11.3: Fluorescence Correlation Spectroscopy
- Page ID
- 294323
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
Fluorescence correlation spectroscopy (FCS) allows one to measure diffusive properties of fluorescent molecules, and is closely related to FRAP. Instead of measuring time-dependent concentration profiles and modeling the kinetics as continuum diffusion, FCS follows the steady state fluctuations in number density of a very dilute fluorescent probe molecule in the small volume observed in a confocal microscope. We measure the fluctuating changes in fluorescence intensity emitted from probe molecules as they diffuse into and out of the focal volume.
- Average concentration of sample: C0 = <10–9 M – 10–7 M.
- This corresponds to an average of ~0.1-100 molecules in the focal volume, although this number varies with diffusion into and out of the volume.
- The fluctuating fluorescence trajectory is proportional to the time-dependent number
density or concentration:
\[ F(t) \propto C(t) \nonumber \]
- How big are the fluctuations? For a Gaussian random process, we expect \(\dfrac{\delta N_{rms}}{N} \sim \dfrac{1}{\sqrt{N}} \)
- The observed concentration at any point in time can be expressed as time-dependent
fluctuations about an average value: \(C(t) = \overline{C} + \delta C(t)\).
To describe the experimental observable, we model the time-dependence of δC(t) from the diffusion equation:
\[ \dfrac{\partial \delta C}{\partial t} = D \nabla^2 \delta C \nonumber \]
\[ \langle \delta C(r,0) \delta C(r',t) \rangle = \dfrac{C_0}{(4\pi Dt)^{3/2}} e^{-(r-r')^2/4Dt} \nonumber \]
The concentration fluctuations can be related to the fluorescence intensity fluctuations as
\[ F(t) = AW(r) C(r,t) \nonumber \]
W(r): Spatial optical profile of excitation and detection
A: Other experimental excitation and detection parameters
Calculate FCS correlation function for fluorescence intensity fluctuations. \(F(t) = \langle F \rangle -\delta F(t) \)
\[ G(t) = \dfrac{\langle \delta F(0) \delta F(t) \rangle}{\langle \delta F \rangle^2} \nonumber \]
For the case of a Gaussian beam with a waist size w0:
\[ G(t) \sim \dfrac{B}{1+t/\tau_{FCS}} \nonumber \]
Where the amplitude is \(B = 4\pi A^2I_0^2\overline{\delta C^2_0} w_0^2\), and the correlation time is related to the diffusion constant by:
\[ \tau_{FCS} = \dfrac{w_0^2}{4D} \nonumber \]
___________________________________
Readings
- P. Schwille and E. Haustein, "Fluorescence Correlation Spectroscopy: An Introduction to its Concepts and Applications" in Biophysics Textbook Online.