Biomedical Engineering Reference
In-Depth Information
steady-state value in the limit of short mRNA lifetimes (9). A more detailed
modeling of this process could include more intermediate processes, such as the
random steps that a ribosome takes along an mRNA, but most turn out to have
little effect when compared in simulations. However, when a repressor or activa-
tor is present, its binding and unbinding might have to be included in the model,
for this can be a major source of noise. It is in this context that the terms shown
in Table 1, type III, are needed. Furthermore, the repressor concentration itself
might be fluctuating, in which case we have to consider the entire system of
genes.
2.5. Linearized Matrix Formulation
The method above can also be used for interacting systems of genes, but
solving it is not straightforward unless the connections are linear. Alternatively,
if the system is at a stable point in steady state, the interaction can be linearized
around the steady state value. A practical way of writing this out is in matrix
form (9). The transition probabilities for species
x
i
are given by
f
i
(
x
1
,
x
2
, ...,
x
n
) for
creation and M
i
for destruction, and
A
and ( are the matrices defined by
s
=
s
f
and (
ij
= H
i
E
ij
. Letting
x
be the vector of chemical species, the lin-
A
i
ij
x
j
x
,
x
...
1
2
(
)
earized macroscopic equations are then given by
x A x
.
Note that in many cases the macroscopic equations include constant crea-
tion terms. If the system is linear it might be necessary to include an additional
variable that is non-fluctuating and allows the inclusion of the constant terms in
compact matrix form. As an illustration of this, the matrices for the single gene
case are
=
(
¯
¯
000
00
000
0
¡
°
¡
°
¡
°
¡
°
Ak
=
¡
(
=
¡
H
0
,
,
[21]
°
°
R
R
¡
°
¡
°
0
k
0
00
H
¡
°
¡
°
¢
±
¢
±
P
P
where the state vector is where
x
T
= (
d
,
r
,
p
) is the gene copy number. This con-
stant state coordinate needs not to represent an actual chemical; for a system
where many species have a constant creation rate, these rates can all be placed in
the first column of
A
(setting
d
= 1 and (
1
j
= (
j
1
= 0). An example of this is the
matrix for the case of two interacting genes, linearized around steady state, with
fixed gene copy numbers
d
1
and
d
2
, respectively, and where the first gene (
r
1
,
p
1
)
represses the second (
r
2
,
p
2
) with transfer function
f
(
p
1
):
