Quantifying membrane binding and diffusion with fluorescence correlation spectroscopy diffusion laws

Abstract

Many transient processes in cells arise from the binding of cytosolic proteins to membranes. Quantifying this membrane binding and its associated diffusion in the living cell is therefore of primary importance. Dynamic photonic microscopies, e.g., single/multiple particle tracking, fluorescence recovery after photobleaching, and fluorescence correlation spectroscopy (FCS), enable non-invasive measurement of molecular mobility in living cells and their plasma membranes. However, FCS with a single beam waist is of limited applicability with complex, non-Brownian, motions. Recently, the development of FCS diffusion laws methods has given access to the characterization of these complex motions, although none of them is applicable to the membrane binding case at the moment. In this study, we combined computer simulations and FCS experiments to propose an FCS diffusion law for membrane binding. First, we generated computer simulations of spot-variation FCS (svFCS) measurements for a membrane binding process combined to 2D and 3D diffusion at the membrane and in the bulk/cytosol, respectively. Then, using these simulations as a learning set, we derived an empirical diffusion law with three free parameters: the apparent binding constant $K_D$, the diffusion coefficient on the membrane $D_{2\mathrm{D}}$, and the diffusion coefficient in the cytosol, $D_{3\mathrm{D}}$. Finally, we monitored, using svFCS, the dynamics of retroviral Gag proteins and associated mutants during their binding to supported lipid bilayers of different lipid composition or at plasma membranes of living cells, and we quantified $K_D$ and $D_{2\mathrm{D}}$ in these conditions using our empirical diffusion law. Based on these experiments and numerical simulations, we conclude that this new approach enables correct estimation of membrane partitioning and membrane diffusion properties ($K_D$ and $D_{2\mathrm{D}}$) for peripheral membrane molecules.

Significance

Many processes in cells start with the binding of a cytosolic protein to the plasma membrane. Accurate determination of the protein binding efficiency as well as the protein diffusion coefficient at the plasma membrane of living cells is of utmost importance to decipher the different processes triggered by the initial binding event. Here we propose a non-invasive novel approach, based on spot-variation fluorescence correlation spectroscopy, to quantitatively determine these two parameters, directly in living cells expressing a fluorescent protein of interest.

Introduction

The binding of cytosolic proteins to a membrane is often the starting point of dynamic processes occurring in the cell. This binding is frequently a key event in signal transduction, metabolism, membrane trafficking (endo/exocytosis), or enveloped virus assembly. Although much has been studied about these processes, quantifying the dynamics of the initial event, i.e., protein binding to and diffusion on the membrane, still remains a challenge in living cells.

Several methods, such as fluorescence recovery after photobleaching (FRAP), fluorescence correlation spectroscopy (FCS), and single (multiple) particle tracking (SPT), have been developed to monitor molecule motions in cells using fluorescence microscopy and have been extensively used in membrane biology (for review see (1)). Each of these dynamic microscopic techniques have pros and cons related to their timescale and statistics. FCS is sensitive on the millisecond to second timescale, corresponding to the diffusion characteristic time of a fluorescently labeled molecule in a lipid mixture, through an illumination focus with a waist of approximately 200 nm. FCS has therefore provided a convenient way to investigate the motions of lipids and membrane proteins in living cells. FCS usually consists in computing the autocorrelogram function (ACF) from the temporal recording of fluorescence intensity fluctuations in the illumination beam. The ACF is then fitted with an analytical expression accounting for a given type of motion; e.g., fractional or normal Brownian motion in 2D or 3D. This fit yields the mobility parameters as estimates of its free parameters. Interestingly, FCS was originally developed to determine the kinetic constants of chemical reactions at equilibrium (2). Recently, FCS has been proposed as an efficient method to monitor more complex dynamics such as reaction-diffusion processes occurring in the case of transcription factor binding to DNA in cells (3) or in the embryo (4). However, single-spot FCS, in which the beam waist is constant, is usually not applicable to discriminate and quantify complex motions. In the case of reaction-diffusion processes for example, there is no analytical solution available to retrieve the kinetic parameters from the fit of the autocorrelogram. Approximated expressions can be derived by simplifying the kinetics (e.g., neglecting diffusion of the bound molecule) or distinguishing between simplification regimes (e.g., reaction kinetic dominant, diffusion dominant) (3). However, even in the case of molecules diffusing in the membrane, single-spot FCS measurements often result in inaccurate estimates of diffusion coefficient, mainly because the single spot size is not sufficient to probe the heterogeneity of the environment where the molecules diffuse.

One way to circumvent that issue is to perform FCS measurements at different beam waists. Plotting the half-time of the decorrelation $\left({\tau }_{1/2}\right)$ as a function of the surface probed by the illumination ($w^2$, where $w$ is the beam waist, or spot size), one obtains so-called svFCS diffusion laws. Wawrezinieck et al. (5) showed that svFCS diffusion laws are a powerful tool for analyzing complex diffusions. They developed a svFCS experiment and, based on numerical simulations, managed to successfully associate svFCS diffusion laws with different types of complex, heterogeneous environments. The same approach has also been used experimentally to characterize molecular motions in membranes, below the diffraction limit (6,7,8) and recently in Imaging-FCS (Im-FCS) (9,10). In terms of FCS diffusion laws ${\tau }_{1/2}=f\left(w^2\right)$, a pure Brownian motion (free diffusion) leads to ${\tau }_{1/2}\to 0$ at $w^2\to 0$; i.e., the extrapolated value of ${\tau }_{1/2}$ for vanishing $w^2$ is expected to be zero. Moreover, one predicts a linear relationship between ${\tau }_{1/2}$ and $w^2$ with a slope that is inversely proportional to the free diffusion coefficient. Interestingly, the extrapolation of ${\tau }_{1/2}$ for $w^2\to 0$, ${\tau }_{1/2}\left(0\right)$, gives information about the nature of the heterogeneity probed by the molecule. For example, molecules experiencing dynamic partitioning between liquid ordered and liquid disordered (Lo/Ld) lipid phases will have a positive ${\tau }_{1/2}\left(0\right)$ value, whereas molecules restricted in their diffusion by sub-membrane fences, such as cortical actin cytoskeleton, are expected to show a ${\tau }_{1/2}\left(0\right)<0$ (5,11). However, lipids exhibiting dynamic partition in solid/liquid disordered phase (S/Ld) also display ${\tau }_{1/2}\left(0\right)<0$ (12), showing that the ${\tau }_{1/2}\left(0\right)$ value alone is not sufficient to correctly characterize the motion (13).

Recently, different types of svFCS diffusion laws have been derived and characterized for different types of heterogeneous environments based on numerical simulations (14,15), mostly for complex motions related to anomalous sub-diffusion. In this article, we explored the applicability of svFCS diffusion laws to the quantification of membrane binding kinetics where the fluorescent molecule diffuses both at the 2D membrane and in the 3D bulk. Our main objective was to test the suitability of such experimental diffusion laws to quantify the main parameters: the apparent binding constant $\left(K_D\right)$, the membrane-bound diffusion coefficient $\left(D_{2\mathrm{D}}\right)$, and the cytosolic/bulk diffusion coefficient $\left(D_{3\mathrm{D}}\right)$ (see Table 1 for the definitions of the different parameter use here).

Table 1.

List of parameters used in this work

Variables	Parameters details
Confocal volume calibration parameters
$w_{xy}$	width of the observation area in x, y direction
$s=w_z/w_{xy}$	shape factor of the 3D Gaussian excitation point spread function
Parameters used in ACF fit
$G\left(\tau \right)$	auto-correlation function (ACF)
${\tau }_{1/2}$	characteristic time of half temporal decorrelation; i.e., $G\left({\tau }_{1/2}\right)=G\left(\infty \right)+\left(G\left(0\right)-G\left(\infty \right)\right)/2$
${\tau }_d$	characteristic diffusion time of particles in the confocal volume, ${\tau }_d={\tau }_{1/2}$ for 2D diffusion
$N$	effective number of particles in confocal volume
$T$	fraction of particles in triplet state
${\tau }_T$	characteristic residence time in triplet state
Parameters of the 2D/3D diffusion and binding kinetics
$D_{2\mathrm{D}}$	diffusion coefficient of membrane-bound molecules
$D_{3\mathrm{D}}$	diffusion coefficient of free molecules
$k_{\text{on}}$	association constant of particles to the membrane
$k_{\text{off}}$	dissociation constant of particles from the membrane
$K_D=k_{\text{off}}/k_{\text{on}}$	binding affinity of the proteins to the membrane
$K_P={\left[F\right]}_{\mathrm{e}\mathrm{q}}/{\left[B\right]}_{\mathrm{e}\mathrm{q}}$	Experimental apparent partition coefficient (with ${\left[F\right]}_{\mathrm{e}\mathrm{q}}$ and ${\left[B\right]}_{\mathrm{e}\mathrm{q}}$ the equilibrium concentrations of free and bound proteins, respectively).

