Biomedical Engineering Reference
In-Depth Information
where
k
1
+
B
1
is the maximum creation rate,
Y
1/2
is the half induction point,
h
is
the Hill coefficient, and
B
1
is the basal transcription level. Assuming that the
internal noise for the second gene alone has the same form,
(
)
q
2
x
2
H
y
b
+
1
,
[48]
1
1
1
the variance and correlation can be explicitly written as
2
h
2
2
¬ ¬
y
y
h
H
-
-
(
)
(
)
2
1
0
-
-
1
E
y
=
2
b
+ +
1
y
b
+
1
,
-
-
-
--
0
1
1
-
Y
2
k
y
®
®
1/2
1
0
h
¬
y
h
H
-
2
(
)
0
-
EE
yy
=
y
-
-
b
+
1
.
[49]
10
1
0
®
Y
2
k
1/2
1
Note that we need the parameters of the macroscopic equations plus an "inter-
nal" parameter for each gene,
b
i
=
k
Pi
/H
Ri
, which depends on the parameters of the
macroscopic equations for each gene.
4.
DISCUSSION AND CONCLUSIONS
We thus have a versatile toolbox of modeling approaches at our disposal,
each suitable for different situations and with different levels of approximation
and scalability. A direct, full master equation approach provides every detail of
the distribution in the few cases where it can be analytically solved. The proper-
ties of the noise as encoded by the variance and correlations can be obtained
explicitly in a wide range of cases from the equations obtained from the generat-
ing-function version of the master equation, in some cases as a function of time.
If we are only interested in the steady-state noise, the linearized matrix formula-
tion of these equations provides a compact way of treating more complex sys-
tems. For a few variables this can be done easily, but increasing the size of the
system can lead to cumbersome algebra. This matrix approach can be easily
solved using numerical methods or matrix manipulation software, but then some
of the insight might be lost. The Langevin approach provides an alternate,
straightforward way of obtaining the noise characteristics that easily incorporate
the effects of larger systems and other sources of noise.
An additional tool that we have not covered but is worth keeping in mind is
the possibility of performing detailed Monte Carlo simulations of the system.
Methods of varying degrees of approximation and efficiency have been devel-
oped recently (24,25), based on Gillespie's stochastic simulation algorithm (26).
This algorithm is essentially exact, but depending on the complexity of the sys-
tem and the computing power available, a suitable level of detail can be chosen
