Fluorescence correlation spectroscopy
Fluorescence correlation spectroscopy is a statistical analysis, via time correlation, of stationary fluctuations of the fluorescence intensity. Its theoretical underpinning originated from L. Onsager's regression hypothesis. The analysis provides kinetic parameters of the physical processes underlying the fluctuations. One of the interesting applications of this is an analysis of the concentration fluctuations of fluorescent particles in solution. In this application, the fluorescence emitted from a very tiny space in solution containing a small number of fluorescent particles is observed. The fluorescence intensity is fluctuating due to Brownian motion of the particles. In other words, the number of the particles in the sub-space defined by the optical system is randomly changing around the average number. The analysis gives the average number of fluorescent particles and average diffusion time, when the particle is passing through the space. Eventually, both the concentration and size of the particle are determined. Both parameters are important in biochemical research, biophysics, and chemistry.
FCS is such a sensitive analytical tool because it observes a small number of molecules in a small volume. In contrast to other methods FCS has no physical separation process; instead, it achieves its spatial resolution through its optics. Furthermore, FCS enables observation of fluorescence-tagged molecules in the biochemical pathway in intact living cells. This opens a new area, "in situ or in vivo biochemistry": tracing the biochemical pathway in intact cells and organs.
Commonly, FCS is employed in the context of optical microscopy, in particular confocal microscopy or two-photon excitation microscopy. In these techniques light is focused on a sample and the measured fluorescence intensity fluctuations are analyzed using the temporal autocorrelation. Because the measured property is essentially related to the magnitude and/or the amount of fluctuations, there is an optimum measurement regime at the level when individual species enter or exit the observation volume. When too many entities are measured at the same time the overall fluctuations are small in comparison to the total signal and may not be resolvable – in the other direction, if the individual fluctuation-events are too sparse in time, one measurement may take prohibitively too long. FCS is in a way the fluorescent counterpart to dynamic light scattering, which uses coherent light scattering, instead of fluorescence.
When an appropriate model is known, FCS can be used to obtain quantitative information such as
- diffusion coefficients
- hydrodynamic radii
- average concentrations
- kinetic chemical reaction rates
- singlet-triplet dynamics
History
Signal-correlation techniques were first experimentally applied to fluorescence in 1972 by Magde, Elson, and Webb, who are therefore commonly credited as the inventors of FCS. The technique was further developed in a group of papers by these and other authors soon after, establishing the theoretical foundations and types of applications.Around 1990, with the ability of detecting sufficiently small number of fluorescence particles, two issues emerged: A non-Gaussian distribution of the fluorescence intensity and the three-dimensional confocal [|Measurement Volume] of a laser-microscopy system. The former led to an analysis of distributions and moments of the fluorescent signals for extracting molecular information, which eventually became a collection of methods known as [|Brightness Analyses]. See Thompson for a review of that period.
Beginning in 1993, a number of improvements in the measurement techniques—notably using confocal microscopy, and then two-photon microscopy—to better define the measurement volume and reject background—greatly improved the signal-to-noise ratio and allowed single molecule sensitivity. Since then, there has been a renewed interest in FCS, and as of August 2007 there have been over 3,000 papers using FCS found in Web of Science. See Krichevsky and Bonnet for a review. In addition, there has been a flurry of activity extending FCS in various ways, for instance to laser scanning and spinning-disk confocal microscopy, in using cross-correlation between two fluorescent channels instead of autocorrelation, and in using Förster Resonance Energy Transfer instead of fluorescence.