We first performed extensive numerical simulations of svFCS diffusion laws for this system, exploring different regimes by varying $K_D$, $D_{2\mathrm{D}}$, and $D_{3\mathrm{D}}$. Using part of these simulations as a learning set, we could derive an empirical svFCS diffusion law and use the other part of the simulations as a test set to confirm that the diffusion law can indeed be used to estimate $K_D$ and $D_{2\mathrm{D}}$ of synthetic data with good accuracy. We then used this empirical diffusion law to fit experimental svFCS measurements of HIV-1 or HTLV-1 retroviral Gag proteins binding to either supported lipid membranes or living HEK 293T cells plasma membranes. HIV-1 and HTLV-1 Gag proteins are multi-domain proteins binding the plasma membrane of host cells and are essential for viral assembly. HIV-1 Gag, for example, contains three main domains, each having distinct roles during viral assembly. The matrix domain (MA) is responsible for membrane binding to the PI(4,5)P₂ lipids, present in the inner leaflet of the plasma membrane, thanks to a highly basic region (HBR). Membrane binding is reinforced by the N-terminal myristate MA substitution that inserts into the inner leaflet. The capsid domain (CA) is involved in Gag-Gag interactions during Gag self-assembly, the initial stage in the generation of a new virion. The nucleocapsid domain (NC) also contains enriched basic motifs that are involved in RNA binding to permit encapsidation of the RNA in the new virion. NMR data and coarse-grained molecular dynamics have shown that the HBR region of MA as well as other polybasic motifs bind to anionic lipids (PI(4,5)P₂ but also PS) (16,17,18,19). Conversely, the lack of myristate has been shown to decrease membrane binding of HIV-1 in living cells (20). Finally, Fogarty et al. (21) have shown that HTLV-1 Gag has a higher affinity for cell plasma membranes than HIV-1 Gag.

Although these results from the literature do not consist in a precise quantification of the binding affinities, they provide us with ordering relations between them; e.g., the binding affinity of HIV-1 in the presence of myristate is higher than in its absence. We used them to challenge the capacity of our svFCS diffusion law to retrieve the correct estimations from experimental svFCS diffusion laws and to correctly estimate $K_D$ and $D_{2\mathrm{D}}$ in vitro, using supported lipid bilayers (SLBs) of various composition and in cellulo by expressing Gag chimera proteins in HEK-293T cells.

On the synthetic test dataset, we show that our svFCS diffusion law yields precise estimates for $D_{2\mathrm{D}}$ and $K_D$, with an accuracy around 12%. The accuracy of our method on real experimental measurements cannot be assessed directly, but our results confirm the capacity of our 2D/3D diffusion and binding svFCS diffusion law to correctly estimate $D_{2\mathrm{D}}$ and $K_D$ in the case of HIV-1 and HTLV-1 Gag proteins binding to model lipid SLB and to HEK-293T cells. Notably, our estimates confirm previous results obtained in the literature with other methods by different teams. Therefore, our results highlight the benefits of our empirical svFCS diffusion law for the determination of $D_{2\mathrm{D}}$ and $K_D$ in the membrane binding problem with diffusion both at the membrane and in the cytosol.

Materials and methods

Lipids and plasmids

Egg phosphatidylcholine (PC), brain phosphatidylserine (PS), brain phosphatidylinositol (4,5) bisphosphate₂ (PI(4,5)P₂), 1,2-dioleoyl-sn-glycero-3-phosphoethanolamine-N-(cyanine 5.5) (PE-Cy5.5), and 1-palmitoyl-2-(dipyrrometheneboron difluoride)-undecanoyl-sn-glycero-3-phosphocholine (TopFluor-PC) were purchased from Avanti Polar Lipids (Alabama, USA). Atto647N-PI(4,5)P₂ is a gift from Pr. Christian Eggeling (Jena, Germany). The plasmid encoding HIV-1-myr(−)Gag-GFP is a gift of Dr. H. de Rocquigny and described in (22). The plasmid encoding HTLV-1 GagYFP (named pHTLV-GagYFP) was obtained from Dr. D. Derse’s laboratory and described previously in (23). The pHIV-1-Gag-mCherry was obtained from Dr. N. Jouvenet (24).

Gag purification and labeling

Purified Gag proteins were produced and provided by Pr. J. Mak’s lab (25). Protein stock concentration was measured at 1.2 mg/mL using a NanoPhotometer (Implen), and 100 $\mu L$ were incubated overnight under agitation and at 4°C with 1 $\mu L$ of Alexa Fluor 488 C5-Maleimide (Invitrogen) at 20 mM in DMSO. The reaction mix was transferred in Slide-A-Lyzer MINI Dialysis Device, 0.5 mL (Thermo Scientific) and dialyzed for 6 h under agitation at 4°C in 15 mL of buffer; Tris (50mM), NaCl (1M), pH = 8. Dialyzing buffer was then renewed, and the reaction mix was dialyzed again overnight under agitation at 4°C, collected, and stored at −20°C.

Large unilamellar vesicles processing

Liposomes were processed by dissolving in chloroform the following lipids, aiming at a 1 mg/mL concentration: egg-PC, brain-PS, brain-PI(4,5)P₂, and PE-Cy5.5 in the ratios described Fig. 2 a. Solvent was then evaporated for 20 min in a rotary evaporator and 10 min more in a desiccator. Lipids were rehydrated in 500 $\mu L$ Na citrate buffer; Na citrate (10 mM), NaCl (100mM), EGTA (0.5mM), pH 4.6. The mixture was then, five consecutive times, frozen for 30 s in liquid nitrogen, water bathed at 37°C for 30 s, and vortexed. Multi lamellar vesicles (MLVs) processed as such can be stored at −20°C. Large unilamellar vesicles (LUVs) are subsequently produced by diluting liposomes 1:5 in Na citrate buffer. This solution is then passed through a 100-nm Nucleopore Track-Etched Membrane in an extruder (Avanti Polar Lipids), and sonicated.

Figure 2.

FCS diffusion laws on model membranes: (a) Different composition of the SLB used here. (b) These SLBs were spread on glass coverslips and set in the microscope to perform svFCS measurements. Fluorescent myr(−)Gag was introduced in the buffer at the beginning of the experiment. Example of normalized correlograms recorded during an experiment at two different laser waists (210 and 480 nm) showing increasing ${\tau }_{1/2}$ values with increasing waists. In order to establish the diffusion laws, the ${\tau }_{1/2}$ values are obtained at different waists by fitting the correlogram with Eq. 3. (c) Experimental FCS diffusion laws of myr(−)Gag obtained at the surface of the different SLB composition (dots) and their fits (lines) using Eq. 6 with $K_P$ and $D_{2\mathrm{D}}$ as free parameters (see Table 1 for color code). In gray, FCS diffusion law of myr(−)Gag in the buffer (dots) and its fit using a linear model (line). In red, FCS diffusion law of Atto647N-PIP₂ in the supported lipid bilayer (dots) and its fit with a linear model (line). (d) Comparison of the $K_P$ values obtained from LUV binding assay (full bars) and svFCS experiments fits (dashed bars). Values are mean $\pm$ SD obtained from the fit in the case of svFCS and mean $\pm$ SD of n = 2 LUV binding experiments. To see this figure in color, go online.

LUV binding experiments

Proteins at a final 40 nM concentration were incubated for 30 min at room temperature with LUVs compositions described Fig. 2 a at a final 0.8 mg/mL concentration. The 100-$\mu L$ samples were then centrifuged at 42,000 rpm for 30 min at 4°C (Beckman TLA 100 rotor). Two fractions were isolated from these samples: 90 $\mu L$ were collected from the supernatants, and the pellets were resuspended with 80 $\mu L$ of Tris NaCl buffer; Tris HCl (10 mM), NaCl (150 mM), pH 7.4. Equal volumes from each fraction were loaded in SDS-PAGE gels and analyzed by western blot. Gag proteins were detected using an anti-p24 as primary antibody and a secondary antibody coupled to horseradish peroxidase (HRP). Membranes were imaged by enhanced chemiluminescence (ECL) in the ChemiDoc (Bio-Rad, USA) and relative intensities of bands were quantified using ImageJ. $K_D$ values were then calculated as in Eq. 1; i.e., the ratio of the band’s signal corresponding to supernatants $\left(S\right)$ over the band’s signal corresponding to pellets $\left(P\right)$ corrected by the pellet obtained with Gag alone $P_{GA}$, which account for spontaneous self-assembly. According to the mass action law, the amount of self-assembly was considered proportional to the total amount of Gag $\left(T=P+S\right)$ and normalized to this total amount when Gag was alone $\left(T_{GA}\right)$. To correctly estimate $K_D$, we then performed the following calculation:

$$K_D=\frac{S}{P_{LU{V}_{Binding}}}=\frac{S}{\left(P-P_{GA}\frac{T}{T_{GA}}\right)}$$ (1)

SLB processing

Glass coverslips (25 mm no. 1.5 H cover glass (Marienfeld)) were treated for 30 min with ozone in Ossila UV Ozone Cleaner and rinsed thoroughly with ultrapure water. The sample was delimited by a 7-mm diameter plastic cylinder fixed to the coverslip using Twinsil (Picodent). Then 100 $\mu L$ of 0.2 mg/mL small unilamellar vesicles (SUVs) of the different mix compositions were spread on the coverslip and incubated for 40 min at 37°C. Formed SLBs were washed four times with filtered Tris NaCl buffer to remove non-fused vesicles. Proteins at a final 10 nM concentration were injected 30 min before the beginning of the experiment.

