Principles of Fluorescence Spectroscopy

800 FLUORESCENCE CORRELATION SPECTROSCOPY
24.2. THEORY OF FCS
The equations for FCS are moderately complex, and
there are many subtle points that result from the unique
aspects of the measurements. Different authors use slightly
different terminology for FCS. Hence it is valuable to
review the theory to provide a basis for understanding
the data and the factors that influence the measured correlation
functions.
The autocorrelation function of the fluorescence intensity
is given by the product of the intensity at time t, F(t),
with the intensity after a delay time τ, F(t + τ), averaged
over a large number of measurements. 17–21 The time t refers
to the actual time as the intensities are observed. Data collection
times range from seconds to minutes. The delay
time τ is the difference in real time between measurement
of F(t) and F(t+ τ), typically in the range from 10 –2 to 10 2
ms. If the intensity fluctuations are slow compared to τ,
then F(t) and F(t+ τ) will be similar in magnitude. That is,
if F(t) is larger than the average intensity , then F(t + τ)
is likely to be larger than . If the intensity fluctuations
are fast relative to τ, then the value of F(t) and F(t + τ) will
not be related. The value of the autocorrelation for a time
delay τ is given by the average value of the products
R(τ)
1
T T
0 F(t)F(t τ)dt
(24.2)
where T is the data accumulation time. The factor 1/T normalizes
the value of the products F(t)F(t + τ) for the data
accumulation time. This mathematical process is typically
performed using dedicated correlation boards that perform
the needed operations in real time as the fluctuating intensity
is observed. With modern electronics it is now also possible
to record the time (t)-dependent intensities for the
duration of the data collection (T) and then perform the calculations.
Additional information can be obtained from the
complete list of time-dependent intensities, so it seems likely
that this approach will eventually be standard in FCS
experiments.
The quantities of interest are the fluctuations in F(t)
around the mean value
δF(t) F(t)
(24.3)
The autocorrelation function for the fluorescence intensities,
normalized by average intensity squared, is given by
G’(τ)
1
2
(24.4)
In this expression we have replaced t with 0. The delay time
τ is always relative to a datapoint at an earlier time so that
only the difference τ is relevant. It is important to remember
the meaning of t and τ in FCS, which is different from
their meaning in time-resolved fluorescence. In timeresolved
fluorescence t refers to the time after the excitation
pulse and τ refers to the decay time. In FCS t refers to real
time and τ refers to a time difference between two intensity
measurements. In the FCS literature there are occasional
inaccuracies in the words used to describe the correlation
functions in eqs. 24.2 and 24.4. Equation 24.2 is the autocorrelation
function of the fluorescence intensities. The
term δF(t) in eq. 24.3 is the variance, so that the last term
on the right side of eq. 24.4 is actually the autocovariance
of F(t), but is conventionally referred to as the "autocorrelation
function." A more accurate term is "the autocorrelation
function of the fluorescence fluctuations." 17 Another
confusing point is the definition of the autocorrelation function.
Some authors use eq. 24.4 and other authors use
G(τ)
2
(24.5)
which is the autocorrelation function of fluorescence fluctuations.
We will use eq. 24.5 since this avoids inclusion of
one (1) in all the subsequent equations. For simplicity we
will refer to eq. 24.5 as the correlation or autocorrelation
function.
In order to interpret the FCS data we need a theoretical
model to describe the fluctuations. The data are typically
the number of photon counts in a given time interval, typically
about a microsecond, which are due to a small number
of fluorophores. The intensity is dependent on the number
of photons detected from each fluorophore in a given
period of time. For FCS it is convenient to use a parameter
called the brightness, which is given by
B qσQ
(24.6)