Typical setup
The typical FCS setup consists of a laser line, and from 690–1100 nm ), which is reflected into a microscope objective by a dichroic mirror. The laser beam is focused in the sample, which contains fluorescent particles in such high dilution, that only a few are within the focal spot. When the particles cross the focal volume, they fluoresce. This light is collected by the same objective and, because it is red-shifted with respect to the excitation light it passes the dichroic mirror reaching a detector, typically a photomultiplier tube, an avalanche photodiode detector or a superconducting nanowire single-photon detector. The resulting electronic signal can be stored either directly as an intensity versus time trace to be analyzed at a later point, or computed to generate the autocorrelation directly. The FCS curve by itself only represents a time-spectrum. Conclusions on physical phenomena have to be extracted from there with appropriate models. The parameters of interest are found after fitting the autocorrelation curve to modeled functional forms.Measurement volume
The measurement volume is a convolution of illumination and detection geometries, which result from the optical elements involved. The resulting volume is described mathematically by the point spread function, it is essentially the image of a point source. The PSF is often described as an ellipsoid of few hundred nanometers in focus diameter, and almost one micrometer along the optical axis. The shape varies significantly depending on the quality of the optical elements. In the case of confocal microscopy, and for small pinholes, the PSF is well approximated by Gaussians:where is the peak intensity, r and z are radial and axial position, and and are the radial and axial radii, and. This Gaussian form is assumed in deriving the functional form of the autocorrelation.
Typically is 200–300 nm, and is 2–6 times larger. One common way of calibrating the measurement volume parameters is to perform FCS on a species with known diffusion coefficient and concentration. Diffusion coefficients for common fluorophores in water are given in a later section.
The Gaussian approximation works to varying degrees depending on the optical details, and corrections can sometimes be applied to offset the errors in approximation.
Autocorrelation function
The autocorrelation function is the correlation of a time series with itself shifted by time, as a function of :where is the deviation from the mean intensity. The normalization here is the most commonly used for FCS, because then the correlation at, G, is related to the average number of particles in the measurement volume.
As an example, raw FCS data and its autocorrelation for freely diffusing Rhodamine 6G are shown in the figure to the right. The plot on top shows the fluorescent intensity versus time. The intensity fluctuates as Rhodamine 6G moves in and out of the focal volume. In the bottom plot is the autocorrelation on the same data. Information about the diffusion rate and concentration can be obtained using one of the models described below.
For a Gaussian illumination profile, the autocorrelation function is given by the general master formula
where the vector denotes the stochastic displacement in space of a fluorophore after time.
The expression is valid if the average number of fluorophores in the focal volume is low and if dark states, etc., of the fluorophore can be ignored. In particular, no assumption was made on the type of diffusive motion under investigation. The formula allows for an interpretation of as a return probability for small beam parameters and the moment-generating function of if are varied.
Interpreting the autocorrelation function
To extract quantities of interest, the autocorrelation data can be fitted, typically using a nonlinear least squares algorithm. The fit's functional form depends on the type of dynamics.Normal diffusion
The fluorescent particles used in FCS are small and thus experience thermal motions in solution. The simplest FCS experiment is thus normal 3D diffusion, for which the autocorrelation is:where is the ratio of axial to radial radii of the measurement volume, and is the characteristic residence time. This form was derived assuming a Gaussian measurement volume. Typically, the fit would have three free parameters—G,, and —from which the diffusion coefficient and fluorophore concentration can be obtained.
With the normalization used in the previous section, G gives the mean number of diffusers in the volume
where the effective volume is found from integrating the Gaussian form of the measurement volume and is given by:
Anomalous diffusion
If the diffusing particles are hindered by obstacles or pushed by a force the dynamics is often not sufficiently well-described by the normal diffusion model, where the mean squared displacement grows linearly with time. Instead the diffusion may be better described as anomalous diffusion, where the temporal dependence of the MSD is non-linear as in the power-law:where is an anomalous diffusion coefficient. "Anomalous diffusion" commonly refers only to this very generic model, and not the many other possibilities that might be described as anomalous. Also, a power law is, in a strict sense, the expected form only for a narrow range of rigorously defined systems, for instance when the distribution of obstacles is fractal. Nonetheless a power law can be a useful approximation for a wider range of systems.
The FCS autocorrelation function for anomalous diffusion is:
where the anomalous exponent is the same as above, and becomes a free parameter in the fitting.
Using FCS, the anomalous exponent has been shown to be an indication of the degree of molecular crowding.