A Quantitative Approach to Analyze Binding Diffusion Kinetics by Confocal FRAP

Abstract

Most of the important types of interactions that occur in cells can be characterized as binding-diffusion type processes, and can be quantified by kinetic rate constants such as diffusion coefficients (D) and binding rate constants (k_on and k_off). Confocal FRAP is a potentially important tool for the quantitative analysis of intracellular binding-diffusion kinetics, but how to dependably extract accurate kinetic constants from such analyses is still an open question. To this end, in this study, we developed what we believe is a new analytical model for confocal FRAP-based measurements of intracellular binding-diffusion processes, based on a closed-form equation of the FRAP formula for a spot photobleach geometry. This approach incorporates a binding diffusion model that allows for diffusion of both the unbound and bound species, and also compensates for binding diffusion that occurs during photobleaching, a critical consideration in confocal FRAP analysis. In addition, to address the problem of parametric multiplicity, we propose a scheme to reduce the number of fitting parameters in the effective diffusion subregime when D's for the bound and unbound species are known. We validate this method by measuring kinetic rate constants for the CAAX-mediated binding of Ras to membranes of the endoplasmic reticulum, obtaining binding constants of k_on ∼ 255/s and k_off ∼ 31/s.

Introduction

Many of the most important types of interactions that occur in cells, such as the binding of transcription factors to DNA, can be described as binding-diffusion processes. One of the most widely used approaches to study diffusion and reaction-diffusion processes in cells is a microscopy-based technique known as fluorescence recovery after photobleaching (FRAP). In FRAP, a group of fluorescently tagged molecules in a defined region of interest (ROI) is rapidly photobleached. Exchange of the bleached molecules with unbleached molecules from the surrounding region is followed over time. This results in a characteristic recovery curve that contains information about the kinetic rate constants of the fluorescently tagged molecules as well as the fraction of molecules free to diffuse. Classically, FRAP measurements were performed using a focused, static laser beam to bleach molecules, and the earliest applications of FRAP focused primarily on characterizing the lateral diffusion of fluorescently tagged proteins at the plasma membrane (1). Today, FRAP has become a common technique that can be performed on most confocal laser-scanning microscopes (LSMs (2)), and with the advent of green fluorescent protein, the applications of FRAP have become broadly extended to include the study of intracellular protein dynamics (3–10).

Given these technological advances, confocal FRAP can now be used to study binding-diffusion kinetics of proteins within their native environments, an important goal in light of the universality of reaction and diffusion processes in cells (11–18). However, kinetic analysis of binding by confocal FRAP is still at its early stages, and no consensus has yet been reached regarding the best experimental methodology and analytical approaches to obtain robust diffusion coefficients and rate constants from such measurements (19–25). In addition, most recent studies of binding diffusion have focused on analysis of nuclear proteins, whereas other examples of such behavior, such as the reversible binding of peripheral membrane proteins (26), have not been as well studied by quantitative confocal FRAP analysis. Thus, more work is still needed to generate more robust methods for analyzing binding-diffusion kinetics by confocal FRAP that can be broadly applied to a variety of intracellular compartments.

To achieve this goal, one factor that requires further consideration is how to account for the finite time required for photobleaching in confocal FRAP, during which significant diffusion and binding of the molecules of interest can occur. Without adjusting for diffusion during photobleaching, the apparent D measured for freely diffusing proteins such as enhanced green fluorescent protein (EGFP) depends on the bleaching spot sizes and number of bleach scans, and the resulting magnitude of D also tends to be underestimated (22,27–29). For molecules undergoing binding- and diffusion-type behavior, both events occur during the photobleaching event, further complicating analysis of confocal FRAP data. In addition, many previous models of binding-diffusion behavior have made the simplifying assumption that the diffusion of the bound species is negligible (7,23,29–32). However, for many biologically relevant interactions, such as the reversible binding of peripheral membrane proteins, significant diffusional mobility of both the unbound and bound species occurs. Finally, analysis of binding kinetics by FRAP data fitting may not necessarily yield well-defined parameter sets. Thus, strategies to reduce the number of fitting parameters will also be needed as models of increasing complexity are developed.

To address these needs, in this study, we develop what we believe is a new analytical model for the quantitative analysis of intracellular binding-diffusion processes by confocal FRAP. To validate this approach, we quantify the transient associations of signaling proteins with intracellular membranes, focusing on the small GTPase Ras as a model protein. Ras is a peripheral membrane protein that can switch between a stably membrane-bound form and a form that can undergo reversible membrane binding as a result of the presence of multiple membrane-binding motifs within the protein (33–35). These include a CAAX motif (where C stands for cysteine, A for aliphatic amino acid, and X for any amino acid) and either one or two palmitoylation sites or a polybasic domain, depending on the specific Ras isoform.

In a previous study, we used confocal FRAP to analyze the dynamics of EGFP-tagged HRas and NRas mutants that lack functional palmitoylation sites but contain an intact CAAX motif (26). We showed in that study that palmitoylation mutants of HRas and NRas appear to reversibly bind both endoplasmic reticulum (ER) and Golgi membranes, as assessed by confocal FRAP. Using fluorescence correlation spectroscopy (FCS), we confirmed the presence of two diffusing species of protein in the ER region of the cell, the first having a diffusion coefficient of ∼45 μm²/s, essentially identical to that obtained for soluble EGFP, and the other with a diffusion coefficient of ∼2.5 μm²/s, slightly higher than that measured previously for Ras at the plasma membrane by confocal FRAP (36). In the study described here, we chose one of these proteins, EGFP-HRas C181S C184S, as a representative model to quantitatively determine the binding constants associated with the reversible binding of these proteins to ER membranes. Here, we provide the first measurements, to our knowledge, of the binding constants for the CAAX-mediated association of Ras with the ER: k_on = 255 (s⁻¹) and k_off = 31 (s⁻¹).

