Definitions

# Fluorescence correlation spectroscopy

Fluorescence correlation spectroscopy (FCS) is a common technique used by physicists, chemists, and biologists to experimentally characterize the dynamics of fluorescent species (e.g. single fluorescent dye molecules in nanostructured materials, autofluorescent proteins in living cells, etc.). Although the name indicates a specific link to fluorescence, the method is used today also exploring other forms of luminescence (like reflections, luminescence from gold-beads or quantum dots or phosphorescent species). The "spectroscopy" in the name is not readily found as in common usage a spectrum is generally understood to be a frequency spectrum. The autocorrelation is a genuine form of spectrum, however: It is the time-spectrum generated from the power spectrum (via inverse fourier transform).

Commonly, FCS is employed in the context of optical microscopy, in particular confocal or two photon microscopy. In these techniques light is focused on a sample and the measured fluorescence intensity fluctuations (due to diffusion, physical or chemical reactions, aggregation, etc.) 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 (or turn on and off in the 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 (incoherent) fluorescence.

When an appropriate model is known, FCS can be used to obtain quantitative information such as

• diffusion coefficients
• average concentrations
• kinetic chemical reaction rates
• singlet-triplet dynamics

Because fluorescent markers come in a variety of colors and can be specifically bound to a particular molecule (e.g. proteins, polymers, metal-complexes, etc.), it is possible to study the behavior of individual molecules (in rapid succession in composite solutions). With the development of sensitive detectors such as avalanche photodiodes the detection of the fluorescence signal coming from individual molecules in highly dilute samples has become practical. With this emerged the possibility to conduct FCS experiments in a wide variety of specimens, ranging from materials science to biology. The advent of engineered cells with genetically tagged proteins (like green fluorescent protein) has made FCS a common tool for studying molecular dynamics in living cells.

## History

Signal-correlation techniques have first been 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. 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 (wavelengths ranging typically from 405 - 633 nm (cw), and from 690 - 1100 nm (pulsed)), which is reflected into a microscope objective by a dichroic mirror. The laser beam is focused in the sample, which contains fluorescent particles (molecules) in such high dilution, that only few are within the focal spot (usually 1 - 100 molecules in one fL). 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 reaching a detector, typically a photomultiplier tube or avalanche photodiode 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 (which requires special acquisition cards). 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. The setup is shown in Figure 1.

## The Measurement Volume

The measurement volume is a convolution of illumination (excitation) and detection geometries, which result from the optical elements involved. The resulting volume is described mathematically by the point spread function (or PSF), it is essentially the image of a point source. The PSF is often described as an ellipsoid (with unsharp boundaries) of few hundred nanometers in focus diameter, and almost one micrometre along the optical axis. The shape varies significantly (and has a large impact on the resulting FCS curves) depending on the quality of the optical elements (it is crucial to avoid astigmatism and to check the real shape of the PSF on the instrument). In the case of confocal microscopy, and for small pinholes (around one Airy unit), the PSF is well approximated by Gaussians:

$PSF\left(r,z\right)= I_0 e^\left\{- 2 r^2/omega_\left\{xy\right\}^2\right\} e^\left\{-2 z^2/omega_\left\{z\right\}^2\right\}$
where $I_0$ is the peak intensity, r and z are radial and axial position, and $omega_\left\{xy\right\}$ and $omega_\left\{z\right\}$ are the radial and axial radii, and $omega_\left\{z\right\} > omega_\left\{xy\right\}$. This Gaussian form is assumed in deriving the functional form of the autocorrelation.

Typically $omega_\left\{xy\right\}$ is 200-300 nm, and $omega_\left\{z\right\}$ 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 $tau$:
$G\left(tau\right)=frac\left\{langledelta I\left(t\right)delta I\left(t+tau\right)rangle\right\}\left\{langle I\left(t\right) rangle^2\right\}=frac\left\{langle I\left(t\right)I\left(t+tau\right) rangle\right\}\left\{langle I\left(t\right)rangle^2\right\}-1$

where $delta I\left(t\right)=I\left(t\right)-langle I\left(t\right)rangle$ 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 can be fitted, typically using a nonlinear least squares algorithm. The fit's functional form depends on the type of dynamics (and the optical geometry in question).

### 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:

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

where $a=omega_z/omega_\left\{xy\right\}$ is the ratio of axial to radial $e^\left\{-2\right\}$ radii of the measurement volume, and $tau_\left\{D\right\}$ is the characteristic residence time. This form was derived assuming a Gaussian measurement volume. Typically, the fit would have three free parameters--G(0), $G\left(infty\right)$, 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:

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

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

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

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

### Anomalous diffusion

If the diffusing particles are hindered by obstacles or pushed by a force (molecular motors, flow, etc.) the dynamics is often not sufficiently well-described by the normal diffusion model, where the mean squared displacement (MSD) grows linearly with time. Instead the diffusion may be better described as anomalous diffusion, where the temporal dependenc of the MSD is non-linear as in the power-law:

$MSD= 6 D_a t^alpha$

where $D_a$ 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:

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

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:

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

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 $tau_\left\{D,i\right\}$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: $G\left(tau\right)=G\left(0\right)frac\left\{1\right\}\left\{\left(1+\left(tau/tau_\left\{D\right\}\right)\right)\left(1+a^\left\{-2\right\}\left(tau/tau_\left\{D\right\}\right)\right)^\left\{1/2\right\}\right\} times exp\left[-\left(tau/tau_v\right)^2 times frac\left\{1\right\}\left\{1+tau/tau_D\right\}\right] +G\left(infty\right)$

where $tau_v=omega_\left\{xy\right\}/v$ 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:

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

where

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

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

$G\left(0\right) = frac\left\{1\right\}\left\{langle Nrangle\right\} frac\left\{k_\left\{on\right\}\right\}\left\{k_\left\{off\right\}\right\} = frac\left\{1\right\}\left\{langle Nrangle\right\} K$

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 (or other non-radiative decaying states) 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:

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

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.
Fluorescent dye $D$ (x10-10 m2 s-1) Excitation wavelength (nm) Reference
Rhodamine 6G 2.8, 3.0 514 ,
Rhodamine 110 2.7 488
Tetramethyl rhodamine 2.6 543
Cy3 2.8 543
Cy5 2.5 633
carboxyfluorescein 3.2 488
Alexa-488 1.96 488

## 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):

$tau_D=frac\left\{3piomega_\left\{xy\right\}^2eta\right\}\left\{2kT\right\}\left(M\right)^\left\{1/3\right\}$

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 (in biology, for example, 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.