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Analysis of Molecular Mobility by Fluorescence Recovery After
Photobleaching in Living Cells
C. Klein1, 2, 3, 5 and F. Waharte4, 5,
1
Centre d'Imagerie Cellulaire et de Cytomètrie (CICC), Centre de Recherche des Cordeliers, Université Pierre et Marie
Curie – Paris6, UMR S 872, Paris, F-75006 France.
2
Université Paris Descartes, UMR S 872, Paris, F-75006 France.
3
INSERM, U872, Paris, F-75006 France.
4
Cell and Tissue Imaging Facility (PICT-IBiSA), Institut Curie-CNRS UMR144, Paris, France.
5
RTmfm (Multidimensional Fluorescence Microscopy Technological Network), MRCT - UPS 2274 CNRS, France.
The development of fluorescent probes and photomodulable fluorescent proteins was a revolution in cell biology, as it
enabled the visualization and tracking of proteins in live cells by time-lapse fluorescence microscopy. Beyond simple
fluorescence imaging, the need for more advanced techniques arose due to the emergence of in vivo experiments.
It is case of FRET (Förster Resonance Energy Transfer) imaging, intensively developed in the last ten years, which helps
to get information about protein-protein interactions. Other techniques address the determination of the molecular mobility
from which interaction properties can also be deduced. Single Particle Tracking (SPT), Fluorescence Correlation
Spectroscopy (FCS) and Fluorescence Recovery After Photobleaching (FRAP)/Photoactivation are some of those.
Established in the seventies, FRAP is a method to monitor the lateral mobility of fluorescently labelled membrane
(proteins or lipids) in cell membranes. Briefly, the steady state distribution of membrane molecules is perturbed by
irreversible photobleaching of a small volume within the cell. The relaxation of the perturbation has a characteristic time
that is dependent on different factors, such as diffusion, flow and/or association/dissociation kinetics of multi-protein
complexes. FRAP is currently used to analyze the mobility and diffusion of molecules within or between cell
compartments. The technique is commonly used in conjunction with green fluorescent protein (GFP) fusion proteins.
FRAP became a very popular method in the mid nineties with the availability of fluorescent proteins and confocal laser
scanning microscopes (CLSM). Lately, the method gained further interest with the development of new technologies such
as widefield or spinning-disk confocal microscopes equipped with photo-perturbation modules combining lasers diodes
and galvanometric mirrors. These new modules overcome the limiting speed of acquisition of the CLSM. Numerous
biophysical models are used to analyse FRAP data, their number and their complexity can be confusing for biologists and
may appear as a major obstacle to initiate such quantitative approaches. Here we describe FRAP principles and explain
technical issues and limitations. We also review a few of the most relevant mathematical models to analyse FRAP data in a
quantitative manner.
Keywords FRAP; photobleaching; confocal laser scanning microscopy; molecular mobility; diffusion; molecular
interactions; biophysical models.
1. Introduction
Molecular mobility is an important parameter in the understanding of cell physiology, as it participates in the regulation
of numerous biological processes such as intra- and inter-cellular signalling. It can regulate the amplitude and the time
span of cellular signals. As mobility is dependent on molecular interactions and diffusion, measuring mobility is also a
way to evaluate molecular interactions. A limited number of methods are available to analyse molecular mobility,
namely: Single Particle Tracking (SPT), Fluorescence Correlation Spectroscopy (FCS) and Fluorescence Recovery
After Photobleaching (FRAP) or related techniques such as photoactivation using photoactivable GFP (PA-GFP).
Single Particle Tracking or Single Molecule Tracking is the most informative and has the best resolution as it follows
by time-lapse microscopy the trajectories of single molecules, enabling the detection of individual events, such as
space-dependent variations of the mobility behaviour and confinement. However the use of GFP is uneasy with this
technique because of the low photo-stability of fluorescent proteins. Thus it is mainly applied to the study of
extracellular membrane molecules labelled with organic dyes or Quantum dots. Moreover, SPT is a time consuming
method, because it requires the tracking and analysis of many individual molecular trajectories [1, 2].
