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Fluorescence Correlation Spectroscopy (FCS) probes molecular diffusion and interaction. It works best with nanomolar concentrations of protein. Typical experiments involve sample volumes as low as a few microliters. The measurements can be performed in solution and living cells. FCS is a statistical method. It is based on the analysis of fluorescence fluctuations. Typically, they originate form Brownian motion of dye labeled molecules through a small laser spot. The mean time these molecules stay within the laser spot depends on their size. If a small, dye tagged molecule binds to a larger one, it slows down and emits photons for a longer time during its diffusion through the laser spot. A sensitive detector records single photons emitted by the dye molecules. The result is an intensity trace representing random noise. Correlation functions are used to extract information about diffusion times (= size) and number of molecules (=concentration) in the sample. Physical models describe the source of the fluctuations and are fitted to the correlated data to quantify these information. Beyond Brownian motion, FCS can analyze other sources of fluorescence fluctuations, including electronic properties of dyes (e.g. triplet states), restricted diffusion, active transport and changes in FRET signals due to conformational changes of molecules. If only one type of fluorescent dye and one detector are used, the method is called autocorrelation. To distinguish between two different species of molecules or a small molecule bound to a larger one a difference of mass of at least 1.4 is required. To increase the flexibility of the method, two dyes and detectors can be used. This method is called crosscorrelation. Other common methods to use fluorescence fluctuations to probe molecular interactions include Photon Counting Histograms (PCH) and Fluorescence Intensity Distribution Analysis (FIDA). Coincidence Analysis is used to probe rare events in the femtomolar range. At the Stowers Institute we are using a ConfoCor 3 manufactured by Carl Zeiss. It is attached to a confocal microscope LSM 510 META NLO. 
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Free DiffusionThe movement of small particles in solution is a random process described as Brownian Motion. Whenever a molecule tagged with a fluorescent dye diffuses through a small laser spot it will emit a burst of photons. The probability C of finding a particle at position r, at a time t, when the particle was at the origin r = 0 at time t = 0 is D is the diffusion coefficient. This is a solution of Fick's second law of diffusion: The Einstein relation links D to the absolute Temperature T, the viscosity of the medium η and the hydrodynamic radius r of the molecule. k = 1.38 x 10^{23} J/K is the Boltzman constant.
The viscosity of liquids is temperature dependent:
For globular molecules the hydrodynamic radius r can be estimated as with molecular weight m and Avogadro's number N_{A} = 6.023x 10^{23} mol^{1}. The mean density ρ of molecules can be estimated for most applications with^{1}
Literature values for the diffusion constant of some molecules in water are:
For rodlike molecules the diffusion coefficient D is given by L is the length of the molecule (e.g. 3.8 Å per DNA nucleotide) and d it's diameter (23.8 Å for double stranded DNA). A is a correction factor given as 
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The smaller the volume the larger the correlation amplitudeFluorescence Correlation Spectroscopy (FCS) is based on the measurement of fluctuations. The fluctuation signal is higher with lower number of molecules. To measure higher concentrations a measurement volume as small as possible is created. There are about 6 molecules within 1 femtoliter (10^{15} liter) of a 10 nanomolar (10 x 10^{9} Mol) solution. Confocal VolumeMost FCS setups use confocal optics to create a measurement volume. It's size matches physiological concentrations very well. It is easy to place within structures like cells. A laser beam is focused to a diffraction limited spot, defining the xy size of the volume in the focal plane. A confocal pinhole limits the detection of photons to light originating from the focal plane. The optical setup is identical to a confocal Laser Scanning Microscope (LSM). Scanning mirrors are not required. Sometimes they are added for scanning FCS and easier positioning of the measurement spot within cells. The diameter (FWHM = Full Width Half Maximum) of the diffraction limited excitation spot depends on the wavelength λ, the numerical aperture NA of the lens used and how much of the Gausian shaped laser beam fits into the back aperture of the objective (T = Truncation factor): There is only a small dependence of the xy dimensions on the diameter of the pinhole. The pinhole diameter defines mainly the extend of the measurement volume along the optical axis (z axis): MultiPhoton ExcitationTwo or Multiphoton excitation is an other way to define a measurement volume. Two (or more) photons are required to reach the dye molecule at the same time (within about 10^{15} seconds). This happens only at the focal spot, thus no outoffocus excitation occurs. A pinhole is not required. The setup is similar to a twophoton microscope. Other SetupsAny arrangement which creates a small enough measurement volume can be used for FCS. The original setup^{1} used a small cell to limit the measurement volume along the optical axis. Other setups use Total Internal Reflection (TIRF) to create a small volume. 
