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| general:fluorescence_correlation_spectroscopy-_a_short_introduction [2021/01/12 14:59] – [The autocorrelation function] doerr | general:fluorescence_correlation_spectroscopy-_a_short_introduction [2023/02/17 12:39] – admin |
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| {{tag>SPT64 MT200 MircoTime LSM Theory SymPhoTime FCS Correlation}} | {{tag>MT200 MircoTime LSM Theory SymPhoTime FCS Correlation}} |
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| ~~TOC~~ | ~~TOC~~ |
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| \begin{equation} | \begin{equation} |
| G(\tau)=\frac{\langle F(t) F(t+\tau) \rangle}{\langle F \rangle^2}=1+\frac{\langle \delta F(t) \delta F(t+\tau) \rangle}{\langle F \rangle^2} | G(\tau)=\frac{\langle F(t) \cdot F(t+\tau) \rangle}{\langle F \rangle^2}=1+\frac{\langle \delta F(t) \cdot \delta F(t+\tau) \rangle}{\langle F \rangle^2} |
| \end{equation}, | \end{equation}, |
| $$ | $$ |
| ==== Translational Diffusion - Confocal Volume ==== | ==== Translational Diffusion - Confocal Volume ==== |
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| In this short article only the case of point, single color FCS with a confocal set up will be examined. For FCS of freely diffusing fl uorescent molecules in a confocal setup the origin of the intensity fluctuations, will be the variation of the number of molecules inside the detection (assuming constant excitation settings). In <imgref figure1>A and <imgref figure1>B two different scenarios are illustrated. In <imgref figure1>A on average only 1 fluorescent molecule resides in the detection volume. As molecule diffuse in and out the relative fluctuations of the fluorescent intensity are large (no intensity when the molecule is out, high intensity when the molecule is in) in comparison with the average value. In <imgref figure1>B a few molecules reside in the detection volume on average, and thus the relative fluctuations will be smaller (always there are a few molecules in focus) in comparison to the average. This difference in the relative fluctuations is reflected in the correlation function. As it is seen in <imgref figure1>C the amplitude of the correlation function is different. | In this short article only the case of point, single color FCS with a confocal set up will be examined. For FCS of freely diffusing fluorescent molecules in a confocal setup the origin of the intensity fluctuations, will be the variation of the number of molecules inside the detection (assuming constant excitation settings). In <imgref figure1>A and <imgref figure1>B two different scenarios are illustrated. In <imgref figure1>A on average only 1 fluorescent molecule resides in the detection volume. As molecule diffuse in and out the relative fluctuations of the fluorescent intensity are large (no intensity when the molecule is out, high intensity when the molecule is in) in comparison with the average value. In <imgref figure1>B a few molecules reside in the detection volume on average, and thus the relative fluctuations will be smaller (always there are a few molecules in focus) in comparison to the average. This difference in the relative fluctuations is reflected in the correlation function. As it is seen in <imgref figure1>C the amplitude of the correlation function is different. |
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| <imgcaption figure1|>{{ :general:fcs1.png|Basic concepts of FCS. (A)Slowly diffusing fluorescent species at low concentrations give rise to large signal fluctuations around the mean fluorescence intensity value, (B) Fast diffusing molecules at high concentration produce small signal fluctuations, (C) The autocorrelation curves from (A) and (B). In case (B) the correlation curve will have a shorter diffusional correlation time and smaller G(0) in comparision to case (A). In general, slower diffusion "shifts" the curve to longer time scales and higer concentration results in reduced G(0). (D) An exemplary correlation curve of an emitter illustrating the various processess that could occur in different time scales and thus be manifested in different parts of the correlation curve.}}</imgcaption> | <imgcaption figure1|>{{ :general:fcs1.png|Basic concepts of FCS. (A)Slowly diffusing fluorescent species at low concentrations give rise to large signal fluctuations around the mean fluorescence intensity value, (B) Fast diffusing molecules at high concentration produce small signal fluctuations, (C) The autocorrelation curves from (A) and (B). In case (B) the correlation curve will have a shorter diffusional correlation time and smaller G(0) in comparision to case (A). In general, slower diffusion "shifts" the curve to longer time scales and higer concentration results in reduced G(0). (D) An exemplary correlation curve of an emitter illustrating the various processess that could occur in different time scales and thus be manifested in different parts of the correlation curve.}}</imgcaption> |
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| Assuming that the mean number of fl uorescent molecules, $N$ within the observation volume follows a Poisson distribution and that the observation volume is described by a Gaussian ellipsoid, the autocorrelation function describing translational diffusion of fluorescent molecules through the detection volume will be given by: | Assuming that the mean number of fluorescent molecules, $N$ within the observation volume follows a Poisson distribution and that the observation volume is described by a Gaussian ellipsoid, the autocorrelation function describing translational diffusion of fluorescent molecules through the detection volume will be given by: |
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| \end{equation}, | \end{equation}, |
| $$ | $$ |
| where $\tau_{Diff}$ is the characteristic correlation time due to translational diffusion | where $\tau_{Diff}$ is the characteristic correlation time due to translational diffusion |
| and $k$ the ellipticity of the detection volume. The effective detection volume as "seen" by FCS will be a Gaussian ellipsoid with characteristic radius $\omega$ along the short axis. | and $k$ the ellipticity of the detection volume. The effective detection volume as "seen" by FCS will be a Gaussian ellipsoid with characteristic radius $\omega$ along the short axis. |
| \\ The effective detection volume as "seen" by FCS is related to the confocal volume: | \\ The effective detection volume as "seen" by FCS is related to the confocal volume: |
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| By fit ting the model function presented in equation $\ref{eq2}$ to the experimental autocorrelation curves obtained by correlating the fluorescence intensity as shown in <imgref figure1> the diffusional correlational time can be obtained. | By fitting the model function presented in equation $\ref{eq2}$ to the experimental autocorrelation curves obtained by correlating the fluorescence intensity as shown in <imgref figure1> the diffusional correlational time can be obtained. |
| If the diffusion coefficient $D$ of the molecular species is known then the geomentrical factors $w$ and $k$ can be obtained and the effective confocal volume can be determined ($\ref{eqVeff}$). After the calibration of the detection volume one can use the diffusional correlation time to determine the diffusion coefficient of an unknown species. | If the diffusion coefficient $D$ of the molecular species is known then the geomentrical factors $w$ and $k$ can be obtained and the effective confocal volume can be determined ($\ref{eqVeff}$). After the calibration of the detection volume one can use the diffusional correlation time to determine the diffusion coefficient of an unknown species. |
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