Department of Biochemistry and Molecular Biology, University of Southern Denmark, Campusvej 55, Odense M, DK-5230, Denmark

Biomedical Imaging Group, Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, CH-1015, Switzerland

Institute for Mathematics and Computer Science (IMADA), University of Southern Denmark, Odense M, DK-5230, Denmark

Department of Physics, Chemistry and Pharmacy, MEMPHYS Center for Biomembrane Physics, University of Southern Denmark, Odense M, DK-5230, Denmark

Abstract

Background

Fluorescence loss in photobleaching (FLIP) is a widely used imaging technique, which provides information about protein dynamics in various cellular regions. In FLIP, a small cellular region is repeatedly illuminated by an intense laser pulse, while images are taken with reduced laser power with a time lag between the bleaches. Despite its popularity, tools are lacking for quantitative analysis of FLIP experiments. Typically, the user defines regions of interest (ROIs) for further analysis which is subjective and does not allow for comparing different cells and experimental settings.

Results

We present two complementary methods to detect and quantify protein transport and aggregation in living cells from FLIP image series. In the first approach, a stretched exponential (StrExp) function is fitted to fluorescence loss (FL) inside and outside the bleached region. We show by reaction–diffusion simulations, that the StrExp function can describe both, binding/barrier–limited and diffusion-limited FL kinetics. By pixel-wise regression of that function to FL kinetics of enhanced green fluorescent protein (eGFP), we determined in a user-unbiased manner from which cellular regions eGFP can be replenished in the bleached area. Spatial variation in the parameters calculated from the StrExp function allow for detecting diffusion barriers for eGFP in the nucleus and cytoplasm of living cells. Polyglutamine (polyQ) disease proteins like mutant huntingtin (mtHtt) can form large aggregates called inclusion bodies (IB’s). The second method combines single particle tracking with multi-compartment modelling of FL kinetics in moving IB’s to determine exchange rates of eGFP-tagged mtHtt protein (eGFP-mtHtt) between aggregates and the cytoplasm. This method is self-calibrating since it relates the FL inside and outside the bleached regions. It makes it therefore possible to compare release kinetics of eGFP-mtHtt between different cells and experiments.

Conclusions

We present two complementary methods for quantitative analysis of FLIP experiments in living cells. They provide spatial maps of exchange dynamics and absolute binding parameters of fluorescent molecules to moving intracellular entities, respectively. Our methods should be of great value for quantitative studies of intracellular transport.

Background

Quantitative fluorescence microscopy witnesses an increasing demand for computational methods allowing for interpretation of the complex data generated by this imaging technique. To determine intracellular transport dynamics of proteins and lipids tagged with suitable fluorophores, one often relies on perturbing the steady state distribution of the probe and following the dynamics of re-establishing a steady state. One example for this approach are pulse-chase experiments, where a fluorescent ligand, like a dye-tagged transferrin (Tf) or fluorescent lipoprotein binds to its receptor at the cell surface at the start of the experiment (

An alternative way to perturb the steady state does not rely on pulse-labeling molecules in a particular compartment and is often used to elucidate transport dynamics along the secretory pathway or between nucleus and cytoplasm

A method related to FRAP is fluorescence loss in photobleaching (FLIP). In FLIP, a region is repeatedly illuminated by an intense laser pulse, while images are taken with reduced laser power between the bleaches. A pause between the laser pulses allows for some recovery in the bleached region. The duration of the pause can be set by the user, who thereby can control the time resolution of the experiment. Repeating this protocol several times creates a sink for the fluorescent molecules in the local environment being in continuous exchange with the bleached region

We present two new approaches to quantify FLIP experiments reliably; either on an image basis to detect areas of different probe mobility, or by physical modelling of FL from moving entities. In the first section, we demonstrate that a stretched exponential (StrExp) function can be fitted to the decaying intensity at each pixel position in the bleached and non-bleached region of the cell. This is used to detect diffusion-limited depletion zones around the bleached area in cells expressing enhanced green fluorescent protein (eGFP). Pixel-wise fitting of the StrExp function to FLIP image sets also allows us to determine local heterogeneity in nucleocytoplasmic transport of eGFP. In the second section, we measure FL kinetics in moving inclusion bodies (IB’s) by combining FLIP with single-particle tracking. From that data, we infer exchange parameters of mutant Huntingtin tagged with eGFP (eGFP-mtHtt) by multi-compartment (MC) modelling of binding/release and fluorescence attenuation due to photobleaching. The presented methods should find wide application in quantitative cell biology and especially in analysis of protein aggregation in neurodegenerative diseases.

