The autocorrelation function, G(Δt), calculated in Fluorescence Correlation Spectroscopy (FCS) measurements represents the correlation coefficient between the intensity at time t = 0, I(0), and the intensity at all times, t, later, I(t). The autocorrelation 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(t) is the time-resolved fluorescence intensity with M + 1 data points spanning from t = 0 to t = Mτ. <I> is the mean intensity over all values of t.
These following sections explain the concept of autocorrelation:
FCS measures fluctuations of the fluorescence signal intensity about the mean in a small detection volume (See What is the Confocal Volume?). Suppose that we take fluorescence intensity measurements over 100 ns intervals (100 ns is referred to as the counting interval) and thus generate a stream of data as a function of time. Figure 1 shows what such data might look like.
Suppose we were to ask whether the data stream in Figure 1 is correlated with itself. This might seem a foolish question, but let's ask it nonetheless. Following what we did in the example in What is Correlation?, we plot, in Figure 2, the intensity against itself and then calculate the correlation coefficient R, which we are not surprised to find, is 1.00. There is perfect correlation.
If we examine the data in Figure 1 closely, we see something interesting. The fluctuations do not appear perfectly random. The widths of the peaks and valleys appear to favor a characteristic time scale. The amplitude of the intensity fluctuations may be randomly distributed, but the distribution of times over which they occur is not perfectly random.
The characteristic fluctuation time scale raises an interesting question. Suppose that we were to plot each intensity value, not against itself, but against the intensity value one counting interval later. This is shown in Figure 3.
We see that the data are still correlated, but that the R value has decreased to R = 0.966. We can, of course continue this process.
Figure 4 shows the correlation shifted by two counting intervals, that is by 200 ns, Figure 5 shows the correlation when the intensities are shifted by 8 counting intervals or 800 ns, and Figure 6 shows the correlation when the intensities are shifted by 16 counting intervals or 1600 ns.
Progressively as the shift in time increases, the degree of correlation decreases. This is further illustrated in Figure 7 where we plot R as a function of the shift in time. Defined in this way, R becomes explicitly a function of Δt and R(Δt) represents a probability.
R(Δt) represents the probability that the intensity stream will still be on the same rising or falling fluctuation some time, Δt, later. Another way to look at it is that the system has memory of where it has been, and the correlation coefficient, R(Δt), expresses how long this memory lasts.
In the section General Intensity Autocorrelation of Mathematics of Autocorrelation, we derive the FCS autocorrelation equation, G(Δt). Below is a summary of the conversion of the R(Δt) probabilities shown above to the function G(Δt).
The formula for calculating the correlation coefficient between two parameters is as follows
Above, we concluded that R(Δt) represents the probability that the intensity stream will still be on the same rising or falling fluctuation some time, Δt, later. Let us formally express R(Δt) in terms of the intensities. Let us assume, as in the example above, that Δt is an integer multiple, m, of some counting interval, τ. In this case, xi becomes I(iτ) and yi becomes I(iτ + mτ), such that
where Δt = mτ (0 ≤ m < M) and I(t) is the time-resolved fluorescence intensity with M + 1 data points spanning from t = 0 to t = Mτ.
R(Δt) is an autocorrelation function. It expresses the correlation between the fluctuation from the mean intensity at time 0 with the fluctuation from the mean intensity at later times. It has a characteristic constant, τD, referred to as the correlation time, which is determined by the underlying physics, chemistry, and biology that are causing the intensity fluctuations.
The normalization of dividing by the standard deviation squared turns R(Δt) into a probability that goes from one to zero as shown in the example above. As it turns out, this choice of normalization results in a loss of some very useful information from the autocorrelation function, namely the number of fluorescent species detected. (In FCS, this represents the number of molecules in the confocal volume.) To retain this information, we normalize, instead, by dividing by the square of the mean intensity. We shall refer to this function as g(Δt), expressed as follows:
Above, we expressed the autocorrelation, g(Δt), which represents the correlation coefficient between the fluctuation from the mean intensity at time t = 0 and the fluctuation from the mean intensity at some time later.
Computationally, calculating g(Δt) is made difficult because it requires maintaining a running measure of the mean intensity, <I>. As a result, it is more convenient to calculate G(Δt), which represents the correlation between the intensity at time t = 0, I(0), and the intensity at some time later, I(t):
Figure 8 illustrates what a correlation function described by the G(Δt) equation might look like. In most modern FCS systems, data curves are shown on a logarithmic time scale because the measurements are made over seven or more decades of time resolution.