Dissecting the Contribution of Diffusion and Interactions to the Mobility of Nuclear Proteins

DNA constructs, protein purification, cell lines, and cell culture

H2B-diHcRed was described in Gerlich et al. (27). β-galactosidase-diHcRed was made by fusing the entire coding sequence of β-galactosidase (28) 5′ to diHcRed (27), generating a LPDPPVAT linker between the two proteins. pEGFP-RCC1 was made by fusing the entire coding sequence of RCC1 (a generous gift from Iain Mattaj) 3′ to EGFP (Clontech Laboratories, Palo Alto, CA) generating a SGLRS linker between the two proteins. PAGFP-RCC1 was made by replacing the coding sequence of EGFP by PAGFP (11). H2B-PAGFP was made by fusing the entire coding sequence of H2B 5′ to PAGFP generating a DPP linker between the two proteins. PAGFP-hERα was made using EGFP-hERα (29) and replacing EGFP by PAGFP. H1.1, H1.2, H1.3, H1.4, and H1.5-PAGFP were made by fusing the entire coding sequence of the different H1 5′ to PAGFP generating the linker PGIHRPVAT between the proteins. PAGFP-SUV39H1 and PAGFP-SUV39H1-H320R were made by fusing the entire coding sequence of SUV39H1 and the mutant H320R (25) 3′ to PAGFP generating a YSDLEGGRDYKDDDDKGGR linker between the two proteins. EGFP-HP1β was made by fusing the entire coding sequence of HP1β (a generous gift from Howard Worman) 3′ to EGFP, generating a SGLRSLE linker between the two proteins.

Normal rat kidney (NRK) cells stably expressing H1.1-PAGFP, H1.2-PAGFP, H1.3-PAGFP, H1.4-PAGFP, H1.5-PAGFP, PAGFP-SUV39H1, H2B-PAGFP, and H2B-diHcRed were selected according to standard protocols and maintained in DMEM/10% FCS/0.5 mg/ml G418. Transfections were done using FuGene 6 (Roche, Mannheim, Germany). PAGFP-RCC1, PAGFP-hERα, and PAGFP-SUV39H1-H320R were transiently transfected and experiments were performed 24 h or 48 h after transfection for PAGFP-RCC1and PAGFP-hERα, and one week after transfection in the case of PAGFP-SUV39H1-H320R. PAGFP-hERα photoactivation was performed in the presence of 10 nM of β-estradiol (Sigma, Taufkirchen, Germany).

For microscopy, cells were cultured in No. 1 LabTekII chambered cover glasses (Nalge Nunc International, Naperville, IL) and maintained at 37°C on the microscope as described (30). In all experiments either H2B-DiHcRed or β-galactosidase-diHcRed were used as nuclear reference for image alignment.

PAGFP-RCC1 was histidine-tagged and purified on a nickel-agarose column (Qiagen, Valencia, CA). It was immobilized in a 30% acrylamide gel cast between a coverslip and a microscope slide to test photoactivation settings.

Photoactivation and three-dimensional imaging

Photoactivation experiments were performed on a Zeiss LSM 510 or a LSM 510 Meta (Carl Zeiss, Jena, Germany). PAGFP was photoactivated using a 80-mW Kr, 413-nm laser (Coherent, Dieburg, Germany) on the LSM 510 or a 20-mW, 405-nm laser diode (Point Source, Hamble, UK) on the LSM 510 Meta and observed at 488 nm. The 488-nm laser line was used at low power to prevent any residual photobleaching or photoactivation. A 40× PlanApochromat IRIS NA 1.0 oil objective (Zeiss) was used and the pinhole was wide-open to ensure imaging the entire nucleus. All images were 128 × 128 pixels to allow short frame acquisition times (100 ms). For all experiments, half of the nucleus was photoactivated and the whole nucleus was imaged. In the case of PAGFP alone, the cytoplasmic pool of protein was photoactivated and subsequently photobleached to prevent any contribution from the cytoplasm to the measurements. The experiment was then performed within the next 30 s on the nucleus to minimize the equilibration between nucleus and cytoplasm. H2B-PAGFP was imaged using an autofocus and tracking macro as described (31).

Simulations in three dimensions were performed starting from stacks of confocal sections of real nuclei with 0.3*0.3*0.35 μm³ (xyz) voxel sizes.

