Diffusion and binding analyzed with combined point FRAP and FCS

Authors


Correspondence to: Malte Wachsmuth, Cell Biology & Biophysics Unit, European Molecular Biology Laboratory, Meyerhofstrasse 1, 69123 Heidelberg, Germany. E-mail: malte.wachsmuth@embl.de

Abstract

To quantify more precisely and more reliably diffusion and reaction properties of biomolecules in living cells, a novel closed description in 3D of both the bleach and the post-bleach segment of fluorescence recovery after photobleaching (FRAP) data acquired at a point, i.e., a diffraction-limited observation area, termed point FRAP, is presented. It covers a complete coupled reaction–diffusion scheme for mobile molecules undergoing transient or long-term immobilization because of binding. We assess and confirm the feasibility with numerical solutions of the differential equations. By applying this model to free EYFP expressed in HeLa cells using a customized confocal laser scanning microscope that integrates point FRAP and fluorescence correlation spectroscopy (FCS), the applicability is validated by comparison with results from FCS. We show that by taking diffusion during bleaching into consideration and/or by employing a global analysis of series of bleach times, the results can be improved significantly. As the point FRAP approach allows to obtain data with diffraction-limited positioning accuracy, diffusion and binding properties of the exon–exon junction complex (EJC) components REF2-II and Magoh are obtained at different localizations in the nucleus of MCF7 cells and refine our view on the position-dependent association of the EJC factors with a maturating mRNP complex. Our findings corroborate the concept of combining point FRAP and FCS for a better understanding of the underlying diffusion and binding processes. © 2013 International Society for Advancement of Cytometry

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.

Materials and Methods

Cell Culture

We established stable cell lines expressing EYFP. HeLa cells (ATCC CCL-2) were transiently transfected using FuGENE6 (Roche Diagnostics, Seoul, Republic of Korea) with the EYFP plasmid according to the manufacturer's instructions and treated with 800 μg/ml G418 for selection. Stable clones were cultured in Dulbecco's modified Eagle's media (DMEM) containing 10% fetal bovine serum, 1% penicillin/streptomycin, and 200 μg/ml G418. Human MCF7 cells stably expressing EGFP-REF2-II and EGFP-Magoh were cultured as described previously [42]. For the experiments, cells were plated in LabTek chambered cover glasses (Fisher Scientific, Seoul, Republic of Korea) and allowed to grow for 48 h at 37°C in 5% CO2. Live cell observations were performed in phenol-free DMEM containing 20 mM HEPES, pH 7.0.

Modification of a Commercial CLSM for Point FRAP and FCS

For the FRAP and FCS experiments, we modified a commercial CLSM with FCS functionality (Leica SP2 AOBS FCS2; Leica Microsystems, Mannheim, Germany) resulting in an experimental setup conceptually similar to previous studies [7, 9], and described in detail in the Supporting Information.

Point FRAP Data Acquisition and Analysis

The work flow for point FRAP and FCS data acquisition in living cells started with recording a confocal image of a cell followed by the selection of a measurement spot where the beam was parked. The photon stream was acquired with a sampling rate of 1 MHz. For each bleach time we performed 50 separate measurements for free EYFP in HeLa cells and 10–15 separate measurements for each cellular localization of REF2-II and Magoh. The signals were resampled with two different frequencies after photobleaching; a high sampling frequency below 1 ms post-bleach time and a low sampling frequency above 1 ms. For free EYFP in HeLa cells, the frequencies were 50 kHz and 500 Hz, respectively; for the other proteins, the frequencies were 10 kHz and 500 Hz, respectively. They were normalized individually to the mean pre-bleach value and then averaged to get a single FRAP curve. The averaged curves were fitted to the model functions described below using Origin (OriginLab, Northampton, MA, USA).

FCS Data Acquisition and Analysis

The data for FCS processing and evaluation were extracted from the post-bleach segments of the FRAP recordings from regions where the fluorescence intensity had reached a plateau, or they were recorded separately. The correlation functions were computed as described in the theory section using the software “Fluctuation Analyzer” written in our laboratory, which allows to correct for slow processes. The resulting correlation curves were fitted to the 3D anomalous diffusion model, Eq. (20), with one or two independent components using Origin.