Materials and Methods

DNA constructs and cell transfections

COS-7 cells (ATCC, CRL-1651) were maintained in 5% CO₂ at 37°C in Dulbecco's modified Eagle's medium supplemented with 10% fetal calf serum. To transiently transfect cells, FuGENE 6 (Roche Diagnostics, Indianapolis, IN) was used according to the manufacturer's protocol. Cells were imaged one day after transfection. A plasmid for the EGFP-HRas palmitoylation mutant (EGFP-HRas C181S, C184S) was as previously described (37).

Confocal FRAP

Cells were imaged using an inverted confocal laser-scanning microscope (LSM 510, Carl Zeiss MicroImaging, Thornwood, NY). The 488-nm line of an argon laser was used to excite EGFP. Emission was detected with a longpass (LP) 505 or 530 or a bandpass (BP) 505–530 filter. For all samples, a Zeiss Plan-Neofluar 40X/1.3 oil immersion lens was used for imaging. Phenol-red-free Dulbecco's modified Eagle's medium containing 10% fetal calf serum and 50 mM HEPES was used to maintain the cells for live-cell imaging. FRAP experiments were performed at room temperature. For confocal FRAP measurements, 40- or 60-pixel squares were chosen as ROIs. Within the squares, circles 10 or 20 pixels in diameter were photobleached for either 10 or 20 scan iterations using 100% transmission of the 488-nm-wavelength laser. We will refer to the 40 × 40- or 60 × 60-pixel ROIs as observational ROIs, the circles of 10- or 20-pixel diameter in the middle of the observational ROIs as the nominal bleach regions or bleach ROIs, and the radii of the nominal bleach regions as nominal radii ($r_n=$0.6 μm and 1.1 μm: 1 pixel = 0.11 μm) from here on.

Ten FRAP experiments were performed for four different setups. Experiments were repeated on two different days to test the reproduceability of the results. FRAP recovery curves were obtained by measuring the average fluorescence intensity in the nominal bleach region at the center of either 40 × 40- or 60 × 60-pixel square ROIs. Since detector blinking has been reported as negligible in circular-region photobleaching (23), no adjustment was made to FRAP data to correct for this effect. FRAP data were normalized by prebleach fluorescence intensity. All fits were performed on averaged normalized FRAP curves (n = 10 cells) and the resulting fit parameters are reported as the mean ± SD for two independent experiments (i.e., a total of n = 20 cells).

Effective radius measurement and pure-diffusion FRAP model

The effective bleach radius (r_e) was measured as described elsewhere (28,38). When data fitting to a pure-diffusion FRAP model was required, we followed the protocol described in our previous study (28) using the FRAP model described in section S1.1 in the Supporting Material.

Numerical evaluation of the theoretical FRAP models, data fitting, and weighted residuals

Theoretical FRAP models (F(t)) were programmed in MATLAB (The MathWorks, Natick, MA). More details about the numerical evaluation of the theoretical FRAP models, as well as MATLAB code, are provided in section S1 in the Supporting Material. Data fitting was performed by minimizing the weighted residual described in our previous study (28). For data fitting, a standard Levenberg-Marquardt least-weighted residual algorithm was used by modifying nlinfit.m in the MATLAB statistics toolbox (The MathWorks). In addition, for one- and two-parameter fittings, an exhaustive search algorithm was used to generate the weighted residual profiles across a range of binding-rate constants and to reinforce the Levenberg-Marquardt scheme.

Theory

Binding-diffusion model

Conventionally, the binding component of a binding-diffusion process is described by the following chemical equation:

$$U+S\begin{array}{c}{\overline{k}}_{\text{on}}\\ \rightleftharpoons B\\ k_{\text{off}}\end{array},$$ (1)

where U and S denote unbound molecules and specific binding sites or receptors, respectively, and B represents bound complexes (US). Assuming the density of binding sites is high enough, we can describe the binding-diffusion kinetics in Eq. 1 using first-order reaction-diffusion equations in terms of the concentration of free molecules (u) and the concentration of bound molecules (b) on binding sites:

$$\{\begin{array}{cc}\frac{\partial u}{\partial t}=& D_1{\nabla }^{2}u-k_{\text{on}}u+k_{\text{off}}b\\ \frac{\partial b}{\partial t}=& D_2{\nabla }^{2}b+k_{\text{on}}u-k_{\text{off}}b,\end{array}$$ (2)

where ${\nabla }^2=\frac{{\partial }^2}{\partial x^2}+\frac{{\partial }^2}{\partial y^2}$. Here, D₁ and D₂ are diffusion coefficients of U and B, and k_on ($\equiv {\overline{k}}_{\text{on}}S$) and k_off are on and off binding-rate constants. We assume that Eq. 2 is defined in ${\mathrm{R}}^2$, that D₁, D₂, k_on, and k_off are positive constants, and that D₁ > D₂. Initial conditions are defined by the postbleach fluorescence intensity profiles:

$$\{u\left(x,y,0\right)=\frac{k_{\text{off}}}{k_{\text{on}}+k_{\text{off}}}C\left(x,y,0\right)b\left(x,y,0\right)=\frac{k_{\text{on}}}{k_{\text{on}}+k_{\text{off}}}C\left(x,y,0\right),$$ (3)

where

$$C\left(x,y,0\right)=C_{\text{i}}\text{exp}\left[-K\text{exp}\left(-\frac{2\left(x^2+y^2\right)}{r_{\text{e}}^2}\right)\right].$$ (4)

Compensation for binding and diffusion during the photobleaching process

To estimate the diffusion and binding events that occur during photobleaching, the photobleaching process was mathematically modeled as following first-order photobleaching kinetics with bleaching rate k_b (${\text{s}}^{-1}$),

$$\{\begin{array}{c}\frac{\partial u}{\partial t}=-k_b{I}_{rn}\left(x,y\right)u+D_1{\nabla }^{2}u-k_{\text{on}}u+k_{\text{off}}b\\ \frac{\partial b}{\partial t}=-k_b{I}_{rn}\left(x,y\right)b+D_2{\nabla }^{2}b+k_{\text{on}}u-k_{\text{off}}b,\end{array}$$ (5)

where initial conditions are given by the prebleach steady state of Eq. 2 (i.e., $u\left(x,y,0\right)=k_{\text{off}}{C}_{\text{i}}/\left(k_{\text{on}}+k_{\text{off}}\right)$and $b\left(x,y,0\right)=k_{\text{on}}{C}_{\text{i}}/\left(k_{\text{on}}+k_{\text{off}}\right)$). Eq. 5 was solved numerically, since no explicit solution for Eq. 5 is known. $I_{\text{rn}}\left(x,y\right)=\frac{2I_0}{\pi r_{\text{n}}^2}\exp \left(-\frac{2\left(x^2+y^2\right)}{r_{\text{n}}^2}\right)$was chosen because some studies (29,39) indicated that photobleaching scanning profile of confocal LSMs can be approximated collectively by a Gaussian laser profile for a small circular bleaching spot.