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While passing the measurement volume, the chromophore attached to the protein of interest emits very view photons. In solution about 5 of them are detected. The number of photons depend on the quantum efficiency of the chromophore, the size of the measurement volume and the diffusion constant of the protein  dye complex. To detect the photons very sensitive photon counting detectors are required. Typically Avalanche Photo Detectors (APD) are used. Adding all pulses created by photons in one time window (binning) gives traces of random noise, representing the fluorescence fluctuations within the measurement volume. You can use our FCS data viewer and view raw data acquired with a Carl Zeiss ConfoCor 2 system to see the single photon events detected during one experiment. 
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To determine the hidden time structure of the fluctuation signal the autocorrelation function of the measured data is calculated. The autocorrelation function transforms the data from the measured time domain (=how long it took to acquire the data) to the correlation time domain (=how fast the fluctuations are, e.g. how long a molecule stayed in the confocal volume). It extracts the average behavior of ensembles from the random behavior of individual molecules. In essence, the correlation function is a "memory" function that measures how long a signal stays the same over time. To calculate the autocorrelation function, one compares the measured data with a timeshifted version (the lag time τ) of itself. If there is no timeshift, both data traces are identical  the correlation is high. If the shift is large, the two traces are very different  the correlation is low (this is true as long as the signal has no periodicity). Mathematically this comparison is done by integration over the measurement time t from start of the experiment (t = 0) to it's end (t = T, T = total measurement time). The result is the autocorrelation function for one given lag time τ: The second term is an alternative spelling of the integral often used in FCS literature. The measured intensity can be described as a constant term plus fluctuations, resulting in an alternative form of the autocorrelation function: It is common to normalize the autocorrelation function: The information about the measured fluctuations is contained in δI(t). The constant term I reduces the signal. Thus it is important to increase δI(t) (e.g. reducing measurement volume) and decrease I (e.g. reducing background fluorescence). The term "+1" is sometimes omitted. 
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There are typically two basic pieces of information that can be gleaned from the correlation function: the average time it takes for a molecule to move through the focal volume, and the number of particles in the focal volume. For simple diffusion in two and three dimensions, this relationship is given as follows: Here τ_{d} is the average time required for a molecule to traverse the radial dimension of the focal volume, γ is a shape factor for the focal volume, N is the average number of molecules in the focus, and r is the ratio between the axial and radial dimensions of the focal volume. Often the ratio γ/N is referred to as G(0) because it describes the amplitude of the correlation function at zero time lag. The diffusion time is directly related to the radial dimension of the focal volume as follows: Here ω_{0} is referred to as the "beam waist" of the focal volume. It is related to the full width half maximum (FWHM) as follows: This analysis assumes a 3D Gaussian laser focus. This is a good approximation only for a small pinhole (see above). Mathematically this focal shape is given as follows: Please try our FCS simulator (link in right margin) to explore how this method works in more detail.