Image segmentation, alignment, and quantitation

Segmentation of three-dimensional stacks and quantitation of fluorescence intensities for the simulation were performed using in-house developed plugins for ImageJ (http://rsb.info.nih.gov/ij/). Photoactivation sequences were aligned as follows: first, each sequence was aligned using the nuclear diHcRed signal using a registration plug-in on ImageJ (http://bigwww.epfl.ch/thevenaz/turboreg/); then the same transformation was applied on the PAGFP signal using a in-house modified version of the plugin.

Quantitation for data fitting was performed with the LSM 2.8 software using an in-house developed macro to measure and format the fluorescence intensities for simulation and parameter optimization. Background was subtracted and intensities were normalized to the total intensity. Typically, nuclei were divided in six regions distributed along the main fluorescence redistribution direction, covering the whole nucleus.

Quantitation of fluorescence recovery was performed with ImageJ. Fluorescence of a bleached or a nonphotoactivated region was background-subtracted and normalized to the total intensity, also background-subtracted. This measurement was then normalized to 0 for the image taken right after photobleaching or photoactivation and to 100 for the steady-state distribution of fluorescence, yielding a direct readout of the percentage of recovery.

Computer simulation and curve fitting

Partial differential equations were simulated numerically using a finite difference approach. The following equation,

becomes, in finite differences in three dimensions,

with p_i, p_j, and p_k the spatial steps in the three dimensions and and the local concentration of free and bound fluorescent proteins at the position (i,j,k) in the grid.

is with i^st (i,j,k) the steady-state intensity at the position (i,j,k) in the grid and the sum of all the i^st (i,j,k) dividing by the number of grid elements that are inside the nucleus.In two dimensions, the equation becomes

with p_i and p_j the spatial steps in the two dimensions. For our simulations, we chose p_i = p_j.

In three dimensions, the nucleus was subdivided in cuboids (Fig. 2 A), all of identical size with a typical side length of 0.3 μm (doubling the size did not significantly change the simulation result). As proteins were assumed not to cross the nuclear envelope, fluxes were fixed to zero across nuclear boundaries. The system of ordinary differential equations derived from the discretization was simulated and fitted to data using the solver Berkeley Madonna (www.berkeleymadonna.com). Initial conditions were measured from images taken right after photoactivation. Steady-state distributions of fluorescence were measured from images either before photoactivation at 413 or 405 nm or when the redistribution was complete at 488 nm. Integration was performed using a Runge-Kutta fourth-order algorithm with an adaptive step-size. The curve fit used the downhill simplex method. The six measured regions were fitted to the simulation except in the case of SUV39H1-H320R, where only two measured regions were fitted (see Results). Residuals were calculated by subtracting the fit from the data and normalizing it to the values of the fit at the end of the simulation.

The simplified two-dimensional model, and potentially other unidentified systematic experimental errors, induced small but systematic deviation of the residuals from zero (see Supplementary Material, Fig. S3A), which makes the statistical analysis of the fit and of the confidence interval by classical methods not conclusive (data not shown). We therefore broadened this confidence interval by using an empirical cutoff at two times the square of the residuals obtained by the best fit (see one example in Supplementary Material, Fig. S3B).

A general reaction-diffusion model in the geometry of the nucleus

A general three-dimensional reaction-diffusion model for chromatin interacting proteins

We therefore developed a model that included both diffusion and chemical interactions to simulate the fluorescence redistribution after photoactivation and fit the simulation to the experiments to determine the amount of bound protein, its residence time and, where possible, the diffusion coefficient of the unbound protein. The model was initially built in three dimensions based on the real geometry of the nucleus and distribution of chromatin of each observed cell.

We assumed that proteins are immobile when bound to chromatin, because chromatin does not show large-scale movements over 1 h in mammalian cells (34–36) and that free molecules can normally diffuse within the whole nucleus with a single diffusion coefficient D following Fick's second law,

(1)

where is the local concentration of unbound fluorescent molecules. We also assumed that chromatin occupies a negligible volume and that all binding sites are equally accessible. The interaction between proteins and chromatin was modeled as a simple first-order chemical reaction,

(2)

where free is the free protein, C is the free binding site on chromatin, bound is the bound protein, and k_on and k_off are the on- and off-rates of the interaction. Combining diffusion and interaction kinetics, changes in the local concentrations of fluorescent proteins during fluorescence redistribution can then be described by the partial differential equations of