Derivation of an explicit FRAP formula for the binding diffusion model that considers binding and diffusion during photobleaching

If the postbleach profile of the binding-diffusion model is approximated by an exponential of a Gaussian, then the FRAP formula for the binding-diffusion model correcting for binding diffusion during photobleaching can be derived explicitly based on the previous studies (28), as shown in the section S1.2 in the Supporting Material.

$$\{\overline{F}\left(t\right)=\frac{k_{\text{off}}{e}^{-k_{\text{on}}t}\mathrm{ℱ}\left(D_{1}t\right)+k_{\text{on}}{e}^{-k_{\text{off}}t}\mathrm{ℱ}\left(D_{2}t\right)}{k_{\text{on}}+k_{\text{off}}}+\frac{t}{D_1-D_2}\underset{D_2}{\overset{D_1}{\int }}\frac{e^{\lambda t-\mu st}\sqrt{k_{\text{on}}{k}_{\text{off}}}}{k_{\text{on}}+k_{\text{off}}}\left({\mathrm{ℐ}}_1\left(s,t\right)+{\mathrm{ℐ}}_2\left(s,t\right)\right)\mathrm{ℱ}\left(st\right)ds\mathrm{ℱ}\left(Dt\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(-K\right)}^n}{n!}·\frac{1}{\left(1+n\left[\frac{r_{\text{n}}^2}{r_{\text{e}}^2}+8\frac{Dt}{r_e^2}\right]\right)},$$ (6)

where $\mathrm{ℱ}\left(Dt\right)$ describes FRAP due to pure diffusion in ${\mathrm{R}}^2$ and

$$\{\lambda =\frac{k_{\text{on}}{D}_2-k_{\text{off}}{D}_1}{D_1-D_2},\mu =\frac{k_{\text{on}}-k_{\text{off}}}{D_1-D_2},$$

and for the modified Bessel functions, ${\mathbf{I}}_0$and ${\mathbf{I}}_1$,

$$\{\begin{array}{c}{\mathrm{ℐ}}_1\left(s,t\right)=\left[k_{\text{off}}\sqrt{\frac{s-D_2}{D_1-s}}+k_{\text{on}}\sqrt{\frac{D_1-s}{s-D_2}}\right]{\mathbf{I}}_1\left(\frac{2t\sqrt{k_{\text{on}}{k}_{\text{off}}\left(D_1-s\right)\left(s-D_2\right)}}{D_1-D_2}\right)\\ {\mathrm{ℐ}}_2\left(s,t\right)=2\sqrt{k_{\text{on}}{k}_{\text{off}}}{\mathbf{I}}_0\left(\frac{2t\sqrt{k_{\text{on}}{k}_{\text{off}}\left(D_1-s\right)\left(s-D_2\right)}}{D_1-D_2}\right).\end{array}$$

For FRAP data that do not recover to their prebleach steady state, the data can be fitted by

$$F\left(t\right)=\overline{F}\left(t\right)\mathrm{\times }MF+\left(1-MF\right)F_0,$$ (7)

where $\overline{F}\left(t\right)$ is as in Eq. 6. The mobile fraction (MF) is defined in terms of a prebleach steady-state fluorescence intensity ($F_{\text{i}}$), a postbleach steady-state fluorescence intensity ($F_{\mathrm{\infty }}$), and an initial postbleach fluorescence intensity (F₀) as $MF=\left(F_{\mathrm{\infty }}-F_0\right)/\left(F_i-F_0\right)$.

Results

Choice of model protein, geometry of photobleach region, and bleaching conditions for confocal FRAP analysis

In our previous analysis of the diffusional mobility of EGFP-HRas C181S C184S in the ER (26), we performed confocal FRAP measurements using a rectangular bleach region centered on ER membranes, and we analyzed the postbleach fluorescence intensities using a program that simulates diffusional recoveries to obtain measures of apparent diffusion coefficients (40). However, in this study, we wished to attempt to extract rate constants for the binding and release of Ras from endomembranes using recently developed analytical approaches (28,41), which are more readily implemented assuming circular bleach geometries. We therefore used a FRAP protocol in which we photobleached a circular bleach spot with nominal radius r_n (Fig. 1, A and B). To be able to correct for diffusion and binding during the photobleaching event in confocal FRAP, we monitored recoveries in a square ROI slightly larger than the bleach region itself (28,38). This enabled us to analyze the distribution of bleached molecules immediately after the photobleach (Fig. 1 C). Finally, to test for the contribution of diffusion to the recovery, as well as the robustness of our corrections for diffusion and binding during the bleach and subsequent kinetic analysis, we performed FRAP for two different-sized bleach regions (r_n = 0.6 and 1.1 μm) and two different bleaching conditions (photobleaching using either 10 or 20 repetitive scans of the bleach region).

Figure 1.

