| HOME | HELP | FEEDBACK | SUBSCRIPTIONS | ARCHIVE | SEARCH | TABLE OF CONTENTS |
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
* Laboratory of Receptor Biology and Gene Expression, NCI, and Division of Bioengineering & Physical Science, ORS, National Institutes of Health, Bethesda, Maryland; and Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania
Correspondence: Address reprint requests to James G. McNally, Tel.: 301-402-0209; E-mail: mcnallyj{at}exchange.nih.gov.
 |
ABSTRACT |
|---|
170 ms). These FRAP analysis tools will be important for measuring key cellular binding parameters necessary for a complete and accurate description of the networks that regulate cellular behavior. |
INTRODUCTION |
|---|
We are using FRAP to assay the in vivo binding interactions of a transcription factor with its promoter target site. We employ a cell line (1
) containing a tandem array of mouse mammary tumor virus (MMTV) promoters and the GFP-tagged form of one of this promoter's cognate transcription factors, the glucocorticoid receptor (GFP-GR). This tandem array can be visualized as a single bright region of GFP-GR binding within the nucleus. We have shown that the FRAP recovery within this spatially localized cluster of specific binding sites is rapid (2), and that it contains information about the in vivo binding interactions of GFP-GR with the MMTV promoters (3). These transient binding interactions have challenged the notion that a stable transcriptional complex forms at a promoter. However, a detailed understanding of these transient interactions requires a model that first explains what the FRAP curve reflects, and then quantifies the underlying processes.
Quantitative analyses of FRAP have been performed for binding sites that are homogeneously and globally distributed throughout a cellular compartment (4–9), but fewer studies (10–14) have tackled the problem for heterogeneously distributed binding sites such as that which occurs at a spatially localized cluster of binding sites, despite its obvious biological importance. The primary reason is that the analysis is more complicated compared to homogeneously and globally distributed binding sites. A cluster of specific binding sites is always embedded in a larger domain throughout which molecules may freely diffuse or interact weakly with uniformly distributed nonspecific sites. Thus spatial variability must be considered in any reasonable model for localized binding. To simplify such an analysis, most previous FRAP models for spatially localized binding have presumed that diffusion plays a negligible role. In those cases where diffusion has been ignored, the consequences of this assumption have not been tested. In addition, previous analyses have been tailored to the specific problem under study, and general principles about expected FRAP behaviors at a spatially localized cluster of binding sites have not been elucidated.
Here we investigate a FRAP model that also incorporates diffusion within and around a spatially localized cluster of binding sites. We show that completely ignoring the localization of binding sites will introduce serious errors into the estimation of binding parameters. However, we find that a reasonable approximation can often be achieved by assuming a cylindrical column of binding sites that accounts for spatially localized binding in every xy cross section of the cell, but not along z. The resultant model is considerably easier to implement and faster to compute, and so should be more widely useful. We also identify limiting behaviors for FRAPs at spatially localized binding sites where either diffusion or binding dominate, and show how ignoring diffusion can also lead to serious errors in the estimation of binding parameters at the localized binding site cluster. Finally, we apply these methods to the analysis of GFP-GR binding at the MMTV promoter, and obtain a first approximation to an upper bound on the GR residence time of 170 ms. This figure underscores the transient nature of this binding interaction, and has broader implications for how assembly of the transcription complex may occur.
|
MATERIALS AND METHODS |
|---|
). Cells were grown and prepared for fluorescence imaging as previously described (3). Thirty minutes prior to imaging, GFP-GR translocation to the nucleus was induced by the addition of 100 nM dexamethasone (synthetic hormone).
FRAP protocol
FRAP experiments were carried out on a Zeiss 510 confocal microscope (Carl Zeiss, Thornwood, NY) with a 100x/1.3 NA oil-immersion objective. Bleaching was performed using the 488- and 514-nm lines from a 40-mW argon laser operating at 75% laser power. Bleaching was done with a single scan that lasted 17 ms. Fluorescence recovery was monitored at low laser intensity (0.2% of the 40 mW laser) at 78 ms intervals. Normalized FRAP curves were generated from the raw data exactly as described in Stavreva and McNally (15).