Living cell svFCS experiments

Human embryonic kidney cell line (HEK-293T) was seeded on 25-mm no.1.5 H cover glass (Marienfeld) coated with poly-L-lysine (Sigma), and 0.15 million cells were maintained in 2 mL of DMEM (GIBCO) for 24 h at 37°C. Transfection was performed using ${\mathrm{C}\mathrm{a}\mathrm{C}\mathrm{l}}_2$ (250 mM) and HBS2X, with 1 $\mu$g of pHIV-Gag-mCherry, 0.3 $\mu$g of pHIV-GagG2A-GFP, and 1 $\mu$g of pHTLV-GagYFP. Fifteen hours post transfection, medium was replaced with phenol-red-free medium L15 (GIBCO) supplemented with 20 mM HEPES, for svFCS experiments on live cells.

Acquisition and fit of the experimental diffusion laws

FCS setup

FCS was performed on a Zeiss LSM 780 microscope equipped with the variable pupil coverage system using water immersion (numerical aperture, NA = 1.2). The underfilling of the objective back-aperture leads to the increase of the beam waist.

Beam waist calibration

$w^2$ was calibrated using highly diluted rhodamine-6G (R6G) and tetramethylrhodamine (TMR; a gift from Dr. M. May) solutions excited at 488 and 561 nm respectively. At least 30 autocorrelogram functions (ACFs) were obtained at T = 37$°$C (for cell experiments) or 20$°$C (for SLB binding experiments) from 5-s fluctuation intensity acquisition. Laser irradiance of the sample is adjusted for the different waists in order to get constant molecular brightness. The free diffusion time is obtained by fitting the auto-correlation function of the measured time-trace intensity with an analytical model of a 3D free diffusion including a triplet model (T + 3D).

$$G\left(\tau \right)=1+\left(1+\frac{T}{1-T}{e}^{-t/{\tau }_T}\right)\frac{1}{N}\frac{1}{1+\tau /{\tau }_d}\frac{1}{{\left(1+\tau /\left(s^2{\tau }_{\mathrm{d}}\right)\right)}^{1/2}}$$ (2)

Rhodamine triplet time was measured from the smallest waist correlogram to be 4 $\mu s$, and was therefore kept fixed to fit all the bigger waist calibration correlogram; only T, the triplet fraction was allowed to change. $s$ is set to 5. $w^2$ is calculated for ${\tau }_d$ using $w^2=4D{\tau }_d$. The diffusion coefficients of R6G and TMR solutions were set to D = 360 $\mu$m².${\mathrm{s}}^{-1}$ for the SLB measurements and D = 550 $\mu$m².${\mathrm{s}}^{-1}$ (respective values of D for R6G at 20 and 37$°$C according to https://www.picoquant.com/images/uploads/page/files/7353/appnote_diffusioncoefficients.pdf, based on (26)). We measured excitation waists varying from 240 to 1120 nm (488-nm excitation) or from 290 to 1220 nm (561-nm excitation) depending on objective’s back-aperture coverage.

FCS diffusion law acquisition on SLBs and cell

The SLB z = 0 axial position was retrieved thanks to the Cy5.5 PE fluorescence, and ACFs were recorded at 20°C. Cell intensity fluctuations and associated ACF correlograms were acquired at 37°C by parking the laser in the z axis at the level of the bottom membrane, in order to avoid deformation of the laser shape due to index mismatch that will occur at the top membrane. The position of the laser spot within the cell was systematically controlled with a z-scan intensity profile before and after each waist of correlogram acquisition. In both cases, FCS measurements were done for seven different beam waists, and we recorded at least 50 series of time-trace intensity of 10-s duration per waist by successive series of 10. Again, the laser irradiance is adjusted over the different waists to keep constant and independent of the waist probed a significant photon count per molecules in the range of 2–5 kHz but also kept low enough to avoid important photobleaching. This is controlled a posteriori by plotting $N$, the number of molecules as a function of $V_{\mathrm{e}\mathrm{f}\mathrm{f}}={\pi }^{3/2}s{w}^3$ (see Fig. S3). In the case of GFP- and YFP-labeled proteins as well as for the Alexa Fluor 488-labeled Gag, each ACF correlogram function was fitted (from 10 $\mu$ s to 1.7 s) with an analytical model for Brownian 2D diffusion.

$$G\left(\tau \right)=1+\frac{1}{N}\left(\frac{1}{1+t/{\tau }_d}\right)$$ (3)

FCS measurements of mCherry-labeled proteins were done with a 561-nm laser line. mCherry is known to exhibit flickering, which leads to dark-state population with typical ground-state recovery kinetics (27). As we do not have an analytical model for this particular photophysics, we used a classical triplet state + 2D diffusion model:

$$G\left(\tau \right)=1+\left(1+\frac{T}{1-T}{e}^{-t/{\tau }_T}\right)\frac{1}{N}\frac{1}{1+\tau /{\tau }_{\mathrm{d}}}$$ (4)

with ${\tau }_T\ll {\tau }_d$. Average ${\tau }_{1/2}$ and their standard deviation (SD) values found for each waist in each condition (either for the SLB experiments or for the live cells), and the corresponding number of ACF correlograms fitted to determine these average ${\tau }_{1/2}$ values are given in Tables S1, S2, S3, and S4. ${\tau }_T$ and $T$ values are given in Table S5.

Numerical simulations of (sv)FCS experiments

Algorithm

We emulated FCS experiments by simulating the binding reaction and diffusion of 100 individual molecules in a cubic nucleus of size $L^3\mu {\mathrm{m}}^3$ during 11 s. Unless otherwise indicated, we use $L=6\mu \mathrm{m}$ in this work. The plasma membrane was defined as the $z=0$ face of the cell, with the cytoplasmic bulk as the $z>0$ 3D volume. At the start of the simulation, some of the individual molecules were positioned at random 3D positions (uniform distribution) within the cytoplasm and the reminder were located at random 2D positions (uniform distribution) on the plasma membrane (i.e., with coordinates $z=0$) (Fig 1 a1). The fraction of molecules initially free in the cytoplasm was set to its theoretical equilibrium fraction; i.e., to $k_{\text{off}}/\left(k_{\text{on}}+k_{\text{off}}\right)$.

Figure 1.

Simulation-based derivation of a phenomenological diffusion law. (a1) We used the numerical simulations of svFCS described under section “materials and methods” to generate synthetic data of an svFCS experiment where HIV-1 Gag molecules diffuse freely in 3D in the bulk/cytoplasm or in 2D as bound molecules at the plasma membrane, with binding and unbinding events driven by rate constants $k_{\text{on}}$ and $k_{\text{off}}$, respectively. (a2) The algorithm outputs the corresponding auto-correlation function, which we quantify by its full width at half maximum (FWHM), ${\tau }_{1/2}$. (b)- The diffusion laws, i.e., the changes of ${\tau }_{1/2}$ with beam width ${w_{xy}}^2$, are first fitted with the phenomenological diffusion law Eq. 6 to the training set in order to estimate the six fit parameters of the law (see text). We show a selection of the synthetic diffusion laws (full circles) and their fits (full lines). The values of $K_D$, $D_{2\mathrm{D}}$, and $D_{3\mathrm{D}}$ for each simulation are given in the legend. (c1) Fitting Eq. 6 to the synthetic data of the test set is then used to estimate $K_D$ and $D_{2\mathrm{D}}$ ($D_{3\mathrm{D}}$ is considered known; see text). The panel illustrates the obtained fits for a randomly chosen sample of simulations. (c2 and c3) The estimated values of $K_D=k_{\text{off}}/k_{\text{on}}$ or $D_{2\mathrm{D}}$ are compared with their real values. The accuracy of these estimations is quantified using the median symmetric accuracy (MSA). To see this figure in color, go online.

Reaction-diffusion simulation was processed by successive iterations of time step $\Delta t$ (we used $\Delta t=1\mu \mathrm{s}$). The position ${\mathbf{r}}_i$ of every individual molecule $i=\left\{1,\dots ,100\right\}$ was first moved according to independent Brownian dynamics: ${\mathbf{r}}_i\left(t+\Delta t\right)={\mathbf{r}}_i\left(t\right)+\sqrt{2D_i\Delta t}\mathit{\xi }$ where $\xi$ is a 3D vector of i.i.d. Normal random numbers and $D_i=D_{2\mathrm{D}}$ or $D_{3\mathrm{D}}$, for membrane-bound or free molecules in the bulk, respectively. Membrane-bound molecules were kept at the membrane at this step (i.e., we reset their coordinate $z$ to zero).