Results

The stretched exponential function as empirical decay law for transport studies

The StrExp function is an empirical decay function with broad applications in modelling of physical, photochemical and biophysical data

**Figure S1.** Simulation of the StrExp function and distance-dependence of fitting to homogenous diffusion. Figure S2. Bleach profile of the Argon laser at 488 nm for various objectives. Figure S3. 3D simulation of a FLIP experiment with heterogeneous diffusion in the unit sphere. Figure S4. FLIP experiment of BODIPY-cholesterol in CHO cells and fitting with the StrExp function. Figure S5. Simulation of the homogeneous compartment model and fitting with the StrExp function. Figure S6. Effect of additive image noise on fitting performance. Figure S7. Effect of time noise on fitting performance. Figure S8. Correlation between amplitude and time constant maps in the StrExp fit to the spatially Figure S9. Pixel-wise FLIP analysis of eGFP in the nucleus.heterogeneous model.

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Analysis of diffusion-limited FLIP experiments using the StrExp function

When the pause between individual bleaches is short compared to the molecular transport rates, the measured FL becomes limited by diffusion of the molecules towards the bleached area. Developing a physical model for this situation requires taking the exact location of the bleach ROI into account. Assuming that binding/dissociation events are very fast compared to molecular diffusion the process is governed by Equation 6 which can be solved (see Methods and Appendix 1). The diffusion model assumes cylindrical cell geometry and cylindrical bleach in the centre of the cell within 3 μm around the origin performed over the whole cell height (Figure _{1} < _{2} in Figure

Simulation of homogenous diffusion and fitting with the StrExp function

**Simulation of homogenous diffusion and fitting with the StrExp function.****A**, Sketch of the FLIP experiment with the cell attached to a surface and filled with eGFP (green) and the cylindrical laser beam focused in the cell center (yellow). **B**, Geometry of the analytical model for the reaction diffusion system in Eq. 6 to model the FLIP experiment. The cell is assumed to be a flat cylinder with a radius, r_{2} = 12 μm. The central bleached region with radius r_{1} = 3 μm is also cylindrical covering the whole cell height (grey shaded area). **C-E**, The model was solved analytically and simulated for two positions outside the bleached area at a distance of 5 μm (red dot in B and red symbols in **C-E**) and 10 μm (blue dot in B and blue symbols in **C-E**) from the origin, respectively. Simulations were performed with a rate constant for the intended bleaching process of ^{-1} and diffusion constants of D = 0.1 μm^{2}/sec (**C**), D = 1 μm^{2}/sec (**D**) and D = 10 μm^{2}/sec (**E**). A non-linear regression with the StrExp function (black lines) was performed in SigmaPlot (upper panels) including the residuals of the fit (lower panels). **F**, **G**, time courses were simulated for D = 0.1 μm^{2}/sec as a function of distance from the origin and fitted to the StrExp function. Fitted parameters including standard deviation of the fit are plotted for the rate constant (‘_{fit}’; F) and stretching parameter (‘_{fit}’; G). **H**, rate coefficients calculated according to Eq. S4 for the parameters in panels **F**, **G** as function of distance from bleach ROI (starting at 4 μm from origin and indicated as ‘** r**’ on the ordinate in H) over time. The scale bar shows rate coefficients color-coded using a FIRE-LUT

Diffusion is a process in space and time, while the StrExp function provides kinetic parameters only at one location (i.e., it does not depend on position; we just map the function over all pixel positions

**Scenario**

**Parameters recovered from the fit of the StrExp function to experimental fluorescence loss in FLIP image sets**

**Stretching ( h)**

**Time constant ( τ)**

**Rate coefficient ( k(t))**

**Explanation**

^{*}In that order as estimated by parameter sensitivity. ^{#} ‘Binding’ is used synonymous for binding-/barrier-limited FLIP.

**
Diffusion-limited FL
**

Diffusion of molecules to the bleached ROI is slower than the bleaching process. This results in a ’depletion zone’ around the bleached area. Binding and release, if any, are faster than diffusion.