|
MODELS |
|---|
200 copies of a basic repeat unit: the MMTV promoter (1 kb) followed by a ras reporter gene and additional sequence from the bovine papilloma virus (totaling 9 kb). With six binding sites for the glucocorticoid receptor (GR) at each of the 200 MMTV promoters, the MMTV array is a cluster of 1200 specific binding sites (Fig. 1, A and B). FRAP experiments at the MMTV array are typically performed by bleaching it with a circular spot equal to the diameter of the MMTV array (Fig. 1 C), with the axial extent of the bleach reasonably approximated as a cylinder (Fig. 1 D).
|
2.0 µm, the average nuclear radius was 7.5 µm, and the average nuclear height was 4.5 µm (Fig. 1 E). We measured the average radial coordinate of the array (i.e., distance from the center of the nucleus), and found that most arrays occupied radial positions between 3.3 and 6.3 µm, with a mean radial coordinate of 4.8 µm. Finally, we measured the distance of the center of mass of arrays to the closest edge of nucleoli (dark regions where GFP-GR is largely absent), and found an average distance of 1.1 µm, with some arrays as far as 3.5 µm from a nucleolus.
Away from the MMTV array, GFP-GR is distributed rather uniformly throughout the rest of the nucleus, again with the exception of nucleoli (Fig. 1 A). To define GR interactions throughout the nucleoplasm, FRAP experiments can be performed at some distance from both the MMTV sites and nucleoli. These nucleoplasmic FRAP recoveries are consistently faster than FRAP recoveries at the MMTV array (3), suggesting that GR associates more strongly with the array sites than with the nucleoplasmic sites. Analysis of these nucleoplasmic FRAPs predicts that GFP-GR interacts predominantly with a single binding state in the nucleoplasm (8). Moreover, for bleach spots >1 µm, the FRAP recovery exhibits effective diffusion, which means that the free and bound GR in the nucleoplasm behave as if they were a single species diffusing at a rate much slower than free diffusion. The molecular identity of this nucleoplasmic binding state is unknown, but it may reflect nonspecific DNA binding, or some other generic association of this transcription factor with chromatin.
Mathematical model for FRAP at a single cluster of localized binding sites
Based on the above features, a minimal model for FRAP at the MMTV array must incorporate two types of binding: the nucleoplasmic sites, S1, and the MMTV array or spatially localized sites, S2 (for a listing of all model variables and parameters; see the Appendix, Tables 3 and 4). The S1 sites are found throughout the nucleus including within the array (since 90% of the DNA at the MMTV array is not promoter sequence). However, the array specific sites S2 are restricted to just the array (Fig. 2 A).
|
|
|
 | (1) |
is the radius of the zone containing the localized S2 binding sites (presumed to be circular), and 
is the half-height of this same zone. The nucleus is modeled as a cylinder with radial boundary at 
and half-height of 
and the array in this first idealization is presumed to be located at the center of the nucleus and surrounded by a homogeneous region without potential complications introduced by the presence of nucleoli.
To model a FRAP recovery, we write differential equations to describe the chemical kinetics for these binding reactions and the diffusion of the unbound molecules. As in other analyses of FRAP (4–6,8,9), we make two simplifying assumptions. First, we presume that the binding sites are immobile relative to the time- and lengthscale of the FRAP experiment. This is reasonable for the MMTV array because time-lapse movies have shown that this array moves very little during the 1-min FRAP recovery. This simplification eliminates a diffusion term for bound molecules (C1 or C2). Second, we assume that the system is at equilibrium before the bleach. This is satisfied if the total amount of the GFP-fusion protein does not change appreciably during the recovery period. This is also a reasonable assumption since the FRAP experiments were performed 18 h after induction of GFP-GR expression and 30 min after hormone induction has led to a constant nuclear concentration of GFP-GR. This simplification means that throughout the FRAP recovery the concentration of free binding sites for each binding state is constant and spatially uniform (namely S1eq and S2eq), since bleaching alters the fluorescence, but not the total concentration of either the free or bound GFP-GR.
With these two assumptions, we obtain two sets of differential equations that can be used to compute the FRAP recovery. Inside the array region, we use the equations for a homogeneous distribution of two binding states, while outside the array region we use the corresponding equations for a single binding state (see Sprague et al. (8)):
 | (2) |
is the Laplacian operator in radial and axial coordinates, Df is the diffusion coefficient of the free molecules, 
is an association rate constant (also known as a pseudo first order on rate or effective first order rate constant) for the nucleoplasmic binding sites, 
is an association rate constant for the MMTV specific binding sites, and 
and 
are the concentrations of the free and bound unbleached fluorescent molecules, F, 
and C2, respectively.
