High Probe Intensity Photobleaching Measurement of Lateral Diffusion in Cell Membranes

Lateral diffusion measurements, most commonly accomplished through Fluorescence Photobleaching Recovery (FPR or FRAP), provide important information on cell membrane molecules' size, environment and participation in intermolecular interactions. However, serious difficulties arise when these techniques are applied to weakly expressed proteins of either of two types: fusions of membrane receptors with visible fluorescent proteins or membrane molecules on autofluorescent cells. To achieve adequate sensitivity in these cases, techniques such as interference fringe FPR are needed. However, in such measurements, cytoplasmic species contribute to the fluorescence recovery signal and thus yield diffusion parameters not properly representing the small number of surface molecules. A new method helps eliminate these difficulties. High Probe Intensity (HPI)-FPR measurements retain the intrinsic confocality of spot measurements to eliminate interference from fluorescent cytoplasmic species. However, HPI-FPR methods lift the previous requirement that FPR procedures be performed at probe beam intensities low enough to not induce bleaching in samples during measurements. The high probe intensities now employed provide much larger fluorescence signals and thus more information on molecular diffusion from each measurement. We report successful measurement of membrane dynamics by this technique.

Appendices

APPENDIX 1: SMALL EXTENTS OF PROBE AND PULSE BLEACHING IN A GAUSSIAN SPOT

Consider a sample examined in a probe beam of constant peak intensity I _p where the ongoing rate of bleaching is small, i.e. that $BI_{\rm p} \ll 1$, and where a brief pulse of high peak intensity I _b and duration Δt quickly bleaches a small “crater” in the sample. We can without loss of generality replace c with 1−c _b where c _b = c _b(r, t) represents the fraction of irreversible photobleaching at any instant. Then, after the conclusion of the bleaching pulse and since the extent of bleaching is small, Eq. (2) becomes

$$ - D\nabla ^2 c_{\rm b} + \frac{{\partial c_{\rm b} }}{{\partial t}} = Be^{ - g^2 r^2 } I_{\rm p} (1 - c_{\rm b} )$$

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We insist that c _b and exp (−g ² r ²) can both be represented by Bessel-Fourier expansions.

$$\begin{array}{l}c_{\rm b} (r,t) = \displaystyle\int_{k = 0}^\infty a (k,t)J_0 (kr)kdk \\e^{ - g^2 r^2 } = \displaystyle\int_{k = 0}^\infty {\left( {\frac{1}{{2g^2 }}e^{ - k^2 /4g^2 } } \right)} J_0 (kr)kdk\\\end{array}$$

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Inserting these expressions into Eq. (3) we obtain

$$Dk^2 a (k,t) + \frac{{da (k,t)}}{{dt}} = \frac{{BI_{\rm p} }}{{2g^2 }}e^{ - k^2 /4g^2 }$$

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Then, observing that a _k (0) = 0, the solution of Eq. (5) can be written by inspection as

$$a (k,t) = \frac{{BI_{\rm p} }}{{2Dg^2 k^2 }}e^{ - k^2 /4g^2 } \left( {1 - e^{ - k^2 Dt} } \right)$$

(16)

Equation (16) effectively represents the Bessel-Fourier expansion of the first term of Eq. (4). While an analytical expression for the actual fluorophore distribution can, with sufficient effort, be obtained, we actually require only the time-dependent normalized fluorescence depletion signal ΔF = F(0)−F(t) which is given by

$$\Delta F/F(0) = 2g^2 \int\nolimits_{r = 0}^\infty {c_{\rm b} (r,t)e^{ - g^2 r^2 } I_{\rm p} }\, rdr\\ = \frac{{BI_{\rm p} }}{D}\int\nolimits_{r = 0}^\infty e^{ - g^2 r^2 } \left( \int\nolimits_{k = 0}^\infty \frac{1}{{k^2 }}e^{ - k^2 /4g^2 } \right.\\ \times\,\left. \big( 1 -e^{ - k^2 Dt} \big)J_0 (kr)kdk \right) rdr\\ = \frac{{BI_{\rm p} }}{{2g^2 D}}\int\nolimits_{k = 0}^\infty {e^{ - k^2 /2g^2 } \left( {1 - e^{ - k^2 Dt} } \right)} \frac{{dk}}{k}$$

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Now, we substitute z for k ²/2g ² and t′ for 2g ² Dt to obtain

$$\Delta F/F(0) = \displaystyle\frac{{BI_{\rm p} }}{{4g^2 D}}\sum\limits_{n = 1}^\infty {\frac{{( - 1)^{n - 1} t'^n }}{{n!}}} \int_{z = 0}^\infty {z^{n - 1} e^{ - z} dz}\\ = \displaystyle\frac{{BI_{\rm p} }}{{4g^2 D}}\sum\limits_{n = 1}^\infty {\frac{{( - 1)^{n - 1} t'^n }}{n}}\\ = \displaystyle\frac{{BI_{\rm p} }}{{4g^2 D}}\ell n\left( {1 + t'} \right)$$