**Inside ROI**

1 <

Dependent on bleach rate constant and diffusion constant*.

Decreasing over time.

Fluorophores get depleted by the bleaching acting as diffusion-limited reaction.

**Outside ROI**

~0.5 <

Dependent on diffusion constant and bleach rate constant*.

Increasing over time.

Compressed decay, because diffusion to the bleach ROI causes delayed response.

**
Bleach-limited FL
**

Diffusion of molecules to the bleached ROI as well as eventual binding and release are faster than the bleaching process. The only process causing FL is the repeated bleaching inside the ROI.

**Inside ROI**

~ 1

Dependent on bleach rate constant only.

Constant over time and equal to 1/τ.

Approx. mono-exponential decay determined by the bleach rate.

**Outside ROI**

~ 1

Dependent on bleach rate constant only.

Constant over time and equal to 1/τ.

Approx. mono-exponential decay determined by the bleach rate.

**
Binding
**

Diffusion is fast but molecules are hindered by binding to cellular organelles or by obstacles. Any spatial fluorescence gradient is rapidly equilibrated and binders/barriers become visible.

**Inside ROI**

~ 1

Dependent on bleach rate constant only.

Constant over time and equal to 1/τ.

Approx. mono-exponential decay determined by the bleach rate.

**Outside ROI**

~0.8 <

Dependent on release rate constant and bleach rate constant*.

Increasing over time.

Compressed decay, because slow release causes delayed response.

To account for heterogeneous intracellular diffusion in FLIP experiments, we performed next a numerical simulation of a circular section of a cell with 2 different diffusion constants (Figure _{1} = 0.2 μm^{2}/sec, while on the right half, we had _{2} = 0.8 μm^{2}/sec. Continuity across the boundary ensures that the probe can diffuse across the boundary region. Bleaching occurred within a radius of 0.5 μm around the cell center with a rate constant of _{
b
} = 10 sec^{-1}. The simulation was implemented in FEniCS and fitted on a pixel-by-pixel basis to a StrExp function (see Methods). Fluorescence dropped first on the right half of the simulated cell due to fast recruitment of molecules to the bleached area. The empirical fit using the StrExp function accurately recovers the simulated data set, as shown in Figure ^{2} parameter, a measure of the weighted sum of the squared errors between data and models. Calculating that parameter on a pixel-by-pixel basis reveals that the fit is less accurate in the bleached ROI (i.e., χ^{2} ~25; Figure _{2}, on the right half of the cell, FL was faster, as indicated by the lower time constants (Figure

**Movie 1.** Numerical simulation of FLIP experimentwith space-dependent diffusion coefficient.

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Simulation of heterogenous diffusion and fitting with the StrExp function

**Simulation of heterogenous diffusion and fitting with the StrExp function.** A 2-dimensional bleaching experiment was simulated on a disk with a circular bleached area of radius _{1}=0.5 _{μm} using FeniCS, an automated computational modelling suite (^{-1} and the diffusion coefficient is _{1}=0.2 μm^{2}/sec and _{2}=0.8 μm^{2}/sec on the left and right half disk, respectively (A). Fluorescence loss inside and outside the bleached region was fitted at every pixel position with the StrExp function. For this purpose the PixBleach plugin to ImageJ was used **B**), the time constant distribution color-coded between 1.0 to 15.0 sec (**C**) and the χ^{2}-map color-coded between 0 to 50 (**E**). A FIRE-LUT was used for color-coding, where dark blue and yellow indicate lowest and highest values, respectively. The reconstructed stack exactly resembles the simulated data set, as seen in **D** (upper row, simulated data (‘sim‘), lower row, regression (‘fit‘)). **F**, profile of time constants, as estimated from fitting the StrExp function to the simulated FLIP data along the line shown in panel **C**. **G-L**, rate coefficients were calculated on a pixel-by-pixel basis according to **G**, shows montage of all time points; **H**, first frame of montage with some regions of interest (boxes 1 to 4) used for analysis in panel **I-L**. Rate coefficient as function of time averaged for box 1 (**I**), box 2 (**J**), box 3 (**K**) and box 4 (**L**) from the whole sequence.