In contrast to SPT, FCS and FRAP give bulk measurement of the average mobility of many molecules in a small
volume or area. In FCS, fluorescence fluctuations caused by molecules crossing an illuminated confocal volume are
measured for some time and the autocorrelation of the recorded signal is computed. Diffusion coefficients and
concentrations of different species can be obtained using biophysical models applied to the experimental autocorrelation
curves, even for fast diffusing species. However, this implies a dedicated instrument and very specific expertise in the
technique to really exploit the results [3-5].
FRAP or PA (Photo-Activation) [6-8] experiments are simpler and widely accessible on a variety of microscopes, in
particular on confocal laser scanning microscopes. They are in some way similar to pulse chase experiments. The
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principle of FRAP is to switch off, by a photophysical process called photobleaching, the fluorescence of a labelled
molecule, in a small region of the sample. GFP fused proteins are generally used in the case of living cells. Then the
mobility of the fluorescent molecule is evaluated from the recovery of fluorescence due to the exchange between
photobleached and intact molecules. In the case of PA, proteins are fused to the PA-GFP (or another photo-switchable
protein). If the photoactivated molecules are mobile, they will redistribute to the whole available volume. For both
methods, after proper normalisation and correction of the data, the parameters of the molecular mobility, like the
diffusion coefficient, can be determined by non-linear least square fit of a mathematical model to the kinetics of the
fluorescence redistribution.
Here we will focus on the FRAP technique but the PA shares the same theoretical background. We will first describe
the principle of FRAP and how to quantify and to interpret the fluorescence recovery curves. Then we will discuss how
acquisitions should be done in order to get data that can be used for quantitative analysis. Finally, we will present a few
mathematical models that may be used for data quantification and interpretation.
2. Principles of FRAP
The technique is composed of three phases (see Fig. 1):
- During the prebleach phase, fluorescence is simply monitored (Fig.1 first image on the left), giving the initial
intensity before photobleaching Fi (see Fig. 2);
- During the photobleaching phase (Fig.1 second image on the left), intense excitation light is applied to a spot or a
region of the sample in order to suppress a variable fraction of the fluorescence (depending on the laser power, the
stability of the fluorophore and the photobleaching phase duration);
- During the postbleach phase, fluorescence is again monitored to follow the redistribution of fluorescence from F0
to F∞, at low excitation to avoid observational photobleaching.
a)
b)
Fig. 1 Image sequences showing the principle of the FRAP technique. a) Bleaching of the nucleus of a cell expressing GFP. The
quantification of the nuclear envelope permeability is obtained by analysing the recovery kinetics. b) Mobility of histone H1-GFP
measured by FRAP. The bleached region is a strip across the nucleus.
On Fig.1, the FRAP technique with photobleaching in a region with 2D images acquired over time is shown. This is
not the only way of doing FRAP.
One alternative is to target a single spot, allowing shorter photobleaching phase and higher acquisition frequency.
Only the fluorescence intensity is recorded and no images are made [7, 9]. In a second alternative, images are recorded
and photobleaching is done on one or several spots. In a third alternative, 3D image acquisition is performed to get the
total fluorescence intensity at every time point for a better quantification.
Different shapes for the photobleached area can be used: a disk, a line, a strip (band) or more complex patterns (fringes
or arbitrary shapes).
FRAP gives qualitative information about the mobility of the molecules: do they move, at least at the time scale of
the experiment? Are they confined in a cellular compartment (or a cell itself) or do they exchange, for example between
the nucleus and the cytosol as seen in Fig. 1a? Is the redistribution of fluorescence isotropic or not? The answer to these
questions can already bring some relevant and useful biological information.
