Biomedical Engineering Reference
In-Depth Information
tively). After synthesis the number of templates is not changed. The correspond-
ing reaction is therefore:
A
¶l
k
AB
+
.
[1]
time)
-1
).
The master equation describes how the probability to be in state [
n
1
, n
2
] (
n
1
A
molecules,
n
2
B
molecules) at time
t
changes in time. For the reaction above:
Molecule
A
produces molecule
B
at rate
k
(in units of (concentration
×
p n n t
(, ,)
=
knpn n t
(, ,)
+
knpn n
(,
,)
t
.
[2]
12
1
12
1
12
The first term reflects a transition from state [
n
1
,
n
2
] to state [
n
1
,
n
2
+ 1] and
therefore leads to a decrease in
p
(
n
1
n
2
,
t
). The second term denotes the transition
[
n
1
,
n
2
- 1] [
n
1
,
n
2
] and leads to increased
p
(
n
1
,
n
2
,
t
). The master equation
above is linear and can be solved for the moments by constructing the moment-
generating function. In general for
N
system variables the moment-generating
function is given by:
=
zz zpnn nt
,
[3]
F zz zt
(
,
,...,
, )
nn
...
n
(
,
,...,
, )
1
2
N
12
N
12
N
12
N
nn n
,
,...,
12
N
where the sum runs over all possible states for each
n
i
(in this case, from 0 to ).
This function has the following useful properties:
2
2
s
F
s
F
s
F
F
=
1,
=
n
,
=
n n
(
1)
,
=
n n
,
[4]
i
2
i
i
i
j
1
s
z
s
z
s s
z
z
i
i
i
j
1
1
1
where |
1
means that the function is evaluated at
z
j
= 1 for all
j
. These expressions
justify the name "moment generating": we can obtain the moments of the prob-
ability distribution by evaluating the partial derivatives of the function.
Multiplying the master equation above by
z z
nn
on both sides gives
1 2
12
z zpnnt k zznpnnt k zznpnn t
.
[5]
nn
(,,)
=
nn
(,,)
+
nn
(, ,)
1
2
1
2
1
2
12 12
121 12
121 12
nn
,
nn
,
nn
,
12
12
12
This equation can be simplified significantly by realizing that
s
F
s
F
(
)
=
nz
n
1
pn n t
( , ,)
º
z
=
nz pn n t
n
( , ,)
1
1
11
1 2
1
11
1 2
s
z
s
z
(
n
n
1
1
1
1
)
nz z pn n
nn
(, ,)
t
=
z
nz
n n
1
z pn n
(, ,)
t
[6]
1
2
1
2
11 2
1 2
1
11
2
1 2
nn
==
0,
0
nn
==
0,
1
1
2
1
2