(3)

It is important to note that photobleaching/activation perturbs the fluorescence only and is assumed not to modify the chemical interaction of fluorescent molecules. Therefore, the fluorescence perturbation does not affect the distribution of chemical species, seen independently of their fluorescent state. Therefore, as in our case chemical interactions can be considered to be in steady state over the whole experiment, the distribution of free binding sites does not depend on time. Their spatial distribution is nevertheless not known, leading to an unknown parameter that depends on space. Nevertheless, we could show that this spatial dependency can be fully determined from the fluorescence distribution in steady state (see Appendix A), and that Eq. 3 can be rewritten by replacing by a parameter where is the steady-state intensity distribution in space, proportional to the concentration of fluorescent protein (37), is the average steady-state intensity over the whole nucleus and Free is the global fraction of unbound proteins in steady state:

(4)

Although Eq. 3 contains two parameters, D and k_off, that are constant and one parameter, that depends on space, Eq. 4 has the major advantage of containing three parameters, D, k_off, and Free that do not depend on space, and only requires the intensity distribution in steady state, which can easily be measured experimentally.

The reaction-diffusion Eq. 4 was simulated in silico using the real geometry of a nucleus and a typical distribution of chromatin interacting proteins (Fig. 2). As it cannot be solved analytically in such a complex geometry, it was solved numerically using a finite difference approach (Fig. 2 A; see also Materials and Methods, above). Initial conditions are needed for the free and bound fluorescent protein concentrations, but for chromatin proteins the resolution of light microscopy does not allow the discrimination between these two populations, and images just after photoactivation only provide their local sum We thus further assumed that the ratio between free and bound fluorescent proteins after activation is the same as in steady state, which is correct either when the fluorescence perturbation (photoactivation/bleaching) is so fast that no significant free protein movement occurs during the perturbation, or when the interaction is so fast that local free and bound pools immediately equilibrate.

Two-dimensional simplification of the reaction-diffusion model

Most molecular kinetics are too fast to be imaged in three dimensions by acquiring z-stacks of images over time. We therefore performed all image acquisitions in two dimensions on a confocal laser scanning microscope with a wide-open pinhole and a low numerical aperture objective focused in the middle of the nucleus, such that the entire nuclear depth could be illuminated and detected (Fig. 3 A). To directly use this two-dimensional information, and to reduce the computing time, we then tested the validity of a two-dimensional model. In this simplified model we assumed that the steady-state concentration of free proteins was homogeneous and that the distribution of bound and free fluorescent proteins could be considered as two-dimensional (see Materials and Methods, above, for the finite difference equations), whereas, in fact, our two-dimensional observations correspond to the convolution of the reaction-diffusion process in three dimensions with the point-spread function (PSF) axially centered in the middle of the nucleus. To test the simplification we simulated a photoactivation experiment in three dimensions (Fig. 3 B, first row; see Supplementary Material 1 for the details of the procedure and (38) for the assumptions used), that we convolved radially, using the axial profile of Fig. 3 A to project in two dimensions and mimic our observation (Fig. 3 B, second row). We then tested the simplified model with these two-dimensional convolved images by fitting the six regions depicted on the figure (Fig. 3 B, second row, last frame) equally distributed in the direction of the fluorescence gradient. As can be seen on the plots, it can be fitted to the convolved images, and the fitted parameters are similar to the one used for the three-dimensional simulation. This shows that deconvolution of images is not necessary and that a two-dimensional model is sufficient to characterize protein dynamics. To statistically test the two-dimensional simplification and the influence of the real geometry, we simulated photoactivation in three dimensions in seven different nuclei, ignoring the convolution and assuming an homogeneous axial illumination in this case. The quality of the fit and the parameters were the same as when the PSF was considered. Starting from k_off = 0.05 s⁻¹, a low 1% free pool and a diffusion coefficient D of 29 μm² s⁻¹ in three dimensions, and keeping D = 29 μm² s⁻¹ in two dimensions, the two-dimensional fit gave k_off = 0.051 s⁻¹ ± 0.001 s⁻¹ and a free pool of 1.04% ± 0.01%, with residual differences between the two- and three-dimensional simulations below 1%.