Quantitative information can also be obtained from the analysis of fluorescence intensity. Indeed, the variation of the
fluorescence intensity can be measured in a region of interest and can thereafter be analysed and fitted to a model in
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order to extract parameters describing the mobility of the protein. From now, we will focus on this quantitative
approach.
a)
b)
Fig. 2 Typical fluorescence recovery curves in a FRAP experiment. a) Fi, the initial fluorescence is the intensity in bleached region
before photobleaching. Fo is the fluorescence at t0, i.e. immediately after photobleaching. F∞ is the fluorescence intensity after full
recovery. F∞ might be inferior to Fi revealing a pool of immobile molecules. In this case, the mobile fraction M, is thus inferior to
100%. b) Time evolution of the fluorescence profile across a uniform disk bleached region (3µm in diameter) for t/τD = 0.1, 0.5, 1, 2,
5. The gradient of fluorescence attenuates as bleached and fluorescent molecules redistributes by diffusion between the bleached
region and adjacent areas.
Biophysical models describing the redistribution of fluorescence over time highly depend on the sample considered.
However, the fundamental processes are: diffusion, convection (or flow) and reaction (association/dissociation). In most
of the cases, the molecular mobility is dependent on the diffusion which is a physical process describing the trend of
molecules or particles to homogenise their concentration gradient. The coefficient of lateral diffusion (D) characterises
the diffusion of one molecular species in one particular set of conditions (for example temperature or viscosity).
The Stokes-Einstein relation indicates that D is proportional to k the Boltzmann constant, and T the temperature in
Kelvin. D is also inversely proportional to η the viscosity and Rh the hydrodynamic radius of the molecule:
D=
kT
.
6πη Rh
We can notice here that if molecules are assumed to be spheres of volume V=πRh3, and to have same densities, the
diffusion coefficient is proportional to the inverse of the cube root of molecular mass. Thus the diffusion coefficient of a
dimer should be 0.8 times the monomer's D and a tetramer's D should be 0.6 times the monomer's D.
Most of the models used to analyse FRAP experiment will depend on a time constant τD determined by fitting a
mathematical model on the experimental data from which D will be obtained:
w2 ,
D=
2 n τD
where w is the radius of the bleached area and n the spatial dimension, so for 2D diffusion: D =
w2
.
4 τD
It is of common use to determine from the fluorescence recovery curve the half time of recovery (t1/2). This parameter
has in general no relation to the characteristic time of diffusion (τD) and thus has no physical meaning. Only for some
models there is a simple relation between τD and t1/2, which can help to estimate the diffusion coefficient. We however
do not recommend using this parameter for FRAP quantification.
From the analyzed recovery curves, several parameters can be measured, namely F0 the fluorescence intensity
immediately after the photobleaching, F∞ the fluorescence intensity after full recovery and Fi the initial fluorescence
intensity before photobleaching. From these parameters, the percentage of bleaching and the percentage of recovery can
be calculated.
The percentage of bleaching, also named photobleaching depth, is mainly an index indicating the reproducibility of
the experiment and the degree of depletion of fluorescence in the chosen region. This parameter depends also on the
diffusion coefficient, even if it is not convenient to use it directly to determine the D value. It can be calculated using
the following formula:
B=
774
Fi − F0
.
F0
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If this percentage is too high (>80%), the Gaussian approximation of the photobleached region shape (for single-spot
FRAP) is not valid anymore [7]. This can limit the use of several common mathematical models. However, this can be
handled in other analysis methods.
The percentage of recovery, also named mobile fraction, has a more direct biological signification. Indeed the
mobile fraction (M) or more precisely 1-M, the immobile fraction, when it is strictly positive indicates that a fraction of
the pool of molecules is not mobile (on the time scale of the experiment) and thus may be retained by interacting to an
immobile ligand for example. It can be calculated using the following formula:
F ∞ − F0
M=
.
Fi − F0
Moreover, measuring the mobile fraction for different radius of the photobleached area is a method used to
demonstrate the existence of membrane domains where some molecules are retained and isolated from the membrane
phase. If the diameter of the photobleached region is small in comparison to the size of these domains the mobile
fraction is high, while when it is larger, the mobile fraction becomes low because the molecules can only freely diffuse
inside the domains and no exchange is possible between them or with the membrane phase [10, 11].
Care must be taken in estimating the mobile fraction as it depends on the acquisition conditions (if full recovery is
reached or not) and may be reduced if observational photobleaching is present (see Fig. 3).
