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Analysis of Binding Reactions by Fluorescence Recovery after Photobleaching

Brian L. Sprague ^*, Robert L. Pego , Diana A. Stavreva ^* and James G. McNally ^*

^* Laboratory of Receptor Biology and Gene Expression, National Cancer Institute, National Institutes of Health, Bethesda, Maryland; and Department of Mathematics, University of Maryland, College Park, Maryland

Correspondence: Address reprint requests to James G. McNally, E-mail: mcnallyj{at}exchange.nih.gov.

	ABSTRACT

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Fluorescence recovery after photobleaching (FRAP) is now widely used to investigate binding interactions in live cells. Although various idealized solutions have been identified for the reaction-diffusion equations that govern FRAP, there has been no comprehensive analysis or systematic approach to serve as a guide for extracting binding information from an arbitrary FRAP curve. Here we present a complete solution to the FRAP reaction-diffusion equations for either single or multiple independent binding interactions, and then relate our solution to the various idealized cases. This yields a coherent approach to extract binding information from FRAP data which we have applied to the question of transcription factor mobility in the nucleus. We show that within the nucleus, the glucocorticoid receptor is transiently bound to a single state, with each molecule binding on average 65 sites per second. This rapid sampling is likely to be important in finding a specific promoter target sequence. Further we show that this predominant binding state is not the nuclear matrix, as some studies have suggested. We illustrate how our analysis provides several self-consistency checks on a FRAP fit. We also define constraints on what can be estimated from FRAP data, show that diffusion should play a key role in many FRAP recoveries, and provide tools to test its contribution. Overall our approach establishes a more general framework to assess the role of diffusion, the number of binding states, and the binding constants underlying a FRAP recovery.

	INTRODUCTION

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The past few years have witnessed a dramatic increase in the application of fluorescence recovery after photobleaching (i.e., FRAP; reviewed in Meyvis et al., 1999; White and Stelzer, 1999; Reits and Neefjes, 2001; Houtsmuller and Vermeulen, 2001). Upon their development in the 70's, photobleaching techniques attracted a small cadre of biophysicists who utilized these methods primarily to determine diffusion constants of biomolecules in membranes (Liebman and Entine, 1974; Poo and Cone, 1974; Edidin et al., 1976; Schlessinger et al., 1976). Recently, with the advent of GFP fusion protein technology, the number of published FRAP experiments has skyrocketed. With this increase in FRAP studies has come an assortment of interpretations of FRAP data beyond measurement of diffusion constants. Indeed most recent FRAP experiments seek to infer something about how the GFP-tagged protein interacts with binding sites within the cell (e.g., McNally et al., 2000; Phair and Misteli, 2000). Toward this end, much of the current analysis is qualitative, assessing simply whether FRAP recovery curves are slower or faster after specific cellular perturbations (e.g., Dou et al., 2002). Faster or slower recoveries, respectively, are presumed to reflect weaker or tighter binding. In other cases, conclusions are drawn about the shape of a single recovery curve and how it may reflect underlying biological processes. For example, curves with a shoulder have been analyzed as containing fast and slow components that could correspond to diffusion and binding, or to two different binding states (Tardy et al., 1995; Kimura et al., 2002; Dundr et al., 2002; Carrero et al., 2003). More sophisticated quantitative analyses have also been performed to obtain rate constants for binding (Thompson et al., 1981; Kaufman and Jain, 1990; Berk et al., 1997; Bulinski et al., 2001; Coscoy et al., 2002; Dundr et al., 2002), although a number of these have examined regimes where the reaction dominates, presuming that diffusion can be safely ignored. Other attempts to simplify the full reaction-diffusion model have provided approximate solutions that allow the estimation of the binding rate constants but have yet to characterize the appropriateness of the assumptions made and the quality of the estimates (e.g., Carrero et al., 2003).

What is missing from this large compendium of experimental data and interpretive approaches is a comprehensive, analytical treatment of FRAP that provides a straightforward and consistent set of guidelines for how to analyze and interpret photobleaching data when binding interactions are present. This is a prerequisite if FRAP is to become a reliable and widely used approach to understanding and quantifying binding mechanisms within a cell. Various special cases of FRAP with binding have been considered previously, such as when diffusion dominates or when binding dominates (Kaufman and Jain, 1990, 1991; Bulinski et al., 2001), but the full spectrum of behaviors has not been characterized. The same system of equations that describes FRAP forms the theoretical basis of fluorescence correlation spectroscopy, where similar idealized models have also been utilized to determine binding rates (Elson and Magde, 1974; Elson, 1985, 2001). The connections among this disparate set of results are lacking, so no synthesis is available of all predicted FRAP recoveries in the presence of binding.

Here we present a detailed analysis of the simplest realistic case of binding that can be analyzed by FRAP, namely a single binding interaction in the presence of cellular diffusion, and also show how this can be extended to cases with multiple, independent binding interactions. Our goal is to provide a thorough, mathematically rigorous foundation for extracting binding information from FRAP recovery data. As such, we include an extensive Appendix describing in detail the mathematics underlying our analysis. In the main body of the text, we outline the key assumptions leading to the derivation of the FRAP model, and highlight the principal results and conclusions of our analysis.

As an example of the method's biological utility, we apply it to the problem of transcription factor mobility in the nucleus. In recent years, FRAP experiments have revealed that most nuclear proteins, including transcription factors, are highly mobile. Although transient binding interactions are presumed to influence the FRAP recovery of a transcription factor (Misteli, 2001), little is known about what these binding interactions are. Some evidence suggests that one class of transcription factors, the steroid hormone receptors, are bound to the nuclear matrix, an insoluble nuclear compartment devoid of DNA. These conclusions are derived from experiments in which either cellular ATP levels have been depleted (Stenoien et al., 2001) or proteasome activity has been inhibited (Stenoien et al., 2001; Deroo et al., 2002; Schaaf and Cidlowski, 2003) in cells containing GFP-tagged steroid hormone receptors. In either case, slower FRAP recoveries result, and extraction procedures demonstrate association of the steroid hormone receptor with the nuclear matrix (Tang and DeFranco, 1996). This has led to the suggestion that these receptors are normally bound to the nuclear matrix, and that their dissociation is promoted by energy and proteasome activity.

