ZEISS
Left Tool Area and Hardware Control Tools
LSM 880
196
000000-2071-464
10/2014 V_01
5.3.4.3
Model Equations
In the following, the mathematical equations of the correlation functions and the fit equations will be
more detailed. The acquired correlation functions must be fitted to models in order to retrieve
meaningful parameters. It depends on the process, which model is the most appropriate. If the
underlying process is known, the model can be chosen a priori. For example, if one studies diffusion in a
membrane, a 2-D diffusion model should be applied. In other cases, the process is not known, for
example, if one deals with free or anomalous diffusion. In this case, one can screen different potential
models and look for the best fit taken into account the
Χ
2
value. Often two models work nearly the
same, for example, a two component free diffusion model can give you as satisfactorily a fit as a one
component anomalous diffusion model and without prior knowledge on the system it will be impossible
to decide, which is the better one. In principle, models can be ruled out, if the fit does not work,
however, a working model is only a potential candidate but does not signify it to be the correct one. Care
should be taken to minimize the free parameters as much a possible to improve on the fit quality. It does
not make too much sense to fit to three components without fixing parameters of at least one of them.
For example, if the diffusion time of a free ligand can be determined in a pre-experiment, that value
should be fixed to reduce the number of floating parameters for the evaluation of the binding
experiment to its receptor.
The FCS software is designed to be flexible. That means that the user can define or assemble equations
that do not make sense. Care should therefore be exerted and used formulas matched with the ones
from literature to obtain meaningful results. Also, the presence of a model does not automatically mean,
that the recorded data are of a quality that allows its usage. For example, anti-bunching requires a lot of
care in data acquisition like long measurement times and cross-correlation to reduce dead times of the
detectors and elimination of after-pulsing artefacts. It is in the responsibility of the user to set up his
experiments accordingly.
(1)
The correlation function
The auto-correlation function is defined as follows:
( ) (
)
( )
( ) (
)
(
)
( )
(
)
( ) (
)
(
)
( )
(
)
∫
∫
∫
∫
+
⋅
⋅
=
⋅
+
⋅
⋅
=
+
⋅
=
T
T
T
T
I
dt
t
I
dt
t
I
t
I
T
dt
t
I
T
dt
t
I
t
I
T
t
I
t
I
t
I
G
0
2
0
0
2
2
0
2
1
1
)
(
d
t
d
d
d
t
d
d
d
t
d
d
t
d
(1a)
or
( ) (
)
( )
( ) (
)
(
)
( )
( )
( ) (
)
(
)
( )
( )
∫
∫
∫
∫
+
⋅
⋅
=
⋅
+
⋅
⋅
=
+
⋅
=
T
T
T
T
I
dt
t
I
dt
t
I
t
I
T
dt
t
I
T
dt
t
I
t
I
T
t
I
t
I
t
I
G
0
2
0
0
2
2
0
2
1
1
)
(
t
t
t
t
(1b)
where
denotes the time average and
( ) ( )
( )
t
I
t
I
t
I
−
=
d
describes the fluctuations around the mean
intensity.
For long time average of I (no bleaching) the following relation exists:
( )
( )
t
t
d
I
I
G
G
+
=
1
(1c)
Summary of Contents for LSM 880
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