Diffusion-limited FLIP of eGFP in the cytoplasm of McA cells

We aimed for an experimental realization of these predictions in a physiologically relevant setting. Enhanced green fluorescent protein (eGFP) is a small 27 kDa protein which does not bear a nuclear targeting sequence and should not bind to any cellular structures. While its cytoplasmic and nuclear diffusion is very rapid (D ~ 25 μm^{2}/sec), eGFP shuttles between nucleus and cytoplasm by passive bidirectional exchange, which is well described by exponential functions

Diffusion-limited FLIP of eGFP in the cytoplasm of McA cells

**Diffusion-limited FLIP of eGFP in the cytoplasm of McA cells.** McArdle RH7777 cells expressing eGFP in the cytoplasm and nucleus were placed on a temperature-controlled stage of a confocal microscope maintained at 35 ± 1°C. A 10 pixel diameter circular region in the cytoplasm (white circle) was repeatedly bleached with full laser power, while the whole field was scanned with 0.5% laser output between the bleach scans. This gave a total frame rate of 1.6 sec (see Methods for further details). **A**, montage of the time series with every 20^{th} frame shown. **B-D**, fit of the StrExp function to the data giving a map of the stretching parameter (**B**), the time constant map (**C**) and its reciprocal, the rate constant map (**D**). A FIRE-LUT was used for color-coding, where dark blue and yellow indicate lowest and highest values, respectively. The range of values is given below the images with a color bar without units in B, in sec in **C** and in sec^{-1} in **D**. Bar, 5 μm. **E**, profile of stretching parameters along the line shown in B (grey line, data; black line moving average to smooth the profile).

Nucleo-cytoplasmic transport of eGFP in McA cells as example of binding/barrier-limited FLIP experiments

In many experimental situations, diffusion is much faster than the photobleaching process in the bleach ROI. Still, transport to the bleached area in FLIP experiments can be hindered by transient binding events or barriers to diffusion, even though the diffusion of a molecule in the aqueous phase of the cytoplasm, i.e., the cytosol, is fast. For example, exchange of proteins between nucleus and cytoplasm has been shown to be limited by transport through the nuclear pore complex but not by diffusion _{1} and _{−1} reflect nuclear export and import rate constants, respectively, but the same equations would also describe binding/dissociation-limited FLIP experiments. We confirmed in additional simulations, that the StrExp function is able to describe binding/barrier-dominated FLIP experiments (see Additional file

For the purpose of an experimental realization of binding/barrier-limited FLIP, we studied nucleo-cytoplasmic shuttling of eGFP by our FLIP approach. A small region in the cytoplasm of McArdle 7777A cells expressing eGFP was repeatedly bleached with a total frame rate including the bleach of 2.6 sec (see white circle in Figure ^{-1}; box 2: ^{-1}) and an almost mono-exponential decay in the cytoplasm (box 1: ^{-1}). Lower stretching with ^{2} in size) contains comparable little eGFP, and FL in that region occurred with a time constant of 124.38 sec with strongly accelerating FL speed (_{1}) nor compressed FL in the bleached ROI (i.e. C_{2}) in simulations of binding/barrier-dominated FLIP simulations. The fact, that we find this in FLIP experiments of eGFP in cells suggests that the simple compartment model is not adequate to describe local heterogeneity of intracellular eGFP motion. Further evidence for that is provided in the Additional files, especially Additional file

**Movie 2.** Time course of barrier-limited FLIPexperiment of eGFP.

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Barrier-limited FLIP of eGFP shuttling between nucleus and cytoplasm

**Barrier-limited FLIP of eGFP shuttling between nucleus and cytoplasm.** McArdle RH7777 cells expressing eGFP in the cytoplasm and nucleus were placed on a temperature-controlled stage of a confocal microscope maintained at 35 ± 1°C. A 30 pixel diameter circular region in the cytoplasm (white circle) was repeatedly bleached with full laser power, while the whole field was scanned with 0.5% laser output between the bleach scans such that the total frame rate was 2.6 sec. A montage of every 30^{th} frame of the data (upper panel, ‘data’) or of the reconstruction from a pixel-wise fit of data to the StrExp function (lower panel, ‘fit’). **B**, amplitude map; **C**, background map, each with three boxes numbered 1 to 3. **D**, FL in these three boxes along the stack (colored symbols) + fit to StrExp function (colored lines). **E-H**, parameter maps produced by PixBleach for: **E**, the stretching parameter (see Eq. 1; the color code for **F**, χ^{2}-values showing the quality of the regression; **G**, time constant map; H, number of iterations. All values are color-coded using a FIRE-LUT, and the range is given below the images with a color bar. The time constant is given in seconds; all other values are without units. **I**, **J**; histograms of the stretching parameters (**I**) and the time constants (**J**) for the cell, shown in **A-H**. Bar, 5 μm.