An alternative view is that these inhibition conditions induce an abnormal association with the nuclear matrix. Some investigators have suggested that the nuclear matrix is itself an artifact of the extraction conditions used to identify it (Pederson, 2000). An untreated cell examined by FRAP offers the opportunity to assess nuclear matrix binding without any perturbation of the system. More generally, it is of considerable interest to identify how many different binding states for a transcription factor are present in the nucleus of a normal cell and what the percent occupancy of each state is. This information is required as a starting point to understand nuclear mobility and its regulation. Mobility rate must play a key role in determining the search time required for a transcription factor to find its specific promoter amid a multitude of other binding sites in the nucleus.

Here we have applied our theory for FRAP recovery to nuclear mobility of a GFP-tagged glucocorticoid receptor (GFP-GR) in nuclei of both normal and ATP-depleted cells. Our results indicate that GFP-GR diffuses from one binding site to the next with an average time of 13 ms per binding event. Our analysis also suggests that in normal cells nuclear matrix binding at best accounts for a small fraction of bound GFP-GR with most GFP-GR molecules (90%) binding to a heretofore unidentified state.

More generally, our theoretical treatment provides several important insights for all biological FRAP analyses. First, we have defined constraints on what can be estimated from FRAP data. We show in several cases how the data enable evaluation only of the ratio of certain parameters, not their individual values. This is critical information for purely computational analyses, where such mathematical limitations may go unappreciated and lead to poor estimates of the individual parameters. Second, we have clarified the contribution of diffusion, binding, and the number of binding states to a FRAP recovery. We show that fast and slow components of a FRAP curve may sometimes represent weak and tight binding states, but that in many cases this is not true. Finally, we have found that diffusion will typically have to be incorporated in the analysis of many biological FRAP recoveries, even in very slow recoveries that last much longer than the recovery time for free diffusion. Ignoring this contribution will lead to erroneous conclusions.

	METHODS

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Cell lines
Mouse adenocarcinoma cell line 3617 was used for most experiments. The cells stably express GFP-GR under the control of a tetracycline-off system (Walker et al., 1999). For control experiments with GFP-only containing cells, the parental cell line (3134 cells) was transfected with a GFP plasmid (pEGFPC1, Clontech, Palo Alto, CA). Cells were grown and prepared for fluorescence imaging as previously described (Müller et al., 2001).

ATP depletion
The ATP-depletion conditions were similar to those used previously for GR (Tang and DeFranco, 1996), except adapted for microscopy. Cells were treated with 10 mM sodium azide (Sigma, St. Louis, MO) in glucose-minus-DMEM supplemented with 6 mM 2-deoxyglucose (Sigma) for 60 min and then brought to the microscope for FRAP experiments for up to 30 min longer.

Quantification of GFP-GR associated with the nuclear matrix
The nuclear matrix extraction procedure was similar to that previously described by Fey et al. (1986), adapted for visualization by fluorescence microscopy with a subsequent fixation step. Briefly, cells were treated with cytoskeleton buffer for 10 min at 4°C, extracted with 250 mM ammonium sulfate at 4°C for 5 min, and digested with DNase I for 30 min at room temperature. Cells were then fixed with 2% paraformaldehyde and examined by fluorescence microscopy on a Leica DMRA microscope (Leica, Exton, PA) equipped with a Photometrics Sensys charge-coupled device camera (Photometrics, Tucson, AZ) and images of nuclei recorded. Total nuclear fluorescence was measured with Metamorph software (Universal Imaging, Downingtown, PA).

FRAP protocol
FRAP experiments were performed on a Zeiss 510 confocal microscope (Carl Zeiss, Thornwood, NY) with either a 25x/0.8 NA dry objective for GFP-only cells, or a 100x/1.3 NA oil-immersion objective for GFP-GR cells. Cells were kept at 37°C using an air-stream stage incubator (Nevtek, Burnsville, VA). Bleaching was performed with a circular spot using the 488- and 514-nm lines from a 40-mW argon laser operating at 75% laser power. A single iteration was used for the bleach pulse, which lasted 0.8–40 ms depending on the bleach spot size. Fluorescence recovery was monitored at low laser intensity (0.2% of a 40-mW laser) at 0.8–40-ms intervals, depending on the experiment.

FRAP data manipulation
Approximately 10 separate FRAPs were performed and then averaged to generate a single FRAP curve. The temporal resolution was kept constant while measuring recovery, but this led to a very large number of closely spaced points in the second, slower phase of the recovery curve. To alleviate this, we averaged 10–30 adjacent points in this slower part of the curve. This generated roughly equally spaced points along the recovery curve and therefore avoided overly weighting the slower phase of the curve during fitting.

FRAP fitting
The model equations were programmed in Matlab (The Math Works, Natick, MA). The Matlab routine nlinfit was used to fit the models to experimental data. Using simulated FRAP curves, we found that nlinfit reliably converged to the correct fit for recoveries exhibiting either effective diffusion or reaction dominant behavior. In contrast, fits to simulated full model data often failed if the initial guess for one of the parameters was far from the true value. As a result, full model fits with real data were always performed by first sampling a grid of all possible and k_off values in 10^0.1 increments on a log scale to find the pair that yielded the smallest sum of residuals between the full model prediction and the experimental data. Then this ( k_off) pair was used as the initial guess in the nlinfit routine.

We reduced the number of fitted parameters in all full model fits by substituting a value for the free-diffusion constant D_f. This value for GFP-GR (9.2 µm²/s) was estimated from the measured value for GFP only (15.0 µm²/s) by correcting for the additional mass of GR (94 kD for GR vs. 27 kD for GFP). Since D_f M^–1/3, where M is mass, the predicted D_f for GFP-GR is 60% of that for GFP alone.

Error analysis
We report all errors here as 95% confidence intervals. For all parameters estimated in the fit, the confidence intervals were directly produced by the nlinfit routine. Some of the estimates reported here depend not only on these fitted parameters, but also on the bleach spot size. Confidence intervals for the bleach spot size were determined from at least 10 measurements of the apparent spot size either in fixed cells for the smaller bleach spot, or in live cells for the larger bleach spots by measurement immediately (9 ms) after the bleach.

