LB박막에서의 분자확산과 탈착의 동시측정을 위한

Fluorescence Recovery After Photobleaching의 모델

박 상권

동국대학교 화학공학과

A Model of Fluorescence Recovery After Photobleaching

for Simultaneous Measurement of Lateral Diffusion and Desorption

in LB Films

Sangkwon Park

Department of Chemical Engineering, Dongguk University

Introduction

Fluorescence recovery after photobleaching (FRAP) technique has been widely used to directly measure lateral diffusivity in various interfacial systems.  Although several theoretical models of FRAP have been developed for different geometries of the photobleached region,1 these conventional models did not account for molecular loss such as dissolution or desorption often occurring in Langmuir-Blodgett (LB) monolayers or films.  Such concentration decrease of diffusing species makes FRAP analysis difficult.  In this study, a theoretical model of FRAP describing lateral diffusion and irreversible molecular desorption is developed.  A two-dimensional diffusion equation for the FRAP accounting for molecular desorption is solved with proper initial and boundary conditions in terms of the concentration profile of unbleached fluorophore.  The resulting fluorescence recovery is expressed as an integral form of the concentration.  The developed model is applied to the FRAP in the LB films of myristic acid containing probe molecules to obtain the lateral mobility and the desorption rate constant simultaneously.

Development of FRAP Model

The basic principle of FRAP technique is to monitor the fluorescence emitted from homogeneously distributed fluorescent probe molecules inside an illuminated spot before and after a photobleaching laser pulse.  Assuming a spontaneous recovery does not occur, the fluorescence recovery kinetics reflects the rate of diffusive exchange of bleached fluorophores inside with unbleached probe molecules outside.  The lateral diffusion coefficient, [eqn: {D}_{s}], is related to the characteristic diffusion time,[eqn: { tau }_{d}], by the equation

[eqn: {D}_{s}= { { omega}^{2} } over { { tau }_{d} }]	[1]

where [eqn: omega] is the diameter of the photobleached spot.  Figure 1 shows a typical fluorescence recovery curve.  Here [eqn: F(-)], [eqn: F(0)], [eqn: F(t)], [eqn: F( INF )] are defined as the fluorescence intensities before bleaching, at[eqn: t=0], [eqn: t=t] and [eqn: t= INF], respectively.  For simplicity, [eqn: F(t)] is usually normalized to relative fluorescence intensity, [eqn: f(t)=[F(t)-F(0)]/[F( INF )-F(0)]].

Figure 1.  Definition of fluorescence intensity at different time in a typical fluorescence recovery.

A two-dimensional diffusion equation including an irreversible first-order desorption is expressed in terms of[eqn: {C}^{*}(r,t)], the concentration of bleached fluorophore at radial position, [eqn: r], and time, [eqn: t], as follows:

[eqn: { PARTIAL {C}^{*}(r,t)} over { PARTIAL t } = {D}_{s} { DEL }^{2} {C}^{*} (r,t) - k {C}^{*} (r,t)]	[2]

where [eqn: k] is the desorption rate constant.  Initial and boundary conditions are

[eqn: {C}^{*} (r,0) = PMATRIX {{ {C}_{0}[1-exp(-K)]}&{(r <= omega)}&{} # {}& {}& {}# {0}& {(r> omega)}& {}}]	[3]

[eqn: {C}^{*}( INF , t) = 0]	[4]

where [eqn: {C}_{0}] and [eqn: K] are the fluorophore concentration before bleaching and the bleaching parameter.  Here,[eqn: {C}^{*}(r,t)] is related to [eqn: C(r,t)], the concentration of unbleached fluorophore, by the equation:

[eqn: {C}^{*}(r,t) + C(r,t) = {C}_{0} {e}^{-kt }]	[5]

The solution,[eqn: {C}^{*}(r,t)], can be expressed in terms of the solution of the diffusion equation without any chemical reaction, [eqn: C'(r,t)], as follows2:

[eqn: {C}^{*}(r,t) = k INT _{0}^{t} C'(r,t) {e}^{-kt}dt + C'(r,t) {e}^{-kt}]	[6]

Based on the conventional FRAP model3, [eqn: C'(r,t)] is expressed in terms of Bessel functions,[eqn: {J}_{0}] and[eqn: {J}_{1}]:

[eqn: C'(r,t) = {C}_{0} omega (1- {e}^{-K}) INT _{0}^{ INF } {J}_{0}(r rho) {J}_{1}( omega rho ) { e}^{- gamma { rho }^{2 } } d rho]	[7]

where [eqn: rho] and [eqn: gamma] are the dummy variable for integration and the multiplication of[eqn: {D}_{s}] and [eqn: t].3  The fluorescence intensity at time [eqn: t], [eqn: F(t)], is obtained by the spatial integration of the concentration multiplied by the laser beam power density, [eqn: I(r)] (=[eqn: P/pi { omega}^{2}]), inside the illumination spot:

[eqn: F(t) = {q} over {A} INT _{ 0}^{2 pi} INT _{0}^{omega }I(r)C(r,t)rdrd theta]	[8]

where [eqn: P], [eqn: q] and [eqn: A] denote the laser power, the product of all quantum efficiencies of laser light absorption and the attenuation factor of the beam, respectively.  Using the above equations with an integral relationship between the Bessel functions, [eqn: {J}_{0}] and[eqn: {J}_{1}], and the definitions of [eqn: F(-)], [eqn: F(0)] and [eqn: F( INF )], a simple expression of [eqn: F(t)] is obtained.  By differentiating it with respect to [eqn: t] and integrating from [eqn: t] to ∞, the following equation is obtained:

[eqn: F(t) = F( INF ) + F(-) {e}^{-kt} - [F(-)-F(0)] INT _{0}^{ {2 { tau }_{d }} over {t}} { {e}^{- phi -{{2k{tau}_{d}} over {phi}}} over { phi } {I}_{1}} ( phi })d phi]     [9]

where [eqn: phi] and [eqn: {I}_{1}] denote [eqn: 2{tau}_{d}/t] and the modified Bessel function, respectively.  Since the analytical expression for the last integration term of equation [9] is not available, it was numerically calculated and the corresponding fluorescence recovery was obtained by using a computer program.  Figure 2 shows the resulting FRAP curves with different [eqn: k] values when[eqn: { tau }_{d }]=500s.  As [eqn: k] increases, FRAP curve significantly changes, and especially, at large time [eqn: f(t)] decreases as time elapses.

Application of The Developed Model

A myristic acid monolayer containing 1 mol% of the fluorescent probe 4-(hexadecylamino)-7-nitrobenz-2-oxa-1,3-diazole (NBD-hexadecylamine) was deposited onto a cylindrical silica disc by the Langmuir-Blodgett (LB) technique.  FRAP experiments were conducted on the monolayer-coated silica discs at constant RH and ambient temperature.  By analyzing the fluorescence recovery curve according to an error minimization algorithm based on the developed model, the lateral mobility and the desorption rate constant were determined.  As shown in Figure 3, the developed model well

Figure 2.  Effect of [eqn: k] on [eqn: f(t)].

Figure 3. (a) Experimental fluorescence recovery (―) and the conventional FRAP model (--) (b) Experimental fluorescence recovery (―) and the developed FRAP model (--).

described the full fluorescence recovery whereas the conventional one did not.  The resulting lateral mobility and desorption rate constant were found to be 4.70×10-9 cm2/s and 1.81×10-5 s-1, respectively.

References

1. Chary, S. R. and Jain, R. K., Chem. Eng. Comm., 55, 235(1987).

2. Danckwerts, P. V., Trans. Faraday Soc., 47, 1014(1951).

3. Soumpasis, D. M. Biophys. J., 41, 95(1983).