Biomedical Engineering Reference
In-Depth Information
and one of them leaves, the relative change will be only 1/10. This means
that the smaller the ratio fluctuation/average-signal, the larger the number of
molecules in the volume. It can be shown that this ratio is exactly proportional
to the inverse of the number of molecules in the volume of excitation [21]. This
relationship allows the measurements of the number of molecules in a given
volume in the interior of cells [13, 22]. Molecular heterogeneity in the cell or
molecular reactions can also induce the simultaneous presence of molecules of
different kind in the same volume [23]. One important case is when proteins
bind each other forming a molecular aggregate. Let us consider two identi-
cal proteins with one fluorescent probe each forming a molecular dimer. The
dimer is different than the monomer because it carries twice the number of flu-
orescent moieties (Fig. 10.6). When one of these aggregates enters the volume
of excitation, it will cause a larger fluctuation of the intensity than a single
monomer. Clearly, the amplitude of the fluctuation carries information on the
brightness of the diffusing particle (i.e., monomer vs. dimer). Thus, both time
and amplitude structures are affected by underlying molecular species, and
the dynamic processes that cause the change of the fluorescence intensity.
The statistical analysis of the fluctuations of the fluorescence signal has to
recover both information (Fig. 10.6a).
Analysis of the Time Structure of the Fluctuating Signal
The analysis of the time structure of the fluctuating signal is typically done
by autocorrelation analysis. The autocorrelation function, ACF =
G
(
τ
), char-
acterizes the time-dependent decay of the fluorescence fluctuations to their
equilibrium value (Fig. 10.6b). In simple terms, ACF calculates the similarity
between a signal
I
(
t
), and a copy of the same signal shifted by a time lag
τ
,
I
(
t
+
τ
):
2
G
(
τ
)=
I
(
t
)
I
(
t
+
τ
)
−I
(
t
)
.
(10.2)
2
I
(
t
)
The autocorrelation function yields two parameters: the diffusion coef-
ficient (
D
) and the average number of particles in the observation volume
<N>
given by the inverse of
G
(0), multiplied by a constant that depends on
the illumination profile. In the case of identical molecules undergoing random
diffusion in a Gaussian illuminated volume, the characteristic autocorrelation
function is given by the following expression [22]:
1+
8
Dτ
ω
r
−
1
1+
8
Dτ
ω
a
−
1
/
2
G
(
τ
)=
γ
N
,
(10.3)
where
D
is the diffusion coe
cient;
ω
r
and
ω
a
are the beam waist in the radial
and in the axial directions;
N
is the number of molecules;
γ
is the numerical
factor that accounts for the nonuniform illumination of the volume; and
τ
is
the delay time. Other formulas have been derived for the Gaussian-Lorentzian
illumination profile [13] and for molecules diffusing on a membrane [21].
