2.1 Theory of diffusion

Diffusion, self-diffusion, convective flow, pressure-driven flow, etc. are examples of translational transport mechanisms of solvents or solutes. In the absence of external forces, the mechanism governing mass transport is most often diffusion, in particular self-diffusion, which may be considered the most fundamental form of mass transport since it is always present. The first observations of diffusion were made by Brown in the beginning of the 19th century (Brown, Reference Brown1828, Reference Brown1829) and diffusion was theoretically explained by Einstein and Smoluchowski at the beginning of the 20th century (Einstein, Reference Einstein1905; Smoluchowski, Reference Smoluchowski1906). As indicated above, the diffusive process may be divided into diffusion and self-diffusion mechanisms. The modern literature is not always clear about the difference, but here we define diffusion as the process where the total mass of a substance is spread by random motions. The process requires no extra energy other than thermal energy, i.e. it is a passive transport mechanism. Self-diffusion is also inherently a random motion displacement process and, like diffusion, requires no extra energy. It is however a process at thermodynamic equilibrium, i.e. it does not involve shifting the center of gravity of a material mass but rather displacing individual molecules randomly by collisions with surrounding molecules. The time-dependent displacement function is always centered at zero for self-diffusive processes (see Eq. (3) and below). This basic feature makes self-diffusion a very ineffective mass transport mechanism at macroscopic distances but highly effective at the nano or micrometer length scale, where it may be used to reveal intrinsic molecular as well as supramolecular and microstructural properties (Dill & Bromberg, Reference Dill and Bromberg2003; Hawlicka, Reference Hawlicka1995; Price, Reference Price2009).

Starting with the basic description of the diffusion process, we note that the driving force for the diffusive flux J is a gradient in chemical potential ϕ (e.g. temperature, T or concentration, C) that may depend on time t and position r (Chaikin & Lubensky, Reference Chaikin and Lubensky1995):

where D is the diffusion coefficient, which in the limit of thermodynamic equilibrium is a constant. By letting the spatial derivative operate in Eq. (1), and using the continuity equation and assuming that D is constant, it is found that (Crank, Reference Crank1975; Dill & Bromberg, Reference Dill and Bromberg2003):

This is referred to as Fick's second law (Fick, Reference Fick1855), which represents the fundamental differential equation for diffusion and self-diffusion. The solution to Eq. (2) for one-dimensional (1D) diffusion from a point source is (Crank, Reference Crank1975; Cussler, Reference Cussler1997):

where C ₀ is the concentration of solute located at position x = 0 for t = 0.

In the equations above, D represents both diffusion and self-diffusion coefficients. As mentioned, self-diffusion is a translational process at thermodynamic equilibrium, i.e. in the absence of a spatial chemical potential gradient. This fact makes the probability density function, P, be defined as:

a preferred parameter for describing diffusion behavior (Price, Reference Price2009). For the situation with free diffusion in one dimension, the self-diffusion coefficient D _s becomes a constant and Eqs. (3) and (4) combine to the probability density function (Green, Reference Green2005):

This is a Gaussian distribution that describes the probability density of finding a molecule at position x at time t after having started at position x = 0 at time t = 0. From expanding Eq. (5) it may be noted that the second moment is the mean-square displacement $\left\langle {x^2} \right\rangle $ , which is related to the diffusion coefficient by Lenk (Reference Lenk1986) and Dill & Bromberg (Reference Dill and Bromberg2003):

The famous Einstein relation (Einstein, Reference Einstein1905) then links D _s to the ‘mobility’, ν, of a particle:

where k is the Boltzmann constant and T the absolute temperature. The mobility is related to the inverse of the friction coefficient f (Green, Reference Green2005):

The friction coefficient in turn depends on the size and shape of the diffusing specie (Edward, Reference Edward1970) and, for a spherical particle significantly larger than the solvent molecules, Stokes law predicts that the friction coefficient is

where η is the macroscopic dynamic viscosity and r _H is the hydrodynamic radius (i.e. particle radius plus ‘bound’ water) of the diffusing particle (Hansen & McDonald, Reference Hansen and McDonald2006; Stokes, Reference Stokes1856). For other geometries and when the particle and solvent are of similar size, the relation is more complex (Cussler, Reference Cussler1997; Dill & Bromberg, Reference Dill and Bromberg2003; Hansen & McDonald, Reference Hansen and McDonald2006). From Eqs. (7)–(9), the famous Stokes–Einstein relationship is readily obtained:

which relates the self-diffusion constant to the particle hydrodynamic radius at infinite dilution. This well-known principle states that the self-diffusion constant decreases with increasing particle size and solvent viscosity, due to the increased friction between particle and solvent (Dill & Bromberg, Reference Dill and Bromberg2003).

For materials where the microstructure hinders molecules in their translational motion or where molecules bind to the interface in the material, the mean-square displacement may deviate from the linear time behavior in Eq. (6). These situations are most commonly referred to as anomalous diffusion, which results in a mean-square displacement that depends on time according to

where Γ is the transport factor or the so-called time-dependent diffusion coefficient (TDDC). This means that D _s in Eq. (6) is not constant but depends on the observation time. Notably, in the short-time limit, Γ equals the diffusion constant in the absence of effects by the material interface. Γ is constant in the long time limit as well, with a value that depends on the tortuosity of the material. If only thermal energy is exerted on the system, the mean-square displacement in the long time limit is always lower than for free diffusion. In that case, α < 1 (Heitjans & Kärger, Reference Heitjans and Kärger2005; Saxton, Reference Saxton2001), a situation often referred to as sub-diffusive motion. One example of this type of motion is a large polymer diffusing in a crowded polymer solution or gel when studied at short observation times (de Gennes, Reference de Gennes1992). Sub-diffusive behavior is also obtained when the diffusing species, solute or solvent, interacts with the microstructure of the materials. For a lucid account of these phenomena in biological systems, the reader is referred to Saxton (Reference Saxton1996). In certain circumstances, α may be larger than 1, which is the so-called super-diffusive regime. For this mechanism, extra energy in addition to thermal energy is needed, such as directed flow or active transport in cells.

2.2 Diffusion and binding

A special case of sub-diffusive motion is when the solute diffuses in the presence of a binding partner. Information about binding and unbinding events (in combination with diffusion) can be retrieved by FRAP experiments in which the solute is labeled with a fluorophore.

As schematically illustrated in Fig. 4, the binding reaction can be described by:

where S _f is the free fluorescently labeled solute molecule, B its unlabeled binding partner, C _f the (fluorescent) molecular complex between S _f and B, and k _on and k _off the on and off rates of binding. The overall fluorescent signal in a FRAP experiment will have a contribution from both fluorescently tagged species. The FRAP recovery therefore reflects the changing concentrations of the free solute C _S(t) = [S _f] and the solute-binding partner complex C _C(t) = [C _f] (but not the binding partner B, which is unlabeled):

Fig. 4.

Schematic representation of the binding process. A free, fluorescent molecule (green sphere) diffuses and eventually encounters a binding site (blue rectangle). The rate of encounter is given by the on rate k _on. The rate of dissociation is given by the off rate k _off. The binding site may itself diffuse or may be part of an immobilized substrate. In either case, the binding process retards the motion of the fluorescent molecule, and this can be detected and quantified in the FRAP.

In most FRAP-binding models, the binding reaction is assumed to be at equilibrium prior to the photobleach, which then perturbs the equilibrium concentrations of fluorescent S _f and C. The return to equilibrium is described by the following pair of coupled reaction–diffusion equations:

The D's represent the diffusion coefficient for each species, and $k_{{\rm on}}^{\rm *} $ is an association rate defined by the product of the on rate k _on and the equilibrium concentration C _S,eq of unbound molecules S( $k_{{\rm on}}^{\rm *} = k_{{\rm on}} C_{{\rm S,eq}} $ ).

The preceding equations are simplified in many FRAP-binding studies that have considered situations where the binding partner B is immobile on the time scale of the FRAP recovery. Setting D _C = 0 in Eq. (14) yields the FRAP equations for binding to an immobile substrate:

The solutions to either Eqs. (14) or (15) with appropriate initial and boundary conditions yield C _S(t) and C _C(t), which by Eq. (13) yields C _FRAP(t) in the presence of binding interactions. Fitting these equations to experimental FRAP data can yield estimates for the diffusion constant(s) and the binding rates. The latter are often used to calculate the equilibrium concentrations of free molecules, S _eq, and bound molecules, C _eq:

The binding rates are also used to calculate the average time, $t = {{\rm 1} / {k_{{\rm on}}^{\rm *}}} $ , it takes molecules to diffuse from one binding site to the next and the average time, t = 1/k _off, molecules remain bound at a site. Equation (15) has been modified to account for the case of either two independent binding states (Sprague et al., Reference Sprague, Pego, Stavreva and McNally2004) or n independent binding states (Sadegh & Montas, Reference Sadegh and Montas2010). Readers should consult these references for the specific differential equations and their solutions.