Simulation of CP and FRAP Data

Simulated CP and FRAP data were computed using MATLAB (MathWorks, Natick, MA) by applying a finite difference approach to Eq. (9) using volume elements with rotational symmetry as defined by the confocal setup. The resulting concentration distribution was transformed into a fluorescence signal by numerically applying Eq. (3). For the focal geometry we assumed math formula for the lateral math formula radius and the structure parameter, respectively (details regarding the parameters are explained below). Spatial sampling was set to math formula and math formula for the radial and the axial dimension, respectively, in cylindrical coordinates. A volume of 2,500 nm in radial and 7,500 nm in axial direction was considered. The temporal sampling was set to math formula where D is the diffusion coefficient, α is the photobleaching rate in the center of the focus, and kon and koff are the association and dissociation rates with immobilized binding sites. We assumed a constant concentration on the boundary of the simulation volume and ensured the applicability by comparison with constant gradient boundary conditions that yielded virtually the same time course of the fluorescence signal. 5—10% Gaussian noise was added to resulting concentration and fluorescence signal curves before further processing.

Results

Theory

Spatiotemporal fluorescence distribution during photobleaching

In a FRAP experiment, the interpretation of the post-bleach fluorescence time trace depends crucially on the distribution of fluorescent molecules directly after bleaching. Therefore, we treat the spatiotemporal fluorescence distribution for the bleach and the post-bleach segment separately. For the bleach segment, this results in an extension of the quantitative treatment of CP [39, 43, 44], which can be used independently to study binding and diffusion properties. A consistent interpretation of both bleach and post-bleach segment is expected to provide more reliable information about the binding and diffusion processes.

The following assumes confocal laser illumination and fluorescence detection with a detection pinhole diameter of one Airy disk or smaller. Thus, the molecular detection efficiency (MDE) or total point spread function (PSF) is the product of illumination and detection PSF, math formula, and is approximated as 3D Gaussian function [49-52] with further simplifications, see Supporting Information.

CP of immobilized molecules

The differential equation

display math(1)

describes the distribution of fluorescent molecules math formula because of diffusion and photobleaching, with the diffusion coefficient D, the Laplace operator Δ and the intensity-dependent and fluorophore-specific bleach rate α in the center of the PSF. As shown previously in Eq. (1) for the case of immobilized molecules, math formula, and assuming a uniform pre-bleach distribution, i.e., math formula, the fluorophore distribution after photobleaching is

display math(2)

This is the initial distribution for the post-bleach segment. The concentration distribution can be transformed into the detected fluorescence signal by integrating over the PSF:

display math(3)

where F0 is the initial fluorescence signal and math formula the effective focal volume with w0 and z0 as lateral and axial math formula radius, respectively, and the structure parameter math formula. The transition from finite integration limits as defined by the physical size of the cell or nucleus to infinite limits does not affect the results detectably because the PSF decreases below 0.1% already at a distance of 2w0 away from its center. We could show previously [39] that for immobilized molecules, the fluorescence time course can be approximated as

display math(4)
CP of diffusive molecules

It is difficult to find a closed analytical solution of Eq. (1) for CP of diffusive molecules that covers the full range between an almost uniform concentration decaying exponentially for small bleach versus diffusion rates and apparently immobile molecules as described above for large bleach rates. Here, we describe the distribution of molecules as a Gaussian function imprinted into a uniform initial concentration and parametrized by a time-dependent width and depth according to

display math(5)

which approaches Eq. (2) for short times. Under appropriate assumptions, this results in differential equations for p and n and corresponding boundary conditions (see Supporting Information). With a further evaluation of Eqs. (1), (2), and (5) approximate solutions of the differential equations for the signal-relevant central area are

display math(6)

where math formula is the diffusional dwell time of the molecules in the focal volume as defined in FCS theory. Using Eq. (3), the resulting time course of the fluorescence signal is

display math(7)

For sufficiently large diffusion times math formula, the fluorescence time course becomes independent of math formula and is described more appropriately by Eq. (4). Bleaching of the overall mobile pool because of its finite size and because of the deviation of the real PSF from a 3D Gaussian function can be taken into consideration by adding an exponential decay with a pool bleach rate kpool, that is, math formula. This rate is smaller than the focal bleach rate α by a factor corresponding to the ratio of focal-to-pool volume as shown previously both in theory and experiment [39]. Assuming linear dimensions of 1, 5, and 10 μm for the focal, nuclear, and cellular volume, respectively, the volume ratio is 1:100 (nucleus) or smaller. Thus, the bleaching-related processes described here, i.e., those in the focal volume, are at least 100-fold faster than cell- or nucleus-wide pool bleaching. Therefore, we describe the local and the global concentration distributions independently and introduce the pool bleaching behavior only at the end.