The effective diffusion behavior at the nucleoplasmic sites (described above and in Sprague et al. (8)) enables us to reduce the number of equations in Eq. 2 from five to three (see section A in the Appendix). This simplification arises by recognizing that with effective diffusion the free and bound fluorescence in the nucleoplasm behave as a single entity, an apparent "free" species 
that appears to diffuse at a rate given by 
where 
This yields the reduced system:
 | (3) |
To simplify the analysis of these equations, we presume that the bleach spot radius is selected to coincide with the circular zone of MMTV specific binding sites, a constraint that is easily achieved experimentally. Equation 3 is then subject to the initial condition that the bleach depth (the initial fluorescence immediately after the bleach) is normalized to zero in the bleach zone, and therefore at 
and 
for all 
Outside the bleach zone, fluorescence is initially at its equilibrium concentration 
at 
for all 
Note that for simplicity these initial conditions presume a cylindrical bleach (i.e., no z dependence), which is a reasonable first approximation to bleach patterns of the MMTV array cells (Fig. 1 D). Boundary conditions are no flux at 
and 
reflecting the fact that the nuclear membrane is impermeable to GR on the timescale of the FRAP recovery.
Note that solutions to Eq. 3 for spatially localized binding generalize in the following way. At the essence of the geometry described by these equations is a small domain of specific binding sites centered in a much larger domain where molecules diffuse freely. The solutions we obtain below should apply to all such scenarios. The diffusion constant may be that for either free or effective diffusion (as long as there are binding sites present throughout the larger region that exhibit effective diffusion behavior). Here we analyze the solutions to Eq. 3 using a diffusion constant corresponding to the effective diffusion constant for GFP-GR measured under our current conditions (D1eff 1.2 µm2/s) and a bleach-spot radius corresponding to the size of the MMTV array (R0 1.0 µm). Fixing these parameters sets a characteristic time for diffusion (
) within the bleach spot, but does not alter the prototypical behaviors of the solution. Extending the applicability of the present findings to other binding site and ligand systems can be facilitated in future studies by nondimensionalizing the governing equations to remove the dependence upon the particular parameter values employed in the simulations for MMTV and GFP-GR.
Axial and radial binding model
We refer to Eq. 3 as the Axial and Radial Binding Model because it accounts for both the axial and radial localization of the binding site cluster (Fig. 2 A). We solved Eq. 3 numerically using a finite element method (see section B in the Appendix for details) implemented in the commercial software package FEMLAB (COMSOL, Burlington, MA). Running on a Pentium 3 processor, this approach required 5 min to generate a single FRAP curve for each set of model parameters.
To fit FRAP curves with this model, we first generated a series of possible solutions by independently varying 
and 
from 10–5 to 10+5 in increments of 100.5. This sampling produced a best (, ) guess, defined as that yielding the smallest difference in the sum of residuals between the predicted FRAP and the experimental data. This best guess was then used as a starting point for finer sampling (increments of 100.1) within its vicinity. This then yielded the best fitting solution. The entire fitting procedure required 24 h.
Radial binding model
To obtain an analytical solution, Eq. 3 was simplified by assuming that the zone containing specific sites extended as a cylinder throughout the depth of the nucleus. This removes all axial (z) terms from the equations and so yields a model that accounts only for the radial localization of the specific binding sites. We refer to this model as the Radial Binding Model (Fig. 3 B), and solved the corresponding equations using a Laplace transform technique (see section C in the Appendix; final solutions summarized in Table 1). The Laplace transform was numerically inverted in Matlab (The MathWorks, Natick, MA). Running on a Pentium 4 processor, this approach required less than 1 s to generate a single FRAP curve for each set of model parameters.
|
|
1 min.
Off-center radial binding model
To account for the fact that arrays are not located at the center of the nucleus, a numerical version of the Radial Binding Model was implemented. This Off-Center Radial Binding Model still presumed a cylindrical zone of array binding sites, but the numerical analysis was performed on a Cartesian (x,y) grid that enabled us to position the array at arbitrary locations throughout the nucleus (see Fig. 6 B, Section C in the Appendix and Müller (16)), and so determine the consequences of a nearby impermeable boundary such as the nuclear membrane.
|
|
RESULTS |
|---|
Next we asked what the consequences of this difference were for binding parameter estimation. Here we used the Global Binding Model to fit the FRAP data generated by the Axial and Radial Binding Model. Interestingly, we found that we could obtain reasonable fits with the Global Binding Model, but the binding parameters estimated were incorrect by more than an order of magnitude (Fig. 2 D). These results demonstrate that when spatial localization of binding occurs, a localized-binding site model is essential for accurate FRAP analysis of the binding interactions.