Classical boundary conditions, e.g., periodic or reflective, introduce strong correlations in the simulations that are not present in the experiments. We therefore opted for the following scheme. Molecules exiting the cell through the plasma membrane at $z=0$ were changed into bound molecules (thus reset at $z=0$), whereas the upper face $\left(z=L\right)$ was considered reflective. Molecules exiting the cell through any of the four remaining faces were removed from the simulation. To keep the number of molecules constant in the cell, removed molecules were compensated by the addition of an equal number of new free molecules that were located in the bulk at a distance $ϵ=10$ nm of one of the four remaining faces, chosen at random (with uniform distribution).

We then simulated the reaction step. We considered that free bulk molecules located at a distance less than $ϵ=10$ nm from the plasma membrane (i.e., for which $z<0.01\mu \mathrm{m}$) were close enough to bind. Each of these molecules was therefore independently turned into a membrane-bound molecule with probability $p_{\text{on}}$. Conversely, every membrane-bound molecule was independently allowed to unbind; i.e., was changed into a free bulk molecule, with probability $p_{\text{off}}$. Both reaction probabilities were set from the simulation parameters according to $p_{\text{on}}=k_{\text{on}}L\Delta t/ϵ$ and $p_{\text{off}}=k_{\text{off}}\Delta t$.

To emulate fluorescence emission, we used a 3D Gaussian illumination profile centered on the plasma membrane at $z=0$. The probability that a molecule located at position $\left(x,y,z\right)$ at time $t$ emit one photon between $t$ and $t+\Delta t$ was computed as $I\left(x,y,z\right)=I_0\exp \left(\frac{-2\left(x^2+y^2\right)}{{w_{xy}}^2}\right)\exp \left(\frac{-2z^2}{{w_z}^2}\right)\Delta t$, with $w_{xy}$ and $w_z$ the beam width in the $x,y$ or $z$ direction respectively. We set $I_0$ to $15\times {10}^4$ to reproduce the experimentally observed brightness. However, due to low number of molecules in the simulations (to optimize the computing time), this $I_0$ correspond to high molecular brightness values, which permit us to obtain a low level of noise in ACF allowing correct determination of ${\tau }_{1/2}$ (see Fig. S2 a). To achieve correct equilibrium in the molecule locations before recording photon emission, the collection of photon emission times was switched off during the first 500 ms of every simulation run. We counted the total number of photons emitted at each $\Delta t$ time step during the simulation before computing the auto-correlation of the corresponding time series. For each parameter value, we ran 20 independent realisations of this 11-s simulation, and averaged the 20 resulting auto-correlations to get the average auto-correlation $G\left(\tau \right)$.

Simulation of svFCS

To emulate an svFCS experiment, we repeated the above simulations for a selection of 16 values of the beam width $w_{xy}$ between 71 and 632 nm and obtained the auto-correlation $G\left(\tau \right)$ for each of these 16 values. The beam width in the $z$ direction was set to $w_z=s{w}_{xy}$ with a constant value $s=5$ (in agreement with experimental measurements).

Model selection

To determine what model is the best fit of an auto-correlation curve, we used the reduced Akaike information criterion (AICc) (28). This criterion has the advantage of taking into account the number of free parameters of the models, penalizing models with more free parameters. We note $G_i$, the $i^{th}$ value of auto-correlation, measured at a delay ${\tau }_i$ and $m\left({\tau }_i\right)$ the value predicted by model $m$ for this delay. The residual sum of squares $RSS=\sum _{i=1}^n{\left(G_i-m\left({\tau }_i\right)\right)}^2$ is used to compute the criterion $AICc=2k+n\ln \left(RSS\right)+\left(2k^2+2k\right)/\left(n-k-1\right)$ with $k$ the number of free parameters of model $m$ and $n$ the number of auto-correlation values. When comparing two models, the model with the smaller AICc is the best, but the evidence is considered strong enough if the AICc difference is at least 6. In case several models are found to be the best fit (i.e., the evidence for their difference is not strong enough), we selected the model with the smaller number of free parameters $k$.

Code availability

The computer code used in this article is freely available at https://gitlab.inria.fr/hberry/gag_svfcs.

Results

Empirical determination of a phenomenological diffusion law

Single spot size

A first possibility to interpret the results of FCS experiments is to derive an analytical expression for the auto-correlation function $G\left(\tau \right)$ and fit it to the data to estimate the parameters. In practice, this is possible only for the simplest cases. In particular, this is not feasible, to our knowledge, when the problem is not homogeneous or isotropic, as is our case here with a 2D membrane located in a 3D volume at $z=0$. However, one can simplify the problem by considering that the membrane binding and diffusion sites are homogeneously distributed in the 3D bulk, and by neglecting the reduction of dimensionality. The result is a system where the bound molecules diffuse in the same 3D space as the free ones; i.e., a spatially homogeneous reaction-diffusion problem with reaction $\mathrm{F}\underset{{\mathrm{k}}_{\mathrm{off}}}{\overset{{\mathrm{k}}_{\mathrm{on}}}{\leftrightarrows }}\mathrm{B}$ with $F$ the free molecules diffusing with diffusion coefficient $D_{3\mathrm{D}}$ and $B$ the bound molecules diffusing with diffusion coefficient $D_{2\mathrm{D}}$. Textbook reviews and several articles have treated simplified versions of the problem. For instance, the correlation function for the case $D_{2\mathrm{D}}\approx 0$ is derived in Michelman-Ribeiro et al. (3), whereas Krichevsky and Bonnet (29) derive it for $D_{3\mathrm{D}}=D_{2\mathrm{D}}$. However, we could not solve the general case $D_{3\mathrm{D}}>D_{2\mathrm{D}}>0$ (see supporting material SI1).

An alternative approach consists in fitting the auto-correlation curves $G\left(\tau \right)=f\left(\tau \right)$ with a selection of expressions derived for other problems and test on synthetic (simulation) data the accuracy or meaning of the parameters thus estimated. Fig. S1 shows the result of this approach. Here, we simulated diffusion and binding for a range of $k_{\text{on}}$ and $k_{\text{off}}$ values using the algorithm described above with $D_{2\mathrm{D}}=1.7$ and $D_{3\mathrm{D}}=30\mu {\mathrm{m}}^2$.${\mathrm{s}}^{-1}$. For each parameter value, we fitted the obtained auto-correlation function with three models: a single population with 3D Brownian motion,

$$G\left(\tau \right)=1+{\left[N\phi \left(\tau ,D,w_{xy}\right)\sqrt{\phi \left(\tau ,D,w_z\right)}\right]}^{-1}$$

with $\phi \left(\tau ,D,w\right)=1+4D\tau /w^2$, a mix of two populations with 3D Brownian motion,

$$G\left(\tau \right)=1+N^{-1}{\left[\frac{K_D}{1+K_D}\phi \left(\tau ,D_1,w_{xy}\right)\sqrt{\phi \left(\tau ,D_1,w_z\right)}+\frac{1}{1+K_D}\phi \left(\tau ,D_2,w_{xy}\right)\sqrt{\phi \left(\tau ,D_2,w_z\right)}\right]}^{-1}$$

or anomalous diffusion

$$G\left(\tau \right)=1+{\left[N\left(1+{\left(4D\tau /{w_{xy}}^2\right)}^{\alpha }\right)\right]}^{-1}\text{.}$$

The best-fit model was then selected using corrected AICc (see section “materials and methods”) (Fig S1 a). Note that, for the two-population mixture, one could use one population with 3D Brownian motion and the other with 2D. Using this 3D-2D mix in place of the 3D-3D mix used here does, however, not change our results.

We found that the best-fit model is the single Brownian model with low diffusion coefficient for $k_{\text{on}}\gg k_{\text{off}}$, or with large diffusion coefficient for $k_{\text{off}}\gg k_{\text{on}}$. Therefore, in these regimes, the approach correctly spots one of the two diffusion coefficients but is blind to the other. Between these extreme regimes, the best-fit model was found to be the two-population model with one small and one large diffusion coefficient (Fig. S1 b1). Interestingly, the anomalous diffusion fit was never found to be the best-fit model except in one case. For most of the $\left(k_{\text{on}},k_{\text{off}}\right)$ values, this simple procedure is remarkably precise regarding the estimation of the diffusion coefficients (Fig S1 b2). However, the estimation for $K_D$ is very bad, with estimated values that can be one order of magnitude smaller than the real value (Fig. S1 b3). This was, of course, to be expected, given the simplifying hypotheses that support this approach. However, since our experimental interest here is mainly on the estimation of $K_D$, we opted for a strategy based on spot variation.

svFCS

Our motivation for svFCS is based on the following argument. Let $F$ and $B$ be the concentration (or density) of molecules in the cytoplasm or bound at the membrane, respectively. With the $z=0$ plan of the beam located at the membrane, for increasing values of the beam waist $w_{xy}$, the surface of illuminated membrane will grow more slowly than the volume of illuminated cytoplasm, because the former scales as ${w_{xy}}^2$ and the later as ${w_{xy}}^3$. Hence, intuitively, one expects the FCS measurement to be more sensitive to the bound fraction at small beam waists and more sensitive to the free fraction for large beam waists. A back-of-the-envelop calculation is as follows: the number of bound molecules found inside the focal volume is $N_B\propto B{w_{xy}}^2$, whereas the number of free molecules found in the focal volume is $N_F\propto F{w_{xy}}^2{w}_z=Fs{w_{xy}}^3$. The ratio between both is thus $N_B/N_F=B/F\times 1/\left(s{w}_{xy}\right)$. The concentrations $B$ and $F$ are constant. The ratio $s=w_z/w_{xy}$ is also roughly constant in our experimental setup. Therefore, we expect from this simple analysis that the svFCS signal at very small beam widths $w_{xy}$ will mostly be dominated by the bound molecules, whereas free molecules should dominate at very large spot sizes.