We found only one particular case where the two-dimensional simplification is invalid, i.e., when the percentage of free fluorescent proteins is large and the dissociation of proteins from chromatin is slow compared to diffusion. This case represented in Fig. 3 C is characterized by an early phase of diffusion of free proteins when there is no significant contribution of the chemical interaction (Fig. 3 C, zoom of the early part of the plot), followed by a slower phase corresponding mostly to the dynamics of the interaction. The intensity amplitude of the early phase corresponds to the amount of free proteins, which fill the nonactivated half of the nucleus homogeneously according to the assumptions of the two-dimensional simplification (Fig. 3 C, second row, time 1.9 s). On the other hand, the three-dimensional simulation predicts an inhomogeneity due to the variable depth of the nucleus, which makes the two-dimensional simplification invalid when one tries to fit all the regions to the model (Fig. 3 C, plots of second row). Nevertheless we observed that when we fitted only the two regions depicted on the third row of Fig. 3 C, we could estimate the three parameters, D, free pool, and k_off with good precision, with residual differences between three-dimensional simulation and two-dimensional fit below 1%. Starting from D = 29 μm² s⁻¹, Free = 50%, and k_off = 0.020 s⁻¹ in three dimensions, the two-dimensional fit gave D = 32 μm² s⁻¹ ± 1 μm² s⁻¹, Free = 52% ± 1%, and k_off = 0.020 ± 0.001 s⁻¹ for eight nuclei. In all other cases, the two-dimensional simplification was satisfactory for all nuclear regions. When the free pool is small enough, the amplitude of the first phase is too small to contribute to the fit, and when the interaction is fast enough, the first purely diffusive phase is no longer visible.

In conclusion, we could show that within the geometry of the nucleus of intact cells, a simplified two-dimensional reaction-diffusion model can globally describe the mobility of chromatin-interacting proteins with good precision. In cases where the free pool is high and the interaction slow compared to diffusion, the parameter estimation must be restricted to certain regions, but such cases are easy to identify before parameter estimation, as they show clear biphasic redistribution kinetics (see also the representative case of PAGFP-SUV39H1-H320R, below).

Calibration of nuclear viscosity

The model contains three parameters: the diffusion coefficient, the fraction of unbound proteins, and the dissociation rate or its inverse, the residence time of the interaction with chromatin. As can be seen on Fig. 4 A, PA experiments will not always allow us to identify all three parameters, because in some cases, different combinations of parameters fit the data equally well. In such cases, we calculated the diffusion coefficient independently from the model based on the molecular weight of the protein and the viscosity of the nucleoplasm. The apparent viscosity of the nucleoplasm was calibrated using PAGFP alone as an inert probe regarding binding interactions in the nucleus.

Nucleoplasmic PAGFP diffusion was probed by photoactivating half of the nucleus and measuring its redistribution over time as described above (Fig. 4 B). The pure diffusion model Eq. 1 was simulated in two dimensions, using an empirical method proposed by Siggia et al. (39) that we validated for the nucleus the same way as in the previous section (see Supplementary Material 2 and Fig. S2 for details). Nuclear PAGFP photoactivation was performed on 67 nuclei. The example shown in Fig. 4 B shows that the fit is qualitatively very close to the data, with residuals below 6%. On average, we measured a two-dimensional diffusion coefficient of 40.6 μm² s⁻¹ ± 3.8 μm² s⁻¹. Given that the GFP diffusion coefficient in water is 87 μm² s⁻¹ at room temperature (40,41) and that water viscosity drops from 1.00 to 0.69 cP between 20°C and 37°C (42), we found the apparent viscosity of the nucleoplasm to be ∼3.1 times higher than water, consistent with the literature (43–45).

Knowing the apparent viscosity, the diffusion coefficient of chromatin-interacting proteins was then calculated from the mass of the protein, using the Stokes-Einstein relation,

(5)

D is the diffusion coefficient, k the Boltzmann constant, T the temperature, η the viscosity, and R the spherical radius of the molecule. We approximated the radius to be proportional to the cubic root of its mass, yielding

(6)

with m_protein as the molecular weight of the protein (Table 1). As all our proteins of interest were of similar size, their diffusion coefficient was estimated to ∼30 μm²s⁻¹. This is only an approximation, which likely contributes an uncertainty of up to a factor of 2 in the diffusion coefficient.

