Fluorescence Correlation Spectroscopy Close to a Fluctuating Membrane Cécile Fradin, Asmahan Abu-Arish, Rony Granek, Michael Elbaum  Biophysical Journal  Volume 84, Issue 3, Pages 2005-2020 (March 2003) DOI: 10.1016/S0006-3495(03)75009-7 Copyright © 2003 The Biophysical Society Terms and Conditions

Figure 1 Geometry of the considered system. Fluorescent molecules are represented by spheres. Dark gray spheres correspond to molecules in the detection volume, whose fluorescence will be detected by the optical system. The membrane separates space into a region (x<〈W〉) containing fluorophores with an average concentration 〈c〉, and a region (x>〈W〉) containing no fluorophores. Biophysical Journal 2003 84, 2005-2020DOI: (10.1016/S0006-3495(03)75009-7) Copyright © 2003 The Biophysical Society Terms and Conditions

Figure 2 (a) Autocorrelation functions calculated for different values of 〈W〉/w0, normalized by their values at t=0, as a function of t/τ, where τ=w02/4D is the free-diffusion residence time. The curves have been calculated for a typical value of the detection volume aspect ratio S=6.4. For illustration, the characteristic residence time τd, defined by Gd(〈W〉,τd)=Gd(〈W〉,0)/2, is indicated for one of the curves. (b) Value of the characteristic residence time τd as a function of 〈W〉/w0 (continuous thick line), compared with the values of the residence time estimated by fitting the autocorrelation functions calculated from Eq. 19 by a free-diffusion autocorrelation function (Eq. 6, continuous thin line) or by an anomalous-diffusion autocorrelation function (Eq. 7, dashed thin line). All curves have been normalized by the free-diffusion residence time τ=w02/4D. (c) Anomalous exponent γ obtained by fitting the autocorrelation functions calculated from Eq. 19 using an anomalous-diffusion model (Eq. 7). Note that for b and c, the x axis has been inverted, so that the plot is read from left to right as the detection volume approaches the membrane. Biophysical Journal 2003 84, 2005-2020DOI: (10.1016/S0006-3495(03)75009-7) Copyright © 2003 The Biophysical Society Terms and Conditions

Figure 3 Diffusion coefficients (normalized by the actual diffusion coefficient) calculated from the residence times obtained by fitting the autocorrelation functions calculated with Eq. 19 using a hindered-diffusion model (thick black line), a free-diffusion model (thin black line), or an anomalous-diffusion model. In this last case, the quantity plotted is Datγ−1, for t=1s (dashed black line), and for t=τa (dashed gray line). Biophysical Journal 2003 84, 2005-2020DOI: (10.1016/S0006-3495(03)75009-7) Copyright © 2003 The Biophysical Society Terms and Conditions

Figure 4 Part of the autocorrelation function due to the fluctuations of the membrane, calculated for qmin=0.1μm−1 (thick line) and for qmin=1⁡μm−1 (thin line). In both cases, K=10⁡kBT and qmax=4⁡μm−1. The time has been normalized by 1/ω(qmin) and the amplitude of the curve by the value of Gf(〈W〉,0) (Eq. 33). The characteristic decay time τf of the autocorrelation function calculated for qmin=0.1⁡μm−1 is indicated. Biophysical Journal 2003 84, 2005-2020DOI: (10.1016/S0006-3495(03)75009-7) Copyright © 2003 The Biophysical Society Terms and Conditions

Figure 5 (a) Amplitude of the two separate contributions (gray line: diffusion term, Eq. 24; black line: membrane fluctuations term, Eq. 33) and of the complete autocorrelation function (dashed line). The curves have been calculated for w0=230 nm, S=8, 〈c〉=15 nM, K=1000kBT, and qmin=0.1⁡μm−1. (b) Relative amplitude of the membrane fluctuation term Gf(0) to the diffusion term Gd(0), calculated for the same set of parameters as in a. Note that the x axis has been inverted as in Figs. 2 and 3. Biophysical Journal 2003 84, 2005-2020DOI: (10.1016/S0006-3495(03)75009-7) Copyright © 2003 The Biophysical Society Terms and Conditions