3.1 Introduction

The ability of fluorescence microscopy to detect specific molecules at very low concentrations, and in general to allow implementation of analytical methods such as FRAP, strongly depends on the performance of the fluorescent molecules being used. Fluorescent dyes featuring high extinction coefficients, high quantum yield and a large Stokes shift are now widely available. Fluorescence emitted by almost all fluorescent dyes fades during observation involving a photochemical modification of the dye and resulting in the irreversible loss of its ability to fluoresce. This phenomenon is commonly known as photobleaching or dye photolysis and is the result of the chemical interaction of an excited fluorophore with free oxygen or other biomolecules, such as lipids or proteins. Unfortunately, the details of the photobleaching mechanism following single or multi-photon excitation are not fully clear and some aspects still remain obscure. In this section, we will refer to photobleaching as an irreversible loss of fluorescence, although in some cases a fluorescent molecule can be switched on again. For instance, the commonly used Green Fluorescent Protein (GFP) variants (eGFP, eYFP, Citrine, etc.) exhibit reversible photobleaching and should be carefully tested for reversible photobleaching effects when used for FRAP experiments, as this can lead to artifacts (Daddysman & Fecko, Reference Daddysman and Fecko2013; Mueller et al. Reference Mueller, Morisaki, Mazza and McNally2012; Sinnecker et al. Reference Sinnecker, Voigt, Hellwig and Schaefer2005).

3.2 Photobleaching process

To get a better understanding of the photobleaching process in relation to FRAP experiments, it is instructive to consider the simplest fluorophore model involving the ground singlet state, S ₀, the excited singlet state, S ₁ and the triplet state, T ₁. When fluorescent molecules are irradiated with light of suitable energy, hν, a transition can occur from the ground energy level (S ₀) to the singlet excited energy level (S ₁), as illustrated in the Jablonski diagram in Fig. 5. Such a temporary excess of energy can be dissipated in several ways including by the emission of fluorescence, through radiation-less processes such as internal conversion and intersystem crossing to the excited triplet state (T ₁).

Fig. 5.

Jablonsky diagram of a fluorophore with a three-level system. After excitation from the ground state to the excited singlet state, the excess energy can be dissipated through several processes, each with a specific rate constant k. Photobleaching of the fluorophore mainly happens from the long-lived triplet state T ₁.

Initially, the molecules are predominantly in the ground state obeying the Boltzmann distribution law. As the excitation intensity is augmented, the ground state population decreases in favor of the excited states. The T ₁ characteristic decay time can range from micro to milliseconds, whereas S ₁ is characterized by a decay within the nanosecond range. Fluorophores in the excited state can easily react with other excited molecules or with free oxygen in the environment, which can lead to a non-fluorescent chemically modified product, i.e. photobleaching. Since the triplet state is a long-lived state compared with the singlet excited state, photobleaching is proportional to the number of fluorophores in the triplet state. It is therefore of interest to consider the kinetics of the triplet-state population during the highly intense photobleaching phase of a FRAP experiment. The set of differential rate equations according to the system in Fig. 5 is given by:

where k ₁, k ₂, k ₃ and k ₄ represent the rate constants of the transitions as indicated in Fig. 5. Evidently, k ₁ is a function of the illumination intensity. If the excitation light has an irradiance H (W m⁻²) and the fluorophore has an absorption cross-section σ _a (m² per molecule), then the rate of photon absorption is

where λ is the wavelength of the incident light, h is Planck's constant and c is the speed of light.

The steady-state condition according to Eq. (17), which is obtained when the population does not undergo any further change, is shown as a function of the irradiance in Fig. 6 for two different fluorophores, fluorescein and GFP. The steady-state solution of the triplet-state population is given by:

which approaches the constant value of k ₃/(k ₃ + k ₄) for high irradiance values. This means that, when the excitation intensity is increased, the triplet-state population increases until it saturates at a constant value. Since the amount of photobleaching is proportional to the number of fluorophores in the triplet level, an increase in the excitation intensity in the saturation regime does not result in faster photobleaching. As will be discussed in Section 3.4, this effect has important consequences for FRAP experiments. Note that, under saturation conditions, higher-order excited triplet states, T _n, can occur when two successive photon absorptions occur at high irradiance. It should also be remembered that many factors, such as the molecular environment, may affect the photobleaching mechanism, and thus the reaction order and the rates of photobleaching.

Fig. 6.

Molecules in S ₀, S ₁ and T ₁ as a function of the excitation intensity: case of fluorescein and GFP. The behavior for T ₁ shows that, under saturation conditions, an increased excitation intensity no longer corresponds to an increased triplet-state population. Inspired by Diaspro (Reference Diaspro2010).

3.3 Photobleaching in FRAP modeling

It is of critical importance for most FRAP models to have accurate knowledge of the concentration distribution of fluorophores after the photobleaching step (cfr. Section 4). However, due to the complexity of the 3D intensity distribution of the excitation beam and the photobleaching mechanism, some simplifications are typically introduced in order to arrive at a closed form solution of the recovery process. Here we will discuss the implications of these frequently used simplifications.

In FRAP models, the fluorescence decay by photobleaching is commonly assumed to be an irreversible first-order reaction:

Let C be the concentration of fluorophores and I _b(r) the photobleaching illumination intensity distribution in the sample. The decay in fluorescence according to Eq. (20) is described by

leading to an exponential decay of the fluorophore concentration with time t:

where C ₀ is the homogeneous fluorophore concentration before photobleaching and α is a fluorophore-dependent photobleaching parameter such that αI _b( r )is the rate constant of the first-order reaction in Eq. (20). The photobleaching parameter can also be written as α = σq, where σ is the photon absorption cross-section and q the photobleaching quantum efficiency, which is the probability that a fluorophore molecule becomes photobleached after absorption of an incoming photon. Similarly, the photobleaching process under two-photon excitations can be represented as (Patterson & Piston, Reference Patterson and Piston2000):

where α _2p is the two-photon bleach rate and $\left\langle {I_{\rm b}^{\rm 2}} \right\rangle $ the time-average bleaching intensity distribution. It should be noted, however, that this first-order description of photobleaching is a highly simplified approach that does not take into account that photobleaching is proportional to the triplet-state population, which can saturate at high illumination intensities, as discussed in the previous section.

The concentration distribution after photobleaching can thus be estimated from Eqs. (22) or (23) for a given intensity distribution of the focused laser beam. A closed form solution is available for the intensity distribution in the focal plane, for example, known as the Fraunhofer-Airy formula (Born & Wolf, Reference Born and Wolf1999):

where w is the radial resolution of the lens, J ₁ is the first-order Bessel function and P is the power of the light in the focus of the objective. As is depicted in Fig. 7A, the Fraunhofer-Airy pattern consists of a central bright spot, also known as the Airy disc, which is surrounded by higher-order diffraction rings with decreasing intensity. The Fraunhofer-Airy formula is difficult to work with mathematically, however, so a much simpler description is usually adopted for FRAP modeling. A Gaussian intensity distribution can quite accurately capture the intensity distribution of the Airy disc, while neglecting the low-intensity higher-order diffraction rings (see Fig. 7A). The same reasoning is often applied to the third dimension, so that one of the most common descriptions used for modeling the bleaching beam is represented by the 3D Gaussian approximation of the intensity distribution:

where r _b and z _b are the e ⁻² dimensions in the radial and axial orientation. Note that the same Gaussian approximation is equally used for two-photon photobleaching (Brown et al. Reference Brown, Wu, Zipfel and Webb1999; Mazza et al. Reference Mazza, Cella, Vicidomini, Krol and Diaspro2007, Reference Mazza, Braeckmans, Cella, Testa, Vercauteren, Demeester, De Smedt and Diaspro2008). The 3D concentration distribution of fluorophores after photobleaching with a stationary light beam for a time t can be readily obtained from Eqs. (22) and (25):

Fig. 7.

(A) The illumination intensity distribution in the focal plane has been calculated using the Fraunhofer-Airy formula for a 10× objective (NA 0·25). The line profile of the illumination intensity distribution (black line) shows that the central peak can be well approximated by the corresponding Gaussian intensity distribution (red line). A focused laser beam of 10 mW was used to photobleach spots into a sample of immobilized fluorophores. The result is shown for bleaching times of 100 ms (B), 500 ms (C) and 1000 ms (D). The higher-order diffraction rings play a more important role as the triplet state saturation increases, leading to larger photobleached spots. Effective bleaching intensity distributions (BID) are calculated that would give rise to these bleach profiles if one would assume a simple first-order bleach process. As indicated by the dashed lines, the effective BID can still be described by a Gaussian distribution, but with a variable effective resolution r _b. Inspired by Braeckmans et al. (Reference Braeckmans, Stubbe, Remaut, Demeester and De Smedt2006).