The Axial and Radial Binding Model, however, required 5 min to generate a single FRAP curve and up to a day to generate the hundreds of curves required to fit experimental FRAP data (see "Axial and Radial Binding Model" in the Models section). Thus, we wondered whether the computationally less expensive Radial Binding Model (Fig. 3 B) could be substituted in some cases. This model requires 1 s to generate a single FRAP curve and 1 min to fit experimental FRAP data, but presumes that the localized sites form a cylindrical column spanning the height of the nucleus, thereby ignoring the spatial variation of binding sites only along the z axis. To evaluate the utility of this model, we first generated FRAPs with it and compared the curves to those generated with the same parameters for the Axial and Radial Binding Model. We found that for smaller values of k*2on/k2off (weaker binding at the specific sites), the two models yielded FRAP curves that were quite similar (Fig. 3 C), while for larger values of k*2on/k2off (tighter binding at the specific sites), the two models disagreed (Fig. 3 D).
A significant contributing factor to this difference was the additional flux of fluorescence that could enter the bleached zone in the Axial and Radial Binding Model from the regions above or below the localized binding sites (Fig. 3 E). This flux is completely absent in the Radial Binding Model because it lacks an axial component. The axial flux generated in the Axial and Radial Binding Model increased as the binding at the localized sites became tighter, as did the discrepancy between the FRAP curves predicted by this model and the simpler Radial Binding Model (data not shown). We conclude that a restricted axial height for the localized binding sites leads to detectable differences in the FRAP recovery, with greater differences arising when binding is tighter at the localized sites, and that much of this difference arises due to the axial flux of fluorescence into the zone of localized binding.
To determine the consequences of ignoring the restricted axial height of the localized binding sites on fitting FRAP data, we generated FRAP curves using the Axial and Radial Binding Model and then attempted to fit them with the Radial Binding Model. In some cases, like that in Fig. 3 C, the restricted axial height of the localized binding sites has little effect on the FRAP curve. As expected, we found for these cases that the Radial Binding Model yielded excellent fits to FRAP data generated by the Axial and Radial Binding Model with perfect estimates of the binding parameters (data not shown). Interestingly, for cases where the Axial and Radial Binding Model FRAP recovery was noticeably different from that produced by the simpler Radial Binding Model (such as Fig. 3 D), the simpler model could nevertheless produce reasonable fits to data generated by the Axial and Radial Binding Model (Fig. 3 F).
Of course in these cases, despite the good fit, the Radial Binding Model yielded incorrect estimates for the rate constants. However these estimates, either for k*2on, k2off, or their ratio k*2on/k2off (see Fig. 3 F for an example), were always within an order of magnitude of the "true" values used in the Axial and Radial Binding Model to generate the FRAP curve to be fit. We conclude that ignoring the axial flux of fluorescence into the localized sites affects parameter estimation when binding at the localized sites is strong, but good fits can still be obtained yielding errors smaller than an order of magnitude for these binding parameters. Thus the Radial Binding Model, though not completely accurate, may often be a useful tool for fitting FRAP data.
Different limiting FRAP behaviors at a cluster of localized binding sites
Several previous studies have demonstrated that FRAPs could sometimes be reduced to simpler equations, defining limiting behaviors in the general equations (4,5,8,9,14). Identifying when these limiting behaviors occur is important because the simpler equations can then be used to most efficiently extract binding information, and for certain behaviors there are limitations on exactly what binding information can be extracted. In addition, these domains introduce self-consistency checks (8,17).
To identify these domains for FRAPs at localized binding sites, we created simplified versions of the Axial and Radial Binding Model corresponding to each of the expected limiting behaviors (see section B in the Appendix). These behaviors are: 1), when binding at the cluster of localized sites is negligible; 2), when binding is not negligible, but the expected time to diffuse across the bleach spot is much shorter than the expected time to begin binding (reaction-dominant behavior); and 3), when binding is not negligible, but the expected time to diffuse across the bleach spot is much longer than the expected time to begin binding (local-equilibrium behavior).