Figure 6 (a) Variation of the relative amplitude of the two terms Gf(0)/Gd(0) with the bending rigidity of the membrane K (all other parameters as in Fig. 5), for two particular distances of the detection volume to the membrane: 〈W〉/w0=1 (gray line) and 〈W〉/w0=0.5 (black line). The dashed line indicates the value of K used for the curves of Fig. 5. (b) Variation of Gf(0)/Gd(0) with the fluorophore concentration (other parameters values as in Fig. 5), again for 〈W〉/w0=1 (gray line) and 〈W〉/w0=0.5 (black line). The dashed line indicates the value of 〈c〉 used for the curves of Fig. 5. Biophysical Journal 2003 84, 2005-2020DOI: (10.1016/S0006-3495(03)75009-7) Copyright © 2003 The Biophysical Society Terms and Conditions

Figure 7 (a) DIC photograph of one of the vesicles used for the presented experiments. The radius of the vesicle is 14.4μm, giving qmin=6.9×104m−1 if qmin=1/R. (b) Measured average intensity profile coming from the fluorescence of the Cy3-streptavidin present outside the vesicle, as a function of the distance 〈W〉 of the detection volume from the membrane (circles). The line is only a guide to the eye. The maximum signal intensity is IM=12.5kHz, and the relative amount of noise is r=0.1. Note that the scales of figures a and b are different and that the x axis in b runs from positive to negative values. Biophysical Journal 2003 84, 2005-2020DOI: (10.1016/S0006-3495(03)75009-7) Copyright © 2003 The Biophysical Society Terms and Conditions

Figure 8 Typical measured autocorrelation functions, for a 6.8-nM concentration of Cy3-streptavidin outside a vesicle, and for different position of the membrane relative to the detection volume: membrane infinitely remote (black dots), 〈W〉=0.44⁡w0 (empty circles), 〈W〉=0.28⁡w0 (empty triangles), and 〈W〉=0.09⁡w0 (empty squares). These positions have been calculated from the measured average intensity using Eq. 10. As explained in the text, continuous black lines are fits obtained using Eq. 36. Biophysical Journal 2003 84, 2005-2020DOI: (10.1016/S0006-3495(03)75009-7) Copyright © 2003 The Biophysical Society Terms and Conditions

Figure 9 Amplitude of the fluorophore diffusion contribution of the autocorrelation function, as a function of 〈W〉/w0, for a 6.8-nM concentration of streptavidin (filled circles), and 23.8-nM (open squares) and 490-nM (open triangles) concentrations of dextran. The best fits using Eq. 24 are indicated (bold continuous line, continuous line, and dotted line, respectively). Biophysical Journal 2003 84, 2005-2020DOI: (10.1016/S0006-3495(03)75009-7) Copyright © 2003 The Biophysical Society Terms and Conditions

Figure 10 Amplitude of the membrane fluctuations contribution in the autocorrelation function as a function of 〈W〉/w0, for a 6.8-nM concentration of streptavidin (filled circles), and 23.8-nM (open squares) and 490-nM (open triangles) concentrations of dextran. The radii of the vesicles were, respectively, 14.4μm, 9.1μm, and 7.3μm. The best fits using Eq. 33 are indicated (bold continuous line, continuous line, and dotted line). The small filled lozenge symbols along the horizontal axis correspond to the measured amplitude of the fluctuation term in the case of a fixed, rigid wall. Biophysical Journal 2003 84, 2005-2020DOI: (10.1016/S0006-3495(03)75009-7) Copyright © 2003 The Biophysical Society Terms and Conditions

Figure 11 Autocorrelation function (on a log-log scale) measured for a 6.8-nM concentration of Cy3-streptavidin outside a vesicle at 〈W〉/w0=0.6 (gray open symbols), and best fit obtained using the complete model derived in this paper (thick black line). The diffusion part Gd (thin black line) and fluctuation part Gf (thin gray line) of the fit are also shown. The diffusion coefficient retrieved from the fit is D=43.8±1.5⁡μm2/s, whereas the bending rigidity is K=720±400kBT (assuming qmin=1/R). Biophysical Journal 2003 84, 2005-2020DOI: (10.1016/S0006-3495(03)75009-7) Copyright © 2003 The Biophysical Society Terms and Conditions

Figure 12 Log-log plot of the quantity 1−Gf(t)/Gf(0) as calculated exactly with our model from Eqs. 32 and 33 (thick continuous line), and approximated expression using the t2/3 dependence of the transversal MSD of a membrane in the time range 1/ω(qmax)⁡≪⁡t⁡≪⁡1/ω(qmin) (thin continuous line). The dashed lines indicate the respective positions of 1/ω(qmax) and 1/ω(qmin). Biophysical Journal 2003 84, 2005-2020DOI: (10.1016/S0006-3495(03)75009-7) Copyright © 2003 The Biophysical Society Terms and Conditions