Fluorescence Intensity Autocorrelation Function

The Bloch equations (Eq. 5) can be solved under different conditions. The transient solution yields an expression for 0-22 (0> time-dependent population of the excited singlet state S. It will be discussed in detail in Section 1.2.4.3 in connection with the fluorescence intensity autocorrelation function. Here we are interested in the steady state solution (an = 0-22 = < 33 = di2 = 0) which allows to compute the line-shape and saturation effects. A detailed description of the steady state solution for a three level system can be found in [35]. From those the appropriate equations for the intensity dependence of the excitation linewidth Avfwhm (FWHM full width at half maximum) and the fluorescence emission rate R for a single absorber can be easily derived [10] ... 

The time distribution of the fluorescence photons emitted by a single dye molecule reflects its intra- and intermolecular dynamics. One example are the quantum jumps just discussed which lead to stochastic fluctuations of the fluorescence emission caused by singlet-triplet quantum transitions. This effect, however, can only be observed directly in a simple fluorescence counting experiment when a system with suitable photophysical transition rates is available. By recording the fluorescence intensity autocorrelation function, i.e. by measuring the correlation between fluorescence photons at different instants of time, a more versatile and powerful technique is available which allows the determination of dynamical processes of a single molecule from nanoseconds up to hundreds of seconds. It is important to mention that any reliable measurement with this technique requires the dynamics of the system to be stationary for the recording time of the correlation function. 

When we measure photons emitted by a single molecule using a quantum detector such as a photomultiplier tube, we interpret /(/) to be the photon counting rate (fluorescence intensity at time i) and we can deduce the fluorescence intensity autocorrelation function by counting the number of photon pairs separated by... 

Figure 14. Simulation of the fluorescence intensity autocorrelation function (t) for a single pentacene molecule for different values of the Rabi-frequency 2 ...

Figure 16. Normalized fluorescence intensity autocorrelation function for a single terrylene molecule in p-terphenyl (site X2) at 2 K measured over 9 orders of magnitude in time ( ). The drawn line is a simulation of the correlation function using appropriate photophysical parameters (from Ref. 27).

Figure 18. Fluorescence intensity autocorrelation function at short times for a single tenylene molecule in p-terphenyl (site Xt) at saturating intensities. The solid lines are fits of Eq. 14 to the data ( ) (a) 2 K (b) 4 K. It is clearly seen that by raising the temperature the Rabi oscillations become subject to stronger damping which is caused by pure optical dephasing (from Ref. 27).

Figure 19. (a) Decay of the fluorescence intensity autocorrelation function for a single penta-cene molecule in / -terphenyl at long times T = 1.4K). The solid line is a fit of Eq. 15 to the data, (b) Plot of X versus logarithmic intensity for three different pentacene molecules. The solid lines are fits of Eq. 16 to the data. For high intensities X —> 23/2) it can be seen that the population rate varies strongly from molecule to molecule (from Ref. 36). 

In a standard FCS measurement, fluorescence fluctuations arise from translational diffusion, as the fluorescent molecules are diffusing into and out of the confocal observation volume, providing information about the translational diffusion coefficients and the average number of molecules residing simultaneously in the observation volume. In the absence of any other kinetic process affecting the fluorescent molecules the time-dependent normalized intensity autocorrelation function (ACF) can be written as ... 

Figure 5 (a) The structure of T4 lysozyme with the two dye labels schematically shown, (b) Fluorescence intensity trajectories of the TM R donor (blue) and the Texas Red acceptor (red) of a single T4 lysozyme in the presence of E coli B cell wall, (c) Distribution of the decay rate constants (/t) of the donor intensity autocorrelation functions. Reproduced with permission from Y. Chen D. Hu E. R. Vorpagel H. P. Lu, J. Phys. Chem. B. 2003,107, 7947-7956. Copyright (2003) American... 

Fluorescence intensity detected with a confocal microscope for the small area of diluted solution temporally fluctuates in sync with (i) motions of solute molecules going in/out of the confocal volume, (ii) intersystem crossing in the solute, and (hi) quenching by molecular interactions. The degree of fluctuation is also dependent on the number of dye molecules in the confocal area (concentration) with an increase in the concentration of the dye, the degree of fluctuation decreases. The autocorrelation function (ACF) of the time profile of the fluorescence fluctuation provides quantitative information on the dynamics of molecules. This method of measurement is well known as fluorescence correlation spectroscopy (FCS) [8, 9]. 

The autocorrelation function, G(x), of the temporal fluctuation of the fluorescence intensity at the confocal volume is analytically represented by the following equation [8, 9] ... 

Fig. 11.10. Schematic illustration of fluorescence correlation spectroscopy. The autocorrelation function characterises the fluctuations of the fluorescence intensity its decay time expresses the average duration of a...

For a single fluorescent species undergoing Brownian motion with a translational diffusion coefficient Dt (see Chapter 8, Section 8.1), the autocorrelation function, in the case of Gaussian intensity distribution in the x, y plane and infinite dimension in the z-direction, is given by... 

Triplet state kinetics can also be studied by FCS (Widengren et al., 1995). In fact, with dyes such as fluoresceins and rhodamines, additional fluctuations in fluorescence are observed when increasing excitation intensities as the molecules enter and leave their triplet states. The time-dependent part of the autocorrelation function is given by... 

When the excitation light is polarized and/or if the emitted fluorescence is detected through a polarizer, rotational motion of a fluorophore causes fluctuations in fluorescence intensity. We will consider only the case where the fluorescence decay, the rotational motion and the translational diffusion are well separated in time. In other words, the relevant parameters are such that tc rp, where is the lifetime of the singlet excited state, zc is the rotational correlation time (defined as l/6Dr where Dr is the rotational diffusion coefficient see Chapter 5, Section 5.6.1), and td is the diffusion time defined above. Then, the normalized autocorrelation function can be written as (Rigler et al., 1993)... 

The fluorescence intensity trajectories of the donor (/d(f)) and acceptor (/a(t)) give autocorrelation times (Fig. 24.2b) indistinguishable from fitting an exponential decay to the autocorrelation functions, (A/d (0) A/d (t)) and (A/a (0) A/a (t)), where A/d(t) is /d(t) — (Id), (Id) is the mean intensity of the overall trajectory of a donor, and A/a(t) has the same definition for an intensity trajectory of an acceptor. In contrast, the cross-correlation function between the donor and acceptor trajectories, (A/d (0) A/d (t)), is anticorrelated with the same decay time (Fig. 24.2b) which supports our assignment of anticorrelated fluctuations of the fluorescence intensities of the donor and acceptor to the spFRET process. 

Fluorescence correlation spectroscopy analyses the temporal fluctuations of the fluorescence intensity by means of an autocorrelation function from which translational and rotational diffusion coefficients, flow rates and rate constants of chemical processes of single molecules can be determined. For example, the dynamics of complex formation between /3-cyclodextrin as a host for guest molecules was investigated with singlemolecule sensitivity, which revealed that the formation of an encounter complex is followed by a unimolecular inclusion reaction as the rate-limiting step.263... 

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