To determine when these limiting behaviors of the Axial and Radial Binding Model were valid approximations to the true FRAP behavior, we compared the FRAPs they predicted to those produced using the full version of the model. Comparisons were performed over a wide range of biologically relevant rate constants at the localized sites (
), while fixing the effective diffusion constant and the bleach-spot radius at values typical for our biological FRAPs. We identified three large domains of k*2on, k2off where at least one of the approximate solutions was reasonably accurate, and a fourth domain where only the full solution described the FRAP recovery (Fig. 4 A).
|
|
Interestingly, the resultant distribution of idealized domains for FRAPs at localized binding sites resembles that previously identified for FRAPs at homogeneously and globally distributed binding sites (compare Fig. 4, A and B, to Fig. 2 D in Sprague et al. (8)), even though the two models yield entirely different FRAP curves for the same k*2on, k2off (Fig. 2 C). This similarity in domain structure was not necessarily expected because the spatially localized binding model is based on an entirely different geometry, namely a cluster of one type of binding site overlaid on a much larger region throughout which diffusion (or effective diffusion) occurs. In contrast, the situation for globally distributed binding sites is much simpler. There is only one type of site, and it is distributed uniformly throughout the entire region. Nevertheless, despite the increase in complexity accompanying spatially localized binding, the distribution of simplified domains as a function of rate constants is preserved.
Given this similarity of both the domains and their distribution between the spatially localized and globally distributed binding models, a number of the general conclusions drawn about global binding (8,17) can now be extended to FRAPs at a spatially localized cluster of binding sites. These are: 1), the contribution of diffusion to a FRAP recovery can be safely ignored only in the reaction-dominant domain, where the expected time for diffusion across the bleach spot is much faster than the expected time for binding (and so diffusion is effectively instantaneous on the timescale of the FRAP recovery). This idealized behavior occupies 1/3 of the region of rate-constant parameter space where binding parameters can be estimated (Fig. 4, A and B). Thus diffusion is expected to contribute to the majority (2/3) of FRAP recoveries at a cluster of binding sites, given typical values for effective diffusion and bleach spot size. Although these proportions are rough estimates because they depend on the specific values for the diffusion constant and bleach-spot size, the results nevertheless point to the importance of carefully considering diffusion's role in a FRAP at a cluster of binding sites. 2), Long FRAP recoveries do not necessarily reflect reaction-dominant behavior, as is often presumed. By computing the time for complete recovery as a function of k*2on, k2off, we find that FRAP recoveries of up to 4 min. can occur at localized binding sites in domains where diffusion contributes substantively (Fig. 4 C), even though a FRAP of a freely diffusing molecule would be complete in 1 s. 3), Fits to FRAP data can be subjected to a consistency check based on the estimated rate constants. Essentially, the predicted rate constants should lie in the appropriate domain in k*2on, k2off parameter space consistent with the type of model used to fit the data (see Table 2 for details). These constraints may be used to rule out incorrect models for a cluster of binding sites that nevertheless yield a good fit. 4), FRAP recoveries lying in the local-equilibrium domain will not yield independent estimates for 
, but rather only the ratio 
This is because the local-equilibrium solution for a cluster of binding sites depends only on the ratio 
(see Table 1; fourth row and the definition for 
in Table 4).
|
To calculate errors introduced by improperly ignoring diffusion, we generated FRAP curves using the Axial and Radial Binding Model in domains where the recovery was expected to depend on diffusion (the local-equilibrium domain and the full-model domain as defined in Fig. 4, A and B). We then attempted to fit these FRAP data with a reaction-dominant form of the Axial and Radial Binding Model produced simply by setting the free diffusion constant to a very large value (see section B in the Appendix). This reaction-dominant form is therefore equivalent to models that presume diffusion is so rapid that it can be ignored. When improperly applied to FRAP data that actually depended on diffusion, this reaction-dominant model nevertheless yielded fairly respectable fits to the diffusion-dependent data, but the binding parameters estimated were off by almost two orders of magnitude (Fig. 5 A). These poor estimates often produced k*2on estimates that were too large to be in the reaction-dominant domain (see Fig. 5 A legend and Table 2 for details), and so could be ruled out by this consistency check, if it were applied. We conclude that improperly ignoring diffusion's role in localized binding can lead to serious errors that might otherwise be identified by applying the appropriate consistency check.
|
Effects of ignoring proximity to an impermeable boundary
Both the Axial and Radial Binding Model and the simpler Radial Binding Model presume that the binding site cluster is at the center of the nucleus. This is, however, rarely the case either for the MMTV array (Fig. 1 E) or for other typical binding site clusters. To evaluate the effects of an off-center location, we used the Off-Center Radial Binding Model to position the cluster of binding sites at an arbitrary position within the nucleus (Fig. 6, A and B).