Estimated value of M depends also on the total intensity decrease due to the photobleaching phase. Indeed, in a
closed system like an isolated cell, if the fraction of photobleached molecules is more than a few percents of the total
initial number of fluorescent molecules, the maximal value that the intensity can reach will be below 100 % even if all
molecules are fully mobile. This can be avoided by carefully setting the acquisition parameters (in particular using a
small photobleached volume compared to the whole volume occupied by the fluorescent molecules) and by data
correction and normalisation after the acquisition (see section 3.3).
Fig. 3 Effect of the mobile fraction on the fluorescence recovery. Curves for M=1 show the effect of the observational
photobleaching (increasing value from the continuous line to the small dashed line).
3. Data acquisition and normalisation
3.1Acquisition
Several parameters have to be carefully optimized during a FRAP experiment:
- Illumination during image acquisition has to be as low as possible to avoid observational photobleaching. This
process will tend to lower the recovery giving erroneous values for τD and mobile fraction.
- The bleach duration has to be as short as possible, because molecules diffuse during the bleaching process.
Otherwise it will cause bias in the analysis. This process is sometimes called the corona effect [12]. This can forbid the
use of many mathematical models for data analysis (see below).
- The rate of image acquisition must be relatively high in regard to τD, in order to acquire a sufficient number of
points during the first part of the recovery. The acquisition speed can be increased by using a crop of the full image or
pixel binning on a CCD camera, at the expend of spatial resolution. Empirically, a frequency of 1/(10 τD) should be
sufficient. However this frequency can be reduced if desired during the second part of the recovery, to reduce
observational photobleaching.
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- The duration of the acquisition must be of around 7 to 10 τD in order to have a good determination of the
different parameters of the fit, in particular F∞.
- The beam radius should be large enough to get a reasonable signal to noise ratio but kept small compared to the
size of the system studied to avoid border effects on the recovery (i.e. influence of the cell walls on the molecular
mobility by reflection) and to limit the total loss of fluorescence.
- The choice of the microscope objective is also of importance while considering the geometry of the
photobleached volume and the geometry of the sample itself. It may be useful to choose a low magnification/NA
objective for sample thicker than ~ 10 µm because the photobleached volume can then be considered as a cylinder
crossing the sample on its whole depth. Thus the diffusion can be considered to be 2D and this simplifies the further
analysis.
- On contrary, the use of two-photon excitation allows to photobleach the molecules in a well-defined volume for
3D diffusion measurements and facilitates further modelling of the fluorescence recovery.
FRAP measurement of highly mobile molecules will be limited by the maximum acquisition frequency of the
instrument used because τD depends on w² and D and furthermore an acquisition frequency of at least 1/(10 τD) is
needed (for example a frequency of 10 frames/s allows to measure a maximum value of D ~ 1 µm2.s-1).
When
the
maximum imaging rate is reached, the only solution is to increase the radius of the bleached region (w) at the cost of
decreased spatial resolution (see Fig. 4). However, problems can arise in the case of a closed system like a cell, because
of border effects and increased ratio of photobleached molecules over total fluorescent molecules. A larger
photobleached region may also lead to strong variations in the fluorescence recovery because of the heterogeneity of the
sample that cannot be avoided.
Fig. 4 Frame rate as function of the coefficient of diffusion of the molecule studied for different photobleached region diameter.
Optimal imaging rate is fixed at the empirical value of 0.1 τD = 0.1 w²/4D. The frame rate limits the highest values of D measurable.
As an example, for a photobleached region of 4µm in diameter, if D is superior to 1 µm².s-1, imaging rate should be at least 10
frames/s (100 ms/frame). Signal to Noise Ratio (SNR) was not considered here, but poor SNR will limit further the highest values of
D quantifiable with a good precision.
Table 1 indicates typical values of the diffusion coefficient for different types of molecules that can be encountered
in a biological sample. This large range of values implies that experimental conditions have to be carefully adjusted to
each case as well as the type of FRAP experiment (single spot, strip…).