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We can replace g ² with $2/r_0^2$ and t′ with t/t _1/2 to obtain.

$$\Delta F/F(0) = \frac{{BI_{\rm p} t_{1/2} }}{2}\ell n\left( {1 + t/t_{1/2} } \right)$$

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Equation (19) provides a convenient closed-form approximation of the evolution of fluorescence signal during bleaching from the probe beam and, in Fig. 5, this approximate baseline is shown as a dotted line. The amount of bleaching in this example is too high for Eq. (19) to apply quantitatively, since eight terms in Eq. (4) were needed to represent the recovery after the bleaching pulse and Eq. (19) is derived from only the first such term. Nonetheless, the plot makes clear that the baseline to which fluorescence recovers in the presence of bleaching by the probe beam never becomes flat or even linear. Hence, data analysis needs to explicitly deal with probe bleaching if diffusion parameters are to be recovered accurately.

Because the extent of bleaching by both probe and pulse is assumed to be small, the recovery of fluorescence signal after a brief bleaching pulse of intensity I _b and duration Δt and occurring at a time t _b after initial exposure of the sample to light, can, as a first-order approximation, be calculated independently. For derivation of this recovery, see, for example, Axelrod et al. [1]. The final result becomes

$$F/F(0) = 1 - \displaystyle\frac{{BI_{\rm p} t_{1/2} }}{2}\ell n\left( {1 + t/t_{1/2} } \right)\quad (t < t_{\rm b} )\\ = 1 - \displaystyle\frac{{BI_{\rm p} t_{1/2} }}{2}\ell n\left( {1 + t/t_{1/2} } \right)\\ - \displaystyle\frac{{BI_{\rm b} \Delta t}}{2}\frac{1}{{1 + (t - t_{\rm b} )/t_{1/2} }}\quad (t \ge t_{\rm b} )$$

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APPENDIX 2: SMALL EXTENTS OF PROBE AND PULSE BLEACHING IN A UNIFORM CIRCULAR SPOT

In certain situations, for example when photobleaching an extended spot in a confocal microscope, a circular region of radius r₀ may be illuminated effectively uniformly with light of intensity I _p. This allows a approximate solution of the diffusion equation using Laplace transforms by solving the diffusion equations inside and outside the spot separately and then insisting the fluorophore concentration be continuous across the spot boundary. In fact, only the fluorophore concentration inside the spot is required as only from there does fluorescence signal arise. The key elements in this approach are demonstrated in Ozişik's examples 8–1 and 7–22 [15]. We begin with separate equations inside and outside the spot, designating c ₁ and c ₂ the fluorophore concentrations inside and outside, respectively. For convenience we denote bleaching rate constant BI _p by the symbol h

$$D\nabla ^2 c_1 (r,t) - hc_1 (r,t) = \displaystyle\frac{{dc_1 (r,t)}}{{dt}}\quad (0 \le r \le b)\\ D\nabla ^2 c_2 (r,t) = \displaystyle\frac{{dc_2 (r,t)}}{{dt}}\quad (b \le r < \infty )$$

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We apply the Laplace transform with respect to time, insist that both c ₁ and c ₂ are everywhere 1 at t = 0 and denote the transform variable by s and the transforms of c ₁ and c ₂ by θ ₁ and θ ₂, respectively. There results

$$D\nabla ^2 \theta _1 - (h + s)\theta _1 = 0\quad (0 \le r < r_0 )\\ D\nabla ^2 \theta _2 - s\theta _2 = 0\quad (r_0 < r < \infty )$$

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The solutions of Eq. set (22) are linear combinations of modified Bessel functions I ₀ and K ₀. Since θ ₁ is finite for r = 0, θ ₁ contains no K ₀. Likewise, since θ ₂ is unity at r = ∞ for all t, θ ₂ contains no I ₀. It is convenient to denote (s/D)^1/2 by m and ((s+h)/D)^1/2 by n. Then

$$\theta _1 = UI_0 (nr)\\ \theta _2 = VK_0 (mr)$$

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where U and V are constants. Insisting that θ ₁ and θ ₂, and their first derivatives as well, agree at r = r ₀ for all t allows evaluation of U and V and hence of θ ₁ and θ ₂.

