t: time v: scanning speed of laser beam w: radius in the focal plane of a circular bleached region zb: Gaussian axial width at the e-2 level of the photobleaching beam zi: Gaussian axial width at the e-2 level of the imaging PSF y: distance between two scanning  : photobleaching parameter; also used in some equations as diffusion anomaly coefficient : typical length scales in the microstructure : standard deviation  : dynamic viscosity : transport coefficient, time-dependent diffusion : wavelength : chemical potential 1. Introduction 1.1 Principle of FRAP Fluorescence recovery after photobleaching (FRAP) is a widely established and powerful fluorescence microscopy based method for obtaining information on the dynamics of mobile fluorescent molecules at the micrometer scale. Specifically, FRAP can be used to obtain quantitative information on the micrometer scale of the diffusion process, transient binding events and directed transport.
where rb and zb are the dimensions in the radial and axial orientation. Note that the same Gaussian approximation is equally used for two-photon photobleaching (Brown et al., 1999, Mazza et al., 2008, Mazza et al., 2007). The 3-D concentration distribution of fluorophores after photobleaching with a stationary light beam for a time t can be readily obtained from Eq. and : . The very same expression holds for the two-photon case: It was experimentally tested whether Eqs. and accurately describe the 3-D bleach profile of a focused laser beam. Figure 8 shows the radial and axial sections of the 3-D distribution of photobleached molecules in single photon excitation (Figure 8A and B) and two-photon excitation (Figure 8C and D). Figure 8E and F show the comparison between the 3-D effective bleached volumes (black lines) and the corresponding 3-D model based on the Gaussian approximation (red lines). It is clear that the 3-D 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 3-D bleached region. In single-photon FRAP, the Gaussian description should be used only in the 2-D case of lateral diffusion. Figure 7 shall be inserted here. 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 2-D case as well,
Figure 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., 2007. Figure 9.The correction constant 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., 1976. Figure 10. A two-dimensional pattern is photobleached by scanning the laser beam with speed v along adjacent line segments that are spaced with a distance Δy. The laser intensity is zero outside the boundaries of the pattern . Reprinted with permission of Braeckmans et al. (2003). Figure 11. (A) A FRAP experiment performed on a solution of FITC-dextran with a molecular weight of 150 kDa. The first image shows the sample before bleaching. The photobleaching of the disk is illustrated in the second image. The subsequent images show the fluorescence recovery. The scale bar represents 20 µm. (B) The fluorescence recovery inside the photobleached disk extracted from the image sequence (●). A best fit of Eq. yields the diffusion coefficient D. Reprinted with permission of Braeckmans et al., 2003. Figure 12. A rectangular area with width lx and height ly is photobleached by scanning the laser beam with speed v along adjacent line segments that are spaced with a distance Δy. The laser intensity is zero outside the rectangular area.