The very same expression holds for the two-photon case:

It was experimentally tested whether Eqs. (26) and (27) accurately describe the 3D bleach profile of a focused laser beam. Figure 8 shows the radial and axial sections of the 3D distribution of photobleached molecules in single-photon excitation (Figs 8A and 8B) and two-photon excitation (Figs 8C and 8D). Figures 8E and 8F show the comparison between the 3D effective bleached volumes (black lines) and the corresponding 3D model based on the Gaussian approximation (red lines). It is clear that the 3D Gaussian approximation of the effective bleaching intensity distribution works better in the two-photon excitation regime than in the single-photon excitation one, since the confinement of the excitation volume due to the two-photon absorption process minimizes effects from the side lobes, thus reducing the extent of the 3D bleached region. In single-photon FRAP, the Gaussian description should be used only in the 2D case of lateral diffusion.

Fig. 8.

Photobleaching was performed on an immobile fluorescent sample made using a polyelectolyte gel labeled with Alexa 555. Images show the radial (A, C) and axial profiles (B, D) of the effective bleached volumes for single (A, B) and two-photon excitation (C, D). Images (E) and (F) show the comparison between the experimental bleaching volume (black line) and the corresponding Gaussian approximation (red line) for 1P and 2P excitation, respectively. Inspired by Mazza et al. (Reference Mazza, Cella, Vicidomini, Krol and Diaspro2007).

3.4 Effective bleaching intensity distribution

As stated above, the Gaussian description is not suitable for the axial dimension in the case of single-photon photobleaching. It is worthwhile to take a closer look at the 2D case as well, since the simple first-order approximation according to Eq. (20) does not take saturation of the triplet level into account, which might occur for high bleaching intensities as are used in a typical FRAP experiment (cfr. Section 3.2). In order to appreciate the effect of triplet-state saturation, it is best to return to the Fraunhofer–Airy pattern according to Eq. (24) and as illustrated in Fig. 7A. Most of the light is confined within the Airy disc. At high laser intensity, according to the discussion in Section 3.2, this means that the fluorophores within this region will quickly be pumped into the triplet excited state, leading to triplet-state saturation. Although the higher-order diffraction rings have relatively low intensity, there is still a significant amount of energy being delivered to the fluorophores in those regions during the highly intense bleaching phase. The result is that fluorophores within the higher-order diffraction rings can also reach the triplet state and become photobleached. Since the fluorophores in the Airy disc are all in the triplet state, the amount of photobleaching in and outside the Airy disc is no longer proportional to the illumination intensity. Instead, the amount of photobleaching in the higher-order diffraction rings will approach the amount of photobleaching within the Airy disc, and the resulting bleach profile will be substantially more extended than expected by the Gaussian approximation, which takes only the Airy disc into account. This was shown explicitly for fluorescein, a commonly used fluorophore for FRAP (Braeckmans et al. Reference Braeckmans, Stubbe, Remaut, Demeester and De Smedt2006). Large dextran molecules (2 MDa) labeled with fluoresceinisothiocyanate were immobilized in an acrylamide gel. A stationary-focused laser beam was used to photobleach spots at different laser intensities. As is clear from Figs 7B to 7D, at the high laser intensities used for photobleaching in FRAP, the higher-order diffraction rings take part in the photobleaching process and cause a substantial broadening of the bleached area as compared to the Airy disc alone. The effective bleach profiles, however, can still be considered to be produced through first-order kinetics by a Gaussian beam, but with a varying effective resolution, r _b, depending on the experimental conditions, including the photon flux, the type of fluorophore and its photobleaching mechanism, which also depends on the local environment. The effect of spot broadening was also observed for 2PE (Mazza et al. Reference Mazza, Braeckmans, Cella, Testa, Vercauteren, Demeester, De Smedt and Diaspro2008) and for different types of fluorophores (Braeckmans et al. Reference Braeckmans, Remaut, Vandenbroucke, Lucas, De Smedt and Demeester2007). In conclusion, the Gaussian description of the focused laser beam can be used for FRAP modeling but only if the width (radially and axially) is considered as an experiment-dependent parameter (instead of being equal to the theoretical optical resolution). Determination of the effective width of the bleach distribution can either be done through calibration measurements under similar conditions as the actual experiment (Braeckmans et al. Reference Braeckmans, Remaut, Vandenbroucke, Lucas, De Smedt and Demeester2007) or by including it as a fitting parameter in the FRAP data analysis (Deschout et al. Reference Deschout, Hagman, Fransson, Jonasson, Rudemo, Lorén and Braeckmans2010; Smisdom et al. Reference Smisdom, Braeckmans, Deschout, van de Ven, Rigo, De Smedt and Ameloot2011).

4.1 Introduction

FRAP is a very powerful technique that can give accurate quantitative information if the FRAP data are analyzed with a suitable model. A wide variety of FRAP models have been developed over the years, each with their own specific assumptions and criteria that must be fulfilled in the experiment. Unfortunately, in many cases, the specific assumptions in the mathematical derivation are not clearly laid out and their implication for experimental conditions often remains unclear. This in turn leads to the inappropriate use of FRAP models and results with low accuracy. To aid in the selection of suitable FRAP models, this section reviews a selection of quantitative FRAP models with a critical discussion of the specific conditions in which they can be used.

4.2 Empirical FRAP methods

Considering the typical shape of a recovery curve, the simplest way to describe this is through a simple exponential function (Kapitza et al. Reference Kapitza, Mcgregor and Jacobson1985):

where F ₀ is the initial fluorescence intensity, F(t) is the fluorescence in the photobleached spot at a time t after photobleaching, τ is a characteristic recovery time that is related to the recovery half-time through t _1/2 = τln2 and b is the fraction of fluorophores that are bleached. By fitting Eq. (28) to experimental recovery curves, it is possible to compare half-times and make relative statements about the rate of diffusion. Note, however, that the use of Eq. (28) is a pragmatic approach with no general theoretical basis. It should therefore be used with care since the recovery half-time generally depends on the size and shape of the photobleaching region, as well as the amount of photobleaching. This comparative type of analysis should be used with care, knowing that the recovery times only have qualitative meaning and can only be compared between experiments if they are obtained with exactly the same bleach region and percentage of photobleaching.

Keeping in mind the caveats discussed above, it is possible to estimate the diffusion coefficient from the recovery half-time as mentioned in the seminal paper by Axelrod et al. (Reference Axelrod, Koppel, Schlessinger, Elson and Webb1976a, Reference Axelrod, Ravdin, Koppel, Schlessinger, Webb, Elson and Podleskib) for a Gaussian spot and a uniform circular spot. If w is the size of the spot, then the diffusion coefficient D can be estimated according to

where γ is a correction constant that depends on the bleaching area and amount of photobleaching. For a uniform circular spot, γ will have a constant value of 0·88. For a spot bleached by a laser beam with Gaussian intensity distribution, however, γ varies between 1.1 and 3 depending on the amount of bleaching produced in the sample. This can be seen in Fig. 9, where the correctional constant γ is shown as a function of the photobleaching parameter, K = αI(0)t _bleach, for both flow and diffusion. The amount of bleaching, b, is related to K according to b = 1−e^−K .

Fig. 9.

The correction constant γ is shown for a Gaussian spot photobleaching experiment as a function of the bleaching parameter K for both flow and diffusion. Reprinted with permission of Axelrod et al. (Reference Axelrod, Koppel, Schlessinger, Elson and Webb1976).

It is clear from Eq. (29) that the estimated diffusion coefficient strongly depends on the precision with which the size of the bleached spot w is known. As discussed in Section 3, this might be difficult to know exactly due to photobleaching saturation so that, contrary to what was suggested by Axelrod et al., w should not be taken to be equal to the size of the laser beam (e.g. as can be predicted by the Rayleigh criterion (Inoué, Reference Inoué and Pawley2006)).

The conclusion to be drawn from this is that the empirical FRAP methods may be useful for getting an initial estimate of the diffusion rate, but they will rarely yield a very accurate value.

4.3 Closed form models

To obtain quantitative information on diffusion from a FRAP experiment, the recovery data must be analyzed with a theoretical FRAP model. The FRAP methods that will be discussed in this section all rely on a theoretical description of the full FRAP process leading to a closed form expression describing the recovery process after photobleaching. A closed form expression is attractive from a practical point of view since it allows for fast and convenient data analysis using standard fitting routines. The most important assumptions that are made in the theoretical derivation will be discussed for each method. Since the experiments have to be carried out in accordance with these assumptions, it is important to have a good understanding of what they are. It is the failure to adhere to these assumptions that has led to a discrepancy in reported D values (Kang et al. Reference Kang, Day, Drake, Kenworthy and DiBenedetto2009).

