376 8.5 Advanced In Silico Analysis Tools
for example, because the local nearest-neighbor separation distance was less than the optical
resolution, but also may often comprise dimmer, faster moving spots that are individual
subunits of the molecular complexes that are brighter and diffuse more slowly and thus are
more likely to be detected in tracking analysis.
Convolution models can quantify this diffusive pool component on a cell-by-cell basis.
This is valuable since it enables estimates to be made of the total number of fluorescently
labeled molecules in a given cell if combined with information from the number and stoi
chiometry of tracked distinct fluorescent spots. In other words, the distribution of copy
number for that molecule across a population of cell can be quantified, which is useful
in its own right, but which can also be utilized to develop models of gene expression
regulation.
The diffusive pool fluorescence can be modeled as a 3D convolution integral of the
normalized PSF, P, of the imaging system, over the whole cell. Every pixel on the camera
detector of the fluorescence microscope has a physical area ΔA with equivalent area dA in the
conjugate image plane of the sample mapped in the focal plane of the microscope:
(8.119)
dA
A
M
= ∆
where M is the total magnification between the camera and the sample. The measured inten
sity, I′, in a conjugate pixel area dA is the summation of the foreground intensity I due to dye
molecules plus any native autofluorescence (Ia) plus detector noise (Id). I is the summation of
the contributions from all of the nonautofluorescence fluorophores in the whole of the cell:
(8.120)
I x y z
A
E I P x
x y
y z
x
i
i
i
i
0
0
0
1
0
0
0
,
,
,
,
(
)
=
−
−
−
(
)
=∑
d
`
Allcell voxels
s
where
Is is the integrated intensity of a single dye (i.e., its brightness)
ρ is the dye density in units of molecules per voxel (i.e., one pixel volume unit)
E is a function that represents the change in the laser profile excitation intensity over a cell
In a uniform excitation field, E = 1, for example, as approximated in wide-field illumination
since cells are an order of magnitude smaller than the typical Gaussian sigma width of the
excitation field at the sample. For narrow-field microscopy, the excitation intensity is uni
form with height z but has a 2D Gaussian profile in the lateral xy plane parallel to the focal
plane. Assuming a nonsaturating regime for fluorescent photon emission of a given dye, the
brightness of that dye, assuming simple single-photon excitation, is proportional to the local
excitation intensity; thus,
(8.121)
E x y z
x
y
xy
, ,
(
) =
−
+
exp
2
2
2
2σ
where σxy is the Gaussian width of the laser excitation field in the focal plane (typically a few
microns). In Slimfield, there is an extra small z dependence also with Gaussian sigma width,
which is ~2.5 that of the σxy value (see Chapter 3). Thus,
(8.122)
I x y z
A
I
x
y
i
All cell voxels
xy
0
0
0
1
2
2
2
,
,
(
)
=
−
+
=∑
d
exp
s
ρ
σ
−
−
−
(
)
P x
x y
y z
z
i
i
i
0
0
0
,
,