In the svFCS literature, the evolution of ${\tau }_{1/2}$, the full width at half maximum (FWHM) of $G\left(\tau \right)$ (FWHM; i.e., the value of $\tau$ where $G\left(\tau \right)$ is half its maximum value; see Fig 1 a2) is referred to as a diffusion law. A first phenomenological diffusion law for our case therefore consists in expressing ${\tau }_{1/2}$ as the sum of a 2D and a 3D contribution, adding the constraint that the 2D bound fraction should dominate for $B/F\to \infty$ or $w_{xy}\to 0$ (i.e., ${\tau }_{1/2}$ should converge to ${w_{xy}}^2/\left(4D_{2\mathrm{D}}\right)$ at these limits), whereas the 3D free fraction should dominate for $B/F\to 0$ and $w_{xy}\to \infty$ (i.e., ${\tau }_{1/2}$ should converge to ${w_{xy}}^2/\left(4D_{3\mathrm{D}}\right)$ at these limits). A simple ansatz that respects these constraints is:

$${\tau }_{1/2}=\left[\frac{1}{D_{2\mathrm{D}}}+\frac{4K_{D}s{w}_{xy}}{3D_{3\mathrm{D}}}\right]\frac{{w_{xy}}^2}{4\left(1+4/3K_{D}s{w}_{xy}\right)}$$ (5)

Unfortunately, Eq. 5 was not found to yield correct estimates of $D_{2\mathrm{D}}$ or $K_D$ on synthetic data. For instance, we found that the prefactor of the $s{w}_{xy}$ terms in Eq. 5 could not be constant since it was decreasing with increasing $D_{2\mathrm{D}}$ and decreasing $D_{3\mathrm{D}}$, and was increasing with $K_D$. Likewise, we found that the dependence of the second addend in the first term of the right side of Eq. 5 was actually non-linear in $s{w}_{xy}$. All this information was included in Eq. 5, yielding a generalized, more complex phenomenological expression:

$${\tau }_{1/2}=\left[\frac{1}{D_{2\mathrm{D}}}+\psi \left(D_{2\mathrm{D}},D_{3\mathrm{D}},K_D\right)\frac{{\left(s{w}_{xy}\right)}^{n_3}}{D_{3\mathrm{D}}}\right]\frac{{w_{xy}}^2}{4\left(1+\psi \left(D_{2\mathrm{D}},D_{3\mathrm{D}},K_D\right)\times {\left(s{w}_{xy}\right)}^{n_3}\right)}$$ (6)

with

$$\psi \left(D_{2\mathrm{D}},D_{3\mathrm{D}},K_D\right)=\frac{\alpha }{{\kappa }_{2D}+D_{2\mathrm{D}}}\frac{D_{3\mathrm{D}}}{{\kappa }_{3D}+D_{3\mathrm{D}}}{K_D}^{n_1}\exp \left({\left(n_2\ln \left(K_D\right)\right)}^2\right)\text{.}$$ (7)

In addition to the simulation parameters $\left(D_{2\mathrm{D}},D_{3\mathrm{D}},K_D,s\right)$, Eq. 6 comprises six free fit parameters: $\alpha ,n_1,n_2,n_3,{\kappa }_{2D}$, and ${\kappa }_{3D}$. We next estimated their values using a simulation-based approach.

We ran a total of 200 svFCS simulations, each with different values for $D_{2\mathrm{D}}$, $D_{3\mathrm{D}}$, $k_{\text{on}}$, and $k_{\text{off}}$, thus obtaining 200 diffusion laws ${\tau }_{1/2}=f\left({w_{xy}}^2\right)$ (see Fig. S2 for examples of simulated fluorescence intensity traces and ACFs at different waists). For each simulation, $K_D$ was set by random sampling with uniform distribution in the $\left[\mathrm{0.1,1}\right]$ interval for 30% of the simulations (60 out of 200), and in $\left[\mathrm{1,10}\right]$ for the rest. Likewise, $k_{\text{on}}$ was set by random sampling (uniform distribution) in $\left[\mathrm{8.3,167}\right]$. $k_{\text{off}}$ was set using $k_{\text{off}}=K_D{k}_{\text{on}}$. The diffusion coefficients for each simulation were also chosen at random with uniform distribution in $\left[\mathrm{1.5,5}\right]\mu {\mathrm{m}}^2.s^{-1}$ for $D_{2\mathrm{D}}$ and in $\left[\mathrm{25,45}\right]\mu {\mathrm{m}}^2.s^{-1}$ for $D_{3\mathrm{D}}$. These 200 simulations were then distributed at random (uniformly) into two groups: 150 simulations for the “training set” and 50 for the “test set.” No quadruplet of parameter values $\left(k_{\text{on}},k_{\text{off}},D_{2\mathrm{D}},D_{3\mathrm{D}}\right)$ were common between the two sets.

In a first stage (training), we considered the values of $D_{2\mathrm{D}}$, $D_{3\mathrm{D}}$, and $K_D$ as known, and fitted the 150 diffusion laws of the training set with $\alpha ,n_1,n_2,n_3,{\kappa }_{2D}$, and ${\kappa }_{3D}$ as free parameters. Fig. 1 b gives a random selection of the diffusion laws and the fits that were obtained. To improve robustness of the parameter estimation and to obtain variation ranges, we used a “bootstrap aggregation” (bagging) process (30): we generated 100 variants of the training set by randomly sampling, with replacement, the initial training set. Each variant contained 150 simulations, like the original training set. For each variant, we then fitted Eq. 6 to the 150 simulations of the variant with $\alpha ,n_1,n_2,n_3,{\kappa }_{2D}$, and ${\kappa }_{3D}$ as free parameters and considering $D_{2\mathrm{D}}$, $D_{3\mathrm{D}}$, and $K_D$ as known. This yielded 100 estimations for free parameters, which we averaged. The resulting values were (mean $\pm$ SD): $\alpha =1.162\pm 0.110$, ${\kappa }_{2D}=0.723\pm 0.103$, ${\kappa }_{3D}=16.84\pm 4.18$, $n_1=1.104\pm 0.017$, $n_2=0.202\pm 0.031$, and $n_3=1.372\pm 0.008$.

In a second stage, we fixed the values of $\alpha ,n_1,n_2,n_3,{\kappa }_{2D}$, and ${\kappa }_{3D}$ to their mean values above and fitted the phenomenological diffusion law Eq. 6 to the 50 diffusion laws of the test set using $D_{2\mathrm{D}}$ and $K_D$ as free fit parameters. We considered that $D_{3\mathrm{D}}$ can easily be determined from independent experiments (see below) and therefore considered it as known. Note again that the values of $D_{2\mathrm{D}}$, $D_{3\mathrm{D}}$, $k_{\text{on}}$, and $k_{\text{off}}$ used for this test set have never been used in the training set. A random choice of the obtained fits is shown in Fig. 1 c1. For each of these 50 simulations, we compare the estimates of $K_D=k_{\text{off}}/k_{\text{on}}$ and $D_{2\mathrm{D}}$ thus obtained with their real values on Fig. 1 c2 and c3, respectively. In these panels, the main diagonals (red line) represent perfect estimations. In both cases, the estimation is most of the time close to the diagonal, indicating a good estimation. Although $K_D$ was varied over two orders of magnitude, its estimation is correct over the whole range. We quantified the accuracy of these estimates by the median symmetric accuracy (MSA) (31): $\text{MSA}=100\exp \left(M\left(\left|\ln \left(y^{\text{pred}}/y^{\text{real}}\right)\right|\right)-1\right)$, where $M$ designates the median function and $y^{\text{pred}}$ or $y^{\text{real}}$ are the predicted or real values of the parameters. We found an estimation error of 11.8% and 11.5% for $K_D$ and $D_{2\mathrm{D}}$, respectively. We therefore considered that the phenomenological diffusion law Eq. 6 provides us with estimates that are correct enough to be tested using experimental data.

