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Fluorescence cross-correlation spectroscopy (FCCS) is a spectroscopic technique that examines the interactions of fluorescent particles of different colours as they randomly diffuse through a microscopic detection volume over time, under steady conditions.[1] Eigen and Rigler first introduced fluorescence cross-correlation spectroscopy (FCCS) in 1994, and then it was experimentally implemented by Schwille in 1997.[2,3] FCCS is a technique that extends the fluorescence correlation spectroscopy (FCS) method by using two differently coloured molecules instead of one. Essentially, FCCS measures the coincident green and red intensity fluctuations of distinct molecules, which correlate if green and red labeled particles move together through a predefined confocal volume. As a result, FCCS provides a highly sensitive measurement of molecular interactions independent of diffusion rate. This is an important advancement, given that diffusion rate of a molecular complex is weakly dependent on its size.[2] FCCS utilizes two species which are independently labeled with two different fluorescent probes in colour. These fluorescent probes are excited and detected by two different laser light sources and detectors usually labeled as "green" and "red". Typically a microscope is used to provide overlapping green and red focal volumes for excitation. By combining FCCS with a confocal microscope, the technique's capabilities are highlighted, as it becomes possible to detect fluorescence molecules in femtoliter volumes within the nanomolar range, with a high signal-to-noise ratio, and at a microsecond time scale and more importantly to provide overlapping green and red focal volumes for excitation.[4]

The normalized cross-correlation function is defined for two fluorescent species, G and R, which are independent green and red channels, respectively:

$$\ G_{GR}(\tau)=1+\frac{\langle \delta I_G(t)\delta I_R(t+\tau)\rangle }{\langle I_G(t)\rangle\langle I_R(t)\rangle}=\frac{\langle I_G(t)I_R(t+\tau)\rangle}{\langle I_G(t)\rangle \langle I_R(t)\rangle}$$

where differential fluorescent signals $$\ \delta I_G$$ at a specific time,$$\ t$$ and $$\ \delta I_R$$ at a delay time, $$\ \tau$$ later is correlated with each other. In the absence of spectral bleed-through -when the fluorescence signal from an adjacent channel is visible in the channel being observed-, the cross-correlation function is zero for non-interacting particles. In contrast to FCS, the cross-correlation function increases with increasing numbers of interacting particles.

FCCS is mainly used to study bio-molecular interactions both in living cells and in vitro. It allows for measuring simple molecular stoichiometries and binding constants. It is one of the few techniques that can provide information about protein-protein interactions at a specific time and location within a living cell. Unlike fluorescence resonance energy transfer, FCCS does not have a distance limit for interactions making it suitable for probing large complexes. However, FCCS requires active diffusion of the complexes through the microscope focus on a relatively short time scale, typically seconds.

Modeling
The mathematical function used to model cross-correlation curves in FCCS is slightly more complex compared to that used in FCS. one of the primary differences is the effective superimposed observation volume, denoted as $$\ V_{eff, RG}$$ in which the G and R channels form a single observation volume:

$$\ V_{eff, RG}=\pi^{3/2}(\omega_{xy,G}^2+\omega_{xy,R}^2)(\omega_{z,G}^2+\omega_{z,R}^2)^{1/2}/2^{3/2}$$

where $$\ \omega_{xy,G}^2$$ and$$\ \omega_{xy,R}^2$$ are radial parameters and $$\ \omega_{z,G}$$ and$$\ \omega_{z,R}$$ are the axial parameters for the G and R channels respectively.

The diffusion time, $$\ \tau_{D,GR}$$ for a doubly (G and R) fluorescent species is therefore described as follows:

$$\ \tau_{D,GR}=\frac{\omega_{xy,G}^2+\omega_{xy,R}^2}{8D_{GR}}$$

where $$\ D_{GR}$$ is the diffusion coefficient of the doubly fluorescent particle.

The cross-correlation curve generated from diffusing doubly labelled fluorescent particles can be modelled in separate channels as follows:

$$\ G_G(\tau)=1+\frac{(Diff_k(\tau)+Diff_k(\tau))}{V_{eff, GR}(+)^2}$$

$$\ G_R(\tau)=1+\frac{(Diff_k(\tau)+Diff_k(\tau))}{V_{eff, GR}(+)^2}$$

In the ideal case, the cross-correlation function is proportional to the concentration of the doubly labeled fluorescent complex:

$$\ G_{GR}(\tau)=1+\frac{Diff_{GR}(\tau)}{V_{eff}(+)(+)}$$

with $$\ Diff_k(\tau)=\frac{1}{(1+\frac{\tau}{\tau_{D,i}})(1+a^{-2}(\frac{\tau}{\tau_{D,i}})^{1/2}}$$

The cross-correlation amplitude is directly proportional to the concentration of double-labeled (red and green) species [5].