In a correlation spectroscopy experiment, it is the fluctuations in the fluorescent signal that are recorded. For a signal (single channel), the normalised correlation curve is defined to be (Bacia and Schwille, 2007; Dross
et al., 2009),
(4-1) Where represents the lag time, and denotes averaging over time:
(4-2)
It can be shown that (Section 4.2.3.1), for freely diffusing fluorophores, the correlation can be described by,
(4-3)
Here is the average number of fluorescent molecules in the confocal volume, represents the nonfluorescent component due to transitions of the fluorescent molecules into the triplet state, is the index for the diffusive species i.e. species diffusing at rate , is the relative fraction of each diffusive species, and is the structural parameter, which depends on the observation volume and is given by,
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(4-4)
where is the axial and the lateral dimension of the confocal volume. The characteristic diffusion time(s) is related to the diffusion coefficient and is given by (for 2-dimensional diffusion),
(4-5)
When characterising FCS data, the goal is to fit for unknown parameters in Eq. (4-3). A useful training exercise is to begin by measuring parameters for a known standard to become familiar with the method. Following the protocol given by Kim et al. (Kim et al., 2007), an initial configuration experiment was carried out to determine the lateral radius of the microscopy set up used. Rhodamine 6G at a of 10 nM was excited with 514nm laser light and emission collected between 530 and 560nm, with 10 lots of 10 s runs used for each measurement. A Zeiss LSM710 with confocor 3 mounted on an Axio observer Z1 microscope with a 63x C-apochromat 1.2 NA water-immersion objective were utilised. Zen 2010B was used for data collection and analysis. Data were collected in intervals of 20 ns, and a data binning time of 200 ns applied. The resulting average autocorrelation curve is shown in Figure 4-1.
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Figure 4-1 Free diffusion of Rhodamine 6G in solution.
Mean and SEM of autocorrelation curves (n=5). Inset is an example fluorescence fluctuation trace (fluorescence count in kHz) over a 1 s period. All measurements were carried out at 37˚C using a water immersion objective with a numerical aperture (NA) of 1.2, and a Rhodamine concentration of 10 nM was measured.
Rhodamine 6G is a freely diffusing dye conforming to single-component Brownian motion and so a reduced form of Eq. (4-3) can be fit to,
(4-6)
For confocal microscope, a structural parameter of 3-6 is suggested by Kim et al. Fitting to Eq. (4-6) with a free structural parameter, the estimated value for across 20 repetitions (mean ± SEM) was found to be 4.9 ± 0.18. A structural parameter of 5 was thus assumed for subsequent fitting.
1 1.002 1.004 1.006 1.008 1.01 1.012 1E-05 0.0001 0.001 0.01 0.1 1 C o rrel a tion Lag Time (s) 0 50 100 150 200 250 300 350 400 0 0.5 1
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It is highlighted here that in order to obtain a meaningful fit to the data, and owing to the highly sensitive nature of FCS measurements, it was first necessary to sort FCS curves individually (from each 10s run) in order that they meet necessary criteria, as outlined in Kim et al. For example, that no greater than 10% photobleaching of fluorescence signal is observed or that spikes in fluorescence due to aggregation are discarded. The FCS curve from such an individual run was hence discarded from the final analysis; Kim et al. suggest that up to three runs can be discarded for a 10 x 10s measurement. For the Rhodamine data, physically meaningless estimates of equal to 1000 could be obtained for certain FCS curves. It is noted here, for example, that the gradual loss of fluorescence can have the appearance of a slow diffusing species in the autocorrelation curve (see Section 4.2.5.1). In such cases, a multi-component fit can yield more meaningful fits to data however the context of what is being fitted needs to be considered. For a freely diffusing dye like Rhodamine, a single diffusion component fit should be sufficient, whereas intracellular data will typically require a multi-component fit due to the complex nature of the cellular environment.
Fixing the structural parameter at 5, a one-component fit value of 46.5±0.13 µs (mean ± SEM) across 20 repetitions was obtained for . The diffusion rate of Rhodamine 6G at 37°C is characterised as between 3-5x10-10 m2 s-1 (Kim et al., 2007). Thus from Eq. (4-5), the lateral radius was estimated to be approximately 230-300 nm. This compares well with the theoretical diffraction limit. The peak excitation and emission wavelengths for Rhodamine 6G are 524 and 550 nm respectively (Behlke et al., 2005) and so the Rayleigh limit for peak emission is:
(4-7)
It is noted that the peak emission wavelengths of the different coloured fluorescent proteins vary from about 500 to 600 nm, giving an approximate
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resolution range of 250-300 nm. However, when conducting dual-colour correlation studies, the pinhole size (set to be 1 Airy unit) is determined by the longer wavelength, and so a lateral radius of 300 nm is assumed across experiments. It is noted here that this lateral radius is determined through measurement of Rhodamine 6G in aqueous solution. In living cells, there is a refractive index mismatch that the distorts the confocal volume and which cannot be corrected for in a straightforward way. With a refractive index of the cellular environment of 1.36 ± 0.004, this leads to an absolute error in estimated diffusion coefficients of 10% for a focal depth of less than 20 micrometers (Curl et al., 2005; Dross et al., 2009).
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4.2.2Measuring molecule concentration