Principles of Fluorescence Spectroscopy

802 FLUORESCENCE CORRELATION SPECTROSCOPY
based on the detection profile p(r). The autocorrelation
function for the intensity fluctuation is given by
B
G(τ)
2 p(r)p(r’) dVd’V
BCp(r)dV 2
(24.10)
where r is the position of the fluorophore at t = 0 and r' is
its position at t = τ. This formidable expression can be
understood as follows. The denominator contains the average
intensity, which is given by the product of the brightness
B, mean concentration , and the detection profile of
the instrument integrated over the observed volume. The
concentration term and brightness appear outside the integral
because Q is independent of position, and only the
average concentration is needed to calculate the average
intensity. The detection profile p(r) accounts for the excitation
and detection efficiency. The numerator calculates the
intensity fluctuations from the concentration fluctuations at
each point in the sample, which is then integrated over the
observed volume. When the fluorophore moves from r to a
new location r', its brightness becomes proportional to the
detection profile at this position p(r'). The integral extends
over the observed volume for all the fluorophores present in
the volume. The term B2 C
is again outside the integral
because the intrinsic brightness of the fluorophore does not
depend on position. Remarkably, the correlation function
(but not the S/N ratio) is independent of fluorophore brightness,
which cancels in eq. 24.10. This makes sense because
we are measuring the correlation between fluorophore locations,
which should not depend on the brightness of the fluorophore.
It is instructive to consider the number of fluorophores
in a typical FCS experiment. A diffraction-limited volume
will typically have a diameter of 0.5 µm and a total length
of 2 µm. For this size and shape the effective volume is 0.35
fl. The effective volume is not equal to the geometric volume
of an ellipsoid because the detection profile does not
have sharp boundaries. Figure 24.6 shows calculations of
the number of fluorophores in this volume. Occupation
numbers of 2 to 20 are expected for fluorophore concentrations
from 9.6 to 96 nM. The width of the distribution is
given by √N and increases with the occupation number, as
characteristic for a Poisson distribution.
Equation 24.10 is not limited to diffusion and can be
used to derive a correlation function for any process that
results in the intensity fluctuations. Chemical or photochemical
processes that change the brightness of a fluo-
Figure 24.6. Poisson distribution in an ellipsoidal volume for various
fluorophore concentrations. The volume of a geometric ellipsoid with
s = 0.25 µm and u = 1.0 µm is 0.26 fl.
rophore can also be studied. In place of δC(r,τ) eq. 24.10 is
then used with a different r' model to derive the expected
autocorrelation function.
24.2.1. Translational Diffusion and FCS
Perhaps the most common application of FCS is to measure
translational diffusion. The rate of diffusion depends on the
size of the molecule and its interactions with other molecules.
The correlation function for diffusion in three dimensions
is given by 18
C (4πDτ) 3/2 exp|r r’| 2 /4Dτ
(24.11)
where D is the diffusion coefficient. Insertion of eq. 24.11
in 24.10, using the Gaussian detection profile (eq. 24.9) and
some complex mathematics, yields the correlation function
for three-dimensional diffusion:
G(τ) G(0) ( 1 4Dτ
s2 ) 1
( 1 4Dτ
u2 ) 1/2
G(0)D(τ)
(24.12)
where G(0) is the amplitude at τ = 0. For future convenience
we define D(τ) as the portion of the correlation function