CP of molecules with exchange between immobilized binding sites and a diffusive pool

In a previous study [39], we have described the coupled reaction–diffusion case where the molecules A experience a continuous exchange between immobilized binding sites B and a diffusive pool

display math(8)

as a solution of the coupled differential equations for the two fractions

display math
display math(9)

for the frequently encountered situation that diffusion dwell-times of soluble molecules within the confocal volume are typically of the order of 10 ms or less. The association and dissociation rates are referred to as cBkon and koff, respectively, where cB is the concentration of the binding sites. The case of bleach rates higher than the binding rates and similar to the diffusion rates is covered by the above-mentioned considerations for the diffusive and the immobile fraction, respectively. The case of similar diffusion and binding rates follows the same rationale as before [39] and results effectively in replacing the expression for the mobile fraction with Eq. (5). The case of bleach rates similar to both the binding and the diffusion rates can be rewritten as Eq. (S19), see Supporting Information.

If the association is fast compared to diffusion, i.e., math formula [28], a free and a bound fraction cannot be distinguished but are considered as a single fraction with a reduced diffusion coefficient math formula. The time course of the concentration and of the fluorescence signal are thus treated as described above.

Spatiotemporal fluorescence redistribution after photobleaching

In order to describe the redistribution of fluorescent molecules after photobleaching, that is, during the post-bleach segment of a FRAP experiment, the concept of Green's functions is employed to solve the differential equation under the respective boundary conditions, see Supporting Information. The most frequently used and model-independent parameters to describe a recovering fluorescence signal math formula are the so-called immobilized fraction math formula and the half-time of recovery math formula, after which the post-bleach signal has increased by 50% [6]. From math formula, an apparent diffusion time math formula can be computed [1].

Redistribution of diffusive molecules after CP of an apparently immobile pool

In FRAP experiments, it is usually assumed that the bleach rate α is significantly higher than all other rate constants such as the dissociation rate from immobilized binding sites koff and the diffusion rate math formula, the reciprocal dwell time of the molecules in the observation volume. Thus, the initial post-bleach distribution obeys Eq. (2) for math formula (the bleach time or length of the bleach segment), and one obtains

display math(10)

for the subsequent time course of the molecular distribution as well as

display math(11)

for the time course of the fluorescence signal with math formula, that is, the fluorescence signal recovers to the maximum signal defined by the remaining available pool. We use the decaying functions

display math(12)

normalized to unity to describe the time dependence.

Redistribution of diffusive molecules after CP of a diffusive pool

However, the assumption of large bleach rates is not always fulfilled because of insufficient laser intensity used for photobleaching or because of high mobility of the fluorescent molecules. Then, the initial post-bleach distribution obeys Eqs. (5) and (6) for math formula, and one obtains

display math(13)

for the time course of the molecular distribution as well as

display math(14)

for the time course of the fluorescence signal. For typical values of the structure parameter of math formula, the half-time of recovery can be approximated as math formula with an error of <3%.

Approximation for short bleach times

For short bleach times, i.e., math formula, both Eqs. (2) and (5) approach

display math(15)

and the fluorescence recovery is simplified to

display math(16)

i.e., the half-time of recovery is approximately math formula. An FRAP experiment with math formula is obviously impossible; nevertheless, the parameters of interest can be extrapolated from series of bleach times for math formula.

Redistribution of molecules with exchange between immobilized binding sites and a diffusive pool

In general, Eq. (S18) provides an appropriate description of the initial post-bleach distribution for this case. Fortunately, as shown by experimental data (see below), the typical bleach rate α in a FRAP experiment can be chosen to be significantly higher than the rate koff of dissociation from immobilized binding sites so that Eq. (2) can be used.

To describe the coupled redistribution of transiently immobilized and diffusive molecules we follow the rationale established for FCS theory [3] but extend the approximation given for math formula to the regime of math formula, see Supporting Information. Assuming the same bleach behavior for both fractions and the post-bleach intensity F0 as defined in Eqs. (4), (7), or (S18), we obtain for the fluorescence recovery

display math(17)

where sd,e are factors weighting between the two approximations for the diffusion and the reaction contribution, respectively. Here, math formula is the decay function as defined above that simplifies to math formula for short bleaching and corresponds to math formula for diffusive molecules, whereas math formula is a decay function with a reduced amplitude and retarded decay as compared to math formula that is defined as

display math(18)

and simplifies to math formula for short bleaching and corresponds to math formula for bleaching of diffusive molecules. Thus, the recovery is composed of a coupled apparently diffusive component with a reduced diffusion coefficient, a purely diffusive component, and a pure dissociation component.

