A new approximate FRAP model for use on any standard CLSM is presented here. The novelty of this method is that it takes into account diffusion during bleach and is valid for use with objective lenses with high NA. The method can be readily applied by anyone familiar with a CLSM, as the mathematical expressions are straightforwardly programmed with standard fitting programs.

The GFP is a 28-kDa protein with a Stokes' radius of 2.35 nm (

Ribbeck and Görlich, 2001 ). This corresponds to an expected diffusion coefficient in water at 37°C of 139.2 μm

^{2} s

^{-1}. Confocal spot photobleaching recovery measurements performed with a specifically modified microscopic system yielded GFP diffusion coefficients of 87 ± 3 μm

^{2} s

^{-1} in aqueous solution and 24 ± 2 μm

^{2} s

^{-1} in the cytoplasm of

*Dictyostelium discoideum* cells kept at room temperature (

Potma *et al*., 2001 ). In our previous work, we used a standard CLSM to perform quantitative FRAP, and we estimated an apparent diffusion coefficient of 5 μm

^{2} s

^{-1} for GFP in the nucleus of HeLa cells (

Calapez *et al*., 2002 ). We believe that this value was significantly underestimated because the mathematical model applied considered that fluorescence recovery during the bleach phase is negligible, an assumption that is not met by the experimental conditions in the case of highly mobile molecules (). Using the same CLSM instrument and the new FRAP model, we estimate a diffusion coefficient of 33.3 ± 3.6 μm

^{2} s

^{-1} for GFP in the nucleus of HeLa cells maintained at 37°C.

To develop a more accurate practical approach that can be readily applied by cell biologists interested in performing quantitative FRAP analysis with a standard CLSM, we first analyzed experimentally the fluorescence profile generated by bleaching a region of interest with a scanning laser beam. FITC-labeled dextrans were immobilized in a polyacrylamide gel, and the bleached volume was imaged along the z-direction. The results show that, for bleaching spots up to ~1 μm in radius, the observed fluorescence profiles can be effectively approximated by the theoretical models that consider bleaching generated by a stationary Gaussian laser. Our data thus validate the application of the Gaussian approximation to postbleach radial fluorescence profiles generated with the CLSM, as assumed previously (

Calapez *et al*., 2002 ). However, our results further indicate that in order to apply the Gaussian approximation the bleach region should not exceed ~1 μm in radius (the maximum radius for which fitting is valid depends on the numerical aperture of the objective lens). For much larger bleach regions, a better approximation is the uniform disk profile, as predicted from the model proposed by

Braeckmans *et al*., 2003 .

To date, most FRAP models assume that both bleaching and image acquisition are sufficiently fast to avoid diffusion during those periods (

Braeckmans *et al*., 2003 ). As a rule, for diffusion to be neglected, the total bleaching time should be at least 15 times smaller than the recovery time (

Meyvis *et al*., 1999 ). If we consider a molecule that diffuses at 0.5 μm

^{2} s

^{-1} and a bleaching region with a radius of 1 μm, then the characteristic recovery time will be ~0.5 s. The bleaching time should therefore be <33 ms. In practice, most cell biological FRAP applications using a similar bleach ROI size use bleach periods of ~100 ms or longer (

Phair and Misteli, 2000 ;

Shimi *et al*., 2004 ). One may therefore predict that in most FRAP experiments the assumption that negligible fluorescence recovery occurs during bleaching is not correct.

A direct demonstration that before the first postbleach image is acquired significant diffusion takes place (namely, for a 40-kDa dextran) is depicted in . Diffusion during bleach is obviously less important for slower moving molecules (i.e., a 500-kDa dextran).

Using the approximations mentioned above. it was possible to derive analytical formulas for the normalized fluorescence recovery curve. The three-dimensional correction is important for measurements in bulk solutions. In cellular measurements, it will contribute significantly only if the cell thickness is larger than the axial extent of the bleach volume. In a thin sample, the 2D variant should be used instead.

