Biomedical Engineering Reference
In-Depth Information
requires a 5 x 5 matrix system as shown, so it gets cumbersome for larger sys-
tems even though most entries are zero. Using symbolic matrix manipulation
software it is straightforward to obtain the desired expressions, so for known
parameters this is a good method for obtaining values without further approxi-
mations.
3.
THE LANGEVIN APPROACH
An alternate approach that allows for a more straightforward interpretation
and scales easily to different levels of detail is to use a Langevin equation. The
Langevin approach consists essentially of adding a noise term to the determinis-
tic equations. This noise term can represent the effect of the intrinsic fluctua-
tions (20) or the external inputs of the system (21).
For
x
, the concentration of some chemical species,
,
[27]
x
=
fx
()
l
x
=+
fx
()
qx t
()()
F
where the random variable
tF
is determined by its statistical properties. For-
mally, this can be any random process, but in practice we assume white-noise
statistics, which will give approximate values for the first two moments. The
conditions for white noise are
()
,
[28]
F
()
t
=
0,
F F U E U
() (
t
t
+
)
=
( ),
where denotes an ensemble average. Since we are interested in the steady
state fluctuations, we will assume the coefficient of the noise term to be con-
stant,
3
i.e., evaluated at
s
x
.
For the case of our basic model of the single gene, we have two macro-
scopic equations representing mRNA and protein creation, respectively:
,
rk
=
l
H
r rk
=
H
rq
+
F
R
R
R
R
r
r
,
[29]
p
=
kr
l
H
p
p
=
kr
H
p q
+
F
P
P
P
P
p
p
where the coefficients of the noise terms are to be determined. Clearly, <
r
> =
k
R
/H
R
and <
p
> =
k
P
<
r
>/H
P
from the condition of zero mean for the noise term.
The difference with the steady state
rr r
follows the equation
E
=
.
[30]
EHE F
r
+=
r
q
R
rr
Fourier transforming, we obtain