For pure or effective diffusion fits, the error in the bleach spot size could be directly incorporated into the final estimate using the formula for _D (Eq. 8) or k*_on/k_off (Eq. 9) and the rules for convolution of errors. For full model fits, the bleach spot size enters as a term in the Laplace transform (Eq. 6), which is then inverted before fitting. Thus to estimate the impact of the bleach spot size error for full model fits, we produced estimates of k*_on and k_off using values for the bleach spot size at the endpoints of its 95% confidence interval. The resultant range in k*_on and k_off values was much larger than the 95% confidence interval computed by the nlinfit routine, an error based solely on the noise in the FRAP recovery data. Thus the error in the bleach spot size contributed more significantly, and so for this full model case, errors on k*_on and k_off values were taken as the endpoints produced from the bleach spot size errors.

	MODEL

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General equation: diffusion plus binding
We outline first the general equations underlying FRAP for a single binding reaction. Our nomenclature for the reaction is

(1)

where F represents free proteins, S represents vacant binding sites, C represents bound [FS] complexes, and k_on and k_off are the on- and off-rates, respectively. The equations for FRAP describing the preceding binding reaction must also incorporate diffusion. The most general case is a set of three coupled reaction-diffusion equations where f = [F], s = [S], and c = [C],

(2)

Here ² is the Laplacian operator and D represents the diffusion coefficient for each of the three species. The remaining terms reflect the standard chemical kinetics for the binding reaction in Eq. 1.

These equations can be simplified considerably by two assumptions that are applicable in many biological situations. The first simplifying assumption is that the biological system has reached equilibrium before photobleaching. For GFP fusion proteins, this means that the total amount of both GFP fusion protein and its binding sites remains constant over the time course of the fluorescence recovery. This is reasonable since most biological FRAPs recover on a timescale of seconds to several minutes, whereas GFP-fusion protein expression changes over a time course of hours, and is typically at a constant level by the time the FRAP experiment is performed. Therefore we assume equilibrium, and denote the corresponding equilibrium concentrations of F, S, and C by F_eq, S_eq, and C_eq. Although the act of bleaching changes the number of visible free and complexed molecules (F or C), it does not change the number of free binding sites. Therefore s = S_eq is a constant throughout the photobleaching recovery. This eliminates the second equation in Eq. 2, and also enables us to replace the variable s in the remaining two equations with a constant S_eq. As a result, we can define a pseudo-first-order rate constant given by k_onS_eq = k*_on. In what follows, we refer to this value as the pseudo-on rate.

The second simplifying assumption is that the binding sites are part of a large, relatively immobile complex, at least on the time- and length-scale of the FRAP measurement. This is a widely used approximation for FRAPs of, for example, either cytoskeletal or DNA binding proteins (Bulinski et al., 2001; Coscoy et al., 2002; Dundr et al., 2002). Ignoring diffusion of the bound complex results in D_c = 0 in the expressions in Eq. 2.

With these two assumptions, the expressions in Eq. 2 reduce to

(3)

Before the bleach, as noted above, the system is at equilibrium, and F and C have achieved steady-state values, F_eq and C_eq, so that the ratio of free/bound molecules is determined by

(4)

Total equilibrium, as given above, is not altered by photobleaching, but the equilibrium for bleached and unbleached molecules is disturbed. Specifically, at the site of bleaching, the concentration of fluorescent molecules is significantly reduced, and so the return to equilibrium concentrations of fluorescence will be governed by the expressions in Eq. 3. Thus in what follows, f and c represent the concentration of fluorescent molecules after a photobleach that occurs at t = 0. The measured FRAP recovery data is the sum of free and bound fluorescence, averaged over the bleach spot: frap(t) = avg(f(t)) + avg(c(t)).

In what follows, we presume that the experimental data can be normalized such that the FRAP recovery ranges from 0 to 1. This normalization is acceptable if individual FRAP curves are analyzed one cell at a time. Typically, however, curves from multiple cells must be averaged to obtain smooth data, and in these circumstances care is required in pooling data. It is critical that cells of comparable fluorescent intensities be averaged if the data are subsequently normalized. Otherwise, different cells will have different levels of expressed fusion proteins, yielding different fractions of bound and free molecules in each cell. This in turn will lead to different FRAP recoveries if binding interactions are present, a feature that can in fact be exploited to obtain evidence for such interactions (Icenogle and Elson, 1983; Safranyos et al., 1987).

The final height of the FRAP recovery equals the sum of the equilibrium concentrations F_eq + C_eq, and so normalization to 1 sets F_eq + C_eq = 1 (which presumes that the bleach spot is small relative to the total cell volume, otherwise some measurable fraction of fluorescence will be lost after the bleach). Combining the preceding equality with Eq. 4 yields the following relationships for the equilibrium concentrations:

(5)

We derive our solutions to the expressions in Eq. 3 for the case of a circular bleach spot. Circular bleach spots are now feasible and widely used in FRAP experiments performed on current scanning confocal microscopes. We also presume that there is a homogeneous distribution of fluorescence in the cell, and that the bleach spot is small relative to the size of the fluorescent compartment. Our model cannot be directly applied to rectangular bleach spots. Nor should it be applied to complicated cellular geometries such as endoplasmic reticulum, or to any highly heterogeneous distribution of fluorescence.

Finally, we adopt the convention of previous theoretical FRAP analyses and assume for simplicity two-dimensional diffusion in the plane of focus. This assumption is appropriate when the bleaching area forms a near-cylindrical shape through the cell, as occurs for a circular bleach spot of reasonable diameter. In this case, axial terms disappear from the Laplacian (²) in the expressions in Eq. 3 and only the radial component remains.

Analysis of the full reaction-diffusion equations
A strategy for obtaining a solution to the full reaction-diffusion system (Eq. 3) is to perform a Laplace transform. By analogy with the heat conduction problem between two concentric cylinders (Carslaw and Jaeger, 1959), a solution involving Bessel functions can be devised. Starting from the expressions in Eq. 3, we derive in the Appendix the general solution for the FRAP recovery within a circular bleach spot. We show there that the average of the Laplace transform of the fluorescent intensity within the bleach spot is given by

(6)

where q depends on k*_on, k_off, and D_f (see Eq. 15 in the Appendix), w is the radius of the bleach spot, I₁ and K₁ are modified Bessel functions of the first and second kind, and p is the Laplace variable that inverts to yield time. Thus the inverse transform of Eq. 6 can be computed numerically to yield the predicted FRAP recovery as a function of time. The numerical inversion requires 1 s with the Matlab routine invlap.m (Hollenbeck, 1998) running on a PC. This permits ready evaluation of how the predicted recovery depends on each of the parameters associated with the FRAP model.

