Principles of Fluorescence Spectroscopy

PRINCIPLES OF FLUORESCENCE SPECTROSCOPY 803
which contains the diffusion coefficient. This expression is
also written as
G(τ) G(0) ( 1 τ ) τD 1
( 1 ( s
u
G(0)D(τ)
where the diffusion time is
τ D s 2 /4D
(24.13)
(24.14)
The origin of eq. 24.14 is understood by recalling that the
mean distance a molecule diffuses in a time τ is proportional
to . The form of eq. 24.12 is a result of the shape of
the detection profile (eq. 24.9) being integrated with the
concentration correlation function. Different detection profiles
will result in different forms of eq. 24.12.
Figure 24.7 shows the correlation functions expected
for a single freely diffusing species, that is, a homogeneous
population, with diffusion coefficients ranging from 10 –5 to
10 –7 cm2 /s. We assumed the observed volume was an ellipsoid
with radius s = 0.25 µm and half-length u = 1 µm. The
correlation function G(τ) is usually plotted using a logarithmic
τ axis. The amplitude of G(τ) decreases as τ increases.
G(τ) approaches zero because at long times the fluorophores
have no memory of their initial position. The value
of τD is usually determined by least-squares fitting of the
simulated curve with the measured data. A diffusion coefficient
of 10 –6 cm2 √Dτ
/s results in τD = 0.156 ms. As the diffusion
coefficient decreases the correlation function shifts to
longer τ values, which reflects slower intensity fluctuations
as the fluorophores diffuse more slowly into and out of the
observed volume. As we will discuss below, a 100-fold
change in diffusion coefficient would require a 10,000-fold
change in molecular weight, so that the curves in Figure
24.7 are more shifted than is typical in an FCS experiment.
The autocorrelation function is also expected to depend on
the fluorophore concentration.
The autocorrelation function is that it provides a measurement
of the average number of molecules N in the
observed volume, even if the bulk concentration is not
known. The number of molecules is given by the inverse of
the intercept at τ = 0, G(0) = 1/N. Hence the amplitude of
the correlation function is larger for a smaller number of
molecules. This is the reason why FCS was relatively
unused in the 1970s and 1980s. Without confocal optics to
) 2
τ
τ D
) 1/2
Figure 24.7. Simulated autocorrelation functions for diffusion in three
dimensions (eq. 24.12) for diffusion coefficients of 10 –5 to 10 –7 cm 2 /s
(top) and for a mixture of two diffusing species (bottom). The
observed volume was assumed to be an ellipsoid with s = 0.25 µm and
u = 1 µm.
reduce the observed volume the number of molecules was
large and the amplitudes of the correlation functions were
too small for reliable measurements, even with long data
acquisition times. Another important feature of G(τ) for a
single species is that G(τ) does not depend on the brightness
of the fluorophore. The brightness B affects the S/N of the
measurement and the ability to see the fluorophore over the
background. However, the correlation function itself is
independent of the brightness and G(0) depends only on the
average number of observed fluorophores.
At first glance it is difficult to see from eqs. 24.5 and
24.10 why the value of G(0) is equal to 1/N. Since N is proportional
to c the denominator contains a factor N2. The
numerator also has two concentration terms as a product,
suggesting a factor of N2 in the numerator. However, for a
Poisson distribution the variance is proportional to the
square root of the mean, so that δC is proportional to √N .
Hence the numerator contains a factor N, and the ratio gives
G(0) = 1/N.
One might think that if G(τ) reveals the number of fluorophores
then it should be possible to determine the fluorophore
concentration. This is only partially correct. The