Monitoring HIV-1 Gag partition using FCS diffusion laws on model membranes

In the following, we use equivalently and interchangeably the equilibrium constant $K_D=k_{\text{off}}/k_{\text{on}}$ and the partition coefficient $K_P={\left[F\right]}_{eq}/{\left[B\right]}_{eq}$, since they are equivalent for the simple binding equilibrium $\mathrm{F}\underset{{\mathrm{k}}_{\mathrm{off}}}{\overset{{\mathrm{k}}_{\mathrm{on}}}{\rightleftarrows }}\mathrm{B}$. To assess the ability of our empirical analytical expression of FCS diffusion law (Eq. 6) to correctly estimate the $K_P$ and the membrane-bound diffusion coefficient $D_{2\mathrm{D}}$ in the case of a 2D/3D binding-unbinding kinetics, we performed svFCS of HIV-1-myr(−)-Gag protein kinetics in the presence of SLBs as depicted in Fig. 2 a. HIV-1-Gag is known to interact with lipid membranes thanks to a bipartite motif consisting of a myristate and a polybasic domain (HBR) (for review see (32)). Electrostatic interactions of HIV1-Gag with negatively charged lipids (such as phosphatidylserine (PS) and phosphatidylinositolbisphosphate (${\mathrm{P}\mathrm{I}\mathrm{P}}_2$) have been shown using NMR (17,18).

We monitored differences occurring in membrane binding using four different lipid compositions containing decreasing amounts of negatively charged lipids (namely mix 1 to 4; see the table in Fig. 2 a for the composition and the molar proportion of each lipid). Interaction with the plasma membrane is a key step in the initiation of HIV-1 Gag self-assembly, occurring during the generation of a new virion (33). However, HIV-1 Gag self-assembly only occurs above a critical concentration of HIV-1 Gag. (34) showed that, for HIV-1 Gag concentration far below 50 nM, no self-assembly was observed on SLB with composition equivalent to our mix 1. To avoid self-assembly, in order to stay in the limit of our kinetic model, fluorescent HIV-1 Gag was injected in the bulk phase, above the SLB, to a final concentration of 10 nM. In parallel, fluorescent PE-Cy5.5 was added to a negligible molar proportion into the lipid composition in order to precisely locate the SLB in the axial direction and correctly focus the laser to ensure maximal $w_{xy}$ value at the SLB. Correlograms were then collected at different waists, as illustrated in Figs. 2 b and S2 b, to determine the mean value of ${\tau }_{1/2}$ for each waist, in order to establish the svFCS diffusion laws. Fig. 2 c shows the svFCS diffusion laws $\left({\tau }_{1/2}=f\left({w_{xy}}^2\right)\right)$ obtained for the four different lipid compositions, from the highest charged one (mix 1, blue dots) to the neutral one (mix 4, pink dots). Each ${\tau }_{1/2}$ value plotted on these diffusion laws represent the mean $\pm$ SD of $30<n<110$ correlograms measured on $2<n<3$ different SLBs for each ${w_{xy}}^2$. Each ${w_{xy}}^2$ value is itself a mean $\pm$ SD of 60 different measurements on R6G standard (see supporting materials Table SI.1 for details). Interestingly, from Fig. 2 c, it can be seen that each of these experimental diffusion laws is flanked by two different diffusion laws. One, the red dots, is the diffusion law of Atto647N-PI(4,5)P₂ inserted in the mix 1 SLB with its linear fit for free diffusion model (red line). The other, in gray dots, was obtained by measuring ${\tau }_{1/2}$ decorrelation times of labeled myr(−)Gag in the buffer, far away (z = 15 $\mu$m) from the mix 1 SLB, with its linear fit for free diffusion (gray line). These linear fits lead to the following diffusion coefficients: in the case of Atto647N-PI(4,5)P₂, $D_{2\mathrm{D}}$ = 2.0 $\pm$ 0.2 $\mu$m².s${}^{-1}$ (mean $\pm$ SEM) and, in the case of labeled myr(−)Gag, $D_{3\mathrm{D}}$ = 24.3 $\pm$ 1.5 $\mu$m².s${}^{-1}$. In comparison, each of the diffusion laws obtained for labeled myr(−)Gag when focusing the laser at the SLB does not seem to have a linear tendency. As in our simulated diffusion laws, they exhibit a curvature that depend on the lipid composition (see Fig. 1 b1, b2, and c1), suggesting a 2D/3D diffusion plus binding/unbinding process. These diffusion laws were fitted with Eq. 6 in order to extract $D_{2\mathrm{D}}$, $K_P$ and $D_{3\mathrm{D}}$ values. Fitting the experimental diffusion laws leaving the three parameters free systematically led to highly erroneous $D_{3\mathrm{D}}$ (see Fig. S3 for details). For this reason, we fixed $D_{3\mathrm{D}}$ = 24 $\mu$m².${\mathrm{s}}^{-1}$ in our fit, as experimentally determined. In this case, we measured respectively $D_{2\mathrm{D}}$ = 2.4 $\pm$ 0.3 $\mu$m².s${}^{-1}$ and $D_{2\mathrm{D}}$ = 2.4 $\pm$ 0.1 $\mu$m².${\mathrm{s}}^{-1}$ for myr(−)Gag in the case of PI(4,5)${\mathrm{P}}_2$ containing SLBs (mix 1 and 2). Interestingly this value is similar to the diffusion coefficient value measured for Atto647N-PI(4,5) ${\mathrm{P}}_2$ in mix 1 using Brownian svFCS diffusion laws. These $D_{2\mathrm{D}}$ then have higher values in SLBs lacking PI(4,5)${\mathrm{P}}_2$ and increase with decreasing surface charges from 4.2 $\pm$ 0.4 to 4.4 $\pm$ 1.4 $\mu$m².${\mathrm{s}}^{-1}$. These determinations therefore are in perfect agreement with the previous literature on the effect of PI(4,5)${\mathrm{P}}_2$ on myr(−)Gag. Our main goal here was to assess the ability to measure $K_P$ using svFCS. To compare our quantification with results independently obtained with a standard method, we performed LUV binding experiments with the four different lipid mixtures of Fig. 2 (see also Fig. S3). Fig. 2 d shows the $K_P$ values obtained respectively by LUV binding experiments (full bars) and by fitting the FCS diffusion laws (dashed bars) with our empirical expression in Eq. 6. For every lipid mixture, the $K_P$ determined from our svFCS measurements were systematically larger than those obtained by LUV binding assays. As $K_P$ depends on the number of accessible lipids, the different experimental procedures can lead to the systematic deviation observed here. Indeed, although our experiments were designed in such a way that the protein to lipid (P/L) ratio is similar in both approaches (LUV binding and svFCS on SLBs), SLB preparation necessitates additional steps for small unilamellar vesicle preparation and fusion, where a lot of lipids could be lost, which could lead to systematic overestimate of the $K_P$ values.

However, overall, our estimation of $K_P$ with the empirical svFCS diffusion law follows exactly the same trend as those obtained with LUV binding assay. Indeed, only mix 4, which has no negatively charged lipids, exhibits a $K_P>1$ (i.e., a partition of myr(−)Gag in favor of the bulk phase), whereas all the other compositions (mix 1 to mix 3) are found to have myr(−)Gag mainly partitioning at the lipid membrane $\left(K_P<1\right)$. Fradin et al. (35) previously showed that, close to a fluctuating membrane, parallel to the z axis of measurement, the experimental $D_{3\mathrm{D}}$ value obtained from FCS could be underestimated by 40%. Using $D_{3\mathrm{D}}$ values measured far from the membrane might therefore lead to wrong estimation of $D_{2\mathrm{D}}$ and $K_P$. However, in our experimental conditions (membrane perpendicular to the z axis), we found here $D_{2\mathrm{D}}$ and $K_P$ to be very robust to $D_{3\mathrm{D}}$ variation (see Fig. S2).

Quantifying different retroviral Gag proteins binding at the plasma membrane of HEK-293T living cells

We then explored the capability of our svFCS experiments to measure $K_P$ and $D_{2\mathrm{D}}$ values in living cells. With this aim, we transfected HEK-293T cells with plasmids expressing either HIV-1-Gag-mCherry or HIV-1-myr(−)GagGFP or HTLV1-GagYFP chimera proteins. Fig. 3 a shows representative confocal fluorescence microscopy images of the HEK-293T cells expressing these three proteins. As can be seen from the fluorescence intensities in the images, fluorescent protein expression among the cells was heterogeneous. We selected the cells with the lowest fluorescence intensity to perform the svFCS experiments as HTLV-1 (poorly) as well as HIV-1 Gag (mostly) proteins might self-aggregate at concentration higher than 500 nM (36). As for model membrane experiments, this self-aggregation will strongly affect our measurements and lead to a kinetics reaction scheme totally different from the one we used to develop our numerical simulations and our empirical analytical solution (Eq. 6). Based on our ACF fits with Eq. 3 or Eq. 4 (see Fig. S5 a and b for examples of intensity traces and their associated ACFs), we found the average number of molecules (N) in the smallest waist to be between 3 and 10, which corresponds with apparent concentrations ranging between 35 and 120 nM (the average number of molecules found in the different confocal volumes are shown in Fig. S6 a).