Experimentally determined postbleach fluorescence intensity profiles and recovery curves for EGFP-HRas C181S C184S in the ER. (A) A representative image of EGFP-HRas C181S C184S in a COS7 cell. The two white boxes represent 40 × 40- and 60 × 60-pixel square observation ROIs in the ER area. (B) Time points from a representative FRAP experiment for EGFP-HRas C181S C184S in the ER area of COS7 cells for $r_{\text{n}}=0.6\mu \text{m}$ and $r_{\text{n}}=1.1\mu \text{m}$ (dashed white circles) in 40 × 40- and 60 × 60-pixel squares (4.4 × 4.4 $\mu {\text{m}}^2$ and 6.6 × 6.6 $\mu {\text{m}}^2$). (C) Representative averaged initial postbleach profiles of EGFP-HRas C181S C184S from the ER area of COS7 cells (n = 10) for r_n = 0.6 μm and r_n = 1.1 μm after 10 or 20 photobleaching scans. Profiles were fitted by exponentials of Gaussians (Eq. 4) to obtain effective radii. (D) The effective radii (r_e) found for different FRAP-setup data were compared with the nominal radii (r_n) and bleaching-depth parameter (K). (E) Representative averaged FRAP curves (n = 10) for r_n = 0.6 μm and r_n = 1.1 μm after 10 or 20 photobleaching scans. Error bars show standard errors. Solid black lines are fit by the effective-diffusion model. (F) Comparison of $D_{\text{eff}}$ values for FRAP data obtained for the above experimental r_n conditions.

When expressed in COS-7 cells, EGFP-HRas C181S C184S localizes to the Golgi complex, ER, and cytoplasm (Fig. 1 A). Typical examples of regions selected for FRAP analysis are shown in Fig. 1 A, and representative images from FRAP data collected for two different sizes of nominal bleaching spot but an identical number of bleach scans (Fig. 1 B). For each set of FRAP experiments, we analyzed the distribution of fluorescence immediately after the photobleach to measure the size of the effective bleach radius, r_e. Under the conditions of our experiments, the effective radii were measured as 2.2 ≤ r_e ≤ 2.6 (μm) and 3.1 ≤ r_e ≤ 3.6 (μm) for r_n = 0.6 and 1.1 μm, respectively (Fig. 1 D). When more bleaching iterations were used, larger r_e values and bleaching-depth parameters (K) were obtained (Fig. 1, C and D). We next compared the FRAP curves (measured from bleach ROIs with the nominal radius, r_n) from different experimental conditions. FRAP data from the smaller bleach ROIs tended to be noisier than those from the larger ROIs, regardless of the bleaching iterations (Fig. 1 E). The half-time of recovery ranged between 0.12 and 0.28 s, depending on r_n and bleaching scan iterations, with the FRAP data showing slower recovery rates for larger effective radii (Fig. 1 E). In contrast, the mobile fraction remained relatively constant between 0.85 and 0.9, regardless of the experimental setup (Table S1). Thus, both $r_{\text{e}}$ and the kinetics of the FRAP recoveries show a strong dependence on the specific bleaching conditions.

Experimentally determined postbleach intensity profiles and FRAP curves for the EGFP-HRas palmitoylation mutant are consistent with the effective diffusion model

The binding-diffusion model (Eq. 2) can be categorized into four regimes (submodels): diffusion-dominant, binding-dominant, full binding diffusion, and effective diffusion (3,4). We therefore next sought to determine into which parametric regime of binding diffusion the EGFP-HRas C181S C184S FRAP data fall. We addressed this question in two ways. First, we analyzed the postbleach fluorescence intensity profiles, which contain information about diffusion and binding that have occurred during the photobleach. To determine to what extent the initial postbleach profiles differ for each submodel, we numerically solved Eq. 5 assuming a constant value of D₁ = 30 μm²/s and D₂ = 1 μm²/s, but for different values of k_on and k_off. Inspection of the resulting initial postbleach profiles shows that all the profiles can be fitted by either a single exponential function of a Gaussian or a linear combination of two exponential functions of Gaussians (Fig. S1). Whereas the initial postbleach profile of the effective-diffusion submodel (k_on = ${10}^{3.5}$/s, k_off = 1/s) could be fitted by a single-exponential function of a Gaussian (Fig. S1 D), the diffusion-dominant submodel (k_on = ${10}^{-2}$/s, k_off = 10/s) required two component fits, as illustrated in Fig. S1 A. In the other submodel regimes, the initial postbleach profiles could be approximated reasonably well as a single-exponential function of a Gaussian (Fig. S1, B–D), although a linear combination of two Gaussian components provides better fitting results. This suggests that the initial postbleach profiles can be used to distinguish the diffusion-dominant submodel from the binding-dominant, full-binding-diffusion, and effective-diffusion models.

We then analyzed the postbleach fluorescence intensity profiles for EGFP-HRas C181S C184S, and found that they could be fitted by a single-exponential function of a Gaussian, regardless of number of photobleaching scan iterations or photobleaching spot size (Eq. 4, Fig. 1 C). By comparison with the results of the numerical simulations (Fig. S1), this implies that the behavior of EGFP-HRas C181S C184S does not fall into the diffusion-dominant submodel regime.

Next, we analyzed the FRAP recovery curves themselves to determine with which reaction-diffusion regime they are most consistent. Depending on the relative values of D of the unbound species, k_on, and k_off, several different types of behavior are predicted to occur. For example, when the unbound pool is large, or when the on/off binding kinetics are fast compared to diffusion, the FRAP recovery curves can be approximated by a pure diffusion model. However, D of the latter (effective diffusion (D_eff)) is much smaller than that of the former (diffusion-dominant), which has a D similar to that of the full-binding-diffusion model. On the other hand, when on and off binding kinetics are slow compared to diffusion, the corresponding FRAP curve is predicted to show biphasic characteristics, resulting from fast diffusion followed by slow binding kinetics. In this case, because the time interval for which diffusion occurs is so short, the recovery is almost entirely due to binding kinetics. Therefore, this is also referred to as a reaction-dominant or binding-dominant model (4). Here, FRAP curves are predicted to be independent of the size of the bleach ROI due to the relative unimportance of the diffusion term (4).