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Given the simple relationship between the correlation function amplitude and the average number of particles, it seems straightforward to immediately calculate the particle concentration. All that is missing is the volume of the focus, right? Unfortunately, the answer is not that simple. Theoretically, the volume of the focus is given by the integral of the 3D Gaussian laser focus as follows: Here we already run into problems. In principle, the z_{0} parameter in the above equation can be determined from the axial/radial ratio defined above, but in practice, this measurement is only weakly sensitive to the actual axial dimension of the focus. In fact, recent papers (Hess and Webb) recommend fixing this parameter to a value of 5 in the analysis. The rational for this lies in the poor z resolution inherent to optical measurement techniques. During the fluctuation measurement, the changes in intensity as the molecule moves radially are much greater than those accompanying axial motion. As a result, one can essentially omit information about the third dimension from FCS analysis and still get reasonable results. Further complications are given by the nature of the correlation amplitude measurement itself. While FCS is uniquely sensitive to the fluctuations of single molecules, its amplitude is notoriously dependent on background fluorescence intensity. In fact, every confocal measurement has significant amounts of out of focus light that contaminate every measurement. Since this light spreads from the focal volume in a diffuse way, it contains essentially no fluctuations and as a result is similar to background. This significantly reduces the apparent correlation amplitude and results in an overestimation of the concentration observed or conversely an underestimation of the apparent focal volume. The only way to accurately calibrate the concentration is to perform a titration from a known dye concentration. We typically use fluorescein in 0.1 M NaOH which has an extinction coefficient of 75,000 M^{1} cm^{1}. The concentration can be accurately measured using a UV/Vis spectrophotometer in the uM concentration range. Subsequent careful dilutions allow for calibration of the confocal volume for FCS measurements. In our hands this method yeilds a focal volume approximately 2.3 times larger than that obtained using the measured beam waist and an axial to radial ratio of 5. 
Stowers Links: Web Links: Literature: Unruh, J. R., G. Gokulrangan, G. S. Wilson, and C. K. Johnson (2005). "Fluorescence Properties of Fluorescein, Tetramethylrhodamine and Texas Red Linked to a DNA Aptamer." Photochem. Photobiol. 81: 682–690. Rüttinger, S., V. Buschmann, B. Krämer, R. Erdmann, R. MacDonald, and F. Koberling (2008). "Comparison and accuracy of methods to determine the confocal volume for quantitative fluorescence correlation spectroscopy." J. Microscopy. 232(2): 343–352. 



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The γ factor has been the source of great confusion for those learning the fundamentals of FCS. Perhaps part of this confusion has been generated by the fact that many publications omit the γ factor all together. This omission is justified by a change in definition of focal volume as follows: Given our previous discussion about the tenuous relationship between the focal volume z dimension and the calculation of concentration, this omission seems quite reasonable. Nevertheless, it is wise to retain the γ factor so that the apparent number of particles is consistant with the number measured from intensity distribution methods like PCH or FIDA. Given our need for the γ factor, it is wise to define it: This definition seems rather obtuse, but we can wrap our heads around it with a simple thought experiment. Consider a square focal volume. Inside the focus, the intensity is 1 and outside it is 0. If we square the intensity of this focal volume, it is unchanged because the 1^{2} = 1. Therefore the γ factor is 1. Now we simply slope the edges of the focal volume a bit. If we square this focal volume, its size changes. For example, in the region where the intensity is 0.5, the squared intensity is 0.25. As a result the squared focal volume is smaller than the focal volume itself and the γ factor is less than 1. Therefore the γ factor is a measure of the sharpness of the edges of the focal volume. For one photon excitation, the γ factor is 0.3536, while for two photon excitation it turns out to be 0.076. This is due to the fact that the two photon focal volume decays to zero very slowly compared to the one photon focal volume. 
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While most FCS studies are focused on the mobility or concentration of the molecules involved, it is important to note that there are fluctuations other than molecular motion to contend with. The most commonly observed of these is triplet behavior. Fluorescence microscopy and spectroscopy typically probe the singlet excited state of the molecule, but a small fraction of the molecules also enter the triplet state. This state is lower in energy than the singlet excited state and transitions to that state have low probability. The triplet state can lead to the emission of phosphorescence rather than fluorescence. Phosphorescence has lower energy than fluorescence (different wavelength) and is emitted over microseconds rather than nanoseconds like fluorescence. This longer lifetime is due to the low probability of transitions to and from the triplet state. In FCS measurements, the triplet state simply appears as a "dark" state that lasts for a few microseconds. This is typically much faster than the diffusion time of molecules and gives rise to a hump at short time scales in the FCS curve. This hump is fit well by an exponential function as follows: The population of the triplet state is dependent on laser power. Triplet behavior is generally avoided by using low laser powers. There are several reasons for this. Firstly, in the absence of triplet dynamics, the analysis of FCS curves is much simpler. Secondly, the power dependence of triplet behavior leads to saturation of the focal volume since molecules in the center of the focus are more likely to enter the triplet state than those at the edges. This leads to distortion of the focal volume and nonstandard diffusion behavior. Finally, the triplet state is often a pathway to photobleaching. Photobleaching causes slow decreases in the fluorescence signal leading to nonstandard correlation curves. In addition, photobleaching can lead to a shortening of the apparent diffusion time because the molecules dissappear before they have traversed the focal volume. Despite attempts to avoid triplet behavior, some molecules like fluorescent proteins have such dominant triplet (or tripletlike) behavior that it is impossible to eliminate. Therefore, it is important that fcs analysis software incorporate the ability to fit triplet dynamics for appropriate analysis of FCS data. 