Table 1 Molecular mobility of typical biological molecules
776
Molecular species
D (µm².s-1)
Freely diffusing protein
10-100
Membrane proteins
0.1-0.001
Membrane lipids
0.1-1
Nuclear proteins
10-0.01
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3.2 Artefacts
During the acquisition of the fluorescent signal, several phenomena can induce unwanted intensity variations. We list
here some of them:
- Laser power fluctuations
- Movements of the whole sample: z drift, cell migration or shape change.
- Instrumental response to laser pulse: PMT blinding [13].
- Reversible photobleaching [14].
- Non-constant number of fluorescent molecules caused by exchange with other cells or neosynthesis.
Table 2 summarises the potential artefacts that must be carefully checked and solved if possible in order to obtain data
that can be quantified and interpreted.
Table 2 Some potential artefacts in FRAP experiments
Experimental problem
How to control/resolve the problem
Laser power fluctuations
Movements of the sample
Instrumental response to the laser bleach pulse
Reversible photobleaching
Measure the baseline and correct data
Use focus stabilization or autofocus, image registration
Check with fixed sample or whole cell photobleaching
Check with fixed sample or increase the bleached region size
3.3 Data normalization and correction
Before analysis, data have to be normalized and corrected in order to compute average values or compare and classify
fluorescence recovery curves. As carefully the experiments are performed, photobleaching can still occur during the
acquisition. Furthermore the extent of the photobleaching although limited to a small region of the cell, can become non
negligible in regard to the total pool of mobile molecules.
There are several ways to correct this. First an internal correction [15] can be made by measuring the total
fluorescence (I0) in the region of interest (ROI) as well as in the whole cell (T0) before photobleaching and during the
recovery (resp. It and Tt). The ratio It/Tt will correct for the observational photobleaching while the ratio T0/Tt will
correct for total loss of molecules. Data are then corrected and prebleach acquisition normalized to 1 by calculating F(t)
= T0 It /Tt I0
Alternatively, a second acquisition can be made after full recovery with the same acquisition conditions to measure
the observational photobleaching. By calculating the ratio of the total intensity of these two acquisitions, the
observational photobleaching will be corrected [13]. In this case the loss of fluorescent molecules is not corrected and
thus the photobleached area must be as small as possible.
Other methods exist, using for example sample immobilised by chemical fixation to estimate the photobleaching rate.
This parameter is introduced in the mathematical model for data fitting [9].
When performing 3D acquisitions over time, it is possible to fully correct the observational photobleaching since the
total fluorescence intensity is measured [16]. This method should be favoured when possible.
4. Models
Once data are corrected they have to be fitted to a model, which is often a solution of the diffusion equation or of a set
of coupled diffusion-reaction equations. Numerous models can be found in the literature and we do not intend here to be
exhaustive but to discuss a few relevant models. The choice of the suitable model is usually the main obstruction to
newcomers in the field. First, models can be separated by the biophysical processes considered: models that deal only
with diffusion or that consider other processes such as molecular interactions. Then a second classification can be made
in regard to the geometry of the bleached region (spot, disk, line, strip). Finally models can be classified as analytical
models or numerical ones. It is also important to bear in mind that for most of the models several assumptions are often
made to simplify the problem:
•Molecules are mobile in a domain of infinite size.
•The total amount of molecules does not vary during the time of the experiment.
•The bleaching is short so that the diffusion occurring during the bleaching is negligible.
•The observational photobleaching is negligible or is introduced as a parameter of the model.
We propose here a decision tree that may help to choose the most appropriate type of model to analyze data with
example of published models (Fig. 5).
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Fig. 5 Decision tree to choose the appropriate analytical mathematical model for FRAP experiments analysis. This is not an
exhaustive list of models, see text for additional references.
4.1
Analytical models
The analytical models are based on a solution (often an approximation) of the equation governing the evolution of the
system studied (e.g. diffusion equation). They provide an expression of the fluorescence as a function of time for the
post-photobleaching phase that can be used directly to fit the experimental data. This is a big advantage of this approach
and makes the estimation of the parameters of the model quite straightforward. The main limitation is that the
expression is valid only under precise hypothesis and geometrical consideration (photobleached volume and sample)
and may not apply easily to the various experimental conditions. They thus must be used with great care and hypothesis
must be checked for validity, which can sometimes be a difficult task.