$$\theta _1 = \displaystyle\frac{1}{{s + h}} + \frac{{\left({\textstyle{1 \over s}} - {\textstyle{1 \over {s + h}}}\right)\left[ {mK_1 (mr_0 )I_0 (nr)} \right]}}{{mI_0 (nr_0 )K_1 (mr_0 ) + nI_1 (nr_0 )K_0 (mr_0 )}};\\ \theta _2 = \frac{1}{s} - \frac{{({\textstyle{1 \over s}} - {\textstyle{1 \over {s + h}}})\left[ {mK_0 (mr_0 )I_1 (nr)} \right]}}{{mI_0 (nr_0 )K_1 (mr_0 ) + nI_1 (nr_0 )K_0 (mr_0 )}}$$

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Inversion of the Laplace transform involves use of asymptotic expansions for the modified Bessel functions and their derivatives valid for large s and, hence, for small times t. These expansions can be found in Abramowitz and Stegun sections 9.7.1–9.7.4 [16] and yield the Laplace transform as a series of terms of the form e ^{−√ s}/s ^{n /2}. These terms can be inverted to obtain a series solution valid for r≠0 and for short times t. We present here only the lowest-order term exhibiting coupling of diffusion and bleaching by the probe beam.

$$c_1 = e^{\!-\! ht} \left( 1 - \frac{h}{{4D}}\sqrt {\frac{{r_0 }}{r}} \left\{ 2(r_0 - r)\sqrt {\frac{{Dt}}{\pi }} e^{ - {\raise0.7ex\hbox{${(r_0 - r)^2 }$} \!\mathord{\left/{\vphantom {{(r_0 - r)^2 } {4Dt}}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{${4Dt}$}}}\right.\right.\\ \left.\left. + \left[ {(r_0 - r)^2 + 2Dt} \right]\hbox{\it erfc}\frac{{r_0 - r}}{{2\sqrt {Dt} }} \right\} + \cdots \right)\quad (0 < r \le r_0 ) \\ c_2 = 1 - \frac{h}{{4D}}\sqrt {\frac{{r_0 }}{r}} \left\{ - 2(r - r_0 )\sqrt {\frac{{Dt}}{\pi }} e^{ - {\raise0.7ex\hbox{${(r - r_0 )^2 }$} \!\mathord{\left/{\vphantom {{(r - r_0 )^2 } {4Dt}}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{${4Dt}$}}} \right.\\ \left. + \left[ {(r - r_0 )^2 + 2Dt} \right]\hbox{\it erfc}\frac{{r - r_0 }}{{2\sqrt {Dt} }} \right\} + \cdots \quad (r_0 \le r < \infty )\\ $$

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However, since our interest actually lies only in the fluorescence signal, we can return to Eq. (4) and integrate across the illuminated area to obtain the average Laplace transform, proportional to the total fluorescence signal.

$$\langle \theta _1 \rangle = \frac{1}{{s + h}} + \frac{2}{{nb}}\frac{{({\textstyle{1 \over s}} - {\textstyle{1 \over {s + h}}})\left[ {mK_1 (mb)I_1 (nb)} \right]}}{{mI_0 (nb)K_1 (mb) + nI_1 (nb)K_0 (mb)}}$$

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Again using series expansion of the modified Bessel functions and inverting, we find that the fluorescence signal can be represented as

$$F = F_0 e^{ - ht} \left( {1 + \frac{{4h}}{{3r_0 }}\sqrt {\frac{{Dt^3 }}{\pi }} + \cdots } \right)$$

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Fig. 6.

Concentration profiles for a sample continuously bleached by a uniform circular probe beam. The inset shows how decay of sample fluorescence during observation in the presence and absence of diffusion.

Figure 6 shows the evolution of fluorescence from a uniformly illuminated region experiencing coupled bleaching and diffusion as described in Eq. (25) for a sample with $D = 0.02r_0^2 /pt$ and $h = 0.02/pt$. The coupling of diffusion and bleaching is clear from the evolution over time of the shape of the bleached region which is initially flat but soon becomes a rounded depression. The actual fluorescence signal for D = 0.02 and D = 0 is plotted in the inset of Fig. 6. As is the case for a Gaussian spot, a plot of the fluorescence versus time shows a steeper initial portion and then a decreasing negative slope as diffusion into the bleached area becomes more significant. It is clear that the shape of the fluorescence baseline, in the absence of a bleaching pulse, will continue to evolve over time and so must be explicitly incorporated in analysis of FPR data accompanied by any noticeable amount of bleaching by the probe beam. One should note that this series treatment is only an approximation valid for short times and so does not itself provide a closed-form function suitable for projecting an extended fluorescence baseline in the absence of a bleaching pulse. For this purpose, numerical solution, as described above, of the differential equation describing coupled beaching and diffusion is necessary.