4.4 Transform methods

A common problem for the closed form models in Section 4.3 is that initial values are needed to solve the diffusion equation. Therefore, specific approximations and assumptions are made in the mathematical description of the FRAP process, which imposes restrictions on the experimental procedure. By working in the spatial frequency domain, the so-called transform methods overcome the major limitation of requiring precise knowledge and control over the initial bleaching profile. In general, two transform methods have been reported so far. One uses the Fourier transform and the other uses the Hankel transform. The idea is to transform the partial differential diffusion equation into an ordinary differential equation in transform space. As will be discussed in more detail below, the transform methods need no assumptions on spatial fluorophore distribution after bleaching, which means that they are independent of the bleach size and shape, as well as of the bleaching resolution and diffusion during bleaching. Nevertheless, the transform methods have their own limitations and drawbacks.

4.5 Algorithmic approaches

While the FRAP analysis methods discussed in Sections 4.2–4.4 are quite convenient and fast for analysis of the recovery data, they all rely on a number of specific assumptions that cannot always be followed in actual practice. One example is the case of samples containing heterogeneities at length scales of the same order as the measurement region. In such a case, a completely numerical approach might be the only solution for interpreting the FRAP data. This was done by Snieckers & van Donkelaar (Reference Snieckers and van Donkelaar2005), for example, who considered inhomogeneities as zones with different apparent diffusion coefficients. When the inhomogeneities can be defined spatially beforehand, finite element analysis in combination with Fick's law of diffusion allows determination of diffusion coefficients in inhomogeneous tissues (Snieckers & van Donkelaar, Reference Snieckers and van Donkelaar2005).

Another example where numerical calculations were used, is to account for diffusion during bleaching. This was considered for diffusion in one dimension (Waharte et al. Reference Waharte, Brown, Coscoy, Coudrier and Amblard2005), two dimensions (e.g. Vinnakota et al. Reference Vinnakota, Mitchell, Deschenes, Wakatsuki and Beard2010; González-Pérez et al. Reference González-Pérez, Schmierer, Hill and Sear2011) and three dimensions (Kubitscheck et al. Reference Kubitscheck, Wedekind and Peters1998) by solving the combined equation for bleaching and diffusion. A more pragmatic approach is to consider the first post-bleach image as the starting condition of the recovery diffusion process. This was done for the pixelwise Gaussian profile FRAP method (cfr. Section 4.3.6), for instance, which was extended to include arbitrary circularly symmetric initial bleaching profiles (Jonasson et al. Reference Jonasson, Hagman, Lorén, Bernin, Nydén and Rudemo2010).

Although the most versatile, the numerical approaches’ major drawback is that they require dedicated programming and are rather slow computationally. Their use has been and most likely will be limited to very specific cases.

4.6 FRAP for diffusion and binding kinetics

The models described so far all focus on pure diffusion measurements. However, apart from diffusion, the FRAP technique is also used to extract information on binding kinetics (simultaneous with the transport parameters). Most of these binding models have only been developed in the past decade, and there is still intensive, ongoing research in this area. Nonetheless, even within this period, many of the approaches discussed above for analysis of diffusion by FRAP have also been applied to the analysis of diffusion and binding by FRAP. These include approaches with closed form solutions for either uniform discs (Sprague et al. Reference Sprague, Pego, Stavreva and McNally2004) or more arbitrary bleach patterns (Beaudouin et al. Reference Beaudouin, Mora-Bermudez, Klee, Daigle and Ellenberg2006), approaches that either account only for the average intensity within the bleached region (Hinow et al. Reference Hinow, Rogers, Barbieri, Pietenpol, Kenworthy and DiBenedetto2006) or that utilize spatial information from the collected images (Stasevich et al. Reference Stasevich, Mueller, Brown and McNally2010a, Reference Stasevich, Mueller, Michelman-Ribeiro, Rosales, Knutson and McNallyb), approaches that utilize a Fourier transform method that can account for anisotropic diffusion (Travascio & Gu, Reference Travascio and Gu2011), approaches that account for diffusion and binding not only during the recovery phase but also during the bleaching phase (Kang et al. Reference Kang, Day, DiBenedetto and Kenworthy2010) and numerical approaches that can account for either specimen complexities or more elaborate binding models or both (Beaudouin et al. Reference Beaudouin, Mora-Bermudez, Klee, Daigle and Ellenberg2006; Montero Llopis et al. Reference Montero Llopis, Sliusarenko, Heinritz and Jacobs-Wagner2012).

Fitting these various models to FRAP data typically generates estimates for not only the diffusion constant but also for the association and dissociation rates of binding. The inverse of these latter estimates can be used to calculate the time it takes a molecule to move from one binding site to the next and the time that a molecule resides at a binding site. Typical binding residence times measured by FRAP range from tens of milliseconds to tens of seconds, with recent approaches utilizing the scanning phase of a CLSM during photobleaching to generate residence time estimates as short as 2 ms (Erdel & Rippe, Reference Erdel and Rippe2012).

Table 1 summarizes some of the key features of published solutions for FRAP recoveries where binding has been considered. The table is divided into two parts corresponding to whether the binding sites are essentially immobile, thereby effectively immobilizing bound molecules, or whether the binding sites are mobile, thereby altering the diffusion constant of bound molecules. Other key features noted in the table and described in more detail below are the dimensionality of the solution, the nature of the photobleaching profile, the form of the boundary condition and whether the solution is closed form or numerical.

Table 1.

Summary of different FRAP models that incorporate diffusion and binding and their distinguishing features

One feature noted in Table 1 is whether the compartment in which the photobleaching is performed is presumed to be much larger than the size of the bleach spot. In this case, the specimen boundary is set at infinity. Otherwise, it is necessary to use a finite boundary corresponding to the size of the diffusible compartment within the specimen. A second defining feature of each solution is its dimensionality and symmetry. Most of the solutions obtained to date are for 2D systems with some form of symmetry in the specimen plane, thus reducing the differential equations to one spatial variable that has to be considered in the FRAP and binding model.

Related to the symmetry of the solution is a third defining feature, the bleach spot geometry, which must be consistent with the symmetry presumed. The most common form of symmetry considered is radial, allowing for radially symmetric bleach profiles such as Gaussian or circular bleaches. Solutions have also been devised for linear symmetry. Here, the bleach spot is a narrow strip, and the spatial variable runs perpendicular to this strip. Note that the 2D models with symmetry are readily applicable to thin specimens, such as a cell membrane. The models can also be applied to thicker 3D specimens if the photobleaching profile shows negligible axial variation and the specimen is also relatively homogeneous in the axial direction. For example, a cylindrical photobleach of a specimen with a homogeneous distribution of fluorescence sets up no fluorescence gradient axially, and so a model for 2D diffusion with radial symmetry and one dependent variable can still be appropriate.

However, real photobleach profiles have some axial dependence, and many specimens may not be well approximated by a 2D model. Consequently, several groups have developed approaches to incorporate more complex and realistic geometries. These models can allow not only for complex specimen features but also for accurate representations of the photobleach profile. A final defining feature is the form of the solution. In general, the more complex geometries require either a numerical solution of the differential equations or a Monte Carlo analysis, while most of the simpler geometries have yielded analytical solutions that typically involve infinite series, integrals and/or inverse Laplace transforms.

Since these models for binding are all relatively recent, a central issue in this field has been whether the binding estimates obtained from the various models are accurate. A gold standard has been lacking here, as most of the measurements have been made of binding in live cells, where such measurements had never been made before. Some comparison of methods has been done and this has provided some confidence in these approaches (Müller et al. Reference Müller, Erdel, Caudron-Herger, Marth, Fodor, Richter, Scaranaro, Beaudouin, Wachsmuth and Rippe2009; Renz & Langowski, Reference Renz and Langowski2008; Schmiedeberg et al. Reference Schmiedeberg, Weisshart, Diekmann, Meyer, Hoerste and Hemmerich2004; Stasevich et al. Reference Stasevich, Mueller, Brown and McNally2010a, Reference Stasevich, Mueller, Michelman-Ribeiro, Rosales, Knutson and McNallyb). Recently, a three-way comparison of binding estimates made under identical conditions by FRAP, FCS and SPT has been performed, and reasonable consensus was demonstrated (Mazza et al. Reference Mazza, Abernathy, Golob, Morisaki and McNally2012). The FRAP model for diffusion and binding (Mueller et al. Reference Mueller, Wach and McNally2008) that was validated in this comparison is outlined below. Key features of this model include correctly accounting for the role of diffusion in the FRAP recovery and an accurate accounting for the amount of photobleached fluorescence, both of which proved to be critical for generating accurate binding estimates. These same features are also present in many of the other models included in Table 1, suggesting that many of these models are also likely to yield accurate binding estimates, but further cross comparisons of different approaches will still be valuable.