Figure 3.

svFCS diffusion of different viral Gag proteins in HEK-293T cells. (a) Typical confocal images obtained in HEK 293-T cells expressing, from top to bottom, HTLV-1-GagYFP, HIV-1-GagmCherry, and HIV-1-myr(−)GagGFP. Scale bar, 10 μm for the three images. (b) Experimental diffusion laws obtained in cells expressing the different fluorescent viral Gag proteins. Red squares are HTLV-1-GagYFP data, blue circles are HIV-1GagmCherry data, and green diamonds are HIV-1-myr(−)GagGFP data with their respective fits (dots, dashed, dot-dashed lines) using Eq. 6 with $K_P$ and $D_{2\mathrm{D}}$ as free parameters. Pink dots and line represent svFCS data and their fit using a linear model obtained at the plasma membrane of TopFluor-PC lipid-labeled HEK-293T cells (n = 2). Black dots and line represent svFCS data and their fit using a linear model of a cytosolic HTLV-1-GagYFP. Each value in the graph represents the mean ± SD as error bars. (c) Comparison of the $K_P$ and $D_{2\mathrm{D}}$ values obtained from the fit of the svFCS diffusion laws for the three different viral Gag expressed in HEK-293T cells. On the left panel ($K_P$ values for the three proteins), the gray area is the area where ${\left[B\right]}_{eq}$ >${\left[F\right]}_{eq}$ and blue area is the opposite. On the right panel ($D_{2\mathrm{D}}$ values for the three proteins), the pink line depicts the value of the TopFluor-PC 2D-diffusion coefficient. Pink squares are mean and white dots are median of the distribution. To see this figure in color, go online.

As we did for model membranes, we first fitted our svFCS data with Eq. 6, leaving the three parameters $K_P$, $D_{2\mathrm{D}}$ and $D_{3\mathrm{D}}$ free. As found previously, these fits systematically led to erroneous values of $D_{3\mathrm{D}}$ ($D_{3\mathrm{D}}={10}^4\mu$m².${\mathrm{s}}^{-1}$ in the case of HIV-1-Gag-mCherry). To circumvent that pitfall, we therefore measured $D_{3\mathrm{D}}$ directly in the cell, by performing svFCS with the laser focused in the cytosol, far from the plasma membrane (see black line and dots in Figs. 3 b and S6 for examples). Linear fit of the ${\tau }_{1/2}$ values led to an estimated average $D_{3\mathrm{D}}$ value of 37.2 $\pm$ 3 $\mu$m².${\mathrm{s}}^{-1}$ (mean $\pm$ SEM, n = 4 cells). $D_{3\mathrm{D}}$ was therefore fixed to 37 $\mu$m².${\mathrm{s}}^{-1}$ and the data were fitted with $K_P$ and $D_{2\mathrm{D}}$ parameters left free.

Fig. 3b shows the experimental diffusion laws obtained in four cells expressing HTLV-1-GagYFP (redish squares), 12 cells expressing HIV-1-GagmCherry (blueish circles), and five cells expressing HIV-1-myr(−)GagGFP (greenish diamonds) with their respective fits as described above. As for model membranes, each of these diffusion laws is flanked by the protein cytosolic free diffusion laws (black dots and line) and by TopFluor PC-labeled plasma membrane (pink dots and line). In these plots, one sees three distinct sets of curves, corresponding to the three different proteins. These differences are confirmed when looking at the $K_P$ and $D_{2\mathrm{D}}$ values obtained from the fits (Fig. 3 c).

Using a linear fit of the diffusion law of TopFluor PC in the membrane of living HEK-293T cells, we found $D_{2\mathrm{D}}$ = 2.7 $\pm$ 0.3 $\mu$m².${\mathrm{s}}^{-1}$. Since this value can appear relatively high for a membrane lipid in cells, we compared it with the value obtained in (11), after replotting of their initial diffusion law with correction of the waist using the most recent R6G diffusion coefficient value (see section “materials and methods” for details). In this case, we found $D_{2\mathrm{D}}$ = 2.1 $\pm$ 0.2 $\mu$m².${\mathrm{s}}^{-1}$ for the TopFluor PC inserted in Cos-7 cells membrane (Fig. S7), showing the consistency of our value. Contrary to our SLB experiments results, each of the three proteins exhibit membrane diffusion coefficients that are 1.5–2 times higher than the diffusion coefficient of TopFluor PC. HTLV-1-GagYFP was found to have the lowest diffusion coefficient of the three, $D_{2\mathrm{D}}$ = 4.1 $\pm$ 0.3 $\mu$m².${\mathrm{s}}^{-1}$ (mean $\pm$ SEM, n = 4 cells), whereas HIV-1-GagmCherry and HIV-1-myr(−)GagGFP exhibited roughly the same value: 6.2 $\pm$ 0.6 (mean $\pm$ SEM, n = 12 cells) and 6.6 $\pm$ 0.3 $\mu$m².${\mathrm{s}}^{-1}$ (mean $\pm$ SEM, n = 5 cells), respectively. Different reasons could account for this systematic overestimation of Gag $D_{2\mathrm{D}}$ in cells, compared with the values obtained in SLBs or with that for TopFluor PC, including physiological ones. Unlike our SLB experiments, the plasma membrane of the cells was not labeled with a fluorescent lipid. The z = 0 position (interception of the laser with the plane of the plasma membrane) was therefore roughly determined by inspecting the fluorescence intensity distribution along the z axis (Figs. 3 b and S9). Performing numerical simulations for different $K_P$, with fixed $D_{3\mathrm{D}}$ and $D_{2\mathrm{D}}$, but shifting the focus position above the plasma membrane inside the cells (z > 0), we observed that the shift in focus induced a systematic overestimation of $D_{2\mathrm{D}}$ (see Fig. S8 b). The higher the displacement, the strongest the overestimation. Moreover, as the beam needs to be positioned again at the plasma membrane for each waist tested, it is always prone to higher uncertainty at large waists since the fluorescence intensity profile does not depict sharp peaks anymore (see Fig. S9 for an experimental fluorescence intensity z profile).

The partition of HIV-1-myr(−)GagGFP between cytosol and membrane was found to be much larger than the other two proteins. We found $K_P$ = 3.1 $\pm$ 0.2 (mean $\pm$ SEM, n = 5) for HIV-1-myr(−)Gag, which implies that, on average, only 24% ([21,28] %, (mean $\pm$ 95% confidence interval (CI)) of the HIV-1-myr(−)Gag is bound to the plasma membrane of HEK293T cells. By contrast, the $K_P$ of HIV-1-GagmCherry was found to be 1.2 $\pm$ 0.2 (mean $\pm$ SEM, n = 12), corresponding to 45% of the total HIV-1-GagmCherry bound to the membrane ([38,57] %, (mean $\pm$ 95% CI)). HTLV-1-GagYFP was the most heavily bound protein as on average 52% ([40,79] %, (mean$\pm$ 95% CI)) of the HTLV-1-GagYFP was found bound to HEK-293T cells plasma membrane ($K_P$ = 0.9 $\pm$ 0.2). Interestingly, unlike $D_{2\mathrm{D}}$, we found that our estimation method for $K_P$ was very robust to inaccurate $z=0$ plan positioning in computer simulations (Fig. S8 c).

Discussion

Monitoring and quantifying molecular motions using FCS mainly rely on the ability to derive analytical solutions in order to fit the autocorrelograms obtained from single-spot FCS measurements. Except in simple cases, such as flow or free diffusion, single-point FCS often fails to correctly determine and quantify molecular motions in heterogeneous and non-isotropic environments. This failure is frequently (and sometimes abusively) circumvented by the addition of an $\alpha$ exponent, signing for anomalous sub-diffusion motion in the heterogeneous media. However, anomalous sub-diffusion may occur from many different processes that theoretically cannot be uniquely identified or quantified by the sole value of $\alpha$ (see (37,38) and references therein). In the case of reaction-diffusion dynamics, it has been shown that no simple analytical solutions could be derived to fit single-spot FCS experiments (3). This is also what we found here. Deriving an expression systematically required us to simplify the dynamics process with different hypotheses, as was also observed in (3). In the 2D/3D diffusion and binding dynamics, we expected to have two limit regimes, namely pure 3D free diffusion (membrane unbound molecule, $k_{\text{off}}\gg k_{\text{on}}$) and pure 2D diffusion (membrane-bound molecule, $k_{\text{off}}\ll k_{\text{on}}$) as well as different intermediate regimes where the system dynamically equilibrates. Interestingly, the fit of the autocorrelograms with different analytical expressions (3D free diffusion, 2D free diffusion, two-component diffusion, and anomalous diffusion) show that this type of dynamics is hardly correctly fitted by anomalous diffusion if, in order to select the best-fitting expression (as is the case with information-theory-derived criteria), the number of free parameters to fit is accounted for. In addition, although $K_D$ could be extracted from single-point FCS, we found its estimated value to be at least 10 times lower than its real value over a large range of $k_{\text{off}}\mathrm{a}\mathrm{n}\mathrm{d}{k}_{\text{on}}$ values.

A powerful method to correctly probe and characterize the dynamics of molecules in complex media is the spot size variation method (svFCS) (5,11), which establishes FCS diffusion laws. We therefore used computer simulations to generate synthetic FCS diffusion laws for 2D/3D diffusion plus binding in different regimes. We first controlled that these simulated diffusion laws were flanked by the two expected limit regimes, namely 2D and 3D diffusion, that both lead to linear FCS diffusion laws. We then generated a set of synthetic diffusion laws in the intermediate regime by varying the values of $D_{2\mathrm{D}}$, $D_{3\mathrm{D}}$, and $K_D$ in our simulations. In these intermediate regimes, we systematically obtained non-linear synthetic FCS diffusion laws. We therefore derived an empirical non-linear analytical expression (Eq. 6) that, when fitted to our synthetic diffusion laws, provides a quantitative estimation of $K_D$ and $D_{2\mathrm{D}}$, with relative precision around 12% in the range $0.1<K_D<10$ and $1.5<D_{2\mathrm{D}}<5\mu {\mathrm{m}}^2.s^{-1}$.