Measuring chromatin interactions with the model

We then applied the model to simulate photoactivation experiments of PAGFP fusion proteins to analyze the interactions of RCC1, SUV39H1, and its hyperactive mutant SUV39H1-H320R, and the five human isoforms of H1 with chromatin. We found two classes of behavior. The most common, instantaneous interaction with chromatin is illustrated in detail by PAGFP-RCC1. Despite large differences in the timescales of redistribution, the dynamics of SUV39H1 and H1 isoforms were extremely similar to that of RCC1. The second class of behavior, a noninstantaneous interaction with chromatin, is illustrated by the hyperactive mutant of SUV39H1.

PAGFP-RCC1 interacts instantaneously with chromatin and 2% of the protein pool is unbound in steady state

RCC1 interacts with the core histones H2A and H2B (24). FRAP experiments on GFP-RCC1 have already shown that its association with chromatin is dynamic: the half-time of recovery was found to be ∼2 s for a bleached spot in U2OS cells (46), and ∼10 s for a bleached stripe in tsBN2 (47), and one study fitted the recovery with a diffusive model with an apparent diffusion coefficient of 0.5 μm² s⁻¹ in 3T3 cells (48). We performed the experiment as described before for PAGFP alone, using transient expression of PAGFP-RCC1 in NRK cells (Fig. 5 A). The model using Eq. 4 was fitted to the measured intensities using D = 30 μm² s⁻¹ (Table 1), yielding as an initial set of parameters a dissociation rate k_off of 2 s⁻¹, i.e., a residence time of 0.5 s, and an unbound pool Free of 2.9%. The fit was qualitatively in good agreement with the data (Fig. 5 A) and residuals did not exceed 8% (Fig. 5 A and Supplementary Material, Fig. S3 A, for the details of the residuals).

We then tested the certainty of the parameters for PAGFP-RCC1 by plotting the sum of the square of the differences between experimental data and simulation for different combinations of k_off and Free. Fig. 5 B shows the importance of exploring parameter space in this manner: the dark-blue region represents the area of parameters that fit the data qualitatively well (see Supplementary Material, Fig. S3 B, for simulations with parameters of this area), and this region is infinite toward the high values of k_off. By contrast, the percentage of free molecules could be identified as 2.9% ± 0.2% (white bounded region on Fig. 5 B) (could be up to 4.3%; isolated white bounded region on Fig. 5 B) and the lower limit of the dissociation rate k_off could be determined as 0.4 s⁻¹, i.e., a maximum residence time of 2.5 s (could be as low 0.2 s⁻¹).

From the chemical point of view, this means that the interaction of PAGFP-RCC1 is too transient to be accurately measured by fluorescence perturbation methods such as PA/FRAP. At first glance, this is counterintuitive, as the fluorescence redistribution of PAGFP-RCC1 was very slow compared to PAGFP alone (compare Fig. 1, A and C, and Figs. 4 B and 5 A). The speed of redistribution of RCC1, however, does not reflect the length of the interaction but rather its affinity. The small fraction of free molecules, 2.9%, are very often trapped on their ubiquitous binding sites, and although they reside in the bound state for short times, they can therefore not diffuse efficiently over long distances and need long times to redistribute across the nucleus.

In such a case, the reaction-diffusion model Eq. 4 can be strongly simplified because the interaction can be considered as instantaneous. For homogeneously distributed binding sites, it has been shown that the model then becomes effectively diffusive (49), and also for an inhomogeneous distribution of binding sites we can rewrite the model to (Appendix B)

(7)

with the local fluorescence intensity. For relatively homogeneous distributions of binding sites Eq. 7 is approximately a diffusive equation with an effective diffusion coefficient equal to the product of the diffusion coefficient with the percentage of free proteins, D*Free. Thus, the speed of redistribution depends only on the diffusion coefficient of the protein and on the percentage of protein available for diffusion in steady state, but not on the kinetics of the interaction. If the unbound fraction is low, the fluorescence redistribution is slow. Fig. 5 C shows a fit performed on the same data as on Fig. 5 A using Eq. 7. The residuals are as expected identical and the estimated fraction of free protein is very similar, 2.8%. It should be noted that this would lead to an effective diffusion coefficient of D*Free = 0.8 μm² s⁻¹, close to already published values (48), but the correct interpretation possible by the model is that 97% of RCC1 is bound to chromatin and that the interaction can be considered as instantaneous. The analysis was performed on 19 nuclei and led to an average percentage of free molecules of 2.1% ± 0.6% (Table 1), with the uncertainty linked to our estimation of the diffusion coefficient (see above).