4.1.1
Diffusion
When the number of fluorescent molecules in the sample is high enough to make the assumption of a continuous
medium, a molecular concentration can be defined and diffusion can be described by a partial differential equation. The
diffusion equation in 2D can be written for pure, isotropic diffusion in a homogeneous medium, as:
 ∂ 2C(x, y, t) ∂ 2C(x, y, t) 
∂C(x, y, t)

= D
+
∂t
∂x 2
∂y 2


Solutions of this equation for a particular set of initial conditions give a function of space and time C(x,y,t) describing
the variation of the fluorescent molecules concentration. Models used for FRAP analysis are usually the integral of one
particular solution on the region of photobleaching. Thus the geometry of the photobleached region is important in the
design of the experiment and the choice of the model used to analyze data.
•Gaussian initial profile
Axelrod et al. [7] described an analytical solution of the 2D diffusion equation in the case of an initial gaussian
profile obtained by focussing the laser in a single diffraction limited spot. It is the case for example in spot FRAP on a
CLSM. This solution is then integrated over the radius of the bleached region and gives the normalized fluorescence
recovery function:
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



∞
1 − e −K
F (t ) − F0
1
(−K) n 

f (t) = ∑
, with the normalization f ( t ) =
and F0 = Fi
.
 2t  
n! 
K
F ∞ − F0
n =0
1+ n1+  
 τD 

w2
Then the diffusion coefficient can be computed using D =
.
4τ D
Practically, only the first terms of the series in the expression of f(t) need to be computed and this model can be
easily implemented in any scientific spreadsheet or data analysis software.
Its main drawback is the evaluation of the parameter K, which is non-linearly related to the bleach ratio F0/Fi. Those
parameters as well as w the region radius are difficult to evaluate because at the first time point after bleaching,
diffusion may have occurred, broadening the profile. This is a recurrent problem in analysis of FRAP experiment, but
this method is especially sensitive to it. This model is thus often not suitable for analysis because of experimental
conditions necessary to get a sufficient photobleaching depth, which lead to long photobleaching phase compared to the
characteristic time of diffusion.
Braga et al. [17] have proposed an interesting extension of the model that takes into account diffusion during the
photobleaching phase by measuring the actual intensity profile after photobleaching. They also propose a model for 3D
diffusion including the same correction.
Further development of this method for 3D diffusion can also be found in other studies [18, 19]. However, in most of
the cases, because of the extension of the photobleaching in the z axis, diffusion along the z axis can be neglected and
2D model such as Axelrod's can reasonably be used. It must be pointed out that in this case the analysis should be made
on optical sections focussed at the same level than the laser used to photobleach. This approximation does not hold
anymore in multiphoton excitation where the photobleaching is restricted to the vicinity of the focal plane. In this case a
3D model must be used.
An alternative to the use of integral model is to measure the width of the fluorescence profile during the recovery as
it is related to D [20, 21]. One advantage of this method is to avoid the determination of t0 (start of recovery), critical for
some models, since it deals only with the postbleach phase and relies on the extraction of intensity profile. The
evolution of the profile width over time allows the estimation of the diffusion coefficient and even the dimensionality of
diffusion (1, 2 or 3D). However this requires more computations, as profiles at each time step of the recovery have to be
fitted to a Gaussian function. This approach is very elegant and should see further developments in a near future
because it gives access to spatial information and not only the fluorescence recovery signal. It completely addresses the
problem of diffusion during photobleaching since it considers the real experimental intensity profile after
photobleaching.
•Uniform disk bleaching
Another situation is obtained by defocusing the bleaching laser when it is possible with the equipment used or by
scanning a relatively large circular region (at least 3µm in diameter) with a confocal laser scanning microscope
(CLSM). In this case the photobleaching intensity is theoretically uniform or quasi uniform with Gaussian edges. The
normalized integral of the 2D solution of diffusion equation is then [22]:
τD
  2τ D 
 2τ D 
 I 0  t  + I1  t  , with I0, I1 being the Bessel functions.