For simultaneously high values of the immobile fraction and of the diffusion coefficient the capability of the 3D (or 2D) method to correctly estimate these parameters is reduced. This happens because the mobile fraction concentration profile is not directly identified, being

*w*_{M} estimated from the fluorescence profile of the mixture of the mobile and immobile fractions. Thus, a higher level of immobilized molecules leads to an underestimation of the mobile fraction profile radius, and consequently, to an underestimation of the diffusion coefficient. A way to circumvent this problem was devised, by making a fitting procedure (with

Eq. 10) in two steps. The first run would be used to estimate the immobile fraction only, computing then the mobile fraction fluorescence profile (

Eq. 11), and in the second run corrected profile width values would be used. It is important to highlight that the approach describe here requires a new calibration every time there is a change in bleaching disk radius, laser beam power, or fluorophore.

To be sure that these expressions were adequate for analysis of real FRAP experiments, we generated recovery curves from the simulations and fitted those curves with the formula. The estimated diffusion coefficients and immobile fractions were always close to the parameters used in the simulations ().

The effect of noise was tested by generating several curves with the same level of Gaussian noise added to a simulated recovery curve. This procedure led to the conclusion that noise has a relevant role in the quality of the estimates, introducing some variability in the estimation. This was especially noticeable when simulating molecules with high diffusion coefficients. The best way to improve the estimation was to average the highest number possible of experiments.

As a first biological application, the new FRAP method was used to compare the diffusion rates of different size macromolecules in aqueous solution and in the nucleus of living HeLa cells. FITC-dextrans were either directly imaged in solution or microinjected into the nucleus. A rather homogeneous fluorophore distribution was observed, suggesting that the dextrans spread freely throughout both the aqueous sample and the nucleoplasm. In aqueous solution, the ratios between the diffusion coefficient values theoretically expected from the Stokes-Einstein equation and those estimated by FRAP both at 37 and 22°C were close to 1. As expected, in the nucleoplasm the diffusion coefficient values decreased relative to water. Noteworthy, higher molecular weight dextrans were proportionally more retarded in the nucleoplasm, at 37 and 22°C, than smaller molecular weight molecules (). This means that nucleoplasm deviates from an ideal liquid behavior, with the effective viscosity increasing with the size of the molecules. A similar observation was reported for FITC-dextrans injected into the eye vitreous, but not for dextrans diffusing in cystic fibrosis sputum (

Braeckmans *et al*., 2003 ), where the decrease in diffusion coefficient values of the FITC-dextrans seemed rather independent of their size. In contrast with the aqueous sputum, the vitreous is composed of a meshwork formed by polymers of hyaluronic acid. Most probably, this meshwork causes a sterical hindrance that is stronger for larger molecules (

Braeckmans *et al*., 2003 ). A similar situation is likely to occur in the nucleoplasm, where sterical hindrance is caused by the meshwork composed of chromatin and nonchromatin nuclear components. Most interestingly, the sterical hindrance effect inside the nucleus is similar at 37 and at 22°C (1.7± 0.3 and 1.5 ± 0.3 μm

^{2} s

^{-1} for 500-kDa dextrans). Yet, decreasing the temperature from 37 to 22°C reduces the expected mobility rate of the dextrans in aqueous solution by ~30% (from 32.8 ± 1.9 to 23.2 ± 1.1 μm

^{2} s

^{-1} for 500-kDa dextrans). We have previously observed that when cells are depleted of ATP or incubated at 22°C, messenger ribonucleoprotein (mRNP) particles show significantly reduced mobility rates in the nucleus, whereas large-molecular-weight dextrans are not much affected (

Calapez *et al*., 2002 ). The results reported here reinforce the view that spatial constraints to diffusion of dextrans inside the nucleus are insensitive to temperature and therefore energy independent. In contrast, energy-dependent processes are possibly involved in facilitating the diffusion of mRNP complexes in transit to the cytoplasm (

Calapez *et al*., 2002 ;

Carmo-Fonseca *et al*., 2002 ).

Inside the cell, binding events are expected to slow down the diffusion dynamics of macromolecules. In the case that the binding reaction is faster than the typical times involved in the diffusive process, an effective diffusion coefficient can be defined (

Crank, 1975 ). Such effective diffusion coefficient integrates the absolute diffusion coefficient and the rates of association and dissociation of the macromolecule. However, further work is needed to address the complex interplay between macromolecular diffusion and binding events that takes place in the living cell (

Phair *et al*., 2004 ;

Sprague *et al*., 2004 ).