FCSXpert Solutions: Fluorescence Correlation Spectroscopy Simplified!.
## FCS Classroom

#### What is Cross-correlation?

### Understanding Cross-correlation in Timeback to top

#### Cross-correlation of Intensity Signals

#### Using Cross-correlation to Measure Particle Interactions

### Expressing Cross-correlation as Useful Functionsback to top

#### Expressing the Cross-correlation Function, r(Δt)

#### Expressing the Cross-correlation Function, R(Δt)

### Sample Cross-correlation Databack to top

#### Fast Time-Decay, No Cross-correlation

#### Slow Time-Decay, No Cross-correlation

#### Slow Time-Decay, Cross-correlation

The cross-correlation function, R(Δt), calculated in Fluorescence Correlation Spectroscopy (FCS) measurements represents the correlation coefficient between two fluorescence signals at time t = 0, I(0), and all times, t, later, I(t). The cross-correlation function can be expressed as

where m is an integer multiple of a time interval, τ, such that
Δt = mτ (where 0 ≤ m < M). I_{1}(t) and I_{2}(t)
are the time-resolved fluorescence intensity curves from channels 1 and 2, respectively. Both intensity curves have M
+ 1 data points spanning from t = 0 to t
= Mτ. <I_{1}> and <I_{2}> are the mean intensities of the signals from channels 1 and 2, respectively.

These following sections explain the concept of cross-correlation:

In direct analogy with the way we described autocorrelation in Understanding Autocorrelation in Time, we may describe
cross-correlation. Here we read the fluorescence signals at two distinct wavelength bands (call the two intensities
I_{1} and I_{2}) and compare the two signals. Our goal is to determine whether the
two signals are correlated (they fluctuate in concert) or not correlated (they fluctuate independently).

In Understanding Autocorrelation in Time, we saw that the correlation of a signal with itself (autocorrelation) decays from perfect correlation at time zero to no correlation at infinite time. For the correlation between two signals (cross-correlation), this is only true if the fluctuations in the two signals are caused by the same source. If the two signals fluctuate independently, correlation between them is zero at all times.

The power of cross-correlation lies in its ability to detect molecular complexing. Cross-correlation extends the capabilities of standard autocorrelation FCS by introducing two different fluorescent probes (e.g. one oligonucleotide and one antibody; two oligonucleotides; or two antibodies) with distinct excitation and/or emission properties, which can be detected in the same confocal volume.

Cross-correlation temporally correlates the intensity fluctuations of the two distinguishable labels. The advantage of using cross-correlation is that both false-positives and false-negatives, which may occur in autocorrelation, are reduced or eliminated, while single particle detection limits are maintained. In cross-correlation mode, only pairs of coincident photon counts from two distinct channels will be recorded as a positive result.

At a simplistic level, cross-correlation analysis is coincidence analysis. However, simultaneous analysis of the autocorrelation functions in the two channels and the cross-correlation function between the channels enables determination of several mathematical properties that can further enable exclusion reduction of false positives and/or negatives.

We define the cross-correlation function, r(Δt), in a way that is analogous to the way we defined the autocorrelation function, g(Δt) in Expressing Autocorrelation as a Useful Function:

where m is an integer multiple of a time interval, τ, such that
Δt = mτ (where 0 ≤ m < M). I_{1}(t) and I_{2}(t)
are the time-resolved fluorescence intensity curves from channels 1 and 2, respectively. Both intensity curves have M
+ 1 data points spanning from t = 0 to t
= Mτ. <I_{1}> and <I_{2}> are the mean intensities of the signals from channels 1 and 2, respectively.

As with g(Δt) (see Expressing Autocorrelation as a Useful Function), calculating r(Δt) is made difficult because it requires maintaining a running measure of the mean intensities, <I_{1}> and <I_{2}>. As a result, it is more convenient to calculate R(Δt) as follows:

We note that R(Δt) can be either positive or negative. For instance in fluorescence resonance energy transfer (FRET), binding quenches donor fluorescence and enhances acceptor fluorescence; so the two signal intensities are negatively correlated.

In Figure 1, Panel A, we have two fluorescent tags, one shown as red star, the other as yellow. Each of these tags shows a rapidly- decaying autocorrelation function (shown to the right in corresponding colors). However, because the two tags move independently of one another, we see no cross-correlation (orange curve to the right).

In Figure 1, Panel B, both the red tag and the yellow tag have bound to their respective targets. Because the targets are significantly larger than the probes, the autocorrelation functions decay more slowly (curves shift to the right or longer time). Because the movements of the targets remain independent, no cross-correlation of the two signals observed.

In Figure 1, Panel C, the two tags bind to their respective and independent sites on the same target molecule or particle. Here the autocorrelation functions show similar decay to those in panel B, but the two signals are now highly cross-correlated because they move in concert.