Chapter 8 – Theoretical Biophysics 377
Defining C(x0,y0,z0) as a numerical convolution integral and also comprising the Gaussian
excitation field dependence over a cell, and assuming the time-averaged dye density is uni
form in space, we can calculate the dye density, in the case of a simple single internal cellular
compartment, as
(8.123)
ρ = (
)
(
)
=
(
) −
+
(
)
′
I x y z
A
C x y z
I
I
x y z
I
I
C x y z
0
0
0
0
0
0
0
0
0
0
0
0
,
,
,
,
,
,
,
,
d
s
a
d
((
)Is
The mean value of the total background noise (Ia + Id) can usually be calculated from parental
cells that do not contain any foreign fluorescent dye tags. In a more general case of several
different internal cellular compartments, this model can be extended:
(8.124) I x y z
A
I
x
y
xy
j
Number of
comp
0
0
0
2
2
1
2
,
,
(
)
=
−
+
=
d
exp
s
σ
artments
i
Compartment
voxels
j
i
i
i
P x
x y
y z
z
∑
∑
=
−
−
−
(
)
1
0
0
0
ρ
,
,
where ρj is the mean concentration of the jth compartment, which can be estimated from this
equation by least squares analysis. An important feature of this model is that each separate
compartment does not necessarily have to be modeled by a simple geometrical shape (such
as a sphere) but can be any enclosed 3D volume provided its boundaries are well-defined to
allow numerical integration.
Distributions of copy numbers can be rendered using objective KDE analysis similar to
that used for stoichiometry estimation in distinct fluorescent spots. The number of protein
molecules per cell has a typically asymmetrical distribution that could be fitted well using a
random telegraph model for gene expression that results in a gamma probability distribution
p(x) that has similarities to a Gaussian distribution but possesses an elongated tail region:
(8.125)
p x
x
x b
b
a
a
a
( ) =
−
(
)
( )
−1exp
/
Γ
where Γ(a) is a gamma function with parameters a and b determined from the two moments
of the gamma distribution by its mean value m and standard deviation, such that a = m2/σ2
and b = σ2/m.
8.5.5 BIOINFORMATICS TOOLS
Several valuable computational bioinformatics tools are available, many developed
by the academic community and open source, which can be used to investigate pro
tein structures and nucleotide sequences of nucleic acid, for example, to probe for the
appearance of the same sequence repeat in different sets of proteins or to predict sec
ondary structures from the primary sequences. These algorithms operate on the basis
that amino acids can be pooled in different classes according to their physical and chem
ical properties (see Chapter 2). Thus, to some extent, there is interchangeability between
amino acids within the same class such that an ultimate secondary, tertiary, and quater
nary structure may be similar. These tools are also trained using predetermined struc
tural data from EM, NMR, and x-ray diffraction methods (see Chapter 5), enabling likely
structural motifs to be identified from their raw sequence data alone. The most common
algorithm used is the basic alignment search tool (BLAST).
Homology modeling (also known as comparative modeling) is a bioinformatics tool for
determining atomic-level resolution molecular structures. It is most commonly performed
on proteins operating by interpolating a sequence, usually the primary structure of amino