favard, Cyril.
2018. "Numerical Simulation and FRAP Experiments Show That the Plasma Membrane Binding Protein PH-EFA6 Does Not Exhibit Anomalous Subdiffusion in Cells" *Biomolecules* 8, no. 3: 90.
https://doi.org/10.3390/biom8030090