The equations generated by Mueller et al. (Reference Mueller, Wach and McNally2008) are for binding to immobile sites distributed homogeneously within a circular domain (designed to approximate a cell nucleus). The circular domain has a finite radius, R _N, and circular symmetry is presumed. The photobleach is performed at the center of the circle, with a radially symmetric profile given by I _b(r). The recovery is measured in a circle of arbitrary radius, R _M, again positioned at the center of the circle. The average fluorescence in the measurement region is then:

with the constants given by:

The J's are Bessel functions, and a _j  = χ _j /R _N with χ _j the jth zero of the Bessel function of the first kind. The series expansion can typically be truncated at 500 terms for fitting most FRAP curves. The fit of Eq. (55) to the experimental FRAP recovery curve yields estimates for the unknown parameters, namely the diffusion constant D and the association rate $k_{{\rm on}}^{\rm *} $ and the dissociation rate k _off of binding. It should be noted that, since the preceding solution presumes a finite domain, the resultant FRAP curve will not recover to 100% because the photobleach will permanently destroy some of the fluorescence. This has implications for both how observational photobleaching is corrected when processing the FRAP data, and also for how the size of the domain, R _N , is determined for real cells whose shape is more complicated than a simple circle. The reader is referred to Mueller et al. (Reference Mueller, Wach and McNally2008) for more detail.

The preceding solution, as well as those in all of the other references in Table 1, provides general solutions that explicitly depend on all three unknown parameters, D, $k_{{\rm on}}^{\rm *} $ and k _off. However, under some conditions, simplified solutions exist which depend on fewer parameters or combinations of parameters; see Fig. 14. Such simplified solutions are important for two reasons. They generally provide simpler equations to fit the FRAP data and, more importantly, they typically limit the unknown parameters that should be independently estimated.

Fig. 14.

Simplified domain space for FRAP with binding. A ‘full model’ equation can describe all FRAP curves with diffusion and binding, but simpler equations suffice for some combinations of association k _on* and dissociation k _off rates. In ‘pure diffusion’, binding is so weak that the FRAP curve is well approximated by a model for just diffusion, thus yielding no information about the binding rates. In the three remaining domains, binding is strong enough to be measured by FRAP. In ‘reaction dominant’, the association rate is low and thus most molecules entering the bleach spot bind only once, yielding a FRAP that separates into two phases, diffusion followed by binding. The rate of recovery in the binding phase is directly related to the off rate. In ‘effective diffusion’, the association rate is high and thus most molecules entering the bleach spot bind many times, yielding a FRAP that is an iterative series of diffusion and binding steps. This mimics a slowed diffusion, and in this case only the ratio of association and dissociation rates can be estimated by the FRAP fit. In ‘full model’, association rates are intermediate and thus most molecules entering the bleach spot bind a few times, yielding a FRAP that cannot be fit by any of the simplified models. In this domain, both the association and dissociation rates, as well as the free diffusion constant, can be estimated independently. The domain boundaries shift up or down depending on the diffusion coefficient and bleach spot size. The plot shown here is for D = 30 μm² s⁻¹ with a bleach spot diameter of 1 μm. Inspired by Sprague et al. (Reference Sprague, Pego, Stavreva and McNally2004).

A number of groups have explored these simplified solutions for the case of immobile binding sites, where two different simplified domains have been identified: one where diffusion is fast relative to binding (reaction dominant domain) and the other where binding is fast relative to diffusion (effective diffusion domain); see Fig. 14. Various groups have come to different conclusions about whether fast or slow binding is determined by the association rate, the dissociation rate or a combination of these two (Carrero et al. Reference Carrero, Crawford, Hendzel and de Vries2004; Kaufman & Jain, Reference Kaufman and Jain1990; Lele et al. Reference Lele, Oh, Nickerson and Ingber2004; Sprague et al. Reference Sprague, Muller, Pego, Bungay, Stavreva and McNally2006; Travascio & Gu, Reference Travascio and Gu2011; Tsibidis & Ripoll, Reference Tsibidis and Ripoll2008). A direct empirical evaluation by Sprague et al. (Reference Sprague, Pego, Stavreva and McNally2004) argues that the key parameter is the association rate. When this is fast enough, the time to bind at a site is short compared with the time to diffuse across the bleach spot, and so fluorescent molecules repeatedly bind and unbind as they progress toward the center of the bleach spot. This looks like a slowed diffusion, known as effective diffusion, with the effective diffusion constant given by $D_{{\rm eff}} = {D / {\left( {1 + {{k_{{\rm on}}^{\rm *}} / {k_{{\rm off}}}}} \right)}}$ . In contrast, when the association rate is slow, the time to bind at a site is long compared with the time to diffuse across the bleach spot and so the FRAP is separable into a diffusive phase and a binding phase. The diffusive relaxation often occurs quickly, such that virtually all of the FRAP recovery reflects binding, leading to the name reaction dominant for this type of behavior. The binding phase is governed by ordinary differential equations yielding an exponential solution that is given as ${\rm e}^{ - k_{{\rm off}} t} $ (Sprague et al. Reference Sprague, Pego, Stavreva and McNally2004; Tsibidis, Reference Tsibidis2009). In some cases, these ordinary differential equations have been solved numerically (Phair & Misteli, Reference Phair and Misteli2001). Hallen & Layton (Reference Hallen and Layton2010) and Michelman-Ribeiro et al. (Reference Michelman-Ribeiro, Mazza, Rosales, Stasevich, Boukari, Rishi, Vinson, Knutson and McNally2009) consider a more general diffusion-phase binding-phase solution in which both phases are measured, and thus the solution is a sum of a diffusive recovery plus a binding recovery.

As a specific example of these simplified equations, we describe the solutions for the effective diffusion and reaction dominant cases for the same example discussed above, namely FRAP in a finite circular domain with radial symmetry. The effective diffusion solution is identical to the solution for a molecule diffusing with a diffusion constant given by $D_{{\rm eff}} = {D / {\left( {1 + {{k_{{\rm on}}^{\rm *}} / {k_{{\rm off}}}}} \right)}}$ :

where the constants U _j are given by:

In many cases, D can be estimated by FRAP in a separate experiment. Thus, with estimates of D _eff and D, only the ratio of binding rates ${{k_{{\rm on}}^{\rm *}} / {k_{{\rm off}}}} $ can be determined. Similarly, the reaction-dominant solution is given by:

where φ = C(0)/C ₀ is the fraction of total fluorescence destroyed by the photobleach and θ is the starting point of the FRAP recovery. In general, it is difficult to measure θ accurately since the free fraction equilibrates very rapidly in a reaction-dominant FRAP. As a result, in general, the preceding equation can only be used to get an accurate estimate of the off rate, k _off.

Simplified solutions also exist for the case of binding to mobile binding sites (Braga et al. Reference Braga, McNally and Carmo-Fonseca2007; Dushek et al. Reference Dushek, Das and Coombs2008). At this time, these have not been extensively applied. The reader should consult Dushek et al. (Reference Dushek, Das and Coombs2008) for more details.

5.1 Fluorescent probes for FRAP

One of the keystones in a successful FRAP experiment is the choice of fluorescent probes (Cheng et al., Reference Cheng, Prud'homme and Thomas2002). Either fluorophores are used directly as model molecules or they are coupled to a specific molecule of interest. In the latter case, it is important that the fluorescent probe does not significantly influence the original molecule's diffusion and interaction properties. Control experiments should preferably be carried out to ascertain that the system is not influenced by the fluorescent label. FRAP experiments can then be used to obtain information on the mobility of the labeled molecule or on the material structure in which the molecule is diffusing. In addition, for FRAP, the probe should preferably have intermediate photostability as on the one hand it should be not too difficult to bleach and on the other hand should be sufficiently photostable to allow imaging of the recovery process. Table 2 gives a number of fluorophores that are frequently used for FRAP measurements.

Table 2.

Examples of fluorophores often used for FRAP experiments

FITC, FluoresceinIsothiocyanate; NBD, nitrobenzoxadiazole; DTAF, fluorescein dichlorotriazine.

