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y 1
y 2
y 3
Genes:
x 1
x 2
x 3
Products:
Fig. 7.4: Example 3.
y 2
(
t
)
x 1
x 2
y 2 (
t
+
dt
)=
) +
,
(7.7)
2
y 1 (
t
)+
y 2 (
t
y 2 (
t
)+
y 3 (
t
)
y 3 (
t
)
x 2
x 3
y 3 (
y 3 (
t
+
dt
)=
) +
2
y 2 (
t
)+
y 3 (
t
t
)
The steady-state solution limited to positive gene expression values is
(
x 1 ,
x 2 ,
x 3 ) (
y 1 =
x 1
x 2 +
x 3 ,
y 2 =
x 2
x 3 ,
y 3 =
x 3 ) .
(7.8)
0 and the equations become x 1 ,
x 2 .If x 3 =
x 2 +
If x 2
x 3 then y 2 =
0
,
0the
solution becomes that of Example 1: ( x 1 , x 2 ,0)
x 2 , x 2 ,0).
Similar to Example 1, if product x 1 is consumed, gene y 1 is expressed. If only
products x 1 and x 2 are consumed equally, then gene y 2 is expressed. The underlying
mechanisms remain the same as Example 1.
However, unlike Example 2, this configuration now has an efficient configura-
tion for the case where products x 1 , x 2 , and x 3 are equally consumed. With equal
consumption genes y 1 and y 3 are expressed equally (1, 1, 1)
( x 1
(1, 0, 1) and gene
y 2 expression turned off. This expression pattern most efficiently replaces this con-
sumption pattern with the least amount of extraneous products.
This case demonstrates that information travels indirectly “through” the promot-
ers based on gene structures. Given equal consumption of x 1 and x 2 , expression of
y 1 is determined by consumption of x 3 through y 3 .If x 3 is not consumed (its value
is 0), then gene y 1 is not expressed. If x 3 is consumed, y 1 becomes active. However,
x 3 is not directly affected by y 1 , and the product affected by y 1
(
)
is not directly
expressed by y 3 . Thus genes can cooperate and function in groups, choosing the best
single gene or gene combination that efficiently replaces the consumption pattern.
x 1
7.3.3 Composition by Infinite Chains
What are the limits of these interactions? The behavior of an infinitely large num-
ber of genes with overlapping products is analyzed. No matter how many genes
are linked, this genetic model attempts to match the product consumption with the
most efficient gene expression configuration. To demonstrate this, gene networks are
composed of chained subunits linked at infinitum. The promoters and genes interact
indirectly by transferring their dynamic activation through the chain.
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