Fluorescence correlation spectroscopy

Fluorescence correlation spectroscopy (FCS) is a common technique used by physicists, chemists, and biologists to experimentally characterize fluorescent species (proteins, biomolecules, pharmaceuticals, etc.) and their dynamics. Using confocal or two photon microscopy, light is focused on a sample and the measured fluorescence intensity fluctuations (due to diffusion, chemical reactions, aggregation, etc.) are analyzed using the temporal autocorrelation. FCS is the fluorescent counterpart to dynamic light scattering, which uses coherent light scattering, instead of (incoherent) fluorescence.

FCS obtains quantitative information such as
 * diffusion coefficients
 * hydrodynamic radii
 * average concentrations
 * kinetic chemical reaction rates.

Because fluorescent markers come in a variety of colors and can be specific to a particular molecule, molecules can be studied individually and in rapid succession in composite solutions. The advent of engineered cells with genetically tagged proteins (like green fluorescent protein) has made FCS a common tool for studying living cells.

History
Fluorescence correlation spectroscopy was invented in 1972 by Magde, Elson, and Webb, and was further developed in a group of papers by these and other authors soon after, establishing the theoretical foundations and types of applications. See Thompson (1991) 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 and allowed single molecule sensitivity. Since then, there has been a renewed interest in FCS, and as of August 2007 there has been over 3,000 papers using FCS found in Web of Science. See Krichevsky and Bonnet for a recent 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 (from a stationary, single point measurement), in using cross-correlation (FCCS) between two fluorescent channels instead of autocorrelation, and in using Förster Resonance Energy Transfer (FRET) instead of fluorescence.

Typical FCS setup
The typical FCS setup consists of a laser line (typically 450-633 nm), which is reflected into the objective by a dichroic mirror. The objective focuses the laser beam in the sample, causing the particles to fluoresce. The fluorescent light then passes through a series of lenses and a pinhole before reaching a detector, typically a photomultiplier tube or avalanche photodiode detector. A computer with a digital correlator card computes the autocorrelation function from the fluorescence intensity signal (averaging appropriately) and stores the result. The parameters of interest are found after fitting the autocorrelation to known functional forms. The setup is shown in Figure 1.

The Measurement Volume
The objective lens focuses the laser beam into the sample and collects the emitted fluorescence light. Because of diffraction, the region of the sample where light is collected--the measurement volume--is football shaped and does not have sharp boundaries. In imaging contexts, this volume is referred to as the point spread function (or PSF), and as its name suggests, is the image of a point source. (Note that the excitation volume has a similar shape to the PSF but extends further, meaning more fluorescent particles are excited than detected.) For small pinholes, the PSF is well approximated by Gaussians:


 * $$ PSF(r,z)= I_0 e^{- 2 r^2/\omega_{xy}^2} e^{-2 z^2/\omega_{z}^2} $$

where $$I_0$$ is the peak intensity, r and z are radial and axial position, and $$\omega_{xy}$$ and $$\omega_{z}$$ are the radial and axial radii, and $$ \omega_{z} > \omega_{xy}$$. This Gaussian form is assumed in deriving the functional form of the autocorrelation.

Typically $$\omega_{xy}$$ is 200-300 nm, and $$\omega_{z}$$ 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 (see below). 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 (temporal) autocorrelation function is the correlation of a time series with itself shifted by time $$\tau$$, as a function of $$\tau$$:
 * $$\ G(\tau)=\frac{\langle\delta I(t)\delta I(t+\tau)\rangle}{\langle I(t) \rangle^2}=\frac{\langle I(t)I(t+\tau) \rangle}{\langle I(t)\rangle^2}-1$$

where $$\delta I(t)=I(t)-\langle I(t)\rangle $$ is the deviation from the mean intensity. The normalization (denominator) here is the most commonly used for FCS, because then the correlation at $$\tau=0$$, G(0), is related to the average number of particles in the measurement volume.