This full model describes all possible behaviors of a FRAP recovery for a single binding reaction in the presence of diffusion. Therefore the model can be used to fit any FRAP recovery that involves a single binding reaction. In previous theoretical analyses of FRAP, three simplified cases of our full model solution have been considered. We refer to these as pure-diffusion dominant, effective diffusion, and reaction dominant behaviors. The pure-diffusion dominant solution is well understood and appreciated. The effective diffusion solution is also well understood, but virtually unknown in most of the FRAP community. The reaction dominant solution is not widely known, nor has it been completely developed. In what follows, we wish to determine when the full model is adequately described by one of these simplified scenarios. As a precursor, we explain each of these simplified scenarios and for each, either review or develop the complete solution for the FRAP recovery.

Pure-diffusion dominant
A first simplifying scenario arises when most of the fluorescent molecules are free. Under these conditions, FRAP measures primarily free diffusion of the fluorescently tagged molecule. For this free fraction, binding can be ignored and the expressions in Eq. 3 reduce to the diffusion equation,

(7)

The FRAP response for free diffusion has been extensively analyzed (Axelrod et al., 1976). For a circular bleach spot a closed form solution exists involving modified Bessel functions (Soumpasis, 1983) as

where

(8)

Thus when binding is almost nonexistent and most molecules are free, the FRAP recovery curve should be fit by one parameter _D in Eq. 8, thereby determining the diffusion constant D_f.

Although solutions for the case of pure diffusion are well known, the relationship of the fitted diffusion constant to the molecular diffusion process underlying it is less clear. The complex geometry of cellular compartments and subcellular space influences the measured macroscopic diffusion constant (Feder et al., 1996; Siggia et al., 2000). For our purposes, the pure-diffusion constant measured by FRAP of a nonbinding protein, such as GFP unfused to a target protein, is sufficient to account for the contribution of diffusive processes in the FRAP recovery. Knowledge of this macroscopic diffusion behavior then enables us to extract the binding information contained in a FRAP recovery curve for a protein that both diffuses and binds.

Effective diffusion
The second simplified case for the expressions in Eq. 3 arises when the reaction process is much faster than diffusion. This implies that at any location within the bleach spot, the binding reaction rapidly achieves a local equilibrium. Under these conditions, Crank (1975) has shown that reaction-diffusion equations reduce to a simple diffusion equation but with a different diffusion constant, known as the effective diffusion constant, D_eff. (Note the same term has been used by some authors to refer to diffusion in the cellular milieu—White and Stelzer, 1999; Siggia et al., 2000; Carrero et al., 2003—and this may or may not relate to the effective diffusion defined by Crank.) Here, we use the term effective diffusion to mean the slowed diffusion due to binding with

(9)

where D_f is the diffusion constant of the molecule in the absence of binding, and the off- and pseudo-on rates are as defined above. Note that the ratio of is the pseudo-equilibrium constant, which is the ratio of bound/free molecules (Eq. 4). D_f can be determined by first measuring FRAP recoveries for GFP. The diffusion constant of the GFP fusion protein can then be calculated by allowing for its extra mass relative to GFP alone, and using the fact that, in the simplest scenario, (see Methods). Thus, determination of D_eff yields

Since effective diffusion is governed by the standard diffusion equation, D_eff can be determined by fitting the FRAP recovery curve with the diffusion model (Eq. 8). The fit will yield a value for _D as

(10)

This gives an estimate for D_eff, and therefore the ratio from Eq. 9. Note that the previous case, pure-diffusion dominant, is a subset of effective diffusion in which the binding is very weak, and so D_eff = D_f. For practical reasons, we have distinguished pure-diffusion dominant from effective diffusion because pure-diffusion dominant behavior provides no useable information about binding, whereas effective diffusion does.

Reaction dominant
The third simplified scenario arises when diffusion is very fast compared both to binding and to the timescale of the FRAP measurement. Free molecules instantly equilibrate after the bleach, so that diffusion is not detected in the FRAP recovery. Unlike the pure and effective diffusion scenarios, a complete solution has not been developed for the case where the binding reaction dominates. Previously, Bulinski et al. (2001) demonstrated analytically that the rate constant for FRAP recovery is identical to the dissociation rate constant, k_off. We have extended the analysis to enable the estimation of both k*_on and k_off from the FRAP recovery curve. As shown in detail in the Appendix, we find the following solution, which describes the total fluorescence recovery, f(t)+c(t), over time:

(11)

C_eq depends only on the off- and pseudo-on rates (Eq. 5), so the fit to the FRAP recovery in the reaction dominant case yields k*_on and k_off.

A surprising result of the preceding analysis is that the rate of FRAP recovery depends only on the off-rate. A similar result has been obtained in a less general context by Bulinski et al. (2001) and in a Fourier space analysis by Kaufman and Jain (1991). In all cases, the pseudo-on rate disappears from the exponential term for FRAP recovery because of the well-mixed assumption.

	RESULTS OF SIMULATION AND MATHEMATICAL ANALYSIS

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Evaluation of (k*_on, k_off) parameter space for the full reaction-diffusion equations
Given solutions for the full model and the three idealized cases of this model, we now investigate when these simplifications hold. Four parameters influence the solution to the general FRAP recovery as given by Eq. 6. These are the free diffusion constant (D_f), the size of the bleach spot (w), the pseudo-on rate (k*_on), and the off-rate (k_off). Two of these four parameters are fairly similar from one laboratory to the next, namely the diffusion constant and the bleach spot size. Measured values for the cellular diffusion of free GFP range from 15–40 µm²/s (Swaminathan et al., 1997; Arrio-Dupont et al., 2000; Coscoy et al., 2002), and in most FRAP experiments, the bleach spot size is on the order of 1 µm in diameter. On the other hand, reported values for binding-association and binding-dissociation rate constants can vary over a much larger range (10⁶-fold). Thus to investigate the range of behaviors exhibited by the full model (Eq. 6), we set the diffusion constant D_f equal to 30 µm²/s and the bleach spot radius to 0.5 µm, and then varied the off- and pseudo-on rates over a 10¹⁰-fold range. Since w²/D_f defines the timescale of the recovery, choosing particular values for w and D_f does not prevent us from observing the entire range of behavior of this system.