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Often, molecules exist in more than one mobility state. This is especially true of proteins in vivo. All proteins at some point in the cell cycle have a fraction associated with recycling vessicles. Therefore, they show both a fast and a slow mobility component. In addition, many proteins interact with complexes of other proteins and cytoskeletal elements resulting in low mobility. There are two general classes of heterogeneous diffusion that is observed. The first is the case where all mobility components have the same molecular brightness and therefore the same oligomerization state. Under those circumstances the correlation functions simply add together as follows: The second case is where different mobility components have different molecular brightness (oligomerization state). Many proteins exhibit oligomerization in complexes, though this process is poorly understood because of the limited nature of biochemical oligomerization measurements. Under these circumstances, the amplitudes of the different correlation components are no longer additive. They depend both on the molecular brightness of the particles and the average number of each type of particle. The amplitude of each correlation curve is then given by: Here i indexes the species of interest, f is the fractional intensity, and ε is the molecular brightness. It is easy to see from this expression that it is impossible to accurately calculate the number of molecules if the molecular brightnesses of each species are not known before hand. The section on brightness methods contains more information on finding those brightnesses. 
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The best focal volume shape for Two Photon FCS is a gaussianlorentzian squared. This is important for intensity distribution methods, but for FCS measurements we can get by simply squaring the focal volume shape as follows: It is not necessary to change the form of the correlation function. We must simply change the definition of the diffusion time: In addition, the gamma factor changes from 0.3536 to 0.076 as mentioned above. The result is that the amplitude of two photon correlation functions is significantly less than that of one photon correlation functions. 
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One of the most promising advances of fluorescence correlation spectroscopy has been the development of fluorescence cross correlation spectroscopy (FCCS). In FCCS, two detectors are used each probing a different color. Typical experiments use red and green colors, so I will use subscripts g and r to denote the two channels. Under these circumstances, the cross correlation function is measured along with the green and red autocorrelation functions. The normalized cross correlation function is calculated as follows: Here I have omitted the "+1" as discussed earlier. The cross correlation function is a measure of how synchronized the fluctuations in intensity are between the green and red channels. Typically, the red and green species are diffusing at approximately the same rate, so the time dependence of the cross correlation function is the same as that of the aucorrelation functions. The amplitude of the cross correlation function, however, is dependent on the coincidence of the two channels. For each molecule, the cross correlation amplitude is dependent on its brightness in both channels as follows: With appropriately chosen filters and fluorophores, this function is close to zero when green and red particles are not interacting but much larger when interactions are present. As a result, this becomes a method to determine molecular codiffusion and therefore heterointeractions. The relationship between the correlation amplitudes and the amount of interaction present in a sample can be complex. Typically, one measures five things related to molecular brightness and interaction stoichiometry in an FCCS experiment: green G(0), red G(0), crosscorrelation G(0), green intensity, and red intensity. As a result, five characteristics of the system can be pulled out of such a measurement. Three of these characteristics are typically the numbers of green unbound species, red unbound species, and interacting species. The other characteristics can either be stoichiometries or the brightnesses of the unbound species. We have devised several online calculators (link in right margin) to allow for calculation of these parameters. Please try our dual color FCS simulator (link in right margin) to explore how this method works in more detail. 
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