 
w2
Then, as before, the diffusion coefficient can be computed using D =
.
4τ D
f (t) = e
−2
t
It must be noticed that this model does not depend on the photobleaching depth factor (K), and thus is less sensible to
experimental bias.
The spatial resolution of this method is of course not as good as in the spot photobleaching method because of the
size of the photobleached area. However this method is interesting for FRAP by CLSM, because the region is
sufficiently large and give comparatively large values of τD that can be sampled optimally despite the low speed of
CLSM [23, 24].
•Strip bleaching
The last common way to perform FRAP experiment is the strip photobleaching method. In this case, the bleached
region covers the full width of the cell, thus diffusion along x and z axis is negligible and mobility can be reduced to 1D
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diffusion along y axis. This method is easily implemented on a CLSM. Exact solutions for the time course of integrated
fluorescence can be found in [25, 26, 13] and an approximated solution is given in [27, 28].
f (t) = 1 −
w2
.
w2 + 4πD
(
)
This last model is quite appealing because of the simplicity of its mathematical expression. However as it is based on
the approximation of an infinite domain and as bleached strip tend to be large in regard to the cell size, border effects
can arise. In addition special care of phototoxicity must be taken because of the large size of the bleached region.
Mathematical solutions are given in [25, 26] depending on the domain size. Their solutions require the absence of
immobile fraction. Hinow et al. & Mueller et al. [26, 13] recommend to perform repetitive bleaching to check for this.
When a high acquisition frequency is needed, it is possible to reduce the size of the strip to a single line. Braeckmans et
al. have established a model for line-FRAP experiments when diffusion is the main process [29].
4.1.2
Diffusion-reaction
When fully simplified, molecular interactions can be taken into account in FRAP experiment modelling. Usually, the
problem is reduced to a single diffusing species (F) interacting with immobile binding sites (S) to form a complex (C).
Such models are especially used in the context of nuclear proteins mobility.
In summary, we have the reaction: F + S ⇔ C, with: kon [F][S] = koff [C] at the equilibrium.
The problem can then be expressed in form of a set of coupled equations:
 ∂f
2
*
 ∂t = D f ∇ f − k on f + k off c

 ∂c = k * f − k c
on
off
 ∂t
, where kon* = kon S , as S is constant.
Sprague et al. [30, 31] clearly established that for disk bleaching four domains of mobility can exist depending on kon
and koff values:
•Free diffusion: binding is negligible if kon is very low and koff is very high, the mobility depends only on diffusion
and the Soumpasis model can be used.
•Reaction dependent: diffusion does not contribute at all to the mobility. Mobility depends only on exchange
reaction of bleached/fluorescent molecules on their ligand. It must be pointed out that recovery time will be the
same for different areas of bleaching and that the gradient of fluorescence will remain constant while it tends
to become homogeneous when diffusion occurs. Exponential association models can be used. However this
situation should be rather rare in living cells [32] and it seems that diffusion will always have an impact on
mobility even if recovery is apparently slow.
•Effective diffusion: it is when the exchange of molecules on their ligand is fast in comparison to the diffusion
process and can be considered to be at steady state (instantaneous reaction). Thus mobility is diffusion
dependent but limited to the pool of free molecules. Diffusion model can be used but D will be slower than its
theoretical value. The measured D is called Deff (effective diffusion coefficient). This process is also called
slowered diffusion.
•Reaction-diffusion: finally it exists a range of kon and koff values where diffusion and reaction have similar
characteristic times and no simplification can be made. Reaction-diffusion models must then be used [30].
Reaction-diffusion models have also been described in the case of strip bleaching experiments [25, 26, 13] and spot
bleaching [39] with similar observations.
