## Introduction

Fluorescence recovery after photobleaching (FRAP) [1, 2] and fluorescence correlation spectroscopy (FCS) [3-5] were first introduced in the 1970s and are now widely used methods to investigate diffusion and binding properties of biomolecules in living cells [6]. In a FRAP experiment the steady state signal from fluorescent molecules in a region of interest (ROI) of a cell, for example, is recorded, followed by the rapid irreversible photoinduced bleaching of the fluorescence in the ROI with laser light of high intensity. The resulting unbalanced distribution of a usually large number of molecules gradually relaxes back to the steady state because of diffusion of bleached molecules out of the ROI and of fluorescent molecules into the ROI as well as the replacement of bleached with fluorescent molecules at immobilized binding sites, which is recorded with again highly attenuated laser light. Initially, FRAP experiments were carried out with a stationary bleach beam merged with a point or imaging fluorescence detector [7-10], referred to as point FRAP [1, 11-13]. Nowadays, owing to their widespread availability, mostly confocal laser scanning microscopes (CLSMs) are used where a diffraction-limited laser beam is scanned across the sample and toggled appropriately between high and low intensity [14-18], termed imaging FRAP [19-24].

To obtain the desired parameters characterizing the underlying diffusion and binding processes, a quantitative analysis is required based on analytical or numerical solutions of coupled reaction–diffusion models as described with sets of partial differential equations. Depending on diffusion coefficients and reaction rates, almost all FRAP curves can be categorized as one of the following cases, namely, pure diffusion, effective diffusion, separated reaction and diffusion kinetics, and the case requiring coupled reaction–diffusion modeling [21, 25-27]. In general, it is difficult to dissect clearly the contributions of diffusion and binding. Moreover, the results depend intricately on the experimental conditions, which may lead to fundamental misinterpretation and inconsistent data [28-33], for example, when neglecting diffusion during the bleach segment, when undersampling the data because of insufficient time resolution or when falsely estimating the three-dimensional (3D) shape of the bleach ROI.

FCS is a different relaxation technique where the focus of a confocal setup like a CLSM is fixed at a position of interest and the steady state concentration fluctuations of small numbers of fluorescent molecules because of thermally induced Brownian motion are recorded. From a temporal correlation analysis of the fluorescence signal, concentrations and diffusion properties of free molecules and larger complexes can be determined [34-37]. Typical and in particular commercial FCS setups are based on a confocal laser illumination and fluorescence detection scheme and are often integrated into a CLSM [38, 39].

Not least because of the rather standardized experimental conditions, FCS is well-described theoretically and allows to determine quantitatively diffusion coefficients and absolute concentrations and to distinguish different modes of diffusion [40, 41] by fitting analytical model functions to experimental data. In contrast to FRAP, FCS is conceptually “blind” to immobilized molecules. The correlation time from an FCS experiment typically characterizes diffusion, whereas a major readout of a FRAP experiment, the half-time of recovery, is often determined by convoluted diffusion and binding, making FCS and FRAP a promising pair of complementary methods.

The exon–exon junction complex (EJC) is formed via association of proteins during splicing of mRNA in a defined manner. Its organization provides a link between biogenesis, nuclear export, and translation of the transcripts. The EJC proteins accumulate in nuclear speckles alongside most other splicing-related factors and show both a mobile component with diffusion properties similar to inert fluorescent proteins and a fraction of reduced nuclear mobility when complexed with RNA [42], providing a model system with intricate diffusion and interaction properties to be studied with point FRAP and FCS.

In this study, we present an integrated approach where point FRAP is described theoretically, implemented employing the components also used for FCS on a CLSM, and applied experimentally. The theoretical treatment aims first at an extension of the established point FRAP formalism [1, 12] to 3D. By extending the concept of confocal continuous fluorescence photobleaching (CP) [10, 43-45] to higher bleach rates, we establish an explicit consideration of diffusion and binding during the bleach segment of a FRAP experiment, which has only rarely been accounted for so far [28, 29, 46]. This is expected to improve the diffusion coefficients resulting from a corresponding fit of the recovery curve and to agree with FCS results. In addition, we present a closed expression describing the coupled reaction–diffusion-induced redistribution after photobleaching covering all regimes usually treated separately. We corroborate the idea that point FRAP and FCS can be combined synergistically [8, 47, 48]: the diffraction-limited size of the bleach spot well below 1 μm and a time resolution in the range of microseconds in a point FRAP experiment is an advantage because the diffusional recovery is very fast compared to imaging FRAP. In this way, the limit where diffusion and binding are effectively uncoupled is pushed to higher rates. Moreover, the diffusional contribution can be measured simultaneously but independently with FCS. In order to assess and confirm our theoretical treatments, which are based on appropriate assumptions and approximations, we solve the reaction–diffusion equations numerically and compare the outcome to the analytical expressions. Finally, we demonstrate and evaluate the applicability of our approach by using a modified commercial FCS/CLSM system for the basic case of freely diffusive fluorescent proteins in living cells. Furthermore, we obtain a quantitative description of the more complex mobility of the EJC components Magoh and REF2-II at different localizations of mammalian cell nuclei.