When the redistribution of the diffusive molecules is fast enough to be equilibrated before an effective exchange with the bound pool sets in, then math formula and both fractions can be treated independently, that is, diffusion and binding are considered uncoupled and Eq. (17) simplifies to the sum of a purely diffusive fraction with a diffusion coefficient D and a bound fraction recovering exponentially with koff as expected. The free one follows Eqs. (10) or (13) and the bound one is simply described as

display math(19)

At the other end, when the rate constants are sufficiently large, math formula and Eq. (17) reduces to a purely diffusive recovery with a reduced diffusion coefficient math formula, that is, the two fractions cannot be distinguished.

Fluorescence correlation spectroscopy

In an FCS experiment, fluorescence photons from a confocal detection volume are recorded on a photon-counting detector for an acquisition length Θ with a typical time resolution of 1 μs or better, resulting in a time course of the fluorescence signal math formula. A temporal autocorrelation analysis is then applied to characterize the concentration fluctuations of the fluorescent molecules in the detection volume. Assuming 3D anomalous diffusion and molecular blinking as sources of fluctuations, the rather general model function

display math(20)

is obtained [41, 53], where cγ represents the average number of molecules in the observation volume, ρ the fraction of molecules in a nonfluorescent state with a lifetime τblink, and α the anomaly parameter of diffusion, resulting in a mean dwell time τdiff of the molecules in the focus as defined above. The anomalous diffusion model is very general, covers also free diffusion, and provides a good parametrization of FCS data without further knowledge of the underlying diffusion process. Typical diffusional correlation times do not exceed 100 ms so that with an acquisition length of a few 10 s, averaging over several hundred or thousand diffusional fluctuations is achieved. The impact of additional short-term binding on the autocorrelation function is described in the Supporting Information and results in Eq. (S28) for the diffusional contribution to math formula.

Correction for slow fluctuations

A frequently encountered problem of FCS especially in living samples are slow signal fluctuations because of bulk photobleaching as well as translocations of larger intracellular structures or entire cells. FCS fluctuations are weighted with the square of their brightness so that often slow fluctuations obscure completely the contributions from single diffusing molecules and render a further evaluation for example using Eq. (20) impossible. To overcome this, we have introduced a sliding average approach where the autocorrelation function is calculated over a small time window Θwin and then averaged over the complete length Θ according to

display math(21)

The window length must be chosen carefully so that the slow fluctuations are eliminated but the usually faster processes of interest are not affected. Here, this is the case for a window of 1 s, which was used throughout this study. This way, the relatively slowly recovering fluorescence signal of a FRAP post-bleach segment as well as the slowly decaying signal of a CP experiment can be used for correlation analysis.

Experimental Results

Comparison of simulated CP and recovery curves with analytical expressions

Bleach segment—CP

To assess the usability of the approximated analytical expression for CP of diffusive molecules at bleach rates comparable to the diffusion rate, we fitted Eq. (7) to the simulated CP curves computed for a set of bleach rates math formula with a diffusion time of math formula, that is, a diffusion coefficient of math formula, so that math formula, see Figure 1a. For math formula the diffusion time could be retrieved from the fit within ∼10% of the values used for the simulations, see Supporting Information Table S1. For the bleach rate this held true only for math formula, whereas for math formula a reasonable fit with Eq. (4), i.e., bleaching of immobilized molecules, could be found. Since the absolute bleach rate is usually neither well defined experimentally nor of major interest, Eq. (7) presents a good description of the bleach segment of a FRAP experiment in the case of diffusion during bleaching and of a CP experiment with moderate rather than small bleach rates, respectively (see also Supporting Information Supporting Movies). The validity of the analytical expressions for transiently or permanently immobilized molecules was shown previously [39] so that we have now a complete coverage of all combinations of bleach, binding, and diffusion rates for FRAP bleach segments and CP.

Figure 1.

Simulated FRAP data and fits with analytical model functions. (a) Simulated time course of the bleach segment intensity for math formula (black, in order of decreasing final intensity), math formula, math formula and the fits (gray) with Eq. [7]. (b) Simulated time course of the post-bleach recovery intensity for math formula, math formula, math formula and math formula (black, in order of decreasing initial intensity), fits (gray) with Eq. [14], resulting diffusion times normalized to the value at math formula from the fit with Eq. [11] (squares) and with Eq. [14] (circles). (c) Simulated time course of the post-bleach recovery intensity for math formula, math formula, math formula, math formula and math formula, math formula, math formula, and math formula (black, in order of decreasing intermediate intensity) with linear and logarithmic timescale and the fits (gray) with Eq. [17].