We varied k*2on, k2off over a wide range of values, and then compared FRAP curves for a 2-µm-diameter binding site cluster located either at the nuclear center or 5 µm distant from the center (i.e., 2.5 µm from the nuclear membrane). The two FRAP curves were essentially identical for values of k*2on, k2off lying within the reaction-dominant domain, but became increasingly different as k*2on increased and the FRAP curves entered first the full model domain and eventually the local equilibrium domain (Fig. 6 C). Within these domains, clusters closer to the impermeable nuclear membrane always showed slower recoveries (Fig. 6 D), consistent with a previous analysis of the effects of an impermeable boundary on a purely diffusive FRAP recovery (6). These slower recoveries reflect a decrease in the diffusive flux into the bleach spot due to the nearby impermeable boundary (6). However, for a reaction-dominant scenario, this difference in diffusive flux should be restricted to only the earliest time points, and so as we observed for these recoveries, the effect of proximity to the impermeable boundary should be negligible. In sum, our results indicate that only FRAP recoveries in the full model or local equilibrium regimes will be significantly retarded by their proximity to an impermeable boundary.
We evaluated the consequences of ignoring this boundary effect by using the analytical solution of the Radial Binding Model to fit the FRAP recoveries generated by the Off-Center Radial Binding Model for the worst-case scenario, namely FRAP data in the local equilibrium domain. As expected, when the displacement from the nuclear center was zero, the Off-Center Radial Binding Model yielded an identical FRAP recovery as the Radial Binding Model, and the fit of the Radial Binding Model yielded the same rate constants as the "true" values used to generate the FRAP curve by the Off-Center Radial Binding Model (data not shown). However, as displacement from the nuclear center increased and the binding site cluster came closer to the nuclear membrane, the fits became less accurate as did the estimates of the binding parameters (Fig. 6, E and F). This error in estimating k*2on/k2off was 50% for a case corresponding to the average MMTV array location at 5 µm from the nuclear center (Fig. 6 F). Despite the fact that errors arise due to ignoring an impermeable boundary, the errors are considerably smaller than those produced by other simplifications, such as ignoring either the role of diffusion, the constraints of the local equilibrium domain, or the localization of specific binding sites.
Application to GFP-GR FRAP recoveries at the MMTV promoter sites
To fit experimental FRAP data from the MMTV array, we used both the Axial and Radial Binding Model (Fig. 3 A and section B in the Appendix) and the Radial Binding Model (Fig. 3 B and section C in the Appendix). Calculation of a single FRAP curve required 5 min for the Axial and Radial Binding Model and 1 s for the Radial Binding Model. Unfortunately, finding a good fit took longer because we generated a large number of FRAP curves (at least 500) to find the one producing the closest match to the experimental data (see Models section). For this reason, the fitting procedure required 1 min with the Radial Binding Model, and 1 day with the Axial and Radial Binding Model. These times could be dramatically shortened if a more efficient fitting procedure were identified.
Using GFP-GR FRAP data from the MMTV array, we achieved a good fit with the local-equilibrium version of the Axial and Radial Binding Model, using a single free parameter, namely the ratio k*2on/k2off = 2.3 (Fig. 7 A). We also achieved a good fit with the local equilibrium form of the Radial Binding Model, again just using a single free parameter, in this case yielding the estimate k*2on/k2off = 2.5 (Fig. 7 B). Note that as expected from our analysis above, the difference in this estimate of k*2on/k2off (2.3 vs. 2.5) is well within an order of magnitude.
|
These self-consistent results from the modeling analysis were satisfying; however, we wondered how the estimates of k*2on/k2off might be affected by "real-life" complications, specifically the off-center position of the array and the presence of nucleoli near the array. The Axial and Radial Binding Model and the Radial Binding Model both presume a central location of the array surrounded by a homogeneous distribution of "nonspecific" GR binding sites. In reality, MMTV arrays are distributed throughout the nucleus with an average radial location of 5 µm from the nuclear center, and are often very near nucleoli.
