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In-Depth Information
these equations with engineering control theory nomenclature, feedback regulation
is “negative,” stabilizing feedback.
⎛
⎝
⎞
⎠
y
a
(
)
n
a
∑
i
t
y
a
(
)
n
a
∑
i
t
x
i
y
a
(
t
+
dt
)=
f
i
=
(7.3)
∑
y
j
(
t
)
∈
N
a
∈
N
a
j
∈
M
i
7.2.4 Stability
Stability of related equations has been previously analyzed [7]. If nonlinear equa-
tions are bounded and well behaved locally, they remain stable. In this model, all
variables are limited to positive values. Thus the values of
y
cannot become negative
and have a lower bound of 0. The upper values of
y
are bounded as well. The expres-
sion value of gene
y
a
will be greatest if all of its promoters
f
i
are maximized. The
promoters will be maximized if genes coactivated by that promoter are not active.
Assuming this is the case then the equation simplifies to
y
a
(
1
n
a
t
)
·
x
max
1
n
a
x
max
·
n
a
∑
∑
y
a
(
t
+
Δ
t
)
≤
=
x
max
≤
=
x
max
(7.4)
y
a
(
t
)
n
a
i
∈
N
a
i
∈
N
a
If maximum consumption
x
max
is bounded by 1, then
y
a
expression is bounded by
1. The values are bounded by positive numbers between zero and the consumption
level. Thus they satisfy boundary conditions and are well behaved. Furthermore as
dt
→
0, Lyapunov functions can be written. This indicates that the networks will set-
tle to a steady state and not display chaotic oscillations. Numerical simulations also
show the equations are well behaved and several cases of gene-product interactions
are demonstrated.
7.3 Results
This system attempts to replace consumed products through a minimum amount of
overall gene expression. Several configurations of genes are analyzed to illustrate
how the system interacts with different patterns of product consumption.
7.3.1 Composition by Overlap of Nodes
7.3.1.1 Complete Overlap
Given that two genes lead to the same product but one of them also leads to another
product, how do they respond to consumption patterns to minimize expression? In a
