# Fluorescence cross-correlation spectroscopy

## Introduction

Fluorescence cross-correlation spectroscopy (FCCS) was introduced by Eigen and Rigler in 1994 and extends the fluorescence correlation spectroscopy (FCS) procedure by introducing high sensitivity for distinguishing fluorescent particles which have a similar diffusion coefficient. FCCS uses two species which are independently labelled with two spectrally separated fluorescent probes. These fluorescent probes are excited and detected by two different laser light sources and detectors commonly known as green and red respectively. Both laser light beams are focused into the sample and tuned so that they overlap to form a superimposed confocal observation volume.

The normalized cross-correlation function is defined for two fluorescent species ${\displaystyle \ G}$ and ${\displaystyle \ R}$ which are independent green, G and red, R channels as follows:

${\displaystyle \ G_{GR}(\tau )=1+{\frac {<\delta I_{G}(t)\delta I_{R}(t+\tau )>}{}}}$

where differential fluorescent signals ${\displaystyle \ \delta I_{G}}$ at a specific time, ${\displaystyle \ t}$ and ${\displaystyle \ \delta I_{R}}$ at a delay time, ${\displaystyle \ \tau }$ later is correlated with each other.

## Modeling

Cross-correlation curves are modeled according to a slightly more complicated mathematical function than applied in FCS. First of all, the effective superimposed observation volume in which the G and R channels form a single observation volume, ${\displaystyle \ V_{eff,RG}}$ in the solution:

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

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

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

${\displaystyle \ \tau _{D,GR}={\frac {\omega _{xy,G}^{2}+\omega _{xy,R}^{2}}{8D_{GR}}}}$

where ${\displaystyle \ D_{GR}}$ 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:

${\displaystyle \ G_{G}(\tau )=1+{\frac {(Diff_{k}(\tau )+Diff_{k}(\tau ))}{V_{eff,GR}(+)^{2}}}}$

${\displaystyle \ G_{R}(\tau )=1+{\frac {(Diff_{k}(\tau )+Diff_{k}(\tau ))}{V_{eff,GR}(+)^{2}}}}$

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

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

with ${\displaystyle \ Diff_{k}(\tau )={\frac {1}{(1+{\frac {\tau }{\tau _{D,i}}})(1+a^{-2}({\frac {\tau }{\tau _{D,i}}})^{1/2}}}}$

Contrary to FCS, the intercept of the cross-correlation curve does not yield information about the doubly labelled fluorescent particles in solution.