To identify values of the rate constants where the idealized cases hold (i.e., pure-diffusion dominant, effective diffusion, or reaction dominant), we used the full model to compute a FRAP recovery for a particular value of (k*_on, k_off). We then used each of the three idealized models to generate a predicted FRAP recovery curve by substituting (k*_on, k_off) into the equations for the idealized cases. The degree of fit between the full model and each idealized model was assessed. Depending on the particular values of (k*_on, k_off), four outcomes were obtained: The full model was well fit by both pure and effective diffusion (Fig. 1 A); or it was fit only by effective diffusion (Fig. 1 B); or it was fit only by reaction dominant (Fig. 1 C); or it was fit by none of these simplifications (Fig. 1 D). This indicates that for particular values of (k*_on, k_off) the idealized models can accurately predict the FRAP recovery of a single binding reaction.

View larger version (34K): [in this window] [in a new window]   FIGURE 1  Model-predicted FRAP recovery curves. The FRAP recovery predicted by each idealized model is compared to that predicted by the full model (Df = 30 µm2/s, w = 0.5 µm). Domains are identified where the full model is well fit by an idealized model (A–C), but this is not always the case (D). (A) With very weak binding, the pure-diffusion dominant model (red line) provides a good fit (sum of residuals = 0.08) to the full model (bold black line). Note that the effective diffusion model result is obscured by the pure-diffusion dominant curve and also provides a good fit; pure-diffusion dominant is a particular case of the effective diffusion model, namely Deff = Df. The reaction dominant model (orange line) provides a very poor fit. (B) With a high the effective diffusion model (green line) alone provides a reasonable fit to the full model (sum of residuals = 0.89). (C) With low and low koff, the reaction dominant model (orange line) provides a good fit with the full model (sum of residuals = 0.66). (D) For certain values of and koff, none of the idealized models result in a good fit to the full model (pure-diffusion dominant sum of residuals = 16.6; effective diffusion sum of residuals = 8.30; and reaction dominant sum of residuals = 8.54).

View larger version (29K): [in this window] [in a new window]   FIGURE 2  Contour plots of model fits. For each idealized model, a contour plot of the sum of residuals between the idealized model and the full model demonstrates the degree of fit as a function of and koff (Df = 30 µm2/s, w = 0.5 µm). (A) The pure-diffusion dominant model results in a good fit (i.e., the sum of residuals becomes small) as becomes small relative to koff (lower-right corner). (B) The effective diffusion model domain is made up of the pure-diffusion dominant regime plus the area with large values of (C) The reaction dominant model matches the full model when both and koff are small (lower-left corner). (D) Superimposing the contour lines of the sum of residuals equal to 1 from parts A–C reveals the domains for each idealized model, as well as a region where none of the idealized models fit (full model only).

We also characterized the rate-constant parameter space using two practical features easily assayed in FRAP experiments. These are the dependence on bleach spot size and the time required for full recovery. A dependence on bleach spot size is an experimental approach to assess whether diffusion contributes to the FRAP recovery. Using the full model we simulated this experiment by generating FRAP curves with the radius of the bleach spot set to either 1 µm or 2 µm. For the larger spot size, the time for 99% recovery was determined. Then over this time span, the recovery curves for the two spot sizes were calculated. The difference between the two curves was measured at 200 equally spaced time points, and the sum of the residuals plotted as a contour plot in (k*_on, k_off) parameter space (Fig. 3 A). The region with low residuals corresponds to the domain where the recovery is independent of bleach spot size, and therefore independent of diffusion. As expected, this region is the reaction dominant regime where diffusion is presumed to be so rapid that it can be neglected. Observe, however, that the majority of the rate-constant parameter space is dependent on bleach spot size. This shows that diffusion plays a role in most recoveries, at least in the absence of constraints on the values for the off- and pseudo-on rates.

View larger version (35K): [in this window] [in a new window]   FIGURE 3  Practical observations from FRAP simulations. (A) A simulated test for bleach spot size dependence revealed a low sum of residuals between full model predictions when both and koff were small (Df = 30 µm2/s, w1 = 1 µm, and w2 = 2 µm). This corresponds well with the reaction dominant regime in Fig 2 D, the only domain independent of diffusion. (B) The contour plot shows the time (t99) predicted by the full model for FRAP recovery to 99% of original fluorescence (Df = 30 µm2/s, w = 0.5 µm). The contour lines of the sum of residuals equal to 1 for the idealized models (Fig. 2 D; colored lines) are overlaid on top of the recovery times (black lines). Recovery times for typical FRAP experiments are generally between 1 and 1000 s, indicating that reaction dominant, effective diffusion, or full model behavior are all possible. (C) The boundaries between the idealized model domains can be approximated by three lines (see text; bold lines in figure), which are superimposed on the sum-of-residuals-equal-to-1 contour lines for each idealized model (Df = 30 µm2/s and w = 0.5 µm). (D) The precise locations of the idealized model domains are dependent upon the bleach spot size and cellular-diffusion constant of the fluorescently labeled protein. This contour plot shows how the idealized model domains shift as the bleach spot radius, w, and diffusion constant, Df, are changed (dash-dot line, Df = 35 µm2/s, w = 0.4 µm; solid line, Df = 30 µm2/s, w = 0.5 µm; and dashed line, Df = 25 µm2/s, w = 0.6 µm. The w2/Df values have units of s).

The contour plot of recovery times (Fig. 3 B) also provides some additional insights into the three idealized domains of pure-diffusion dominant, effective diffusion, and reaction dominant. The pure-diffusion dominant regime corresponds approximately to the domain with recovery times of <1 s. In this case binding is so weak that it does not slow down the FRAP recovery. The effective diffusion regime corresponds approximately to the region where the contour lines for recovery times are parallel with slope equal to 1. This follows from the fact that the effective diffusion constant depends only on the ratio of k*_on/k_off (see Eq. 9), so constant values of this ratio yield the same recovery time. This generates a family of lines with slope 1 in the log/log rate-constant parameter space. Finally the reaction dominant domain corresponds approximately to the region where the contour lines for recovery times are parallel and vertical. Here the reaction dominant idealization results in a FRAP recovery rate that depends only on the off-rate (Eq. 11), so for constant k_off, the recovery time is the same.