Interpreting the Autocorrelation Function
To extract quantities of interest, the autocorrelation data must be fit, 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:

$$\ G(\tau)=G(0)\frac{1}{(1+(\tau/\tau_{D}))(1+a^{-2}(\tau/\tau_{D}))^{1/2}} +G(\infty)$$

where $$ a=\omega_z/\omega_{xy}$$ is the ratio of axial to radial $$e^{-2}$$ radii of the measurement volume, and $$\tau_{D}$$ is the characteristic residence time. This form was derived assuming a Gaussian measurement volume. Typically, the fit would have three free parameters--G(0), $$G(\infty)$$, and $$\tau_D$$--from which the diffusion coefficient and fluorophore concentration can be obtained.

With the normalization used in the previous section, G(0) gives the mean number of diffusers in the volume , or equivalently--with knowledge of the observation volume size--the mean concentration:

$$\ G(0)=\frac{1}{}=\frac{1}{V_{eff}}$$,

where the effective volume is found from integrating the Gaussian form of the measurement volume and is given by:

$$\ V_{eff}=\pi^{3/2}\omega_{xy}^2\omega_z $$.

$$\tau_D$$ gives the diffusion coefficient: $$\ D=\omega_{xy}^2/{4\tau_D}.$$

Anomalous diffusion
If the diffusing particles are hindered by obstacles, the dynamics may not be well described by normal diffusion, where the mean squared displacement (MSD) grows linearly with time. Instead the diffusion may be better described as anomalous diffusion, where the MSD grows in time as a power law:

$$\ MSD= 6 D_a t^\alpha $$

where $$D_a$$ is an anomalous diffusion coefficient (having units that depend on $$\alpha$$). "Anomalous diffusion" refers only to this one particular model, and not the many other possibilities that might colloquially be described as anomalous. Also, a power law is 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 seems to be a useful approximation for a range of systems.

The FCS autocorrelation function for anomalous diffusion is:

$$ G(\tau)=G(0)\frac{1}{(1+(\tau/\tau_{D})^\alpha)(1+a^{-2}(\tau/\tau_{D})^\alpha)^{1/2}} +G(\infty)$$,

where the anomalous exponent $$ \alpha $$ 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 (it is less than one and smaller for greater degrees of crowding).

Polydisperse diffusion
If there are diffusing particles with different sizes (diffusion coefficients), it is common to fit to a function that is the sum of single component forms:

$$\ G(\tau)=G(0)\sum_i \frac{\alpha_i}{(1+(\tau/\tau_{D,i}))(1+a^{-2}(\tau/\tau_{D,i}))^{1/2}} +G(\infty)$$

where the sum is over the number different sizes of particle, indexed by i, and $$\alpha_i$$ gives the weighting, which is related to the quantum yield and concentration of each type. This introduces new parameters, which makes the fitting more difficult as a higher dimensional space must be searched. Nonlinear least square fitting typically becomes unstable with even a small number of $$\tau_{D,i}$$s. A more robust fitting scheme, especially useful for polydisperse samples, is the Maximum Entropy Method.

Diffusion with flow
With diffusion together with a uniform flow with velocity $$v$$ in the lateral direction, the autocorrelation is : $$\ G(\tau)=G(0)\frac{1}{(1+(\tau/\tau_{D}))(1+a^{-2}(\tau/\tau_{D}))^{1/2}} \times \exp[-(\tau/\tau_v)^2 \times \frac{1}{1+\tau/\tau_D}] +G(\infty)$$

where $$\tau_v=\omega_{xy}/v$$ is the average residence time if there is only a flow (no diffusion).

Chemical relaxation
A wide range of possible FCS experiments involve chemical reactions that continually fluctuate from equilibrium because of thermal motions (and then "relax"). In contrast to diffusion, which is also a relaxation process, the fluctuations cause changes between states of different energies. One very simple system showing chemical relaxation would be a stationary binding site in the measurement volume, where particles only produce signal when bound (e.g. by FRET, or if the diffusion time is much faster than the sampling interval). In this case the autocorrelation is:

$$\ G(\tau)=G(0) \exp(-\tau/\tau_B) +G(\infty)$$

where

$$\ \tau_B=(k_{on}+k_{off} )^{-1}$$

is the relaxation time and depends on the reaction kinetics (on and off rates), and:

$$ G(0) = \frac{1}{\langle N\rangle} \frac{k_{on}}{k_{off}} = \frac{1}{\langle N\rangle} K$$

is related to the equilibrium constant K.