To account for the off-center location of the array, we used the analysis of FRAP recoveries produced by the Off-Center Radial Binding Model. Comparing this model to the Radial Binding Model, we showed above (Fig. 6 F) that the Radial Binding Model will overestimate k*2on/k2off by 50% for a binding site cluster that is 5 µm from the nuclear center. This example (Fig. 6 F) corresponds precisely to the array data, as both the Radial Binding Model in Fig. 6 F and the array data yield k*2on/k2off = 2.5 instead of the "true" k*2on/k2off = 1.7. Thus to account for the array's off-center location, the ratio k*2on/k2off = 2.5 should be corrected to 1.7.
All of the models analyzed in this study ignore the presence of nucleoli in the vicinity of the array, and instead presume that the concentration of fluorescence surrounding the array is uniform before the bleach and that the diffusion and binding properties of the nucleoplasm surrounding the array are homogeneous. However, compared to the rest of the nucleoplasm, nucleoli contain little or no GFP-GR. This could arise because there may be very few binding sites for GR within nucleoli, or because nucleoli might be fairly impermeable to GR, or some combination of these two factors. Fewer GR binding sites within nucleoli should lead to faster FRAP recoveries near nucleoli because GR diffusion through the neighboring nucleolus would be less retarded by binding interactions. Conversely, a less permeable nucleolus should lead to slower FRAP recoveries near nucleoli due to slower diffusion of GR through the nucleolus.
To test the impact of nucleoli on GR recoveries at the MMTV array, we performed FRAPs first on one set of arrays whose centers were 0.5–1.0 µm away from nucleoli, and second on another set of arrays whose centers were 1.7–3.0 µm away from nucleoli. Since arrays are 1 µm in radius, all of the arrays from the first data set directly contacted nucleoli while none of the arrays in the second data set did. We expected to detect some difference between these two sets of FRAP data, at least at early time points, if the nucleolus either slows down or speeds up the FRAP recovery. However, the measured recoveries from the two data sets were within experimental error at all time points (Fig. 7 C). This suggests that the nucleolus has at best a modest effect on the FRAP recovery, perhaps because the properties of the nucleolus counteract each other, or because cell-to-cell variability overshadows small differences arising from nucleoli. We conclude that the estimate of k*2on/k2off = 1.7 for the MMTV array is not significantly altered by the presence of nucleoli.
This estimate of k*2on/k2off for the array enables an estimate of an in vivo binding constant. Using GFP-tagged viral particles, we have previously determined that on average 900 of the possible 1200 GFP-GR binding sites at the MMTV array are occupied (18). Therefore, there are 300 free binding sites at the array, which permits an estimate of S2eq, the equilibrium concentration of free binding sites at the array. Using our measurements of the array volume (
µm3), we obtain S2eq 220 nM. Since k*2on/k2off = (k2on/k2off)S2eq = (1/Kd)S2eq we arrive at an order of magnitude estimate for the binding dissociation constant, Kd 10–7 M for GFP-GR binding to MMTV.
Although we cannot extract independent estimates of k*2on or k2off from these FRAP data, we can use the structure of the domain space (Fig. 4, A and B) to obtain a rough upper-bound estimate of the GFP-GR residence time at the MMTV promoter. This is because local-equilibrium behavior imposes a lower bound of 6 s–1 for k2off (see Fig. 7 D). Since the residence time is given by 
R = 1/k2off, a lower bound of 6 s–1 for k2off yields an upper bound for the GFP-GR residence time at the MMTV promoter of 170 ms.
|
DISCUSSION |
|---|
5 min to generate a FRAP curve. A second disadvantage of this approach is that it also requires specialized numerical-analysis software (FEMLAB).
Even though the Radial Binding Model presumed an axial zone of localized binding extending throughout the entire nucleus, we could still use this model to achieve "good" fits of simulated FRAP data generated from the Axial and Radial Binding Model, which accounted for not only the radial but also the axial restriction in localized binding. By "good" fit, we mean that we could find a FRAP curve using the Radial Binding Model that nearly overlapped the one produced by the Axial and Radial Binding Model. The parameters estimated from this fit were in all cases less than an order of magnitude different from the true parameters used to produce the FRAP curve in the Axial and Radial Binding Model. This suggests that the faster and more easily implemented Radial Binding Model might still be used in many cases to achieve order of magnitude estimates of binding parameters, even though it ignores the restricted height of the localized binding zone. This result is consistent with the observation that a two-dimensional FRAP analysis also yields reasonable estimates for binding parameters obtained for the heterogeneous three-dimensional distribution of certain chromatin associated proteins (14). Our results, however, demonstrate that there are limits to this simplification, for example as the binding strength increases so too do the errors in parameter estimation. An additional cautionary note is that our results are only valid for a single binding site cluster like the MMTV array that occupies 50% or more of the nuclear height. Further numerical analysis would be required to quantify the errors introduced for a smaller cluster, such as a centrosome, which may occupy <10% of the nuclear height, or for multiple clusters such as nuclear pores, which may be close enough together to influence each other.