PAGFP-SUV39H1 interacts instantaneously with chromatin and 36% of the protein pool is unbound in steady state

SUV39H1 is a methyltransferase that specifically methylate lysine 9 of histone H3 (25), a key epigenetic modification involved in gene silencing. SUV39H1 binds to core histones without apparent preferences in vitro (50).

Photoactivation experiments to probe SUV39H1 interaction with chromatin were performed in NRK cells stably expressing PAGFP-SUV39H1. Models were simulated and fit to the data with a diffusion coefficient of 30 μm² s⁻¹. Fluorescence redistribution could be fit equally well by the reaction-diffusion model Eq. 4 (data not shown) and by the instantaneous reaction model Eq. 7 (Fig. 6, A–C). Thus, similar to PAGFP-RCC1, the percentage of free protein and only the lower limit of the dissociation rate could be determined. The analysis of 20 nuclei led to an average free protein pool of 36% ± 8% and a lower limit of dissociation rate of 0.9 s⁻¹, i.e., a residence time of, at most, 1.1 s determined from the parameter space analysis (Fig. 6 B).

Five isoforms of H1-PAGFP interact instantaneously with chromatin and have <0.1% of unbound protein in steady state

Histones H1 are components of the nucleosomal subunits that play an important role in chromatin structure and function (26). The dynamics of some isoforms have already been investigated by FRAP (3,51), which gave a recovery in the range of 1 min for a small bleached spot, a time that was interpreted as the residence time.

We stably expressed five isoforms of H1-PAGFP, H1.1-5, in NRK cells. Models were simulated using a diffusion coefficient of 34 μm² s⁻¹. All isoforms showed very similar behavior, which is illustrated for H1.1-PAGFP in Fig. 6 D. The behavior of fluorescence redistribution is similar to PAGFP-RCC1 and PAGFP-SUV39H1. It could be fit by the reaction-diffusion model Eq. 4 (data not shown) as well as by the instantaneous reaction model Eq. 7 (Fig. 6 F), yielding an average fraction of free protein of 0.09% ± 0.04% (n = 19). Due to the stiffness of the reaction-diffusion model Eq. 4 with such a low percentage of unbound molecules, we could not explore parameter space as exhaustively as for PAGFP-RCC1 and PAGFP-SUV39H1 at reasonable computational cost. We therefore only plotted the sum of the residual squares for different values of dissociation rate k_off, fixing the percentage of free proteins that best fit the data (Fig. 6 E). Because of the curved distribution of best-fitting k_off values (see Figs. 5 B and 6 B), this procedure could lead to an underestimation of the lowest possible limit of k_off of a factor of 2.9 for PAGFP-RCC1 and 2.4 for PAGFP-SUV39H1 (Supplementary Material, Fig. S4). Because the lower limit of k_off is 0.017 s⁻¹, according to the plot (Fig. 6 E), we can estimate it to 6*10⁻³ s⁻¹—i.e., a residence time of no more than 170 s, with a correction factor of 2.7.

The results for the four other isoforms are summarized in Table 1. The percentages of free protein are all similar to H1.1-PAGFP and, as the levels of noise are comparable, the lower limits of the dissociation rate are also expected to be similar (see Discussion).

It should be noted that the model shows a complete recovery of fluorescence, contrary to what has been published for H1°-GFP and H1c-GFP (3). It should also be noted that our timescale of fluorescence redistribution is much longer than the published ones (3,51), notably because our photoactivated regions are much larger, which directly affects a diffusion-limited redistribution (see Discussion).

Hyperactive PAGFP-SUV39H1-H320R is bound for 200 s on chromatin

The hyperactive mutant of PAGFP-SUV39H1-H320R is mutated in the catalytic SET domain of the enzyme, unexpectedly resulting in an increase of activity (25). It was transiently expressed in NRK cells. Likely due to the toxicity of the mutant in living cells, only very low levels of expression could be observed, which explains the lower signal/noise ratio of the images compared to the other proteins studied here (Fig. 7). In contrast to the wild-type protein, most nuclei exhibited a high percentage of diffusive proteins that led to a fast early redistribution followed by a slower phase limited by the kinetics of the chemical interaction (Fig. 7, plots), as already mentioned for validation of the two-dimensional model. In this case, all three parameters of the reaction-diffusion model could be identified using the two regions depicted on the images of Fig. 7. The fit yielded an average diffusion coefficient of 5.7 μm² s⁻¹ ± 1.6 μm² s⁻¹, a percentage of free diffusive proteins of 57% ± 17% and a dissociation rate of 5.1*10⁻³ s⁻¹ ± 2.2*10⁻³ s⁻¹, i.e., a residence time of 196 s (n = 8).