Most systems with chemical relaxation also show measureable diffusion as well, and the autocorrelation function will depend on the details of the system. If the diffusion and chemical reaction are decoupled, the combined autocorrelation is the product of the chemical and diffusive autocorrelations.

Triplet State Correction
The autocorrelations above assume that the fluctuations are not due to changes in the fluorescent properties of the particles. However, for the majority of (bio)organic fluorophores--e.g. green fluorescent protein, rhodamine, Cy3 and Alexa Fluor dyes--some fraction of illuminated particles are excited to a triplet state and then do not emit photons for a characteristic relaxation time $$\tau_F$$. Typically $$\tau_F$$ is on the order of microseconds, which is usually smaller than the dynamics of interest (e.g. $$\tau_D$$) but large enough to be measured. A multiplicative term is added to the autocorrelation account for the triplet state. For normal diffusion:

$$\ G(\tau)=G(0)\frac{1-F+Fe^{-\tau/\tau_F}}{1-F}\frac{1}{(1+(\tau/\tau_{D,i}))(1+a^{-2}(\tau/\tau_{D,i}))^{1/2}}+G(\infty)$$

where $$\ F$$ is the fraction of particles that have entered the triplet state and $$\ \tau_F$$ is the corresponding triplet state relaxation time. If the dynamics of interest are much slower than the triplet state relaxation, the short time component of the autocorrelation can simply be truncated and the triplet term is unnecessary.

Common fluorescent probes
The fluorescent species used in FCS is typically a biomolecule of interest that has been tagged with a fluorophore (using immunohistochemistry for instance), or is a naked fluorophore that is used to probe some environment of interest (e.g. the cytoskeleton of a cell). The following table gives diffusion coefficients of some common fluorophores in water at room temperature, and their excitation wavelengths.

Variations of FCS
FCS almost always refers to the single point, single channel, temporal autocorrelation measurement, although the term "fluorescence correlation spectroscopy" out of its historical scientific context implies no such restriction. FCS has been extended in a number of variations by different researchers, with each extension generating another name (usually an acronym).

Fluorescence Cross-Correlation Spectroscopy (FCCS)
FCS is sometimes used to study molecular interactions using differences in diffusion times (e.g. the product of an association reaction will be larger and thus have larger diffusion times than the reactants individually); however, FCS is relatively insensitive to molecular mass as can be seen from the following equation relating molecular mass to the diffusion time of globular particles (e.g. proteins):

$$\ \tau_D=\frac{3\pi\omega_{xy}^2\eta}{2kT}(M)^{1/3}$$

where $$\ \eta$$ is the viscosity of the sample and $$\ M$$ is the molecular mass of the fluorescent species. In practice, the diffusion times need to be sufficiently different--a factor of at least 1.6--which means the molecular masses must differ by a factor of 4. Dual color fluorescence cross-correlation spectroscopy (FCCS) measures interactions by cross-correlating two or more fluorescent channels (one channel for each reactant), which distinguishes interactions more sensitively than FCS, particularly when the mass change in the reaction is small.

FRET-FCS
Another FCS based approach to studying molecular interactions uses fluorescence resonance energy transfer (FRET) instead of fluorescence, and is called FRET-FCS. With FRET, there are two types of probes, as with FCCS; however, there is only one channel and light is only detected when the two probes are very close--close enough to ensure an interaction. The FRET signal is weaker than with fluorescence, but has the advantage that there is only signal during a reaction (aside from autofluorescence).

Image Correlation Spectroscopy (ICS)
When the motion is slow (e.g. diffusion in a membrane), getting adequate statistics from a single point FCS experiment may take a prohibitively long time. More data can be obtained by performing the experiment in multiple spatial points in parallel, using a laser scanning confocal microscope. This approach has been called Image Correlation Spectroscopy (ICS). The measurements can then be averaged together.

