Analysis of membrane-localized binding kinetics with FRAP

Abstract

Interactions between plasma membrane-associated proteins on interacting cells are critical for many important biological processes. Few experimental techniques, however, can accurately determine the association and the dissociation rates between such interacting pairs when the two molecules diffuse on apposing membranes or lipid bilayers. In this study, we give a theoretical description of how and when fluorescence recovery after photobleaching (FRAP) experiments can be used to quantify these reaction rates. We analyze the effect of binding on FRAP recovery curves with a reaction–diffusion model and systematically identify different regimes in the parameter space of the association and the dissociation constants for which the full model simplifies into equivalent one-parameter models. Based on this analysis, we propose an experimental protocol that may be used to identify the kinetic parameters of binding in the appropriate parameter regime. We present simulated experiments illustrating our protocol and lay down guidelines for parameter estimation.

Appendix: evaluation of theoretical FRAP recovery curves

Simple FRAP analysis

In typical surface photobleaching experiments, the photobleached area is much smaller than the total area of the plasma membrane. Therefore, we can assume that the domain is effectively infinite in extent without introducing appreciable error. For the parameter regimes that reduce to simple diffusion (cases 1–3), the FRAP recovery curve is determined by Eq. 9 with initial conditions

$$ u(t=0) = 1- \left[H(x + X_b/2) - H(x - X_b/2)\right] \left[H(y + Y_b/2) - H(y - Y_b/2) \right] $$

(13)

where H(x) is the Heaviside step-function. This is easily solved by Fourier transform, and the desired FRAP recovery curve is then obtained by integrating u over the monitoring region (−X _m /2 ≤ x ≤  X _m /2,−Y _m /2 ≤ y ≤ Y _m /2), to obtain the following expression for the theoretical FRAP recovery curve (Dushek and Coombs 2007):

$$ \begin{aligned} U(t,D) & = 1 - \,\frac{\sqrt{Dt}}{2\pi X_m Y_m} \left\{\sqrt{\pi} \left[X_+ \hbox{erf} (X_{+}) - X_{-} \hbox{erf} (X_{-})\right] + \left[\exp(-X_{+}^{2}) - \exp(-X_{-}^{2}) \right] \right\} \\ & \quad \times \left\{\sqrt{\pi} \left[Y_{+} \hbox{erf}(Y_{+}) - Y_{-} \hbox{erf}(Y_{-})\right] + \left[\exp(-Y_{+}^{2}) - \exp(-Y_{-}^{2}) \right] \right\} \end{aligned} $$

(14)

where we define

$$ X_{\pm} = \,\frac{X_b \pm X_m}{4\sqrt{Dt}} \quad \hbox{and} \quad Y_{\pm} = \,\frac{Y_{b} \pm Y_{m}}{4\sqrt{Dt}}. $$

(15)

This expression can be fit to data to determine the parameter D.

FRAP with binding kinetics

We are now concerned with solving Eqs. 3 and 4. Taking the Fourier transform (denoted with hats), we obtain the linear system

$$ \partial {\hat{f}} / \partial t = D_{\rm f} (-4\pi^2 q^2_x - 4\pi^2 q^2_y) {\hat{f}} - k_{\rm on}^{*} {\hat{f}} + k_{\rm off} {\hat{c}}, $$

(16a)

$$ \partial {\hat{c}} / \partial t = D_{\rm c} (-4\pi^2 q^2_x - 4\pi^2 q^2_y) {\hat{c}} + k_{\rm on}^{*} {\hat{f}} - k_{\rm off} {\hat{c}}. $$

(16b)

We define q _f = 4π² D _f (q ² _x  + q ² _y ) + k ^*_on and q _c = 4π² D _c (q ² _x  + q ² _y ) + k _off to obtain

$$ {{\partial {\hat{f}}}\over {\partial t}} = -q_f {\hat{f}} + k_{\rm off} {\hat{c}}, $$

(17a)

$$ \,\frac{\partial {\hat{c}}}{\partial t} = k_{\rm on}^{*}\,{\hat{f}} - q_c {\hat{c}}, $$

(17b)

whose solutions are

$$ \begin{aligned} {\hat{f}} & = \,\frac{1}{v} \left[(1/2) {\hat{f}} (t=0) \left(\exp(D_{1}t)(q_c - q_f + v) + \exp(D_{2} t)(-q_c + q_p + v)\right) \right. \\ & \quad \left. + k_{\rm off} {\hat{c}}(t=0) (\exp(D_{1}t) - \exp(D_{2}t)) \right], \end{aligned} $$