Because the FRAP curves for EGFP-HRas C181S C184S were dependent on the size of the bleach ROI (Fig. 1 D), we ruled out the possibility that the data fall within the binding-dominant regime. In addition, the results of the postbleach fluorescence intensity profile analysis were inconsistent with the diffusion-dominant submodel, leaving the full-binding-diffusion model and effective-diffusion models as possible candidate regimes. In principle, these two regimes can be distinguished by the fact that the effective-diffusion model, but not the full-binding-diffusion model, can be well described by a pure diffusion model with an effective-diffusion coefficient, $D_{\text{eff}}$ (3,4). We therefore tested whether a pure-diffusion FRAP model (section S1.1 in the Supporting Material) could describe the EGFP-HRas palmitoylation mutant recovery curves. To adjust for diffusion during photobleaching, we used the experimentally determined values of $r_{\text{e}}$ measured from the initial postbleach profiles. In all cases, the EGFP-HRas C181S C184S FRAP data could be fitted by the pure diffusion model (Fig. 1 E, black lines). The bleach-depth-related parameter, K, increased for the higher number of photobleaching iterations (Fig. 1, C and D and Table S1). The K value is likely to be larger in the condition of larger nominal bleaching radius, because the time required for the fluorescence molecule to refill the ROI volume is shorter for a smaller $r_{\text{n}}$ (28). In contrast, the diffusion coefficient was ∼7.2 μm²/s, regardless of the size of the bleach region or number of bleach scans (Fig. 1 F, solid black lines, and Table S1). Based on these findings, we conclude that the binding-diffusion kinetics of EGFP-HRas C181S C184S fall into the effective-diffusion regime.

The effective diffusion coefficient can provide the on/off rate-constant ratio and the fractions of soluble and membrane-bound HRas

In the effective-diffusion regime, the timescale of binding kinetics is much faster than the timescale of diffusion kinetics. Under these conditions, Eq. 2 can be approximated as the effective-diffusion submodel with an effective-diffusion coefficient $D_{\text{eff}}$, where

$$D_{\text{eff}}=\frac{k_{\text{off}}{D}_1+k_{\text{on}}{D}_2}{k_{\text{on}}+k_{\text{off}}}=\frac{D_1+K_{\text{P}}{D}_2}{1+K_{\text{P}}},$$ (8)

for a partition coefficient $K_{\text{P}}=k_{\text{on}}/k_{\text{off}}$. From this relation, if FRAP data can be fitted by the effective-diffusion model with diffusion coefficient $D_{\text{eff}}$, then a partition coefficient can be calculated from D₁, D₂, and $D_{\text{eff}}$ by

$$K_{\text{P}}=\frac{D_1-D_{\text{eff}}}{D_{\text{eff}}-D_2},$$ (9)

which implies that $D_2<D_{\text{eff}}<D_1$ given that $K_{\text{P}}>0$ and $D_1>D_2$.

In a previous study, we measured D₁ and D₂ as 43.6 $\mu {\text{m}}^2/\text{s}$ and 2.8 $\mu {\text{m}}^2/\text{s}$, respectively, by FCS (26). Using those values, as well as $D_{\text{eff}}$ determined in this study by confocal FRAP, the $K_{\text{P}}$ of EGFP-HRas C181S C184S was estimated as ∼8.3 (Table S1).

From the steady-state solution of Eq. 2, $K_{\text{P}}$ can be regarded as the fraction of steady-state fluorescence intensities between soluble and membrane-bound HRas, because

$$\frac{b_{\mathrm{\infty }}}{u_{\mathrm{\infty }}}=\frac{k_{\text{on}}}{k_{\text{off}}}=K_{\text{P}}.$$ (10)

From the computed $K_{\text{P}}\simeq 8.3$, the fractions of soluble pool ($u_{\mathrm{\infty }}$) and membrane-bound pool ($b_{\mathrm{\infty }}$) can be estimated using

$$\frac{u_{\mathrm{\infty }}}{u_{\mathrm{\infty }}+b_{\mathrm{\infty }}}=\frac{1}{1+K_{\text{P}}}\simeq 0.11,\frac{b_{\mathrm{\infty }}}{u_{\mathrm{\infty }}+b_{\mathrm{\infty }}}=\frac{K_{\text{P}}}{1+K_{\text{P}}}\simeq \mathrm{0.89.}$$ (11)

This indicates that ∼10% of HRas C181S C184S exists in a soluble state, whereas the majority of the protein (∼90%) exists in a membrane-bound state in the ROIs.

Next, we estimate the order of binding rate constants from the effective-diffusion model approximation.

The order of binding rate constants can be estimated by the effective-diffusion model approximation

Having established that diffusion and binding of HRas to ER membranes is consistent with the effective-diffusion regime, and having obtained an estimate of $K_{\text{P}}$ as the k_on/k_off ratio, we next sought to determine k_on and k_off. To do so, we first took advantage of the fact that the effective-diffusion submodel parametric regime satisfies k_on × r_e² ≫ D₁, or

$$r_{\mathrm{e}}^2/D_1\gg 1/k_{\mathrm{on}},$$ (12)

where r_e²/D₁ and 1/k_on measure the timescales of diffusion and binding, respectively (4).

Since the binding-diffusion kinetics of EGFP-HRas C181S C184S can be approximated by the effective diffusion submodel, k_on has to satisfy Eq. 12. Given that the effective radius, r_e, was measured between 2.2 and 2.6 μm² for $r_{\mathrm{n}}$ = 0.55 μm², assuming that D₁ = 43.6 μm²/s, and D₂ = 2.8 μm²/s, we use Eq. 12 to estimate that $k_{\text{on}}\gg 10$${\text{s}}^{-1}$.