Another variation of ICS performs a spatial autocorrelation on images, which gives information about the concentration of particles. The correlation is then averaged in time.

A natural extension of the temporal and spatial correlation versions is spatio-temporal ICS (STICS). In STICS there is no explicit averaging in space or time (only the averaging inherent in correlation). In systems with non-isotropic motion (e.g. directed flow, asymmetric diffusion), STICS can extract the directional information. A variation that is closely related to STICS (by the Fourier transform) is k-space Image Correlation Spectroscopy (kICS).

There are cross-correlation versions of ICS as well.

Scanning FCS variations
Some variations of FCS are only applicable to serial scanning laser microscopes. Image Correlation Spectroscopy and its variations all were implemented on a scanning confocal or scanning two photon microscope, but transfer to other microscopes, like a spinning disk confocal microscope. Raster ICS (RICS), and position sensitive FCS (PSFCS) incorporate the time delay between parts of the image scan into the analysis. Also, low dimensional scans (e.g. a circular ring) --only possible on a scanning system--can access time scales between single point and full image measurements. Scanning path has also been made to adaptively follow particles.

Spinning disk FCS, and spatial mapping
Any of the image correlation spectroscopy methods can also be performed on a spinning disk confocal microscope, which in practice can obtain faster imaging speeds compared to a laser scanning confocal microscope. This approach has recently been applied to diffusion in a spatially varying complex environment, producing a pixel resolution map of diffusion coefficient. . The spatial mapping of diffusion with FCS has subsequently been extended to TIRF system. Spatial mapping of dynamics using correlation techniques had been applied before, but only at sparse points or at coarse resolution.

Total internal reflection FCS
Total internal reflection fluorescence (TIRF) is a microscopy approach that is only sensitive to a thin layer near the surface of a coverslip, which greatly minimizes background fluorscence. FCS has been extended to that type of microscope, and is called TIR-FCS. Because the fluorescence intensity in TIRF falls off exponentially with distance from the coverslip (instead of as a Gaussian with a confocal), the autocorrelation function is different.

Other fluorescent dynamical approaches
There are two main non-correlation alternatives to FCS that are widely used to study the dynamics of fluorescent species.

Fluorescence recovery after photobleaching (FRAP)
In FRAP, a region is briefly exposed to intense light, irrecoverably photobleaching fluorophores, and the fluorescence recovery due to diffusion of nearby (non-bleached) fluorophores is imaged. A primary advantage of FRAP over FCS is the ease of interpreting qualitative experiments common in cell biology. Differences between cell lines, or regions of a cell, or before and after application of drug, can often be characterized by simple inspection of movies. FCS experiments require a level of processing and are more sensitive to potentially confounding influences like: rotational diffusion, vibrations, photobleaching, dependence on illumination and fluorescence color, inadequate statistics, etc. It is much easier to change the measurement volume in FRAP, which allows greater control. In practice, the volumes are typically larger than in FCS. While FRAP experiments are typically more qualitative, some researchers are studying FRAP quantitatively and including binding dynamics. A disadvantage of FRAP in cell biology is the free radical perturbation of the cell caused by the photobleaching. It is also less versatile, as it cannot measure concentration or rotational diffusion, or co-localization. FRAP requires a significantly higher concentration of fluorophores than FCS.

Particle tracking
In particle tracking, the trajectories of a set of particles are measured, typically by applying particle tracking algorithms to movies. Particle tracking has the advantage that all the dynamical information is maintained in the measurement, unlike FCS where correlation averages the dynamics to a single smooth curve. The advantage is apparent in systems showing complex diffusion, where directly computing the mean squared displacement allows straightforward comparison to normal or power law diffusion. To apply particle tracking, the particles have to be distinguishable and thus at lower concentration than required of FCS. Also, particle tracking is more sensitive to noise, which can sometimes affect the results unpredictably.