The diffusion coefficient of 5.7 μm² s⁻¹ is nevertheless smaller than 30 μm² s⁻¹ expected from the size of the protein. The diffusion coefficient of the early redistribution, may therefore also correspond to an effective diffusion coefficient, reduced by a relatively large fraction of protein bound in an instantaneous interaction, like in the case of wild-type PAGFP-SUV39H1. The redistribution of fluorescence would then be due to two types of interactions: one fast, compared to diffusion that led to an apparent diffusive process at early times similar to the wild-type protein; and one slow, with a dissociation rate of 5.1*10⁻³ s⁻¹ and a percentage of free proteins, i.e., proteins that are not bound in the second interaction, of 57%. To understand the estimated parameters in the context of two reactions, we wrote the reaction-diffusion equations for a model where the second interaction is a stabilization of the first:

If the first interaction can be considered as instantaneous, then we can write the differential equations describing the system in a similar way to Eq. 4 (see Appendix C). The interpretation of the dissociation rate remains the same: it corresponds to k_off2. What we call here the free pool of 57% is actually the ratio The fraction of free proteins regarding only the first reaction is the ratio of the apparent diffusion coefficient with the real D of the free proteins (see Appendix C), i.e., 19% assuming a diffusion coefficient of 30 μm² s⁻¹. The real fraction of free proteins, is therefore the product of these 19% with the apparent free pool of 57%, i.e., 11%.

Comparison to other FRAP experiment and analysis methods

One of the main experimental approaches used to quantitatively characterize diffusion (12,52) and reactions limited by diffusion (18) consists in bleaching a small spot and analytically analyzing the fluorescence recovery in this spot. Compared to this classical FRAP approach, the method presented here requires more computational skills and more time, but offers several advantages. First, the theoretical analysis of the spot-bleaching technique has always assumed an infinite system, a questionable assumption as most cellular compartments are not very large compared to the bleached spot. By contrast, our method takes the complete geometry of the sample, including boundaries, into account. Moreover, the spot technique requires the size and shape of the bleaching intensity profile to be known (12), which requires nontrivial optical calibrations. Our method is independent of the geometry of bleaching/photoactivation, and does not require the characterization of the amount of bleaching, as in certain cases for the spot technique (12). Furthermore, spots are typically chosen small to make the rest of the sample infinite, as possible leading to noisy data. Here, we measure the intensity in the whole nucleus, offering a much better precision in model validation and parameter estimation. Finally, fluorescence distributions are typically considered as homogeneous in space to simplify the analysis, whereas here such simplification is not needed: the distributions of both bound and unbound molecules are taken into account, allowing higher precision and a convincing validation of the model.

Limits of residence times that can be measured by FRAP/PA

Three of the proteins we examined, RCC1, H1, and wild-type SUV39H1 illustrate that for generic chromatin-interacting proteins, the binding reaction can appear instantaneous in FRAP/PA experiments and that only a lower limit for k_off can be determined from such experiments. Exploiting the possibility to do in silico experiments with our reaction-diffusion model, we wanted to generally test the dependence of the ability to identify the dissociation rate on the percentage of free proteins, which reflects the affinity of the interaction in steady state.

To this end, we used a simplified one-dimensional reaction-diffusion model with a homogeneous distribution of binding sites and a hypothetical protein with a diffusion coefficient of 30 μm² s⁻¹. For given percentages of free protein, we tested at what dissociation rate the reaction-diffusion model can no longer be discriminated from an instantaneous reaction model with a tolerance of 5% between the two models (see Supplementary Material 3 for details). Fig. 8 shows that the method has the best sensitivity for high dissociation rates if ∼30% of the protein is unbound, then dissociation rates up to ∼30 s⁻¹— i.e., residence times as short as 33 ms can be identified. For amounts of free protein from 0 to 30%, the limit that can be estimated for the dissociation rate increases proportionally with the free fraction. For higher percentages of free proteins, the identifiable limit of dissociation rate decreases rapidly again with the free fraction, presumably because the contribution of the bound fraction to the fluorescence equilibration diminishes significantly. These in silico results are in good agreement with our experimental data on the instantaneous interactions of SUV39H1, RCC1, and H1.1 with chromatin. For all three proteins, the limit of the identifiable dissociation rate increased with the free fraction (Fig. 8, vertical lines).