Postbleach segment—recovery of diffusive molecules

Next, we computed the redistribution of molecules and the resulting recovery after photobleaching with different bleach rates math formula and different bleach times math formula, see Figure 1b. The recovery curves could be fitted well both with Eqs. (11) and (14). However, in the first case, i.e., when neglecting diffusion during bleaching, the retrieved diffusion time matched the value used for the simulations only for very short bleach times and increased with increasing bleach time, see Figure 1b and Supporting Information Table S1. In contrast, in the second case, i.e., when allowing for diffusion during bleaching, the fitted diffusion time did not deviate by more than ∼10% from the expected value independent of the bleach time. Thus, taking diffusion during bleaching into consideration can improve the outcome of a FRAP experiment significantly, and especially when using commercial equipment the bleach intensity cannot always be increased enough for diffusion during bleaching to be neglected. In addition, Figure 1b shows that an extrapolation to a virtual zero bleach time allows to retrieve the diffusion time independent of the assumptions made for the processes during bleaching.

Figure 2.

FRAP and FCS of EYFP in HeLa cells. (a) Confocal fluorescence image of HeLa cells expressing EYFP; scale bar 10 μm. (b) Post-bleach intensity time course of EYFP in HeLa cells for math formula(black) and fits with an exponential recovery, Eq. [11] and Eq. [14] (green, blue, red). (c) Post-bleach intensity time course for math formula (black, green, blue, orange) and fits with Eq. [14] (red). (d) Half-times of recovery (experiment: black squares; fit: dashed lines) and initial post-bleach intensities (experiment: blue squares; fit: continuous lines) of the bleach time series in (c) based on global fits assuming diffusive (red) or immobilized molecules during bleaching (green). (e) Fluorescence intensity trace of a FRAP experiment showing the segments used for FRAP and for FCS. (f) FCS curve obtained from the corresponding segment in (e). [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Postbleach segment—recovery of transiently immobilized molecules

Furthermore we computed the redistribution of molecules and the resulting recovery after strong photobleaching ( math formula) of molecules undergoing diffusion ( math formula, math formula) as well as association to and dissociation from homogeneously distributed immobile binding sites with different rates covering math formula and math formula, see Figure 1c. First, we fitted the curves with the expression derived above, Eq. (17), to obtain the globally optimized weighting parameters math formula and math formula that we used subsequently to obtain the diffusion time and the association and dissociation rate from fits. As one may educe already from the logarithmic timescale plot in Figure 1c, for chemical relaxation rates math formula a single diffusive component was sufficient to fit the data, that is, the system was in the effective diffusion regime, and only with the knowledge of the initially used ratio math formula the real diffusion time could be well retrieved. With decreasing R a biphasic behavior of the recovery curve emerged and a fit of Eq. (17) allowed to retrieve the diffusion time and the chemical reaction rates with a mean deviation of ∼10% for all math formula, see Supporting Information Table S1. Nevertheless, providing the diffusion time (e.g., from FCS) could improve the fit especially in the intermediate regime of comparable reaction and diffusion rates. In summary, our model describes well the regimes of both fully coupled reaction–diffusion processes and separated binding and diffusion.

The mobility of EYFP in HeLa cells

FRAP of EYFP with varying bleach times

We investigated the practical feasibility of the different models for the bleach segment and the description of the redistribution process by performing FRAP experiments in HeLa cells expressing free EYFP (Fig. 2a). Figure 2b shows the recovery curve for a bleach time of 100 μs and the fit curves using an exponential recovery, Eqs. (11) and (14). One can hardly distinguish the fit curves over the whole course of the experiment except for a region at ∼90% of recovery, and the sums of squares of the residuals (2.45·10−2, 2.54·10−2, and 2.50·10−2, respectively) also do not allow a distinction of the quality of the fits. The resulting diffusion times of 520, 560, and 450 μs, respectively (see Table 1), corresponding to diffusion coefficients of 21.2, 19.7, and 24.5 μm2s–1, agreed reasonably and showed together with immobilized fractions of <1% that free diffusion of a single component could be considered as the source of redistribution but also that even for such a short bleach time, some diffusion must have occurred during the bleach segment.