Since this precision was acceptable, we challenged the capacity of our empirical expression to estimate the change of the membrane apparent partition coefficient $K_P$ and the membrane diffusion $D_{2\mathrm{D}}$ occurring when HIV-1 and HTLV-1 Gag proteins and an HIV-1 derivative (myr(−)) binds either to model lipid membranes, with controlled composition, or at the plasma membrane of living HEK-293T cells. In both cases, fitting the experimental FCS diffusion law with Eq. 6 did not provide correct estimates when we tried to determine the three parameters of the model simultaneously, namely $D_{3\mathrm{D}}$, $D_{2\mathrm{D}}$, and $K_P$. The simplest explanation is that the number of beam waists to fit is not sufficient to achieve a good estimation of the three parameters at the same time.

Increasing the number of waists monitored should help in having better fits, but, although it is easy to achieve numerically, it remains illusive experimentally. Indeed, correct determination of ${\tau }_{1/2}$ needs avoiding photobleaching during successive experiments, which cannot be done if we drastically increase the number of waists probed in the same cell. To circumvent that issue, we directly measured the $D_{3\mathrm{D}}$ in the bulk (in the case of model membranes) or in the cytosol (in the case of HEK293T cells). Once $D_{3\mathrm{D}}$ is determined, we could successfully fit the experimental svFCS diffusion laws the two free parameters remaining ($K_P$ and $D_{2\mathrm{D}}$). Interestingly, although Fradin et al. (35) showed that the apparent $D_{3\mathrm{D}}$ close to a membrane could differ from the $D_{3\mathrm{D}}$ measured far from it (as we do here), we found that our estimation of $D_{2\mathrm{D}}$ and, to a lesser extent, $K_P$, were robust over a large range of $D_{3\mathrm{D}}$ values. Note that, as small beam waist values are more sensitive to diffusion of the proteins bound on the membrane, stimulated emission depletion (STED)-FCS, which addresses much smaller beam waists than classical confocal FCS, could be another way to circumvent this issue by estimating $D_{2\mathrm{D}}$ instead of $D_{3\mathrm{D}}$ as the fixed parameter. Combination of the two approaches will certainly help in increasing the accuracy of the method.

We first used SLB model membranes made of PC, PS, and PI(4,5)${\mathrm{P}}_2$ with decreasing surface negative charges by tuning the molar ratio of PS and PI(4,5)P₂. We found the membrane diffusion $D_{2\mathrm{D}}$ of HIV-1-myr-Gag to be equivalent to that of Atto-647N-PI(4,5)${\mathrm{P}}_2$ in the SLB. PI(4,5)${\mathrm{P}}_2$ is known to be the specific target lipid of HIV-1-Gag association to the plasma membrane (33) and has been shown to be trapped by HIV-1-Gag in model membranes (34) as well as in living T cells (39). However, PS is also involved in binding of Gag to the membrane (18,19). This could explain why we only measured $K_P$ > 1 for SLBs lacking negatively charged lipids. However, surprisingly, as we expected to obtain increasing values of $K_P$ with decreasing amount of PI(4,5)${\mathrm{P}}_2$, the $K_P$ values obtained with svFCS diffusion laws exhibited exactly the same trend than those obtained by LUV binding experiments (i.e., $K_P$ > 1 only observed with neutral lipids). We also previously showed that lack of myristate strongly decreases the specificity for PI(4,5)${\mathrm{P}}_2$ (19). In addition, in a 2:1 PC:PS mol:mol lipid composition, (40) reported no significant change in membrane binding for molar proportion of PI(4,5)${\mathrm{P}}_2$ varying from 0% to 2%, in agreement with our own measurements.

Finally, we examined the ability of our method to quantify membrane binding and diffusion of retroviral Gag proteins and their derivatives in HEK-293T cells. In this case, we found $D_{2\mathrm{D}}$ to be 4.1 $\pm$ 0.3 $\mu$m².${\mathrm{s}}^{-1}$ and 6.2 $\pm 0.6\mu$m².${\mathrm{s}}^{-1}$ for HTLV-1 and HIV-1 Gag, respectively, questioning the origin of such high $D_{2\mathrm{D}}$ values. Unlike in SLB experiments, where the lipid membrane is labeled, making the z axis laser parking easier, the cell plasma membrane is not labeled and its location is defined thanks to z axis fluorescence intensity profile of the protein. This can lead to a high inaccuracy in the laser z positioning, which in turns strongly affects the estimation of $D_{2\mathrm{D}}$ in the sense of a strong overestimation. Fortunately, we found that this inaccuracy has low impact in the determination of $K_P$. However, if one aims at correctly estimating $D_{2\mathrm{D}}$ values of membrane binding proteins in cells, a stable fluorescent labeling of the plasma membrane (using MemBright, for example (41)) will definitely help.

Using a combination of total internal reflection fluorescence microscopy and fluorescence fluctuation spectroscopy, (21) showed that HTLV-1-Gag has a higher affinity than HIV-1-Gag for the plasma membrane of HeLa cells. Using svFCS diffusion laws, we also measured a lower $K_P$ for HTLV-1-Gag compared with HIV-1-Gag. This reflects the higher affinity of HTLV-1-Gag versus HIV-1-Gag for the plasma membrane of HEK-293T cells and illustrates again the ability of our method to correctly determine this parameter. We also showed that removing myristate from HIV-1-Gag resulted in the re-localization of the Gag protein toward the cytosol, with only 24% $\pm$ 3% of the total HIV-1-myr(−)Gag proteins bound to the membrane. Again, this value is in good agreement with the results previously obtained by (21). Importantly, the results we obtained here perfectly in line with previous results obtained by different groups using different approaches/techniques.

It was not the main aim of this study to establish a molecular mechanism for retroviral Gag proteins by accurately quantifying their binding to lipid membranes in different conditions. However, we believe it is worth stressing that we have not only strong differences in the $K_P$ values obtained for HIV-1-myr(−)Gag in model versus cell lipid membranes but also a lower $K_P$ in the case of HIV-1-Gag in cells versus HIV-1-myr(−)Gag on negatively charged SLBs. Several reasons could account for such discrepancies between model lipid membranes and living cell plasma membranes:

• •The accessibility of PI(4,5)P₂ in the plasma membrane of the cells, and more generally of negatively charged lipids, might be strongly decreased compared with model membranes, as many other proteins present in the cells are also known to interact with these lipids. This would result in a screening of the targeted lipids and, consequently, an increase of apparent $K_P$ values as the total concentration of available binding sites decreases.
• •In the presence of RNA (which are highly present in cells), HIV-1 Gag has been shown to adopt a horseshoe configuration where both MA and NC domains bind to the RNA (42). The membrane binding process is then mediated by the HBR domain of the MA interaction with PI(4,5)P₂, inducing the release of the RNA, which stays bound to the NC domain (43). This screening of HIV-1-Gag membrane binding domain in cells might be more important than in our model membrane experiments, where RNA is hardly present, resulting again in an increase of K.
• •In the absence of RNA, the NC domain has been shown to exhibit significant affinity for negatively charged lipids (44). In our model membranes, where lipids are in large excess compared with RNA, this second lipid binding motif in the NC domain of HIV-1-Gag could compensate the lack of myristate and favor stronger binding to the SLB (lower $K_P$), although it will be screened in cells by the cytosolic/cellular RNAs.

In this study, we have demonstrated the ability of svFCS diffusion laws to better estimate (apparent) membrane binding coefficients and membrane diffusion coefficients than single-spot FCS can. Using a numerical simulation-based approach, we have derived an empirical analytical expression that we then used to fit experimental svFCS data obtained on retroviral Gag proteins binding either to model membranes or to plasma membranes of living cells. Overall, the results we obtained with our method are in perfect line with the ones reported in previous literature, obtained with different methods. This method has been shown to be stable over a large range of $K_P$ values, covering cases where the bound fraction at equilibrium varies from 90% down to 5% of the total protein. We conclude that our results provide a non-invasive and direct way to fairly estimate, in the same experiment, membrane binding and diffusion coefficients (with a correct z positioning) in living cells.

Author contributions

C.F., H.B., and D.M. designed the project and the experiments. H.B. developed the computer code, performed the numerical simulations, and derived the empirical expression. A.M., E.B., and J.N. performed the FCS experiments. A.M., E.B., and J.N. fitted experimental data. R.D. and C.A. prepared the cell samples, and E.B. the SLBs. J.M. purified and provided the myr(−)Gag protein. H.B. and C.F. drafted the manuscript assisted by A.M., E.B., C.A., and D.M.

Editor: Erdinc Sezgin.