To estimate the approximate order of k_on in an alternate way, we compared the experimentally determined FRAP curve for EGFP-HRas C181S C184S ($n=10$) with theoretical curves calculated using the binding-diffusion FRAP model. We first determined $D_{\text{eff}}$ by fitting the FRAP curves for EGFP-HRas to the free-diffusion FRAP model (section S1.1 in the Supporting Material and Fig. 2 A), and then computed $K_{\text{P}}$ using Eq. 9. Finally, theoretical FRAP curves generated by the binding-diffusion FRAP model (Eqs. 6 and 7) were plotted for different values of k_on, with the ratio of k_on to $k_{\text{off}}$ fixed as $K_{\text{P}}$ ($k_{\text{on}}/k_{\text{off}}=K_{\text{P}}$; Fig. 2 B). As k_on increased, FRAP curves generated by the binding diffusion FRAP formula (Eq. 6) converged to the FRAP curve predicted by the effective-diffusion submodel ($D_{\text{eff}}$; Fig. 2 B). For the binding-diffusion model to fit the averaged FRAP data with a degree of accuracy similar to that of the effective-diffusion submodel, binding rate constants of k_on ≥ 150/s and k_off ≥ 10/s were required (Fig. 2, B and C). Moreover, for k_on ≥ 300/s, the FRAP curves of the binding-diffusion model were almost indistinguishable from those obtained assuming a pure-diffusion FRAP curve with $D_{\text{eff}}$ (Fig. 2, B and C).

Figure 2.

Fitting the FRAP data with an effective-diffusion model and determination of the approximate order of binding rate constants. (A) A representative averaged FRAP data set (solid circles) for EGFP-HRas C181S C184S in the ER area of COS7 cells ($r_{\text{n}}$ = 1.1 $\mu \text{m}$, 10 bleach scan iterations) was fitted by the effective-diffusion submodel with $D_{\text{eff}}=7.3\mu \text{m}/{\text{s}}^2$ (solid black line). (B) To obtain approximate orders of the rate constants, theoretical FRAP curves were plotted using Eqs. 6 and 7 for different $k_{\text{on}}$ (1, 10, 50, 100, 200, and 400 (${\text{s}}^{-1}$)) for k_off = k_on/K_p, where K_p = 8.0. (C) The residuals between the averaged FRAP data and theoretical FRAP curves in B are plotted. The binding rate constants, k_on > 150/s, produce residuals similar to the effective-diffusion model (solid black line). Stars represent the minimal residual points and squares the residuals at $k_{\text{on}}=$1, 10, 50, 100, 200, and 400 (${\text{s}}^{-1}$).(Insets) Zooms of the boxed areas not shown to scale to emphasize the differences.

Since this analysis provides only the approximate order and not the actual value of k_on, we next considered how to extract the rate constants accurately.

One parameter fitting provides a robust and consistent estimation of rate constants

In the binding-diffusion FRAP model (Eqs. 6 and 7), a set of parameters {D₁, D₂, k_on, $k_{\text{off}}$} defines a FRAP curve. Conversely, a unique set of parameters {D₁, D₂, k_on, $k_{\text{off}}$} should ideally be obtained from fits to a given binding-diffusion FRAP model. However, this is not necessarily true in practice. For example, binding-diffusion FRAP data in the regimes of diffusion-dominant submodel and effective-diffusion submodel both yield curves that can be easily fitted by a pure-diffusion FRAP model (3,4,41,42), indicating that a binding-diffusion model can produce almost identical FRAP curves for different sets of parameters. In addition, FRAP data are characterized by intrinsic noise that varies depending on the specific experimental setup (Fig. 1 E). We therefore sought to test how well kinetic constants could be determined from our FRAP data given these potential sources of parametric multiplicity.

To do so, we analyzed FRAP data collected over the four different experimental conditions investigated, using our FRAP model corrected for binding and diffusion that occur during the photobleach. As a starting point for extracting kinetic constants, we first took advantage of the fact that our data fall in the effective-diffusion regime, allowing for reduction of the number of fitting parameters when D₁ and D₂ are known. In particular, the introduction of $K_{\text{P}}$ by the effective-diffusion approximation allows us to choose $k_{\text{on}}$ as a single fitting parameter by an additional constraint, k_off = k_on/K_p. Using values of D₁ = 43.6 $\mu {\text{m}}^2/\text{s}$ and D₂ = 2.8 $\mu {\text{m}}^2/\text{s}$ (26) and fitting for k_on, we were able to obtain one-parameter fits to the EGFP-HRas C181S C184S FRAP data that provide robust and consistent values of binding rate constants, without being influenced by bleaching spot sizes and number of bleaching scan iterations (Fig. 3). k_on and k_off were found to be in the ranges 185–376/s and 21–46/s, respectively. It is important to note that all of the parameters determined by this approach satisfied the a priori estimation for k_on ($\ge 150$ (Eq. 12)). Moreover, comparison of the FRAP curves from one-parameter fitting and from the effective diffusion model show very little difference and the weighted residuals of two methodologies were comparable (see Fig. S2). Thus, a one-parameter fit provides reasonable and internally consistent values of k_on and k_off that are also relatively insensitive to experimental conditions.

Figure 3.

Binding rate constants from one-parameter fitting. (A) $k_{\text{on}}$ found from one-parameter fitting for different FRAP setups. (B) $k_{\text{off}}$ determined from $k_{\text{off}}=k_{\text{on}}/K_{\text{P}}$ for $K_{\text{P}}$ determined using Eq. 9. Error bars show standard deviations. (C) The best-fitting curve from one-parameter fitting (solid black line, $k_{\text{on}}=214.8/\text{s}$, $k_{\text{off}}=24.8/\text{s}$, and $K_{\text{P}}=8.4$) was compared with a representative FRAP data set ($r_{\text{n}}$ = 1.1 $\mu \text{m}$, 10 bleach scan iterations) and an FRAP curve generated by the effective-diffusion submodel (solid gray line, $D_{\text{eff}}=7.3\mu \text{m}/{\text{s}}^2$).