Pixel-wise rate coefficients of fluorescence loss kinetics of eGFP

**Pixel-wise rate coefficients of fluorescence loss kinetics of eGFP.** Using the parameter maps of the FLIP analysis for the cell in Figure **A**, the resulting 32-bit image stack was color-coded using a FIRE-LUT, and selected frames of the rate coefficients were plotted (blue and yellow-white indicate low and high rate coefficients, respectively). Areas with accelerating speed of FL turn from blue to yellow-white over time. A’ few regions are highlighted with boxes (numbered ‘1’ to ‘5’) for further analysis. Boxes 1 and 2 were placed in the nucleus, while boxes 3 to 5 were in the cytoplasm (inset shows a zoom for the area containing boxes 3 and 4 close to the right edge of the cell). **B**, rate coefficients as function of time plotted for box 1 to 4. **C**,

Heterogeneous transport of mutant huntingtin detected by quantitative FLIP analysis

A group of neurodegenerative disorders is characterized by expansion of poly-glutamine (polyQ) repeats in aggregation prone proteins. For example, more than 36 polyQ repeats in mutant huntingtin (mtHtt) causes aggregation of the protein in the cytoplasm of affected cells causing Huntington disease ^{-3} μm^{2}/s (red line in Figure ^{-1}. Our tracking program 'SpotTracker' could reliably extract the FL kinetics: a fit to the extracted FL profile recovers the bleach rate constant approximately, (fitted _{b} = 0.0156 sec^{-1}) and the residual term is a good measure of the background noise (~60 intensity units). For these tests, we implemented random walk simulations in ImageJ using self-programmed macros. Particle bleaching during movement was simulated using a single exponential decay of intensity in the presence of noise, and the intensity value was updated for every new position along an image sequence. Since ImageJ by default allows only for generating random numbers from a uniform distribution, the Box-Muller method was implemented to transform these random numbers to a normal (Gaussian) distribution

Quantitative FLIP and single particle tracking of mobile IB’s in the cytoplasm

**Quantitative FLIP and single particle tracking of mobile IB’s in the cytoplasm.** A FLIP experiment with a bleach region of 10 pixels in diameter was performed in CHO cells expressing eGFP-Q73 on a temperature-controlled stage of a confocal microscope maintained at 35 ± 1°C. The total frame rate was 1.45 sec. **A**, montage of every 20^{th} frame of the data. Arrows point to an inclusion body (IB) in the cytoplasm. **B-E**, pixel-wise fit of the data to a StrExp function with **B**, amplitude map; **C**, map of the stretching parameter; **D**, time constant map; E, χ^{2}-values showing the quality of the regression. All values are color-coded using a FIRE-LUT, and the range is indicated by the respective color bar. The time constant is given in seconds; all other values are without units. F, first frame of the time series is shown in green, while selected subsequent frames are overlayed in red (box in F shows movement of the IB). **G**, FL of eGFP-Q73 in the IB (red symbols) with fit to the StrExp function (black line). **H**, x,y-plot of the trajectory of the IB during the FLIP experiment. I, mean square displacement (MSD) calculated from the trajectory and linear fit to the five intital data points (red line). **J**, **K**, simulation and tracking of a particle experiencing FL with a rate constant, ^{-1}. **J**, trajectory separated in x- (straight lines) and y-direction (dotted lines) for a particle undergoing bleaching (black lines, simulated trajectory; red lines, tracked trajectory). The tracked trajectory coincided with the simulated trajectory until to 180 sec (or frames). **K**, intensity of the tracked particle (grey lines) compared to a mono-exponential fit with residual. Bar, 5 μm.