4.2
Numerical models
When it is not possible to solve analytically the evolution equation of the system, it is often possible to do it using
numerical methods. They do not provide an expression that can be used directly for data fitting, but a computed curve
that can be compared to the experimental one. Using optimization algorithms, it is possible to vary the parameters of the
model to iteratively calculate the numerical solution and find the curve fitting the best the experimental data.
The major advantage of numerical methods is that they do not need fixed initial conditions. In contrast to analytical
methods, initial conditions can be measured from experimental data without bias and evolution of the profile can be
precisely computed. Complex situations with irregular bleaching geometry can be solved. Conditions in the boundaries
(flux, no flux, sink, sources...) can be applied as well. This is convenient to simulate for example reaction between
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Microscopy: Science, Technology, Applications and Education
A. Méndez-Vilas and J. Díaz (Eds.)
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membrane receptors and a pool of cytoplasmic molecules [12]. Also, the geometry of the sample can be taken into
example), which is usually not possible in analytical approaches.
Solution of the evolution equation has to be calculated for each time point for given initial conditions by numerical
methods. This computation can be done mainly by two ways: finite differences or finite elements.
•Finite differences
Values of the concentration of fluorescent molecules after photobleaching at time step t + dt is computed from the
values at t according to the finite difference form of the evolution equation. For example, for 1D diffusion it can be
written as follows:
 C t − 2Cxt + Cxt −1 
Cxt +1 − Cxt
= D x +1
.
∆t
∆x 2


Rearranging this equation gives the compact formulation:
Cxt +1 = λCxt +1 + (1 − 2 λ )Cxt + λCxt −1 , where λ =
D ∆t
,
∆x 2
which is the so-called explicit scheme for 1D diffusion. Refinements of this method are detailed elsewhere [33]. Finitedifference schemes can be implemented in mathematical software such as Matlab or Octave, and has been used to
analyze FRAP experiments [34, 35, 12, 32].
•Finite elements
Finite element is the method of choice to solve numerically problems involving a set of coupled partial differential
equations. It consists in dividing the geometry of the problem to be solved in simpler elements (for example triangles in
2D) and applying approximation of the equation on each element, often using a variational formulation. This method is
widespread in physics and engineering and has countless application in industrial research and development. Compared
to other methods its advantages are speed and the capacity to solve very large problems. It is also easier to implements
problems with complex boundary geometry such as image-based problems. The finite element method has been used to
model FRAP experiments in recent years [36-38] and will become for sure a standard for the analysis of FRAP or PA
experiment and more generally image based molecular and cellular dynamics problems. Finite elements methods can be
extended to 2D or 3D problems and reaction terms can be mixed as well [39].
4.3 FRAP simulations
When more complex situations have to be studied, it may be even not possible to write an evolution equation for the
biological system or to solve them. It is then necessary to use a different approach such as simulations.
A known example is the Monte-Carlo simulation method that is based on the following principle: trajectories of a
pool of single molecules are computed according to rules describing the dynamics of particles. Highly complex
behaviours can be implemented such as the existence of several pools of molecules with different affinities for a ligand,
membrane compartment versus cytosolic compartiments [40, 41].
The main limitations of these simulations are the high computational cost and the fact that fluorescence recovery
curves generated cannot usually be used for direct data fitting because of noise on the simulated data due to the inherent
random process.
Sbalzarini et al. [42] proposed an original approach based on the Particle Strength Exchange technique that allows
simulations on complex geometry (like the endoplasmic reticulum in this study) and produce more accurate simulated
curves. This method has two main advantages: it is not based on a grid for space discretization so it is more adapted to
complex geometries and it needs a small number of particles, so a much lower computational cost, that makes it usable
for quantitative FRAP analysis.
The flexibility of such approaches is very high and should bring more people to use it in the context of interpretation
of FRAP experiments. It can also be used in combination with other approaches to estimate the influence of a particular
process on the fluorescence recovery (like reactions) and guide the development of models and the interpretation of
data.
Acknowledgements. The authors would like to thank Dr S. Coscoy, H. Fohrer-Ting and E. Devêvre for critical reading of the
manuscript. Support from the MRCT- UPS 2274 CNRS is gratefully acknowledged.
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