A simplified view of rate-constant parameter space
The basic geometry of the rate-constant parameter space determined above can be explained with a few rules. The boundaries dividing the four separate regions of parameter space can be approximated by three lines (Fig. 3 C). The pure-diffusion dominant regime occurs when the ratio of bound to free molecules is 1% or less. From Eq. 4, Hence the border of the pure-diffusion dominant regime in log/log parameter space is approximated by a line of slope 1 and y-intercept of –2 (red line in Fig. 3 C). As noted above, the pure-diffusion dominant regime is actually a subdomain of effective diffusion. Therefore the practical effective diffusion regime is bounded on the right by the pure-diffusion dominant regime, and approximately defined by the remaining region above the line log (k*_on) = 3 (green line in Fig. 3 C). Similarly, the reaction dominant regime is bounded below by the pure-diffusion dominant regime, and approximately defined by the remaining region below the line log (k*_on) = 0 (orange line in Fig. 3 C). The remaining domain requires the full model (Eq. 6). The exact location of the boundaries between these regimes is dependent upon the applicable values of w and D_f (Fig. 3 D).

These boundaries as defined empirically clarify apparently conflicting theoretical approximations of reaction-diffusion equations reported in the literature. Crank (1975) concludes that boundaries between different regimes depend on the magnitude of k_off. Kaufman and Jain (1990) assert that the boundaries depend on the magnitude of k*_on. Elson and co-workers conclude that the sum of k*_on + k_off is critical (Elson and Magde, 1974; Elson and Reidler, 1979). All of these constraints can be applied to explain subregimes of our rate-constant parameter space, but they cannot account for the complete parameter space. Rather our rules detailed above provide the simplest explanation, namely that for a large free pool (C_eq/F_eq 0.01), pure diffusion dominates, otherwise the magnitude of k*_on partitions the remainder of the space into reaction dominant, full model, or effective diffusion.

To confirm these empirical observations, we show mathematically in the Appendix that our full model solution reduces to each of the idealized cases by applying the constraints outlined in Fig. 3 C. Namely, the full model (Eq. 6) simplifies to:

1. The pure-diffusion dominant model (Eq. 8) when the free pool is large ;
   
2. The effective-diffusion model (Eqs. 8–10) when the pseudo-on rate is large compared to the characteristic diffusion time ; and
   
3. The reaction-dominant model (Eq. 11) when the pseudo-on rate is small compared to the characteristic diffusion time and C_eq is significantly large
   

See Derivation of Idealized Solutions from the Full Model in the Appendix for the derivations.

In the Appendix, we also examine mathematically the transition between the pure-diffusion dominant regime and the reaction dominant regime, as well as the transition between the reaction dominant regime and the effective diffusion regime. The crossover from pure-diffusion dominant to reaction dominant occurs in the lower half of the plot in Fig. 3 C, namely for With this constraint, we show in the Appendix that the full model solution can be simplified to the sum of two terms, one representing the pure-diffusion component, multiplied by the size of the free pool, plus a second representing the reaction component, multiplied by the size of the bound pool (see Eq. 41 and its derivation in the Appendix). Thus in this domain of (k*_on, k_off) parameter space, the FRAP recovery curve is always the sum of two independent processes, diffusion plus reaction, each of which contributes to the total recovery based on both the size of the free and bound pools and on the characteristic times for recovery w²/D_f and 1/k_off. Typically, one process dominates, and this leads to the zones we have called either pure-diffusion dominant or reaction dominant. We see a sharp transition between these zones in Fig. 3 C because we used 99% recovery as an arbitrary threshold for full recovery. When the free pool falls below 99%, then the bound pool with its typically much slower recovery timescale of 1/k_off becomes a significant component of the recovery to 99% of final fluorescence. As a result, recovery times slow suddenly upon crossover to the reaction dominant regime.

The crossover from reaction dominant to effective diffusion occurs in the left half of the plot of Fig. 3 C, where goes from small to large. This transition is also analyzed mathematically in the Appendix. A simplified solution can be obtained when the free fraction, F_eq, is small (). This leads to a reduced model equation which depends on only two parameters, k_off and the ratio (Eq. 43). We call this the hybrid model and show in the Appendix that it involves a somewhat complicated combination of reaction-like and diffusion-like terms (Eq. 46). Thus in this regime, reaction and diffusion are coupled and not separable. For large the hybrid model reduces to effective diffusion, whereas for small it reduces to reaction dominant (see Appendix). The significance of this hybrid model is that it occupies a large portion of our full model domain (see Fig. 8 in Appendix), and therefore in this region the full model is capable of predicting only the ratio rather than unique values for each parameter. Additionally, the complex combination of reaction and diffusion terms in the hybrid model solution indicates that in this regime, it is inappropriate to expect or assign fast and slow components to the FRAP recovery, as is often attempted in the analysis of FRAP results (Kimura et al., 2002). Rather, it should be recognized that both the hybrid model (Eq. 46) and the effective diffusion solution (Eqs. 8–10) may appear by eye to contain two separate recovery phases, despite the fact that they cannot be separated into discrete reaction and diffusion processes.

View larger version (20K): [in this window] [in a new window]   FIGURE 8  Hybrid model results. We have found a simplified model that depends on only two parameters, Df / and koff, which provides a good fit (sum of residuals <1) to the full model when koff << (area above solid line; Df = 30 µm2/s, w = 0.5 µm). This hybrid model describes the crossover between the reaction and effective diffusion regimes (compare to Fig. 3 C), and mathematically reduces to either the reaction dominant equation or the effective diffusion equation with small or large, respectively. Practically, the hybrid model demonstrates that a large portion (shaded area) of the previously identified full-model-only zone (see Fig. 3 C) is dependent on only two parameters: Df / and koff. Therefore it is important that Df be determined before attempting to extract the rate constants from the full model.

Based on these model equations, we have performed a detailed analysis of the two-binding-state system. As detailed in the Appendix, we show that the same three idealized cases are still good approximations to the full model under certain conditions. This means in particular that in the presence of significant binding interactions, either the two-state effective diffusion model or the two-state reaction dominant model will provide excellent fits for some FRAP recoveries. Thus the model equations we derive for these scenarios will find practical application. We also show once again that there is a domain in rate-constant parameter space in which only the full model solution is valid, and so our complete Laplace transform solution for two binding states will also find practical application.