Table 1. Mobility of EYFP in HeLa cells
MethodBleach time T (ms)Half-time of recovery τ1/2 (ms)Immobilized fraction fimmo (%)Diffusion time τdiff (ms)
Weak bleaching approximately (2/3 τ1/2)Immobilized during bleachingDiffusion during bleachingAuto-correlation function
  1. Mobility of EYFP in HeLa cells

FRAP5.06.23 ± 0.42<19.76 ± 0.241.58 ± 0.16
 1.02.23 ± 0.20<12.84 ± 0.080.72 ± 0.11
 0.51.25 ± 0.17<11.22 ± 0.040.63 ± 0.12
 0.31.03 ± 0.13<10.96 ± 0.040.63 ± 0.13
 0.10.78 ± 0.09<10.56 ± 0.020.45 ± 0.09
 Extrapolated to 0.00.78 ± 0.13<10.52 ± 0.090.49 ± 0.180.50 ± 0.05
 Global fit<11.61 ± 1.610.45 ± 0.05
FCS0.46 ± 0.07

To confirm and better assess this effect, we carried out FRAP experiments with a series of bleach times ranging from 0.1 to 5 ms under otherwise constant conditions as shown in Figure 2c. As expected, the half-time of recovery increased with increasing bleach time. For all three models, the (apparent) diffusion time also depended on the bleach time; however, this effect was smallest when taking diffusion during bleaching explicitly into account, i.e., when fitting with Eq. (14) as shown in Figure 2d and Table 1. The degree of independence of the diffusion time from the bleach time serves as a measure how well the different models describe the data, showing that the best description is obtained for Eq. (14), i.e., when taking diffusion during bleaching into consideration.

In order to rule out reversible photobleaching, i.e., the return of apparently photobleached molecules into a fluorescent state, we performed point FRAP experiments in HeLa cells expressing a core histone H2A-EYFP fusion protein. Except for a fraction of ∼10% recovering in a diffusion-like manner, the fluorescence remained depleted during the time course of the experiments.

Global analysis of bleach time series

We extrapolated linearly the diffusion time to a virtual zero bleach time resulting in 520, 490, and 500 μs, respectively (see Table 1). Thus, we obtained consistent results for the diffusion time in a model-independent way by eliminating the effect of diffusion during bleaching.

For a further comparison of the models with and without diffusion during bleaching taken into account, we applied a global analysis of the respective models. The results are presented in Figure 2d where fits of the predicted to the measured half-times of recovery and initial post-bleach intensities as a function of the bleach time are shown. When including diffusion during bleaching, i.e., when employing Eqs. (6), (7), and (14), the fit matches the experimental data much better yielding a diffusion time of 450 ms and a bleach rate of 3670 s−1 as compared to assuming immobilized molecules during bleaching, Eq. (11), with 1.61 ms and 9850 s−1. This corroborates further that our explicit consideration of simultaneous diffusion and bleaching is valid and suggests the use of multiple bleach times.

FCS of EYFP in HeLa cells

To determine the diffusion time of EYFP molecules in HeLa cells independently but using the same samples and raw data, we computed the autocorrelation function of the fluorescence signal taken from the tail of the post-bleach segment of the FRAP experiments by employing Eq. (21). Figure 2e shows a single intensity time trace with the regions used for FRAP and FCS, respectively. Figure 2f shows the resulting autocorrelation function and the fit curve of an anomalous diffusion model, Eq. (20). The diffusion process is characterized by a diffusion time of 460 μs and an anomaly parameter of 0.72, see Table 1. This confirms well the FRAP results, earlier observations [41, 54, 55], and the assumption that the EYFP molecules are diffusive. Thus, point FRAP can be used to quantify even the very fast process of diffusion of freely mobile molecules in living cells. The somewhat larger value as obtained with FRAP most likely follows from the deviation from free diffusion as expected in the interior of a living cell.

The mobility of the EJC components Magoh and REF2-II in MCF7 cells

FRAP and FCS of Magoh and REF2-II in MCF7 cells

The distribution of two components of the EJC, the outer shell protein REF2-II and of the core protein Magoh, can be classified as a quite homogeneous nuclear background and an additional accumulation in so-called speckles, initially identified as accumulation sites of the splicing factor SC-35 [56], see Figures 3a and 3b. We studied their mobilities in the nucleoplasm and in speckles with FRAP and FCS. Previously, we could identify two components with position-dependent contributions and mobilities [42]. However, the information about the nature and quantitative characteristics of the different mobility states remained rather speculative. In order to further elucidate this, we performed FRAP experiments at the different localizations with a bleach time of 1 ms (Figs. 3c and 3d), yielding half-times of recovery of 20 ± 3 ms and 60 ± 4 ms for REF2-II in the nucleoplasm and in speckles, respectively, as well as 4.2 ± 0.7 ms and 7.1 ± 0.8 ms for Magoh.