Multi-compartment modeling of FLIP data reveals dynamics of eGFP-Q73 in inclusion bodies

**Multi-compartment modeling of FLIP data reveals dynamics of eGFP-Q73 in inclusion bodies.** A FLIP experiment was performed in CHO cells expressing eGFP-Q73 as described in legend to Figure **A**, montage of every 20^{th} frame of the data. Inset is a zoom of the small box pointing to an IB; a FIRE LUT is used for visualization purposes. A’, sketch of the multi-compartment (MC) model used for determining binding/dissociation parameters. **B**, FL kinetics for the IB (blue dots) and for a region in the cytoplasm (green dots) with overlayed fit to the MC model for the IB (compartment 1; dark blue line) and for the cytoplasm (compartment 2; dark green line). **C**, another FLIP sequence imaged with a total frame rate of 2.4 sec. First frame of the time series is shown in green, while selected subsequent frames are overlayed in red. This color coding visualizes movement of two IB’s at the cell edge (large one up and small one below; arrows). Tracking of the mobile IB’s and measurement of their intensity was performed using the SpotTracker plugin for ImageJ **D**, x,y-plot of the trajectories of the IB’s; **E**, mean square displacement (MSD) calculated from the trajectories; **F**, step length distribution between subsequent steps; **G**, fluorescence intensity of eGFP-Q73 in the IB’s (blue dots = large IB; red dots = small IB) and in the cytoplasm (green dots) and fit with the MC model for the IB’s (compartments 1, dark blue and red line, respectively) as well as for the cytoplasm (compartment 2, dark green line).

Multi-compartment (MC) modeling of eGFP-mtHtt exchange between cytoplasm and IB’s

Although we were able to quantify FL of eGFP-mtHtt in moving IB’s using the strategy outlined above, this method does not provide cell-independent measures of eGFP-mtHtt dynamics. This is, since the FL depends not only on the protein exchange dynamics but also on the total cellular pool size of eGFP-mtHtt. In other words, for a larger cell the same FLIP imaging settings would give slower FL from a given IB than for a smaller cell, simply because it would take longer to deplete the whole protein pool in the larger cell. To directly compare exchange dynamics of eGFP-mtHtt between cytoplasm and IB’s from various cells, we developed an analytical MC model providing association and dissociation rate constants (see Figure _{1}) and the cytoplasm (compartment 2, C_{2}) takes place with rate constants _{1} and _{−1} (Figure ^{2} > 0.98) and gave a half-time for FL in the cytoplasm of t_{1/2}= 51.7 sec. Note that the FL in the cytoplasm is solely a consequence of the bleach protocol. In contrast, the FL in the IB is a result of both, the dissociation of eGFP-Q73 from that protein aggregate and the subsequent photodestruction of the released eGFP-Q73 in the bleach ROI. Using the FL kinetics in the cytoplasm (see above) and the MC model, eGFP-mtHtt dissociation could be calculated to occur here with a half time of t_{1/2}= ln2/_{1} =73.8 sec. Another example of a cell with 2 IB’s of varying mobility is shown in Figure _{1/2}= 200.6 sec and t_{1/2}= 142.3 sec for the large and small IB, respectively). We tracked additional IB’s in the cytoplasm of several cells and found mean off- and on-rate constants for eGFP-Q73 of _{1} = 0.0127 ± 0.004 sec^{-1} and _{
−1} = 0.016 ± 0.006 sec^{-1} (n=6; mean ± SE). Thus, the average half-time of eGFP-Q73 dissociation from IB’s is 101.2 ± 30 sec in our experiments. The observed heterogeneity in exchange dynamics of eGFP-mtHtt between cytoplasm and IB’s is in line with earlier qualitative FLIP and FRAP experiments

Discussion

Fluorescence loss in photobleaching (FLIP) is a dynamic imaging technique with the potential to include spatial information into analysis of protein dynamics, but this potential has not been explored. By treating images as data arrays rather than pictures, we present two FLIP analysis methods for assessing intracellular transport dynamics of fluorescent proteins. Our first approach comprises fitting a StrExp function to FL kinetics on a pixel-by-pixel basis. The rationale behind this idea is that the time-dependent rate coefficient of the StrExp function is suitable to describe transport under conditions, where the “well-stirred compartment” assumption fails. Diffusion gradients or topological constraints cause deviation from the concept of compartment homogeneity, and that can be modelled with differential equations having time-dependent rate coefficients