(18a)

$$ \begin{aligned} {\hat{c}} & = \,\frac{1}{v} \left[ (1/2) {\hat{c}} (t=0) \left(\exp(D_{1}t)(-q_c + q_f + v) + \exp(D_{2}t) (q_c - q_f + v)\right) \right. \\ & \quad \left. + k_{\rm on}^{*} {\hat{f}}(t=0) (\exp(D_{1}t) - \exp(D_{2}t)) \right] \end{aligned} $$

(18b)

where v = [(q _c −q _f )² + 4k ^*_on k _off]^1/2, D ₁ =  (−q _c −q _f  + v)/2, D ₂ = (−q _c −q _f −v)/2. ${\hat{f}}(t=0)\;\hbox{and} \;{\hat{c}}(t=0)$ are Fourier transforms of the initial conditions,

$$ {\hat{f}}(t=0) = -F_{\rm eq} \hbox{sinc}(\pi X_b q_x) \hbox{sinc}(\pi Y_b q_y) + F_{\rm eq} \delta(q_x) \delta(q_y) $$

(19a)

$$ {\hat{c}}(t=0) = -C_{\rm eq} \hbox{sinc}(\pi X_b q_x) \hbox{sinc}(\pi Y_b q_y) + C_{\rm eq} \delta(q_x) \delta(q_y). $$

(19b)

where F _eq = k _off/(k ^*_on  + k _off) and F _eq = k ^*_on /(k ^*_on  + k _off). We obtain these relationships by assuming that the system is in equilibrium prior to the FRAP experiment (k ^*_on F _eq = k _off C _eq) and the FRAP recovery can be normalized to unity (F _eq + C _eq = 1), see Sprague et al. (2004).

It is not possible to obtain an analytical inverse Fourier transform of these equations, and therefore theoretical FRAP recovery curves must be numerically computed. These are obtained by integrating the sum f + c over the monitoring region, and dividing by the area of the monitoring region to obtain the mean fluorescence intensity over this area. Thus, for a rectangular monitoring region of dimensions X _m (≤ X _b ) by Y _m (≤ Y _b ), the recovery curve is given by

$$ G_{\rm fm}(t) = \left(\int\limits^{X_m/2}_{-X_m/2} {\rm d}x \int\limits^{Y_m/2}_{-Y_m/2} {\rm d}y \left[f(t,x,y) + c(t,x,y) \right]\,\right)\left/ \left(X_m Y_m\right)\right., $$

(20)

and, by definition of the inverse Fourier transform,

$$ \begin{aligned} G_{\rm fm}(t) & = \,\frac{1}{4 X_m Y_m} \int\limits^{X_m}_{-X_m}{\rm d}x \int\limits^{Y_m}_{-Y_m} {\rm d}y \int\limits^{\infty}_{-\infty} {\rm d}q_x \int\limits^{\infty}_{-\infty} {\rm d}q_y \left[{\hat{f}} (t, q_x, q_y) + {\hat{c}}(t, q_x, q_y)\right] \\ & \quad \times \exp(2 \pi \imath q_x x) \exp(2\pi \imath q_y y) \\ & = \int\limits^{\infty}_{-\infty} {\rm d}q_x \int\limits^{\infty}_{-\infty} {\rm d}q_y \left[{\hat{f}}(t, q_x, q_y) + {\hat{c}}(t, q_x, q_y) \right] \hbox{sinc}(\pi q_x X_m) \hbox{sinc}(\pi q_y Y_m) \\ & = 4 \int\limits^{\infty}_{0} {\rm d}q_x \int\limits^{\infty}_{0} {\rm d}q_y \left[{\hat{f}}(t, q_x, q_y) + {\hat{c}}(t, q_x, q_y)\right] \hbox{sinc}(\pi q_x X_m) \hbox{sinc}(\pi q_y Y_m), \end{aligned} $$

(21)

which we compute by numerical integration. We found integrating to 15–20 in q _x and q _y yields sufficient accuracy for our purposes. Using the Matlab function quadl , the above integral can be evaluated 100 times (typical number of images in FRAP experiments) in approximately 3 s, allowing parameter fits in a reasonable timeframe using a standard fitting procedure. All fits presented in this work were carried out using the Matlab function lsqcurvefit .