Figure 3.

FRAP and FCS in MCF7 cells expressing EGFP-REF2-II and EGFP-Magoh. Confocal fluorescence image of (a) EGFP-REF2-II- and (b) EGFP-Magoh-expressing cell; scale bar 2 μm. Post-bleach intensity time course for math formula of (c) EGFP-REF2-II and (d) EGFP-Magoh in nucleoplasm (black) and speckles (blue) and fits (red) as described in the text. (e) FCS curve of EGFP-REF2-II in nucleoplasm (black) and speckle (blue). [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

In the REF2-II-expressing cells, we acquired FCS data at the said localizations (Fig. 3e; FCS of Magoh could not be recorded because of high-concentration levels), resulting in weighted average diffusion times, calculated as math formula, of 17 ± 6 ms and 23 ± 9 ms in the nucleoplasm and in speckles, respectively, see Table 2.

Table 2. Diffusion times and free, transiently bound, and immobile fractions of REF2-II- and Magoh-GFP in MCF-7 cells from FRAP and FCS
Method (bleach time)SiteREF2-IIMagoh
τdiff,1 (ms)τdiff,2 (ms)koff (s-1)f1 (%)f2 (%)fimmo (%)τdiff,1 (ms)τdiff,2 (ms)f1 (%)f2 (%)fimmo (%)
  1. a

    Fit with two diffusive and an immobile fraction.

  2. b

    Fit with a diffusive, a transiently bound, and an immobile fraction.

  3. c

    Fit with two diffusive fractions, see also [42].

FRAPa(1 ms)Nucleoplasm3.8 ± 1.370.4 ± 28.963.5 ± 11.335.0 ± 11.41.5 ± 0.11.7 ± 0.248.6 ± 0.773.2 ± 2.224.2 ± 2.22.6 ± 0.1
Speckle4.4 ± 0.6333.2 ± 29.148.7 ± 2.144.4 ± 2.26.9 ± 0.11.1 ± 0.276.3 ± 9.149.9 ± 2.332.2 ± 2.417.9 ± 0.1
FRAP, global fit (0.5–10 ms)Nucleoplasma5.0 ± 0.792.7 ± 4.546.3 ± 1.149.1 ± 1.24.6 ± 1.61.9 ± 0.447.7 ± 6.556.9 ± 3.033.2 ± 2.89.9 ± 4.1
Specklea5.6 ± 0.4364.8 ± 6.135.5 ± 0.447.5 ± 0.417.0 ± 0.61.6 ± 0.281.8 ± 3.042.5 ± 0.427.0 ± 0.430.5 ± 0.4
Speckleb21.5 ± 0.90.53 ± 0.0139.8 ± 2.843.9 ± 2.916.3 ± 0.6
FCScNucleoplasm1.9 ± 0.561.3 ± 37.175.0 ± 14.0
Speckle2.4 ± 0.983.4 ± 29. 874.0 ± 11.0

To transform the half-times of recovery into diffusion times we acquired a series of FRAP experiments with bleach times of 0.5, 1, 5, 10, 30, 50, and 100 ms, see Figure 4, and determined the diffusion time according to math formula after extrapolation to math formula, yielding 14 ± 5 ms and 90 ± 9 ms for REF2-II and 4.7 ± 1.0 ms and 5.1 ± 1.0 ms for Magoh in the nucleoplasm and in speckles, respectively. While the numbers for REF2-II in the nucleoplasm and for Magoh at both localizations indicate that the molecules are mobile, the remaining discrepancy for REF2-II in speckles must be because of additional binding.

Figure 4.