Without doubt, the most commonly used probes for FRAP are green fluorescent protein (GFP) (Tsien, Reference Tsien1998) and fluorescein. FluoresceinIsoThioCyanate (FITC) is the reactive form of fluorescein, due to the isothiocyanate group, and it is commonly attached to larger molecules such as dextrans or other large complexes in order to label specific molecules. GFP or mutant fluorescent proteins are commonly used in cell biology to tag specific proteins in living cells. In this case, the fluorescent proteins are not added to the cells as such; rather, the gene sequence is added to the genome. One drawback of the fluorescent proteins for FRAP is that some of them have been shown to exhibit reversible photobleaching, which can complicate the analysis of the fluorescence recovery (Dayel et al. Reference Dayel, Hom and Verkman1999; Sinnecker et al. Reference Sinnecker, Voigt, Hellwig and Schaefer2005; Swaminathan et al. Reference Swaminathan, Hoang and Verkman1997). Two recent papers have shown that the reversible effects of GFP can be significant and have described methods to either avoid or correct the impact of this reversible photobleaching on FRAP (Daddysman & Fecko, Reference Daddysman and Fecko2013; Mueller et al. Reference Mueller, Morisaki, Mazza and McNally2012). An attractive alternative approach which still uses a genetically encoded fusion protein that does not exhibit reversible photobleaching is the HaloTag protein combined with the fluorescent ligand tetramethylrhodamine (Morisaki & McNally, Reference Morisaki and McNally2014).

For quantitative FRAP, it is essential that the observed fluorescence is directly proportional to the fluorophore concentration. On the one hand, the concentration should not be too low, as this will result in very weak fluorescence, poor SNR and number fluctuations will start to dominate the signal. On the other hand, too high a concentration will result in inner filtering effects (Herman, Reference Herman1998; van Oostveldt & Bauwens, Reference Van Oostveldt and Bauwens1990). The first inner filtering effect comes from the laser light that needs to pass ‘layers’ of probe molecules before reaching the focal plane. For each ‘layer’ of fluorophores, some of the excitation light is absorbed and the number of photons reaching the next ‘layer’ will be reduced. This is a well-known phenomenon in fluorescence microscopy that causes a drop in observed fluorescence when focusing deeper into the sample. The other filtering effect is caused by the overlap between the fluorophore's absorption and emission spectrum. When a photon is emitted by a fluorophore, there is a wavelength-dependent probability that it will be reabsorbed when it reaches another fluorophore on its way out of the sample. The result is that the observed fluorescence intensity from the focal plane drops. Even if the secondary excitation causes a new photon to be emitted, it will not be detected since the confocal pinhole removes out of focus light. Both filtering effects will be more and more pronounced with increasing probe concentration and will cause a deviation from fluorescence linearity at high fluorophore concentrations depending on the depth in the sample. By making a series of dilutions, it is usually possible to check for the linearity of the fluorescence. Typically one then chooses to work with the highest concentration in the linear concentration range. This is not possible in cells expressing fluorescent proteins but, in that case, cells should be selected that express only moderate amounts of the fluorescent protein, which is also preferable from a biological point of view.

In recent decades, there has been particular interest in the development of new variants of GFPs and in the discovery of red or infrared shifted proteins whose spectral properties can be controlled and modified by light (Lukyanov et al. Reference Lukyanov, Chudakov, Lukyanov and Verkhusha2005). A step forward in this field was the generation of fluorophores able to switch from a quiescent dark state to a bright one (photoactivation) or convertible from a fluorescent emission spectral band to a different one (photoconversion). Such optical highlighters offer a powerful tool for tracking and investigating protein dynamics in living cells (Lippincott-Schwartz et al. Reference Lippincott-Schwartz, Altan-Bonnet and Patterson2003) and have facilitated the development of super-resolution imaging techniques (Betzig et al. Reference Betzig, Patterson, Sougrat, Lindwasser, Olenych, Bonifacino, Davidson, Lippincott-Schwartz and Hess2006). The light-induced change in the spectral properties of the fluorophore allows photoselection of a subset of molecules in the population and, since only a selected sub-population of photoactivated molecules exhibits fluorescence emission in a defined spectral range, their behavior can be followed independently, offering a valuable alternative to photobleaching-based techniques, such as FRAP and FLIP. Similar to regular FRAP, where contrast is achieved by destroying the fluorescence in a defined region, photo activation enables generation of contrast by either turning on the fluorescence or by inducing a shift in color in the case of photo conversion. Apart from that, the principle of performing diffusion analysis is just like that of regular FRAP. The reader is referred to Mazza et al. (Reference Mazza, Braeckmans, Cella, Testa, Vercauteren, Demeester, De Smedt and Diaspro2008) for an example of adapting regular FRAP equations for the analysis of photo activation experiments. Several fluorescent proteins with photochromic spectral properties have recently been developed and they can be divided, according to the transition mechanism (as shown in Fig. 15), into three main groups: Photoactivatable fluorescent proteins, Photoconvertible green-to-red fluorescent proteins and Irreversibly Photoconvertible Fluorescent proteins. Table 3 reports some basic properties.

Fig. 15.

Transition mechanism behind photoactivation, photoconversion and photoswitching of fluorescent proteins. (a) Photoactivation process: conversion of the chromophore from a neutral to an anionic state induced by decarboxylation of Glu222 is responsible for photoactivation of pa-GFP and PS-CFP2. (b) Green-to-red photoconversion induced by UV–vis radiation for Kaede, KikGR, Dendra and Eos is characterized by cleavage between the amide nitrogen and the α-carbon atoms in the His62. (c) Cis–trans photoisomerization leads to the subsequent photoswitching process of Dronpa: alternated 405 and 488 nm illumination drives the protein from the fluorescent to the dark state and back. Inspired by Shaner et al. (Reference Shaner, Patterson and Davidson2007).

Table 3.

Basic properties of photoactivatable, photoconvertible and photoswitchable fluorescent proteins

5.2 Experimental considerations

A number of FRAP methods were discussed in Section 4, each with their own set of advantages and limitations. Their main assumptions, advantages and drawbacks are summarized in Table 4. In general, the following assumptions are common to most models:

• • Infinite homogeneous media;

• • Absence of flow or other type of directed motion;

• • Free and unrestricted diffusion;

• • Initial uniform distribution of fluorophores;

• • Absence of diffusion during bleaching.

Table 4.

Summary of the extra assumptions, advantages and drawbacks of the selected FRAP methods

Any exceptions to these assumptions are explicitly stated in Table 4. Note that Table 4 only includes methods used for measurement of diffusion by FRAP, not methods for measurement of binding by FRAP, since the latter methods are still being developed and no set of methods has yet appeared in widespread use. The reader is referred to Table 1 in Section 4 for a tabulation of current methods for binding analysis and some of their assumptions.

The assumptions, advantages and drawbacks of the different FRAP models in connection to the properties of the material and the available equipment are important for the correct choice of FRAP model. Apart from that, there are a number of general considerations that should be kept in mind, as will be discussed in the following sections.

5.3 The effect of heterogeneous media, binding and anomalous diffusion on FRAP measurements

6.1 FRAP in cell biology

The first FRAP measurement dates from 1974 when Peters et al. demonstrated the mobility of membrane proteins in erythrocyte ghosts. FRAP has been closely associated with cell biology ever since. Some of the major contributions of FRAP to cell biology are discussed here.

The work of Peters et al. (Reference Peters, Peters, Tews and Bähr1974) originated in an era in which the structure and function of the cellular membrane and its components were extensively studied (Saffman & Delbruck, Reference Saffman and Delbruck1975; Singer & Nicolson, Reference Singer and Nicolson1972). Compared with cell fusion experiments (Frye & Edidin, Reference Frye and Edidin1970), FRAP offered a straightforward alternative that, in comparison to FCS, seemed more appropriate for covering the time scale at which membrane protein diffusion occurs (Koppel et al. Reference Koppel, Axelrod, Schlessinger, Elson and Webb1976). Furthermore, the utilization of a fluorescent probe to specifically label membrane components, even in live cells, was another asset that matched the research questions at that time. As a result, FRAP has since its inception been frequently used for studying the diffusion of membrane molecules.

The combination of a fluorescent probe with a natural ligand of a targeting receptor enabled the study of various receptors with high specificity, including the acetylcholine receptor (Axelrod et al. Reference Axelrod, Koppel, Schlessinger, Elson and Webb1976a, Reference Axelrod, Ravdin, Koppel, Schlessinger, Webb, Elson and Podleskib), Fc receptors (Schlessinger et al. Reference Schlessinger, Webb, Elson and Metzger1976), and insulin and epidermal growth factor receptor (Schlessinger et al. Reference Schlessinger, Shechter, Cuatrecasas, Willingham and Pastan1978). In addition to these proteins, the fluidity of the membrane itself could also be studied by membrane lipid probes (Wu et al. Reference Wu, Jacobson and Papahadjopoulos1977).