Since D₁ and D₂ may not necessarily be independently available for the proteins under study, we also investigated how closely one might be able to estimate k_on and k_off using reasonable guesses for D₁ and D₂ (Fig. S2). To do so, we considered a range of D₁ and D₂ values similar to those reported in the literature for soluble globular proteins and membrane-bound proteins, respectively (2). We then used these values to calculate $K_{\text{P}}$ for a range of possible D₁ and D₂ combinations and, in turn, to calculate k_on and k_off from one of the experimentally determined FRAP curves ($r_{\text{n}}$ = 1.1 μm, 10 bleach scan iterations, $D_{\text{eff}}$ = 7.3 $\mu {\text{m}}^2/\text{s}$). This analysis yielded k_on values ranging from 125 to 250/s, and k_off values ranging between 20 and 45/s for $40\le D_1\le 50$ and $2\le D_2\le 3$, respectively (Fig. S2). Thus, using the estimated values of D₁ and D₂, the resulting k_on and k_off values were within ranges similar to those obtained using experimentally determined D₁ and D₂ values and a one-parameter fit (k_on = 214.8/s and k_off = 24.8/s (Fig. 3)).

Adjustments for binding and diffusion during photobleaching and reduction in fitting parameters are required to obtain reasonable rate constants

Our finding that the one-parameter fit yields rate constants that are constant across experimental conditions suggests that the correction for binding and diffusion during the photobleach is effective. To investigate this further, we directly assessed the consequence of not correcting for diffusion and binding during photobleaching in the FRAP analysis for binding-diffusion kinetics. To do so, we followed the computation for a partition coefficient in the previous sections, except that we measured $D_{\text{eff}}^{\text{n}}$ assuming $r_{\text{n}}=r_{\text{e}}$ in Eq. 6, and computed $K_{\text{P}}$ from $D_{\text{eff}}^{\text{n}}$. We found that in the absence of the correction for $r_{\text{e}}$, $D_{\text{eff}}^{\text{n}}$ = 1.0 ± 0.4 $\mu {\text{m}}^2/\text{s}$ ($n=8$) was obtained, a value considerably smaller than $D_{\text{eff}}$ = 7.3 $\mu {\text{m}}^2/\text{s}$ for $r_{\text{e}}>r_{\text{n}}$. This is in good agreement with a previous study in which we showed that when diffusion during photobleaching is ignored, diffusion coefficients are underestimated and vary with experimental setup (28). Even worse, we found that because $D_{\text{eff}}\mathrm{<}{D}_2\mathrm{<}{D}_1$ here, the $K_{\text{P}}$ values calculated by Eq. 9 are negative numbers, which implies that either k_on or k_off is a negative number. This nonphysiological relationship between $D_{\text{eff}}^{\text{n}}$ and binding rate constants illustrates the critical importance of incorporating such corrections into binding-diffusion type analyses of FRAP data as well.

In the approach outlined above, we took advantage of the fact that in the effective diffusion regime, if D₁ and D₂ are known, then $k_{\text{on}}$ and $k_{\text{off}}$ can be determined from a one-parameter fit by virtue of the relationship between $K_{\text{P}}$, $k_{\text{on}}$, and $k_{\text{off}}$ ($k_{\text{off}}$ = $k_{\text{on}}/K_{\text{P}}$). However, this represents a special case, and it often may be necessary to attempt to extract k_on and k_off independent of one another, even if D₁ and D₂ are known. Even more generally, D₁ and D₂ may not be known a priori, necessitating a fitting scheme that simultaneously solves for D₁, D₂, k_on, and k_off as independent parameters. Given the inherent noise in the FRAP data, combined with the possibilities for parametric multiplicity (i.e., that multiple combinations of D₁, D₂, k_on, and k_off values could potentially yield similar FRAP curves), we were curious to what extent these less well constrained fitting approaches could describe the data, and whether they could accurately recapitulate the results of the one-parameter fit. To test this, we reanalyzed the FRAP data using either a four-parameter fit by varying D₁, D₂, k_on, and k_off separately, or using a two-parameter fit by setting the two diffusion coefficients at D₁ = 43.6 $\mu {\text{m}}^2/\text{s}$ and D₂ = 2.8 $\mu {\text{m}}^2/\text{s}$, as in the previous study (26), but allowing k_on and k_off to vary separately. In each case, we incorporated the correction for binding and diffusion during the bleach, since we had found that this was essential to obtain positive rate constants for the one-parameter fit. Neither approach was able to recover consistent rate constants, likely due to parametric multiplicity (For a detailed discussion, see section S2.1 in the Supporting Material).

Discussion

Confocal FRAP offers the potential to quantitatively analyze intracellular reaction-diffusion kinetics, but how to best obtain accurate kinetic constants from such analyses is still an open question (19–23). Toward this goal, in the study presented here, we developed a new (to our knowledge) analytical model for the quantitative analysis of intracellular binding-diffusion processes that incorporates several new mathematical and experimental considerations. First, we utilized a closed analytic form of the FRAP formula for binding-diffusion kinetics, based on a spot photobleach geometry (41). It is important to note that this formalism allows for the bound species to undergo diffusion and thus is an attractive model to describe the behavior of proteins that exist in both a freely diffusing soluble state and a mobile bound state. Second, we developed what we believe is a new FRAP formalism to compensate for diffusion and binding that occur during photobleaching, based on an extension of our recently proposed confocal FRAP model, which incorporates an effective bleach radius measured from the first postbleach image (28). Third, using a combination of postbleach profile analysis, FRAP measurements as a function of bleach-spot size, and fitting of the resulting FRAP curves, we tested which subregime of the binding-diffusion model best accounts for the observed recoveries. Finally, we developed a strategy to address the problem of parametric multiplicity by reducing the number of fitting parameters in the effective diffusion subregime. The ultimate success of the analytical scheme developed in this study depended on a priori knowledge of D₁ and D₂ for unbound and bound proteins and, by extension, $K_{\text{P}}$. The reduction to one-parameter fitting was only made possible by this additional information, rather than a scheme that independently solves the problem of parametric multiplicity. We validated this binding-diffusion confocal FRAP model by quantifying the CAAX-mediated binding of Ras to the ER. The implications of these findings for Ras biology are discussed elsewhere (see section S2.2 in the Supporting Material)

Closed-form reaction-diffusion FRAP model corrected for diffusion and binding during the photobleach