Insights provided by a knowledge of limiting FRAP behaviors at localized sites
We also investigated several additional assumptions that are often made in fitting FRAP data at localized binding sites. A role for diffusion is frequently ignored, first because FRAP recoveries of many GFP fusion proteins are much slower than those for unconjugated GFP, and second because the spatially localized cluster of specific binding sites occupies a small domain in which diffusion should occur rapidly. We found however that these presumptions are often incorrect: FRAP recovery times within a cluster of localized binding sites may last several minutes and still involve diffusion. Based on typical parameters for cellular binding, bleach-spot diameter and diffusion, we estimate that a sizeable proportion (perhaps as much as 2/3) of FRAP recoveries at a cluster of binding sites are likely to involve diffusion. Moreover, we demonstrate here that improperly ignoring this role for diffusion at the binding site cluster will have serious consequences leading to very poor estimates of the binding parameters.
A second common assumption in all FRAP models at localized sites is that independent estimates of the on and off rates of binding can be recovered. Using the models for a binding site cluster, we showed that in a simplified domain that we called local equilibrium, the FRAP recovery depended only on the ratio of these rates. If we improperly ignored this constraint, we found that the models could produce vastly different estimates for these rates. These radically different estimates still yielded a constant ratio, which is the only parameter that can be accurately estimated in the local-equilibrium domain. This domain can be recognized using the inequalities in Table 2, or simply by checking to see if a series of equally good fits can be achieved for different on and off rates that are constrained to yield a constant ratio. Our results underscore the importance of determining when FRAP data are in the local-equilibrium domain before assigning unique values to the on and off rates.
Importantly, these preceding features echo those that we found for FRAPs at homogeneously and globally distributed binding sites (8), and they are also similar to observations of Beaudouin et al. (14) for a heterogeneous distribution of binding sites. This suggests that these features are generic to all FRAP analyses.
Comparison of analytical and numerical approaches
In this study, we employed parallel numerical and analytical approaches. For ultimate precision, numerical models are preferable since they can incorporate all essential features of the real experiment. We used our numerical analysis to account both for the effects of a restricted axial height of localized binding sites and for the effects of proximity to an impermeable boundary. In a more general numerical approach, Beaudoin et al. (14) account for FRAPs arising from an arbitrary distribution of binding sites that can be measured directly from the image data.
These numerical methods can also be used for additional future tests of the many assumptions still present in most FRAP models.
TOP
ABSTRACT
INTRODUCTION
using the FRAP model for globally distributed binding sites described in Sprague et al. (8
from the cell center. (C) Comparison of FRAP curves predicted by these two models shows little difference when the FRAP recovery is in the reaction-dominant domain (RD), but increasing disparities arise through the full model domain (FM) reaching a plateau in the local equilibrium domain. Boundaries between the domains are shown with dotted lines extracted from 
again at 
and twice again at 
over all z in each instance. This procedure eliminated the discrepancy between the models except when the binding at the localized sites was extremely tight (k*2on/k2off > 
For the reaction-dominant fit here, 
µm2/s and 
µm. Therefore, the estimated value for 
must satisfy 
s–1, which is violated by the 
which was estimated by the fit to be 2.3. All other parameters in the model were fixed at measured values (see 
(C) Experimental FRAPs on a set of arrays whose centers were either 0.5–1.0 µm away from nucleoli or 1.7–3.0 µm away from nucleoli. No significant difference was detected. Error bars have been omitted for clarity, but are comparable to those in panels A and B. (D) Although only the ratio k*2on/k2off can be estimated from these local equilibrium data, a lower bound on the off rate is available based on the structure of the domain space. The estimate k*2on/k2off = 1.7 defines a series of points along the thick black line, each of which is consistent with the experimental FRAP data. At the star, this line intersects the contour line specifying the boundary separating the local-equilibrium domain from the full-model domain (see also