FRAP of EGFP-REF2-II and EGFP-Magoh. Post-bleach intensity time course for math formula (black, green, blue, orange) and fits (red) with Eqs. (14) or (17) of (a) EGFP-REF2-II and (b) EGFP-Magoh in nucleoplasm as well as (c) EGFP-REF2-II and (d) EGFP-Magoh in speckles. Half-times of recovery as a function of bleach time in nucleoplasm (black) and in speckles (blue) and interpolating splines (red) for (e) EGFP-REF2-II and (f) EGFP-Magoh. (g) Model for the nuclear mobility of REF2-II and Magoh. Apparent diffusion coefficients were calculated for math formula and averaged from FRAP and FCS data (REF2-II) or taken from FRAP data (Magoh). The reaction rates of transient immobilization of REF2-II in speckles were determined from FRAP. The exchange between the fast and the slowly diffusive fraction was not observed in FCS and FRAP so that we estimated math formula for the longest math formula (dashed straight reaction arrows). The exchange between the diffusive and the immobilized pool of the mRNP complex was not observed in FRAP so that we estimated math formula as defined by the duration of the post-bleach segments (dashed curved reaction arrows). [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Fit with two-component diffusion and complete diffusion-reaction model

Therefore, we fitted the FCS and the FRAP data with a two-component diffusion model, see Table 2, for 1 ms bleach time as well as globally for all bleach times as described for EYFP. We could identify for both REF2-II and Magoh at both localizations a fast component with a diffusion time of a few microseconds and a slow component with several 10–100 ms. For REF2-II, the diffusion times showed a good agreement in the nucleoplasm but not in speckles so that we repeated the FRAP fits with Eq. (17), that is, the complete model for a single diffusive component (two could not be resolved) and an association/dissociation process. Only for REF2-II in speckles, the fit converged with math formula yielding an effective diffusion time of 21.5 ± 0.9 ms probably representing the diffusive molecules, especially when compared to 23 ms as obtained with FCS, and a dissociation rate of 0.53 ± 0.01 s–1 characterizing transient immobilization. From the fractions of bound and free molecules, the law of mass action allowed us to determine an association rate of 0.66 ± 0.05 s–1. For REF2-II in the nucleoplasm and for Magoh at both localizations, we could only find diffusive fractions (Table 2).

These observations in combination with the position-dependent fully immobile fractions support and extend quantitatively the model of the mobility of EJC components [42], see Figure 4g. From a diffusive pool of proteins either in monomeric state or in small complexes (represented by the fast diffusion time) present in the whole nucleus, the inner core EJC factor Magoh as well as the outer shell factor REF2-II bind to pre-mRNA forming a 20- to 40-fold larger yet diffusive mRNP complex (represented by the slow diffusion time). In the process of EJC formation, the mRNPs may become immobilized predominantly in speckles (represented by the immobile fraction) whereas the outer shell factor REF2-II binds transiently to the mRNP for a mean association time of ∼1.9 s until it is either released or the EJC is formed properly and the mRNP complex becomes mobile again and ready for nuclear export.

Discussion

Very often FCS and FRAP data as well as FRAP data acquired at different time- and length scales feature apparent discrepancies of resulting diffusion and reaction properties. Therefore, in this study we extend the existing formalism applied to point FRAP [1, 12] with an explicit consideration of diffusion during bleaching in 3D, which includes the previously disregarded case of CP with high-bleach rates [39]. Moreover, we develop an analytical expression that solves the coupled reaction–diffusion scheme for a point FRAP post-bleach segment and that includes the different regimes of transport-to-reaction-rate ratios, which usually require an a priori classification [21, 25-27]. It is expressed in a way that makes the results immediately comparable to typical parameters obtained from FCS experiments. The outcome is assessed and confirmed with numerical solutions of the underlying differential equations. To acquire point FRAP and FCS data in living cells, a CLSM/FCS setup is modified such that it can be used for both FCS and point FRAP at the same time with microsecond time resolution. The applicability of theory and setup is evaluated and shown with EYFP in living cells, resulting in a more precise measurement even of rather high-diffusion coefficients in good agreement with FCS data and leading to an experimental way to reduce the effect of diffusion during bleaching by acquiring bleach time series. In principle, even FCS and point FRAP assess different scales, namely, fluctuations seen with FCS are driven by diffusion inside the observation volume, whereas the redistribution in a FRAP experiment is driven by diffusion inside as well as in the vicinity of the observation volume. We obtain very comparable diffusion properties when using the confocal volume as observation volume, which suggests that the diffusion properties are very similar on these different scales. Subsequently, the novel formalism and experimental approach is applied to the EJC components REF2-II and Magoh that are known to have complex contributions of diffusion and binding to their mobilities at different localizations in the cell nucleus [42]. Different diffusion and reaction components are identified and characterized quantitatively, yielding a refined view on the spatiotemporal association pattern of outer shell and core EJC components with the maturating mRNP and thus supporting the concept of combined point FRAP and FCS experiments.

Acknowledgments

We would like to thank Leica Microsystems for technical support.

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