Thus, for cases where interactions appear instantaneous, FRAP/PA experiments are suitable to measure the amount of free protein but give limited information on the kinetics of the interaction. As illustrated over the parameter space in Fig. 8, combinations of dissociation rate and unbound fraction that can be physiologically expected can easily be found in the half-space, where k_off cannot be identified. Thus there is clearly a need for complementary methods to measure transient biochemical interactions in living cells. Fluorescence correlation spectroscopy may be a good alternative to access this information as the timescales accessible by this technique are orders-of-magnitude shorter than with FRAP/PA.

General implications for the interpretation of FRAP and PA experiments

Deriving equilibrium dissociation constant and association rate in living cells

From the reaction-diffusion Eq. 3 it can be seen that k_on cannot be estimated from photobleaching/activation without information on the concentration of free binding sites. Likewise the equilibrium dissociation constant K_D, which is the product of Free/(1-Free) and the concentration of free binding sites, cannot be determined without the latter.

PAGFP-RCC1 interacts with histones, which in cells are assembled into nucleosomes, therefore

where is the concentration of free nucleosomes available for RCC1 binding. Here,

The amount of free nucleosomes available for RCC1 binding is nevertheless not known and cannot be estimated from our experiments as nucleosomes can be occupied by RCC1 tagged and untagged, as well as by many other nucleosome binding proteins, whose concentrations and affinities are unknown. We can therefore only estimate its upper limit, assuming that all nucleosomes are free. Given that a rat cell contains ∼6*10⁹ basepairs of DNA and nucleosomes repeat at intervals of ∼200 basepairs, we can estimate that a rat cell contains ∼3*10⁷ nucleosomes (54). As the nuclear volume is ∼0.7 pl, the total concentration of nucleosomes is ∼70 μM. If we assume one binding site per nucleosome, the dissociation constant K_D is then smaller than 1.4 μM. The real value is the product of this upper limit by the fraction of free binding sites, a number difficult to estimate. It should be noticed that K_D has been estimated to ∼5 nM in vitro (55), suggesting that only ∼0.5% of nucleosomes are available for RCC1 binding in steady state.

The case of H1 is very similar: as one linker histone H1 binds to one histone octamer, like RCC1, this leads to an upper limit for the dissociation constant K_D of 70 nM. Like in the case of RCC1, the real dissociation constant is the product of this upper limit with the fraction of free histones. In vitro H1 binds nucleosomes with a K_D of 7.4 nM (56), suggesting that ∼10% of nucleosomes are available for binding to H1 in steady state.

Implications for dynamics of protein-DNA interactions

The problem of specific protein-DNA recognition has been a challenging issue since 1970 when the Escherichia coli lac repressor was found to find its target at a much higher rate than predicted for a diffusion-controlled process (57). It was therefore suggested that more elaborate mechanisms than simple three-dimensional diffusional collisions should occur (58,59). Notably it has been proposed that proteins can interact with nonspecific sequences of DNA at low affinity and then diffuse along the DNA molecule in one dimension, restricting the volume which has to be searched by the protein and resulting in more efficient encounters with specific sites. Most FRAP studies on chromatin interacting proteins have been taken to suggest that the three-dimensional diffusional collision process was universal in living eukaryotic cells (60). The suggestion that chromatin interacting proteins can diffuse all over the nucleus before interacting with a binding site is in contradiction with the very high association rates found for the lac repressor. The interpretation of FRAP data was based on the assumption that diffusion does not limit fluorescence recovery, an assumption not validated in most studies. Here we could see that in the cases of RCC1, Suv39H1 and H1, this assumption is not valid. In the context of a three-dimensional reaction-diffusion model these individual proteins will therefore reassociate with a binding site in close proximity, which could be compatible with a one-dimensional diffusion along DNA molecules. Our present study shows that FRAP studies showing rapid nuclear protein mobilities do not provide per se grounds to rule out the one-dimensional diffusion hypothesis for eukaryotic cells. It might therefore be worth considering testing it with appropriate methods.