Biomedical Engineering Reference
In-Depth Information
s
F
(
)
=
z
n z
n
1
z
n
'
+
1
p n n
(,' ,)
t
=
s
z z
,
1
2
1
1 1
2
1
2
1 2
z
nn
==
0,
'
0
1
1
2
where the change in the lower limit of the sum for
n
2
is allowed since
p
(
n
1
, -1,
t
)
= 0. This leads to
s
F
Fzz t kzz
(,,) ( 1)
=
s
.
[7]
12 12
z
1
In the special case of synthesis from a fixed number of templates (
n
1
=
n
), the
equation for the moment-generating function reduces to
F zt knz F
.
(,) ( )
=
[8]
2
2
This equation can be explicitly solved, but in itself it does not represent the full
process. We therefore will obtain the expressions for the other terms before
combining them to model a real situation.
2.2. Degradation
Now consider the degradation reaction
B
¶l
0
.
[9]
This reaction can represent two different processes: degradation, where mole-
cule
B
is converted into a species that is not part of the subset of interest, and
dilution, where it is physically separated from the volume of interest. In the lat-
ter context H is the degradation rate and ln(2)/H the half-life of the molecule.
The master equation for this reaction is
p nt
(,)
=
H
npnt n pn t
.
(,) ( 1)( ,)
+++
H
[10]
1
1
1
1
1
Using the same strategy as above the time evolution of the moment-generation
function yields:
s
F
Fzt
(,) ( 1)
=
s
H
z
.
[11]
1
1
z
1