For larger pauses between the bleaches, eventual hindrance to diffusion due to transient binding or barriers can be detected on the sub-cellular level by pixel-wise FLIP analysis without pre-selection of ROIs (Figures

**Movie 3.** Time evolution of rate coefficients forbarrier-limited FLIP experiment of eGFP.

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Using our quantitative FLIP analysis, we could also visualize local diffusion barriers for eGFP in the nucleus (see Figures

To the best of our knowledge, there is only one full publication published this year, which also applies a pixel-wise regression of a decay function to FL data of FLIP image sets

Quantitative FLIP microscopy should also be a method to compare results from different cells. This, however, is not possible with the pixel-based FLIP analysis, because the kinetic parameters recovered from the StrExp fit to data are a function of the total amount of fluorophore in a given cell, the diameter of the bleach spot and the length of the pauses between bleaches. The empirical StrExp fitting function only provides information about the nature of the observed transport process (e.g. diffusion- or binding-limited, see Table _{2}) in our MC model, but that recovered a mono-exponential decay with

Conclusions

We present two new approaches for quantitative analysis of FLIP experiments in living cells. Pixel-wise fitting of a StrExp function to FLIP image sets allows for detecting areas of different probe mobility, while physical modelling of FLIP data in the second method provides for the first time dissociation parameters of fluorescent proteins from moving entities. Our methods are easy to apply to other transport problems, where a fluorescent biomolecule is soluble in the nucleus or cytoplasm and binds to or partitions into static structures (for pixel-based FLIP quantification) or dynamic structures (for combined SPT and MC modeling). Typical other applications could be transient binding of ras-protein to the Golgi apparatus and plasma membrane

Methods

Reagents and cell culture

Fetal calf serum and DMEM were from GIBCO BRL (Life Technologies, Paisley, Scotland). 3,3,3’,3’-tetramethylindocarbocyanine perchlorate (DiIC12) was purchased from Molecular Probes (Eugene, Oregon, USA). All other chemicals were from Sigma Chemical (St. Louis, MO). Medium 1 contained 150 mM NaCl, 5 mM KCl, 1 mM CaCl_{2}, 1 mM MgCl_{2}, 5 mM glucose and 20 mM HEPES (pH 7.4). McArdle RH7777 (McA) cells expressing enhanced green fluorescent protein (EGFP) have been reported previously

Confocal laser scanning fluorescence microscopy

Confocal microscopy was performed using a laser scanning inverted fluorescence microscope (Zeiss LSM 510 META, Zeiss, Jena, Germany) equipped with a 63x 1.4 NA plan Apochromat water immersion objective and a 37°C temperature control (Zeiss, Jena, Germany). Fluorescein and eGFP fluorescence was collected with a 505–530 bandpass filter after excitation with a 25-milliwatt argon laser emitting at 488 nm. FLIP experiments were performed by first defining regions of interest (ROI), which were repeatedly bleached, while an image was acquired with reduced laser power (0.5% output) at the start of the experiment and after each bleach. Either one iteration (for eGFP-Huntingtin constructs in CHO cells) or five iterations (for eGFP in McA cells) with 100% laser power were used for the bleaching pulses. An eventual pause between the bleaches ensured some recovery in the ROI. Images were acquired using the time-lapse function of the Zeiss LSM510 Meta confocal system. The microscope was located at a nitrogen-floated table to prevent vibrations and focus drift and contained a temperature-controlled stage maintained at 35 ± 1°C. For spatial registration of image stacks, a plugin to ImageJ named “StackReg” developed by Dr. Thevenaz at the Biomedical Imaging Group, EPFL, Lausanne, Switzerland was used

Non-linear regression of a stretched exponential function to experimental and synthetic FLIP data

Pre- and post-bleach images were imported into ImageJ and combined into a single stack in 16-bit format. Image time series were smoothed with a Gaussian filter (standard deviation = 0.5) in the spatial and temporal domain. Fluorescence loss

using “PixBleach” our recently developed image fitting program implemented in ImageJ software (download at: _{0} is the pre-bleach intensity of the dynamic fluorescence pool, _{∞} is the remaining intensity at infinite time, resembling either an immobile fraction or autofluorescence of the cells (most often, _{∞} approximates the background noise level). The decay time constant