This study utilized a closed-form analytical expression of the FRAP formula for our recently developed reaction-diffusion model (41). This model is based on an exact and explicit solution of the reaction-diffusion model described in the mathematical literature (43,44). When coupled with Eq. 2, our reaction-diffusion FRAP formula offers several advantages over previously published approaches. First, it provides solutions for both a uniform circular laser profile and a Gaussian laser profile. It can thus be readily incorporated into our proposed FRAP model to correct for either diffusion (41) or diffusion and binding (this study) during the photobleach. As discussed further below, this is critical, because failure to account for diffusion during the photobleach is one of the factors that contributes to the dependence of kinetic parameters on experimental set-ups (23). Second, our new formalism allows for the diffusion of bound complexes. This could potentially be important for studies of proteins that interact with soluble macromolecular complexes or for studies of peripheral membrane proteins that diffuse along the membrane in their bound state. By comparison, many of the published studies using confocal FRAP to analyze binding-diffusion events have regarded the bound fraction of proteins as immobile (3, 4,7,8). One exception is a recent study by Dushek et al. (45), which considered mobile binding sites. However, this approach is based on the conventional FRAP model with the uniform laser profile and the Fourier transform solution. In contrast, our previous study (41) introduced an analytic FRAP equation for the binding-diffusion model that also allows for the mobility of both free and bound-state proteins. Here, we have improved on that analytic FRAP equation (Eq. 2) for the binding-diffusion model by extending it to a form that is applicable to confocal FRAP analysis.

Our model also incorporates both diffusion and binding events during the photobleach. In contrast, if no correction were made for diffusion and binding during photobleaching and the data were instead analyzed using the nominal bleach radius, physically unrealistic rate constants (even negative numbers) were obtained. This demonstrates the necessity of incorporating these corrections into FRAP analysis to better quantify kinetic constants. Indeed, we speculate that failure to make such corrections could at least partially account for discrepancies in binding parameters currently reported in the literature for various transcription factors (21,23).

Accounting for parametric multiplicity

An intrinsic property of binding-diffusion behavior is that a given recovery curve is not necessarily defined by a unique set of kinetic constants, resulting in so-called parametric multiplicity. We addressed this problem in several ways. First, we took advantage of special properties of the effective diffusion regime, which enabled us not only to estimate the magnitudes of the binding rate constants but also to reduce the number of fitting parameters by virtue of the relationship between $K_{\text{P}}$, k_on, and k_off, thus making it possible to determine k_on when D₁ and D₂ are known. Furthermore, to test the consistency of our analysis method, we analyzed FRAP data obtained under four different experimental conditions and using three different sets of fitting parameters (one-parameter, two-parameter, and four-parameter fits; section S2.1 in the Supporting Material). We found that under the conditions of our experiments, it was important to reduce the number of fitting parameters to obtain robust values of k_on and k_off, as well as to perform data fitting on averaged curves rather than individual FRAP recoveries. In many cases, it may be possible to experimentally determine D₁ and/or D₂ using alternative methods including FCS, or to estimate these values in other ways (see section S2.3 in the Supporting Material). It is important to note that the procedure introduced here is also applicable when the bound species is immobile ($D_2=0$), as is often assumed for many DNA binding proteins (4,20; see section S2.3 in the Supporting Material for more details).

Influence of ER membrane geometry

In this study, a uniform distribution of ER membranes was assumed for simplicity. However, the ER has a convoluted structure compared to a flat sheet of plasma membrane, which may impact our current FRAP-based kinetic analysis in several ways. First of all, the complex geometry of the ER slows the apparent diffusion of ER-localized molecules ((46) and references therein). This in turn would be expected to affect the magnitude of $K_{\text{P}}$ computed from Eq. 9. Second, the spatial density of ER may vary within a given cell, and this may cause binding rate constants to vary from region to region or from cell to cell. For these reasons, more realistic binding rate constants should be understood as a function of location and binding-site density. Finally, when the unbound species is present at concentrations far below saturation, the value of $K_{\text{P}}$ defined here will vary linearly with the density of binding sites and hence with the local spatial density of ER membranes. To this point, assuming S is location independent, we note that in the conversion of the binding-diffusion model (Eq. 1) in terms of a partial differential equation (Eq. 2), it was assumed that the density of binding sites is very high compared to the HRas concentration, so that the density of S ([S]) can be regarded as a constant. A study in progress suggests that this approximation is valid if the magnitude of S is approximately more than 10-fold greater than that of unbound protein in the ROIs (M. Kang, unpublished observations). Under this assumption, k_on in Eq. 2 is indeed k_on = k_on[S] and the association rate constants found in this study have to be understood in this context. However, in reality, the density of binding sites may not necessarily be high enough to make this assumption. For this reason, a large variation in k_on (Fig. 3 A) could potentially arise from the nonidentical ER membrane densities from different cells.

Perspectives

In summary, we have developed a new (to our knowledge) approach to quantify binding-diffusion kinetics by confocal FRAP that takes advantage of newly developed mathematical models for FRAP analysis and corrects for a common problem associated with confocal FRAP diffusion and binding during the photobleaching process. Using this approach, we show that we can extract from FRAP data rate constants associated with a reversible binding event for a protein that undergoes diffusion in both the bound and unbound form. Although the methods described here have been specifically applied to study the association of Ras with ER membranes, they are more generally applicable to any protein exhibiting binding-diffusion type behavior. This is important because in addition to Ras, a number of peripheral membrane proteins have the capacity to cycle on and off membranes. Furthermore, our current analysis is readily transferrable to the study of molecules that interact with filamentous structures that allow one-dimensional diffusion of transiently bound molecules, as well as to nuclear proteins such as transcription factors, for which a growing FRAP literature already exists (3–5,7–9,21,24,25,31,32). By providing quantitative measures of both diffusion coefficients and binding constants in the context of the native environment of the cell, the approach described here should be a powerful tool to help build better models of how signaling networks and their emergent properties are regulated over space and time (47).