By an extensive exploration of rate-constant parameter space for the two-binding-state model, we find that the full model domain is larger compared to the one-state model (see discussion in the Appendix). The increase in size of the full model domain occurs at the expense of the reaction dominant and effective diffusion domains. We show that these regions shrink because full model behavior typically results whenever binding reactions from different regimes are combined (Fig. 4). For example, if one binding state has rate constants characteristic of effective diffusion (for the one-state model), and the other state has rate constants characteristic of reaction dominant (for the one-state model), the combination will typically produce full model behavior for the two-state model. As this is generally true for any combination of reactions drawn from different regimes, the full model regime becomes progressively larger as the number of binding states increases.

View larger version (40K): [in this window] [in a new window]   FIGURE 4  Model-predicted FRAP recovery curves for two binding reactions. (A, D, and G) Matching domains: When both individual reactions exhibit the same idealized behavior, then the FRAP recovery predicted by the two-reaction full model (bold black line) is fit by the two-reaction model of that idealized behavior (colored lines). For example in A, given that and k1off place one reaction in the single-reaction effective diffusion domain, and that and k2off place the other reaction in the single-reaction effective diffusion domain, the two-reaction full model FRAP recovery is well fit by the two-reaction effective diffusion model (see Eq. 59). (B, E, and H) Different domains: When the individual reactions do not exhibit the same idealized behavior, then none of the two-reaction idealized models will match the FRAP recovery predicted by the full two-reaction model. In this case, the full two-reaction model is required (Eq. 55). (C and F) Exceptions to the rule: When either the equilibrium concentration or the time for recovery is much greater for one of the reactions than the other, then the two-reaction full model result will be fit by the two-reaction idealized model of the reaction with the larger concentration or longer time. (Df = 30 µm2/s, w = 0.5 µm. ED, effective diffusion model; RD, reaction dominant model; and FM, full model.)

Expected FRAP behaviors
Given the description of rate-constant parameter space for the one-binding-state model (Fig. 3 C), and the rules described above and in the Appendix for the two- (or more) binding-state models, it is of interest to estimate which types of behavior typical biological FRAPs should exhibit. In the absence of binding interactions, the FRAP recovery of the fusion protein reflects the pure-diffusion scenario, with a recovery rate very similar to that of GFP alone (differing only in proportion to the added mass of the fusion protein). In cases where significant differences exist between the recovery of the fusion protein and GFP alone, binding interactions are implicated. Considering typical ranges for the binding rate constants provides some feel for the likelihood of possible FRAP outcomes. Typical off-rates range from 10¹ s^–1 for nonspecific DNA binding to 10^–6 s^–1 for many types of specific binding. Thus most biological FRAPs should occupy the left half of Fig. 3 C where reaction dominant, full model, or effective diffusion should occur.

To see which of these behaviors are possible or likely, on-rates must be considered. The diffusion-limited on-rate is 10⁶ M^–1 s^–1, although this is not an absolute upper bound. Typical on-rates range from 10²–10⁸ M^–1 s^–1. It is the pseudo-on rate, namely the product of the on-rate with the equilibrium concentration of bound sites, that determines location in the rate-constant parameter space of Fig. 3 C. Thus for diffusion-limited on-rates (10⁶ M^–1 s^–1), a 1-mM concentration of bound sites will yield pseudo-on rates of 10³ s^–1 or effective diffusion behavior, whereas a 1-µM concentration of bound sites will yield pseudo-on rates of 10⁰ s^–1 near the boundary between the full model and reaction dominant. Thus the reaction dominant regime can be entered by a combination of slower on-rates and low concentrations of binding sites. Considering a specific case, DNA binding in a mammalian nucleus of 5-µm radius and 6 x 10⁹ basepairs of DNA, the concentration of basepair sites is 20 mM. Others have estimated the concentration of DNA in the nucleus as high as 100 mM (Lieberman and Nordeen, 1997). In this latter case, the smallest possible value for k*_on is 10¹ s^–1, altogether eliminating reaction-dominant as a possible behavior. However, this value for DNA concentration is probably an upper limit, since the number of available sites might well be reduced by constraints that limit access to some subset of sites. Nevertheless, these rough calculations suggest that many FRAPs should exhibit behavior that depends on diffusion either via the full model or effective diffusion. This is underscored by the analysis of models with two or more binding states which indicate that the domain occupied by the full model increases (see the preceding section). In sum, reaction dominant, effective diffusion, and full model behavior are all possible outcomes of FRAP experiments, but the latter two behaviors with their dependence on diffusion should be more common than currently appreciated.

	EXPERIMENTAL RESULTS AND DISCUSSION

TOP ABSTRACT INTRODUCTION METHODS MODEL RESULTS OF SIMULATION AND... EXPERIMENTAL RESULTS AND... APPENDIX ACKNOWLEDGEMENTS REFERENCES

 
To test the utility of the preceding theoretical development, we applied the method to the problem of transcription factor mobility. We first evaluated our experimental protocol for FRAP using mouse adenocarcinoma cells transfected with GFP only. The average recovery was well fit by Eq. 8 for pure-diffusion dominant behavior (Fig. 5 A) yielding a cellular diffusion constant D_f for GFP of 15.0 µm²/s, within the range observed in other studies (Swaminathan et al., 1997; Arrio-Dupont et al., 2000; Coscoy et al., 2002). The good fit and accurate determination of D_f demonstrate that our FRAP protocol satisfies the basic assumptions of our model.

View larger version (37K): [in this window] [in a new window]   FIGURE 5  Control GFP-GR results. (A) The FRAP recovery of GFP (, n = 12) is well fit by the pure-diffusion dominant model (solid line), with Df = 15.0 µm2/s (w = 2.7 µm). In this and all other figures, the error bars represent the 95% confidence interval of each data point. For clarity not all error bars are shown for early time points. (B) The pure-diffusion dominant model (solid line) was simulated using Eq. 8 with w = 1.1 µm and Df = 9.2 µm2/s, which is the predicted diffusion constant for GFP-GR in the nucleus based on the measured diffusion constant for GFP and the size difference between GFP and GFP-GR. The observed FRAP recovery of GFP-GR (, n = 10) was significantly slower, indicating the presence of binding interactions. (C) FRAP of GFP-GR with different bleach spot sizes reveals a dependence on spot size and, therefore, diffusion (n = 10 in each case). (D) Consistent with the dependence on spot size and diffusion, the reaction dominant idealized model (solid line) provides a poor fit to the GFP-GR FRAP recovery (w = 1.1 µm). (E) The FRAP recovery of GFP-GR (w = 1.1 µm) is well fit by the effective diffusion model (dashed line, obscured by FM fit curve), with Deff = 1.3 µm2/s. This yields an estimate of the pseudo-equilibrium constant, /koff = 6.0, via Eq. 9, with Df = 9.2 µm2/s. The full model (solid line) also provides a good fit to the data, with a similar value of the pseudo-equilibrium constant (/koff = 5.8) calculated from the full model estimates of = 500 s–1 and koff = 86.4 s–1. (F) With the bleach spot size reduced to 0.5 µm, the effective diffusion (dashed line) and full model (solid line) provide reasonably good fits. However, the full model fit yields a clear improvement in the sum of residuals (see Table 1), in contrast to the larger-spot-size experiment where the sum of residuals does not change appreciably between the effective diffusion and full model fits (see Table 1 and E). The smaller spot size has therefore moved the FRAP recovery to the boundary between effective diffusion and the full model, enabling independent estimates of and koff = 78.6 s–1. These estimates yield which is in reasonable agreement with that found in the larger-spot-size experiment shown in E.