These first FRAP measurements allowed for the comparison of the mobility of proteins in the plasma membrane of different cell types (Cherry, Reference Cherry1979), revealing differences between cell types, and even within cells (Schlessinger et al. Reference Schlessinger, Elson, Webb, Yahara, Rutishauser and Edelman1977). Through the use of chemical agents such as metabolic inhibitors, studies clearly proved that the motion of proteins in the cell membrane relies in part on Brownian motion, although interaction with intracellular components was suggested. In addition, early FRAP observations indicated that the apparent mobile fraction of a lipid probe in the cell membrane was unexpectedly dependent on the size of the bleached region, even though this area was significantly smaller than that of the cell surface (Yechiel & Edidin, Reference Yechiel and Edidin1987). This effect arises because the cell membrane contains a variety of molecular complexes and domains, characterized by different composition and spatial arrangement of the membrane-constituting lipids, which implies constraints on the diffusion of the membrane components (Kusumi et al. Reference Kusumi, Koyama-Honda and Suzuki2004). As a result, the cell membrane is seen as a non-ideal liquid mixture of molecules with variable degrees of mutual miscibility (Kusumi et al. Reference Kusumi, Koyama-Honda and Suzuki2004).

Apart from flask-shaped invaginations of the plasma membrane (caveolae), it has been suggested that membrane inhomogeneities comprise so-called corrals and lipid rafts. Corrals refer to compartments composed of the membrane-associated actin cytoskeleton (fences) and by rows of transmembrane proteins anchored to it (pickets). Lipid rafts can be seen as sub-micron ‘liquid-ordered’ (Lo) membrane domains enriched in cholesterol, glycosphingolipids and phospholipids with saturated acyl chains (Kusumi et al. Reference Kusumi, Koyama-Honda and Suzuki2004; Pike, Reference Pike2006). The use of FRAP in exploring these putative lipid rafts in living cells was discussed by Kenworthy (Reference Kenworthy2007). The coexistence of different lipid phases in the apical plasma membrane of epithelial cells has been demonstrated in FRAP experiments performed at different temperatures (Meder et al. Reference Meder, Moreno, Verkade, Vaz and Simons2006). On the other hand, it has been found through FRAP experiments that GFP-tagged raft and non-raft proteins move similarly, and several explanations have been suggested (Kenworthy et al. Reference Kenworthy, Nichols, Remmert, Hendrix, Kumar, Zimmerberg and Lippincott-Schwartz2004). For a more complete overview of FRAP applications in the context of membrane organization, the reader is referred to the review by Kang et al. (Reference Kang, Day, Drake, Kenworthy and DiBenedetto2009).

FRAP is not exclusively applied to study membrane organization. For example, it allowed Maxfield et al. (Reference Maxfield, Willingham, Pastan, Dragsten and Cheng1981) to demonstrate that the thyroid hormone does not passively diffuse into cells, but that it is actively internalized upon binding to its receptor at the cell membrane. The broader potential of lateral diffusion measurements and accompanying mobile fraction was comprehensively discussed by Axelrod (Reference Axelrod1983). He named a wide range of applications including, among others, receptor clustering and homeostasis at synapses, immunological response upon antigen binding, receptor down-regulation upon ligand binding, viral infection of cells, cell–cell contact specificity and the enhanced reaction rate due to surface diffusion (Axelrod, Reference Axelrod1983).

For applications involving cytosolic proteins, the necessity of labeling molecules of interest with a fluorescent label has been a burden. However, this did not render them impossible. Microinjection offered a solution to this problem, albeit a labor intensive one. This allowed researchers to study the dynamics of tubulin in microtubule networks (Saxton et al. Reference Saxton, Stemple, Leslie, Salmon, Zavortink and McIntosh1984). The ability to create chimeras comprising the protein of interest and a fluorescent protein like GFP sparked a range of new applications involving cytosolic proteins (Lippincott-Schwartz & Patterson, Reference Lippincott-Schwartz and Patterson2003). These new applications were further boosted by an augmented availability of commercial CLSMs. Such new applications covered various fields, including nuclear proteins (Houtsmuller, Reference Houtsmuller2005; Mueller et al. Reference Mueller, Mazza, Stasevich and McNally2010; Raghuram et al. Reference Raghuram, Carrero, Th'ng and Hendzel2009), nucleocytoplasmic shuttling (Koster et al. Reference Koster, Frahm and Hauser2005), intracellular trafficking of proteins like aquaporin (Luu & Maurel, Reference Luu and Maurel2013), assembly and dynamics of the cytoskeleton (Walczak et al. Reference Walczak, Rizk and Shaw2010; Watanabe et al. Reference Watanabe, Yamashiro, Vavylonis and Kiuchi2013), endoplasmic reticulum dynamics (Costantini & Snapp, Reference Costantini and Snapp2013; Gonzalez-Gonzalez et al. Reference Gonzalez-Gonzalez, Jaskolski, Goldberg, Ashby and Henley2012), protein aggregate degradation (Holmberg et al. Reference Holmberg, Staniszewski, Mensah, Matouschek and Morimoto2004; Stenoien et al. Reference Stenoien, Mielke and Mancini2002), assembly and dynamics of myofibrils (Sanger et al. Reference Sanger, Wang, Fan, White and Sanger2010), RNA metabolism (Nickerson, Reference Nickerson2009), and pre- and postsynaptic scaffolds (Ziv & Fisher-Lavie, Reference Ziv and Fisher-Lavie2014).

The advent of these new applications was accompanied by a strong interest in measuring binding in living cells as well. When photobleached, many of the fusion proteins inside cells recover much more slowly than would be expected based on diffusion rates of GFP alone inside of cells (Sprague & McNally, Reference Sprague and McNally2005). This retardation is most likely due to binding of the protein of interest to immobile cellular substrates, such as DNA or the cytoskeleton. This can be directly demonstrated by mutation of the appropriate binding site in the protein, which then results in faster recoveries. The models summarized in Section 4.6 were all developed to exploit the fact that FRAP recoveries of GFP fusion proteins typically contain information about in vivo binding. There is now a relatively large literature dealing with FRAP to assay an assortment of nuclear processes where proteins interact with DNA, including transcription, DNA repair and DNA replication (Hager et al. Reference Hager, McNally and Misteli2009). In addition to this extensive body of work on binding within the nucleus, FRAP has been used in a wide variety of cell biology studies to investigate transient binding and dynamic processes on microfilaments and microtubules (Al Tanoury et al. Reference Al Tanoury, Schaffner-Reckinger, Halavatyi, Hoffmann, Moes, Hadzic, Catillon, Yatskou and Friederich2010; Hallen et al. Reference Hallen, Ho, Yankel and Endow2008a, Reference Hallen, Liang and Endowb) and to proteins within cell membranes (Haj et al. Reference Haj, Sabet, Kinkhabwala, Wimmer-Kleikamp, Roukos, Han, Grabenbauer, Bierbaum, Antony, Neel and Bastiaens2012; Kang et al. Reference Kang, Day, DiBenedetto and Kenworthy2010).

Other techniques such as FCS and SPT have also found their way to biological laboratories. However, despite their availability, the interest in FRAP remains high, as is illustrated by the large number of publications (refer to Fig. 2) and recently published protocols (Costantini & Snapp, Reference Costantini and Snapp2013; Gonzalez-Gonzalez et al. Reference Gonzalez-Gonzalez, Jaskolski, Goldberg, Ashby and Henley2012; Hardy, Reference Hardy2012; Zuleger et al. Reference Zuleger, Kelly and Schirmer2013). The simplicity of the fundamental idea of FRAP combined with less stringent technical requirements, comprehensible data analysis and the ability to cover long time scales offer a good explanation, although this is difficult to prove.

Finally, the availability of other techniques allowed for cross-validations with FRAP, yielding in some cases large discrepancies (Cherry et al. Reference Cherry, Smith, Morrison and Fernandez1998; Wachsmuth, Reference Wachsmuth2014). At least for systems involving binding and diffusion, Mazza et al. (Reference Mazza, Abernathy, Golob, Morisaki and McNally2012) elegantly demonstrated that discrepancies are not attributed to technical limitations, but that they arise as a consequence of the selection of improper models and accompanying assumptions for all techniques. However, FRAP is still considered to be a good standard (Groeneweg et al. Reference Groeneweg, van Royen, Fenz, Keizer, Geverts, Prins, de Kloet, Houtsmuller, Schmidt and Schaaf2014) and, even in combination with a single molecule technique, it has proven its added value (e.g. Black et al. Reference Black, Cheney, Campbell and Knowles2014; Watanabe et al. Reference Watanabe, Yamashiro, Vavylonis and Kiuchi2013).

6.2 FRAP in pharmacy and pharmacology

It only took a short while after the introduction of FRAP in the 1970s before the technique was applied to pharmaceutical and biomedical research. FRAP was used to non-destructively study the diffusion of therapeutic macromolecules in a variety of systems such as artificial gels, biological extracellular matrices and living cells. FRAP was also used to obtain information on the binding kinetics between therapeutic macromolecules and receptors inside living cells and on the flow rate of drugs in biological extracellular matrices. Since a detailed review of FRAP in pharmaceutical research was published recently (Deschout et al. Reference Deschout, Raemdonck, Demeester, De Smedt and Braeckmans2013), we will here give only a brief overview of the most recent applications.

