Understanding Cross-correlation in Timeback to top
Cross-correlation of Intensity Signals
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 I1 and I2 ) 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.
Using Cross-correlation to Measure Particle Interactions
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.
Expressing Cross-correlation as Useful Functionsback to
Expressing the Cross-correlation Function, r(Δt)
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 ). I1 (t) and I2 (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. <I1 > and <I2 > are the mean intensities of the signals from channels 1 and 2, respectively.
Expressing the Cross-correlation Function, R(Δt)
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, <I1 > and <I2 >. 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.
Sample Cross-correlation Databack to top
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 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.