View larger version (61K): [in this window] [in a new window]   FIGURE 7  Nuclear matrix extraction. (A) Before the nuclear matrix extraction procedure, cell nuclei appear bright with GFP-GR fluorescence. (B) In control cells almost all fluorescence disappears after nuclear matrix extraction. (C) In cells treated with sodium azide the nuclei retain a large amount of fluorescence after nuclear matrix extraction, indicating that GFP-GR is immobilized at the matrix under these conditions. Bar = 10 µm. (D) The amount of fluorescence lost after nuclear matrix extraction was quantified. In control cells 5% of fluorescence is retained (n = 99), whereas in azide-treated cells 40% of fluorescence is retained (n = 86). The large amount of fluorescence retained at the nuclear matrix in azide-treated cells matches the two-state full model prediction that 44% of GFP-GR in the nucleus is bound in a second reaction state.

Both effective diffusion and the full model yielded good, mutually consistent fits (Fig. 5 E). The full model fit predicted rate constants k*_on = 500 s^–1 and k_off = 86.4 s^–1. As a consistency check, we determined in which regime these rate constants were located (see Fig. 3 C). This was done by using the predicted from the full model fit to calculate (see the rules defining domains in Fig. 3 D). The computed value of 65 was >>1, thereby placing the full model fit in the effective diffusion regime. Since the full model encompasses all simplified regimes, we expect it to agree with the effective diffusion model when this simplified scenario holds. As further proof of self consistency, we found that the predicted rate constants from the full model yielded a ratio of which was similar to that predicted directly from the effective diffusion fit, namely Thus we conclude that the GFP-GR FRAP recovery exhibits effective diffusion when the spot size radius is 1.1 µm.

As Eqs. 9 and 59 show, the same effective diffusion fit may represent one or several different binding states. To assess the number of GFP-GR binding states, we attempted to shift the FRAP recovery from the effective diffusion regime to the full model regime, where the number of binding states can be directly determined. Regime boundaries (Fig. 3 D) are inversely proportional to w²/D. Therefore a sufficiently small bleach spot size (w) should shift the boundary for the full model upward such that it eventually encompasses (k*_on, k_off) for GFP-GR. In an attempt to achieve this, we reduced our bleach spot size to 0.5 µm and then remeasured FRAP recoveries. Both the effective diffusion and full model now yielded reasonably good fits to these data (Fig. 5 F), but the full model fit yielded a clear improvement in the sum of residuals (Table 1). This is in contrast to the larger spot size examined first (1.1 µm), where the sum of residuals did not change appreciably between the effective diffusion and full model fits (Table 1). For the smaller spot size, the difference in the sum of residuals between the full model and effective diffusion fits equaled 0.5 (summed over 64 data points). This placed the recovery in the boundary zone between these two regimes, which we earlier defined operationally as a sum of residuals equal to 1 (but summed over 200 data points, see Fig. 3 C). Very little additional improvement in the sum of residuals was seen with a two-state full model compared to a one-state full model (Table 1), suggesting that normally GFP-GR in the nucleus occupies predominantly one binding state. The one-binding state full model fit for the smaller spot size also yielded independent estimates for k*_on and k_off. Their ratio (5.1 ± 1.1) was in good agreement with that obtained directly via the effective diffusion fit (using Eqs. 9 and 10) for the larger spot size (6.0 ± 0.34). This agreement for different spot sizes is a satisfying confirmation of the experimental and modeling protocols.

The preceding fits illustrate several key points about GFP-GR binding within nuclei. First, they suggest that there is predominantly one binding state for GFP-GR, since the one-binding-state full model yielded a satisfactory fit that was little improved by adding a second state. Second, by using the fitted pseudo-equilibrium binding constant for this predominant binding state, we can calculate, using Eq. 5, that 14% of GFP-GR is free whereas 86% is bound. Before this analysis, it was not appreciated that such a large fraction of GFP-GR is bound in the nucleus. Since GFP-GR is probably overexpressed approximately five times relative to endogenous GR levels (unpublished observations), this suggests that there must be many binding sites of this predominant state within the nucleus. Third, the transient nature of this binding was also not appreciated. The effective diffusion fit for the larger spot size indicates that on average a GFP-GR molecule undergoes multiple binding interactions within the 1.1-µm-radius bleach spot during the FRAP recovery. The average binding time per site is given by or 12.7 ms and the average time for diffusion to the next site is given by or 2.5 ms (Berg, 1986). These parameters underscore the rapid mobility of GFP-GR, indicating that on average each GFP-GR molecule samples 65 binding sites in 1 s. This rapid sampling of sites is likely to be important in the ability of GFP-GR to find and bind its specific DNA target site for transcription initiation.

Based on previous studies, this bound state of GFP-GR should reflect association with the nuclear matrix (Tang and DeFranco, 1996). Since release of GR from the matrix is thought to require ATP (Tang and DeFranco, 1996), a depletion in ATP levels should lead to a smaller k_off value as measured from a fit to the FRAP recovery.

To analyze this hypothesis we performed FRAPs on cells depleted of ATP via sodium azide treatment (Tang and DeFranco, 1996). Consistent with previous observations of reduced mobility of a steroid receptor after ATP depletion (Stenoien et al., 2001), we observed a sharp decrease in the rate of FRAP recovery (Fig. 6 A). However, in contrast to the simple prediction that ATP depletion d