Compartment modeling of fluorophore transport in FLIP experiments

In the classical FLIP experiment, a pause between the intended local photobleaching ensures that emitting flurophores are transported into the bleached area such that the recovery process is not diffusion-limited. In this case and if transient binding events take place, the FLIP experiment can be modeled by a set of ordinary differential equations (see below). The simplest model for transport and bleaching contains two compartments; (1) the region outside the bleached area from which transport to the bleached ROI occurs with the amount of fluorophores, _{1}(_{2}(

Here, _{
F
} is the quantum yield of the fluorophore, ^{-1} cm^{-1}) and

We consider bidirectional transport of fluorophores between compartment 1 and 2 with the forward and backward rate constants _{1} and _{−1}, respectively. The bleaching process within the repeatedly bleached area (compartment 2) is modeled by the rate constant _{2}.

This gives the following system of differential equations:

The time-dependent solutions for both compartments calculated from the eigenvalues λ_{1} and λ_{2} and corresponding eigenvectors of the associated kinetic matrix are:

with

Here, the initial values _{1}
^{0} and _{2}
^{0} describe the initial amounts outside and inside the bleached region, respectively. We have eliminated _{−1} by detailed balance, _{1} · _{1}
^{0} = _{− 1} · _{2}
^{0}. This kinetic compartment model was simulated with varying parameter combinations as function of time using SigmaPlot 9.0 (SPSS Inc, Chicago, IL, USA) or as function of time and pixel coordinates using self-programmed Macros in ImageJ. An extension of that model to an arbitrary number of non-bleached compartments 1 is given in Appendix 2.

Analytical model of FLIP experiments with spatially invariant diffusion coefficient

A FLIP experiment without pause between the individual bleaches results in diffusion-limited transport of fluorophores,

with the boundary conditions that _{1} and _{2.} The solution of this model is given in Appendix 1.

Numerical simulation of FLIP experiments with spatially heterogeneous diffusion

It has been reported that diffusion coefficients can depend on the position within the cells (i.e.,

Here,

is easily implemented in FEniCS (_{2} = 1μm. A circular bleached area of radius with _{1} = 0.5 μm was placed in the disk center. The bleaching rate is set to k = 10 sec^{-1} and the diffusion coefficient is d_{L} = 0.2 μm^{2}/sec and d_{R} = 0.8 μm^{2}/sec on the left and right half circle, respectively.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

Conceived and designed experiments: DW. Performed experiments: DW, LMS, FWL. Analyzed experimental data: DW, FWL. Developed mathematical models and performed simulations: DW, DS, HJS, MAL. Performed non-linear regression to simulated data sets: DW. Wrote the paper: DW (with comments and criticism from all authors). All authors read and approved the final manuscript.

Appendix

Appendix 1

The diffusion problem in Equation 6 reads in Laplace space (_{0} is the initial density and

_{1}

for _{1} where _{
n
},_{
n
}represent modified Bessel functions of the first and second kind, respectively of order _{1},_{2}, _{2} are defined from the boundary conditions (see main text and Eq. 6). We get

where

Appendix 2

A multi-compartment model for binding-limited FLIP experiments can be described by a scheme like

where

_{2}. This gives for FL in compartment 2:

_{1}
^{i}(t) is assumed to satisfy

with the initial amount in the ith compartment of type 1 denoted as N_{1}
^{i,0}, we obtain the following solution

_{2}(t) was fitted to FL in the cytoplasm. The solution for the N_{1}
^{i}(t) was fitted separately to each IB per cell using the value of _{2} obtained from the fit of N_{2}(t) to FL in the cytoplasm, such that the first IB in a cell gives k_{1}
^{1} and N_{1}
^{1}(t), the second k_{1}
^{2} and N_{1}
^{2}(t), and so on.

Acknowledgements

DW acknowledges funding by grants of the Lundbeck Foundation, the Novo Nordisk Foundation and the Danish Research Foundations FNU and FSS. We are grateful to Tanja Christensen for expert technical assistance. Dr. E. Snapp (Department of Anatomy and Structural Biology, Albert Einstein College of Medicine, Bronx, New York, USA) is acknowledged for kindly providing plasmids for mtHtt-constructs used in this study.