Artificial matrices such as solutions and gels are important in drug delivery for transport and controlled release of therapeutic molecules (Lin & Metters, Reference Lin and Metters2006). Studying the molecular transport in these systems can be useful for gaining a better understanding of these materials and improving their design and composition. In that context, FRAP was used for instance to study the mobility of macromolecules in hydrogels consisting of methacrylated dextran (Verheyen et al. Reference Verheyen, van der Wal, Deschout, Braeckmans, de Smedt, Barendregt, Hennink and van Nostrum2011), poly(ethylene glycol) (Brandl et al. Reference Brandl, Hammer, Blunk, Tessmar and Goepferish2010), peptides (Branco et al. Reference Branco, Pochan, Wagner and Schneider2009), konjacglucomannan (Alvarez-Manceñido et al. Reference Alvarez-Manceñido, Braeckmans, De Smedt, Demeester, Landin and Martínez-Pacheco2006) and agarose (Kusto & Deen, Reference Kusto and Deen2004), to name a few.

It is equally important to study the mobility of drugs in extracellular matrices, such as cell interstitium, mucus, vitreous and biofilms, which can form important barriers in drug delivery (De Smedt et al. Reference De Smedt, Remaut, Lucas, Braeckmans, Sanders and Demeester2005; Remaut et al. Reference Remaut, Sanders, De Geest, Braeckmans, Demeester and De Smedt2007). These materials consist of complicated networks of macromolecules that interact with each other, resulting in their viscoelastic properties. Small molecules may diffuse freely in this matrix, but larger macromolecular drugs may be hindered or trapped. It is therefore important to find out which type of extracellular matrices form barriers for drug delivery and how they can be overcome. In the case of cancer treatment, FRAP was used to investigate the diffusion of proteins (Brown et al. Reference Brown, Bouchera, Nasser and Jain2004; Netti et al. Reference Netti, Berk, Swartz, Grodzinsky and Jain2000), liposomes (Pluen et al. Reference Pluen, Boucher, Ramanujan, McKee, Gohongi, di Tomaso, Brown, Izumi, Campbell, Berk and Jain2001) and dextrans (Papadopoulos et al. Reference Papadopoulos, Binder and Verkman2005; Stylianopoulos et al. Reference Stylianopoulos, Diop-Frimpong, Munn and Jain2010; Sullivan et al. Reference Sullivan, Sipprell, Brown and Brown2009) in tumor interstitium. In the context of cystic fibrosis, transport of drugs through the thick and viscous lung mucus is crucial. Here, FRAP was used to measure the diffusion of nucleic acids (Braeckmans et al. Reference Braeckmans, Peeters, Sanders, De Smedt and Demeester2003; Shen et al. Reference Shen, Hu and Saltzman2006) and dextrans (Derichs et al. Reference Derichs, Jin, Song, Finkbeiner and Verkman2011; Salinas et al. Reference Salinas, Haggie, Thiagarajah, Song, Rosbe, Finkbeiner, Nielson and Verkman2005) in mucus.

The mobility of therapeutic compounds inside living cells also needs to be investigated in the case that the drug has to be delivered intracellularly. This is the case for nucleic acid therapies, for example, where therapeutic nucleic acids must be delivered in the cytosol or nucleus. FRAP has been used to study the diffusion of model proteins and dextrans in the cytoplasm (Arrio-Dupont et al. Reference Arrio-Dupont, Foucault, Vacher, Devaux and Cribier2000; Cardarelli et al. Reference Cardarelli, Serresi, Bizzarri, Giacca and Beltram2007; Lukacs et al. Reference Lukacs, Haggie, Seksek, Lechardeur, Freedman and Verkman2000; Reits et al. Reference Reits, Griekspoor, Neijssen, Groothuis, Jalink, van Veelen, Janssen, Calafat, Drijfhout and Neefjes2003; Molecular Therapy 15, pp. 1313–1322) and the nucleus (Braga et al. Reference Braga, Desterro and Carmo-Fonseca2004; Maru et al. Reference Maru, Tobias, Rivers, Caunt, Normana and McArdlea2009; Reits et al. Reference Reits, Griekspoor, Neijssen, Groothuis, Jalink, van Veelen, Janssen, Calafat, Drijfhout and Neefjes2003) of living cells.

6.3 FRAP in food science and food technology

FRAP in foods is important for understanding the mass transport mechanisms controlling the food properties and structure development during manufacturing. Diffusion influences the structure evolution during manufacturing, product shelf-life, the time-dependent development of the properties of the food during cooking, consumer preferences during consumption and the bioavailability of the nutrients (Hermansson et al. Reference Hermansson, Lorén, Nydén, del Pilar Buera, Welti-Chanes, Lillford and Corti2006). FRAP in foods is equally challenging as it is in cell biology and pharmaceuticals because the microstructures of foods are often multiphase materials consisting of gels, emulsions, foams, solutions, crystal networks, amorphous and crystalline areas, etc. and heterogeneous at different length scales.

FRAP is not yet widely used for food applications. However, its ability to determine local diffusion properties in heterogeneous foods with high precision makes it a versatile tool for understanding the mechanisms controlling diffusion in foods, especially if it is combined with stages for controlled surrounding conditions (temperature, moisture) or other more global diffusion measurement techniques such as NMRd. For example, in chocolate pralines, the fat migrates from the filling to the praline surface where it can induce fat bloom that is seen as a grayish haze (Ghosh et al. Reference Ghosh, Ziegler and Anantheswaran2002). FRAP has been used to understand the fat migration mechanisms to enable better control of the fat crystal network in chocolate during manufacturing and storage (Svanberg et al. Reference Svanberg, Ahrné, Lorén and Windhab2011) and triglyceride networks (Marty et al. Reference Marty, Schroeder, Baker, Mazzanti and Marangoni2009). Clark et al. (Reference Clark, Wilde and Wilson1990) used FRAP to evaluate the mechanism of hop iso-α acid stabilization of beer foams. Other studies of stabilization of protein foams and emulsion have been made for β-lactoglobulin and β-casein foams and emulsions (Clark et al. Reference Clark, Mackie, Wilde and Wilson1994; Mackie et al. Reference Mackie, Nativel, Wilson, Ladha and Clark1996).

In other foods, FRAP has been used to determine the diffusion of protease inhibitors in halibut fish to prevent softening (Carvajal-Rondanelli & Lanier, Reference Carvajal-Rondanelli and Lanier2010) and of pectin methylesterases in pectin gels and solutions (Videcoq et al. Reference Videcoq, Steenkeste, Bonnina and Garnier2013). The effects of heterogeneity and differences in micro and macro viscosity on nutrients’ diffusion in arabinoxylan solutions were revealed by FRAP (Shelat et al. Reference Shelat, Vilaplana, Nicholson, Wonga, Gidley and Gilbert2010). Jensen et al. (Reference Jensen, Rolin and Ipsen2010) investigated mechanisms for stabilizing acidified milk drinks and used FRAP as a method to evaluate the mobility of proteins in the samples. The effects of solute size, shape and charge on diffusion have been determined in model cheeses to control ripening (Floury et al. Reference Floury, Madec, Waharte, Jeanson and Lortal2012; Silva et al. Reference Silva, Peixoto, Lortal and Floury2013). Heterogeneity in maize starch granules has been investigated because of its implications for chemical modification reactions and digestion kinetics (Dhitala et al. Reference Dhitala, Shelata, Shrestha and Gidleya2013).

Diffusion properties in gels have been determined using FRAP to reveal transient structures in gelatin (Hagman et al. Reference Hagman, Lorén and Hermansson2010) and correlations between diffusion and microstructure in κ-carrageenan gels (Hagman et al. Reference Hagman, Lorén and Hermansson2012). In addition, effects on the diffusion of capillaries and gel strand orientation were determined in alginate gels (Schuster et al. Reference Schuster, Eckardt, Hermansson, Larsson, Lorén, Altskär and Ström2014a) and of protein and salt concentration, and solute size (Balakrishnan et al. Reference Balakrishnan, Nicolai and Durand2012), and electrostatic interactions (Schuster et al. Reference Schuster, Eckardt, Hermansson, Larsson, Lorén, Altskär and Ström2014b) in β-lactoglobulin gels. FRAP has also been used to analyze packaging and coatings to determine the migration of plasticizers and other unhealthy molecules into the food from packagings and coatings (Karbowiak et al. Reference Karbowiak, Hervet, Léger, Champion, Debeaufort and Voilley2006, Reference Karbowiak, Debeaufort, Voilley and Trystram2009; Pinte et al. Reference Pinte, Joly, Plé, Dole and Feigenbaum2008).